The ATSP polytope can be expressed by an asymmetric polynomial-size linear program.

A model that represents the rate of changes of the population with limited environmental resources can be described by,

\[
\frac{dp}{dt} = p\left(a – {bp}\right) + g(t,p) = p(t_0)= p_0
\]

where \( a \) measures the growth rate in the absence of the restriction force \( b \) and \( \frac{a}{b} \) is called the carrying capacity of the environment. The random perturbation \( g(t,P) \) is generated by random change in the environment. The behavior of the solution of this model for continuous and discrete case when \( g(t,P)=w(t) \) is density independent with a constant random factor \( w \) in a short time interval \([t, t + \delta t)\) will be studied. The stability and the behavior of the equilibrium point will also be investigated. A computational approach to the solution using Excel spreadsheet and Maple will be presented.

For a set \( S \) of two or more vertices in a nontrivial connected graph \( G \) of order \( n \), a collection \(\{T_1, T_2, \ldots, T_\ell\}\) of trees in \( G \) is said to be an internally disjoint set of trees connecting \( S \) if these trees are pairwise edge-disjoint and \( V(T_i) \cap V(T_j) = S \) for every pair \( i, j \) of distinct integers with \( 1 \leq i, j \leq \ell \). For an integer \( k \) with \( 2 \leq k \leq n \), the tree \( k \)-connectivity \( \kappa_k(G) \) of \( G \) is the greatest positive integer \( \ell \) for which \( G \) contains at least \( \ell \) internally disjoint trees connecting \( S \) for every set \( S \) of \( k \) vertices of \( G \). It is shown for every two integers \( k \) and \( r \) with \( 3 \leq k \leq 2r \) that
\[
\kappa_k(K_{r,r}) = r – \left\lceil \frac{k-1}{4} \right\rceil.
\]

This paper investigates the existence of monadic balanced ternary designs (BTDs). A monadic BTD is a BTD where each size \( K \) block contains one element that appears doubly and \( K-2 \) elements that appear singly. The authors show that the conditions

1. \( \rho_1 = 2\rho_2 \),
2. \( \Lambda(V-1) = 10\rho_2 \),
3. \( \Lambda \neq 3 \),

are sufficient for the existence of monadic BTDs \( (V; B; \rho_1, \rho_2, R; 4; \Lambda) \). The authors also give necessary and sufficient conditions for the existence of monadic BTDs where the block size is five and \( \Lambda \) is 3 or 6.

We consider the placement of detection devices at the vertices of a graph \( G \), where a detection device at vertex \( v \) has three possible outputs: there is an intruder at \( v \); there is an intruder at one of the vertices in the open neighborhood \( N(v) \), the set of vertices adjacent to \( v \), but which vertex in \( N(v) \) cannot be determined; or there is no intruder in \( N[v] = N(v) \cup \{v\} \). We introduce the \( 1 \)-step locating-dominating problem of placing the minimum possible number of such detection devices in \( V(G) \) so that the presence of an intruder in \( V(G) \) can be detected, and the exact location of the intruder can be identified, either immediately or when the intruder has moved to an adjacent vertex. Some related problems are introduced.

Recently, four new vertex colorings of graphs (in which adjacent vertices may be colored the same) were introduced for the purpose of distinguishing every pair of adjacent vertices. For each graph and for each of these four colorings, the minimum number of required colors never exceeds the chromatic number of the graph. In this paper, we summarize some of the results obtained on these colorings and introduce some relationships among them.

We address the problem: for which values of \( d \) and \( n \) does there exist a triangle-free regular graph of degree \( d \) on \( n \) vertices? A complete solution is given.

Let \( G = K_{a,b} \), where \( a, b \) are even, or \( G = K_{a,a} – M_{2a} \), where \( a \geq 1 \) is an odd integer and \( M_{2a} \) is a perfect matching in \( K_{a,a} \). It has been shown ([3,4]) that \( G \) is arbitrarily decomposable into closed trails. Billington asked if the graph \( K_{r,s} – F \), where \( s, r \) are odd and \( F \) is a (smallest possible) spanning subgraph of odd degree, is arbitrarily decomposable into closed trails ([2]).

In this article we answer the question in the affirmative.

