For rational functions it is proved that the Julia set contains buried components whenever the Julia set is disconnected and the Fatou set has no completely invariant component. For transcendental entire functions of ...For rational functions it is proved that the Julia set contains buried components whenever the Julia set is disconnected and the Fatou set has no completely invariant component. For transcendental entire functions of finite type it is proved that the Julia set contains unbounded continua of buried points whenever the Fatou set is disconnected.展开更多
The authors study the functional equation f^[m]=1/f and analyze the featuresof its piecewise continuous solutions. All the piecewise continuous real solutions are obtained constructively. The results generalize the on...The authors study the functional equation f^[m]=1/f and analyze the featuresof its piecewise continuous solutions. All the piecewise continuous real solutions are obtained constructively. The results generalize the ones in [2]. Moreover, the conclusion is drawn that there is no circuit iterative roots for those functions not satisfying Babbage equation.展开更多
Let R(z) be a rational function of degree d ≥ 2. Then R(z) has at least one repelling periodic point of given period k ≥ 2, unless k = 4 and d=2, or k= 3 and d ≤ 3, or k=2 and d≤8. Examples show that all exception...Let R(z) be a rational function of degree d ≥ 2. Then R(z) has at least one repelling periodic point of given period k ≥ 2, unless k = 4 and d=2, or k= 3 and d ≤ 3, or k=2 and d≤8. Examples show that all exceptional cases occur.展开更多
The proximal alternating linearized minimization(PALM)method suits well for solving blockstructured optimization problems,which are ubiquitous in real applications.In the cases where subproblems do not have closed-for...The proximal alternating linearized minimization(PALM)method suits well for solving blockstructured optimization problems,which are ubiquitous in real applications.In the cases where subproblems do not have closed-form solutions,e.g.,due to complex constraints,infeasible subsolvers are indispensable,giving rise to an infeasible inexact PALM(PALM-I).Numerous efforts have been devoted to analyzing the feasible PALM,while little attention has been paid to the PALM-I.The usage of the PALM-I thus lacks a theoretical guarantee.The essential difficulty of analysis consists in the objective value nonmonotonicity induced by the infeasibility.We study in the present work the convergence properties of the PALM-I.In particular,we construct a surrogate sequence to surmount the nonmonotonicity issue and devise an implementable inexact criterion.Based upon these,we manage to establish the stationarity of any accumulation point,and moreover,show the iterate convergence and the asymptotic convergence rates under the assumption of the Lojasiewicz property.The prominent advantages of the PALM-I on CPU time are illustrated via numerical experiments on problems arising from quantum physics and 3-dimensional anisotropic frictional contact.展开更多
In this paper, Jurgen's relative result is extended to a uniformly convex Banach space, and the convergence of iterative sequence in an uniformly convex Banach space for asymptotically non-expansive mapping is ...In this paper, Jurgen's relative result is extended to a uniformly convex Banach space, and the convergence of iterative sequence in an uniformly convex Banach space for asymptotically non-expansive mapping is proved.展开更多
In this paper,we consider the fixed point theorem for Proinov mappings with a contractive iterate at a point.In other words,we combine and unify the basic approaches of Proinov and Sehgal in the framework of the compl...In this paper,we consider the fixed point theorem for Proinov mappings with a contractive iterate at a point.In other words,we combine and unify the basic approaches of Proinov and Sehgal in the framework of the complete metric spaces.We consider examples to illustrate the validity of the obtained result.展开更多
In this paper, we establish the dynamical systems of iterated entire aigebroid functions. According to dynamics, we give a classification theorem of entire aigebroid functions. Some typical properties on Julia set and...In this paper, we establish the dynamical systems of iterated entire aigebroid functions. According to dynamics, we give a classification theorem of entire aigebroid functions. Some typical properties on Julia set and Fatou set are proved. And we obtain a similar property on the dstribution of J(f) and Vf.展开更多
This paper introduces a post-iteration averaging algorithm to achieve asymptotic optimality in convergence rates of stochastic approximation algorithms for consensus control with structural constraints. The algorithm ...This paper introduces a post-iteration averaging algorithm to achieve asymptotic optimality in convergence rates of stochastic approximation algorithms for consensus control with structural constraints. The algorithm involves two stages. The first stage is a coarse approximation obtained using a sequence of large stepsizes. Then, the second stage provides a refinement by averaging the iterates from the first stage. We show that the new algorithm is asymptotically efficient and gives the optimal convergence rates in the sense of the best scaling factor and ＇smallest＇ possible asymptotic variance.展开更多
The dynamics of complex quadratic polynomials P_c(z)=z^2+c has been studiedextensively. In this note, we get one upper bound of the radius of the filled-in Juliaset of P(z), by Bieberbach conjecture(Theorem of de Bran...The dynamics of complex quadratic polynomials P_c(z)=z^2+c has been studiedextensively. In this note, we get one upper bound of the radius of the filled-in Juliaset of P(z), by Bieberbach conjecture(Theorem of de Branges) in univalent functionsand gives an answer to the question of Douady. As its application, a lower bound ofthe Hausdorff dimension for the Julia set of P(z) is obtained.展开更多
文摘For rational functions it is proved that the Julia set contains buried components whenever the Julia set is disconnected and the Fatou set has no completely invariant component. For transcendental entire functions of finite type it is proved that the Julia set contains unbounded continua of buried points whenever the Fatou set is disconnected.
