Distribution dependent stochastic differential equations

Due to their intrinsic link with nonlinear Fokker-Planck equations and many other applications, distribution dependent stochastic differential equations (DDSDEs) have been intensively investigated. In this paper, we summarize some recent progresses in the study of DDSDEs, which include the correspondence of weak solutions and nonlinear Fokker-Planck equations, the well-posedness, regularity estimates, exponential ergodicity, long time large deviations, and comparison theorems.     

On the Relative Twist Formula of l-adic Sheaves

We propose a conjecture on the relative twist formula of l-adic sheaves, which can be viewed as a generalization of Kato—Saito's conjecture. We verify this conjecture under some transversal assumptions. We also define a relative cohomological characteristic class and prove that its formation is compatible with proper push-forward. A conjectural relation is also given between the relative twist formula and the relative cohomological characteristic class.     

Self-emulsification in chemical and pharmaceutical technologies

The interest in the low energy self-emulsification techniques has exploded in the recent years, driven by three main trends: by the transition to “greener” technologies in both its aspects—less energy consumption and replacement of the petrochemicals by natural ingredients; by the costly and maintenance demanding equipment for nanoemulsification; and by the quest for efficient and robust self-emulsifying formulations for oral drug delivery. Here, we first present a brief overview of the main known low-energy methods for nanoemulsion formation, focusing on their mechanistic understanding and discussing some recent advances in their development and applications. Next, we review three conceptually new approaches for self-emulsification in chemical technologies, discovered in the last several years. The colloidal features and the specific requirements of the self-emulsifying drug-delivery systems (SEDDS) are also discussed briefly. Finally, we summarize the current trends and the main challenges in this vivid research area.     

Solvent- and Substrate-Induced Chiroptical Inversion in Amphiphilic,Biocompatible Glycoconjugate Supramolecules: Shape-Persistent Gelation,Self-Healing,and Antibacterial Activity

We have shown solvent- and substrate-dependent chiral inversion of a few glycoconjugate supramolecules. (Z)-F-Gluco, in which d -glucosamine has been attached chemically to Cbz-protected l -phenylalanine at the C terminus, forms a self-healing hydrogel through intertwining of the nanofibers wherein the gelators undergo lamellar packing in the β-sheet secondary structures with a single chiral handedness. Dihybrid (Z)-F-gluco nanocomposite gel was prepared by in-situ formation of silver nanoparticles AgNPs in the gel; this enhances the mechanical properties of the composite gel through physical crosslinking without altering the packing pattern. In contrast, (Z)-L-gluco bearing an l -leucine moiety does not form a hydrogel but an organogel. Interestingly, the chiral handedness of the aggregates of (Z)-L-gluco can be reversed by choosing suitable solvents. In addition to self-healing behavior, (Z)-L-gluco gel revealed shape persistency. Further, (Z)-F-gluco hydrogel is benign, nontoxic, non-immunogenic, and non-allergenic in animal cells. AgNP-loaded (Z)-F-gluco hydrogel showed antibacterial activity against both Gram-positive and Gram-negative bacteria.     

A Structural Entropy Measurement Principle of Propositional Formulas in Conjunctive Normal Form

The satisfiability (SAT) problem is a core problem in computer science. Existing studies have shown that most industrial SAT instances can be effectively solved by modern SAT solvers while random SAT instances cannot. It is believed that the structural characteristics of different SAT formula classes are the reasons behind this difference. In this paper, we study the structural properties of propositional formulas in conjunctive normal form (CNF) by the principle of structural entropy of formulas. First, we used structural entropy to measure the complex structure of a formula and found that the difficulty solving the formula is related to the structural entropy of the formula. The smaller the compressing information of a formula, the more difficult it is to solve the formula. Secondly, we proposed a λ-approximation strategy to approximate the structural entropy of large formulas. The experimental results showed that the proposed strategy can effectively approximate the structural entropy of the original formula and that the approximation ratio is more than 92%. Finally, we analyzed the structural properties of a formula in the solution process and found that a local search solver tends to select variables in different communities to perform the next round of searches during a search and that the structural entropy of a variable affects the probability of the variable being flipped. By using these conclusions, we also proposed an initial candidate solution generation strategy for a local search for SAT, and the experimental results showed that this strategy effectively improves the performance of the solvers CCAsat and Sparrow2011 when incorporated into these two solvers.     

Thermoelastic solutions for thermal distributions moving over thin slim rod under memory-dependent three-phase lag magneto-thermoelasticity

Abstract

Enlightened by the Caputo fractional derivative, the present study deals with a novel mathematical model of generalized thermoelasticity to investigate the transient phenomena due to the influence of magnetic field and moving heat source in a rod in the context of three-phase lag (TPL) theory of thermoelasticity. Both ends of the rod are fixed and heat insulated. Employing Laplace transform as a tool, the problem has been transformed into the space-domain and solved analytically. Finally, solutions in the real-time domain are obtained by applying the inverse Laplace transform. Numerical calculation for stress, displacement, and temperature within the rod is carried out and displayed graphically. The effect of moving heat source speed on temperature, stress, and temperature is studied. It is found from the distributions that the temperature, thermally induced displacement and stress of the rod are found to decrease at large source speed. For the better understanding of the effect of moving heat source on all the distributions, three animations are added.     

Multi-variable special polynomials using an operator ordering method

Using an operator ordering method for some commutative superposition operators, we introduce two new multi-variable special polynomials and their generating functions, and present some new operator identities and integral formulas involving the two special polynomials. Instead of calculating complicated partial differential, we use the special polynomials and their generating functions to concisely address the normalization, photocount distributions and Wigner distributions of several quantum states that can be realized physically, the results of which provide real convenience for further investigating the properties and applications of these states.