The existence of an inverse limit of inverse system of measure spaces - a purely measurable case

The existence of an inverse limit of an inverse system of (probability) measure spaces has been investigated since the very beginning of the birth of the modern probability theory. Results from Kolmogorov [10], Bochner [2], Choksi [5], Metivier [14], Bourbaki [3] among others have paved the way of the deep understanding of the problem under consideration. All the above results, however, call for some topological concepts, or at least ones which are closely related topological ones. In this paper we investigate purely measurable inverse systems of (probability) measure spaces, and give a sucient condition for the existence of a unique inverse limit. An example for the considered purely measurable inverse systems of (probability) measure spaces is also given

Analytical Study of Certain Magnetohydrodynamic-alpha Models

In this paper we present an analytical study of a subgrid scale turbulence model of the three-dimensional magnetohydrodynamic (MHD) equations, inspired by the Navier-Stokes-alpha (also known as the viscous Camassa-Holm equations or the Lagrangian-averaged Navier-Stokes-alpha model). Specifically, we show the global well-posedness and regularity of solutions of a certain MHD-alpha model (which is a particular case of the Lagrangian averaged magnetohydrodynamic-alpha model without enhancing the dissipation for the magnetic field). We also introduce other subgrid scale turbulence models, inspired by the Leray-alpha and the modified Leray-alpha models of turbulence. Finally, we discuss the relation of the MHD-alpha model to the MHD equations by proving a convergence theorem, that is, as the length scale alpha tends to zero, a subsequence of solutions of the MHD-alpha equations converges to a certain solution (a Leray-Hopf solution) of the three-dimensional MHD equations.Comment: 26 pages, no figures, will appear in Journal of Math Physics; corrected typos, updated reference

Synthesis and characterisation of pyrene-labelled polydimethylsiloxane networks: towards the in situ detection of strain in silicone elastomers

Pyrene-substituted polyhydromethylsiloxanes (PHMS-Py-x) were synthesised by the hydrosilylation reaction of prop-3-enyloxymethylpyrene with polyhydromethylsiloxane (M-n = 3700). The ratio of pyrene substituent to Si-H unit was varied to afford a range of pyrene-functionalised polysiloxanes. These copolymers were subsequently incorporated into polydimethylsiloxane (PDMS) elastomers by curing via either Pt(0) catalysed hydrosilylation with divinyl-terminated PDMS (M-n = 186) and tetrakis(dimethylsiloxy) silane, or Sn(II) catalysed condensation with alpha,omega-dihydroxyPDMS (M-n = 26 000) and tetraethoxysilane. An alternative method involving the synthesis and integration of [3-(pyren-1-ylmethoxy)propyl]triethoxysilane (Py-TEOS) into PDMS elastomers was also investigated: a mixture of alpha,omega-dihydroxyPDMS (M-n = 26 000), tetraethoxysilane, and Py-TEOS was cured using an Sn( II) catalyst. Certain of the resulting fluorescent pyrene-labelled elastomers were studied by differential scanning calorimetry and dynamic mechanical analysis. No significant changes were observed in the thermal or mechanical properties of the elastomers containing pyrene when compared to otherwise identical samples not containing pyrene. All of the pyrene-containing elastomers were demonstrated to be fluorescent under suitable excitation in a photoluminescent spectrometer. Two of the elastomers were placed in a photoluminescence spectrometer and subjected to cycles of extension and relaxation (strain = 0-16.7%) while changes in the emission spectra were monitored. The resulting spectra of the elastomer containing the PHMS-Py-50 copolymers were variable and inconsistent. However, the emission peaks of elastomers containing Py-TEOS displayed clear and reproducible changes in fluorescence intensity upon stretching and relaxation. The intensity of the monomer and excimer emission peaks was observed to increase with elongation of the sample and decrease upon relaxation. Furthermore, the ratio of the intensities of the excimer : monomer peak decreased with elongation and increased with relaxation. In neither case was there appreciable hysteresis, suggesting that fluorescent labelling of elastomers is a valid approach for the non-invasive in situ monitoring of stress and strain in such materials

