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Publications of the Research Institute for Mathematical Sciences
Publ. Res. Inst. Math. Sci.
PRIMS
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1663-4926
General
10.4171/PRIMS
http://www.ems-ph.org/doi/10.4171/PRIMS
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© Research Institute for Mathematical Sciences, Kyoto University
35
1999
1
Existence Theorems for Ordered Variants of Weyl Quantization
Daniel
Dubin
Open University, MILTON KEYNES, UNITED KINGDOM
Mark
Hennings
Rugby School, RUGBY, UNITED KINGDOM
Thomas
Smith
Open University, MILTON KEYNES, UNITED KINGDOM
We consider some mathematical properties of Weyl-like quantizations based on two families of orderings of et(αP+δQ): the first family, W(λ,0) interpolates between Wick (λ = 1) and antiWick (λ = –1) ordering, while the second family, W(0,μ), interpolates between the Q-(μ=1) and P-((μ=–1)) orderings. The ordering W(0,0) common to both families is the unordered Weyl system. The most important property is that of the existence of quantizations. For all orderings W(0,μ) and for W(λ,0) with –1 ≤ λ ≤ 0 quantization is a well-defined map from the tempered distributions on phase space into the continuous linear operators from S(ℝ) into S(ℝ)'. For the orderings W(λ,0) with –1 ≤ λ ≤ 0 we have to restrict the class of wave functions from S(ℝ) to a certain dense subset of it, and the resulting quantization procedure sends tempered distributions on phase space into sesquilinear forms on this subspace. For Wick ordering itself we have not been able to find any useable quantization scheme, and we doubt whether any one exists that is based on tempered distributions. We also consider questions of boundedness, and determine the matrix coefficients for the quantizations of phase space functions of radius or of angle. In particular, we consider various quantizations of the angle function in phase space.
Quantum theory
Functional analysis
General
1
29
10.2977/prims/1195144188
http://www.ems-ph.org/doi/10.2977/prims/1195144188
Generic and q-Rational Representation Theory
Edward
Cline
University of Oklahoma, NORMAN, UNITED STATES
Brian
Parshall
University of Virginia, CHARLOTTESVILLE, UNITED STATES
Leonard
Scott
University of Virginia, CHARLOTTESVILLE, UNITED STATES
Part I of this paper develops various general concepts in generic representation and cohomology theories. Roughly speaking, we provide a general theory of orders in non-semisimple algebras applicable to problems in the representation theory of finite and algebraic groups, and we formalize the notion of a “generic” property in representation theory. Part II makes new contributions to the non-describing representation theory of finite general linear groups. First, we present an explicipt Morita equivalence connecting GLn(q) with the theory of g-Schur algebras, extending a unipotent block equivalence of Takeuchi [T]. Second, we apply this Morita equivalence to study the cohomology groups em>H*(GLn(q),L), when L is an irreducible module in non-describing characteristic. The generic theory of Part I then yields stability results for various groups H1(GLn(q),L), reminscent of our general theory [CPSK] with van der Kallen of generic cohomology in the describing characteristic case, (in turn, the stable value of such a cohomology group can be expressed in terms of the cohomology of the affine Lie algebra gln(ℂ).) The arguments entail new applications of the theory of tilting modules for q~Schur algebras. In particular, we obtain new complexes involving tilting modules associated to endomorphism algebras obtained from general finite Coxeter groups.
Group theory and generalizations
General
31
90
10.2977/prims/1195144189
http://www.ems-ph.org/doi/10.2977/prims/1195144189
The Plancherel Theorem for Biinvariant Hilbert Spaces
Bernhard
Krötz
Universität Paderborn, PADERBORN, GERMANY
Topological groups, Lie groups
General
91
122
10.2977/prims/1195144190
http://www.ems-ph.org/doi/10.2977/prims/1195144190
2
The Berezin Calculus
Paul Lee
Robinson
University of Florida, GAINESVILLE, UNITED STATES
We present a canonical account of the Berezin integral and associated Berezin expectation over Hilbert spaces of arbitrary dimension. Our account is illustrated by an extensive discussion of Gaussians, by a Berezinian version of the kernel theorem for generalized functions, and by an extension of the Shale–Stinespring theorem on spin transformations.
