We investigate the approximation of quadratic Dirichlet $L$-functions over function fields by truncations of their Euler products. We first establish representations for such $L$-functions as products over prime polynomials times products over their zeros. This is the hybrid formula in function fields. We then prove that partial Euler products are good approximations of an $L$-function away from its zeros and that, when the length of the product tends to infinity, we recover the original $L$-function. We also obtain explicit expressions for the arguments of quadratic Dirichlet $L$-functions over function fields and for the arguments of their partial Euler products. In the second part of the paper we construct, for each quadratic Dirichlet $L$-function over a function field, an auxiliary function based on the approximate functional equation that equals the $L$-function on the critical line. We also construct a parametrized family of approximations of these auxiliary functions and prove that the Riemann hypothesis holds for them and that their zeros are related to those of the associated $L$-function. Finally, we estimate the counting function for the zeros of this family of approximations, show that these zeros cluster near those of the associated $L$-function, and that, when the parameter is not too large, almost all the zeros of the approximations are simple.

Let ${\it\mu}_{1},\ldots ,{\it\mu}_{s}$ be real numbers, with ${\it\mu}_{1}$ irrational. We investigate sums of shifted $k$th powers $\mathfrak{F}(x_{1},\ldots ,x_{s})=(x_{1}-{\it\mu}_{1})^{k}+\cdots +(x_{s}-{\it\mu}_{s})^{k}$. For $k\geqslant 4$, we bound the number of variables needed to ensure that if ${\it\eta}$ is real and ${\it\tau}>0$ is sufficiently large then there exist integers $x_{1}>{\it\mu}_{1},\ldots ,x_{s}>{\it\mu}_{s}$ such that $|\mathfrak{F}(\mathbf{x})-{\it\tau}|<{\it\eta}$. This is a real analogue to Waring’s problem. When $s\geqslant 2k^{2}-2k+3$, we provide an asymptotic formula. We prove similar results for sums of general univariate degree-$k$ polynomials.

This paper presents a model of a 1D–1D dynamic multi-structure, supporting propagation of a transition wave. It is used to explain the recent phenomenon of the collapse of the San Saba bridge. An analytical model is supplied with illustrative numerical simulations.

For a completely regular space X, denote by Cp(X) the space of continuous real valued functions on X, endowed with the pointwise convergence topology. The spaces X and Y are t-equivalent if Cp(X) and Cp(Y) are homeomorphic. It is proved that, for metrizable spaces X, the countable dimensionality is preserved by t-equivalence. It is also shown that this relation preserves absolute Borel classes greater than 2 and all projective classes.

The isotropic constant $L_{K}$ is an affine-invariant measure of the spread of a convex body $K$. For a $d$-dimensional convex body $K$, $L_{K}$ can be defined by $L_{K}^{2d}=\det (A(K))/(\text{vol}(K))^{2}$, where $A(K)$ is the covariance matrix of the uniform distribution on $K$. It is an open problem to find a tight asymptotic upper bound of the isotropic constant as a function of the dimension. It has been conjectured that there is a universal constant upper bound. The conjecture is known to be true for several families of bodies, in particular, highly symmetric bodies such as bodies having an unconditional basis. It is also known that maximizers cannot be smooth. In this work we study bodies that are neither smooth nor highly symmetric by showing progress towards reducing to a highly symmetric case among non-smooth bodies. More precisely, we study the set of maximizers among simplicial polytopes and we show that if a simplicial polytope $K$ is a maximizer of the isotropic constant among $d$-dimensional convex bodies, then when $K$ is put in isotropic position it is symmetric around any hyperplane spanned by a $(d-2)$-dimensional face and the origin. By a result of Campi, Colesanti and Gronchi, this implies that a simplicial polytope that maximizes the isotropic constant must be a simplex.

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

The stability equation of the asymptotic suction boundary layer profile, using a linear stability analysis, is transformed into a generalised hypergeometric equation. The solution of the stability problem may thus be written formally in terms of the relevant generalised hypergeometric functions. An asymptotic analysis is carried out on these functions for large values of the Reynolds number, and the asymptotic representation of the solutions shown to agree with that given by the usual Orr–Sommerfeld analysis.

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].

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

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].

We give a new proof of a recent result of Kohnen–Martin on a characterization of degree 2 Siegel cusp forms by the growth of their Fourier coefficients. Our main tools are Koecher–Maass series, Imai’s converse theorem and the theory of singular modular forms.

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.

The purpose of this article is to generalize some known characterizations of Banach space properties in terms of graph preclusion. In particular, it is shown that superreflexivity can be characterized by the non-equi-bi-Lipschitz embeddability of any family of bundle graphs generated by a non-trivial finitely branching bundle graph. It is likewise shown that asymptotic uniform convexifiability can be characterized within the class of reflexive Banach spaces with an unconditional asymptotic structure by the non-equi-bi-Lipschitz embeddability of any family of bundle graphs generated by a non-trivial $\aleph _{0}$-branching bundle graph. For the specific case of $L_{1}$, it is shown that every countably branching bundle graph bi-Lipschitzly embeds into $L_{1}$ with distortion no worse than $2$.

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

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 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.

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.

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.

We show that under the Eikonal abrasion model, prescribing uniform normal speed in the direction of the inward surface normal, the isoperimetric quotient of a convex shape is decreasing monotonically.