Pablo Diaz

This paper studies the bidiagonal factorization of the collocation matrices of analytic bases using symmetric functions. Explicit formulas for their initial minors are derived in terms of Schur functions. The structure of these formulas permits establishing sufficient conditions for the total positivity of generic systems of analytic functions. In addition, they have been found to lead to generalizations of the Cauchy identity for certain families of functions.

This paper studies the bidiagonal factorization of the collocation matrices of analytic bases using symmetric functions. Explicit formulas for their initial minors are derived in terms of Schur functions. The structure of these formulas permits establishing sufficient conditions for the total positivity of generic systems of analytic functions. In addition, they have been found to lead to generalizations of the Cauchy identity for certain families of functions.

We present a broader framework for the Cauchy identity derived from the determinant expansion of collocation matrices. This approach yields an infinite family of identities, where the original Cauchy identity stands as a particular case. To illustrate the versatility and depth of this approach, we provide a range of compelling examples, showcasing the connections and applications of these novel identities.

The elements of the bidiagonal decomposition (BD) of a totally positive (TP) collocation matrix can be expressed in terms of symmetric functions of the nodes. Making use of this result, and studying the relation between Wronskian and collocation matrices of a given TP basis of functions, we can express the entries of the BD of Wronskian matrices as the values of certain symmetric functions evaluated at a single node. Moreover, in the case of polynomial bases, we obtain compact formulae for the entries of the BD of their Wronskian matrices. Interesting examples illustrate the applications of the obtained formulae.

In this paper, Schur polynomials are used to provide a bidiagonal decomposition of polynomial collocation matrices. The symmetry of Schur polynomials is exploited to analyze the total positivity on some unbounded intervals of a relevant class of polynomial bases. The proposed factorization is used to achieve relative errors of the order of the unit round-off when solving algebraic problems involving the collocation matrix of relevant polynomial bases, such as the Hermite basis. The numerical experimentation illustrates the accurate results obtained when using the findings of the paper.

We study a classical bilocal field theory perturbatively up to second-order. The chosen theory is the simplest which incorporates action-at-a-distance, while keeping nonlocal effects short-ranged. We show that the new degrees of freedom introduced by bilocality can be interpreted as gravitational degrees of freedom in the following sense: solutions of the bilocal system at linear and second-orders contain as a subset, gravitational perturbations (spacetime fluctuations) also to that order. In other words, gravity can be thought to originate in a bilocal field theory. We examine potential implications.

In this paper we construct two infinite sets of self-adjoint commuting charges for a quite general CFT. They come out naturally by considering an infinite embedding chain of Lie algebras, an underlying structure that share all theories with gauge groups U(N), SO(N) and Sp(N). The generality of the construction allows us to carry all gauge groups at the same time in a unified framework, and so to understand the similarities among them. The eigenstates of these charges are restricted Schur polynomials and their eigenvalues encode the value of the correlators of two restricted Schurs. The existence of these charges singles out restricted Schur polynomials among the number of bases of orthogonal gauge invariant operators that are available in the literature.