scholarly journals Sublinear Circuits for Polyhedral Sets

2021 ◽  
Author(s):  
Helen Naumann ◽  
Thorsten Theobald
Keyword(s):  
Exponential Sums ◽  
Polyhedral Combinatorics ◽  
Geometric Inequality ◽  
Matroid Theory ◽  
Polyhedral Sets ◽  
Constraint Sets

AbstractSublinear circuits are generalizations of the affine circuits in matroid theory, and they arise as the convex-combinatorial core underlying constrained non-negativity certificates of exponential sums and of polynomials based on the arithmetic-geometric inequality. Here, we study the polyhedral combinatorics of sublinear circuits for polyhedral constraint sets. We give results on the relation between the sublinear circuits and their supports and provide necessary as well as sufficient criteria for sublinear circuits. Based on these characterizations, we provide some explicit results and enumerations for two prominent polyhedral cases, namely the non-negative orthant and the cube [− 1,1]n.

AM/GM-based optimization : gheometry and generalizations

2021 ◽  
Author(s):  
◽  
Helen Naumann
Keyword(s):  
Duality Theory ◽  
Exponential Sums ◽  
Negative Term ◽  
Polyhedral Sets ◽  
Duality Results ◽  
The Third ◽  
Np Hard Problem ◽  
Constraint Sets ◽  
Extreme Rays

The problem of unconstrained or constrained optimization occurs in many branches of mathematics and various fields of application. It is, however, an NP-hard problem in general. In this thesis, we examine an approximation approach based on the class of SAGE exponentials, which are nonnegative exponential sums. We examine this SAGE-cone, its geometry, and generalizations. The thesis consists of three main parts: 1. In the first part, we focus purely on the cone of sums of globally nonnegative exponential sums with at most one negative term, the SAGE-cone. We ex- amine the duality theory, extreme rays of the cone, and provide two efficient optimization approaches over the SAGE-cone and its dual. 2. In the second part, we introduce and study the so-called S-cone, which pro- vides a uniform framework for SAGE exponentials and SONC polynomials. In particular, we focus on second-order representations of the S-cone and its dual using extremality results from the first part. 3. In the third and last part of this thesis, we turn towards examining the con- ditional SAGE-cone. We develop a notion of sublinear circuits leading to new duality results and a partial characterization of extremality. In the case of poly- hedral constraint sets, this examination is simplified and allows us to classify sublinear circuits and extremality for some cases completely. For constraint sets with certain conditions such as sets with symmetries, conic, or polyhedral sets, various optimization and representation results from the unconstrained setting can be applied to the constrained case.

scholarly journals A note on two-term exponential sum and the reciprocal of the quartic Gauss sums

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Wenpeng Zhang ◽  
Xingxing Lv
Keyword(s):  
Exponential Sums ◽  
Exponential Sum ◽  
Gauss Sums ◽  
Power Mean ◽  
Hybrid Power ◽  
Analytic Methods ◽  
Hybrid Power Mean

AbstractThe main purpose of this article is by using the properties of the fourth character modulo a prime p and the analytic methods to study the calculating problem of a certain hybrid power mean involving the two-term exponential sums and the reciprocal of quartic Gauss sums, and to give some interesting calculating formulae of them.

Importance-Different Charging Scheduling Based on Matroid Theory for Wireless Rechargeable Sensor Networks

2021 ◽  
pp. 1-1
Author(s):  
Wenyu Ouyang ◽  
Mohammad S. Obaidat ◽  
Xuxun Liu ◽  
Xiaoting Long ◽  
Wenzheng Xu ◽  
...  
Keyword(s):  
Sensor Networks ◽  
Matroid Theory ◽  
Wireless Rechargeable Sensor Networks

scholarly journals On Waring’s problem: Three cubes and a sixth power

2001 ◽  
Vol 163 ◽  
pp. 13-53 ◽  
Author(s):  
Jörg Brüdern ◽  
Trevor D. Wooley
Keyword(s):  
Exponential Sums ◽  
Large Integer ◽  
Natural Numbers ◽  
Waring’S Problem ◽  
Smooth Numbers ◽  
Waring's Problem ◽  
Order Of Magnitude ◽  
Almost All ◽  
Hardy Littlewood Method

We establish that almost all natural numbers not congruent to 5 modulo 9 are the sum of three cubes and a sixth power of natural numbers, and show, moreover, that the number of such representations is almost always of the expected order of magnitude. As a corollary, the number of representations of a large integer as the sum of six cubes and two sixth powers has the expected order of magnitude. Our results depend on a certain seventh moment of cubic Weyl sums restricted to minor arcs, the latest developments in the theory of exponential sums over smooth numbers, and recent technology for controlling the major arcs in the Hardy-Littlewood method, together with the use of a novel quasi-smooth set of integers.

Construction of systems of polynomial equations with exact p-Divisibility via the covering method

2014 ◽  
Vol 13 (06) ◽  
pp. 1450013 ◽  
Author(s):  
Francis N. Castro ◽  
Ivelisse M. Rubio
Keyword(s):  
Exponential Sums ◽  
Polynomial Equations ◽  
Prime Field ◽  
Elementary Method ◽  
Covering Method ◽  
Systems Of Polynomial Equations

We present an elementary method to compute the exact p-divisibility of exponential sums of systems of polynomial equations over the prime field. Our results extend results by Carlitz and provide concrete and simple conditions to construct families of polynomial equations that are solvable over the prime field.

Short cubic exponential sums over primes

2017 ◽  
Vol 296 (1) ◽  
pp. 211-233
Author(s):  
Z. Kh. Rakhmonov ◽  
F. Z. Rakhmonov
Keyword(s):  
Exponential Sums ◽  
Exponential Sums Over Primes
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