Advances in Mathematics and Computer Science Vol. 1

A Matrix Vector Transition Net Implementation

2021-07-26T03:25:44+00:00

Aims: Classic Petri nets also known as place transition nets provide many interesting and useful features for system modeling. They are however limited by the place types that are used. A novel approach is presented in this work. A matrix vector transition net model is created and is used to model complex system behavior. This solution extends the modeling power of normal Petri nets.

Proposed Solution: A traditional Petri net is modified to create a matrix vector transition net (MVTN). The idea is to combine the ideas from normal Petri net semantics with a matrix vector approach.

Implementing the Matrix Vector Transition Net: Ordinary Petri net places are replaced with matrices or vectors. The input and output arcs must have a specific function matrix that determines firing. Firing and behavior remain conceptually and functionally similar to that of a Petri net. It is possible to interchange row and column vectors. The behavior of matrix transition nets must elicit similar behavior to that of a place transition net. Instead of normal tokens, matrix elements are used. The matrix vector type of structure increases the modeling power, abstraction capacity and the complexity of the net.

Case Study: To illustrate this work a toy case of an abstract network structure containing processing elements is used to illustrate the use of the matrix vector transition net structure.

Results and Findings: The behavior of matrix transition nets is shown to be similar in principle to that of a place transition net. However instead of tokens, matrix elements are used. It is possible to construct a symbolic marking graph or reachability graph for the system This type of structure definitely increases the modeling power, abstraction capacity and the complexity of the net. The matrix transition net could be useful for certain types of communication system problems and complex system interfacing. Several other uses can be found for this approach in both computing and modeling.

Exponential Latin square, exponential Sudoku matrix and bitwise operations

2021-07-26T03:30:21+00:00

The main thrust of the paper is to provide an interesting and useful for students example of using bitwise operations in programming languages C ++ and Java. As an example, we describe an algorithm for receiving a Latin square of arbitrary order. So we will outline some techniques for the use of bitwise operations. The article considers also the special case when the Latin square is a Sudoku matrix

There is No Standard Model of ZFC and ZFC 2

2021-07-26T03:42:53+00:00

In this Chapter we obtain a contradictions in formal set theories under assumption that these theories have omega-models or nonstandard model with standard part. An posible generalization of Lob’s theorem is considered. Main results are:

Examples of Simply and Multiply Connected Fatou Sets for a Class of Meromorphic Functions

2021-07-26T03:55:47+00:00

Aims: We give some families of functions which are meromorphic outside a compact countable set B of essential singularities. Our aim is to give some examples of the stable set (called the Fatou set) and the unstable set (called the Julia set) since there are not many examples of parametric family of this class of functions (called in the introduction functions of class ) in complex dynamics.
Study design: We study components of the Fatou set and some theorems related with the iteration of functions in class and design a computational program to give examples of the Julia and Fatou sets.
Place and Duration of Study: Facultad de Ciencias F__sico Matem_aticas, Benem_erita Universidad Aut_onoma de Puebla, M_exico between June 2011 and July 2012.
Methodology: We use some theorems of complex dynamics in order to study components of the Fatou set. We program some algorithms in C and get some _gures of the Fatou set.
Results: Given a family of functions in class we get some mathematical results of the Fatou and Julia sets and its _gures for some parameters given.
Conclusion: Taking some families in class we give examples of the Fatou set which can be either simply-connected or multiply-connected in the last case the Julia set is a totally disconnected set.

DRBEM Sensitivity Analysis and Shape Optimization of Rotating Magneto-Thermo-Viscoelastic FGA Structures Using DRBEM-GSS and DRBEM-NGGP Algorithms

2021-07-26T04:01:04+00:00

Aims: A practicalshape optimization technique is developed, using the dual reciprocity boundary element method (DRBEM) with the golden-section search algorithm based on uniform bicubic B-splines, for rotating magneto-thermo-viscoelastic functionally graded anisotropic (FGA) structures subjected to a moving heat source in the context of the Green and Naghdi theory of type III.

Study Design: Original Research Paper.

Place and Duration of Study: Jamoum University College, Mathematics Department, between July 2016 and August 2017.

Methodology: An implicit-implicit staggered algorithm was proposed for use with the DRBEM to obtain the final DRBEM coupled linear system of equations for displacements and temperature that describe the magneto-thermo-viscoelastic structural analysis problem. An implicit differentiation of the discretized dual reciprocity boundary integral equation with respect to design variables is used to calculate shape displacement sensitivities of anisotropic materials with very high accuracy. This method allows the coupling between optimization technique and a dual reciprocity boundary element method. The feasible direction method was developed and implemented for use with the one-dimensional golden-section search technique based on uniform bicubic B-splines, as a numerical optimization method for minimizing weight while satisfying all of the constraints. The DRBEM was developed and implemented for use with the golden-section search (DRBEM-GSS) algorithm and also implemented with the neutrosophic goal geometric programming (DRBEM-NGGP) algorithm as a numerical optimization techniques for minimizing weight while satisfying all of the constraints.

Results: The optimum shape design of fillet in tension bars used as the numerical example in order to verify the formulation and the implementation of the proposed technique. The numerical results show our technique is efficient and precise.

