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We give stabilization and parametrization theorems for a class of singular varieties in the space of polynomials of one variable and generalize the results of Arnol'd and Givental'. The class contains the open swallowtails and the open Whitney umbrella. The parametrization is associated with the singularity of a stable mapping (in the sense of Thom and Mather) of kernel rank one.

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Generalizations of the classical conjugation operator can be defined on a general uniform algebra. In this paper the L 2 weighted problems for the conjugation operators are considered and it is shown that the weights have forms similar to the classical case. The results in this paper have applications to concrete uniform algebras, for example, a polydisc algebra and a uniform algebra which consists of rational functions.

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Operators of Kahler-Dirac type are defined in an abstract infinite dimensional Boson-Fermion Fock space and a path integral representation of their index is established. As preliminaries to this end, some trace formulas associated with "Gibbs states" are derived in both an abstract Boson and Fermion Fock space. This is done by introducing Euclidean Bose and Fermi fields at "finite temperature" in each case. In connection with supersymmetric. quantum field theories, the result gives a path integral formula of the so-called "Witten index" in a model with cutoffs.

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A Dirichlet algebra has always a spectral dilation for any operator representation. We don't know the examples of non-Dirichlet algebras which have spectral dilations for any operator representations. In this paper we give an example of such an algebra.

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Let f and g be holomorphic function germs at 0 in C n x c l = {(x,s)} If dxg ∧ dxf = O and if f(x) = f(x,O) is not a power or a unit, then there exists a germ λ at O in C x C l such that g(x,s) = λ(f(x,s),s) . The result has the implication that the notion of an RL-morphism in the unfolding theory of foliation germs generalizes that of a right-left morphism in the function germ case. The notion of an RL-morphism in the unfolding theory of foliation singularities was introduced in [5] to describe the determinacy results and in [6] the versality theorem for these morphisms is proved. This note, which should be considered as an appendix to [5] or [6], contains a factorization theorem implying that an RL-morphism is a generalization of a right-left morphism in the unfolding theory of function germs. It depends on the Mattei-Moussu factorization theorem ([1]) and is a generalization of a result of Moussu [2].

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We consider the Cauchy problem for the two-dimensional vorticity equation. We show that the solution ω behaves like a constant multiple of the Gauss kernel having the same total vorticity as time tends to infinity. No particular structure of initial data ω = ω(x,0) is assumed except the restriction that 0 the Reynolds number R = ſlω0 ldx/v is small, where v is the kinematic viscosity. Applying a time-dependent scale transformation, we show a stability of Burgers' vortex, which physically implies formation of a concentrated vortex.

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We derive lower bound estimates for gradients of a viscosity solution to a Hamilton-Jacobi equation by studying solutions to the approximate Hamiltonian systems.

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This paper concerns solutions to the Hardy–Hénon equation −Δu = |x|σup in Rⁿ with n ≥ 1 and arbitrary p, σ ∈ R. This equation was proposed by Hénon in 1973 as a model to study rotating stellar systems in astrophysics. Although there have been many works devoting to the study of the above equation, at least one of the following three assumptions p > 1, σ ≥ −2, and n ≥ 3 is often assumed. The aim of this paper is to investigate the equation in other cases of these parameters, leading to a complete picture of the existence/non-existence results for non-trivial, non-negative solutions in the full generality of the parameters. In addition to the existence/non-existence results, the uniqueness of solutions is also discussed.

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This paper studies a fourth-order crystalline curvature ow for a curve represented by the graph of a spatially periodic function. This is a spe-
cial example of general crystalline surface diffusion flow. We consider a special class of piecewise linear functions and calculate its speed. We introduce notion of firmness and prove that the solution stays firm if initially it is firm at least for a short time. We also give an example that a facet (flat part) may split if the initial profile is not firm. Moreover, an example of facet-merging is given as well as several estimates for the speed of each facet.

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Two kinds of weak comparison principles are established for a viscosity sub- and supersolution to a fully nonlinear degenerate parabolic equation with discontinuous source terms.

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We consider a class of anisotropic curvature flows called a crystalline curvature flow. We present a survey on this class of flows with special emphasis on the well-posedness of its initial value problem.

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We introduce a space of vector fields with bounded mean oscillation whose "tangential"and "normal" components to the boundary behave differently. We establish its Helmholtz decomposition when the domain is bounded. This substantially extends the authors' earlier result for a half space.

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We consider an initial-boundary value problem for the time-fractional diffusion equation. We prove the equivalence of two notions of weak solutions, viscosity solutions and distributional solutions.

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We consider a one-Laplace equation perturbed by p-Laplacian with 1 < p < ∞. We prove that a weak solution is continuously differentiable (C1) if it is convex. Note that similar result fails to hold for the unperturbed one-Laplace equation. The main difficulty is to show C1-regularity of the solution at the boundary of a facet where the gradient of the solution vanishes. For this purpose we blow-up the solution and prove that its limit is a constant function by establishing a Liouville-type result, which is proved by showing a strong maximum principle. Our argument is rather elementary since we assume that the solution is convex. A few generalization is also discussed.

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This paper solves a singular initial value problem for a system of ordinary di erential equations describing a polygonal flow called a crystalline flow. Such a problem corresponds to a crystalline flow starting from a general polygon not necessarily admissible in the sense that the corresponding initial value problem is singular. To solve the problem, a self-similar expanding solution constructed by the first two authors with H. Hontani (2006) is effectively used.

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We introduce various spaces of vector fields of bounded mean oscillation (BMO) defined in a domain so that normal trace on the boundary is bounded when its divergence is well controlled. The behavior of "normal" component and "tangential" component may be different for our BMO vector fields. As a result zero extension of the normal component stays in BMO although such property may not hold for tangential components.

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We derive a lower spatially Lipschitz bound for viscosity solutions to fully nonlinear parabolic partial differential equations when the initial datum belongs to the Holder space. The resulting estimate depends on the initial Holder expo-nent and the growth rates of the equation with respect to the first and second order derivative terms. Our estimate is applicable to equations which are possibly singular at the initial time. Moreover, it gives the optimal rate of the regularizing effect for solutions, which occurs for some uniformly parabolic equations and first order Hamilton-Jacobi equations. In the proof of our lower estimate, we con-struct a subsolution and a supersolution by optimally rescaling the solution of the heat equation and then compare them with the solution. For linear equations, the lower spatially Lipschitz bound for solutions can be obtained in a different way if the fundamental solution satisfies the Aronson estimate. Examples include the heat convection equation whose convection term has singularities.

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A general purely crystalline mean curvature flow equation with a nonuniform driving force term is considered. The unique existence of a level set flow is established when the driving force term is continuous and spatially Lipschitz uniformly in time. By introducing a suitable notion of a solution a comparison principle of continuous solutions is established for equations including the level set equations. An existence of a solution is obtained by stability and approximation by smoother problems. A necessary equi-continuity of approximate solutions is established. It should be noted that the value of crystalline curvature may depend not only on the geometry of evolving surfaces
but also on the driving force if it is spatially inhomogeneous.

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We consider the initial boundary value problem for a fully-nonlinear parabolic equation in a half space. The boundary condition we study is a degenerate one in the sense that it does not depend on the normal derivative on the boundary. A typical example is a stationary boundary condition prescribing the value of the time derivative of the unknown function. Our setting also covers the classical Dirichlet boundary condition. We establish a comparison principle for a viscosity sub- and supersolution under a weak continuity assumption on the solutions on the boundary. We also prove existence of solutions and give some examples of solutions under several boundary conditions. We show among other things that, in the sense of viscosity solutions, the stationary boundary condition can be different from the Dirichlet boundary condition which is obtained by integrating the stationary condition.

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An explicit representation of the Gamma limit of a single-well Modica-Mortola functional is given for one-dimensional space under the graph convergence which is finer than conventional L1-convergence or convergence in measure. As an application, an explicit representation of a singular limit of the Kobayashi-Warren-Carter energy, which is popular in materials science, is given. Some compactness under the graph convergence is also established. Such formulas as well as compactness is useful to characterize the limit of minimizers the Kobayashi-Warren-Carter energy. To characterize the Gamma limit under the graph convergence, a new idea which is especially useful for one-dimensional problem is introduced. It is a change of parameter of the variable by arc-length parameter of its graph, which is called unfolding by the arc-length parameter in this paper.

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In this paper, we consider the asymptotic stability for the system of linear delay differential equations. Because of the complicated interactions induced by the delay effects of the system, there are few results of the asymptotic stability for the system of the delay differential equations with multiple delays. Given this fact, we propose the new stability conditions for the system and apply these conditions to some mathematical models for the population dynamics and neural network system described by the system of delay differential equations.

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We study a level-set mean curvature flow equation with driving and source terms, and establish convergence results on the asymptotic behavior of solutions as time goes to infinity under some additional assumptions. We also study the associated stationary problem in details in a particular case, and es- tablish Alexandrov’s theorem in two dimensions in the viscosity sense, which is of independent interest.

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The split Bregman framework for Osher-Sol´e-Vese (OSV) model and fourth or-der total variation flow are studied. We discretize the problem by piecewise con- stant function and compute ∇(−∆av)−1 approximately and exactly. Furthermore, we provide a new shrinkage operator for Spohn’s fourth order model. Numerical experiments are demonstrated for fourth order problems under periodic boundary condition.

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A Bertrand curve is a special class of space curves that the principal normal line of the curve and the principal normal line of another curve are the same. On the other hand, a Mannheim curve is also a special class of space curves that the principal normal line of the curve and the bi-normal line of another curve are the same. By definitions, the other curves are parallel curves to the direction of the principal normal vector. Even if regular cases, the existence conditions of the Bertrand and Mannheim curves seem to be wrong. Moreover, parallel curves may have singular points. As smooth curves with singular points, we consider framed curves in the Euclidean space. Then we define Bertrand and Mannheim curves of framed curves. Moreover, we clarify the Bertrand and Mannheim curves are depend of the moving frame.

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This paper is concerned with a regularity criterion based on vorticity direction for Navier-Stokes equations in a three-dimensional bounded domain under the no-slip boundary condition. It asserts that if the vorticity direction is uniformly continuous in space uniformly in time, there is no type I blow-up. A similar result has been proved for a half space by Y. Maekawa and the rst and the last authors (2014). The result of this paper is its natural but non-trivial extension based on L∞ theory of the Stokes and the Navier-Stokes equations recently developed by K. Abe and the rst author.

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In this paper, we propose a new numerical scheme for a spatially discrete model of constrained total variation flows, which are total variation flows whose values are constrained in a Riemannian manifold. The difficulty of this problem is that the underlying function space is not convex and it is hard to calculate the minimizer of the functional with the manifold constraint. We overcome this difficulty by “localization technique" using the exponential map and prove the finite-time error estimate in general situation. Finally, we show a few numerical results for the cases that the target manifolds are S2 and SO(3).

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This paper is devoted to deterministic discrete game-theoretic interpretations for fully nonlinear parabolic and elliptic equations with nonlinear dynamic boundary conditions. It is known that the classical Neumann boundary condition for general parabolic or elliptic equations can be generated by including reflections on the boundary to the interior optimal control or game interpretations. We study a dynamic version of such type of boundary problems, generalizing the discrete game-theoretic approach proposed by Kohn-Serfaty (2006, 2010) for Cauchy problems and later developed by Giga-Liu (2009) and Daniel (2013) for Neumann type boundary problems.

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We introduce a discrete scheme for second order fully nonlinear parabolic PDEs with Caputo’s time fractional derivatives. We prove the convergence of the scheme in the framework of the theory of viscosity solutions. The discrete scheme can be viewed as a resolvent-type approximation.

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In this paper, the evolution of a polygonal spiral curve by the crystalline curvature flow with a pinned center is considered with two view points, discrete model consist of an ODE system of facet lengths and a level set method. We investigate the difference of these models numerically by calculating the area of the region enclosed by these spiral curves. The area difference is calculated by the normalized L1 norm of the difference of step-like functions which are branches of argx whose discontinuities are only on the spirals. We find the differences of the numerical results considered in this paper are very small even though the evolution laws of these models around the center and the farthest facet are slightly different.

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We combine the total variation flow suitable for crystal modeling and image analysis withthedynamicboundaryconditions. Weanalyzethebehavioroffacetsatthepartsofthe boundarywheretheseconditionsareimposed. Wedevoteparticularattentiontotheradially symmetric data. We observe that the boundary layer detachment actually can happen at concave parts of the boundary.

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Differential operators acting in a general class of L2-functions in Fock spaces with values in tensor products of separable Hilbert spaces are considered. Then, we prove that those operators are closable.

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It is shown that an irreducible representation of the CCR over a dense subspace of a boson Fock space is associated with a quantum system whose space configuration may give rise to Casimir effect in the context of a quantum scalar field and that it is inequivalent to the Fock representation of the same CCR. A quantum scalar field is constructed from the representation. A new feature of the analysis is to treat a singular Bogoliubov transformation, which is different from the usual bosonic Bogoliubov transformation and from which the inequivalent irreducible representation of the CCR is constructed.

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We consider the sharp interface limit of the Allen-Cahn equation with Dirichlet or dynamic boundary conditions and give a varifold characterization of its limit which is formally a mean curvature flow with Dirichlet or dynamic boundary conditions. In order to show the existence of the limit, we apply the phase field method under the assumption that the discrepancy measure vanishes on the boundary. For this purpose, we extend the usual Brakke flow under these boundary conditions by the first variations for varifolds on the boundary.

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In this paper, we study an obstacle problem associated with the mean curvatureflow with constant driving force. Our first main result concerns interior and boundary regularityof the solution. We then study in details the large time behavior of the solution and obtainthe convergence result. In particular, we give full characterization of the limiting profiles in theradially symmetric setting.

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A functional directional differential operator on Boson Fock spaces in the Q-space representation with a Gelfand’s triple is considered. Then, we derive an integration by parts formula on that differential operator by employing the notion of strongly continuous one parameter unitary group.

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We prove the existence of strong time-periodic solutions to the bidomain equations with arbitrary large forces. We construct weak time-periodic solutions by a Galerkin method combined with Brouwer’s fixed point theorem and a priori estimate independent of approximation. We then show their regularity so that our solution is a strong time-periodic solution in L2 spaces. Our strategy is based on the weak-strong uniqueness method.

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In this paper, we prove the existence of asymptotic speed of solutions to fully nonlinear, possibly degenerate parabolic partial differential equations in a general setting. We then give some explicit examples of equations in this setting and study further properties of the asymptotic speed for each equation. Some numerical results concerning the asymptotic speed are presented.

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Consider the anisotropic Navier-Stokes equations as well as the primitive equations. It is shown that the horizontal velocity of the solution to the anisotropic Navier-Stokes equations in a cylindrical domain of height ε with initial data u0 = (v0,w0) ∈ B2−2/p q,p , 1/q + 1/p ≤ 1 if q ≥ 2 and 4/3q+2/3p ≤ 1 if q ≤ 2, converges as ε → 0 with convergence rateO(ε) to the horizontal velocity of the solution to the primitive equations with initial data v0 with respect to the maximal-Lp-Lq-regularity norm. Since the difference of the corresponding vertical velocities remains bounded with respect to that norm, the convergence result yields a rigorous justification of the hydrostatic approximation in the primitive equations in this setting. It generalizes in particular a result by Li and Titi for the L2-L2-setting. The approach presented here does not rely on second order energy estimates but on maximal Lp-Lq-estimates for the heat equation.

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We consider the initial value problem for a fully-nonlinear degenerate parabolic equation with a dynamic boundary condition in a half space. Our setting includes geometric equations with singularity such as the level-set mean curvature flow equation. We establish a comparison principle for a viscosity sub- and supersolution. We also prove existence of solutions and Lipschitz regularity of the unique solution. Moreover, relation to other types of boundary conditions is investigated by studying the asymptotic behavior of the solution with respect to a coefficient of the dynamic boundary condition.

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Consider the 3-d primitive equations in a layer domain Ω = G×(−h,0), G = (0,1)2, subject to mixed Dirichlet and Neumann boundary conditions at z = −h and z = 0, respectively, and the periodic lateral boundary condition. It is shown that this equation is globally, strongly well-posed for arbitrary large data of the form a = a1 + a2, where a1 ∈ C(G;Lp(−h,0)), a2 ∈ L∞(G;Lp(−h,0)) for p > 3, and where a1 is periodic in the horizontal variables and a2 is sufficiently small. In particular, no differentiability condition on the data is assumed. The approach relies on L∞ H Lp z(Ω)-estimates for terms of the form t1/2∥∂zetAσPf∥L∞ H Lp z(Ω) ≤ Cetβ∥f∥L∞ H Lp z(Ω) for t > 0, where etAσ denotes the hydrostatic Stokes semigroup. The difficulty in proving estimates of this form is that the hydrostatic Helmholtz projection P fails to be bounded with respect to the L∞-norm. The global strong well-posedness result is then obtained by an iteration scheme, splitting the data into a smooth and a rough part and by combining a reference solution for smooth data with an evolution equation for the rough part.

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In the classical level set method, the slope of solutions can be very small or large, and it can make it difficult to get the precise level set numerically. In this paper, we introduce an improved level set equation whose solutions are close to the signed distance function to evolving interfaces. The improved equation is derived via approximation of the evolution equation for the distance function. Applying the comparison principle, we give an upper- and lower bound near the zero level set for the viscosity solution to the initial value problem.

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This is essentially a survey paper on a large time behavior of solutions of some simple birth and spread models to describe growth of crystal surfaces. The models discussed here include level-set flow equations of eikonal or eikonal-curvature flow equations with source terms. Large time asymptotic speed called growth rate is studied. As an application, a simple proof is given for asymptotic profile of crystal grown by anisotropic eikonal-curvature flow.

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This article presents the maximal regularity approach to the primitive equations. It is proved that the 3D primitive equations on cylindrical domains admit a unique, global strong solution for initial data lying in the critical solonoidal Besov space B2/p pq for p,q ∈ (1,∞) with 1/p + 1/q ≤ 1. This solution regularize instantaneously and becomes even real analytic for t > 0.

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We study the asymptotic behavior of global solutions to the initial value problem for the generalized KdV-Burgers equation. One can expect that the solution to this equation converges to a self-similar solution to the Burgers equation, due to earlier works related to this problem. Actually, we obtain the optimal asymptotic rate similar to those results and the second asymptotic profile for the generalized KdV-Burgers equation.

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Consider the primitive equations on R2 ×(z0, z1) with initial data a of the form a = a1 +a2, where a1 ∈ BUCσ(R2; L1(z0, z1)) and a2 ∈ L∞σ (R2; L1(z0, z1)) and where BUCσ(L1) and L∞σ (L1) denote the space of all solenoidal, bounded uniformly continuous and all solenoidal, bounded functions on R2, respectively, which take values in L1 (z0, z1). These spaces are scaling invariant and represent the anisotropic character of these equations. It is shown that, if ka2kL∞σ (L1) is sufficiently small, then this set of equations has a unique, local, mild solution. If in addition a is periodic in the horizontal variables, then this solution is a strong one and extends to a unique, global, strong solution. The primitive equations are thus strongly and globally well-posed for these data. The approach depends crucially on mapping properties of the hydrostatic Stokes semigroup in the L∞(L1)-setting and can thus be seen as the counterpart of the classical iteration schemes for the Navier-Stokes equations for the situation of the primitive equations

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A framed surface is a smooth surface in the Euclidean space with a moving frame. The framed surfaces may have singularities. We treat smooth surfaces with singular points, that is, singular surfaces more directly. By using the moving frame, the basic invariants and curvatures of the framed surface are introduced. Then we show that the existence and uniqueness for the basic invariants of the framed surfaces. We give properties of framed surfaces and typical examples. Moreover, we construct framed surfaces as one-parameter families of Legendre curves along framed curves. We give a criteria for singularities of framed surfaces by using the curvature of Legendre curves and framed curves.

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We consider a gradient ow of the total variation in a negative Sobolev space H􀀀s (0 s 1) under the periodic boundary condition. If s = 0, the ow is nothing but the classical total variation ow. If s = 1, this is the fourth order total variation ow. We consider a convex variational problem which gives an implicit-time discrete scheme for the ow. By a duality based method, we give a simple numerical scheme to calculate this minimizing problem numerically and discuss convergence of a forward- backward splitting scheme. Several numerical experiments are given.

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This paper concerns the processes of nonlinear di usion in a mov- ing domain which lies on a moving closed surface. The nonlinear di usion equations and corresponding energy identities are derived by regarding the moving surface as a thin width (thickness) limit of moving thin domains, for which suitable boundary conditions are imposed to insure that there is no exchange of mass between the thin domains and the environments. We also employ an energetic variational approach to derive these nonlinear di usion equations. Most of all, we show that these nonlinear energetic variational procedures can commute with the passing to the zero width limits.

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We show that every bounded subset of an Euclidean space can be approximated by a set that admits a certain vector eld, the so-called Cahn-Ho man vector eld, that is subordinate to a given anisotropic metric and has a square-integrable divergence. More generally, we introduce a concept of facets as a kind of directed sets, and show that they can be approximated in a similar manner. We use this approximation to construct test functions necessary to prove the comparison principle for viscosity solutions of the level set formulation of the crystalline mean curvature ow that were recently introduced by the authors. As a consequence, we obtain the wellposedness of the viscosity solutions in an arbitrary dimension, which extends the validity of the result in the previous paper.

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A Hamilton-Jacobi equation with Caputo's time-fractional deriv- ative of order less than one is considered. The notion of a viscosity solution is introduced to prove unique existence of a solution to the initial value problem under periodic boundary conditions. For this purpose comparison principle as well as Perron's method is established. Stability with respect to the order of derivative as well as the standard one is studied. Regularity of a solution is also discussed. Our results in particular apply to a linear transport equation with time-fractional derivatives with variable coefficients.

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Evolution of convex polygonal spiral with fixed center by crystalline eikonal-curvature flow is considered. In this evolution we consider a new facet of the polygonal curve generates from center when a facet associated with the center evolves with enough length, which is equal to the length of the facet in Wulff shape of energy density function. We prove the existence, uniqueness and intersection free of solution to our formulation globally-in-time. In the proof of the existence we also prove that new facets are generated repeatedly in time. The important property for intersection-free result is monotonicity property such that the normal velocity of every facets are positive after the next new facet is generated, so that the center is always behind of the moving facets.

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We study bidomain equations that are commonly used as a model to represent the electrophysiological wave propagation in the heart. We prove existence, uniqueness and regularity of a strong solution in Lp spaces. For this purpose we derive an L1 resolvent estimate for the bidomain operator by using a contradiction argument based on a blow-up argument. Interpolating with the standard L2-theory, we conclude that bidomain operators generate C0-analytic semigroups in Lp spaces, which leads to construct a strong solution to a bidomain equation in Lp spaces.

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We consider a smooth curve with singular points in the Euclidean space. As a smooth curve with singular points, we have introduced a framed curve or a framed immersion. A framed immersion is a smooth curve with a moving frame and the pair is an immersion. We consider an evolute and a focal surface of a framed immersion in the Euclidean space. The evolutes and focal surfaces of framed immersions are generalizations of each object in the case of a regular space curve. We investigate relationships between singularities of the evolutes and of the focal surfaces. Moreover, we give properties of the evolutes and repeated evolutes.

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In an abstract framework, a new concept on time operator, ultra-weak time operator, is introduced, which is a concept weaker than that of weak time operator. Theorems on the existence of an ultra-weak time operator are established. As an application of the theorems, it is shown that Schr¨odinger operators HV with potentials V obeying suitable conditions, including the Hamiltonian of the hydrogen atom, have ultra-weak time operators. Moreover, a class of Borel measurable functions f : R ! R such that f(HV ) has an ultra-weak time operator is found.

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It is well-known that there exists a unique local-in-time strong solution u of the initial-boundary value problem for the Navier-Stokes sytem in a three-dimensional smooth bounded domain when the initial velocity u0 belongs to critical Besov spaces. A typical space is B = B􀀀1+3=q q;s with 3 < q < 1, 2 < s < 1 satisfying 2=s+3=q 1 or B = B 􀀀1+3=q q;1 . In this paper we show that the solution u is continuous in time up to initial time with values in B. Moreover, the solution map u0 7! u is locally Lip- schitz from B to C ([0; T];B). This implies that in the range 3 < q < 1, 2 < s 1 with 3=q + 2=s 1 the problem is well-posed which is in strong contrast to norm in ation phenomena for B􀀀1 1;s, 1 s < 1.

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We consider weak solutions of the instationary Navier-Stokes system in a smooth bounded domain R3 with initial value u0 2 L2 ( ). It is known that a weak solution is a local strong solution in the sense of Serrin if u0 satis es the optimal initial value condition u0 2 B􀀀1+3=q q;sq with Serrin exponents sq > 2; q > 3 such that 2 sq + 3 q = 1. This result has recently been generalized by the authors to Rweighted Serrin conditions such that u is contained in the weighted Serrin class T 0 ( ku( )kq)s d < 1 with 2 s + 3 q = 1 􀀀 2 , 0 < < 1 2 . This regularity is guaranteed if and only if u0 is contained in the Besov space B􀀀1+3=q q;s . In this article we consider the limit case of initial values in the Besov space B􀀀1+3=q q;1 and in its subspace B 􀀀1+3=q q;1 based on the continuous interpolation functor. Special emphasis is put on questions of uniqueness within the class of weak solutions.

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The purpose of this paper is to shed light on the fact that the global solvability for the quadratically perturbed wave equation with small initial data in two space dimension can be shown by using only a restricted set of vector fields associated with the space-time translation and spatial rotations. As a by-product, we establish almost best possible decay estimates related to the above vector fields, as well as the tangential derivatives to the forward light cones.

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We introduce various spaces of functions of bounded mean oscillations (BMO) defined in a domain by taking into account the behavior of functions near the boundary. Then we establish several equivalences of these spaces. Moreover, we compare our space with a BMO space introduced by Miyachi. As an application we prove that the heat and the Stokes semigroup are analytic in such a type of spaces.

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For singular plane curves, the classical definitions of envelopes are vague. In order to define envelopes for singular plane curves, we introduce a one-parameter family of Legendre curves in the unit tangent bundle over the Euclidean plane and the curvature. Then we give a definition of an envelope for the one-parameter family of Legendre curves. We investigate properties of the envelopes. For instance, the envelope is also a Legendre curve. Moreover, we consider bi-Legendre curves and give a relationship between envelopes.

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A framed curve is a smooth curve in the Euclidean space with a moving frame. We call the smooth curve in the Euclidean space the framed base curve. In this paper, we give an existence condition of framed curves. Actually, we construct a framed curve such that the image of the framed base curve coincides with the image of a given smooth curve under a condition. As a consequence, polygons in the Euclidean plane can be realised as not only a smooth curve but also a framed base curve.

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We introduce a new notion of viscosity solutions for the level set formulation of the motion by crystalline mean curvature in three dimensions. The solutions satisfy the comparison principle, stability with respect to an approximation by regularized problems, and we also show the uniqueness and existence of a level set ow for bounded crystals.

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A family of irreducible representations of the canonical anticommutation rela- tions (CAR) over an abstract Hilbert space indexed by a set of bounded linear operators is presented and a theorem on the mutual equivalence of them is estab- lished. As an application of the theorem, it is proved that quantum Dirac elds of different masses are mutually inequivalent. Moreover, a new class of irreducible representations of the CAR over a Hilbert space, which includes, as a special case, time-zero quantum Dirac elds, is constructed.

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Considered is a family of irreducible Weyl representations of canonical commu- tation relations with in nite degrees of freedom on the abstract boson Fock space over a complex Hilbert space. Theorems on equivalence or inequivalence of the representations are established. As a simple application of one of these theorems, the well known inequivalence of the time-zero eld and conjugate momentum for different masses in a quantum scalar eld theory is rederived with space dimension d 1 arbitrary. Also a generalization of representations of the time-zero eld and conjugate momentum is presented. Comparison is made with a quantum scalar eld in a bounded region in Rd. It is shown that, in the case of a bounded space region with d = 1; 2; 3, the representations for different masses turn out to be mutually equivalent.

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We investigate a model equation in the crystal growth, which is described by a level-set mean curvature ow equation with driving and source terms. We establish the well-posedness of solutions, and study the asymptotic speed. Interestingly, a new type of nonlinear phenomena in terms of asymptotic speed of solutions appears, which is very sensitive to the shapes of source terms.

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We study the mean curvature ow of graphs both with Neumann boundary conditions and transport terms. We derive boundary gradient estimates for the mean curvature ow. As an application, the existence of the mean curvature ow of graphs is presented. A key argument is a boundary monotonicity formula of a Huisken type derived using re ected backward heat kernels. Furthermore, we provide regularity conditions for the transport terms.

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Consider the Stokes equations in a sector-like C3 domain Ω R2. It is shown that the Stokes operator generates an analytic semigroup in Lp (Ω) for p 2 [2;1). This includes domains where the Lp-Helmholtz decomposition fails to hold. To show our result we interpolate results of the Stokes semigroup in VMO and L2 by constructing a suitable non-Helmholtz projection to solenoidal spaces.

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This paper studies the analyticity of the Stokes semigroup in an infinite cylinder or more generally a cylindrical domain with several exits to infinity in the space C0; , the L 1 -closure of all smooth compactly supported solenoidal vector fields. These domains are not strictly admissible in the sense of the first two authors (2014). However, it is shown that these domains are still admissible which yields the analyticity in C0; . A new proof based on a blow-up argument is given to derive an L 1 -type resolvent estimate which enables us to conclude that the analyticity angle of the Stokes semigroup in C0; is =2.

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We study the phase eld method for the volume preserving mean curvature flow. Given an initial C1 hypersurface we proved the existence of the weak solution for the volume preserving mean curvature ow via the reaction diffusion equation with a nonlocal term. We also show the monotonicity formula and the density upper bound for the reaction diffusion equation.

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We consider a one-dimensional Allen{Cahn equation with con- straint from the view-point of numerical analysis. Our constraint is the subdifferential of the indicator function on the closed interval, which is the multivalued function. Therefore, it is very difficult to make numerical ex- periments of our equation. In this paper we approximate our constraint by Yosida approximation. Then, we study the approximating system of our original model numerically. In particular, we give the criteria for the stan- dard forward Euler method to give stable numerical experiments of our approximating equation. Moreover, we give some numerical experiments of approximating equation.

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We consider the Stokes semigroup in a large class of domains including bounded domains, the half-space and exterior domains. We will prove that the Stokes semigroup is analytic in a certain type of solenoidal subspaces of BMO.

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We clarify the notion of well-chosen weak solutions of the instationary Navier-Stokes system recently introduced by the authors and P.-Y. Hsu in the article Initial values for the Navier-Stokes equations in spaces with weights in time, Funkcialaj Ekvacioj (2015). Well-chosen weak solutions have initial values in L2 ( ) contained also in a quasi-optimal space of Besov type of initial values such that nevertheless Serrin’s Uniqueness Theorem cannot be applied. However, we find universal conditions such that a weak solution given by a concrete approximation method coincides with the strong solution in a weighted function class of Serrin type.

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We consider the space of solenoidal vector fields in an unbounded domain Ω ⊂ Rn whose boundary is given as a Lipschitz graph. It is shown that, under suitable functional setting, the space of solenoidal vector fields is isomorphic to the n − 1 product space of the space of scalar functions. As an application, we introduce a natural and systematic reduction of the equations describing the motion of incompressible flows. This gives a new perspective of the derivation of Ukai’s solution formula for the Stokes equations in the half space, and provides a key step for the generalization of Ukai’s approach to the Stokes semigroup in the case of the curved boundary.

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We study the stability of some exact stationary solutions to the two-dimensional Navier-Stokes equations in an exterior domain to the unit disk. These stationary solutions are known as a simple model of the flow around a rotating obstacle, while their stability has been open due to the difficulty arising from their spatial decay in a scale-critical order. In this paper we affirmatively settle this problem for small solutions. That is, we will show that if these exact solutions are small enough then they are asymptotically stable with respect to small L2 perturbations.

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In this paper we set up a rigorous justification for the reinitialization algorithm. Using the theory of viscosity solutions, we propose a well-posed Hamilton-Jacobi equation with a parameter, which is derived from homogenization for a Hamiltonian discontinuous in time which appears in the reinitialization. We prove that, as the parameter tends to infinity, the solution of the initial value problem converges to a signed distance function to the evolving interfaces. A locally uniform convergence is shown when the distance function is continuous, whereas a weaker notion of convergence is introduced to establish a convergence result to a possibly discontinuous distance function. In terms of the geometry of the interfaces, we give a necessary and sufficient condition for the continuity of the distance function. We also propose another simpler equation whose solution has a gradient bound away from zero.

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We study convexity of simple closed frontals in the Euclidean plane by using the curvature of Legendre curves. We show that for a Legendre curve, the simple closed frontal is convex if and only if the sign of both functions of the curvature of the Legendre curve does not change. We also give some examples of convex simple closed frontals.

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We consider the differential geometry of evolutes of singular curves in hyperbolic 2- space and de Sitter 2-space. Firstly, as an application of the basic Legendrian duality theorems, we give the definitions of fronts in hyperbolic 2-space or de Sitter 2-space, respectively. We also give the notions of moving frames along the fronts. By using the moving frames, we define the evolutes of spacelike fronts and timelike fronts, and investigate the geometric properties of these evolutes. As results, these evolutes can be viewed as wavefronts from the viewpoint of Legendrian singularity theory. At last, we study the relationships among these evolutes.

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In order to consider singular curves in the unit sphere, we consider Legendre curves in the unit spherical bundle. By using a moving frame, we de ne the curvature of Legendre curves in the unit spherical bundle. As applications, we give a relationship among Legendre curves in the unit spherical bundle, Legendre curves in the unit tangent bundle and framed curves in the Euclidean space, respectively. Moreover, we de ne not only an evolute of a front, but also an evolute of a frontal in the unit sphere under certain conditions. Since the evolute of a front is also a front, we can take evolute again. On the other hand, the evolute of a frontal if exists, is also a frontal. We give an existence and uniqueness conditions of the evolute of a frontal.

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We study a cell problem arising in homogenization for a Hamilton-Jacobi equation whose Hamiltonian is not coercive. We introduce a generalized notion of effective Hamiltonians by approximating the equation and characterize the solvability of the cell problem in terms of the generalized effective Hamiltonian. Under some sufficient conditions, the result is applied to the associated homogenization problem. We also show that homogenization for non-coercive equations fails in general.

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非線形な力学系では、単純な規則から複雑なフラクタル構造をもつ幾何的対象物が生成される。それでは、それを目の前のコンピュータスクリーン上で見ることは可能であるか。これは一見単純そうな問題であるが、奥深い事実が隠されていることを近年の研究は明らかにした。本論文では実力学系、複素力学系、計算理論の基礎を紹介し、複素力学系のジュリア集合の計算可能性と複雑性についての事実をまとめる[17]。

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A viscoelastic flow in a two-dimensional layer domain is considered. An L2-stability of the Poiseuille-type flow is established provided that both Poiseuille flow and perturbation is sufficiently small. Our analysis is based on a stream function formulation introduced by F.-H. Lin, C. Liu and P. Zhang (2005).

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We consider the Stokes equations in a class of domains that we will call admissible domains including bounded domains, the half space and exterior domains. We will prove new L∞ estimates for derivatives of velocity and pressure. The estimates will be given in terms of a BMO-type norm of the initial data.

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We study a free boundary problem for the curvature ow with driving force for a family of planar curves with two xed contact angles on the x-axis. Our aim of this paper is to analyze the dimension of stable and unstable manifolds of traveling wave solution.

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We present a new Harnack inequality for non-negative discrete supersolutions of fully nonlinear uniformly elliptic difference equations on rectangular lattices. This estimate applies to all supersolutions; instead the Harnack constant depends on the graph distance on lattices. For the proof we modify the proof of the weak Harnack inequality. Applying the same idea to elliptic equations in a Euclidean space, we also derive a Harnack type inequality for non-negative viscosity supersolutions.

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We consider a fourth-order total variation ow, which is studied in image recovery and materials science. In this paper, we characterize a total variation ow in H􀀀1-space with Dirichlet boundary condition. Furthermore, we show an extinction time estimate for the solution of a total variation ow with Dirichlet boundary condition. Giga and Kohn (2011) established the same extinction time estimate in periodic case. Their argument is based on interpolation inequalities. Using extension operators, we derive this type of inequalities, which we apply to the case of Dirichlet boundary condition.

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We study a two fluid system which models the motion of a charged fluid. Local in time solutions of this system were proven by Giga-Yoshida [15]. In this paper, we improve this result in terms of requiring less regularity on the electromagnetic field. We also prove that small solutions are global in time.

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We study the motion of the so-called bent rectangles by the singular weighted mean curvature. We are interested in the curves which can be rendered as graphs over a smooth onedimensional reference manifold. We establish a sufficient condition for that. Once we deal with graphs we can have the tools of the viscosity theory available, like the Comparison Principle. With its help we establish uniqueness of variational solutions constructed by the authors [18]. In addition, we establish a criterion for the mobility coefficient guaranteeing vertex preservation.

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This paper shows that Lp-Helmholtz decomposition is not necessary to establish the analyticity of the Stokes semigroup in C0; , the L 1 -closure of the space of all compactly supported smooth solenoidal vector fields. In fact, in a sector-like domain for which the Lp-Helmholtz decomposition does not hold, the analyticity of the Stokes semigroup in C0; is proved.

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We consider the nonstationary Navier-Stokes system in a smooth bounded domain R3 with initial value u0 2 L2 ( ). It is an important question to determine the optimal initial value condition in order to prove the existence of a unique local strong solution satisfying Serrin's condition. In this paper, we introduce a weighted Serrin condition that yields a necessary and su cient initial value condition to guarantee the existence of R local strong solutions u( ) contained in the weighted Serrin class T 0 ( ku( )kq)s d < 1 with 2 s + 3 q = 1 􀀀 2 , 0 < < 1 2 . Moreover, we prove a restricted weak-strong uniqueness theorem in this Serrin class.

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The geometric model for Dn-Dynkin diagram is explicitly constructed and associated generic singularities of tangent surfaces are classified up to local diffeomorphisms. We observe, as well as the triality in D4 case, the difference of the classification for D3,D4,D5 and Dn(n ≥ 6), and a kind of stability of the classification in Dn for n → ∞. Also we present the classifications of singularities of tangent surfaces for the cases B3,A3 = D3,G2,C2 = B2 and A2 arising from D4 by the processes of foldings and removings.

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We study a singular limit problem of the Allen-Cahn equation with Neumann boundary conditions and general initial data of uniformly bounded energy. We prove that the time-parametrized family of limit energy measures is Brakke’s mean curvature flow with a gen- eralized right angle condition on the boundary.

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Concentration phenomena in statistical ensembles of Young diagrams have been investigated as static models first for the Plancherel ensemble by Vershik–Kerov and Logan–Shepp in 1970s and later for some other group-theoretical ensembles by Biane. On the other hand, a dynamical model of concentration for Young diagrams, which is not directly connected with group representations, was shown by Funaki– Sasada in the framework of hydrodynamic limit. Our aim here is to discuss a dynamical model of concentration for Young diagrams in grouptheoretical ensembles. We especially feature Biane’s approximate factorization property of ensembles as an origin to give rise to concentration. Starting from an initial ensemble yielding concentration and a microscopic dynamics keeping the Plancherel measure invariant, we derive an evolution of rescaled shapes of Young diagrams through hydrodynamic limit. The resulting evolution along macroscopic time is described in terms of the notions of Voiculescu’s free probability theory such as free compression and free convolution of Kerov transition measures.

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The quantum system of a massless charged scalar field with a self-interaction is investigated. By introducing a spacial cut-off function, the Hamiltonian of the system is realized as a linear operator on a boson Fock space. It is proven that the Hamiltonian strongly commutes with the total charge operator. This fact implies that the state space of the charged scalar field is decomposed into the infinite direct sum of fixed total charge spaces. Moreover, under certain conditions, the Hamiltonian is bounded below, self-adjoint and has a ground ground state for an arbitrarily coupling constant. A relation between the total charge of the ground state and a number operator bound is also revealed.

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Since A. M. Turing introduced the notion of computability in 1936, various theories of real number computation have been studied [1][10][13]. Some are of interest in nonlinear and statistical physics while others are extensions of the mathematical theory of computation. In this review paper, we introduce a recently developed computability theory for Julia sets in complex dynamical systems by Braverman and Yampolsky [3].

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Weyl-von Neumann Theorem asserts that two bounded self-adjoint operators A;B on a Hilbert space H are unitarily equivalent modulo compacts, i.e., uAu +K = B for some unitary u 2 U(H) and compact self-adjoint operator K, if and only if A and B have the same essential spectra: ess(A) = ess(B). In this paper we consider to what extent the above Weyl-von Neumann's result can(not) be extended to unbounded operators using descriptive set theory. We show that if H is separable in nite-dimensional, this equivalence relation for bounded self-adjoin operators is smooth, while the same equivalence relation for general self-adjoint operators contains a dense G -orbit but does not admit classi cation by countable structures. On the other hand, apparently related equivalence relation A B , 9u 2 U(H) [u(A 􀀀 i) 􀀀1u 􀀀 (B 􀀀 i) 􀀀1 is compact], is shown to be smooth.

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We have already defined the evolutes and the involutes of fronts without inflection points. For regular curves or fronts, we can not define the evolutes at inflection points. On the other hand, the involutes can be defined at inflection points. In this case, the involute is not a front but a frontal at inflection points. We define evolutes of frontals under conditions. The definition is a generalisation of both evolutes of regular curves and of fronts. By using relationship between evolutes and involutes of frontals, we give an existence condition of the evolute with inflection points. We also give properties of evolutes and involutes of frontals.

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It is well known that the projective duality can be understood in the context of geometry of An-type. In this paper, as D4-geometry, we construct explicitly a flag manifold, its triplefibration and differential systems which have D4-symmetry and conformal triality. Then we give the generic classification for singularities of the tangent surfaces to associated integral curves, which exhibits the triality. The classification is performed in terms of the classical theory on root systems combined with the singularity theory of mappings. The relations of D4-geometry with G2-geometry and B3-geometry are mentioned. The motivation of the tangent surface construction in D4-geometry is provided.

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Let h be a cuspidal Hecke eigenform of half-integral weight, and En/2+1/2 be Cohen’s Eisenstein series of weight n/2+1/2. For a Dirichlet character χ we define a certain linear combination R(χ)(s, h,En/+1/2) of the Rankin-Selberg convolution products of h and En/2+1/2 twisted by Dirichlet characters related with χ. We then prove a certain algebraicity result for R(χ)(l, h,En/2+1/2) with l integers.

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Let K = Q( √ −D) be an imaginary quadratic field with discriminant −D, and χ the Dirichlet character corresponding to the extension K/Q. Let m = 2n or 2n + 1 with n a positive integer. Let f be a primitive form of weight 2k + 1 and character χ for Γ0(D), or a primitive form of weight 2k for SL2(Z) according as m = 2n, or m = 2n + 1. For such an f let Im(f) be the lift of f to the space of Hermitian modular forms constructed by Ikeda. We then give an explicit formula of the Koecher-Maass series L(s, Im(f)) of Im(f). This is a generalization of Mizuno [Mi06

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For a very strong diffusion equation like total variation flow it is often observed that the solution stops at a steady state in a finite time. This phenomenon is called a finite time stopping or a finite time extinction if the steady state is zero. Such a phenomenon is also observed in one-harmonic map flow from an interval to a unit circle when initial data is piecewise constant. However, if the target manifold is a unit two-dimensional

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Ma and Wang derived an equation linking the separation location and times for the boundary layer separation of incompressible uid ows. The equation gave a necessary condition for the separation (bifurcation) point. The purpose of this paper is to generalize the equation to other geometries, and to phrase it as a simple ODE. Moreover we consider the Navier-Stokes equation

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We investigate Lp stability issues of small viscoelastic Poiseuilletype ows in two dimensions stemming from a model considered in Fang- Hua Lin, Chun Liu, and Ping Zhang (2005). We show local existence of perturbed ows of locally-in-time existing Poiseuille-type ows and global existence of the peturbed ows whenever the initial perturbation is small enough. In this case the perturbed ow decays exponentially. In all cases, the perturbations immediately regularize.

