scholarly journals An Extension of Weyl’s Equidistribution Theorem to Generalized Polynomials and Applications

2020 ◽  
Author(s):  
Vitaly Bergelson ◽  
Inger J Håland Knutson ◽  
Younghwan Son
Keyword(s):  
Classical Theorem ◽  
Integer Part ◽  
Generalized Polynomials ◽  
Generalized Polynomial ◽  
Natural Density

Abstract Generalized polynomials are mappings obtained from the conventional polynomials by the use of the operations of addition and multiplication and taking the integer part. Extending the classical theorem of Weyl on equidistribution of polynomials, we show that a generalized polynomial $q(n)$ has the property that the sequence $(q(n) \lambda )_{n \in \mathbf{Z}}$ is well-distributed $\bmod \, 1$ for all but countably many $\lambda \in{\mathbf R}$ if and only if $\lim\nolimits _{\substack{|n| \rightarrow \infty \\ n \notin J}} |q(n)| = \infty $ for some (possibly empty) set $J$ having zero natural density in $\mathbf{Z}$. We also prove a version of this theorem along the primes (which may be viewed as an extension of classical results of Vinogradov and Rhin). Finally, we utilize these results to obtain new examples of sets of recurrence and van der Corput sets.

A Deterministic Global Design Optimization Method for Nonconvex Generalized Polynomial Problems

1990 ◽  
Author(s):  
Chihsiung Lo ◽  
Panos Y. Papalambros
Keyword(s):  
Objective Function ◽  
Design Optimization ◽  
Global Solutions ◽  
Optimization Method ◽  
Original Model ◽  
Two Phase ◽  
Generalized Polynomials ◽  
Generalized Polynomial ◽  
New Variables ◽  
Global Design Optimization

Abstract A new design optimization method is described for finding global solutions of models with a nonconvex objective function and nonlinear constraints. All functions are assumed to be generalized polynomials. By introducing new variables, the original model is transformed into one with a linear objective function, one convex and one reversed convex constraint. A two-phase algorithm that includes global feasible searches and local optimal searches is used for globally optimizing the transformed model. Several examples illustrate the method.

A Deterministic Global Design Optimization Method for Nonconvex Generalized Polynomial Problems

1996 ◽  
Vol 118 (1) ◽  
pp. 75-81 ◽  
Author(s):  
Chihsiung Lo ◽  
P. Y. Papalambros
Keyword(s):  
Objective Function ◽  
Design Optimization ◽  
Global Solutions ◽  
Optimization Method ◽  
Original Model ◽  
Two Phase ◽  
Generalized Polynomials ◽  
Generalized Polynomial ◽  
New Variables ◽  
Global Design Optimization

A new design optimization method is described for finding global solutions of models with a nonconvex objective function and nonlinear constraints. All functions are assumed to be generalized polynomials. By introducing new variables, the original model is transformed into one with a linear objective function, one convex and one reversed convex constraint. A two-phase algorithm that includes global feasible search and local optimal search is used for globally optimizing the transformed model. Several examples illustrate the method.

Limit cycles of the generalized polynomial Liénard differential equations

2009 ◽  
Vol 148 (2) ◽  
pp. 363-383 ◽  
Author(s):  
JAUME LLIBRE ◽  
ANA CRISTINA MEREU ◽  
MARCO ANTONIO TEIXEIRA
Keyword(s):  
Differential Equations ◽  
Limit Cycles ◽  
Averaging Theory ◽  
Integer Part ◽  
Third Order ◽  
Generalized Polynomial

AbstractWe apply the averaging theory of first, second and third order to the class of generalized polynomial Liénard differential equations. Our main result shows that for any n, m ≥ 1 there are differential equations of the form ẍ + f(x)ẋ + g(x) = 0, with f and g polynomials of degree n and m respectively, having at least [(n + m − 1)/2] limit cycles, where [·] denotes the integer part function.

Multi-parameter Singular Integrals. (AM-189)

2017 ◽  
Author(s):  
Brian Street
Keyword(s):  
Partial Differential Equations ◽  
Graduate Students ◽  
General Theory ◽  
Differential Equations ◽  
Singular Integrals ◽  
Main Tool ◽  
Classical Theorem ◽  
Quantitative Version ◽  
Linear Partial Differential Equations

This book develops a new theory of multi-parameter singular integrals associated with Carnot–Carathéodory balls. The book first details the classical theory of Calderón–Zygmund singular integrals and applications to linear partial differential equations. It then outlines the theory of multi-parameter Carnot–Carathéodory geometry, where the main tool is a quantitative version of the classical theorem of Frobenius. The book then gives several examples of multi-parameter singular integrals arising naturally in various problems. The final chapter of the book develops a general theory of singular integrals that generalizes and unifies these examples. This is one of the first general theories of multi-parameter singular integrals that goes beyond the product theory of singular integrals and their analogs. This book will interest graduate students and researchers working in singular integrals and related fields.

scholarly journals On the integer part of the reciprocal of the Riemann zeta function tail at certain rational numbers in the critical strip

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
WonTae Hwang ◽  
Kyunghwan Song
Keyword(s):  
Zeta Function ◽  
Rational Number ◽  
Riemann Zeta Function ◽  
Rational Numbers ◽  
Critical Strip ◽  
Integer Part ◽  
Integral Points ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

Abstract We prove that the integer part of the reciprocal of the tail of $\zeta (s)$ ζ ( s ) at a rational number $s=\frac{1}{p}$ s = 1 p for any integer with $p \geq 5$ p ≥ 5 or $s=\frac{2}{p}$ s = 2 p for any odd integer with $p \geq 5$ p ≥ 5 can be described essentially as the integer part of an explicit quantity corresponding to it. To deal with the case when $s=\frac{2}{p}$ s = 2 p , we use a result on the finiteness of integral points of certain curves over $\mathbb{Q}$ Q .

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