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A note on numbers

arXiv 2021年

作者： Carvalho, Alda Huggan, Melissa A. Nowakowski, Richard J. dos Santos, Carlos Pereira ISEL–IPL CEMAPRE University of Lisbon Portugal Ryerson University Canada Dalhousie University Canada Center for Functional Analysis Linear Structures and Applications University of Lisbon ISEL–IPL Portugal

When are all positions of a game numbers? We show that two properties are necessary and sufficient. These properties are consequences of that, in a number, it is not an advantage to be the first player. One of these properties implies the other. However, checking for one or the other, rather than just one, can often be accomplished by only looking at the positions on the ‘board’. If the stronger property holds for all positions, then the values are *** Codes 91A46 Copyright © 2021, The Authors. All rights reserved.

关键词： Game theory

An action of the cactus group on shifted tableau crystals

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arXiv 2020年

作者： Rodrigues, Inês Center for Functional Analysis Linear Structures and Applications Faculty of Sciences University of Lisbon Portugal

Recently, Gillespie, Levinson and Purbhoo introduced a crystal-like structure for shifted tableaux, called the shifted tableau crystal. We introduce a shifted analogue of the crystal reflection operators, which coincides with the restriction of the shifted Schützenberger involution to any primed interval of two adjacent letters. Unlike type A Young tableau crystals, these operators do not realize an action of the symmetric group on the shifted tableau crystal because braid relations do not hold. We exhibit a natural internal action of the cactus group, realized by restrictions of the shifted Schützenberger involution on primed intervals of the underlying crystal alphabet. Copyright © 2020, The Authors. All rights reserved.

关键词： Crystals

Impartial games with entailing moves

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arXiv 2021年

作者： Larsson, Urban Nowakowski, Richard J. Santos, Carlos P. School of Computing National University of Singpore Singapore Department of Mathematics and Statistics Dalhousie University Canada Center for Functional Analysis Linear Structures and Applications University of Lisbon ISEL IPL Portugal

Combinatorial Game Theory has also been called 'additive game theory', whenever the analysis involves sums of independent game components. Such disjunctive sums invoke comparison between games, which allows abstract values to be assigned to them. However, there are rulesets with entailing moves that break the alternating play axiom and/or restrict the other player's options within the disjunctive sum components. These situations are exemplified in the literature by a ruleset such as nimstring, a normal play variation of the classical children's game dots & boxes, and top entails, an elegant ruleset introduced in the classical work Winning Ways, by Berlekamp Conway and Guy. Such rulesets fall outside the scope of the established normal play theory. Here, we axiomatize normal play via two new terminating games, 8 (Left wins) and 8 (Right wins), and a more general theory is achieved. We define affine impartial, which extends classical impartial games, and we analyze their algebra by extending the established Sprague-Grundy theory, with an accompanying minimum excluded rule. Solutions of nimstring and top entails are given to illustrate the *** Codes 91A46 © 2021, CC BY.

关键词： Game theory

Invertible Elements of the Misère Dicotic Universe

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arXiv 2020年

作者： Fisher, Michael Nowakowski, Richard dos Santos, Carlos Pereira West Chester University Dalhousie University Center for Functional Analysis Linear Structures and Applications University of Lisbon ISEL–IPL

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Ordinal sums, clockwise Hackenbush, and Domino shave

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arXiv 2020年

作者： Carvalho, Alda Huggan, Melissa A. Nowakowski, Richard J. dos Santos, Carlos Pereira ISEL–IPL & CEMAPRE University of Lisbon Portugal Ryerson University Canada Dalhousie University Canada Center for Functional Analysis Linear Structures and Applications University of Lisbon ISEL–IPL Portugal

We present two rulesets, domino shave and clockwise hackenbush. The first is somehow natural and, as special cases, includes stirling shave and Hetyei’s Bernoulli game. Clockwise hackenbush seems artificial yet it is equivalent to domino shave. From the pictorial form of the game, and a knowledge of hackenbush, the decomposition into ordinal sums is immediate. The values of clockwise blue-red hackenbush are numbers and we provide an explicit formula for the ordinal sum of numbers where the literal form of the base is {x|} or {|x}, and x is a number. That formula generalizes van Roode’s signed binary number method for blue-red *** Codes 91A46 Copyright © 2020, The Authors. All rights reserved.

