Symmetric Linear Functions Research Articles

Symmetric linear functionals on the Banach space generated by pseudometrics

In this work we consider the notion of $B$-equivalence of pseudometrics.Two pseudometrics $d_1$ and $d_2$ on a set $X$ are called $B$-equivalent, where $B$ is a subgroup of the group of all bijections on $X,$ if there exists an element $b$ of $B$ such that $d_1(x,y) = d_2(b(x),b(y))$ for every $x,y\in X,$ that is, $d_1$ can be obtained from $d_2$ by permutating elements of $X$ with the aid of the bijection $b.$The group $B$ generates the group $\widehat B$ of transformations of the set of all pseudometricson $X,$ elements of which act as $d(\cdot, \cdot)\mapsto d(b(\cdot),b(\cdot)),$ where $d$ is a pseudometrics on $X$ and $b\in B.$ A function $f$ on the set of all pseudometrics on $X$is called $\widehat B$-symmetric if $f$ is invariant under the action on its argument of elements of the group $\widehat B.$If two pseudometrics $d_1$ and $d_2$ are $B$-equivalent, then $f(d_1)=f(d_2)$ for every $\widehat B$-symmetric function $f.$ In general, the technique of symmetric functions is well-developed for the case of symmetric continuous polynomials and, in particular, for the case of symmetric continuous linear functionals on Banach spaces. To use this technique for the construction of $\widehat B$-symmetricfunctions on sets of pseudometrics, we map these sets to some appropriate Banach space $V$, which is isometrically isomorphic to the Banach space $\ell_1$of all absolutely summing real sequences. Weinvestigate symmetric (with respect to an arbitrary group of symmetry, elements of whichmap the standard Schauder basis of $\ell_1$ into itself) linear continuous functionalson $\ell_1.$ We obtain the complete description of the structure of these functionals.Also we establish analogical results for symmetric linear continuous functionals on the space $V.$ These results are used for the construction of $\widehat B$-symmetric functionals on the set of all pseudometrics on an arbitrary set $X$ for the following case:the group $B$ of bijections on $X,$ that generates the group $\widehat B,$ is such that the set of all $x\in X,$ for which there exists $b\in B$ such that $b(x)\neq x,$is finite.

Inverse problems, Sobolev–Chebyshev polynomials and asymptotics

UDC 517.9 Let ( u , v ) be a pair of quasidefinite and symmetric linear functionals with { P n } n ≥ 0 and { Q n } n ≥ 0 as respective sequences of monic orthogonal polynomial (SMOP). We define a sequence of monic polynomials { R n } n ≥ 0 as follows: P n + 2 ' ( x ) n + 2 + b n P n ' ( x ) n - Q n + 1 ( x ) = d n R n - 1 ( x ) , n ≥ 1. We give necessary and sufficient conditions for { R n } n ≥ 0 to be orthogonal with respect to a quasidefinite linear functional w . In addition, we consider the case where { P n } n ≥ 0 and { Q n } n ≥ 0 are monic Chebyshev polynomials of the first and second kinds, respectively, and study the relative outer asymptotics of Sobolev polynomials orthogonal with respect to the Sobolev inner product 〈 p , q 〉 S = ∫ -1 1 p q ( 1 - x 2 ) - 1 / 2 ⅆ x + λ 1 ∫ -1 1 p ' q ' ( 1 - x 2 ) 1 / 2 ⅆ x + λ 2 ∫ -1 1 p ' ' q ' ' ⅆ μ ( x ) , where μ is a positive Borel measure associated with w and λ 1 , λ 2 > 0 , λ 2 is a linear polynomial of λ 1 .

A Note on the Symmetry of Sequence Spaces

We give a self-contained treatment of symmetric Banach sequence spaces and some of their natural properties. We are particularly interested in the symmetry of the norm and the existence of symmetric linear functionals. Many of the presented results are known or commonly accepted, but are not found in the literature.

SOME REMARKS ON PSEUDO-TRACE FUNCTIONS FOR ORBIFOLD MODELS ASSOCIATED WITH SYMPLECTIC FERMIONS

We propose a method to give pseudo-trace functions for vertex operator algebras satisfying Zhu's finiteness condition and apply our method to the ℤ2-orbifold model associated with d-pairs of symplectic fermions. Pseudo-trace functions are defined by using symmetric linear functions on (higher) Zhu's algebras. But our approach does not use such algebras. For d = 1, we determine the dimension of the space of one-point functions, that is, the one of the triplet [Formula: see text]-algebra. For d > 1, we construct 22d-1 + 3 linearly independent one-point functions and study their values at the vacuum vector.

