Necessary and sufficient conditions for Hölder continuity of Gaussian processes

The continuity of Gaussian processes is an extensively studied topic and it culminates in Talagrand’s notion of majorizing measures that gives complicated necessary and sufficient conditions. In this note we study the Hölder continuity of Gaussian processes. It turns out that necessary and sufficient conditions can be stated in a simple form that is a variant of the celebrated Kolmogorov–Čentsov condition.     

半定矩阵秩极小的非凸精确松弛

本文主要研究半定矩阵秩极小问题(P)的非凸精确松弛及其性质.首先,为求解问题(P),我们引入其Schatten p-范数(0＜p＜1)松弛,记为(Sp).其次,通过定义半定限制等距常数和半定限制正交常数,我们给出了问题(P)有唯—解的充分条件.最后,利用半定限制等距性质,我们给出了问题(P)和(Sp)有相同唯一解的充分条件.特别地,对任意0＜p＜1,我们还得到—个一致的精确恢复条件.     

Comment on “On the Neumann function and the method of images in spherical and ellipsoidal geometry”

In this note, we point out two errors in the article “On the Neumann function and the method of images in spherical and ellipsoidal geometry” by Dassios and Sten. Two corrections are then proposed.     

A Singular Limit Problem for Conservation Laws Related to the Camassa–Holm Shallow Water Equation

We consider a shallow water equation of the Camassa–Holm type, which contains nonlinear dispersive effects as well as fourth order dissipative effects. We prove that as the diffusion and dispersion parameters tend to zero, with a condition on the relative balance between these two parameters, smooth solutions of the shallow water equation converge to discontinuous weak solutions of a scalar conservation law. The proof relies on deriving suitable a priori estimates together with an application of the compensated compactness method in the L ^p setting.     

Effect of incoherent scattering on three-terminal quantum Hall thermoelectrics

A three-terminal conductor presents peculiar thermoelectric and thermal properties in the quantum Hall regime: it can behave as a symmetric rectifier and as an ideal thermal diode. These properties rely on the coherent propagation along chiral edge channels. We investigate the effect of breaking the coherent propagation by the introduction of a probe terminal. It is shown that chiral effects not only survive the presence of incoherence but they can even improve the thermoelectric performance in the totally incoherent regime.     

Nonlinear dynamics for proper ferroelastic transitions for which the landau condition is violated

On the basis of a lattice model the domain structure for ferroelastic transformations is examined. The model accounts for both strongly nonlinear and competing interactions which allow for, in some situations, the propagation of nonlinear excitations. The model can be mostly applied to proper ferroelastic transformations of which In-Tl, Ti-Ni, etc.… are good prototypes. The phonon dispersion of the transverse acoustic waves obtained in the linearized case is discussed and the results show first a phonon softening at nonzero wave-number and next an upwards convexity of the phonon branch near the long-wavelength limit. This can be seen as pre-transitional effects. In a fully nonlinear case we consider vanishing dilatation transformations and the continuum approximation is applied to the one-dimensional version. Then, we investigate nonlinear excitations; three main classes of solution are found: i) quasi-harmonic solutions corresponding to periodically modulated structures in space which is a precursor effect of the elastic transformation, ii) an array of solitons made of periodic arrangements of parent-elastic domains and iii) a moving strain soliton. All the significant results are numerically illustrated by means of the microscopic model. Finally, the similarity to martensitic transformations and some extensions of the model are outlined.     

An efficient stereoselective synthesis of aminocyclohexenes from 7-azanorbornenes via LiAlH4 mediated tandem double bond migration-ring opening sequence

hNav1.7 had received great interest in the past decade owing to its attractive mechanism of actions in pain processing pathway. More recently, we developed a series of efficacious hNav1.7 inhibitors for analgesic, which were characterized with a novel aminocyclohexene as pharmacophore. Herein, we reported our synthetic efforts to construct the unique scaffold, which can be rapidly accessed from readily available 7-azanorbornenes via an unexpected catalyst-free, LiAlH₄ mediated tandem double bond migration-ring opening sequence. A probable mechanism was proposed to illuminate the good regio- and stereoselectivity.     

Application of a biological‐based nanomagnetic catalyst in the synthesis of bis‐pyrazols and pyrano[3,2‐c]pyrazoles

{Fe₃O₄@SiO₂@(CH₂)₃‐thiourea dioxide‐SO₃H/HCl}, a newly reported nanomagnetic core–shell supported solid acid catalyst, was successfully employed in the preparation of 4,4′‐(arylmethylene)bis(1H –pyrazol‐5‐ol) and pyrano[3,2‐c ]pyrazole derivatives. The presented methods are very efficient and high‐yielding. Also, the catalyst exhibited powerful potential for reusability in both types of reactions.     

Coupling influence on the refractive index sensitivity of photonic wire ring resonator

By taking the coupling effect into consideration, we study the resonance condition of a photonic wire microring resonator (PWRR) sensor and compare our results with the previous work. Simulation results show that the resonant wavelength and sensitivity strongly depend on the coupling strength. The difference caused by the coupling effect can be up to tens of nanometers for the resonant peak position and tens of nm/RIU for the sensitivity in a silicon-on-insulator (SOI) PWRR. Such a giant influence from coupling effect cannot be disregarded and should be considered seriously for the design and application of PWRRs. It also shows an alternative tuning technique by controlling the coupling strength.     

Two optimal Dai–Liao conjugate gradient methods

Two adaptive choices for the parameter of Dai–Liao conjugate gradient (CG) method are suggested. One of which is obtained by minimizing the distance between search directions of Dai–Liao method and a three-term CG method proposed by Zhang et al. and the other one is obtained by minimizing Frobenius condition number of the search direction matrix. Global convergence analyses are made briefly. Numerical results are reported; they demonstrate effectiveness of the suggested adaptive choices.