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.

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.

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.

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 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.

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 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.

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.

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).

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.

characterization of the spaces dual to weighted Lorentz spaces are given by means of reverse Hölder inequalities (Theorems 2.1, 2.2). This principle of duality is then applied to characterize weight functions for which the identity operator, the Hardy-Littlewood maximal operator and the Hilbert transform are bounded on weighted Lorentz spaces.

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.

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 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.

The geometry of quadric varieties (hypersurfaces) in finite projective spaces of N dimensions has been studied by Primrose (12) and Ray-Chaudhuri (13). In this paper we study the geometry of another class of varieties, which we call Hermitian varieties and which have many properties analogous to quadrics. Hermitian varieties are defined only for finite projective spaces for which the ground (Galois field) GF(q2) has order q2, where q is the power of a prime. If h is any element of GF(q2), then = hq is defined to be conjugate to h.

Let A denote the Banach algebra of functions analytic in the open unit disc D and continuous in . If f and its first m derivatives belong to A, then the boundary function f(eiθ) belongs to Cm(∂D). The space Am of all such functions is a Banach algebra with the topology induced by Cm(∂D). If all the derivatives of/ belong to A, then the boundary function belongs to C∞(∂D), and the space A∞ all such functions is a topological algebra with the topology induced by C∞(∂D). In this paper we determine the structure of the closed ideals of A∞ (Theorem 5.3).

Beurling and Rudin (see e.g. [7, pp. 82-89; 10]) have characterized the closed ideals of A, and their solution suggests a possible structure for the closed ideals of A∞.

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.