This article handles the problem of validating the results of nested (as opposed to ”flat”) clusterings. It shows that standard external validation indices used for partitioning clustering validation, like Rand statistics, Hubert Γ statistic or F-measure are not applicable in nested clustering cases. Additionally to the work, where F-measure was adopted to hierarchical classification as hF-meas...

Word extraction is one of the important tasks in text information processing. There are mainly two kinds of statisticbased measures for word extraction: the internal measure and the contextual measure. This paper discusses these two kinds of measures for Chinese word extraction. First, nine widely adopted internal measures are tested and compared on individual basis. Then various schemes of com...

We show how multifractal properties of a measure supported by a fractal F ⊆ [0, 1] may be expressed in terms of complementary intervals of F and thus in terms of spectral triples and the Dixmier trace of certain operators. For self-similar measures this leads to a non-commutative integral over F equivalent to integration with respect to an auxiliary multifractal measure.

A function f : Ω→ R is called F -measurable if f−1(B) := {f ∈ B} := {ω ∈ Ω : f(ω) ∈ B} ∈ F for all Borel sets B ⊆ R. Denote by F 0 the collection of sets in F with finite measure, i.e., F 0 := {E ∈ F : μ(E) <∞}. The measure space (Ω,F , μ) is called σ-finite if there exist sets Ei ∈ F 0 such that ⋃∞ i=0Ei = Ω. If needed, these sets may be chosen to additionally satisfy either (a) Ei ⊆ Ei+1 or (...

The uniqueness theorem for the ergodic maximal operator is proved in the continuous case. Let (X, S, μ) be a finite measure space, μ(X) < ∞, (1) and let (Tt)t≥0 be an ergodic semigroup of measure-preserving transformations of (X, S, μ). As usual the map (x, t) → Ttx is assumed to be jointly measurable. For an integrable function f , f ∈ L(X), the ergodic maximal function f∗ is defined by equati...

Let M be a space of weakly*-measurable functions F : ? B(H) on measure (?,?,?), for which the function F*F is Gel'fand integrable and integral d? nuclear operator Hilbert H. We show that H*-module contains an orthonormal basis.

We apply percolation theory to a recently proposed measure of fragmentation F for social networks. The measure F is defined as the ratio between the number of pairs of nodes that are not connected in the fragmented network after removing a fraction q of nodes and the total number of pairs in the original fully connected network. We compare F with the traditional measure used in percolation theo...

Consider a continuous function f on the real line with scalar values. It is said to have bounded support if there is a bounded interval [a, b] of real numbers such that f(x) 6= 0 implies a ≤ x ≤ b. An alternative terminology that is often used is that f is said to have compact support. Let Cc denote the vector space of continuous scalar valued functions on the real line, each of which has compa...

Remark 1.2. Uniform exponential growth for solvable non-virtually nilpotent groups was proved independently by Osin and byWilson. The existence of a uniform constant C(d) as in Theorem 1.1 for solvable groups is very striking since it implies the well known Lehmer conjecture concerning the Mahler measure of algebraic integers. However for virtually solvable non-virtually nilpotent groups we onl...