Average distances on self-similar sets and higher order average distances of self-similar measures

The purpose of this paper is twofold: (1) we study different notions of the average distance between two points of a self-similar subset of ℝ, and (2) we investigate the asymptotic behaviour of higher order average moments of self-similar measures on self-similar subsets of ℝ

Curvature-direction measures of self-similar sets

We obtain fractal Lipschitz-Killing curvature-direction measures for a large class of self-similar sets F in R^d. Such measures jointly describe the distribution of normal vectors and localize curvature by analogues of the higher order mean curvatures of differentiable submanifolds. They decouple as independent products of the unit Hausdorff measure on F and a self-similar fibre measure on the sphere, which can be computed by an integral formula. The corresponding local density approach uses an ergodic dynamical system formed by extending the code space shift by a subgroup of the orthogonal group. We then give a remarkably simple proof for the resulting measure version under minimal assumptions.Comment: 17 pages, 2 figures. Update for author's name chang

A Survey on the Krein-von Neumann Extension, the corresponding Abstract Buckling Problem, and Weyl-Type Spectral Asymptotics for Perturbed Krein Laplacians in Nonsmooth Domains

In the first (and abstract) part of this survey we prove the unitary equivalence of the inverse of the Krein--von Neumann extension (on the orthogonal complement of its kernel) of a densely defined, closed, strictly positive operator, $S\geq \varepsilon I_{\mathcal{H}}$ for some $\varepsilon >0$ in a Hilbert space $\mathcal{H}$ to an abstract buckling problem operator. This establishes the Krein extension as a natural object in elasticity theory (in analogy to the Friedrichs extension, which found natural applications in quantum mechanics, elasticity, etc.). In the second, and principal part of this survey, we study spectral properties for $H_{K,\Omega}$, the Krein--von Neumann extension of the perturbed Laplacian $-\Delta+V$ (in short, the perturbed Krein Laplacian) defined on $C^\infty_0(\Omega)$, where $V$ is measurable, bounded and nonnegative, in a bounded open set $\Omega\subset\mathbb{R}^n$ belonging to a class of nonsmooth domains which contains all convex domains, along with all domains of class $C^{1,r}$, $r>1/2$.Comment: 68 pages. arXiv admin note: extreme text overlap with arXiv:0907.144

On Testing Dependence between Time to Failure and Cause of Failure when Causes of Failure Are Missing

The hypothesis of independence between the failure time and the cause of failure is studied by using the conditional probabilities of failure due to a specific cause given that there is no failure up to certain fixed time. In practice, there are situations when the failure times are available for all units but the causes of failures might be missing for some units. We propose tests based on U-statistics to test for independence of the failure time and the cause of failure in the competing risks model when all the causes of failure cannot be observed. The asymptotic distribution is normal in each case. Simulation studies look at power comparisons for the proposed tests for two families of distributions. The one-sided and the two-sided tests based on Kendall type statistic perform exceedingly well in detecting departures from independence

Multifractal tubes

Tube formulas refer to the study of volumes of $r$ neighbourhoods of sets. For sets satisfying some (possible very weak) convexity conditions, this has a long history. However, within the past 20 years Lapidus has initiated and pioneered a systematic study of tube formulas for fractal sets. Following this, it is natural to ask to what extend it is possible to develop a theory of multifractal tube formulas for multifractal measures. In this paper we propose and develop a framework for such a theory. Firstly, we define multifractal tube formulas and, more generally, multifractal tube measures for general multifractal measures. Secondly, we introduce and develop two approaches for analysing these concepts for self-similar multifractal measures, namely: (1) Multifractal tubes of self-similar measures and renewal theory. Using techniques from renewal theory we give a complete description of the asymptotic behaviour of the multifractal tube formulas and tube measures of self-similar measures satisfying the Open Set Condition. (2) Multifractal tubes of self-similar measures and zeta-functions. Unfortunately, renewal theory techniques do not yield "explicit" expressions for the functions describing the asymptotic behaviour of the multifractal tube formulas and tube measures of self-similar measures. This is clearly undesirable. For this reason, we introduce and develop a second framework for studying multifractal tube formulas of self-similar measures. This approach is based on multifractal zeta-functions and allow us obtain "explicit" expressions for the multifractal tube formulas of self-similar measures, namely, using the Mellin transform and the residue theorem, we are able to express the multifractal tube formulas as sums involving the residues of the zeta-function.Comment: 122 page

