ON UNIFORM APPROXIMATION BY PARTIAL SUMS OF JACOBI SERIES ON ELLIPTIC REGION

In this paper, we discuss the relation between the partial sums of Jacobi serier on an elliptic region and the corresponding partial sums of Fourier series. From this we derive a precise approximation formula by the partial sums of Jacobi series on an elliptic region.

UNIFORM APPROXIMATION BY COMBINATIONS OF BERNSTEIN POLYNOMIALS

We prove a pointwise characterization result for combinations of Bernstein polynomials. The main result of this paper includes an equivalence theorem of H. Berens and G. G. Lorentz as a special case.

RATIONAL FUNCTIONS OF BEST UNIFORM APPROXIMATION AND HOLOMORPHIC CONTINUATION OF FUNCTIONS

Let f be a function, continuous and real valued on the segment △,△ (-∞,∞) and {Rn} be the sequence of the rational functions of best uniform approximation to fon △ of order (n,n). In the present work, the convergence of {Rn} in the complex plane is considered for the special caseswhen the poles (or the zeros, respectively) of {Rn} accumulate in the terms of weak convergence of measures to acompact set of zera capacity.As a consequence, sufficient conditions for the holomorphic and the meromorphic continuability of fare given.

On the Uniform and Simultaneous Approximations of Functions

We consider the relation between the simultaneous approximation of two functions and the uniform approximation to one of these functions. In particular, F1 and F2 are continuous functions on a closed interval [a,b], S is an n-dimensional Chebyshev subspace of C [a,b] and s1* & s2* are the best uniform approximations to F1 and F2 from S respectively. The characterization of the best approximation solution is used to show that, under some restrictions on the point set of alternations of F1−s1* and F2−s2*, s1* or s2* is also a best A(1) simultaneous approximation to F1 and F2 from S with F1≥F2 and n=2.

Revision of single atom local density and capture number varying with coverage in uniform depletion approximation and its effect on coalescence and number of stable clusters

In vapour deposition, single atoms （adatoms） on the substrate surface are the main source of growth. The change in its density plays a decisive role in the growth of thin films and quantum size islands. In the nucleation and cluster coalescence stages of vapour deposition, the growth of stable clusters occurs on the substrate surface covered by stable clusters. Nucleation occurs in the non-covered part, while the total area covered by stable clusters on the substrate surface will gradually increase. Carefully taking into account the coverage effect, a revised single atom density rate equation is given for the famous and widely used thin-film rate equation theory, but the work of solving the revised equation has not been done. In this paper, we solve the equation and obtain the single-atom density and capture number by using a uniform depletion approximation. We determine that the single atom density is much lower than that evaluated from the single atom density rate equation in the traditional rate equation theory when the stable cluster coverage fraction is large, and it goes down very fast with an increase in the coverage fraction. The revised equation gives a higher value for the ＇average＇ capture number than the present equation. It also increases with increasing coverage. That makes the preparation of single crystalline thin film materials difficult and the size control of quantum size islands complicated. We also discuss the effect of the revision on coalescence and the number of stable clusters in vapour deposition.

UNIFORM RECIPROCAL APPROXIMATION SUBJECT TO COEFFICIENT CONSTRAINTS

In this paper best approximation by reciprocals of functions of a subspace U_n=span (u_1,…,u_n)satisfying coefficient constraints is considered.We present a characterization of best approximations.When(u_1,…,u_n)is a Descartes system an explicit characterization of best approximations by equioscillations is given.Existence and uniqueness results are shown. Moreover,the theory is applied to best approximaitons by reciprocals of polynomials.

Uniform Convergence of Extremal Polynomials When Domains Have Corners and Special Cusps on the Boundary

We study the approximation properties of the extremal polynomials in Ap?norm and C?norm. We prove estimates for the rate of such convergence of the sequence of the extremal polynomials on domains with corners and special cusps.

Bifurcation phenomena of photodetached electron flux in parallel external fields

A semiclassical method based on the closed-orbit theory is applied to analysing the dynamics of photodetached electron of H- in the parallel electric and magnetic fields. By simply varying the magnetic field we reveal spatial bifurcations of electron orbits at a fixed emission energy, which is referred to as the fold caustic in classical motion. The quantum manifestations of these singularities display a series of intermittent divergences in electronic flux distributions. We introduce semiclassical uniform approximation to repair the electron wavefunctions locally in a mixed phase space and obtain reasonable results. The approximation provides a better treatment of the problem.

