Partial regularity and smooth topology-preserving approximations of rough domains

For a bounded domain $\Omega\subset\mathbb{R}^m, m\geq 2,$ of class $C^0$, the properties are studied of fields of `good directions', that is the directions with respect to which $\partial\Omega$ can be locally represented as the graph of a continuous function. For any such domain there is a canonical smooth field of good directions defined in a suitable neighbourhood of $\partial\Omega$, in terms of which a corresponding flow can be defined. Using this flow it is shown that $\Omega$ can be approximated from the inside and the outside by diffeomorphic domains of class $C^\infty$. Whether or not the image of a general continuous field of good directions (pseudonormals) defined on $\partial\Omega$ is the whole of $\mathbb{S}^{m-1}$ is shown to depend on the topology of $\Omega$. These considerations are used to prove that if $m=2,3$, or if $\Omega$ has nonzero Euler characteristic, there is a point $P\in\partial\Omega$ in the neighbourhood of which $\partial\Omega$ is Lipschitz. The results provide new information even for more regular domains, with Lipschitz or smooth boundaries.Comment: Final version appeared in Calc. Var PDE 56, Issue 1, 201

Liquid crystal defects in the Landau-de Gennes theory in two dimensions-beyond the one-constant approximation

We consider the two-dimensional Landau-de Gennes energy with several elastic constants, subject to general $k$-radially symmetric boundary conditions. We show that for generic elastic constants the critical points consistent with the symmetry of the boundary conditions exist only in the case $k=2$. In this case we identify three types of radial profiles: with two, three of full five components and numerically investigate their minimality and stability depending on suitable parameters. We also numerically study the stability properties of the critical points of the Landau-de Gennes energy and capture the intricate dependence of various qualitative features of these solutions on the elastic constants and the physical regimes of the liquid crystal system

Uniqueness results for an ODE related to a generalized Ginzburg-Landau model for liquid crystals

We study a singular nonlinear ordinary differential equation on intervals {[}0, R) with R <= +infinity, motivated by the Ginzburg-Landau models in superconductivity and Landau-de Gennes models in liquid crystals. We prove existence and uniqueness of positive solutions under general assumptions on the nonlinearity. Further uniqueness results for sign-changing solutions are obtained for a physically relevant class of nonlinearities. Moreover, we prove a number of fine qualitative properties of the solution that are important for the study of energetic stability

Half-Integer Point Defects in the Q-Tensor Theory of Nematic Liquid Crystals

We investigate prototypical profiles of point defects in two dimensional liquid crystals within the framework of Landau-de Gennes theory. Using boundary conditions characteristic of defects of index $k/2$, we find a critical point of the Landau-de Gennes energy that is characterised by a system of ordinary differential equations. In the deep nematic regime, $b^2$ small, we prove that this critical point is the unique global minimiser of the Landau-de Gennes energy. We investigate in greater detail the regime of vanishing elastic constant $L \to 0$, where we obtain three explicit point defect profiles, including the global minimiser.Comment: 15 pages, 16 figure

Fiber Optic Temperature Sensor Insert for High Temperature Environments

A thermal protection system (TPS) test plug has optical fibers with FBGs embedded in the optical fiber arranged in a helix, an axial fiber, and a combination of the two. Optionally, one of the optical fibers is a sapphire FBG for measurement of the highest temperatures in the TPS plug. The test plug may include an ablating surface and a non-ablating surface, with an engagement surface with threads formed, the threads having a groove for placement of the optical fiber. The test plug may also include an optical connector positioned at the non-ablating surface for protection of the optical fiber during insertion and removal

Genetic and systems level analysis of Drosophila sticky/citron kinase and dFmr1 mutants reveals common regulation of genetic networks

