scholarly journals Gradual transitivity in orthogonality spaces of finite rank

2020 ◽  
Author(s):  
Thomas Vetterlein
Keyword(s):  
Automorphism Group ◽  
Linear Space ◽  
Binary Relation ◽  
Finite Rank ◽  
Positive Definite ◽  
Inner Product ◽  
Orthogonality Space ◽  
One Dimensional ◽  
Finite Dimensional ◽  
Divisible Subgroup

Abstract An orthogonality space is a set together with a symmetric and irreflexive binary relation. Any linear space equipped with a reflexive and anisotropic inner product provides an example: the set of one-dimensional subspaces together with the usual orthogonality relation is an orthogonality space. We present simple conditions to characterise the orthogonality spaces that arise in this way from finite-dimensional Hermitian spaces. Moreover, we investigate the consequences of the hypothesis that an orthogonality space allows gradual transitions between any pair of its elements. More precisely, given elements e and f, we require a homomorphism from a divisible subgroup of the circle group to the automorphism group of the orthogonality space to exist such that one of the automorphisms maps e to f, and any of the automorphisms leaves the elements orthogonal to e and f fixed. We show that our hypothesis leads us to positive definite quadratic spaces. By adding a certain simplicity condition, we furthermore find that the field of scalars is Archimedean and hence a subfield of the reals.

Orthogonality spaces of finite rank and the complex Hilbert spaces

2019 ◽  
Vol 16 (05) ◽  
pp. 1950080 ◽  
Author(s):  
Thomas Vetterlein
Keyword(s):  
Binary Relation ◽  
Hilbert Spaces ◽  
Finite Rank ◽  
Representation Theorem ◽  
Quantum Physics ◽  
Orthogonality Relation ◽  
Quadratic Space ◽  
Orthogonality Space ◽  
Finite Dimensional

An orthogonality space is a set endowed with a symmetric, irreflexive binary relation. By means of the usual orthogonality relation, each anisotropic quadratic space gives rise to such a structure. We investigate in this paper the question to which extent this strong abstraction suffices to characterize complex Hilbert spaces, which play a central role in quantum physics. To this end, we consider postulates concerning the nature and existence of symmetries. Together with a further postulate excluding the existence of nontrivial quotients, we establish a representation theorem for finite-dimensional orthomodular spaces over a dense subfield of [Formula: see text].

Inequalities between means of positive operators

1978 ◽  
Vol 83 (3) ◽  
pp. 393-401 ◽  
Author(s):  
K. V. Bhagwat ◽  
R. Subramanian
Keyword(s):  
Hilbert Space ◽  
Dimensional Space ◽  
Adjoint Operator ◽  
Positive Definite ◽  
Inner Product ◽  
Finite Dimensional Space ◽  
Finite Dimensional ◽  
Definite Operator ◽  
Unit Operator ◽  
Adjoint Operators

One of the most fruitful – and natural – ways of introducing a partial order in the set of bounded self-adjoint operators in a Hilbert space is through the concept of a positive operator. A bounded self-adjoint operator A denned on is called positive – and one writes A ≥ 0 - if the inner product (ψ, Aψ) ≥ 0 for every ψ ∈ . If, in addition, (ψ, Aψ) = 0 only if ψ = 0, then A is called positive-definite and one writes A > 0. Further, if there exists a real number γ > 0 such that A — γI ≥ 0, I being the unit operator, then A is called strictly positive (in symbols, A ≫ 0). In a finite dimensional space, a positive-definite operator is also strictly positive.

scholarly journals Finite Rank Perturbations of Linear Relations and Matrix Pencils

2021 ◽  
Vol 15 (2) ◽  
Author(s):  
Leslie Leben ◽  
Francisco Martínez Pería ◽  
Friedrich Philipp ◽  
Carsten Trunk ◽  
Henrik Winkler
Keyword(s):  
Finite Rank ◽  
Regular Matrix ◽  
One Dimensional ◽  
Linear Relations ◽  
Finite Dimensional ◽  
Matrix Pencils ◽  
The Difference

AbstractWe elaborate on the deviation of the Jordan structures of two linear relations that are finite-dimensional perturbations of each other. We compare their number of Jordan chains of length at least n. In the operator case, it was recently proved that the difference of these numbers is independent of n and is at most the defect between the operators. One of the main results of this paper shows that in the case of linear relations this number has to be multiplied by $$n+1$$ n + 1 and that this bound is sharp. The reason for this behavior is the existence of singular chains. We apply our results to one-dimensional perturbations of singular and regular matrix pencils. This is done by representing matrix pencils via linear relations. This technique allows for both proving known results for regular pencils as well as new results for singular ones.

scholarly journals Smooth representations of unit groups of split basic algebras over non-Archimedean local fields

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Carlos A. M. André ◽  
João Dias
Keyword(s):  
Local Field ◽  
Unit Group ◽  
Basic Algebra ◽  
Local Fields ◽  
Dimensional Representation ◽  
Smooth Representation ◽  
One Dimensional ◽  
Finite Dimensional