A New Third S.N.Bernstein Interpolation Polynomial

In this paper,a new third type S.N.Bernstein interpolation polynomial H n(f;x,r) with zeros of the Chebyshev ploynomial of the second kind is constructed. H n(f;x,r) converge uniformly on [-1,1] for any continuous function f(x) . The convergence order is the best order if \{f(x)∈C j[-1,1],\}0jr, where r is an odd natural number.

Actor-Critic Reinforcement Learning and Application in Developing Computer-Vision-Based Interface Tracking

This paper synchronizes control theory with computer vision by formalizing object tracking as a sequential decision-making process.A reinforcement learning(RL)agent successfully tracks an interface between two liquids,which is often a critical variable to track in many chemical,petrochemical,metallurgical,and oil industries.This method utilizes less than 100 images for creating an environment,from which the agent generates its own data without the need for expert knowledge.Unlike supervised learning(SL)methods that rely on a huge number of parameters,this approach requires far fewer parameters,which naturally reduces its maintenance cost.Besides its frugal nature,the agent is robust to environmental uncertainties such as occlusion,intensity changes,and excessive noise.From a closed-loop control context,an interface location-based deviation is chosen as the optimization goal during training.The methodology showcases RL for real-time object-tracking applications in the oil sands industry.Along with a presentation of the interface tracking problem,this paper provides a detailed review of one of the most effective RL methodologies:actor–critic policy.

Solving two-dimensional Volterra-Fredholm integral equations of the second kind by using Bernstein polynomials

In this paper, we present a numerical method for solving two-dimensional VolterraFredholm integral equations of the second kind（2DV-FK2）. Our method is based on approximating unknown function with Bernstein polynomials. We obtain an error bound for this method and employ the method on some numerical tests to show the efficiency of the method.

Spatio-temporal dynamics of phytohormones in the tomato graft healing process

Graft healing involves a series of cytological and molecular events including wound responses, callus formation and vascular bundle remodelling. Hormones are important signalling molecules regulating plant development and responses to environmental stimuli. However,the detailed dynamics of phytohormones in graft healing remain elusive. In this research, internodes above and below the graft site were harvested from 0 to 168 h after grafting(HAG), and liquid chromatography tandem mass spectrometry(LC-MS/MS) was used to determinate jasmonic acid, auxin, cytokinin, ethylene, salicylic acid, abscisic acid and gibberellin levels during the graft healing process. Uniform manifold approximation and projection(UMAP) and k-means analyses were performed to explore hormone spatio-temporal dynamics. We found the stage-specific and asymmetric accumulation of phytohormones in the tomato graft healing process. At the early healing stage(before vascular bundle reconnection), IAA, cZ, ABA, JA and SA mainly accumulated above the graft site, while tZ and ACC mainly accumulated below the graft site. MEIAA, ICAld and IP mainly accumulated at the later stage. Comminated with the healing process, we suggested that JA is mainly involved in wound responses, IAA is beneficial to the formation of callus and vascular cell development, tZ promotes cell division, and IP is linked to vascular bundle remodelling. In addition, expression of JA-related genes SlMYC2 and SlJAZ2, IAA-related gene SlIAA1, tZ-related genes SlHP2 and SlRR8, and IP-related gene SlRR9 correlated with hormone accumulation. The findings provide important information about the hormones and genes involved in the tomato graft healing process.

Jackson type theorems on complex curves

Let Γ be a closed smooth Jordan curve in the complex plane. In this paper, with the help of a class of fundamental functions of Hermite interpolation, the author introduces a continuous function interpolation which uniformly approximates to f(z) ∈ C(Γ ) with the same order of approximation as that in Jackson Theorem 1 on real interval [1, 1]. The accuracy of the order of approximation is proved. Using the method different from the early works, the author studies simultaneous approximation to function and its derivatives and the desired results analogues to that in Jackson Theorem 2 on real interval [1, 1] are obtained.