Search results for: weak convergence

Authors: Mubarak Alhajri

Abstract:

This paper addresses certain inherent limitations of local priority hysteresis switching logic. Our main result establishes that under persistent excitation assumption, it is possible to relax constraints requiring strict positivity of local priority and hysteresis switching constants. Relaxing these constraints allows the adaptive system to reach optimality which implies the performance improvement. The unconstrained local priority hysteresis switching logic is examined and conditions for global convergence are derived.

Keywords: adaptive control, convergence, hysteresis constant, hysteresis switching

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Authors: J. Jena, Monica Saxena

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In this paper, the propagation of weak nonlinear waves in non-equilibrium flow has been studied in detail using the perturbation method. The expansive action of receding piston undergoing infinite acceleration has been discussed. Central expansion fan, compression waves and shock fronts have been discussed and the solutions up to the first order in the characteristic plane and physical plane have been obtained.

Keywords: Characteristic wave front, weak non-linear waves, central expansion fan, compression waves

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Authors: Saleh Maina

Abstract:

There is an on-going debate among intellectuals and scholars of international relations and world politics over the direction which the world is heading particularly in the current era of globalization. On the one hand are adherents to the convergence thesis which is premised on the assumption that global social order is tending toward universalism which could translate into the possible end of the classical state system and the unification of world societies under a single and common ideological dispensation. The convergence thesis is hinged on the globalization process which is gradually reducing world societies into a 'global village'. On the other hand are intellectuals who hold the view that despite advances made in communication technology which appear to threaten the survival of the classical state system. Invariance, as expressed in the survival of the existing state system and the diverse social traditions in world societies, remain a realistic possibility contrary to the conclusions of the convergence thesis. The invariance thesis has been advanced by scholars like Samuel P. Huntington whose work on clash of civilizations suggests that world peace can only be sustained through the co-habitation of diverse civilizations across the world. The purpose of this paper is to examine both sides of the debate with the aim of making a realistic assessment on where world societies are headed, between convergence and invariance. Using the realist theory of international relations as our theoretical premise the paper argues that while there is sufficient ground to predict the future direction of world societies as headed towards some form of convergence, invariance as expressed in the co-existence of diverse civilizations will for a long time remain a major feature of the international system.

Keywords: convergence, invariance, clash of civilization, classical state system, universalism

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Authors: Mauricio Quintero Angel, Andrés A. Duque Nivia, Carlos H. Fajardo Toro

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This paper analyzes two approaches of sustainability, the weak and strong, considering a case of study of oil palm production for an industry of biodegradable detergent. In this case, a company demand the oil palm as the active element for washing and through its trademark aims to supply 10% of the Colombian market of washing powders. Under each approach the economic and ecological implications of the palm oil production and especially the implications for crop management are described. The crop production under the weak sustainability implies plantations, intensive use of agrochemicals and the inclusion of new areas of cultivation as the market grows. Under the strong sustainability the production system is limited by the productive vocation of the ecosystem, so that new approaches and creativity for making viable the nature conservancy and the business development are require.

Keywords: agriculture, environmental impacts, oil palm, strong sustainability, weak sustainability

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Authors: Francis O. Nwawuru

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The convergence analysis of split monotone inclusion problems and fixed point problems of certain nonlinear mappings are investigated in the setting of real Hilbert spaces. Inertial extrapolation term in the spirit of Polyak is incorporated to speed up the rate of convergence. Under standard assumptions, a strong convergence of the proposed algorithm is established without computing the resolvent operator or involving Yosida approximation method. The stepsize involved in the algorithm does not depend on the spectral radius of the linear operator. Furthermore, applications of the proposed algorithm in solving some related optimization problems are also considered. Our result complements and extends numerous results in the literature.

Keywords: fixedpoint, hilbertspace, monotonemapping, resolventoperators

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Authors: Rashid Ahmed , John N. Avaritsiotis

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Many signal subspace-based approaches have already been proposed for determining the fixed Direction of Arrival (DOA) of plane waves impinging on an array of sensors. Two procedures for DOA estimation based neural networks are presented. First, Principal Component Analysis (PCA) is employed to extract the maximum eigenvalue and eigenvector from signal subspace to estimate DOA. Second, minor component analysis (MCA) is a statistical method of extracting the eigenvector associated with the smallest eigenvalue of the covariance matrix. In this paper, we will modify a Minor Component Analysis (MCA(R)) learning algorithm to enhance the convergence, where a convergence is essential for MCA algorithm towards practical applications. The learning rate parameter is also presented, which ensures fast convergence of the algorithm, because it has direct effect on the convergence of the weight vector and the error level is affected by this value. MCA is performed to determine the estimated DOA. Preliminary results will be furnished to illustrate the convergences results achieved.

