Nearly Comonotone Approximation of Periodic Functions

Suppose that a continuous 27r-periodic function f on the real axis changes its monotonicity at points y_1:-π≤ y_(2s)<y_(2s-1)<… < y_1 <π,s ∈ IN.In this PaPer,for each n≥N,a trigonometric polynomial P_n of order cn is found such that:P_n has the same monotonicity as f,everywhere except,perhaps,the small intervals(y_i-π/n,y_i+π/n)and‖f-P_n‖<c(s)ω_3(f,π/n),where N is a constant depending only on mini=1,...,2s {y_i-y_(i+1)},c,c(s) are constants depending only on s,ω_3(f_1,·) is the modulus of smoothness of the 3-rd order of the function f,and ||·|| is the max-norm.

Robust Topology Optimization of Periodic Multi-Material Functionally Graded Structures under Loading Uncertainties

This paper presents a robust topology optimization design approach for multi-material functional graded structures under periodic constraint with load uncertainties.To characterize the random-field uncertainties with a reduced set of random variables,the Karhunen-Lo`eve(K-L)expansion is adopted.The sparse grid numerical integration method is employed to transform the robust topology optimization into a weighted summation of series of deterministic topology optimization.Under dividing the design domain,the volume fraction of each preset gradient layer is extracted.Based on the ordered solid isotropic microstructure with penalization(Ordered-SIMP),a functionally graded multi-material interpolation model is formulated by individually optimizing each preset gradient layer.The periodic constraint setting of the gradient layer is achieved by redistributing the average element compliance in sub-regions.Then,the method of moving asymptotes(MMA)is introduced to iteratively update the design variables.Several numerical examples are presented to verify the validity and applicability of the proposed method.The results demonstrate that the periodic functionally graded multi-material topology can be obtained under different numbers of sub-regions,and robust design structures are more stable than that indicated by the deterministic results.

LOCAL BIFURCATION OF STEADY ALMOST PERIODIC WATER WAVES WITH CONSTANT VORTICITY

In this paper we investigate the traveling wave solution of the two dimensional Euler equations with gravity at the free surface over a flat bed.We assume that the free surface is almost periodic in the horizontal direction.Using conformal mappings,one can change the free boundary problem into a fixed boundary problem for some unknown functions with the boundary condition.By virtue of the Hilbert transform,the problem is equivalent to a quasilinear pseudodifferential equation for an almost periodic function of one variable.The bifurcation theory ensures that we can obtain an existence result.Our existence result generalizes and covers the recent result in[15].Moreover,our result implies a non-uniqueness result at the same bifurcation point.

Machine Learning Kinetic Energy Functional for a One-Dimensional Periodic System

Kinetic energy(KE) functional is crucial to speed up density functional theory calculation. However, deriving it accurately through traditional physics reasoning is challenging. We develop a generally applicable KE functional estimator for a one-dimensional (1D) extended system using a machine learning method. Our end-to-end solution combines the dimensionality reduction method with the Gaussian process regression, and simple scaling method to adapt to various 1D lattices. In addition to reaching chemical accuracy in KE calculation, our estimator also performs well on KE functional derivative prediction. Integrating this machine learning KE functional into the current orbital free density functional theory scheme is able to provide us with expected ground state electron density.

S-ASYMPTOTICALLY BLOCH TYPE PERIODIC SOLUTIONS TO SOME SEMI-LINEAR EVOLUTION EQUATIONS IN BANACH SPACES

This paper is mainly concerned with the S-asymptotically Bloch type periodicity.Firstly,we introduce a new notion of S-asymptotically Bloch type periodic functions,which can be seen as a generalization of concepts of S-asymptoticallyω-periodic functions and S-asymptoticallyω-anti-periodic functions.Secondly,we establish some fundamental properties on S-asymptotically Bloch type periodic functions.Finally,we apply the results obtained to investigate the existence and uniqueness of S-asymptotically Bloch type periodic mild solutions to some semi-linear differential equations in Banach spaces.

Optical soliton and elliptic functions solutions of Sasa-satsuma dynamical equation and its applications

The Sasa-satsuma(SS)dynamical equation interpret propagation of ultra-short and femto-second pulses in optical fibers.This dynamical model has important physical significance.In this article,two mathematical techniques namely,improved F-expansion and improved aux-iliary methods are utilized to construct the several types of solitons such as dark soliton,bright soliton,periodic soliton,Elliptic function and solitary waves solutions of Sasa-satsuma dynamical equation.These results have imperative applications in sciences and other fields,and construc-tive to recognize the physical structure of this complex dynamical model.The computing work and obtained results show the infuence and effectiveness of current methods.

Global attractors of a degenerate parabolic equation and their error estimates

The existences of the global attractor Ae for a degenerate parabolic equation and of the homogenized attractor A0 for the corresponding homogenized equation are studied, and explicit estimates for the distance between Ae and A0 are given.

Pseudo almost periodic solutions for the systems of differential equations with piecewise constant argument [t]

In this paper, we investigate the existence and uniqueness of new almost periodic type solutions, so-called pseudo almost periodic solutions for the systems of differential equations with piecewise constant argument by means of introducing the notion of pseudo almost periodic vector sequences.

DYNAMICS IN A CLASS OF NEURON MODELS

In this paper, we investigate the dynamics in a class of discrete-time neuron mod-els. The neuron model we discussed, defined by such periodic input-output mapping as a sinusoidal function, has a remarkably larger memory capacity than the conven-tional association system with the monotonous function. Our results show that the orbit of the model takes a conventional bifurcation route, from stable equilibrium, to periodicity, even to chaotic region. And the theoretical analysis is verified by numerical simulations.