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The notions of involutes (also known as evolvents) and evolutes were studied by C. Huygens. For a regular plane curve, an involute of it is the trajectory described by the end of stretched string unwinding from a point of the curve. Even if a regular curve, the involute of the curve have singularities. By using a moving frame of the front and the curvature of the Legendre immersion in the unit tangent bundle, we define an involute of the front in the Euclidean plane and discuss properties of them. We also consider about relationship between evolutes and involutes of fronts without inflection points. As a result, we observe that the evolutes and the involutes of fronts without inflection points are corresponding to the differential and the integral in classical calculus.

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In this note, we establish the estimate on the Lorentz space L(3=2; 1) for vector elds in bounded domains under the assumption that the normal or the tangential component of the vector elds on the boundary vanishing. We prove that the L(3=2; 1) norm of the vector eld can be controlled by the norms of its divergence and curl in the atomic Hardy spaces and the L1 norm of the vector eld itself.

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We establish a Liouville type result for a backward global solution to the Navier- Stokes equations in the half plane with the no-slip boundary condition. No assump- tions on spatial decay for the vorticity nor the velocity eld are imposed. We study the vorticity equations instead of the original Navier-Stokes equations. As an appli- cation, we extend the geometric regularity criterion for the Navier-Stokes equations in the three-dimensional half space under the no-slip boundary condition.

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In this paper we derive a new type of pointwise decay estimates for solutions to the Cauchy problem for the wave equation in 2D, in the sense that one can diminish the weight in the time variable for the forcing term if it is compactly supported in the spatial variables. As an application of the estimate, we also establish an improved decay estimate for the solution to the exterior problem in 2D.

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We prove under certain assumptions that there exists a solution of the Schr¨odinger or the Heisenberg equation of motion generated by a linear operator H acting in some complex Hilbert space H, which may be unbounded, not symmetric, or not normal. We also prove that, under the same assumptions, there exists a time evolution operator in the interaction picture and that the evolution operator enjoys a useful series expansion formula. This expansion is considered to be one of the mathematically rigorous realizations of so called “time-ordered exponential”, which is familiar in the physics literature. We apply the general theory to prove the existence of dynamics for the mathematical model of Quantum Electrodynamics (QED) quantized in the Lorenz gauge, the interaction Hamiltonian of which is not even symmetric or normal.

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In general, before separating from a boundary, the flow moves toward reverse direction near the boundary against the laminar flow direction. Here in this paper, creation of the reverse flow phenomena (in a mathematical sense) is observed. More precisely, in the non-stationary two-dimensional Navier-Stokes equation with circular arc no-slip boundary conditions and diffusing laminar initial data (we de ne them rigorously in the paper), topologically changing flow, namely, instability is observed.

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We consider the surface diffusion flow equation when the curve is given as the graph of a function v(x; t) defined in a half line R+ = {x > 0} under the boundary conditions vx = tan > 0 and vxxx = 0 at x = 0. We construct a unique (spatially bounded) self-similar solution when the angle is sufficiently small.We further prove the stability of this self-similar solution. The problem stems from an equation proposed by W. W. Mullins (1957) to model formation of surface grooves on the grain boundaries, where the second boundary condition vxxx = 0 is replaced by zero slope condition on the curvature of the graph. For construction of a self-similar solution we solves the initial-boundary problem with homogeneous initial data. However, since the problem is quasilinear and initial data is not compatible with the boundary condition a simple application of an abstract theory for quasilinear parabolic equation is not enough for our purpose. We use a semi-divergence structure to construct a solution. 2010 Mathematics Subject Classification: Primary 35C06; Secondary 35G31, 35K59, 74N20. Keywords: Self-similar solution; Surface diffusion flow; Stability; Analytic semigroup; Mild solution.

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Let k and n be positive even integers. For a cuspidal Hecke eigenform h in the Kohnen plus space of weight k¡n/2+1/2 for Γ0(4), let f be the corresponding primitive form of weight 2k ¡ n for SL2(Z) under the Shimura correspondence, and In(h) the Duke-Imamo¯glu-Ikeda lift of h to the space of cusp forms of weight k for Spn(Z). Moreover, let φIn(h),1 be the first Fourier-Jacobi coefficient of In(h) and σn−1(φIn(h),1) be the cusp form in the generalized Kohnen plus space of weight k¡1/2 corresponding to φIn(h),1 under the Ibukiyama isomorphism. We then give an explicit formula for the Koecher-Maass series L(s, σn−1(φIn(h),1)) of σn−1(φIn(h),1) expressed in terms of the usual L-functions of h and f.

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A random dynamical systems model is studied to understand coupled dynamics of auditory area and motor area modulated by external force. We measure transfer entropy of coupled oscillators with the presence of noise to explain results of human brain wave experiments.

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We derive several formulae for the spectra of the second quantization operators in abstract fermionic Fock spaces.

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We extend the theory of viscosity solutions to a class of very singular nonlinear parabolic problems of non-divergence form in a periodic domain of an arbitrary dimension with di usion given by an anisotropic total variation energy. We give a proof of a comparison principle, an outline of a proof of the stability under approximation by regularized parabolic problems, and an existence theorem for general continuous initial data, which extend the results recently obtained by the authors.

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This paper studies stability of the Ekman boundary layer. We utilize a new approach developed by the authors in [12] based on Fourier transformed finite vector Radon measures which yields exponential stability of the Ekman spiral. By this method we can also derive very explicit bounds for solutions of the linearized and the nonlinear Ekman system. For example, we can prove these bounds to be uniform with respect to the angular velocity of rotation which has proved to be relevant for several aspects. Another advantage of this approach is that we obtain well-posedness in classes containing nondecaying vector fields such as almost periodic functions. These outcomes give respect to the nature of boundary layer problems and cannot be obtained by approaches in standard function spaces such as Lebesgue, Bessel-potential, H¨older or Besov spaces.

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A general anisotropic curvature flow equation with singular in- terfacial energy and spatially inhomogeneous driving force is considered for a curve given by the graph of a periodic function. We prove that the initial value problem admits a unique global-in-time viscosity solution for a general periodic continuous initial datum. The notion of a viscosity solution used here is the same as proposed by Giga, Giga and Rybka, who established a compar- ison principle. We construct the global-in-time solution by careful adaptation of Perron's method.

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Given a mean-convex domain Rn with boundary of class C2;1, we provide a representation formula for a boundary integral of the type Z @ f(k(x)) dHn􀀀1 where k 0 is the mean curvature of @ and f is non-increasing and su ciently regular, in terms of volume integrals and defect measure on the ridge set.

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In this paper, we show the existence of real-analytic stationary NavierStokesflows with isotropic streamlines in all latitudes in some simply-connected flow regionon a rotating round sphere. We also exclude the possibility of having a Poiseuille’sflow profile to be one of these stationary Navier-Stokes flows with isotropic streamlines.When the sphere is replaced by a 2-dimensional hyperbolic space, we also givethe analog existence result for stationary parallel laminar Navier-Stokes flows along acircular-arc boundary portion of some compact obstacle in the 2-D hyperbolic space.The existence of stationary parallel laminar Navier-Stokes flows along a straight boundaryof some obstacle in the 2-D hyperbolic space is also studied. In any one of thesecases, we show that a parallel laminar flow with a Poiseuille’s flow profile ceases to bea stationary Navier-Stokes flow, due to the curvature of the background manifold.

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We introduce a new notion of viscosity solutions for a class of very singular nonlinear parabolic problems of non-divergence form in a periodic domain of arbitrary dimension, whose di usion on flat parts with zero slope is so strong that it becomes a nonlocal quantity. The problems include the classical total variation flow and a motion of a surface by a crystalline mean curvature. We establish a comparison principle, the stability under approximation by regularized parabolic problems, and an existence theorem for general continuous initial data.

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We give an explicit formula for the twisted Koecher-Maa? series of the Duke-Imamoglu-Ikeda lift. As an application we prove a certain algebraicity result for the values of twisted Rankin-Selberg series at integers of half-integral weight modular forms.

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We give a moving frame of a Legendre curve (or, a frontal) in the unite tangent bundle and de ne a pair of smooth functions of a Legendre curve like as the curvature of a regular plane curve. The existence and uniqueness for Legendre curves are holded like as regular plane curves. It is quite useful to analyse the Legendre curves. As applications, we consider contact between Legendre curves and the arc-length parameter of Legendre immersions in the unite tangent bundle.

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The evolute of a regular curve in the Euclidean plane is given by not only the caustics of the regular curve, envelope of normal lines of the regular curve, but also the locus of singular loci of parallel curves. In general, the evolute of a regular curve have singularities, since such a point is corresponding to a vertex of the regular curve and there are at least four vertices for simple closed curves. If we repeated an evolute, we cannot define the evolute at a singular point. In this paper, we define an evolute of a front and give properties of such evolute by using a moving frame of a front and the curvature of the Legendre immersion. As applications, repeated evolutes can be well-defined and these are useful to recognize the shape of curves.

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We introduce a new level set method to simulate motion of spirals in a crystal surface governed by an eikonal-curvature ow equation. Our formulation allows collision of several spirals and different strength (different modulus of Burgers vectors) of screw dislocation centers. We represent a set of spirals by a level set of a single auxiliary function u minus a pre-determined multi-valued sheet structure function , which re ects the strength of spirals (screw dislocation centers). The level set equation used in our method for u 􀀀 is the same as that of the eikonal-curvature ow equation. The multi-valued nature of the sheet structure function is only invoked when preparing the initial auxiliary function, which is nontrivial, and in the nal step when extracting information such as the height of the spiral steps. Our simulation enables us not only to reproduce all speculations on spirals in a classical paper by Burton, Cabrera and Frank (1951) but also to nd several new phenomena.

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Detailed study of the characters of S∞(T)S∞(T), the wreath product of compact group T with the infinite symmetric group S∞, is indispensable for harmonic analysis on this big group. In preceding works, we investigated limiting behavior of characters of the finite wreath product Sn(T) as n→∞ and its connection with characters of S∞(T). This paper takes a dual approach to these problems. We study harmonic functions on Y(Tˆ), the branching graph of the inductive system of Sn(T)'s, and give a classification of the minimal nonnegative harmonic functions on it. This immediately implies a classification of the characters of S∞(T), which is a logically independent proof of the one obtained in earlier works. We obtain explicit formulas for minimal nonnegative harmonic functions on Y(Tˆ) and Martin integral expressions for harmonic functions.

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Let H0 and HI be a self-adjoint and a symmetric operator on a complex Hilbert space, respectively, and suppose that H0 is bounded below and the infimum E0 of the spectrum of H0 is a simple eigenvalue of H0 which is not necessarily isolated. In this paper, we present a new asymptotic perturbation theory for an eigenvalue E(λ) of the operator H(λ) := H0 +λHI (λ 2 Rn f0g) satisfying limλ!0 E(λ) = E0. The point of the theory is in that it covers also the case where E0 is a non-isolated eigenvalue of H0. Under a suitable set of assumptions, we derive an asymptotic expansion of E(λ) up to an arbitrary finite order of λ as λ ! 0. We apply the abstract results to a model of massless quantum fields, called the generalized spinboson model (A. Arai and M. Hirokawa, J. Funct. Anal. 151 (1997), 455–503) and show that the ground state energy of the model has asymptotic expansions in the coupling constant λ as λ ! 0.

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We give a direct proof for the analyticity of the Stokes semigroup in spaces of bounded functions. This was recently proved by an indirect argument by the first and second authors for a class of domains called strictly admissible domains including bounded and exterior domains. Invoking the strictly admissibility, our approach is based on an adjustment of a standard resolvent estimate method for general elliptic operators introduced by K. Masuda (1972) and H. B. Stewart (1974). The resolvent approach in particular clarifies the sectorial region, Re z > 0 for z ∈ C for which the Stokes semigroup has an analytic continuation in spaces of bounded functions.

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Some aspects of spectral analysis of an effective Hamiltonian in non-relativistic quantum electrodynamics are reviewed. The Lamb shift of a hydrogen-like atom is derived as the lowest order approximation (in the fine structure constant) of an energy level shift of the effective Hamiltonian. Moreover, a general class of effective operators is presented, which comes from models of an abstract quantum system interacting with a Bose field.

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In the split G2-geometry, we study the correspondence found by E. Cartan between the Cartan distribution and the contact distribution with Monge structure on spaces of five variables. Then the generic classification is given on singularities of tangent surfaces to Cartan curves and to Monge curves via the viewpoint of duality. The geometric singularity theory for simple Lie algebras of rank 2, namely, for A2,C2 = B2 and G2 is established.

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A planar anisotropic curvature flow equation with constant driving force term is considered when the interfacial energy is crystalline. The driving force term is given so that a closed convex set grows if it is sufficiently large. If initial shape is convex, it is shown that a flat part called a facet (with admissible orientation) is instantaneously formed. Moreover, if the initial shape is convex and slightly bigger than the critical size, the shape becomes fully faceted in a finite time provided that the Frank diagram of interfacial energy density is a regular polygon centered at the origin. The proofs of these statements are based on approximation by crystalline algorithm whose foundation was established a decade ago. Our results indicate that the anisotropy of intefacial energy plays a key role when crystal is small in the theory of crystal growth. In particular, our theorems explain a reason why snow crystal forms a hexagonal prism when it is very small.

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We consider Hilbert space representations of a generalization of canonical commutation relations (CCR): [Xj ,Xk] := XjXk ¡ XkXj = iΘjkI (j, k = 1, 2, . . . , 2n), where Xj ’s are elements of an algebra with identity I, i is the imaginary unit, and Θjk is a real number with Θjk = ¡Θkj (j, k = 1, . . . , 2n). Some basic aspects on Hilbert space representations of the generalized CCR (GCCR) are discussed. We define a Schrödinger type representation of the GCCR by analogy with the usual Schrödinger representation of the CCR with n degrees of freedom. Also we introduce a Weyl type representation of the GCCR. The main result of the present paper is a uniqueness theorem on Weyl representations of the GCCR.

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This is the continuation of our previous paper “Contact Geometry of Second Order I” , where we have formulated the contact equivalence of systems of second order partial differential equations for a scalar function as the geometry of PD manifolds of second order. In this paper, we will discuss the Two Step Reduction procedure in Contact Geometry of Second Order. In fact we establish the Second Reduction Theorem for PD manifolds (R;D1,D2) of second order admitting the first order covariant systems ˜N . Utilizing the covariant system ˜N , we construct the intermediate object (W;C:,N), called the IG manifold of corank r, as a submanifold of the Involutive Grassmann bundle Ir(J) over the contact manifold (J,C), where J = R/Ch (D1). We will seek the condition when the equivalence of (R;D1,D2) is reducible to that of (W;C∗,N). Moreover, when Ch (N) is non-trivial, the equivalence of (W;C∗,N) is further reducible to that of (Y ;D∗ N,DN), where Y = W/Ch (N). This theorem gives a sufficient condition for the existence of higher dimensional characteristics of (R;D1,D2). By analyzing the construction parts of the Two Step Reduction procedure, we will show several examples of Parabolic Geometries, which are, through the Second Reduction Theorem, associated with the geometry of PD manifolds of second order.

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The algebraic notion of openings of a map-germ is introduced in this paper. An opening separates the self-intersections of the original mapgerm, preserving its singularities. The notion of openings is different from the notion of unfoldings. Openings do not unfold the singularities. For example, the swallowtail is an opening of the Whitney’s cusp mapgerm from plane to plane and the open swallowtail is a versal opening of them. Openings of map-germs appear as typical singularities in several problems of geometry and its applications. The notion of openings has close relations to isotropic map-germs in a symplectic space and integral map-germs in a contact space. We describe the openings of Morin singularities, namely, stable unfoldings of map-germs of corank one. The relation of unfoldings and openings are discussed. Moreover we provide a method to construct versal openings of map-germs and give versal openings of stable map-germs (R4, 0) ! (R4, 0). Lastly the relation of lowerable vector fields and openings is discussed.

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By taking embedded tangent spaces to a submanifold in an affine space, we obtain a ruled variety, which is called the tangent variety to the submanifold and has non-isolated singularities in general. We explain a method of modifications of map-germs, which we call openings of map-germs, and study the local classification problem of tangent varieties in terms of the opening construction. In particular, we present the general stable classification result of tangent varieties to generic submanifolds of sufficiently high codimension.

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We are concerned with non-local diffusion equations in the presence of a divergence free drift term. By using the classical Nash approach we show the existence of fundamental solutions, together with the continuity estimates, under weak regularity assumptions on the kernel of the diffusion term and the velocity of the drift term. As an application, our result gives the alternative proof of the global regularity for the two-dimensional dissipative quasi-geostrophic equations in the critical case.

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We investigate some non-local diffusion equations in the presence of a divergence free drift term. We derive pointwise upper bounds for fundamental solutions under low regularity assumptions for the velocity of the drift term.

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A random dynamics with two stochastic terms is modeled based on a time series of physiologi- cal experimental data to study synchrony between human heartbeats and pedaling rhythms modulated by music. We investigate reproduced time series, rotation numbers, and invariant densities in the model to explain transitory stag- nation motion of synchrony in the experiments.

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The Stokes semigroup is extended to an analytic semigroup in spaces of bounded functions in an exterior domain with C3 boundary. Some of these spaces include vector fields non-decaying at the space infinity. Moreover, uniform bounds by a sup-norm of initial velocity are established in finite time for second spacial derivatives of velocity and also for gradient of pressure to the Stokes equations.

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In general, before separating from a boundary, the flow moves toward reverse direction near the boundary against the laminar flow direction. Here in this paper, a clue of such reverse flow phenomena (in the mathematical sense) is observed. More precisely, the non-stationary two-dimensional Navier-Stokes equation with an initial datum having a parallel laminar flow (we de ne it rigorously in the paper) is considered. We show that the direction of the material di erentiation is opposite to the initial flow direction and e ect of the material di erentiation (inducing the reverse flow) becomes bigger when the curvature of the boundary becomes bigger. We also show that the parallel laminar flow cannot be a stationary Navier-Stokes flow near a portion of the boundary with nonzero curvature.

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We consider an area minimizing scheme for anisotropic mean curvature flow originally due to Chambolle (2004). We show the convergence of the scheme to anisotropic mean curvature flow in the sense of Hausdorff distance by the level set method provided that no fattening occurs.

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Let μ be a R probability measure on the set R of real numbers and μˆ(t) := R e¡itλdμ(λ) (t 2 R) be the Fourier transform of μ (i is the imaginary unit). Then, under suitable conditions, asymptotic formulae of jˆμ(t/x)j2x in 1/x as x ! 1 are derived. These results are applied to the so-called quantum Zeno effect to establish asymptotic formulae of its occurrence probability in the inverse of the number N of measurements made in a time interval as N ! 1.

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In this paper we consider the curves in the unit 3-sphere. The unit 3-sphere can be canonically embedded in the lightcone and de Sitter 4-space in Loretnz-Minkowski 5- space. We investigate these curves in the framework of the theory of Legendrian dualities between pseudo-spheres in Lorentz-Minkowski 5-space.

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In this paper, we construct one-parameter families of new extrinsic di erential geometries on spacelike submanifolds of codimension two in Lorentz-Minkowski space.

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We consider the Navier-Stokes equations for viscous incompressible flows in the half plane under the no-slip boundary condition. By using the vorticity formulation we prove the (local in time) convergence of the Navier-Stokes flows to the Euler flows outside a boundary layer and to the Prandtl flows in the boundary layer at the inviscid limit when the initial vorticity is located away from the boundary.

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The lightlike geometry of codimension two spacelike submanifolds in Lorentz-Minkowski space has been developed in [15] which is a natural Lorentzian analogue of the classical Euclidean differential geometry of hypersurfaces. In this paper we investigate a special class of surfaces (i.e., marginally trapped surfaces) in Minkowski space-time from the view point of the lightlike geometry.

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Some aspects of spectral analysis of an effective Hamiltonian in non-relativistic quantum electrodynamics are reviewed. The Lamb shift of a hydrogen-like atom is derived as the lowest order approximation (in the fine structure constant) of an energy level shift of the effective Hamiltonian. Moreover, a general class of effective operators is presented, which comes from models of an abstract quantum system interacting with a Bose field.

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We introduce the totally absolute lightcone curvature for a spacelike submanifold with general codimension and investigate global properties of this curvature. One of the consequences is that the Chern-Lashof type inequality holds. Then the notion of lightlike tightness is naturally induced. Moreover, the lightcone Willmore conjecture is proposed.

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The singular point of the Gauss map of a hypersurface in Euclidean space is the parabolic point where the Gauss-Kronecker curvature vanishes. It is well-known that the contact of a hypersurface with the tangent hyperplane at a parabolic point is degenerate. The parabolic point has been investigated in the previous research by applying the theory of Lagrangian or Legendrian singularities. In this paper we give a new interpretation of the singularity of the Gauss map from the view point of the theory of wave front propagations.

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In this paper we introduce the notion of pseudo-spherical evolutes of curves on a spacelike surface in three dimensional Lorentz-Minkowski space which is analogous to the notion of evolutes of curves on the hyperbolic plane. We investigate the the singularities and geometric properties of pseudo-spherical evolutes of curves on a spacelike surface.

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An incompressible two-dimensional flow on a β plane is considered. The β plane is a tangent plane of a sphere to approximately describe fluid motion on a rotating sphere assuming that the Coriolis parameter is a linear function of the latitude. Rossby waves are expected to dominate the β plane dynamics, and here in this paper, a mathematical support for the crucial role of the resonant pairs of the Rossby waves is given.

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Consider the solution u(x; t) of the heat equation with initial data u0. The diffusive sign SD[u0](x) is de ned by the limit of sign of u(x; t) as t ! 0. A sufficient condition for x 2 Rd and u0 such that SD[u0](x) is well-de ned is given. A few examples of u0 violating and ful lling this condition are given. It turns out that this diffusive sign is also related to variational problem whose energy is the Dirichlet energy with a delty term. If initial data is a difference of characteristic function of two disjoint sets, it turns out that the boundary of the set SD[u0](x) = 1 (or 􀀀1) is roughly an equi-distance hypersurface from A and B and this gives a separation of two data sets.

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In this paper we consider curves on a spacelike surface in Lorentz-Minkowski 3-space. We introduce new geometric invariants for these curves. As an application of the unfolding theory of functions, we investigate the local and global propereties of these invariants.

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We construct examples of ill-posedness of solutions of the 2D QG and the 3D Euler equations in the Besov and the logarithmic Lipschitz spaces.

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In this paper we extend the notion of specialization functor to the case of several closed submanifolds satisfying some suitable conditions. Applying this functor to the sheaf of Whitney holomorphic functions we construct different kinds of sheaves of multi-asymptotically developable functions, whose definitions are natural extensions of the definition of strongly asymptotically developable functions introduced by Majima.

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We study the motion of regular bent rectangles driven by singular curvature flow with a driving term. The curvature is being interpreted as a solution to a minimization problem. The evolution equation becomes in a local coordinate a system of Hamilton-Jacobi equations with free boundaries, coupled to a system of ODE’s with nonlocal nonlinearities. We establish local-in-time existence of variational solutions to the flow and uniqueness is proved under additional regularity assumptions on the data.

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We consider the Navier-Stokes equations for viscous incompressible flows in the half plane under the no-slip boundary condition. In this paper we first establish a solution formula for the vorticity equations through the appropriate vorticity formulation. The formula is then applied to establish the asymptotic expansion of vorticity fields at ! 0 that holds at least up to the time c 1=3, where is the viscosity coefficient and c is a constant. As a consequence, we get a natural sufficient condition on the initial data for the vorticity to blow up in the inviscid limit, together with explicit estimates.

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A new notion of a viscosity solution for Eikonal equations in a general metric space is introduced. A comparison principle is established. The existence of a unique solution is shown by constructing a value function of the corresponding optimal control theory. The theory applies to in- nite dimensional setting as well as topological networks, surfaces with singularities.

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We prove a formula of Petersson’s type for Fourier coefficients of Siegel cusp forms of degree 2 with respect to congruence subgroups, and as a corollary, show upper bound estimates of individual Fourier coefficient. The method in this paper is essentially a generalization of Kitaoka’s previous work which studied the full modular case, but some modification is necessary to obtain estimates which are sharp with respect to the level aspect.

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We construct a Poiseuille type ow which is a bounded entire solution of the nonstationary Navier-Stokes and the Stokes equations in a half space with non-slip boundary condition. Our result in particular implies that there is a nontrivial solution for the Liouville problem under the non-slip boundary condition. A review for cases of the whole space and a slip boundary condition is included.

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Mathematical investigations on quantum Zeno effect (QZE) are presented, including the following aspects: (i) QZE by frequent measurements made by an arbitrary partition of a time interval [0, t] (t > 0); (ii) non-occurrence of QZE for vector states which are not in the domain of the Hamiltonian of the quantum system under consideration; (iii) asymptotic behavior of the survival probability characterizing QZE in the number N of divisions of [0, t]; (iv) QZE along a curve in the Hilbert space of state vectors.

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We study the initial-value problem for a Hamilton-Jacobi equation whose Hamiltonian is dis- continuous with respect to state variables. Our motivation comes from a model describing the two dimensional nucleation in crystal growth phenomena. A typical equation has a semicontinu- ous source term. We introduce a new notion of viscosity solutions and prove among other results that the initial-value problem admits a unique global-in-time uniformly continuous solution for any bounded uniformly continuous initial data. We also give a representation formula of the solution as a value function by the optimal control theory with a semicontinuous running cost function.

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Existence of local-in-time unique solution and loss of smoothness of full Magnet-Hydro-Dynamics system (MHD) is considered for periodic initial data. The result is proven using Fujita-Kato's method in l^1 based (for the Fourier coeffcients) functional spaces enabling us to easily estimate nonlinear terms in the system as well as solutions to Maxwells's equations. A loss of smoothness result is shown for the velocity and magnetic field. It comes from the damped-wave operator which does not have any smoothing effect.

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We investigate large time existence of solutions of the Navier-Stokes-Boussinesq equations with spatially almost periodic large data when the density strati cation is su - ciently large. In 1996, Kimura and Herring [16] examined numerical simulations to show a stabilizing e ect due to the strati cation. They observed scattered two-dimensional pancake- shaped vortex patches lying almost in the horizontal plane. Our result is a mathematical justi cation of the presence of such two-dimensional pancakes. To show the existence of solutions for large times, we use `1-norm of amplitudes. Existence for large times is then proven using techniques of fast singular oscillating limits and bootstrapping argument from a global-in-time unique solution of the system of limit equations.

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The first goal of our paper is to give a new type of regularity criterion for solutions u to Navier-Stokes equation in terms of some supercritical function space condition u 2 L1(L ; ) (with 3 4 (1712 􀀀 1) < < 3) and some exponential control on the growth rate of div( u juj ) along the streamlines of u. This regularity criterion greatly improves a previous result of the rst author. However, we also point out that totally new idea which involves the use of the new supercritical function space condition is necessary for the success of our new regularity criterion in this paper. The second goal of our paper is to construct a divergence free vector eld u within a owinvariant tubular region with increasing twisting of streamlines towards one end of a bundle of streamlines. The increasing twisting of streamlines is controlled in such a way that the associated quantities kukLp (2 < p < 3) and k div( u juj )kL6 blow up while preserving the nite energy property u 2 L2 at the same time. We also brie y mention how this construction is related to the regularity criterion proved in our paper.

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In this paper we prove Liouville type theorem for the stationary Navier-Stokes equations on R3. More speci cally, if a solution u 2 _H 1(R3) \ L1(R3) to the stationary Navier-Stokes system satis es additional conditions characterized by the de- cays near in nity and by the oscillation, then we show that u = 0.

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For a smooth domain D containing the origin, we consider a vector eld u 2 C1(Dn f0g;R3) with div u 0 and exclude certain types of possible isolated singularities at the origin, based on the geometry of streamlines that go near that possible singular point.

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A random dynamics is extracted from time series of laminar-turbulent transition in rotating fluid in an open cylinder. We focus on the dynamics of the surface height in the central region and measure switching dynamics between different quasi-stationary states and intensity of underlying turbulence. Density of return map is constructed from an one dimensional map with an stochastic term from the experimental data. It is shown that the random dynamics whose noise amplitude depends on the slow variable describes the observed macroscopic features of rotating fluid in terms of noise-induced phenomena.

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In this paper we consider a hypersurface of the graph of the mean curvature flow with transport term. The existence of the mean curvature flow with transport term was proved by Liu, Sato and Tonegawa by using geometric measure theory techniques. We give a proof of the gradient estimates and the short time existence for the mean curvature flow with transport term by applying the backward heat kernel.

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In this paper we develop a new approach to rotating boundary layers via Fourier transformed finite vector Radon measures. As an application we consider the Ekman boundary layer. By our methods we can derive very explicit bounds for existence intervals and solutions of the linearized and the nonlinear Ekman system. For example, we can prove these bounds to be uniform with respect to the angular velocity of rotation which has proved to be relevant for several aspects (see introduction). Another advantage of our approach is that we obtain well-posedness in classes containing nondecaying vector fields such as almost periodic functions. These outcomes give respect to the nature of boundary layer problems and cannot be obtained by approaches in standard function spaces such as Lebesgue, Bessel-potential, H¨older or Besov spaces.

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A. Kawauchi has introduced the notion of warping degrees of knot diagrams and A. Shimizu has given an inequality for warping degrees and crossing number of knot diagrams in the paper [5]. In this paper, we extend the notion of warping degrees and Shimizu’s inequality to nanowords. Moreover, to describe the condition for the equality, we introduce the new notion on nanowords, ”the alternating nanowards”, which corresponds to the alternating knot diagrams.

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We consider random perturbations of non-singular measur- able transformations S on [0; 1]. By using the spectral decomposition theorem of Komornik and Lasota, we prove that the existence of the invariant densities for random perturbations of S. Moreover the densi- ties for random perturbations with small noise strongly converges to the deinsity for Perron-Frobenius operator corresponding to S with respect to L1([0; 1])-norm.

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Let K = Q( p ¡D) be an imaginary quadratic field with discriminant ¡D, and χ the Dirichlet character corresponding to the extension K/Q. Let m = 2n or 2n + 1 with n a positive integer. Let f be a primitive form of weight 2k+1 and character χ for Γ0(D), or a primitive form of weight 2k for SL2(Z) according as m = 2n, or m = 2n+1. For such an f let Im(f) be the lift of f to the space of modular forms of weight 2k+2n and character det−k−n for the Hermitian modular group Γ(m) K constructed by Ikeda. We then express the period hIm(f), Im(f)i of Im(f) in terms of special values of the adjoint L-functions of f twisted by the character χ. This poves the conjecture concerning the period of the Ikeda lift proposed by Ikeda.

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We study large-time asymptotics for a class of noncoercive Hamilton-Jacobi equations with Dirichlet boundary condition in one space dimension. We prove that the average growth rate of a solution is constant only in a subset of the whole domain and give the asymptotic pro¯le in the subset. We show that the large-time behavior for noncoercive problems may depend on the space variable in general, which is di®erent from the usual results under the coercivity condition. This work is an extension with more rigorous analysis of a recent paper by E. Yokoyama, Y. Giga and P. Rybka, in which a growing crystal model is established and the asymptotic behavior described above is first discovered.

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We give the generic classification on singularities of tangent surfaces to Lengendre curves and to null curves by using the contact-cone duality between the contact 3-sphere and the Lagrange-Grassmannian with cone structure of a symplectic 4-space. As a consequence, we observe that the symmetry on the lists of such singularities is breaking for the contact-cone duality, compared with the ordinary projective duality.

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We investigate the large-time behavior of viscosity solutions of Hamilton- Jacobi equations with noncoercive Hamiltonian in a multidimensional Euclidean space. Our motivation comes from a model describing growing faceted crystals recently discussed by E. Yokoyama, Y. Giga and P. Rybka. Surprisingly, growth rates of viscosity solutions of these equations depend on x-variable. In a part of the space called the effective domain, growth rates are constant but outside of this domain, they seem to be unstable. Moreover, on the boundary of the effective domain, the gradient with respect to x-variable of solutions blows up as time goes to infinity. Therefore, we are naturally led to study singular Neumann problems for stationary Hamilton-Jacobi equations. We establish the existence, stability and comparison results for singular Neumann problems and apply the results for a large-time asymptotic profile on the effective domain of viscosity solutions of Hamilton-Jacobi equations with noncoercive Hamiltonian.

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Cellular automata are discrete dynamical systems whose configurations are determined by local rules acting on each cell in synchronous. Topological entoropy is a tool for measuring the complexity of these dynamical systems. In this paper we estimate topological entropy of a two-dimensional nonlinear cellular automaton. The method we use is that a one-dimensional cellular automaton with positive topological entoropy is “naturally” embedded into the twodimensional cellular automaton. Hence we obtain a multidimensional cellular automaton with infinite topological entoropy.

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The final open part of Strauss’ conjecture on semilinear wave equations was the blow-up theorem for the critical case in high dimensions. This problem was solved by Yordanov and Zhang [17], or Zhou [20] independently. But the estimate for the lifespan, the maximal existence time, of solutions was not clarified in both papers. In this paper, we refine their theorems and introduce a new iteration argument to get the sharp upper bound of the lifespan. As a result, with the sharp lower bound by Li and Zhou [9], the lifespan T(ε) of solutions of utt − Δu = u2 in R4 × [0,∞) with the initial data u(x, 0) = εf(x), ut(x, 0) = εg(x) of a small parameter ε > 0, compactly supported smooth functions f and g, has an estimate exp ( cε −2) ≤ T(ε) ≤ exp ( Cε −2) , where c and C are positive constants depending only on f and g. This upper bound has been known to be the last open optimality of the general theory for fully nonlinear wave equations.

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A Cauchy problem for a parabolic-elliptic system of drift-di usion type is considered. The problem is formally of the form Ut = r (rU 􀀀 Ur(􀀀 )􀀀1U): This system describes a mass-conserving aggregation phenomenon including gravitational collapse and bacterial chemotaxis. Our concern is the asymptotic behavior of blowup solutions when the blowup is type I in the sense that its blowup rate is the same as the corresponding ordinary di erential equation yt = y2 (up to a multiple constant). It is shown that all type I blowup is asymptotically (backward) self-similar provided that the solution is radial, nonnegative when the blowup set is a singleton and the space dimension is greater than or equal to three. 2000 Mathematics Subject Classi cation. 35K55, 35K57, 92C17. 1

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This paper corrects Asakura's observation on semilinear wave equations with non-compactly supported data by showing a sharp blow-up theorem for classical solutions. We know that there is no global in time solution for any power nonlinearity if the spatial decay of the initial data is weak, in spite of nite propagation speed of the linear wave. Our theorem clari es the nal criterion on such a phenomenon.

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We investigate spectral properties of an effective Hamiltonian which is obtained as a scaling limit of the Pauli-Fierz model in nonrelativistic quantum electrodynamics. The Lamb shift of a hydrogen-like atom is derived as the lowest order approximation (in the fine structure constant) of an energy level shift of the effective Hamiltonian.

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Inter alia we prove L1 maximal regularity for the Laplacian in the space of Fourier transformed finite Radon measures FM. This is remarkable, since FM is not a UMD space and by the fact that we obtain Lp maximal regularity for p = 1, which is not even true for the Laplacian in L2. We apply our result in order to construct strong solutions to the Navier-Stokes equations for initial data in FM in a rotating frame. In particular, the obtained results are uniform in the angular velocity of rotation.

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For smooth explicit n-th order ordinary differential equations, there exists a unique solution with initial condition and hence exists an n-parameter family of solutions at least locally. On the other hand, for smooth implicit ordinary differential equations, existence and uniqueness for solutions with initial condition does not hold. In this paper, we give a necessary and sufficient condition for existence of an n-parameter family of geometric solutions in the smooth category. Moreover, we give a sufficient condition that implicit ordinary differential equations have a unique geometric solution with initial condition. As a consequence, we classify completely integrable first and second ordinary differential equations in detail.

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We study three singular parabolic evolutions: the second-order total variation ｆｌow, the fourth-order total variation ow, and a fourth-order surface di usion law. Each has the property that the solution becomes identically zero in nite time. We prove scale-invariant estimates for the extinction time, using a simple argument which combines an energy estimate with a suitable Sobolev-type inequality.

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We consider a one-parameter family of new extrinsic differential geometries on hypersurfaces in Hyperbolic space. Recently, the second author and his collaborators have constructed a new geometry which is called horospherical geometry on Hyperbolic space. There is another geometry which is the famous Gauss-Boryay-Robechevski geometry (i.e., the hyperbolic geometry) on Hyperbolic space. The slant geometry is a one-parameter family of geometries which connect these two geometries. Moreover, we construct a oneparameter family of geometries on spacelike hypersurfaces in de Sitter space.

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In this paper, we consider a special class of the surfaces in 3-sphere de ned by oneparameter families of great circles. We give a generic classi cation of singularities of such surfaces and investigate the geometric meanings from the view point of spherical geometry.

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This paper studies singular diffusion equations whose diffusion effect is so strong that the speed of evolution becomes a nonlocal quantity. Typical examples include the total variation flow as well as crystalline flow which are formally of second order. This paper includes fourth order models which are less studied compared with second order models. A typical example of this model is an H−1 gradient flow of total variation. It turns out that such a flow is quite different from the second order total variation flow. For example, we prove that the solution may instantaneously develop jump discontinuity for the fourth order total variation flow by giving an explicit example.

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We study numerically the long-time evolution of the surface quasi-geostrophic equation with generalised viscosity of the form $(-¥Delta)^¥alpha$, where global regularity has been proved mathematically for the subcritical parameter range $¥alpha ¥geq ¥frac{1}{2}$. Even in the supercritical range, we have found numerically that smooth evolution persists, but with a very slow and oscillatory damping in the long run. A subtle balance between nonlinear and dissipative terms is observed therein. Notably, qualitative behaviours of the analytic properties of the solution do not change in the super and subcritical ranges, suggesting the current theoretical boundary $¥alpha =¥frac{1}{2}$ is of technical nature.

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We consider a quantum system of a Dirac particle interacting with the quantum radiation field, where the Dirac particle is in a 4 £ 4-Hermitian matrix-valued potential V . Under the assumption that the total Hamiltonian HV is essentially self-adjoint (we denote its closure by ¯HV ), we investigate properties of the Heisenberg operator xj(t) := eit ¯HV xje−it ¯HV (j = 1, 2, 3) of the j-th position operator of the Dirac particle at time t 2 R and its strong derivative dxj(t)/dt (the j-th velocity operator), where xj is the multiplication operator by the j-th coordinate variable xj (the j-th position operator at time t = 0). We prove that D(xj ), the domain of the position operator xj , is invariant under the action of the unitary operator e−it ¯HV for all t 2 R and establish a mathematically rigorous formula for xj(t). Moreover, we derive asymptotic expansions of Heisenberg operators in the coupling constant q 2 R (the electric charge of the Dirac particle).

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We give a geometric nonblow up criterion on the direction of the vorticity for the three dimensional Navier-Stokes flow whose initial data is just bounded and may have infinite energy. We prove that under a restriction on behavior in time (type I condition) the solution does not blow up if the vorticity direction is uniformly continuous at place where vorticity is large. This improves the Lipschitz regularity condition for the vorticity direction first introduced by P. Constantin and C. Fefferman (1993) for finite energy (weak) solution. Our method is based on a simple blow up argument which says that the situation looks like two-dimensional under continuity of the vorticity direction. We also discuss about boundary value problems.

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We show a comparison principle for viscosity super- and subsolutions to Hamilton- Jacobi equations with discontinuous Hamiltonians. The key point is that the Hamiltonian depends upon u and it has a special structure. The supersolution must enjoy some additional regularity.

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Let k and n be positive even integers. For a primitive form f in S2k−n(SL2(Z)), let In(f) be the Duke-Imamo¯glu-Ikeda lift of f to Sk(Spn(Z)), and e f the cusp form in Kohnen’s plus subspace of weight k¡n/2+1/2 for Γ0(4) corresponding to f under the Shimura correspondence. We then express the ratio hIn(f), In(f)i h e f, e fi of the period of In(f) to that of e f in terms of special values of certain L-functions of f. This proves the conjecture proposed by Ikeda [Ike06] concerning the period of the Duke-Imamo¯glu-Ikeda lift.

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In this paper, we consider one-parameter families of new extrinsic differential geometries on spacelike hypersurfaces in the lightcone. These geometries are constructed by applying two of one-parameter families of Legendrian dualities between pseudo-spheres in Lorentz-Minkowski space. These Legendrian dualities have been recently given as a part of extensions of the mandala of Legendrian dualities in the previous research of the authors.

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For any bounded (real) initial data it is known that there is a unique global solution to the two dimensional Navier-Stokes equations. This paper is concerned with a bound for the sum of the modulus of amplitudes when initial velocity is spatially almost periodic in 2D. In the case of general dimension, it is bounded on local time of existence shown by Giga, Inui, Mahalov and Matsui in 2005. A class of initial data is given such that the sum of the modulus of amplitudes of a solution is bounded on any nite time interval. It is shown by an explicit example that such a bound may diverge to in nity as the time goes to in nity at least for complex initial data.

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We investigate the differential geometry of spacelike submanifolds of codimension at least two in de Sitter space as an application of the theory of Legendrian singularities. We also discuss related geometric property of spacelike hypersurfaces in de Sitter space. Mathematics Subject Classification (2000): 53A35, 53B30, 58C25. Key words: de Sitter space, spacelike submanifold, spacelike hypersurface.

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We explain how the Bloch-Kato conjecture leads us to the following conclusion: a large prime dividing a critical value of the L-function of a classical Hecke eigenform f of level 1, should often also divide certain ratios of critical values for the standard L-function of a related genus two (and in general vector-valued) Hecke eigenform F. The relation between f and F (Harder’s conjecture in the vector-valued case) is a congruence involving Hecke eigenvalues, modulo the large prime. In the scalar-valued case we prove the divisibility, subject to weak conditions. In two instances in the vector-valued case, we confirm the divisibility using elaborate computations involving special differential operators. These computations do not depend for their validity on any unproved conjecture.

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In this paper we show one-parameter families of Legendrian dualities between pseudo-spheres in Lorentz-Minkowski space which are the extensions of four dualities in the previous research. Moreover, we construct new extrinsic differential geometries on spacelike hypersurfaces in these pseudo-spheres as applications of such extensions of the mandala.

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Using the Legnedrian duarities between surfaces in pseudo-spheres in Lorentz-{Minkow}{ski} 4-space, we study various kind of flat surfaces in pseudo-spheres. We consider a surface in the pseudo-sphere and its dual surface. Flatness of a surface is defined by the degeneracy of the dual surface similar to the case for the Gauss map of a flat surface in the Euclidean space. We study singularities of these flat surfaces and dualities of singularities.

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Let D be a bounded domain in Rn with n >= 2. For a function f on ∂D we denote by HDf the Dirichlet solution of f over D. It is classical that if D is regular, then HD maps the family of continuous boundary functions to the family of harmonic functions in D continuous up to the boundary ∂D. We show that the better continuity of a boundary function f ensures the better continuity of HDf in the context of general modulus of continuity.

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We study the motion of noncompact hypersurfaces moved by their ean curvature obtained by a rotation around x-axis of the graph function y = u(x, t) (defined for all x 2 R). We are interested o estimate its profile when the hypersurface closes open ends at the uenching (pinching) time T. We estimate its profile at the quenching ime from above and below. We in particular prove that u(x, T) » xj¡a as jxj ! 1 if u(x, 0) tends to its infimum with algebraic rate xj¡2a (as jxj ! 1 with a > 0).

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In word and phrase theory of Turaev, we interpret links or virtual inks as equivalences of phrases over an alphabet consisting four letters. V. Turaev onstructed a version of the Jones polynomial for phrases. We study the ell-definedness of the Jones polynomial for phrases in word theory. On the ther hand, M. Khovanov introduced a collection of homology groups which is strictly stronger link invariant than the Jones polynomial and O. Viro reconstructed hese Khovanov homology groups. We construct phrase invariants as the omology groups of certain chain complexes for phrases where the coefficients of he Jones polynomial are the Euler characteristics of these complexes using the iro’s method of Khovanov theory. The invariance of these homology groups is howed in only terminology of Turaev’s theory of phrases. Moreover, we apply he homology groups to getting invariants for an other type of phrases over an lphabet consisting any letters.

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In [8], we gave the normalization of the standard zeta alues for Siegel modular forms, and considered the relationship etween such values and congruence of cuspidal Hecke eigenforms. n this paper we give more reasonable normalization for such values nd improve our previous result.

