A theorem of Mina evaluates the determinant of a matrix with entries D(f(x)). We note the important special case where the matrix entries are evaluated at x = 0 and give a simple proof of it, as well as some special additivity properties that hold in this case, but not in general. Some applications are given also. We then give a short proof of the general case. October 25, 1998 An old theorem o...

We introduce a multi-parameter generalization of the Lambda-determinant of Robbins and Rumsey, based on the cluster algebra with coefficients attached to a T -system recurrence. We express the result as a weighted sum over alternating sign matrices.

We study the problem of expressing permanents of matrices as determinants of (possibly larger) matrices. This problem has close connections to the complexity of arithmetic computations: the complexities of computing permanent and determinant roughly correspond to arithmetic versions of the classes NP and P respectively. We survey known results about their relative complexity and describe two re...

We prove a constant term conjecture of Robbins and Zeilberger (J. Combin. Theory Ser. A 66 (1994), 17–27), by translating the problem into a determinant evaluation problem and evaluating the determinant. This determinant generalizes the determinant that gives the number of all totally symmetric self-complementary plane partitions contained in a (2n)×(2n)×(2n) box and that was used by Andrews (J...

From the irreducible decompositions’ point of view, the structure of the cyclic GLn(C)-module generated by the α-determinant degenerates when α = ± 1 k (1 ≤ k ≤ n − 1) (see [6]). In this paper, we show that − 1 k -determinant shares similar properties which the ordinary determinant possesses. From this fact, one can define a new (relative) invariant called a wreath determinant. Using (GLm, GLn)...

Wendt’s determinant of order m is the circulant determinant Wm whose (i, j)-th entry is the binomial coefficient ( m |i−j| ) , for 1 ≤ i, j ≤ m. We give a formula for Wm, when m is even not divisible by 6, in terms of the discriminant of a polynomial Tm+1, with rational coefficients, associated to (X + 1)m+1 −Xm+1 − 1. In particular, when m = p − 1 where p is a prime ≡ −1 (mod 6), this yields a...

The purpose of this paper is to present the construction of a canonical determinant functional on elliptic pseudodifferential operators (ψdos) associated to the Guillemin-Wodzicki residue trace. The resulting residue determinant functional is multiplicative and unique up to taking powers. It is a local invariant and not defined by a regularization procedure. The residue determinant is consequen...

Determinant evaluations play an important role in mathematics, perhaps most notably in combinatorics, see Krattenthaler’s surveys [K2] and [K3]. Many useful determinant evaluations are rational identities, which rises the question of finding generalizations to the elliptic level. In recent work with Schlosser [RS2], we gave an approach to elliptic determinant evaluations that encompasses most r...

We study a matrix that arises from singular form of the Woodbury identity. present generalized inverse and pseudo-determinant identities for this matrix, which have direct applications Gaussian process regression, specifically its likelihood representation precision matrix. extend definition to Bott-Duffin covariance preserving properties related conditional independence, precision, marginal pr...

To investigate the mechanism underlying neonatal T cell tolerance, we used synthetic peptides to induce tolerance. We found that induction of tolerance to one determinant on a 23-amino acid peptide did not affect the response to an adjacent determinant on the same peptide. There was no evidence of suppression of the response to the second determinant. Furthermore, even small peptides near the m...