Keywords: Direction of Arrival, neural networks, Principle Component Analysis, Minor Component Analysis

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Authors: Sara Mahesar, Saleem M. Chandio, Hira Soomro

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Nonlinear system of equations in two variables is a system which contains variables of degree greater or equal to two or that comprises of the transcendental functions. Mathematical modeling of numerous physical problems occurs as a system of nonlinear equations. In applied and pure mathematics it is the main dispute to solve a system of nonlinear equations. Numerical techniques mainly used for finding the solution to problems where analytical methods are failed, which leads to the inexact solutions. To find the exact roots or solutions in case of the system of non-linear equations there does not exist any analytical technique. Various methods have been proposed to solve such systems with an improved rate of convergence and accuracy. In this paper, a new scheme is developed for solving system of non-linear equation in two variables. The iterative scheme proposed here is modified form of the conventional Newton’s Method (CN) whose order of convergence is two whereas the order of convergence of the devised technique is three. Furthermore, the detailed error and convergence analysis of the proposed method is also examined. Additionally, various numerical test problems are compared with the results of its counterpart conventional Newton’s Method (CN) which confirms the theoretic consequences of the proposed method.

Keywords: conventional Newton’s method, modified Newton’s method, order of convergence, system of nonlinear equations

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Authors: Mahdiyeh Abbasi, Akbar Golchin

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It is well-known that, using principal weak flatness property, some important monoids are characterized, such as regular monoids, left almost regular monoids, and so on. In this article, we define a generalization of principal weak flatness called GP-Flatness, and will characterize monoids by this property of their right (Rees factor) acts. Also we investigate new classes of monoids called generally regular monoids and generally left almost regular monoids.

Keywords: G-left stabilizing, GP-flatness, generally regular, principal weak flatness

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Authors: Gulay Aras, Isil Kutluturk Karagoz, Z. Candan Algun

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Background: Hypermetropia in school-age is a pathology that responds to treatment. In the literature, there has been no study of exercise practice in hypermetropia treatment. Objective: The purpose of this study was to investigate the effects of eye exercises and convergence exercise on visual acuity in school-age children with hypermetropia. Methods: Forty volunteer school-age children with hypermetropia (30 girls, 30 boys, between 7-17 years of age) were included in the study. Sociodemographic information and clinical characteristics were evaluated. 40 participants were randomly divided into two groups: eye exercises and convergence exercises. Home exercise protocols were given to all groups for six weeks, and regular phone calls were made once a week. Individuals performed eye exercises 10 times, convergence exercises 5 min. for two sessions per day for six weeks. The right and left eyes of all the subjects participating in the study were assessed separately by the eye doctor with a Snellen chart. The participants' quality of life was assessed using Pediatric Quality of Life Inventory Version 4.0. Physical health total score (PHTS) and scale total score (STS), which were obtained by evaluating Psychosocial health total score (PSHTS) school, emotional and social functioning, were calculated separately in the scores. At the end of the exercise program, the assessment tests applied at the beginning of the study were reapplied to all individuals. Results: There was no statistically significant difference between the pre- and post-Snellen chart measurements and quality of life in the eye exercises group (p > 0,05). There was a statistically significant difference in visual acuity of right and left eyes (p=0,004, p=0,014) and quality of life in PHTS, PSHTS and STS in the convergence exercise group (p=0,001, p=0,017, p=0,001). Conclusions: In school-age children, convergence exercises were found to be effective on visual acuity and health-related quality of life. Convergence exercises are recommended for the treatment of school-aged children with hypermetropia.

Keywords: convergence exercise, eye exercises, hypermetropia, school-age children

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Authors: Paula Verdugo-Hernandez, Patricio Cumsille

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We analyze a first-class on the convergence of real number sequences, named hereafter sequences, to foster exploration and discovery of concepts through graphical representations before engaging students in proving. The main goal was to differentiate between sequences and continuous functions-of-a-real-variable and better understand concepts at an initial stage. We applied the analytic frame of mathematical working spaces, which we expect to contribute to extending to sequences since, as far as we know, it has only developed for other objects, and which is relevant to analyze how mathematical work is built systematically by connecting the epistemological and cognitive perspectives, and involving the semiotic, instrumental, and discursive dimensions.

