A generalization of the Pólya–Vinogradov inequality

In this paper we consider an approach of Dobrowolski and Williams which leads to a generalization of the Pólya–Vinogradov inequality. We show how the Dobrowolski–Williams approach is related to the classical proof of Pólya–Vinogradov using Fourier analysis. Our results improve upon the earlier work of Bachman and Rachakonda (Ramanujan J. 5:65–71, 2001). In passing, we also obtain sharper explicit versions of the Pólya–Vinogradov inequality.     

Existence of relative integral basis over quadratic fields and Pólya property

Acta Mathematica Hungarica - For $$L/K$$ a finite extension of algebraic number fields, L may or may not have a relative integral basis over K. We show the existence of relative integral basis of a...     

On the Mixed Pólya–Szeg? Principle

In this paper,the mixed Pólya-Szeg? principle is established.By the mixed Pólya-Szeg? principle,the mixed Morrey-Sobolev inequality and some new analytic inequalities are obtained.     

Convex symmetrization and Pólya–Szegö inequality

We prove a Pólya–Szegö inequality involving a convex symmetrization of functions and we investigate the equality case.     

Pólya and Newtonian function fields

Let be a finite function field extension and denote by O _K the integral closure of in K. In this article, we are interested in Pólya fields, that is, fields K, such that the O _K -module Int(O _K ) of integer-valued polynomials over O _K admits a regular basis. We show that the cyclotomic extensions of are Pólya fields, and we characterize some totally imaginary extensions which are Pólya fields. Then, we are interested in Pólya fields K which have a regular basis of the form for some sequences of elements of O _K . For totally imaginary extensions, we show that it is the case if and only if O _K is isomorphic to . This gives a answer to a question raised by Thakur. The author thanks his thesis adviser Jean-Luc Chabert, and Mireille Car for their help, and their valuable advices to do this work. The author thanks also the referee for his valuable remarks.     

Pólya–Schur Inequality and the Green Energy of a Discrete Charge

Doklady Mathematics - The classical Pólya–Schur inequality for the logarithmic energy of a point charge distributed on a circle is generalized to the Green energy with respect to the...     

The Pólya–Chebotarev problem and inverse polynomial images

Consider the problem, usually called the Pólya–Chebotarev problem, of finding a continuum in the complex plane including some given points such that the logarithmic capacity of this continuum is minimal. We prove that each connected inverse image ${\mathcal {T}}_{n}^{-1} ([-1,1])$ of a polynomial  ${\mathcal {T}}_{n}$ is always the solution of a certain Pólya–Chebotarev problem. By solving a nonlinear system of equations for the zeros of ${\mathcal {T}}_{n}^{2}-1$ , we are able to construct polynomials ${\mathcal {T}}_{n}$ with a connected inverse image.     

On One Generalization of the Hardy–Littlewood–Pólya Inequality

We generalize the Hardy–Littlewood–Pólya inequality for numerical sets to certain sets of vectors on a plane.     

On the Pólya Conversion Problem for Permanents and Determinants

Let $ \mathbb{F} $ be a finite field of characteristics different from 2. We show that no bijective map transforms the permanent to the determinant when the cardinality of $ \mathbb{F} $ is sufficiently large. Also we determine Gibson barriers (the maximal and minimal numbers of nonzero elements) for convertible (0, 1)-matrices and solve several related problems in different matrix subspaces. Our results are illustrated by examples. This paper is based on the joint work with G. Dolinar, B. Kuzma, and M. Orel.     

Extensions of the Generalized Pólya Process

Methodology and Computing in Applied Probability - A new self-exciting counting process is here considered, which extends the generalized Pólya process introduced by Cha (Adv Appl Probab...     

Polynomial Ensembles and Pólya Frequency Functions

Journal of Theoretical Probability - We study several kinds of polynomial ensembles of derivative type which we propose to call Pólya ensembles. These ensembles are defined on the spaces of...     

Type II bivariate Pólya–Aeppli distribution

In this paper, we introduce the Type II bivariate Pólya–Aeppli distribution as a compound Poisson distribution with bivariate geometric compounding distribution. The probability mass function, recursion formulas, conditional distributions and some other properties are then derived for this distribution.     

A metric discrepancy result for the Hardy–Littlewood–Pólya sequences

The exact law of the iterated logarithm for discrepancies of the Hardy– Littlewood–Pólya sequences is proved. The exact constant in the law of the iterated logarithm is explicitly computed in the case when the Hardy–Littlewood–Pólya sequence consists of odd numbers.     

The arithmetic of the characteristic Pólya functions

In the framework of the theory of D. Kendall's delphic semigroups are considered problems of divisibility in the semigroup of convex characteristic functions on the semiaxis (0,). Letn ()={:₁¦₁1 or ₁=}, and I_o()={: ₁¦ ₁ N()}. The following results are proved: 1) The semigroup is almost delphic in the sense of R. Davidson. 2) N() is a set of the type G which is dense in (in the topology of uniform convergence on compacta). 3) The class I_o() contains only the function identically equal to one.Translated from Matematicheskie Zametki, Vol. 21, No. 5, pp. 717–725, May, 1977.The author thanks I. V. Ostrovskii for the formulation of the problem and valuable remarks.     

On the shape of random Pólya structures

Panagiotou and Stufler recently proved an important fact on their way to establish the scaling limits of random Pólya trees: a uniform random Pólya tree of size $n$ consists of a conditioned critical Galton–Watson tree $C_n$ and many small forests, where with probability tending to one, as $n$ tends to infinity, any forest $F_n\left(v\right)$, that is attached to a node $v$ in $C_n$, is maximally of size $\left|F_n\left(v\right)\right|=O(logn)$. Their proof used the framework of a Boltzmann sampler and deviation inequalities.In this paper, first, we employ a unified framework in analytic combinatorics to prove this fact with additional improvements for $\left|F_n\left(v\right)\right|$, namely $\left|F_n\left(v\right)\right|=\Theta (logn)$. Second, we give a combinatorial interpretation of the rational weights of these forests and the defining substitution process in terms of automorphisms associated to a given Pólya tree. Third, we derive the limit probability that for a random node $v$ the attached forest $F_n\left(v\right)$ is of a given size. Moreover, structural properties of those forests like the number of their components are studied. Finally, we extend all results to other Pólya structures.     

Functional Limit Theorems for the Pólya Urn

Journal of Theoretical Probability - For the plain Pólya urn with two colors, black and white, we prove a functional central limit theorem for the number of white balls, assuming that the...     

Martingale characterization of Pólya processes and sequences

It is proved that a mixed Poisson process ξ_t is a Pólya process if and only if there exists a nondegenerate linear transform ξ_t → η_t = a(t)ξ_t + b(t) such that η_t is a martingale. A similar result is valid for Pólya sequences.