This paper considers the Lehmer matrix and its recursive analogue. The determinant of the Lehmer matrix is derived explicitly by both its LU and Cholesky factorizations. We further define a generalized Lehmer matrix with \((i,j)\) entries \( g_{ij} = \frac{\text{min} \{u_{i+1}, u_{j+1}\}}{\text{max} \{u_{i+1}, u_{j+1}\}} \) where \( u_n \) is the \( n \)th term of a binary sequence \(\{u_n\}\). We derive both the LU and Cholesky factorizations of this analogous matrix and we precisely compute the determinant.

Random number generators are a small part of any computer simulation project. Yet they are the heart and the engine that drives the project. Often times software houses fail to understand the complexity involved in building a random number generator that will satisfy the project requirements and will be able to produce realistic results. Building a random number generator with a desirable periodicity, that is uniform, that produces all the random permutations with equal probability, and at random, is not an easy task. In this paper we provide tests and metrics for testing random number generators for uniformity and randomness. These tests are in addition to the already existing tests for uniformity and randomness, which we modify by running each test a large number of times on sub-sequences of random numbers, each of length \( n \). The test result obtained each time is used to determine the probability distribution function. This eliminates the random number generator misclassification error. We also provide new tests for uniformity and randomness, the new tests for uniformity test the skewness of each one of the subgroups as well as the kurtosis. The tests for randomness, which include the Fourier spectrum, the phase spectrum, the discrete cosine transform spectrum, and the orthogonal wavelet domain, test for patterns not detected in the raw data space. Finally we provide visual and acoustic tests.

For a connected graph \( G \) of order \( n \), the detour distance \( D(u, v) \) between two vertices \( u \) and \( v \) in \( G \) is the length of a longest \( u-v \) path in \( G \). A Hamiltonian labeling of \( G \) is a function \( c: V(G) \to \mathbb{N} \) such that

\[ |c(u) – c(v)| + D(u,v) \geq n \]

for every two distinct vertices \( u \) and \( v \) of \( G \). The value \( \text{hn}(c) \) of a Hamiltonian labeling \( c \) of \( G \) is the maximum label (functional value) assigned to a vertex of \( G \) by \( c \); while the Hamiltonian labeling number \( \text{hn}(G) \) of \( G \) is the minimum value of a Hamiltonian labeling of \( G \). We present several sharp upper and lower bounds for the Hamiltonian labeling number of a connected graph in terms of its order and other distance parameters.

A graph \( G \) is \( 3 \)-existentially closed (\( 3 \)-e.c.) if each \( 3 \)-set of vertices can be extended in all of the possible eight ways. Results which improve the lower bound of the minimum order of a \( 3 \)-e.c. graph are reported. It has been shown that \( m_{ec}(3) \geq 24 \), where \( m_{ec}(3) \) is defined to be the minimum order of a \( 3 \)-e.c. graph.

In this study, we analyze the structure of the full collineation group of certain Veblen-Wedderburn (VW) planes of orders \( 5^2 \), \( 7^2 \), and \( 11^2 \). We also discuss a reconstruction method using their collineation groups.

A Sarvate-Beam Quad System \( SB(v, 4) \) is a set \( V \) of \( v \) elements and a collection of \( 4 \)-subsets of \( V \) such that each distinct pair of elements in \( V \) occurs \( i \) times for every \( i \) in the list \( 1, 2, \ldots, \binom{v}{2} \). In this paper, we completely enumerate all Sarvate-Beam Quad Systems for \( v = 6 \).

In this paper, we present (by using Cauchy-Schwarz inequalities) some new results amongst the parameters of balanced arrays (B-arrays) with two symbols and having strength four, which are necessary for the existence of such balanced arrays. We then discuss and illustrate their use and applications.

The Oberwolfach problem (OP) asks whether \( K_n \) (for \( n \) odd) or \( K_n \) minus a \( 1 \)-factor (for \( n \) even) admits a \( 2 \)-factorization where each \( 2 \)-factor is isomorphic to a given \( 2 \)-factor \( F \). The order \( n \) and the type of the \( 2 \)-factor \( F \) are the parameters of the problem. For \( n \leq 17 \), the existence of a solution has been resolved for all possible parameters. There are also many special types of \( 2 \)-factors for which solutions to OP are known. We provide solutions to OP for all orders \( n \), \( 18 \leq n \leq 40 \). The computational results for higher orders were obtained using the SHARCNET high-performance computing cluster.

Let \( f_6(n) \) denote the number of partitions of the natural number \( n \) into parts co-prime to \( 6 \). This function was originally studied by Schur. We derive two explicit formulas for \( f_6(n) \), one of them in terms of the partition function \( p(n) \). We also derive three recurrences for \( f_6(n) \).

