Separation of Variables in the Anisotropic Shottky–Frahm Model

We construct separated coordinates for the completely anisotropic Shottky–Frahm model on an arbitrary coadjoint orbit of SO(4). We find explicit reconstruction formulas expressing dynamical variables in terms of the separation coordinates and write the equations of motion in the Abel-type form.     

Continuous inclusions and Bergman type operators in n-harmonic mixed norm spaces on the polydisc

We study anisotropic mixed norm spaces of n-harmonic functions in the unit polydisc of . Bergman type reproducing integral formulas are established by means of fractional derivatives and some continuous inclusions. It gives us a tool to construct corresponding projections and related operators and prove their boundedness on the mixed norm and Besov spaces.     

Asymptotic Analysis and Modeling of the Jointing of a Massive Body with Thin Rods

Asymptotic formulas are obtained for solutions of the anisotropic elasticity problem for a body with cavities into which thin rods are inserted, the outer ends of the rods being rigidly fixed. The surface of the body and the lateral surface of the rods are assumed load-free, but the entire elastic junction is subject to mass forces. The elastic materials are inhomogeneous and the stiffness of the rods may differ greatly from that of the body, their ratio being of the order h with an arbitrary exponent ; for = 0, the junction is homogeneous. Together with the asymptotic formulas, we construct and justify an asymptotic model of the junction. This model is applicable for a wide range of the exponent and preserves the parameter h in the conjugation conditions but is represented by a regularly perturbed problem. Since the leading asymptotic term involves fields with strong singularities, we have to give correct statements of the limit problem for a body with one-dimensional rods. For this purpose, we use the theory of self-adjoint extensions of operators or the technique of weighted spaces with separated asymptotics. The justification of our asymptotic expansions utilizes weighted anisotropic Korn inequalities, which take into account the mutual position of the rods and provide the best possible a priori estimates of the solutions. In contrast to other investigations, we describe, in explicit terms, the dependence of the bounds in the error estimates on the right-hand sides of the original problem. We also discuss the relationship between the asymptotic ansatz formulas and the weighted norms in the asymptotically precise Korn inequality.__________Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 24, pp. 95–214, 2004.     

Rankin-Selberg without unfolding and bounds for spherical Fourier coefficients of Maass forms

We use the uniqueness of various invariant functionals on irreducible unitary representations of in order to deduce the classical Rankin-Selberg identity for the sum of Fourier coefficients of Maass cusp forms and its new anisotropic analog. We deduce from these formulas non-trivial bounds for the corresponding unipotent and spherical Fourier coefficients of Maass forms. As an application we obtain a subconvexity bound for certain -functions. Our main tool is the notion of a Gelfand pair from representation theory.

     

Homogenization of a Ginzburg–Landau functional

We consider a nonlinear homogenization problem for a Ginzburg–Landau functional with a (positive or negative) surface energy term describing a nematic liquid crystal with inclusions. Assuming that sizes and distances between inclusions are of the same order ?, we obtain a limiting functional as $?\to 0$. We generalize the method of mesocharacteristics to show that a corresponding homogenized problem for arbitrary, periodic or non-periodic geometries is described by an anisotropic Ginzburg–Landau functional. We give computational formulas for material characteristics of an effective medium. To cite this article: L. Berlyand et al., C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Killing forms of isotropic Lie algebras

This paper presents a method for computing the Killing form of an isotropic Lie algebra defined over an arbitrary field based on the Killing form of a subalgebra containing its anisotropic kernel. This approach allows for streamlined formulas for many Lie algebras of types $E_6$ and $E_7$ and yields a unified formula for all Lie algebras of inner type $E_6$, including the anisotropic ones.     

