scholarly journals Piecewise Linear Sheaves

2019 ◽  
Author(s):  
Masaki Kashiwara ◽  
Pierre Schapira
Keyword(s):  
Vector Space ◽  
Piecewise Linear ◽  
Building Blocks ◽  
Convex Polyhedra ◽  
Triangulated Category ◽  
Real Vector ◽  
Proper Cone ◽  
Finite Dimensional ◽  
Higher Dimensional

Abstract On a finite-dimensional real vector space, we give a microlocal characterization of (derived) piecewise linear sheaves (PL sheaves) and prove that the triangulated category of such sheaves is generated by sheaves associated with convex polyhedra. We then give a similar theorem for PL $\gamma $-sheaves, that is, PL sheaves associated with the $\gamma $-topology, for a closed convex polyhedral proper cone $\gamma $. Our motivation is that convex polyhedra may be considered as building blocks for higher dimensional barcodes.

scholarly journals On relationships between two linear subspaces and two orthogonal projectors

2019 ◽  
Vol 7 (1) ◽  
pp. 142-212 ◽  
Author(s):  
Yongge Tian
Keyword(s):  
Vector Space ◽  
Matrix Decomposition ◽  
Complex Vector ◽  
Linear Subspaces ◽  
Survey Paper ◽  
Finite Dimensional ◽  
Generic Position ◽  
Ep Matrix ◽  
Orthogonal Projectors

Abstract Sum and intersection of linear subspaces in a vector space over a field are fundamental operations in linear algebra. The purpose of this survey paper is to give a comprehensive approach to the sums and intersections of two linear subspaces and their orthogonal complements in the finite-dimensional complex vector space. We shall establish a variety of closed-form formulas for representing the direct sum decompositions of the m-dimensional complex column vector space 𝔺m with respect to a pair of given linear subspaces 𝒨 and 𝒩 and their operations, and use them to derive a huge amount of decomposition identities for matrix expressions composed by a pair of orthogonal projectors onto the linear subspaces. As applications, we give matrix representation for the orthogonal projectors onto the intersections of a pair of linear subspaces using various matrix decomposition identities and Moore–Penrose inverses; necessary and su˚cient conditions for two linear subspaces to be in generic position; characterization of the commutativity of a pair of orthogonal projectors; necessary and su˚cient conditions for equalities and inequalities for a pair of subspaces to hold; equalities and inequalities for norms of a pair of orthogonal projectors and their operations; as well as a collection of characterizations of EP-matrix.

A characterization of piecewise linear homology manifolds

1983 ◽  
Vol 93 (2) ◽  
pp. 271-274 ◽  
Author(s):  
W. J. R. Mitchell
Keyword(s):  
Piecewise Linear ◽  
Subsequent Paper ◽  
Locally Compact ◽  
Locally Finite ◽  
Finite Dimensional ◽  
Finite Set ◽  
Topological Manifolds ◽  
Homology Manifolds

We state and prove a theorem which characterizes piecewise linear homology manifolds of sufficiently large dimension among locally compact finite-dimensional absolute neighbourhood retracts (ANRs). The proof is inspired by Cannon's observation (3) that a piecewise linear homology manifold is a topological manifold away from a locally finite set, and uses Galewski and Stern's work on simplicial triangulations of topological manifolds, the Edwards–Cannon–Quinn characterization of topological manifolds and Siebenmann's work on ends (3, 6, 4, 13, 14, 15, 16). All these tools have suitable relative versions and so the theorem can be extended to the bounded case. However, the most satisfactory extension requires a classification of triangulations of homology manifolds up to concordance. This will be given in a subsequent paper and the bounded case will be postponed to that paper.

scholarly journals Linear Imbeddings of Self-Dual Homogeneous Cones

1972 ◽  
Vol 46 ◽  
pp. 121-145 ◽  
Author(s):  
I. Satake
Keyword(s):  
Vector Space ◽  
Lie Group ◽  
Isotropy Subgroup ◽  
Real Vector Space ◽  
Real Vector ◽  
Finite Dimensional ◽  
Homogeneous Cones ◽  
Homogeneous Cone

Let G be a reductive algebraic Lie group acting linearly on a (finite-dimensional) real vector-space U with a maximal compact isotropy subgroup K and suppose that the quotient Ω = G/K is a self-dual homogeneous cone in U. Let (G′, K′) be another such pair corresponding to a self-dual homogeneous cone Ω′ in U′.

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.

scholarly journals Criteria for periodicity and an application to elliptic functions

2020 ◽  
pp. 1-11
Author(s):  
Ehud de Shalit
Keyword(s):  
Vector Space ◽  
Elliptic Function ◽  
Function Fields ◽  
Elliptic Functions ◽  
Real Vector Space ◽  
Real Vector ◽  
Finite Dimensional ◽  
Full Discussion ◽  
Rank 2 ◽  
Higher Rank

Abstract Let P and Q be relatively prime integers greater than 1, and let f be a real valued discretely supported function on a finite dimensional real vector space V. We prove that if $f_{P}(x)=f(Px)-f(x)$ and $f_{Q}(x)=f(Qx)-f(x)$ are both $\Lambda $ -periodic for some lattice $\Lambda \subset V$ , then so is f (up to a modification at $0$ ). This result is used to prove a theorem on the arithmetic of elliptic function fields. In the last section, we discuss the higher rank analogue of this theorem and explain why it fails in rank 2. A full discussion of the higher rank case will appear in a forthcoming work.

scholarly journals Another proof for the continuity of the Lipsman mapping

2020 ◽  
Vol 72 (7) ◽  
pp. 945-951
Author(s):  
A. Messaoud ◽  
A. Rahali
Keyword(s):  
Vector Space ◽  
Lie Algebra ◽  
Semidirect Product ◽  
Lie Group ◽  
Inner Product ◽  
Coadjoint Orbits ◽  
Compact Lie Group ◽  
Real Vector ◽  
Finite Dimensional ◽  
Unitary Dual

UDC 515.1 We consider the semidirect product G = K ⋉ V where K is a connected compact Lie group acting by automorphisms on a finite dimensional real vector space V equipped with an inner product 〈 , 〉 . By G ^ we denote the unitary dual of G and by 𝔤 ‡ / G the space of admissible coadjoint orbits, where 𝔤 is the Lie algebra of G . It was pointed out by Lipsman that the correspondence between G ^ and 𝔤 ‡ / G is bijective. Under some assumption on G , we give another proof for the continuity of the orbit mapping (Lipsman mapping) Θ : 𝔤 ‡ / G - → G ^ .

A SYMMETRY PRESERVING REDUCTION SCHEME FOR PERIODIC POINTS NEAR A FIXED POINT OF FAMILIES OF DIFFEOMORPHISMS WITH A COMPACT SYMMETRY GROUP

2006 ◽  
Vol 16 (09) ◽  
pp. 2545-2557
Author(s):  
MARIA-CRISTINA CIOCCI ◽  
JOHAN NOLDUS
Keyword(s):  
Fixed Point ◽  
Normal Form ◽  
Vector Space ◽  
Periodic Points ◽  
Reduction Scheme ◽  
Real Vector ◽  
Linear Action ◽  
One Dimensional ◽  
Finite Dimensional ◽  
Normal Form Theorem

We present a generalized Lyapunov Schmidt (LS) reduction scheme for diffeomorphisms on a finite dimensional real vector space V which transform under real one-dimensional characters χ of an arbitrary compact group with linear action on V. Moreover we prove a normal form theorem, such that the normal form still has the desirable transformation properties with respect to χ.

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