Related papers: The epsilon-delta game and the chess
The usual $\epsilon,\delta$-definition of the limit of a function (whether presented at a rigorous or an intuitive level) requires a "candidate $L$" for the limit value. Thus, we have to start our first calculus course with "guessing"…
We present a new way of organizing the few mathematical statements which form introduction to Calculus: the epsilon-delta characterization of the limit is now d e r i v e d from four simple, intuitive and frequently used statements, which…
Let $f$ be a continuous real function defined in a subset of the real line. The standard definition of continuity at a point $x$ allow us to correlate any given epsilon with a (possibly depending of $x$) delta value. This pairing is known…
We provide a simple reformulation of the $\epsilon$-$\delta$ limit definition introduced in undergraduate calculus courses that enhances its pedagogical value for conceptual understanding and computational skill.
Under certain general conditions, an explicit formula to compute the greatest delta-epsilon function of a continuous function is given. From this formula, a new way to analyze the uniform continuity of a continuous function is given.…
In this paper, we consider the concept of limit, one of the basic concepts of mathematical analysis. At a point $a\in{\mathbb{R}}$, the limit of a function $f$ from $A\subset\mathbb{R}$ to $\mathbb{R}$ is $L\in{\mathbb{R}}$ if and only if…
A weighted game or a threshold function in general admits different weighted representations even if the sum of non-negative weights is fixed to one. Here we study bounds for the diameter of the corresponding weight polytope. It turns out…
We give an example of two $n\times n$ chess positions, $A$ and $B$, such that (1) there is a sequence $\sigma$ of legal chess moves leading from $A$ to $B$; (2) the length of $\sigma$ cannot be less than $\exp \Theta(n)$.
We introduce a game related to the $I_{3322}$ game and analyze a constrained value function for this game over various families of synchronous quantum probability densities.
This paper has a twofold scope. The first one is to clarify and put in evidence the isomorphic character of two theories developed in quite different fields: on one side, threshold logic, on the other side, simple games. One of the main…
We introduce $(\varepsilon, \delta)$-bisimulation, a novel type of approximate probabilistic bisimulation for continuous-time Markov chains. In contrast to related notions, $(\varepsilon, \delta)$-bisimulation allows the use of different…
The deduction game is a variation of the game of cops and robber on graphs in which searchers must capture an invisible evader in at most one move. Searchers know each others' initial locations, but can only communicate if they are on the…
We introduce a refinement of the usual Ehrenfeucht-Fra\"{\i}ss\'e game. The new game will help us make finer distinctions than the traditional one. In particular, it can be used to measure the size formulas needed for expressing a given…
Sequential equilibrium is one of the most fundamental refinements of Nash equilibrium for games in extensive form. However, it is not defined for extensive-form games in which a player can choose among a continuum of actions. We define a…
Quantum phenomena have remained largely inaccessible to the general public. This can be attributed to the fact that we do not experience quantum mechanics on a tangible level in our daily lives. Games can provide an environment in which…
The main purpose of the paper is the proof of a cardinal inequality for a space with points $G_\delta$, obtained with the help of a long version of the Menger game. This result improves a similar one of Scheepers and Tall.
Limit can be defined by two axioms: 1. Strict inequality between limits implies, ultimately, strict inequality between functions. 2. For constant functions limit is trivial. How can basic results on convergence be derived from these axioms?…
We introduce a two-player game, in which each player extends a given sequence by picking a free element in a domain D of the real line. The aim of the players is to control the parity of the number of transpositions necessary to put the…
A famous result in game theory known as Zermelo's theorem says that "in chess either White can force a win, or Black can force a win, or both sides can force at least a draw". The present paper extends this result to the class of all…
Nonlinear PDEs, mean value properties, and stochastic differential games are intrinsically connected. In this short expository note, we will describe how the solutions to certain PDEs (of $p$-Laplacian type) can be interpreted as limits of…