Keywords: convergence, graphical representations, mathematical working spaces, paradigms of real analysis, real number sequences

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Authors: Hassan Samadi, Mustapha El Jarroudi

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In this work, we consider the homogenization of a non-linear problem in periodic medium with two periodic connected media exchanging a heat flux throughout their common interface. The interfacial exchange coefficient λ is assumed to tend to zero or to infinity following a rate λ=λ(ε) when the size ε of the basic cell tends to zero. Three homogenized problems are determined according to some critical value depending of λ and ε. Our method is based on Γ-Convergence techniques.

Keywords: variational methods, epiconvergence, homogenization, convergence technique

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Authors: Badoiu Mihaela Catalina

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Main focus of COHESION policy was reducing social and economic disparities between member states and regions, sustainable development and equal opportunities. In this perspective, the present study intend to analyze the evolution of the European architecture and its direct impact over the real convergence in the member states.

Keywords: cooperation, European union, member states, cohesion policy

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Authors: Young Hee Geum

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We compare optimal quartic order method for the multiple zeros of nonlinear equations illustrating the basins of attraction. To construct basins of attraction effectively, we take a 600×600 uniform grid points at the origin of the complex plane and paint the initial values on the basins of attraction with different colors according to the iteration number required for convergence.

Keywords: basins of attraction, convergence, multiple-root, nonlinear equation

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Authors: Entfellner Manuel, Wannenmacher Helmut, Reisenbauer Josef, Schubert Wulf

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When tunnelling in overstressed weak rocks ("squeezing ground"), two basic design approaches are available: the resistance principle, and the yielding principle. The resistance principle relies on rigid support systems to withstand the ground pressure. Alternatively, the yielding principle prioritizes controlled deformation, allowing the ground to deform without compromising tunnel integrity. This paper highlights the beneficial factors of the yielding principle for conventionally excavated tunnels in overstressed weak rocks. Especially the application of a ductile shotcrete lining with yielding elements is analysed in detail. Construction costs, safety, short- and long-term stabilities are discussed.

Keywords: squeezing ground, yielding principle, yielding element, conventional tunneling

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Authors: Elham Kiyani, S. Mansour Vaezpour, Javad Tavakoli

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In this paper, we first introduce a new distance function in a linear space not necessarily endowed with a topology. The algebraic concepts of interior and closure are useful to study optimization problems without topology. So, we define Q-weak efficient solutions generated by the algebraic interior of a set Q, where Q is not necessarily convex. Studying nonconvex vector optimization is valuable since, for a convex cone K in topological spaces, we have int(K)=cor(K), which means that topological interior of a convex cone K is equal to the algebraic interior of K. Moreover, we used the scalarization technique including the distance function generated by the vectorial closure of a set to characterize these Q-weak efficient solutions. Scalarization is a useful approach for solving vector optimization problems. This technique reduces the optimization problem to a scalar problem which tends to be an optimization problem with a real-valued objective function. For instance, Q-weak efficient solutions of vector optimization problems can be characterized and computed as solutions of appropriate scalar optimization problems. In the convex case, linear functionals can be used as objective functionals of the scalar problems. But in the nonconvex case, we should present a suitable objective function. It is the aim of this paper to present a new distance function that be useful to obtain sufficient and necessary conditions for Q-weak efficient solutions of general optimization problems via scalarization.

Keywords: weak efficient, algebraic interior, vector closure, linear space

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Authors: Bruna Luisa Ramos Prado Vasques, Mariane Rembold Petraglia, Antonio Petraglia

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This paper investigates MIMO (Multiple-Input Multiple-Output) adaptive filtering techniques for the application of supervised source separation in the context of convolutive mixtures. From the observation that there is correlation among the signals of the different mixtures, an improvement in the NSAF (Normalized Subband Adaptive Filter) algorithm is proposed in order to accelerate its convergence rate. Simulation results with mixtures of speech signals in reverberant environments show the superior performance of the proposed algorithm with respect to the performances of the NLMS (Normalized Least-Mean-Square) and conventional NSAF, considering both the convergence speed and SIR (Signal-to-Interference Ratio) after convergence.

