相关论文: Inverting catalecticants of ternary quartics
We consider two (densely defined) involutions on the space of $q\times q$ matrices; $I(x_{ij})$ is the matrix inverse of $(x_{ij})$, and $J(x_{ij})$ is the matrix whose $ij$th entry is the reciprocal $x_{ij}^{-1}$. Let $K=I\circ J$. The set…
We study the problem of maximum likelihood estimation for $3$-dimensional linear spaces of $3\times 3$ symmetric matrices from the point of view of algebraic statistics where we view these nets of conics as linear concentration or linear…
The problem of studying the two seemingly unrelated sets of invariants forming the Segre and the Verlinde series has gone through multiple different adaptations including a version for the virtual geometries of Quot schemes on surfaces and…
It is known that a $2\times 2$ quaternionic matrix has one, two or an infinite number of left eigenvalues, but the available algebraic proofs are difficult to generalize to higher orders. In this paper a different point of view is adopted…
In this paper, we study the symmetric rank of products of linear forms and an irreducible quadratic form. The main result presents a new, non-trivial lower bound for the rank, and the arguments rely on the apolarity lemma. In the special…
While every matrix admits a singular value decomposition, in which the terms are pairwise orthogonal in a strong sense, higher-order tensors typically do not admit such an orthogonal decomposition. Those that do have attracted attention…
We provide formulas for the degrees of the projections of the locus of square matrices with given rank from linear spaces spanned by a choice of matrix entries. The motivation for these computations stem from applications to `matrix…
We investigate some geometric properties of the real algebraic variety $\Delta$ of symmetric matrices with repeated eigenvalues. We explicitly compute the volume of its intersection with the sphere and prove a Eckart-Young-Mirsky-type…
We consider the problem of determining the symmetric tensor rank for symmetric tensors with an algebraic geometry approach. We give algorithms for computing the symmetric rank for $2\times ... \times 2$ tensors and for tensors of small…
In 2007, the first author gave an alternative proof of the refined alternating sign matrix theorem by introducing a linear equation system that determines the refined ASM numbers uniquely. Computer experiments suggest that the numbers…
In this letter we highlight the inherent problems associted with the inverted hierarchical model of neutrinos with only three generations and suggest possible solutions within the MSSM. we discuss the new parametrisation of the solar mixing…
The recent discovery that the exponent of matrix multiplication is determined by the rank of the symmetrized matrix multiplication tensor has invigorated interest in better understanding symmetrized matrix multiplication. I present an…
Given a knot and an SL(n,C) representation of its group that is conjugate to its dual, the representation that replaces each matrix with its inverse-transpose, the associated twisted Reidemeister torsion is reciprocal. An example is given…
We introduce an elementary method to study the border rank of polynomials and tensors, analogous to the apolarity lemma. This can be used to describe the border rank of all cases uniformly, including those very special ones that resisted a…
Two-dimensional linear spaces of symmetric matrices are classified by Segre symbols. After reviewing known facts from linear algebra and projective geometry, we address new questions motivated by algebraic statistics and optimization. We…
These lecture notes are intended as an introduction to several notions of tensor rank and their connections to the asymptotic complexity of matrix multiplication. The latter is studied with the exponent of matrix multiplication, which will…
In symmetric spaces of rank one we obtain a right inverse of the Cartan differential on exact differential forms as an integral operator. As a corollary we extend Gauss linking integral beyond space forms.
The reciprocal Pascal matrix is the Hadamard inverse of the symmetric Pascal matrix. We show that the ordinary matrix inverse of the reciprocal Pascal matrix has integer elements. The proof uses two factorizations of the matrix of super…
This paper discusses the explicit inverse of a class of seven-diagonal (near) Toeplitz matrices, which arises in the numerical solutions of nonlinear fourth-order differential equation with a finite difference method. A non-recurrence…
We study framed translation surfaces corresponding to meromorphic differentials on compact Riemann surfaces, for which a horizontal separatrix is marked for each pole or zero. Such geometric structures naturally appear when studying flat…