Related papers: Exploring unimodality of the plucking polynomial w…
Computing polynomial invariants for knots and links using braid representations relies heavily on finding the trace of Hecke algebra elements. There is no easy method known for computing the trace and hence it becomes difficult to compute…
In this paper, we propose and discuss implications of a general conjecture that there is a canonical action of a rank 1 double affine Hecke algebra on the Kauffman bracket skein module of the complement of a knot $K \subset S^3$. We prove…
We construct motivic invariants of a subvariety of an algebraic torus from its tropicalization and initial degenerations. More specifically, we introduce an invariant of a compactification of such a variety called the "tropical motivic…
Tree-child networks are a recently-described class of directed acyclic graphs that have risen to prominence in phylogenetics (the study of evolutionary trees and networks). Although these networks have a number of attractive mathematical…
We consider the probability of knotting in equilateral random polygons in Euclidean 3-dimensional space, which model, for instance, random polymers. Results from an extensive Monte Carlo dataset of random polygons indicate a universal…
Trees or rooted trees have been generously studied in the literature. A forest is a set of trees or rooted trees. Here we give recurrence relations between the number of some kind of rooted forest with $k$ roots and that with $k+1$ roots on…
The main theme of this dissertation is retooling methods to work for different situations. I have taken the method derived by O'Hara and simplified by Zeilberger to prove unimodality of $q$-binomials and tweaked it. This allows us to create…
We explain how to obtain the set of solutions of a multivariate polynomial equation modulo a power of a prime number. These solutions are determined by a tree, called the trunk, which makes it possible to reconstruct all solutions. We apply…
Comparative analyses of phylogenetic trees typically require identical taxon sets, however, in practice, trees often include distinct but overlapping taxa. Pruning non-shared leaves discards phylogenetic signal, whereas tree completion can…
We derive two formulas for the weighted sums of rooted spanning forests of particular sequence of graphs by using the matrix tree theorem. We consider cycle graphs with edges so called the pendant edges. One of our formula can be described…
This work is divided into three parts. The first part concerns polynomials in one variable with all real roots. We consider linear transformations that preserve real rootedness, as well as matrices that preserve interlacing. The second part…
The W-polynomial is applied in two ways to questions involving the Kauffman bracket of some families of links. First we find a geometric property of a link diagram, which is less than or equal to the twist number, that bounds the Mahler…
Recombining trinomial trees are a workhorse for modeling discrete-event systems in option pricing, logistics, and feedback control. Because each node stores a state-dependent quantity, a depth-$D$ tree naively yields $\mathcal{O}(3^{D})$…
We study monic univariate polynomials whose coefficients are analytic functions of a real variable and whose roots lie in a specified analytic curve. These include characteristic polynomials of unitary and hermitian matrices whose entries…
We report on two machine learning experiments in search of statistical relationships between Dirichlet coefficients and root numbers or analytic ranks of certain low-degree $L$-functions. The first experiment is to construct interpretable…
In this paper, we consider the knot matching problem arising in computational forestry. The knot matching problem is an important problem that needs to be solved to advance the state of the art in automatic strength prediction of lumber. We…
We seek complex roots of a univariate polynomial $P$ with real or complex coefficients. We address this problem based on recent algorithms that use subdivision and have a nearly optimal complexity. They are particularly efficient when only…
Polynomial algebra offers a standard approach to handle several problems in geometric modeling. A key tool is the discriminant of a univariate polynomial, or of a well-constrained system of polynomial equations, which expresses the…
In the present paper we study twisted foldings of root systems which generalize usual involutive foldings corresponding to automorphisms of Dynkin diagrams. Our motivating example is the Lusztig projection of the root system of type $E_8$…
An equivariant characteristic quasi-polynomial is a quasi-polynomial in $q$ consisting of the permutation character on the mod $q$ complement of the corresponding Coxeter arrangement. This concept is a refinement of the conventional…