Let G be a bounded, open, connected, non-empty subset of the complex plane C. We put the usual two dimensional (Lebesgue) area measure on G and consider the Hilbert space L2(G) that consists of the complex-valued, measurable functions defined on G that are square integrable. The inner product on L2(G) is given by the norm ‖h‖2 of a function h in L2(G) is given by ‖h‖2 = (∫G|h|2)1/2.

The Bergman space of G, denoted La2(G), is the set of functions in L2(G) that are analytic on G. The Bergman space La2(G) is actually a closed subspace of L2(G) (see [12 , Section 1.4]) and thus it is a Hilbert space.

Let G denote the closure of G and let C(G) denote the set of continuous, complex-valued functions defined on G.

1. Introduction. One of the most remarkable q-extensions of the classical beta integral was recently introduced by Askey and Wilson [1]

(1.1)

where |q| < 1 and the pairwise products of {a, b, c, d} as a multiset do not belong to the set {qj, j = 0, – 1, – 2, …}. The contour C is the unit circle described in the positive direction, but with suitable deformations to separate the sequences of poles converging to zero from the sequences of poles diverging to infinity. The symbol (A; q)∞ is an infinite product defined by

(1.2)

whenever it converges.

The purpose of this note is to point out some connexions between generalized Hadamard matrices (4, 5) and various tactical configurations such as group divisible designs (3), affine resolvable balanced incomplete block designs (1), and orthogonal arrays of strength two (2). Some constructions for these arrays are also indicated.

A balanced incomplete block design (BIBD) with parameters v, b, r, k, λ is an arrangement of v symbols called treatments into b subsets called blocks of k < v distinct treatments such that each treatment occurs in r blocks and any pair of treatments occurs in λ blocks.

§ 1. An Epstein zeta function (in its simplest form) is a function represented by the Dirichlet's series where a1… ak are real and n1, n2, … nk run through integral values. The properties of this function are well known and the simplest of them were proved by Epstein [2, 3]. The aim of this note is to define a general class of Dirichlet's series, of which the above can be viewed as an instance, and to discuss the problem of analytic continuation of such series.

In some parts of the Geometry of Numbers it is convenient to know that certain affine invariants associated with convex regions attain their upper and lower bounds. A classical example is the quotient of the critical determinant by the content (if the region is symmetrical) for which Minkowski determined the exact lower bound 2–n. The object of this paper is to prove that for continuous functions of bounded regions the bounds are attained. The result is, of course, deduced from the selection theorem of Blaschke, and itself is a compactness theorem about the space of affine equivalence-classes.

Let N be a finite set of n elements. A collection ﹛S1, S2, … , Sm﹜ of subsets of N is called a determining collection if an arbitrary subset T of N is uniquely determined by the cardinalities of the intersections Si ⋂ T, 1 ≤ i ≤ m. The purpose of this paper is to study the minimum value D(n) of m for which a determining collection of m subsets exists.

This problem can be expressed as a coin-weighing problem (1; 7).

In a recent paper Cantor (1) showed that D(n) = O(n/log log n), thus proving a conjecture of N. J. Fine (3) that D(n) = o(n). More recently Erdös and Rényi (2), Söderberg and Shapiro (7), Berlekamp, Mills, and Leo Moser have independently found proofs that D(n) = O(n/log n).

The concept of uniform convexity in a normed linear space is based on the geometric condition that if two members of the unit ball are far apart, then their midpoint is well inside the unit ball. We consider here a generalization of this concept whose geometric significance is that the collection of all chords of the unit ball that are parallel to a fixed direction and whose lengths are bounded below by a positive number has the property that the midpoints of the chords lie uniformly deep inside the unit ball. This notion, called uniform convexity in every direction (UCED), was first used by A. L. Garkavi [5; 6] to characterize normed linear spaces for which every bounded subset has at most one Čebyŝev center. We discuss questions of renorming spaces so as to be UCED and forming products of spaces that are uniformly convex in every direction.

Let the genus of an orientable 2-manifold M be denoted by γ(M). The genus, γ(G), of a graph G is then the smallest of the numbers γ(N) for orientable 2-manifolds N in which G can be embedded. An embedding of G in M is called minimal if γ(G) = γ(M). When each component of the complement of G in M is an open 2-cell, the embedding of G in M is called a 2-cell embedding. In (3), J. W. T. Youngs has shown that each minimal embedding is a 2-cell embedding. It follows from the results of (3) that for each graph G, there is a number d(G), called the regional number, such that for any 2-cell embedding of G in an orientable 2-manifold, the number of (2-cell) complementary domains of G is ⩽d(G), with equality holding if and only if the embedding is minimal.