Keywords: adaptive filtering, multi-rate processing, normalized subband adaptive filter, source separation

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Authors: Rukshana Pervin

Abstract:

We report a detailed study on the transport properties of a 40 nm thick granular aluminum film. As measured by temperature-dependent resistance R(T), a resistance peak is observed before the transition to superconductivity, which indicates that the diffusion channel is subjected to weak localization and electron-electron interaction, and the superconductor channel is subjected to SC fluctuations (SCFs). The zero-magnetic field transport measurement demonstrated that Electron-Electron Interaction (EEI), weak localization, and SCFs are closely related in this granular aluminum film. The characteristic temperature at which SCFs emerge on the sample is determined by measuring the R(T) during cooling. The SCF of the film is studied in terms of the direct contribution of the Aslamazov-Larkin's fluctuation Cooper pair density and the indirect contribution of the Maki-Thomson's quasiparticle pair density. In this sample, the rise in R(T) above the SCF characteristic temperature indicates the WL and/or EEI. Comparative analyses are conducted on how the EEI and WL contribute to the upturn in R(T).

Keywords: fluctuation superconductivity, weak localization, thermal deposition, electron-electron interaction

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Authors: Giorgi Dalakishvili, Goderdzi G. Didebulidze, Maya Todua

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The possibility of sporadic E (Es) layer formation in the mid-latitude nighttime lower thermosphere by horizontal homogeneous and inhomogeneous (vertically and horizontally changing) winds is investigated in 3D by analytical and numerical solutions of continuity equation for dominant heavy metallic ions Fe+. The theory of influence of wind velocity direction, value, and its shear on formation of sporadic E is developed in case of presence the effect of horizontally changing wind (the effect of horizontal convergence). In this case, the horizontal wind with horizontal shear, characterized by compressibility and/or vortices, can provide an additional influence on heavy metallic ions Fe+ horizontal convergence and Es layers density, which can be formed by their vertical convergence caused as by wind direction and values and by its horizontal shear as well. The horizontal wind value and direction have significant influence on ion vertical drift velocity and its minimal negative values of divergence necessary for development of ion vertical convergence into sporadic E type layer. The horizontal wind horizontal shear, in addition to its vertical shear, also influences the ion drift velocity value and its vertical changes and correspondingly on formation of sporadic E layer and its density. The atmospheric gravity waves (AGWs), with relatively smaller horizontal wave length than planetary waves and tidal motion, can significantly influence location of ion vertical drift velocity nodes (where Es layers formation expectable) and its vertical and horizontal shear providing ion vertical convergence into thin layer. Horizontal shear can cause additional influence in the Es layers density than in the case of only wind value and vertical shear only. In this case, depending on wind direction and value in the height region of the lower thermosphere about 90-150 km occurs heavy metallic ions (Fe+) vertical convergence into thin sporadic E type layer. The horizontal wind horizontal shear also can influence on ions horizontal convergence and density and location Es layers. The AGWs modulate the horizontal wind direction and values and causes ion additional horizontal convergence, while the vertical changes (shear) causes additional vertical convergence than in the case without vertical shear. Influence of horizontal shear on sporadic E density and the importance of vertical compressibility of the lower thermosphere, which also can be influenced by AGWs, is demonstrated numerically. For the given wavelength and background wind, the predictability of formation Es layers and its possible location regions are shown. Acknowledgements: This study was funded by Georgian Shota Rustaveli National Science Foundation Grant no. FR17-357.

Keywords: in-homogeneous, sporadic E, thermosphere, wind

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Authors: Mohamed Shalaby, Yasser Kamal, Negm Shawky

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Mutually unbiased bases (MUBs) have an important role in the field of quantum computation and information. A complete set of these bases can be constructed when the system dimension is the power of the prime. Constructing such complete set in composite dimensions is still an open problem. Recently, the concept of weak mutually unbiased bases (WMUBs) in composite dimensions was introduced. A complete set of such bases can be constructed by combining the MUBs in each subsystem. In this paper, we present a comparative study between MUBs and WMUBs in the context of complex projective t-design. Explicit proofs are presented.

