A Lax equivalence theorem for stochastic differential equations

In this paper, a stochastic mean square version of Lax’s equivalence theorem for Hilbert space valued stochastic differential equations with additive and multiplicative noise is proved. Definitions for consistency, stability, and convergence in mean square of an approximation of a stochastic differential equation are given and it is shown that these notions imply similar results as those known for approximations of deterministic partial differential equations. Examples show that the assumptions made are met by standard approximations.     

On the number of lattice points in a small sphere and a recursive lattice decoding algorithm

Let L be a lattice in ${\mathbb{R}^n}$ . This paper provides two methods to obtain upper bounds on the number of points of L contained in a small sphere centered anywhere in ${\mathbb{R}^n}$ . The first method is based on the observation that if the sphere is sufficiently small then the lattice points contained in the sphere give rise to a spherical code with a certain minimum angle. The second method involves Gaussian measures on L in the sense of Banaszczyk (Math Ann 296:625–635, 1993). Examples where the obtained bounds are optimal include some root lattices in small dimensions and the Leech lattice. We also present a natural decoding algorithm for lattices constructed from lattices of smaller dimension, and apply our results on the number of lattice points in a small sphere to conclude on the performance of this algorithm.     

(51)V solid-state NMR investigations and DFT studies of model compounds for vanadium haloperoxidases

Three cis-dioxovanadium(V) complexes with similar N-salicylidenehydrazide ligands modeling hydrogen bonding interactions of vanadate relevant for vanadium haloperoxidases are studied by (51)V solid-state NMR spectroscopy. Their parameters describing the quadrupolar and chemical shift anisotropy interactions (quadrupolar coupling constant C(Q), asymmetry of the quadrupolar tensor eta(Q), isotropic chemical shift delta(iso), chemical shift anisotropy delta(sigma), asymmetry of the chemical shift tensor eta(sigma) and the Euler angles alpha, beta and gamma) are determined both experimentally and theoretically using DFT methods. A comparative study of different methods to determine the NMR parameters by numerical simulation of the spectra is presented. Detailed theoretical investigations on the DFT level using various basis sets and structural models show that by useful choice of the methodology, the calculated parameters agree to the experimental ones in a very good manner.     

Predicting 19F NMR Chemical Shifts: A Combined Computational and Experimental Study of a Trypanosomal Oxidoreductase–Inhibitor Complex

The absence of fluorine from most biomolecules renders it an excellent probe for NMR spectroscopy to monitor inhibitor–protein interactions. However, predicting the binding mode of a fluorinated ligand from a chemical shift (or vice versa) has been challenging due to the high electron density of the fluorine atom. Nonetheless, reliable ¹⁹F chemical-shift predictions to deduce ligand-binding modes hold great potential for in silico drug design. Herein, we present a systematic QM/MM study to predict the ¹⁹F NMR chemical shifts of a covalently bound fluorinated inhibitor to the essential oxidoreductase tryparedoxin (Tpx) from African trypanosomes, the causative agent of African sleeping sickness. We include many protein–inhibitor conformations as well as monomeric and dimeric inhibitor–protein complexes, thus rendering it the largest computational study on chemical shifts of ¹⁹F nuclei in a biological context to date. Our predicted shifts agree well with those obtained experimentally and pave the way for future work in this area.     

Silanylidene and Germanylidene Anions: Valence‐Isoelectronic Species to the Well‐Studied Phosphinidene

The reduction of TipMCl₃ (Tip=2,4,6‐triisopropylphenyl) (M=Si, Ge) with KC₈ in the presence of cyclic alkyl(amino) carbene (cAAC) afforded the acyclic silanylidene and germanylidene anions in the form of potassium salt [K(cAAC)MTip]₂ (M=Si ( 1 ); Ge ( 2 )). The silanylidene and germanylidene anions are valence‐isoelectronic to the well‐studied phosphinidene and are a new class of acyclic anions of Group 14. Compounds 1 and 2 were isolated and well characterized by NMR and single‐crystal X‐ray structure analysis. Furthermore, the structure and bonding of compounds 1 and 2 was investigated by computational methods.     