It has been shown elsewhere (1, 4) that the number of rooted non-separable planar maps with n edges is

In the present paper we improve upon this result by finding the number fi,j of rooted non-separable planar maps with i + 1 vertices and j + 1 faces. We use the definitions of (1).

Among the non-separable planar maps only the loop-map and the link-map have i = 0 or j = 0. We therefore confine our attention to the case in which i and j are both positive.

A reflection in Euclidean n-dimensional space is a particular type of congruent transformation which is of period two and leaves a prime (i.e., hyperplane) invariant. Groups generated by a number of these reflections have been extensively studied [5, pp. 187-212]. They are of interest since, with very few exceptions, the symmetry groups of uniform polytopes are of this type. Coxeter has also shown [4] that it is possible, by Wythoff's construction, to derive a number of uniform polytopes from any group generated by reflections. His discussion of this construction is elegantly illustrated by the use of a graphical notation [4, p. 328; 5, p. 84] whereby the properties of the polytopes can be read off from a simple graph of nodes, branches, and rings.

We consider random i.i.d. samples of absolutely continuous measures on bounded connected domains. We prove an upper bound on the $\infty $-transportation distance between the measure and the empirical measure of the sample. The bound is optimal in terms of scaling with the number of sample points.

A P-space is a topological space in which every Gδ-set is open. P-spaces are fairly rare. For example, the only compact (or even pseudocompact) P-spaces are finite. A larger class of spaces, the almost-P-spaces, consists of those spaces in which G δ-sets have dense interiors. The almost-P-spaces are far less restricted than the P-spaces—for example, there are infinite, compact, connected almost-P-spaces. In this paper, we study almost-P-spaces and raise a number of questions relating to them.

The initial value problem for a semi-linear fractional heat equation is investigated. In the focusing case, global well-posedness and exponential decay are obtained. In the focusing sign, global and non global existence of solutions are discussed via the potential well method.

We introduce and study a remarkable family of real probability measures ${{\pi }_{st}}$ that we call free Bessel laws. These are related to the free Poisson law $\pi $ via the formulae ${{\text{ }\!\!\pi\!\!\text{ }}_{s1}}={{\text{ }\!\!\pi\!\!\text{ }}^{\boxtimes s}}$ and $\text{ }\pi {{\text{ }}_{1t}}=\text{ }\pi {{\text{ }}^{\boxtimes }}^{t}$. Our study includes definition and basic properties, analytic aspects (supports, atoms, densities), combinatorial aspects (functional transforms, moments, partitions), and a discussion of the relation with random matrices and quantum groups.

An odd unbounded (respectively, $p$-summable) Fredholm module for a unital Banach $*$-algebra, $A$, is a pair $(H,D)$ where $A$ is represented on the Hilbert space, $H$, and $D$ is an unbounded self-adjoint operator on $H$ satisfying:

(1) ${{(1+{{D}^{2}})}^{-1}}$ is compact (respectively, Trace $\left( {{\left( 1+{{D}^{2}} \right)}^{-(p/2)}} \right)<\infty $, and

(2) $\{a\in A|\,\,[D,a]\,\,\text{is}\,\,\text{bounded}\}\text{ }$ is a dense $*$- subalgebra of $A$.

If $u$ is a unitary in the dense $*$-subalgebra mentioned in (2) then

$$uD{{u}^{*}}=D+u[D,{{u}^{*}}]=D+B$$

where $B$ is a bounded self-adjoint operator. The path

$$D_{t}^{u}:=(1-t)D+tuD{{u}^{*}}=D+tB$$

is a “continuous” path of unbounded self-adjoint “Fredholm” operators. More precisely, we show that

$$F_{t}^{u}:=D_{t}^{u}{{\left( 1+{{\left( D_{t}^{u} \right)}^{2}} \right)}^{-\frac{1}{2}}}$$

is a norm-continuous path of (bounded) self-adjoint Fredholm operators. The spectral flow of this path $\left\{ F_{t}^{u} \right\}$ (or $\{D_{t}^{u}\}$) is roughly speaking the net number of eigenvalues that pass through 0 in the positive direction as $t$ runs from 0 to 1. This integer,

$$\text{sf}\left( \left\{ D_{t}^{u} \right\} \right):=\text{sf}\left( \left\{ F_{t}^{u} \right\} \right),$$

recovers the pairing of the $K$-homology class $[D]$ with the $K$-theory class $[u]$.

