Intersection Graph of a Module

Let $V$ be a left $R$-module where $R$ is a (not necessarily commutative) ring with unit. The intersection graph \cG(V) of proper $R$-submodules of $V$ is an undirected graph without loops and multiple edges defined as follows: the vertex set is the set of all proper $R$-submodules of $V,$ and there is an edge between two distinct vertices $U$ and $W$ if and only if $U\cap W\neq 0.$ We study these graphs to relate the combinatorial properties of \cG(V) to the algebraic properties of the $R$-module $V.$ We study connectedness, domination, finiteness, coloring, and planarity for \cG (V). For instance, we find the domination number of \cG (V). We also find the chromatic number of \cG(V) in some cases. Furthermore, we study cycles in \cG(V), and complete subgraphs in \cG (V) determining the structure of $V$ for which \cG(V) is planar

Fuzzy Modeling for Uncertainty Nonlinear Systems with Fuzzy Equations

The uncertain nonlinear systems can be modeled with fuzzy equations by incorporating the fuzzy set theory. In this paper, the fuzzy equations are applied as the models for the uncertain nonlinear systems. The nonlinear modeling process is to find the coefficients of the fuzzy equations. We use the neural networks to approximate the coefficients of the fuzzy equations. The approximation theory for crisp models is extended into the fuzzy equation model. The upper bounds of the modeling errors are estimated. Numerical experiments along with comparisons demonstrate the excellent behavior of the proposed method

Uncertain non near system control with Fuzzy Differential Equations and Z-numbers

In this paper, the solutions of fuzzy differential equations (FDEs) are estimated by using two types of Bernstein neural networks. Here, the uncertainties are in the form of Z numbers. Firstly, we transform the FDE to four ordinary differential equations (ODEs) at par with Hukuhara differentiability. After that we develop neural models having the structure of ODEs. By using modified backpropagation technique for Z number variables, the training of neural networks are carried out. The results of the simulation illustrate that these innovative models, Bernstein neural networks, are efficient to approximate the solutions of FDEs which are on the basis of Z-numbers

Numerical Solution of Fuzzy Equations with Z-numbers using Neural Networks

In this paper, the uncertainty property is represented by the Z-number as the coefficients of the fuzzy equation. This modification for the fuzzy equation is suitable for nonlinear system modeling with uncertain parameters. We also extend the fuzzy equation into dual type, which is natural for linear-in-parameter nonlinear systems. The solutions of these fuzzy equations are the controllers when the desired references are regarded as the outputs. The existence conditions of the solutions (controllability) are proposed. Two types of neural networks are implemented to approximate solutions of the fuzzy equations with Z-number coefficients

Quantum Phase Transition in the One-Dimensional Extended Quantum Compass Model in a Transverse Field

Quantum phase transitions in the one-dimensional extended quantum compass model in transverse field are studied by using the Jordan-Wigner transformation. This model is always gapful except at the critical surfaces where the energy gap disappears. We obtain the analytic expressions of all critical fields which drive quantum phase transitions. This model shows a rich phase diagram which includes spin-flop, strip antiferromagnetic and saturate ferromagnetic phases in addition to the phase with anti parallel ordering of spin $y$ component on odd bonds. However we study the universality and scaling properties of the transverse susceptibility and nearest-neighbor correlation functions derivatives in different regions to confirm the results obtained using the energy gap analysis.Comment: 8 Page, 15 Figure