Multigranulation roughness based on soft relations

The original rough set model, developed by Pawlak depends on a single equivalence relation. Qian et al, extended this model and defined multigranulation rough sets by using finite number of equivalence relations. This model provide new direction to the research. Recently, Shabir et al. proposed a rough set model which depends on a soft relation from an universe V to an universe W . In this paper we are present multigranulation roughness based on soft relations. Firstly we approximate a non-empty subset with respect to aftersets and foresets of finite number of soft binary relations. In this way we get two sets of soft sets called the lower approximation and upper approximation with respect to aftersets and with respect to foresets. Then we investigate some properties of lower and upper approximations of the new multigranulation rough set model. It can be found that the Pawlak rough set model, Qian et al. multigranulation rough set model, Shabir et al. rough set model are special cases of this new multigranulation rough set model. Finally, we added two examples to illustrate this multigranulation rough set model.

Mobile Robots with Uncertain Visibility Sensors: Possibility Results and Lower Bounds

We consider mobile robotic entities that have to cooperate to solve assigned tasks. In the literature, two models have been used to model their visibility sensors: the full visibility model, where all robots can see all other robots, and the limited visibility model, where there exists a limit [Formula: see text] such that all robots closer than [Formula: see text] are seen and all robots further than [Formula: see text] are not seen. We introduce the uncertain visibility model, which generalizes both models by considering that a subset of the robots further than [Formula: see text] cannot be seen. An empty subset corresponds to the full visibility model, and a subset containing every such robot corresponds to the limited visibility model. Then, we explore the impact of this new visibility model on the feasibility of benchmarking tasks in mobile robots computing: gathering, uniform circle formation, luminous rendezvous, and leader election. For each task, we determine the weakest visibility adversary that prevents task solvability, and the strongest adversary that allows task solvability. Our work sheds new light on the impact of visibility sensors in the context of mobile robot computing, and paves the way for more realistic algorithms that can cope with uncertain visibility sensors.

p-ideals of BCI-algebras based on neutrosophic N -structures

In this paper, neutrosophic N -structures are applied to p-ideals of BCI-algebras. In fact, we introduce the notion of neutrosophic N -p-ideal in BCI-algebras, and investigate several properties. Further, we present characterizations of neutrosophic N -p-ideal. Moreover, we consider relations between a neutrosophic N -ideal and a neutrosophic N -p-ideal. Also, we provide conditions for a neutrosophic N -ideal to be a neutrosophic N -p-ideal. Furthermore, it is proved that the neutrosophic N -structure Q N G over Q is a neutrosophic N p -ideal of Q ⇔ G is a p-ideal of Q where G is a non-empty subset of a BCI-algebras Q.

Characterizations of Cayley graphs of finite transformation semigroups with restricted range

The transformation semigroup with restricted range [Formula: see text] is the set of all functions from a set [Formula: see text] into a non-empty subset [Formula: see text] of [Formula: see text]. In this paper, we characterize Cayley graphs of [Formula: see text] with the connection set [Formula: see text]. Moreover, the undirected property of Cayley graphs Cay [Formula: see text] is studied.

The Zariski topology-graph of modules over commutative rings II

Let M be a module over a commutative ring R. In this paper, we continue our study about the Zariski topology-graph $$G(\tau _T)$$ G ( τ T ) which was introduced in Ansari-Toroghy et al. (Commun Algebra 42:3283–3296, 2014). For a non-empty subset T of $$\mathrm{Spec}(M)$$ Spec ( M ) , we obtain useful characterizations for those modules M for which $$G(\tau _T)$$ G ( τ T ) is a bipartite graph. Also, we prove that if $$G(\tau _T)$$ G ( τ T ) is a tree, then $$G(\tau _T)$$ G ( τ T ) is a star graph. Moreover, we study coloring of Zariski topology-graphs and investigate the interplay between $$\chi (G(\tau _T))$$ χ ( G ( τ T ) ) and $$\omega (G(\tau _T))$$ ω ( G ( τ T ) ) .

