Overview

2012 ◽  
Author(s):  
Nicholas M. Katz
Keyword(s):  
Finite Field ◽  
Numerical Experiments ◽  
Multiplicative Characters

This chapter provides an overview of the present analysis, which grew out of two questions, raised by Ron Evans and Zeev Rudnick respectively, in May and June of 2003. Evans and Rudnick performed numerical experiments on the sums as χ‎ varies over all multiplicative characters of k, and as χ‎ varies over all nontrivial multiplicative characters of a finite field k of odd characteristic, respectively. Both equidistribution results of Evans and Rudnick are proven.

scholarly journals On the distribution of Jacobi sums

2018 ◽  
Vol 2018 (741) ◽  
pp. 67-86
Author(s):  
Qing Lu ◽  
Weizhe Zheng ◽  
Zhiyong Zheng
Keyword(s):  
Finite Field ◽  
Unit Circle ◽  
Jacobi Sums ◽  
Jacobi Sum ◽  
Multiplicative Characters

Abstract Let \mathbf{F}_{q} be a finite field of q elements. For multiplicative characters \chi_{1},\ldots,\chi_{m} of \mathbf{F}_{q}^{\times} , we let J(\chi_{1},\ldots,\chi_{m}) denote the Jacobi sum. Nicholas Katz and Zhiyong Zheng showed that for m=2 , the normalized Jacobi sum q^{-1/2}J(\chi_{1},\chi_{2}) ( \chi_{1}\chi_{2} nontrivial) is asymptotically equidistributed on the unit circle as q\to\infty , when \chi_{1} and \chi_{2} run through all nontrivial multiplicative characters of \mathbf{F}_{q}^{\times} . In this paper, we show a similar property for m\geq 2 . More generally, we show that the normalized Jacobi sum q^{-(m-1)/2}J(\chi_{1},\ldots,\chi_{m}) ( \chi_{1}\cdots\chi_{m} nontrivial) is asymptotically equidistributed on the unit circle, when \chi_{1},\ldots,\chi_{m} run through arbitrary sets of nontrivial multiplicative characters of \mathbf{F}_{q}^{\times} with two of the sets being sufficiently large. The case m=2 answers a question of Shparlinski.

Beta dynamical systems over the formal fields

2014 ◽  
Vol 51 (4) ◽  
pp. 454-465
Author(s):  
Lu-Ming Shen ◽  
Huiping Jing
Keyword(s):  
Dynamical Systems ◽  
Finite Field ◽  
Haar Measure ◽  
Laurent Series ◽  
Formal Laurent Series ◽  
Almost All

Let \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{F}_q ((X^{ - 1} ))$$ \end{document} denote the formal field of all formal Laurent series x = Σ n=ν∞anX−n in an indeterminate X, with coefficients an lying in a given finite field \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{F}_q$$ \end{document}. For any \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\beta \in \mathbb{F}_q ((X^{ - 1} ))$$ \end{document} with deg β > 1, it is known that for almost all \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$x \in \mathbb{F}_q ((X^{ - 1} ))$$ \end{document} (with respect to the Haar measure), x is β-normal. In this paper, we show the inverse direction, i.e., for any x, for almost all \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\beta \in \mathbb{F}_q ((X^{ - 1} ))$$ \end{document}, x is β-normal.

Controllable Compliance Joint For Human Oriented Robots

2013 ◽  
Vol 43 (1) ◽  
pp. 47-60
Author(s):  
Mihail Tsveov ◽  
Dimitar Chakarov
Keyword(s):  
Numerical Experiments ◽  
Position Control ◽  
Joint Stiffness ◽  
Joint Motion ◽  
Compliance Control ◽  
Leaf Springs ◽  
Antagonistic Actuation

Abstract In the paper, different approaches for compliance control for human oriented robots are revealed. The approaches based on the non- antagonistic and antagonistic actuation are compared. In addition, an approach is investigated in this work for the compliance and the position control in the joint by means of antagonistic actuation. It is based on the capability of the joint with torsion leaf springs to adjust its stiffness. Models of joint stiffness are presented in this paper with antagonistic and non-antagonistic influence of the spring forces on the joint motion. The stiffness and the position control possibilities are investigated and the opportunity for their decoupling as well. Some results of numerical experiments are presented in the paper too.

Valued Field Graph and Some Related Parameters

2020 ◽  
pp. 389-395
Author(s):  
G. Suresh Singh ◽  
P. K. Prasobha
Keyword(s):  
Finite Field ◽  
Independence Number ◽  
Valued Field ◽  
Vertex Set ◽  
Adic Valuation

Let $K$ be any finite field. For any prime $p$, the $p$-adic valuation map is given by $\psi_{p}:K/\{0\} \to \R^+\bigcup\{0\}$ is given by $\psi_{p}(r) = n$ where $r = p^n \frac{a}{b}$, where $p,a,b$ are relatively prime. The field $K$ together with a valuation is called valued field. Also, any field $K$ has the trivial valuation determined by $\psi{(K)} = \{0,1\}$. Through out the paper K represents $\Z_q$. In this paper, we construct the graph corresponding to the valuation map called the valued field graph, denoted by $VFG_{p}(\Z_{q})$ whose vertex set is $\{v_0,v_1,v_2,\ldots, v_{q-1}\}$ where two vertices $v_i$ and $v_j$ are adjacent if $\psi_{p}(i) = j$ or $\psi_{p}(j) = i$. Here, we tried to characterize the valued field graph in $\Z_q$. Also we analyse various graph theoretical parameters such as diameter, independence number etc.

Sign in / Sign up

Export Citation Format

Share Document