scholarly journals The Real Vector Spaces of Finite Sequences are Finite Dimensional

2009 ◽  
Vol 17 (1) ◽  
pp. 1-9 ◽  
Author(s):  
Yatsuka Nakamura ◽  
Artur Korniłowicz ◽  
Nagato Oya ◽  
Yasunari Shidama
Keyword(s):  
Vector Spaces ◽  
Real Vector ◽  
The Real ◽  
Finite Dimensional ◽  
Finite Sequences

scholarly journals Real Vector Space and Related NotionsThis study was supported in part by JSPS KAKENHI Grant Numbers 17K00182 and 20K19863.

2021 ◽  
Vol 29 (3) ◽  
pp. 117-127
Author(s):  
Kazuhisa Nakasho ◽  
Hiroyuki Okazaki ◽  
Yasunari Shidama
Keyword(s):  
Vector Space ◽  
Algebraic Structure ◽  
Linear Independence ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Real Vector ◽  
Type Conversion ◽  
Finite Dimensional ◽  
Finite Sequences ◽  
Complicated Process

Summary. In this paper, we discuss the properties that hold in finite dimensional vector spaces and related spaces. In the Mizar language [1], [2], variables are strictly typed, and their type conversion requires a complicated process. Our purpose is to formalize that some properties of finite dimensional vector spaces are preserved in type transformations, and to contain the complexity of type transformations into this paper. Specifically, we show that properties such as algebraic structure, subsets, finite sequences and their sums, linear combination, linear independence, and affine independence are preserved in type conversions among TOP-REAL(n), REAL-NS(n), and n-VectSp over F Real. We referred to [4], [9], and [8] in the formalization.

Topological equivalence and rigidity of flows on certain solvmanifolds

1999 ◽  
Vol 19 (3) ◽  
pp. 559-569
Author(s):  
D. BENARDETE ◽  
S. G. DANI
Keyword(s):  
Lie Group ◽  
Vector Spaces ◽  
Complex Numbers ◽  
Real Vector ◽  
Orbit Equivalent ◽  
Finite Dimensional ◽  
Generic Class ◽  
Induced Flow ◽  
Simply Connected ◽  
Solvable Lie Group

Given a Lie group $G$ and a lattice $\Gamma$ in $G$, a one-parameter subgroup $\phi$ of $G$ is said to be rigid if for any other one-parameter subgroup $\psi$, the flows induced by $\phi$ and $\psi$ on $\Gamma\backslash G$ (by right translations) are topologically orbit-equivalent only if they are affinely orbit-equivalent. It was previously known that if $G$ is a simply connected solvable Lie group such that all the eigenvalues of $\mathrm{Ad} (g) $, $g\in G$, are real, then all one-parameter subgroups of $G$ are rigid for any lattice in $G$. Here we consider a complementary case, in which the eigenvalues of $\mathrm{Ad} (g)$, $g\in G$, form the unit circle of complex numbers.Let $G$ be the semidirect product $N \rtimes M$, where $M$ and $N$ are finite-dimensional real vector spaces and where the action of $M$ on the normal subgroup $N$ is such that the center of $G$ is a lattice in $M$. We prove that there is a generic class of abelian lattices $\Gamma$ in $G$ such that any semisimple one-parameter subgroup $\phi$ (namely $\phi$ such that $\mathrm{Ad} (\phi_t)$ is diagonalizable over the complex numbers for all $t$) is rigid for $\Gamma$ (see Theorem 1.4). We also show that, on the other hand, there are fairly high-dimensional spaces of abelian lattices for which some semisimple $\phi$ are not rigid (see Corollary 4.3); further, there are non-rigid semisimple $\phi$ for which the induced flow is ergodic.

scholarly journals Cross-Ratio in Real Vector Space

2019 ◽  
Vol 27 (1) ◽  
pp. 47-60
Author(s):  
Roland Coghetto
Keyword(s):  
Vector Space ◽  
Real Line ◽  
Cross Ratio ◽  
Real Vector Space ◽  
Vector Spaces ◽  
Real Vector ◽  
Complex Vector ◽  
The Real ◽  
Mizar Mathematical Library ◽  
The Cross

Summary Using Mizar [1], in the context of a real vector space, we introduce the concept of affine ratio of three aligned points (see [5]). It is also equivalent to the notion of “Mesure algèbrique”1, to the opposite of the notion of Teilverhältnis2 or to the opposite of the ordered length-ratio [9]. In the second part, we introduce the classic notion of “cross-ratio” of 4 points aligned in a real vector space. Finally, we show that if the real vector space is the real line, the notion corresponds to the classical notion3 [9]: The cross-ratio of a quadruple of distinct points on the real line with coordinates x1, x2, x3, x4 is given by: $$({x_1},{x_2};{x_3},{x_4}) = {{{x_3} - {x_1}} \over {{x_3} - {x_2}}}.{{{x_4} - {x_2}} \over {{x_4} - {x_1}}}$$ In the Mizar Mathematical Library, the vector spaces were first defined by Kusak, Leonczuk and Muzalewski in the article [6], while the actual real vector space was defined by Trybulec [10] and the complex vector space was defined by Endou [4]. Nakasho and Shidama have developed a solution to explore the notions introduced by different authors4 [7]. The definitions can be directly linked in the HTMLized version of the Mizar library5. The study of the cross-ratio will continue within the framework of the Klein- Beltrami model [2], [3]. For a generalized cross-ratio, see Papadopoulos [8].

