The existence of the asymptotic probability measure of the Riemann zeta-function was proved in Bohr-Jessen’s classical paper [3] [4].

Let s = σ + it be a complex variable, ζ(s) the Riemann zeta-function, and R an arbitrary rectangle with the edges parallel to the axes. Then, for any σ0 > ½ and T > 0, the set

is Jordan measurable, and we denote the Jordan measure of this set by V(T, R; ξ). Then, Bohr-Jessen’s main result asserts the existence of the limit

The Fock space is a basic structure for the quantum field theory and quantum stochastic calculus. In all the cases, a Fock space can be described as a direct sum of a sequence of some Hilbert spaces, i.e. a Fock space has the form of , where, is the complex field and is a given Hilbert space.

The classification of isoparametric hypersurfaces with four principal curvatures in spheres in [2] hinges on a crucial characterization, in terms of four sets of equations of the 2nd fundamental form tensors of a focal submanifold, of an isoparametric hypersurface of the type constructed by Ferus, Karcher and Münzner. The proof of the characterization in [2] is an extremely long calculation by exterior derivatives with remarkable cancellations, which is motivated by the idea that an isoparametric hypersurface is defined by an over-determined system of partial differential equations. Therefore, exterior differentiating sufficiently many times should gather us enough information for the conclusion. In spite of its elementary nature, the magnitude of the calculation and the surprisingly pleasant cancellations make it desirable to understand the underlying geometric principles.

In this paper, we give a conceptual, and considerably shorter, proof of the characterization based on Ozeki and Takeuchi’s expansion formula for the Cartan-Münzner polynomial. Along the way the geometric meaning of these four sets of equations also becomes clear.

The purpose of this paper is to study the relationship between a certain isomorphism of some rings of functions on Riemann surfaces and a quasi-conformal mapping.

It is well known that two compact Hausdorff spaces are topologically equivalent if and only if their rings of continuous functions are isomorphic. We shall establish an analougous result concerning a function ring on a Riemann surface and the quasi-conformal equivalence.

We define analogs of the Jucys-Murphy elements for the affine Temperley-Lieb algebra and give their explicit expansion in terms of the basis of planar Brauer diagrams. These Jucys-Murphy elements are a family of commuting elements in the affine Temperley-Lieb algebra, and we compute their eigenvalues on the generic irreducible representations. We show that they come from Jucys-Murphy elements in the affine Hecke algebra of type A, which in turn come from the Casimir element of the quantum group . We also give the explicit specializations of these results to the finite Temperley-Lieb algebra.

To a polynomial f over a non-archimedean local field K and a character χ of the group of units of the valuation ring of K one associates Igusa’s local zeta function Z (s, f, χ). In this paper, we study the local zeta function Z(s, f, χ) associated to a non-degenerate polynomial f, by using an approach based on the p-adic stationary phase formula and Néron p-desingularization. We give a small set of candidates for the poles of Z (s, f, χ) in terms of the Newton polyhedron Γ(f) of f. We also show that for almost all χ, the local zeta function Z(s, f, χ) is a polynomial in q−s whose degree is bounded by a constant independent of χ. Our second result is a description of the largest pole of Z(s, f, χtriv) in terms of Γ(f) when the distance between Γ(f) and the origin is at most one.

In this paper we investigate the connection between the u-invariant, u(F), of a formally real field F as defined by Elman and Lam [2] and certain Hasse Principles studied by Elman, Lam and Prestel in [3].

In [10] it was proved that, if X denotes a non singular surface which is a complete intersection in (k an algebraically closed field of characteristic 0) and C an irreducible curve on X, which is a set-theoretic complete intersection in X, then C is actually a complete intersection in X; the key point was to show that Pic (X) modulo the subgroup generated by the class of is torsion-free.

In this paper, the characterization of domains in ℂn by their noncompact automorphism groups are given.

Let G be a finite group and let p be a fixed prime number. If D is any p-subgroup of G, then the problem whether there exists a p-block with D as its defect group is reduced to whether NG(D)/D possesses a p-block of defect 0. Some necessary or sufficient conditions for a finite group to possess a p-block of defect 0 have been known (Brauer-Fowler [1], Green [3], Ito [4] [5]). In this paper we shall show that the existences of such blocks depend on the multiplicative structures of the p-elements of G. Namely, let p be a prime divisor of p in an algebraic number field which is a splitting one for G, o the ring of p-integers and k = o/p, the residue class field.

