Maple and the Putnam Competition

We apply the Maple computer algebra system to the resolution of problems taken from the Putnam Competition. Out of the twelve problems of the 1993 contest, Maple is shown to help the resolution of half of them

Using the Maple Computer Algebra System as a Tool for Studying Group Theory

The purpose of this study was to show that computers can be powerful tools for studying group theory. Specifically the author examined ways that the computer algebra system Maple can be used to assist in the study of group theory. The study consists of four main parts. After a brief introduction in chapter one, chapter two discusses simple procedures written by the author to study small finite groups. These procedures rely on the fact that for small finite groups, the elements can all be stored on a computer and tested for various properties. All of the procedures are contained in the appendix, and each is described in chapter two. The Maple software comes with a built in set of group theory procedures. The procedures work with two types of groups, permutation groups and finitely presented groups. The author discusses all of the procedures dealing with permutation groups in chapter three and the procedures for finitely presented groups in chapter four. The main theoretical tool for permutation groups is a stabilizer chain, and the main tool for finitely presented groups is the Todd-Coxeter algorithm. Both of these methods and their implementations in Maple are discussed in detail. The study is concluded by examining some applications of group theory. The author discusses check digit schemes, RSA encryption, and permutation factoring. The ability to factor a permutation in terms of a set of generators can be used to solve several puzzles such as the Rubik\u27s cube

Точні розв'язки однієї спектральної задачі з диференціальним оператором Шрьодінгера з поліноміальним потенціалом у R²

Вперше розглянуто суттєво двовимірний випадок оператора Шрьодінгера з поліноміальним потенціалом. За допомогою FD-методу та системи комп'ютерної алгебри Maple знайдено чотири точні власні значення для потенціалу конкретного вигляду з шести найменших.Впервые рассмотрен существенно двухмерный случай оператора Шрёдингера с полиномиальным потенциалом. С помощью FD-метода и системы компьютерной алгебры Maple найдены четыре точные собственные значения для потенциала конкретного вида из шести наименьших.The essentially two-dimensional case of the Schrödinger operator with polynomial potential is considered for the first time. Using the FD-method and the Maple computer algebra system, we found four of six lowest eigenvalues for a fixed form of the potential

Calculation of the temperature of boundary layer beside wall with time-dependent heat transfer coefficient

This paper proposes to investigate the changes in the temperature of external wall boundary layers of buildings when the heat transfer coefficient reaches its stationary state in time exponentially. We seek the solution to the one-dimensional parabolic partial differential equation describing the heat transfer process under special boundary conditions. The search for the solution originates from the solution of a Volterra integral equation of the second kind. The kernel of the Volterra integral equation is slightly singular therefore its solution is calculated numerically by one of the most efficient collocation methods. Using the Euler approach an iterative calculation algorithm is obtained, to be implemented through a programme written in the Maple computer algebra system. Changes in the temperature of the external boundary of brick walls and walls insulated with polystyrene foam are calculated. The conclusion is reached that the external temperature of the insulated wall matches the air temperature sooner than that of the brick wall

Improving the Learning and Teaching of Mathematical Logic Elements Using Maple

The study of the elements of mathematical logic and its sections, especially in the process of teaching in high school in general and specialized courses of mathematics or higher educational institutions, is a rather tricky task. The paper demonstrates how one can design and develop a package of interactive applications to help learn, understand, and apply the essential elements of mathematical logic. The research aims to develop and describe an interactive application for studying the order of constructing truth tables and studying and demonstrating the production of principal normal disjunctive and conjunctive forms. The development is based on the methods of the Maple computer algebra system of the Maplet library. In addition, other methods are used to solve the research issues – theoretical and accurate analytical solutions to mathematical logic issues. All constructed material (Maplet) will serve as a good simulator for students, allowing one to perform an infinite number of attempts to analyze, synthesize, and build truth tables

Robust Computer Algebra, Theorem Proving, and Oracle AI

In the context of superintelligent AI systems, the term "oracle" has two meanings. One refers to modular systems queried for domain-specific tasks. Another usage, referring to a class of systems which may be useful for addressing the value alignment and AI control problems, is a superintelligent AI system that only answers questions. The aim of this manuscript is to survey contemporary research problems related to oracles which align with long-term research goals of AI safety. We examine existing question answering systems and argue that their high degree of architectural heterogeneity makes them poor candidates for rigorous analysis as oracles. On the other hand, we identify computer algebra systems (CASs) as being primitive examples of domain-specific oracles for mathematics and argue that efforts to integrate computer algebra systems with theorem provers, systems which have largely been developed independent of one another, provide a concrete set of problems related to the notion of provable safety that has emerged in the AI safety community. We review approaches to interfacing CASs with theorem provers, describe well-defined architectural deficiencies that have been identified with CASs, and suggest possible lines of research and practical software projects for scientists interested in AI safety.Comment: 15 pages, 3 figure