A spectral method for nonlinear elliptic equations

Let Ω be an open, simply connected, and bounded region in \(\mathbb {R}^{d}\), d ≥ 2, and assume its boundary ?Ω is smooth and homeomorphic to \(\mathbb {S}^{d-1}\). Consider solving an elliptic partial differential equation L u = f(?, u) over Ω with zero Dirichlet boundary value. The function f is a nonlinear function of the solution u. The problem is converted to an equivalent elliptic problem over the open unit ball \(\mathbb {B}^{d}\) in \(\mathbb {R}^{d}\), say \(\widetilde {L}\widetilde {u} =\widetilde {f}(\cdot ,\widetilde {u})\). Then a spectral Galerkin method is used to create a convergent sequence of multivariate polynomials \(\widetilde {u} _{n}\) of degree ≤ n that is convergent to \(\widetilde {u}\). The transformation from Ω to \(\mathbb {B}^{d}\) requires a special analytical calculation for its implementation. With sufficiently smooth problem parameters, the method is shown to be rapidly convergent. For \(u\in C^{\infty } \left (\overline {\Omega }\right ) \) and assuming ?Ω is a C ^∞ boundary, the convergence of \(\left \Vert \widetilde {u} -\widetilde {u}_{n}\right \Vert _{H^{1}}\) to zero is faster than any power of 1/n. The error analysis uses a reformulation of the boundary value problem as an integral equation, and then it uses tools from nonlinear integral equations to analyze the numerical method. Numerical examples illustrate experimentally an exponential rate of convergence. A generalization to ?Δu + γ u = f(u) with a zero Neumann boundary condition is also presented.     

Power‐to‐Gas

The power‐to‐gas concept is a promising technology to chemically store energy and therefore a feasible approach to mitigate the challenges of energy transition. Heterogeneous catalysis plays a crucial role in CO₂ conversion to methane using nickel based catalysts. A thorough catalyst characterization facilitates the synthesis of optimized catalyst systems. The determination of reaction kinetics is fundamental for industrial reactor design.     

Adaptive space-time boundary element methods for the wave equation

For the solution of the wave equation a space-time energetic boundary integral formulation is used. The resulting single layer boundary integral equation is discretised by a conforming ansatz space on the lateral boundary. To derive an adaptive scheme an a posteriori error estimator based on the representation formula is used. Finally, numerical examples for a one-dimensional spatial domain are presented. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Fluid Structure Interaction of Blood Vessels in Circle of Willis

This work involves studying the role of the Brain Blood Barriers (BBB) on damping the stress applied on brain tissue through blood pressure that is the main cause of brain aneurysm. The numerical simulations are focused on the geometry of the Anterior Communicating Artery (ACoA) because there is a 30 % higher probability for a brain aneurysm in this location. A linear elastic model is used to model the structure part including BBB and brain tissue. Regarding the fluid properties, blood is assumed to behave as a Newtonian fluid. The stress and deformation of the brain tissue is analyzed. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The coordination chemistry of the proton

It is well known that an acidic hydrogen atom can form hydrogen bonds to a hydrogen bond acceptor, a Lewis base. It is considerably less known that the proton can coordinate two or more atoms conveniently in bonding modes that cannot be described as hydrogen bonding. Agostic interactions, bridging hydrides, 3-centre-2-electron bonds in boranes, bifurcated hydrogen atoms, they are all elements of the coordination chemistry of the proton and, of course, the hydrogen bond comes in more than one facette as well.     

Azobenzene-containing triazatriangulenium adlayers on Au(111): structural and spectroscopic characterization

Adlayers of different azobenzene-functionalized derivatives of the triazatriangulenium (TATA) platform on Au(111) surfaces were studied by scanning tunneling microscopy (STM), X-ray photoelectron spectroscopy (XPS), gap-mode surface-enhanced Raman spectroscopy (gap-mode SERS), and cyclic voltammetry (CV). The chemical composition of the adlayers is in good agreement with the molecular structure, i.e., different chemical groups attached to the azobenzene functionality were identified. Furthermore, the presence of the azobenzene moieties in the adlayers was verified by the vibration spectra and electrochemical data. These results indicate that the molecules remain intact upon adsorption with the freestanding functional groups oriented perpendicularly to the TATA platform and thus also to the substrate surface.     

Tuning the spin-transition properties of pyrene-decorated 2,6-bispyrazolylpyridine based Fe(II) complexes

Two 2,6-bispyrazolylpyridine ligands (bpp) were functionalized with pyrene moieties through linkers of different lengths. In the ligand 2,6-di(1H-pyrazol-1-yl)-4-(pyren-1-yl)pyridine (L1) the pyrene group is directly connected to the bpp moiety via a C-C single bond, while in the ligand 4-(2,6-di(1H-pyrazol-1-yl)pyridin-4-yl)benzyl-4-(pyren-1-yl)butanoate (L2) it is separated by a benzyl ester group involving a flexible butanoic chain. Subsequent complexation of Fe(II) salts revealed dramatic the influence of the nature of the pyrene substitution on the spin-transition behaviour of the resulting complexes. Thus, compound [Fe(L1)(2)](ClO(4))(2) (1) is blocked in its high spin state due to constraints caused by a strong intermolecular π-π stacking in its structure. On the other hand, the flexible chain of ligand L2 in compounds [Fe(L2)(2)](ClO(4))(2) (2) and [Fe(L2)(2)](BF(4))(2)·CH(3)CN·H(2)O (3) prevents structural constraints allowing for reversible spin transitions. Temperature-dependent studies of the photophysical properties of compound 3 do not reveal any obvious correlation between the fluorescence of the pyrene group and the spin state of the spin transition core.     

Proof complexity of propositional default logic

Default logic is one of the most popular and successful formalisms for non-monotonic reasoning. In 2002, Bonatti and Olivetti introduced several sequent calculi for credulous and skeptical reasoning in propositional default logic. In this paper we examine these calculi from a proof-complexity perspective. In particular, we show that the calculus for credulous reasoning obeys almost the same bounds on the proof size as Gentzen??s system LK. Hence proving lower bounds for credulous reasoning will be as hard as proving lower bounds for LK. On the other hand, we show an exponential lower bound to the proof size in Bonatti and Olivetti??s enhanced calculus for skeptical default reasoning.