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In 2005 V. Turaev introduced the theory of topology of words and hrases. Turaev defined an equivalence relation on generalized words and phrases hich is called homotopy. This is suggested by the Reidemeister moves in the not theory. Then Turaev gave the homotopy classification of generalized words ith less than or equal to five letters. In this paper we give the classification of eneralized phrases up to homotopy with less than or equal to three letters. To o this we construct a new homotopy invariant for nanophrases over any α.

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We discuss the motion of noncompact axisymmetric hypersurfaces Tt volved by mean curvature flow. Our study provides a class of hypersurfaces hat share the same quenching time with that of the shrinking ylinder evolved by the flow and prove that they tend to a smooth hypersurface aving no pinching neck and having closed ends at infinity of the xis of rotation as the quenching time is approached. Moreover, they are ompletely characterized by a condition on initial hypersurface.

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We evaluate the maximum value of the unique positive solution to < u + f(u) = 0 in Rn, im x|→∞ (x) = 0, here (u) = ¡ωu + up ¡ uq, ω > 0, q > p > 1. t is known that a positive solution to this problem exists if and only if (u) := u f(s)ds > 0 for some u > 0. Moreover, Ouyang and Shi in 1998 ound that the solution is unique. In the present paper we investigate the aximum value of the solution. The key idea is to examine the function efined from the nonlinearity, which arises from the well-known Pohozaev dentity.

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We consider the problem < u + f(u) = 0 in Rn, im xj!1 (x) = 0, 1) here (u) = ¡ωu + up ¡ uq, ω > 0, q > p > 1. t is R known that a positive solution to (1) exists if and only if F(u) := u f(s)ds > 0 for some u > 0. Moreover, Ouyang and Shi in 1998 found hat the solution is unique if f satifies furthermore the condition that f(u) := (uf u)) (u) ¡ uf u)2 < 0 for any u > 0. In the present

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We consider the problem < A ¡ A + Ap ¡ kA n A2dx = 0 in Rn, im x|→∞ (x) = 0, here p > 1, k > 0 are constants. We classify the existence of all possible ositive solutions to this problem.

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We consider the uniqueness of positive solutions to < u ¡ ωu + up ¡ u2p−1 = 0 in Rn, im x|→∞ (x) = 0. 1) t is known that for fixed p > 1, a positive solution to (1) exists if and only f ω 2 (0, ωp), where ωp := p + 1)2 . We deduce the uniqueness in the ase where ω is close to ωp, from the argument in the classical paper by eletier and Serrin [9], thereby recovering a part of the uniqueness result f Ouyang and Shi [8] for all ω 2 (0, ωp).

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We consider the weighted mean curvature flow with a driving term in the plane. For nisotropy functions this evolution problem degenerates to a first order Hamilton-Jacobi quation with a free boundary. The resulting problem may be written as a Hamilton-Jacobi quation with a spatially non-local and discontinuous Hamiltonian. We prove existence nd uniqueness of solutions. On the way we show a comparison principle and a stability heorem for viscosity solutions.

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We study horo-tight immersions of manifolds into hyperbolic spaces. The main result gives several characterizations of horo-tightness of spheres, answering a question proposed by T. Cecil and P. Ryan. For instance, we prove that a sphere is horo-tight if and only if it is tight in the hyperbolic sense. For codimension bigger than one, it follows that horo-tight spheres in hyperbolic space are metric spheres. We also prove that horo-tight hyperspheres are characterized by the property that both of its total absolute horospherical curvatures attend their minimum value. We also introduce the notion of weak horo-tightness: an immersion is weak horo-tight if only one of its total absolute curvature attends its minimum. We prove a characterization theorem for weak horo-tight hyperspheres.

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In this paper, a discrete version of a reaction-diffusion equation, lso known as coupled map lattice (CML), which corresponds to the Tur- ng model of morphogenesis is studied. It is shown that CML possesses a yperbolic property displaying a type of spatio-temporal chaos. Through- ut a mathematical background of hyperbolic properties in lattice dynamical ystems which are related to spatio-temporal chaos, a mathematical proof is iven that the CML obtained from the Turing model possesses such hyperbolic roperties. Finally, numerical studies of this finding in varying parameters to resent a variety of chaotic behaviors of this system is performed.

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We study in this paper projections of embedded timelike hypersurfaces $M$ in$S^n_1$ along geodesics. We deal in more details with the case of surfaces in $S^3_1$, characterise geometrically the singularities of the projections and prove duality results analogous to those of Shcherbak for central projections of surfaces in $¥mathbb{R}P^3$.

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Let W be a finite irreducible real reflection group, which is a Coxeter roup. We explicitly construct a basis for the module of differential -forms with logarithmic poles along the Coxeter arrangement by sing a primitive derivation. As a consequence, we extend the Hodge iltration, indexed by nonnegative integers, into a filtration indexed by ll integers. This filtration coincides with the filtration by the order f poles. The results are translated into the derivation case.

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We establish a global existence result for the rotating Navier-Stokes equations with ondecaying initial data in a critical space which includes a large class of almost periodic unctions. The scaling invariant function space we introduce is given as the divergence of the pace of 3×3 fields of Fourier transformed finite Radon measures. The smallness condition n initial data for global existence is explicitly given in terms of the Reynolds number. The ondition is independent of the size of the angular velocity of rotation.

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We show six Legendrian dualities between pseudo-spheres in semi-Euclidean space which are basic tools for the study of extrinsic differential geometry of submanifolds in these pseudo-spheres.

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This paper constructs a family of discrete games, whose value functions converge to the unique viscosity solution of the Neumann boundary problem of the curve shortening flow equation. To derive the boundary condition, a billiard semiflow is introduced and its basic properties near the boundary are studied for convex and more general domains. It turns out that Neumann boundary problems of mean curvature flow equations can be intimately connected with purely deterministic game theory.

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In this paper we give the classification of stable equivalence classes of ordered, pointed, riented multi-component curves on surfaces with minimum crossing number less than r equal to 2 which any curve in its equivalent class has no simply closed curves in its omponents. To do this, we use the theory of words and phrases which was introduced y V. Turaev. Indeed we give the homotopy classification of nanophrases with less than r equal 4 letters. This is an extension of the classification of nanophrases of length 2 ith less than or equal to 4 letters which was given by the author in the recent paper.

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We define in this paper the asymptotic, characteristic and principal directions associated to the de Sitter Gauss map on a smooth timelike surface $M$ in the de Sitter space $S^3_1$. We study their properties and determine the local topological configurations of their integral curves. These curves form pairs of foliations on some regions of $M$ and are defined in an analogous way to their classical contrepart on surfaces in the Euclidean 3-space. However, we show that their behaviour is distinct from that of their analogue on surface in the Euclidean 3-space.

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We introduce a new definition of a generalized logarithmic module of multiarrangements by uniting those of the logarithmic derivation and the differential modules. This module is realized as a logarithmic derivation module of an arrangement of hyperplanes with a multiplicity consisting of both positive and negative integers. We consider several properties of this module including Saito's criterion and reflexivity. As applications, we prove a shift isomorphism and duality of some Coxeter multiarrangements by using the primitive derivation.

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The lightlike hypersurfaces in Lorentz-Minkowski space are of special interest in Relativity Theory. In particular, the singularities of these hypersurfaces provide good models for the study of different horizon types. We introduce the notion of flatness for these hypersurfaces and study their singularities. The classification result asserts that a generic classification of flat lightlike hypersurfaces is quite different from that of generic lightlike hypersurfaces.

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Let $p$ be a rational prime, $k$ be a perfect field of characteristic $p$, $W=W(k)$ be the ring of Witt vectors, $K$ be a finite totally ramified extension of $\Frac(W)$ of degree $e$ and $r$ be a non-negative integer satisfying $r<p-1$. Let $V$ be a semi-stable $p$-adic $G_K$-representation with Hodge-Tate weights in $\{0,\dots,r\}$. In this paper, we prove the upper numbering ramification group $G_{K}^{(j)}$ for $j>u(K,r,n)$ acts trivially on the mod $p^n$ representations associated to $V$, where $u(K,0,n)=0$, $u(K,1,n)=1+e(n+1/(p-1))$ and $u(K,r,n)=1-p^{-n}e(K(\zeta_p)/K)^{-1}+e(n+r/(p-1))$ for $r>1$.

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For a semilinear heat equation admitting blow-up solutions a sufficient condition for nonexistence of local-in-time solutions are obtained. In particular, it is shown that if an initial data tends to infinity at space infinity then there is no local-in-time solution. As an application if the solution blows up at space infinity with least blow-up time, the solution cannot be extendable after blow-up time.

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A central arrangement $\A$ of hyperplanes in an $\ell$-dimensional vector space $V$ is said to be totally free}if a multiarrangement if $(\A, m)$ is free for any multiplicity $ m : \A\rightarrow \Z_{> 0}$. It has been known that $\A$ is totally free whenever $\ell \le 2$. In this article, we will prove that there does not exist any totally free arrangement other than the obvious ones, that is, a product of one-dimensional arrangements and two-dimensional ones.

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We define the notions of $S_t^1¥times S_s^2$-nullcone Legendrian Gauss maps and $S^2_+$-nullcone Lagrangian Gauss maps on spacelike surfaces in Anti de Sitter 4-space. We investigate the relationships between singularities of these maps and geometric properties of surfaces as an application of the theory of Legendrian/Lagrangian singularities. By using $S^2_+$-nullcone Lagrangian Gauss maps, we define the notion of $S^2_+$-nullcone Gauss-Kronecker curvatures and show a Gauss-Bonnet type theorem as a global property. We also introduce the notion of horospherical Gauss maps whch has different geometric properties of the above Gauss maps. As a consequence, we can say that Anti de Sitter space has much more rich geometric properties than the other space forms such as Euclidean space, Hyperbolic space, Lorentz-Minkowski space and de Sitter space.

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We define the notions of lightcone Gauss images of spacelike hypersurfaces in de Sitter space. We investigate the relationships between singularities of these maps and geometric properties of spacelike hypersurfaces as an application of the theory of Legendrian singularities. We classify the singularities and give some examples in the generic case in de Sitter 3-space.

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We develop a mathematical theory of time operators of a Hamiltonian with urely discrete spectrum. The main results include boundedness, unboundedness nd spectral properties of them. In addition, possible connections of a time operator f H with regular perturbation theory are discussed.

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We propose a certain formula of Dedekind sum by using two mutually distinct methods. The one is an elementary number theoretic method and the other is a geometric method. Moreover, we re-prove the reciprocity law by the formula.

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We study timelike surfaces in Anti de Sitter 3-space as an application of singularity theory. We define two mappings associated to a timelike surface which are called a {¥it Anti de Sitter nullcone Gauss image} and a {¥it Anti de Sitter torus Gauss map.} We also define a family of functions named the {¥it Anti de Sitter null height function} of the timelike surface. We use this family of functions as a basic tool to investigate the geometric meanings of singularities of the Anti de Sitter nullcone Gauss image and the Anti de Sitter torus Gauss map.

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An inverse scattering problem for a quantized scalar field interacting ith a classical source J is considered. Assuming that J is of he form J(t, x) = JT (t)JX(x), (t, x) 2 R×R3, we represent JX (resp. T ) in terms of JT (resp. JX) and the asymptotic fields of and its onjugate field.

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This paper gives numerical examples showing that non self-similar collapse can occur in the motion of four point vortices on a sphere. It is found when the $4$-vortex problem is integrable, in which the moment of vorticity vector is zero. The non self-similar collapse has significant properties. It is \textit{partial} in the sense that three of the four point vortices collapse to one point in finite time and the other one moves to the antipodal position to the collapse point. Moreover, it is \textit{robust} with respect to perturbation of the initial configuration as long as the system remains integrable. The non self-similar, robust and partial collapse of point vortices is a new phenomenon that has not yet been reported.

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An integral coefficient matrix determines an integral arrangement of hyperplanes n Rm. After modulo q reduction (q ∈ Z>0), the same matrix determines an arrangement q of “hyperplanes” in Zmq In the special case of central arrangements, amiya, Takemura and Terao [J. Algebraic Combin., to appear] showed that the cardinality f the complement of Aq in Zmq s a quasi-polynomial in q ∈ Z>0. Moreover, hey proved in the central case that the intersection lattice of Aq is periodic from ome q on. The present paper generalizes these results to the case of non-central rrangements. The paper also studies the arrangement ˆ B[0,a] of Athanasiadis [J.

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This paper constructs a family of discrete two-person games, whose values converge to the unique viscosity solution of a general curvature flow equation in dimension two. We summarize all of the techniques needed for such second-order games. We introduce barrier games, which can be considered as a combination of the classical barrier argument and game perspectives.

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In this paper we analyze the motion of a network of three planar curves with a speed proportional to the curvature of the arcs, having perpendicular intersections with the outer boundary and a common intersection at a triple junction. As a main result we show that a linear stability criterion due to Ikota and Yanagida is also sufficient for nonlinear stability. We also prove local and global existence of classical smooth solutions as well as various energy estimates. Finally, we prove exponential stabilization of an evolving network starting from the vicinity of a linearly stable stationary network.

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The locally-in-time solvability of the Cauchy problem of the incompressible Navier-Stokes equations is established with nitial velocity U0 of the form U0(x) := u0(x) - Mx, where M is a real-valued matrix and u0 is a bounded function. It is also hown that in 2-dimensional case the Navier-Stokes equations admit unique globally-in-time smooth solution, due to the uniform bound for vorticity. Although the semigroup is not analytic, our mild solution satisfies the Navier-Stokes equations in the classical sense, provided the pressure term is suitably chosen. The form of the pressure is uniquely determined, provided the disturbance of velocity is bounded and the modified pressure is in a certain function space.

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We present evidence on global existence of solutions of De Gregorio's equation, based on numerical computation and a mathematical criterion analogous to the Beale-Kato-Majda theorem. Its meaning in the context of a generalized Constantin-Lax-Majda equation will be discussed. We then argue that the convection term can deplete solutions of blow-up.

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The purpose of this paper is to give the homotopy classification of nanophrases of ength 2 with 4 letters. To do it we construct some new invariants of nanophrases γ, T. he invariant γ defined in this paper is an extension of the invariant γ for nanowords ntroduced in [5]. The invariant T is a new invariant of nanophrases. As a corollary of hese results, we give the classification of two-components pointed, ordered, oriented urves on surfaces with minimum crossing number ≤ 2.

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The asymmetric Burgers vortices are vortex solutions to the three dimensional stationary Navier-Stokes equations for viscous incompressible fluids in the presence of an asymmetric background straining flow. The asymmetry of the straining flow is expressed by a non-negative parameter less than $1$. The Burgers vortices have been used as a model which expresses tube-like structures of concentrated vorticity fields in turbulence, and they are numerically well investigated especially in the case of large circulation numbers. However, their existence was proved mathematically only when either the asymmetry of the straining flow is not so strong or the circulation number is sufficiently small. In this paper we prove the existence of asymmetric Burgers vortices for all circulation numbers and each asymmetry parameter less than $1$. We also obtain their asymptotic expansion at large circulation numbers, which gives an explanation for a symmetrizing effect by a fast rotation.

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Let E be a Banach space on a set X and M(E) the space of multipliers of E. In this paper, we study the space of multipliers of the quotient space E/K where K is a closed M(E) - invariant subspace in E. When E is the classical Hilbert Hardy space, the Nevanlinna and Pick theorem shows M(E/K) is a quotient algebra of M(E).

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We study an abstract nonlinear evolution equation governed by time-dependent operator of subdierential type in a real Hilbert space. In this paper we discuss the convergence of solutions to our evolution equations. Also, we investigate the optimal control problems of nonlinear evolution equations. Moreover, we apply our abstract results to a quasilinear parabolic PDE with a mixed boundary condition.

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We define specific multiplicities on the braid arrangement by using edge-bicolored graphs. To consider their freeness, we introduce the notion of bicolor-eliminable graphs as a generalization of Stanley's classification theory of free graphic arrangements by chordal graphs. This generalization gives us a complete classification of the free multiplicities defined above. As an application, we prove one direction of a conjecture of Athanasiadis on the characterization of the freeness of the deformation of the braid arrangement in terms of directed graphs.

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We study the differential geometry of spacelike surfaces in Anti de Sitter 3-space from the view point of Legendrian singularity theory. We define the timelike Anti de Sitter Gauss image on spacelike surface and investigate the geometric meanings of singularities.

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In this paper, we study when M, I or J is a Fredholm operator on a Hilbert space which satisfies few natural axioms.

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We prove an analogue of Baxter's inequality for fractional Brownian motion-type processes with Hurst index less than 1/2. This inequality is concerned with the norm estimate of the difference between finite- and infinite-past predictor coefficients.

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We study crystalline driven curvature flow with spatially nonuniform driving force term. e assume special monotonicity properties of the driving term, which are motivated by ur previous work on Berg’s effect. We consider special initial data which we call ‘bent ectangles’. We prove existence of solutions for a generic forcing term as well as generic ubclass of bent rectangles. We show the initially flat facets may begin to bend, provided, oosely speaking, they are too large. Moreover, depending on the initial configuration we otice instantaneous loss of regularity of the moving curve.

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Weak Weyl representations of the canonical commutation relation (CCR) with one degree of freedom are considered in relation to the theory of time operator in quantum mechanics. It is proven that there exists a general structure through which a weak Weyl representation can be constructed from a given weak Weyl representation. As a corollary, it is shown that a Weyl representation of the CCR can be constructed from a weak Weyl representation which satisfies some additional property. Some examples are given.

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We study in this paper orthogonal projections of embedded surfaces $M$ in $H^3_+(-1)$ along horocycles to planes. The singularities of the projections capture the extrinsic geometry of $M$ related to the lightcone Gauss map. We give geometric characterisations of these singularities and prove a Koenderink type theorem which relates the hyperbolic curvature of the surface to the curvature of the profile and of the normal section of the surface. We also prove duality results concerning the bifurcation set of the family of projections.

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We study a model of the quantized electromagnetic field interacting ith an external static source in the Feynman (Lorentz) gauge and onstruct the quantized radiation field Aμ (μ = 0, 1, 2, 3) as an operatorvalued istribution acting on the Fock space F with an indefinite metric. y using the Gupta subsidiary condition @μAμ(x)(+) = 0, one can select he physical subspace Vphys. According to the Gupta-Bleuler formalism, phys is a non-negative subspace so that elements of Vphys, called physical tates, can be probabilistically interpretable. Indeed, assuming that the xternal source is infrared regular, i.e., ˆ /|k|3/2 2 L2(R3), we can characterize he physical subspace Vphys and show that Vphys is non-negative. n addition, we find that the Hamiltonian of the model is reduced to the amiltonian of the transversal photons with the Coulomb interaction. We owever prove that the physical subspace is trivial, i.e., Vphys = 0, if and nly if the external source is infrared singular, i.e., ˆ /|k|3/2 62 L2(R3). e also discuss a representation different from the above representation uch that the physical subspace is not trivial under the infrared singular ondition.

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The morphological stability of a growing faceted crystal is discussed. It has been explained hat the interplay between nonuniformity in supersaturation on a growing facet and nisotropy of surface kinetics derived from the lateral motion of steps leads to a faceted nstability. Qualitatively speaking, as long as the nonuniformity in supersaturation on the acet is not too large, it can be compensated by a variation of step density along the facet nd the faceted crystal can grow in a stable manner. The problem can be modeled as a amilton-Jacobi equation for height of the crystal surface. The notion of a maximal stable egion of a growing facet is introduced for microscopic time scale approximation of the riginal Hamilton-Jacobi equation. It is shown that the maximal stable region keeps its hape, determined by profile of the surface supersaturation, with constant growth rate by tudying large time behavior of solution of macroscopic time scale approximation. As a esult, a quantitative criterion for the facet stability is given.

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Consider a scattering problem for the Dirac equation with a nonlocal term including he Hartree type. We show the existence of scattering operators for small initial data n the subcritical and critical Sobolev spaces.

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We present a general approach to the pricing of products in finance and insurance in the multi-period setting. It is a combination of the utility indifference pricing and optimal intertemporal risk allocation. We give a characterization of the optimal intertemporal risk allocation by a first order condition. Applying this result to the exponential utility function, we obtain an essentially new type of premium calculation method for a popular type of multi-period insurance contract. This method is simple and can be easily implemented numerically. We see that the results of numerical calculations are well coincident with the risk loading level determined by traditional practices. The results also suggest a possible implied utility approach to insurance pricing.

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In this paper we mathematically analyse an evolution variational inequality which formulates the double critical-state model for type-II superconductivity in 3D space and propose a finite element method to discretize the formulation. The double critical-state model originally proposed by Clem and Perez-Gonzalez is formulated as a model in 3D space which characterises the nonlinear relation between the electric field, the electric current, the perpendicular component of the electric current to the magnetic flux, and the parallel component of the current to the magnetic flux in bulk type-II superconductor. The existence of a solution to the variational inequality formulation is proved and the representation theorem of subdifferential for a class of energy functionals including our energy is established. The variational inequality formulation is discretized in time by a semi-implicit scheme and in space by the edge finite element of lowest order on a tetrahedral mesh. The fully discrete formulation is an unconstrained optimisation problem. The subsequence convergence property of the fully discrete solution is proved. Some numerical results computed under a rotating applied magnetic field are presented.

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We study the horospherical geometry of submanifolds in hyperbolic space. The main result is a formula for the total absolute horospherical curvature of $M,$ which implies, for the horospherical geometry, the analogues of classical inequalities of the Euclidean Geometry. We prove the horospherical Chern-Lashof inequality for surfaces in $3$-space and the horospherical Fenchel and Fary-Milnor's theorems.

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The Eisenstein polynomial is the weighted sum of all classes of Type II codes of fixed length. In this note, we investigate the ring of the Eisenstein polynomials in genus $2$.

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We consider codimension two spacelike submanifolds with a parallel normal field (i.e. vanishing normal curvature) in Minkowski space. We use the analysis of their contacts with hyperplane and hyperquadrics in order to get some global informations on them. As a consequence we obtain new versions of Carat¥`eodory's and Loewner's conjectures on spacelike surfaces in $4$-dimensional Minkowski space and $4$-flattenings therorems for closed spacelike curves in $3$-dimensional Minkowski space.

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We describe a Brownian ratchet scheme which we use to calculate relative equilibrium configurations of N point vortices of mixed strength on the surface of a unit sphere. We formulate it as a linear algebra problem $A\Gamma = 0$ where $A$ is a $N \times N(N − 1)/2$ non-normal configuration matrix obtained by requiring that all inter-vortical distances on the sphere remain constant, and $\Gamma \in R^N$ is the (unit) vector of vortex strengths which must lie in the nullspace of $A$. Existence of an equilibrium is expressed by the condition $det($A^TA) = 0$, while uniqueness follows if $Rank(A) = N−1$. The singular value decomposition of $A$ is used to calculate an optimal basis set for the nullspace, yielding all values of the vortex strengths for which the configuration is an equilibrium. To home in on an equilibrium, we allow the point vortices to undergo a random walk on the sphere and after each random step we compute the smallest singular value of the configuration matrix, keeping the new arrangement only if it decreases. When the singular value drops below a predetermined convergence threshold, an equilibrium configuration is achieved and we find a basis set for the nullspace of A by calculating the right singular vectors corresponding to the singular values that are zero. For each $N = 4 \rightarrow 10$, we generate an ensemble of 1000 equilibrium configurations which we then use to calculate statistically averaged singular value distributions in order to obtain the averaged Shannon entropy and Frobenius norm of the collection. We show that the statistically averaged singular values produce an average Shannon entropy that closely follows a power-law scaling of the form $< S > \sim N^\beta$, where $\beta \sim 2/3$. We also show that the length of the conserved center-of-vorticity vector clusters at a value of one and the total vortex strength of the configurations cluster at the two extreme values ±1, indicating that the ensemble average produces a single vortex of unit strength which necessarily sits at the tip of the center-ofvorticity vector. The Hamiltonian energy averages to zero reflecting a relatively uniform distribution of points around the sphere, with vortex strengths of mixed sign.

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In this paper, we consider elliptic estimates for a system with mooth variable coeffcients on a domain ­ ½ Rn; n ¸ 2 containing the origin. e first show the invariance of the estimates under a domain expansion de¯ned y the scale that y = Rx, x; y 2 Rn with parameter R > 1, provided that the oeffcients are in a homogeneous Sobolev space. Then we apply these invariant stimates to the global existence of unique strong solutions to a parabolic ystem de¯ned on an unbounded domain.

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It is rather clear that the solution is periodic if it is initially eriodic provided that the evolution equation under study is well-posed and ranslation invariant. It is less obvious to see that almost periodicity is reserved under slightly stronger assumptions. We are interested in evolution f the frequency set when initial data is almost periodic. We consider such problem for the Navier-Stokes equations and other evolution equations. typical result is that an additive group generated by frequency set called odule is contained in the module of initial data. We propose a method mbedding almost periodic problems to periodic problems which yields such result.

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We study structures of derivation modules of Coxeter multiarrangements with quasi-constant multiplicities by using the primitive derivation. As an application, we show that the characteristic polynomial of a Coxeter multiarrangement with quasi-constant multiplicity is combinatorially computable.

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The uniqueness and existence of generalized solutions of `spiral curves' for the mean curvature flow with driving force is studied by an adapted level set formulation. It is shown that the curves which are given by the level set formulation are unique with respect to initial spiral curves. For given spiral curves the method of a construction of an initial datum of a level set equation is also obtained by constructing a branch of the arguments from the centers of spiral curves, which has discontinuity at the given spiral curves.

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Some of the most important results in prediction theory and time series analysis when finitely many values are removed from or added to its infinite past have been obtained using difficult and diverse techniques ranging from duality in Hilbert spaces of analytic functions (Nakazi, 1984) to linear regression in statistics (Box and Tiao, 1975). We unify these results via a finite-dimensional duality lemma and elementary ideas from the linear algebra. The approach reveals the inherent finite-dimensional character of many difficult prediction problems, the role of duality and biorthogonality for a finite set of random variables. The lemma is particularly useful when the number of missing values is small, like one or two, as in the case of Kolmogorov and Nakazi prediction problems. The stationarity of the underlying process is not a requirement. It opens up the possibility of extending such results to nonstationary processes.

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We introduce a class of Gaussian processes with stationary increments which exhibit long-range dependence. The class includes fractional Brownian motion with Hurst parameter H>1/2 as a typical example. We establish infinite and finite past prediction formulas for the processes in which the predictor coefficients are given explicitly in terms of the MA and AR coefficients. We apply the formulas to prove an analogue of Baxter's inequality, which concerns the L^1-estimate of the difference between the finite and infinite past predictor coefficients.

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We give a generalization of Yamaguchi--Yau's result to Walcher's extended holomorphic anomaly equation.

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A class of representations on the Hilbert space $L^2(\R^d)$ ($d\geq 2$) of the quantum plane $\C_q^2$ and the quantum algebra $U_q({\rm sl}_2)$ is presented. The boundedness and the unboundedness of the representations are discussed. A physically interesting example of the representations is shown to appear in a two-dimensional quantum system with a magnetic field concentrated on an infinite lattice.

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The aim of this paper is to characterize variable Lp spaces Lp( · )(Rn) using wavelets with proper smoothness and decay. We obtain conditions for the wavelet characterizations of Lp( · )(Rn) with respect to the norm estimates and modular inequalities.

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We introduce the concept of multiplicity lattices of $2$-multiarrangements, and determine the behaviour of exponents in that lattice.

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For an irreducible root system R, consider a coefficient matrix S of the positive roots with respect to the associated simple roots. Then S defines an arrangement of “hyperplanes” modulo a positive integer q. The cardinality of the complement of this arrangement is a quasi-polynomial of q, which we call the characteristic quasi-polynomial of R. This paper gives the complete list of the characteristic quasi-polynomials of all irreducible root systems, and shows that the characteristic quasi-polynomial of an irreducible root system R is positive at q 2 Z>0 if and only if q is greater than or equal to the Coxeter number of R. Key words: characteristic quasi-polynomial, elementary divisor, hyperplane arrangement, root system.

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The volume preserving fourth order surface diffusion flow has constant mean curvature hypersurfaces as stationary solutions. We show nonlinear stability of certain stationary curves in the plane which meet an exterior boundary with a prescribed contact angle. Methods include semigroup theory, energy arguments, geometric analysis and variational calculus.

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We give a transformation formula for certain infinite series in which some elliptic Dedekind-Rademacher sums arise. In the course of its proof, we also obtain a transformation formula for elliptic Dedekind-Rademacher sums. When a complex parameter $¥tau$ tends to $i ¥infty$, these represent some classical results which include the reciprocity formula for Apostol-Dedekind sums.

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We give a simple proof that the solution of the Allen-Cahn equation converges to Brakke’s motion as a parameter tends to zero by utilizing the recent results of R¨oger and Sch¨atzle. The proof avoids some of the technicalities in Ilmanen’s proof.

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Some properties of distributions f satisfying x ¢ rf 2 Lp(Rn), 1 · p < 1, are studied. The operator x ¢ r is the generator of a semi-group of dilations. We first give Sobolev type inequalities with respect to the operator x ¢r. Using the inequalities, we also show that if f 2 Lp loc(Rn), x ¢rf 2 Lp(Rn) and jxjn=pjf(x)j vanishes at infinity, then f belongs to Lp(Rn). One of the Sobolev type inequalities is shown to be equivalent to the Hardy inequality in L2(Rn).

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We consider the Navier-Stokes equations with the Coriolis force when intial data may not decrease at spatial infinity so that almost periodic data is allowed. We prove that the local-in-time solution is analytic in time when initial data is in $FM_0$, Fourier preimage of the space of all finite Radon measures with no point mass at the origin. When the initial data is almost periodic, this implies that the complex amplitude is analytic in time. In particular, a new mode cannot be created at any positive time.

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In this paper we derive some Sobolev inequalities for radially symmetric functions in ˙H s with 1 2 < s < n 2 . We show the end point case s = 1 2 on the homogeneous Besov space ˙B 12 2;1. These results are extensions of the well-known Strauss’ inequality [11]. Also non-radial extensions are given, which show that compact embeddings are possible in some Lp spaces if a suitable angular regularity is imposed.

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We study the motion of N point vortices with N∈ℕ on a sphere in the presence of fixed pole vortices, which are governed by a Hamiltonian dynamical system with N degrees of freedom. Special attention is paid to the evolution of their polygonal ring configuration called the N-ring, in which they are equally spaced along a line of latitude of the sphere. When the number of the point vortices is N=5n or 6n with n∈ℕ, the system is reduced to a two-degree-of-freedom Hamiltonian with some saddle-center equilibria, one of which corresponds to the unstable N-ring. Using a Melnikov-type method applicable to two-degree-of-freedom Hamiltonian systems with saddle-center equilibria and a numerical method to compute stable and unstable manifolds, we show numerically that there exist transverse homoclinic orbits to unstable periodic orbits in the neighborhood of the saddle-centers and hence chaotic motions occur. Especially, the evolution of the unstable N-ring is shown to be chaotic.

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We study analytically and numerically the stability of the standing waves for a nonlinear Schr¨odinger equation with a point defect and a power type nonlinearity. A main difficulty is to compute the number of negative eigenvalues of the linearized operator around the standing waves, and it is overcome by a perturbation method and continuation arguments. Among others, in the case of a repulsive defect, we show that the standing wave solution is stable in H1 rad(R) and unstable in H1(R) under subcritical nonlinearity. Further we investigate the nature of instability: under critical or supercritical nonlinear interaction, we prove the instability by blow-up in the repulsive case by showing a virial theorem and using a minimization method involving two constraints. In the subcritical radial case, unstable bound states cannot collapse, but rather narrow down until they reach the stable regime (a finite-width instability). In the non-radial repulsive case, all bound states are unstable, and the instability is manifested by a lateral drift away from the defect, sometimes in combination with a finite-width instability or a blowup instability.

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We prove a nonlinear Poisson type formula for the Schrödinger group. Such a formula had been derived in a previous paper by the authors, as a consequence of the study of the asymptotic behavior of nonlinear wave operators for small data. In this note, we propose a direct proof, and extend the range allowed for the power of the nonlinearity to the set of all short range nonlinearities. Moreover, H1-critical nonlinearities are allowed.

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The aim of this paper is to obtain the wavelet expansion in the Besov spaces and the Triebel-Lizorkin spaces coming with Aloc p -weights. After characterizing these spaces in terms of wavelet, we shall obtain unconditional bases and greedy bases.

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We study the reflectional symmetry of a generically embedded 2-dimensional surface M in the hyperbolic or de Sitter 3-dimensional spaces. This symmetry is picked up by the singularities of folding maps that are defined and studied here. We also define the evolute and symmetry set of M and prove duality results that relate them to the bifurcation sets of the family of folding maps.

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The condition for the curvature of a statistical manifold to admit a kind of standard hypersurface is given as a first step of the statistical submanifold theory. A complex version of the notion of statistical structures is also introduced.

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We will derive linear differential relations satis ed by Wirtinger integrals by exploiting classical formulas of Jacobi's theta functions, forgetting that Wirtinger integrals are related to Gauss hypergeometric functions, although these linear differential relations are related to ones satis ed by Gauss hypergeometric functions.

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In this paper, we study closed invariant subspaces under the action of a unilateral shift and a truncated shift in the Hardy space that takes values in a two dimensional Hilbert space. We deal with characteristic functions, unitary equivalence and C∗-algebras on these spaces.

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For a fermion system, an operator theoretic renormalization group method based on the smooth Feshbach map is constructed. By using the fermionic renormalization group method, the closed operator of the form: Hg(θ) = HS ⊗ 1 + eθν1 ⊗ Hf + Wg(θ) is analyzed, where HS is a selfadjoint operator on a separable Hilbert space and bounded from below, Hf denotes the fermionic quantization of the one fermion kinetic energy c|k|ν, k ∈ Rd (c, ν > 0), Wg(θ) is a small perturbation with respect to HS ⊗ 1 + eθν1 ⊗ Hf and θ ∈ C is a complex scaling parameter. The constant g ∈ R denotes a coupling constant such that Wg(θ) → 0(g → 0) in some sense. It is assumed that HS has a discrete simple eigenvalue E ∈ σd(HS), and proved that Hg(θ) has an eigenvalue Eg(θ) close to E for a small coupling constant g. Moreover, the eigenvalue Eg(θ) and the corresponding eigenvector Ψ(θ) is constructed by the process of the operator theoretic renormalization group method.

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Let M be a forward shift invariant subspace and N a backward shift invariant subspace in the Hardy space H2 on the bidisc. We assume that H2 = N M. Using the wandering subspace of M and N, we study the relations between M and N. Moreover we study M and N using several natural operators which are defined by shift operators on H2.

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A fast and accurate algorithm to compute interaction between N point vortices on a sphere is proposed. It is an extension of the fast tree-code algorithm based on the Taylor expansion developed by Draghicescu for the point vortices in the plane. When we choose numerical parameters in the fast algorithm suitably, the computational cost of O(N2) is reduced to O(N(logN)4) and the approximation error decreases like O(1=N) as N ! 1, which are clearly con rmed in the present article. We also apply the fast method to a long-time evolution of two vortex sheets on the sphere. A key device is to embed the sphere into the three-dimensional space, in which we reformulate the equation of motion for the N point vortices.

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Axisymmetric or non-axisymmetric Burgers vortices have been studied numerically as a model of concentrated vorticity elds. Recently, it is rigorously proved that non-axisymmetric Burgers vortices exist for all values of the vortex Reynolds number if an asymmetric parameter is su ciently small. On the other hand, several numerical results suggest that Burgers vortices have simpler structures if the vortex Reynolds number is large, even when the asymmetric parameter is not small. In this paper we give a rigorous explanation for this numerical observation and extend the existence results for high vortex Reynolds numbers.

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In this paper, for a complete discrete valuation field $K$ of mixed characteristic $(0, p)$ and a finite flat group scheme $\mathcal{G}$ of $p$-power order over $\mathcal{O}_K$, we determine the tame characters appearing in the Galois representation $\mathcal{G}(\bar{K})$ in terms of the ramification theory of Abbes and Saito, without any restriction on the absolute ramification index of $K$ or the embedding dimension of $\mathcal{G}$.

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We study from a mathematical point of view a model equation for Bose Einstein condensation, in the case where the trapping potential varies randomly in time. The model is the so called Gross-Pitaevskii equation, with a quadratic potential with white noise fluctuations in time. We prove the existence of solutions in 1D and 2D in the energy space. The blow-up phenomenon is also discussed under critical and super critical nonlinear interactions in the attractive case.

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Burgers vortices have been used as a model which expresses concentrated vorticity fields in turbulence. Several numerical results indicate that as the vortex Reynolds number is increasing, the associated Burgers vortex becomes more radial symmetric and has simpler structures. The purpose of this paper is to give a mathematical explanation for these numerical observations by studying a linearized operator of the equation for Burgers vortices. Based on the previous works by Th. Gallay and C. E. Wayne, we obtain some estimates and spectrum behavior of the linearized operator at large Reynolds numbers which are compatible with the numerical observations.

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We prove the convergence of phase-field approximations of the Gibbs–Thomson law. This establishes a relation between the first variation of the Van-der-Waals–Cahn–Hilliard energy and the first variation of the area functional. We allow for folding of diffuse interfaces in the limit and the occurrence of higher-multiplicities of the limit energy measures. We show that the multiplicity does not affect the Gibbs–Thomson law and that the mean curvature vanishes where diffuse interfaces have collided. We apply our results to prove the convergence of stationary points of the Cahn–Hilliard equation to constant mean curvature surfaces and the convergence of stationary points of an energy functional that was proposed by Ohta– Kawasaki as a model for micro-phase separation in block-copolymers.

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We study central hyperplane arrangements with integral coefficients modulo positive integers q. We prove that the cardinality of the complement of the hyperplanes is a quasi-polynomial in two ways, first via the theory of elementary divisors and then via the theory of the Ehrhart quasi-polynomials. This result is useful for determining the characteristic polynomial of the corresponding real arrangement. With the former approach, we also prove that intersection lattices modulo q are periodic except for a finite number of q’s.

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Recently we discovered a new geometry on submanifolds in hyperbolic $n$-space which is called {\it horospherical geometry}. Unfortunately this geometry is not invariant under the hyperbolic motions (it is invariant under the canonical action of $SO(n)$), but it has quite interesting features. For example, the flatness in this geometry is a hyperbolic invariant and the total curvatures are topological invariants. In this paper, we investigate the {\it horospherical flat surfaces} (flat surfaces in the sense of horospherical geometry) in hyperbolic $3$-space. Especially, we give a generic classification of singularities of such surfaces. As a consequence, we can say that such a class of surfaces has quite a rich geometric structure.

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The Stokes resolvent equations are studied in locally uniform Lp spaces where the domain is an exterior of a bounded domain. The unique existence of a solution of the Stokes resolvent equations is proved with a resolvent estimate. In particular, the analyticity of the Stokes semigroup is established. An interesting aspect of locally uniform Lp spaces is that these spaces contain non-decaying functions.

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For implicit second order differential equations, uniqueness for solutions does not hold in general. In this paper, we give a sufficient condition that implicit second order was ordinary differential equations have the unique geometric solution around a point,

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This paper study the Gauss-Bonnet theorem for Finsler surfaces with smooth boundary. This is a natural generalization of the Gauss-Bonnet theorem for Riemannian surfaces with smooth boundary as well as an extension of the Gauss-Bonnet theorem for boundaryless Finsler surfaces. The paper starts with an introduction in the Finsler geometry of surfaces with emphasis on the Berwald and Landsberg surfaces.

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We show the global existence of solutions of a reactiondiffusion system with the nonlinear terms |x|σjupj vqj . The proof is based on the existence of super-solutions and the comparison principle. We also prove that uniqueness of the global solutions holds in the superlinear case by contraction argument. Our conditions for the global existence are optimal in view of the nonexistence results proved by Yamauchi [12].

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We study in this paper orthogonal projections in a hyperbolic space to hyperhorospheres and hyperplanes. We deal in more details with the case of embedded surfaces $M$ in $H^3_+(-1)$. We study the generic singularities of the projections of $M$ to horospheres and planes. We give geometric characterisations of these singularities and prove duality results concerning the bifurcation sets of the families of projections. We also prove Koendrink type theorems that give the curvature of the surface in terms of the curvatures of the profile and the normal section of the surface.

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The aim of this paper is to give a natural definition of the weighted Besov spaces and the weighted Triebel-Lizorkin spaces. The highlight of this paper is that we form an atomic decomposition for the class even wider than the class Ap due to Muckenhoupt.

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We consider the scattering problem for the nonlinear Klein-Gordon equation. The nonlinear term of the equation behaves like a power term. We show that we can define the scattering operator on a suitable 0-neighborhood of the weighted Sobolev space, which improves the known results in some sense. Our proof is based on the Strichartz type estimates.

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We prove that for the L2-critical nonlinear Schrödinger equations, the wave operators and their inverse are related explicitly in terms of the Fourier transform. We discuss some consequences of this property. In the onedimensional case, we show a precise similarity between the L2-critical nonlinear Schr¨odinger equation and a nonlinear Schr¨odinger equation of derivative type.

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We prove a representation of the partial autocorrelation function (PACF), or the Verblunsky coefficients, of a stationary process in terms of the AR and MA coefficients. We apply it to show the asymptotic behaviour of the PACF. We also propose a new definition of short and long memory in terms of the PACF.

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We consider a financial market model driven by a Gaussian semimartingale with stationary increments. This driving noise process consists of independent components and each component has memory described by two parameters. We extend results of the authors on optimal investment in this market.

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We study the global well-posedness (GWP) and small data scattering of radial solutions of the relativistic Hartree type equations with nonlocal nonlinearity F(u) = ¸(j ¢ j¡° ¤ juj2)u, ¸ 2 R n f0g, 0 < ° < n, n ¸ 3. We establish a weighted L2 Strichartz estimate applicable to non-radial functions and some fractional integral estimates for radial functions.

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We give the first complete classification of free and non-free multiplicities on an arrangement, called the arrangement of type $A_3-1$, which admits both of them.

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We characterize the homogeneous weighted Herz space ˙K α,p q (w1,w2) and the non-homogeneous weighted Herz space Kα,p q (w1,w2) using wavelets in C1(Rn) with compact support. Applying the characterizations, we prove that the wavelet basis forms an unconditional basis in ˙Kα,p q (w1,w2) and in Kα,p q (w1,w2) .

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The addition-deletion theorems for hyperplane arrangements, which were originally shown in [T1], provide useful ways to construct examples of free arrangements. In this article, we prove addition-deletion theorems for multiarrangements. A key to the generalization is the definition of a new multiplicity, called the e-multiplicity, of a restricted multiarrangement. We compute the e-multiplicities in many cases. Then we apply the addition-deletion theorems to various arrangements including supersolvable arrangements and the Coxeter arrangement of type A3 to construct free and non-free multiarrangements.

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We study analytic smoothing effect of solutions to the Schr odinger equation with Cauchy data decaying exponentially at in nity. The do- main of analyticity in the space variables of solutions is described under weight conditions on the data in the terms of the corresponding supporting functios. The domain of analyticity in the time variable is characterized by means of weight conditions of Gaussian type on the data. A general- ization of various isometrical identities related to the analytic smoothing effect is introduced.

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We prove the ¡-convergence of the Allen-Cahn action functional in the sharp-interface limit. In previous work, good lower bounds were developed under the assumption of single-multiplicity, but the bounds deteriorated in the case of higher-multiplicity interfaces. We develop improved bounds by working directly with the limiting energy measures.

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We consider Garsia-Haiman modules for the symmetric groups, a doubly graded generalization of Springer modules. Our main interest lies in singly graded submodules of a Garsia-Haiman module. We show that these submodules satisfy a certain combinatorial property, and verify that this property is implied by a behavior of Macdonald polynomials at roots of unity.

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Dynamics of information flow in adaptively interacting stochastic processes is studied. We give an extended form of game dynamics for Markovian processes and study its behavior to observe information flow through the system. Examples of the adaptive dynamics for two stochastic processes interacting through matching pennies game interaction are exhibited along with underlying causal structure.

