Related papers: From Chio Pivotal Condensation to the Matrix-Tree …
We provide a formula for the Poincar\'e dual of the Chern-Schwartz-MacPherson (CSM) cycle of a matroid in the Chow ring of the matroid. We derive the formula from the case of matroids realizable over the complex numbers and prove that it…
In this paper we extend notions of the core inverse, core EP inverse, DMP inverse, and CMP inverse over the quaternion skew-field ${\mathbb{H}}$ and get their determinantal representations within the framework of the theory of column-row…
We introduce a model of evolving preferential attachment trees where vertices are assigned weights, and the evolution of a vertex depends not only on its own weight, but also on the weights of its neighbours. We study the distribution of…
We compute two parametric determinants in which rows and columns are indexed by compositions, where in one determinant the entries are products of binomial coefficients, while in the other the entries are products of powers. These results…
We use the exterior product of double forms to reformulate celebrated classical results of linear algebra about matrices and bilinear forms namely the Cayley-Hamilton theorem, Laplace expansion of the determinant, Newton identities and…
In this paper we provide an identity between determinant and generalized matrix function. Also, a criterion of positive semi-definite matrices affirming the permanent dominant conjecture is given. As a consequence, infinitely many infinite…
Topological invariance is a powerful concept in different branches of physics as they are particularly robust under perturbations. We generalize the ideas of computing the statistics of winding numbers for a specific parametric model of the…
This article gives a summary of the author's Ph.D. dissertation (arXiv:1609.06297). In addition to an overview of notions and results, it also provides sketches of various proofs and simplified presentations of certain abstract results of…
We define quantum determinants in Quantum Matrix Algebras, related to couples of compatible braidings following the scheme from [G]. We establish relations between these determinants and the so-called column-(row-)determinants, often used…
We suggest a generalization of \pi_0 for topological groupoids, which encodes incidence relations among the strata of the associated quotient object, and argue for its utility by example, starting from the orbit categories of the theory of…
We consider a number of generalizations of the $\beta$-extended MacMahon Master Theorem for a matrix. The generalizations are based on replacing permutations on multisets formed from matrix indices by partial permutations or derangements…
We give generalizations of a finite version of Euler's pentagonal number theorem and of a q-identity of Gauss.
This work builds on Varchenko et al's introduction of bilinear forms for hyperplane arrangements, where the determinant of the associated matrices factorizes into simple components. While one of the determinant formula developed by…
A binomial coefficient identity due to Zhi-Wei Sun is the subject of half a dozen recent papers that prove it by various analytic techniques and establish a generalization. Here we give a simple proof that uses weight-reversing involutions…
In this paper, we establish an analogue of the Fundamental Theorem of Algebra for polynomial matrix equations, where both the coefficient matrices and the unknown matrix are $Q$-circulant matrices. This result generalizes Abramov's result…
In this paper we deal with the noteworthy Sylvester's determinantal identity and some of its generalizations. We report the formulae due to Yakovlev, to Gasca, Lopez--Carmona, Ramirez, to Beckermann, Gasca, M\"uhlbach, and to Mulders in a…
We give an elementary proof of an interesting combinatorial identity which is of particular interest in graph theory and its applications. Two applications to enumeration of forests with closed-form expressions are given.
Suppose a map $\phi$ on the set of positive definite matrices satisfies $\det(A+B)=\det(\phi(A)+\phi(B))$. Then we have $${\rm tr}(AB^{-1}) = {\rm tr}(\phi(A){\phi(B)}^{-1}).$$ Through this viewpoint, we show that $\phi$ is of the form…
We show a general theorem of existence of temporal foliations in a general causal set, under mild constraints. Then we study automorphisms of infinite causal sets (which satisfy further requirements) and show that they fall under one of two…
We use the equivariant cohomology ring of the permutohedral variety to study matroids and their invariants. Investigating the pushforward of matroid Chern classes defined by A. Berget, C. Eur, H. Spink and D. Tseng to the product space…