The Commuting Graphs on Dicyclic Groups

For a group G and a non-empty subset Ω of G, the commuting graph [Formula: see text] of Ω is a graph whose vertex set is Ω and any two vertices are adjacent if and only if they commute in G. Define [Formula: see text], the dicyclic group of order [Formula: see text] [Formula: see text], which is also known as the generalized quaternion group. We mainly investigate the properties and metric dimension of the commuting graphs on the dicyclic group [Formula: see text].

INDEPENDENT DOMINATION NUMBER (IDN) FOR SOME SPECIAL NETWORKS IN ADAPTIVE MESH REFINEMENT (AMR)-WENO SCHEME

Let G(V,E) be a graph, V has a subset C, this set is an non-empty subset of V and the vertices in C is adjacent to the minimum of one vertex of the set V, then G has the dominating set C. If there is no adjacency between the vertices of C, then G has an independent dominating set C and so the number of vertices present in the set C represents the IDN, the minimum cardinality of the sets C. Here in our research, we find the same for some special networks, namely the polygons with nine, ten and eleven sides by above mentioned Scheme.

A non-increasing tree growth process for recursive trees and applications

We introduce a non-increasing tree growth process $((T_n,{\sigma}_n),\, n\ge 1)$ , where T n is a rooted labelled tree on n vertices and σ n is a permutation of the vertex labels. The construction of (T n , σ n ) from (Tn−1, σn−1) involves rewiring a random (possibly empty) subset of edges in Tn−1 towards the newly added vertex; as a consequence Tn−1 ⊄ T n with positive probability. The key feature of the process is that the shape of T n has the same law as that of a random recursive tree, while the degree distribution of any given vertex is not monotone in the process. We present two applications. First, while couplings between Kingman’s coalescent and random recursive trees were known for any fixed n, this new process provides a non-standard coupling of all finite Kingman’s coalescents. Second, we use the new process and the Chen–Stein method to extend the well-understood properties of degree distribution of random recursive trees to extremal-range cases. Namely, we obtain convergence rates on the number of vertices with degree at least $c\ln n$ , c ∈ (1, 2), in trees with n vertices. Further avenues of research are discussed.

Multi-bump solutions for quasilinear elliptic equations with variable exponents and critical growth in ℝN

In this paper, we are concerned with the existence of multi-bump solutions for the following class of [Formula: see text]-Laplacian equations: [Formula: see text] where [Formula: see text] and [Formula: see text] are two real parameters, the nonlinearity [Formula: see text] is a continuous function with subcritical growth, [Formula: see text], the exponent [Formula: see text] can be equal to the critical exponent [Formula: see text] at some points of [Formula: see text] including at infinity and the potentials [Formula: see text], [Formula: see text] are continuous functions verifying some conditions. We show that if the zero set of [Formula: see text] has several isolated connected components [Formula: see text] such that the interior of [Formula: see text] is not empty and [Formula: see text] is smooth, then for [Formula: see text] large enough there exists, for any non-empty subset [Formula: see text], a bump solution trapped in a neighborhood of [Formula: see text]. The proofs are based on variational and topological methods.

Zero-sum subsets of decomposable sets in Abelian groups

A subset D of an abelian group is decomposable if ∅≠D⊂D+D. In the paper we give partial answers to an open problem asking whether every finite decomposable subset D of an abelian group contains a non-empty subset Z⊂D with ∑Z=0. For every n∈N we present a decomposable subset D of cardinality |D|=n in the cyclic group of order 2n−1 such that ∑D=0, but ∑T≠0 for any proper non-empty subset T⊂D. On the other hand, we prove that every decomposable subset D⊂R of cardinality |D|≤7 contains a non-empty subset T⊂D of cardinality |Z|≤12|D| with ∑Z=0. For every n∈N we present a subset D⊂Z of cardinality |D|=2n such that ∑Z=0 for some subset Z⊂D of cardinality |Z|=n and ∑T≠0 for any non-empty subset T⊂D of cardinality |T|<n=12|D|. Also we prove that every finite decomposable subset D of an Abelian group contains two non-empty subsets A,B such that ∑A+∑B=0.