Linear and multilinear algebra; matrix theory
Functional analysis
General
123
194
10.2977/prims/1195143948
http://www.ems-ph.org/doi/10.2977/prims/1195143948
Polynomial Weyl Representations
Paul Lee
Robinson
University of Florida, GAINESVILLE, UNITED STATES
For the canonical commutation relations in infinite dimensions, we offer an explicit direct construction of Weyl representations generated from the Fock representation by polynomial transformations of arbitrary degree, solving a problem posed by Proksch, Reents and Summers. Our solution employs new approaches to Hilbert–Schmidt polynomials and their Wick ordering.
Quantum theory
Functional analysis
Measure and integration
Special functions
195
222
10.2977/prims/1195143949
http://www.ems-ph.org/doi/10.2977/prims/1195143949
D-Modules Associated to the Group of Similitudes
Philibert
Nang
Université Pierre et Marie Curie, PARIS CEDEX 05, FRANCE
We classify regular holonomic D-modules whose characteristic variety is the union of the conormal bundles of the orbits of the group of similitudes of a non degenerate quadratic form.
Partial differential equations
Global analysis, analysis on manifolds
General
223
247
10.2977/prims/1195143950
http://www.ems-ph.org/doi/10.2977/prims/1195143950
Canonical Isomorphism of Two Lie Algebras Arising in CR-geometry
Vladimir
Ezhov
The University of Adelaide, ADELAIDE SA, AUSTRALIA
Alexander
Isaev
Australian National University, CANBERRA, AUSTRALIA
We show that the maximal prolongation of a certain algebra associated with a non-degenerate Hermitian form on ℂn x ℂn with values in ℝk is canonically isomorphic to the Lie algebra of infinitesimal holomorphic automorphisms of the corresponding quadric in ℂn+k. This fact creates a link between different approaches to the equivalence problem for Levi-nondegenerate strongly uniform CR-manifolds.
Several complex variables and analytic spaces
Nonassociative rings and algebras
General
249
261
10.2977/prims/1195143951
http://www.ems-ph.org/doi/10.2977/prims/1195143951
Automorphic Forms on the Expanded Symmetric Domain of Type IV
Hiroki
Aoki
Kyoto University, KYOTO, JAPAN
We introduce a lifting from a given Jacobi form of index 1 to an automorphic form on an expanded domain of type F , introduced by Saito [20, 22]. The method is a generalization of Gritsenko [10, 11] for symmetric domain of type IV. We constract a lifting function satisfying a certain translation formula on the expanded domain.
Number theory
Algebraic geometry
General
263
283
10.2977/prims/1195143952
http://www.ems-ph.org/doi/10.2977/prims/1195143952
Propagation of Singularities in the Ramified Cauchy Problem for a Class of Operators with Non-involutive Multiple Characteristics
Katsuju
Igari
Kyoto University, KYOTO, JAPAN
In the ramified Cauchy problem, analytic continuation of holomorphic solutions has been mainly studied. In this paper, we prove the propagation of singularities for a class of linear partial differential equations with non-involutive multiple characteristics.
Partial differential equations
General
285
307
10.2977/prims/1195143953
http://www.ems-ph.org/doi/10.2977/prims/1195143953
The Furuta Inequality and an Operator Equation for Linear Operators
Chia-Shiang
Lin
Bishop’s University, LENNOXVILLE (QUEBEC), CANADA
Hp-2rn/2(n+1) T (Hp+2r/(n+1) T)n Hp-2rn/2(n+1) = Kp. This result also generalizes Lemma 1 in [3] which is about the operator equation T (H1/n T)n = K. A new characterization of the Löwner–Heinz formula and some applications are given.