A MATRIX REALIZATION OF THE QUANTUM GROUP 𝔤p,q

In this paper we shall find a matrix realization of the quantum group 𝔤p,q introduced in [B. L. Feigin et al., Nucl. Phys. B757 (2006) 303–343]. For this purpose, we construct all primitive idempotents and a basis of 𝔤p,q. We determine the action of elements of the basis on indecomposable projective modules, that gives rise to a matrix realization of 𝔤p,q. By using this result, we obtain a basis of the space of symmetric linear functions on 𝔤p1,p2 and express the symmetric linear functions obtained by the left integral, the balancing element, and the center of 𝔤p,q in terms of this basis.

Some remarks on symmetric linear functions and pseudotrace maps

Let $A$ be a finite-dimensional associative algebra and $\phi$ a symmetric linear function on $A$. In this note, we will show that the pseudotrace maps defined in [6] are obtained as special cases of well-known symmetric linear functions on the endomorphism rings of projective modules. As an application of our approach, we will give proofs of several propositions and theorems in [6] for an arbitrary finite-dimensional associative algebra.

A construction of symmetric linear functions on the restricted quantum group $\bar{U}_{q}(\mathit{sl}_{2})$

In this article we construct all the primitive idempotents o f the restricted quantum group Uq (sl2) and also describe Uq (sl2) as the subalgebra of the direct sum of matrix algebras. By using this result we construct a basis of the spa ce of symmetric linear functions of Uq (sl2) and determine the decomposition of the integral of the dual of Uq (sl2) twisted by the balancing element to the basis of the space of symmetric linear functions.

Continuous symmetrized Sobolev inner products of order N (I)

Given a symmetrized Sobolev inner product of order N, the corresponding sequence of monic orthogonal polynomials { Q n } satisfies that Q 2 n ( x ) = P n ( x 2 ) , Q 2 n + 1 ( x ) = x R n ( x 2 ) for certain sequences of monic polynomials { P n } and { R n } . In this paper, we deduce the integral representation of the inner products such that { P n } and { R n } are the corresponding sequences of orthogonal polynomials. Moreover, we state a relation between both inner products which extends the classical result for symmetric linear functionals.

Reconsidering Nonlinearity and Asymmetry in Customer Satisfaction and Loyalty Models: An Empirical Study in Three Retail Service Settings

The use of symmetric linear functional forms to describe the relationships among customer evaluative judgments is omnipresent in services research. However, according to several researchers, the use of symmetric linear functions to model the relationships among customer evaluative judgments leads to serious model misspecification. Although several asymmetric nonlinear functional forms have been proposed, only modest empirical evidence is available on the superiority of these more complex model specifications. Based on a formal empirical comparison in multiple service settings, we find that asymmetric nonlinear functions do not possess superior explanatory power compared to symmetric linear functional forms.

Generating functions and companion symmetric linear functionals

In this contribution we analyze the generating functions for polynomials orthogonal with respect to a symmetric linear functional u, i.e., a linear application in the linear space of polynomials with complex coefficients such that \(u\left( {x^{2n + 1} } \right) = 0\). In some cases we can deduce explicitly the expression for the generating function $${\mathcal{P}}\left( {x,w} \right) = \sum\limits_{n = 0}^\infty {c_n P_n \left( x \right)w^n ,} $$ where {Pn}n is the sequence of orthogonal polynomials with respect to u.

On some ( n−1)-symmetric linear functionals

Many authors have dealt with problems related to the symmetrization of sequences of orthogonal polynomials on a real line or on a unit circle. Particular aspects are treated for quadratic or cubic decompositions. In this paper, we present a technique that unifies the treatment of these topics for an arbitrary positive order of decomposition n by considering a more general definition of orthogonality. This technique is based on the decomposition with respect to the cyclic group of order n. The main theorem is applied to the orthogonality on ( n−1)-symmetric curves. Some particular cases are singled out. These results may be useful in studying Hermite–Padé approximations, vector continued fractions and dynamical systems.