Embedding an Affine Space in a Vector Space

2011 ◽  
pp. 85-101
Author(s):  
Jean Gallier
Keyword(s):  
Vector Space ◽  
Affine Space

Integrable spreads and spaces of constant curvature

1988 ◽  
Vol 109 (3-4) ◽  
pp. 225-229 ◽  
Author(s):  
H. R. Farran ◽  
S. A. Robertson
Keyword(s):  
Vector Bundle ◽  
Vector Space ◽  
Constant Curvature ◽  
Affine Space ◽  
Sufficient Conditions ◽  
Spaces Of Constant Curvature ◽  
Space Of Constant Curvature

SynopsisThis paper is a continuation of [2], where we introduced the notion of global k-spreads on manifolds. Here we show that the space of all k-spreads on a manifold has the structure of an affine space, modelled on the vector space of sections of a certain vector bundle. We give some sufficient conditions for a manifold admitting an integrable k-spread to be a space of constant curvature and answer one of the questions raised in [2].

scholarly journals Permutations of zero-sumsets in a finite vector space

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Giovanni Falcone ◽  
Marco Pavone
Keyword(s):  
Vector Space ◽  
Affine Space ◽  
Galois Field ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional ◽  
Finite Vector ◽  
Finite Vector Space ◽  
The Family

AbstractIn this paper, we consider a finite-dimensional vector space {{\mathcal{P}}} over the Galois field {\operatorname{GF}(p)}, with p being an odd prime, and the family {{\mathcal{B}}_{k}^{x}} of all k-sets of elements of {\mathcal{P}} summing up to a given element x. The main result of the paper is the characterization, for {x=0}, of the permutations of {\mathcal{P}} inducing permutations of {{\mathcal{B}}_{k}^{0}} as the invertible linear mappings of the vector space {\mathcal{P}} if p does not divide k, and as the invertible affinities of the affine space {\mathcal{P}} if p divides k. The same question is answered also in the case where the elements of the k-sets are required to be all nonzero, and, in fact, the two cases prove to be intrinsically inseparable.

The Space Groups of Two Dimensional Minkowski Space

1978 ◽  
Vol 30 (5) ◽  
pp. 1103-1120
Author(s):  
George Maxwell
Keyword(s):  
Vector Space ◽  
Minkowski Space ◽  
Bilinear Form ◽  
Orthogonal Group ◽  
Affine Space ◽  
Symmetric Bilinear Form ◽  
Two Dimensional ◽  
Space Groups ◽  
Dimensional Minkowski Space ◽  
Real Affine

LetEbe an n-dimensional real affine space,Vits vector space of translations andA(E)the affine group ofE.Suppose that (. , .) is a nondegenerate symmetric bilinear form on F of signature(n —1, 1), O(V) its orthogonal group andS(V)its group of similarities.

Indicator Sets in an Affine Space of any Dimension

1979 ◽  
Vol 31 (1) ◽  
pp. 211-224 ◽  
Author(s):  
F. A. Sherk
Keyword(s):  
Vector Space ◽  
Translation Plane ◽  
Affine Space ◽  
Finite Translation ◽  
Finite Translation Plane ◽  
Indicator Sets

It is well known that a translation plane can be represented in a vector space over a field F where F is a subfield of the kernel of a quasifield which coordinatizes the plane [1; 2; 4, p.220; 10]. If II is a finite translation plane of order qr (q = pn, p any prime), then II may be represented in V2r(q), the vector space of dimension 2r over GF(q), as follows:(i) The points of II are the vectors in V = V2r(q)(ii) The lines of II are(a) A set of qr + 1 mutually disjoint r-dimensional subspaces of V.(b) All translates of in V.(iii) Incidence is inclusion.

scholarly journals Haas' theorem revisited

2017 ◽  
Vol Volume 1 ◽  
Author(s):  
Benoît Bertrand ◽  
Erwan Brugallé ◽  
Arthur Renaudineau
Keyword(s):  
Vector Space ◽  
Algebraic Curve ◽  
Affine Space ◽  
Linear Subspace ◽  
Vector Subspace ◽  
Original Result ◽  
Affine Subspaces ◽  
Trivalent Graph ◽  
Real Algebraic Curve ◽  
Topological Surface

Haas' theorem describes all partchworkings of a given non-singular plane tropical curve $C$ giving rise to a maximal real algebraic curve. The space of such patchworkings is naturally a linear subspace $W_C$ of the $\mathbb{Z}/2\mathbb{Z}$-vector space $\overrightarrow \Pi_C$ generated by the bounded edges of $C$, and whose origin is the Harnack patchworking. The aim of this note is to provide an interpretation of affine subspaces of $\overrightarrow \Pi_C $ parallel to $W_C$. To this purpose, we work in the setting of abstract graphs rather than plane tropical curves. We introduce a topological surface $S_\Gamma$ above a trivalent graph $\Gamma$, and consider a suitable affine space $\Pi_\Gamma$ of real structures on $S_\Gamma$ compatible with $\Gamma$. We characterise $W_\Gamma$ as the vector subspace of $\overrightarrow \Pi_\Gamma$ whose associated involutions induce the same action on $H_1(S_\Gamma,\mathbb{Z}/2\mathbb{Z})$. We then deduce from this statement another proof of Haas' original result. Comment: 22 pages, 14 figures

scholarly journals On the contravariant of homogeneous forms arising from isolated hypersurface singularities

2016 ◽  
Vol 27 (12) ◽  
pp. 1650097 ◽  
Author(s):  
A. V. Isaev
Keyword(s):  
Vector Space ◽  
Affine Space ◽  
Upper Bounds ◽  
Product Formula ◽  
Affine Variety ◽  
Hypersurface Singularities

Let [Formula: see text] be the vector space of homogeneous forms of degree [Formula: see text] on [Formula: see text], with [Formula: see text]. The object of our study is the map [Formula: see text], introduced in earlier papers by J. Alper, M. Eastwood and the author, that assigns to every form for which the discriminant [Formula: see text] does not vanish the so-called associated form lying in the space [Formula: see text]. This map is a morphism from the affine variety [Formula: see text] to the affine space [Formula: see text]. Letting [Formula: see text] be the smallest integer such that the product [Formula: see text] extends to a morphism from [Formula: see text] to [Formula: see text], one observes that the extended map defines a contravariant of forms in [Formula: see text]. In this paper, we obtain upper bounds for [Formula: see text] thus providing estimates for the contravariant’s degree.

On Semi - Pre irresolute Topological Vector Space

2018 ◽  
Vol 14 (3) ◽  
pp. 184-192
Author(s):  
Radhi Ali ◽  
◽  
Jalal Hussein Bayati ◽  
Suhad Hameed
Keyword(s):  
Vector Space ◽  
Topological Vector Space
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