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Given a multiarrangement of hyperplanes we define a series by sums of the Hilbert series of the derivation modules of the multiarrangement. This series turns out to be a polynomial. Using this polynomial we define the characteristic polynomial of a multiarrangement which generalizes the characteristic polynomial of an arragnement. The characteristic polynomial of an arrangement is a combinatorial invariant, but this generalized characteristic polynomial is not. However, when the multiarrangement is free, we are able to prove the factorization theorem for the characteristic polynomial. The main result is a formula that relates ‘global’ data to ‘local’ data of a multiarrangement given by the coefficients of the respective characteristic polynomials. This result gives a new necessary condition for a multiarrangement to be free. Consequently it provides a simple method to show that a given multiarrangement is not free.

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We study the global Cauchy problem for nonlinear Schrödinger equations with cubic interactions of derivative type in space dimension n ¸ 3. The global existence of small classical solutions is proved in the case where every real part of the first derivatives of the interaction with respect to first derivatives of wavefunction is derived by a potential function of quadratic interaction. The proof depends on the energy estimate involving the quadratic potential and on the endpoint Strichartz estimates.

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As a way to draw a picture of a manifold in the plane, we consider to take the image of the manifold through a stable map. We call the image, paired with the critical values, the planar portrait of a manifold. The cusped fan is its basic local con guration. In this article, we focus on the breing structure over the cusped fan, and give its characterisation. As application, the source manifolds of certain planar portraits are characterised, and stable maps of closed manifolds such as the projective plane kP 2 (k = R;C;H), regular complex toric surfaces, and some sphere bundles over spheres etc. are constructed. As a biproduct, we obtain an in nite to one correspondence of projections up to right-left equivalence, of a xed manifold to a planar portrait. Further applications on characterising manifolds by planar portraits are left to our forthcomming papers.

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Let T be a Toeplitz operator on the one variable Hardy space H2. We show that if T has a nontrivial invariant subspace in the set of invariant subspaces of Tz then belongs to H1. In fact, we also study such a problem for the several variables Hardy space H2.

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We investigate the boundary growth of positive superharmonic functions on a bounded domain satisfying a certain nonlinear elliptic inequality. The result is applied to show the Harnack inequality for such superharmonic functions. Also, we study the existence of positive solutions, with singularity on the boundary, of elliptic equations with nonlinear term conditioned by the generalized Kato class.

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We consider initial value problems for the semirelativistic Hartree equations with cubic convolution nonlinearity $F(u) = (V * |u|^2)u$. Here $V$ is a sum of two Coulomb type potentials. Under a specified decay condition and a symmetric condition for the potential $V$ we show the global existence and scattering of solutions.

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We consider the motion of an elastic closed curve with constant enclosed area. This motion is governed by a system involving fourth order parabolic equations. We shall prove that this system has a unique classical solution for all time and the solution converges uniformly to a stationary solution together with its derivatives of any order.

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We prove that suitable wavelets and scaling functions give characterizations and unconditional bases of the weighted Sobolev space Lp,s(w) with Ap or Aloc p weights. In the case of w 2 Ap, we use only wavelets with proper regularity. If we consider the case of w 2 Aloc p , we obtain the results by applying wavelets and scaling functions in Cs+1 comp(Rn). We also construct the greedy bases for Lp,s(w) by normalizing the unconditional bases in both of two cases.

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This paper is concerned with the boundary behavior of solutions of the Helmholtz equation in $\mathbb{R}^n$. In particular, we give a Littlewood-type theorem to show that the approach region introduced by Kor\'anyi and Taylor (1983) is best possible.

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Runge gave the ring of genus three Siegel modular forms as a quotient ring, 3=hJ(3)i. R3 is the genus three ring of code polynomials and J(3) is the di®erence of he weight enumerators for the e8 © e8 and d+ 6 codes. Freitag and Oura gave a degree 24 elation, R(4) , of the corresponding ideal in genus four; R(4) is also a linear combination of eight enumerators. We take another step toward the ring of Siegel modular forms in genus our. We explain new techniques for computing with Siegel modular forms and actually ompute six new relations, classifying all relations through degree 32. We show that the local odimension of any irreducible component de¯ned by these known relations is at least 3 and hat the true ideal of relations in genus four is not a complete intersection. Also, we explain ow to generate an in¯nite set of relations by symmetrizing ¯rst order theta identities and ive one example in degree 32. We give the generating function of R5 and use it to reprove esults of Nebe [25] and Salvati Manni [37].

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It is known that the weight enumerator of a self-dual doubly-even code in genus g = 1 can be uniquely written as an isobaric polynomial in certain homogeneous polynomials with integral coeffcients. We settle the case where g = 2 and prove the non-existence of such polynomials under some conditions.

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The first half of this paper deals with the structure of the twisted homology group ssociated to the Wirtinger integral. A basis of the first homology group is given, and the vanishing f the other homology groups is proved (Theorem 1). The second half deals with the structure of the wisted cohomology groups associated to the Wirtinger integral. The isomorphism between the twisted ohomology groups and the cohomology groups associated to a subcomplex of the de Rham complex is stablished, and a basis of the first cohomology group of this subcomplex (therefore, of the first twisted ohomology group) is given (Theorem 2).

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In this paper, we consider a certain Dirichlet series of Rankin-Selberg type associated with two Siegel cusp forms with respect to $Sp_n(\bf Z)$, which can be viewed as a generalization of the Rankin-Selberg convolution series for two elliptic modular forms. In particular, we give the explicit formula for the Dirichlet series associated with the so-called Ikeda lifting of cuspidal Hecke eigenforms with respect to $SL_2(\bf Z)$. We also comment on a contribution to the Ikeda's conjecture on periods of the lifting.

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In \cite{T}, Terao introduced an admissible map of chambers of a real central arrangement, and completely classified it. An admissible map is a generalization of a social welfare function and Terao's classification is that of Arrow's impossibility theorem in economics. In this article we consider an admissible map not of chambers but faces, and show that an admissible map of faces is a projection to a component if an arrangement is indecomposable and its cardinality is not less than three. From the view point of Arrow's theorem, our result corresponds to the impossibility theorem of a welfare function which permits the ''tie" choice.

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Closed expressions are obtained for sums of products of Kronecker's double series. Corresponding results are derived for functions which are an elliptic analogue of the periodic Euler polynomials. As corollaries, we reproduce the formulas for sums of products of Bernoulli numbers, Bernoulli polynomials, Euler numbers, and Euler polynomials, which were given by K. Dilcher.

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We consider a singular perturbation problem of Modica-Mortola functional as the thickness of diffused interface approaches to zero. We assume that sequence of functions have uniform energy and square-integral curvature bounds in two dimension. We show that the limit measure concentrate on one rectifiable set and has square integrable curvature.

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We study isometric composition operators C between two weighted ardy spaces H2( ) and H2(μ) when is a radial measure. The isometric C is related o a moment sequence and such a is studied by the Nevanlinna counting function of hen μ is the normalized Lebesgue measure on the unit circle.

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We shall discuss a free boundary problem for viscous incompressible fluids which is considered as the relaxation of a two phase free boundary problem with surface tension on the interface. Our relaxation ensures the regularity of the interface, and we shall construst a unique time-local solution of the problem. One of the keys is to obtain the optimal regularity of the velocity in tangential direction to the interface.

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We briefly describe the so-called Monge-Kantorovich Problem (MKP for short) which is often referred to as an optimal mass transportation problem and study the stochastic optimal control problem (SOCP for short) with fixed initial and terminal distributions. In particular, we study the so-called Duality Theorem for the SOCP where continuous semimartingales under consideration have a variable diffusion matrix and then discuss the relation between the MKP and the SOCP. We also study the so-called Nelson's Problem, as the SOCP with fixed marginal distributions at each time, to which we give a new approach from the Duality Theorem. We finally consider a class of deterministic variational problems with fixed marginal distributions which is related to the SOCP by extending a class of measures under consideration.

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Let ${\mathcal A}$ be a nonempty real central arrangement of hyperplanes and ${\rm \bf Ch}$ be the set of chambers of ${\mathcal A}$. Each hyperplane $H$ makes a half-space $H^{+} $ and the other half-space $H^{-}$. Let $B = \{+, -\}$. For $H\in {\mathcal A}$, define a map $\epsilon_{H}^{+} : {\rm \bf Ch} \to B$ by $ \epsilon_{H}^{+} (C) = + _*_\text{(if_*_} C\subseteq H^{+}) \, \text{_*_and_*_} \epsilon_{H}^{+} (C) = - _*_\text{(if_*_} C \subseteq H^{-}).$ Define $ \epsilon_{H}^{-}=-\epsilon_{H}^{+}.$ Let ${\rm \bf Ch}^{m} = {\rm \bf Ch} \times{\rm \bf Ch}\times\dots\times{\rm \bf Ch} \,\,\,(m\text{_*_times}).$ Then the maps $\epsilon_{H}^{\pm}$ induce the maps $\epsilon_{H}^{\pm} : {\rm \bf Ch}^{m} \to B^{m} $. We will study the admissible maps $\Phi : {\rm \bf Ch}^{m} \to {\rm \bf Ch}$ which are compatible with every $\epsilon_{H}^{\pm}$. Suppose $|{\mathcal A}|\geq 3$ and $m\geq 2$. Then we will show that ${\mathcal A}$ is indecomposable if and only if every admissible map is a projection to a component. When ${\mathcal A}$ is a braid arrangement, which is indecomposable, this result is equivalent to Arrow's impossibility theorem in economics. We also determine the set of admissible maps explicitly for every nonempty real central arrangement.

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Motivated by the importance and universal character of phase singularities which are clarified recently, we study the local structure of equi-phase loci near the dislocation locus of complex valued planar and spatial waves, from the viewpoint of singularity theory of differentiable mappings, initiated by H. Whitney and R. Thom. The classification of phase-singularities are reduced to the classification of planar curves by radial transformations due to the theory of A. du Plessis, T. Gaffney and L. Wilson. Then fold singularities are classified into hyperbolic and elliptic singularities. We show that the elliptic singularities are never realized by any Helmholtz waves, while the hyperbolic singularities are realized in fact. Moreover, the classification and realizability of Whitney’s cusp, as well as its bifurcation problem are considered in order to explain the three points bifurcation of phase singularities. In this paper, we treat the dislocation of linear waves mainly, developing the basic and universal method, the method of jets and transversality, which is applicable also to non-linear waves.

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We study the evolution of N-point vortices in ring formation embedded in a background flowfield that initially corresponds to solid-body rotation on a sphere. The evolution of the point vortices are tracked numerically as an embedded dynamical system along with the M contours which separate strips of constant vorticity. The full system is a discretization of the Euler equations for incompressible flow on a rotating spherical shell, hence a ‘barotropic’ model of the one-layer atmosphere. We describe how the coupling creates a mechanism by which energy is exchanged between the ring and the background, which ultimately serves to break-up the structure. When the center-of-vorticity vector associated with the ring is initially misaligned with the axis of rotation of the background field, it sets up the propagation of Rossby-Haurwitz waves around the sphere which move retrograde to the solid-body rotation. These waves pass energy to the ring (in the case when the solid-body field and the ring initially co-rotate), or extract energy from the ring (when the solid-body field and the ring initially counter-rotate), hence the Hamiltonian and the center-of-vorticity vector associated with the N-point vortices are no longer conserved as they are for the one-way coupled model described in Newton & Shokraneh (2006a). In the first case, energy is transferred to the ring, the length of the center-of-vorticity vector increases, while its tip spirals in a clockwise manner towards the North Pole. The ring stays relatively intact for short times but ultimately breaks-up on a longer timescale. In the later case, energy is extracted from the ring, the length of the center-of-vorticity vector decreases while its tip spirals towards the North Pole and the ring loses its coherence more quickly than in the co-rotating case. The special case where the ring is initially oriented so that its center-of-vorticity vector is perpendicular to the axis of rotation is also examined as it shows how the coupling to the background field breaks this symmetry. In this case, both the length of the center-of-vorticity vector and the Hamiltonian energy of the ring achieve a local maximum at roughly the same time.

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We introduce new classes of domains, i.e., semi-uniform domains and inner emi-uniform domains. Both of them are intermediate between the class of John domains nd the class of uniform domains. Under the capacity density condition, we show that the armonic measure of a John domain D satisfies certain doubling conditions if and only if is a semi-uniform domain or an inner semi-uniform domain.

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We prove existence of global regular solutions for the 3D Navier-Stokes quations with (or without) Coriolis force for a class of initial data u0 in he space FM¾;± , i.e. for functions whose Fourier image bu0 is a vector-valued adon measure and that are supported in sum-closed frequency sets with istance ± from the origin. In our main result we establish an upper bound or admissible initial data in terms of the Reynolds number, uniform on the oriolis parameter ­. In particular this means that this upper bound is inearly growing in ±. This implies that we obtain global in time regular olutions for large (in norm) initial data u0 which may not decay at space nfinity, provided that the distance ± of the sum-closed frequency set from he origin is sufficiently large.

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We prove that the constant anisotropic mean curvature immersion of sphere $S^2$ in $\Bbb R^3$ is unique, provided that anisotropy is weak in the sense that the energy density function is close to isotropic one.

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A Q-algebra can be represented as an operator algebra on an infinite dimensional Hilbert space. However we don't know whether a finite n-dimensional Q-algebra can be represented on a Hilbert space of dimension n except n = 1, 2. It is known that a two dimensional Q-algebra is just a two dimensional commutative operator algebra on a two dimensional Hilbert space. In this paper we study a finite n-dimensional semisimple Q-algebra on a finite n-dimensional Hilbert space. In particular we describe a three dimensional Q-algebra of the disc algebra on a three dimensional Hilbert space. Our studies are related to the Pick interpolation problem for a uniform algebra.

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We consider the semi-relativistic Hartree type equation with nonlocal nonlinearity $F(u) = \lambda (|x|^{-\gamma} * |u|^2)u, 0 < \gamma < n, n \ge 1$. In \cite{chooz2}, the global well-posedness (GWP) was shown for the value of $\gamma \in (0, \frac{2n}{n+1}), n \ge 2$ with large data and $\gamma \in (2, n), n \ge 3$ with small data. In this paper, we extend the previous GWP result to the case for $\gamma \in (1, \frac{2n-1}n), n \ge 2$ with radially symmetric large data. Solutions in a weighted Sobolev space are also studied.

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We consider the Cauchy problem for the reactiondiffusion ystem with the nonlinear terms |x|σjupj vqj . In this system, he exponents p1 and q2 play a crucial role to determine the ehavior of the solutions. Using an ODE method, we prove the ujita-type nonexistence results for p1, q2 < 1, for q2 < 1 < p1 or or p1, q2 > 1. Moreover, we also show the nonexistence results for arge initial data.

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We introduce a $B_2$-type arrangements as a generalization of the classical Coxeter arrangement of type $B_2$, and consider the stability and the freeness of it. We show their (semi)stability is determined by the combinatorics. Moreover, we give a partial answer to the $4$-shift problem, which is the conjecture on the combinatorics and geometry induced from $B_2$-type arrangements.

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Let H(D) be an algebra of all holomorphic functions on the open nit disc D and X a subspace of H(D). When g is a function in H(D), put g(f)(z) = z ( )g0( )d and Ig(f)(z) = z 0( )g( )d (z 2 D) or f in X. In this paper, we study J[X] = {g 2 H(D) ; Jg(f) 2 X for all f in X} and [X] = {g 2 H(D) ; Ig(f) 2 X for all f in X}. We apply the results to concrete spaces. or example, we study J[X] and I[X] when X is a weighted Bloch space, a Hardy space r a Privalov space.

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This paper studies the heat convection equations when the convection term has some singularities at time zero. We shall establish the pointwise estimates for fundamental solutions from below by the Gaussian-like functions. As an application, we prove the existence and uniqueness of the mild solutions of the equations.

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This paper deals with an operator theory of compressed shifts on the Hardy space over the bidisk. We give commutant lifting type theorems and some interpolation theorems in two variables.

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We consider the scattering problems for two types of nonlinear Klein-Gordon equations. One is the equation of the Hartree type, and the other one is the equation with power nonlinearity. We show that the scattering operator for the equation of the Hartree type converges to that for the one with power nonlinearity in some sense. Our proof is based on some inequalities in the Lorentz space, and a strong limit of Riesz potentials.

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In this paper we introduce an elliptic analogue of the generalized Dedekind-Rademacher sums which satisfy reciprocity laws. In these sums, Kronecker's double series play a role of elliptic Bernoulli functions. This paper gives an answer to the problem of S. Fukuhara and N. Yui concerning the elliptic Apostol-Dedekind sums. We also mention a relation between the generating function of Kronecker's double series and that of the (Debye) elliptic polylogarithms studied by A. Levin.

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The Lorentzian space form with the positive curvature is called de Sitter space which is an important subject in the theory of relativity. In this paper we consider spacelike curves in de Sitter $3$-space. We define the notion of lightlike surfaces of spacelike curves in de Sitter $3$-space. We investigate the geometric meanings of the singularities of such surfaces.

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The problem of hyponormality for Toeplitz operators with (trigonometric) polynomial symbols is studied. We give a necessary and sufficient condition using the zeros of the analytic polynomial induced by the Fourier coefficients of the symbol.

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We study a mathematically rigorous derivation of a quantum mechanical Hamiltonian in a general framework. We derive such a Hamiltonian by taking a scaling limit for a generalization of the Nelson model, which is an abstract interaction model between particles and a Bose field with some internal degrees of freedom. Applying it to a model for the field of the nuclear force with an isospin, we obtain a Schrödinger Hamiltonian with a matrix valued potential, describing an effective interaction between nucleons.

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We say that a surface in Minkowski $3$-space is a {\it lightlike developable} if any pseudo-normal vector of the regular part of the surface is lightlike. We show that such a surface is a part of a lightlike plane, the lightcone, the tangent surface of a spacelike curve in a lightlike plane, the tangent surface of a lightlike curve or the glue of such surfaces. The most interesting surfaces in such the class of surfaces is the tangent surface of a lightlike curve. We give a classification of the singularities for the tangent surface of a generic lightlike curve. As a consequence, the $H_3$ type singularity appears in generic.

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Consider the nonlinear wave equation with zero mass in two space dimensions. hen it comes to the associated Cauchy problem with small initial data, the known existence esults are already sharp; those require the data to decay at a rate k ¸ kc, where kc is a critical ecay rate that depends on the order of the nonlinearity. However, the known scattering results reat only the supercritical case k > kc. In this paper, we prove the existence of the scattering perator for the full optimal range k ¸ kc.

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We consider the motion of four vortex points on sphere, which defines a Hamiltonian dynamical system. When the moment of vorticity vector, which is a conserved quantity, is zero at the initial moment, the motion of the four vortex points is integrable. The present paper gives a description of the integrable system by reducing it to a three-vortex problem. At the same time, we discuss if the vortex points collide self-similarly in finite time.

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We consider a gradient flow system of total variation with constraint. Our system is often used in the color image processing to remove a noise from picture. In particular, we want to preserve the sharp edges of picture and color chromaticity. Therefore, the values of solutions to our model is constrained in some fixed compact Riemannian manifold. By this reason, it is very difficult to analyze such a problem, mathematically. The main object of this paper is to show the global solvability of a constrained singular diffusion equation associated with total variation. In fact, by using abstract convergence theory of convex functions, we shall prove the existence of solutions to our models with piecewise constant initial and boundary data.

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We introduce the notion of the lightcone Gauss-Kronecker curvature for a spacelike submanifold of codimention two in Minkowski space which is a generalization of the ordinary notion of Gauss curvature of hypersurfaces in Euclidean space. In the local sense, this curvature describes the contact of such submanifolds with lightlike hyperplanes. We study geometric properties of such curvatures and show a Gauss-Bonnet type theorem. As examples we have hypersurfaces in hyperbolic space, spacelike hypersurfaces in the lightcone and spacelike hypersurfaces in de Sitter space.

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We study the Navier-Stokes equations for compressible {\it barotropic} fluids in a bounded or unbounded domain $\Omega $ of $ \mathbf{R}^3$. The initial density may vanish in an open subset of $\Omega$ or to be positive but vanish at space infinity. We first prove the local existence of solutions $(\rho^{(j)}, u^{(j)})$ in $C([0,T_* ]; H^{2(k-j)+3} \times D_0^1 \cap D^{2(k-j)+3} (\Omega ) )$, $0 \le j \le k, k \ge 1$ under the assumptions that the data satisfy compatibility conditions and that the initial density is sufficiently small. To control the nonnegativity or decay at infinity of density, we need to establish a boundary value problem of $(k+1)$-coupled elliptic system which may not be in general solvable. The smallness condition of initial density is necessary for the solvability, which is not necessary in case that the initial density has positive lower bound. Secondly, we prove the global existence of smooth radial solutions of {\it isentropic} compressible Navier-Stokes equations on a bounded annulus or a domain which is the exterior of a ball under a smallness condition of initial density.

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We study the global Cauchy problem and scattering problem for the semi-relativistic equation in $\mathbb{R}^n, n \ge 1$ with nonlocal nonlinearity $F(u) = \lambda (|x|^{-\gamma} * |u|^2)u, 0 <\gamma < n$. We prove the existence and uniqueness of global solutions for $0 < \gamma < \frac{2n}{n+1}, n \ge 2$ or $\gamma > 2, n \ge 3$ and the non-existence of asymptotically free solutions for $0 < \gamma \le 1, n\ge 3$. We also specify asymptotic behavior of solutions as the mass tends to zero and infinity.

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We study the inverse scattering problem for the nonlinear Schr\"odinger equation and for the nonlinear Klein-Gordon equation with the generalized Hartree type nonlinearity. We reconstruct the nonlinearity from knowledge of the scattering operator, which improves the known results.

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Let $A$ and $B$ be commuting bounded linear operators on a Hilbert space. In this paper, we study spectral area estimates for norms of $A^*B - BA^*$ when $A$ is subnormal or $p$-hyponormal.

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Consider the nonlinear wave equation with zero mass and a time-independent otential in three space dimensions. When it comes to the associated Cauchy problem, it s already known that short-range potentials do not a®ect the existence of small-amplitude olutions. In this paper, we focus on the associated scattering problem and we show that he situation is quite di®erent there. In particular, we show that even arbitrarily small and apidly decaying potentials may a®ect the asymptotic behavior of solutions.

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We study nonlinear Schr\"odinger equation with a delta-function impurity in one space dimension. Global well-posedness is proved for the Cauchy problem in $L^2(\R)$ under subcritical nonlinearity, as well as under critical nonlinearity with smallness assumption on the data. In the attractive case, orbital stability and instability of the ground state is proved in $H^1(\R)$.

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We consider extrinsic differential geometry on spacelike hypersurfaces in Minkowski pseudo-spheres (hyperbolic space, de Sitter space and the lightcone). In the previous paper we have shown a basic Legendrian duality theorem between pseudo-spheres. We define the spacelike parallels by using the basic Legendrian duality theorem. This definition unifies the notions of parallels of spacelike hypersurfaces in pseudo-spheres. We also define the evolute as the locus of singularities of the spacelike parallels. These notions are investigated as applications of Lagrangian or Legendrian singularity theory. We consider geometric properties of non-singular spacelike hypersurfaces corresponding to singularities of spacelike parallels or evolutes.

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The new class of weights called $A_p^{\dy ,m}$ weights is introduced. We prove that a characterization and an unconditional basis of the weighted $L^p$ space $L^p(\R^n , w(x)dx)$ with $w \in A_p^{\dy ,m}$ $(1<p<\infty)$ are given by the Haar wavelets and the Haar scaling function. As an application of these results, we establish a greedy basis by using the Haar wavelets and the Haar scaling function again.

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Let $\theta (x)$ denote Jacobi's theta function. We show that the function $F_\xi (x) = (\theta '(0) \theta (x+\xi) )/ (\theta (x) \theta (\xi))$ satisfies functional equations, which is a generalization of the harmonic relations for multiple zeta values.

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We study the existence and scattering of global small amplitude solutions to generalized Boussinesq (Bq) and improved modified Boussinesq (imBq) equations with nonlinear term $f(u)$ behaving as a power $u^p$ as $u \to 0$ in $\mathbb{R}^n, n \ge 1$.

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We study simple cases of crystalline driven curvature flow with spatially nonuniform riving force term. We assume special monotonicity properties of the driving term, which re motivated by our previous work on Berg’s effect. We show that initially flat facets, if oosely speaking they are to large, then they begin to bend.

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In a series of papers, we have proposed a dynamical model for gap junction-coupled networks of class I^* neurons, and investigated its dynamic characters. We found various dynamic states in a model neural network with diffusively coupled class I¤ neuron models, called μ-models. Among others, hierarchies of intermittent transitions attracted attention in relation with real brain dynamics. This paper is devoted to report a mechanism of the first transition appeared in the intermittenly transitory dynamics among an all-synchronized state, various metachronal waves and a weakly chaotic state. We clarify that this intermittent transition is described as an in-out intermittency.

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We prove time-local existence and uniqueness of solutions to a boundary layer roblem in a rotating frame around the stationary solution called Ekman spiral. Initial ata we choose in the vector-valued homogeneous Besov space _ B01 1; (R2;Lp(R+)) for < p < 1. Here the Lp-integrability is imposed in the normal direction, while we ay have no decay in tangential components, since the Besov space _ B01 1 contains ondecaying functions such as almost periodic functions. A crucial ingredient is theory or vector-valued homogeneous Besov spaces. For instance we provide and apply an perator-valued bounded H1-calculus for the Laplacian in _ B01 1(Rn; E) for a general anach space E.

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A blowing up solution of the semilinear heat equation $u_t =\Delta u+f(u) $ with $f$ satisfying $\liminf f(u)/u^p >0$ for some $p>1$ is considered when initial data $u_0 $ satisfies $u_0 \le M$, $u_0 \not\equiv M$ and $\lim_{m\to \infty } $ $ \inf_{x\in B_m } u_0 (x) =M$ with sequence of ball $\{ B_m \} $ whose radius diverging to infinity. It is shown that the solution blows up only at space infinity. A notion of blow-up direction is introduced. A characterization for blow-up direction is also established.

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Both the boundary Harnack principle and the Carleson estimate describe the boundary behavior of positive harmonic functions vanishing on a portion of the boundary. These notions are inextricably related and have been obtained simultaneously for domains with specific geometrical conditions. The main aim of this paper is to show that the boundary Harnack principle and the Carleson estimate are equivalent for arbitrary domains.

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We are concerned with the system of the $N$ vortex points on a sphere with two fixed vortex points at the both poles. This article gives a reduction method of the system to invariant dynamical systems. It is accomplished by using the invariance of the system with respect to the shift and the pole reversal transformations, for which the polygonal ring configuration of the $N$ vortex points at the line of latitude, called ``$N$-ring'', remains unchanged. We prove that there exists the $2p$-dimensional invariant dynamical system reduced by the $p$-shift transformation for arbitrary factor $p$ of $N$, and the $p$-shift invariant system is equivalent to the $p$-vortex points system generated by the averaged Hamiltonian on the sphere with the modified pole vortices. It is also shown that the system can be reduced by the pole reversal transformation when the pole vortices are identical. Since the reduced dynamical systems are defined by the linear combination of the eigenvectors obtained in the linear stability analysis for the $N$-ring, we obtain the inclusion structure among the invariant reduced dynamical systems, which allows us to decompose the system of the large vortex points into a collection of small reduced systems.

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For a nonnegative integrable weight function w on the unit circle , we provide an expression for p = 2, in terms of the series coefficients of he outer function of w, for the weighted Lp distance inff 􀀀T 1 − f|pwdμ, here μ is the normalized Lebesgue measure and f ranges over trigonometric olynomials with frequencies in [{. . . ,−3,−2,−1}\{−n}]∪{m}, m ≥ 0, n ≥ 2. he problem is open for p = 2.

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We present here classes of parabolic geometries arising naturally from Seashi’s principle to form good classes of linear differential equations of finite type, which generalize the cases of second and third order ODE for scalar function. We will explicitly describe the symbols of these differential equations. The model equations of these classes admit nonlinear contact transformations and their symmetry algebras (the Lie algebra of infinitesimal contact transformations preserving these equations) become finite dimensional and simple.

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We study smooth mappings with patterns which given by certain divergence diagrams of smooth mappings. The divergent diagrams of smooth mappings can be regard as smooth mappings from manifolds with singular foliations. Our concerns are generic differential topology and generic smooth mappings with patterns. We give a generic semi-local classification of surfaces with singularities and patterns as an application of singularity theory.

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Let 1 < p < ∞ . In this paper, for a measurable function v and a weight function w, the generalized Riesz projection P v is defined by P vf = vP(v -1f). (f ∈ L p(w)). If P0 is the self-adjoint projection from L2 (w) onto H2 (w), then P0 = P α for some outer function α satisfying w = |α| -2 . In this paper, P v on L p (w) is studied. As an application, the invertibility criterion for the generalized Toeplitz operator Tφv and the generalized singular integral operator φPv+Qv, Qv = I - Pv are investigated using the weighted norm inequality. The operator norm inequality for the generalized Hankel operator Hφv is also presented.

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We present a one-phase quasi-steady Stefan problem with Gibbs-Thomson and the kinetic effects when the interfacial energy is singular so that the equilibrium shape is a cylinder. We derive this model to describe crystal growth from vapor or solution. We summarize mathematical results on this model. Among other results we prove that a cylindrical shape is preserved if the initial cylindrical shape of a crystal is close to the equilibrium shape. Our formulation allows the possibility that cylindrical shape may break.

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The goal of this work is to investigate the relation of the no-response approach to some other non-iterative reconstruction schemes. We will derive several equivalence statements and dependency results. For simplicity we consider the obstacle reconstruction problem from far field data. In particular, we investigate two versions of the no-response test (NRT) for the inverse scattering problem. The first version is the pure NRT, the second combines the NRT with a rangetest element. We show convergence for these two versions without any eigenvalue assumption about the scatterer. Second, we state the natural formulation of the probe method for far field data and reformulate the singular sources method. We show that these statements of the two methods coincide and they form one face of the first version of the no-response test. Third, we prove that the convergence of the linear sampling method implies the convergence of the second version of the no-response test. Precisely, we show that we can use the blowup sequence of the linear sampling method to create the blowup sequence of the second version of the no response test. Fourth, we show that the two versions of the no response method are equivalent with respect to their convergence. Thus, the convergence of the linear sampling method also implies the convergence of the pure no-response test.

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An inverse boundary value problem for identifying the coefficient of some second order hyperbolic equation by one boundary measurement is considered. The problem is transformed to a minimization problem of a functional. By computing the Gateaux derivative of the functional, an algorithm for identifying the coefficient is given based on the projected gradient method. A numerical result is given testing the algorithm.

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We consider suspension semi-flows of angle-multiplying maps on the circle. Under a $C^r$generic condition on the ceiling function, we show that there exists an anisotropic Sobolev space¥cite{BT} contained in the $L^2$ space such that the Perron-Frobenius operator for the time-$t$-map act on it and that the essential spectral radius of that action is bounded by the square root of the inverse of the minimum expansion rate. This leads to a precise description on decay of correlations, which extends the result of M. Pollicott¥cite{Po}.

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This is a half survey on the classical results of extrinsic differential geometry of hypersurfaces in Euclidean space from the view point of Lagrangian or Legendrian singularity theory. Many results in this paper have been already obtained in some articles. However, we can discover some new information of geometric properties of hypersurfaces from this point of view.

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A scaling limit for the generalized spin-boson (GSB) model is considered. We derive a scaling limit of the Hamiltonian of the GSB model independently of whether or not the quantum scalareld has a mass. Applying it to a model for the eld of the nuclear force with isospin, we obtain an effective potential of the interaction between nucleons. Also, we get some applications to condensed matter physics.

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We consider the scattering problem for the Klein-Gordon equation with cubic convolution nonlinearity. We give some estimates for the nonlinearity, and prove the existence of the scattering operator, which improves the known results in some sense. Our proof is based on the Strichartz estimates for the inhomogeneous Klein-Gordon equation.

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In this paper, we consider the Cauchy problem of Schr\"{o}dinger-IMBq equations in $\mathbb{R}^n, n \ge 1$. We first show the global existence and blowup criterion of solutions in the energy space for the 3 and 4 dimensional system without power nonlinearity under suitable smallness assumption. Secondly the global existence is established to the system with $p$-powered nonlinearity in $H^s(\mathbb{R}^n), n = 1, 2$ for some $\frac n2 < s < \min(2,p)$ and some $p > \frac n2$. We also provide a blowup criterion for $n = 3$ in Triebel-Lizorkin space containing BMO space naturally.

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We introduce an elliptic analogue of the Bernoulli functions, which we call elliptic Bernoulli functions. They are defined by using the modified generating function of the elliptic polylogarithms. By degeneration of the elliptic Bernoulli functions, we obtain standard properties and new identities for the Bernoulli functions.

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We study the Navier-Stokes equations for heat-conducting incompressible fluids in a domain $\Omega \subset \mathbf{R}^3$ whose viscosity, heat conduction coefficients and specific heat at constant volume are in general functions of density and temperature. We prove the local existence of the unique strong solution, provided the initial data satisfy a natural compatibility condition. For the strong regularity, we do not assume the positivity of initial density; it may vanish in an open subset (vacuum) of $\Omega$ or decay at infinity when $\Omega$ is unbounded.

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We construct the local mild solutions of the Cauchy problem for the incompressible homogeneous Navier-Stokes equations in the $d$-dimensional Eucledian space with initial data in uniformly local $ L^{p} $ (=$ L^{p}_{uloc}) spaces where $ p $ is larger than or equal to $d$. As an application, we show that the mild solution associated with $ L^{d}_{uloc} $ almost periodic initial data at time zero becomes uniformly local almost periodic (=$ L^{\infty}-almost periodic ) in any positive time.

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The Cauchy problem for a coupled system of Schr\"odinger and improved Boussinesq equations is studied. Local well-posedness is proved in $L^2(\R^n)$ for $n\le 3$. Global well-posedness is proved in the energy space for $n\le 2$. Under smallness assumption on the Cauchy data, the local result in $L^2$ is proved for $n=4$.

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We consider Green polynomials at roots of unity. We obtain a recursive formula for Green polynomials at appropriate roots of unity, which is described in a combinatorial manner. The coefficients of the recursive formula are realized by the number of permutations satisfying a certain condition, which leads to interpretation of a combinatorial property of certain graded modules of the symmetric group in terms of representation theory.

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The convergence of solutions of anisotropic Allen-Cahn equations is studied when the interface thickness parameter(denoted by $\varepsilon$) tends to zero. It is shown that the convergence to a level set solution of the corresponding anisotropic interface equations is uniform with respect to the derivatives of a suface energy density function. As an application a cryatalline motion of interfaces in shown to be approximated by anisotropic Allen-Cahn equations.

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We consider the polaron model in the relativistic quantum electrodynamics(QED). We prove that the ground state energy of the model is finite for all values of the fine-structure constant and the ultraviolet cutoff ¤. Moreover we give an upper bound and a lower bound of the ground state energy.

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The unique local existence is established for the Cauchy problem of the incompressible Navier-Stokes equations with the Coriolis force. The Coriolis operator restricted to divergence free vector fields is a zero order pseudodifferential operator with the skew-symmetric matrix symbol related to the Riesz operator. It leads to the additional term in the Navier-Stokes equations which has real parameter being proportional to the speed of rotation. For initial data as Fourier preimage of the space of all finite Radon measures with no point mass at the origin we prove uniform estimate for the existence time in the speed of rotation.

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We find and prove sharp estimates for the Green function and 3G inequalities in uniform cones. Estimates are applied to give equivalent conditions for measures to satisfy the generalized Cranston-McConnell inequality, and to show the existence of infinitely many continuous solutions to nonlinear Schr¨odinger problems.

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We consider an investment model with memory in which the prices of n risky assets are driven by an Rn-valued Gaussian process with stationary increments that is different from Brownian motion. The driving process consists of n independent components, and each component is characterized by two parameters describing the memory. For the model, we explicitly solve the problem of maximizing the expected growth rate as well as that of maximizing the probability of overperforming a given benchmark.

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Generalizing the theorem for Goursat flags, we will characterize those flags which are obtained by “Rank 1 Prolongation” from the space of 1 jets for 1 independent and m dependent variables.

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A circular surface is a one-parameter family of standard circles in R3. In this paper some corresponding properties of circular surfaces with classical ruled surfaces are investigated. Singularities of circular surfaces are also studied.

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We consider Green polynomials at roots of unity, corresponding to partitions which we call $l$-partitions. We obtain a combinatorial formula for Green polynomials corresponding to $l$-partitions at primitive $l$-th roots of unity. The formula is rephrased in terms of representation theory of the symmetric group.

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We consider an inverse scattering problem for multiple obstacles $D=\cup_{j=1}^ND_j\subset {R}^3$ with different types of boundary of $D_j$. By constructing an indicator function from the far-field pattern of scattered wave, we can firstly determine the boundary location for all obstacles, then identify the boundary type for each obstacle, as well as the boundary impedance in case of Robin-type obstacles. The reconstruction procedures for these identifications are also given. Comparing with the existing probing method which is applied to identify one obstacle in generally, we should analyze the behavior of both the imaginary part and the real part of the indicator function so that we can identify the type of multiple obstacles.

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We study Martin boundary points of cones generated by spherical John regions. In particular, we show that such a cone has a unique (minimal) Martin boundary point at the vertex, and also at infinity. We also study a relation between ordinary thinness and minimal thinness, and the boundary behavior of positive superharmonic functions.

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A comparison estimate for the product of the Green function and the Martin kernel is given in a uniform domain. As its application, we show the equivalence of ordinary thinness and minimal thinness of a set contained in a non-tangential cone. We also give a comparison estimate for the Martin kernels with distinct singularities.

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Given two intersecting domains, we investigate the boundary behavior of the quotient of Martin kernels of each domain. To this end, we give a characterization of the minimal thinness for a difference of two subdomains in terms of Martin kernels of each domain. As a consequence of our main theorem (Theorem 2.1), we obtain the boundary growth of the Martin kernel of a Lipschitz domain, which corresponds to earlier results for the boundary decay of the Green function for a Lipschitz domain investigated by Burdzy, Carroll and Gardiner.

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We show the existence of a semimartingale of which one-dimensional marginal distributions are given by the solution of the Fokker-Planck equation with the $p$-th integrable drift vector ($p>1$).

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We study the existence and scattering of global small amplitude solutions to modified improved Boussinesq equations in one dimension with nonlinear term $f(u)$ behaving as a power $u^p$ as $u \to 0$. Solutions in $H^s$ space are considered for all $s > 0$. According to the value of $s$, the power nonlinearity exponent $p$ is determined. Liu \cite{liu} obtained the minimum value of $p$ greater than $8$ at $s = \frac32$ for sufficiently small Cauchy data. In this paper, we prove that $p$ can be reduced to be greater than $\frac92$ at $s > \frac85$ and the corresponding solution $u$ has the time decay such as $\|u( t)\|_{L^\infty} = O(t^{-\frac25})$ as $t \to \infty$. We also prove nonexistence of nontrivial asymptotically free solutions for $1 < p \le 2$ under vanishing condition near zero frequency on asymptotic states.

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We consider weak type $ (1,1) $ type estimates of Hardy-Littlewood maximal operators on a compact metric space with Radon measure, and also on a $ \sigma $-compact metric space with Radon measure. We show that the analogus results with M. Trinidad Menarguez and F. Sorias' hold in these settings if we impose some conditions on metric measure spaces.

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The strong comparison principle for semicontinuous viscosity solutions of some nonlinear elliptic equations are considered For linear elliptic equations it is well known that the strong comparison principle is equivalent to the strong maximum principle However for nonlinear equations the strong maximum principle may not imply the strong comparison principle We establish a strong comparison principle for some nonlinaer elliptic equations including the minimal surface equation

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In this paper, we consider the relation between the special values of the standard zeta functions and the congruence of cuspidal Hecke eigenforms with respect to $Sp_n(\bf Z)$. In particular, we propose a conjecture concerning the congruence of Saito-Kurokawa lift, and prove it under certain condition. Furthermore, we give exact values of the standerd zeta function for cuspidal Hecke eigenforms with respect to $Sp_2(\bf Z)$.

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This paper studies the large time asymptotic behavior of derivatives of the vorticity solving the two-dimensional vorticity equations equivalent to the Navier-Stokes equations. It is well-known by now that the vorticity behaves asymptotically as the Oseen vortex provided that the initial vorticity is integrable. This paper shows that each derivative of the vorticity also behave asymptotically as that of the Oseen vortex. For the proof new spatial decay estimates for derivatives are established. These estimates control behavior at the space infinity. The convergence result follows from a rescaling and compactness argument.

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Let $H$ be a self-adjoint operator on a Hilbert space ${\cal H}$, $T$ be a symmetric operator on ${\cal H}$ and $K(t)$ ($t\in \R$) be a bounded self-adjoint operator on ${\cal H}$. We say that $(T,H,K)$ obeys the {\it generalized weak Weyl relation} (GWWR) if $e^{-itH}D(T) \subset D(T)$ for all $t \in \R$ and $Te^{-itH}\psi=e^{-itH}(T+K(t))\psi, \forall \psi \in D(T)$ ( $D(T)$ denotes the domain of $T$). In the context of quantum mechanics where $H$ is the Hamiltonian of a quantum system, we call $T$ a {\it generalized time opeartor} of $H$. We first investigate, in an abstract framework, mathematical structures and properties of triples $(T,H,K)$ obeying the GWWR. These include the absolute continuity of the spectrum of $H$ restricted to a closed subspace of ${\cal H}$, an uncertainty relation between $H$ and $T$ (a \lq\lq{time-energy uncertainty relation}"), the decay property of transition probabilities $\left|\lang \psi,e^{-itH}\phi\rang \right|^2$ as $|t| \to \infty$ for all vectors $\psi$ and $\phi$ in a subspace of ${\cal H}$. We describe methods to construct various examples of triples $(T,H,K)$ obeying the GWWR. In particular we show that there exist generalized time operators of second quantization operators on Fock spaces (full Fock spaces, boson Fock spaces, fermion Fock spaces) which may have applications to quantum field models with interactions.

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We study a singular perturbation problem arising in the scalar two-phase field model. Given a sequence of functions with a uniform bound on the surface energy, assume the Sobolev norms W1,p of the associated chemical potential fields are bounded uniformly, where p > n 2 and n is the dimension of the domain. We show that the limit interface as ε tending to zero is an integral varifold with the sharp integrability condition on the mean curvature.

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We study global properties of closed spacelike curves in Minkowski 3-space.

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We propose an extended set of differential operators for local mirror symmetry. If X is Calabi-Yau such that dimH4(X,Z) = 0, then we show that our operators fully describe mirror symmetry. In the process, a conjecture for intersection theory for such X is uncovered. We also find new operators on several examples of type X = KS through similar techniques. In addition, open string PF systems are considered.

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We characterize horo-tight immersions into Dm in terms of a family of real valued functions parametrized by Sm−1. By means of such functions we provide an elementary proof that horo-tightness and tightness are equivalent properties in the class of immersions from S1 into hyperbolic space.

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In this paper, we compute the contributions from double cover maps to genus 0 degree 2 Gromov-Witten invariants of general type projective hypersurfaces. Our results correspond to a generalization of Aspinwall-Morrison formula to general type hypersurfaces in some special cases.

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The Helfrich variational problem is the minimizing problem of the bending energy among the closed surface with the prescribed area and enclosed volume. This is one of models for shape transformation theory of human red blood cell. Here the associated gradient flow, called the Helfrich flow, is studied. The existence of this geometric flow is proved locally for arbitrary initial data, and globally near spheres. Furthermore its center manifold near spheres is investigated.

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Weighted Strichartz estimates with homogeneous weights with critical exponents are proved for the wave equation without support restriction on the forcing term. The method of proof is based on the expansion by spherical harmonics and on the Sobolev space over the unit sphere, by which the required estimates are reduced to the radial case. As an application of the weighted Strichartz estimates, the existence and uniqueness of self-similar solutions to nonlinear wave equations is proved up to 5 space dimensions.

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We consider the motion of the $N$-vortex points that are equally spaced along a line of latitude on sphere with fixed pole vortices, called ``$N$-ring''. In particular, we focus on the evolution of the odd unstable $N$-ring. Since the eigenvalues that determine the stability of the odd $N$-ring are double, each of the unstable and stable manifolds corresponding to them is two-dimensional manifold. Accordingly, it is generally difficult to describe the global structure of the manifolds. In this article, based on the linear stability analysis, we propose a projection method to show the structure of the iso-surfaces of the Hamiltonian, in which the orbit of the vortex points exist. Then, applying the projection method to the motion of the $3$-ring and $5$-ring, we discuss the existence of the high-dimensional homoclinic and heteroclinic connections in the phase space, which characterize the evolution of the unstable $N$-ring.

