Kyushu Journal of Mathematics

HODOGRAPH TRANSFORMATIONS AND THE CHERN-MOSER INVARIANTS ON THE BOUNDARY OF TUBE DOMAINS

Eisuke KOIZUMI

2005 Volume 59 Issue 1 Pages 1-23
Published: 2005
Released on J-STAGE: June 28, 2005

DOIhttps://doi.org/10.2206/kyushujm.59.1

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By using a hodograph transformation, we calculate a part of the Chern-Moser invariants on the boundary of tube domains. As an application of this result, we give another proof of a result of Yang on the characterization of tube domains with a spherical boundary.

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A SCALING LIMIT FOR t-SCHUR MEASURES

Sho MATSUMOTO

2005 Volume 59 Issue 1 Pages 25-38
Published: 2005
Released on J-STAGE: June 28, 2005

DOIhttps://doi.org/10.2206/kyushujm.59.25

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We introduce a new measure on partitions. We assign to each partition λ a probability S_λ(x; t)s_λ(y)/Z_t where s_λ is the Schur function, S_λ(x; t) is a generalization of the Schur function defined by Macdonald (Symmetric Functions and Hall Polynomials, 2nd edn. Oxford University Press, Oxford, 1995) and Z_t is a normalization constant. This measure, which we call the t-Schur measure, is a generalization of the Schur measure (A. Okounkov. Infinite wedge and random partitions. Selecta Math. (N.S.) 7 (2001), 57-81) and is closely related to the shifted Schur measure studied by Tracy and Widom (A limit theorem for shifted Schur measures. Preprint (2002), math.PR/0210255) for a combinatorial viewpoint.
We prove that a limit distribution of the length of the first row of a partition with respect to t-Schur measures is given by the Tracy-Widom distribution, i.e. the limit distribution of the largest eigenvalue suitably centered and normalized in the Gaussian unitary ensemble.

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SPECIAL VALUES OF THE SPECTRAL ZETA FUNCTION OF THE NON-COMMUTATIVE HARMONIC OSCILLATOR AND CONFLUENT HEUN EQUATIONS

Takashi ICHINOSE, Masato WAKAYAMA

2005 Volume 59 Issue 1 Pages 39-100
Published: 2005
Released on J-STAGE: June 28, 2005

DOIhttps://doi.org/10.2206/kyushujm.59.39

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We study the special values at s = 2 and 3 of the spectral zeta function ζ_Q(s) of the non-commutative harmonic oscillator Q(x, D_x) introduced in A. Parmeggiani and M. Wakayama (Proc. Natl Acad. Sci. USA 98 (2001), 26-31; Forum Math. 14 (2002), 539-604). It is shown that the series defining ζ_Q(s) converges absolutely for Re s > 1 and further the respective values ζ_Q(2) and ζ_Q(3) are represented essentially by contour integrals of the solutions, respectively, of a singly confluent Heun ordinary differential equation and of exactly the same but an inhomogeneous equation. As a by-product of these results, we obtain integral representations of the solutions of these equations by rational functions.

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SELF-HOMOTOPY OF THE DOUBLE SUSPENSION OF THE REAL 6-PROJECTIVE SPACE

Toshiyuki MIYAUCHI, Juno MUKAI

2005 Volume 59 Issue 1 Pages 101-116
Published: 2005
Released on J-STAGE: June 28, 2005

DOIhttps://doi.org/10.2206/kyushujm.59.101

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Let Pⁿ be the real n-dimensional projective space. We determine the group structure of the self-homotopy set of the double suspension of Pⁿ where n is 3, 4, 5 and 6 using the ideas and methods of the second author (The suspension order of the real even dimensional projective space, J. Math. Kyoto Univ. 43(4) (2003), 755-769).

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A GENERALIZATION OF THE BETA INTEGRAL ARISING FROM THE KNIZHNIK-ZAMOLODCHIKOV EQUATION FOR THE VECTOR REPRESENTATIONS OF TYPES B_n, C_n AND D_n

Katsuhisa MIMACHI, Takashi TAKAMUKI

2005 Volume 59 Issue 1 Pages 117-126
Published: 2005
Released on J-STAGE: June 28, 2005

DOIhttps://doi.org/10.2206/kyushujm.59.117

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We evaluate the integral appearing in the solution of the Knizhnik-Zamolodchikov equation associated with the vector representation of the simple Lie algebra g of type B_n, C_n or D_n. This integral, which can be considered as a generalization of the Beta integral, is expressed in terms of an alternating product of the Gamma function.

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ON STRUCTURE OF LOCALLY CONVEX *-ALGEBRAS WITH NORMAL UNBOUNDED C*-NORMS

Atsushi INOUE, Naoto TAKESHITA

2005 Volume 59 Issue 1 Pages 127-144
Published: 2005
Released on J-STAGE: June 28, 2005

DOIhttps://doi.org/10.2206/kyushujm.59.127

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The purpose of this paper is to define the notion of normal unbounded C*-norms on locally convex *-algebras and to study the structure of locally convex *-algebras equipped with such norms.

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SPACES OF FREE LOOPS ON REAL PROJECTIVE SPACES

Kohhei YAMAGUCHI

2005 Volume 59 Issue 1 Pages 145-153
Published: 2005
Released on J-STAGE: June 28, 2005

DOIhttps://doi.org/10.2206/kyushujm.59.145

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We study the homotopy types of the spaces of free algebraic loops on real projective spaces. In particular, we show that the Morse theoretic principle holds for these spaces in the stable homotopy category.

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ON THE HYPOELLIPTICITY WITH A BIG LOSS OF DERIVATIVES

Cesare PARENTI, Alberto PARMEGGIANI

2005 Volume 59 Issue 1 Pages 155-230
Published: 2005
Released on J-STAGE: June 28, 2005

DOIhttps://doi.org/10.2206/kyushujm.59.155

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We study, for a model class of classical pseudodifferential operators with symplectic characteristics of multiplicity k, necessary and sufficient conditions for the hypoellipticity with loss of r + k/2 derivatives (r > 0).

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