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4 results

Invitation to Linear Programming and Game Theory
  • David C. Vella
  • Published online:
    11 March 2021
    Print publication:
    11 March 2021
  • Written in a conversational tone, this classroom-tested text introduces the fundamentals of linear programming and game theory, showing readers how to apply serious mathematics to practical real-life questions by modelling linear optimization problems and strategic games. The treatment of linear programming includes two distinct graphical methods. The game theory chapters include a novel proof of the minimax theorem for 2x2 zero-sum games. In addition to zero-sum games, the text presents variable-sum games, ordinal games, and n-player games as the natural result of relaxing or modifying the assumptions of zero-sum games. All concepts and techniques are derived from motivating examples, building in complexity, which encourages students to think creatively and leads them to understand how the mathematics is applied. With no prerequisite besides high school algebra, the text will be useful to motivated high school students and undergraduates studying business, economics, mathematics, and the social sciences.

  • 1 - Preliminaries

  • David C. Vella, Skidmore College, New York
  • Book:
    Invitation to Linear Programming and Game Theory
    Published online:
    11 March 2021
    Print publication:
    11 March 2021, pp 1-62

  • Summary

    We introduce the mathematical modeling process. We also set the stage for the rest of the book by discussing systems of linear equations and their solutions, matrices, Gauss-Jordan elimination, linear combinations of vectors, basis vectors of Euclidean space, and their connection to basic solutions of a linear system. We conclude with simple optimization problems with quadratic functions.

  • Optimal lower bounds for multiple recurrence

  • Part of
  • SEBASTIÁN DONOSO, ANH NGOC LE, JOEL MOREIRA, WENBO SUN
  • Journal:
    Ergodic Theory and Dynamical Systems / Volume 41 / Issue 2 / February 2021
    Published online by Cambridge University Press:
    07 October 2019, pp. 379-407
    Print publication:
    February 2021

  • Let $(X,{\mathcal{B}},\unicode[STIX]{x1D707},T)$ be an ergodic measure-preserving system, let $A\in {\mathcal{B}}$ and let $\unicode[STIX]{x1D716}>0$. We study the largeness of sets of the form

    $$\begin{eqnarray}S=\{n\in \mathbb{N}:\unicode[STIX]{x1D707}(A\cap T^{-f_{1}(n)}A\cap T^{-f_{2}(n)}A\cap \cdots \cap T^{-f_{k}(n)}A)>\unicode[STIX]{x1D707}(A)^{k+1}-\unicode[STIX]{x1D716}\}\end{eqnarray}$$
    for various families $\{f_{1},\ldots ,f_{k}\}$ of sequences $f_{i}:\mathbb{N}\rightarrow \mathbb{N}$. For $k\leq 3$ and $f_{i}(n)=if(n)$, we show that $S$ has positive density if $f(n)=q(p_{n})$, where $q\in \mathbb{Z}[x]$ satisfies $q(1)$ or $q(-1)=0$ and $p_{n}$ denotes the $n$th prime; or when $f$ is a certain Hardy field sequence. If $T^{q}$ is ergodic for some $q\in \mathbb{N}$, then, for all $r\in \mathbb{Z}$, $S$ is syndetic if $f(n)=qn+r$. For $f_{i}(n)=a_{i}n$, where $a_{i}$ are distinct integers, we show that $S$ can be empty for $k\geq 4$, and, for $k=3$, we found an interesting relation between the largeness of $S$ and the abundance of solutions to certain linear equations in sparse sets of integers. We also provide some partial results when the $f_{i}$ are distinct polynomials.

  • A removal lemma for linear configurations in subsets of the circle

  • Part of
  • Pablo Candela, Olof Sisask
  • Journal:
    Proceedings of the Edinburgh Mathematical Society / Volume 56 / Issue 3 / October 2013
    Published online by Cambridge University Press:
    28 June 2013, pp. 657-666
    • Article
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  • We obtain a removal lemma for systems of linear equations over the circle group, using a similar result for finite fields due to Král′, Serra and Vena, and we discuss some applications.