arXiv subjects

Hoon Hong

Publications and source records attributed to Hoon Hong.

41 records · Page 3Linked to original sources

arXiv2014

The Secant-Newton Map is Optimal Among Contracting $n^{th}$ Degree Maps for $n^{th}$ Root Computation

Consider the problem: given a real number $x$ and an error bound $ε$, find an interval such that it contains the $\sqrt[n]{x}$ and its width is less than $ε$. One way to solve the problem is to start with an initial interval and to repeatedly update it by applying an interval refinement map on it until it becomes narrow enough. In this paper, we prove that the well known Secant-Newton map is optimal among a certain family of natural generalizations.

cs.SC↗

arXiv2013

Special Algorithm for Stability Analysis of Multistable Biological Regulatory Systems

We consider the problem of counting (stable) equilibriums of an important family of algebraic differential equations modeling multistable biological regulatory systems. The problem can be solved, in principle, using real quantifier elimination algorithms, in particular real root classification algorithms. However, it is well known that they can handle only very small cases due to the enormous computing time requirements. In this paper, we present a special algorithm which is much more efficient than the general methods. Its efficiency comes from the exploitation of certain interesting structures of the family of differential equations.

cs.SC↗

arXiv2011

Maximum Gap in (Inverse) Cyclotomic Polynomial

Let $g(f)$ denote the maximum of the differences (gaps) between two consecutive exponents occurring in a polynomial $f$. Let $Φ_n$ denote the $n$-th cyclotomic polynomial and let $Ψ_n$ denote the $n$-th inverse cyclotomic polynomial. In this note, we study $g(Φ_n)$ and $g(Ψ_n)$ where $n$ is a product of odd primes, say $p_1 < p_2 < p_3$, etc. It is trivial to determine $g(Φ_{p_1})$, $g(Ψ_{p_1})$ and $g(Ψ_{p_1p_2})$. Hence the simplest non-trivial cases are $g(Φ_{p_1p_2})$ and $g(Ψ_{p_1p_2p_3})$. We provide an exact expression for $g(Φ_{p_1p_2}).$ We also provide an exact expression for $g(Ψ_{p_1p_2p_3})$ under a mild condition. The condition is almost always satisfied (only finite exceptions for each $p_1$). We also provide a lower bound and an upper bound for $g(Ψ_{p_1p_2p_3})$.

math.NT↗

arXiv2008

Sylvester's Double Sums: the general case

In 1853 Sylvester introduced a family of double sum expressions for two finite sets of indeterminates and showed that some members of the family are essentially the polynomial subresultants of the monic polynomials associated with these sets. A question naturally arises: What are the other members of the family? This paper provides a complete answer to this question. The technique that we developed to answer the question turns out to be general enough to charactise all members of the family, providing a uniform method.

math.AC↗

arXiv2006

An Elementary Proof of Sylvester's Double Sums for Subresultants

In 1853 Sylvester stated and proved an elegant formula that expresses the polynomial subresultants in terms of the roots of the input polynomials. Sylvester's formula was also recently proved by Lascoux and Pragacz by using multi-Schur functions and divided differences. In this paper, we provide an elementary proof that uses only basic properties of matrix multiplication and Vandermonde determinants.

math.AC↗