Convergence rates for sparse chaos approximations of elliptic problems with stochastic coefficients

^** Email: todor{at}math.ethz.ch^*** Corresponding author. Email: schwab{at}math.ethz.ch A scalar, elliptic boundary-value problem in divergence formwith stochastic diffusion coefficient a(x, ) in a bounded domainD ^d is reformulated as a deterministic, infinite-dimensional,parametric problem by separation of deterministic (x D) andstochastic ( ) variables in a(x, ) via Karhúnen–Loèveor Legendre expansions of the diffusion coefficient. Deterministic,approximate solvers are obtained by projection of this probleminto a product probability space of finite dimension M and sparsediscretizations of the resulting M-dimensional parametric problem.Both Galerkin and collocation approximations are considered.Under regularity assumptions on the fluctuation of a(x, ) inthe deterministic variable x, the convergence rate of the deterministicsolution algorithm is analysed in terms of the number N of deterministicproblems to be solved as both the chaos dimension M and themultiresolution level of the sparse discretization resp. thepolynomial degree of the chaos expansion increase simultaneously.     

On the spectral norm of algebraic numbers

In this paper we continue to study the spectral norms and their completions ([4]) in the case of the algebraic closure $ \overline {\mathbb Q} $ of ? in ?. Let $ \widetilde{\overline{\mathbb{Q}}} $ be the completion of $ \overline {\mathbb Q} $ relative to the spectral norm. We prove that $ \widetilde{\overline{\mathbb{Q}}} $ can be identified with the R‐subalgebra of all symmetric functions of C(G), where C(G) denotes the ?‐Banach algebra of all continuous functions defined on the absolute Galois group G = Gal$ {\overline {\mathbb Q}} / {\mathbb Q} $. We prove that any compact, closed to conjugation subset of ? is the pseudo‐orbit of a suitable element of $ \widetilde{\overline{\mathbb{Q}}} $. We also prove that the topological closure of any algebraic number field in $ \widetilde{\overline{\mathbb{Q}}} $ is of the form $\widetilde{\mathbb{Q}[x]}$ with x in $ \widetilde{\overline{\mathbb{Q}}} $.     

On the Global Regularity for a Wave-Klein—Gordon Coupled System

In this paper we consider a coupled Wave-Klein—Gordon system in 3D, and prove global regularity and modified scattering for small and smooth initial data with suitable decay at infinity. This...     

Connections on the Total Space of a Holomorphic Lie Algebroid

The main purpose of the paper is the study of the total space of a holomorphic Lie algebroid E. The paper is structured in three parts. In the first section, we briefly introduce basic notions on holomorphic Lie algebroids. The local expressions are written and the complexified holomorphic bundle is introduced. The second section presents two approaches on the study of the geometry of the complex manifold E. The first part contains the study of the tangent bundle \(T_{\mathbb {C}}E=T'E\oplus T''E\) and its link, via the tangent anchor map, with the complexified tangent bundle \(T_{\mathbb {C}}(T'M)=T'(T'M)\oplus T''(T'M)\). A holomorphic Lie algebroid structure is emphasized on \(T'E\). A special study is made for integral curves of a spray on \(T'E\). Theorem 2.8 gives the coefficients of a spray, called canonical, obtained from a complex Lagrangian on \(T'E\). In the second part of section two, we study the holomorphic prolongation \(\mathcal {T}'E\) of the Lie algebroid E. In the third section, we study how a complex Lagrange (Finsler) structure on \(T'M\) induces a Lagrangian structure on E. Three particular cases are analysed by the rank of the anchor map, the dimensions of manifold M, and those of the fibres. We obtain the correspondent on E of the Chern–Lagrange nonlinear connection from \(T'M\).     

Transition‐Metal‐Free Reductive Hydroxymethylation of Isoquinolines

A transition‐metal‐free reductive hydroxymethylation reaction has been developed, enabling the preparation of tetrahydroisoquinolines bearing C4‐quaternary centers from the corresponding isoquinolines. Deuterium labelling studies and control experiments enable a potential mechanism to be elucidated which features a key Cannizzaro‐type reduction followed by an Evans–Tishchenko reaction. When isoquinolines featuring a proton at the 4‐position are used, a tandem methylation‐hydroxymethylation occurs, leading to the formation of 2 new C?C bonds in one pot.     

