一类特殊级数的和函数

利用k次单位根及其正交性得到级数∑∞n=0xkn+l/(kn+l)!的和函数.它与利用微分方程理论来求级数的和有很大区别.作为应用,得到了一些特殊级数的和.     

A Nanometer Aerosol Size Analyzer (nASA) for Rapid Measurement of High-concentration Size Distributions

We have developed a fast-response nanometer aerosol size analyzer (nASA) that is capable of scanning 30 size channels between 3 and 100 nm in a total time of 3 s. The analyzer includes a bipolar charger (Po²¹⁰), an extended-length nanometer differential mobility analyzer (Nano-DMA), and an electrometer (TSI 3068). This combination of components provides particle size spectra at a scan rate of 0.1 s per channel free of uncertainties caused by response-time-induced smearing. The nASA thus offers a fast response for aerosol size distribution measurements in high-concentration conditions and also eliminates the need for applying a de-smearing algorithm to resulting data. In addition, because of its thermodynamically stable means of particle detection, the nASA is useful for applications requiring measurements over a broad range of sample pressures and temperatures. Indeed, experimental transfer functions determined for the extended-length Nano-DMA using the tandem differential mobility analyzer (TDMA) technique indicate the nASA provides good size resolution at pressures as low as 200 Torr. Also, as was demonstrated in tests to characterize the soot emissions from the J85-GE engine of a T-38 aircraft, the broad dynamic concentration range of the nASA makes it particularly suitable for studies of combustion or particle formation processes. Further details of the nASA performance as well as results from calibrations, laboratory tests and field applications are presented below.     

Lower critical solution temperature sensitivity to structural changes in poly(N‐isopropyl acrylamide) homopolymers

The effect of the molecular weight on the lower critical solution temperature (LCST) has been discussed extensively, where LCST increased with molar mass, decreased or kept constant, which leads to confusion. This work is focused on the preparation of poly(N‐isopropyl acrylamide) homopolymers, obtained in a wide molecular weights range. The LCST behavior is analyzed by calorimetry and rheology, and a deep study of molecular features is carried out for a better knowledge of the influence of various parameters involved on LCST. Finally, the molecular weight trend is observed, and its influence on LCST is compared with the effect of other parameters as polymer concentration in water, end‐group effect, and tacticity. It is observed that other parameters such tacticity and end‐group effect will affect the LCST behavior over molecular weight, if this one is not high enough. Furthermore, the study of the LCST ranges will be a useful tool for analyzing the molecular weight trends. © 2019 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2019 , 57, 1386–1393     

Analysis of several subcycling schemes in partitioned simulations of a strongly coupled fluid-structure interaction

Fluid-structure interaction (FSI) simulations are used extensively to calculate the vibration of structures subjected to an internal or external flow. In the case of partitioned FSI simulations, separate flow and structure solvers are used, which requires some kind of coupling between both. The time step in both solvers is typically taken the same, but this unnecessarily leads to long calculation times when the time step is small due to stability reasons in one of the two solvers. Subcycling, the procedure where the time step of one solver is chosen smaller than the time step used in the other solver, may reduce the computational cost of the FSI simulation. The subcycling procedure can be either explicit or implicit, the latter implying the use of coupling iterations in each time step. Contrary to explicit subcycling, no stability analyses of implicit subcycling schemes are found in the literature. In this paper, the temporal stability of the implicit subcycling procedure is investigated. The one-dimensional flow in an elastic cylindrical tube is studied analytically. The results of this analysis are subsequently compared to a partitioned two-dimensional axisymmetric FSI calculation with implicit coupling between the flow and structure solvers.     

Nonlocal stochastic differential equations with time-varying delay driven by G-Brownian motion

This article studies a class of nonlocal stochastic differential equations driven by G-Brownian motion (G-NSDEs for short). We show the existence and uniqueness results of solutions by means of fixed point theorem. In addition, exponential estimation of (1) has been discussed. Furthermore, we present global solution to Equation (1) with the help of G-Lyapunov functional and ψ-type function.     

Torsional vibration of functionally graded nano-rod under magnetic field supported by a generalized torsional foundation based on nonlocal elasticity theory

Abstract

This article contains the nonlocal elasticity theory to capture size effects in functionally graded (FG) nano-rod under magnetic field supported by a torsional foundation. Torque effect of an axial magnetic field on an FG nano-rod has been defined using Maxwell’s relation. The material properties were assumed to vary according to the power law in radial direction. The Navier equation and boundary conditions of the size-dependent FG nano-rod were derived by the Hamilton’s principle. These equations were solved by employing the generalized differential quadrature method (GDQM). Presented model has the ability to turn into the classical model if the material length scale parameter is taken to be zero. The effects of some parameters, such as inhomogeneity constant, magnetic field and small-scale parameter, were studied. As an important result of this study can be stated that an FG nano-rod model based on the nonlocal elasticity theory behaves softer and has smaller natural frequency.     

Bio‐based comb‐like copolymers derived from alkyl 10‐undecenoates and maleic anhydride

In this article, 10‐undecenoic acid, based on castor oil, was used a raw material for the synthesis of alternating copolymers. ω‐Unsaturated fatty esters as alkyl 10‐undecenoates were prepared by the esterification reaction of 10‐undecenoic acid with alkyl alcohol. A series of comb‐like copolymers were synthesized by free radical polymerization from maleic anhydride and alkyl 10‐undecenoates copolymers with different length of alkyl side chains in a toluene solution. These copolymers were investigated by ¹H and ¹³C nuclear magnetic resonance, Fourier transform infrared spectroscopy, gel permeation chromatography, thermogravimetric analysis, differential scanning calorimetry (DSC). The copolymers were obtained in a low molecular weight in a range 3370–12,240 g mol⁻¹ and their structural characterization indicated the formation of alternating copolymers. DSC characterization revealed that these comb‐like copolymers showed amorphous to semicrystalline behavior by increasing the length of side chains. The bio‐based comb‐like copolymers allow for the development of new polymeric materials for several applications. © 2018 Wiley Periodicals, Inc. J. Polym. Sci., Part A: Polym. Chem. 2018 , 56, 1039–1045     

不确定中立型线性随机时滞系统的鲁棒稳定性

建立了中立型随机微分时滞方程的LaSalle不变原理,然后应用LaSalle不变原理讨论了不确定中立型随机时滞系统的随机渐近稳定和几乎必然指数稳定的代数判据, 同时给出示例说明结果的有效性．     

Smooth and Nonsmooth Lipschitz Controls for a Class of Vector Differential Equations

This paper is devoted to the study of a class of control problems associated to a nonlinear second-order vector differential equation with pointwise state constraints. The control is realized via a function of the state. We extend the results of Akkouchi, Bounabat, and Goebel to vector differential equations and furthermore consider the more general case. Under proper conditions, we prove the existence of optimal controls in the class of Lipschitz functions and obtain an optimality condition which looks somehow like the Pontryagin maximum principle for a smooth optimal control function. For a nonsmooth optimal control function, we derive a suboptimality condition by means of the Ekeland variational principle.Communicated by M. J. BalasThis work was supported by 985 Project of Jilin University. The author thanks Professor Yong Li for valuable suggestions. He also thanks Professor M. J. Balas and the anonymous referees for their comments.