In 2005 Bourgain gave the first explicit construction of a two-source extractor family with min-entropy rate less than $1/2$. His approach combined Fourier analysis with innovative but inefficient tools from arithmetic combinatorics and yielded an unspecified min-entropy rate which was greater than $0.499$. This remained essentially the state of the art until a 2015 breakthrough by Chattopadhyay and Zuckerman in which they gave an alternative approach which produced extractors with arbitrarily small min-entropy rate. In the current work, we revisit the Fourier analytic approach. We give an improved analysis of one of Bourgain’s extractors which shows that it in fact extracts from sources with min-entropy rate near $\frac{21}{44}=0.477\ldots .$ Moreover, we construct a variant of this extractor which we show extracts from sources with min-entropy rate near $4/9=0.444\ldots .$ While this min-entropy rate is inferior to Chattopadhyay and Zuckerman’s construction, our extractors have the advantage of exponential small error, which is important in some applications. The key ingredient in these arguments is recent progress connected to the restriction theory of the finite field paraboloid by Rudnev and Shkredov. This in turn relies on a Rudnev’s point–plane incidence estimate, which in turn relies on Kollár’s generalization of the Guth–Katz incidence theorem.

We provide primitive recursive bounds for the finite version of Gowers’ $c_{0}$ theorem for both the positive and the general case. We also provide multidimensional versions of these results.

Erdoső has asked whether there exists a positive constant c < 1 such that

where pn is the n-th prime. In 1943 Selberg «3» proved, on the assumption of the Riemann Hypothesis, that

uniformly for H > 0. By an elementary extension of Selberg's method we prove the following two results free of any unproved assumptions.

Let x, y, z and n denote positive integers with x < y < z and (x, y, z) = 1. The purpose of this note is to prove two theorems, the first of which is

THEOREM 1. for some positive number Co, and if

then n is less than C, a number which is effectively computable in terms of Co.

In this note we prove that every convex Borel set in a finite dimensional real Banach space can be obtained, starting from the compact convex sets, by the iteration of countable increasing unions and countable decreasing intersections. This question was first raised by V. Klee [1, p. 451]. It was answered affirmatively by Klee for R2 in [2, pp. 109–111] and for R3 by D. G. Larman in [4]. C. A. Rogers has given an equivalent formulation of the question for Rn in [6].

The result of my paper with the title given above (Mathematika, 26 (1979), 72–75) is not new; it was proved by Delone, see [2] and [3]. I have not been able to refer to either of these papers, but the result is given in [1], A similar problem was considered by Kitaoka in [5]. Professor Kneser of Göttingen tells me that he solved the problem about ten years ago in an unpublished manuscript; also that Schering's result [4] is less general and weaker than mine. I am obliged to Professor Peters of Münster for a copy of [1], giving the references [2] to [4], and also for the reference [5].

Prime number theory is concerned with the distribution of primes in sequences of natural numbers, such as arithmetic progressions or polynomial sequences. An extensive range of such questions is embraced by

Hypothesis H (Schinzel, see [10]. Let f1, …, fgbe distinct irreducible polynomials in Z[x] (with positive leading coefficients) and suppose that f1 …fghas no fixed prime divisors. Then there exist infinitely many integers n such that each fi (n) (i = 1, …, g) is prime.

The main object of this paper is to show that an indefinite nonsingular quadratic form which is incommensurable (that is, is not a constant multiple of a form with integral coefficients) takes infinitely many distinct small values, for a suitable interpretation of the word small. This proves a conjecture made by Dr. Chalk in conversation with the writer. I believe that the theorems proved are new and of interest, though they are easy deductions from known results.

In several of his papers, Mordell has developed a method which, in certain cases, leads to an inequality connecting the critical determinant of an n dimensional star body with that of a related n—1 dimensional star body. The purpose of this paper is to exhibit the underlying principle in a general form and to show that the same principle can sometimes be carried further.

