Let A be a Noetherian local ring with the maximal ideal m and an m-primary ideal J .Let F ={I_n}_n≥0 be a good filtration of ideals in A. Denote by F_J (F) =⊕_n≥0(I_n/JI_n)tⁿ the fiber cone of F with respect to J. In this paper we characterize the multiplicity and the Cohen-Macaulayness of F_J (F) in terms of minimal reductions of F.

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In this paper we investigate special generalized Bernoulli polynomials that generalize classical Bernoulli polynomials and numbers. We call them poly-Bernoulli polynomials. We prove a collection of extremely important and fundamental identities satisfied by our poly-Bernoulli polynomials and numbers. These properties are of arithmetical nature.

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This article follows recent work of Miyajima on the complex-analytic approach to deformations of the regular part (i.e. the punctured smooth neighbourhood) of isolated singularities. Attention has previously focused on stably-embeddable infinitesimal deformations as those which correspond to standard algebraic deformations of the germ of a variety, and which also lead to convergent series solutions of the Kodaira-Spencer integrability equation. The emphasis of the present paper, however, is on the subspaces Z₀ of first cohomology classes containing infinitesimal deformations with vanishing Kodaira-Spencer bracket, and W₀, consisting more broadly of deformations for which the bracket represents the trivial second cohomology class. Deformations representing classes in Z₀ are automatically integrable, regardless of their analytic behaviour near the singular point. Classes in W₀ are those for which only the first formal obstruction to integrability is overcome. After some preliminary results on cohomology, the main theorem of this paper gives a partial description of the analytic geometry of Z₀ and W₀ for affine cones of arbitrary dimension.

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In this paper we shall locate a class of fundamental identities for the gamma function and trigonometric functions in the chart of functional equations for the zeta-functions as a manifestation of the underlying modular relation. We use the beta-transform but not the inverse Heaviside integral. Instead we appeal to the reciprocal relation for the Euler digamma function which gives rise to the partial fraction expansion for the cotangent function. Through this we may incorporate basic results from the theory of the digamma (and gamma) function, thereby dispensing also with the beta-transform. Section 4 could serve as a foundation of the theory of the gamma function through the digamma function.

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Let ₃E₂(a₁,a₂,a₃;b₁,b₂) denote the generalized hypergeometric differential equation of rank three with parameters (a₁,a₂,a₃,b₁,b₂), which is defined on the projective line P¹(x) with regular singularities at x =0, 1, ∞. Any set of linearly independent solutions defines a multi-valued map from P¹ -{0, 1, ∞}to the projective plane P² called the Schwarz map. The image of this map is locally a curve, determined uniquely up to projective transformations. The condition for the image curve to be a cubic curve is written in terms of the parameters, and we list those curves and study the action of the monodromy groups on them.

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We show that any finite graph without loops can be realized as the Reeb graph of a smooth function on a closed manifold with finitely many critical values, but possibly with positive dimensional critical point set. We also show that such a function can be chosen as the height function on a surface immersed in 3-space, provided that the graph has no isolated vertices.

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The aim of this paper is twofold. One is to give a definition of the Euler characteristic of infinite acyclic categories with filtrations and the other is to prove the invariance of the Euler characteristic under the subdivision of finite categories.

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Lichtenbaum has conjectured (Ann of Math. (2) 170(2) (2009), 657-683) the existence of a Grothendieck topology for an arithmetic scheme X such that the Euler characteristic of the cohomology groups of the constant sheaf Z with compact support at infinity gives, up to sign, the leading term of the zeta function ζ_X(s) at s = 0. In this paper we consider the category of sheaves X_L on this conjectural site for X = Spec(Ο_F) the spectrum of a number ring. We show that X_L has, under natural topological assumptions, a well-defined fundamental group whose abelianization is isomorphic, as a topological group, to the Arakelov-Picard group of F . This leads us to give a list of topological properties that should be satisfied by X_L. These properties can be seen as a global version of the axioms for the Weil group. Finally, we show that any topos satisfying these properties gives rise to complexes of étale sheaves computing the expected Lichtenbaum cohomology.

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Gauss hypergeometric functions with a dihedral monodromy group can be expressed as elementary functions, since their hypergeometric equations can be transformed to Fuchsian equations with cyclic monodromy groups by a quadratic change of the argument variable. This paper presents general elementary expressions of these dihedral hypergeometric functions, involving finite bivariate sums expressible as terminating Appell' s F₂ or F₃ series. Additionally, trigonometric expressions for the dihedral functions are presented, and degenerate cases (logarithmic, or with the monodromy group Z/2Z)are considered.

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In this paper we show a two-dimensional variant of the classical Jacobi formula between a theta constant and the Gauss hypergeometric function. We use the family of algebraic curves given in the form w⁴ = z²(z-1)²(z-λ₁)(z-λ₂) with two complex parameters λ₁,λ₂ and the modular functions for them. Our result is an exact extension of the classical formula that is contained as a degenerated case. As an application we give a new proof for the extended Gauss arithmetic geometric mean theorem in two variables obtained by Koike and Shiga (J. Number Theory 128 (2008), 2029-2126).

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For a character χ of a finite group G, it is known that the product sh(χ) = Π_l∈L(χ(1) - l) is a multiple of |G|,where L is the image of χ on G -{1}. χ is said to be a sharp character of type L if sh(χ) =|G|. This is a generalization of the permutation characters of sharp permutation groups. Without loss of generality, we may assume that (χ, 1_G)_G = 0. In this paper, we classify the finite groups with sharp characters of type {l, l + p} for an odd prime p under the additional hypothesis Z(G) > 1 and (χ, χ)_G = p.

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