Let X be a locally compact and σ-compact abelian group and be the dual group of X. We denote by ξ a fixed Haar measure on X and by the Haar measure on associated with ξ.

Let N denote the set of non-negative integers. An affine semigroup is a finitely generated submonoid S of the additive monoid Nm for some positive integer m. Let k[S] denote the semigroup ring of S over a field k. Then one can identify k[S] with the subring of a polynomial ring k[t1, …, tm] generated by the monomials .

We describe the irreducible components of the jet schemes with origin in the singular locus of a two-dimensional quasi-ordinary hypersurface singularity. A weighted graph is associated with these components and with their embedding dimensions and their codimensions in the jet schemes of the ambient space. We prove that the data of this weighted graph is equivalent to the data of the topological type of the singularity. We also determine a component of the jet schemes (equivalent to a divisorial valuation on $\mathbb{A}^{3}$), that computes the log-canonical threshold of the singularity embedded in $\mathbb{A}^{3}$. This provides us with pairs $X\subset \mathbb{A}^{3}$ whose log-canonical thresholds are not computed by monomial divisorial valuations. Note that for a pair $C\subset \mathbb{A}^{2}$, where $C$ is a plane curve, the log-canonical threshold is always computed by a monomial divisorial valuation (in suitable coordinates of $\mathbb{A}^{2}$).

Let E be an elliptic curve defined over the field Q of rational numbers, then the torsion subgroup of the Mordell-Weil group E(Q) is finite and it is known that there exist the elliptic curves whose torsion subgroups E(Q)t are of the following types: (1), (2), (3), (2, 2), (4), (5), (2, 3), (7), (2, 4), (8), (9), (2, 5), (2, 2, 3), (3, 4) and (2, 8). It has been conjectured from various reasons that E(Q)t is exhausted by the above types only. If E has a torsion point of order precisely n, then it is known that E has an n-isogeny, that is to say, an isogeny of degree n.

In a recent paper [9] the author showed (among other results) estimates on the asymptotic behaviour of the representation numbers of positive definite integral ternary quadratic forms, in particular, that for n in a fixed square class tZ2 and lattices L, K in the same spinor genus one has . The main tool utilized for the proof was the theory of modular forms of weight 3/2, especially Shimura’s lifting from the space of cusp forms of weight 3/2 to the space of modular forms of weight 2.

Let q be a finite field, G = GLn(q), θ be the outer automorphism of G, suitably normalized. Consider the non-connected group G ⋊ {1, θ} and its connected component = Gθ. We know two ways to produce functions on , with complex values and invariant by conjugation by G: on one hand, let π be an irreducible representation of G we can and do extend to a representation π+ of G ⋊ {1, θ}, then the restriction trace to of the character of π+ is such a function; on the other hand, Lusztig define character-sheaves a, whose characteristic functions ϕ(a) are such functions too. We consider only “quadratic-unipotent” representations. For all such representation π, we define a suitable extension π+, a character-sheave f(π) and we prove an identity trace = γ(π)ϕ(f(π)) with an explicit complex number γ(π).

Given a local ring containing a field, we define and investigate a family of invariants that includes the Lyubeznik numbers but captures finer information. These generalized Lyubeznik numbers are defined in terms of D-modules and are proved well defined using a generalization of the classical version of Kashiwara’s equivalence for smooth varieties; we also give a definition for finitely generated K-algebras. These new invariants are indicators of F-singularities in characteristic p > 0 and have close connections with characteristic cycle multiplicities in characteristic zero. We characterize the generalized Lyubeznik numbers associated to monomial ideals and compute examples of those associated to determinantal ideals.

In this paper, based on a Harnack-type estimate and a local Sobolev constant bounded for the Calabi flow on closed surfaces, we extend author’s previous results and show the long-time existence and convergence of solutions of 2-dimensional Calabi flow on closed surfaces. Then we establish the uniformization theorem for closed surfaces.

Let g(f) be a Littlewood-Paley square function of f, which belongs to Campanato spaces . We prove that if g(f)(x0) exists (i.e. g(f)(x0) < ∞) for a single point x0 ∈ Rn, then g(f)(x) exists almost everywhere in Rn and . Thus we give an improvement of some earlier results such as in [8], where it is always needed to assume g(f)(x) exists in a set of positive measure in order to get the a.e. existence and boundedness of g(f)(x).

We study the monomial crystal defined by the second author. We show that each component of the monomial crystal can be embedded into a crystal of an extremal weight module introduced by Kashiwara. And we determine all monomials appearing in the components corresponding to all level 0 fundamental representations of quantum affine algebras except for some nodes of . Thus we obtain explicit descriptions of the crystals in these examples. We also give those for the corresponding finite dimensional representations. For classical types, we give them in terms of tableaux. For exceptional types, we list up all monomials.

