• Communications of the Korean Mathematical Society
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• v.29 no.1
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• pp.205-212
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• 2014

This paper investigates mean square exponential dissipativity of singularly perturbed stochastic delay differential equations. The L-operator delay differential inequality and stochastic analysis technique are used to establish sufficient conditions ensuring the mean square exponential dissipativity of singularly perturbed stochastic delay differential equations for sufficiently small ${\varepsilon}$ > 0. An example is presented to illustrate the efficiency of the obtained results.

• Kyungpook Mathematical Journal
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• v.45 no.4
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• pp.503-510
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• 2005

Let $\left{K_{\alpha}\right}_{{\alpha}{\in}{\mathbb{R}}}$ be the multifractal spectrums of a perturbed Cantor set K. We find the set of values ${\alpha}$ of nonempty set $K_{\alpha}$ by using the Birkhoff ergodic theorem. And we also show that such $K_{\alpha}$ is a fractal set in the sense of Taylor [12].

• The Transactions of The Korean Institute of Electrical Engineers
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• v.62 no.8
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• pp.1144-1150
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• 2013

The stabilizing controller design problem of time-delay singularly perturbed systems is considered. The proposed approach is based on the $H_{\infty}$ norm and the composite control method. A sufficient condition for the stability of the time-delay slow subsystem is presented. Using this condition, we can construct the composite control law for the time-delay singularly perturbed system and analysis the system by the matrix Lambert W function. Illustrated examples are presented to demonstrate the validity and applicability of the proposed method.

• Journal of applied mathematics & informatics
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• v.39 no.5_6
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• pp.623-641
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• 2021

In this paper, numerical treatment of singularly perturbed parabolic delay differential equations is considered. The considered problem have small delay on the spatial variable of the reaction term. To treat the delay term, Taylor series approximation is applied. The resulting singularly perturbed parabolic PDEs is solved using Crank Nicolson method in temporal direction with non-standard finite difference method in spatial direction. A detail stability and convergence analysis of the scheme is given. We proved the uniform convergence of the scheme with order of convergence O(N^-1 + (∆t)²), where N is the number of mesh points in spatial discretization and ∆t is mesh length in temporal discretization. Two test examples are used to validate the theoretical results of the scheme.

• Journal of applied mathematics & informatics
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• v.14 no.1_2
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• pp.329-341
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• 2004

We consider the rate of convergence for a class of perturbed hemivariational inequalities in reflexive Banach Spaces. Our results can be viewed as an extension and refinement of some previous known results in this area.

• Computational Structural Engineering
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• v.7 no.3
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• pp.115-122
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• 1994

Design sensitivity analysis of the second order perturbed eigenproblems for random structural system is presented. Dynamic response of random system including uncertainties for the design variable is calculated with the first order and second order perturbation method to original governing equation. In optimal design methods, there is fundamental requirement for design gradients. A method for calculating the sensitivity coefficients is developed using the direct differentiation method for the governing equation and first order and second order perturbed equation.

• The Transactions of The Korean Institute of Electrical Engineers
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• v.58 no.9
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• pp.1821-1826
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• 2009

An adaptive neural controller for perturbed strict-feedback nonlinear system is proposed. All the previous adaptive neural (or fuzzy) controllers are based on the backstepping scheme where the universal approximators are employed in every design steps. These schemes involve virtual controls and their time derivatives that make the stability analysis and implementation of the controller very complex. This fact is called 'explosion of complexty ' since the complexity grows exponentially as the system dynamic order increases. The proposed adaptive neural control scheme adopt the backstepping design procedure only for determining ideal control law and employ only one neural network to approximate the finally selected ideal controller, which makes the controller design procedure and stability analysis considerably simple compared to the previously proposed controllers. It is shown that all the time-varing signals containing tracking error are stable in the Lyapunov viewpoint.

• The Pure and Applied Mathematics
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• v.13 no.4 s.34
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• pp.261-267
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• 2006

A local convergence analysis of Newton's method for perturbed generalized equations is provided in a Banach space setting. Using center Lipschitzian conditions which are actually needed instead of Lipschitzian hypotheses on the $Fr\'{e}chet$-derivative of the operator involved and more precise estimates under less computational cost we provide a finer convergence analysis of Newton's method than before [5]-[7].

• Genomics & Informatics
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• v.7 no.2
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• pp.122-130
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• 2009

A systems biology approach for the identification of perturbed molecular functions is required to understand the complex progressive disease such as breast cancer. In this study, we analyze the microarray data with Gene Ontology terms of molecular functions to select perturbed molecular functional modules in breast cancer tissues based on the definition of Gene ontology Functional Code. The Gene Ontology is three structured vocabularies describing genes and its products in terms of their associated biological processes, cellular components and molecular functions. The Gene Ontology is hierarchically classified as a directed acyclic graph. However, it is difficult to visualize Gene Ontology as a directed tree since a Gene Ontology term may have more than one parent by providing multiple paths from the root. Therefore, we applied the definition of Gene Ontology codes by defining one or more GO code(s) to each GO term to visualize the hierarchical classification of GO terms as a network. The selected molecular functions could be considered as perturbed molecular functional modules that putatively contributes to the progression of disease. We evaluated the method by analyzing microarray dataset of breast cancer tissues; i.e., normal and invasive breast cancer tissues. Based on the integration approach, we selected several interesting perturbed molecular functions that are implicated in the progression of breast cancers. Moreover, these selected molecular functions include several known breast cancer-related genes. It is concluded from this study that the present strategy is capable of selecting perturbed molecular functions that putatively play roles in the progression of diseases and provides an improved interpretability of GO terms based on the definition of Gene Ontology codes.

• Journal of Mechanical Science and Technology
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• v.16 no.10
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• pp.1229-1238
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• 2002

In this paper, a method to obtain the sensitivity of eigenvalues and the random responses of the structure with uncertain parameters is proposed. The concept of the proposed method is that the perturbed equation of each uncertain substructure is obtained using the finite element method, and the perturbed equation of the overall structure is obtained using the mode synthesis method. By this way, the reduced order perturbed equation of the uncertain system can be obtained. And the response of the uncertain system is obtained using probability method. As a numerical example, a simple piping system is considered as an example structure. The damping and spring constants of the support are considered as the uncertain parameters. Then the variations of the eigenvalues, the correlation function and the power spectral density function of the responses are calculated. As a result, the proposed method is considered to be useful technique to analyze the sensitivities of eigenvalues and random response against random excitation in terms of the accuracy and the calculation time.