With any hypothesis class one can associate a bipartite graph whose vertices are the hypotheses H on one side and all possible labeled examples X on the other side, and an hypothesis is connected to all the labeled examples that are consistent with it. We call this graph the hypotheses graph. We prove that any hypothesis class whose hypotheses graph is mixing cannot be learned using less than 2...

Abstract: The purpose of this paper is to discuss the properties of regularity and strong regularity of fuzzy measure on metric spaces following the previous results. Some properties are defined with the help of nulladditivity such as inner\outer regularity and the regularity of fuzzy measure. We define the strong regularity of fuzzy measures and show our result that the null-additive fuzzy mea...

The problem of finding heavy hitters and approximating the frequencies of items is at the heart of many problems in data stream analysis. It has been observed that several proposed solutions to this problem can outperform their worst-case guarantees on real data. This leads to the question of whether some stronger bounds can be guaranteed. We answer this in the positive by showing that a class ...

We investigate the strength of set existence axioms needed for separable Banach space theory. We show that a very strong axiom, Π1 comprehension, is needed to prove such basic facts as the existence of the weak-∗ closure of any norm-closed subspace of `1 = c0. This is in contrast to earlier work [6, 4, 7, 23, 22] in which theorems of separable Banach space theory were proved in very weak subsys...

In this paper, we introduce the SP -integral and the SPα-integral defined on an interval in the n-dimensional Euclidean space Rn. We also investigate the relationship between these two integrals.

Let X be a finite dimensional Banach space and let Y ⊂ X be a hyperplane. Let L Y = {L ∈ L (X, Y ) : L |Y = 0}. In this note, we present sufficient and necessary conditions on L0 ∈ L Y being a strongly unique best approximation for given L ∈ L (X). Next we apply this characterization to the case of X = l ∞ and to generalization of Theorem I.1.3 from [12] (see also [13]).

In this paper we formulate in general terms an approach to prove strong consistency of the Empirical Risk Minimisation inductive principle applied to the prototype or distance based clustering. This approach was motivated by the Divisive Information-Theoretic Feature Clustering model in probabilistic space with Kullback-Leibler divergence, which may be regarded as a special case within the Clus...

In this paper we formulate in general terms an approach to prove strong consistency of the Empirical Risk Minimisation inductive principle applied to the prototype or distance based clustering. This approach was motivated by the Divisive Information-Theoretic Feature Clustering model in probabilistic space with Kullback-Leibler divergence which may be regarded as a special case within the Clust...

We identify the phase-space structures that regulate atomic double ionization in strong ultrashort laser pulses. The emerging dynamical picture complements the recollision scenario by clarifying the distinct roles played by the recolliding and core electrons, and leads to verifiable predictions on the characteristic features of the "knee", a hallmark of the nonsequential process.

We investigate the occurrence and nature of naked singularities in the Szekeres space-times. These space-times represent irrotational dust. They do not have any Killing vectors and they are generalisations of the Tolman-Bondi-Lemâıtre space-times. It is shown that in these space-times there exist naked singularities that satisfy both the limiting focusing condition and the strong limiting focus...