An (N-1)-dimensional convex compact set gives an N-dimensional traveling front in the Allen--Cahn equation

This paper studies traveling fronts to the Allen–Cahn equation in RN for N ≥ 3. Let (N −2)-dimensional smooth surfaces be the boundaries of compact sets in RN−1 and assume that all principal curvatures are positive everywhere. We define an equivalence relation between them and prove that there exists a traveling front associated with a given surface and that it is asymptotically stable for given initial perturbation. The associated traveling fronts coincide up to phase transition if and only if the given surfaces satisfy the equivalence relation

Global exponential convergence to variational traveling waves in cylinders

We prove, under generic assumptions, that the special variational traveling wave that minimizes the exponentially weighted Ginzburg-Landau functional associated with scalar reaction-diffusion equations in infinite cylinders is the long-time attractor for the solutions of the initial value problems with front-like initial data. The convergence to this traveling wave is exponentially fast. The obtained result is mainly a consequence of the gradient flow structure of the considered equation in the exponentially weighted spaces and does not depend on the precise details of the problem. It strengthens our earlier generic propagation and selection result for "pushed" fronts.Comment: 23 page

Existence and Uniqueness of Solutions to a Nonlocal Equation with Monostable Nonlinearity

Let $J \in C(\mathbb{R})$, $J\ge 0$, \int_{\tiny\mathbb{R}} J = 1 and consider the nonlocal diffusion operator $\mathcal{M}[u] = J \star u - u$. We study the equation $\mathcal{M} u + f(x,u) = 0$, $u \ge 0$, in $\mathbb{R}$, where $f$ is a KPP-type nonlinearity, periodic in $x$. We show that the principal eigenvalue of the linearization around zero is well defined and that a nontrivial solution of the nonlinear problem exists if and only if this eigenvalue is negative. We prove that if, additionally, $J$ is symmetric, then the nontrivial solution is unique

Entropy-Dominated Dissipation in Sapphire Shock-Compressed up to 400 GPa (4 Mbar)

Sapphire (single-crystal Al2O3) is a representative Earth material and is used as a window and/or anvil in shock experiments. Pressure, for example, at the core-mantle boundary is about 130 gigapascals (GPa). Defects induced by 100-GPa shock waves cause sapphire to become opaque, which precludes measuring temperature with thermal radiance. We have measured wave profiles of sapphire crystals with several crystallographic orientations at shock pressures of 16, 23, and 86 GPa. At 23 GPa plastic-shock rise times are generally quite long (~100 ns) and their values depend sensitively on the direction of shock propagation in the crystal lattice. The long rise times are probably caused by the high strength of inter-atomic interactions in the ordered three-dimensional sapphire lattice. Our wave profiles and recent theoretical and laser-driven experimental results imply that sapphire disorders without significant shock heating up to about 400 GPa, above which Al2O3 is amorphous and must heat. This picture suggests that the characteristic shape of shock compression curves of many Earth materials at 100 GPa pressures is caused by a combination of entropy and temperature.Comment: 12 pages, 4 figure

Lack of increases in methylation at three CpG-rich genomic loci in non-mitotic adult tissues during aging

<p>Abstract</p> <p>Background</p> <p>Cell division occurs during normal human development and aging. Despite the likely importance of cell division to human pathology, it has been difficult to infer somatic cell mitotic ages (total numbers of divisions since the zygote) because direct counting of lifetime numbers of divisions is currently impractical. Here we attempt to infer relative mitotic ages with a molecular clock hypothesis. Somatic genomes may record their mitotic ages because greater numbers of replication errors should accumulate after greater numbers of divisions. Mitotic ages will vary between cell types if they divide at different times and rates.</p> <p>Methods</p> <p>Age-related increases in DNA methylation at specific CpG sites (termed "epigenetic molecular clocks") have been previously observed in mitotic human epithelium like the intestines and endometrium. These CpG rich sequences or "tags" start unmethylated and potentially changes in methylation during development and aging represent replication errors. To help distinguish between mitotic versus time-associated changes, DNA methylation tag patterns at 8–20 CpGs within three different genes, two on autosomes and one on the X-chromosome were measured by bisulfite sequencing from heart, brain, kidney and liver of autopsies from 21 individuals of different ages.</p> <p>Results</p> <p>Levels of DNA methylation were significantly greater in adult compared to fetal or newborn tissues for two of the three examined tags. Consistent with the relative absence of cell division in these adult tissues, there were no significant increases in tag methylation after infancy.</p> <p>Conclusion</p> <p>Many somatic methylation changes at certain CpG rich regions or tags appear to represent replication errors because this methylation increases with chronological age in mitotic epithelium but not in non-mitotic organs. Tag methylation accumulates differently in different tissues, consistent with their expected genealogies and mitotic ages. Although further studies are necessary, these results suggest numbers of divisions and ancestry are at least partially recorded by epigenetic replication errors within somatic cell genomes.</p

