We characterize all bounded Hankel operators $\unicode[STIX]{x1D6E4}$ such that $\unicode[STIX]{x1D6E4}^{\ast }\unicode[STIX]{x1D6E4}$ has finite spectrum. We identify spectral data corresponding to such operators and construct inverse spectral theory including the characterization of these spectral data.

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.

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

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.

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).

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 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 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.

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

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.

We show that for self-adjoint Jacobi matrices and Schrödinger operators, perturbed by dissipative potentials in $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\ell ^1({\mathbb{N}})$ and $L^1(0,\infty )$ respectively, the finite section method does not omit any points of the spectrum. In the Schrödinger case two different approaches are presented. Many aspects of the proofs can be expected to carry over to higher dimensions, particularly for absolutely continuous spectrum.

Building on recent work of Bhargava, Elkies and Schnidman and of Kriz and Li, we produce infinitely many smooth cubic surfaces defined over the field of rational numbers that contain rational points.

Let $P^{+}(n)$ denote the largest prime factor of the integer $n$ and $P_{y}^{+}(n)$ denote the largest prime factor $p$ of $n$ which satisfies $p\leqslant y$. In this paper, first we show that the triple consecutive integers with the two patterns $P^{+}(n-1)>P^{+}(n)<P^{+}(n+1)$ and $P^{+}(n-1)<P^{+}(n)>P^{+}(n+1)$ have a positive proportion respectively. More generally, with the same methods we can prove that for any$J\in \mathbb{Z}$, $J\geqslant 3$, the $J$-tuple consecutive integers with the two patterns $P^{+}(n+j_{0})=\min _{0\leqslant j\leqslant J-1}P^{+}(n+j)$ and $P^{+}(n+j_{0})=\max _{0\leqslant j\leqslant J-1}P^{+}(n+j)$ also have a positive proportion, respectively. Second, for $y=x^{\unicode[STIX]{x1D703}}$ with $0<\unicode[STIX]{x1D703}\leqslant 1$ we show that there exists a positive proportion of integers $n$ such that $P_{y}^{+}(n)<P_{y}^{+}(n+1)$. Specifically, we can prove that the proportion of integers $n$ such that $P^{+}(n)<P^{+}(n+1)$ is larger than 0.1356, which improves the previous result “0.1063” of the author.

Suppose that is a distribution of N points in U0, the closed disc of unit area and centred at the origin 0. For every measurable set B in ℝ2, let Z[; B] denote the number of ponts of in B, and write

where µ denotes the usual measure in ℝ2