Keywords: complex projective t-design, finite quantum systems, mutually unbiased bases, weak mutually unbiased bases

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Authors: Kuniyoshi Abe

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Bi-conjugate gradient (Bi-CG) is a well-known method for solving linear equations Ax = b, for x, where A is a given n-by-n matrix, and b is a given n-vector. Typically, the dimension of the linear equation is high and the matrix is sparse. A number of hybrid Bi-CG methods such as conjugate gradient squared (CGS), Bi-CG stabilized (Bi-CGSTAB), BiCGStab2, and BiCGstab(l) have been developed to improve the convergence of Bi-CG. Bi-CGSTAB has been most often used for efficiently solving the linear equation, but we have seen the convergence behavior with a long stagnation phase. In such cases, it is important to have Bi-CG coefficients that are as accurate as possible, and the stabilization strategy, which stabilizes the computation of the Bi-CG coefficients, has been proposed. It may avoid stagnation and lead to faster computation. Motivated by a large number of processors in present petascale high-performance computing hardware, the scalability of Krylov subspace methods on parallel computers has recently become increasingly prominent. The main bottleneck for efficient parallelization is the inner products which require a global reduction. The resulting global synchronization phases cause communication overhead on parallel computers. The parallel variants of Krylov subspace methods reducing the number of global communication phases and hiding the communication latency have been proposed. However, the numerical stability, specifically, the convergence speed of the parallel variants of Bi-CGSTAB may become worse than that of the standard Bi-CGSTAB. In this paper, therefore, we compare the convergence speed between the standard Bi-CGSTAB and the parallel variants by numerical experiments and show that the convergence speed of the standard Bi-CGSTAB is faster than the parallel variants. Moreover, we propose the stabilization strategy for the parallel variants.

Keywords: bi-conjugate gradient stabilized method, convergence speed, Krylov subspace methods, linear equations, parallel variant

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Authors: Hudson Akewe

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This research presents complementary results with wider applications on convergence and rate of convergence of classical fixed point theory in Banach spaces to the world of the theory of fixed points of mappings defined in classes of spaces of measurable functions, known in the literature as modular function spaces. The study gives a comprehensive survey of various iterative fixed point results for the classes of multivalued ρ-contractive-like, ρ-quasi-contractive-like, ρ-quasi-contractive, ρ-Zamfirescu and ρ-contraction mappings in the framework of modular function spaces. An example is presented to demonstrate the applicability of the implicit-type iterative schemes to the system of ordinary differential equations. Furthermore, numerical examples are given to show the rate of convergence of the various explicit Kirk-type and implicit Kirk-type iterative schemes under consideration. Our results complement the results obtained on normed and metric spaces in the literature. Also, our methods of proof serve as a guide to obtain several similar improved results for nonexpansive, pseudo-contractive, and accretive type mappings.

Keywords: implicit Kirk-type iterative schemes, multivalued mappings, convergence results, fixed point

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Authors: Agrinë Baraku

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This paper focuses on the convergence of the internet, fake news, and democracy. This paper will examine the convergence of these concepts, the tenets of democracy which are affected by the ever-increasing exposure to fake news, and whether the impact strengthens or can further weaken countries with fragile democracies. To demonstrate the convergence and the impact and to further the discussion about this topic, the case of Kosovo is explored. Its position in the Western Balkans makes it even more susceptible to the pressure stemming from geopolitical interests, which intersect with the generation of fake news by different international actors. Domestically, through data generated by Kantar (Index) Kosova Longitudinal Study on Media Measurement Survey (MMS), which focused on media viewership, the trend among Kosovar citizens is traced and then inserted into a bigger landscape, which is compounded by tenuous circumstances and challenges that Kosovo faces. Attention will be paid to what this can tell about where Kosovo currently is and the possibilities of what can be done regarding the phenomenon that is taking place.

Keywords: democracy, disinformation, internet, social media, fake news

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Authors: Reza Mollapourasl, Majid Haghi

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In this study, we develop local meshfree methods known as radial basis function-generated finite difference (RBF-FD) method and Hermite finite difference (RBF-HFD) method to design stencil weights and spatial discretization for Helmholtz equation. The convergence and stability of schemes are investigated numerically in three dimensions with irregular shaped domain. These localized meshless methods incorporate the advantages of the RBF method, finite difference and Hermite finite difference methods to handle the ill-conditioning issue that often destroys the convergence rate of global RBF methods. Moreover, numerical illustrations show that the proposed localized RBF type methods are efficient and applicable for problems with complex geometries. The convergence and accuracy of both schemes are compared by solving a test problem.