The present paper is a continuation of [10] where we have proved the principle of limiting absorption for uniformly propagative systems with perturbations of long-range class. In this paper, we consider the Maxwell equation in crystal optics as an important example of non-uniformly propagative systems and, under the same assumptions on perturbations as in [10], we prove the principle of limiting absorption for the stationary problem associated with this equation by using a way similar to that in [10]. We here restrict our consideration to a very special class of non-uniformly propagative systems, but the method developed in this paper will be applicable to more general systems for which non-zero roots of characteristic equations of unperturbed systems are at most double. For another works on the spectral and scattering problems for non-uniformly propagative systems with perturbations of short-range class, see [1], [5], [6], [7] and [8], etc.

In this paper, we prove that, for an integer r with (r, 6) = 1 and 0 < r < 24 and a nonnegative even integer s, the set

is isomorphic to

as Hecke modules under the Shimura correspondence. Here Ms(1) denotes the space of modular forms of weight is the space of newforms of weight 2k on Γ0 (6) that are eigenfunctions with eigenvalues €2 and €3 for Atkin-Lehner involutions W2 and W3, respectively, and the notation ⊕(12/.) means the twist by the quadratic character (12/-). There is also an analogous result for the cases (r, 6) = 3.

In his famous lectures [7] Painlevé investigates general solutions of algebraic differential equations which depend algebraically on some of arbitrary constants. Although his discussions are beyond our understanding, the rigorous and accurate interpretation to make his intuition true would be possible. Successful accomplishments have been done by some authors, for example, Kimura [1], Umemura [8, 9]. From differential algebraic viewpoint in [5] the author introduces the notion of rational dependence on arbitrary constants of general solutions of algebraic differential equations, and in [6] clarifies the relation between it and the notion of strong normality. Here we aim at generalizing to higher order case the result in [4] that in the first order case solutions of equations depend algebraically on those of equations free from moving singularities which are determined uniquely as the closest ones to the given. Part of our result can be seen in [7].

We slightly modify the definitions of q-Hurwitz ζ-functions and q-L-series constructed by J. Satoh. By using these modified functions, we give some relations for the ordinary Dirichlet L-series. Especially we give an elementary proof of Katsurada’s formula on the values of Dirichlet L-series at positive integers.

In this paper, we prove the functional equations for the zeta functions in two variables associated with prehomogeneous vector spaces acted on by maximal parabolic subgroups of orthogonal groups. Moreover, applying the converse theorem of Weil type, we show that elliptic modular forms of integral or half integral weight can be obtained from the zeta functions.

Let K be a commutative ring, A a K-algebra, and B a K-subalgebra of A. The object of this paper is to prove some results on higher derivations (in the sense of Jacobson [4]) of B into A. In § 1 we introduce a notion of equivalence among higher derivations. With this notion of equivalence, we prove in § 2 (Theorem 1) that the equivalence classes of higher K-derivations of B into A are in one-one correspondence with the isomorphism classes of certain filtered B ⊗ KA°-modules, where A° denotes the opposite algebra of A.

For a hyper-Kähler manifold deformation equivalent to a generalized Kummer manifold, we prove that the action of the automorphism group on the total Betti cohomology group is faithful. This is a sort of generalization of a work of Beauville and a more recent work of Boissière, Nieper-Wisskirchen, and Sarti, concerning the action of the automorphism group of a generalized Kummer manifold on the second cohomology group.

This paper deals with cluster sets. While cluster sets can be considered in a more abstract setting, we shall limit ourselves to the study of functions f defined in the open unit disc D of the complex plane and taking their values on the Riemann sphere C̅. For p a point of the unit circle C, we denote by C(f,p) the cluster set of f at p, i.e., the set of all values w ∈ C̅ for which there is a sequence {zn}, zn ∈ D, such that zn → p and f(zn) → w. The point p is called a point of determination for f if C(f, p) is a singleton. In Section 1 we characterize the set of points of determination of a function f defined in D.

A probability measure μ is called unimodal if there is a point α such that the distribution function of μ is convex on (− ∞, α) and concave on (α, ∞). The point α is called a mode of μ. When μ is unimodal, the mode of μ is not always unique; the set of modes is a one point set or a closed interval. If μ is a unimodal distribution with finite variance, Johnson and Rogers [6] give a bound

where m and v are mean and variance of μ (see also [11]).

A projective log variety (X, D) is called a log Fano manifold if X is smooth and if D is a reduced simple normal crossing divisor on Χ with − (KΧ + D) ample. The n-dimensional log Fano manifolds (X, D) with nonzero D are classified in this article when the log Fano index r of (X, D) satisfies either r ≥ n/2 with ρ(X) ≥ 2 or r ≥ n − 2. This result is a partial generalization of the classification of logarithmic Fano 3-folds by Maeda.