scholarly journals Numerical Range of the Derivation of an Induced Operator

2007 ◽  
Vol 82 (3) ◽  
pp. 325-344
Author(s):  
Randall R. Holmes ◽  
Chi-Kwong Li ◽  
Tin-Yau Tam
Keyword(s):  
Convex Hull ◽  
Symmetric Group ◽  
Irreducible Character ◽  
Product Space ◽  
Numerical Range ◽  
Inner Product Space ◽  
Inner Product ◽  
Complex Numbers ◽  
Derivation Operator ◽  
Induced Operator

AbstractLet V be an n–dimensional inner product space over , let H be a subgroup of the symmetric group on {l,…, m}, and let x: H → be an irreducible character. Denote by (H) the symmetry class of tensors over V associated with H and x. Let K (T) ∈ End((H)) be the operator induced by T ∈ End(V), and let DK(T) be the derivation operator of T. The decomposable numerical range W*(DK(T)) of DK(T) is a subset of the classical numerical range W(DK(T)) of DK(T). It is shown that there is a closed star-shaped subset of complex numbers such that⊆ W*(DK(T)) ⊆ W(DK(T)) = con where conv denotes the convex hull of . In many cases, the set is convex, and thus the set inclusions are actually equalities. Some consequences of the results and related topics are discussed.

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 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.

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.

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