Operator theory
General
309
313
10.2977/prims/1195143954
http://www.ems-ph.org/doi/10.2977/prims/1195143954
Characterization of the Pull-Back of D-Modules
Takayuki
Takahashi
Kyoto University, KYOTO, JAPAN
Several complex variables and analytic spaces
Associative rings and algebras
General
315
319
10.2977/prims/1195143955
http://www.ems-ph.org/doi/10.2977/prims/1195143955
Correction to 'Existence theorems for ordered variants of Weyl quantization'
Correction to Vol. 35, No. 1: Daniel A. DUBIN, Mark A. HENNINGS and Thomas B. SMITH, “Existence theorems for ordered variants of Weyl quantization”, pp. 1-29. page 1; the first line of the footnote should be replaced by Communicated by T. Kawai, October 28, 1997. Revised April 7, 1998. We sincerely apologize for the misprint. Editors
General
0
0
10.2977/prims/1195143956
http://www.ems-ph.org/doi/10.2977/prims/1195143956
3
On Defining Relations of Affine Lie Superalgebras and Affine Quantized Universal Enveloping Superalgebras
Hiroyuki
Yamane
Osaka University Graduate School of Science, OSAKA, JAPAN
Nonassociative rings and algebras
Associative rings and algebras
General
321
390
10.2977/prims/1195143607
http://www.ems-ph.org/doi/10.2977/prims/1195143607
Energy Decay for a Degenerate Hyperbolic Equation with a Dissipative Term
Fumihiko
Hirosawa
University of Tsukuba, IBARAKI, JAPAN
Partial differential equations
General
391
406
10.2977/prims/1195143608
http://www.ems-ph.org/doi/10.2977/prims/1195143608
Representations of the Quantum Toroidal Algebra on Highest Weight Modules of the Quantum Affine Algebra of Type glN
Kouichi
Takemura
Kyoto University, KYOTO, JAPAN
Denis
Uglov
Kyoto University, KYOTO, JAPAN
A representation of the quantum toroidal algebra of type slN is defined on every integrable irreducible highest weight module of the quantum affme algebra of type glN. The q-version of the level-rank duality giving the reciprocal decomposition of the q-Fock space with respect to mutually commutative actions of U'q(glN) of level L and U'g(slL) of level N is described.
Nonassociative rings and algebras
Quantum theory
General
407
450
10.2977/prims/1195143609
http://www.ems-ph.org/doi/10.2977/prims/1195143609
Some Limit Transitions between BC Type Orthogonal Polynomials Interpreted on Quantum Complex Grassmannians
Mathijs
Dijkhuizen
Kobe University, KOBE, JAPAN
Jasper
Stokman
University of Amsterdam, AMSTERDAM, NETHERLANDS
The quantum complex Grassmannian Uq/Kq of rank l is the quotient of the quantum unitary group Uq = Uq(n) by the quantum subgroup Kq = Uq(n–l) x Uq(l). We show that (Uq, Kq) is a quantum Gelfand pair and we express the zonal spherical functions, i.e. Kq-biinvariant matrix coefficients of finite-dimensional irreducible representations of Uq, as multivariable little q-Jacobi polynomials depending on one discrete parameter. Another type of biinvariant matrix coefficients is identified as multivariable big q-Jacobi polynomials. The proof is based on earlier results by Noumi, Sugitani and the first author relating Koornwinder polynomials to a one-parameter family of quantum complex Grassmannians, and certain limit transitions from Koornwinder polynomials to multivariable big and little q-Jacobi polynomials studied by Koornwinder and the second author.
