Mathematics, Volume 6, Issue 2 (February 2018) – 15 articles

We propose the so-called fuzzy semi-metric space in which the symmetric condition is not assumed to be satisfied. In this case, there are four kinds of triangle inequalities that should be considered. The purpose of this paper is to study the common coincidence points and common fixed points in the newly proposed fuzzy semi-metric spaces endowed with the so-called ⋈-triangle inequality. The other three different kinds of triangle inequalities will be the future research, since they cannot be similarly investigated as the case of ⋈-triangle inequality. Full article

Fuzzy graph theory is a conceptual framework to study and analyze the units that are intensely or frequently connected in a network. It is used to study the mathematical structures of pairwise relations among objects. An m-polar fuzzy (mF, for short) set is a useful notion in practice, which is used by researchers or modelings on real world problems that sometimes involve multi-agents, multi-attributes, multi-objects, multi-indexes and multi-polar information. In this paper, we apply the concept of mF sets to hypergraphs, and present the notions of regular mF hypergraphs and totally regular mF hypergraphs. We describe the certain properties of regular mF hypergraphs and totally regular mF hypergraphs. We discuss the novel applications of mF hypergraphs in decision-making problems. We also develop efficient algorithms to solve decision-making problems. Full article

In this paper, we introduce a new definition for nilpotent fuzzy subgroups, which is called the good nilpotent fuzzy subgroup or briefly g-nilpotent fuzzy subgroup. In fact, we prove that this definition is a good generalization of abstract nilpotent groups. For this, we show that a group G is nilpotent if and only if any fuzzy subgroup of G is a g-nilpotent fuzzy subgroup of G. In particular, we construct a nilpotent group via a g-nilpotent fuzzy subgroup. Finally, we characterize the elements of any maximal normal abelian subgroup by using a g-nilpotent fuzzy subgroup. Full article

In this paper, the existence of fixed point for Pata type Zamfirescu mapping in a complete metric space is proved. Our result give existence of fixed point for a wider class of functions and also prove the existence of best proximity point to the result on “A fixed point theorem in metric spaces”, given by vittorino Pata. Full article

The static problem for elastic shallow cables suspended at points at different levels under general vertical loads is addressed. The cases of both suspended and taut cables are considered. The funicular equation and the compatibility condition, well known in literature, are here shortly re-derived, and the commonly accepted simplified hypotheses are recalled. Furthermore, with the aim of obtaining simple asymptotic expressions with a desired degree of accuracy, a perturbation method is designed, using the taut string solution as the generator system. The method is able to solve the static problem for any distributions of vertical loads and shows that the usual, simplified analysis is just the first step of the perturbation procedure proposed here. Full article

This manuscript reveals both the full experimental and methodical details of a most-recent patent that demonstrates a much-desired goal of rotational maneuvers via angular exchange momentum, namely extremely high torque without mathematical singularity and accompanying loss of attitude control while the angular momentum trajectory resides in the mathematical singularity. The paper briefly reviews the most recent literature, and then gives theoretical development for implementing the new control methods described in the patent to compute a non-singular steering command to the angular momentum actuators. The theoretical developments are followed by computer simulations used to verify the theoretical computation methodology, and then laboratory experiments are used for validation on a free-floating hardware simulator. A typical 3/4 CMG array skewed at 54.73° yields 0.15H. Utilizing the proposed singularity penetration techniques, 3H momentum is achieved about yaw, 2H about roll, and 1H about pitch representing performance increases of 1900%, 1233%, and 566% respectfully. Full article

In this paper, we prove the generalized nonlinear stability of the first and second of the following $\rho$ -functional equations, $G\left(\right|a|{\Delta }_{\mathcal{A}}^{*}\left|b\right|){\Delta }_{\mathcal{B}}^{*}G\left(\right|a|{\Delta }_{\mathcal{A}}^{**}\left|b\right|)-G\left(\right|a\left|\right){\Delta }_{\mathcal{B}}^{**}G\left(\right|b\left|\right)=\rho (2\left[G,\left({\displaystyle \frac{\left|a\right|{\Delta }_{\mathcal{A}}^{*}\left|b\right|}{2}}\right),{\Delta }_{\mathcal{B}}^{*},G,\left({\displaystyle \frac{\left|a\right|{\Delta }_{\mathcal{A}}^{**}\left|b\right|}{2}}\right)\right]-G\left(\right|a\left|\right){\Delta }_{\mathcal{B}}^{**}G\left(\right|b\left|\right))$ , and $2\left[G,\left({\displaystyle \frac{\left|a\right|{\Delta }_{\mathcal{A}}^{*}\left|b\right|}{2}}\right),{\Delta }_{\mathcal{B}}^{*},G,\left({\displaystyle \frac{\left|a\right|{\Delta }_{\mathcal{A}}^{**}\left|b\right|}{2}}\right)\right]-G\left(\right|a\left|\right){\Delta }_{\mathcal{B}}^{**}G\left(\right|b\left|\right)=\rho \left(G\left(\right|a|,{\Delta }_{\mathcal{A}}^{*},\left|b\right|),{\Delta }_{\mathcal{B}}^{*},G\left(\right|a|,{\Delta }_{\mathcal{A}}^{**},\left|b\right|),-,G,\left(\right|a\left|\right),{\Delta }_{\mathcal{B}}^{**},G,\left(\right|b\left|\right)\right)$ in latticetic random Banach lattice spaces, where $\rho$ is a fixed real or complex number with $\rho \ne 1$ . Full article