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Conservation laws of the charge and of the energy are proved for nonlinear Schr\"odinger equations with nonlinearities of gauge invariance in a way independent of approximate solutions.

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We analyze the sharp-interface limit of the action minimization problem for the stochastically perturbed Allen-Cahn equation in one space dimension. The action is a deterministic functional which is linked to the behavior of the stochastic process in the small noise limit. Pre- viously, heuristic arguments and numerical results have suggested that the limiting action should \count" two competing costs: the cost to nucleate interfaces and the cost to propagate them. In addition, con- structions have been used to derive an upper bound for the minimal action which was proved optimal on the level of scaling. In this paper, we prove that for d = 1, the upper bound achieved by the constructions is in fact sharp. Furthermore, we derive a lower bound for the func- tional itself, which is in agreement with the heuristic picture. To do so, we characterize the sharp-interface limit of the space-time energy mea- sures. The proof relies on an extension of earlier results for the related elliptic problem.

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Let $Tf(x,t) = e^{2\pi it\phi(D)}f$ be the solution of of the general dispersive equation with the phase function $\phi$ and initial data $f$ in the Schwartz class. In case that the phase $\phi$ has a suitable growth rate at the infinity and the origin and $f$ is a finite linear combination of radial and spherical harmonic functions, we have global $L^p$ estimates of maximal operator defined by taking the supremum w.r.t. $t$. In particular, we obtain a global estimate at the end point left open.

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In this note we observe an interesting link between an intersection of two Legendrian manifolds in $ST^\ast\R^2$ - or dangerous self-tangencies, on the one hand and hyperbolic Morse transition of immersed curves on a torus on the other hand. The connection between these two arises naturally in the study of one-parameter families of conflict sets.

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We give a weighted L^p version of the Sobolev-Lieb-Thirring inequality for suborthonormal functions.

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We give a sufficient condition that non-radial H1-solutions to the Haraux-Weissler equation should belong to the weighted Sobolev space H1 (Rn), where ρ is the weight function exp(jxj2=4). Our result provides, in some sense, a connection between the solutions obtained by ODE method and those by variational approach in the space H1 (Rn).

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The effect of inhomogenity of nonlinear medium is discussed concerning the stability of standing waves eiωtφω(x) for a nonlinear Schrödinger equation with an inhomogeneous nonlinearity V (x)|u|p-1u, where V(x) is proportional to the electron density. Here, ω > 0 and φω(x) is a ground state of the stationary problem. When V(x) behaves like |x|-b at in nity, where 0 < b < 2, we show that eiωtφω(x) is stable for p < 1 + (4 - 2b)=n and sufficiently small ω > 0. The main point of this paper is to analyze the linearized operator at standing wave solution for the case of V (x) = |x|-b. Then, this analysis yields a stability result for the case of more general, inhomogeneous V (x) by a certain perturbation method.

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Exponential decay estimates are obtained for complex-valued solutions to non- linear elliptic equations in Rn, where the linear term is given by Schr odinger operators H = -Δ + V with nonnegative potentials V and the nonlinear term is given by a single power with subcritical Sobolev exponent in the attractive case. We descrive speci c rates of decay in terms of V , some of which are shown to be optimal. Moreover, our estimates provide a uni ed understanding of two distinct cases in the available literature, namely, the vanishing potential case V = 0 and the harmonic potential case V (x) = |x|2.

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We study the instability of standing waves eiωtφω(x) for a nonlinear Schr odinger equation with an inhomogeneous nonlinearity V (x)jujp-1u. Here, ω > 0 and φω(x) is a ground state of the stationary problem. When V (x) behaves like |x|-b at in nity, where 0 < b < 2, we show that eiωtφω(x) is unstable for p > 1 + (4 - 2b)=n and sufficiently small ω > 0. Due to the inhomogeneous medium, the unstable effect occurs in the region 1 + (4 - 2b)=n < p < 1 + 4=n which is the stable region in the case where V (x) 1.

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We construct a basic framework for the study of extrinsic differential geometry on timelike hypersurfaces from the view point of the theory of Legendrian singularities. As an application, we study the contact of timelike hypersurfaces with flat totally umbilic timelike hypersurfaces in de Sitter space.

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Recently there are so many mathematical models which describe nonlinear phenomena. In some phenomena, the free energy functional is not convex. So, the existence-uniqueness question is sometimes difficult. In order to study such phenomena, let us introduce the new class of abstract nonlinear evolution equations governed by timedependent operators of subdifferential type. In this paper we shall show the existence and uniqueness of solution to nonlinear evolution equations with time-dependent constraints in a real Hilbert space. Moreover we apply our abstract results to a parabolic variational inequality with time-dependent double obstacles constraints.

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A class of graded representations of the symmetric group, concerning with the cohomology ring of the corresponding flag variety, are considered. We point out a certain combinatorial property of the Poincaré polynomial of these graded representations, and interpret it in the language of representation theory of the symmetric group.

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We study the equation for improper (parabolic) affine spheres from the view point of contact geometry and provide the generic classification of singularities appearing in geometric solutions to the equation as well as their duals. We also show the results for surfaces of constant Gaussian curvature and for developable surfaces. In particular we confirm that generic singularities appearing in such a surface are just cuspidal edges and swallowtails.

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In this paper we study the partial regularity of a functional on BV space proposed by Chambolle and Lions [3] for the purposes of image restoration. The functional designed to smooth corrupted images using isotropic diffusion via the Laplacian where the gradients of the image are below a certain threshold \epsilon and retain edges where gradients are above the threshold using the total variation. Here we prove that if the solution $u \in BV$ of the model minimization problem, defined on an open set \Omega, is such that the Lebesgue measure of the set where the gradient of $u$ is below the threshold \epsilon is positive, then ther exists a non-empty open region $E$ for which $u \in C^{1,\alpha}$ on $E$ and $|\nabla u|<\epsilon$, and $|\nabla u| \geq \epsilon $ on $\Omega\setminus E $ a.e. Thus we indeed have smoothing where $|\nabla u|<\ \epsilon$.

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We will introduce the $k$ times modified centered and uncentered Hardy-Littlewood maximal operators on nonhomogeneous spaces for $k>0$. We will prove the $k$ times modified centered Hardy-Littlewood maximal operator is weak type $(1,1)$ bounded with constant $1$ when $k \ge 2$ if the Radon measure of the space has ``continuitiy'' in some sense. In the proof, we will use the outer measure associated with the Radon measure. We will also prove other results of Hardy-Littlewood maximal operators on homogeneous spaces and on the real line by using outer measures.

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We establish the existence, uniqueness and $L^q$ estimates of weak solutions to the stationary Stokes equations with rotation effect both in the whole space and in exterior domains. The equation arises from the study of viscous incompressible flows around a body that is rotating with a constant angular velocity, and it involves an important drift operator with unbounded variable coefficient that causes some difficulties.

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We solve optimal transportation problem using stochastic optimal control theory. Indeed, for a super linear cost at most quadratic at infinity, we prove Kantorovich duality theorem by a zero noise limit (or vanishing viscosity) argument.. We also obtain a characterization of the support of an optimal measure in Monge-Kantorovich minimization problem (MKP) as a graph. Our key tool is a duality result for a stochastic control problem which naturally extends (MKP).

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In this paper we study a global in time existence of classical so- lutions of semilinear systems of 3-dim. wave equations. We see that one component of the global in time solution can be arbitrarily large if another component is small enough according to some balance of each amplitude. Also its sharpness is discussed. This is a speci c nature of strongly coupled systems.

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We give a global existence theorem to systems of quasilinear wave equations in three space dimensions, especially for the multiple-speed cases. It covers a wide class of quadratic nonlinearities which may depend on unknowns as well as their first and second derivatives. Our proof is achieved through total use of pointwise and L2-estimates concerning unknowns and their first and second derivatives.

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We propose a regularized optimization problem for computing numerical di®erentiation for the second order deriva- tive for functions with two variables from the noisy values of the function at scattered points, and give the proof of the existence and uniqueness of the solution of this problem. The reconstruc- tion scheme is also given during the proof, which is based on bi- harmonic Green function. The convergence estimate of the regular- ized solution to the exact solution for the regularized optimization problem as the regularized parameter and discrepancy of noisy data tending to zero is provided under a simple choice of regularization parameter. In the end we give the numerical examples and analyze the computational results.

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Let L2 = L2(D, rdrdq/p) be the Lebesgue space on the open unit disc D and let L2 a = L2 \Hol(D) be a Bergman space on D. In this paper, we are interested in a closed subspace M of L2 which is invariant under the multiplication by the coordinate function z, and a Hankel type operator from L2 a toM?. In particular, we study an invariant subspace M such that there does not exist a finite rank Hankel type operator except a zero operator. 2

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In this note, we consider a maximal operator supt2R ju(x; t)j = supt2R jeit­(D)f(x)j, where u is the solution to the initial value problem ut = i­(D)u, u(0) = f for a C2 function ­ with some growth rate at in¯nity. We prove that the operator supt2R ju(x; t)j has a mapping property from a fractional Sobolev space H 1 4 with additional angular regularity to L2 loc.

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We study the convolution operator $T^z$ with the distribution kernel given by analytic continuation from the function $$ \widetilde{K}^z(y,s,t)= \left\{\begin{array}{ll} (t^2-s^2-|y|^2)_+^z/\Gamma(z+1)\quad &\mbox{if}\quad t>0\\ 0 \quad&\mbox{if} \quad t\le 0\end{array}\right\}, \quad Re(z)>-1 $$ where $(y,s,t)\in \mathbb R^{n-1}\times\mathbb R\times \mathbb R$. We obtain some improvement upon the previous known estimates for $T^z$. Then we extend the result of the cone multiplier of negative order on $\mathbb{R}^3$ \cite{lee1} to the case of general $\mathbb{R}^{n+1},\, n \ge 2$.

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A unique classical solution of the Cauchy problem for the Navier-Stokes equa- tions is considered when the initial velocity is spatially almost periodic. It is shown that the solution is always spatially almost periodic at any time provided that the solution exists. No restriction on the space dimension is imposed. This fact follows from continuous dependence of the solution with respect to initial data in uniform topology. Similar result is also established for Cauchy problem of the three- dimensional Navier-Stokes equations in a rotating frame.

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We study geometric transitions on Calabi-Yau manifolds from the perspective of the B model. Looking toward physically motivated predictions, it is shown that the traditional conifold transition is too simple a case to yield meaningful results. The mathematics of a nontrivial example is worked out, and the expected equivalence is demonstrated.

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A definition of open string period integrals for noncompact Calabi- Yau manifolds is given. It is shown that the open string Picard-Fuchs operators, originally derived through physical considerations, follow from these period integrals. Also, we find that the natural extension to the compact case does not yield the expected results.

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We show the blow-up of smooth solution of viscous heat-conducting flow when the initial density is compactly supported. This is an ex- tension of Z. Xin's result[4] to the case of positive heat conduction coe±cient but we do not need any information for the lower bound of the entropy. We control the lower bound of second moment by total energy.

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Consider a Stefan-like problem with Gibbs-Thomson and kinetic effects as a model of crystal growth from vapor. The equilibrium shape is assumed to be a regular circular cylinder. Our main concern is a problem whether or not a surface of cylindrical crystals (called a facet) is stable under evolution in the sense that its normal velocity is constant over the facet. If a facet is unstable, then it breaks or bends. A typical result we establish is that all facets are stable if the evolving crystal is near the equilibrium. The stability criterion we use is a variational principle for selecting the correct Cahn-Hoffman vector. The analysis of the phase plane of an evolving cylinder (identi ed with points in the plane) near the unique equilibrium provides a bound for ratio of velocities of top and lateral facets of the cylinders. 1

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We consider a Dirac operator H acting in the Hilbert space L2(IR3;C4) ○C2, which describes a Hamiltonian of the chiral quark soliton model in nuclear physics. The mass term of H is a matrix-valued function formed out of a function F : IR3 ! IR, called a pro le function, and a vector eld n on IR3, which xes pointwise a direction in the iso-spin space of the pion. We rst show that, under suitable conditions, H may be regarded as a generator of a supersymmetry. In this case, the spetra of H are symmetric with respect to the origin of IR. We then identify the essential spectrum of H under some condition for F. For a class of pro le functions F, we derive an upper bound for the number of discrete eigenvalues of H. Under suitable conditions, we show the existence of a positive energy ground state or a negative energy ground state for a family of scaled deformations of H. A symmetry reduction of H is also discussed. Finally a unitary transformation of H is given, which may have a physical interpretation.

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We give a simple proof of the duality theorem for the Monge-Kantorovich problem in the Euclidean setting. The selection lemma which is useful in the theory of stochastic optimal controls plays a crucial role.

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We study the Navier-Stokes equations for compressible barotropic fluids in a bounded or unbounded domain ­ of R3. We ¯rst prove the local existence of solutions (p u) in C([0; T*]; (p∞ +H3(Ω))×(D10 ∩ \D3)(Ω)) under the assumption that the data satis¯es a natural compatibility condition. Then deriving the smoothing effect of the velocity u in t > 0, we conclude that (p, u) is a classical solution in (0; T**) × Ω for some T** ∈ (0, T*]. For these results, the initial density needs not be bounded below away from zero and may vanish in an open subset (vacuum) of Ω.

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We consider the full Navier-Stokes equations for viscous polytropic fluids with nonnegative thermal conductivity. We prove the existence of unique local strong solutions for all initial data satisfying some compatibility condition. The initial density need not be positive and may vanish in an open set. Moreover our results hold for both bounded and unbounded domains.

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In this paper, we use the no-response test idea, introduced in ([L-P], [P1]) for the inverse obstacle problem, to identify the interface of the discontinuity of the coefficient of the equation ∇· (x)∇+c(x) with piecewise regular and bounded function c(x). We use infinitely many Cauchy data as measurement and give a reconstructive method to localize the interface. We will base this multiwave version of the no-response test on two different proofs. The first one contains a pointwise estimate as used by the singular sources method. The second one is built on an energy (or an integral) estimate which is the basis of the probe method. As a conclusion of this, the no response can be seen as a unified framework for the probe and the singular sources method. As a further contribution, we provide a formula to reconstruct the values of the jump of (x), x ∈ @D at the boundary.

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We show four Legendrian dualities between pseudo-spheres in Minkowski space as a basic theorem. We can apply such dualities for constructing extrinsic differential geometry of spacelike hypersurfaces in pseudo-spheres. In this paper we stick to spacelike hypersurfaces in the lightcone and establish an extrinsic differential geometry which we call the lightcone differential geometry.

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In this paper we consider 2 x 2 matrix functions analytic on the upper half plane associated to the hypergeometric function of Gauss, and establish transformations of these matrix functions under some modular transformations. The matrix functions studied here are obtained as the lifts of the local solutions of the matrix hypergeometric differential equation of SL type (i.e., whose image of monodromy representation is contained in S£(2, C)) at 0, 1, oo to the upper half plane by the lambda function (§2). Each component of the matrix functions is represented by a definite integral with a power product of theta functions as integrand. Such an integral was invented by Wirtinger in order to uniformize the hypergeometric function of Gauss to the upper half plane ([5)). In this paper we call it Wirtinger integral (cf. (1.2)). In spite of many possibilities of application of the Wirtinger integral, there seems to be very few examples of application of the Wirtinger integral in literature. One of the advantages of exploiting the matrix functions above in the study of the hypergeometric function is that the monodromy property and the connection relations of the hypergeometric function are all translated as transformations of those matrix functions under modular transformations of the independent variable (§3). Moreover we can derive such transformations by exploiting classical formulas of theta functions without need to use any monodromy property or connection formula of the hypergeometric function. That is to say, this gives another new derivation of the monodromy property and the connection formulas of the hypergeometric function of Gauss.

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The purpose of this note is to study the coefficients of the polynomials if we express the weight enumerator as the polynomial the fixed generators.

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A nonnegative blowing up solution of the semilinear heat equation ut = ¢u + up with p > 1 is considered when initial data u0 satisfies lim|x|→∞ 0 = M > 0, 0 ≤ M and u0 ≢ M. It is shown that the solution blows up only at space infinity and that lim|x|→∞ u(x, t) is the solution of the ordinary di®erential equation vt = vp with v(0) = M.

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Periodic orbits forhomeomorphisms on the plane give mathematical braids, which are topologically classified into three types by Thurston-Nielsen (T-N) theory; (1) periodic, (2) reducible, and (3) pseudo-Anosov (pA). If the braid is pA, then the homeomorphism must have an infinitely many number of pe-riodic orbits of distinct periods. This kind of complexity induced by the pA braid is called “topological chaos”, which was introduced by Boyland et. al [4] recently. We investigate numerically the topological chaos embedded in the particle mixing by the blinking vortex system introduced by Aref [1]. It has already been known that the system generates the chaotic advection due to the homoclinic chaos, but the chaotic mixing region is restricted locally in the vicinity of the vortex points. In the present study, we propose an in-genious operation of the blinking vortex system that defines a mathematical braid of pA type. The operation not onlygenerates the chaotic mixing region due to the topological chaos, but also ensures global particle mixing in the whole plane. We give a mathematical explanation for the phenomenon by the T-N theory and some numerical evidences to support the explanation. More-over, we makemention of the relation between the topological chaos and the homoclinic chaos in the blinking vortex system.

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We prove $L^p-L^q$ maximal regularity estimates for the Stokes equations in spatial regions with moving boundary. Our result includes bounded and unbounded regions. The method relies on a reduction of the problem to an equivalent nonautonomous system on a cylindrical space-time domain. By applying suitable abstract results for nonautonomous Cauchy problems we show maximal regularity of the associated propagator which yields the result. The abstract results, also proved in this note, are a modified version of a nonautonomous maximal regularity result of Y. Giga, M. Giga, and H. Sohr and a suitable perturbation result. Finally we describe briefly the application to the special case of rotating regions.

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We study an abstract doubly nonlinear evolution equations associated with elliptic-parabolic free boundary problems. In this paper we show the existence and uniqueness of solution for the doubly nonlinear evolution equation. Moreover we apply our abstract results to an elliptic-parabolic free boundary problem.

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We are concerned with variational properties of a fold energy for a unit, dilation-invariant gradient field (called a cluster) in the unit disk. We show that boundedness of a fold energy implies L1-compactness of clusters. We also show that a fold energy is L1-lower semicontinuous. We characterize absolute minimizers. We also give a sequence of stationary states and discuss its stability. Surprisingly, the stability depends upon q, the power of modulus of the jump discontinuities, in the definition of the fold energy.

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We study the center map of an equiaffine immersion which is introduced using the equiaffine support function. The center map is a constant map if and only if the hypersurface is an equiaffine sphere. We investigate those immersions for which the center map is affine congruent with the original hypersurface. In terms of centroaffine geometry, we show that such hypersurfaces provide examples of hypersurfaces with vanishing centroaffine Tchebychev operator. We also characterize them in equiaffine differential geometry using a curvature condition involving the covariant derivative of the shape operator. From both approaches, assuming the dimension is 2 and the surface is definite, a complete classification follows.

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Three-dimensional rotating Navier-Stokes equations are considered with a constant Coriolis parameter $\Omega$ and initial data nondecreasing at infinity. In contrast to the non-rotating case ($\Omega=0$), it is shown for the problem with rotation ($\Omega \neq 0$) that Green's function corresponding to the linear problem (Stokes + Coriolis combined operator) does not belong to $L^1({\mathbb R}^3)$. Moreover, the corresponding integral operator is unbounded in the space $L^{\infty}_{\sigma}({\mathbb R}^3)$ of solenoidal vector fields in ${\mathbb R}^3$ and the linear (Stokes+Coriolis) combined operator does not generate a semigroup in $L^{\infty}_{\sigma}({\mathbb R}^3)$. Local in time, uniform in $\Omega$ unique solvability of the rotating Navier-Stokes equations is proven for initial velocity fields in the space $L^{\infty}_{\sigma,a}({\mathbb R}^3)$ which consists of $L^{\infty}$ solenoidal vector fields satisfying vertical averaging property such that their baroclinic component belongs to a homogeneous Besov space ${\dot B}_{\infty,1}^0$ which is smaller than $L^\infty$ but still contains various periodic and almost periodic functions. This restriction of initial data to $L^{\infty}_{\sigma,a}({\mathbb R}^3)$ which is a subspace of $L^{\infty}_{\sigma}({\mathbb R}^3)$ is essential for the combined linear operator (Stokes + Coriolis) to generate a semigroup. The proof of uniform in $\Omega$ local in time unique solvability requires detailed study of the symbol of this semigroup and obtaining uniform in $\Omega$ estimates of the corresponding operator norms in Banach spaces. Using the rotation transformation, we also obtain local in time, uniform in $\Omega$ solvability of the classical 3D Navier-Stokes equations in ${\mathbb R}^3$ with initial velocity and vorticity of the form $\mbox{\bf{V}}(0)=\tilde{\mbox{\bf{V}}}_0(y) + \frac{\Omega}{2} e_3 \times y$, $\mbox{curl} \mbox{\bf{V}}(0)=\mbox{curl} \tilde{\mbox{\bf{V}}}_0(y) + \Omega e_3$ where $\tilde{\mbox{\bf{V}}}_0(y) \in L^{\infty}_{\sigma,a}({\mathbb R}^3)$.

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We present new criteria for a self-adjoint operator to have a ground state. As an application, we consider models of ``quantum particles'' coupled to a massive Bose field and prove the existence of a ground state of them, where the particle Hamiltonian does not necessarily have compact resolvent.

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We derive a class of macroscopic differential equations that describe collective adaptation, starting from a discrete-time stochastic microscopic model. The behavior of each agent is a dynamic balance between adaptation that locally achieves the best action and memory loss that leads to randomized behavior. We show that, although individual agents interact with their environment and other agents in a purely self-interested way, macroscopic behavior can be interpreted as game dynamics. Application to several familiar, explicit game interactions shows that the adaptation dynamics exhibits a diversity of collective behaviors, including stable limit cycles, quasiperiodicity, intermittency, and deterministic chaos. The simplicity of the assumptions underlying the macroscopic equations suggests that these behaviors should be expected broadly in collective adaptation. We also analyze the adaptation dynamics from an information-theoretic viewpoint and discuss self-organization induced by information flux between agents, giving a novel view of collective adaptation.

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We construct a binary market model with memory that approximates a continuous-time market model driven by a Gaussian process equivalent to Brownian motion. We give a sufficient conditions for the binary market to be arbitrage-free. In a case when arbitrage opportunities exist, we present the rate at which the arbitrage probability tends to zero as the number of periods goes to infinity.

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We give a new proof of the global existence theorem of Klainerman for the Cauchy problem of quasi-linear wave equations in space dimensions n ¸ 4. In addition to the Klainerman-Sideris inequality, a space-time L2-estimate plays a key role in the proof. We answer a question raised by Metcalfe in [11].

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We shall give a new proof of temporally global existence of small solutions for systems of semi-linear wave equations. Our proof uses the Klainerman-Sideris inequality and a space-time L^2-estimate. We shall also discuss whether the scale-invariant version of the space-time L^2-estimates can hold, and obtain some related estimates. Among other things, we prove that the Keel-Smith-Sogge estimate actually holds in all space dimensions.

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We shall consider the inverse scattering problem for time dependent version of Hartree-Fock equation and nonlinear Klein-Gordon equation.The uniqueness theorem on identifying the cubic convolution nonlinearity from the knowledge of the scattering operator will be shown.

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We study the linear filtering problem for systems driven by continuous Gaussian processes V1 and V2 with memory described by two parameters. The processes Vj have the virtue that they possess stationary increments and simple semimartingale representations simultaneously. It allows for straightforward parameter estimations. After giving the semimartingale representations of Vj by innovation theory, we derive Kalman-Bucy-type filtering equations for the systems. We apply the result to the optimal portfolio problem for an investor with partial observations. We illustrate the tractability of the filtering algorithm by numerical implementations.

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We consider an operator $Q(V)$ of Dirac type with a meromorphic potential given in terms of a function $V$ of the form $V(z)=\lambda V_1(z)+\mu V_2(z), \ z\in \BbbC\setminus\{0\}$, where $V_1$ is a complex polynomial of $1/z$, $V_2$ is a polynomial of $z$, and $\lambda$ and $\mu$ are non-zero complex parameters. The operator $Q(V)$ acts in the Hilbert space $L^2(\BbbR^2;\BbbC^4)=\oplus^4L^2(\BbbR^2)$. The main results we prove include: (i) the (essential) self-adjointness of $Q(V)$; (ii) the pure discreteness of the spectrum of $Q(V)$ ; (iii) if $V_1(z)=z^{-p}$ and $4 \leq \deg V_2 \leq p+2$, then $\ker Q(V)\not=\{0\}$ and $\dim \ker Q(V)$ is independent of $(\lambda,\mu)$ and lower order terms of $\partial V_2/\partial z$; (iv) a trace formula for $\dim \ker Q(V)$.

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We consider the Cauchy problem for some semi-linear wave equations in three space dimensions and prove global or almost global existence of the Klainerman-Machedon radial solutions. The proof is carried out by a contraction-mapping argument based on a refined version of the Keel-Smith-Sogge estimate, together with the Morawetz-type inequality.

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Let $W$ be a nonnegative summable function whose logarithm is also summable with respect to the Lebesgue measure on the unit circle. For $0 < p < \infty,_*_H^p(W)$ denotes a weighted Hardy space on the unit circle. When $W \equiv 1,_*_H^p(W)$ is the usual Hardy space $H^p$. We are interested in $H^p(W)_+$ the set of all nonnegative functions in $H^p(W)$. If $p \geq 1/2,_*_H^p_+$ consists of constant functions. However $H^p(W)_+$ contains a nonconstant nonnegative function for some weight $W$. In this paper, if $p \geq 1/2$ we determine $W$ and describe $H^p(W)_+$ when the linear span of $H^p(W)_+$ is of finite dimension. Moreover we show that the linear span of $H^p(W)_+$ is of infinite dimension for arbitrary weight $W$ when $0 < p < 1/2$.

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We give a duality theorem for the stochastic optimal control problem with a convex cost function and show that the minimizer can be characterized by a class of forward-backward stochastic differential equations. As an application, we give an approach, from the duality theorem, to h-path processes for diffusion processes.

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In this paper we study the inverse scattering problem of determining the potential for the two dimensional Schr\"odinger operator of the form -\delta u(x)+i \sqrt{E} b(x)u(x)=Eu(x), E>0 which is derived from the dissipative wave equation w_{tt}(x, t)-\delta w(x, t)+b(x)w_t =0 The uniqueness theorem will be shown without assuming the smallnes condition on b(x) under the low energy.

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In the previous paper, we give a characterization of backward shift invariant subspaces of the Hardy space in the bidisc which satisfy the doubly commuting condition S_z S_w^* = S_w^* S_z for the compression operators S_z and S_w. In this paper, we give a characterization of backward shift invariant subspaces satisfying S_z^2 S_w^* = S_w^* S_z^2.

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For a given sector a selfsimilar expanding solution to a crystalline flow is constructed. The solution is shown to be unique. Because of selfsimilarity the problem is reduced to solve a system of algebraic equations of degree two. The solution is constructed by a method of continuity and obtained by solving associated ordinary differential equations. The selfsimilar expanding solution is useful to construct a crystalline flow from an arbitrary polygon not necessarily admissible.

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This paper deals with a dynamical inverse problem for a composite beam formed by two connected beams. The vibrations of the composite beam are governed by a differential system where a coupling takes place between longitudinal and bending motions. In this paper, we neglect bending motions and we only deal with the longitudinal motions. These motions are governed by a two-by-two second order system coupled in the lower order terms by the shearing stiffness coefficient, which models the connection between the two beams and which contains direct information on the integrity of the system. We prove that the shearing stiffness coefficient can be reconstructed from the frequency response function of the system evaluated at one end of the beam.

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We establish the interior gradient estimate for general 1-D anisotropic curvature flow. The estimate depends only on the height of the graph and not on the gradient at initial time.

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We consider the finite-past predictor coefficients of stationary time series, and establish an explicit representation for them, in terms of the MA and AR coefficients. The proof involves the alternate iteration of projection operators associated with the infinite past and the infinite future. We provide several applications, which include rates of convergence of the finite predictor coefficients, an equality of Baxter-type for long memory processes, and a simple representation of the partial autocorrelation function α(·). We use the last result to obtain the precise asymptotic behavior of α(·) with remainder, for the fractional ARIMA processes.

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Let us consider a nonlinear evolution equation associated with time-dependent subdifferential in a separable Hilbert space. In this paper we treat an asymptotically periodic system which means that time-dependent terms converge to some time-periodic ones as time goes to +∞. Then we consider the large-time behaviour of solutions without uniqueness. In such a situation the corresponding dynamical systems are multivalued. In fact we discuss the stability of multivalued semiflows from the view-point of attractors. Namely, the main object of this paper is to construct a global attractor for the asymptotically periodic multivalued dynamical system, and to discuss the relationship to one for the limiting periodic system

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We show Rouch e type theorems using a theorem of Adamyan, Arov and Krein. As applications, we obtain a certain characterization of self-maps of the unit disc in terms of the location of the Denjoy-Wolf point and we study a function in the Smirnov class whose real part is positive.

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Steady solution and asymptotic behaviour of corresponding nonsteady solution are studied for the Navier-Stokes equations under general Navier slip boundary condition. It is proved that the existence of a unique stationary solution and that this solution is asymptotically stable under some restrictions of the data.

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We study the motion of a polygonal ring consists of identical vortex points that are equally spaced at a line of latitude on a sphere with vortex points fixed at the both poles. First, we calculate explicitly all the eigenvalues and the eigenvectors corresponding to them for the linearized problem, from which we consider the stability of the polygonal vortex ring in the presence of the pole vortices. Next, when the number of the vortex points is even in particular, the equations of the vortex points are reduced to those for a pair of two vortex points by assuming a special symmetry. Studying the reduced system mathematically and numerically, we describe an universal transition of global periodic motion of the perturbed polygonal ring. Moreover, we also discuss the stability of the periodic motion.

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We study analytic smoothing effects for solutions to the Cauchy problem for the Schr\"odinger equation with interaction described by the integral of the intensity with respect to one direction in two space dimensions. The only assumption on the Cauchy data is the weight condition of exponential type and no regularity assumption is imposed.

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We study some geometrical properties associated to the contact of submanifolds with hyperhorospheres in hyperbolic n-space as an application of the theory of Legendrian singularities.

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A gradient system of total variation is considered for a mapping from the unit disk to the unit sphere in R3, For a class of initial data it is shown that a solution of its Dirichlet problem loses its smoothness in finite time.

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We prove that the Stokes operator with Robin boundary conditions is the generator of a bounded holomorphic semigroup on L^\infty_\sigma({\mathbb R}^n_+), which is even strongly continuous on the space \BUC_\sigma({\mathbb R}^n_+). Contrary to that result it is also proved that there exists no Stokes semigroup on L^1_\sigma({\mathbb R}^n_+), except if we assume the special case of Neumann boundary conditions. Nevertheless, we also obtain gradient estimates for the solution of the Stokes equations in L^1_\sigma({\mathbb R}^n_+) for all types of Robin boundary conditions.

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The spectra of the Toeplitz operators on the weighted Hardy space Hp(Wd =2 ) are studied. For example, the theorems of Brown-Halmos type and Hartman- Wintner type are proved. These generalize results in the previous paper which were proved for p = 2. 2

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We are concerned with a quasi-steady Stefan type problem with Gibbs-Thomson relation and the mobility term which is a model for a crystal growing from supersaturated vapor. The evolving crystal and the Wulff shape of the interfacial energy are assumed to be (right-circular) cylinders. In pattern formation deciding what are the conditions which guarantee that the speed in the normal direction is constant over each facet, so that the facet does not break, is an important question. We formulate such a condition with an aid of a convex variational problem with a convex obstacle type constraint. We derive necessary and sufficient conditions for the non-breaking of facets in terms of the size and the supersaturation at space infinity when the motion is self-similar.

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We shall propose a new predator-prey system model which not only answeres D'Ancona's question but also explains Gause's experiment.

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For some 3-dimensional system of electrial circuits, the uniformly ultimate boundedness of solutions is proved, and consequently existing unstable manifolds around equilibrium points is bounded.

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We study the initial-boundary value problem for the Stokes equations with Robin boundary conditions in the half-space Rn+. It is proved that the associated Stokes operator is sectorial and admits a bounded H∞-calculus on Lq0(Rn+). As an application we prove also a local existence result for the nonlinear initial value problem of the Navier-Stokes equations with Robin boundary conditions.

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We give an algorithm for computing the special values of twisted standard zeta functions of elliptic modular forms by using the pullback formula for the Siegel Eisenstein series of degree 2.

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In this paper let us consider double obstacle problems, which includes regional economic growth models. By prescribed time-dependent obstacles, our problems are nonautonomous systems and it is impossible to show the uniqueness of solutions. Therefore the associated dynamical systems are multivalued. In this paper from the viewpoint of attractors we shall consider the periodic stability for the double obstacle problem with asymptotically periodic data. Namely, assuming that time-dependent data converges to time-periodic ones as time goes to infinity, we shall construct the global attractor for the asymptotically periodic multivalued dynamical system. Moreover we shall discuss the relationship to the attractor for the limiting periodic problem.

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In this paper, we study submodules over R2. We will give a Lax-type of theorem and a result analogous ti Helson's theory.

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We are interested in extremal functions in a Hardy space Hp(Tn) (1 p 1). For example, we study extreme points of the unit ball of H1(Tn) and give a factorization theorem. In particular, we show that any rational function can be factorized.

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A numerical method for obtaining a crystalline flow starting from a general polygon is presented. A crystalline flow is a polygonal flow and can be regarded as a discrete version of a classical curvature flow. In some cases, new facets may be created instantaneously and their facet lengths are governed by a system of singular ordinary differential equations(ODEs). The proposed method solves the system of the ODEs numerically by using expanding selfsimilar solutions for newly created facets. The computation method is applied to a multi-scale analysis of a contour figure.

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In this paper, we deal with the inverse spectral problem for the equation ¡(pu0)0+qu = ¸½u on a finite interval (0; h). We give some uniqueness results on q and ½ from the Gelfand spectral data, when the coefficients p and ½ are piecewise Lipschitz and q is bounded. We also prove an equivalence result between the Gelfand spectral data and the Borg-Levinson spectral data. As a consequence, we have similar uniqueness results if we consider the Borg-Levinson spectral data. Finally, we consider the inverse problem from the nodes and give uniqueness results on ½ and in the

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We propose two new formulations of inverse scattering problems for ocean acoustics and give the reconstruction formula for them. Both of them use near field data instead of the far field pattern of the scattered wave.

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In this paper we classify the singular fibres of a stable maps from a closed 4-manifolds to a 3-manifolds up to the right-left equivalence. Furthermore, we obtain several results on the co-existence of the singular fibres of such maps. As a consequence, we show that Euler characteristic of the source 4-manifold with the suitable condition, has the same parity as the total number of specified singular fibres. In orientable case, the crucial result is obtained by O.Saeki [14]. The main theorem of this paper is a generalization of his theorem.

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The purpose of this paper is to collect computations related to the weight enumerators and to present some relationships among invariant rings. The latter is done by combining two maps, the Brou´e–Enguehard map and Igusa’s ρ homomorphism. For the completeness of the story, some formulae are given which are not necessarily used in the present manuscript. Sections 1 and 2 contain no new result.

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In this paper, we explicitly derive the generalized mirror transformation of quantum cohomology of general type projective hypersurfaces, proposed in our previous article, as an effect of coordinate change of the virtual Gauss-Manin system.

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In this papar we consider an important class of first order partial differential equations (or, holonomic systems). The notion of general Clairaut type equations is one of the generalized notions of classical Clairaut equations. We give a generic classification of holonomic systems of general Clairaut type as an application of the theory of complete Legendrian unfoldings.

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Let (znQ ) be a sequence of points in the open unit disc D and ½n = m6=n j(zn ¡zm)(1¡ ¯zmzn)¡1j > 0. Let a = (aj)1j =1 be a sequence of positive numbers and `s(a) = f(wj) ; (ajwj) 2 `sg where 1 · s · 1. When 1 · p · 1 and 1=p + 1=q = 1, we show that f(f(zn)) ; f 2 Hpg ¾ `s(a) if and only if there exists a finite positive constant ° such that ( 1X n=1 (an½n)¡t(1 ¡ jznj2)tjf(zn)jt )1=t · °kfkq (f 2 Hq), where 1=s+1=t = 1. As results, we show that f(f(zj)) ; f 2 Hpg ¾ `1(a) if and only if sup n (an½n)¡1(1¡jznj2)1=p < 1, and f(f(zn)) ; f 2 H1g ¾ `1(a) if and only if X n (an½n)¡1(1 ¡ jznj2)±zn is finite measure on D. These are also proved in the case of weighted Hardy spaces.

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A local unique solvability is established for viscous conservation laws when the initial data may grow at the space infinity with a natural order. It is also shown that such a classical solution can be extended to a global-in-time solution proved that the growth order of the initial data is less than the critical order1.

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We give the characterization of Arnol’d-Mather type for stable singular Legendre immersions. The most important building block of the theory is providing a module structure on the space of infinitesimal integral deformations by means of the notion of natural liftings of differential systems and of contact Hamiltonian vector fields.

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We consider the motion of a vortex sheet on the surface of a unit sphere in the presence of point vortices fixed on north and south poles. Analytic and numerical research revealed that a vortex sheet in two-dimensional space has the following three properties. First, the vortex sheet is linearly unstable due to Kelvin-Helmholtz instability. Second, the curvature of the vortex sheet diverges in finite time. Last, the vortex sheet evolves into a rolling-up doubly branched spiral, when the equation of motion is regularized by the vortex method. The purpose of this article is to investigate how the curvature of the sphere and the presence of the pole vortices affect these three properties mathematically and numerically. We show that some low spectra of disturbance become linearly stable due to the pole vortices and thus the singularity formation tends to be delayed. On the other hand, however, the vortex sheet, which is regularized by the vortex method, acquires complex structure of many rolling-up spirals.

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A detailed description is given on the large time behavior of scattering solutions to the Cauchy problem for nonlinear Schrodinger equations with repulsive interac­tions in the short-range case without smallness condition on the data.

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We propose a framework of neurocognitive experiments that clarifies the mathematical structure of experiments and can be used to analyze experimental results and to determine the limitation of their possible interpretation. In contrast to the conventional analysis that employs simple Boolean logic, the present analysis treats classification in terms of higher-order functions. We also predict the existence of a previously unidentified type of neuron.

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A 1¡harmonic map flow equation, a gradient system of total variation where values of unknowns are constrained in a compact manifold in RN is formulated by use of subdifferentials of a singular energy - the total variation. An abstract convergence result is established to show that solutions of approximate problem converge to a solution of the limit problem. As an application of our convergence result a local-in-time solution of 1¡harmonic map flow equation is constructed as a limit of the solutions of p¡harmonic (p > 1) map flow equation, when the initial data is smooth with small total variation under periodic boundary condition.

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We give a weighted version of the Sobolev-Lieb-Thirring inequality for suborthonormal functions. In the proof of our result we use '-transform of Frazier-Jawerth.

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We prove endpoint Strichartz estimates for the Klein-Gordon and wave equations in mixed norms on the polar coordinates in three spatial dimensions. As an application, global wellposedness of the nonlinear Dirac equation is shown for small data in the energy class with some regularity assumption for the angular variable.

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We study a cylindrical crystalline flow in three dimensions coupled to a diffusion field. This system arises in modeling crystals grown from supersaturated vapor. We show existence of self-similar solutions to the system under a special choice of interfacial energy and kinetic coefficients.

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We study Hamilton-Jacobi equations with upper semicontinuous initial data without convexity assumptions on the Hamiltonian. We analyse the behavior of generalized u.s.c. solutions at the initial time t = 0, and find necessary and sufficient conditions on the Hamiltonian such that the solution attains the initial data along a sequence (right accessibility).

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There is a class of nonlinear evolution equations with singular diffusivity, so that diffusion effect is nonlocal. A simplest one-dimensional example is a diffusion equation of the form ut = ±(ux)uxx for u = u(x; t), where ± denotes Dirac’s delta function. This lecture is intended to provide an overview of analytic aspects of such equations, as well as various applications. Equations with singular diffusivity are applied to describe several phenomena in the applied sciences, and to provide several devices in technology, especially image processing. A typical example is a gradient flow of the total variation of a function, which arises in image processing, as well as in material science to describe the motion of grain boundaries. In the theory of crystal growth the motion of a crystal surface is often described by an anisotropic curvature flow equation with a driving force term. At low temperature the equation includes a singular diffusivity, since the interfacial energy is not smooth. Another example is a crystalline algorithm to calculate curvature flow equations in the plane numerically, which is formally written as an equation with singular diffusivity. Because of singular diffusivity, the notion of solution is not a priori clear, even for the above one-dimensional example. It turns out that there are two systematic approaches. One is variational, and applies to divergence type equations. However, there are many equations like curvature flow equations which are not exactly of divergence type. Fortunately, our approach based on comparision principles turns out to be succesful in several interesting problems. It also asserts that a solution can be considered as a limit of solution of an approximate equation. Since the equation has a strong diffusivity at a particular slope of a solution, a flat portion with this slope is formed. In crystal growth ploblems this flat portion is called a facet. The discontinuity of a solution (called a shock) for a scalar conservation law is also considered as a result of singular diffusivity in the vertical direction.

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We prove the weighted Strichartz estimates for the wave equation in even space dimensions with radial symmetry in space. Although the odd space dimensional cases have been treated in our previous paper [4], the lack of the Huygens principle prevents us from a similar treatment in even space dimensions. The proof is based on the two explicit representations of solutions due to Rammaha [10] and Takamura [13] and to Kubo-Kubota [5]. As in the odd space dimensional cases [4], we are also able to construct self-similar solutions to semilinear wave equations on the basis of the weighted Strichartz estimates.

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We study the existence of self-similar solutions to the Cauchy problem for semilinear wave equations with power type nonlinearity. Radially symmetric self-similar solutions are obtained in odd space dimensions when the power is greater than the critical one that are widely referred to in other existence problems of global solutions to nonlinear wave equations with small data. This result is a partial generalization of [11] to odd space dimensions. To construct self-similar solutions, we prove the weighted Strichartz estimates in terms of weak Lebesgue spaces over space-time.

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In this article, the behavior of solutions to the free wave equation with homogeneous Cauchy data are considered. In particular, the propagation of singularities are observed explicitly. Such Cauchy data are of special interest in view of applications to self-similar solutions to nonlinear wave equations.

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We study the smoothing effect in space and asymptotic behavior in time of solutions to the Cauchy problem for the nonlinear Schr¨odinger equation with interaction described by the integral of the intensity with respect to one direction in two space dimensions. A detailed description is given on the phase modification of scattering solutions by taking into account the long range effect of the interaction.

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An upper bound of the life-span of smooth solutions to the complex Ginzburg­Landau equation with periodic boundary conditon in one space dimension is given explicitly in terms of an integral mean of the Cauchy data in the case where the interaction is focusing.

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A fourth order equation with singular diffusivity, which is a model of relaxation dynamics for crystalline surfaces driven by surface diffusion, is formulated. The notion of subdifferentials enables us to formulate the singular diffusion equation mathematically as a gradient flow equation in the Sobolev space of negative power H-1. The subdifferential of the singular energy in H-1 is calculated. Moreover, the speed of a special profile is calculated for one dimensional problem. It turns out that a seemingly natural free boundary formulation with facets is inconsistent with a subdifferential formulation which can be approximated by a smooth energy.

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We consider a singular perturbation problem arising in the scalar phase field model which [-converges to the area func­ tional. Assuming the stability of the critical points for s-problcms, we show that the interface regions converge to a generalized stable minimal hypcrsurfacc as s --'t 0. The limit has L2 generalized second fundamental form and the stability condition is expressed in terms of the corresponding inequality satisfied by stable minimal hypcrsurfaccs. We show that the limit is a finite number of lines with no intersections when the dimension of the domain is 2.

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We study the asymptotic behavior, in the zero noise limit, of solu­tions to Schrodinger's functional equations and that of h-pass pro­cesses, and give a new proof of the existence of the minimizer of Monge's problem with a quadratic cost.