scholarly journals Vector and Ordered Variational Inequalities and Applications to Order-Optimization Problems on Banach Lattices

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Linsen Xie ◽  
Jinlu Li ◽  
Wenshan Yang
Keyword(s):  
Fixed Point ◽  
Variational Inequality ◽  
Variational Inequalities ◽  
Fixed Point Theorems ◽  
Optimization Problems ◽  
Banach Lattices ◽  
Variational Inequality Problems ◽  
Vector Spaces ◽  
Real Vector ◽  
Finite Dimensional

We investigate the connections between vector variational inequalities and ordered variational inequalities in finite dimensional real vector spaces. We also use some fixed point theorems to prove the solvability of ordered variational inequality problems and their application to some order-optimization problems on the Banach lattices.

scholarly journals The fundamental theorems of affine and projective geometry revisited

2016 ◽  
Vol 19 (05) ◽  
pp. 1650059
Author(s):  
Shiri Artstein-Avidan ◽  
Boaz A. Slomka
Keyword(s):  
Three Dimensional ◽  
Real Space ◽  
Vector Spaces ◽  
Real Vector ◽  
Injective Mapping ◽  
Finite Dimensional ◽  
Fixed Set ◽  
Dimensional Projective Space ◽  
Projective Lines ◽  
Fundamental Theorems

The fundamental theorem of affine geometry is a classical and useful result. For finite-dimensional real vector spaces, the theorem roughly states that a bijective self-mapping which maps lines to lines is affine-linear. In this paper, we prove several generalizations of this result and of its classical projective counterpart. We show that under a significant geometric relaxation of the hypotheses, namely that only lines parallel to one of a fixed set of finitely many directions are mapped to lines, an injective mapping of the space must be of a very restricted polynomial form. We also prove that under mild additional conditions the mapping is forced to be affine-additive or affine-linear. For example, we show that five directions in three-dimensional real space suffice to conclude affine-additivity. In the projective setting, we show that [Formula: see text] fixed projective points in real [Formula: see text]-dimensional projective space, through which all projective lines that pass are mapped to projective lines, suffice to conclude projective-linearity.

On suns and cosuns in finite dimensional normed real vector spaces

1985 ◽  
Vol 45 (1-2) ◽  
pp. 53-68 ◽  
Author(s):  
L. Hetzelt
Keyword(s):  
Vector Spaces ◽  
Real Vector ◽  
Finite Dimensional

scholarly journals HYPERBOLIC GROUPS AND NON-COMPACT REAL ALGEBRAIC CURVES

2021 ◽  
Author(s):  
SERGEY NATANZON ◽  
ANNA PRATOUSSEVITCH
Keyword(s):  
Discrete Group ◽  
Algebraic Curves ◽  
Hyperbolic Groups ◽  
Vector Spaces ◽  
Fuchsian Groups ◽  
Connected Components ◽  
Connected Component ◽  
Real Vector ◽  
Real Algebraic Curves ◽  
Finite Dimensional

AbstractIn this paper we study the spaces of non-compact real algebraic curves, i.e. pairs (P, τ), where P is a compact Riemann surface with a finite number of holes and punctures and τ: P → P is an anti-holomorphic involution. We describe the uniformisation of non-compact real algebraic curves by Fuchsian groups. We construct the spaces of non-compact real algebraic curves and describe their connected components. We prove that any connected component is homeomorphic to a quotient of a finite-dimensional real vector space by a discrete group and determine the dimensions of these vector spaces.

scholarly journals Real Dimension Groups

2013 ◽  
Vol 56 (3) ◽  
pp. 551-563 ◽  
Author(s):  
David Handelman
Keyword(s):  
Direct Limit ◽  
Polynomial Rings ◽  
Vector Spaces ◽  
Real Vector ◽  
Product Decomposition ◽  
Finite Dimensional ◽  
Dimension Groups ◽  
Ordered Vector Spaces ◽  
Direct Limits ◽  
Directed Set

AbstractDimension groups (not countable) that are also real ordered vector spaces can be obtained as direct limits (over directed sets) of simplicial real vector spaces (finite dimensional vector spaces with the coordinatewise ordering), but the directed set is not as interesting as one would like; for instance, it is not true that a countable-dimensional real vector space that has interpolation can be represented as such a direct limit over a countable directed set. It turns out this is the case when the group is additionally simple, and it is shown that the latter have an ordered tensor product decomposition. In an appendix, we provide a huge class of polynomial rings that, with a pointwise ordering, are shown to satisfy interpolation, extending a result outlined by Fuchs.

scholarly journals On the space of matrices of given rank

1989 ◽  
Vol 32 (1) ◽  
pp. 99-105
Author(s):  
M. C. Crabb ◽  
D. L. Gonçalves
Keyword(s):  
Grassmann Manifold ◽  
Vector Spaces ◽  
Real Vector ◽  
Finite Dimensional ◽  
Linear Maps ◽  
Open Subspace

Let V and W be finite dimensional real vector spaces, k≧0 an integer. We write L(V, W) for the space of all linear maps V→W and Lk(V, W) for the subspace of maps with kernel of dimension k; in particular, L0(V, W) is the open subspace of injective linear maps. Thus Lk(ℝn, ℝn) is the space of n × n-matrices of rank n – k in the title. We also need the notation Gk(V) for the Grassmann manifold of K-dimensional subspaces of V.

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