Inspired by Gale's proof of Farkas's lemma, this expository note aims to give a very simple and intuitive proof of Gordan's theorem and the equivalence of this theorem to Farkas's lemma and some formulations of the separating hyperplane theorems. The tool we have employed is limited to only very simple linear algebra over the eld of real numbers.
The goal of this paper is the geometrical and numerical study of the main sizes of the mathematical models of prey-predator interactions which are important in determining long-time dynamics, based on the application of various notions from the theory of dynamical systems to the numerical approximation of initial value problems over long-time intervals. The numerical methods are widely used for the study of complicated temporal behavior of dynamical systems, in order to approximate different types of invariants sets or invariant manifolds and also to extract statistical information on the dynamical behavior in the computation of natural invariant measures or almost invariants sets. The present study is a interplay between dynamical systems geometrical theory and computational calculus of dynamical systems, knowing that the theory provides a framework for interpreting numerical observations and foundations for efﬁcient numerical algorithms.
This research work sought to estimate and analyze the causes and trend of crime in Ho Municipality. The research was carried out in Ho central of the Municipality where 102 respondents made up of police officers and prison officers were selected to complete a questionnaire asking them to indicate the level of importance attached to listed original factors considered to be the causes of crime in the Municipality. The raw data consists of 13 original factors subjected to correlation analysis to identify new composite factors that can explain the causes of crime in the Municipality. Using time series data (total quarterly crime for the major crime categories) covering the period of 2004 to 2014 obtained from the Regional crime unit of the Volta Regional command of the Ghana police service, and the prison service, an impact assessment model was obtained to determine the trend of crime in the municipality using time series analysis, the data was also used to forecast for the next six years. Information obtained from the field data was also used to determine the sex and age group that mostly engaged in crime in the Municipality.
At the end, five factors were identified to be the major causes of crime, these are; parental neglect, poverty, unemployment, peer pressure and drug abuse. It has also been revealed that trend of crime is in the increase in the Municipality. Males and the age group of 16-35 years are found to engage in crime in the Municipality. Finally, we propose an alternative strategy to control crime by enhancing police efficiency that is by introducing volunteer reinforcement either by enlarging its size or by updating its technology, it is hoped that the findings of this research would prompt society to be mindful of criminal activities in the Municipality.
In the paper the local density and the local weak density of topological spaces are investigated. It is proved that for a π-irreducible mapping f of a topological space X onto a topological space Y the followings hold: d(X) = d(Y ), wd(X) = wd(Y ), ld(X) ≤ ld(Y ), lwd(X) ≤ lwd(Y ). Moreover, it is showed that the functor of probability measures of nite supports Pn, the functor of the permutation degree and the functor expnpreserve the cardinality of k-networks of in nite compacts.
Integrated Water Management System is studied from the dynamical system perspective. Risk factors, management factors and total water consumption of area are considered to be three interaction variable of the dynamical system. Chaotic dynamics characteristics of the system is studied by using Lyapunov exponents, bifurcation diagrams and Poincare mappings. Abundant chaotic dynamics characteristics of the system are showed. An Integrated Water Management attractor is obtained. Linear feedback controller is used to stabilize and control the system to a limit cycle.
The steady flow of a viscous incompressible fluid between two coaxial discs, one rotating and other stationary with suction is analysed. We propose a semi-numerical method in which recurrence relations derived allow to generate large number of universal coefficients in small Reynolds number perturbation series of the solution function. The convergence of the series is restricted by a simple pole using some special techniques the region of validity of the series is increased. The results provided are in excellent agreement with pure numerical studies.
An n-independent set in two dimensions is a set of nodes admitting (not necessary unique) bivariate interpolation with polynomials of total degree at most n: For an arbitrary n-independent node set X we are interested with the property that each node possesses an n-fundamental polynomial in form of products of linear or quadratic factors. In the present paper we show that each node of X has an n-fundamental polynomial, which is a product of lines, if #X ≤ 2n + 1: Next we prove that each node of X has an n-fundamental polynomial, which is a product of lines or conics, if #X ≤ 2n+[n/2]+1. We bring a counterexample in each case to show that the results are not valid in general if #X ≥ 2n + 2 and #X ≥ 2n + [n/2] + 2; respectively. At the end we bring an algorithm for the construction of above mentioned fundamental polynomials. This, in view of the Lagrange formula, can be used to obtain readily also the interpolation polynomials.