Infinite games on finite graphs using grossone

2020 ◽  
Vol 24 (23) ◽  
pp. 17509-17515 ◽  
Author(s):  
Louis D’Alotto
Keyword(s):  
1993 ◽  
Vol 65 (2) ◽  
pp. 149-184 ◽  
Author(s):  
Robert McNaughton
Keyword(s):  

Author(s):  
J.M BUDD ◽  
Y. VAN GENNIP

An emerging technique in image segmentation, semi-supervised learning and general classification problems concerns the use of phase-separating flows defined on finite graphs. This technique was pioneered in Bertozzi and Flenner (2012, Multiscale Modeling and Simulation10(3), 1090–1118), which used the Allen–Cahn flow on a graph, and was then extended in Merkurjev et al. (2013, SIAM J. Imaging Sci.6(4), 1903–1930) using instead the Merriman–Bence–Osher (MBO) scheme on a graph. In previous work by the authors, Budd and Van Gennip (2020, SIAM J. Math. Anal.52(5), 4101–4139), we gave a theoretical justification for this use of the MBO scheme in place of Allen–Cahn flow, showing that the MBO scheme is a special case of a ‘semi-discrete’ numerical scheme for Allen–Cahn flow. In this paper, we extend this earlier work, showing that this link via the semi-discrete scheme is robust to passing to the mass-conserving case. Inspired by Rubinstein and Sternberg (1992, IMA J. Appl. Math.48, 249–264), we define a mass-conserving Allen–Cahn equation on a graph. Then, with the help of the tools of convex optimisation, we show that our earlier machinery can be applied to derive the mass-conserving MBO scheme on a graph as a special case of a semi-discrete scheme for mass-conserving Allen–Cahn. We give a theoretical analysis of this flow and scheme, proving various desired properties like existence and uniqueness of the flow and convergence of the scheme, and also show that the semi-discrete scheme yields a choice function for solutions to the mass-conserving MBO scheme.


1985 ◽  
Vol 50 (1) ◽  
pp. 110-122
Author(s):  
Howard Becker

For any A ⊂ R, the Banach game B(A) is the following infinite game on reals: Players I and II alternately play positive real numbers a1; a2, a3, a4,… such that for n > 1, an < an−1. Player I wins iff ai exists and is in A.This type of game was introduced by Banach in 1935 in the Scottish Book [15], Problem 43. The (rather vague) problem which Banach posed was to characterize those sets A for which I (II) has a winning strategy in B(A). (There are three parts to Problem 43. In the first, Mazur defined a game G**(A) for every set A ⊂ R and conjectured that II has a winning strategy in G**(A) iff A is meager and I has a winning strategy in G**(A) iff A is comeager in some neighborhood; this conjecture was proved by Banach. Presumably Banach had this result in mind when he asked the question about B(A), and hoped for a similar type of characterization.) Incidentally, Problem 43 of the Scottish Book appears to be the first time infinite games of any sort were studied by mathematicians.This paper will not provide the reader with any answer to Banach's question. I know of no nontrivial way to characterize when player I (or II) wins, and I suspect there is none. This paper is concerned with a different (also rather vague) question: For which sets A is the Banach game B(A) determined? To say that B(A) is determined means, of course, that one of the players has a winning strategy for B(A).


2016 ◽  
Vol 32 (6) ◽  
pp. 2575-2589
Author(s):  
Seongmin Ok ◽  
R. Bruce Richter ◽  
Carsten Thomassen

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