Regularity of optimal sets for some functional involving eigenvalues of an operator in divergence form

In this paper we consider minimizers of the functional$\min \{{\lambda }_1(\mathrm{\Omega })+\cdots +{\lambda }_k(\mathrm{\Omega })+\mathrm{\Lambda }|\mathrm{\Omega }|,:\mathrm{\Omega }\subset D\text{ open}\}$ where $D\subset {\mathbb{R}}^d$ is a bounded open set and where $0<{\lambda }_1(\mathrm{\Omega })\le \cdots \le {\lambda }_k(\mathrm{\Omega })$ are the first k eigenvalues on Ω of an operator in divergence form with Dirichlet boundary condition and with Hölder continuous coefficients. We prove that the optimal sets ${\mathrm{\Omega }}^{⁎}$ have finite perimeter and that their free boundary $\partial {\mathrm{\Omega }}^{⁎}\cap D$ is composed of a regular part, which is locally the graph of a $C^{1,\alpha }$-regular function, and a singular part, which is empty if $d<d^{⁎}$, discrete if $d=d^{⁎}$ and of Hausdorff dimension at most $d-d^{⁎}$ if $d>d^{⁎}$, for some $d^{⁎}\in \{5,6,7\}$.     

Chemical bonding in representative astrophysically relevant neutral,cation, and anion HCnH chains

Most existing studies assign a polyynic and cumulenic character of chemical bonding in carbon-based chains relying on values of the bond lengths. Building on our recent work, in this paper we add further evidence on the limitations of such an analysis and demonstrate the significant insight gained via natural bond analysis. Presently reported results include atomic charges, natural bond order and valence indices obtained from ab initio computations for representative members of the astrophysically relevant neutral and charged HC_2k/2k+1H chain family. They unravel a series of counter-intuitive aspects and/or help naive intuition in properly understanding microscopic processes, e.g., electron removal from or electron attachment to a neutral chain. Demonstrating that the Wiberg indices adequately quantify the chemical bonding structure of the HC_2k/2k+1H chains—while the often heavily advertised Mayer indices do not—represents an important message conveyed by the present study.     

An analysis of over-relaxation in a kinetic approximation of systems of conservation laws

The over-relaxation approach is an alternative to the Jin–Xin relaxation method in order to apply the equilibrium source term in a more precise way. This is also a key ingredient of the lattice Boltzmann method for achieving second-order accuracy. In this work, we provide an analysis of the over-relaxation kinetic scheme. We compute its equivalent equation, which is particularly useful for devising stable boundary conditions for the hidden kinetic variables.     

Fluid‐structure coupling of linear elastic model with compressible flow models

Cavitation erosion is caused in solids exposed to strong pressure waves developing in an adjacent fluid field. The knowledge of the transient distribution of stresses in the solid is important to understand the cause of damaging by comparisons with breaking points of the material. The modeling of this problem requires the coupling of the models for the fluid and the solid. For this purpose, we use a strategy based on the solution of coupled Riemann problems that has been originally developed for the coupling of 2 fluids. This concept is exemplified for the coupling of a linear elastic structure with an ideal gas. The coupling procedure relies on the solution of a nonlinear equation. Existence and uniqueness of the solution is proven. The coupling conditions are validated by means of quasi‐1D problems for which an explicit solution can be determined. For a more realistic scenario, a 2D application is considered where in a compressible single fluid, a hot gas bubble at low pressure collapses in a cold gas at high pressure near an adjacent structure.     

Sufficient Conditions for a Digraph to be Supereulerian

A (di)graph is supereulerian if it contains a spanning eulerian sub(di)graph. This property is a relaxation of hamiltonicity. Inspired by this analogy with hamiltonian cycles and by similar results in supereulerian graph theory, we analyze a number of sufficient Ore type conditions for a digraph to be supereulerian. Furthermore, we study the following conjecture due to Thomassé and the first author: if the arc‐connectivity of a digraph is not smaller than its independence number, then the digraph is supereulerian. As a support for this conjecture we prove it for digraphs that are semicomplete multipartite or quasitransitive and verify the analogous statement for undirected graphs.     

13C NMR investigations of molecular order of rod-like,bent-core,and thiophene mesogens

In this review, methods to obtain the orientational order of topologically variant molecular mesogens using by one- and two-dimensional (2D) solid-state ¹³C nuclear magnetic resonance (NMR) spectroscopy are described. Besides ¹³C chemical shifts, the ¹³C─¹H dipolar couplings measured from 2D-separated local field (SLF) technique are used for computing the order parameters of a variety of mesogens. The investigated molecules are composed of a variable number of rings in the core, that is, core ranging from simply one ring to five rings. Among the mesogens investigated, a special focus has been placed on mesogens with thiophene rings, which are gaining popularity as liquid crystalline organic semiconductors. The replacement of a phenyl ring by thiophene in the core has a dramatic influence on molecular topology, as observed from the measured order parameters. The review highlights the advantages of the 2D SLF method for understanding the local dynamics and for mapping the topology of mesogens through the measured order parameters. SLF NMR studies of as many as 24 molecular mesogens that vary in terms of the molecular structure as well as topology are covered in the review. Order parameters of the rings have been estimated from the ¹³C─¹H dipolar couplings in the nematic, smectic A, smectic C, and tilted hexatic phases as well as in B₁ and B₂ mesophases of various mesogens. It is anticipated that, in the years to come, the 2D SLF method would provide advanced molecular information on structurally complex mesogens that are emerging in liquid crystal science through the incessant efforts of synthetic chemists. The mini review covers the orientational order of topologically variant molecular mesogens determined by 1D and 2D solid-state ¹³C NMR spectroscopy. Accordingly, rod-like, bent-core, and thiophene mesogens were subjected to 2D SLF measurements to get the order parameters from which the topology was established. The replacement of phenyl ring by thiophene and its influence on order parameters as well as on molecular topology is also discussed.     

Purcell Effect and Nonlinear Behavior of the Emission in a Periodic Structure Composed of InAs Monolayers Embedded in a GaAs Matrix

Enhancement of spontaneous emission in a resonant Bragg quantum well (QW) structure with 60 periods of triple InAs monolayers embedded in a GaAs matrix is studied experimentally and theoretically. From measurements of the time‐resolved photoluminescence, besides the QW exciton at 1.47 eV, a specific super‐radiant (SR) emission demonstrating nonlinear properties is found. The SR mode shows a near‐quadratic dependence of intensity on excitation power, while its energy position follows the Bragg condition. It is revealed that the SR mode shows a peculiar non‐monotonic dependence of intensity on direction, with a maximum observed at approximately 40°. The enhancement in the SR emission at a specific direction is correlated well with suggested theoretical consideration of the modal Purcell factor for periodic quantum well structures.     

Decay rate for viscoelastic wave equation of variable coefficients with delay and dynamic boundary conditions

In this paper, we consider a viscoelastic wave equation of variable coefficients in the presence of past history with nonlinear damping and delay in the internal feedback and dynamic boundary conditions. Under suitable assumptions, we establish an explicit and general decay rate result without imposing restrictive assumption on the behavior of the relaxation function at infinity by Riemannian geometry method and Lyapunov functional method.