We consider the problem of relocating a sensor node in its neighborhood so that the connectivity of the network is not altered. In this context, we introduce the notion of \in-free and out-free regions to capture the set of points where the node can be relocated by conserving connectivity. We present a characterization of maximal free-regions that can be used for identifying the position where the node can be moved to increase the reliability of the network connectivity. In addition, we prove that the free-region computation problem has a lower bound \(\Omega(n\log n)\) in the comparison tree model of computation, and also present two approximation algorithms for computing the free region of a sensor node in time \(O(k)\) and \(O(k\log k)\).

Chain integrator backstepping is a recursive design tool that has been used in nonlinear control systems. The complexity of the computation of the chain integrator backstepping control law makes inevitable the use of a computer algebra system. A recursive algorithm is designed to compute the integrator backstepping control process. A computer algebra program (Maple procedure) is developed for symbolic computation of the control function using a newly developed recursive algorithm. We will present some demonstrative examples to show the stability of the control systems using Lyapunov functions.

Chain integrator backstepping is a recursive design tool that has been used in nonlinear control systems. The complexity of the computation of the chain integrator backstepping control law makes inevitable the use of a computer algebra system. A recursive algorithm is designed to compute the integrator backstepping control process. A computer algebra program (Maple procedure) is developed for symbolic computation of the control function using a newly developed recursive algorithm. We will present some demonstrative examples to show the stability of the control systems using Lyapunov functions.

Let \( G \) be a graph with vertex set \( V(G) \) and edge set \( E(G) \), and let \( A \) be an abelian group. A labeling \( f: V(G) \to A \) induces an edge labeling \( f^*: E(G) \to A \) defined by \( f^*(xy) = f(x) + f(y) \) for each \( xy \in E(G) \). For each \( i \in A \), let \( v_f(i) = \text{card}\{v \in V(G) \mid f(v) = i\} \) and \( e_f(i) = \text{card}\{e \in E(G) \mid f^*(e) = i\} \). Let \( c(f) = \{\lvert e_f(i) – e_f(j) \rvert \mid (i, j) \in A \times A\} \). A labeling \( f \) of a graph \( G \) is said to be \( A \)-friendly if \( \lvert v_f(i) – v_f(j) \rvert \leq 1 \) for all \( (i, j) \in A \times A \). If \( c(f) \) is a \( (0, 1) \)-matrix for an \( A \)-friendly labeling \( f \), then \( f \) is said to be \( A \)-cordial. When \( A = \mathbb{Z}_2 \), the friendly index set of the graph \( G \), \( FI(G) \), is defined as \( \{\lvert e_f(0) – e_f(1) \rvert \mid \text{the vertex labeling } f \text{ is } \mathbb{Z}_2\text{-friendly}\} \). In \([15]\) the friendly index set of a cycle is completely determined. We consider the friendly index sets of broken wheels with three spokes.

The intractability of the traditional discrete logarithm problem (DLP) forms the basis for the design of numerous cryptographic primitives. In \([2]\) M. Sramka et al. generalize the DLP to arbitrary finite groups. One of the reasons mentioned for this generalization is P. Shor’s quantum algorithm \([4]\) which solves efficiently the traditional DLP. The DLP for a non-abelian group is based on a particular representation of the group and a choice of generators. In this paper, we show that care must be taken to ensure that the representation and generators indeed yield an intractable DLP. We show that in \(\text{PSL}(2,p) = \langle \alpha, \beta \rangle\) the generalized discrete logarithm problem with respect to \((\alpha,\beta)\) is easy to solve for a specific representation and choice of generators \(\alpha\) and \(\beta\). As a consequence, such representation of \(\text{PSL}(2,p)\) and generators should not be used to design cryptographic primitives.

Beautifully Ordered Balanced Incomplete Block Designs, \(\text{BOBIBD}(v, k, \lambda, k_1, \lambda_1)\), are defined and the proof is given to show that necessary conditions are sufficient for the existence of BOBIBDs with block size \(k = 3\) and \(k = 4\) for \(k_1 = 2\) except possibly for eleven exceptions. Existence of BOBIBDs with block size \(k = 4\) and \(k_1 = 3\) is demonstrated for all but one infinite family and the non-existence of \(\text{BOBIBD}(7, 4, 2, 3, 1)\), the first member of the unknown series, is shown.