Diverse roles of androgen receptor (AR) domains in AR-mediated signaling

Androgens control male sexual development and maintenance of the adult male phenotype. They have very divergent effects on their target organs like the reproductive organs, muscle, bone, brain and skin. This is explained in part by the fact that different cell types respond differently to androgen stimulus, even when all these responses are mediated by the same intracellular androgen receptor. To understand these tissue- and cell-specific readouts of androgens, we have to learn the many different steps in the transcription activation mechanisms of the androgen receptor (NR3C4). Like all nuclear receptors, the steroid receptors have a central DNA-binding domain connected to a ligand-binding domain by a hinge region. In addition, all steroid receptors have a relatively large amino-terminal domain. Despite the overall structural homology with other nuclear receptors, the androgen receptor has several specific characteristics which will be discussed here. This receptor can bind two types of androgen response elements (AREs): one type being similar to the classical GRE/PRE-type elements, the other type being the more divergent and more selective AREs. The hormone-binding domain has low intrinsic transactivation properties, a feature that correlates with the low affinity of this domain for the canonical LxxLL-bearing coactivators. For the androgen receptor, transcriptional activation involves the alternative recruitment of coactivators to different regions in the amino-terminal domain, as well as the hinge region. Finally, a very strong ligand-induced interaction between the amino-terminal domain and the ligand-binding domain of the androgen receptor seems to be involved in many aspects of its function as a transcription factor. This review describes the current knowledge on the structure-function relationships within the domains of the androgen receptor and tries to integrate the involvement of different domains, subdomains and motifs in the functioning of this receptor as a transcription factor with tissue- and cell-specific readouts

Multiscale Theory of Finite Size Bose Systems: Implications for Collective and Single-Particle Excitations

Boson droplets (i.e., dense assemblies of bosons at low temperature) are shown to mask a significant amount of single-particle behavior and to manifest collective, droplet-wide excitations. To investigate the balance between single-particle and collective behavior, solutions to the wave equation for a finite size Bose system are constructed in the limit where the ratio \varepsilon of the average nearest-neighbor boson distance to the size of the droplet or the wavelength of density disturbances is small. In this limit, the lowest order wave function varies smoothly across the system, i.e., is devoid of structure on the scale of the average nearest-neighbor distance. The amplitude of short range structure in the wave function is shown to vanish as a power of \varepsilon when the interatomic forces are relatively weak. However, there is residual short range structure that increases with the strength of interatomic forces. While the multiscale approach is applied to boson droplets, the methodology is applicable to any finite size bose system and is shown to be more direct than field theoretic methods. Conclusions for Helium-4 nanodroplets are drawn.Comment: 28 pages, 5 figure

Stochastic Reaction-diffusion Equations Driven by Jump Processes

We establish the existence of weak martingale solutions to a class of second order parabolic stochastic partial differential equations. The equations are driven by multiplicative jump type noise, with a non-Lipschitz multiplicative functional. The drift in the equations contains a dissipative nonlinearity of polynomial growth.Comment: See journal reference for teh final published versio

Retinoic Acid Mediates Long-Paced Oscillations in Retinoid Receptor Activity: Evidence for a Potential Role for RIP140

Mechanisms that underlie oscillatory transcriptional activity of nuclear receptors (NRs) are incompletely understood. Evidence exists for rapid, cyclic recruitment of coregulatory complexes upon activation of nuclear receptors. RIP140 is a NR coregulator that represses the transactivation of agonist-bound nuclear receptors. Previously, we showed that RIP140 is inducible by all-trans retinoic acid (RA) and mediates limiting, negative-feedback regulation of retinoid signaling.Here we report that in the continued presence of RA, long-paced oscillations of retinoic acid receptor (RAR) activity occur with a period ranging from 24 to 35 hours. Endogenous expression of RIP140 and other RA-target genes also oscillate in the presence of RA. Cyclic retinoid receptor transactivation is ablated by constitutive overexpression of RIP140. Further, depletion of RIP140 disrupts cyclic expression of the RA target gene HOXA5. Evidence is provided that RIP140 may limit RAR signaling in a selective, non-redundant manner in contrast to the classic NR coregulators NCoR1 and SRC1 that are not RA-inducible, do not cycle, and may be partially redundant in limiting RAR activity. Finally, evidence is provided that RIP140 can repress and be induced by other nuclear receptors in a manner that suggests potential participation in other NR oscillations.We provide evidence for novel, long-paced oscillatory retinoid receptor activity and hypothesize that this may be paced in part, by RIP140. Oscillatory NR activity may be involved in mediating hormone actions of physiological and pathological importance

Testing the FMR1 Promoter for Mosaicism in DNA Methylation among CpG Sites, Strands, and Cells in FMR1-Expressing Males with Fragile X Syndrome

Variability among individuals in the severity of fragile X syndrome (FXS) is influenced by epigenetic methylation mosaicism, which may also be common in other complex disorders. The epigenetic signal of dense promoter DNA methylation is usually associated with gene silencing, as was initially reported for FMR1 alleles in individuals with FXS. A paradox arose when significant levels of FMR1 mRNA were reported for some males with FXS who had been reported to have predominately methylated alleles. We have used hairpin-bisufite PCR, validated with molecular batch-stamps and barcodes, to collect and assess double-stranded DNA methylation patterns from these previously studied males. These patterns enable us to distinguish among three possible forms of methylation mosaicism, any one of which could explain FMR1 expression in these males. Our data indicate that cryptic inter-cell mosaicism in DNA methylation can account for the presence of FMR1 mRNA in some individuals with FXS