scholarly journals A Classification of Symmetric (1, 1)-Coherent Pairs of Linear Functionals

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 213
Author(s):  
Herbert Dueñas Ruiz ◽  
Francisco Marcellán ◽  
Alejandro Molano
Keyword(s):  
Positive Definite ◽  
Linear Functionals ◽  
Definite Case ◽  
Coherent Pairs

In this paper, we study a classification of symmetric ( 1 , 1 ) -coherent pairs by using a symmetrization process. In particular, the positive-definite case is carefully described.

scholarly journals (1,1)-Coherent Pairs on the Unit Circle

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Luis Garza ◽  
Francisco Marcellán ◽  
Natalia C. Pinzón-Cortés
Keyword(s):  
Orthogonal Polynomials ◽  
Unit Circle ◽  
Linear Functional ◽  
Linear Functionals ◽  
Spectral Transformation ◽  
Coherent Pair ◽  
Coherent Pairs

A pair(𝒰,𝒱)of Hermitian regular linear functionals on the unit circle is said to be a(1,1)-coherent pair if their corresponding sequences of monic orthogonal polynomials{ϕn(x)}n≥0and{ψn(x)}n≥0satisfyϕn[1](z)+anϕn-1[1](z)=ψn(z)+bnψn-1(z),an≠0,n≥1, whereϕn[1](z)=ϕn+1′(z)/(n+1). In this contribution, we consider the cases when𝒰is the linear functional associated with the Lebesgue and Bernstein-Szegő measures, respectively, and we obtain a classification of the situations where𝒱is associated with either a positive nontrivial measure or its rational spectral transformation.

scholarly journals On the converse of Weyl’s conformal and projective theorems

2013 ◽  
Vol 94 (108) ◽  
pp. 55-65 ◽  
Author(s):  
Graham Hall
Keyword(s):  
Positive Definite ◽  
The Other ◽  
Dimensional Problem ◽  
Projective Tensor ◽  
Definite Case

This note investigates the possibility of converses of the Weyl theorems that two conformally related metrics on a manifold have the same Weyl conformal tensor and that two projectively related connections on a manifold have the same Weyl projective tensor. It shows that, in all relevant cases, counterexamples to each of Weyl?s theorems exist except for his conformal theorem in the 4-dimensional, positive definite case, where the converse actually holds. This (conformal) 4-dimensional problem is then solved completely for the other possible signatures.

scholarly journals A classification of isotropic affine hyperspheres

2016 ◽  
Vol 27 (09) ◽  
pp. 1650074 ◽  
Author(s):  
Marilena Moruz ◽  
Luc Vrancken
Keyword(s):  
Acta Math ◽  
Complete Classification ◽  
Positive Definite ◽  
Affine Hypersurface ◽  
Affine Hypersphere ◽  
Difference Tensor ◽  
Affine Spheres

We study affine hypersurfaces [Formula: see text], which have isotropic difference tensor. Note that, any surface always has isotropic difference tensor. In case that the metric is positive definite, such hypersurfaces have been previously studied in [O. Birembaux and M. Djoric, Isotropic affine spheres, Acta Math. Sinica 28(10) 1955–1972.] and [O. Birembaux and L. Vrancken, Isotropic affine hypersurfaces of dimension 5, J. Math. Anal. Appl. 417(2) (2014) 918–962.] We first show that the dimension of an isotropic affine hypersurface is either [Formula: see text], [Formula: see text], [Formula: see text] or [Formula: see text]. Next, we assume that [Formula: see text] is an affine hypersphere and we obtain for each of the possible dimensions a complete classification.

Free group C*-algebras associated with ℓp

2014 ◽  
Vol 25 (07) ◽  
pp. 1450065 ◽  
Author(s):  
Rui Okayasu
Keyword(s):  
Free Group ◽  
Positive Definite ◽  
Positive Definite Functions ◽  
Linear Functionals ◽  
C Algebra ◽  
Tracial State ◽  
C Algebras ◽  
Positive Linear Functionals ◽  
The Ideal

For every p ≥ 2, we give a characterization of positive definite functions on a free group with finitely many generators, which can be extended to positive linear functionals on the free group C*-algebra associated with the ideal ℓp. This is a generalization of Haagerup's characterization for the case of the reduced free group C*-algebra. As a consequence, the canonical quotient map between the associated C*-algebras is not injective, and they have a unique tracial state.

scholarly journals Classification of all δ-Coherent Pairs

2000 ◽  
Vol 9 (1) ◽  
pp. 1-18 ◽  
Author(s):  
I. Area ◽  
E. Godoy ◽  
F. Marcellán
Keyword(s):  
Coherent Pairs

scholarly journals Classification of positive definite lattices

2000 ◽  
Vol 105 (3) ◽  
pp. 525-567 ◽  
Author(s):  
Richard E. Borcherds
Keyword(s):  
Positive Definite
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