Convergence of solutions for the fractional Cahn–Hilliard system

This paper deals with the Cauchy–Dirichlet problem for the fractional Cahn–Hilliard equation. The main results consist of global (in time) existence of weak solutions, characterization of parabolic smoothing effects (implying under proper condition eventual boundedness of trajectories), and convergence of each solution to a (single) equilibrium. In particular, to prove the convergence result, a variant of the so-called ?ojasiewicz–Simon inequality is provided for the fractional Dirichlet Laplacian and (possibly) non-analytic (but $C^1$) nonlinearities.     

Investigation of mononuclear,dinuclear, and trinuclear transition metal (II) complexes derived from an asymmetric Salamo‐based ligand possessing three different coordination modes

A new asymmetric Salamo‐based ligand H₂L was synthesized using 3‐tert‐butyl‐salicylaldehyde and 6‐methoxy‐2‐[O‐(1‐ethyloxyamide)]‐oxime‐1‐phenol. By adjusting the ratio of the ligand H₂L and Cu (II), Co (II), and Ni (II) ions, mononuclear, dinuclear, and trinuclear transition metal (II) complexes, [Cu(L)], [{Co(L)}₂], and [{Ni(L)(CH₃COO)(CH₃CH₂OH)}₂Ni] with the ligand H₂L possessing completely different coordination modes were obtained, respectively. The optical spectra of ligand H₂L and its Cu (II), Co (II) and Ni (II) complexes were investigated. The Cu (II) complex is a mononuclear structure, and the Cu (II) atom is tetracoordinated to form a planar quadrilateral structure. The Co (II) complex is dinuclear, and the two Co (II) atoms are pentacoordinated and have coordination geometries of distorted triangular bipyramid. The Ni (II) complex is a trinuclear structure, and the terminal and central Ni (II) atoms are all hexacoordinated, forming distorted octahedral geometries. Furthermore, optical properties including UV–Vis, IR, and fluorescence of the Cu (II), Co (II), and Ni (II) complexes were investigated. Finally, the antibacterial activities of the Cu (II), Co (II), and Ni (II) complexes were explored. According to the experimental results, the inhibitory effect was found to be enhanced with increasing concentrations of the Cu (II), Co (II), and Ni (II) complexes.     

Crosslinking network of bio‐based bis‐functional epoxides derived from trans‐limonene oxide

The thiol‐ene reaction between trans‐limonene oxide (trans‐LO) and ethane‐1,2‐dithiol in the presence of triethylborane affords a bio‐based bis‐functional epoxide (bis‐trans‐LO). The crosslinking reaction of bis‐trans‐LO with branched polyethyleneimine (BPEI; M_n = 600; BPEI₆₀₀) at a feed ratio of bis‐trans‐LO/BPEI₆₀₀ = 57/43 (wt/wt) yields the corresponding network polymer with T_d10 (10% thermal decomposition temperature) of 304.7 °C in 98% yield. In contrast, negligible amounts of network polymer are obtained by the reaction of bis‐LO (bis‐functional epoxide derived from cis and trans‐LO) and BPEI₆₀₀ regardless of the feed ratio. The mechanical strengths as measured by direct tensile tests of the network polymers derived from bis‐trans‐LO and BPEI_600,1800 (M_n = 600 and 1800) were approximately 16 and 11 times higher than that of bis‐LO and BPEI₁₈₀₀, respectively. The tensile shear strengths of the metal‐to‐metal adhesive bonds induced by bis‐trans‐LO and BPEI_600,1800 were 9.5 and 14.1 MPa, respectively. DMA revealed that the storage modulus of the network polymer derived from bis‐trans‐LO and BPEI₁₈₀₀ in the rubber region was higher than that of the material prepared from bis‐LO and BPEI₁₈₀₀, indicating higher crosslink density of the bis‐trans‐LO/BPEI₁₈₀₀ system. © 2019 Wiley Periodicals, Inc. J. Polym. Sci., Part A: Polym. Chem. 2019 , 57, 2466–2473     

The energy-critical nonlinear wave equation with an inverse-square potential

We study the energy-critical nonlinear wave equation in the presence of an inverse-square potential in dimensions three and four. In the defocussing case, we prove that arbitrary initial data in the energy space lead to global solutions that scatter. In the focusing case, we prove scattering below the ground state threshold.     

为什么要引入溶剂的渗透因子

阐明了引入溶剂渗透因子的必要性,较详细地介绍了两种渗透因子及它们间的关系。     

Decay rate for viscoelastic wave equation of variable coefficients with delay and dynamic boundary conditions

In this paper, we consider a viscoelastic wave equation of variable coefficients in the presence of past history with nonlinear damping and delay in the internal feedback and dynamic boundary conditions. Under suitable assumptions, we establish an explicit and general decay rate result without imposing restrictive assumption on the behavior of the relaxation function at infinity by Riemannian geometry method and Lyapunov functional method.     

The 4D Klein-Gordon, Dirac and Quantization Equations from 5D Null Paths

Results from 5D induced-matter and membrane theory with null paths are extended to show that a particle obeys the 4D Klein-Gordon equation but with a variable mass. The Dirac equation also follows, but raises concerns about 4D quantization in the two natural 5D gauges, and reopens the question of a Regge-like trajectory for the spin angular momenta and squared masses of gravitationally-dominated systems.     

ON A PAIR OF NON-ISOMETRIC ISOSPECTRAL DOMAINS WITH FRACTAL BOUNDARIES AND THE WEYL-BERRY CONJECTURE

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Boundedness of commutators on homogeneous Herz spaces

The boundedness on homogeneous Herz spaces is established for a large class of linear commutators generated by BMO(R ⁿ ) functions and linear operators of rough kernels which include the Calderón-Zygmund operators and the Ricci-Stein oRfiUatory singular integrals with rough kernels. Project supponed in pan by the National h’atural Science Foundation of China (Grant No. 19131080) and the NEDF of China.     

Integral Kernel Operators on Fock Space – Generalizations and Applications to Quantum Dynamics

A general theory of operators on Boson Fock space is discussed in terms of the white noise distribution theory on Gaussian space (white noise calculus). An integral kernel operator is generalized from two aspects: (i) The use of an operator-valued distribution as an integral kernel leads us to the Fubini type theorem which allows an iterated integration in an integral kernel operator. As an application a white noise approach to quantum stochastic integrals is discussed and a quantum Hitsuda–Skorokhod integral is introduced. (ii) The use of pointwise derivatives of annihilation and creation operators assures the partial integration in an integral kernel operator. In particular, the particle flux density becomes a distribution with values in continuous operators on white noise functions and yields a representation of a Lie algebra of vector fields by means of such operators.