Atomic and molecular decompositions of anisotropic Besov spaces

In this work we develop the theory of weighted anisotropic Besov spaces associated with general expansive matrix dilations and doubling measures with the use of discrete wavelet transforms. This study extends the isotropic Littlewood- Paley methods of dyadic -transforms of Frazier and Jawerth [19, 21] to non-isotropic settings.Several results of isotropic theory of Besov spaces are recovered for weighted anisotropic Besov spaces. We show that these spaces are characterized by the magnitude of the -transforms in appropriate sequence spaces. We also prove boundedness of an anisotropic analogue of the class of almost diagonal operators and we obtain atomic and molecular decompositions of weighted anisotropic Besov spaces, thus extending isotropic results of Frazier and Jawerth [21].The author was partially supported by the NSF grant DMS-0441817.     

Young walls and graded dimension formulas for finite quiver Hecke algebras of type A^{(2)}_{2\ell } and D^{(2)}_{\ell +1}

We study graded dimension formulas for finite quiver Hecke algebras \(R^{\Lambda _0}(\beta )\) of type \(A^{(2)}_{2\ell }\) and \(D^{(2)}_{\ell +1}\) using combinatorics of Young walls. We introduce the notion of standard tableaux for proper Young walls and show that the standard tableaux form a graded poset with lattice structure. We next investigate Laurent polynomials associated with proper Young walls and their standard tableaux arising from the Fock space representations consisting of proper Young walls. Then, we prove the graded dimension formulas described in terms of the Laurent polynomials. When evaluating at \(q=1\) , the graded dimension formulas recover the dimension formulas for \(R^{\Lambda _0}(\beta )\) described in terms of standard tableaux of strict partitions.     

Theory of Semi-Instantiation in Abstract Argumentation

We study instantiated abstract argumentation frames of the form (S, R, I), where (S, R) is an abstract argumentation frame and where the arguments x of S are instantiated by I(x) as well formed formulas of a well known logic, for example as Boolean formulas or as predicate logic formulas or as modal logic formulas. We use the method of conceptual analysis to derive the properties of our proposed system. We seek to define the notion of complete extensions for such systems and provide algorithms for finding such extensions. We further develop a theory of instantiation in the abstract, using the framework of Boolean attack formations and of conjunctive and disjunctive attacks. We discuss applications and compare critically with the existing related literature.     

Tensor product type subspace splittings and multilevel iterative methods for anisotropic problems

We describe tensor product type techniques to derive robust solvers for anisotropic elliptic model problems on rectangular domains in ^d . Our analysis is based on the theory of additive subspace correction methods and applies to finite element and prewavelet schemes. We present multilevel- and prewavelet-based methods that are robust for anisotropic diffusion operators with additional Helmholtz term. Furthermore, the resulting convergence rates are independent of the discretization level. Beside their theoretical foundation, we also report on the results of various numerical experiments to compare the different methods.     

Die mehrstellenformeln für den Laplaceoperator

Summary Difference methods for the numerical solution of linear partial differential equations may often be improved by using a weighted right hand side instead of the original right hand side of the differential equation. Difference formulas, for which that is possible, are called Mehrstellenformeln or Hermitian formulas. In this paper the Hermitian formulas for the approximation of Laplace's operator are characterized by a very simple condition. We prove, that in two-dimensional case for a Hermitian formula of ordern at leastn+3 discretization points are necessary. We give examples of such optimal formulas of arbitrary high-order.

     

On geometric representations of modular ortholattices

A pre-orthogonality on a projective geometry is a symmetric binary relation, ⊥, such that for each point ${p, p^{\perp} = \{q | p \perp q \}}$ is a subspace. An orthogonality is a pre-orthogonality such that each p ^⊥ is a hyperplane. Such ⊥ is called anisotropic iff it is irreflexive. For projective geometries with an anisotropic pre-orthogonality, we show how to find a (large) projective subgeometry with a natural embedding for the lattices of subspaces and with an orthogonality induced by the given pre-orthogonality. We also discuss (faithful) representations of modular ortholattices within this context and derive a condition which allows us to transform a representation by means of an anisotropic pre-orthogonality into an anisotropic orthogeometry by means of an anisotropic orthogonality.     