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Let A be a uniform algebra on a compact Hausdorff space X and m a probability measure on X. Let HP (m) be the norm closure of A in £P (m) with 1 :Sp< oo and H∞ (m) the weak * closure in L∞ (m). In this paper, we describe a closed ideal of A and a closed invariant subspace of HP (m) which is of finite codimension.

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This paper studies a growth rate of a solution blowing up at time T of the semilinear heat equation Ut - £:iu - lulP-1u = 0 in a convex domain D in Rn with zero-boundary condition. For a subcritical p E (1, (n + 2)/(n - 2)) a growth rate estimate lu(x, t)I :S C(T- t)-1/(p-l), x E D, t E (0, T) is established with C independent oft provided that D is uniformly 02. The estimate applies to sign-changing solutions. The same estimate has been recently established when D = Rn by authors. The proof is similar but we need to establish Lh - Lk estimate for a time-dependent domain because of the presence of the boundary.

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For every invariant subspace NI in the Hardy spaces H2 (f2 ), let Vz and Vw be mulitplication operators on AL Then it is known that the condition Vz V; v;vz on NI holds if and only if J;I is a Demling type invariant subspace. For a backward shift invariant subspace N in H2(f2), two operators Sz and Sw on N are defined by Sz = PN LzPN and Sw = PN Lw PN, where PN is the orthogonal projection from L2(f2) onto N. It is given a characterization of N satisfying szs1:J = s1:JsZ on N.

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We study numerically the anisotropic bunching effect in crystal growth under curvature and a singular vertical diffusive regularization. Our assump­tion is that the mobility of the growth depends on the height of the given crystal. This assumption may result in overhanging crystals if approached in a naive way. Instead, we embed the profile of the crystal as the zero level set of a continuous function and study the corresponding level set evolution. To prevent "overhanging", we regularize the equation with a singular diffu­sion that vanishes everywhere except at the fonnation of "overhanging". In addition, we add the mean curvature regularization to keep the convexity of the level sets.

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An essential bounded function </> gives a continuous linear functional on the Hardy space H1 on the bitorus. In this paper, we consider extremal problems on H1 when ¢ is a rational function, ¢ is a product of one variable functions or </> = If 1/.f for some outer function f in H1 such that f ( z, w) has a good property with respect to w for a.e. z.

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The locally-in-time solutions to the Navier-Stokes equations in H%-1(Rn ) are regular for t > 0. The spatial analyticity is established by deriving rate estimates of higher order derivatives of the solutions. The solutions or the initial velocities need not be small. The estimates also provide the decay estimate on derivatives for large time. Although the basic strategy to derive estimates is similar to our previous results with Y. Giga for Ln space, we are forced to apply several tools from harmonic analysis since our space 1{%-1 is more complicated.

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Considered is a quantum system of N(?:_ 2) charged particles moving in the plane R2 under the influence of a perpendicular magnetic field. Each particle feels the magnetic field concenrated in the positions of the other particles. The gauge potential which gives this magnetic field is called a winding gauge potential. Properties of the momentum operators with the winding gauge potential are investigated. The momentum operators with the winding gauge potential are represented by the fibre direct integral of Arni's momentum operators [1]. Using this fibre direct integral decomposition, commutation properties of the momentum operators are investigated. A notion of local quantization of the magnetic flux is introduced to characterize the strong commutativity of the momentum operators. Aspects of the representation of the canonical commutation relations (CCR) are discussed. There is an interesting relation between the representation of the CCR with respect to this system and Arni's representation. Some applications of those results are also discussed.

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To consider phenomena of physics and economics, we need solu­tions in weak sense, one of which is the viscosity solution. Its definition has many variations to indicate the uniqueness and the existence of solutions. This paper attempts to give a framework which unifies viscosity solutions synthetically.

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For the implicit systems of first order ordinary differential equa­tions on the plane there is presented the complete local classification of generic singularities of family of its phase curves up to smooth orbital equivalence. Besides the well known singularities of generic vector fields on the plane and the singularities described by a generic first order implicit differential equations, there exists only one generic singularity described by the implicit first order equation supplied by Whitney umbrella surface generically embedded to the space of direc­tions on the plane.

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We consider an inverse boundary value problem for identifying the inclusion inside a known anisotropic conductive medium. We give a reconstruction procedure for identifying the in­clusion from the Dirichlet-Neumann map or the Neumann-Dirichlet map associated with the mixed type boundary condition.

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Let A be a uniform algebra on X and P a set of all probability measures on X. For each µ in P, H2 (µ) is the closure of A in L2 (µ) and Tt is a Toeplitz operator on H2 (µ) for a continuous function cf> on X. In this paper we study the invertibility and the spectrum of Tip = L EB Tt. We show that if Tip is invertible then the index of cf> is zero and if the converse is true for an arbitrary continuous function cf> then A is a Dirichlet algebra on X. Moreover we study the spectrum of Tip.

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We consider dynamical systems generated by partially hyperbolic surface endomorphisms of class er with one-dimensional strongly unstable subbundlc. As the main result, we prove that such a dynamical system generically admits finitely many ergodic physical measures whose union of basins of attraction has total Lebesgue measure, provided that r 2: 19.

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We study the differential geometry of hypersurfaces in hyperbolic space. As an applica­tion of the theory of Lagrangian singularities, we investigate the contact of hypersurfaces with families of hyperspheres or equidistant hyperplanes.

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We classify singularities of lightlike hypersurfaces in Minkowski 4-space via the contact invariants for corresponding spacelike surfaces and lightcones.

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We construct a new framework for the study of multiplane gravitational lensing from the view point of Symplectic geometry. Symplectic relations are used to compose the systems and weaker Lagrangian equivalence is applied for classifying the caustics of mul­tiplane gravitational lensing.

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We show that the Gauss-Bonnet type theorem holds for the hyperbolic Gauss-Kronecker curvature of a closed orientable even dimensional hypersurface in hyperbolic space. We also give detailed studies for surfaces.

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We study some geometrical properties associated to the contacts of surfaces with hyperhorospheres in Ht ( -1). We introduce the concepts of osculating hyperhorospheres, horobinormals, horoasymptotic directions and horospherical points and provide conditions ensuring their existence. We show that totally semiumbilical surfaces have orthogonal horoasymp­totic directions.

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Suppose that Tq, is a Toeplitz operator with a symbol 1> on the Hardy space H2 on the bidisc. Let N be a backward shift invariant subspace of H2, that is, N is an invariant subspace under r; and T:U. Let P be the orthogonal projection from H2 onto N. For 1> in H00 put Sq, = PTq,[N. In this paper, we give a characterization of a , backward shift invariant subspace which satisfies SzS􀀄 S􀀄Sz.

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We consider a contingent claim in a jump-diffusion model of complete market. Given initial wealth less than the replicating cost, we explic­itly solve the problem of minimizing the expected value of hedging loss weighted by power functions. We show that the optimal portfolio is the difference between the perfect hedging portfolio of the contingent claim and the optimal portfolio of a utility minimization problem. We also give an explicit formula for the value function. These results hold for every European-type contingent claim.

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We construct a random crystalline ( or polyhedral) approximation of a convexified Gauss curvature flow of bounded open sets in an anisotropic external field. We also show that a weak solution to the PDE which describes the motion of a bounded open set is unique and is a viscosity solution of it.

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We introduce the notion of strong supercommutativity of self-adjoint operators on a Z2-graded Hilbert space and give some basic properties. We clarify that strong super­commutativity is a unification of strong commutativity and strong anticommutativity. We also establish the theory of super quantization. Applications to supersymmetric quantum field theory and a fermion-boson interaction system are discussed.

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It is known that an Ln-valued continuous solution in time inter­val (0, T) of the Kavier-Stokes equations in Rn is regular for positive time. In this paper regularizing rate estimates similar to a solution of the heat equation arc established. The estimates also provide analyticity in space variables as well as decay estimates on derivatives for large time. The solu­tions need not be small. Our results are obtained by estimating the integral equation with a new version of the Gronwall type inequality originally ob­tained in [2].

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Let X be an analytic subvariety of complex Euclidean space with isolated singularity at the origin, and let r7 be a smooth form of type ( 1. 1) defined on X \ { 0}. The main result of this note is a criterion for solubility of the equation Dau r7. This implies a criterion for triviality of a Hermitian holomorphic line bundle (L, h) ----+ X \ {O} in a neighbourhood of the origin.

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Let α and β be measurable functions on the unit circle T, and let W be a positive function on T such that the Ricsz projection P+ is bounded on the weighted space L2(W) on T. The singular integral operator Sa,/3 is defined by Sa,f3f = nI'+f + pI'_f, (f E L2 (W)) where I'_ = I-I'+ · Leth be an outer function such that TV= 1h12 , and let cp be an unimodular function such that cp = Ti/h. In this paper, the norm of Sa ,rJ on L2 (TV) is calculated in general, using α,β and cp. Moreover, if o; and (} are constant functions, then we give the another proof of the Feldman-Krupnik-Yiarcus theorem. If αβ belongs to the Hardy space H00 , we give the theorem which is similar to the Feldman-Krupnik-Marcus theorem.

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We propose a recipe for determination of the partition function of N 4 ADE gauge theory on K3 by generalit1ing our previous results of the SU(N) case. The resulting partition function satisfies Montonen-Olive duality for ADE gauge group.

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In this article, we obtain the uniqueness of solutions ( u, p) of the N avier-Stokes equations in the class u E L∞((0, T)×Rn), p E L1loc([0,T);BMO(Rn)) for initial data in L∞ (Rn). Although there are a few results which treats the uniqueness without decay assumption as |x| →∞([5], [15], [14]), our result gives the another characterization of condition on p.

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In the first part (§2, §3), we give a survey of the recent results on application of singularity theory for curves and surfaces in Hyperbolic space. After that we define the hyperbolic canal surface of a hyperbolic space curve and apply the results of the first part to get some geometric relations between the hyperbolic canal surface and the center curve.

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We study some properties of spacelike submanifolds in Minkowski n­space all whose points are umbilic with respect to some normal field. As a consequence of these and some results contained in [l] we obtain that being v-umbilic with respect to a parallel lightlike normal field implies conformal flatness for submanifolds of dimension n 􀀩 2 2: 3. In the case of surfaces we relate the umbilicity condition to that of total semiumbilicity (degeneracy of the curvature ellipse at every point). Moreover, if the considered normal field is parallel we show that it is everywhere timelike, spacelike or lightlike if and only if the surface is included in a hyperbolic 3-space, a de Sitter 3-space or a 3-dimensional light cone respectively. We also give characterizations of total semiumbilicity for surfaces contained in hyperbolic 4-space, de Sitter 4-space and 4-dimensional light cone.

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We study the geometry of the spacclike surfaces in Minkowski 4-space through their generic contact with lightlike hyperplanes.

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A line congruence is a two parameter family of lines in IR3. In this paper we study singu­larities of line congruences. We show that generic singularities of general line congruences are the same as those of stable mappings between three dimensional manifolds. Moreover, we also study singularities of normal congruences and equiaffine normal congruences from the view point of the theory of Lagrangian singularities.

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We define new special curves in Euclidean 3-space which we call slant helices and conical geodesic curves. Those notions are generalizations of the notion of cylin­drical helices. One of the results in this paper gives a classification of special de­velopable surfaces under the condition of the existence of such a special curve as a geodesic. As a result, we consider geometric invariants of space curves. By using these invariants, we can estimate the order of contact with those special curves for general space curves. All arguments in this paper are elementary and classi­cal. However, there have been no papers which have investigated slant helices and conical geodesic curves so far as we know.

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We consider a singular perturbation problem arising in the scalar phase field model with Keumann boundary conditions on convex domains, and establish a monotonicity formula for general critical points. This gives the Hausdorff distance convergence of the phase boundaries as the parameters tend to ,1ero. We apply the result to stable critical points defined on strictly convex domains, showing that the limit interfaces for stable critical points arc necessarily connected.

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A Neumann problem for the Laplace equation is considered outside a three dimensional straight cylinder. The value of a solution O" at space infinity is prescribed. The Neumann data aO" / an ( n is the outer normal of the cylinder) is assumed to be independent of the spatial variables on the top and the bottom and also on the lateral part of the boundary of the cylinder. The behavior of the value of O" on the boundary is studied. In particular, it is shown that O" is an increasing function of the distance from the center of the top ( respectively, the bottom) if a(J" / an > o on the lateral part and a(J" / an is the same constant on the top and (respectively, the bottom). An analogous statement is shown for O" on the lateral part. In the theory of crystal growth O" is interpreted as a supersaturation and cylinder is a crystal. The value aO" / an is the growth speed. The main contribution of this paper is considered as the first rigorous proof of Berg's effect when the crystal shape is a cylinder.

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In this paper we introduce a new level set model for the growth of spirals on the surface of a crystal. Since the conventional level set method cannot express a spiral curve having orientation, we modify the level set method by using a sheet structure function. Since the model equation we obtain is degenerate parabolic, we need to consider a notion of weak solution. Our goal is to prove the existence and the uniqueness of the solution for our model in the sence of viscosity solutions.

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q'J is called a Rudin's ( orthogonal) function if q'J is a function in Hx and the different nonnegative powers of cf> arc orthogonal in H2. \Yhcn cf> is a multiple of an inner function and ¢(0) = 0, cf> is a Iludin's function. Sundberg and Bishop showed that a Iludin's function is not necessarily a multiple of an inner function. \Ye study a Iludin's function which is a linear combination of two inner functions or a polynomial of an inner function.

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Let N+ denote the Srnirnov class on the open unit disc D. It is easy to sec that for any outer function g in N +, there exists a function G in N + such that lgl 􀀢 IleG on DD. We describe such a G. In general, G may not be outer. In this paper, a necessary and sufficient condition on g is given for the existence of an outer function G such that |g| <- ReG. When g belongs to the Hardy space H1 , G is trivially given as the Herglotz integral of |g|.

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We give an estimate for the moments of the negative eigenvalues of elliptic operators on ]Rn in low dimensions. The estimate is a generalization of the Lieb-Thirring inequalities in one or two dimensions. We use the cp-transform decomposition of Frazier and J awerth.

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This paper studies the problem or minimizing coherent risk measures or shortfall for general discrete-time financial models with cone-constrained trading strategies, as developed by Pham and Touzi (1999) and Pham (1999). We show that the optimal strategy is obtained by super-hedging a contingent claim, which is represented as a .Xeyman-Pearson-type random variable.

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A non standard dynamic boundary condition for a Hamilton-Jacobi equa­tion in one space dimension is studied in the context of viscosity solutions. A comparison principle and, hence, uniqueness is prm·ed by consideration of an equivalent notion of vis­cosity solution for an alternative formulation of the boundary condition. The relationship with a l\eumann condition is established. Global existence is obtained by consideration of a related parabolic approximation with a dynamic boundary condition. The problem is motivated by applications in superconductivity and interface evolution.

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A time-local solution is constructed for the Cauchy problem of the n­dimensional l'\avier-Stokes equations when the initial velocity belongs to Besov spaces of non positive order. The space contains L∞ in some exponents, so our solution may not decay at space infinity. In order to use iteration scheme we have to establish the Holder type inequality for estimating bilinear term by dividing the sum of Besov norm with respect to levels of frequency. Moreover, by regularizing effect our solutions belongs to L∞ for any positive time.

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The non-relativistic (scaling) limit of a particle-field Hamiltonian H, called a Dirac-Maxwell operator, in relativistic quantum electrodynamics is considered. It is proven that the non-relativistic limit of H yields a self-adjoint extension of the Pauli-Fierz Hamiltonian with spin 1/2 in non-relativistic quantum electrodynamics. This is done by establishing in an abstract framework a general limit theorem on a family of self-adjoint operators partially formed out of strongly anticommuting self-adjoint operators and then by applying it to H.

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H∞ and H2 denote the Hardy spaces on the open unit disc: D. Let cf> be a function in H∞ and 11</Jll ∞ = 1. If cf> is an inner function and ¢(0) = 0, then { cpn ; n = 0, 1, 2, · · ·} is orthogonal in H2 . \\7.Rudin asked if the converse is true and C.Sundberg and C.Bishop showed that the converse is not true. Therefore there exists a function c/J such that c/J is not an inner function and { c/Jn } is orthogonal in H2. In this paper, the following is shown : { q'Jn } is orthogonal in H2 if and only if there exists a uniqueprobability measure v0 on [0,1] with 1 E supp v0 such that N4,(z) - S log r/|z| dv0(r) for nearly all z in D where Nc/J is the Nevanlinna counting function of c/J. If 6 is an inner function, then v0 is a Dirac: measure at r = 1.

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We derive the partition function of N = 4 supersymmetric Yang-Mills theory on orbifold-T4 /Z2 for SU(N). We generalize our previous work for SU(2) to the SU(N) case. These partition functions can be factorized into product of bulk contribution of quotient space T4 /Z2 and of blow-up formula including AN-1 theta functions with level N.

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We shmv the existence of ,veak solutions to the PDE which de­scribes the motion by R-curvaturc in Rd , by the continuum limit of a class of infinite particle systems. We also show that weak solutions of the PDE arc viscosity solutions and give the uniqueness result on both weak and viscosity solutions.

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In this paper, we discuss some applications of Givental 's differential equations to enumer­ative problems on rational curves in projective hypersurfaces. Using this method, we prove some of the conjectures on the structure constants of quantum cohomology of projective hy­persurfaces, proposed in our previous article. Moreover, we clarify the correspondence between the virtual structure constants and Givental's differential equations when the projective hyper­surface is Calabi-Yau or general type.

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Let a, (J, 1, () be complex numbers such that 􀀉;fJ =J 0. If A and B arc bounded linear operators on the Hilbert space H such that 1A + fJB is right invertible then we study the operator norm of (aA + /JB)bA + fJB)-1 using the angle (p between two subspaces ran A and ran B or the angle '􀀕J = '􀀕J(A, B) between two operators A and B where c:os'􀀕J(A,B) = sup{l(Af,Bf)I / (IIA.fll · IIB.fll); f EH, Af =J 0, Bf =JO}.

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We prove a representation of the partial autocorrelation function α(・) of a stationary process { Xn : n ∈ Z}, in terms of the AR(∞) and MA(∞) coefficients. We apply it to show what α(n) looks like for large n, especially, when {Xn} is a long-memory process. For example, if {Xn} is a fractional ARIMA(p. d. q) process, then we have a(n)～d/n as n → ∞.

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In this article, for a residual Galois representation defined m·er an arbitrary finite field, Gouvca's conjecture ,vhich says that the universal deformation ring is isomorphic to a certain Hecke algebra is proven in the unobstructed case.

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We introduce two types of finite difference methods to compute the L­solution [14] and the proper viscosity solution [13] recently proposed by the second author for semi-discontinuous solutions to a class of Hamilton-Jacobi equations. By regarding the graph of the solution as the zero level curve of a continuous function in one dimension higher, we can treat the correspond­ing level set equation using the viscosity theory introduced by Crandall and Lions [7]. However, we need to pay special attention both analytically and numerically to prevent the zero level curve from overturning so that it can be interpreted as the graph of a function. We demonstrate our Lax-Friedrichs type numerical methods for computing the L-solution using its original level set formulation. In addition, we couple our numerical methods with a singu­lar diffusive term which is essential to computing solutions to a more general class of HJ equations that includes conservation laws. With this singular viscosity, our numerical methods do not require the divergence structure of equations and do apply to more general equations developing shocks other than conservation laws. These numerical methods are generalized to higher order accuracy using WENO Local Lax-Friedrichs methods [21]. We verify that our numerical solutions approximate the proper viscosity solutions of [ 13]. Finally, since the solution of scalar conservation law equations can be constructed using existing numerical techniques, we use it to verify that our numerical solution approximates the entropy solution.

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We rigorously prove that the solution surface of the intermediate surface diffusion flow converges to that of the averaged mean curva­ture flow locally in time as the diffusion coefficient tends to infinity. As an application of this convergence result, we show that the inter­mediate surface diffusion flow can drive embedded hypersurfaces into self-intersections.

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We answer a question posed by Ilmanen on the integrality of varifolds which appear as the singular perturbation limit of the Allen-Cahn equation. We show that the density of the limit measure is integer multiple of the surface constant 1-1,n-1-a.e. for a.e. time. This shows that limit measures obtained via the Allen­Cahn equation and those via Brakke's construction share the same integrality property as well as being weak solutions for the mean curvature flow equation.

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Gonda and Gorni (T.Gonda, H.Gomi, Ann. Glaciology, 6 (1985), 222 224) have grown large elongated ice crystals from supersaturated vapor. Theoretrically this problem may be recast in a framework similar to that used by Seeger (A.Seeger, Philos. Mag., ser. 7, 44, no 348, (1953) 1-13) for studies of planar crystals. The resulting set of equations is of Stefan type. We also include the Gibbs­Thomson relation on the crystal surface. In order to make this system tractable mathematically we assume that the \\\1lff crystal is a fixed cylinder. Subsequently we study a weak form of our system. We show local in time existence of solutions assuming that the initial shape is an arbitrary cylinder. We comment on properties of weak solutions.

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The presence of steps associated with screw dislocations plays a key role for the growth of crystal surfaces. In geometric model the motion of curves describing location of steps is governed by curvature flow equations with a driving force term. We show the existence of spiral-shaped solutions for such an equation when anisotropic effect is small. Such a spiral-shaped solution is ahown to be stable and unique up to translation of the time.

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We show the following: Let T3 be a two dimensional commutative Banach algebra with identity. If T3 satisfies TE B, \IT\I :'S 1 􀀉 \\f(T)I\ :'S 1 whenever f is a polynomial satisfying \f(z)\ :::; 1 (\z\ :'S 1) then T3 is isometric to a subalge­bra of the algebra B(H) of all bounded linear operators on some Hilbert space H, and T3 satisfies Tk E !3, IITklJ :'S 1 (k = 1, ... , n) 􀀉 llf(T1, ... , Tn) II :'S 1 whenever f is a polynomial inn variables satisfying lf(z1, ... , Zn)I :'S 1 (\zki :'S 1, k = 1, ... , n), for all n.

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This paper introduces a class of AR( oo )-type models for mean-square continuous processes with stationary increments. The models allow for short- or long-memory dynamics in the processes. Their solutions are shown to have a semimartingale representation. The models are used to describe the dynamics of asset prices, which reduce to the traditional Black-ScholPB model as a special case. It is shown that there exists an equivalent martingale measure under which the behaviour of the discounted price process is equal to that in the Black-Scholes environment. As a result, the European option price is given by the Black-Scholes formula. The variance of the log price ratio is also obtained.

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If we have a finite number of sections of a complex vector bundle E over a manifold M, certain Chern classes of E are localized at the singular set S, i.e., the set of points where the sections fail to be linearly independent. When S is compact, the localizations define the residues at each connected component of S by the Alexander duality. If M itself is compact, the sum of the residues is equal to the Poincare dual of the corresponding Chern class. This type of theory is also developed for vector bundles over a possibly singular subvariety in a complex manifold. Explicit formulas for the residues at an isolated singular point are also given, which expresses the residues in terms of Grothendieck residues relative to the subvariety.

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We introduce covariance kernels for Borel probability measures on Rd, and study the relation between the central limit theorem in the total variation distance and the convergence of covariance kernels.

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Abstract. Let HP be the Hardy space on the bidisc and 1 < p < oo. For a function </J in £00 , we study the norm of the Hankel operator H</> on HP and the invertibility of the Toeplitz operator T</> on HP. The latter is strongly related with a weighted norm inequalities on the bidisc.

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The idea of efficient hedging has been introduced by Follmer and Leuk­ert (2000). They defined the shortfall risk as the expectation of the short­fall weighted by a loss function, and looked for strategies that minimize the shortfall risk under a capital constraint. In this paper, to rneasme the shortfall risk, we use the coherent risk measures introduced by Artzner, Delbaen, Eber and Heath (1999). We show that, for a given contingent claim H, the optimal strategy consists in hedging a modified claim cpil for some randomized test <p. This is an analogou of lhe results by Follmer and Leukert (2000).

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We study continuous coherent risk measures on LP, in particular, the worst conditional expectations. We show some representation theorems for them, extending the results of Artzner, Delbaen. Ehn, Heath. and Kusuoka.

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A solution of single nonlinear first order equations may develop jump discontinuities even if initial data is smooth. Typical examples include a crude model equation describing some bunching phenomena observed in epitaxial growth of crystals as well as conservation laws where jump discontinuities are called shocks. Conventional theory of viscosity solutions does not apply. We introduce a notion of proper (viscosity) solutions to track whole evolutions for such equations in multi-dimensional spaces. We establish several versions of comparison principles. We also study the vanishing viscosity method to construct a unique global proper solution at least when the evolution is monotone in time or the initial data is monotone in some sense under additional technical assumptions. In fact, we prove that the graph of approximate solutions converges to that of a proper solution in the Hausdorff distance topology. Such a convergence is also established for conservation laws with monotone data. In particular, local uniform convergence outside shocks is proved.

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We derive the partition function of N = 4 supersymmetric Yang-Mills theory on orbifold-T4 /Z2 . In classical geometry, K3 surface is constructed from the orbifold-T4 /Z2 . Along the same way as the orbifold construction, we construct the partition function of K3 surface from orbifold-T4 /Z2 . The partition function is given by the product of the contribution of the untwisted sector of T4 /Z2 , and that of the twisted sector of T4 /Z2 i.e., 0(-2) curve blow-up formula.

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We will characterize the boundedness and compactness of the composition operators on weighted Bloch space B10g = {f E H(D) : supzEn(l -!zl2) (log 1_fzl") lf'(z)I < +oo}, where H ( D) be the class of all analytic functions on D.

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We consider the contact between curves and horospheres in Hyperbolic 3-space as an application of singularity theory of functions. We define the osculating horosphere of the curve. We also define the horospherical surface of the curve whose singular points correspond to the locus of polar vectors of osculating horospheres of the curve. One of the main results is to give a generic classification of singularities of horospherical surface of curves.

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We study generic properties of cylindrical helices and Bertrand curves as applications of singularity theory for plane curves and spherical curves.

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In this paper we adopt the Hyperboloid in Minkowski space as the model of Hyperbolic space. We define the hyperbolic Gauss map and the hyperbolic Gauss indicatrix of a hypersurface in Hyperbolic space. The hyperbolic Gauss map has been introduced by Epstein[7] in the Poincare ball model which is very useful for the study of constant mean curvature ·surfaces. However, it is very hard to proceed the calculation because it has an intrinsic form. Here, we give an extrinsic definition and we study singularities of these. In the study of singularities of the hyperbolic Gauss map (indicatrix), we understand that the hyperbolic Gauss indicatrix is much easier to proceed the calculation. We introduce the notion of hyperbolic Gauss-Kronecker curvature whose zero sets correspond to the singular set of the hyperbolic Gauss map (indicatrix). We also develop a local differential geometry of hypersurfaces concerning on contact with hyperhorospheres.

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In order to study singularities of Haefliger foliation, we define the notion of transver­sally Whitney C∞-topology modulo a regular foliation on the set of C∞-mappings into a foliated manifold. We prove a kind of transversaHty theorem with respect to this new topology. All arguments we use here are analogous to those· of the theory for the ordi­nary Whitney C∞-topology. However, this is the first attempt for the study of generic properties of Haefliger foliations

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The developable surface in JR.3 has the unique (singular or non-singular) Legendrian lift to the projective cotangent bundle on JR.3 • In this paper we show that the converse assertion holds for singular rule<:! surfaces. We call such a surface a ruled front. We give an explicit form of the generating family of the Legendrian lift of a developable surface ( a ruled front) and study singularities and their stability.

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We study cylindrical helices and Bertrand curves as curves on ruled surfaces. Some results in this paper clarify that the cylindrical helix is related to the Gaussian curvature and the Bertrand curve is related to the mean curvature of the ruled surface. We also study the singularities of the principal normal surfaces of generic curves.

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We study some asymptotic behavior of phase interfaces with variable chemical potential under the uniform energy bound. The problem is motivated by the Cahn-Hilliard equation, where one has a control of the total energy and chemical potential. We show that the limit interface is an integral varifold with generalized LP mean curvature. The convergence of interfaces as c -+ 0 is in the Hausdorff distance sense.

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We consider a multi-component nonlinear partial differential equation in the two­dimensional space-time R2 which unifies Schrodinger and Klein-Gordon equations in R 2• By supersymmetric methods connected with shape-invariant potentials, we show that a class of soliton-type solutions to the equation can be constructed from solutions to nonlinear equations of some types for superpotentials. Moreover we present new soliton-type solutions which are written in terms of q-deformed hyperbolic functions.

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A perturbed Dirac operator Q(a) on the abstract Boson-Fermion Fock space is considered, where a E C is a perturbation ( coupling) parameter and the un­perturbed operator Q(0) is taken to be a free infinite dimensional Dirac operator introduced by the author ( A. Arai, J. Funct. Anal. 105(1992), 342-408). The following results are reported: (i) Under some conditions, the kernel of Q(a) is one dimensional for all a f:. ao with some ao f:. 0 and degenerate at a = a0, while, under another condition, the kernel of Q(a) is one dimensional for all a E C. (ii) There are cases where, for all sufficiently large la I with a < 0, Q( a) has infinitely many non-zero eigenvalues even if Q(0) has no non-zero eigenvalues. This is a strong cou­pling effect. (iii) Fredholm property of Q (a) also depends on the coupling parameter a.

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We will present an introductory review of the mathematics for investigating the interfacial motion in crystal growth problems. Anisotropy is quite an important factor in such problems. There are two types of anisotropy - the kinetic anisotropy and the one of curvature effect. The main theme of this article is how the kinetic anisotropy determines the growth form of the crystal and how the curvature effect works on it.

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Let A be a uniform algebra and M(A) the maximal ideal space of A. A sequence { an }n in M(A) is called .e1-interpolating if for every sequence (an) in f1 there exists a function f in A such that f (an ) = an for all n. In this paper, an f1-interpolating sequence is studied for an arbitrary uniform algebra. For some special uniform algebras, an f1-interpolating sequence is equivalent to an £<)()-interpolating sequence which is familiar for us. However, in general these two interpolating sequences may be different from each other.

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Let {Xn : ri E Z} be a fractional ARIMA{p, d, q) process with partial autocorrelation functiono:(·). In this paper, we prove that if d E ( 􀀙1/2, 0) then jo:(n)I rv jdj/n as n ---+ oo. This extends the previous result for the case O < d < 1/2.

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A global-in-time unique smooth solution is constructed for the Cauchy problem of the Navier-Stokes equations in the plane when initial velocity field is merely bouncled not necessary square-integrable. The proof is based on a uniform bound for the vorticity which is only valid for planar flows. The uniform bound for the vorticity yields a coarse globally-in­time a priori estimate for the maximum norm of the velocity which is enough to extend a local solution. A global existence of solution for a q-th integrable initial velocity field is also established when q > 2.

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We consider a model of quantum particles coupled to a massless quantum scalar field, called the massless Nelson model, in a non-Fock representation of the time­zero fields which satisfy the canonical commutation relations. We show that the model has a ground state for all values of the coupling constant even in the case where no infrared cutoff is made. The non-Fock representation used is inequivalent to the Fock one if no infrared cutoff is made.

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We study dynamical systems generated by skew products T : S1 × R → S1 × R, T(x, y) = (lx, λy + f(x)) where l >= 2, 1/l < λ < 1 and f is a C2 function on S1 We show that the SBR measure for Tis absolutely continuous for almost every f.

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We construct a mathematical model for the dynamic behavior of hip­pocampus. The model is described by the skew product transformation in terms of chaotic dynamics and contracting dynamics. In the contracting subspace, fractal objects are generated. We show that such fractal objects are characterized by a code of a temporal sequence generated by chaotic dynamics.

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We give a rigorous proof that the solution curve of the intermediate surface diffusion fl.ow equation converges to that 􀀗f the averaged curvature fl.ow equation locally in time as the diffusion coefficient D goes to infinity. As an application of this convergence result, we also prove that a self-intersection of curves can be developed by the intermediate surface diffusion fl.ow for any positive D.

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Anisotropic curvature flow equations with singular interfacial energy are important for good unders_tanding of motion of phase-boundaries. If the energy and the interfacial surface were smooth, then the speed of the interface would be equal to the gradient of the energy. However, this is not so simple in the case of non-smooth crystalline energy. But it's well-known that a unique gradient charac­terization of the velocity is possible if the interface is a curve in the two-dimensional space. In this paper we propose a notion of solution in the three-dimensional space by introducing geometric subdifferentials and characterizing the speed. We also give a counterexample to a problem concerning the Cahn-Hoffman vector field on a facet, a flat portion of the interface.

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Suppose T<I> is a Toeplitz operator with a symbol <p on the Hardy space on the bidisc, H2. Let N be a backward shift invariant subspace of H2, that is, N is an invariant subspace under r; and r;.Let P be the orthogonal projection from H2 onto N. For </Jin H00 , put S</> = PT<t>JN. In this paper, we are interested in backward shift invariant subspaces which satisfy BzS:V = S!Sz. In particular, we show that SzS:V = S:VSz if N is of finite dimension.

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We study the optimal control problem for Rd-valued absolutely continuous stochastic processes with given marginal distributions at every time. This can be considered as a generalization of Monge-Kantorovich problem in which they fix marginal distributions only at an initial and a terminal times. When d = 1, we show that there exists a minimizer which is a function of a time and an initial point, that is, an optimal mass transportation. When d > 1, we only show that minimizers satisfy the same ordinary differential equation.

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Using the concepts of chaotic itinerancy and Cantor coding, we present an interpretation of dynamic neural activity found in cortical and subcortical areas. The discovery of chaotic itinerancy and Cantor coding in high- dimensional dynamical systems has motivated a new interpretation of this dynamic neural activity, cast in terms of the high-dimensional transitory dynamics among quasi-attractors.

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Motivated by the interf aciology proposed by Otto Rossler, we have attempted to construct a framework of internal logic of the mind and brain. We propose a functional equation as an abstract form representing mental processes. We consider a method by which such in­ternal logic can be interpreted and understood by an (external) observer. For this purpose, we propose a theory for cognitive experiments. Applying this theory to simple deductive inference processes exhibited by animal subjects in an experimental setting, with the as­sumption that syllogism is expressed as a composite mapping corresponding to the product operation of two implications A-t Band B -t C, an interpretation of the neural activity associated with the behavior in these experiments is obtained. This theory is consistent with the internal description hypothesized by Rob Rosen.

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In this article, Gouvea's conjecture on controlling the conductor is proven in a special case.

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Suppose F is a nonzero function in the Hardy space H1. We study the set {f ; f is outer and !Fl ::; Re f a.e. on 8D} where 8D is a unit circle. When F is a strongly outer function in H1 and 'Y is a positive constant, we describe the set {! ; f is outer, IFI ::; 'Y Re f and IF-1 I ::; 'Y Re u-1) a.e. on 8D}. Suppose w is a Helson-Szego weight. As an application, we parametrize real valued functions v in L∞(∂D) such that the difference between log W and the harmonic conjugate function v_*_ of v belongs to L∞(∂D) and llvll∞ is strictly less than π/2 using a contractive function α in H∞ such that (1+α)/(1-α) is equal to the Herglotz integral of W.

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We construct unramified Galois extensions over maximal abelian extensions of algebraic number fields by using division points of abelian varieties which have everywhere semistable reduction. Further, by using division points of elliptic curves, we construct infinitely many linearly inde­pendent unramified Galois extensions of Q((p= )ab having SL2(Zv) as the Galois group over Q( (p= )ab.

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We prove a simple asymptotic formula for partial autocorrelation functions of fractional ARIMA processes.

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We classify the R-operator, which is a solution of the quantum Yang­Baxter equation on a function space.

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An essentially bounded function</> on the unit circle gives a continuous linear functional T,p on the Hardy space H1 • p( </>) denotes a set of all complex numbers s such that there exists at least one function which attains the norm of T¢,-s· In a previous paper, we showed that C\p(</>) is empty or an open disc. Unfortunately we did not know when p( </>) is open or closed. In this paper, we study when p( </>) is open or closed. Moreover the functions in the Smirnov class N+ whose real parts are nonnegative on the unit circle are described and studied. Then we give new characterizations of exposed points in the unit ball of H1 and we determine when th􀀕 sum of two inner functions is outer. As an result, we can describe all functions which have their Denjoy-Wolff points on the unit circle.

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We consider a class of representations {(Hv, Fv )}v of the Heisenberg commutation relation (HvFv - FvHv = -i) which are deformations of the Schrodinger representation, where the index parameter V is in a class of real­valued C00-functions on R. The Schrodinger representation is given by the case V = l. We classify the representations in terms of properties of V and show that they are divided into three subclasses. All elements of one of these subclasses are unitarily equivalent to the Schrodinger representation. But the others contain no elements unitarily equivalent to the Schrodinger represen­tation. In particular a subclass consists of representations in each of which Hv has no self-adjoint extension. Moreover, we obtain exact solutions to the Heisenberg equation and the Schrodinger equation associated with Hv. We show that, even in the case where Hv has no self-adjoint extension, exact solutions local in time can be constructed.

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In this paper we give a classification of simple stable singularities of Lagrange projections of the first open Whitney umbrella, the simplest singularity of Lagrange varieties. Our classification extends the ADE-classification, due to Arnold, of simple stable singularities of Lagrange projections of smooth Lagrange submanifolds. For the classification, we also prove a criterion of equivalence of stable Lagrange projections of open Whitney umbrellas which is analogous to Mather's fundamental theorem on stable map-germs.

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Two dimensional commutative Banach algebra B with a unit has a simple form : a+ bB for some fixed B in B and for a, bin C. When B is an operator algebra on a Hilbert space, we show that the norm on B is explicitly determined and then B is a Q-algebra. Moreover, we describe completely two dimensional Q-algebras with their norms.

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We consider two kinds of stability ( under a perturbation) of the ground state of a . self-adjoint operator: the one is concerned with the sector to which the ground state belongs and the other is about the uniqueness of the ground state. As an application to the Wigner-Weisskopf model which describes one mode fermion coupled to a quantum scalar field, we prove in the massive case the following: (a) For a value of the coupling constant, the Wigner-Weisskopf model has degenerate ground states ; (b) for a value of the coupling constant, the Wigner-Weisskopf model has a first excited state with energy level below the bottom of the essential spectrum. These phenomena are nonperturbative.

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Recently, a level set formulation is extended by the authors to handle evolution of curves driven by singular interfacial energy including crystalline energy. In this paper as an application of this theory a general convergence result is established for a crystalline algorithm for a general anisotropic curvature flow.

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Let F be a nonzero function in H2 (Dn ) such that if <pis a function in L (Tn ) and </>Fis in H2 (Dn ), then <p belongs to H00 (Dn ). We study the set of multipliers of an invariant subspace M of H2 (Dn) whose common zero set of M is just a zero set of F.

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We introduce the notion of affine parallels to convex plane curves and study these bifurcations.

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We consider the stationary processes that have completely mono­tone autocovariance functions R(·). We prove that regular variation of R(·) implies an asymptotic formula for the prediction error.

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Let B be a von Neumann algebra and Pa selfdjoint projection. For A and B in B, set S A,B = AP + BQ where Q = I - P. The operator S A,B will be called a singular integral operator. When B = L00(T) where LQ(.'(T) is the usual Lebesgue space on the unit circle and Pis an analytic projection, in [6] we established formulae for norms of SA,B and (SA,B)-1• In this paper, if A= {D E B : PDP = D} and (B, A, P) has a lifting property, then we will establish formulae of norms of S A,B and ( S A,B )-1. These formulae are operator theoretic and different from the previous ones. There are several examples such that (B, A, P) has a lifting property. As result, we give several interesting inequalities.

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We characterize the Bloch space and the Besov spaces on the open unit disc D by using many kinds of oscillations. We give new characterizations with known ones. For example, we use the oscillation and the mean oscillation as the following :

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A new notion of solution (called £-solution) is introduced so that the Cauchy problem for the Hamilton-Jacobi equation with semicontinuous initial data is globally solvable. No convexity assumptions on Hamiltonians are imposed. The solution is interpreted as the level set of an auxiliary problem. It turns out that our£­solution is consistent with a classical discontinuous viscosity solution and a bitateral viscosity solution. Moreover, our L-solution is unique and enjoy the comparison principle. The condition that initial data is really attained is also discussed.

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We prove that the sharp interface model for a three-phase boundary motion by surface diffusion proposed by H. Garcke and A. Novick-Cohen admits a unique global solution provided the initial data fulfills a certain symmetric criterion and is also close to a minimizer of the energy under an area-constraint. This minimizer is also a stationary solution of the present model. Moreover we prove that the global solution converges to the minimizer of the energy as time goes to infinity.

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The local and global well-posedness for the Cauchy problem for a class of nonlinear wave equations is studied. The global well-posedness of the problem is proved in the homogeneous Sobolev space jp = il8(Rn) of fractional order s > n/2 under the following assump­tions: (1} Concerning the Cauchy data (</>,'l/;) E if.s = jJs E9 jJs-1, ll(</>,'l/;);if.11211 is relatively small with respect to 11(</>,'l/;);if.a ll for any fixed a with n/2 < a s; s. (2) Concerning the nonlinearity f, f(u) behaves as a conformal power u1+4/(n-l) near zero and has an arbitrary growth rate at infinity.

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The local and global well-posedness for the Cauchy problem for a class of nonlinear Schrodinger equations is studied. The global well­posedness of the problem is proved in the Sobolev space H8 = H8(Rn) of fractional order s > n/2 under the following assumptions: (1) Concerning the Cauchy data <p E H8 , II¢; L2 II is relatively small with re­spect to II¢; jfa II for any fixed a with n/2 < a :=:; s. (2) Concerning the nonlinearity f, f(u) behaves as a conformal power u1+4/n near zero and has an arbitrary growth rate at infinity.

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The main purpose of this paper is to introduce several graphical methods of describing deductive hyperdigraphs and explains its sig­nificance for description of complex systems.

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This paper is concerned with a system of nonlinear wave equations in three space dimensions ∂2tui - c2i Δui = Fi(u, ∂u, ∂2u), i = 1,2,…,m, where O < c1 < c2 < · · · < Cm. We prove the global existence of classical solutions to the system with small initial data, provided pi satisfy "Null condition".

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In the present paper we study the geometric properties of the multivalued solutions to the eikonal equation and we give the appropriate classification theorems. Our motivation stems from geometrical optics for approximating high frequency waves in stratified media. We consider the case of a fixed Hamiltonian imposed by the medium, and we present the geometric framework that describes the geometric solutions, using the notion of Legendrian immersions with an initial point source or an initial smooth front. · Then, we study the singularities of the solutions in the case of a smooth or piecewise Hamiltonian in a boundaryless stratified medium. Finally, we study the singularities of the solutions in a domain with a boundary that describes the propagating field in a waveguide.

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We give all the meromorphic functions defined near the origin 0 E C satisfying a functional equation investigated by Bruschi and Calogero [1], [2].

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R. Jordan, D. Kinderlehrer and F. Otto proposed the discrete-time approximation of the Fokker-Planck equation by the variational formulation. It is determined by the Wasserstein metric, an energy functional and the Gibbs-Boltzmann entropy func­tional. In this paper we study the asymptotic behavior of the dynamical systems which describe their approximation of the Fokker-Planck equation and characterize the limit as a solution to a class of variational problems. We also give a simple approach in a one-dimensional case.

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Let o: and /3 be bounded measurable functions on the unit circle T. The singular integral operator Sa.,() is defined by Sa.,rd = o:P f + /3Qf (f E L2 (T)) where P is an analytic projection and Q is a co-analytic projection. In the previous paper, the norm of Sa.,() was calculated in general, using o:, /3 and o:fJ + H00 where H00 is a Hardy space in L00 (T). In this pa.per, the essential norm JJSa.,(Jlle of Sa.,() is calculated in general, using o:fJ + H00 + C where C is a set of all continuous functions on T. Hence if o:fJ is in H∞ + C then IISa.,(Jlle = max(llo:Jl00, Jl/31100). This gives a known result when o:, /3 are in C.

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We explicitly determine the minimal, self-dual centroaffine surfaces in JR.4 \ {O} by giving a representation formula. Moreover, we describe the self-dual centroaffine surfaces with affine mean curvature -1.

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We study singularities of ruled surfaces in JR3. The main result asserts that only cross­caps appear as singularities for generic ruled surfaces.

