• Title, Summary, Keyword: local Lipschitz condition

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THE CONVERGENCE BALL OF INEXACT NEWTON-LIKE METHOD IN BANACH SPACE UNDER WEAK LIPSHITZ CONDITION

  • Argyros, Ioannis K.;George, Santhosh
    • Journal of the Chungcheong Mathematical Society
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    • v.28 no.1
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    • pp.1-12
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    • 2015
  • We present a local convergence analysis for inexact Newton-like method in a Banach space under weaker Lipschitz condition. The convergence ball is enlarged and the estimates on the error distances are more precise under the same computational cost as in earlier studies such as [6, 7, 11, 18]. Some special cases are considered and applications for solving nonlinear systems using the Newton-arithmetic mean method are improved with the new convergence technique.

LOCAL CONVERGENCE OF NEWTON-LIKE METHODS FOR GENERALIZED EQUATIONS

  • Argyros, Ioannis K.
    • East Asian mathematical journal
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    • v.25 no.4
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    • pp.425-431
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    • 2009
  • We provide a local convergence analysis for Newton-like methods for the solution of generalized equations in a Banach space setting. Using some ideas of ours introduced in [2] for nonlinear equations we show that under weaker hypotheses and computational cost than in [7] a larger convergence radius and finer error bounds on the distances involved can be obtained.

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REAL-VARIABLE CHARACTERIZATIONS OF VARIABLE HARDY SPACES ON LIPSCHITZ DOMAINS OF ℝn

  • Liu, Xiong
    • Bulletin of the Korean Mathematical Society
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    • v.58 no.3
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    • pp.745-765
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    • 2021
  • Let Ω be a proper open subset of ℝn and p(·) : Ω → (0, ∞) be a variable exponent function satisfying the globally log-Hölder continuous condition. In this article, the author introduces the "geometrical" variable Hardy spaces Hp(·)r (Ω) and Hp(·)z (Ω) on Ω, and then obtains the grand maximal function characterizations of Hp(·)r (Ω) and Hp(·)z (Ω) when Ω is a strongly Lipschitz domain of ℝn. Moreover, the author further introduces the "geometrical" variable local Hardy spaces hp(·)r (Ω), and then establishes the atomic characterization of hp(·)r (Ω) when Ω is a bounded Lipschitz domain of ℝn.

CONE VALUED LYAPUNOV TYPE STABILITY ANALYSIS OF NONLINEAR EQUATIONS

  • Chang, Sung-Kag;Oh, Young-Sun;An, Jeong-Hyang
    • Journal of the Korean Mathematical Society
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    • v.37 no.5
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    • pp.835-847
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    • 2000
  • We investigate various ${\Phi}$(t)-stability of comparison differential equations and we obtain necessary and/or sufficient conditions for the asymptotic and uniform asymptotic stability of the differential equations x'=f(t, x).

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A COMPARATIVE STUDY BETWEEN CONVERGENCE RESULTS FOR NEWTON'S METHOD

  • Argyros, Ioannis K.;Hilout, Said
    • The Pure and Applied Mathematics
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    • v.15 no.4
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    • pp.365-375
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    • 2008
  • We present a new theorem for the semilocal convergence of Newton's method to a locally unique solution of an equation in a Banach space setting. Under a gamma-type condition we show that we can extend the applicability of Newton's method given in [12]. We also provide a comparative study between results using the classical Newton-Kantorovich conditions ([6], [7], [10]), and the ones using the gamma-type conditions ([12], [13]). Numerical examples are also provided.

