scholarly journals Divisor Problems Related to Hecke Eigenvalues in Three Dimensions

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Jing Huang ◽  
Huafeng Liu
Keyword(s):  
Upper Bounds ◽  
Three Dimensions ◽  
Asymptotic Formulas ◽  
Hecke Eigenvalues ◽  
Divisor Problems

In this paper, we consider divisor problems related to Hecke eigenvalues in three dimensions. We establish upper bounds and asymptotic formulas for these problems on average.

Extension of the Spatial PDI Model to Three Dimensions

1997 ◽  
Vol 84 (1) ◽  
pp. 176-178
Author(s):  
Frank O'Brien
Keyword(s):  
Population Density ◽  
Dimensional Space ◽  
Finite Lattice ◽  
Three Dimensional ◽  
Upper Bounds ◽  
Three Dimensions ◽  
Lower And Upper Bounds ◽  
Density Index ◽  
Three Dimensional Space

The author's population density index ( PDI) model is extended to three-dimensional distributions. A derived formula is presented that allows for the calculation of the lower and upper bounds of density in three-dimensional space for any finite lattice.

scholarly journals Almost tight upper bounds for the single cell and zone problems in three dimensions

1995 ◽  
Vol 14 (4) ◽  
pp. 385-410 ◽  
Author(s):  
D. Halperin ◽  
M. Sharif
Keyword(s):  
Single Cell ◽  
Upper Bounds ◽  
Three Dimensions

scholarly journals On some upper bounds for the zeta-function and the Dirichlet divisor problem

2016 ◽  
Vol 12 (08) ◽  
pp. 2231-2239
Author(s):  
Aleksandar Ivić
Keyword(s):  
Zeta Function ◽  
Error Term ◽  
Upper Bounds ◽  
Mean Value ◽  
Divisor Problem ◽  
Asymptotic Formulas ◽  
Number Of Divisors ◽  
The Riemann Zeta Function ◽  
Dirichlet Divisor Problem ◽  
Riemann Zeta

Let [Formula: see text] be the number of divisors of [Formula: see text], let [Formula: see text] denote the error term in the classical Dirichlet divisor problem, and let [Formula: see text] denote the Riemann zeta-function. Several upper bounds for integrals of the type [Formula: see text] are given. This complements the results of [A. Ivić and W. Zhai, On some mean value results for [Formula: see text] and a divisor problem II, Indag. Math. 26(5) (2015) 842–866], where asymptotic formulas for [Formula: see text] were established for the above integral.

scholarly journals The Largest Complete Bipartite Subgraph in Point-Hyperplane Incidence Graphs

10.37236/8253 ◽  
2020 ◽  
Vol 27 (1) ◽  
Author(s):  
Thao Do
Keyword(s):  
Upper Bounds ◽  
Three Dimensions ◽  
Lower And Upper Bounds ◽  
Incidence Graph ◽  
Five Dimensions ◽  
Bipartite Subgraph ◽  
Complete Bipartite Subgraph

Given $m$ points and $n$ hyperplanes in $\mathbb{R}^d$ ($d\geqslant 3)$, if there are many incidences, we expect to find a big cluster $K_{r,s}$ in their incidence graph. Apfelbaum and Sharir found lower and upper bounds for the largest size of $rs$, which match (up to a constant) only in three dimensions. In this paper we close the gap in four and five dimensions, up to some polylogarithmic factors.

scholarly journals APPROXIMATION OF CLASSES OF POISSON INTEGRALS BY REPEATED FEJER SUMS

2020 ◽  
Vol 8 (2) ◽  
pp. 114-121
Author(s):  
O. Rovenska
Keyword(s):  
Fourier Series ◽  
Real Variable ◽  
Upper Bounds ◽  
Partial Sums ◽  
Asymptotic Formulas ◽  
Periodic Functions ◽  
Arithmetic Means ◽  
Poisson Integrals ◽  
Fourier Sums ◽  
Uniformly Convergent

The paper is devoted to the approximation by arithmetic means of Fourier sums of classes of periodic functions of high smoothness. The simplest example of a linear approximation of continuous periodic functions of a real variable is the approximation by partial sums of the Fourier series. The sequences of partial Fourier sums are not uniformly convergent over the class of continuous periodic functions. A significant number of works is devoted to the study of other approximation methods, which are generated by transformations of Fourier sums and allow us to construct trigonometrical polynomials that would be uniformly convergent for each continuous function. Over the past decades, Fejer sums and de la Vallee Poussin sums have been widely studied. One of the most important direction in this field is the study of the asymptotic behavior of upper bounds of deviations of linear means of Fourier sums on different classes of periodic functions. Methods of investigation of integral representations of deviations of trigonometric polynomials generated by linear methods of summation of Fourier series, were originated and developed in the works of S.M. Nikolsky, S.B. Stechkin, N.P. Korneichuk, V.K. Dzadyk and others. The aim of the work systematizes known results related to the approximation of classes of Poisson integrals by arithmetic means of Fourier sums, and presents new facts obtained for particular cases. In the paper is studied the approximative properties of repeated Fejer sums on the classes of periodic analytic functions of real variable. Under certain conditions, we obtained asymptotic formulas for upper bounds of deviations of repeated Fejer sums on classes of Poisson integrals. The obtained formulas provide a solution of the corresponding Kolmogorov-Nikolsky problem without any additional conditions.

