New systems of Laplace (Casimir) operators for the orthogonal and symplectic Lie algebras are constructed. The operators are expressed in terms of paths in graphs related to matrices formed by the generators of these Lie algebras with the use of some properties of the noncommutative symmetric functions associated with a matrix. The decomposition of the Sklyanin determinant into a product of qua...

Voting (or rank aggregation) is a general method for aggregating the preferences of multiple agents. One important voting rule is the Slater rule. It selects a ranking of the alternatives (or candidates) to minimize the number of pairs of candidates such that the ranking disagrees with the pairwise majority vote on these two candidates. The use of the Slater rule has been hindered by a lack of ...

We study the total positivity of the kernel 1/(x+2 cos(πα)xy+y). The case of infinite order is characterized by an application of Schoenberg’s theorem. We then give necessary conditions for the cases of any given finite order with the help of Chebyshev polynomials of the second kind. Sufficient conditions for the finite order cases are also obtained, thanks to Propp’s formula for the Izergin-Ko...

We prove that the determinantal complexity of a hypersurface of degree d > 2 is bounded below by one more than the codimension of the singular locus, provided that this codimension is at least 5. As a result, we obtain that the determinantal complexity of the 3×3 permanent is 7. We also prove that for n > 3, there is no nonsingular hypersurface in Pn of degree d that has an expression as a dete...

We give matrices of which their determinants are the Jack polynomials expanded in terms of the monomial basis. The top row of this matrix is a list of monomial functions, the entries of the sub-diagonal are of the form −(rα + s), with r and s ∈ +, the entries above the sub-diagonal are nonnegative integers, and below all entries are 0. The quasi-triangular nature of this matrix gives a recursio...

This paper develops an abstract framework for constructing “seminormal forms” for cellular algebras. That is, given a cellular R–algebra A which is equipped with a family of JM–elements we give a general technique for constructing orthogonal bases for A, and for all of its irreducible representations, when the JM–elements separate A. The seminormal forms for A are defined over the field of frac...

D.V. Chudnovsky and G.V. Chudnovsky [CH] introduced a generalization of the FrobeniusStickelberger determinantal identity involving elliptic functions that generalize the Cauchy determinant. The purpose of this note is to provide a simple essentially non-analytic proof of this evaluation. This method of proof is inspired by D. Zeilberger’s creative application in [Z1]. AMS Subject Classificatio...

Synthesis is a process for producing reflection responses from more general sources or from prescribed incident waves by combining common-shot data gathers. Synthesis can provide survey-wide data sets, similar in that regard to common-offset data gathers, but with the added advantage that each synthesized data set is a solution to a single wave equation. A common-offset data set does not have t...

We report on the development of a new theoretical tool that allows for isospin projection of Slater determinants and we present its first applications. In particular, we determine the isospin mixing in ground states of N = Z nuclei and discuss its dependence on the size of the harmonic-oscillator basis used in the calculations. We also discuss the unphysical contribution to the isospin mixing c...

In the early seventies D. Ray and I. Singer [17] introduced the notion of zeta-regularized determinants. They used it to define the analytic version of Reidemeister torsion as an alternating product of determinants. One way to understand analytic torsion is to consider it as a ”multiplicative index” of an elliptic complex. By the L2-index theorem of M. Atiyah [1] this analogy suggests that one ...