Convergence analysis of a modified weak Galerkin finite element method for Signorini and obstacle problems

In this article, we apply a modified weak Galerkin method to solve variational inequality of the first kind which includes Signorini and obstacle problems. Optimal order a priori error estimates in the energy norm are derived. We also provide some numerical experiments to validate the theoretical results.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1459–1474, 2017     

Representation and approximation of convex dynamic risk measures with respect to strong–weak topologies

We provide a representation for strong-weak continuous dynamic risk measures from L^p into L^p_t spaces where these spaces are equipped respectively with strong and weak topologies and p is a finite number strictly larger than one. Conversely, we show that any such representation that admits a compact (with respect to the product of weak topologies) sub-differential generates a dynamic risk measure that is strong--weak continuous. Furthermore, we investigate sufficient conditions on the sub-differential for which the essential supremum of the representation is attained. Finally, the main purpose is to show that any convex dynamic risk measure that is strong-weak continuous can be approximated by a sequence of convex dynamic risk measures which are strong--weak continuous and admit compact sub-differentials with respect to the product of weak topologies. Throughout the arguments, no conditional translation invariance or monotonicity assumptions are applied.     

The Weak 3D Topological Insulator Bi12Rh3Sn3I9

Topological insulators (TIs) gained high interest due to their protected electronic surface states that allow dissipation-free electron and information transport. In consequence, TIs are recommended as materials for spintronics and quantum computing. Yet, the number of well-characterized TIs is rather limited. To contribute to this field of research, we focused on new bismuth-based subiodides and recently succeeded in synthesizing a new compound Bi₁₂Rh₃Sn₃I₉, which is structurally closely related to Bi₁₄Rh₃I₉ – a stable, layered material. In fact, Bi₁₄Rh₃I₉ is the first experimentally supported weak 3D TI. Both structures are composed of well-defined intermetallic layers of _∞²[(Bi₄Rh)₃I]²⁺ with topologically protected electronic edge-states. The fundamental difference between Bi₁₄Rh₃I₉ and Bi₁₂Rh₃Sn₃I₉ lies in the composition and the arrangement of the anionic spacer. While the intermetallic 2D TI layers in Bi₁₄Rh₃I₉ are isolated by _∞¹[Bi₂I₈]²⁻ chains, the isoelectronic substitution of bismuth(III) with tin(II) leads to _∞²[Sn₃I₈]²⁻ layers as anionic spacers. First transport experiments support the 2D character of this material class and revealed metallic conductivity.     

Weak interactions in imidazole-containing zinc(II)-based metal–organic frameworks

The self-assembly of the two zinc(II) metal–organic frameworks, [Zn₂(L)(bdc)₂]·3MeOH·4H₂O}_n ( 1 , L = 2-(pyridin-4-yl)-3H-imidazo[4,5-c]pyridine, H₂bdc = 1,4-benzenedicarboxylic acid) and [Zn₂(L)(bdc)₂]·2DMF·H₂O}_n ( 2 ), was achieved under mild reaction conditions. Both compounds 1 and 2 were structurally characterized by single-crystal X-ray diffraction analysis. Interestingly, the coordination modes of the ligand L in two structures are entirely different. Compounds 1 and 2 were made up of paddle wheel-shaped {Zn₂(O₂C)₄} secondary building unit (SBU) clusters, which adopted three-dimensional structures with a pcu topology. Rich weak interactions were observed in the structures of both 1 and 2 . The uncoordinated imidazole and pyridine moieties exhibited electron donor–acceptor interactions, π–π stacking, hydrogen bonding, and CH–π interactions. These interactions also facilitated the abilities of the framework to adsorb CO₂ molecules. Gas adsorption studies revealed that compound 1 selectively adsorbed CO₂ (131.1 cm³/g) over N₂ (23.5 cm³/g) and H₂ (36.5 cm³/g) at a pressure of 1 atm.     

Stable sub-Gaussian models constructed by Poisson processes

In this paper, we first prove that one-parameter standard α-stable sub-Gaussian processes can be approximated by processes constructed by integrals based on the Poisson process with random intensity. Then we extend this result to the two-parameter processes. At last, we consider the approximation of the subordinated fractional Brownian motion.     

Moment Propagation for Weak Solutions to the Vlasov–Poisson System

We consider weak solutions to the Cauchy problem for the three dimensional Vlasov–Poisson system of equations. We obtain a propagation result for any velocity moment of order > 2 as well as a uniqueness statement in ?³. In the periodic case, we show that velocity moments of order > 14/3 are propagated.     

Analysis of the Brinkman-Forchheimer equations with slip boundary conditions

In this work, we study the Brinkman–Forchheimer equations driven under slip boundary conditions of friction type. We prove the existence and uniqueness of weak solutions by means of regularization combined with the Faedo-Galerkin approach. Next, we discuss the continuity of the solution with respect to Brinkman’s and Forchheimer’s coefficients. Finally, we show that the weak solution of the corresponding stationary problem is stable.