Let f be a meromorphic mapping of the n-dimensional complex plane Cn into the N-dimensional complex projective space PN(C). We denote by T(r,f) the characteristic function of f and by N(r,f*H) the counting function for a hyperplane H ⊂ PN(C). The purpose of this paper is to establish the following results.

An Lp operator norm estimate of the difference between the Kac operator and the Schrödinger semigroup is proved and used to give a variant of the Trotter product formula for Schrödinger operators in the Lp operator norm. The method of the proof is probabilistic based on the Feynman-Kac formula. The problem is discussed in the relativistic as well as nonrelativistic case.

One of the most useful theorems in classical representation theory is a result due to N. Ito, which can be stated using the classification of the finite simple groups in the following way.

THEOREM (N. Ito, G. Michler). Let Irr (G) be the set of all irreducible complex characters of the finite group G and q be a prime number. Then if and only if G has a normal, abelian Sylow-q-subgroup.

In connection with a Gaussian system X = {X(x); x ∈ M} called Lévy’s Brownian motion (Definition 1), we shall introduce two integral transformations of special type—one is a generalized Radon transform R on a measure space (M, m), and the other is a dual Radon transform R* on another measure space (H, v) such that H ⊂ 2M, the set of all subsets of M (Definition 2). To each Lévy’s Brownian motion X, there is attached a distance d(x, y):= E[(X(x) — X(y)2] on M having a notable property named L1-embeddability. The above measure v on H is then chosen to satisfy

where Δ stands for the symmetric difference.

One of the most challenging problems in complex differential geometry is the following conjecture of Frankel [3].

(F-n) A compact Kaehler manifold M of dimension n with positive sectional (or more generally, positive holomorphic bisectional) curvature is biholomorphic to the complex projective space Pn(C).

Let M be any number field. We let DM, dM, hu, , AM and RegM be the discriminant, the absolute value of the discriminant, the class-number, the Dedekind zeta-function, the ring of algebraic integers and the regulator of M, respectively.

we set If q is any odd prime we let (⋅/q) denote the Legendre’s symbol.

The asymptotic distribution of eigenvalues has been studied by many authors for the Schrõdinger operators —Δ+V with scalar potential growing unboundedly at infinity. Let N(λ) be the number of eigenvalues less than λ of —Δ + V on L2Rnx). Under suitable assumptions on V(x), N(λ) obeys the following asymptotic formula:

We study in this paper a generalization of the notion of a discrete hypergroup with particular emphasis on the relation with systems of orthogonal polynomials. The concept of a locally compact hypergroup was introduced by Dunkl [8], Jewett [12] and Spector [25]. It generalizes convolution algebras of measures associated to groups as well as linearization formulae of classical families of orthogonal polynomials, and many results of harmonic analysis on locally compact abelian groups can be carried over to the case of commutative hypergroups; see Heyer [11], Litvinov [17], Ross [22], and references cited therein. Orthogonal polynomials have been studied in terms of hypergroups by Lasser [15] and Voit [31], see also the works of Connett and Schwartz [6] and Schwartz [23] where a similar spirit is observed.

In 2005, using a famous lemma of Atkin and Swinnerton-Dyer (Some properties of partitions, Proc. Lond. Math. Soc. (3) 4 (1954), 84–106), Yesilyurt (Four identities related to third order mock theta functions in Ramanujan’s lost notebook, Adv. Math. 190 (2005), 278–299) proved four identities for third order mock theta functions found on pages 2 and 17 in Ramanujan’s lost notebook. The primary purpose of this paper is to offer new proofs in the spirit of what Ramanujan might have given in the hope that a better understanding of the identities might be gained. Third order mock theta functions are intimately connected with ranks of partitions. We prove new dissections for two rank generating functions, which are keys to our proof of the fourth, and the most difficult, of Ramanujan’s identities. In the last section of this paper, we establish new relations for ranks arising from our dissections of rank generating functions.

Dans ce mémoir nous démontrons que le groupe d’automorphismes d’un espace fibre principal analytique complexe à base compacte est un groupe de Lie complexe par la topologie de la convergence compacte et ensuite nous étudierons quelques relations entre le groupe d’automorphismes d’un espace fibre principal analytique complexe et le groupe d’homéomorphisme analytiques de sa base.

Let B be a ring and A a subring of B with the common identity element 1. If the residue A-module B/A is inversible as an A-A- bimodule, i.e. B/A ⊗A HomA(B/A, A) ≈ HomA(B/A, A) ⊗A B/A ≈ A, then B is called a quadratic extension of A. In the case where B and A are division rings, this definition coincides with in P. M. Cohn [2]. We can see easily that if B is a Galois extension of A with the Galois group G of order 2, in the sense of [3], and if is a quadratic extension of A. A generalized crossed product Δ(f, A, Φ, G) of a ring A and a group G of order 2, in [4], is also a quadratic extension of A.

