Periodic stationary processes and the Pólya-Vinogradov inequality

Inequalities for the moments of the maximum of the sums of elements of a random stationary sequence are suggested. The consequences of the results obtained include the Pólya-Vinogradov inequality for sums of the Dirichlet characters.     

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.     

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...     

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 Norm of a Self-Adjoint Operator and a New Bilinear Integral Inequality

In this paper, the expression of the norm of a self-adjoint integral operator T ： L^2（0, ∞） → L^2 （0, ∞） is obtained. As applications, a new bilinear integral inequality with a best constant factor is established and some particular cases are considered.     

On a Trigonometric Inequality of Vinogradov

In this paper, we improve an inequality of Vinogradov by analysis method.     

On a Problem of Hayman

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On the Equation and the Number of Solution of Znam's Problem

§1. In 1972,St.Znam posed the problem whether for every s>1 thereexist integers x_i>1,i=1,…,s such that x_i is a proper divisor of the numberx_1…x_(i-1)x_(i 1)…x_s 1 for i=1,…, s.Without loss of generality, we may assume1相似文献   

On a theorem of G. Pólya

f(z), :f(n)=0 (n=0, ±1, ±2, ...). ((n)} L _p ,p>1, .     

On a Generalization of Martins’ Inequality

 Let be an increasing nonconstant sequence of positive real numbers. Under certain conditions on this sequence we prove the following inequality

On a Generalization of Martins’ Inequality

 Let be an increasing nonconstant sequence of positive real numbers. Under certain conditions on this sequence we prove the following inequality

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.     

An Elementary Proof of Pólya-Vinogradov's Inequality

Periodica Mathematica Hungarica -     

On the Number and Locations of the Eigenvalues of Generalized Eigenvalue Problem

The gerleralized eigenvalue problem Aχ=λBχ (1) where A and B are teal symmetric n×n matrices,is one 0f the basie Problems in matrix theory,which apPears in many fields,Such as physics,mechanics and engineering.When B is a positiVe definite matrix(1)can be solved by reducing it to the form     

On Development of a Ky Fan Inequality of the Complementary A-G Type

Two new proofs of a discrete Ky Fan inequality of the complementary A-G type are given. Its continuous version and determinantal analogueon a set of pairwise commutative positive definite matrices are establishbed. Afurther extension concerning general positive definite matrices of the inequalityis also suggested.     

On the Problem of Heilbron Type

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On an Extension and a Refinement of Van der Corput's Inequality

By introducing a parameterα,we give an extension of Van der Corput's inequal- ity.Also a new strengthened version of it is considered.