On quadrics through five real points

1967 ◽  
Vol 63 (2) ◽  
pp. 369-388
Author(s):  
R. H. F. Denniston
Keyword(s):  
Projective Space ◽  
Fixed Points ◽  
Complex Case ◽  
Real Case ◽  
Quadric Surface ◽  
Real Projective Space ◽  
The Real ◽  
Homology Groups

Let Q1,…, Q5 be five fixed points (no four coplanar) of the real projective space S3: let s be a variable quadric surface through these points. The set of all such quadrics can be represented by the points of a real S4, in which there is a quartic primal that represents cones. The geometry of this threefold is well known in the complex case, but has hardly been considered at all in the real case: and one object of the present paper is to describe the real threefold and determine its homology groups.

Projection constants of symmetric spaces and variants of Khintchine's inequality

1999 ◽  
Vol 1999 (511) ◽  
pp. 1-42 ◽  
Author(s):  
Hermann König ◽  
Carsten Schütt ◽  
Nicole Tomczak-Jaegermann
Keyword(s):  
Symmetric Spaces ◽  
Dimensional Space ◽  
Complex Case ◽  
Real Case ◽  
The Real ◽  
Symmetric Basis ◽  
Projection Constant ◽  
Khintchine’S Inequality ◽  
Averaging Techniques

Abstract The projection constants of the lpn-spaces for 1 ≦ p ≦ 2 satisfy with in the real case and in the complex case. Further, there is c < 1 such that the projection constant of any n-dimensional space Xn with 1-symmetric basis can be estimated by . The proofs of the results are based on averaging techniques over permutations and a variant of Khintchine's inequality which states that

scholarly journals Generalized Givens Rotations Applied to Complex Joint Eigenvalue Decomposition

2021 ◽  
Vol 1 (1) ◽  
pp. 58-62
Author(s):  
Ammar Mesloub
Keyword(s):  
Complex Case ◽  
Eigenvalue Decomposition ◽  
Real Case ◽  
Joint Diagonalization ◽  
The Real ◽  
Givens Rotation ◽  
Givens Rotations ◽  
Simulation Results ◽  
Complex Joint

This paper shows the different ways of using generalized Givens rotations in complex joint eigenvaluedecomposition (JEVD) problem. It presents the different schemes of generalized Givens rotation, justifies the introducedapproximations and focuses on the process of extending an algorithm developed for real JEVD to the complex JEVD.Several Joint Diagonalization problem use generalized Givens rotations to achieve the solution, many algorithmsdeveloped in the real case exist in the literature and are not generalized to the complex case. Hence, we show herein asimple and not trivial way to get the complex case from the real one. Simulation results are provided to highlight theeffectiveness and behaviour of the proposed techniques for different scenarios.

scholarly journals An irreducible Heegaard diagram of the real projective 3-spacep3

2000 ◽  
Vol 23 (2) ◽  
pp. 123-129 ◽  
Author(s):  
Young Ho Im ◽  
Soo Hwan Kim
Keyword(s):  
Projective Space ◽  
Real Projective Space ◽  
Heegaard Diagrams ◽  
The Real ◽  
Heegaard Diagram ◽  
Genus 2

We give a genus 3 Heegaard diagramHof the real projective spacep3, which has no waves and pairs of complementary handles. So Negami's result that every genus 2 Heegaard diagram ofp3is reducible cannot be extended to Heegaard diagrams ofp3with genus 3.

Sequential motion planning algorithms in real projective spaces: An approach to their immersion dimension

2018 ◽  
Vol 30 (2) ◽  
pp. 397-417 ◽  
Author(s):  
Natalia Cadavid-Aguilar ◽  
Jesús González ◽  
Darwin Gutiérrez ◽  
Aldo Guzmán-Sáenz ◽  
Adriana Lara
Keyword(s):  
Positive Integer ◽  
Projective Space ◽  
Motion Planning ◽  
Product Space ◽  
Projective Spaces ◽  
Topological Complexity ◽  
Real Projective Space ◽  
Binary Expansion ◽  
The Real

AbstractThes-th higher topological complexity{\operatorname{TC}_{s}(X)}of a spaceXcan be estimated from above by homotopical methods, and from below by homological methods. We give a thorough analysis of the gap between such estimates when{X=\operatorname{\mathbb{R}P}^{m}}, the real projective space of dimensionm. In particular, we describe a number{r(m)}, which depends on the structure of zeros and ones in the binary expansion ofm, and with the property that{0\leq sm-\operatorname{TC}_{s}(\operatorname{\mathbb{R}P}^{m})\leq\delta_{s}(% m)}for{s\geq r(m)}, where{\delta_{s}(m)=(0,1,0)}for{m\equiv(0,1,2)\bmod 4}. Such an estimation for{\operatorname{TC}_{s}(\operatorname{\mathbb{R}P}^{m})}appears to be closely related to the determination of the Euclidean immersion dimension of{\operatorname{\mathbb{R}P}^{m}}. We illustrate the phenomenon in the case{m=3\cdot 2^{a}}. In addition, we show that, for large enoughsand evenm,{\operatorname{TC}_{s}(\operatorname{\mathbb{R}P}^{m})}is characterized as the smallest positive integer{t=t(m,s)}for which there is a suitable equivariant map from Davis’ projective product space{\mathrm{P}_{\mathbf{m}_{s}}}to the{(t+1)}-st join-power{((\mathbb{Z}_{2})^{s-1})^{\ast(t+1)}}. This is a (partial, but conjecturally complete) generalization of the work of Farber, Tabachnikov and Yuzvinsky relating{\operatorname{TC}_{2}}to the immersion dimension of real projective spaces.

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