On the complexity of the closed fragment of Japaridze’s provability logic

We consider the well-known provability logic GLP. We prove that the GLP-provability problem for polymodal formulas without variables is PSPACE-complete. For a number n, let \({L^{n}_0}\) denote the class of all polymodal variable-free formulas without modalities \({\langle n \rangle,\langle n+1\rangle,...}\) . We show that, for every number n, the GLP-provability problem for formulas from \({L^{n}_0}\) is in PTIME.     

Perturbation of transportation polytopes

We describe a perturbation method that can be used to reduce the problem of finding the multivariate generating function (MGF) of a non-simple polytope to computing the MGF of simple polytopes. We then construct a perturbation that works for any transportation polytope. We apply this perturbation to the family of central transportation polytopes of order $kn\times n$, and obtain formulas for the MGFs of the feasible cone of each vertex of the polytope and the MGF of the polytope. The formulas we obtain are enumerated by combinatorial objects. A special case of the formulas recovers the results on Birkhoff polytopes given by the author and De Loera and Yoshida. We also recover the formula for the number of maximum vertices of transportation polytopes of order $kn\times n$.     

Semigroups associated to generalized polynomials and some classical formulas

We study operator semigroups associated with a family of generalized orthogonal polynomials with Hermitian matrix entries. For this we consider a Markov generator sequence, and therefore a Markov semigroup, for the family of orthogonal polynomials on related to the generalized polynomials. We give an expression of the infinitesimal generator of this semigroup and under the hypothesis of diffusion we prove that this semigroup is also Markov. We also give expressions for the kernel of this semigroup in terms of the one-dimensional kernels and obtain some classical formulas for the generalized orthogonal polynomials from the correspondent formulas for orthogonal polynomials on .     

Parabolic boundary integral operators symbolic representation and basic properties

In this paper we develop a theory of parabolic pseudodifferential operators in anisotropic spaces. We construct a symbolic calculus for a class of symbols globally defined on ⁿ⁺¹× ⁿ⁺¹, and then develop a periodisation procedure for the calculus of symbols on the cylinder ×. We show Gårding's inequality for suitable operators and precise estimates for the essential norm in anisotropic Sobolev spaces. These new mapping properties are needed in localization arguments for the analysis of numerical approximation methods.     

Hall Numbers and the Composition Algebra of the Kronecker Algebra

We present formulas for the structure constants (Hall numbers) of the Hall algebra associated to the Kronecker algebra. The formulas which in some cases involve the classical Hall polynomials enable us to determine every Hall number. Using again these formulas we construct new PBW-bases with simple structure constants for the composition algebra , making possible the definition of the generic composition algebra via Hall polynomials.Presented by C. Ringel.     

Approximation of an optimal control problem in coefficient for variational inequality with anisotropic <Emphasis Type="Italic">p</Emphasis>-Laplacian

We study an optimal control problem for a variational inequality with the so-called anisotropic p-Laplacian in the principle part of this inequality. The coefficients of the anisotropic p-Laplacian, the matrix A(x), we take as a control. The optimal control problem is to minimize the discrepancy between a given distribution \({y_d \in L^{2}(\Omega)}\) and the solutions \({y \in K \subset W^{1,p}_{0}(\Omega)}\) of the corresponding variational inequality. We show that the original problem is well-posed and derive existence of optimal pairs. Since the anisotropic p-Laplacian inherits the degeneracy with respect to unboundedness of the term \({|(A(x)\nabla y, \nabla y)_{\mathbb{R}^N}|^{\frac{p-2}{2}}}\), we introduce a two-parameter model for the relaxation of the original problem. Further we discuss the asymptotic behavior of relaxed solutions and show that some optimal pairs to the original problem can be attained by the solutions of two-parametric approximated optimal control problems.