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We consider a model of a quantum mechanical system coupled to a (massless) Bose field, called the generalized spin-boson model ( A. Arai and M. Hirokawa, J. Funct. Anal. 151 (1997), 455-503), without infrared regularity condition. We define a regularized Hamilt􀀺nian H(v) with a parameter v 2:: 0 such that H = H(0) is the Hamiltonian of the original model. We clarify a relation between ground states of H(v) and those of H by formulating sufficient conditions under which weak limits, as v---+ 0, of the ground states of H(v )'s are those of H. We also establish existence theorems on ground states of H(v) and H under weaker conditions than in the previous paper mentioned above.

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We study hyperbolic invariants of hyperbolic plane curves as applications of the singularity theory of smooth functions

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Let G be a finite group and S a finite G-monoid. A crossed G-set over S is a finite G-set equipped with a G-map into S called a weight function. A crossed Burnside ring XO(G, S) is the Grothendieck ring of the category of crossed G-scts with respect to disjoint unions and tensor products. In this paper, we prove the fundamental theorem of crossed Burnside rings and an idempotent fornmla at characteristic 0.

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The notion of the generic knots in contact 3-manifolds is introduced, in this paper, as an extension of that of the transversal knots. We show that any generic knots in contact 3-manifolds are constructed by perturbations of some Legendrian knots. As for the classification problem, the situation is completely different from the cases of the transversal and Legcndrian knots. Two generic knots in a contact 3-manifold arc generically isotopic if and only if they have the same number of non­transversal points and-belong to the same topological knot class. We treat, in this paper, not only trivial knots in tight contact 3-manifolds but also non-trivial knots and those in overtwisted contact 3-manifolds.

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Let µ be a finite positive Borel measure on the open unit disc D and H a set of all analytic functions on D. For each a in D, put r(µ, a)= sup lf(a)J2 where .f EH and klfl2dµ ::; 1. Unless the support set ofµ is a finite set, fnr(µ, a)dµ(a) ∞. However zED sup }f Dt(z) r(µ, a)dµ(a) < ∞ may happen where Dt(z) denotes the Bergman disc in D. We study when this is possible. When vis a descrete measure such that dv = I:s(µ,a)8a,zED sup/, D1(z) r(µ, a)dv(a) Under some condition onµ, we show that zsup eDJf D1(z) r(µ,a)dv(a) < ∞ for a finite positive Borel measure v on D if and only if ( v, µ )-Carleson inequality is valid.

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We consider arithmetical aspects of analysis on Fock spaces (Boson Fock space, Fermion Fock space, and Boson-Fermion Fock space) with applications to analytic number theory.

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We present new examples of non-singular developables hypersurfa<;:es, which are algebraic and homogeneous, in real projective spaces. Moreover we give a characterization of compact homogeneous developable hypersurfaces, using the theory of isoparametric hyper­surfaces.

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In this survey article we explain the recent results on the singu­larities appearing in the tangent developables of space curves via the projective duality. First we examine, from our viewpoint, the classical local classification and Hartman-Nirenberg's theorem on developable surfaces. Then we show results on the local differentiable and topological classification of tangent developables of space curves. Lastly we present some related problems and questions, for instance, questions on the finite determinacy.

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We study the Hamiltonian H(V, v) of a Dirac particle (a relativistic charged particle with spin 1/2) minimally coupled to the quantized radiation field, acting in the Hilbert space :F := [EB4L2(R3)]@ :Frad, where :Frad is the Fock space of the quantized radiation field in the Coulomb gauge, V is an external field in which the Dirac particle moves, and v is a momentum cutoff function for the interaction between the Dirac particle and the quantized radi­ation field. We first discuss the self-adjointness problem of H(V, v). Then we investigate in detail properties of H := H(O, v), the Hamiltonian in the case V = 0. In this case a unitary transform it of H has a direct integral decomposition it= f3.3 H(p)dp, where H(p) is an operator on EB4:Frad, physically the polaron Hamiltonian of the Dirac particle with total momentum p E R3We define a one parameter family {H.r(P )}TE[O,l) of deformations of H(p) such that H1 (p) = H(p). On the operator HT(p), we discuss the following aspects: (i) properties of the ground-state energy; (ii) existence of a ground state; (iii) the spectrum.

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Let L􀀃(n,dud0/21r) be a complete weighted Bergman space on the open unit disc n where du is a positive finite Borel measure on [O, 1). We show the following : When cp is a continuous function on the closed unit disc D, T</J is compact if and only if cp = 0 on an.

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We establish general theorems on locating the essential spectrum of a self-adjoint operator of the form A® I+ I® df(S) + Hr on the tensor product 1-{ ® Fb(K) of a Hilbert space 1-l and the abstract Boson Fack space Fb(K) over a Hilbert space K, where A is a self-adjoint operator on 1-l bounded from below, df(S) is the second quantization of a nonnegative self-adjoint operator S on K and Hr is a symmetric operator on 1-l ® Fb(K). We then apply the theorems to the generalized spin-boson model (A. Arai· and M. Hirokawa, J. Funct. Anal. 151 (1997), 455-503) and a general class of models of quantum particles coupled to a Bose field including the Pauli-Fierz model in nonrelativistic quantum electrodynamics.

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In this paper we propose a two dimensional random crystalline algorithm for Gauss cur­vature flow.

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Let L(H) be the algebra of all bounded linear operators on a Hilbert space H and let A be a uniform algebra. In this paper we study the following questions. When is a unital bounded homomorphism q, of A in L(H) completely bounded ? When is the norm 11-I>II of q, equal to the completely bounded norm 11-I>llcb ? In some special cases we answer this question. Suppose q, is p-contractive (0 < p < oo) where 4> is contractive if p == 1. We show that if A is a Dirichlet algebra or dim A/ ker q, == 2 then q, has a p-dilation. If q, is a p-contractive homomorphism then 1\-I>\\ == max(l, p) and if it has a p-dilation then 11-I>llcb == max(l, p). Moreover we give a new example of a hypo-Dirichlet algebra in which a unital contractive homomorphism has a contractive dilation.

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We prove the existence of absolutely continuous invariant measures for expanding piecewise linear maps on bounded polyhedral domains of arbitrary dimensions.

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We define a bijection from Littlewood-Richardson tableaux to rigged configurations and show that it preserves the appropriate sta­tistics. This proves in particular a quasi-particle expression for the generalized Kostka polynomials' K􀀸R(q) labeled by a partition >. and a sequence of rectangles R. The generalized Kostka polynomials are q-analogues of multiplicities of the irreducible GL(n, C)-module V􀀸 of highest weight >. in the tensor product yRi ® ••• ® yR£.

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A class of models of quantized, massless Bose fields, called the generalized spin­boson model (A. Arai and M. Hirokawa, J. Funct. Anal. 151 (1997), 455-503) is considered. Theorems on the absence of ground states and the other eigenvectors of the model without infrared cutoff (but with ultraviolet cutoff) are established with conditions in terms of correlation functions for some operators.

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We present an example of stationary process with long-time memory for which we can calculate explicitly the prediction error from the finite part of past. The long-time behavior of the prediction error is discussed.

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Suppose Tt/i is a Toeplitz operator on the Bergman space of the open unit disc. When the symbol <p is bounded and radial, it is known that T¢ is compact if and only if lim - </>(r)dr = 0. In this paper, we study this type theorem for arbitrary bounded symbol <p.

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Let L2 L2( D, rdrd0 / 1r) be the Lebesgue space on the open unit disc and L􀀁 = L2 n 1iol(D) be the Bergman space. Let P be the orthogonal projection of L2 onto L􀀃 and let Q be the orthogonal projection onto L! 0 = {g E L2 ; g EL􀀃, g(O) = O}. Then I -P 􀀖 Q. The big Hankel operator and the small Hankel operator on L􀀃 are defined as the following : For</> in L00 , H!i9(!) = (I - P)( </>f) and H;,mall(f) = Q( </>f) (f E L􀀃). In this paper, the finite rank intermediate Hankel operators between H!ig and H;,ma/1 are studied. We are working on the more general space, that is, the weighted Bergman space.

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We prove Abel-Tauber theorems for Hankel and Fourier transforms. For example, let f be a locally integrable function on [O, oo) which is eventually decreasing to zero at infinity. Let p = 3, 5, 7, · · · and £ be slowly varying at infinity. We characterize the asymptotic behavior f(t) 􀀕 l(t)t-P as t -+ oo in terms of the Fourier cosine transform of f. Similar results for sine and Hankel transforms are also obtained. As an application, we give an answer to a problem of R. P. Boas on Fourier series.

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Let G be a finite cyclic subgroup of the mapping class group of order m. We prove the Morita-Mumford classes restricted to G admit a certain kind of periodicity whose period is given by the Euler function ¢(m). Using this periodicity theorem, we compute the Morita-Mumford classes on arbitrary finite cyclic subgroups of the automorphism group of Klein's quartic curve.

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We give a way of constructing a compact, closed, strictly convex hypersurface which loses convexity without developing singularities when it moves by its surface diffusion for a short time.

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This paper is a sequel to the author's paper [Yos 98]. We construct some operations on categories and functors, for example, summation, multiplication, derivation, partial derivation, exponentia­tion, substitution. These operations are correspond to those of species. The main results of this paper are two theorems (Theorem 5.8, 6.5) which characterizing strict Krull-Schmidt categories(KS-categories). Especially, Theorem 6.5 states that for a skeletally small and locally finite category £ with finite coproducts and its full subcategory C, if the exponential formula E(t) = exp(C(t)) implies, under a techni­cal assumption, that £ is a strict Krull-Schmidt category. Further­more, we define the substitution T(S) of functors S : £---+ S and T : F--+ Seti, where S has finite coproducts. These operations satisfy formulas similar to such operations on the usual power series. There are some applications.

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We study the spectral problem associated to a Ruijsenaars-type ( q-)difference version of the one-dimensional Schrodinger operator with Poschl­Teller potential. The eigenfunctions are constructed explicitly with the aid of the inverse scattering theory of reflectionless Jacobi operators. As a result, we arrive at combinatorial formulas for basic hypergeometric deformations of zonal spherical functions on odd-dimensional hyperboloids and spheres.

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The system of a particle minimally coupled to a radiation field with an 1iltraviolet cut-off is considered. The Hamiltonian of the system is defined with non-perturbative approach; any assumptions on the magnitude of the absolute value of a coupling con­stant in the Hamiltonian is not required. It. is established that the ground state of the system, if exists, is unique and overlapps with the ground state of a decoupled Hamiltonian.

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The system of one charged non-relativistic particle with a class of external po­tentials minimally coupled to a massless quantized radiation field without the dipole approximation is considered. An ultraviolet cut-off is imposed on the quantized radia­tion field, the charged particle has spin 1/2 and the class of external potentials contains the Coulomb potential. It is shown that the ground states of the system exist provided that a coupling constant is in a region, and an expression of the ground state energy of the system is given by a functional integral containing a vector-v,1.lucd stochastic integral.

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We study afliue iuvariauts of plauc curves from the view point of the singu­larity theory of smooth functions. We describe how affine vertices and afliue inflexions arc created and destroyed.

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We present a method to locate the essential spectrum of a self-adjoint operator on the tensor product of a Hilbert space and the abstract boson Fack space, and discuss its application to Hamiltonians of quantum field models.

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We prove Strichartz estimates for wave equations in the homoge­neous Besov space. The main purpose in this paper is to present a unified way to derive Strichartz estimates given by Bak-McMichael­Oberlin (1, Theorem 6'], Ginibre-Velo [3, Proposition 3.1], Harmse (5, Theorem 2.3] and Oberlin (10, Theorem 3]. Our argument proceeds under the abstract setting once we have used the stationary phase esti­mate for wave equations, and the main tool is the complex interpolation method, by which we shall obtain new Strichartz estimates.

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Combinatorial objects called rigged configurations give rise to q-analogues of certain Littlewood-Richardson coefficients. The Kostka-Foulkes polynomials and two-column Macdonald-Kostka polynomials occur as special cases. Conjecturally these polynomials coin­cide with the Poincare polynomials of isotypic components of certain graded G L( n )-modules supported in a nilpotent conjugacy class closure in gl(n).

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We propose a two dimensional frame-invariant phase field model of grain im­pingement and coarsening. One dimensional analytical solutions for a stable grain boundary in a bicrystal are obtained, and equilibrium energies are com­puted. We are able to calculate the rotation rate for a free grain between two grains of fixed orientation. For a particular choice of functional dependencies in the model the grain boundary energy takes the same analytic form as the microscopic (dislocation) model of Read and Shockley [11]. 64.70.Dv, 64.60.-i, 81.30.Fb, 05.70.Ln

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We define the binormal indicatrix and focal developable of a nonlightlike curve in Minkowski 3-space. We establish the relationships between singularities of these subjects and geometric invariants of curves under the action of Lorentzian group.

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Recently the models of the faceted crystal growth and of the grain boundary were proposed based on the gradient system with nondifferentiable energy. In this article, we study their most basic forms given by the equations ut = (ux/|ux|)x and ut = a/1(a * ux / ux)x where both of the related energy include luxl term of power one which is nondiffer­ entiable at Ux = 0. The first equation is spatially homogeneous while the second one is spatially inhomogeneous when a depends on x. These equations naturally express non-local interactions through their singular diffusivities ( Ux = 0) which make the profiles of the solutions completely flat. Mathematical basis for justifying and analyzing these equations will be explained, and also it will be shown how the solutions of the equations evolve by theoretical and numerical approach.

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In this paper we study singularities of certain surfaces and curves associated with the family of rectifying planes along space curves. We establish the relationships between singularities of these subjects and geometric invariants of curves which are deeply related to the order of contact with helices.

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In this paper we construct an exact forward self-similar solution, rep­resenting a stagnation flow, for the Navier-Stokes equations by solving a third order ordinary differential equations.

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In this paper, we study formal power series with exponents in a category. For example, the generating function of a category E with finite Hom-sets is defined by E(t) = :Etx /IAut(X)I, where the summation is taken over all isomorphism classes of obejects of E. We can use such power series to enumerate the number of £-structures along a faithful functors(Theorem 4.6). Our theory is closely related to the thory of species (Joyal 1981). A species can be identified with a faithful functor from a groupoid to the category of finite sets (The­orem 3.6). We use mainly the concept of faithful functors with finite fibers instead of that of species, by which we can separate the roles which categories play and which functors play. For example, the ex­ponential formula E(t) = exp(Con(E)(t)) means the unique coproduct decomposition property(Theorem 5.8). In the final section, we give some applications of our theory to rather classical enumerations.

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For 1 <_ r < oo, we construct a piecewise er expanding map F : D - D on the domain D = (0, 1) x (-1, 1) c R2 with the following property: there exists an open set B in D such that the diameter of pn (B) converges to Oas n - oo and the empirical measure n-1E􀀁:i 8Fk(x) converges to the point measure 8p at a point p as n→∞ for any point x EB.

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We prove the existence of absolutely continuous invariant measures for piecewise real-analytic expanding maps on bounded regions in the plane.

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We define the notion of lightcone Gauss maps, lightcone pedal curves and lightcone developables of spacelike curves in Minkowski 3-space and establish the relationships between singularities of these subjects and geometric invariants of curves under the action of Lorentz group.

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We rigorously prove that there exists a simple, strictly convex, smooth closed curve which loses convexity but stays simple without developing singularities when it moves by its surface diffusion for a short time.

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In this paper, we,study an operator A on a Hilbert space H which satisfies one of the following inequalities For some ,\ with O ::; ,\ ::; 1 l(Ay, y)I ::; AIIYll2 + (1 - ,\)IIAYll2 (y EH) or AIIAYll2 + (1 - ,\)l(Ay, y)I ::; IIYll2 (y EH). These two inequalities can be regarded as special cases of generalized numerical ranges. If A has a p-dilation with p > 0, then it satisfies one of them. We show that the operator radii wp(A) of A are calculated using l(Ay,y)I and IIAYII- Several applications are given.

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Let a finite group G act on a compact Riemann surface C in a faithful and orientation preserving way. Then we describe the Morita-Mumford classes en (Ca) E H2n (G;Z) of the homotopy quotient (or the Borel construction) Ca of the action in terms of fixed-point data. This fixed-point formula is deduced from a higher analogue of the classical Riemann-Hurwitz formula based on computations of Miller [Mi] and Morita [Mo].

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A system consisting of finitely many charged non-relativistic particles bound by some external scalar potential and minimally coupled to a massless quantized radia­tion field without the dipole approximation is considered. An ultraviolet cut-off on the quantized radiation field is imposed. An interaction Hamiltonian of this system is defined as a self-adjoint. operator in a Hilbert space. The existence of the ground states of the interaction Hamiltonians is established. It is also shown that asymptotic limits of annihilation operators and creation operators exist. Asymptotic creation operators and a ground state of the interaction Hamiltonian provide for a closed subspace in the Hilbert space which reduces the interaction Hamiltonian. It is discovered that the reduced part of the interaction Hamiltonian is equivalent to a well known self-adjoint operator. Hence the absolutely continuous spectrum of the interaction Hamiltonian is specified.

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We discuss the existence of a global small solution to the Cauchy problem for a system of quasilinear wave equations in three space dimensions, when its nonlinear term have a critical exponent. Global existence is established on the null condition which is extended to the condition for systems of wave equations with different propagation speeds.

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In this paper, the following is proved. When !al $ 1 and lbl $ 1, A = [ ;  ・] is a p-contraction if and only if lcl2 + la-bl2 < inf {p + (1 -p )a(}{p + (1 -p )b(} -abl(l2 where D is the open unit disc.

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We give a simple proof for monotonicity of topological entropy in the quadratic family. We use a spectral property of a Ruelle operator.

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In this paper we discuss a class of Markov marginal problems (MMP). By MMP, we mean the problem to construct a Markov process with given (marginal) constraints on the path space. As an application we consider Markov optimal control problems.

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A skeleton dynamics for the self-replicating patterns (SRP) of reaction diffusion system is presented. Self­replicating dynamics can be regarded as a transient process from a localized trigger to a stable Turing pattern or oscillatory Turing pattern. It looks like a reverse process of usual coarsening phenomena, i.e., the number of unit localized pattern increases until the domain is filled by them completely. SRP was found in several chemical reaction models, for instance, the Gray-Scott model as well as in real experi­ments. The most difficult point to describe SRP lies in the fact that it is truly a transient phenomenon in the sense that it can be captured neither as a definite object in dynamical system theory like an attractor nor an orbit itinerating among saddle points in the phase space. To our knowledge, it is not known that what kind of dynamical framework is suitable to clarify the behavior of SRP. The aim is to give a new point of view to describe such a transient dynamics of SRP on a finite interval. Especially we concen­trate on the basic mechanism causing SRP from a bifurcational view point by employing a new model system and its finite-dimensional con:.partment model which shares common qualitative features with the Gray-Scott model. By a careful anatomy of global bifurcation diagrams, the skeleton dynamics of SRP comes from a hierarchy structure of the subcritical bifurcating loops of oscillatory branches of pulse type. It should be noted that these loops themselves do not constitute the skeleton dynamics of SRP, but the ruins of them do it. In other words, the aftereffect of the hierarchy structure manifests the dynamics of SRP. The most important ingredient of an organizing center from which the whole hierarchy structure of SRP emerges is Bogdanov-Takens-Turing singularity as well as the existence of stable equilibrium point, which indicates universality of the above structure in the class of nonlinearities sharing this character.

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The set P* (I) of finite multisets on a set J has a natural quantale structure. In this note we study the quantic nuclei of this quantale, and show that they are parametrized by those subsets of the infinite multisets P** (I) whose elements are mutually uncomparable. We also study the quantic nuclei of another natural quantale structure on P*(I) with respect to the opposite order.

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The category of simulations of nondeterministic dynamical systems is shown to be symmetric monoidal closed category with a subobject classi­fyer. (AMS Classification:18Dl5,68Ql0,03F50)

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We consider the Cauchy problem for nonlinear wave equations in the homogeneous Sobolev space ifl'(Rn ), where n >_ 2 and O <_ µ < n/2 using the generalized Strichartz estimates given by J.Ginibre and G.Velo [10].

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Let a and {3 be bounded measurable functions on the unit circle T. Then the singular integral operator Sa,/3 is defined by Sa,f3f = aP+f + f3P-f, (J E L2 (T)) where P+ is an analytic projection and P_ is a co-analytic projection. In this paper, the norms of Sa,/3 and its inverse operator on the Hilbert space L2 (T) are calculated in general, using a, {3 and a􀀡 + H00 Moreover, the relations between these and the norms of Hankel operators are established. As an application, in some special case in which a and /3 are nonconstant functions, the norm of Sa,/3 is calculated in a completely explicit form. If a and {3 are constant functions, then it is well known that the norm of Sa,/3 on L2 (T) is equal to max {lal, 1/31}. If a and /3 are nonzero constant functions, then it is also known that S ,{3 on L2 (T) has an inverse operator Sa-1,13whose norm is equal to max { alal-1, 1/31-1 }.

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Investigated are some representation-theoretic aspects of a two-dimensional quan­tum system of a charged particle in a vector potential A which may be singular on an infinite discrete subset D of R2 . For each vector v in a set V(D) C R2 \ {O}, the projection Pv of the physical momentum operator P := p - aA to the di­rection of vis defined by Pv := v · P as an operator acting in L2 (R2 ), where p = (-iDx, -iDy ) [(x, y) E R2] with Dx (resp. Dy ) being the generalized par­tial differential operator in the variable x (resp. y) and a E R is a parameter denoting the charge of the particle. It is proven that Pv is essentially self-adjoint and an explicit formula is derived for the strongly continuous one-parameter uni­tary group { eitPv heR generated by the self-adjoint operator E'v ( the closure of Pv ), i.e., the magnetic translation to the direction of the vector v. The mag­netic translations along curves in R2 \ D are also considered. Conjugately to Pv and Pw [w E V(D)], a self-adjoint multiplication operator Qv,w is introduced, which is a linear combination of the position operators x and y, such that, if A is flat on R2 \ D, then 1r¢".w := {Qv,w,Qw,v,Pv,Pw} gives a representa­tion of the canonical commutation relations (CCR) with two degrees of freedom. Properties of the representation 1r¢".w are analyzed. In particular, established is a necessary and sufficient condition for 1r¢".w to be unitarily equivalent ( or inequivalent) to the Schrodinger representation of CCR. The case where rr􀀊 w is inequivalent to the Schrodinger representation corresponds to the Aharon􀀍v­Bohm effect. Quantum algebraic structures [quantum plane and the quantum group Uq(sl2)] associated with the pair {Pv,.Pw} are also discussed. Moreover, for every A in a class of vector potentials having singularities on the infinite lattice L(w1,w2) := {mw1 + nw2 lm, n E Z} [the case D = L(w1,w2)], where w E R2 and w2 E R2 are linearly independent, it is shown that the magnetic translations eiPwi, j = 1, 2, with A replaced by a modified vector potential are reduced by the Hilbert space £2(L(w1,w2)) identified with a closed subspace of L2(R2 ). This result, which may be regarded as one of the most important novel results of the present paper, establishes a connection of continuous quantum systems in vector potentials to lattice ones.

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We study asymptotic stability of the planar rarefaction wave in one or two space dimensional scalar viscous conservation law for nonsmooth initial data.

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We show the local in time solvability of the Cauchy problem for non­linear wave equations in the Sobolev space of critical order with nonlinear term of exponential type.

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A new estimate for the ground state energy of Schrodinger operators on 2 L (Rn) (n 2: 1) is presented. As a corollary, it is shown that the ground state energy of the Schrodinger operator with a scalar potential V is more than the classical lower bound ess.inf xeRn V( x) if V is essentially bounded from below in a certain manner ( enhancement of the ground state energy due to quantization). As an application, it is proven that the ground state energy of the Hamiltonian of the hydrogen-like atom is enhanced under a class of perturbations given by scalar potentials (vanishing at infinity).

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A functional integral representation of a heat semigroup acting on a Hilbert space is constructed. Its generator describes an interaction between particles and a quantized field. By using the functional integral representation, for the Nelson model with a nonnegative mass we investigate a weak coupling limit which removes an ultraviolet cut-off. As a result, it is shown that a many-body Schrodinger Hamiltonian with the many-body Coulomb potential (or the many-body Yukawa potential) appears in the limit.

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We give a rigorous proof for formation of I pinching· of evolving curves moved by surface diffusion.

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For a non-zero function f in H1 , the classical Hardy space on the unit disc, we put Sf= {g E H1 : argf(i8 ) = argg(ei0) a.e. 0}. The intersection of Sf and the unit sphere in H1 is just a set of solutions of some extremal problem in H1 It is known that Sf can be represented in the form Sf = S x g0, where β is a Blaschke product and g0 is a function in H1 with S90 = {Λ· g0 : Λ> O}. Also it is known that the linear span of Sf is of finite dimensional if and only if β is a finite Blaschke product, and when β is a finite Blaschke product, we can describe completely the set Sβ and the zeros of functions in Sβ. In this paper, we study the set of zeros of functions in Sβ when β is an infinite Blaschke product whose set of singularities is not the whole circle. Especially we study the behavior of zeros of functions in Sβ in the sectors of the form: Δ = { reiQ : 0 < r <_ 1, c1 < 0 < c2} on which the zeros of B has no accumulation points, and establish a convergence order theorem of zeros in Δ of functions in Sβ .

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Let V be Euclidean space. Let WC GL(V) be a finite irreducible reflection group. Let A be the corresponding Coxeter arrangement. Let S be the algebra of polynomial functions on V. For H E A choose O'.H E V* such that H = ker( O'.H ). The arrangement A is known to be free: the derivation module D(A) = {8 E Ders I 8(aH) E SaH} is a free S-module with generators of degrees equal to the exponents of W. In this paper we prove an analogous theorem for the submodule E(A) of D(A) defined by E(A) = {8 E Ders I 8(aH) E Sa.1:,-}. The degrees of the basis elements are all equal to the Coxeter number. The module E(A) may be considered a deformation of the derivation module for the Shi arrangement, which is conjectured to be free. The proof is by explicit construction using a derivation introduced by K. Saito in his theory of flat generators.

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In this paper, we propose a new method to characterize in terms of "quasi -topology" the image of Gelfand transform of commutative Banach algebras, which were inspired by the works of R. Doss in 1976-77. Generalizations of Bochner-Schoenberg-Eberlein theorem to general commutative Banach algebras ( which play very important roles in this paper) are also considered. Our method is applied in this paper to some concrete Banach algebras such as, group algebras L1 (G), ideals and quotient algebras of L1 (G), Segal algebras L1 (G) n £P(G) (1 < p < oo) for locally compact abelian groups G and C*-algebra C0(0) on locally compact Hausdorff spaces O etc.

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The spectra of the Toeplitz operators on the weighted Hardy space H2(Wd0/21r) and the Hardy space HP(d0/21r), and the singular·integral operators on the Lebesgue space L2 (d0/21r) are studied. For example, the theorems of Brown-Halmos type and Hartman-Wintner type are studied.

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A general stability and convergence theorem is established for generalized solutions of a family of nonlinear evolution equations with non­local diffusion in one space dimension. As the first application motion by nonlocal weighted curvature is approximated by solutions of regular problem, when initial curve is given as the graph of a continuous periodic function. This justifies the motion by crystalline energy as a limit of regularized prob­lems. As the second application the motion by crystalline energy is shown to approximate the motion by regular interfacial energy if the crystalline energy approximates the regular energy. This gives the convergence of crys­talline algorithm for general curvature flow equations. Our general results are also important to explain that geometric evolution of crystals depends continuously on temperature even if facets appear.

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The Cauchy problem for the nonlinear Schrodinger equations is considered in the Sobolev space Hnf2(Rn) of critical order n/2, where the embedding into L00(Rn) breaks down and any power behavior of interaction works as a subcritical nonlinearity. Under the interaction of exponential type the existence and uniqueness is proved for global Hn/2-solutions with small Cauchy data.

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This paper is concerned with the initial value problem of a system owning a hyperbolic equation with respect to unknown function u(x, t) and an elliptic one with respect to unknown function q(x, t) in one space dimension. This system originates in the dynamics of a radiating gas. The purpose in the present paper is to give the results on global existence and asymptotic behaviour of EV-solutions to the present system for the two cases: the first case is that initial data uo(x) decay as lxl → ∞ and the second one is that initial data uo(x) tend to two given constants 'U± with u_ < u+ as x→ ±∞. In the first case, we prove that the present problem is well-posed in EV, that is, for any EV-initial datum, there exists a unique EV-solution. We also show that if initial data uo are small in a certain sence, then the solutions u(·, t) decay in the order O(t-(I-I/p)/2) as t→ ∞ in V(R) with p E [1, ∞]. Furthermore, in the second case, we prove that the present problem is well-posed in ro+EV, where ro(x) is u_ when x < 0 and u+ when x > 0. Finally, we prove the main result of the present paper that if both iu+ - u-1 and initial EV-perturbations to ro are small in a certain sense, then the rarefaction waves of the inviscid Burgers equation are stable for the present system and the convergence rates of u( ·, t) to the rarefaction waves as t→ ∞ are O(c{I-I/p)/2) in V(R) with p E (1, ∞].

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In models of a quantum harmonic oscillator coupled to a quantum field with a quadratic interaction, embedded eigenvalues of the unperturbed system may be unstable under the perturbation given by the interaction of the oscillator with the quantum field. A general mathematical structure underlying this phenomenon is clarified in terms of a class of Fock space representations of the *-algebra of the canonical commuta­tion relations over a Hilbert space. It is also shown that each of the representations is given as a composition of a proper Bogoliubov ( canonical) transformation and a partial isometry on the Fock space of the representation.

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A generalization of the standard spin-boson model is considered. The Hamil­tonian H (a) of the model with a coupling parameter a E R acts in the tensor product 1-l ® :Fb of a Hilbert space 1-l and the boson (symmetric) Fack space :Fb over L2 (R11). The existence and uniqueness of ground states of H(a) are investigated. The degeneracy of the ground states is also discussed. The results obtained are nonperturbative. The methods used are those of constructive quantum field theory and the min-max principle. Also an exact asymptotic formula for the ground state energy of H(a) as lal → ∞ is established.

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We study affine invariants of space curves from the view point of the singularity theory of smooth functions. By the aid of the singularity theory we define a new equi-afline frame for space curves. We also introduce two surfaces associated with this equi-afline frame and give a generic classification of the singularities of those surfaces.

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In this paper we give equivalent conditions on the central limit theorem in total variation norm for a sequence of probability measures on R. This generalizes Cacoullos, Papathana­siou and Utev's central limit theorem in L1-norm for a sequence of probability density functions on R. We also give equivalent conditions on the central limit theorem in weak convergence and those on the local limit theorem.

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For any finite subgroup G of S£(2, C) of order n, we consider HilbG(A.2 ) "a Hilbert scheme" parametrising G-invariant 0-dimensional subschemes of length n. HilbG (A.2 ) is a minimal resolution of a simple singularity A2 /G in the canonical manner. \Ve study HilbG(A2 ) in the cases of E5 1 E1 and Es in detail. As a consequence we have a new explana­tion for the ( classical) two-dimensional McKay correspondence in these cases as in the same manner as in [IN2].

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We consider the scattering problem for the nonlinear Schrodinger equations with interactions behaving as a. power p at zero. In the critical and subcritical cases (s >_ n/2 - 2/(p - 1) >_ 0), we prove the existence and asymptotic completeness of wa.ve operators in the sense of Sobolev norm of orders on a set of asymptotic states with small homogeneous norm of order n/2 - 2/(p - 1) in space dimension n >_ 1.

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We present a neural model for a singular-continuous nowhere-differentiable (SCND) attractors. This model shows various characteristics originated in attractor's nowhere­differentiability, in spite of a differentiable dynamical system. SCND attractors are still unfamiliar in the neural network studies and have not yet been observed in both artificial and biological neural systems. vVith numerical calculations of various kinds of statisti­cal quantities in artificial neural network, dynamical characters of SCND attractors are strongly suggested to be observed also in neural systems experiments. vVe also present possible information processings with these attractors.

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Motivating the study of constructing a formal neuron in computer, we propose a logic-based dynamical theory for a genesis of biological threshold which specific pro­ teins like ion channel proteins or their network can be responsible for. By viewing such a protein or a protein network as a computation machine, the statements concerning the states of reaction chains which eventually activate or inactivate the protein are treated. Introducing dynamical systems, associated with an inference process on the statements with continuous truth values, we investigate invariant characters of such a dynamics, thereby we obtain a sigmoidal function for an invariant distribution func­tion of the truth values. The domain of solutions of functional equations regarded as a representation of the self-description of proteins or protein networks as a machine indicates the emergence of threshold, namely the realization of dyadic value, 0 or 1, b􀁃d on the continuous truth values takes place. The results obtained may highlight the mechanism of neuronal threshold in the framework different from population dy­namics. The derived dynamical systems may also provide a simple model of" demon" rectifying the thermal fluctutions to drive unidirectional movements.

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We study the scattering problem for the Hartree equation X { u(i&tu 0,x) = = -(1/uo(x), 2)􀀯u + x JE (Rluln2. )u, (t,x) ER where J(lul2) = V * lul2 , V(x) = ..\lxl- A ER, n 2: 2. We prove that for any U0 E H0,'Hr with 1/2 < r < n/2 1 such that the value e = ||U0||0,r+||U0||, is sufficiently small, there exists unique U +- < H0 ^ H0 with 1/2 < 0 < γ , such that for all |t| >= 1 U(t) is the free Schrodinger evolution group and Hm,s is the weighted Sobolev space defined by Hm

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In this paper we prove instant extinction of the solutions to Dirichlet and Neumann boundary value problem for some quasilinear parabolic equations whose diffusion coefficient is singular when the spatial gradient of unknown function is zero.

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When the thickness of the interface ( denoted by c) tends to zero, any stable stationary internal layered solutions to a class of reaction diffusion systems cannot have a smooth limiting interfacial configuration. This means that if the limiting configuration of the interface has a smooth limit, it must become unstable for small c, which makes a sharp contrast with one-dimensional case as in [5]. This suggests that stable layered patterns must become very fine and complicated in this singular limit. In fact we can formally derive that the rate of shrinking of stable patterns is of order c1/3 as well as the rescaled reduced equation which determines the morphology of magnified patterns. A variational.characterization of the critical eigenvalue combined with the matched asymptotic expansion method is a key ingredient for the proof, although the original system is not of gradient type.

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We investigate a certain class of invariant subspaces of subdiagonal algebras which contains the both cases of (extended) weak-*Dirichle algebras and analytic crossed products. We show a version of the Beurling-Lax-Halmos theorem.

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This paper presents a rigorous derivation of an N-body Schrodinger Hamiltonian from taking a weak coupling limit of a renormalized Hamiltonian describing an interac­tion of N-spineless particles and a massive scalar field at the same time as an ultraviolet cut-off is removed. In particular, in the case where the space dimension equals three, the Yukawa potential appears in the N-body Schrodinger Hamiltonian.

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In this paper, we consider the initial value problems to systems of quasilinear wave equations with different speeds in two space dimensions. Applying John-Shatah obser­vations to our problem, we introduce the null condition for the system with different speeds. Moreover, we prove a global existence theorem for a class satisfying the null condition.

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For any finite subgroup G of SL(2, C) of order n, we consider a certain subscheme HilbG(A2 ) of Hilbn(A2 ) consisting of G-invariant 0-dimensional subschemes of length n. We prove without using the classification of finite subgroups in SL(2,C) that HilbG(A2 ) is a minimal resolution of a simple singularity A2 /G in the canonical manner. Any point of the exceptional set is a G-invariant 0-dimensional subscheme of A2 with support the origin, to which we associate the ideal sheaf I defining the subscheme. A minimal G-submodule of I generating the O A2-module I is a one dimensional trivial G-module plus one or two nontrivial mutually distinct irreducible G-modules. This gives a bijective correpondence between the set of all the irreducible components of the exceptional set and the set of all the equivalence classes of irreducible G-modules, which turns out to be the ( classical) McKay correspondence.

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For bounded Lebesgue measurable functions α, β on the unit circle, Sα,β = αP+ + βP_ is called a singular integral operator, where P + is an analytic projection and P_ is a co-analytic projection. We study one-weighted norm inequalities of Sα,β on L2(W). We introduce a class HSr of weights with r = |α-β|/||α-β||∞ in order to characterize those weights. For example, we show that Sα,β is bounded with respect to a weight W if and only if W belongs to HSr or |α-β|W ≡ 0. If r is a nonzero constant, then HSr is just a well known class of weights due to Helson and Szego. Moreover we study the Koosis type problem of two weights of Sα,β and get very simple necessary and sufficient conditions for such weights.

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We study asymptotic behaviors as t -+ ±oo of solutions to the nonlinear wave equation Utt-Δu = |u|p (p > 1) in x E Rn , -∞ < t < ∞ for p larger than a critical value P0(n). These asymptotic behaviors guarantee the existence of the scattering operator. We prove the radially sym­metric small solutions exist and are asymptotic to the solutions of the homogeneous wave equations, provided n >_ 5.

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The purpose of the present note is to announce our recent results on the cohomology of the moduli space of curves or equivalently ( over the rationals) the cohomology of the mapping class group of orientable surfaces. Our main results are twofold. First we construct explicit group cocycles for any of the known stable characteristic classes ( the Mumford-Morita-Miller classes) of the moduli spaces. Secondly, by combining our result with that of Hain in [H2), we show that the "continuous part" of the cohomology of the moduli space (see §5 for the definition) is exactly equal to the subalgebra generated by the above stable classes. This second result may be considered as a supporting evidence for the conjecture that the stable cohomology of the moduli spaces would be equal to the polynomial algebra generated by the Mumford­Morita-Miller classes. The details of the results sketched in this note will appear elsewhere.

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We prove Abel-Tauber theorems which link the asymptotics of a func­tion and its Fourier-Stieltjes coefficients. Both cosine and sine coefficients are studied. The results in the cosine case can be applied to stationary time series with long-time memory. The analogues for Fourier-Stieltjes transforms are also given.

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It is shown that a class of unitary transformations of the canonical momentum operator in a direct sum of L2 (Rd ) is given by a class of operator-valued Lorentz trans­formations of perturbed canonical momentum operators. As an application, analysis of the quantum theory of spin-½ charged particles in an external electromagnetic field is given.

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This paper presents a scaling limit of Hamiltonians which describe interactions of N-nonrelativistic charged particles in a scalar potential and a quantized radiation field in the Coulomb gauge with the dipole approximation. The scaling limit defines effective potentials. In one-nonrelativistic particle case, the effective potentials have been known to be Gaussian transformations of the scalar potential [J.Math.Phys.34(1993)4478- 4518]. However it is shown that the effective potentials in the case of N-nonrelativistic particles are not necessary to be Gaussian transformations of the scalar potential.

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A new type of self-organized dynamics is presented, in relation with chaos in neural networks. One is chaotic itinerancy and the other is chaos-driven contraction dynam­ics. The former is addressed as a universal behavior in high-dimensional dynamical systems. In particuiar, it can be viewed as one possible form of memory dynamics in brain. The latter gives rise to singular-continuous nowhere-differentiable attractors. These dynamics can be related to each other in the context of dimentionality and of chaotic information processings. Possible roles of these oomplex dynamics in brain are also discussed.

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Solids can exist in polygonal shapes with boundaries unions of flat -pieces· called· facets. Analyzing the- growth -of such crystalline shapes is an important problem in materials science. In this paper we derive equa­tions that govern the evolution of such shapes; we formulate the correspon­ding initial-value problem variationally; and we use this formulation to establish a comparison principle for crystalline evolutions. This principle as­serts that two evolving crystals one initially inside the other will remain in that configuration for all time.

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We study various local invariants associated with a singular holomorphic foliation on a complex surface admitting a possibly singular invariant curve. We establish the relation among them and prove/reprove formulas relating the total sum of these invariants to some global invariants of the foliation and the invariant curve.

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The spectrum a(T¢) of a Toeplitz operator Tt/> on the open unit disc n for a unimodular symbol <Pis studied. Exactly, many sufficient conditions for a(T¢) .􀀍 an or a(T¢) = lJ are given. If <Pis a unimodular function in H00 + C, then a(T¢) 􀀍 an or a(T¢ ) = D.

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We consider singular degenerate parabolic equations including the p-Laplace diffusion equation. We establish a comparison priciple which is a natural extension of the paper [12] by Ishii and Souganidis. Once we get a comparison priciple we can construct the unique global-in-time solution to the Cauchy problem for the p-Laplace diffution equation. The solution is bounded, uniformly continuous [0,T）× R N if the initial data is bounded, unformaly continuous on R N.

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A new notion of solutions is introduced to study degenerate non­linear parabolic equations in one space dimension whose diffusion effect is so strong at particular slopes of unknowns that the equation is no longer a partial differential equation. Extending the theory of viscosity solutions comparison principle is established. For a periodic continuous initial data a unique global continuous solution (periodic in space) is constructed. The theory applies to motion of interfacial curves by crystalline energy or more generally by anisotropic interlacial energy with corners when the curves are the graphs of functions. Even if the driving force term exists, the initial value problem is solvable for general nonadmissible continuous (periodic) initial data.

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We study the visions for contour lines of surfaces when one looks at it from a distant view in some direction. The study of such a landscape (i.e. so-called "topography") is reduced to the study of a certain divergent diagram of the smooth mappings JR. +- M -+ JR.2 , where M is a smooth surface. We give a generic semi-local classification of such divergent diagrams.

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It is proven that the spin 1/2 Schrodinger operator iI with a con­stant magnetic field is the square of a sum of mutually strongly anti­commuting self-adjoint operators. As an application, by using this for­mula, the essential spectrum of iI with a vector potential in a class is identified. The class contains a vector potential to which Shigekawa's theorem (I. Shigekawa, J. Funct. Anal., 101:255-285, 1991) cannot be applied.

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We consider the Cauchy problem for the nonlinear Schrodinger equation in one space dimension with interaction satisfying null gauge condition. We prove the local well-posedness of the problem in the Sobolev space H1l2• The method depends on the nonlinear gauge transformation and on sharp smoothing estimates for the null gauge form.

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We develop an abstract theory on factorization of nonnegative self-adjoint operators in terms of abstract Dirac operators. As an application, we obtain a necessary and sufficient condition for a second quantization on the abstract Boson-Fermion Fock space to be the square of a Dirac type operator and a result on uniqueness of such Dirac type operators.

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In this note we review an infinitesimal approach to the stable coho­ mology of the moduli of compact Riemann surfaces by means of complex analytic Gel'fand-Fuks cohomology of open Riemann surfaces. Under a hypothesis (a certain kind of the Frobenius Reciprocity Laws) we prove the (p, q)-equivariant cohomology of the dressed moduli of compact Riemann surfaces coincides with the polynomial al­gebra generated by the Morita-Mumford classes en = K.n (n 􀀁 1) [Mo] [Mu] for p 􀀭 q. This suggests it is reasonable to conjecture that the stable cohomology algebra of the moduli of compact Riemann surfaces would be generated by the Morita-Mumford classes en's. For a more detailed description the reader is referred to [Kal,2,4,5].

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In one variable, an outer function has several important properties and a function with one of the properties is an outer function. In two variables, the situation is very different and we study these functions.

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We extend the theory of viscosity solutions for singular parabolic equa­tions including, for example, axisymmetrized level set equation for mean curvature flow equation. We establish a comparison principle for viscosity solutions of singular degenerate parabolic equations including such an equation . We discuss the rela­tion between axisymmetric viscosity solutions of original level set equation for mean curvature flow equation and the viscosity solution of axisymmetrized one.

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Let N C M be an irreducible inclusion of type type 111 factors with finite Jones index. We shall introduce the notion of normality for intermediate subfactors of the inclusion N C M. If the depth of N C M is 2, then an intermediate subfactor K for N C M is normal in N C M if and only if the depths of N C K and K C M are both 2. In particular, if M is the crossed product N ><l G of a finite group G, then K = N ><l His normal in NC M if and only if His a normal subgroup of G.

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We classify the bifurcation of generic local pictures of shadows for one­parameter families of surfaces in the Euclidean 3-space.

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Motion of curves by crystalline energy is often considered for "admissible" piecewise linear curves. This is because the evolution of such curves can be described by a simple system of ordinary differential equations. Recently, a generalized notion of solutions based on comparison principle is introduced by the authors. In this note we show that a classical admissible solution is always a generalized solution in our sense.