Special functions
Quantum theory
Nonassociative rings and algebras
General
451
500
10.2977/prims/1195143610
http://www.ems-ph.org/doi/10.2977/prims/1195143610
Large Time Behavior of Solutions for Derivative Cubic Nonlinear Schrödinger Equations
Nakao
Hayashi
Tokyo University of Science, TOKYO, JAPAN
Pavel
Naumkin
Universidad Michoacana, MORELIA, MICHOACÁN, MEXICO
Hidetake
Uchida
Tokyo University of Science, TOKYO, JAPAN
We study the asymptotic behavior in time and scattering problem for the solutions to the Cauchy problem for the derivative cubic nonlinear Schrodinger equations of the following form (A) iut+uxx = N(u, u, ux, ux), t ∈ ℝ, x ∈ ℝ; u(0, x) = u0(x), x ∈ ℝ, where N(u, u, ux, ux) = ℋ1|u|2 u + iℋ2|u|2 ux + iℋ3u2 ux + ℋ4|ux|2 u + ℋ5uux2 + iℋ6|ux|2 ux, ℋ1 = ℋ1(|u|2), ℋ1(z) ∈ C3(ℝ+); ℋ1(z) = λ1 + O(z), as z → +0, ℋ1, ℋ6 are real valued functions. Here the parameters λ1, λ6 ∈ ℝ, and λ2, λ3, λ4, λ5 ∈ ℂ are such that λ2 – λ3 ∈ ℝ and λ4 – λ5 ∈ ℝ. If ℋ5(z) = λ5/(1+μz) and λ5 = μ = ∓ 1, ℋ1 = ℋ2 = ℋ3 = ℋ4 = ℋ5 = ℋ6 = 0 equation (A) appears in the classical pseudospin magnet model [9]. We prove that if u0 ∈ H3,0 ∩ H2,1 and the norm ‖u0‖3,0 + ‖u0‖2,1 = ε is sufficiently small, then the solution of (A) exists globally in time and satisfies the sharp time decay estimate ‖u(t)‖2,0, ∞ ≤ Cε(1+|t|)-1/2, where ‖φ‖m,s,p = ‖(1+x2)s/2(1−∂x2)m/2φ‖Lp, Hpm,s ={φ ∈ S'; ‖φ‖m,s,p < ∞}. Furthermore we prove existence of modified scattering states and nonexistence of nontrivial scattering states. Our method is based on a certain gauge transformation and an appropriate phase function.
Partial differential equations
General
501
513
10.2977/prims/1195143611
http://www.ems-ph.org/doi/10.2977/prims/1195143611
Blowing Ups of 3-dimensional Terminal Singularities
Takayuki
Hayakawa
Kanazawa University, KANAZAWA, JAPAN
We study the blowing up π : X -> X of a 3-dimensional terminal singularity X of index m > 2 such that the exceptional locus of π consists of a prime divisor E with discrepancy 1/m. A complete classification of such blowing ups is given and it is proved that these correspond to weighted blow ups by a certain kind of maximal weights except for the case where X is of type (cD/2). We shall treat the (cD/2) case later. These also give examples of contractions of extremal rays which contract a divisor to a point.
Algebraic geometry
Several complex variables and analytic spaces
General
515
570
10.2977/prims/1195143612
http://www.ems-ph.org/doi/10.2977/prims/1195143612
4
Multiple Poles at Negative Integers for ∫Afλ☐ in the Case of an Almost Isolated Singularity
Daniel
Barlet
Université Henri Poincaré, VANDOEUVRE CEDEX, FRANCE
We give a necessary and sufficient topological condition on A ∈ H0({f ≠ 0},ℂ), for a real analytic germ f : (ℝn+1,0) → (ℝ,0), whose complexification has an isolated singularity relatively to the eigenvalue 1 of the monodromy, in order that the meromorphic continuation of ∫Afλ☐ has a multiple pole at sufficiently “large” negative integers. We show that if such a multiple pole exists, it occurs already at λ = −(n + 1) with its maximal order which is computed topologically.
Several complex variables and analytic spaces
Algebraic geometry
General
571
584
10.2977/prims/1195143493
http://www.ems-ph.org/doi/10.2977/prims/1195143493
Dérivations et Idéaux Réels Invariants
Herwig
Hauser
Universität Innsbruck, INNSBRUCK, AUSTRIA
Jean-Jacques
Risler
Université Pierre et Marie Curie, PARIS CEDEX 05, FRANCE
Several complex variables and analytic spaces
Algebraic geometry
Nonassociative rings and algebras
General
585
597
10.2977/prims/1195143494
http://www.ems-ph.org/doi/10.2977/prims/1195143494
Régularité des Ondes élastiques dans la Région Glancing des Ondes P
Tatsushi
Morioka
Osaka University, OSAKA, JAPAN
In this paper, we show that the outgoing solutions of the isotropic elastic equation with Dirichlet condition at the boundary have the Gevrey 3 regularity in the glancing region of the longitudinal waves, when the dimension of the space is 3, the domain is exterior and the Gaussian curvature is positive. It is an analogy of the work by Lebeau [14] concerning the wave equation, for the isotropic elastic equation.
