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.     

Linearizability of planar polynomial Hamiltonian systems

Isochronicity and linearizability of two-dimensional polynomial Hamiltonian systems are revisited and new results are presented. We give a new computational procedure to obtain the necessary and sufficient conditions for the linearization of a polynomial system. Using computer algebra systems we provide necessary and sufficient conditions for linearizability of Hamiltonian systems with homogeneous non-linearities of degrees 5, 6 and 7. We also present some sufficient conditions for systems with nonhomogeneous nonlinearities of degrees two, three and five.     

Existence of multi-bump solutions for a system with critical exponent in RN

We consider the following system with critical exponent in ${\mathbb{R}}^N$:$\{\begin{array}{c}?\mathrm{\Delta }u=K_1(y)u^{2^{?}?1}+\frac{p}{2^{?}}V(y)u^{p?1}{v}^q\text{in}{\mathbb{R}}^N,\hfill \\ ?\mathrm{\Delta }v=K_2(y)v^{2^{?}?1}+\frac{q}{2^{?}}V(y)u^p{v}^{q?1}\text{in}{\mathbb{R}}^N,\hfill \\ u,v>0,y\in {\mathbb{R}}^N,\hfill \end{array}$ where $N\ge 5$, $p,q>1$ and $p+q=2^{?}=\frac{2N}{N?2}$. Using finite dimensional reduction method, we prove the existence of multi-bump solutions. Their bumps can be placed on arbitrarily many or even infinitely many lattice points in ${\mathbb{R}}^N$. Since $p<2$ or $q<2$, we introduce two new norms to avoid singularity.     

Observation of field induced anomalous quantum criticality in Ce0.6Y0.4NiGe2 compound

We report the results of our investigation of magnetization and heat capacity on a series of compounds Ce${}_{1?x}$Y_xNiGe₂ ($x=0.1,0.2\text{ and }0.4$) under the influence of external magnetic field. Our studies of the thermodynamic quantity $?\mathrm{d}M/\mathrm{d}T$ on these compounds indicate that magnetic frustration persists in Ce_0.9Y_0.1NiGe₂, as also reported for the parent compound CeNiGe₂. The weak signature of this frustration is also noted in Ce_0.8Y_0.2NiGe₂, whereas, it is suppressed in Ce_0.6Y_0.4NiGe₂. Heat capacity studies on Ce_0.9Y_0.1NiGe₂ and Ce_0.8Y_0.2NiGe₂ indicate the presence of a new magnetic anomaly at high field which indicates that quantum criticality is absent in these compounds. However, for Ce_0.6Y_0.4NiGe₂ such an anomaly is not noted. For this later compound, the magnetic field (H) and temperature (T) dependence of heat capacity and magnetization obey $H/T$ scaling above critical fields. However, the obtained scaling critical parameter (δ) is 1.6, which is away from mean field value of 3. This deviation suggests the presence of unusual fluctuations and anomalous quantum criticality in these compounds. This unusual fluctuation may arise from disorderness induced by Y-substitution.     

Synthesis of heterobiaryls via Suzuki‐Miyaura coupling reaction of potassium aryltrifluoroborates with heteroaryl halides in aqueous systems

A variety of heterobiaryl compounds have been synthesized by the Suzuki‐Miyaura coupling reactions of heteroaryl halides with potassium aryltrifluoroborates. Pd (OAc)₂ was found to be highly efficient for the Suzuki‐Miyaura coupling reactions of various heteroaryl halides with potassium aryltrifluoroborates in aqueous systems, delivering the corresponding heterobiaryl compounds in good to excellent yields.     

Entropic effect implication for change in polymer coils swelling state in the demixing enthalpy recovery of aqueous poly(vinyl methyl ether) solutions

Time‐dependent demixing enthalpy recovery behavior of aqueous poly(vinyl methyl ether) (PVME) solutions exhibits distinct recovery characteristics in three concentration regions. The absence of recovery behavior below a water concentration of 38.3 wt % indicates that the PVME coil is in a globular state. The typically sigmoidal recovery behavior of demixing enthalpy above 38.3 wt % is ascribed to the reswelling of the collapsed polymer coils induced by the entropic effect. The increase in difference between the upper and lower limits indicates the continued swelling of the PVME coils. Above 65 wt %, a dominant diluting effect can be observed, and a much longer phase separation time is needed to reach the expected lower limit. In contrast, the recovery of demixing enthalpy in a wide range of water concentration (from 38.3 to 90 wt %) exhibits the same feature. The infrared spectroscopy results are in agreement with the above macroscopic differential scanning calorimetry results. © 2018 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2019 , 57, 142–151