Keywords: radial basis functions, Hermite finite difference, Helmholtz equation, stability

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Authors: Philippe Gugler

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This contribution reflects some important questions regarding inter alia the economic development occurring in the light of the ASEAN’s goal of creating the ASEAN Economic Community (AEC) by 2015. We observe a continuing economic growth of GDP per capita over recent years despite the negative effects of the world economic crisis. IMF forecasts indicate that this trend will continue. The paper focuses on the analysis and comparison of economic growth trends of ASEAN countries.

Keywords: ASEAN, convergence, divergence, economic growth, globalization, integration

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Authors: Herjuno Bagus Wicaksono, Emma Almira Fauni, Salma Amelia Dina

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The Efficient Market Hypothesis (EMH) states that, in an efficient capital market, price fully reflects the information available in the market. This theory has influenced many investors behavior in trading in the stock market. Advanced researches have been conducted to test the efficiency of the stock market in particular countries. Indonesia, as one of the emerging countries, has performed substantial growth in the past years. Hence, this paper aims to examine the efficiency of Islamic stock market in Indonesia in its weak form. The daily stock price data from Indonesia Sharia Stock Index (ISSI) for the period October 2015 to October 2016 were used to do the statistical tests: Run Test and Serial Correlation Test. The results show that there is no serial correlation between the current price with the past prices and the market follows the random walk. This research concludes that Indonesia Islamic stock market is weak form efficient.

Keywords: efficient market hypothesis, Indonesia sharia stock index, random walk, weak form efficiency

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Authors: Safeer Hussain Khan

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In this paper, we prove a strong convergence result using a recently introduced iterative process with contractive-like operators. This improves and generalizes corresponding results in the literature in two ways: the iterative process is faster, operators are more general. In the end, we indicate that the results can also be proved with the iterative process with error terms.

Keywords: contractive-like operator, iterative process, fixed point, strong convergence

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Authors: Ramin Javadzadeh

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The particle swarm optimization are Meta heuristic optimization method, which are used for clustering and pattern recognition applications are abundantly. These algorithms in multimodal optimization problems are more efficient than genetic algorithms. A major drawback in these algorithms is their slow convergence to global optimum and their weak stability can be considered in various running of these algorithms. In this paper, improved Particle swarm optimization is introduced for the first time to overcome its problems. The fuzzy cellular automata is used for improving the algorithm efficiently. The credibility of the proposed approach is evaluated by simulations, and it is shown that the proposed approach achieves better results can be achieved compared to the Particle swarm optimization algorithms.

Keywords: cellular automata, cellular learning automata, local search, optimization, particle swarm optimization

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Authors: Malika El Kyal, Ahmed Machmoum

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We prove the superlinear convergence of asynchronous multi-splitting methods applied to differential equations. This study is based on the technique of nested sets. It permits to specify kind of the convergence in the asynchronous mode.The main characteristic of an asynchronous mode is that the local algorithm not have to wait at predetermined messages to become available. We allow some processors to communicate more frequently than others, and we allow the communication delays to be substantial and unpredictable. Note that synchronous algorithms in the computer science sense are particular cases of our formulation of asynchronous one.

Keywords: parallel methods, asynchronous mode, multisplitting, ODE

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Authors: Helmi Temimi

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In this paper, we study the super-convergence properties of the discontinuous Galerkin (DG) method applied to one-dimensional mth-order ordinary differential equations without introducing auxiliary variables. We found that nth−derivative of the DG solution exhibits an optimal O (hp+1−n) convergence rates in the L2-norm when p-degree piecewise polynomials with p≥1 are used. We further found that the odd-derivatives and the even derivatives are super convergent, respectively, at the upwind and downwind endpoints.

Keywords: discontinuous, galerkin, superconvergence, higherorder, error, estimates

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Authors: Chinedu Peter, Dashrath Singh

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Membrane computing is a computability model which abstracts its structures and functions from the biological cell. The main ingredient of membrane computing is the notion of a membrane structure, which consists of several cell-like membranes recurrently placed inside a unique skin membrane. The emergence of several variants of membrane computing gives rise to the notion of a P system. The paper presents a variant of P systems for arithmetic operations on non-negative integers based on the weak priorities for rule application. Consequently, we obtain deterministic P systems. Two membranes suffice. There are at most four objects for multiplication and five objects for division throughout the computation processes. The model is simple and has a potential for possible extension to non-negative integers and real numbers in general.

Keywords: P system, binary operation, determinism, weak rule priority

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