If G is a torsion–free group and F is a field, is the group ring F[G] a ring without zero divisors? This is true if G is an ordered group or various generalizations thereof - beyond this the question remains untouched. This paper proves a related result.

Non-commutative measure theory embraces measure theory on cr-fields of subsets of a set, on projection lattices of von Neumann algebras or JBW-algebras and on hypergraphs alike [20], [27], [33], [37], [39], [40], [41]. Due to the unifying structure of an orthoalgebra concepts can easily be transferred from one branch to the other. Additional conceptual inpetus is obtained from the logico-probabilistic foundations of quantum mechanics (see [6], [19], [21]).

In the late seventies the author studied the Jordan-Hahn decomposition of measures on orthomodular posets and certain graphs. These investigations revealed an interesting geometrical aspect of this decomposition in that the Jordan-Hahn property of the convex set of probability charges on a finite orthomodular poset can be characterized in terms of the extreme points of the unit ball of the Banach space dual of the base normed space of Jordan charges.

Let σ be a finite positive singular Borel measure defined on Euclidean space R N . For w ∈ R N and y > 0, its Poisson integral is defined by the formula

where CN is chosen so that

Since σ is singular, almost everywhere with respect to Lebesgue measure on R N . On the other hand, almost everywhere dσ. It follows that for all sufficiently small y,

is a non-empty open subset of R N . If σ has compact support then |E y | → 0 as y → 0, where |E y | denotes the Lebesgue measure of E y . In this paper we give a lower bound on the rate at which |E y | may go to zero. The lower bound depends on the smoothness of the measure; the smoother the measure, the more slowly |E y | may approach 0.

The linearized theory of surface waves leads to several mixed boundary value problems which have been investigated by various methods. As the physical background of the theory has been repeatedly discussed, it will suffice to deal here mainly with the mathematical aspect of the question.

We explore the geometric notion of prolongations in the setting of computational algebra, extending results of Landsberg and Manivel which relate prolongations to equations for secant varieties. We also develop methods for computing prolongations that are combinatorial in nature. As an application, we use prolongations to derive a new family of secant equations for the binary symmetric model in phylogenetics.

Our motivation for this note originates with consideration of a subset A of Euclidean w-space, Rn, which contains only part of its boundary. The part contained is t h a t part of the closure of A which cannot be “bettered“ within A with respect to the preference associated with a fixed closed convex cone Γ. Here b is preferred to a if and only if a — b ∊ Γ; if, for instance, Γ is the non-negative orthant of Rn, this preference is ordinary vector inequality.

Let ${{S}_{k}}(\Gamma )$ be the space of holomorphic cusp forms of even integral weight $k$ for the full modular group. Let ${{\lambda }_{f}}(n)$ and ${{\lambda }_{g}}(n)$ be the $n$ -th normalized Fourier coefficients of two holomorphic Hecke eigencuspforms $f(z),\,g(z)\,\in \,{{S}_{k}}(\Gamma )$ , respectively. In this paper we are able to show the following results about higher moments of Fourier coefficients of holomorphic cusp forms.

(i)For any $\varepsilon \,>\,0$ , we have

$$\sum\limits_{n\le x}{\lambda _{f}^{5}(n){{\ll }_{f,\varepsilon }}}{{x}^{\frac{15}{16}+\varepsilon }}\text{and}\sum\limits_{n\le x}{\lambda _{f}^{7}(n){{\ll }_{f,\varepsilon }}}{{x}^{\frac{63}{64}+\varepsilon }}.$$

(ii)If $\text{sy}{{\text{m}}^{3\,}}{{\pi }_{f}}\,\ncong \,\text{sy}{{\text{m}}^{3\,}}{{\pi }_{g}}\,$ , then for any $\varepsilon \,>\,0$ , we have

$$\sum\limits_{n\le x}{\lambda _{f}^{3}(n)\lambda _{g}^{3}(n){{\ll }_{f,\varepsilon }}}{{x}^{\frac{31}{32}+\varepsilon }};$$