In an earlier paper, we gave a proof of the conjecture of the pinching of the bisectional curvature mentioned in those two papers of Hong et al. of 1988 and 2011. Moreover, we proved that any compact Kähler–Einstein surface M is a quotient of the complex two-dimensional unit ball or the complex two-dimensional plane if (1) M has a nonpositive Einstein constant, and (2) at each point, the average holomorphic sectional curvature is closer to the minimal than to the maximal. Following Siu and Yang, we used a minimal holomorphic sectional curvature direction argument, which made it easier for the experts in this direction to understand our proof. On this note, we use a maximal holomorphic sectional curvature direction argument, which is shorter and easier for the readers who are new in this direction. Full article

The meticulous study of finite automata has produced many important and useful results. Automata are simple yet efficient finite state machines that can be utilized in a plethora of situations. It comes, therefore, as no surprise that they have been used in classic game theory in order to model players and their actions. Game theory has recently been influenced by ideas from the field of quantum computation. As a result, quantum versions of classic games have already been introduced and studied. The $PQ$ penny flip game is a famous quantum game introduced by Meyer in 1999. In this paper, we investigate all possible finite games that can be played between the two players Q and Picard of the original $PQ$ game. For this purpose, we establish a rigorous connection between finite automata and the $PQ$ game along with all its possible variations. Starting from the automaton that corresponds to the original game, we construct more elaborate automata for certain extensions of the game, before finally presenting a semiautomaton that captures the intrinsic behavior of all possible variants of the $PQ$ game. What this means is that, from the semiautomaton in question, by setting appropriate initial and accepting states, one can construct deterministic automata able to capture every possible finite game that can be played between the two players Q and Picard. Moreover, we introduce the new concepts of a winning automaton and complete automaton for either player. Full article

In this article, three numerical iterative schemes, namely: Jacobi, Gauss–Seidel and Successive over-relaxation (SOR) have been proposed to solve a fuzzy system of linear equations (FSLEs). The convergence properties of these iterative schemes have been discussed. To display the validity of these iterative schemes, an illustrative example with known exact solution is considered. Numerical results show that the SOR iterative method with $\omega =1.3$ provides more efficient results in comparison with other iterative techniques. Full article

Graph theory has numerous applications in various disciplines, including computer networks, neural networks, expert systems, cluster analysis, and image capturing. Rough neutrosophic set (NS) theory is a hybrid tool for handling uncertain information that exists in real life. In this research paper, we apply the concept of rough NS theory to graphs and present a new kind of graph structure, rough neutrosophic digraphs. We present certain operations, including lexicographic products, strong products, rejection and tensor products on rough neutrosophic digraphs. We investigate some of their properties. We also present an application of a rough neutrosophic digraph in decision-making. Full article

Pulsed-field gradient (PFG) diffusion experiments can be used to measure anomalous diffusion in many polymer or biological systems. However, it is still complicated to analyze PFG anomalous diffusion, particularly the finite gradient pulse width (FGPW) effect. In practical applications, the FGPW effect may not be neglected, such as in clinical diffusion magnetic resonance imaging (MRI). Here, two significantly different methods are proposed to analyze PFG anomalous diffusion: the effective phase-shift diffusion equation (EPSDE) method and a method based on observing the signal intensity at the origin. The EPSDE method describes the phase evolution in virtual phase space, while the method to observe the signal intensity at the origin describes the magnetization evolution in real space. However, these two approaches give the same general PFG signal attenuation including the FGPW effect, which can be numerically evaluated by a direct integration method. The direct integration method is fast and without overflow. It is a convenient numerical evaluation method for Mittag-Leffler function-type PFG signal attenuation. The methods here provide a clear view of spin evolution under a field gradient, and their results will help the analysis of PFG anomalous diffusion. Full article

Solving differential equations of fractional (i.e., non-integer) order in an accurate, reliable and efficient way is much more difficult than in the standard integer-order case; moreover, the majority of the computational tools do not provide built-in functions for this kind of problem. In this paper, we review two of the most effective families of numerical methods for fractional-order problems, and we discuss some of the major computational issues such as the efficient treatment of the persistent memory term and the solution of the nonlinear systems involved in implicit methods. We present therefore a set of MATLAB routines specifically devised for solving three families of fractional-order problems: fractional differential equations (FDEs) (also for the non-scalar case), multi-order systems (MOSs) of FDEs and multi-term FDEs (also for the non-scalar case); some examples are provided to illustrate the use of the routines. Full article

In this paper, after a brief review of the physical notion of quality factor in viscoelasticity, we present a complete discussion of the attenuation processes emerging in the Maxwell–Prabhakar model, recently developed by Giusti and Colombaro. Then, taking profit of some illuminating plots, we discuss some potential connections between the presented model and the modern mathematical modelling of seismic processes. Full article