Abstract We consider smooth representations of the unit group G = A × G=\mathcal{A}^{\times} of a finite-dimensional split basic algebra 𝒜 over a non-Archimedean local field. In particular, we prove a version of Gutkin’s conjecture, namely, we prove that every irreducible smooth representation of 𝐺 is compactly induced by a one-dimensional representation of the unit group of some subalgebra of 𝒜. We also discuss admissibility and unitarisability of smooth representations of 𝐺.

scholarly journals The Action of the Weyl Group on the $$E_8$$ Root System

2021 ◽  
Author(s):  
Rosa Winter ◽  
Ronald van Luijk
Keyword(s):  
Automorphism Group ◽  
Weyl Group ◽  
Root System ◽  
Sufficient Conditions ◽  
Maximal Clique ◽  
Necessary And Sufficient Conditions ◽  
Inner Product ◽  
Colored Graphs ◽  
Necessary And Sufficient ◽  
Root Polytope

AbstractLet $$\varGamma $$ Γ be the graph on the roots of the $$E_8$$ E 8 root system, where any two distinct vertices e and f are connected by an edge with color equal to the inner product of e and f. For any set c of colors, let $$\varGamma _c$$ Γ c be the subgraph of $$\varGamma $$ Γ consisting of all the 240 vertices, and all the edges whose color lies in c. We consider cliques, i.e., complete subgraphs, of $$\varGamma $$ Γ that are either monochromatic, or of size at most 3, or a maximal clique in $$\varGamma _c$$ Γ c for some color set c, or whose vertices are the vertices of a face of the $$E_8$$ E 8 root polytope. We prove that, apart from two exceptions, two such cliques are conjugate under the automorphism group of $$\varGamma $$ Γ if and only if they are isomorphic as colored graphs. Moreover, for an isomorphism f from one such clique K to another, we give necessary and sufficient conditions for f to extend to an automorphism of $$\varGamma $$ Γ , in terms of the restrictions of f to certain special subgraphs of K of size at most 7.

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 Auerbach Bases and Minimal Volume Sufficient Enlargements

2011 ◽  
Vol 54 (4) ◽  
pp. 726-738
Author(s):  
M. I. Ostrovskii
Keyword(s):  
Linear Space ◽  
Isometric Embedding ◽  
Normed Linear Space ◽  
Convex Set ◽  
Linear Projection ◽  
Minimal Volume ◽  
Finite Dimensional ◽  
Dimensional Normed Space ◽  
Closed Convex Set

AbstractLet BY denote the unit ball of a normed linear space Y. A symmetric, bounded, closed, convex set A in a finite dimensional normed linear space X is called a sufficient enlargement for X if, for an arbitrary isometric embedding of X into a Banach space Y, there exists a linear projection P: Y → X such that P(BY ) ⊂ A. Each finite dimensional normed space has a minimal-volume sufficient enlargement that is a parallelepiped; some spaces have “exotic” minimal-volume sufficient enlargements. The main result of the paper is a characterization of spaces having “exotic” minimal-volume sufficient enlargements in terms of Auerbach bases.

AUTOMORPHISMS OF CERTAIN NILPOTENT LIE ALGEBRAS OVER COMMUTATIVE RINGS

2007 ◽  
Vol 17 (03) ◽  
pp. 527-555 ◽  
Author(s):  
YOU'AN CAO ◽  
DEZHI JIANG ◽  
JUNYING WANG
Keyword(s):  
Automorphism Group ◽  
Commutative Ring ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Commutative Rings ◽  
Nilpotent Lie Algebras ◽  
Simple Lie Algebra ◽  
Finite Dimensional ◽  
Positive Roots ◽  
Nilpotent Subalgebra

Let L be a finite-dimensional complex simple Lie algebra, Lℤ be the ℤ-span of a Chevalley basis of L and LR = R⊗ℤLℤ be a Chevalley algebra of type L over a commutative ring R. Let [Formula: see text] be the nilpotent subalgebra of LR spanned by the root vectors associated with positive roots. The aim of this paper is to determine the automorphism group of [Formula: see text].

scholarly journals Identifying the heat sink

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
J. D. Audu ◽  
A. Boumenir ◽  
K. M. Furati ◽  
I. O. Sarumi
Keyword(s):  
Inverse Problem ◽  
Heat Equation ◽  
Temperature Profile ◽  
Spectral Methods ◽  
Heat Sink ◽  
Evolution Equations ◽  
Identification Problem ◽  
One Dimensional ◽  
Finite Dimensional ◽  
Pseudo Spectral

<p style='text-indent:20px;'>In this paper we examine the identification problem of the heat sink for a one dimensional heat equation through observations of the solution at the boundary or through a desired temperature profile to be attained at a certain given time. We make use of pseudo-spectral methods to recast the direct as well as the inverse problem in terms of linear systems in matrix form. The resulting evolution equations in finite dimensional spaces leads to fast real time algorithms which are crucial to applied control theory.</p>

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