Gene products of chromosome 11q and their association with CCND1 gene amplification and tamoxifen resistance in premenopausal breast cancer

Introduction: The amplification event occurring at chromosome locus 11q13, reported in several different cancers, includes a number of potential oncogenes. We have previously reported amplification of one such oncogene, namely CCND1, to be correlated with an adverse effect of tamoxifen in premenopausal breast cancer patients. Over-expression of cyclin D-1 protein, however, confers tamoxifen resistance but not a tamoxifen-induced adverse effect. Potentially, co-amplification of an additional 11q13 gene, with a resulting protein over-expression, is required to cause an agonistic effect. Moreover, during 11q13 amplification a deletion of the distal 11q region has been described. In order to assess the potential impact of the deletion we examined a selected marker for this event. Method: Array comparative genomic hybridization analysis was employed to identify and confirm changes in the gene expression of a number of different genes mapping to the 11q chromosomal region, associated with CCND1 amplification. The subsequent protein expression of these candidate genes was then examined in a clinical material of 500 primary breast cancers from premenopausal patients who were randomly assigned to either tamoxifen or no adjuvant treatment. The protein expression was also compared with gene expression data in a subset of 56 breast cancer samples. Results: Cortactin and FADD (Fas-associated death domain) over-expression was linked to CCND1 amplification, determined by fluorescence in situ hybridization, but was not associated with a diminished effect of tamoxifen. However, deletion of distal chromosome 11q, defined as downregulation of the marker Chk1 (checkpoint kinase 1), was associated with an impaired tamoxifen response, and interestingly with low proliferative breast cancer of low grade. For Pak1 (p21-activated kinase 1) and cyclin D-1 the protein expression corresponded to the gene expression data. Conclusions: The results indicate that many 11q13 associated gene products are over-expressed in conjunction with cyclin D-1 but not linked to an agonistic effect of tamoxifen. Finally, the deletion of distal 11q, linked to 11q13 amplification, might be an important event affecting breast cancer outcome and tamoxifen response

Trigger finger: etiology, evaluation, and treatment

Trigger finger is a common finger aliment, thought to be caused by inflammation and subsequent narrowing of the A1 pulley, which causes pain, clicking, catching, and loss of motion of the affected finger. Although it can occur in anyone, it is seen more frequently in the diabetic population and in women, typically in the fifth to sixth decade of life. The diagnosis is usually fairly straightforward, as most patients complain of clicking or locking of the finger, but other pathological processes such as fracture, tumor, or other traumatic soft tissue injuries must be excluded. Treatment modalities, including splinting, corticosteroid injection, or surgical release, are very effective and are tailored to the severity and duration of symptoms

Ultrasound-assessed perirenal fat is related to increased ophthalmic artery resistance index in HIV-1 patients

<p>Abstract</p> <p>Background</p> <p>The introduction of highly active antiretroviral therapy (HAART) has dramatically changed the prognosis of human immunodeficiency virus (HIV) infection, with a significant decline in morbidity and mortality.</p> <p>Changes in body fat distribution are a common finding in individuals with HIV infection being treated with antiretrovirals, and this condition (collectively termed lipodystrophy syndrome) is associated with depletion of subcutaneous fat, increased triglycerides and insulin resistance. Obesity, particularly visceral obesity, is associated with increased risk of cardiovascular disease. Therefore, estimating visceral fat distribution is important in identifying subjects at high risk for cardiovascular disease.</p> <p>The aim of our study was to evaluate whether perirenal fat thickness (PRFT), a parameter of central obesity, is related to ophthalmic artery resistance index (OARI), an index of occlusive carotid artery disease in HIV-1 infected patients.</p> <p>Methods</p> <p>We enrolled 88 consecutive HIV-1-infected patients receiving highly active antiretroviral therapy for more than 12 months, in a prospective cohort study. Echographically measured PRFT and OARI, as well as serum metabolic parameters, were evaluated. PRFT and OARI were measured by 3.75 MHz convex and 7.5 MHz linear probe, respectively.</p> <p>Results</p> <p>The means of PRFT and OARI in HIV-1-infected patients with visceral obesity was considerably higher than in patients without it (p < 0.0001 and p < 0.001, respectively). Using the average OARI as the dependent variable, total serum cholesterol level, HDL, triglycerides, glycemia, sex, blood pressure, age and PRFT were independent factors associated with OARI. A PRFT of 6.1 mm was the most discriminatory value for predicting an OARI > 0.74 (sensitivity 78.9%, specificity 82.8%).</p> <p>Conclusions</p> <p>Our data indicate that ultrasound assessment of PRFT may have potential as a marker of increased endothelial damage with specific involvement of the ocular vascular region in HIV-1-infected patients.</p