关键词： Clocks

BOUNDING GAME TEMPERATURE USING CONFUSION INTERVALS

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arXiv 2019年

作者： Huntemann, Svenja Nowakowski, Richard J. Dos Santos, Carlos Pereira School of Mathematics and Statistics Carleton University Ottawa Canada Dept. of Mathematics and Statistics Dalhousie University Halifax Canada Center for Functional Analysis Linear Structures and Applications University of Lisbon ISEL–IPL Lisbon Portugal

For a combinatorial games, temperature is a measure of the volatility, that is, by how much the advantage can change. Typically, the temperature has been measured for individual positions within specific games. In this paper, we give the first general upper bounds on the temperature for any class of games. For a position G, the closure of the set of numbers {g} such that G − g is a first player win, is called the confusion interval of G. Let (G) be the length of this interval. Our first main result is: For a class of games S , if there are constants J and K such that (GL), (GR) ≤ J and (G) ≤ K for for every G ∈ S , then the temperature of every game is bounded by K/2 + J. We give an example to show that this bound is tight. Our second main result is a method to find a bound for the confusion intervals. In GL − G when Left gets to go first, the number of passing moves required by Right to win gives an upper bound on (G). This is the first general upper bound on temperature. As examples of the bound and the method, we give upper bounds on the temperature of subclasses of Domineering and *** Codes 91A46 Copyright © 2019, The Authors. All rights reserved.

关键词： Game theory

Using wolfram mathematica in spectral theory 3

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Using wolfram mathematica in spectral theory

3rd International Conference on Numerical and Symbolic Computation: Developments and applications, SYMCOMP 2017

作者： Conceicao, Ana C. Pereira, Jose C. Center for Functional Analysis Linear Structures and Applications Faculdade de Ciências e Tecnologia Universidade Do Algarve Faro8005-139 Portugal

ISBN: (纸本)9789899941038

The development of the spectral theory is motivated by the need to solve problems emerging from several fields in Mathematics and Physics. Some progress has been achieved for some classes of singular integral operators (SIOs) whose properties allow the use of particular strategies in the study of the spectral problem but there is still no general and explicit method for obtaining the spectrum of any given SIO. Furthermore, the existing algorithms allow, in general, to study the spectrum of some kind of SIOs but they are not designed to be implemented on a computer. The main goal of this work is to present our spectral algorithms [ASpec-Hankel], [ASpecPaired-Scalar], and [ASpecPaired-Matrix] which use the symbolic and numeric computation capabilities of the computer algebra system Mathematica to explore the spectra of some classes of SIOs, defined on the unit circle. These analytical algorithms allow us to check, for each considered SIO, if a complex number (chosen arbitrarily) belongs to its spectrum. © ECCOMAS, Portugal.

关键词： Computation theory

The geometer dog who did not know calculus

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College Mathematics Journal 2017年第5期48卷 339-345页

作者： Carvalho, Alda Dos Santos, Carlos Pereira Silva, Jorge Nuno High Institute of Engineering of Lisbon Center for Applied Mathematics and Economics University of Lisbon Portugal Center for Functional Analysis Linear Structures and Applications University of Lisbon Portugal Department of history and philosophy of science University of Lisbon Portugal

We present a purely geometric solution for Elvis's path problem. The geometric approach brings, for a constructible ratio of speeds, an easy compass-and-straightedge construction that visually explains some extrem... 详细信息

关键词： College mathematics Calculus Ratios Triangles Swimming Applied mathematics Dogs Running Bones

Progress on misère dead ends: Game comparison, canonical form, and conjugate inverses

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arXiv 2018年

作者： Larsson, Urban Milley, Rebecca Nowakowski, Richard Renault, Gabriel Santos, Carlos Industrial Engineering and Management Technion - Israel Institute of Technology Israel Computational Mathematics Grenfell Campus Memorial University Canada Dept of Mathematics and Statistics Dalhousie University Canada Center for Functional Analysis Linear Structures and Applications University of Lisbon Portugal

This paper addresses several significant gaps in the theory of restricted misère play (Plambeck, Seigel 2008), primarily in the well-studied universe of dead-ending games, ϵ (Milley, Renault 2013);if a player run out of moves in X ∈ ϵ, then they can never move again in any follower of X. A universe of games is a class of games which is closed under disjunctive sum, taking options and conjugates. We use novel results from absolute combinatorial game theory (Larsson, Nowakowski, Santos 2017) to show that ϵ and the universe D ⊂ ϵ of dicot games (either both, or none of the players can move) have 'options only' test for comparison of games, and this in turn is used to define unique reduced games (canonical forms) in ϵ. We develop the reductions for ϵ by extending analogous work for D, in particular by solving the problem of reversibility through ends in the larger universe. Finally, by using the defined canonical forms in ϵ and D, we prove that both of these universes, as well as the subuniverse of impartial games, have the conjugate property: every inverse game is obtained by swapping sides of the players. Copyright © 2018, The Authors. All rights reserved.

关键词： Game theory