Given a probability space (Ω,A, P ) and some σ-fields F ,G ⊂ A, we consider iterated conditional distributions of the form (P [ · | F ])[ · | G]. Thus, in the second step, we form the conditional distribution with respect to G, using P [ · |F ] instead of P as the underlying probability measure. Under suitable regularity conditions, we show that conditioning with respect to F and G commute. The...

In this paper, we provide inequalities of Jensen-Ostrowski type, by investigating the magnitude of the quantity ∫ Ω (f ◦ g) dμ− f(ζ)− ∫ Ω (g − ζ)f ′ ◦ g dμ+ 1 2 λ ∫ Ω (g − ζ) dμ, for various assumptions on the absolutely continuous function f : [a, b] → C, ζ ∈ [a, b], λ ∈ C and a μ-measurable function g on Ω. Special cases are considered to provide some inequalities of Jensen type, as well as O...

Let (Ω,Σ, μ) be a measure space and let 1 < p < ∞. The weak L space L(Ω,Σ, μ) consists of all measurable functions f such that ‖f‖ = sup t>0 t 1 p f∗(t) < ∞, where f∗ is the decreasing rearrangement of |f |. It is a Banach space under a norm which is equivalent to the expression above. In this paper, we pursue the problem of classifying weak L spaces isomorphically when (Ω,Σ, μ) is purely atomi...

We investigate the problem of estimating a smooth invertible transformation f when observing independent samples X1, . . . , Xn ∼ P ◦ f where P is a known measure. We focus on the two dimensional case where P and f are defined on R. We present a flexible class of smooth invertible transformations in two dimensions with variational equations for optimizing over the classes, then study the proble...

Let F be a nonarchimedean local field, let GL(n) = GL(n, F ) and let ν denote Plancherel measure for GL(n). Let Ω be a component in the Bernstein variety Ω(GL(n)). Then Ω yields its fundamental invariants: the cardinality q of the residue field of F , the sizes m1, . . . ,mt, exponents e1, . . . , et, torsion numbers r1, . . . , rt, formal degrees d1, . . . , dt and the Artin conductors f11, . ...

In this paper n, k are natural numbers, X is a non empty set, and S is a σ-field of subsets of X. Next we state several propositions: (1) Let M be a σ-measure on S, F be a function from N into S, and given n. Then {x ∈ X: ∧ k (n ≤ k ⇒ x ∈ F (k))} is an element of S. (2) Let F be a sequence of subsets of X and n be an element of N. Then (the superior set sequence of F )(n) = ⋃ rng(F ↑ n) and (th...

We present a theoretical analysis of F -measures for binary, multiclass and multilabel classification. These performance measures are non-linear, but in many scenarios they are pseudo-linear functions of the per-class false negative/false positive rate. Based on this observation, we present a general reduction of F measure maximization to cost-sensitive classification with unknown costs. We the...

We study the ergodic properties of the additive Euclidean algorithm f defined in R+. A natural extension of f is obtained using the action of SL(2,Z) on a subset of SL(2,R). We prove that even though f is ergodic and has an infinite invariant measure equivalent to the Lebesgue measure, such a measure is not unique; (in fact there is a continuous family of such measures). While it is folklore th...

Given two points x; y 2 S 1 randomly chosen independently by a mixing absolutely continuous invariant measure of a piecewise expanding and smooth map f of the circle, we consider for each " > 0 the point process obtained by recording the times n > 0 such that jf n (x) ? f n (y)j ". With the further assumption that the density of is bounded away from zero, we show that when " tends to zero the a...

We propose a new approach to cluster the web pages. Utilizing an iterative reinforced algorithm, the model extracts semantic feature vectors from user click-through data. We then use LSA (Latent Semantic Analysis) to reduce the feature dimension and K-means algorithm to cluster documents. Compared to the traditional way of feature extraction (lexical binomial model), our new model has better pu...