Continuum-based models and concepts for the transport of nanoparticles in saturated porous media: A state-of-the-science review

Environmental applications of nanoparticles (NP) increasingly result in widespread NP distribution within porous media where they are subject to various concurrent transport mechanisms including irreversible deposition, attachment/detachment (equilibrium or kinetic), agglomeration, physical straining, site-blocking, ripening, and size exclusion. Fundamental research in NP transport is typically conducted at small scale, and theoretical mechanistic modeling of particle transport in porous media faces challenges when considering the simultaneous effects of transport mechanisms. Continuum modeling approaches, in contrast, are scalable across various scales ranging from column experiments to aquifer. They have also been able to successfully describe the simultaneous occurrence of various transport mechanisms of NP in porous media such as blocking/straining or agglomeration/deposition/detachment. However, the diversity of model equations developed by different authors and the lack of effective approaches for their validation present obstacles to the successful robust application of these models for describing or predicting NP transport phenomena. This review aims to describe consistently all the important NP transport mechanisms along with their representative mathematical continuum models as found in the current scientific literature. Detailed characterizations of each transport phenomenon in regards to their manifestation in the column experiment outcomes, i.e., breakthrough curve (BTC) and residual concentration profile (RCP), are presented to facilitate future interpretations of BTCs and RCPs. The review highlights two NP transport mechanisms, agglomeration and size exclusion, which are potentially of great importance in controlling the fate and transport of NP in the subsurface media yet have been widely neglected in many existing modeling studies. A critical limitation of the continuum modeling approach is the number of parameters used upon application to larger scales and when a series of transport mechanisms are involved. We investigate the use of simplifying assumptions, such as the equilibrium assumption, in modeling the attachment/detachment mechanisms within a continuum modelling framework. While acknowledging criticisms about the use of this assumption for NP deposition on a mechanistic (process) basis, we found that its use as a description of dynamic deposition behavior in a continuum model yields broadly similar results to those arising from a kinetic model. Furthermore, we show that in two dimensional (2-D) continuum models the modeling efficiency based on the Akaike information criterion (AIC) is enhanced for equilibrium vs kinetic with no significant reduction in model performance. This is because fewer parameters are needed for the equilibrium model compared to the kinetic model. Two major transport regimes are identified in the transport of NP within porous media. The first regime is characterized by higher particle-surface attachment affinity than particle-particle attachment affinity, and operative transport mechanisms of physicochemical filtration, blocking, and physical retention. The second regime is characterized by the domination of particle-particle attachment tendency over particle-surface affinity. In this regime although physicochemical filtration as well as straining may still be operative, ripening is predominant together with agglomeration and further subsequent retention. In both regimes careful assessment of NP fate and transport is necessary since certain combinations of concurrent transport phenomena leading to large migration distances are possible in either case

Somatic mosaicism and common genetic variation contribute to the risk of very-early-onset inflammatory bowel disease

Abstract: Very-early-onset inflammatory bowel disease (VEO-IBD) is a heterogeneous phenotype associated with a spectrum of rare Mendelian disorders. Here, we perform whole-exome-sequencing and genome-wide genotyping in 145 patients (median age-at-diagnosis of 3.5 years), in whom no Mendelian disorders were clinically suspected. In five patients we detect a primary immunodeficiency or enteropathy, with clinical consequences (XIAP, CYBA, SH2D1A, PCSK1). We also present a case study of a VEO-IBD patient with a mosaic de novo, pathogenic allele in CYBB. The mutation is present in ~70% of phagocytes and sufficient to result in defective bacterial handling but not life-threatening infections. Finally, we show that VEO-IBD patients have, on average, higher IBD polygenic risk scores than population controls (99 patients and 18,780 controls; P < 4 × 10−10), and replicate this finding in an independent cohort of VEO-IBD cases and controls (117 patients and 2,603 controls; P < 5 × 10−10). This discovery indicates that a polygenic component operates in VEO-IBD pathogenesis