New representation and factorizations of the higher-order ultraspherical-type differential equations

The paper deals with the class of linear differential equations of any even order 2α+4$2\alpha +4$, α∈_N0$\alpha \in {\mathbb{N}}_0$, which are associated with the so-called ultraspherical-type polynomials. These polynomials form an orthogonal system on the interval [−1,1]$[-1,1]$ with respect to the ultraspherical weight function (1−x²)^α${(1-x^2)}^{\alpha }$ and additional point masses of equal size at the two endpoints. The differential equations of “ultraspherical-type” were developed by R. Koekoek in 1994 by utilizing special function methods. In the present paper, a new and completely elementary representation of these higher-order differential equations is presented. This result is used to deduce the orthogonality relation of the ultraspherical-type polynomials directly from the differential equation property. Moreover, we introduce two types of factorizations of the corresponding differential operators of order 2α+4$2\alpha +4$ into a product of α+2$\alpha +2$ linear second-order operators.     

A Quantitative Regularity Estimate for Nonnegative Supersolutions of Uniformly Parabolic Equations

This note establishes an interior quantitative lower bound for nonnegative supersolutions of fully nonlinear uniformly parabolic equations. The result may be interpreted as a quantitative version of a growth lemma established by Krylov and Safonov for nonnegative supersolutions of linear uniformly parabolic equations in nondivergence form. Our approach is different, and follows from an application of a reverse Holder inequality. The result is the parabolic analogue of an elliptic regularity estimate established by Caffarelli, Souganidis, and Wang in the stochastic homogenization of fully nonlinear uniformly elliptic equations.     

Weak Euler Approximation for Itô Diffusion and Jump Processes

This article studies the rate of convergence of the weak Euler approximation for Itô diffusion and jump processes with Hölder-continuous generators. It covers a number of stochastic processes including the nondegenerate diffusion processes and a class of stochastic differential equations driven by stable processes. To estimate the rate of convergence, the existence of a unique solution to the corresponding backward Kolmogorov equation in Hölder space is first proved. It then shows that the Euler scheme yields positive weak order of convergence.     

First-principles calculation of structural stability,lattice dynamic and thermodynamic properties of BeX (X = S,Se and Te) compounds under high pressure

The phase transitions, lattice dynamical and thermodynamic properties of BeS, BsSe and BeTe at high pressure have been investigated with the density functional theory. The calculated equilibrium structural parameters agree well with the available experimental and theoretical values. The phase transition pressures from the zinc-blende (ZB) to the nickel arsenide (NiAs) phase of these compounds are determined. The calculated phonon dispersion curves of these compounds in ZB phase at zero pressure do not show any anomaly or instability. Dynamically, the ZB phase of BeS, BeSe and BeTe is found to be stable near transition pressures P_T. Within the quasiharmonic approximation, the thermodynamic properties including the thermal expansion coefficient, heat capacity at constant volume, heat capacity at constant pressure and entropy are predicted.     

Structure‐Based Design of Inhibitors of the Aspartic Protease Endothiapepsin by Exploiting Dynamic Combinatorial Chemistry

Structure‐based design (SBD) can be used for the design and/or optimization of new inhibitors for a biological target. Whereas de novo SBD is rarely used, most reports on SBD are dealing with the optimization of an initial hit. Dynamic combinatorial chemistry (DCC) has emerged as a powerful strategy to identify bioactive ligands given that it enables the target to direct the synthesis of its strongest binder. We have designed a library of potential inhibitors (acylhydrazones) generated from five aldehydes and five hydrazides and used DCC to identify the best binder(s). After addition of the aspartic protease endothiapepsin, we characterized the protein‐bound library member(s) by saturation‐transfer difference NMR spectroscopy. Cocrystallization experiments validated the predicted binding mode of the two most potent inhibitors, thus demonstrating that the combination of de novo SBD and DCC constitutes an efficient starting point for hit identification and optimization.     