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EXTENDING THE APPLICABILITY OF INEXACT GAUSS-NEWTON METHOD FOR SOLVING UNDERDETERMINED NONLINEAR LEAST SQUARES PROBLEMS

  • Argyros, Ioannis Konstantinos;Silva, Gilson do Nascimento
    • Journal of the Korean Mathematical Society
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    • v.56 no.2
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    • pp.311-327
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    • 2019
  • The aim of this paper is to extend the applicability of Gauss-Newton method for solving underdetermined nonlinear least squares problems in cases not covered before. The novelty of the paper is the introduction of a restricted convergence domain. We find a more precise location where the Gauss-Newton iterates lie than in earlier studies. Consequently the Lipschitz constants are at least as small as the ones used before. This way and under the same computational cost, we extend the local as well the semilocal convergence of Gauss-Newton method. The new developmentes are obtained under the same computational cost as in earlier studies, since the new Lipschitz constants are special cases of the constants used before. Numerical examples further justify the theoretical results.

Euler-Maruyama Numerical solution of some stochastic functional differential equations

  • Ahmed, Hamdy M.
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • v.11 no.1
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    • pp.13-30
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    • 2007
  • In this paper we study the numerical solutions of the stochastic functional differential equations of the following form $$du(x,\;t)\;=\;f(x,\;t,\;u_t)dt\;+\;g(x,\;t,\;u_t)dB(t),\;t\;>\;0$$ with initial data $u(x,\;0)\;=\;u_0(x)\;=\;{\xi}\;{\in}\;L^p_{F_0}\;([-{\tau},0];\;R^n)$. Here $x\;{\in}\;R^n$, ($R^n$ is the ${\nu}\;-\;dimenional$ Euclidean space), $f\;:\;C([-{\tau},\;0];\;R^n)\;{\times}\;R^{{\nu}+1}\;{\rightarrow}\;R^n,\;g\;:\;C([-{\tau},\;0];\;R^n)\;{\times}\;R^{{\nu}+1}\;{\rightarrow}\;R^{n{\times}m},\;u(x,\;t)\;{\in}\;R^n$ for each $t,\;u_t\;=\;u(x,\;t\;+\;{\theta})\;:\;-{\tau}\;{\leq}\;{\theta}\;{\leq}\;0\;{\in}\;C([-{\tau},\;0];\;R^n)$, and B(t) is an m-dimensional Brownian motion.

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Estimation of error variance in nonparametric regression under a finite sample using ridge regression

  • Park, Chun-Gun
    • Journal of the Korean Data and Information Science Society
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    • v.22 no.6
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    • pp.1223-1232
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    • 2011
  • Tong and Wang's estimator (2005) is a new approach to estimate the error variance using least squares method such that a simple linear regression is asymptotically derived from Rice's lag- estimator (1984). Their estimator highly depends on the setting of a regressor and weights in small sample sizes. In this article, we propose a new approach via a local quadratic approximation to set regressors in a small sample case. We estimate the error variance as the intercept using a ridge regression because the regressors have the problem of multicollinearity. From the small simulation study, the performance of our approach with some existing methods is better in small sample cases and comparable in large cases. More research is required on unequally spaced points.

DNA Watermarking Method based on Random Codon Circular Code (랜덤 코돈 원형 부호 기반의 DNA 워터마킹)

  • Lee, Suk-Hwan;Kwon, Seong-Geun;Kwon, Ki-Ryong
    • Journal of Korea Multimedia Society
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    • v.16 no.3
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    • pp.318-329
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    • 2013
  • This paper proposes a DNA watermarking method for the privacy protection and the prevention of illegal copy. The proposed method allocates codons to random circular angles by using random mapping table and selects triplet codons for embedding target with the help of the Lipschitz regularity value of local modulus maxima of codon circular angles. Then the watermark is embedded into circular angles of triplet codons without changing the codes of amino acids in a DNA. The length and location of target triplet codons depend on the random mapping table for 64 codons that includes start and stop codons. This table is used as the watermark key and can be applied on any codon sequence regardless of the length of sequence. If this table is unknown, it is very difficult to detect the length and location of them for extracting the watermark. We evaluated our method and DNA-crypt watermarking of Heider method on the condition of similar capacity. From evaluation results, we verified that our method has lower base changing rate than DNA-crypt and has lower bit error rate on point mutation and insertions/deletions than DNA-crypt. Furthermore, we verified that the entropy of random mapping table and the locaton of triplet codons is high, meaning that the watermark security has high level.