Genomic Characterization of Methanomicrobiales Reveals Three Classes of Methanogens

BACKGROUND:Methanomicrobiales is the least studied order of methanogens. While these organisms appear to be more closely related to the Methanosarcinales in ribosomal-based phylogenetic analyses, they are metabolically more similar to Class I methanogens. METHODOLOGY/PRINCIPAL FINDINGS:In order to improve our understanding of this lineage, we have completely sequenced the genomes of two members of this order, Methanocorpusculum labreanum Z and Methanoculleus marisnigri JR1, and compared them with the genome of a third, Methanospirillum hungatei JF-1. Similar to Class I methanogens, Methanomicrobiales use a partial reductive citric acid cycle for 2-oxoglutarate biosynthesis, and they have the Eha energy-converting hydrogenase. In common with Methanosarcinales, Methanomicrobiales possess the Ech hydrogenase and at least some of them may couple formylmethanofuran formation and heterodisulfide reduction to transmembrane ion gradients. Uniquely, M. labreanum and M. hungatei contain hydrogenases similar to the Pyrococcus furiosus Mbh hydrogenase, and all three Methanomicrobiales have anti-sigma factor and anti-anti-sigma factor regulatory proteins not found in other methanogens. Phylogenetic analysis based on seven core proteins of methanogenesis and cofactor biosynthesis places the Methanomicrobiales equidistant from Class I methanogens and Methanosarcinales. CONCLUSIONS/SIGNIFICANCE:Our results indicate that Methanomicrobiales, rather than being similar to Class I methanogens or Methanomicrobiales, share some features of both and have some unique properties. We find that there are three distinct classes of methanogens: the Class I methanogens, the Methanomicrobiales (Class II), and the Methanosarcinales (Class III)

Localized Plasticity in the Streamlined Genomes of Vinyl Chloride Respiring Dehalococcoides

Vinyl chloride (VC) is a human carcinogen and widespread priority pollutant. Here we report the first, to our knowledge, complete genome sequences of microorganisms able to respire VC, Dehalococcoides sp. strains VS and BAV1. Notably, the respective VC reductase encoding genes, vcrAB and bvcAB, were found embedded in distinct genomic islands (GEIs) with different predicted integration sites, suggesting that these genes were acquired horizontally and independently by distinct mechanisms. A comparative analysis that included two previously sequenced Dehalococcoides genomes revealed a contextually conserved core that is interrupted by two high plasticity regions (HPRs) near the Ori. These HPRs contain the majority of GEIs and strain-specific genes identified in the four Dehalococcoides genomes, an elevated number of repeated elements including insertion sequences (IS), as well as 91 of 96 rdhAB, genes that putatively encode terminal reductases in organohalide respiration. Only three core rdhA orthologous groups were identified, and only one of these groups is supported by synteny. The low number of core rdhAB, contrasted with the high rdhAB numbers per genome (up to 36 in strain VS), as well as their colocalization with GEIs and other signatures for horizontal transfer, suggests that niche adaptation via organohalide respiration is a fundamental ecological strategy in Dehalococccoides. This adaptation has been exacted through multiple mechanisms of recombination that are mainly confined within HPRs of an otherwise remarkably stable, syntenic, streamlined genome among the smallest of any free-living microorganism