perturbation

摘要： We present the results of an investigation into the behavior of the unsteady flow of a Casson Micropolar nanofluid over a shrinking/stretching curved surface,together with a heat transfer analysis of the same problem.The body force acting perpendicular to the surface wall is in charge of regulating the fluid flow rate.Curvilinear coordinates are used to account for the considered curved geometry and a set of balance equations for mass,momentum,energy and concentration is obtained accordingly.These are turned into ordinary differential equations using a similarity transformation.We show that these equations have dual solutions for a number of different combinations of various parameters.The stability of such solutions is investigated by applying perturbations on the steady states.It is found that high values of the Micropolar and Casson parameters cause the flow to move more slowly.However,when compared to a shrunken surface,a stretched surface produces a greater Micro-rotation flux.

摘要： This paper examines the performance of five algorithms for numerically inverting the Laplace transform, in standard, 16-digit and multi-precision environments. The algorithms are taken from three of the four main classes of numerical methods used to invert the Laplace transform. Because the numerical inversion of the Laplace transform is a perturbed problem, rounding errors which are generated in numerical approximations can adversely affect the accurate reconstruction of the inverse transform. This paper demonstrates that working in a multi-precision environment can substantially reduce these errors and the resulting perturbations exist in transforming the data from the s-space into the time domain and in so doing overcome the main drawback of numerically inverting the Laplace transform. Our main finding is that both the Talbot and the accelerated Gaver functionals perform considerably better in a multi-precision environment increasing the advantages of using Laplace transform methods over time-stepping procedures in solving diffusion and more generally parabolic partial differential equations.

摘要： Federated Learning(FL)is a new computing paradigm in privacy-preserving Machine Learning(ML),where the ML model is trained in a decentralized manner by the clients,preventing the server from directly accessing privacy-sensitive data from the clients.Unfortunately,recent advances have shown potential risks for user-level privacy breaches under the cross-silo FL framework.In this paper,we propose addressing the issue by using a three-plane framework to secure the cross-silo FL,taking advantage of the Local Differential Privacy(LDP)mechanism.The key insight here is that LDP can provide strong data privacy protection while still retaining user data statistics to preserve its high utility.Experimental results on three real-world datasets demonstrate the effectiveness of our framework.

摘要： Equivariant eta forms and equivariant differential K-theory Bo Liu Abstract In this paper, for a compact Lie group action, we prove the anomaly formula and the functoriality of the equivariant Bismut-Cheeger eta forms with perturbation operators when the equivariant family index vanishes.

摘要： The importance of perturbation theory in many fields is very clear through almost a century or even more. Its importance was exemplified in solving many problems in physics and other applied fields. A great deal of applications arose in dealing with eigenvalue problems especially in quantum mechanics in conjunction with the field of atomic physics. Accordingly, it came to our mind to write a brief review article on the subject. At the beginning, we give some important definitions to do with various eigenvalue problems;then we introduce concepts that have to do with perturbation theory and the techniques used in such a theory, beginning with the algebraic perturbation theory giving a good number of examples from the literature on the use of the theory in solving integral equation, algebraic equations and differential equations. Few applications are then given in applied fields such as classical mechanics, quantum mechanics and fluid mechanics. Finally, a concluding discussion is given which is related to the use of the theory.

摘要： This paper is concerned with the Navier-stokes equations with nonlinear perturbation in R^2,which studies the existence of solution,and gets the existence of the attractors.Finally,we discuss with limit-behavior of the Navier-stokes equation4 with nonlinear per-turbation,asα→0.

摘要： Perturbation to Noether quasi-symmetry and adiabatic invariants for the nonholonomic system on time scales are studied. Firstly, some properties of time scale calculus are reviewed. Secondly, the differential equations of motion for the nonholonomic system on time scales, Noether quasi-symmetry and conserved quantity are given. Thirdly, perturbation to Noether quasi-symmetry and adiabatic invariants, which are the main results of this paper, are investigated. The main results are achieved by two steps, the first step is to obtain adiabatic invariants without transforming the time, and the next is to obtain adiabatic invariants under the infinitesimal transformations of both the time and the coordinates. And in the end, an example is given to illustrate the methods and results.

摘要： This paper is concerned with a stochastic predator-prey system with Beddington-DeAngelis functional response and time delay. Firstly, we show that this system has a unique positive solution as this is essential in any population dynamics model. Secondly, the validity of the stochastic system is guaranteed by stochastic ultimate boundedness of the analyzed solution. Finally, by constructing suitable Lyapunov functions, the asymptotic moment estimation of the solution was given. These properties of the solution can provide theoretical support for biological resource management.

摘要： The development of theoretical models for crystals has led to the evolution of computational methods with which much more thorough investigations than previously possible can be done, including studies of the nonlinear optical properties. There has recently been a rise in interest in 2-dimensional materials;unfortunately, measurements of the nonlinear susceptibility of these materials in the wavelength range of the order of hundreds of nanometers by traditional methods are difficult. Studies of second-harmonic generation (SHG) from the transition-metal dichalcogenides (TMDCs), MoS2 and MoSe2, have been reported;however, SHG from other typical van der Waals crystals such as GaSe and other transition metal monochalcogenides (TMMCs) has rarely been studied under the same conditions. In this study, the 211 (i = 2, j = 1, k = 1) elements in the susceptibility matrices of GaSe, InSe, MoS2 and WS2 were calculated and compared. A tendency for the SHG intensity to weaken as the wavelength increases from 500 nm to 1000 nm was observed for GaSe and InSe, and, apart from some periodic fluctuations, no clear increase could be seen for these two materials in the SHG response curve in the near infrared. By comparison, MoS2 and WS2 have obvious peaks in both the visible and infrared bands. Calculations of the SHG response show peaks at around 500 nm (for GaSe), 570 (for InSe), 660 nm, 980 nm (for MoS2) and 580 nm, 920 nm (for WS2). Moreover, similarities between the SHG curves for GaSe and InSe and for MoS2 and WS2 were revealed, which may be due to the similarities found for these two groups of crystals.

摘要： This paper uses the Ansatz method to solve for exact topological soliton solutions to sine-Gordon type equations. Single, double, and triple sine-Gordon and sine-cosine-Gordon equations are investigated along with dispersive and highly dispersive variations. After these solutions are found, strong perturbations are added to each equation and the new solutions are found. In solving both the perturbed and unperturbed sine-Gordon type equations, constraints are imposed on the parameters. The novel contributions of the authors are the soliton solutions to the strongly perturbed sine-Gordon equation and its variations. These results are important to the study of Josephson junctions, crystal dislocations, ultra-short optical pulses, relativistic field theory, and elementary particles.