scholarly journals Spinors and canonical hermitian forms

1988 ◽  
Vol 30 (3) ◽  
pp. 263-270 ◽  
Author(s):  
P. L. Robinson
Keyword(s):  
Clifford Algebra ◽  
Product Space ◽  
Hermitian Form ◽  
Inner Product Space ◽  
Inner Product ◽  
Spin Representation ◽  
Hermitian Forms ◽  
Real Inner Product Space ◽  
Adjoint Operation

The spaceSof spinors associated to a2m-dimensional real inner product space (V, B) carries a canonical Hermitian form 〈 〉 determined uniquely up to real multiples. This form arises as follows: the complex Clifford algebraC(V) of (V, B) is naturally provided with an antilinear involution; this induces an involution on EndSvia the spin representation; this is the adjoint operation corresponding to 〈 〉.

scholarly journals Inner Bogoliubov automorphisms of the minimal C* Weyl algebra

1992 ◽  
Vol 34 (3) ◽  
pp. 263-270
Author(s):  
P. L. Robinson
Keyword(s):  
Clifford Algebra ◽  
Product Space ◽  
Clifford Algebras ◽  
Weyl Algebra ◽  
Inner Product Space ◽  
Inner Product ◽  
Spin Group ◽  
Group Of Units ◽  
Orthogonal Geometry ◽  
Real Inner Product Space

Within the context of orthogonal geometry, isometries of a real inner product space induce Bogoliubov automorphisms of its associated Clifford algebras. The question whether or not such automorphisms are inner is of considerable interest and importance. Inner Bogoliubov automorphisms were fully characterized for the C* Clifford algebra by Shale and Stinespring [14] and for the W* Clifford algebra by Blattner [2]: each case engenders a corresponding notion of spin group, constructed as a group of units inside the Clifford algebra [4].

On Closed Subsets of Root Systems

1994 ◽  
Vol 37 (3) ◽  
pp. 338-345 ◽  
Author(s):  
D. Ž. Doković ◽  
P. Check ◽  
J.-Y. Hée
Keyword(s):  
Weyl Group ◽  
Root System ◽  
Irreducible Component ◽  
Product Space ◽  
Root Systems ◽  
Inner Product Space ◽  
Inner Product ◽  
Closed Sets ◽  
Finite Dimensional ◽  
Real Inner Product Space

AbstractLet R be a root system (in the sense of Bourbaki) in a finite dimensional real inner product space V. A subset P ⊂ R is closed if α, β ∊ P and α + β ∊ R imply that α + β ∊ P. In this paper we shall classify, up to conjugacy by the Weyl group W of R, all closed sets P ⊂ R such that R\P is also closed. We also show that if θ:R —> R′ is a bijection between two root systems such that both θ and θ-1 preserve closed sets, and if R has at most one irreducible component of type A1, then θ is an isomorphism of root systems.

scholarly journals Thick sets and quasisymmetric maps

1994 ◽  
Vol 135 ◽  
pp. 121-148 ◽  
Author(s):  
Jussi Väisälä ◽  
Matti Vuorinen ◽  
Hans Wallin
Keyword(s):  
Positive Integer ◽  
Convex Hull ◽  
Product Space ◽  
Finite Sequence ◽  
Closed Ball ◽  
Inner Product Space ◽  
Inner Product ◽  
Real Inner Product Space

1.1. Thickness. Let E be a real inner product space. For a finite sequence of points a0, . . . ,ak in E we let a0. . . ,ak denote the convex hull of the set {a0, . . . , ak}. If these points are affinely independent, the set Δ = a0. . .ak is a k-simplex with vertices a0. . . ,ak. It has a well-defined k-volume written as mk(Δ) or briefly as m(Δ). We are interested in sets A ⊂ E which are “nowhere too flat in dimension k”. More precisely, suppose that A ⊂ E, q > 0 and that k: is a positive integer. We let denote the closed ball with center x and radius r. We say that A is (q, k)-thick if for each x ∈ A and r> 0 such that A\ ≠ there is a k-simplex Δ with vertices in A ∩ such that mk(Δ) ≥ qr.

scholarly journals On Orthogonally Additive Functions With Big Graphs

2017 ◽  
Vol 31 (1) ◽  
pp. 57-62
Author(s):  
Karol Baron
Keyword(s):  
Topological Space ◽  
Product Space ◽  
Linear Topological Space ◽  
Inner Product Space ◽  
Inner Product ◽  
Additive Functions ◽  
Big Graphs ◽  
Tychonoff Topology ◽  
Real Inner Product Space ◽  
Orthogonally Additive

Abstract Let E be a separable real inner product space of dimension at least 2 and V be a metrizable and separable linear topological space. We show that the set of all orthogonally additive functions mapping E into V and having big graphs is dense in the space of all orthogonally additive functions from E into V with the Tychonoff topology.

scholarly journals Fuzzy Inner Product Space: Literature Review and a New Approach

Mathematics ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 765
Author(s):  
Lorena Popa ◽  
Lavinia Sida
Keyword(s):  
Literature Review ◽  
Product Space ◽  
Inner Product Space ◽  
Inner Product ◽  
Product Spaces ◽  
Comprehensive Overview ◽  
New Approach ◽  
Inner Product Spaces ◽  
Product Function ◽  
Suitable Definition

The aim of this paper is to provide a suitable definition for the concept of fuzzy inner product space. In order to achieve this, we firstly focused on various approaches from the already-existent literature. Due to the emergence of various studies on fuzzy inner product spaces, it is necessary to make a comprehensive overview of the published papers on the aforementioned subject in order to facilitate subsequent research. Then we considered another approach to the notion of fuzzy inner product starting from P. Majundar and S.K. Samanta’s definition. In fact, we changed their definition and we proved some new properties of the fuzzy inner product function. We also proved that this fuzzy inner product generates a fuzzy norm of the type Nădăban-Dzitac. Finally, some challenges are given.

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