Abstract Define the k -th Radon number $$r_k$$ r k of a convexity space as smallest (if it exists) for which any set points can be partitioned into parts whose convex hulls intersect. Combining recent abstract fractional Helly theorem Holmsen and Lee with earlier method...

A non-erasing morphism σ is said to be recognizable on x if there exists ` such that, for each m ∈ Cσ(x), m′ ∈ Z, y[m−`,m+`) = y[m′−`,m′+`) implies that m′ ∈ Cσ(x), with y = σ(x). Is the Thue-Morse substitution σ : a 7→ ab, b 7→ ba recognizable on σ∞(a)? Same question for the Fibonacci substitution τ : a 7→ ab, b 7→ a. Let σ be a primitive morphism that is recognizable on some x and injective o...

The maximum absolute value of integral weights sufficient to represent any linearly separable Boolean function is investigated. It is shown that upper bounds exhibited by Muroga (1971) for rational weights satisfying the normalized System of inequalities also hold for integral weights. Therewith, the previous best known upper bound for integers is improved by approximately a factor of 1/2.

There are not many new results concerning the linear independence of numbers. Exceptions in the last decade are, e.g., the result of Sorokin [8] which proves the linear independence of logarithmus of special rational numbers, or that of Bezivin [2] which proves linear independence of roots of special functional equations. The algebraic independence of numbers can be considered as a generalizati...

Let T be a linear operator defined on a complex vector space X and let n be a positive integer. Kaplansky [4] proved that T is algebraic of degree at most n if and only if for every x ∈ X the vectors x, Tx, . . . , Tnx are linearly dependent. One consequence of Kaplansky’s result is that if X is a Banach space and T : X → X a bounded linear operator, then T is algebraic if and only if for every...

For a finite alphabet A, we define a class of embeddings of A∗ into an infinite-dimensional feature space X and show that its finitely supported hyperplanes define regular languages. This suggests a general strategy for learning regular languages from positive and negative examples. We apply this strategy to the piecewise testable languages, presenting an embedding under which these are precise...