Equilibrium Diffusion on the Cone of Discrete Radon Measures

Let \({\mathbb {K}(\mathbb {R}^{d})}\) denote the cone of discrete Radon measures on \(\mathbb {R}^{d}\). There is a natural differentiation on \(\mathbb {K}(\mathbb {R}^{d})\): for a differentiable function \(F:\mathbb {K}(\mathbb {R}^{d})\to \mathbb {R}\), one defines its gradient \(\nabla ^{\mathbb {K}}F\) as a vector field which assigns to each \(\eta \in \mathbb {K}(\mathbb {R}^{d})\) an element of a tangent space \(T_{\eta }(\mathbb {K}(\mathbb {R}^{d}))\) to \(\mathbb {K}(\mathbb {R}^{d})\) at point η. Let \(\phi :\mathbb {R}^{d}\times \mathbb {R}^{d}\to \mathbb {R}\) be a potential of pair interaction, and let μ be a corresponding Gibbs perturbation of (the distribution of) a completely random measure on \(\mathbb {R}^{d}\). In particular, μ is a probability measure on \(\mathbb {K}(\mathbb {R}^{d})\) such that the set of atoms of a discrete measure \(\eta \in \mathbb {K}(\mathbb {R}^{d})\) is μ-a.s. dense in \(\mathbb {R}^{d}\). We consider the corresponding Dirichlet form

$$\mathcal{E}^{\mathbb{K}}(F,G)={\int}_{\mathbb K(\mathbb{R}^{d})}\langle\nabla^{\mathbb{K}} F(\eta), \nabla^{\mathbb{K}} G(\eta)\rangle_{T_{\eta}(\mathbb{K})}\,d\mu(\eta). $$

Integrating by parts with respect to the measure μ, we explicitly find the generator of this Dirichlet form. By using the theory of Dirichlet forms, we prove the main result of the paper: If d ≥ 2, there exists a conservative diffusion process on \(\mathbb {K}(\mathbb {R}^{d})\) which is properly associated with the Dirichlet form \(\mathcal {E}^{\mathbb {K}}\).

     

Structure and mechanical property variations in Mg–Gd–Y–Zn–Zr alloy depending on its composition and processing conditions

An effect of alloying element content on mechanical properties and precipitate formation in Mg–RE alloys was studied for Mg–8Gd–4Y–1Zn–0.4Zr (wt%) and Mg–10Gd–5Y–1.8Zn–0.4Zr (wt%). It is shown that small variations in the alloying element concentration can be used to manipulate the alloy microstructure and precipitate formation towards eliminating the asymmetry (tension/compression) and anisotropy of yield stress.     

Homochiral Ketals in Organic Synthesis. Enantioselective Synthesis of (4S, 5S)-4,5-Dihydroxycyclopent-2-En-1-One Isopropylidine Ketal

The title compound has been synthesized in 17% yield over six steps from 2-cyclopenten-1-one. The key step is catalytic osmylation of 4-bromo-2-cyclopenten-1-one 1,4-di-O-benzyl-D-threitol ketal to produce a chromatographically separable mixture of diastereomeric diols.     

Phosphorinane and Enol Rings in One Molecule. Evidence for Reciprocal Stabilization of Half-Chair Conformations

Abstract.

The X-ray crystal structure of 2-(2′,4′-dioxo-3′-pentyl)-5,5-dimethyl-2-oxo-1,3,2-dioxaphosphorinane (2) reveals significant half-chair distortion of the axially oriented cisenol ring. The molecule also undergoes in-plane deformations. R(O...O) = 2.410 Å in the enol moiety indicates a very strong hydrogen bonding. The enol content, δ_OH and thermodynamic parameters for the axial-equatorial conformational and keto-enol equilibriums were obtained from ¹H, ³¹P NMR and IR measurements in comparison with the planar 4,6-dimethyl isomer (1) containing equatorially oriented enol ring. The X-ray single crystal structure of 5,5-dimethyl-2-(methoxycarbonyl-3′-oxo-2′-butyl)-2-oxo-1,3,2-dioxaphosphorinane (3) reveals the unusual half-chair conformation of the dioxaphosphorinane cycle disposed a trans-enol ring substituent. ¹H, ³¹P NMR and IR solution data support the same structure displays a strong conformational preference while the minor forms are chair conformers with an axial or equatorial cis-enol ring.     

NOTE: Synthesis and Characterization of a Poly(acrylamide) with Pendant 1,4-piperazine-2,5-dione Moieties via Post-polymerization Cyclization

A poly(acrylamide) was synthesized from N ^α -Boc-N ^? -acrolyl-l-lysylglycine methyl ester via radical polymerization. This polymer typically had Mn ～ 100,000 g/mol, Mw ～ 300,000 g/mol, and a T_g of 93°C. Removal of Boc with TFA and cyclization with DABCO? in DMSO at 65°C afforded a soluble piperazinedione-containing polymer that had a T_g of 157°C and thermal stability up to 300°C. These results demonstrate a viable and efficient synthetic route to piperazinedione-containing polyacrylamides of high molecular weight. Related polymers that incorporate substituted indane moieties could be useful high T_g materials for fabrication of LC and NLO devices.     

Computational Insights into the Catalytic Mechanism of Is-PETase: An Enzyme Capable of Degrading Poly(ethylene) Terephthalate

Is-PETase has become an enzyme of significant interest due to its ability to catalyse the degradation of polyethylene terephthalate (PET) at mesophilic temperatures. We performed hybrid quantum mechanics and molecular mechanics (QM/MM) at the DSD-PBEP86-D3/ma-def2-TZVP/CHARMM27//rev-PBE-D3/dev2-SVP/CHARMM level to calculate the energy profile for the degradation of a suitable PET model by this enzyme. Very low overall barriers are computed for serine protease-type hydrolysis steps (as low as 34.1 kJ mol⁻¹). Spontaneous deprotonation of the final product, terephthalic acid, with a high computed driving force indicates that product release could be rate limiting.     

The Scaling Limit of the Critical One-Dimensional Random Schrödinger Operator

We consider two models of one-dimensional discrete random Schrödinger operators

$(H_n\psi)_\ell =\psi_{\ell -1}+\psi_{\ell +1}+v_\ell \psi_\ell$

, \({\psi_0=\psi_{n+1}=0}\) in the cases \({ v_k=\sigma \omega_k/\sqrt{n}}\) and \({ v_k=\sigma \omega_k/ \sqrt{k}}\) . Here ω _k are independent random variables with mean 0 and variance 1.

We show that the eigenvectors are delocalized and the transfer matrix evolution has a scaling limit given by a stochastic differential equation. In both cases, eigenvalues near a fixed bulk energy E have a point process limit. We give bounds on the eigenvalue repulsion, large gap probability, identify the limiting intensity and provide a central limit theorem.In the second model, the limiting processes are the same as the point processes obtained as the bulk scaling limits of the β-ensembles of random matrix theory. In the first model, the eigenvalue repulsion is much stronger.     

Fast electro-optical mode in photo-aligned reflective deformed helix ferroelectric liquid crystal cells

The electro-optical behavior of deformed helix ferroelectric liquid crystal in reflective mode is described in this paper. The electrically controlled reflectance has been measured at subkilohertz driving voltage frequency for different polarizations of the incident light and compared quite successfully with the simulation results.