Global analysis, analysis on manifolds
Partial differential equations
General
599
619
10.2977/prims/1195143495
http://www.ems-ph.org/doi/10.2977/prims/1195143495
On Totally Characteristic Type Non-linear Partial Differential Equations in the Complex Domain
Hua
Chen
Wuhan University, WUHAN, HUBEI, CHINA
Hidetoshi
Tahara
Sophia University, TOKYO, JAPAN
The paper deals with a singular non-linear partial differential equation t∂u/∂t = F(t, x, u, ∂u/∂x) with two independent variables (t,x) ∈ ℂ2 under the assumption that F(t, x, u, v) is holomorphic and F(0,x,0,0) = 0. Set γ(x) = (∂F/∂v)(0,x,0,0). In case γ(x) = 0 the equation was investigated quite well by Gerard-Tahara [3]. In case γ(0) = 0 and Reγ' < 0 the existence of holomorphic solution was proved in Chen–Tahara [2] under a non-resonance condition. The present paper proves the existence of holomorphic solution under the same non-resonance condition but using the following weaker condition: γ(0) = 0 and γ'(0) ∈ ℂ\[0, ∞). The result is extended to higher order equations.
Partial differential equations
General
621
636
10.2977/prims/1195143496
http://www.ems-ph.org/doi/10.2977/prims/1195143496
Correspondance d'Andreotti–Norguet et D-Modules
Jean-Louis
Frot
Université Pierre et Marie Curie, PARIS CEDEX 05, FRANCE
We calculate the integral transform of a D-module of rank >1, locally free outside the zero section of the cotangent space to the complex projective space ℙn. This allows us to complete some results of Andreotti–Norguet and Barlet: in particular we prove that the image of the integral transform obtained by integrating holomorphic forms along the linear cycles of ℙn\ℙn-p-1 (where 0 < p < n - 1), is the space of holomorphic functions on the variety of cycles Cp(ℙn\ℙn-p-1) which are annihilated by a family of differential operators of order four, that we determine explicitly.
Several complex variables and analytic spaces
Nonassociative rings and algebras
Partial differential equations
Integral transforms, operational calculus
637
677
10.2977/prims/1195143497
http://www.ems-ph.org/doi/10.2977/prims/1195143497
Méthodes à N Corps pour un Problème de Milieux Pluristratifiés Perturbés
Yves
Dermenjian
Université de Provence, MARSEILLE CEDEX 13, FRANCE
Viorel
Iftimie
Romanian Academy, BUCHAREST, ROMANIA
On étudie l'opérateur H := ∇*p∇ → V dans L2(X), X espace euclidien r�el de dimension finie, oú V est un potentiel du type "N corps" associé à une famille finie ℒ de sous-espaces vectoriels de X et p admet une décomposition suivant ℒ compatible avec celle de V. Chaque composante de V, respectivement p, est une somme de perturbations de type "courte portée" et "longue portée". En utilisant une variante de la méthode de Mourre [14] ainsi que des idées de la théorie du problème à N corps de la mécanique quantique, on fait l'analyse spectrale de l'opérateur H et on prouve un principe d'absorption limite.
Partial differential equations
Operator theory
Quantum theory
General
679
709
10.2977/prims/1195143498
http://www.ems-ph.org/doi/10.2977/prims/1195143498
5
A Theorem of Tits, Normalizers of Maximal Tori and Fibrewise Bousfield–Kan Completions
Frank
Neuman
Georg-August-Universität Göttingen, GÖTTINGEN, GERMANY
We use a theorem of Tits on the presentation of the normalizer of a maximal torus of a connected compact semisimple Lie group in terms of generators and relations to give several equivalent conditions for the splitting of the associated normalizer group extension and interprete them in terms of p-adic fibrewise homotopy theory.
