Harmonic analysis onn-dimensional vector spaces over local fields

This chapter introduces the concept of stable completion and provides a concrete representation of unit vector Mathematical Double-Struck Capital A superscript n in terms of spaces of semi-lattices, with particular emphasis on the frontier between the definable and the topological categories. It begins by constructing a topological embedding of unit vector Mathematical Double-Struck Capital A superscript n into the inverse limit of a system of spaces of semi-lattices L(Hsubscript d) endowed with the linear topology, where Hsubscript d are finite-dimensional vector spaces. The description is extended to the projective setting. The linear topology is then related to the one induced by the finite level morphism L(Hsubscript d). The chapter also considers the condition that if a definable set in L(Hsubscript d) is an intersection of relatively compact sets, then it is itself relatively compact.

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Orbit-Finite-Dimensional Vector Spaces and Weighted Register Automata

2021 36th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS) ◽

10.1109/lics52264.2021.9470634 ◽

2021 ◽

Author(s):

Mikolaj Bojanczyk ◽

Bartek Klin ◽

Joshua Moerman

Keyword(s):

Vector Spaces ◽

Dimensional Vector ◽

Finite Dimensional

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A note on extensions of multilinear maps defined on multilinear varieties

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091521000055 ◽

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.

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The genus of graphs associated with vector spaces

Journal of Algebra and Its Applications ◽

10.1142/s0219498820500863 ◽

2019 ◽

Vol 19 (05) ◽

pp. 2050086 ◽

Cited By ~ 1

Author(s):

T. Tamizh Chelvam ◽

K. Prabha Ananthi

Keyword(s):

Finite Field ◽

Vector Space ◽

Undirected Graph ◽

Vector Spaces ◽

Dimensional Vector ◽

Vertex Connectivity ◽

Dimensional Vector Space ◽

Nonzero Component ◽

Vertex Set ◽

Component Graph

Let [Formula: see text] be a k-dimensional vector space over a finite field [Formula: see text] with a basis [Formula: see text]. The nonzero component graph of [Formula: see text], denoted by [Formula: see text], is a simple undirected graph with vertex set as nonzero vectors of [Formula: see text] such that there is an edge between two distinct vertices [Formula: see text] if and only if there exists at least one [Formula: see text] along which both [Formula: see text] and [Formula: see text] have nonzero scalars. In this paper, we find the vertex connectivity and girth of [Formula: see text]. We also characterize all vector spaces [Formula: see text] for which [Formula: see text] has genus either 0 or 1 or 2.

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