Narrow Results

For a strongly connected network, the network may achieve synchronization by coupling of nodes. However, as the size of the network increases, it is difficult for one node with higher degree to couple with all its neighbors, especially when the dimension of the isolated node’s dynamics is more than one. In order to solve this problem, a novel hybrid coupling strategy is proposed to make a complex dynamical network achieve synchronization, where only one random dimension of each node is coupled with its all neighbors, and the other dimension of the node is coupled with only one node. Several numerical simulations verify the effectiveness of the proposed strategy.

A semigroup S is said to be ℓ-threshold k-testable if it satisfies all identities u = v where u, v is an arbitrary pair of words over a finite alphabet Σ such that they simultaneously belong or fail to belong to any ℓ-threshold k-testable (regular) language. We give an asymptotic formula for the free spectrum of the variety of all ℓ-threshold k-testable semigroups, thereby providing an asymptotic upper bound on the size of an arbitrary finitely generated locally threshold testable semigroup. The combinatorial interpretation of this task yields an enumeration problem for particular edge labelings of de Bruijn graphs.

Humanity’s efforts are manifested in the creation of novel solutions to complex problems in diverse fields. Traditional mathematical methods fail to solve real-world problems due to their complexity. Researchers have come up with new mathematical theories like fuzzy set theory and rough set theory to help them figure out how to model the uncertainty in these fields. Soft set theory is a novel approach to real-world problem solving that does not require the membership function to be specified. This aids in the resolution of a wide range of issues, and significant progress has recently been made. After Jun et al. came up with a hybrid system that combined fuzzy and soft set concepts, many people came up with hybrid ideas in different algebraic structures. In this paper, we introduce the concepts of subsystem and strong subsystem of a hybrid finite state machine (HFSM) and investigate a portion of their significant properties. We also provide an example that shows that every subsystem does not need to be a strong subsystem. Additionally, we study the cyclic subsystem of HFSMs and also obtain their equivalent results and examples. Finally, we define the notions of homomorphism of subsystems and strong subsystems of HFSMs.

Traditional approaches to connectivity in sensor networks are based on the omnidirectional antenna model which relies on the assumption that the sensors send and receive in all directions. Current technologies make possible the utilization of sensors with directional antenna capabilities whereby the sensors send and/or receive along a sector of a predefined angle (or beam-width). Although several researchers in the scientific literature have investigated the impact of directional antennae on network throughput, energy consumption, as well as security very little is known concerning the effect of directional antennae on its connectivity. In this paper, we introduce for the first time a new sensor model with each sensor being able to transmit in any one of k directions, for some fixed k, and explore the algorithmic limits and potential of such a directional antenna model. More specifically, given a set of n sensors in the plane, we consider the problem of establishing a strongly connected ad hoc network from these sensors using directional antennae. In particular, we prove that given such set of sensors, each equipped with k, 1 ≤ k ≤ 5, directional antennae with any angle of transmission, these antennae can be oriented in such a way that the resulting communication structure is a strongly connected digraph spanning all n sensors. Moreover, the transmission range of the antennae is at most times the optimal range (a range necessary to establish a connected network on the same set of sensors using omnidirectional antennae). The algorithm which constructs this orientation runs in O(n) time provided a minimum spanning tree on the set of sensors is given. We show that our solution can be used to give a tradeoff on the range and angle when each sensor has one antenna. Further, we also prove that for two antennae it is NP-hard to decide whether such an orientation exists if both the transmission angle and range are small for each antennae.

Given an edge-weighted undirected hypergraph $K=(V_K,E_K)$ and an even-sized set of vertices $R\subseteq V_K$, a T-cut is a partition of $V_K$ into two parts $Q$ and $\bar{Q}:=V_K\setminus Q$ such that $|Q\cap R|$ is odd. A T-join in $K$ for $R$ is a set of hyperedges $M\subseteq E_K$ such that for every T-cut $(Q,\bar{Q})$ there is a hyperedge $e\in M$ intersecting both $Q$ and $\bar{Q}$.

A directed hypergraph has for every hyperedge exactly one vertex, called the head, and several vertices, that are tails. A directed hypergraph is strongly connected if there exists at least one directed path between any two vertices of the hypergraph, where a directed path is defined to be a sequence of vertices and hyperedges for which each hyperedge has as one of its tails the vertex preceding it and as its head the vertex following it in the sequence. Orienting an undirected hypergraph means choosing a head for each hyperedge.

We prove that every edge-weighted undirected hypergraph that admits a strongly connected orientation has a T-join of total weight at most $7/8$ times the total weight of all the edges of the hypergraph, and sketch an improvement to $4/5$. We also exhibit a series of example showing that one cannot improve the constant above to $2/3-𝜖$.