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 \( \lambda \) copies, where necessarily \( \lambda \geq 2 \), of the complete graph \( K_v \), into so-called “2-petal”, “stem-infinity”, “barbell”, and “box-edge” graphs, all with four vertices and five edges.

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:

1. M. Behzad is the sole author of the Total Chromatic Number Conjecture.
2. The wrong referrals provided by numerous authors over the last forty-four years, to indicate Vizing’s authorship, must be brought to the attention of the authors and researchers, by appropriate means, as soon as possible.

Let \(\mathcal{G}_n\) be the set of all simple loopless undirected graphs on \(n\) vertices. Let \(T\) be a linear mapping, \(T : \mathcal{G}_n \rightarrow \mathcal{G}_n\) for which the independence number of \(T(G)\) is the same as the independence number for \(G\) for any \(G \in \mathcal{G}_n\). We show that \(T\) is necessarily a vertex permutation. Similar results are obtained for mappings preserving the matching number of bipartite graphs, the vertex cover number of undirected graphs, and the edge independence number of undirected graphs.

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) \( T \) with parameters \( (m, N, t) \) and index set \( \underline{\mu}’ = \{\mu_0, \mu_1, \ldots, \mu_t\} \) is a matrix with \( m \) rows, \( N \) columns, and with two elements (say, \( 0 \) and \( 1 \)) such that in every \( (t \times N) \)-submatrix \( T^* \) (clearly, there are \( \binom{m}{t} \) such submatrices) of \( T \), the following combinatorial condition is satisfied: every \( (t \times 1) \) vector \( \underline{\alpha} \) of \( T^* \) with \( i \) (\( 0 \leq i \leq t \)) ones in it appears the same number \( \mu_i \) (say) times. \( T \) is called a B-array of strength \( t \). Clearly, an orthogonal array (O-array) is a special case of a B-array. These combinatorial arrays have been extensively used in information theory, coding theory, and design of experiments. In this paper, we restrict ourselves to arrays with \( t = 4 \) and \( t = 6 \). We derive some inequalities involving \( m \) and \( \mu_i \), using the concept of coincidences amongst the columns of \( T \), which are necessary conditions for B-arrays to exist. We then use these inequalities to study the existence of these arrays and to obtain the bounds on the number of rows (also called constraints) \( m \), for a given value of \( \underline{\mu}’ \).

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 \( p \) for a bit to be corrupted. We also provide a variable number of protection depending on the importance of certain bits. In addition, we provide a variable packet size depending on the noise, in order to decrease the probability of automatic repeat request. The preferred protocol to be used with our algorithm is the User Datagram Protocol (UDP) fortified with our dynamic redundancy check algorithm, a packet sequence number, number of redundancy bits, signal group size as part of the packet header. Our algorithm has two parts. The first one is noise detection and noise quantization. The second part is redundancy bit adjustment and packet size adjustment to maximize the transmission throughput. In this paper, we present the analytics of keeping the correctable groups of bits in each transmission until the whole packet is received.

Beautifully Ordered Balanced Incomplete Block Designs, BOBIBD( \( v, k, \lambda, k_1, \lambda_1 \) ), were introduced by Chan and Sarvate along with some existence results for block size \( 3 \) and \( 4 \). We have shown that necessary conditions are sufficient for the existence of BOBIBDs with \( k = 5 \) for \( k_1 = 2 \) and \( 3 \) along with partial results for \( k_1 = 4 \). We also claim the nonexistence of cyclic solutions for certain BOBIBDs. The existence of the previously unknown BOBIBD(\( v, 4, 2, 3, 1 \)), \( v \equiv 1 \pmod{6} \), is demonstrated for all \( v \geq 19 \).