In this paper, we use the theta function method [10] to give explicit Doi-Naganuma type maps associated to an imaginary quadratic field K, lifting cusp forms on any congruence subgroup of SL(2, Z) to forms on SL(2, C) automorphic with respect to an appropriate arithmetic discrete subgroup. The case of class number one, and form modular with respect to group Γ0(D) and character χ0 = (–D/*), where –D is the discriminant of K has been treated by Asai [1]. In order to complete his discussion, we must first introduce a more general theta function associated to an indefinite quadratic form (here of type (3, 1)), which we regard as a specialization of a symplectic theta function (see also [4]).

We consider a group G of finite order g = pag′ where p is a prime number and (p, g′) = 1. Let Ω be the algebraic number field which contains the p-th roots of unity. Let K1, K2,…, Kn be the classes of conjugate elements in G and the first m(≦n) classes be p-regular. There exist n distinct (absolutely) irreducible characters x1, x2,…, xn of G.

In this paper we consider the birational classification of pairs (S, ℒ), with S a rational surface and ℒ a linear system on S. We give a classification theorem for such pairs, and we determine, for each irreducible plane curve B, its Cremona minimal models, that is, those plane curves which are equivalent to B via a Cremona transformation and have minimal degree under this condition.

In this paper we study the map properties of the homogeneous fractional integral operator TΩ, α on Lp(ℝn) for n/α ≤ p ≤ ∞.

We prove that if Ω satisfies some smoothness conditions on Sn−1 then TΩ, α is bounded from Ln/α(ℝn) to BMO(ℝn), and from Lp(ℝn) (n/α < p ≤ ∞) to a class of the Campanato spaces l, λ (ℝn), respectively. As the corollary of the results above, we show that when Ω satisfies some smoothness conditions on Sn−1 the homogeneous fractional integral operator TΩ, α is also bounded from Hp(ℝn) (n/(n + α) ≤ p ≤ 1) to Lq(ℝn) for 1/q = 1/p-α/n. The results are the extensions of Stein-Weiss (for p = 1) and Taibleson-Weiss’s (for n/(n + α) ≤ p < 1) results on the boundedness of the Riesz potential operator Iα on the Hardy spaces Hp(ℝn).

In [2, Section 1.6] Peskine and Szpiro prove a theorem on adic approximations of finite free resolutions over local rings which, together with M. Artin’s Approximation Theorem [1], allows them to “descend” modules of finite projective dimension over the completions of certain local rings to modules of finite projective dimension over finite étale extensions of those rings. In this note we will prove a more general result, which deals with the change in homology under an adic approximation of any complex of finitely generated modules over a noetherian ring, and which allows one to descend not only modules of finite projective dimension, but also the Euler characteristic or intersection multiplicity of two such modules.

Let D be an open set in the 2-plane, C its boundary, z0 a point on C, and f(z) a one-valued meromorphic function in D.

In this paper, using techniques of value distribution theory, we give a uniqueness theorem for meromorphic mappings of ℙm into ℙPn with (3n+1) moving targets and truncated multiplicities.

Let k be an algebraic number field of finite degree over Q, the field of rationals, and K be an extension of finite degree over k. By the use of the class number of algebraic tori, we can introduce an arithmetical invariant E(K/k) for the extension K/k. When k = Q and K is quadratic over Q, the formula of Gauss on the genera of binary quadratic forms, i.e. the formula where = the class number of K in the narrow sense, the number of classes is a genus of the norm form of K/Q and tK = the number of distinct prime factors of the discriminant of K, may be considered as an equality between E(K/Q) and other arithmetical invariants of K.

The first two Hilbert coefficients of a primary ideal play an important role in commutative algebra and in algebraic geometry. In this paper we give a complete algebraic structure of the Sally module of integrally closed ideals $I$ in a Cohen–Macaulay local ring $A$ satisfying the equality $\text{e}_{1}(I)=\text{e}_{0}(I)-\ell _{A}(A/I)+\ell _{A}(I^{2}/QI)+1,$ where $Q$ is a minimal reduction of $I$, and $\text{e}_{0}(I)$ and $\text{e}_{1}(I)$ denote the first two Hilbert coefficients of $I,$ respectively, the multiplicity and the Chern number of $I.$ This almost extremal value of $\text{e}_{1}(I)$ with respect to classical inequalities holds a complete description of the homological and the numerical invariants of the associated graded ring. Examples are given.