We use I.M. Singer's idea (as did E. Getzler in the $\theta $-summable case) to consider the operator $B$ as a parameter in the Banach manifold, ${{B}_{\text{sa}}}\left( H \right)$, so that spectral flow can be exhibited as the integral of a closed 1-form on this manifold. Now, for $B$ in our manifold, any $X\,\in \,{{T}_{B}}({{B}_{\text{sa}}}(H))$ is given by an $X$ in ${{B}_{\text{sa}}}(H)$ as the derivative at $B$ along the curve $t\,\mapsto \,B\,+\,tX$ in the manifold. Then we show that for $m$ a sufficiently large half-integer:

$$\alpha (X)=\frac{1}{{{{\tilde{C}}}_{m}}}\text{Tr}\left( X{{\left( 1+{{\left( D+B \right)}^{2}} \right)}^{-m}} \right)$$

is a closed 1-form. For any piecewise smooth path $\left\{ {{D}_{t}}=D+{{B}_{t}} \right\}$ with ${{D}_{0}}$ and ${{D}_{1}}$unitarily equivalent we show that

$$\text{sf}\left( \left\{ {{D}_{t}} \right\} \right)=\frac{1}{{{{\tilde{C}}}_{m}}}\int_{\text{0}}^{\text{1}}{\text{T}}\text{r}\left( \frac{d}{dt}\left( {{D}_{t}} \right){{\left( 1+D_{1}^{2} \right)}^{-m}} \right)dt$$

the integral of the 1-form $\alpha $. If ${{D}_{0}}$ and ${{D}_{1}}$ are not unitarily equivalent, wemust add a pair of correction terms to the right-hand side. We also prove a bounded finitely summable version of the form:

$$\text{sf}\left( \left\{ {{F}_{t}} \right\} \right)=\frac{1}{{{C}_{n}}}\int_{\text{0}}^{\text{1}}{\text{T}}\text{r}\left( \frac{d}{dt}\left( {{F}_{t}} \right){{\left( 1-{{F}_{t}}^{2} \right)}^{n}} \right)dt$$

for $n\ge \frac{p-1}{2}$ an integer. The unbounded case is proved by reducing to the bounded case via the map $D\mapsto F=D{{(1+{{D}^{2}})}^{-\frac{1}{2}}}$ We prove simultaneously a type II version of our results.

For $c\,>\,1$ we denote by ${{\pi }_{c}}\left( x \right)$ the number of integers $n\,\le \,x$ such that $\left\lfloor {{n}^{c}} \right\rfloor $ is prime. In 1953, Piatetski-Shapiro has proved that ${{\pi }_{c}}\left( x \right)\,\sim \,\frac{x}{c\,\log \,x},\,x\to \,+\infty $ holds for $c\,<\,12/11$. Many authors have extended this range, which measures our progress in exponential sums techniques. In this article we obtain $c\,<\,1.16117\ldots $.

The algebraic structure of a topological algebra influences its topological structure in a way which is profound but not well understood. (See § 7 below for various examples.) Here we examine this influence rather generally, and give a fairly complete analysis of one of the many forms it can take, namely, the influence of the identities of on the group identities obeyed by the homotopy group (or groups of the components) of .

A homogeneous real polynomial $p$ is hyperbolic with respect to a given vector $d$ if the univariate polynomial $t\,\mapsto \,p(x\,-\,td)$ has all real roots for all vectors $x$. Motivated by partial differential equations, Gårding proved in 1951 that the largest such root is a convex function of $x$, and showed various ways of constructing new hyperbolic polynomials. We present a powerful new such construction, and use it to generalize Gårding’s result to arbitrary symmetric functions of the roots. Many classical and recent inequalities follow easily. We develop various convex-analytic tools for such symmetric functions, of interest in interior-point methods for optimization problems over related cones.

In [5] C. Akemann and G. Pedersen defined four concepts of semicontinuity for elements of A**, the enveloping W*-algebra of a C*-algebra A. For three of these the associated classes of lower semicontinuous elements are , and (notation explained in Section 2), and we will call these the classes of strongly lsc, middle lsc, and weakly lsc elements, respectively. There are three corresponding concepts of continuity: The strongly continuous elements are the elements of A itself, the middle continuous elements are the multipliers of A, and the weakly continuous elements are the quasi-multipliers of A. It is natural to ask the following questions, each of which is three-fold.

(Q1) Is every lsc element the limit of a monotone increasing net of continuous elements?

(Q2) Is every positive lsc element the limit of an increasing net of positive continuous elements?

(Q3) If h ≧ k, where h is lsc and k is usc, does there exist a continuous x such that h ≧ x ≧ k?

In this paper we prove a certain ${{L}^{2}}$-estimate for multilinear Fourier multiplier operators with multipliers of limited smoothness. As a consequence, we extend the result of Calderón and Torchinsky in the linear theory to the multilinear case. The sharpness of our results and some related estimates in Hardy spaces are also discussed.