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 \({roaming-region}\) \( R(i) \) of a Delaunay node \( v_i \) is such that the Delaunay graph does not change as long as \( v_i \) remains within \( R(i) \). A node \( v_i \) with a large roaming region \( R(i) \) such that \( v_i \) is positioned near the center of \( R(i) \) is identified as a reliable node. Two types of roaming regions called (i) \({lateral\; roaming\; region}\) \( LR(i) \) and (ii) \({radial\; roaming\; region}\) \( RR(i) \) are distinguished to develop the algorithm. The roaming region itself is expressed as the intersection of \( RR(i) \) and \( LR(i) \). For nodes inside the convex hull, called \({deep\; internal\; nodes}\), we present an \( O(n^2) \) time algorithm for computing their roaming region, where \( n \) is the number of nodes in the Delaunay triangulation. We finally discuss generalization and extension of the proposed algorithm.

A pencyclic graph on \( v \) vertices is called pancyclic if it contains cycles of every length from \( 3 \) to \( v \). In this paper we address the question: what is the minimum number of edges in a pancyclic graph? We present a simple analysis using chord patterns.

Software interaction test suites serve two complementary roles. They are employed to systematically verify that, for some strength \( t \), no \( t \)-way interaction of a system’s parameters causes a fault. They are also employed to locate a faulty configuration when at least one interaction fault remains. Algorithms to find such test suites employing a number of tests close to the minimum have been extensively explored, in order to test all \( t \)-way interactions. However, when faults remain, the expected number of tests needed to reveal an interaction fault is also important. One might anticipate that the test suites of minimum size also have the lowest expected time to detection of an interaction fault; or, at the very least, that some test suite of minimum size does. However, in this paper it is shown that minimum test suite size and lowest expected time to fault detection are incompatible objectives. This underlies a challenging problem of how to generate test suites that have early coverage of \( t \)-way interactions, in order to reduce time to fault detection. A hybrid approach is developed that combines a simple greedy algorithm with heuristic search to construct one test at a time while attempting to maximize the number of \( t \)-way interactions covered by the earliest tests.

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 \( n \)-way combinatorial coverage of inter-window interactions. While greedy algorithms that generate Combinatorial Interaction Test suites enumerate and track the coverage of all possible \( t \)-tuples and constraints, we have noticed that our user-session-based test suites often do not contain every possible \( t \)-tuple, and we can take advantage of this in our algorithm by only storing \( t \)-tuples that appear in the test suite. Our empirical study shows both time and memory usage associated with our algorithm for 3-way inter-window parameter-value interaction coverage. Further, we conduct an empirical study where we compare 2-way and 3-way combinatorial coverage of inter-window parameter interactions in terms of the rate of fault detection for a web application called Schoolmate and a user-session-based test suite. Our results show that the rate of fault detection for 2-way and 3-way prioritization are within \(1\%\) of each other, but 2-way provides a slightly better result. A closer look at the characteristics of the web application, test cases, and faults reveals that most faults are triggered by 2-way interactions. We motivate the need for future work to examine a larger set of empirical studies to identify characteristics of web applications that benefit from prioritization with higher strength inter-window event interaction coverage.

In this work, we present a greedy algorithm for covering the set of incomplete STRIPS planning domain interpretations by \( t \)-strength diagnoses. We present a greedy algorithm to cover the incomplete domain model interpretations with a set of plans by iteratively generating plans so that each additional plan is biased to cover at least one new interpretation not previously covered. We also present a second greedy algorithm to construct a set of plans that covers all \( t \)-strength diagnoses of plan failure for plans in the incomplete domain model. We show that covering domain interpretations by \( t \)-strength diagnoses leads to increased coverage by a set of plans despite potentially lower coverage per plan because covering by \( t \)-strength diagnoses leads to a more scalable approach to planning where more plans can be found.

The article presents the compatibility matrix method and illustrates it with the application to the \( \text{P} \) vs \( \text{NP} \) problem. The method is a generalization of descriptive geometry: in the method, we draft problems and solve them utilizing the image creation technique. The method reveals: \( \text{P} = \text{NP} = \text{PSPACE} \subseteq \text{P/poly} \), etc.

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 \( K_{2n} \), deriving new proofs to a number of results on the enumeration of regular graphs with excluded cycles through the configuration model [3]. Here we extend this from excluded cycles to some excluded balanced subgraphs, and derive asymptotic results on the probability that a random regular multigraph from the configuration model contains at least one from a family of balanced subgraphs in question.

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.