搜索结果：Mathematics

I discuss some general aspects of the creation, interpretation, and reception of mathematics as a part of civilization and culture.

In this article we will use Minecraft to experimentally approximate the values of four different mathematical constants. The mathematical constants that we will approximate are $\sqrt{2}, π$, Euler's number $e$, and Apéry's constant $ζ(3)$. We will begin each section with a brief history of the number being approximated and describe where it appears in mathematics. We then explain how we used Minecraft mechanics to approximate the constant. At the end of each section, we provide some ideas for how to apply our techniques to the approximation of other mathematical constants in Minecraft or elsewhere. This article is a proof of concept that Minecraft can be used in higher education. We should note that the goal of this article is not to have the most accurate approximations possible, the goal is to inspire people to have fun while learning about various mathematical topics. We hope you learn something new in this article and feel inspired to try some of these techniques on your own.

We propose a mathematical model for the spread of Japanese encephalitis, with emphasis on environmental effects on the aquatic phase of mosquitoes. The model is shown to be biologically well-posed and to have a biologically and ecologically meaningful disease free equilibrium point. Local stability is analyzed in terms of the basic reproduction number and numerical simulations presented and discussed.

We analyze a mathematical model to understand the dynamics of bullying in schools. The model considers a population divided into four groups: susceptible individuals, bullies, individuals exposed to bullying, and violent individuals. Transitions between these states occur at rates designed to capture the complex interactions among students, influenced by factors such as romantic rejection, conflicts with peers and teachers, and other school-related challenges. These interactions can escalate into bullying and violent behavior. The model also incorporates the role of parents and school administrators in mitigating bullying through intervention strategies. The results suggest that bullying can be effectively controlled if anti-bullying programs implemented by schools are sufficiently robust. Additionally, the conditions under which bullying persists are explored.

We propose and analyse a mathematical model for cholera considering vaccination. We show that the model is epidemiologically and mathematically well posed and prove the existence and uniqueness of disease-free and endemic equilibrium points. The basic reproduction number is determined and the local asymptotic stability of equilibria is studied. The biggest cholera outbreak of world's history began on 27th April 2017, in Yemen. Between 27th April 2017 and 15th April 2018 there were 2275 deaths due to this epidemic. A vaccination campaign began on 6th May 2018 and ended on 15th May 2018. We show that our model is able to describe well this outbreak. Moreover, we prove that the number of infected individuals would have been much lower provided the vaccination campaign had begun earlier.

We provide an historical overview of how advances in technology influenced high school and university mathematical competitions in the United States and at the International Mathematical Olympiad. While students are not allowed the usage of technological aids during mathematical competitions, the developments in technology (especially graphing technology) throughout the past century and the increasing employment of such aids in the classroom have affected both the nature of the proposed problems and their expected solutions. We examine several interesting examples from competitions going back several decades.

We give a brief survey on aspects of the local index theory as developed from the mathematical works of V. K. Patodi. It is dedicated to the 70th anniversary of Patodi.

A comprehensive review will be given about the rich mathematical structure of mean field spin glass theory, mostly developed, until now, in the frame of the methods of theoretical physics, based on deep physical intuition and hints coming from numerical simulation. Central to our treatment is a very simple and yet powerful interpolation method, allowing to compare different probabilistic schemes, by using convexity and positivity arguments. In this way we can prove the existence of the thermodynamic limit for the free energy density of the system, a long standing open problem. Moreover, in the frame of a generalized variational principle, we can show the emergency of the Derrida-Ruelle random probability cascades, leading to the form of free energy given by the celebrated Parisi \textit {Ansatz}. All these results seem to be in full agreement with the mechanism of spontaneous replica symmetry breaking as developed by Giorgio Parisi.

A visual type theory is a cognitive tool that has much in common with language, and may be regarded as an exceptional form of spatial text adjunct. A mathematical visual type theory, called NPM, has been under development that can be viewed as an early-stage project in mathematical knowledge management and mathematical user interface development. We discuss in greater detail the notion of a visual type theory, report on progress towards a usable mathematical visual type theory, and discuss the outlook for future work on this project.

We start recalling with critical eyes the mathematical methods used in gauge theory and prove that they are not coherent with continuum mechanics, in particular the analytical mechanics of rigid bodies or hydrodynamics, though using the same group theoretical methods and despite the well known couplings existing between elasticity and electromagnetism (piezzoelectricity, photoelasticity, streaming birefringence). The purpose of this paper is to avoid such contradictions by using new mathematical methods coming from the formal theory of systems of partial differential equations and Lie pseudogroups. These results finally allow to unify the previous independent tentatives done by the brothers E. and F. Cosserat in 1909 for elasticity or H. Weyl in 1918 for electromagnetism by using respectively the group of rigid motions of space or the conformal group of space-time. Meanwhile we explain why the Poincaré "duality scheme" existing between "geometry " and "physics" has to do with homological algebra and algebraic analysis. We insist on the fact that these results could not have been obtained before 1975 as the corresponding tools were not known before and are still not known today by p