Narrow Results

The algebraic composition of an element $x$ of a commutative binary structure ${ℬ}$ is modeled by its divisor (pseudo)graph ${\mathrm{\Gamma }}_x$ (which can be regarded as a simple graph if ${ℬ}$ is an abelian group), whose vertices are the elements of ${ℬ}$ such that two vertices $a$ and $b$ are adjacent if and only if $ab=x$. With respect to the action of a group of symmetries on the set of divisor pseudographs of ${ℬ}$, the numbers of orbits are considered, and these numbers are shown to yield groupoid-isotopy invariants when ${ℬ}$ is a finite abelian group. The minimum number of orbits is computed for every abelian group of order at most 11, and it is also determined for all finite cyclic groups. Moreover, systems of pseudographs that can be realized as those of finite abelian groups are completely characterized. In fact, recurrence relations are given for constructing systems of divisor pseudographs of finite cyclic groups, and all commutative binary structures that are isotopic to a finite abelian group are established.

A key issue in performing tree structured parallel computations is to distribute process components of a parallel program over processors in a parallel computer at run time such that both the maximum load and dilation are minimized. The main contribution of this paper is the application of recurrence relations in studying the performance of a dynamic tree embedding algorithm in hypercubes. We develop recurrence relations that characterize the expected load in randomized tree embeddings where, a tree grows by letting its nodes to take random walks of short distance. By using these recurrence relations, we are able to calculate the expected load on each processor. Therefore, for constant dilation embeddings, we are able to evaluate expected loads numerically and analytically. The applicability of recurrence relations is due to the recursive structure of trees and the fact that embeddings of the subtrees of a process node are independent of each other. Our methodology does not depend on the hypercube topology. Hence, it can be applied to studying dynamic tree growing in other networks.

By using the Hamiltonian identity, we present a generalized hypervirial theorem for the D dimensional single-particle system with arbitrary potential. It is shown that this generalized hypervirial theorem is powerful in deriving the Blanchard's and Kramers' recurrence relations among the matrix elements. We apply those recurrence relations to some physical systems, exactly solvable and unsolvable, such as the pseudoharmonic oscillator, the Morse, the modified Pöschl-Teller, the Lennard-Jones, the Buckingham and the Yukawa potentials. The Blanchard's and Kramers' recurrence relations in two dimensions are also briefly mentioned.

Let ${{𝒫}{𝒪}}_n$ be the semigroup of order-preserving partial transformations on an $n$-chain. We establish several results for the number $F(n,m)$ of elements of ${{𝒫}{𝒪}}_n$ with $m$ fixed points, including recurrence relations and generating functions. We also prove that the probability that, for large $n$, a randomly chosen map in ${{𝒫}{𝒪}}_n$ has exactly $m$ fixed points is asymptotically $2\sqrt{2}(2m+1)/n$.

We use the Legendre polynomials and the Hermite polynomials as two examples to illustrate a simple and systematic technique on deriving asymptotic formulas for orthogonal polynomials via recurrence relations. Another application of this technique is to provide a solution to a problem recently raised by M. E. H. Ismail.

We characterize all the multiple orthogonal three-fold symmetric polynomial sequences whose sequence of derivatives is also multiple orthogonal. Such a property is commonly called the Hahn property and it is an extension of the concept of classical polynomials to the context of multiple orthogonality. The emphasis is on the polynomials whose indices lie on the step line, also known as $2$-orthogonal polynomials. We explain the relation of the asymptotic behavior of the recurrence coefficients to that of the largest zero (in absolute value) of the polynomial set. We provide a full characterization of the Hahn-classical orthogonality measures supported on a $3$-star in the complex plane containing all the zeros of the polynomials. There are essentially three distinct families, one of them $2$-orthogonal with respect to two confluent functions of the second kind. This paper complements earlier research of Douak and Maroni.

A well-known recurrence relation for the 6j-symbol of the quantum group su_q(2) is realized as a tridiagonal, symmetric eigenvalue problem. This formulation can be used to implement an efficient numerical evaluation algorithm, taking advantage of existing specialized numerical packages. For convenience, all formulas relevant for such an implementation are collected in Appendix A. This realization is a byproduct of an alternative proof of the recurrence relation, which generalizes a classical (q = 1) result of Schulten and Gordon and uses the diagrammatic spin network formalism of Temperley–Lieb recoupling theory to simplify intermediate calculations.

We derive the general series-product identities from which we deduce several applications, including an identity of Gauss, the generalization of Winquist's identity by Carlitz and Subbarao, an identity for , the quintuple product identity, and the octuple product identity.

Families of polynomials associated to arithmetic functions $g(n)$ are studied. The case $g(n)=\sigma (n)$, the divisor sum, dictates the non-vanishing of the Fourier coefficients of powers of the Dedekind eta function. The polynomials $P_n^g(X)$ are defined by $n$-term recurrence relations. For the case that $g(x)$ is a polynomial of degree $d$, we prove that at most a $d+2$ term recurrence relation is needed. For the special case $g(x)=x$, we obtain explicit formulas and results.

The focus of this paper is to study the two-variable Kauffman polynomials $L$ and $F$, and the one-variable BLM/Ho polynomial $Q$ of $(2,n)$-torus link as the Fibonacci-type polynomials and to express the Kauffman polynomials in terms of the BLM/Ho polynomial. For this purpose, we prove that each of the examined polynomials of $(2,n)$-torus link can be determined by a third-order recurrence relation and give the recursive properties of them. We correlate these polynomials with the Fibonacci-type polynomials. By using the relations between the BLM/Ho polynomials and Fibonacci-type polynomials, we express the Kauffman polynomials in terms of the BLM/Ho polynomials.

If a Slinky suspended in a U-shape is rotated horizontally, it will have the shape of an upside-down $\mathrm{\Omega }$, like $\mathrm{\Omega }$, due to the centrifugal force in the horizontal direction acting on each point of the Slinky. In this paper, we discuss the shape of a rotating Slinky in both theoretical and experimental aspects. The rotating Slinky has the shape described by a multi-valued function if the angular velocity of the rotation satisfies $\omega >\pi \sqrt{K/M}$. Our theoretical results are in good agreement with experimental observations. The video of the experiments can be viewed at https://youtu.be/rpu9Ol2jdAM.

We use the Legendre polynomials and the Hermite polynomials as two examples to illustrate a simple and systematic technique on deriving asymptotic formulas for orthogonal polynomials via recurrence relations. Another application of this technique is to provide a solution to a problem recently raised by M. E. H. Ismail.