Spectral gaps for water waves above a corrugated bottom

In this paper, the essential spectrum of the linear problem on water waves in a water layer and in a channel with a gently corrugated bottom is studied. We show that, under a certain geometric condition, the essential spectrum has spectral gaps. In other words, there exist intervals in the positive real semi-axis that are free of the spectrum but have their endpoints in it. The position and the length of the gaps are found out by applying an asymptotic analysis to the model problem in the periodicity cell

On multi-scale asymptotic structure of eigenfunctions in a boundary value problem with concentrated masses near the boundary

We construct two-term asymptotics ?? k = ?m?2(M + ??k + O(?3/2)) of eigenvalues of a mixed boundary-value problem in ? R2 with many heavy (m > 2) concentrated masses near a straight part of the boundary ? . ? is a small positive parameter related to size and periodicity of the masses; k ? N. The main term M > 0 is common for all eigenvalues but the correction terms ?k , which are eigenvalues of a limit problem with the spectral Steklov boundary conditions on , exhibit the effect of asymptotic splitting in the eigenvalue sequence enabling the detection of asymptotic forms of eigenfunctions. The justification scheme implies isolating and purifying singularities of eigenfunctions and leads to a new spectral problem in weighed spaces with a "strongly" singular weight.This research work has been partially supported by Spanish MINECO, MTM2013-44883-P. Also, the research work of the first author has been partially supported by Russian Foundation of Basic research (project 15–01–02175)

A gap in the spectrum of the Neumann-Laplacian on a periodic waveguide

We will study the spectral problem related to the Laplace operator in a singularly perturbed periodic waveguide. The waveguide is a quasi-cylinder with contains periodic arrangement of inclusions. On the boundary of the waveguide we consider both Neumann and Dirichlet conditions. We will prove that provided the diameter of the inclusion is small enough in the spectrum of Laplacian opens spectral gaps, i.e. frequencies that does not propagate through the waveguide. The existence of the band gaps will verified using the asymptotic analysis of elliptic operators.Comment: 26 pages, 6 figure

Asymptotic analysis of an elastic rod with rounded ends

We derive a one-dimensional model for an elastic shuttle, that is, a thin rod with rounded ends and small fixed terminals, by means of an asymptotic procedure of dimension reduction. In the model, deformation of the shuttle is described by a system of ordinary differential equations with variable degenerating coefficients, and the number of the required boundary conditions at the end points of the one-dimensional image of the rod depends on the roundness exponent m is an element of(0,1). Error estimates are obtained in the case m is an element of(0,1/4) by using an anisotropic weighted Korn inequality, which was derived in an earlier paper by the authors. We also briefly discuss boundary layer effects, which can be neglected in the case m is an element of(0,1/4) but play a crucial role in the formulation of the limit problem for m >= 1/4.Peer reviewe

Restrictions and extensions of semibounded operators

We study restriction and extension theory for semibounded Hermitian operators in the Hardy space of analytic functions on the disk D. Starting with the operator zd/dz, we show that, for every choice of a closed subset F in T=bd(D) of measure zero, there is a densely defined Hermitian restriction of zd/dz corresponding to boundary functions vanishing on F. For every such restriction operator, we classify all its selfadjoint extension, and for each we present a complete spectral picture. We prove that different sets F with the same cardinality can lead to quite different boundary-value problems, inequivalent selfadjoint extension operators, and quite different spectral configurations. As a tool in our analysis, we prove that the von Neumann deficiency spaces, for a fixed set F, have a natural presentation as reproducing kernel Hilbert spaces, with a Hurwitz zeta-function, restricted to FxF, as reproducing kernel.Comment: 63 pages, 11 figure

Altered metabolic landscape in IDH‐mutant gliomas affects phospholipid, energy, and oxidative stress pathways

Heterozygous mutations in NADP‐dependent isocitrate dehydrogenases (IDH) define the large majority of diffuse gliomas and are associated with hypermethylation of DNA and chromatin. The metabolic dysregulations imposed by these mutations, whether dependent or not on the oncometabolite D‐2‐hydroxyglutarate (D2HG), are less well understood. Here, we applied mass spectrometry imaging on intracranial patient‐derived xenografts of IDH‐mutant versus IDH wild‐type glioma to profile the distribution of metabolites at high anatomical resolution in situ. This approach was complemented by in vivo tracing of labeled nutrients followed by liquid chromatography–mass spectrometry (LC‐MS) analysis. Selected metabolites were verified on clinical specimen. Our data identify remarkable differences in the phospholipid composition of gliomas harboring the IDH1 mutation. Moreover, we show that these tumors are characterized by reduced glucose turnover and a lower energy potential, correlating with their reduced aggressivity. Despite these differences, our data also show that D2HG overproduction does not result in a global aberration of the central carbon metabolism, indicating strong adaptive mechanisms at hand. Intriguingly, D2HG shows no quantitatively important glucose‐derived label in IDH‐mutant tumors, which suggests that the synthesis of this oncometabolite may rely on alternative carbon sources. Despite a reduction in NADPH, glutathione levels are maintained. We found that genes coding for key enzymes in de novo glutathione synthesis are highly expressed in IDH‐mutant gliomas and the expression of cystathionine‐β‐synthase (CBS) correlates with patient survival in the oligodendroglial subtype. This study provides a detailed and clinically relevant insight into the in vivo metabolism of IDH1‐mutant gliomas and points to novel metabolic vulnerabilities in these tumors

Effective Rheology of Bubbles Moving in a Capillary Tube

We calculate the average volumetric flux versus pressure drop of bubbles moving in a single capillary tube with varying diameter, finding a square-root relation from mapping the flow equations onto that of a driven overdamped pendulum. The calculation is based on a derivation of the equation of motion of a bubble train from considering the capillary forces and the entropy production associated with the viscous flow. We also calculate the configurational probability of the positions of the bubbles.Comment: 4 pages, 1 figur