Related papers: An $O(n\log^2n)$ Algorithm for Computing Hankel De…
We study the Gram matrix determinants for the groups $S_n,O_n,B_n,H_n$, for their free versions $S_n^+,O_n^+,B_n^+,H_n^+$, and for the half-liberated versions $O_n^*,H_n^*$. We first collect all the known computations of such determinants,…
Let f be analytic in D={z:|z|<1} with f(0)=0 and f'(0)=1. We give sharp bounds for the second Hankel determinant of f, when f is starlike of order alpha in D.
We propose a novel parallel numerical algorithm for calculating the smallest eigenvalues of highly ill-conditioned matrices. It is based on the {\it LDLT} decomposition and involves finding a $k \times k$ sub-matrix of the inverse of the…
Erd\H{o}s-Ginzburg-Ziv theorem is a famous theorem in additive number theory, which states any sequence of $2n-1$ integers contains a subsequence of $n$ elements, with their sum being a multiple of $n$. In this article, we provide an…
We show that every n-by-n matrix is generically a product of [n/2] + 1 Toeplitz matrices and always a product of at most 2n+5 Toeplitz matrices. The same result holds true if the word "Toeplitz" is replaced by "Hankel", and the generic…
We present an algorithm to compute the Hecke operators on the equivariant cohomology of an arithmetic subgroup $\Gamma$ of the general linear group $\mathrm{GL}_n$. This includes $\mathrm{GL}_n$ over a number field or a finite-dimensional…
Hankel matrices are an important class of highly-structured matrices, arising across computational mathematics, engineering, and theoretical computer science. It is well-known that positive semidefinite (PSD) Hankel matrices are always…
G{\"o}del's completeness theorem for classical first-order logic is one of the most basic theorems of logic. Central to any foundational course in logic, it connects the notion of valid formula to the notion of provable formula.We survey a…
In the paper, we give the recurrent equations of the Hankel determinants of the Cantor sequence, and show that the Hankel determinants as a double sequence is 3-automatic. With the help of the Hankel determinants, we prove that the…
The six-vertex model with Domain Wall Boundary Conditions, or square ice, is considered for particular values of its parameters, corresponding to 1-, 2-, and 3-enumerations of Alternating Sign Matrices (ASMs). Using Hankel determinant…
For each element of certain families of integer sequences, we study the term-wise ratios of the Hankel transforms of three sequences related to that element by series reversion. In each case, the ratios define well-known sequences, and in…
In this paper, we present an improvement for the problem of deterministically finding an element of large multiplicative order modulo some integer $N$. This problem arises as a key subroutine in current deterministic factoring algorithms,…
In this paper, we investigate the sharp bounds of the second Hankel determinant of Logarithmic coefficients for the starlike and convex functions with respect to symmetric points in the open unit disk.
We study the horizontally regular curves in the Heisenberg groups $H_n$. We show the fundamental theorem of curves in $H_n$ $(n\geq 2)$ and define the concept of the orders for horizontally regular curves. We also show that the curve…
We present a simple and fast algorithm for computing the $N$-th term of a given linearly recurrent sequence. Our new algorithm uses $O(\mathsf{M}(d) \log N)$ arithmetic operations, where $d$ is the order of the recurrence, and…
This paper addresses the problem of efficiently computing higher-order variational integrators in simulation and trajectory optimization of mechanical systems as those often found in robotic applications. We develop $O(n)$ algorithms to…
In this paper, we provide an estimation for the sharp bound of the third Hankel determinant of starlike functions of order $\alpha$, where $\alpha$ ranges in the interval $[0, 1/6]\cup \{1/2\}$ and thereby extending the result of Rath et…
We consider the product of determinantal point processes (DPPs), a point process whose probability mass is proportional to the product of principal minors of multiple matrices, as a natural, promising generalization of DPPs. We study the…
In this (mostly expository) paper I want to share some observations prompted by a class of matrices whose determinants are Catalan numbers. Considering different methods of proof we obtain some generalizations and q-analogues and…
Martin Aigner introduced Catalan-like numbers as elements of the first column of admissible matrices and studied Hankel determinants of their forward shifts. In this paper we collect some properties of the Hankel determinants of the other…