A nonlocal multi-point singular fractional integro-differential problem of Lane–Emden type

In this paper, using Riemann–Liouville integral and Caputo derivative, we study a nonlinear singular integro-differential equation of Lane–Emden type with nonlocal multi-point integral conditions. We prove the existence and uniqueness of solutions by application of Banach contraction principle. Also, we prove an existence result using Schaefer fixed point theorem. Then, we present some examples to show the applicability of the main results.     

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.     

Reduced dimensionality (3,2)D NMR experiments and their automated analysis: implications to high‐throughput structural studies on proteins

Protein NMR spectroscopy has expanded dramatically over the last decade into a powerful tool for the study of their structure, dynamics, and interactions. The primary requirement for all such investigations is sequence‐specific resonance assignment. The demand now is to obtain this information as rapidly as possible and in all types of protein systems, stable/unstable, soluble/insoluble, small/big, structured/unstructured, and so on. In this context, we introduce here two reduced dimensionality experiments – (3,2)D‐hNCO canH and (3,2)D‐hN coCA nH – which enhance the previously described 2D NMR‐based assignment methods quite significantly. Both the experiments can be recorded in just about 2–3 h each and hence would be of immense value for high‐throughput structural proteomics and drug discovery research. The applicability of the method has been demonstrated using alpha‐helical bovine apo calbindin‐D9k P43M mutant (75 aa) protein. Automated assignment of this data using AUTOBA has been presented, which enhances the utility of these experiments. The backbone resonance assignments so derived are utilized to estimate secondary structures and the backbone fold using Web‐based algorithms. Taken together, we believe that the method and the protocol proposed here can be used for routine high‐throughput structural studies of proteins. Copyright © 2014 John Wiley & Sons, Ltd.     

含非完整界面的功能梯度压电材料Ⅲ型裂纹问题

本文研究了含非完整界面的功能梯度压电复合材料的Ⅲ型裂纹问题．此裂纹垂直于非完整界面，采用弹簧型力电耦合界面模型模拟非完整界面．界面两侧材料的性质，如弹性模量、压电常数和介电常数均假定呈指数函数形式且沿着裂纹方向变化．运用积分变换法将裂纹面条件转换为奇异积分方程，并使用Gauss-Chebyshev方法对其进行数值求解．根据算例结果讨论了一些退化问题并分析了裂纹尖端强度因子与材料的非均匀系数和非完整界面参数的关系．     

Double‐crosslinked reversible redox‐responsive hydrogels based on disulfide–thiol interchange

We present novel redox‐responsive hydrogels based on poly(N‐isopropylacrylamide) or poly(acrylamide), consisting of a reversible disulfide crosslinking agent N,N′‐bis(acryloyl)cystamine and a permanent crosslinking agent N,N′‐methylenebisacrylamide for microfluidic applications. The mechanism of swelling/deswelling behavior starts with the cleavage and reformation of disulfide bonds, leading to a change of crosslinking density and crosslinking points. Raman and ultraviolet‐visible spectroscopy confirm that conversion efficiency of thiol–disulfide interchange up to 99%. Rheological analysis reveals that the E modulus of hydrogel is dependent on the crosslinking density and can be repeatedly manipulated between high‐ and low‐stiffness states over at least 5 cycles without significant decrease. Kinetic studies showed that the mechanical strength of the gels changes as the redox reaction proceeds. This process is much faster than the autonomous diffusion in the hydrogel. Moreover, cooperative diffusion coefficient (D_coop) indicates that the swelling process of the hydrogel is affected by the reduction reaction. Finally, this reversibly switchable redox behavior of bulky hydrogel could be proven in microstructured hydrogel dots through short‐term photopatterning process. These hydrogel dots on glass substrates also showed the desired short response time on cyclic swelling and shrinking processes known from downsized hydrogel shapes. Such stimuli‐responsive hydrogels with redox‐sensitive crosslinkers open a new pathway in exchanging analytes for sensing and separating in microfluidics applications. © 2019 Wiley Periodicals, Inc. J. Polym. Sci., Part A: Polym. Chem. 2019 , 57, 2590–2601     

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.     

Explicit maximal and minimal curves over finite fields of odd characteristics

In this work we present explicit classes of maximal and minimal Artin–Schreier type curves over finite fields having odd characteristics. Our results include the proof of Conjecture 5.9 given in [1] as a very special subcase. We use some techniques developed in [2], which were not used in [1].