Gavrilov [2] has shown that a holomorphic function f(z) in the unit disc |z|<1 is normal, in the sense of Lehto and Virtanen [5, p. 86], if and only if f(z) does not possess a sequence of ρ-points in the sense of Lange [4]. Gavrilov has also obtained an analagous result for meromorphic functions by introducing the property that a meromorphic function in the unit disc have a sequence of P-points. He has shown that a meromorphic function in the unit disc is normal if and only if it does not possess a sequence of P-points.

Let k be an infinite field and A a standard G-algebra. This means that there exists a positive integer n such that A = R/I where R is the polynomial ring R := k[Xv …, Xn] and I is an homogeneous ideal of R. Thus the additive group of A has a direct sum decomposition A = ⊕ At where AiAj ⊆ Ai+j. Hence, for every t ≥ 0, At is a finite-dimensional vector space over k. The Hilbert Function of A is defined by

Let A(m), B(n) be positive definite integral matrices and suppose that B is represented by A over each p-adic integers ring Zp. Using the circle method or theory of modular forms in case of n = 1, B, if sufficiently large, is represented by A provided that m ≥ 5. The approach via the theory of modular forms has been extended by [7] to Siegel modular forms to obtain a partial result in the particular case when n = 2, m ≥ 7.

Let (ω: Z2r→Cr be an isomorphism of the free additive group of rank 2r into the complex vector n-space such that the quotient group T = Crω/(Z2r) is compact, i.e., Tω is a complex torus.

W. Sierpisnki [3] demonstrated 82 propositions, called C1-C82, with the aid of the continuum hypothesis. D. A. Martin and R. M. Solovay remarked in [2] that 48 of these propositions followed from Martin’s axiom (MA), 23 were refuted by and three were independent of But the relation of the remaining eight propositions to has been unsettled.

A first order stochastic differential equation is any equation which can be expressed symbolically in the form

(1. 1)

m and σ are called the drift and diffusion coefficients and z( · ) is usually a Brownian motion process.

0. Let p be a prime number or zero and let g be a non-negative integer. Then there is a coarse moduli space Mg for complete non-singular irreducible curves of genus g defined over fields of characteristic p, which is an irreducible variety over the algebraic closure F̅p of the prime field Fp. (Especially, F0 is also denoted by Q as usual.) ([8], [2]). The curve corresponding to a generic point of Mg over F̅p is called a generic curve of genus g.

The aim of this paper is to study a property of a special kind of exact functors and give some applications to projective modules and injective modules.

In section 1 we introduce the notion of faithfully exact functors [Definition 1] as a generalization of the functor T(X) = X⊗M, where M is a faithfully flat module, and give a property of this class of functors [Theorem 1.1]. Next, applying this general theory to functors ⊗ and Horn, we define the notion of faithfully projective modules [Definition 2] and faithfully injective modules [Definition 3]. In the commutative case “faithfully projective” means, however, simply “projective and faithfully flat” [Proposition 2.3]. In section 2, equivalent conditions for a projective module P to be faithfully projective are given [Theorem 2.2, Proposition 2. 3 and 2.4]. And a simpler proof is given to Y. Hinohara’s result [6] asserting that projective modules over an indecomposable weakly noetherian ring are faithfully flat [Proposition 2.5]. In section 3, we consider faithfully injective modules.

In the paper [1] mentioned in the title, the authors attempted to determine the Schur multipliers of the five simple Mathieu groups. In rechecking the calculations, we find that an error was made, leading to incorrect results for M12 and M22. Our purpose here is to compute again the multipliers of M12 and M22, which turn out to be cyclic groups of orders 2 and 6 respectively. The multipliers of M11M23, M24 were originally (and correctly) determined to be trivial.

A 𝔻-semiclassical weight is one which satisfies a particular linear, first-order homogeneous equation in a divided-difference operator 𝔻. It is known that the system of polynomials, orthogonal with respect to this weight, and the associated functions satisfy a linear, first-order homogeneous matrix equation in the divided-difference operator termed the spectral equation. Attached to the spectral equation is a structure which constitutes a number of relations such as those arising from compatibility with the three-term recurrence relation. Here this structure is elucidated in the general case of quadratic lattices. The simplest examples of the 𝔻-semiclassical orthogonal polynomial systems are precisely those in the Askey table of hypergeometric and basic hypergeometric orthogonal polynomials. However within the 𝔻-semiclassical class it is entirely natural to define a generalization of the Askey table weights which involve a deformation with respect to new deformation variables. We completely construct the analogous structures arising from such deformations and their relations with the other elements of the theory. As an example we treat the first nontrivial deformation of the Askey–Wilson orthogonal polynomial system defined by the q-quadratic divided-difference operator, the Askey–Wilson operator, and derive the coupled first-order divided-difference equations characterizing its evolution in the deformation variable. We show that this system is a member of a sequence of classical solutions to the q-Painlevé system.