If $\text{sy}{{\text{m}}^{2}}\,{{\pi }_{f}}\,\ncong \,\text{sy}{{\text{m}}^{2}}\,{{\pi }_{g}}$ , then for any $\varepsilon \,>\,0$ , we have

$$\sum\limits_{n\le x}{\lambda _{f}^{4}(n)\lambda _{g}^{2}(n)}=cx\log x+{c}'x+{{O}_{f,\varepsilon }}({{x}^{\frac{31}{32}+\varepsilon }});$$

If $\text{sy}{{\text{m}}^{2}}\,{{\pi }_{f}}\,\ncong \,\text{sy}{{\text{m}}^{2}}\,{{\pi }_{g}}$ and $\text{sy}{{\text{m}}^{4}}{{\pi }_{f}}\,\ncong \,\text{sy}{{\text{m}}^{4}}{{\pi }_{g}}$ , then for any $\varepsilon \,>\,0$ , we have

$$\sum\limits_{n\le x}{\lambda _{f}^{4}(n)\lambda _{g}^{4}(n)}=xP(\log x)+{{O}_{f,\varepsilon }}({{x}^{\frac{127}{128}+\varepsilon }}),$$

where $P\left( x \right)$ is a polynomial of degree 3.

We investigate mean convergence of Lagrange interpolation at the zeros of orthogonal polynomials pn (W 2, x) for Erdös weights W 2 = e -2Q . The archetypal example is Wk,α = exp(—Qk,α ), where

α > 1, k ≥ 1, and is the k-th iterated exponential. Following is our main result: Let 1 < p < ∞, Δ ∊ ℝ, k > 0. Let Ln [f] denote the Lagrange interpolation polynomial to ƒ at the zeros of pn (W 2, x) = pn (e -2Q , x). Then for

to hold for every continuous function ƒ: ℝ —> ℝ satisfying

it is necessary and sufficient that

We develop a method for deriving integral representations of certain orthogonal polynomials as moments. These moment representations are applied to find linear and multilinear generating functions for $q$ -orthogonal polynomials. As a byproduct we establish new transformation formulas for combinations of basic hypergeometric functions, including a new representation of the $q$ -exponential function $\text{ }{{\varepsilon }_{q}}$ .

The genus series for maps is the generating series for the number of rooted maps with a given number of vertices and faces of each degree, and a given number of edges. It captures topological information about surfaces, and appears in questions arising in statistical mechanics, topology, group rings, and certain aspects of free probability theory. An expression has been given previously for the genus series for maps in locally orientable surfaces in terms of zonal polynomials. The purpose of this paper is to derive an integral representation for the genus series. We then show how this can be used in conjunction with integration techniques to determine the genus series for monopoles in locally orientable surfaces. This complements the analogous result for monopoles in orientable surfaces previously obtained by Harer and Zagier. A conjecture, subsequently proved by Okounkov, is given for the evaluation of an expectation operator acting on the Jack symmetric function. It specialises to known results for Schur functions and zonal polynomials.

We discuss the subRiemannian geometry induced by two noncommutative vector fields which are left invariant on the Lie group ${{\mathbb{S}}^{3}}$ .

We shall be concerned throughout this paper with the Sobolev space Wm,p (G) and the existence and compactness (or lack of it) of its imbeddings (i.e. continuous inclusions) into various LP spaces over G, where G is an open, not necessarily bounded subset of n-dimensional Euclidean space En . For each positive integer m and each real p ≧ 1 the space Wm,p (G) consists of all u in LP (G) whose distributional partial derivatives of all orders up to and including m are also in LP (G). With respect to the norm

1.1

Wm,p (G) is a Banach space. It has been shown by Meyers and Serrin [9] that the set of functions in Cm (G) which, together with their partial derivatives of orders up to and including m, are in LP (G) forms a dense subspace of Wm,p (G).