<p>Abstract</p> <p>Background</p> <p>In <it>Drosophila</it>, the genes <it>sticky </it>and <it>dFmr1 </it>have both been shown to regulate cytoskeletal dynamics and chromatin structure. These genes also genetically interact with Argonaute family microRNA regulators. Furthermore, in mammalian systems, both genes have been implicated in neuronal development. Given these genetic and functional similarities, we tested <it>Drosophila sticky </it>and <it>dFmr1 </it>for a genetic interaction and measured whole genome expression in both mutants to assess similarities in gene regulation.</p> <p>Results</p> <p>We found that <it>sticky </it>mutations can dominantly suppress a <it>dFmr1 </it>gain-of-function phenotype in the developing eye, while phenotypes produced by RNAi knock-down of <it>sticky </it>were enhanced by <it>dFmr1 </it>RNAi and a <it>dFmr1 </it>loss-of-function mutation. We also identified a large number of transcripts that were misexpressed in both mutants suggesting that <it>sticky </it>and <it>dFmr1 </it>gene products similarly regulate gene expression. By integrating gene expression data with a protein-protein interaction network, we found that mutations in <it>sticky </it>and <it>dFmr1 </it>resulted in misexpression of common gene networks, and consequently predicted additional specific phenotypes previously not known to be associated with either gene. Further phenotypic analyses validated these predictions.</p> <p>Conclusion</p> <p>These findings establish a functional link between two previously unrelated genes. Microarray analysis indicates that <it>sticky </it>and <it>dFmr1 </it>are both required for regulation of many developmental genes in a variety of cell types. The diversity of transcripts regulated by these two genes suggests a clear cause of the pleiotropy that <it>sticky </it>and <it>dFmr1 </it>mutants display and provides many novel, testable hypotheses about the functions of these genes. As both of these genes are implicated in the development and function of the mammalian brain, these results have relevance to human health as well as to understanding more general biological processes.</p

Orientability and energy minimization in liquid crystal models

Uniaxial nematic liquid crystals are modelled in the Oseen-Frank theory through a unit vector field $n$. This theory has the apparent drawback that it does not respect the head-to-tail symmetry in which $n$ should be equivalent to -$n$. This symmetry is preserved in the constrained Landau-de Gennes theory that works with the tensor $Q=s\big(n\otimes n- \frac{1}{3} Id\big)$.We study the differences and the overlaps between the two theories. These depend on the regularity class used as well as on the topology of the underlying domain. We show that for simply-connected domains and in the natural energy class $W^{1,2}$ the two theories coincide, but otherwise there can be differences between the two theories, which we identify. In the case of planar domains we completely characterise the instances in which the predictions of the constrained Landau-de Gennes theory differ from those of the Oseen-Frank theory

Energy Dissipation and Regularity for a Coupled Navier-Stokes and Q-Tensor System

We study a complex non-newtonian fluid that models the flow of nematic liquid crystals. The fluid is described by a system that couples a forced Navier-Stokes system with a parabolic-type system. We prove the existence of global weak solutions in dimensions two and three. We show the existence of a Lyapunov functional for the smooth solutions of the coupled system and use the cancellations that allow its existence to prove higher global regularity, in dimension two. We also show the weak-strong uniqueness in dimension two

Glycolysis Upregulation Is Neuroprotective As A Compensatory Mechanism In Als

Amyotrophic Lateral Sclerosis (ALS), is a fatal neurodegenerative disorder, with TDP-43 inclusions as a major pathological hallmark. Using a Drosophila model of TDP-43 proteinopathy we found significant alterations in glucose metabolism including increased pyruvate, suggesting that modulating glycolysis may be neuroprotective. Indeed, a high sugar diet improves locomotor and lifespan defects caused by TDP-43 proteinopathy in motor neurons or glia, but not muscle, suggesting that metabolic dysregulation occurs in the nervous system. Overexpressing human glucose transporter GLUT-3 in motor neurons mitigates TDP-43 dependent defects in synaptic vesicle recycling and improves locomotion. Furthermore, PFK mRNA, a key indicator of glycolysis, is upregulated in flies and patient derived iPSC motor neurons with TDP-43 pathology. Surprisingly, PFK overexpression rescues TDP-43 induced locomotor deficits. These findings from multiple ALS models show that mechanistically, glycolysis is upregulated in degenerating motor neurons as a compensatory mechanism and suggest that increased glucose availability is protective