A convex compact subset of ℝd is called a convex body. The (Euclidean) surface area and volume of a convex body K are denoted s(K) and v(K) respectively. The support function of a convex body K is denned by h(K, x) = maxy∈K xty and the polar dual of K is given by K0 = {x: |xty|1, y∈K}. Double vertical bars shall denote the Euclidean length of a vector , and S shall denote the unit sphere (the Euclidean unit ball): S = {x: ║x║≤1}. We use for the mixed volume

We say that a system ∑ of equal spheres S1S2, … covers a proportion θ of n-dimensional space, if the limit, as the side of the cube C tends to infinity, of the ratio

of the volume of C covered by the spheres to the volume of C, exists and has the value θ. We say that such a system ∑ has density δ, if the corresponding ratio

has the limit δ as the side of the cube C tends to infinity. We confine our attention to systems ∑ for which both limits exist. It is clear that δ = θ, if no two spheres of the system overlap, i.e. if we have a. packing; and that, in general, the difference δ-θ is a measure of the amount of overlapping.

Let W denote a positive, increasing and continuous function on [1, ∞]. We write to denote the Dirichlettype space of functions f that are holomorphic in the unit disc and for which

Where If W(x) = x for all x, then is the classicial Dirichlet space for which Note also that for every so, by Fatu's theoreum, every function in . ha finite radial(and angular) limits a.e. on the boundary of U. The question of the existence a.e. on ∂U of certain tangential limits for functions in has been considered in [6,11], but we shall be concerned here with the radial variation

i.e., the length of the image of the ray from 0 to eiθ under the mapping w = f(z), and, in particular, with the size of the set of values of θ for which Lf(θ) can be infinite when

The integral representation for the solution of the 2-D Dirichlet problem for harmonic functions with boundary data on closed and open curves is obtained. The solution is expressed as a sum of potentials, the density of which obeys the uniquely solvable Fredholm integral equation of the second kind.

The time-dependent motion of water waves with a parametrically defined free surface is mapped to a fixed time-independent rectangle by an arbitrary transformation. The emphasis is on the general properties of transformations. Special cases are algebraic transformations based on transfinite interpolation, conformal mappings, and transformations generated by nonlinear elliptic partial differential equations. The aim is to study the effect of transformation on variational principles for water waves such as Luke’s Lagrangian formulation, Zakharov’s Hamiltonian formulation, and the Benjamin–Olver Hamiltonian formulation. Several novel features are exposed using this approach: a conservation law for the Jacobian, an explicit form for surface re-parameterization, inner versus outer variations and their role in the generation of hidden conservation laws of the Laplacian. Also some of the differential geometry of water waves becomes explicit. The paper is restricted to the case of planar motion, with a preliminary discussion of the extension to three-dimensional water waves.

Abstract. We consider the structure of the Kac modules V(Λ) for dominant integral doubly atypical weights Λ of the Lie superalgebra s1(2/2). Primitive vectors of V(Λ) are constructed and it is shown that the number of composition factors of V(Λ) for such Λ is in exact agreement with the conjectures of [HKV]. These results are used to show that the extended Kac-Weyl character formula which was proved in [VHKTl] for singly atypical simple modules of s1(m/n), and conjectured to be valid for all finite dimensional irreducible representations of sl(m/n) in [VHKT2] is in fact valid for all finite-dimensional doubly atypical simple modules of s1(2/2).

We show that the $\unicode[STIX]{x1D703}=\infty$ conjecture implies the Riemann hypothesis.

The flow induced by an oscillating circular cylinder which may perform transverse, torsional and axial vibrations is considered. The steady streaming associated with purely transverse vibrations of the cylinder may be significantly modified by the presence of, and interaction with, torsional oscillations. Similarly the interaction between the transverse and axial vibrations introduces a modification to the axial flow, which results in a steady streaming motion in the axial direction.

We define s-dimensional Hausdorff outer measure on subsets of ℝn by

Here | | denotes the diameter of a set. The Souslin sets, which include the Borel sets, are Hs-measurable for any s ≥ 0.

A (convex) d-polytope is the convex hull of a finite set of points in Euclidean d–space Ed. The (Minkowski) sum of two polytopes P1 and P2 is defined by

.

Kan and Thurston, in their paper [5], asked whether each smooth closed manifold other than S2 or RP2 has the same integral homology as a closed aspherical manifold. F. E. A. Johnson in [3], [4] is concerned with the answer to this question when the smooth closed manifold is an n-dimensional sphere Sn. He asked whether there exist aspherical manifolds Xг which have the homology of Sn.