Narrow Results

This paper studies mathematical properties of reaction systems that were introduced by Ehrenfeucht and Rozenberg as computational models inspired by biochemical reaction in the living cells. In particular, we continue the study on the generative power of functions specified by minimal reaction systems under composition initiated by Salomaa. Allowing degenerate reaction systems, functions specified by minimal reaction systems over a quarternary alphabet that are permutations generate the alternating group on the power set of the background set.

Using a directed acyclic graph (dag) model of algorithms, we solve a problem related to precedence-constrained multiprocessor schedules for array computations: Given a sequence of dags and linear schedules parametrized by n, compute a lower bound on the number of processors required by the schedule as a function of n. In our formulation, the number of tasks that are scheduled for execution during any fixed time step is the number of non-negative integer solutions d_n to a set of parametric linear Diophantine equations. We present an algorithm based on generating functions for constructing a formula for these numbers d_n. The algorithm has been implemented as a Mathematica program. Example runs and the symbolic formulas for processor lower bounds automatically produced by the algorithm for Matrix-Vector Product, Triangular Matrix Product, and Gaussian Elimination problems are presented. Our approach actually solves the following more general problem: Given an arbitrary r× s integral matrix A and r-dimensional integral vectors b and c, let d_n(n=0,1,…) be the number of solutions in non-negative integers to the system Az=nb+c. Calculate the (rational) generating function ∑_{n≥ 0}d_ntⁿ and construct a formula for d_n.

We present a surface area lemma to characterize the surface area of a product graph in terms of those of its factors via a generating function approach. We then apply this lemma to derive surface area related results for meshes and tori. Moreover, we also provide explicit formulas for the average distances of these networks.

This work is continuation of a stochastic quantization program reported earlier. In this paper, we propose a consistent scheme of doing computations directly in four dimensions using conventional quantum field theory methods.

A new discrete Lax operator involving discrete canonical variable is introduced which generate new integrable system, and is analyzed in the light of the new concept of canonical Bäcklund transformation and classical r-matrix. The generating function of the transformation is explicitly deduced. The second half of the paper deals with the quantization problem where an explicit form of the Bethe equations are deduced.

In an earlier paper, it has been shown that the ultra violet divergence structure of anomalous $U(1)$ axial vector gauge model in the stochastic quantization scheme is different from that in the conventional quantum field theory. Also, it has been shown that the model is expected to be renormalizable. Based on the operator formalism of the stochastic quantization, a new approach to anomalous $U(1)$ axial vector gauge model is proposed. The operator formalism provides a convenient framework for analysis of ultra violet divergences, but the computations in a realistic model become complicated. In this paper a new approach to do computations in the model is formulated directly in four dimensions. The suggestions put forward here will lead to simplification in the study of applications of the axial vector gauge theory, as well as those of other similar models.

In this paper, we constructed characteristic identities for the 3-split (polarized) Casimir operators of simple Lie algebras in the adjoint representations $ad$ and deduced a certain class of subrepresentations in ${ad}^{\otimes 3}$. The projectors onto invariant subspaces for these subrepresentations were directly constructed from the characteristic identities for the 3-split Casimir operators. For all simple Lie algebras, universal expressions for the traces of higher powers of the 3-split Casimir operators were found and dimensions of the subrepresentations in ${ad}^{\otimes 3}$ were calculated. All our formulas are in agreement with the universal description of (irreducible) subrepresentations in ${ad}^{\otimes 3}$ for simple Lie algebras in terms of the Vogel parameters.

The investigation of random walks is central to a variety of stochastic processes in physics, chemistry and biology. To describe a transport phenomenon, we study a variant of the one-dimensional persistent random walk, which we call a zero-one-only process. It makes a step in the same direction as the previous step with probability p, and stops to change the direction with 1 − p. By using the generating-function method, we calculate its characteristic quantities such as the statistical moments and probability of the first return.

We have derived rigorously the generating function for the number of pyramid-like polyhedra g(a,b,c) on the cubic lattice with width a, depth b, and height c.

We have derived the five-variable generating functions for the numbers of pyramid and staircase polyhedra on the cubic lattice with given values of width, depth, height, area and volume.

Stochastic context-free grammars are an important tool in syntactic pattern analysis and other applications as well. This paper discusses major results in single-type and multitype branching processes used to study a grammar’s stochastic derivations. Probability generating functions are well-established as a tool in this area and are used extensively here.

We study the evolution of degree distributions of susceptible-infected-susceptible (SIS) model on random networks, where susceptible nodes are capable of being infected, and infected nodes can spread the disease further. The network of contacts is modeled as a configuration model featuring heterogeneous degree distribution. We derive systematically the (excess) degree distributions among susceptible and infected individuals by using the probability generating function formalism.

The paper considers the analytical solution methods of the maximizing entropy or minimizing variance with fixed orness level problems and the maximizing orness with fixed entropy or variance value problems together. It proves that both of these two kinds of problems have common necessary conditions for their optimal solutions. The optimal solutions have the same forms and can be seen as the same OWA (ordered weighted averaging) weighting vectors from different points of view. The problems of minimizing orness problems with fixed entropy or variance constraints and their analytical solutions are proposed. Then these conclusions are extended to the corresponding RIM (regular increasing monotone) quantifier problems, which can be seen as the continuous case of OWA problems with free dimension. The analytical optimal solutions are obtained with variational methods.

We discover a family of probability measures μ_a, 0 < a ≤ 1,

We give a free probabilistic interpretation of the multiplicative renormalization method. As a byproduct, we give a short proof of the Asai–Kubo–Kuo problem on the characterization of the family of measures for which this method applies with h(x) = (1 - x)^-1 which turns out to be the free Meixner family. We also give a representation of the Voiculescu transform of all free Meixner laws (even in the non-freely infinitely divisible case).

Consider the Lévy–Meixner one-mode interacting Fock space {Γ_LM, 〈 ⋅, ⋅ 〉_LM}. Inspired by a derivative formula appearing in 〈 ⋅, ⋅ 〉_LM, we define scalar products 〈 ⋅, ⋅ 〉_{LM, n} in symmetric n-particle spaces. Then, we introduce a class of one-mode type interacting Fock spaces naturally associated to the one-dimensional infinitely divisible distributions with Lévy–Meixner type {μ_r; r > 0}. The Fourier transform in generalized joint eigenvectors of a family of Lévy–Meixner Jacobi fields provides a way to explicit a unitary isomorphism 𝔘_LM between and the so-called Lévy–Meixner white noise space . We derive a chaotic decomposition property of the quadratic integrable functionals of the Lévy–Meixner white noise processes in terms of an appropriate Wick tensor product. For their stochastic regularity, we give explicit form and sharp estimate of the associated Donsker's delta function.

In this paper, utilizing the moments representations of Hahn polynomials, we show how to derive their bilinear, trilinear and multilinear generating functions. Moreover, from Euler's finite q-differences, we deduce the q-Chu–Vandermonde formula and consider its generalizations by the moments method.

A formula for the sandwiched relative $\alpha$-entropy

An important and interesting parameter of an interconnection network is the number of vertices of a specific distance from a specific vertex. This is known as the surface area or the Whitney number of the second kind. In this paper, we give explicit formulas for the surface areas of the (n, k)-star graphs and the arrangement graphs via the generating function technique. As a direct consequence, these formulas will also provide such explicit formulas for the star graphs, the alternating group graphs and the split-stars since these graphs are related to the (n, k)-star graphs and the arrangement graphs. In addition, we derive the average distances for these graphs.