scholarly journals The deligne complex of a real arrangement of hyperplanes

1993 ◽  
Vol 131 ◽  
pp. 39-65 ◽  
Author(s):  
Luis Paris
Keyword(s):  
Vector Space ◽  
Finite Family ◽  
Real Vector Space ◽  
Real Vector ◽  
Arrangement Of Hyperplanes

Let V be a real vector space. An arrangement of hyperplanes in V is a finite family of hyperplanes of V through the origin. We say that is essential if ∩H ∊H = {0}

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].

On structure theory of pre-Hilbert algebras

2009 ◽  
Vol 139 (2) ◽  
pp. 303-319 ◽  
Author(s):  
José Antonio Cuenca Mira
Keyword(s):  
Vector Space ◽  
Associative Algebra ◽  
Real Vector Space ◽  
Inner Product ◽  
Structure Theory ◽  
Classical Theorem ◽  
Real Vector ◽  
Algebraic Identity ◽  
Hilbert Algebras

Let A be a real (non-associative) algebra which is normed as real vector space, with a norm ‖·‖ deriving from an inner product and satisfying ‖ac‖ ≤ ‖a‖‖c‖ for any a,c ∈ A. We prove that if the algebraic identity (a((ac)a))a = (a2c)a2 holds in A, then the existence of an idempotent e such that ‖e‖ = 1 and ‖ea‖ = ‖a‖ = ‖ae‖, a ∈ A, implies that A is isometrically isomorphic to ℝ, ℂ, ℍ, $\mathbb{O}$,\, $\stackrel{\raisebox{4.5pt}[0pt][0pt]{\fontsize{4pt}{4pt}\selectfont$\star$}}{\smash{\CC}}$,\, $\stackrel{\raisebox{4.5pt}[0pt][0pt]{\fontsize{4pt}{4pt}\selectfont$\star$}}{\smash{\mathbb{H}}}$,\, $\stackrel{\raisebox{4.5pt}[0pt][0pt]{\fontsize{4pt}{4pt}\selectfont$\star$}}{\smash{\mathbb{O}}}$ or ℙ. This is a non-associative extension of a classical theorem by Ingelstam. Finally, we give some applications of our main result.

scholarly journals Axioms for convexity

1993 ◽  
Vol 47 (2) ◽  
pp. 179-197 ◽  
Author(s):  
W.A. Coppel
Keyword(s):  
Vector Space ◽  
Convex Sets ◽  
Real Vector Space ◽  
Real Vector ◽  
Complete Set

The basic elementary results about convex sets are derived successively from various properties of segments. The complete set of properties is shown to form a natural set of axioms characterising the convex sets in a real vector space.

scholarly journals Duality between D(X) and with its application to picard sheaves

1981 ◽  
Vol 81 ◽  
pp. 153-175 ◽  
Author(s):  
Shigeru Mukai
Keyword(s):  
Vector Space ◽  
Fourier Transformation ◽  
Real Vector Space ◽  
Real Vector ◽  
Dual Vector

As is well known, for a real vector space V, the Fourier transformation gives an isometry between L2(V) and L2(Vv), where Vv is the dual vector space of V and < , >: V×Vv → R is the canonical pairing.

The quadratic form positive definite on a linear manifold

1951 ◽  
Vol 47 (1) ◽  
pp. 1-6 ◽  
Author(s):  
S. N. Afriat
Keyword(s):  
Quadratic Form ◽  
Vector Space ◽  
Linear Manifold ◽  
Sufficient Conditions ◽  
Positive Definite ◽  
Necessary And Sufficient Conditions ◽  
Real Vector Space ◽  
Real Vector ◽  
Necessary And Sufficient ◽  
Real Quadratic

1. Introduction. Necessary and sufficient conditions are established for a real quadratic form to be positive definite on a linear manifold, in a real vector space, explicit in terms of the dual Grassmann coordinates for the manifold.

Differentiation of n-Dimensional Additive Processes

1981 ◽  
Vol 33 (3) ◽  
pp. 749-768 ◽  
Author(s):  
M. A. Akcoglu ◽  
A. Del Junco
Keyword(s):  
Vector Space ◽  
Lebesgue Measure ◽  
Cartesian Product ◽  
Initial Point ◽  
Positive Cone ◽  
Real Vector Space ◽  
Real Vector ◽  
One Dimensional ◽  
Additive Processes ◽  
Dimensional Lebesgue Measure

Let n ≧ 1 be an integer and let Rn be the usual n-dimensional real vector space, considered together with all its usual structure. The usual n-dimensional Lebesgue measure on Rn is denoted by λn. The positive cone of Rn is Rn+ and the interior of Rn + is Pn. Hence Pn is the set of vectors with strictly positive coordinates. A subset of Rn is called an interval if it is the cartesian product of one dimensional bounded intervals. If a, b ∊ Rn then [a, b] denotes the interval {u|a ≦ u ≦ b|. The closure of any interval I is of the form [a, b]; the initial point of I will be defined as the vector a. The class of all intervals contained in Rn+ is denoted by . Also, for each u ∊ Pn, let be the set of all intervals that are contained in the interval [0, u] and that have non-empty interiors. Finally let en ∊ Pn be the vector with all coordinates equal to 1.

scholarly journals Power-associative regular real normed algebras

1966 ◽  
Vol 6 (2) ◽  
pp. 193-209 ◽  
Author(s):  
Emanuel Strzelecki
Keyword(s):  
Vector Space ◽  
Real Number ◽  
Real Vector Space ◽  
Real Vector ◽  
Real Algebra ◽  
Image Position

Definition 1. A real algebra A is a real vector space in which an operation of multiplication is defined satisfying the following conditions: for arbitrary x, y, z ∈ A and any real number α.

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