A Theoretical Model for Release Dynamics of an Antifungal Agent Covalently Bonded to the Chitosan

The aim of the study was to create a mathematical model useful for monitoring the release of bioactive aldehydes covalently bonded to the chitosan by reversible imine linkage, considered as a polymer–drug system. For this purpose, two hydrogels were prepared by the acid condensation reaction of chitosan with the antifungal 2-formyl-phenyl-boronic acid and their particularities; influencing the release of the antifungal aldehyde by shifting the imination equilibrium to the reagents was considered, i.e., the supramolecular nature of the hydrogels was highlighted by polarized light microscopy, while scanning electron microscopy showed their microporous morphology. Furthermore, the in vitro fungicidal activity was investigated on two fungal strains and the in vitro release curves of the antifungal aldehyde triggered by the pH stimulus were drawn. The theoretical model was developed starting from the hypothesis that the imine-chitosan system, both structurally and functionally, can be assimilated, from a mathematical point of view, with a multifractal object, and its dynamics were analyzed in the framework of the Scale Relativity Theory. Thus, through Riccati-type gauges, two synchronous dynamics, one in the scale space, associated with the fungicidal activity, and the other in the usual space, associated with the antifungal aldehyde release, become operational. Their synchronicity, reducible to the isomorphism of two SL(2R)-type groups, implies, by means of its joint invariant functions, bioactive aldehyde compound release dynamics in the form of “kink–antikink pairs” dynamics of a multifractal type. Finally, the theoretical model was validated through the experimental data.     

Vapour pressure and excess Gibbs energy of binary 1,2-dichloroethane + cyclohexanone,chloroform + cyclopentanone and chloroform + cyclohexanone mixtures at temperatures from 298.15 to 318.15 K

The vapour pressures of the binary systems 1,2-dichloroethane + cyclohexanone, chloroform + cyclopentanone and chloroform + cyclohexanone mixtures were measured at temperatures between 298.15 and 318.15 K. The vapour pressures vs. liquid phase composition data for three isotherms have been used to calculate the activity coefficients of the two components and the excess molar Gibbs energies, G^E, for these mixtures, using Barker's method. Redlich–Kister, Wilson, NRTL and UNIQUAC equations, taking into account the vapour phase imperfection in terms of the 2-nd virial coefficient, have represented the G^E values. No significant difference between G^E values obtained with these equations has been observed. Our data on vapour–liquid equilibria (VLE) and excess properties of the studied systems are examined in terms of the DISQUAC and modified UNIFAC (Dortmund) predictive group contributions models.     

Synthesis,Characterisation, and in vitro Cytotoxic Activity of Dithiocarbamato Glycoconjugate Half-Sandwich Ruthenium and Osmium Complexes

The synthesis by transmetallation and in-depth characterisation by IR spectroscopy of five half-sandwich ruthenium and osmium dithiocarbamato isonipecotamide glycoconjugate complexes are presented herein. The evaluation of their in vitro cytotoxicity and of the zinc precursors’ in vitro cytotoxicity towards ovarian cancer (A2780 and A2780cisR) and normal prostate (PNT2) cells is reported. While the cytotoxicity of the compounds is rather limited, some selectivity is observed.     

Phase Behavior of a Designed Cyclopropyl Analogue of Monoolein: Implications for Low‐Temperature Membrane Protein Crystallization

Lipidic cubic phases (LCPs) are used in areas ranging from membrane biology to biodevices. Because some membrane proteins are notoriously unstable at room temperature, and available LCPs undergo transformation to lamellar phases at low temperatures, development of stable low‐temperature LCPs for biophysical studies of membrane proteins is called for. Monodihydrosterculin (MDS) is a designer lipid based on monoolein (MO) with a configurationally restricted cyclopropyl ring replacing the olefin. Small‐angle X‐ray scattering (SAXS) analyses revealed a phase diagram for MDS lacking the high‐temperature, highly curved reverse hexagonal phase typical for MO, and extending the cubic phase boundary to lower temperature, thereby establishing the relationship between lipid molecular structure and mesophase behavior. The use of MDS as a new material for LCP‐based membrane protein crystallization at low temperature was demonstrated by crystallizing bacteriorhodopsin at 20 °C as well as 4 °C.