Conclusion: From the research that has been performed, it is possible to conclude that the optimal shape of the top half of the fillet under stress constraint based on magneto-thermo-viscoelasticity is crucial when magneto-thermoviscoelastic field is sensitive to boundary shape. Also from this knowledge of the variation of the displacements and temperature sensitivities with time for magneto-thermo-viscoelastic FGA structures, we can design various magneto-thermoviscoelastic structures to meet specific engineering requirements and utilize within which to place new information can be more effective.

On the Proof Complexities of Strongly Equal Non-classical Tautologies

2021-07-26T04:05:01+00:00

The strong equality of classical tautologies and their proof complexities comparative analysis in certain proof systems were given by ﬁrst author in previous studies. Here we introduce the analogous notions of strong equality for non-classical (intuitionistic and minimal) tautologies and investigate the relations between the proof complexity measures of strongly equal non-classical tautologies in some proof systems. We prove that 1) the strongly equal tautologies have the same proof complexities in some proof systems and 2) there are such proof systems, in which some measures of proof complexities for strongly equal tautologies are the same, while the other measures diﬀer from each other only as a function of the sizes of tautologies.

Recursive Computation of Binomial and Multinomial Coefficients and Probabilities

2021-07-26T04:18:14+00:00

This chapter studies a prominent class of recursively-defined combinatorial functions, namely, the binomial and multinomial coefficients and probabilities. The chapter reviews the basic notions and mathematical definitions of these four functions. Subsequently, it characterizes each of these functions via a recursive relation that is valid over a certain two-dimensional or multi-dimensional region and is supplemented with certain boundary conditions. Visual interpretations of these characterizations are given in terms of regular acyclic signal flow graphs. The graph for the binomial coefficients resembles a Pascal Triangle, while that for trinomial or multinomial coefficients looks like a Pascal Pyramid, Tetrahedron, or Hyper-Pyramid. Each of the four functions is computed using both its conventional and recursive definitions. Moreover, the recursive structures of the binomial coefficient and the corresponding probability are utilized in an iterative scheme, which is substantially more efficient than the conventional or recursive evaluation. Analogous iterative evaluations of the multinomial coefficient and probability can be constructed similarly. Applications to the reliability evaluation for two-valued and multi-valued k-out-of-n systems are also pointed out.

A New Definition of Limit of Periodic Function and Periodic g-Contractive Mapping at Infinity

2021-07-26T04:25:39+00:00

Limit is a basic concept of calculus. However, according to the updated definition, the limit of periodic function at infinity is not in existence. This conclusion of description does not suit with the periodic phenomenon. For example, the temperature on earth is changed periodically every year since the birth of the earth (viewed as t=0). Today (viewed as t →∞) the temperature on earth is continuing. Continuation means that the limit exists. In this paper, a new definition of limit of periodic function and periodic g-contractive mapping at infinity is defined by the value of its initial point based on transformation of variables. Similar definition is made for g- contractive ratio of periodic g-contractive mapping with k-related fixed points. These definitions can be used to describe the k-polar problems and calculation the limit of combinations of periodic functions at infinity. Furthermore, the new definition on contractive ratio of periodic iterative g-contractive mapping at infinity can help us to find the constant G and improves the application of the periodic iterative g-contractive mapping theorem.

The Effect of Anisotropy on the Structure Optimization Using BEM-GSS and BEM-NGGP Algorithms

2021-07-26T04:42:06+00:00

Aims: A shape optimization technique is developed, using the boundary element method, for two-dimensional anisotropic structures to study the effects of anisotropy on the displacements and stresses, then minimize weight while satisfying certain constraints upon stresses and geometry.

Study Design: Original Research Paper.

Place and Duration of Study: Jamoum University College, Mathematics Department, between June 2016 and July 2017.

Methodology: The shape design sensitivity analysis of a two-dimensional anisotropic structure using a singular formulation of the boundary element method is investigated to study the effects of anisotropy on the displacements and stresses. An Implicit differentiation technique of the discretized boundary integral equations is performed to produce terms that contain derivatives of the fundamental solutions employed in the analysis. This technique allows the coupling between optimization technique and numerical boundary element method (BEM) to form an optimum shape design algorithm that yields shape design sensitivities of the displacement and stress fields for anisotropic materials with very high accuracy. The fundamental solutions of displacements and tractions in terms of complex variables employed in the analysis. The boundary element method was developed and implemented for use with the golden-section search (GSS) algorithm and neutrosophic goal geometric programming (NGGP) algorithm as a numerical optimization technique for minimizing weight while satisfying all of the constraints.

Results: The proposed method has been verified by using the two-dimensional plate with an elliptical hole as the numerical example. The numerical results show that the proposed method is suitable and effective tool for the computer implementation of the solution.

Conclusion: From the research that has been performed, it is possible to conclude that the optimal shape of the two-dimensional plate with an elliptical hole is crucial when elastic field is sensitive to boundary shape. Also from this knowledge of the effects of anisotropy on the displacements and stresses, we can design various anisotropic structures to meet specific engineering requirements and utilize within which to place new information can be more effective.