基金the Youth Foundation of the Educational Department of Sichuan Province(No.072B042).
文摘The authors study the functional equation f^[m]=1/f and analyze the featuresof its piecewise continuous solutions. All the piecewise continuous real solutions are obtained constructively. The results generalize the ones in [2]. Moreover, the conclusion is drawn that there is no circuit iterative roots for those functions not satisfying Babbage equation.
文摘Let R(z) be a rational function of degree d ≥ 2. Then R(z) has at least one repelling periodic point of given period k ≥ 2, unless k = 4 and d=2, or k= 3 and d ≤ 3, or k=2 and d≤8. Examples show that all exceptional cases occur.
基金supported by National Natural Science Foundation of China(Grant Nos.12125108,11971466,11991021,11991020,12021001 and 12288201)Key Research Program of Frontier Sciences,Chinese Academy of Sciences(Grant No.ZDBS-LY-7022)CAS(the Chinese Academy of Sciences)AMSS(Academy of Mathematics and Systems Science)-PolyU(The Hong Kong Polytechnic University)Joint Laboratory of Applied Mathematics.
文摘The proximal alternating linearized minimization(PALM)method suits well for solving blockstructured optimization problems,which are ubiquitous in real applications.In the cases where subproblems do not have closed-form solutions,e.g.,due to complex constraints,infeasible subsolvers are indispensable,giving rise to an infeasible inexact PALM(PALM-I).Numerous efforts have been devoted to analyzing the feasible PALM,while little attention has been paid to the PALM-I.The usage of the PALM-I thus lacks a theoretical guarantee.The essential difficulty of analysis consists in the objective value nonmonotonicity induced by the infeasibility.We study in the present work the convergence properties of the PALM-I.In particular,we construct a surrogate sequence to surmount the nonmonotonicity issue and devise an implementable inexact criterion.Based upon these,we manage to establish the stationarity of any accumulation point,and moreover,show the iterate convergence and the asymptotic convergence rates under the assumption of the Lojasiewicz property.The prominent advantages of the PALM-I on CPU time are illustrated via numerical experiments on problems arising from quantum physics and 3-dimensional anisotropic frictional contact.
文摘In this paper, Jurgen's relative result is extended to a uniformly convex Banach space, and the convergence of iterative sequence in an uniformly convex Banach space for asymptotically non-expansive mapping is proved.
文摘In this paper,we consider the fixed point theorem for Proinov mappings with a contractive iterate at a point.In other words,we combine and unify the basic approaches of Proinov and Sehgal in the framework of the complete metric spaces.We consider examples to illustrate the validity of the obtained result.
基金Project supported by the Tianyuan Foundation of China
文摘In this paper, we establish the dynamical systems of iterated entire aigebroid functions. According to dynamics, we give a classification theorem of entire aigebroid functions. Some typical properties on Julia set and Fatou set are proved. And we obtain a similar property on the dstribution of J(f) and Vf.
基金supported by the U.S. Army Research Office (No. W911NF-12-1-0223)
文摘This paper introduces a post-iteration averaging algorithm to achieve asymptotic optimality in convergence rates of stochastic approximation algorithms for consensus control with structural constraints. The algorithm involves two stages. The first stage is a coarse approximation obtained using a sequence of large stepsizes. Then, the second stage provides a refinement by averaging the iterates from the first stage. We show that the new algorithm is asymptotically efficient and gives the optimal convergence rates in the sense of the best scaling factor and ＇smallest＇ possible asymptotic variance.
文摘The dynamics of complex quadratic polynomials P_c(z)=z^2+c has been studiedextensively. In this note, we get one upper bound of the radius of the filled-in Juliaset of P(z), by Bieberbach conjecture(Theorem of de Branges) in univalent functionsand gives an answer to the question of Douady. As its application, a lower bound ofthe Hausdorff dimension for the Julia set of P(z) is obtained.
