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Ho-Hon Leung

Publications and source records attributed to Ho-Hon Leung.

14 recordsLinked to original sources
arXiv2022

Numerical computing in engineering mathematics

The rapid advances in technology over the last decade have significantly altered the nature of engineering knowledge and skills required in the modern industries. In response to the changing professional requirements, engineering institutions have updated their curriculum and pedagogical practices. However, most of the changes in the curriculum have been focused on the core engineering courses without much consideration for the auxiliary courses in mathematics and sciences. In this paper, we aim to propose a new, augmented mathematics curriculum aimed at meeting the requirements of the modern, technology-based engineering workplace. The proposed updates require minimal resources and can be seamlessly integrated into the existing curriculum.

math.HO↗
arXiv2020

Game of Pure Chance with Restricted Boundary

We consider various probabilistic games with piles for one player or two players. In each round of the game, a player randomly chooses to add $a$ or $b$ chips to his pile under the condition that $a$ and $b$ are not necessarily positive. If a player has a negative number of chips after making his play, then the number of chips he collects will stay at $0$ and the game will continue. All the games we considered satisfy these rules. The game ends when one collects $n$ chips for the first time. Each player is allowed to start with $s$ chips where $s\geq 0$. We consider various cases of $(a,b)$ including the pairs $(1,-1)$ and $(2,-1)$ in particular. We investigate the probability generating functions of the number of turns required to end the games. We derive interesting recurrence relations for the sequences of such functions in $n$ and write these generating functions as rational functions. As an application, we derive other statistics for the games which include the average number of turns required to end the game and other higher moments.

math.CO↗
arXiv2019

On Horadam quaternions by using matrix method

In this paper, we give several matrix representations for the Horadam quaternions. We derive several identities related to these quaternions by using the matrix method. Since quaternion multiplication is not commutative, some of our results are non-commutative analogues of the well known identities for the Fibonacci-like integer sequences. Lastly, we derive some binomial-sum identities for the Horadam quaternions as an application of the matrix method.

math.NT↗
arXiv2019

A Probabilistic Two-Pile Game

We consider a game with two piles, in which two players take turn to add $a$ or $b$ chips ($a$, $b$ are not necessarily positive) randomly and independently to their respective piles. The player who collects $n$ chips first wins the game. We derive general formulas for $p_n$, the probability of the second player winning the game by collecting $n$ chips first and show the calculation for the cases $\{a,b\}$ = $\{-1,1\}$ and $\{-1,2\}$. The latter case was asked by Wong and Xu \cite{WX}. At the end, we derive the general formula for $p_{n_1,n_2}$, the probability of the second player winning the game by collecting $n_2$ chips before the first player collects $n_1$ chips.

math.CO↗
arXiv2019

A note on congruence properties of the generalized bi-periodic Horadam sequence

In this paper, we consider a generalization of Horadam sequence {w_n} which is defined by the recurrence w_n = aw_n-1 + cw_n-2; if n is even, w_n = bw_n-1 + cw_n-2; if n is odd with arbitrary initial conditions w_0, w_1 and nonzero real numbers a, b, and c. We investigate some congruence properties of the generalized Horadam sequence {w_n}.

math.NT↗
arXiv2018

On a generalization of the Pentagonal Number Theorem

We study a generalization of the classical Pentagonal Number Theorem and its applications. We derive new identities for certain infinite series, recurrence relations and convolution sums for certain restricted partitions and divisor sums. We also derive new identities for Bell polynomials.

math.CO↗
arXiv2018

Determinants containing powers of polynomial sequences

We derive identities for the determinants of matrices whose entries are (rising) powers of (products of) polynomials that satisfy a recurrence relation. In particular, these results cover the cases for Fibonacci polynomials, Lucas polynomials and certain orthogonal polynomials. These identities naturally generalize the determinant identities obtained by Alfred, Carlitz, Prodinger, Tangboonduangjit and Thanatipanonda.

math.CO↗
arXiv2017

Derivations on four dimensional genetic Volterra algebra

In this paper, we describe all derivations on four dimensional genetic Volterra algebras. We show that any local derivation is a derivation of the algebra. It is a positive answer to a conjecture made by Ganikhodzhaev, Mukhamedov, Pirnapasov and Qaralleh.

math.RA↗
arXiv2016

Equivariant structure constants for Hamiltonian-$T$-spaces

If there exists a set of canonical classes on a compact Hamiltonian-$T$-spaces in the sense of Goldin and Tolman, we derive some formulas for certain equivariant structure constants in terms of other equivariant structure constants and the values of canonical classes restricted to some fixed points. These formulas can be regarded as a generalization of Tymoczko's results.

math.SG↗
arXiv2013

Divided difference operators in equivariant $KK$-theory

Let $G$ be a compact connected Lie group with a maximal torus $T$. Let $A$, $B$ be $G$-$\mathrm{C}^\ast$-algebras. We define certain divided difference operators on Kasparov's $T$-equivariant $KK$-group $KK_T(A,B)$ and show that $KK_G(A,B)$ is a direct summand of $KK_T(A,B)$. More precisely, a $T$-equivariant $KK$-class is $G$-equivariant if and only if it is annihilated by an ideal of divided difference operators. This result is a generalization of work done by Atiyah, Harada, Landweber and Sjamaar.

math.KT↗
arXiv2013

K-theory of weight varieties

Let $T$ be a compact torus and $(M,ω)$ a Hamiltonian $T$-space. We give a new proof of the $K$-theoretic analogue of the Kirwan surjectivity theorem in symplectic geometry by using the equivariant version of the Kirwan map introduced in one of R. Goldin's papers. We compute the kernel of this equivariant Kirwan map, and hence give a computation of the kernel of the Kirwan map. As an application, we find the presentation of the kernel of the Kirwan map for the $T$-equivariant $K$-theory of flag varieties $G/T$ where $G$ is a compact, connected and simply-connected Lie group. In the last section, we find explicit formulae for the $K$-theory of weight varieties.

math.KT↗