von Neumann’s trace inequality for Hilbert–Schmidt operators

von Neumann’s inequality in matrix theory refers to the fact that the Frobenius scalar product of two matrices is less than or equal to the scalar product of the respective singular values. Moreover, equality can only happen if the two matrices share a joint set of singular vectors, and this latter part is hard to find in the literature. We extend these facts to the separable Hilbert space setting, and provide a self-contained proof of the “latter part”.     

Renormalization group second‐order approximation for singularly perturbed nonlinear ordinary differential equations

We consider a 2 time scale nonlinear system of ordinary differential equations. The small parameter of the system is the ratio ϵ of the time scales. We search for an approximation involving only the slow time unknowns and valid uniformly for all times at order O(ϵ²). A classical approach to study these problems is Tikhonov's singular perturbation theorem. We develop an approach leading to a higher order approximation using the renormalization group (RG) method. We apply it in 2 steps. In the first step, we show that the RG method allows for approximation of the fast time variables by their RG expansion taken at the slow time unknowns. Next, we study the slow time equations, where the fast time unknowns are replaced by their RG expansion. This allows to rigorously show the second order uniform error estimate. Our result is a higher order extension of Hoppensteadt's work on the Tikhonov singular perturbation theorem for infinite times. The proposed procedure is suitable for problems from applications, and it is computationally less demanding than the classical Vasil'eva‐O'Malley expansion. We apply the developed method to a mathematical model of stem cell dynamics.     

Singular Hamiltonian elliptic systems with critical exponential growth in dimension two

We will focus on the existence of nontrivial solutions to the following Hamiltonian elliptic system where are numbers belonging to the interval [0, 2), V is a continuous potential bounded below on by a positive constant and the functions f and g possess exponential growth range established by Trudinger–Moser inequalities in Lorentz–Sobolev spaces. The proof involves linking theorem and a finite‐dimensional approximation.     

Non‐self‐adjoint singular second‐order dynamic operators on time scale

In this study, maximal dissipative second‐order dynamic operators on semi‐infinite time scale are studied in the Hilbert space , that the extensions of a minimal symmetric operator in limit‐point case. We construct a self‐adjoint dilation of the dissipative operator together with its incoming and outgoing spectral representations so that we can determine the scattering function of the dilation as stated in the scheme of Lax‐Phillips. Moreover, we construct a functional model of the dissipative operator and identify its characteristic function in terms of the Weyl‐Titchmarsh function of a self‐adjoint second‐order dynamic operator. Finally, we prove the theorems on completeness of the system of root functions of the dissipative and accumulative dynamic operators.     

The double deflating technique for irreducible singular M-matrix algebraic Riccati equations in the critical case

As is known, Alternating-Directional Doubling Algorithm (ADDA) is quadratically convergent for computing the minimal nonnegative solution of an irreducible singular M-matrix algebraic Riccati equation (MARE) in the noncritical case or a nonsingular MARE, but ADDA is only linearly convergent in the critical case. The drawback can be overcome by deflating techniques for an irreducible singular MARE so that the speed of quadratic convergence is still preserved in the critical case and accelerated in the noncritical case. In this paper, we proposed an improved deflating technique to accelerate further the convergence speed – the double deflating technique for an irreducible singular MARE in the critical case. We proved that ADDA is quadratically convergent instead of linearly when it is applied to the deflated algebraic Riccati equation (ARE) obtained by a double deflating technique. We also showed that the double deflating technique is better than the deflating technique from the perspective of dimension of the deflated ARE. Numerical experiments are provided to illustrate that our double deflating technique is effective.     

Improving the gas barrier,mechanical and thermal properties of poly(vinyl alcohol) with molybdenum disulfide nanosheets

New multifunctional materials with both high structural and gas barrier performances are important for a range of applications. Herein we present a one‐step mechanochemical process to prepare molybdenum disulfide (MoS₂) nanosheets with hydroxy functional groups that can simultaneously improve mechanical strength, thermal conductivity, and gas permittivity of a polymer composite. By homogeneously incorporating these functionalized MoS₂ nanosheets at low loading of less than 1 vol %, a poly(vinyl alcohol) (PVA) polymer exhibits elongation at break of 154%, toughness of 82 MJ/m³, and in‐plane thermal conductivity of 2.31 W/m K. Furthermore, this composite exhibits significant gas barrier performance, reducing the permeability of helium by 95%. Under fire condition, the MoS₂ nanosheets form thermally stable char, thus enhancing the material's resistance to fire. Hydrogen bonding has been identified as the main interaction mechanism between the nanofillers and the polymer matrix. The present results suggest that the PVA composite reinforced with 2D layered nanomaterial offers great potentials in packaging and fire retardant applications. © 2019 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2019 , 57, 406–414     

三维位势问题的梯度边界积分方程的新解法

三维位势问题的边界元分析中,关于坐标变量的边界位势梯度的计算是一个困难的问题. 已有一些方法着手解决这个问题,然而,这些方法需要复杂的理论推导和大量的数值计算. 本文提出求解一般边界位势梯度边界积分方程的辅助边值问题法. 该方法构造了与原边界值问题具有相同解域的辅助边值问题,该辅助边值问题具有已知解,因此通过求解此辅助边值问题,可获得梯度边界积分方程对应的系统矩阵,然后将此系统矩阵应用于求解原边值问题,求解过程非常简单,只需求解一个线性系统即可获得原边值问题的解. 值得注意的是,在求解原边值问题时,不再需要重新计算系统矩阵,因此辅助边值问题法的效率并不很差. 辅助边值问题法避免了强奇异积分的计算,具有数学理论简单、程序设计容易、计算精度高等优点,为坐标变量梯度边界积分方程的求解提供了一个新的途径. 3个标准的数值算例验证了方法的有效性.      

Ocimum basilicum seed mucilage reinforced with montmorillonite for preparation of bionanocomposite film for food packaging applications

Use of nanocomposites is a well-established approach in enhancing the mechanical and barrier properties of bionanocomposite film for food packaging applications. The seed mucilage of Ocimum basilicum was employed for the preparation of bionanocomposite films with montmorillonite (MMT) as nanofiller. The films were prepared by solvent-casting method at varied solution pH (1, 3, 5 and 9) and MMT loading (1%, 3%, 5%, 10%, 15% and 20%). The films were characterized for physical, mechanical and barrier properties in addition to microstructure and X-ray diffraction pattern. XRD analysis revealed the exfoliated dispersion of MMT at pH 9, confirming its effective interaction with the bionanocomposite film. Maximum film tensile strength was achieved at a lower MMT load of 5%. Water vapour permeability reduced with increase in MMT loading up to 5%, followed by an increase at higher MMT loadings. Film formed at pH 9 showed tensile strength of 17.3 ± 0.33 MPa and reduced water vapour permeability (WVP) of 0.21 g mm.m⁻².hr⁻¹.kPa⁻¹.