1. Definitions and notation. A generalized Steiner system (t-design, tactical configuration) with parameters t, λt, k, v is a system (T, B), where T is a set of v elements, B is a set of blocks each of which is a k-subset of T (but note that blocks bi and bj may be the same k-subset of T) and such that every set of t elements of T belongs to exactly λt of the blocks. If we put λt = u we denote by Su(t, k, v) the collection of all systems with these parameters. Thus Q ∈ Su(t, k, v) means Q = (T, B) is a system with the given parameters. If λt = u = 1, we write S(t, k, v) instead of S 1(t, k, v) and refer to the system as a Steiner system. If t = 2, the system is called a balanced incomplete block design.

We consider a generalization of a trace formula identity of Jacquet, in the context of the symmetric spaces GL(2n)/GL(/n) × GL(n) and G′/H′. Here G′ is an inner form of GL(2n) over F with a subgroup H′ isomorphic to GL(n, E) where E/F is a quadratic extension of number field attached to a quadratic idele class character η of F. A consequence of this identity would be the following conjecture: Let π be an automorphic cuspidal representation of GL(2n). If there exists an automorphic representation π′ of G′ which is related to π by the Jacquet-Langlands correspondence, and a vector ø in the space of π′ whose integral over H′ is nonzero, then both L(1/2, π) and L(1/2,π ⊗ η) are nonvanishing. Moreover, we have L(1/2, π)L(1/2, π ⊗ η) > 0. Here the nonvanishing part of the conjecture is a generalization of a result of Waldspurger for GL(2) and the nonnegativity of the product is predicted from the generalized Riemann Hypothesis. In this article, we study the corresponding local orbital integrals for the symmetric spaces. We prove the "fundamental lemma for the unit Hecke functions" which says that unit Hecke functions have "matching" orbital integrals. This serves as the first step toward establishing the trace formula identity and in the same time it provides strong evidence for what we proposed.

The problem of expanding an arbitrary function in a series of characteristic solutions of the ordinary differential equation

(1.1)

and the boundary relations

(1.2)

is well known. The various discussions are distinguished by the manner in which a parameter λ appears in the differential system and by the number of points at which the boundary conditions apply. The case in which the boundary conditions apply at intermediate as well as at the end points of a fundamental interval has been considered by Wilder (3).

By a d-polytope we shall mean a d-dimensional convex polytope. We shall denote a j-dimensional face (or j-face) of a polytope by Fj . Every d-polytope P has proper j-faces for 0 ≦j ≦d — 1 and we shall also say that P is a d-face of itself. Observe that every face of a polytope is again a polytope. The collection of all convex polytopes shall be denoted by .

Un anneau commutatif unitaire est dit réflexif si tous ses idéaux sont divisoriels [3; 6]. On dira qu'un anneau commutatif unitaire et intègre est un anneau de Mori, si ses idéaux divisoriels entiers vérifient la condition de chaîne ascendante [8]. Enfin, on appellera provisoirement, M-anneau, un anneau de Mori, tel que tout idéal engendré par deux éléments est divisoriel. Un anneau noethérien réflexif est un M -anneau; dans la première partie de cette note, nous établissons la réciproque. La troisième partie étudie la clôture intégrale de certains anneaux locaux réflexifs non noethériens.

Let X be a Hausdorff space. Let 2X denote the set of all non-empty closed subsets of X. For a subset A of X, we set 2A = {F 6 2X : F ⊆ A}. Recall that the finite topology on 2X is that topology having as a sub-basis the family {2G : G is open in X} U }2X — 2F : F is closed in X). When endowed with this topology, 2X is referred to as the hyper space of X. For the fundamental properties of hyperspaces, we refer the reader to [6; 7]. Following [6], we adopt the following notation: If A0, A1, … , An are subsets of X, we set and for all .

Let K be a field and Mn﹛K) denote the vector space of n X n matrices over K. Marcus [4] posed the following general problem: Let W be a subspace of Mn(K) and S a subset of W. Describe the set L(S, W) of all linear transformations T on W such that T(S) is contained in S.

We formulate a conjecture analogous to Gross' refinement of the Stark conjectures on special values of abelian L-series at s = 0. Some evidence for the conjecture can be obtained, thanks to the fundamental ideas of F. Thaine.