Synthesis of a polyrotaxane-based macroporous polymer as stationary phase for capillary electrochromatography via host-guest complexation of N,N '-ethylenedianilinediacrylamide with statistically methylated beta-cyclodextrin

A synthetic route to acrylamide-based monolithic stationary phases for CEC with rotaxane-type immobilized derivatized beta-CD was explored. N,N'-Ethylenedianilinediacrylamide was synthesized as the water-insoluble crosslinker forming water-soluble inclusion complexes with statistically methylated beta-CD. Mixed-mode stationary phases were synthesized by free radical copolymerization of the bisacrylamide-CD host-guest complex with water-soluble monomers and an additional water-soluble crosslinker in aqueous solution. Complex formation in solution and inclusion of the pseudorotaxane into the polymeric network (formation of a polyrotaxane architecture) were studied by means of (1)H-NMR chemical shift analysis, CD modified micellar EKC (CD-MEKC), 2D-NOESY spectroscopy, and solid state( 13)C-NMR spectroscopy. The presence of a mixed-mode selectivity of the stationary phase based on hydrophobic and hydrophilic interaction was confirmed by CEC with neutral polar and nonpolar solutes.     

Synthesis and Characterization of Gd2[Pt2(SO4)4(HSO4)2](HSO4)2

Red single crystals of Gd₂[Pt₂(SO₄)₄(HSO₄)₂](HSO₄)₂ (triclinic, , Z = 1, a = 844.02(9), b = 908.50(9), c = 939.49(8) pm, α = 107.73(1)°, β = 112.10(1)°, γ = 103.53(1)°) were obtained by the reaction of [Gd(NO₃)(H₂O)₇][PtCl₆]·4H₂O with sulfuric acid at 320 °C in a sealed glass ampoule. In the crystal structure, Pt₂ dumbbells are coordinated by four chelating sulfate groups and two monodentate hydrogensulfate ions. Two further HSO₄^? ions are not bonded to the Pt₂ dumbbell. The Gd³⁺ ions are eightfold coordinated by oxygen atoms. The IR data of Gd₂[Pt₂(SO₄)₄(HSO₄)₂](HSO₄)₂ are typical for these type of compounds. The thermal decomposition of the compound leads to elemental platinum and Gd₂O₃.     

Hexachloroplatinates of the Lanthanides: Syntheses and Thermal Decomposition of [M(NO3)2(H2O)6]2[PtCl6]·2H2O (M = La,Pr) and [M(NO3)(H2O)7][PtCl6]·4H2O (M = Gd,Dy)

The reaction of the nitrates M(NO₃)₃·6H₂O (M = La, Pr) and (H₃O)₂PtCl₆ led to yellow single crystals of [M(NO₃)₂(H₂O)₆]₂[PtCl₆]·2H₂O (M = La, Pr) (monoclinic, P2₁/c, Z = 2, La/Pr: a = 697.4(3)/695.5(1), b = 1654.5(1)/1652.5(2), c = 1317.7(6)/1318.5(3) pm, β = 93.97°(7)/93.93°(2), R_all = 0.0169/0.0659) while the reaction of M(NO₃)₃·5H₂O (M = Gd, Dy) and (H₃O)₂PtCl₆ yielded yellow single crystals of [M(NO₃)(H₂O)₇][PtCl₆]·4H₂O (monoclinic, P2₁/n, Z = 4, Gd/Dy: a = 838.72(3)/838.40(2), b = 2131.98(6)/2139.50(7), c = 1142.63(3)/1143.10(3) pm, β = 95.670(4)/95.698(3), R_all = 0.0475/0.0337). The crystal structures consist of octahedral [PtCl₆]^2? anions and complex [M(NO₃)₂(H₂O)₆]₂⁺ and [M(NO₃)(H₂O)₇]²⁺ cations, respectively. The thermal decomposition of both types of compounds leads via various steps to elemental platinum and the oxide chlorides MOCl (M = La, Pr, Gd, Dy).     

Visualization of structural changes in cellulosic substrates during enzymatic hydrolysis using multimodal nonlinear microscopy

Enzymatic hydrolysis of cellulose provides a renewable source of monosaccharides for production of variety of biochemicals and biopolymers. Unfortunately, the enzymatic hydrolysis of cellulose is often incomplete, and the reasons are not fully understood. We have monitored enzymatic hydrolysis in terms of molecular density, ordering and autofluorescence of cellulose structures in real time using simultaneous CARS, SHG and MPEF microscopy with the aim of contributing to the understanding and optimization of the enzymatic hydrolysis of cellulose. Three cellulose-rich substrates with different supramolecular structures, pulp fibre, acid-treated pulp fibre and Avicel, were studied at microscopic level. The microscopy studies revealed that before enzymatic hydrolysis Avicel had the greatest carbon-hydrogen density, while pulp fibre and acid-treated fibre had similar density. Monitoring of the substrates during enzymatic hydrolysis revealed the double exponential SHG decay for pulp fibre and acid-treated fibre indicating two phases of the process. Acid-treated fibre was hydrolysed most rapidly and the hydrolysis of pulp fibre was spatially non-uniform leading to fractioning of the particles, while the hydrolysis of Avicel was more than an order of magnitude slower than that of both fibres.