Decompositions of spaces of measures

Let M be the Banach space of sigma-additive complex-valued measures on an abstract measurable space. We prove that any closed, with respect to absolute continuity norm-closed, linear subspace L of M is complemented and describe the unique complement, projection onto L along which has norm 1. Using this fact we prove a decomposition theorem, which includes the Jordan decomposition theorem, the generalized Radon-Nikodym theorem and the decomposition of measures into decaying and non-decaying components as particular cases. We also prove an analog of the Jessen-Wintner purity theorem for our decompositions.

Decay measures on locally compact abelian topological groups

Source: PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH SECTION A-MATHEMATICS Volume: 131 Pages: 1257-1273 Part: Part 6 Published: 2001 Times Cited: 5 References: 23 Citation MapCitation Map beta Abstract: We show that the Banach space M of regular sigma-additive finite Borel complex-valued measures on a non-discrete locally compact Hausdorff topological Abelian group is the direct sum of two linear closed subspaces M-D and M-ND, where M-D is the set of measures mu is an element of M whose Fourier transform vanishes at infinity and M-ND is the set of measures mu is an element of M such that nu is not an element of MD for any nu is an element of M \ {0} absolutely continuous with respect to the variation \mu\. For any corresponding decomposition mu = mu(D) + mu(ND) (mu(D) is an element of M-D and mu(ND) is an element of M-ND) there exist a Borel set A = A(mu) such that mu(D) is the restriction of mu to A, therefore the measures mu(D) and mu(ND) are singular with respect to each other. The measures mu(D) and mu(ND) are real if mu is real and positive if mu is positive. In the case of singular continuous measures we have a refinement of Jordan's decomposition theorem. We provide series of examples of different behaviour of convolutions of measures from M-D and M-ND.

Responses of calves to acute stress: Individual consistency and relations between behavioral and physiological measures

The present study examined the consistency over time of individual differences in behavioral and physiological responsiveness of calves to intuitively alarming test situations as well as the relationships between behavioral and physiological measures. Twenty Holstein Friesian heifer calves were individually subjected to the same series of two behavioral and two hypothalamo-pituitary-adrenocortical (HPA) axis reactivity tests at 3, 13 and 26 weeks of age. Novel environment (open field, OF) and novel object (NO) tests involved measurement of behavioral, plasma cortisol and heart rate responses. Plasma ACTH and/or cortisol response profiles were determined after administration of exogenous CRH and ACTH, respectively, in the HPA axis reactivity tests. Principal component analysis (PCA) was used to condense correlated measures within ages into principal components reflecting independent dimensions underlying the calves' reactivity. Cortisol responses to the OF and NO tests were positively associated with the latency to contact and negatively related to the time spent in contact with the NO. Individual differences in scores of a principal component summarizing this pattern of inter-correlations, as well as differences in separate measures of adrenocortical and behavioral reactivity in the OF and NO tests proved highly consistent over time. The cardiac response to confinement in a start box prior to the OF test was positively associated with the cortisol responses to the OF and NO tests at 26 weeks of age. HPA axis reactivity to ACTH or CRH was unrelated to adrenocortical and behavioral responses to novelty. These findings strongly suggest that the responsiveness of calves was mediated by stable individual characteristics. Correlated adrenocortical and behavioral responses to novelty may reflect underlying fearfulness, defining the individual's susceptibility to the elicitation of fear. Other independent characteristics mediating reactivity may include activity or coping style (related to locomotion) and underlying sociality (associated with vocalization). (c) 2005 Elsevier Inc. All rights reserved.

Local network-based measures to assess the inferability of different regulatory networks

The purpose of this study is to compare the inferability of various synthetic as well as real biological regulatory networks. In order to assess differences we apply local network-based measures. That means, instead of applying global measures, we investigate and assess an inference algorithm locally, on the level of individual edges and subnetworks. We demonstrate the behaviour of our local network-based measures with respect to different regulatory networks by conducting large-scale simulations. As inference algorithm we use exemplarily ARACNE. The results from our exploratory analysis allow us not only to gain new insights into the strength and weakness of an inference algorithm with respect to characteristics of different regulatory networks, but also to obtain information that could be used to design novel problem-specific statistical estimators.

Verteporfin Photodynamic Therapy Cohort Study Report 2: Clinical Measures of Vision and Health-Related Quality of Life

Purpose: To quantify decreases in health-related quality of life (HRQoL) for given deterioration in clinical measures of vision; to describe the shape of these relationships; and to test whether the gradients of these relationships change with duration of visual loss.

Information theoretic measures of UHG graphs with low computational complexity

We introduce a novel graph class we call universal hierarchical graphs (UHG) whose topology can be found numerously in problems representing, e.g., temporal, spacial or general process structures of systems. For this graph class we show, that we can naturally assign two probability distributions, for nodes and for edges, which lead us directly to the definition of the entropy and joint entropy and, hence, mutual information establishing an information theory for this graph class. Furthermore, we provide some results under which conditions these constraint probability distributions maximize the corresponding entropy. Also, we demonstrate that these entropic measures can be computed efficiently which is a prerequisite for every large scale practical application and show some numerical examples. (c) 2007 Elsevier Inc. All rights reserved.