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In this review, we introduce an elliptic R-operator, which is a solution of the Yang­Baxter equation on some function space, and show the vertex-face correspondence for the eliiptic R-operator. As a result, the factorized L-operators for the elliptic R-operator are constructed. Moreover we explain that Belavin's R-matrix, the vertex-face correspondence and the factorized L-operators for it are produced from the elliptic R-operator.

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A two-dimensional quantum system of a charged particle interacting with a vector potential determined by the Weierstrass Zeta-function is considered. The position and the physical momentum operators give a representation of the canonical commutation relations (CCR) with two degrees of freedom. If the charge of the particle is not an integer (the case corresponding to the Aharonov-Bohm effect), then the representation is inequivalent to the Schrodinger representation. It is shown that the inequivalent representation induces infinite dimensional Hilbert space representations of the quantum group Uq(.s[2 ). Some properties of these representations of Uq(s[2 ) are investigated.

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We show that the first exit times of small random perturbations of dynamical systems from a bounded domain D(c Rd) weakly converge to the explosion time of an explosive diffusion process and that the mean first exit times converge to the mean explosion time, as random perturbations disappear, when they are appropriately scaled. We consider the case when D contains only one equilibrium point o of dynamical systems and when o is polynomially unstabe and is repulsive.

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Let D2 be the closed bidisc and T2 be its distinguished boundary. For (a, /3) E D2, let <I>ar, be a slice map, that is, (<I>ar,J)(>.) = f(a>., /3>.) for >. E D and f E H2(D2). Then ker<I>ar, is an invariant subspace, and it is not difficult to describe ker<I>ar, and M(ker<I>ar,) = {<PE L00(T2) : cpker<I>ar, C H2(D2)}. In this paper, we study the set M(M) of all multipliers for an invariant subspace M such that the common zero set of M contains that of ker<I>af3 ·

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A dynamical system with two stable equilibrium points will show a flip-flop motion between the neighborhoods of the two points when it is perturbed by small random noises. A typical example is the stochastic disk dynamo model where the two equilibrium points correspond to the two polarities of the earth's magnetic field. We will prove what has been suggested by a computer simulation, that is, the counting process of the flips or the reversals of the earth's field converges to a standard Poisson process if the time is suitably scaled.

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Let :E0,1 be an oriented compact surface of genus g with 1 boundary component, and r g,1 the mapping·class group of :Eg,1 · We determine the stable cohomology group of r 0,1 with coefficients in H1 (:Eg,1; Z)®n, n 􀀥 l, explicitly modulo the stable cohomology group with trivial coefficients. As a corollary the rational stable cohomology algebra of the semi-direct product r g,1 1>< H1 (:E0,1; Z) (which we call the extended mapping class group) is proved to be freely generated by the generalized Morita-Mumford classes in;J 's (i 􀀵 0, j 􀀵 1, i + j 􀀵 2) [Ka) over the rational stable cohomology algebra of the group r g,1 •

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We study affine invariants of plane curves from the view point of the singularity theory of smooth functions

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The existence and uniqueness of ground states of the spin-boson Hamiltonian HsB are considered. The main results in the case of massive bosons include: (i)(existence) there exists a ground state without restriction for the strength of the coupling constant ; (ii)(uniqueness) under a mild (nonperturbative) condition for the parameters contained in HsB, HsB has only one ground state; (iii) (degeneracy) under a certain condition for the parameters of HsB which is weaker than that of (ii), the number of the ground states is at most two. In the case of massless bosons, the existence of a ground state of H SB is shown as a limit of ground states of the massive case. The methods used are nonperturbative.

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We consider a representation of canonical commutation relations (CCR) appear­ing in a (non-Abelian) gauge theory on a non-simply connected region in the two­dimensional Euclidean space. A necessary and sufficient condition for the representa­tion to be equivalent to the Schrodinger representation of CCR is given in terms of Wilson loops. A representation inequivalent to the Schrodinger representation gives a mathematical expression for the (non-Abelian) Aharonov-Bohm effect. Some aspects of the Dirac-Weyl operator associated with the representation of CCR are discussed.

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For singular foliations defined by meromorphic functions on com­plex surfaces, we compute the Baum-Bott residues and describe the residue theorem. Applying these to the case of foliations arising from polynomials in two variables, we obtain various formulas including a generalization of a formula of Le and a "Milnor number formula" for a holomorphic map with non-reduced fibers.

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Passing from regular variation of a function f to regular variation of a Mellin convo­lution k* f with kernel k is an Abelian problem; its converse, under suitable Tauberian con­ditions, a Tauberian one. In either case, one has a comparison statement ( k * f)( x) / f ( x) → c (x → ∞), in which c is the Mellin transform k(p) of the kernel k at the index p of regular variation. Passing from a comparison statement to a regular-variation statement is a Merce­rian problem. The prototype results here are the Drasin-Shea theorem (fork non-negative) and Jordan's theorem (for kernels which may change sign). In each case, the Mellin trans­forms k( s) is assumed absolutely convergent in the relevant strip in the complex s-plane. We extend Jordan's theorem to cases where k( s) is only conditionally convergent, at the price of restricting from general to special (Fourier cosine and sine, and Hankel) kernels - and considerable extra complication in an already long proof. We need Korenblum's extension to the Wiener Tauberian theory, and results of Weber, Sonine and Gegenbauer on Bessel functions. 1 2

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This paper studies the Cauchy problem both at finite and infinite times for a class of nonlinear Schrodinger equations with coupling of derivative type. The proof uses gauge transformations which reduce the original equations to systems of equations without coupling of derivative type. Concerning the Cauchy problem at finite times, we give sufficient conditions for the global well-posedness in the energy space. Concerning the Cauchy problem at infinity, we construct modified wave operators on small and sufficiently· regular asymptotic states.

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This paper is a continuation of the paper [15] and [16]. Ginzburg-Landau equation with the magnetic effect is studied in the case of a superconductor which is not simply-connected. There are multiple stable steady state solutions, which correspond to many kinds of permanent current of electrons.

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This paper reviews the recent progress of the analytic theory for surface­energy-driven motion of curves when the interface energy is not smooth. The theory of viscosity solutions is extended for the nonsmooth energy including crystalline energy.

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We prove the existence of the solutions which converge in c0 to a har­monic map for an elliptic system depending on a large parameter.

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In this article we define and compute an index for a holomorphic vector field on a (possibly singular) subvariety of a complex manifold, provided the subvariety is a local complete intersection. This index reduces to the usual Poincare-Hopf index in case the subvariety is smooth, and is equal more generally to the index defined in [GSV] and [SS].

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We study singularities of solution surfaces of characteristic Cauchy problem for quasilinear first order partial differential equations as an application of the previous result on vector fields near a generic submanifold.

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The system of all the closure operators on a set V forms a lattice [3]. This lattice is isomorphic to the lattice of all the topped intersection structures on V. This paper discribes basic properties of these lattices, and gives a method of listing up all the members of these lattices.

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By unitary transformations (gauge transformation, Bogoliubov transformation) and the strong Trotter product formula, diamagnetic inequalities for the Pauli-Fierz model of QED and the Nelson model are derived. In the Nelson model, the unitary transfor­mation defines effective potentials. Moreover, the infimum of the spectrum of Hamilto­nians for these models are estimated and some abstract Kata's inequalities are obtained.

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We define a Rohlin property for one-parameter automorphism groups of unital simple C*-algebras and show that for such an automorphism group any cocycle is almost a coboundary. We apply the same method to the single automorphism case and show that if an automorphism of a unital simple C* -algebra with a certain condition has a central sequence of approximate eigen-unitaries for any complex number of modulus one, then any cocycle is almost a coboundary, or the automor­phism has the stability. We also show that if a one-parameter automorphism group of a unital separable purely infinite simple C* -algebra has the Rohlin property then the crossed product is simple and purely infinite.

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Simple characterizations of a pseudosphere or a plane in Minkowski 3-space by the existence of pseudocircles are given.

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Let :E9,1 be an orientable compact surface of genus g with 1 boundary component, and r g,1 the mapping class group of :E9,1. We define a bigraded series of cohomology classes mi,j E H2i+i-2 (f9,1 ;/\/ H1 (:E9,1;Z)), 2i+j-2 ;:=: 1, i,j ;:=: 0. When j = 0, the class mi+1 ,o is the i-th Morita-Mumford class [Mo][Mu]. It is proved that Hr (r9,1;/\8 H1 (:E9,1;Q)) is generated by m;,j's for the case r + s = 2 and the case g ;:=: 5 and (r, s) = (1, 3). Especially the Johnson homomorphism extended to the whole mapping class group by Morita [Mo3] has an implicit representation by the classes mo,3 and mo,2m1,1 over Q.

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This article presents functional integral representations for the heat semigroups with the infinitesimal generators given by self-adjoint Hamiltonians describing an interaction of a. non-relativistic charged particle and a quantized radiation field in the Coulomb gauge without the dipole approximation. Special attention is paid to definition of the "time-ordered Hilbert space-valued stochastic integrals associated with a family of isometries from a Hilbert space to another one" and semigroup techniques. Some inequalities are derived, which are infinite degree versions of those known for finite dimensional Schrodinger operators with classical vector potentials.

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We give examples of bounded solutions whose gradient blow up in a finite time but it stays bounded on the boundary for a class of quasilinear parabolic equations with zero boundary data. The method reflects a geometric argument for curve evolution equations.

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Cellular automata induced by quantization of coupled map lattice are investigated. The following results were obtained by the numerical experiments. The 29 states on each cell is sufficient for equivalent dy­namics, and at least 26 states are necessary for qualitative dynamics. We also develop a theory to assure the conditions.

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For an automorphism a of a UHF algebra it is shown that a has the Rohlin property if and only if am is uniformly outer for any m =/ 0. It is also shown that the automorphisms of a UHF algebra with the Rohlin property are outer conjugate with each other.

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The author has proved that an outer function in the Hardy space can be factored into a product in which one factor is strongly outer and the other is the sum of two inner functions. In an endeavor to understand better the latter factor, we introduce a class of functions containing sums of inner functions as a special case . Using it, we describe the solutions of extremal problems in the Hardy spaces HP for 1 ::; p < ∞.

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For a self-adjoint extension of the Dirac-Weyl operator with a singular vector potential, the strong coupling limit of the zero-energy-state density (ZESD) is computed. The result shows that a limit theorem of L.Erdos ( Lett.Math.Phys.29:219-240,1993) con­cerning the ZESD of the Dirac-Weyl operator with a "regular" magnetic field does not hold in the present singular case.

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Let µ be a finite positive Borel measure on the closed unit disc D. For each a in D, put S(a) = inf k If IP dµ where f ranges over all analytic polynomials with f(a) = 1. This upper semicontinu­ous function S(a) is called a Riesz's function and studied in detail. Moreover several applications are given to weighted Bergman and Hardy spaces.

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We discuss Gaussian generalized random fields indexed by smooth sec­tions of vector bundles with respect to Markov properties. We propose a new set-up which is suitable for the present question and within which new phenomena are de­tected naturally. In particular, we give a counterexample to the belief that locality in the RKHS implies germ Markov property. We also prove the close connection between Markov property and cokernels of local operators. Furthermore we prove the Markov property for a very degenerate Gaussian random field.

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We consider small random perturbation$ of dynamical systems {Xe(t, x )} t􀀞o,xeRd (e > 0) on a cl-dimensional Euclidean space Rd when the origin o E Rd is a repulsive equilibrium point of unperturbed dynamical systems and when o is not exponentially unstable. The weak law of large numbers, as e 􀀫 0, on the first exit time r1,(x) of {Xe(t, x )}t􀀞o from a bounded domain D(􀀯 o) of Rd and the asymptotic behavior of E[r1,(x)], as e 􀀫 0, are given.

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A family of strongly commuting self-adjoint operators associated with some objects in the d-dimensional Minkowski space is introduced and operator calculi concerning these self-adjoint operators and the canonical momentum operator p = (p0, PI, …, Pd-I) are developed. It is shown that a class of unitary transformations of Pﾂｵ is given by a class of operator-valued Lorentz transformations of perturbed Pﾂｵ 's. Moreover, the integral kernels of the unitary groups of perturbed d' Alembertians are explicitly computed. As an application, a detailed analysis of the quantum theory of a charged spinless relativistic particle in an external electromagnetic field is given. The present analysis clarifies a genﾂｭeral mathematical structure behind Schwinger's proper-time method for the external field problem in quantum field theory.

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Let Δ be a simplicial complex of dimension d - 1 on the vertex set V and write Ei for the number of "minimal" subsets er C V with U(er) = i + 1 and er <t Δ We discuss what can be said about the combinatorial sequence (E1, E2 , •• ,Ed) associated with Δ

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Let P( v, d) be a stacked d-polytope with v vertices, .6.(P( v, d)) the boundary complex of P( v, d), and k[.6.(P( v, d))] = A/ IA(P(v,d)) the Stanley-Reisner ring of .6.(P( v, d)) over a field k. We comﾂｭpute the Betti numbers which appear in a minimal free resolution of k[.6.(P(v,d))] over A, and show that every Betti number depends only on v and d and is independent of the base field k.

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We give a combinatorial formula for the Betti numbers which ap­pear in a minimal free resolution of the Stanley-Reisner ring k[􀀁(P)]= A/ h('P) of the boundary complex 􀀁(P) of an odd-dimensional cyclic polytope P over a field k. A corollary to the formula is that the Betti number sequence of k[􀀁(P)] is unimodal and does not depend on the base field k.

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We study the Betti numbers which appear in a minimal free res­olution of the Stanley-Reisner ring k[􀀃] = A/ft:. of a simplicial com­plex 􀀃 over a field k. It is known that the second Betti number of k[􀀃] is independent of the base field k. We show that, when the ideal ft:. is generated by square-free monomials of degree two, the third and fourth Betti numbers are also independent of k. On the other hand, we prove that, if the goemetric realization of􀀃 is homeomorphic to either the 3-sphere or the 3-ball, then all the Betti numbers of k[􀀃] are independent of the base field k.

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Simple characterizations of a pseudosphere or a time-like plane in Minkowski 3-space by the existence of light-like lines are given.

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The Ginzburg Landau equation with a large parameter is studied in a bounded domain with the Neumann B.C. It is shown that if n = 2 or n = 3 and the domain is not simply connected, many kinds of non-constant stable equilibrium solutions exist.

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A quantum system of a particle interacting with a (non-Abelian) gauge field on the non­simply-connected domain M = R2 \ { an}􀀢=l is considered, where an, n = 1, · · · , N, are fixed isolated points in R 2• The gauge potential A of the gauge field is a p x p anti-Hermitian matrix-valued 1-form on Mand may be strongly singular at the points an, n = 1, · · · , N. If A is fl.at, then the (non-canonical) momentum and the position operators {Pi, qi lJ= 1 of the particle satisfy the canonical commutation relations (CCR) with two degrees of freedom on a suitable dense domain of the Hilbert space L2(R2; CP). A necessary and sufficient condition for this representation to be the Schrodinger 2-system is given in terms of the Wilson loops of the rectangles not intersecting an, n = 1, · · · , N. This gives also a characterization for the representaion to be non-Schrodinger. It is proven that, for a class of gauge potentials, which is not necessarily fl.at, Pi is essentially self-adjoint. Moreover, an example, which gives a class of non-Schrodinger represenations of the CCR with two degrees of freedom, is discussed in some detail.

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Operator-theoretical analysis is made on ( unbounded) representations, in Hilbert spaces, of a supersymmetry (SUSY) algebra coming from a supersymmetric quantum field theory in two-dimensional space-time. A basic idea for the analysis is to apply the theory of strongly anticommuting self-adjoint operators. A theorem on integrability of a representation of the SUSY algebra is established. 1foreover, it is shown that strong anticommutativity of self-adjoint operators is a natural and suitable concept in analyzing representations of the SUSY algebra in Hilbert spaces.

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This paper reviews the theory of generalized solutions by the level set method for the curve shortening equation in a domain subject to boundary conditions. An exam­ple of fattening is provided for the curve shortening equation with right angle boundary condition. The example provides infinitely many reasonable solutions after onset of singu­larities.

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We give a sufficient condition for the uniqueness of solutions to Cauchy problem for a class of integro-partial differential operators related to the stochastic analysis, by way of the large deviations technique.

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We study the Cauchy problem for the equations describing the resonant interactions between long and short waves. Global well-posedness of the problem is proved in the space H1 12 x £2 , the first and second comp on en t of which correspond to the short and long waves, respectively.

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We consider the Cauchy problem for the nonlinear Schrodinger equation with interaction described by the integral of the intensity with respect to one direction in two space dimensions. Concerning the problem with finite initial time, we prove the global well-posedness in the largest space L2(!R2 ). Concerning the problem with infinite initial time, we prove the existence of modified wave operators on a dense set of small and sufficiently regular asymptotic states.

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We prove an Abel-Tauberian theorem for Fourier sine transforms. It can be considered as the analogue of the Abel-Tauber theorem of Pitman in the boundary case. We apply it to Fourier sine series as well as to the tail behavior of a probability distribution.

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We define the concept of the copula field, and give the applications to the stochastic quantization and to the stochastic control.

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We discuss Gaussian generalized random fields indexed by smooth sec­tions of vector bundles with respect to Markov properties. We propose a new set-up which is suitable for the present question and within which new phenomena are de­tected naturally. In particular, we give a counterexample to the belief that locality in the RKHS implies germ Markov property. We also prove the close connection between Markov property and cokernels of local operators.

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A difference scheme is introduced for computing the motion of level surfaces moved by the mean curvature. This scheme is proved to be stable in the maximum norm so that the computation can be completed without over-flow.

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Trace formulas for the heat semi-groups of second quantization operators and their perturbations in the abstract Boson Fock space are given in terms of path (functional) integral representations. As applications, an inequality of Golden-Thompson type and a classical Hmit are derived for the trace of the heat semi-group of a perturbed second quatizatidn operator. The abstract results are applied to a model of P( </> )-type in quantum field theory.

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The mutual information of two random variables plays fundamental roles in many areas of applied mathematics. This quantity can be generalized to finite sets of random variables by (1). In contrast to the non-negativity of the usual mutual information, the triple mutual information can be negative as well as positive and its sign gives us a rough indication of the mode of mutual dependency among three random variables. The purpose of this note is to determine the range of the triple mutual information and to exarnin when the extremals are attained.

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Let {Xn}􀁈1 be a Eyraud-Farlie-Gumbel-Morgenstern process. Put Sn = E􀀅=l Xk. In this paper we prove the large deviations theorem for Bn/n, and the central limit theorem for Bn/n112 , as n - ∞.

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We prove Tauberian theorems for Fourier cosine transforms, which can be consid­ered as analogues of the theorem of Soni and Soni for the boundary cases. They involve both II-variation and improper integrals.

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In this paper we study smoothing effects of solutions to the Benjamin-Ono equation (BO) 8tu + u8zu + H8􀀌u = O, u(O) = </J, :z: E IR, (t,:z:) E IR X IR, where H is the Hilbert transform defined by (Hf)(z) == p.v.-1 / f--dy) y. We prove that if </J E H4 and (z8z )4</J, then the solution u of (BO) belongs to Lf:c(JR\{O};H8 •-4), where

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Let v and µ be finite positive measures on the open unit disk D. We say that v and µ satisfy the ( v ,µ )-Carleson inequality, if there is a constant C > 0 such that 111 dv 􀀊 C 􀀇 D 111 dµ for all analytic polynomials 1 . In this paper, we study the necessary and sufficient condition for the ( v , µ )-Carleson inequality. We establish it when v or µ is an absolutely continuous measure with respect to the Lebesgue area measure which satisfies the (A )-condition. Moreover, many concrete examples of such measures are given.

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We reexamine the mechanism of smoothing effects of the Schrodinger evolution group in the weighted Sobolev spaces by using the generator of space-time dilations instead of Galilei transformations.

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We prove an above and a below differential inequality for the solutions of semilinear heat equation.From them,we obtain the estimates of critical initial date,which guarantee that the solution will blow up or be bounded globally.We also get the estimates of the growth rate of blowing up solution and the blowing up time.

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In this paper, we are concerned with the existence of periodic solutions of a quasilinear parabolic equation

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We study the Betti numbers which appear in a minimal free res­olution of the Stanley-Reisner ring k[.6.] of a Buchsbaum complex .6. and find a characterization of Buchsbaum complexes .6. for which a minimal free resolution of k[.6.] is linear.

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A characterization of a sharp form of Trudinger's inequality is established in terms of the Gagliardo-Nirenberg inequality in the limiting case for Sobolev's imbeddings.

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We consider small random perturbations of dynamical systems {Xe (t)}o9 (0 < c) on a cl-dimensional Euclidean space Rd when the origin o E Rd is a hyperbolic equilibrium point of unperturbed dynamical systems. The first exit time Tv of {X£ (t)}o9 from a bounded domain D(3 o) of Rd obeys the large deviations phenominon with a variable decay rate.

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We present three curious phenomena in coupled chaotic systems with different time scales: a copy, an itinerant motion, and 'firework'. The phenomena obtained may provide possible implication of interacting two macroscopic systems, namely 'observing' and 'observed' systems.

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This paper proves the global existence of small radially symmetric solu­tions to the nonlinear Schrodinger equations of the form { i8,u + ½.6.u = F(u, Vu, ii, Vu), (t, :z:) nE 􀀇 x 􀀇n, u(O, :z:) = eo¢(1:z:I), :z: E 􀀇 , where n >= 3, eo is sufficiently small, l:z:I = ( E :z:])1 12 , L >-au°'lu°'2 lo$lal91 +L >.a13u°'lu°'2( L l8;ul2)131( L (8;u)2)132 with >.a,>..013 E C, l1,l2,l3 E N, lo = 3 for n = 3,4, and lo = 2 for n 􀀠 5. The method depends on the combination of a gauge transformation and generalized energy estimtes and does not require the condition such that 8vu F is pure imaginary which is needed for the classical energy method.

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It is shown that every two dimensional representation of a uniform algebra has a dilation, which extends the result by Paulsen [6]. We also prove some dilation result for a representation of the disk algebra.

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We introduce a new type of stochastic dynamics as stochastic renewal of maps, relating to the neurodynamics of cortical memory process. This stochastic dynamics can be reformulated by a skew product transformation of two kinds of variables, one of which describes an underlying dynamical system and the other describes chaotic dynamics, say, Bernoulli shift. The feature of orbits in phase space is investigated in the particular case of neurodynamics model for cortical chaotic memories. A new computational result on the functional role of cortical chaos is obtained. We also present a neurobiological interpretation of psychological perception and memories by means of the notion of chaotic itinerancy.

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We apply Thom-Mather theory to the diagram of smooth map germs associated to first order partial differential equations. This reduces the problem of function moduli of infinite dimension for PDE as well as divergent diagrams of map germs to the theory of deformation of functions on singular varieties.

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The Ginzburg Landau equations in a rotational domain in l13 are studied. Rotational solutions are constructed and proved to be stable by the spectral analysis on the linearized equation.

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We give a combinatorial classification of Cohen-Macaulay partially ordered sets P for which a minimal free resolution of the Stanley­Reisner ring k[D.(P)] of the order complex D.(P) of P is pure.

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A general parabolic evolution equation is considered for a closed hy­persurface in Euclidean space. All stationary solutions are shown to be Lyapunov unstable if the normal velocity of a hypersurface depends only on its normal and sec­ond fundamental form and is independent of its position. Instability of time periodic solution is also discussed.

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Evolution equations of curvatures of convex curves are considered by the Gauss map parametrization. A time periodic unstable solution is constructed for a reason­able class of time periodic data. Our solution is arranged to satisfy a constraint so that it yields closed, embedded, convex curves moving periodically in time (up to . translation) .whose normal speed equals the curvature minus a given time periodic function depending on curves only through its normals. For curvatures of periodically evolving curves a priori lower and upper bounds depending only on periodic data are obtained. A new penalty method is introduced so that our solution satisfies the constraint. Solutions of penalized equations are constructed by adapting the degree theory.

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Let A and B be two ant1commut1ng self-adjoint operators and V ( 11,) be a sym­metric operator in a Hilbert space, where 11, > 0 is a parameter. We prove that, under some conditions for V ( K), the resolvents of KA + 11,2 i3 ± 11,2 I BI + V ( K) converge as K -+ oo. This solves in a most general form the nonrelativistk-limit problem of Dirac operators.

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It is shown that a class of Dirac operators acting in the abstract Boson-Fermion Fock space, which were introduced in a previous paper (A.Arai, J.Funct.Anal. 105(1992), 342-408), is essentially self-adjoint. The result is applied to the Wess-Zumino models of supersymmet­ric quantum field theory to prove the essential self-adjointness of their supercharges and Hamiltonians.

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The Betti numbers dimk.ToriA(k[􀀁],k) with i > v - d of the Stanley-Reisner ring k[􀀁] = A/I􀀂 of a Buchsbaum complex 􀀁 of dimension d - 1 ouer a field k with v uertices are studied.

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We study overdetermined systems of first order partial differential equations with singular solutions: The main result gives a characterization of such systems and asserts that the singular solution is equal to the contact singular set. vVe can also give a local normal form of such systems up to contact diffeomorphism.

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We give a survey of some recent results on bt·(k[6]) = dimk Torf(k[6], k) of the face ring (Stanley-Reisner ring) k[6] of a simplicial complex 6. In particular, we study Cohen-Macaulay types and canonical modules of partially ordered sets.

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Jerzy vVierzbicki Abstract. vVe show that for a triple I{ C N C M of type II1 factors the depth of the inclusion "K C M" is not greater than the maximum of depths of the inclusions" K C N" and" N CM", provided there is such a factor P, that the P C lvl diagram U U is commuting and co-commuting square ( or a non-degenerate K C N commuting square) oftype II1 factors. vVe give also a characterization of the above condition.

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A new characterization of anticommutativity of (unbounded) self-adjoint operators is presented in connection with Clifford algebra. Some consequences of the characterization and applications are discussed.

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Quantization of angle-variables in the classical Hamilton mechanics with one degree of freedom is considered in a mathematically rigorous way. The method taken in this paper is that of the Weyl quantization, so that the quantized angles are given by pseudo-differential operators of the Weyl type which may be singular. Some operator-theoretical aspects of the quantized angles are discussed. It is shown that the relation between a classical Hamiltonian and an angle-variable of it, which is given by a Poisson bracket relation, is not preserved in general to the quantized version where the Poisson bracket relation is replaced by a commutation relation; an "anomaly" may occur in the commutation relation of the quantized Hamiltonian with the quantized angle. The anomaly may be regarded as a quantum effect. Special attention is paid to the case of the harmonic oscillator. It is proven that the quantized angle eh of the harmonic oscillator, where 1i > 0 is a parameter denoting the Planck constant divided by 211", is represented as an integral operator of the Carleman type. The integral operator representation enables one to expand eh as a power series of n-1, which describes the aymptotic behavior of eh as 1i -+ oo. Also it is shown that the classical limit 1i-+ 0 of eh is given by a modified Hilbert transform.

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Abstract. For a class of long range potentials, sharp propagation estimates of the corre­sponding Schrodinger evolution groups are obtained without low-energy cut-off technique. Instead of low-energy cut-off, an explicit condition is given on the vanishing order in the L sense at zero energy of initial states.

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Some useful and remarkable property are derived from a representa­tion formula of a radially symmetric solution to the Cauchy problem for a homogeneous wave equation in odd space dimensions. These prop­erty provide us with enough information to consider the semilinear case, namely, the associated integral equation with the problem will be con­sidered on a weighted L00-space. This formulation enable us to deal with the problem for slowly decaying initial data.

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In this paper we study the global existence and asymptotic behavior of solutions for the Zakharov equations in three space dimensions with the final data given at t = +oo. This leads to the construction of the wave operator for certain scattered data. It is also shown that the initial value problem of the Zakharov equations in three space dimensions has the global solutions in time for a certain class of initial data, which includes not only small data but also some large data.

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This article gives a solution to an open problem in the paper: A.Arai; J.Math.Phys.31, 2653(1990). We present an abstract asymptotic theory of families of unitary operators {U(11:)},.>o and self-adjoint operators {H,.},.>o acting in the tensor product of two Hilbert spaces. We prove that H;en (V, 11: ), which represents a scaled total Hamiltonian of a coupled system of a one elec­tron atom and a quantized radiation :field, with parameters O 􀀖 e 􀀖 1, 11: > 0, and the electron mass renormlized, is unitarily equivalent to an operator fI;en (V, 11:), which can be regarded as a decoupled Hamiltonian. Applying the abstract asymptotic theory and the unitary equiva­lence, we prove that the resolvent of H;en (v, 11:) strongly converges as 11: -+ oo to an operator which de:fines an effective potential of the electron. We compare the effective potential with that obtained in the paper mentioned above.

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This paper is concerned with the initial value problem for nonlinear Schrodinger equations of the form where 8 = 8z = 8 / 8:i:1 ..\, A1, ..\2 E Iffi., and 2 ;£ Pl < P2 < 5. It is shown that if q> E H1(Iffi.) and then there exists a unique global solution 'Ip of ( *) such that In this paper we introduce a new method to obtain the result. Key words. Derivative Nonlinear Schrodinger Equations, Gauge Transformations. AMS(MOS) subject classifications. 35Q55, 35Q60.

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For any nonzero invariant subspace M in H2 (T2), set M x = [ U z nM] n [ U w nM] then Mx is also an invariant subspace of H2 (T2) that contains M .. If M is of finite codimension in H2 (T2 ) then M>< = H2 (T2 ) and if M = qH2 (T2 ) for some inner function q then M>< = M. In this paper invariant subspaces with M:>< =Mare studied. If M = q1H2 (T2 )nq2 H2 (T2 ) and q1, q2 are inner functions then Mx = M. However in general this invariant subspace may not be of the form : qH2 (T2 ) for some inner function q. Put /ll(M) = {¢<: L00 :¢M􀀌 H2 (T2 )} then hz(M) is described and rJZ(M) = )n.(M'J< ) is shown. This is the set of all multipliers of Min the title. A necessary and sufficient condition for /11(M) = t0(T2 ) is given. It is noted that the kernel of a Hankel operator is an invariant subspace M with Mx = M. The argument applies to the polydisc case.

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Schrodinger operators with time-dependent potentials are studied. Nec­essary and sufficient conditions for existence of ordinary and Dollard-type modified wave operators are obtained. Sharp results for potentials with a specified leading term are obtained. Applications are given to the surfboard Schrodinger equation and to Stark Hamiltonians. In the latter case the discrepancy between classical and quantum scattering in dimension one is resolved.

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We prove the unitary equivalence between the Dirac Hamiltonian Hv for a rela­tivistic spin 1/2 neutral particle with an anomalous magnetic moment in a two-dimensional electrostatic field E = (E1 , E2 ) and the direct sum of the Dirac-Weyl operators D(±A) for a spin 1/2 charged particle in two-dimensional magnetic fields ±dA with the vector potential A = E2 dx1 - E1 dx2 , ( x1 , x2 ) E IR2• As applications, we investigate the ground state and the spectra of Hv .

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We reduce some normal forms, which have functional moduli, for a certain class of mapping diagrams. This class is associated to completely integrable first order ordi­nary differential equations. The reduction is given relative to the equivalence relation among the differential equations under the group of point transformations in the sense of S. Lie.

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An asymptotic Hankel opera tor on the Lebesgue space L 2 is defined and a distance formula is obtained. From it, two distance formulas on the Hardy space H 2 follow. One of them is known and concerned with Hankel operators. Another one is new and related with a commutator ideal of Toeplitz operators.

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We investigate the ground state structure of the Schrodinger operator (Pauli Hamiltonian) H with a magnetic field b for a spin 1/2 particle in 􀀈2d ,...., Cd . We consider the case where bis given by the complex exterior derivative of a function W on Cd of the form = i( 8 + 8)( 8 8)W. We found that dim ker H is related to the asymptotic behavior of W at infinity. More precisely, if there exists a constant C E 􀀈 such that W ( z) rv -C log I z I as z -, oo, then dim ker H is equal to the number of all monomials f in d variables such that the deg;ree off is smaller than ICI - d. Moreover we clarify the structure of ker H.

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It is known that the N =2 Wess-Zumino supersymmetric quantum mechanical model with the superpotential V(z) = >.eaz (>. E C \ {O}, a > 0) has infinitely many bosonic zero­energy ground states and no fermionic zero-energy ground states [A. Arai, J. Math. Phys. 30 (1989), 1164]. In this paper, we extend these results to a more general model. The main results include the following: (1) identification of the spectra of the Hamiltonian H of the model; (2) non-Fredholmness of a supercharge of the model, which is a Dirac type operator; (3) existence of infinitely many bosonic zero-energy states of H; ( 4) non-existence of fermionic zero-energy states of H.

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The symbols of invertible Toeplitz operators from 0 (wd0 /2-a) to L ° (wd0 /2-a )/e-'9 H0 (wd0 /2-a) are described completely where H° (wd0/2rr) denotes a weighted Hardy space. The result is strongly related with a weighted norm inequality. If the weight w satisfies the con- dition (Ao) then L 0 (wd0/2rr)/e- 18 H0 (wd0/2rr) = H° (wd0/2rr) with equiva- lent norms.

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The present paper studies the lifespan of solutions to quasi-linear wave equations in two space dimensions. We shall show a lower bound for the lifespan. We shall also show that if the non-linear term satisfies "null­condition", the equations have global solutions. Our basic idea is to solve ordinary differential equations which are constructed from wave equations.

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A characterization of systems of first order differential equations with (classical) complete solutions is given. Systems with affine complete solutions are also charac­terized.

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Asymptotic expansions of solutions of the wave equations in even dimensional spaces are obtained with the initial data of non-compact support. A relationship is proved between the vanishing order at the origin of the Fourier transform of the data and the decay rate of the corresponding solutions in semi-infinite cylinders or along rays inside the forward light cone.

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We discuss the unfolding and determinacy theories for codimension one complex analytic foliation germs. The first order unfoldings of such a germ are clas­sified and the versality theorem which asserts that an infinitesimally versa] unfolding is versa] is stated together with a proof and various applications. Some types of finite determinacy results are also explained.

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We give a characterization of the notion of complete integrability for overdeter­mined systems of first order partial differential equations of real valued functions.

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For a non-zero function f in H1 , the classical Hardy space on the unit circle, put S11111 = {g E H1: Jlglli = 1, argl(eit) = argg(eit) a.e.t },then S1!1/ 1 is the set of extremal functions of a well known linear extremal problem in H1 . It is known and easy to see that if 1-1 belongs to H1 then the dimension of < SIJl/f > , the linear span of S11111, is one. A simple example shows that even if 1-1 belongs to HP for some p (0 < p < l), the dimension of < SIJI/ 1 > may be infinite. On the other hand, a sophisticated example ( will be shown in this paper) shows that even if 1-1 locally belongs to H1 on the unit circle except a finite set, the dimension of < S1!1/ 1 > may be infinite. In this paper it is shown that if I E H1 has the properties such that 1-1 locally belongs to H1 on the unit circle except a finite set and that 1-1 E HP for some p > 0, then the dimension of< SIJl/f > is finite.

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Geometric evolutions of curves represented by graphs are studied when the interface energy is not necessarily smooth. The theory of nonlinear semigroups yields a unique global solution even if the initial curve is not necessarily admissible. Usual evolution law for the crystalline interface energy is justified.

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We consider the global Cauchy problem for the nonlinear Schrodinger equations with quadratic nonlinearities. Concerning a class of nonlinearities which ensures the global existence of solutions, we prove that Klainerman's condition and Hayashi's condition are equivalent. An explicit solution to Strauss' equation is also given.

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We compute Cannes-Stormer relative entropy H(MIN) for two subfactors Mand N of the type II1 factor without assuming N C M. If they form a commuting square, then we have H(MIN) = H(MjMnN). If their commutants form a commuting square, then we have H(MIN) = H(M V NIN).

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We prove a comparison theorem for viscosity solutions of singular degenerate parabolic equations with the Neumann boundary condition on a domain not necessarily convex. Our result applies to various level set equations including the Neumann problem for the mean curvature flow equations where every level set of solutions moves by its mean curvature and perpendicularly intersects the boundary of the domain.

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For general surfaces of finite type, probability measures for parallel transport are con­structed. Relations to the topology of the surface are pointed out. We also discuss possible loop invariants.

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Infinite dimensional analysis is developed on an abstract Boson-Fermion Fock space. A general class of Dirac operators acting there is introduced and properties of them are inves­tigated. An index theorem for the Dirac operators is established in terms of a path integral on ,a loop space. It is shown that the abstract formalism presented here gives a mathematical unification for some models of supersymmetric quantum field theory.

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Considered is a quantum system of a charged particle moving in the plane R 2 under the influence of a perpendicular magnetic field concentrated on some fixed isolated points in R 2• Such a magnetic field is represented as a finite linear combination of the two­dimensional Dirac delta distributions and their derivatives, so that the gauge potential of the magnetic field also may be strongly singular at those isolated points. Properties of the Dirac-Weyl operator with such a singular gauge potential are investigated. It is seen that some of them depend on whether the magnetic flux is locally quantized or not. Particular attention is paid to the zero-energy state. For each of self-adjoint realizations of the Dirac­Weyl operator, the number of the zero-energy states is computed. It is shown that, in the present case, a theorem of Aharonov and Casher [Phys.Rev. A 19, 2461(1979)], which relates the total magnetic flux to the number of zero-energy states, does not hold. It is also proven that the spectrum of every self-adjoint extension of the minimal Dirac-Weyl operator is equal to R.

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We say that a Cohen-Macaulay poset (partially ordered set) is "superior" if euery open intenrnl ( x , y) of P "' with µp.-.(x,y) :it: 0 is doubly Cohen-Macaulay. For e2-rnmple, if L = P "' is a modular lattice, then the Cohen-Macaulay poset P is superior. We present a formula for the computation of the Cohen-Macaulay type of the Stanley-Reisner ring of the order compleH of a Cohen-Macaulay poset which is superior.

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A formula for the Euler number of a generic singular surface in a 3-manifold is given. This formula not only unifies the previous results but also allows some new applications.

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Commutation properties of two-dimensional momentum operators with gauge potentials are investigated. A notion of local quantization of magnetic flux is introduced to charac­terize physically the strong commutativity of the momentum operators. In terms of the notion, a necessary and sufficient condition is given for the position and the momentum operators to be equivalent to the Schrodinger representation of the canonical commutation relations.

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We prove a comparison theorem for viscosity solutions of degenerate parabolic equations which is singular at finite directions of derivatives. We apply our theorem to construct a global generalized evolution for interfaces equations with a certain class of the interface energy not necessarily 02 •

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We present a new form of Trudinger type inequality, which shows an explicit dependence of functions in the Sobolev space of critical order. Moreover, we give a proof of the Brezis-Gallouet-Wainger inequality which is independent of the Fourier representation, thereby arriving at a solution to Brezis' problem.

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Let N be a irreducible subfactor of a type I factor M. If Jones index [M:NJ is finite, then the set fa.t(NcM) of the intermediate subfactors for the inclusion N c M form a finite lattice. The (co-) commuting square conditions for intermediate subfactors are related with the modular identity in the lattice fai(NcN). In particular simplicity of a finite group G is characterized in terms of (co-) commuting square conditions of intermediate subfactors for N c M = N ><I G.

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A global-in-time weak solution of the nonstationary two-phase Stokes flow is constructed for arbitrary given initial domains (under periodic boundary condition) when two viscosities are close. Our solution tracks the evolution of the interface after it develops singularities. The theory of viscosity solutions is adapted to solve the interface equation. Surface tension effects are ignored here.

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We consider the initial value problem for nonlinear Schodinger equations :｛i8tu+ ½82u = F(u,8u,'ii,8'ii), (t,:z:) E ]R+ X IR, u(O,:z:)=uo(:z:), :z:EIR,｝where 8 = 82 = 8/8:z:, F : C4 -+ C is a polynomial having no constant nor linear terms. Without smallness condition on the data uo, it is shown that (t) have a unique local solution in time if uo is in H3•0 n H2•1 , where Hm,• = {/ E S'; 11/llm,a = 11(1 + :z:2 )½(1- 􀀸)Zf' /ll2 < oo}, m,s E !R.

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The symbols of hyponormal Toeplitz operators are completely described and those are also studied, being related with the extremal problems of Hardy spaces. Moreover we discuss the Halmos's question about a subnormal Toeplitz operator when the selfcornmutator is finite rank.

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We are concerned with the deformation theorem in the geometric mea­sure theory. We shall prove the theorem for "weighted" mass to know its essence and to apply it to some variational problem. Our basic idea is to devide the space into suitable cubes on which we can treat the weighted mass as usual one.

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An interpolation inequality for the total variation of the gradient of a com­posite function has been derived by applying the coarea formula. A bound for the pressure integral has been studied by establishing an a priori estimate for a solution of the Grad-Shafranov equation of plasma equilibrium. A weak formulation of the Grad­Shafranov equation has been given to include singular current profiles.

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We give a relation between Euler characteristics of a generic closed Legendrian surface and its wavefront.

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A duality representation of a measure f ( z, µ) for a finite dimensional vector valued Radon measureµ is established ior a continuous function/= /(z,p) which is convex and of linear growth in p. Besides technical new aspects of the result the proof given here is simpler than those based on the theory of convex functionals. As an application, a new proof for a representation /(z, Vu) by the graph of u is given when u is a mapping of bounded variation.

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Two types of fundamental spaces of differential forms on infinite dimensional topological vector spaces are considered; one is a fundamental space of Hida's type and the other is one of Malliavin's. It is proven that the former space is smaller than the latter. Moreover, it is shown that, under some conditions, the fundamental space of Hida's type is nuclear as a complete countably normed space, while that of Malliavin's in the L2 sense is not.

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It is given a classification of generic vector fields near a generic submanifold. The normal forms are linear vector fields near the local model of the submanifold. Similar results are obtained for vector fields near a hypersurface with boundary and near a piecewise-smooth hypersurface.

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We study classifications of holonomic systems of first order differential equations for one real valued functions by the equivalence relation under the group of point transformations in the sense of Sophus Lie. In order to pursue the classification, we introduce the notion of one-parameter Legendrian unfoldings which induces a special class of divergent diagrams of map-germs which are called integral diagrams. Our normal forms are represented by integral diagrams.

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We discuss two topics related with combinatorial study of canonical modules of Stanley-Reisner rings, viz., (i) some linear Inequalities on the number of faces of a matrold compleH and (ii) a formula to compute the Cohen-Macaulay type of the Stanley-Reisner ring of ,a finite distributive lattice.

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We consider some properties about completely integrable first order differential equations for real-valued functions. In order to study this subject, we introduce the theory of Legendrian unfoldings. One of our theorems asserts that the set of equations with singular solution is an open set in the space of completely integrable equations even though such a set is thin in the space of all equations.

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An operator theoretical analysis is made on the representation of the su­persymmetry(SUSY) algebra originating from the relativistic supersymmetry in the two­ dimensional space-time with a proper treatment of anticommutatvity and commutativity of operators in the SUSY algebra. It is shown that, given on a Hilbert space .fj two self­adjoint supercharges anticommuting in the strong sense, one can construct a representation of the SUSY algebra on .fj such that the supersymmetric Hamiltonian and the momentum operator commute in the strong sense. As an application, a general class of representations of the SUSY algebra is constructed on an abstract Boson-Fermion Fock space.

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It is proven that, for every pair {A, B} of anticommuting self-adjoint operators, iAB is essntially self-adjoint on a suitable domain and its closure O(A, B) anticommutes with A and B. For every self-adjoint opearlor S, a partial isometry Us is defined by the polar decomposition S = Us lSI. Let Ps be the orthogonal projection onto (Ker S)l. . The commutation properties of' the operators UA, UB, Uc(A,B), PA , PB , and PAPB are investigated. These operators multiplied by some constants satisfy a set of' commutation, relations, which may be regarded as an extension of that satisfied by the standard basis of the Lie algebra .au(2, C) of' the special unitary group SU(2). It is shown that there exists a Lie algebra ro? associated with those operators and that, if' A and B are injective, then ro? gives a completely reducible representation of su(2, C) with the heighest weight of' each irreducible component being 1/2. Moreover, the "diagonalization" of' A+ B is given.

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Let M be an invariant subspace of L2 (T2) on the bidisc. v1 and v2 denote the multiplication operators on M by coordinate functions z and w, respectively. In this paper we study the relation between M and the commutator of and V2. For example, M is studied when the commutator is self-adjoint or finite rank.