Let G_k be a split reductive group over a field k of ch(k) =0 corresponding to a compact Lie group G.Then the mod p motivic cohomology H ^∗,∗´ (G_k;Z/p) is isomorphic to just a tensor product of the usual mod p cohomology of G and the mod p motivic cohomology of the point, when G =SU_n, SO_n,G₂,F₄, or E₆.Moreover, the Kunneth formula holds for cohomology of these groups. However, when G_k is not split over k, the situation is quite different.

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Let (A, m) be a Noetherian local ring. Let F ={I_n}_n≥0 be a good filtration of ideals in A. Denote by R(F) = ⊕_n≥0 I_ntⁿ the Rees algebra of F. In this paper we investigate the mixed multiplicities of F and the relationship between the multiplicity of R(F) and the mixed multiplicities of F. As an application, we obtain several multiplicity formulas that express the multiplicity of R(F) as a sum of Hilbert-Samuel multiplicities of some quotient rings of A.

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We investigate the moduli sets of central extensions of H-spaces enjoying inversivity, power associativity and Moufang properties. By considering rational H-extensions, it turns out that there is no relationship between the first and the second properties in general.

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Fisher’s solution of the Ising model on a planar graph G relies on converting it to a dimer model on a related planar graph G^*, which in turn can be solved using Pfaffians. In this paper we show that for cubic graphs G the number of perfect matchings of the graph G^*. equals 2^{|E(G)|-|V(G)|+1}. Our proof is based on two combinatorial ideas from statistical physics, namely the high-temperature expansion of the Ising model, and the Fisher construction that maps the Ising model to a dimer model. We give several applications, including two concrete three-dimensional crystal structures whose entropy can be computed exactly using our result.

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We construct explicit solutions to the discrete motion of discrete plane curves that has been introduced by one of the authors recently. Explicit formulas in terms of the τ function are presented. Transformation theory of the motions of both smooth and discrete curves is developed simultaneously.

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In this paper, we derive several connection formulas for Appell’s F₂, from which we obtain a monodromy representation of it. We also determine a necessary and sufficient condition for its irreducibility.

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For 1 ≤ p, r <+∞, f (≠ 0) ∈ L_p (R, dx), and g (≠ 0) ∈ L_r (R, dx), the sequence space Λ_p (f) with metric d^f_p(a,b) was introduced in a previous paper and we discussed the inclusion relations between l_p and Λ_p (f), and the linearity of Λ_p (f).The purpose of this paper is to discuss the topological structures of (Λ_p (f), d^f_p).First we show that the space (Λ_p (f), d^f_p) is a complete separable metric group. Next we show that if Λ_p (f) is a linear space, then (Λ_p (f), d^f_p) is a topological linear space. On the other hand, we give a necessary and sufficient condition for the inclusion Λ_p (f) ⊂ Λ_r (g). Furthermore, we show that the inclusions among the sequence spaces Λ_p (f), Λ_r (g) and l_r are continuous. The fact that Λ_p (f) ⊂ Λ_r (g) as sets implies the continuity of the inclusion (Λ_p (f), d^f_p) → (Λ_r (g), d^g_r) is emphasized.

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For a real hypersurface M of class C² in Cⁿ , n ≥ 2, an explicit formula for the Levi form of the distance function δ_M to M is given in terms of the Hermitian matrix (∂²ρ/∂zi∂zj) and the symmetric matrix (∂²ρ/∂zi∂zj) determined by the defining function ρ with || grad ρ || = 1 of M.

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In this article we determine the structure of a twisted first cohomology group of the first homology of a trivalent graph with a coefficient associated with the quantum Clebsch-Gordan condition. As an application we give a characterization of a combinatorial property, the external edge condition, which is defined in the study of the Heisenberg representation on the module constructed in the framework of topological quantum field theory, i. e. TQFT-module.

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In this paper, we give an elementary proof of Thomae formula for Z₃ curves. Let C(λ) be a cyclic triple covering parameterized by the set λ of its branch points. We explicitly give the zero divisors of the pullbacks θ_m(P) of theta functions with some characteristics m under the Abel-Jacobi map from C(λ) to its Jacobian. We take a symplectic basis for the homology group of C(λ) so that we can see the action of a covering transformation ρ on θ_m(P).

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We consider the zeta functions satisfying the functional equation with multiple gamma factors and prove a far-reaching theorem,an intermediate modular relation, which gives rise to many (including many of the hitherto found) arithmetical Fourier series as a consequence of the functional equation.Typical examples are the Diophantine Fourier series considered by Hardy and Littlewood and one considered by Hartman and Wintner, which are reciprocals of each other, in addition to our previous work. These have been thoroughly studied by Li, Ma and Zhang. Our main contribution is to the effect that the modular relation gives rise to the Fourier series for the periodic Bernoulli polynomials and Kummer’s Fourier series for the log sin function, thus giving a foundation for a possible theory of arithmetical Fourier series based on the functional equation.

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We improve C. Fefferman’s result on double singular integrals (Studia Math. 44 (1972), 1-15) by relaxing the regularity condition on the kernels.

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W. Wirtinger, in 1902, introduced an integral representation for the Gauss hypergeometric function in terms of theta functions. We give a generalization of this integral representation. More concretely, we study a definite integral of a power product of theta functions which has zeros at N-torsion points (N is a natural number greater than one) on a one-dimensional complex torus. We show that this integral gives a solution of a system of Fuchsian differential equations on the modular curve of level N and determine the spectral types of the system at its regular singularities.

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In analogy with the theory of integral lattices, the concept of the deep hole is defined for binary linear codes.In this paper we study combinatorial configurations that come from the deep hole cosets in the first-order Reed-Muller codes of lengths 2^m (m = 3, 4, 5, 6). As consequences we obtain the Hamming association subschemes of various orders.

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We study the asymptotic stability of nonlinear waves for damped wave equations with a non-convex convection term on the half line. In the case where the convection term satisfies the convexity and sub-characteristic conditions at the origin, it is shown by our previous work [Osaka J. Math. To appear] that the solution tends to the stationary wave. In this paper, we deal with the damped wave equations with convection term which is not necessarily the convexity at the origin, and we show not only the asymptotic stability of the non-degenerate stationary wave but also for the degenerate stationary wave. Furthermore, for the non-degenerate case, we also show that the time convergence rate is polynomially (respectively exponentially) fast if the initial perturbation decays polynomially (respectively exponentially) as x goes up to infinity. Our proofs are based on a combination of the antiderivative method and the L² weighted energy method.

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For a pair of graphs, Lovász introduced a polytopal complex called the Hom complex in order to estimate topological lower bounds for chromatic numbers of graphs. The definition is generalized to hypergraphs. Given an r-graph H , we compare the Hom complex of the complete r-partite r-graph and H with the box complex of H invented by Alon, Frankl and Lovász.We verify that both complexes which are equipped with right actions of the symmetric group on r letters Σ_r, are of the same simple Σ_r-homotopy type.

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In this paper, an n-dimensional complete open manifold with nonnegative Ricci curvature and collapsing volume has been investigated. If its radial sectional curvature is bounded from below, it shows that such a manifold is of finite topological type under some restrictions.

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