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The k-resultant modulus set problem on algebraic varieties over finite fields
Covert, D.,Koh, D.,Pi, Y. Academic Press 2017 Finite fields and their applications Vol.48 No.-
We study the k-resultant modulus set problem in the d-dimensional vector space F<SUB>q</SUB><SUP>d</SUP> over the finite field F<SUB>q</SUB> with q elements. Given E@?F<SUB>q</SUB><SUP>d</SUP> and an integer k≥2, the k-resultant modulus set, denoted by Δ<SUB>k</SUB>(E), is defined asΔ<SUB>k</SUB>(E)={@?x<SUP>1</SUP>+/-x<SUP>2</SUP>+/-...+/-x<SUP>k</SUP>@?@?F<SUB>q</SUB>:x<SUP>j</SUP>@?E, j=1,2,...,k}, where @?α@?=α<SUB>1</SUB><SUP>2</SUP>+...+α<SUB>d</SUB><SUP>2</SUP> for α=(α<SUB>1</SUB>,...,α<SUB>d</SUB>)@?F<SUB>q</SUB><SUP>d</SUP>. In this setting, the k-resultant modulus set problem is to determine the minimal cardinality of E@?F<SUB>q</SUB><SUP>d</SUP> such that Δ<SUB>k</SUB>(E)=F<SUB>q</SUB> or F<SUB>q</SUB><SUP>@?</SUP>. This problem is an extension of the Erdos-Falconer distance problem. In particular, we investigate the k-resultant modulus set problem with the restriction that the set E@?F<SUB>q</SUB><SUP>d</SUP> is contained in a specific algebraic variety. Energy estimates play a crucial role in our proof.
LONG PATHS IN THE DISTANCE GRAPH OVER LARGE SUBSETS OF VECTOR SPACES OVER FINITE FIELDS
BENNETT, MICHAEL,CHAPMAN, JEREMY,COVERT, DAVID,HART, DERRICK,IOSEVICH, ALEX,PAKIANATHAN, JONATHAN Korean Mathematical Society 2016 대한수학회지 Vol.53 No.1
Let $E{\subset}{\mathbb{F}}^d_q$, the d-dimensional vector space over the finite field with q elements. Construct a graph, called the distance graph of E, by letting the vertices be the elements of E and connect a pair of vertices corresponding to vectors x, y 2 E by an edge if ${\parallel}x-y{\parallel}:=(x_1-y_1)^2+{\cdots}+(x_d-y_d)^2=1$. We shall prove that the non-overlapping chains of length k, with k in an appropriate range, are uniformly distributed in the sense that the number of these chains equals the statistically correct number, $1{\cdot}{\mid}E{\mid}^{k+1}q^{-k}$ plus a much smaller remainder.
Implications of Air Pollution Effects on Athletic Performance
Pierson, W.E.,Covert, D.S.,Koenig, J.Q.,Namekata, T.,Kim, Y.S. Korean Society of Environmental Health 1985 한국환경보건학회지 Vol.11 No.2
There are a large number or chemical compounds that are present in a polluted atmosphere and that alone or in combination are important to consider for their potential effect on the respiratory system and impact on athletic performance. A general categorization or description of the level of pollution in terms of the concentration of one or more compounds or by type such as oxidizing compounds is inadequate and misleading. A useful initial categorization of pollutant compounds according to their mechanism of production, primary or secondary, is often made. For health effects, consideraiions of the physical state, gaseous or particulate, and the solublity and reactivity of the pollutant is also important. Pollutant compounds or substances that are emitted directly from a source and that undergo little or no chemical change in the atmosphere from source to receptor are termed primary pollutants.
Long paths in the distance graph over large subsets of vector spaces over finite fields
Michael Bennett,Jeremy Chapman,David Covert,Derrick Hart,Alex Iosevich,Jonathan Pakianathan 대한수학회 2016 대한수학회지 Vol.53 No.1
Let $E \subset {\mathbb F}_q^d$, the $d$-dimensional vector space over the finite field with $q$ elements. Construct a graph, called the distance graph of $E$, by letting the vertices be the elements of $E$ and connect a pair of vertices corresponding to vectors $x,y \in E$ by an edge if $||x-y||:={(x_1-y_1)}^2+\dots+{(x_d-y_d)}^2=1$. We shall prove that the non-overlapping chains of length $k$, with $k$ in an appropriate range, are uniformly distributed in the sense that the number of these chains equals the statistically correct number, $1 \cdot {|E|}^{k+1}q^{-k}$ plus a much smaller remainder.