
Here presented is a unified expression of Stirling numbers and their generalizations by using generalized factorial functions and generalized divided difference. Previous well-known extensions of Stirling numbers due to Riordan, Carlitz, Howard, Charalambides-Koutras, Gould-Hopper, Hsu-Shiue, Tsylova, Todorov, and Ahuja-Enneking are included as particular
cases of our generalization. Four algorithms for calculating the Stirling numbers and their generalizations based on our unified form are also given, which include two comprehensive algorithms using the characterization of Riordan arrays.
We give necessary and sufficient conditions to decompose
The total chromatic number conjecture, which has appeared in a few hundred articles and in numerous books thus far, is now one of the classic mathematical unsolved problems. It appears that many authors coincidentally have attributed it to Professor M. Behzad and/or to Professor V.G. Vizing. Eventually, after four decades, Professor A. Soifer investigated the origin of this conjecture; published his findings in *The Mathematical Coloring Book* (2009); and stated that, “In my opinion this unquestionably merits the joint credit to Vizing and Behzad.” After checking all the arguments presented and the blames cited, I decided to investigate the controversy stated in this book on my own. My findings, which are presented in this report, specifically signify the following two points:
Let
We will study the random perturbation on a linear differential equation as a nowhere differentiable function. The noise in the historical Langevin stochastic differential equation will be treated as a nowhere differentiable model for Brownian motion. A short introduction of Wiener process leading to It\^o’s calculus will be used in derivation of the mean and variance of the solutions to the Langevin Equation. Computational algorithms were developed and applied to study the numerical solutions to linear stochastic differential equations. Symbolic computation and simulation of a computer algebra system will be used to demonstrate the behavior of the solution to the Langevin Stochastic Differential Equation when the perturbation is density independent.
A bi-level balanced array (B-array)
The typical real-time wireless video-audio digital transmission process consists of capturing the signal, digitizing it, compressing it, adding cryptography to it (crypto it), adding redundancy to enable the receiver to detect and correct a number of bit errors, packetizing it, and then transmitting it. Transmitting the signal via the Transmission Control Protocol (TCP-IP) provides a fixed number of redundancy bits, and a very rigid transmission process that could result in a large number of automatic repeat requests and denial of services. In this research, we develop a dynamic transmission algorithm, whereby the degree of redundancy is a function of the noise and the probability
Beautifully Ordered Balanced Incomplete Block Designs, BOBIBD(
Delaunay graphs have been used in CAD/CAM, sensor networks, and geographic information systems. We investigate the reliability properties of nodes in Delaunay graphs. For measuring the reliability, we formulate the concept of roaming-region for nodes. The
A pencyclic graph on
Software interaction test suites serve two complementary roles. They are employed to systematically verify that, for some strength
Faults in software systems often occur due to interactions between parameters. Several studies show that faults are caused by 2-way through 6-way interactions of parameters. In the context of test suite prioritization, we have studied prioritization by 2-way inter-window interaction coverage and found that this criterion is effective at finding faults quickly in the test execution cycle. However, since faults may be caused by interactions between more than 2 parameters, in this paper, we provide a greedy algorithm for test suite prioritization by
In this work, we present a greedy algorithm for covering the set of incomplete STRIPS planning domain interpretations by
The article presents the compatibility matrix method and illustrates it with the application to the
Our previous paper [9] applied a lopsided version of the Lovász Local Lemma that allows negative dependency graphs [5] to the space of random matchings in
We prove nonexistence of circulant weighing matrices with parameters from seven previously open entries of the updated Strassler’s table. The method of proof utilizes some modular constraints on circulant weighing matrices with multipliers.
1970-2025 CP (Manitoba, Canada) unless otherwise stated.