Controlling Covalent Connection and Disconnection with Light

The on‐going need for feature miniaturization and the growing complexity of structures for use in nanotechnology demand the precise and controlled formation of covalent bonds at the molecular level. Such control requires the use of external stimuli that offer outstanding spatial, temporal, as well as energetic resolution. Thus, photoaddressable switches are excellent candidates for creating a system that allows reversible photocontrol over covalent chemical connection and disconnection. Here we show that the formation of covalent bonds between two reagents and their scission in the resulting product can be controlled exclusively by illumination with differently colored light. A furyl‐substituted photoswitchable diarylethene was shown to undergo a reversible Diels–Alder reaction with maleimide to afford the corresponding Diels–Alder adduct. Our system is potentially applicable in any field already relying on the benefits of reversible Diels–Alder reactions.     

Selector‐Induced Dynamic Deracemization of a Selectand‐Modified Tropos BIPHEPO‐Ligand: Application in the Organocatalyzed Asymmetric Double‐Aldol‐Reaction

Stereolabile interconverting catalysts open up the possibility of directing enantioselectivity in asymmetric synthesis by formation of diastereomeric complexes with chiral auxiliaries and deracemization. However, the stoichiometrically used auxilliaries can significantly limit the potential applications of such systems. We synthesized a new BIPHEPO tropos ligand containing achiral selectands in the backbone, which forms transient diastereomeric associates with amylose‐tris‐3,5‐dimethylphenyl carbamate as a selector and thus deracemizes. The enantiomerically enriched BIPHEPO obtained was successfully used in the organocatalytic asymmetric double aldol addition of substituted methyl ketones to form benzaldehyde. This strategy combines an on‐column deracemization with the high stereoinduction of chiral biarylphosphineoxides and opens up new possibilities in the field of self‐amplified asymmetric syntheses.     

A Tetrameric Cage with D2h Symmetry through Alkyne Metathesis

Shape‐persistent covalent organic polyhedrons (COPs) with ethynylene linkers are usually prepared through kinetically controlled cross‐coupling reactions. The high‐yielding synthesis of ethynylene‐linked rigid tetrameric cages via one‐step alkyne metathesis from readily accessible triyne precursors is presented. The tetrameric cage contains two macrocyclic panels and exhibits D_2h symmetry. The assembly of such a COP is a thermodynamically controlled process, which involves the initial formation of macrocycles as key intermediates followed by the connection of two macrocycles with ethynylene linkages. With a large internal cavity, the cage exhibits a high binding selectivity toward C₇₀ (K=3.9×10³ L mol^?1) over C₆₀ (no noticeable binding).     

Diastereo‐ and Enantioselective anti‐Selective Hydrogenation of α‐Amino‐β‐keto Ester Hydrochlorides and Related Compounds Using Transition‐Metal–Chiral‐Bisphosphine Catalysts

This review describes our recent works on the diastereo‐ and enantioselective synthesis of anti‐β‐hydroxy‐α‐amino acid esters using transition‐metal–chiral‐bisphosphine catalysts. A variety of transition metals, namely ruthenium (Ru), rhodium (Rh),iridium (Ir), and nickel (Ni), in combination with chiral bisphosphines, worked well as catalysts for the direct anti‐selective asymmetric hydrogenation of α‐amino‐β‐keto ester hydrochlorides, yielding anti‐β‐hydroxy‐α‐amino acid esters via dynamic kinetic resolution (DKR) in excellent yields and diastereo‐ and enantioselectivities. The Ru‐catalyzed asymmetric hydrogenation of α‐amino‐β‐ketoesters via DKR is the first example of generating anti‐β‐hydroxy‐α‐amino acids. Complexes of iridium and axially chiral bisphosphines catalyze an efficient asymmetric hydrogenation of α‐amino‐β‐keto ester hydrochlorides via dynamic kinetic resolution. A homogeneous Ni–chiral‐bisphosphine complex also catalyzes an efficient asymmetric hydrogenation of α‐amino‐β‐keto ester hydrochlorides in an anti‐selective manner. As a related process, the asymmetric hydrogenation of the configurationally stable substituted α‐aminoketones using a Ni catalyst via DKR is also described.