# Accumulation Point

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### Cornelis Roos – 1st expert on this subject based on the ideXlab platform

• ##### Convergence of Full-Newton Step Infeasible Interior-Point Methods for Linear Optimization
, 2020
Co-Authors: A. Asadi, G. Gu, Cornelis Roos

Abstract:

Roos [SIAM J. Optim., 16(4):1110‐1136, 2006] proposed a new primal-dual infeasible interior-Point method for Linear Optimization. This method can be viewed as a homotopy method, which diers from the classical infeasible interior-Point methods in that the new method uses only full-Newton steps (instead of damped steps). In this paper we investigate convergence properties of this method. We show that the homotopy path has precisely one Accumulation Point in the optimal set. Moreover, this Accumulation Point is the analytic center of a subset of the optimal set and depends on the starting Point of the infeasible interior-Point method.

• ##### Convergence of the homotopy path for a full-Newton step infeasible interior-Point method
Operations Research Letters, 2010
Co-Authors: A. Asadi, G. Gu, Cornelis Roos

Abstract:

Roos [C. Roos, A full-Newton step O(n) infeasible interior-Point algorithm for linear optimization. SIAM J. Optim. 16 (4) (2006) 1110-1136 (electronic)] proposed a new primal-dual infeasible interior-Point method for linear optimization. This new method can be viewed as a homotopy method. In this work, we show that the homotopy path has precisely one Accumulation Point in the optimal set. Moreover, this Accumulation Point is the analytic center of a subset of the optimal set and depends on the starting Point of the infeasible interior-Point method.

### Yuqiu Zhao – 2nd expert on this subject based on the ideXlab platform

• ##### Uniform asymptotics for discrete orthogonal polynomials on infinite nodes with an AccumulationPoint
Analysis and Applications, 2016
Co-Authors: Xiaobo Wu, Shuaixia Xu, Yuqiu Zhao

Abstract:

In this paper, we develop the Riemann–Hilbert method to study the asymptotics of discrete orthogonal polynomials on infinite nodes with an Accumulation Point. To illustrate our method, we consider the Tricomi–Carlitz polynomials [Formula: see text] where [Formula: see text] is a positive parameter. Uniform Plancherel–Rotach type asymptotic formulas are obtained in the entire complex plane including a neighborhood of the origin, and our results agree with the ones obtained earlier in [W. M. Y. Goh and J. Wimp, On the asymptotics of the Tricomi–Carlitz polynomials and their zero distribution. I, SIAM J. Math. Anal. 25 (1994) 420–428] and in [K. F. Lee and R. Wong, Uniform asymptotic expansions of the Tricomi–Carlitz polynomials, Proc. Amer. Math. Soc. 138 (2010) 2513–2519].

• ##### uniform asymptotics for discrete orthogonal polynomials on infinite nodes with an AccumulationPoint
Analysis and Applications, 2016
Co-Authors: Xiaobo Wu, Shuaixia Xu, Yuqiu Zhao

Abstract:

In this paper, we develop the Riemann–Hilbert method to study the asymptotics of discrete orthogonal polynomials on infinite nodes with an Accumulation Point. To illustrate our method, we consider the Tricomi–Carlitz polynomials fn(α)(z) where α is a positive parameter. Uniform Plancherel–Rotach type asymptotic formulas are obtained in the entire complex plane including a neighborhood of the origin, and our results agree with the ones obtained earlier in [W. M. Y. Goh and J. Wimp, On the asymptotics of the Tricomi–Carlitz polynomials and their zero distribution. I, SIAM J. Math. Anal. 25 (1994) 420–428] and in [K. F. Lee and R. Wong, Uniform asymptotic expansions of the Tricomi–Carlitz polynomials, Proc. Amer. Math. Soc. 138 (2010) 2513–2519].

• ##### uniform asymptotics for discrete orthogonal polynomials on infinite nodes with an AccumulationPoint
arXiv: Classical Analysis and ODEs, 2014
Co-Authors: Xiaobo Wu, Shuaixia Xu, Yuqiu Zhao

Abstract:

In this paper, we develop the Riemann-Hilbert method to study the asymptotics of discrete orthogonal polynomials on infinite nodes with an Accumulation Point. To illustrate our method, we consider the Tricomi-Carlitz polynomials $f_n^{(\alpha)}(z)$ where $\alpha$ is a positive parameter. Uniform Plancherel-Rotach type asymptotic formulas are obtained in the entire complex plane including a neighborhood of the origin, and our results agree with the ones obtained earlier in [{\it SIAM J.\;Math.\;Anal} {\bf 25} (1994)] and [{{\it Proc.\;Amer.\;Math.\;Soc.\,}{\bf138} (2010)}].

### O B Isaeva – 3rd expert on this subject based on the ideXlab platform

• ##### period tripling AccumulationPoint for complexified henon map
arXiv: Chaotic Dynamics, 2005
Co-Authors: O B Isaeva, Sergey P Kuznetsov

Abstract:

Accumulation Point of period-tripling bifurcations for complexified Henon map is found. Universal scaling properties of parameter space and Fourier spectrum intrinsic to this critical Point is demonstrated.

• ##### Properties of Fourier spectrum of the signal, generated at the AccumulationPoint of period-tripling bifurcations
arXiv: Chaotic Dynamics, 2005
Co-Authors: O B Isaeva

Abstract:

Universal regularities of the Fourier spectrum of signal, generated by complex analytic map at the period-tripling bifurcations Accumulation Point are considered. The difference between intensities of the subharmonics at the values of frequency corresponding to the neighbor hierarchical levels of the spectrum is characterized by a constant $\gamma=21.9$ dB?, which is an analogue of the known value $\gamma_F=13.4$ dB, intrinsic to the Feigenbaum critical Point. Data of the physical experiment, directed to the observation of the spectrum at period-tripling Accumulation Point, are represented.

• ##### effect of noise on the dynamics of a complex map at the period tripling AccumulationPoint
Physical Review E, 2004
Co-Authors: O B Isaeva, Sergey P Kuznetsov, A H Osbaldestin

Abstract:

As shown recently [O.B. Isaeva et al., Phys. Rev. E 64, 055201 (2001)], the phenomena intrinsic to dynamics of complex analytic maps under appropriate conditions may occur in physical systems. We study scaling regularities associated with the effect of additive noise upon the period-tripling bifurcation cascade generalizing the renormalization group approach of Crutchfield et al. [Phys. Rev. Lett. 46, 933 (1981)] and Shraiman et al. [Phys. Rev. Lett. 46, 935 (1981)], originally developed for the period doubling transition to chaos in the presence of noise. The universal constant determining the rescaling rule for the intensity of the noise in period tripling is found to be γ=12.206 640 9… . Numerical evidence of the expected scaling is demonstrated.