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.

For a graph \( G = (V, E) \), a non-empty set \( S \subseteq V \) is a global offensive alliance (respectively, global strong offensive alliance) if for every vertex \( v \in V – S \), at least half of the vertices in its closed neighborhood are in \( S \) (respectively, a strict majority of its closed neighborhood are in \( S \)). The global offensive alliance number \( \gamma_o(G) \) (respectively, global strong offensive alliance number \( \gamma_{\hat{o}}(G) \)) is the minimum cardinality of a global offensive alliance (respectively, global strong offensive alliance) of \( G \). In this paper, we determine an upper bound on each parameter for bipartite graphs without isolated vertices. More precisely, we show that \( \gamma_o(G) \leq \frac{n – \ell + s}{2} \) and \( \gamma_{\hat{o}}(G) \leq \frac{n + \ell}{2} \), where \( n \), \( \ell \), and \( s \) are the order, the number of leaves, and the support vertices of \( G \), respectively. Moreover, extremal trees attaining each bound are characterized.

There is a hypothesis that a non-self-centric radially-maximal graph of radius \( r \) has at least \( 3r – 1 \) vertices. Moreover, if it has exactly \( 3r – 1 \) vertices, then it is planar with minimum degree \( 1 \) and maximum degree \( 3 \). Using an enhanced exhaustive computer search, we prove this hypothesis for \( r = 4, 5 \).

A vertex set \( X \) of a simple graph is called OO-irredundant if for each \( v \in X \), \( N(v) – N(X – \{v\}) \neq \emptyset \). Basic results for maximal OO-irredundant sets of a graph are obtained.