Group theory and generalizations
Linear and multilinear algebra; matrix theory
Topological groups, Lie groups
Manifolds and cell complexes
711
723
10.2977/prims/1195143419
http://www.ems-ph.org/doi/10.2977/prims/1195143419
Coincidence Points for Perturbations of Linear Fredholm Maps of Index Zero
Ravi
Agarwal
Texas A&M University, KINGSVILLE, UNITED STATES
Donal
O'Regan
National University of Ireland, GALWAY, IRELAND
Coincidence points for single and set valued maps are discussed in this paper. We show if F is essential and F ≅ G then G has a coincidence point.
Algebraic topology
Ordinary differential equations
General
725
736
10.2977/prims/1195143420
http://www.ems-ph.org/doi/10.2977/prims/1195143420
An Integral Transformation and its Applications to Harmonic Analysis on the Space of Solutions of the Heat Equation
Soon-Yeong
Chung
Sogang University, SEOUL, SOUTH KOREA
Yongjin
Yeom
Seoul National University, SEOUL, SOUTH KOREA
We introduce an integral transformation T defined by (Tf)(x,t) = ∫ e−ixy−t|y|2 f(y)dy, f ∈ L1(ℝn) in order to do harmonic analysis on the space of C∞ solutions of the heat equation. First, the Paley–Wiener type theorem for the transformation T will be given for the C∞ functions and distributions with compact support. Secondly, as an application of the transformation the solutions of heat equation given on the torus T* will be characterized. Finally, we represent solutions of the heat equation as an infinite series of Hermite temperatures, which are to be defined as the images of Hermite polynomials under the transformation T.
Partial differential equations
Fourier analysis
Functional analysis
General
737
755
10.2977/prims/1195143421
http://www.ems-ph.org/doi/10.2977/prims/1195143421
Hilbert C*-Module Representation on Haagerup Tensor Products and Group Systems
Jaeseong
Heo
Seoul National University, SEOUL, SOUTH KOREA
Using the Hilbert C*-module representation associated with completely multi-positive linear maps [Heo], we give another representation on Haagerup tensor product without the bridging maps. We also construct covariant representations of covariant group systems on Hilbert C*-modules.
Functional analysis
General
757
768
10.2977/prims/1195143422
http://www.ems-ph.org/doi/10.2977/prims/1195143422
Sierpiński Gasket as a Martin Boundary II — (The Intrinsic Metric)
Manfred
Denker
Georg-August-Universität Göttingen, GÖTTINGEN, GERMANY
Hiroshi
Sato
Kyushu University, FUKUOKA, JAPAN
It is shown in [DS] that the Sierpiński gasket S ∈ ℝN can be represented as the Martin boundary of a certain Markov chain and hence carries a canonical metric pM induced by the embedding into an associated Martin space M. It is a natural question to compare this metric pM with the Euclidean metric. We show first that the harmonic measure coincides with the normalized H=(log(N+l)/log2)-dimensional Hausdorff measure with respect to the Euclidean metric. Secondly, we define an intrinsic metric p which is Lipschitz equivalent to pM and then show that p is not Lipschitz equivalent to the Euclidean metric, but the Hausdorff dimension remains unchanged and the Hausdorff measure in p is infinite. Finally, using the metric p, we prove that the harmonic extension of a continuous boundary function converges to the boundary value at every boundary point.
Probability theory and stochastic processes
Potential theory
General
769
794
10.2977/prims/1195143423
http://www.ems-ph.org/doi/10.2977/prims/1195143423
Algebraic Coset Conformal Field Theories II
Feng
Xu
University of California, RIVERSIDE, UNITED STATES
Some mathematical questions relating to coset conformal field theories (CFT) are considered in the framework of algebraic quantum field theory as developed previously by us. We consider the issue of fix point resolution in the diagonal cosets of type A. We show how to decompose certain reducible representations into irreducibles, and prove that the coset CFT gives rise to a unitary modular category and therefore may be used to construct 3-manifold invariants. We prove that if the coset inclusion satisfies certain conditions which can be checked in examples, the Kac–Wakimoto Hypothesis (KWH) is equivalent to the Kac–Wakimoto Conjecture (KWC), a result which seems to be hard to prove by purely representation considerations. Examples are also presented.
