Wyniki wyszukiwania

Sortuj według:

Ogranicz wyniki do:

Universality of the μ-predictor

100%

Hardin C.

Fundamenta Mathematicae

2013

tom 220

nr 3

227-241

For suitable topological spaces X and Y, given a continuous function f:X → Y and a point x ∈ X, one can determine the value of f(x) from the values of f on a deleted neighborhood of x by taking the limit of f. If f is not required to be continuous, it is impossible to determine f(x) from this information (provided |Y| ≥ 2), but as the author and Alan Taylor showed in 2009, there is nevertheless a means of guessing f(x), called the μ-predictor, that will be correct except on a small set; specifically, if X is T₀, then the guesses will be correct except on a scattered set. In this paper, we show that, when X is T₀, every predictor that performs this well is a special case of the μ-predictor.

Minimal predictors in hat problems

63%

Hardin C., Taylor A.

Fundamenta Mathematicae

2010

tom 208

nr 3

273-285

We consider a combinatorial problem related to guessing the values of a function at various points based on its values at certain other points, often presented by way of a hat-problem metaphor: there are a number of players who will have colored hats placed on their heads, and they wish to guess the colors of their own hats. A visibility relation specifies who can see which hats. This paper focuses on the existence of minimal predictors: strategies guaranteeing at least one player guesses correctly, regardless of how the hats are colored. We first present some general results, in particular showing that transitive visibility relations admit a minimal predictor exactly when they contain an infinite chain, regardless of the number of colors. In the more interesting nontransitive case, we focus on a particular nontransitive relation on ω that is elementary, yet reveals unexpected phenomena not seen in the transitive case. For this relation, minimal predictors always exist for two colors but never for ℵ₂ colors. For ℵ₀ colors, the existence of minimal predictors is independent of ZFC plus a fixed value of the continuum, and turns out to be closely related to certain cardinal invariants involving meager sets of reals.

Ograniczanie wyników

2 Fundamenta Mathematicae

2 Hardin C.

1 Taylor A.

1 2013

1 2010

Wyniki wyszukiwania

Universality of the μ-predictor

Minimal predictors in hat problems