Related papers: Computing the determinant of links through Fourier…

We unify Linear Algebra by proposing a definition of determinants via one equation that implies all known properties of them:\\ 1. Cramer's Rule,\\ 2. Cofactor expansion,\\ 3. Antisymmetry of determinants,\\ 4. Linearity of determinants,\\…

The integral of a function $f$ defined on a symmetric space $M \simeq G/K$ may be expressed in the form of a determinant (or Pfaffian), when $f$ is $K$-invariant and, in a certain sense, a tensor power of a positive function of a single…

We reconfigure the Milnor invariant of links in terms of central group extensions and unipotent Magnus embeddings. We also develop a diagrammatic computation of the invariant and compute the first non-vanishing invariants of the Milnor link…

An important yet challenging problem in numerical linear algebra is finding a principal submatrix with maximum determinant from a given symmetric positive semidefinite matrix. This problem arises in experimental design, statistics, and…

We give constructions to realize an odd number, which is representable as sum of two squares, as determinant of an achiral knot, thus proving that these are exactly the numbers occurring as such determinants. Later we study which numbers…

We define a grid presentation for singular links i.e. links with a finite number of rigid transverse double points. Then we use it to generalize link Floer homology to singular links. Besides the consistency of its definition, we prove that…

We calculate the exceptional points of the eigenvalues of several parameter-dependent Hamiltonian operators of mathematical and physical interest. We show that the calculation is greatly facilitated by the application of the discriminant to…

We study (a generalization of) the notion of linked determinantal loci recently introduced by the second author, showing that as with classical determinantal loci, they are Cohen-Macaulay whenever they have the expected codimension. We…

In the article we develop Euler-Lagrange method and calculate all the roots of an arbitrary complex polynomial $P(z)$ on the base of calculation (similar to the Bernoulli-Aitken-Nikiporets methods) of the limits of ratios of Hadamard…

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…

We define the notions of trace, determinant and, more generally, Berezinian of matrices over a (Z_2)^n graded commutative associative algebra. The applications include a new approach to the classical theory of matrices with coefficients in…

A direct solver is introduced for solving overdetermined linear systems involving nonuniform discrete Fourier transform matrices. Such matrices can be transformed into a Cauchy-like form that has hierarchical low rank structure. The rank…

Based on the recent works of Radziwill-Soundararajan and Roberts, we establish a rate of convergence in Bourgade's central limit theorem for shifted Dirichlet $L$-functions. Our results also indicate that the dependence structure in the…

We observe that the ratio of determinants of $2d$ Laplacians which appear in the duality transformation relating two sigma models with abelian isometries can be represented as a torsion of an elliptic (DeRham) complex. As a result, this…

We define and study an $\ell$-adic Fourier transform for a relative version of Banach-Colmez spaces (over a perfectoid space which is not necessarily a geometric point), which can be thought of as some analytic analogue of the $\ell$-adic…

We show how the multivariable signature and Alexander polynomial of a colored link can be computed from a single symmetric matrix naturally defined from a colored link diagram. In the case of a single variable, it coincides with the matrix…

Given commutative, unital rings $A$ and $B$ with a ring homomorphism $A\to B$ making $B$ free of finite rank as an $A$-module, we can ask for a "trace" or "norm" homomorphism taking algebraic data over $B$ to algebraic data over $A$. In…

Through a brute-force approach to calculating the higher derivatives of the falling factorial function, a number of interesting quantities were obtained and analyzed. In particular, it was found that a quantity that can be described as the…

We present a simple and accessible method which uses contour integration methods to derive formulae for functional determinants. To make the presentation as clear as possible we illustrate the general ideas using the Laplacian with…

It is well known how the linking number and framing can be extracted from the degree 1 part of the (framed) Kontsevich integral. This note gives a general formula expressing any product of powers of these two invariants as combination of…