Functional analysis
Quantum theory
General
795
824
10.2977/prims/1195143424
http://www.ems-ph.org/doi/10.2977/prims/1195143424
6
Strong Unique Continuation Property for the Dirac Equation
Laura
De Carli
Università degli Studi di Napoli Federico II, NAPOLI, ITALY
Takashi
Ōkaji
Kyoto University, KYOTO, JAPAN
We discuss some Carleman type inequalities which lead to a strong unique continuation property for Dirac operators with potential dominated by the Coulomb singularity.
Partial differential equations
General
825
846
10.2977/prims/1195143357
http://www.ems-ph.org/doi/10.2977/prims/1195143357
Note on the Paper “Strong Unique Continuation Property for the Dirac Equation” by De Carli and Ōkaji
Hubert
Kalf
Universität München, MÜNCHEN, GERMANY
Osanobu
Yamada
Kyoto University, KYOTO, JAPAN
Partial differential equations
General
847
852
10.2977/prims/1195143358
http://www.ems-ph.org/doi/10.2977/prims/1195143358
On the Spherically Symmetric Solution to the Mixed Problem for a Weakly Hyperbolic Equation of Second Order
Akisato
Kubo
Fujita Health University, AICHI, JAPAN
Partial differential equations
General
853
870
10.2977/prims/1195143359
http://www.ems-ph.org/doi/10.2977/prims/1195143359
Determinant Formula for Solutions of the Quantum Knizhnik–Zamolodchikov Equation Associated with Uq(sln) at |q| = 1
Tetsuji
Miwa
Kyoto University, KYOTO, JAPAN
Yoshihiro
Takeyama
Kyoto University, KYOTO, JAPAN
Vitaly
Tarasov
Steklov Mathematical Institute, ST. PETERSBURG, RUSSIAN FEDERATION
We construct the hypergeometric solutions for the quantized Knizhnik–Zamolodchikov equation with values in a tensor product of vector representations of Uq(sln) at |q| = 1 and give an explicit formula for the corresponding determinant in terms of the double sine function.
Special functions
Nonassociative rings and algebras
Quantum theory
General
871
892
10.2977/prims/1195143360
http://www.ems-ph.org/doi/10.2977/prims/1195143360
Divergence Property of Formal Solutions for Singular First Order Linear Partial Differential Equations
Masaki
Hibino
Meijo University, NAGOYA, AICHI, JAPAN
This paper is concerned with the study of the convergence and the divergence of formal power series solutions of the following first order singular linear partial differential equation with holomorphic coefficients at the origin: P(x,D)u(x) = ∑di=1al(x)Dlu(x)+b(x)u(x) = f(x), with f(x) holomorphic at the origin. Here the equation is said to be singular if aj(0)=0 (j = 1, ...,d). In this case, it is known that under the so-called Poincare condition, if {al(x)}dl=1 generates a simple ideal, every formal solution is convergent. However if we remove these conditions, we shall see that the formal solution, if it exists, may be divergent. More precisely, we will characterize the rate of divergence of formal solutions via Gevrey order of formal solutions determined by a Newton Polyhedron, a generalization of Newton Polygon which is familiar in the study of ordinary differential equations with an irregular singular point.
Partial differential equations
General
893
919
10.2977/prims/1195143361
http://www.ems-ph.org/doi/10.2977/prims/1195143361
A Generalization of the Radon–Nikodym Property in Dual Banach Spaces, Fragmentedness, and Differentiability of Convex Functions
Minoru
Matsuda
Shizuoka University, SHIZUOKA, JAPAN
For non-empty bounded subsets A of Banach spaces, we introduce the notion of the A-Radon–Nikodym property in dual Banach spaces, a slight generalization of the Radon–Nikodym property in such spaces. Making the effective use of this notion and a weak*-measurable function constructed here, we give a direct study of some related properties (especially, A-fragmentedness) of weak*-compact subsets of dual Banach spaces.
Functional analysis
General
921
933
10.2977/prims/1195143362
http://www.ems-ph.org/doi/10.2977/prims/1195143362