scholarly journals Sharp extension theorems and Falconer distance problems for algebraic curves in two dimensional vector spaces over finite fields

2012 ◽  
pp. 157-178 ◽  
Author(s):  
Doowon Koh ◽  
Chun-Yen Shen
Keyword(s):  
Finite Fields ◽  
Algebraic Curves ◽  
Vector Spaces ◽  
Two Dimensional ◽  
Dimensional Vector ◽  
Extension Theorems

scholarly journals A note on extensions of multilinear maps defined on multilinear varieties

2021 ◽  
pp. 1-26
Author(s):  
W. T. Gowers ◽  
L. Milićević
Keyword(s):  
Finite Fields ◽  
Vector Spaces ◽  
Restriction Map ◽  
Zero Set ◽  
Dimensional Vector ◽  
Prime Field ◽  
Finite Dimensional ◽  
Multilinear Maps ◽  
Multilinear Map

Abstract Let $G_1, \ldots , G_k$ be finite-dimensional vector spaces over a prime field $\mathbb {F}_p$ . A multilinear variety of codimension at most $d$ is a subset of $G_1 \times \cdots \times G_k$ defined as the zero set of $d$ forms, each of which is multilinear on some subset of the coordinates. A map $\phi$ defined on a multilinear variety $B$ is multilinear if for each coordinate $c$ and all choices of $x_i \in G_i$ , $i\not =c$ , the restriction map $y \mapsto \phi (x_1, \ldots , x_{c-1}, y, x_{c+1}, \ldots , x_k)$ is linear where defined. In this note, we show that a multilinear map defined on a multilinear variety of codimension at most $d$ coincides on a multilinear variety of codimension $O_{k}(d^{O_{k}(1)})$ with a multilinear map defined on the whole of $G_1\times \cdots \times G_k$ . Additionally, in the case of general finite fields, we deduce similar (but slightly weaker) results.

scholarly journals Restriction Operators Acting on Radial Functions on Vector Spaces over Finite Fields

2014 ◽  
Vol 57 (4) ◽  
pp. 834-844
Author(s):  
Doowon Koh
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Algebraic Varieties ◽  
Test Functions ◽  
Radial Functions ◽  
Zero Radius ◽  
Intersection Points ◽  
Field Setting

AbstractWe study Lp → Lr restriction estimates for algebraic varieties V in the case when restriction operators act on radial functions in the finite field setting. We show that if the varieties V lie in odd dimensional vector spaces over finite fields, then the conjectured restriction estimates are possible for all radial test functions. In addition, assuming that the varieties V are defined in even dimensional spaces and have few intersection points with the sphere of zero radius, we also obtain the conjectured exponents for all radial test functions.

scholarly journals On $k$-simplexes in $(2k-1)$-dimensional vector spaces over finite fields

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Le Anh Vinh
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Finite Fields ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Nous Montrons ◽  
International Audience

International audience We show that if the cardinality of a subset of the $(2k-1)$-dimensional vector space over a finite field with $q$ elements is $\gg q^{2k-1-\frac{1}{ 2k}}$, then it contains a positive proportional of all $k$-simplexes up to congruence. Nous montrons que si la cardinalité d'un sous-ensemble de l'espace vectoriel à $(2k-1)$ dimensions sur un corps fini à $q$ éléments est $\gg q^{2k-1-\frac{1}{ 2k}}$, alors il contient une proportion non-nulle de tous les $k$-simplexes de congruence.

Harmonic Analysis Related to Homogeneous Varieties in Three Dimensional Vector Spaces over Finite Fields

2012 ◽  
Vol 64 (5) ◽  
pp. 1036-1057 ◽  
Author(s):  
Doowon Koh ◽  
Chun-Yen Shen
Keyword(s):  
Finite Fields ◽  
Three Dimensional ◽  
Vector Spaces ◽  
Extension Problem ◽  
Dimensional Vector ◽  
Complete Mapping ◽  
Plane Passing ◽  
Size Condition ◽  
Distance Problem ◽  
Homogeneous Varieties

Abstract In this paper we study the extension problem, the averaging problem, and the generalized Erdős-Falconer distance problem associated with arbitrary homogeneous varieties in three dimensional vector spaces over finite fields. In the case when the varieties do not contain any plane passing through the origin, we obtain the best possible results on the aforementioned three problems. In particular, our result on the extension problem modestly generalizes the result by Mockenhaupt and Tao who studied the particular conical extension problem. In addition, investigating the Fourier decay on homogeneous varieties enables us to give complete mapping properties of averaging operators. Moreover, we improve the size condition on a set such that the cardinality of its distance set is nontrivial.

scholarly journals On the Number of Orthogonal Systems in Vector Spaces over Finite Fields

10.37236/907 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Le Anh Vinh
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Finite Fields ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Graph Theoretic ◽  
Theoretic Proof ◽  
Orthogonal Systems

Iosevich and Senger (2008) showed that if a subset of the $d$-dimensional vector space over a finite field is large enough, then it contains many $k$-tuples of mutually orthogonal vectors. In this note, we provide a graph theoretic proof of this result.

Examples of Gibbs States of Mechanical Systems with Symmetries

2020 ◽  
Vol 58 ◽  
pp. 55-79
Author(s):  
Charles-Michel Marle ◽  
Keyword(s):  
Gibbs State ◽  
Three Dimensional ◽  
Cross Product ◽  
Symplectic Manifolds ◽  
Gibbs States ◽  
Vector Spaces ◽  
Two Dimensional ◽  
Dimensional Vector ◽  
Hamiltonian Action ◽  
French Mathematician

In a previous paper, the notion of Gibbs state for the Hamiltonian action of a Lie group on a symplectic manifold was given, together with its applications in Statistical Mechanics, and the works in this field of the French mathematician and physicist Jean-Marie Souriau were presented. Using an adaptation of the cross product for pseudo-Euclidean three-dimensional vector spaces, we present several examples of such Gibbs states, together with the associated thermodynamic functions, for various two-dimensional symplectic manifolds, including the pseudo-spheres, the Poincar{\'e} disk and the Poincar{\'e} half-plane.

scholarly journals On nonzero component graph of vector spaces over finite fields

2017 ◽  
Vol 16 (01) ◽  
pp. 1750007 ◽  
Author(s):  
Angsuman Das
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Clique Number ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional ◽  
Nonzero Component ◽  
Component Graph

In this paper, we study nonzero component graph [Formula: see text] of a finite-dimensional vector space [Formula: see text] over a finite field [Formula: see text]. We show that the graph is Hamiltonian and not Eulerian. We also characterize the maximal cliques in [Formula: see text] and show that there exists two classes of maximal cliques in [Formula: see text]. We also find the exact clique number of [Formula: see text] for some particular cases. Moreover, we provide some results on size, edge-connectivity and chromatic number of [Formula: see text].

scholarly journals A theorem on two-dimensional vector spaces.

1957 ◽  
Vol 4 (3) ◽  
pp. 257-259
Author(s):  
John S. Griffin ◽  
J. E. McLaughlin
Keyword(s):  
Vector Spaces ◽  
Two Dimensional ◽  
Dimensional Vector
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