scholarly journals Approximation of the classes $W^{r}_{\beta,\infty}$ by three-harmonic Poisson integrals

2019 ◽  
Vol 11 (2) ◽  
pp. 321-334 ◽  
Author(s):  
U.Z. Hrabova ◽  
I.V. Kal'chuk
Keyword(s):  
Fourier Series ◽  
Function Class ◽  
Upper Bounds ◽  
Extremal Problems ◽  
Asymptotic Formulas ◽  
Specific Method ◽  
Triangular Matrices ◽  
Differentiable Functions ◽  
Poisson Integrals ◽  
Functional Classes

In the paper, we solve one extremal problem of the theory of approximation of functional classes by linear methods. Namely, questions are investigated concerning the approximation of classes of differentiable functions by $\lambda$-methods of summation for their Fourier series, that are defined by the set $\Lambda =\{{{\lambda }_{\delta }}(\cdot )\}$ of continuous on $\left[ 0,\infty \right)$ functions depending on a real parameter $\delta$. The Kolmogorov-Nikol'skii problem is considered, that is one of the special problems among the extremal problems of the theory of approximation. That is, the problem of finding of asymptotic equalities for the quantity $$\mathcal{E}{{\left( \mathfrak{N};{{U}_{\delta}} \right)}_{X}}=\underset{f\in \mathfrak{N}}{\mathop{\sup }}\,{{\left\| f\left( \cdot \right)-{{U}_{\delta }}\left( f;\cdot;\Lambda \right) \right\|}_{X}},$$ where $X$ is a normalized space, $\mathfrak{N}\subseteq X$ is a given function class, ${{U}_{\delta }}\left( f;x;\Lambda \right)$ is a specific method of summation of the Fourier series. In particular, in the paper we investigate approximative properties of the three-harmonic Poisson integrals on the Weyl-Nagy classes. The asymptotic formulas are obtained for the upper bounds of deviations of the three-harmonic Poisson integrals from functions from the classes $W^{r}_{\beta,\infty}$. These formulas provide a solution of the corresponding Kolmogorov-Nikol'skii problem. Methods of investigation for such extremal problems of the theory of approximation arised and got their development owing to the papers of A.N. Kolmogorov, S.M. Nikol'skii, S.B. Stechkin, N.P. Korneichuk, V.K. Dzyadyk, A.I. Stepanets and others. But these methods are used for the approximations by linear methods defined by triangular matrices. In this paper we modified the mentioned above methods in order to use them while dealing with the summation methods defined by a set of functions of a natural argument.

scholarly journals Existence of generalized solutions for Keller-Segel-Navier-Stokes equations with degradation in dimension three

2021 ◽  
Vol 4 (5) ◽  
pp. 1-25
Author(s):  
Kyungkeun Kang ◽  
◽  
Dongkwang Kim
Keyword(s):  
Convergence Rates ◽  
Stokes Equations ◽  
Upper Bounds ◽  
Three Dimensions ◽  
Generalized Solutions ◽  
Navier Stokes ◽  
Navier Stokes Equations

<abstract><p>We construct generalized solutions for the Keller-Segel system with a degradation source coupled to Navier Stokes equations in three dimensions, in case that the power of degradation is smaller than quadratic. Furthermore, if the logistic type source is purely damping with no growing effect, we prove that solutions converge to zero in some norms and provide upper bounds of convergence rates in time.</p></abstract>

scholarly journals On the zeros of Epstein zeta functions

2014 ◽  
Vol 26 (6) ◽  
Author(s):  
Yoonbok Lee
Keyword(s):  
Quadratic Form ◽  
Lower Bounds ◽  
Critical Line ◽  
Zeta Functions ◽  
Upper Bounds ◽  
Positive Definite ◽  
Asymptotic Formulas ◽  
Positive Definite Quadratic Form ◽  
Number Of Zeros ◽  
Existence Of Zeros

AbstractWe investigate the zeros of Epstein zeta functions associated with a positive definite quadratic form with rational coefficients. Davenport and Heilbronn, and also Voronin, proved the existence of zeros of Epstein zeta functions off the critical line when the class number of the quadratic form is bigger than 1. These authors give lower bounds for the number of zeros in strips that are of the same order as the more easily proved upper bounds. In this paper, we improve their results by providing asymptotic formulas for the number of zeros.

On general divisor problems involving Hecke eigenvalues

2011 ◽  
Vol 135 (1-2) ◽  
pp. 148-159 ◽  
Author(s):  
Guangshi Lü
Keyword(s):  
Hecke Eigenvalues ◽  
Divisor Problems
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