Related papers: Generalisation of Scott permanent identity

We prove an identity relating the permanent of a rank $2$ matrix and the determinants of its Hadamard powers. When viewed in the right way, the resulting formula looks strikingly similar to an identity of Carlitz and Levine, suggesting the…

Consider a space X with the singular locus, Z=Sing(X), of positive dimension. Suppose both Z and X are locally complete intersections. The transversal type of X along Z is generically constant but at some points of Z it degenerates. We…

Consider a space X with the singular locus of positive dimension, Z=Sing(X). Suppose both Z and X are locally complete intersections at each point. The transversal type of X along Z is generically constant but at some points of Z it…

In the stable general linear group over an arbitrary field, we prove that every element with determinant $\pm 1$ is the product of three involutions, and of no less in general. We also obtain several results of the same flavor, with…

Let k be a definable L-cardinal. Then there is a set of reals X, class-generic over L, such that L(X) and L have the same cardinals, X has size k in L(X) and some pi-1-2 formula defines X in all set-generic extensions of L(X). Two…

We provide upper bounds on the total number of irreducible factors, and in particular irreducibility criteria for some classes of bivariate polynomials $f(x,y)$ over an arbitrary field $\mathbb{K}$. Our results rely on information on the…

Significant research has been carried out in the past half-century on defining generalised determinants for transformations between (typically real) vector spaces of different dimensions. We review three different generalisations of the…

Let $X$ be a normal, excellent, noetherian scheme over $\operatorname{Spec}\mathbb{Q}$ with a dualizing complex. In this note, we find an alternate characterization of the multiplier ideal of $X$, as defined by de Fernex-Hacon, by…

In 2009, Grant, Shallit, and Stoll constructed a large family of pseudorandom sequences, called generalized Rudin--Shapiro sequences, for which they established some results about the average of discrete correlation coefficients of order 2…

A generalization of the Catalan numbers is considered. New results include binomial identities, recursive relations and a close formula for the multivariate generating function. A simple expression for the Catalan determinant is derived.

Let $X$ be a variety defined over an algebraically closed field $k$ of characteristic $0$, let $N\in\mathbb{N}$, let $g:X\dashrightarrow X$ be a dominant rational self-map, and let $A:\mathbb{A}^N\to \mathbb{A}^N$ be a linear transformation…

Let $G$ be a group. If an equation $x^n = y^n$ in $G$ implies $x = y$ for any elements $x$ and $y$, then $G$ is called an $R$--group. It is completely understood which knot groups are $R$--groups. Fay and Walls introduced $\bar{R}$--group…

We characterize sequences of positive integers $(a_1,a_2,\ldots,a_n)$ for which the $2\times2$ matrix $\left( \begin{array}{cc} a_n&-1 1&0 \end{array} \right) \left( \begin{array}{cc} a_{n-1}&-1 1&0 \end{array} \right) \cdots \left(…

I show that assuming PFA, every proper Scott set is the standard system of a model of PA. A Scott set X is proper if it is arithmetically closed and the quotient Boolean algebra X/Fin is a proper partial order.

We prove a generalization of the polarization identity of linear algebra expressing the inner product of a complex inner product space in terms of the norm, where the field of scalars is extended to an associative algebra equipped with an…

Paper deals with the singular Sturm-Liouville expressions $$l(y) = -(py')' + qy$$ on a finite interval with coefficients $$q = Q', \quad 1/p, Q/p, Q^2/p \in L_1,$$ where derivative of the function $Q$ is understood in the sense of…

The generalized sequence of numbers is defined by W_{n}=pW_{n-1}+qW_{n-2} with initial conditions W_{0}=a and W_{1}=b for a,b,p,q\inZ and n\geq2, respectively. Let W_{n}=circ(W_{1},W_{2},...,W_{n}). The aim of this paper is to establish…

Starting from the expression for the superdeterminant of $ (xI-M)$, where $M$ is an arbitrary supermatrix , we propose a definition for the corresponding characteristic polynomial and we prove that each supermatrix satisfies its…

In this work we generalise previous results connecting (rational) Gaudin magnet models and classical separation of variables. It is shown that the connection persists for the case of linear r-matrix algebra which corresponds to the…

Assume that there is a set of monic polynomials $P_n(z)$ satisfying the second-order difference equation $$ A(s) P_n(z(s+1)) + B(s) P_n(z(s)) + C(s) P_n(z(s-1)) = \lambda_n P_n(z(s)), n=0,1,2,..., N$$ where $z(s), A(s), B(s), C(s)$ are some…