• Bulletin of the Korean Mathematical Society
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• v.44 no.4
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• pp.683-689
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• 2007

Let $\mathcal{B}$ be a certain Banach space consisting of analytic functions defined on a bounded domain G in the complex plane. Let ${\varphi}$ be an analytic polynomial or a rational function and let $M_{\varphi}$ denote the operator of multiplication by ${\varphi}$. Under certain condition on ${\varphi}$ and G, we characterize the commutant of $M_{\varphi}$ that is the set of all bounded operators T such that $TM_{\varphi}=M_{\varphi}T$. We show that $T=M_{\Psi}$, for some function ${\Psi}$ in $\mathcal{B}$.

• International Journal of Fuzzy Logic and Intelligent Systems
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• v.5 no.3
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• pp.253-258
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• 2005

In this study, we introduce a new category of fuzzy inference systems based on information granulation to carry out the model identification of complex and nonlinear systems. Informal speaking, information granules are viewed as linked collections of objects (data, in particular) drawn together by the criteria of proximity, similarity, or functionality. To identify the structure of fuzzy rules we use genetic algorithms (GAs). Granulation of information with the aid of Hard C-Means (HCM) clustering algorithm help determine the initial parameters of fuzzy model such as the initial apexes of the membership functions and the initial values of polynomial functions being used in the premise and consequence part of the fuzzy rules. And the initial parameters are tuned effectively with the aid of the genetic algorithms and the least square method (LSM). The proposed model is contrasted with the performance of the conventional fuzzy models in the literature.

• Proceedings of the Korean Institute of Information and Commucation Sciences Conference
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• 2009.05a
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• pp.433-436
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• 2009

Many researchers had made a diversity of attempts for generating pseudorandom sequences such as the method of using LFSR whose characteristic polynomial is a primitive polynomial, of using Cellular Automata and of using quadratic functions. In this paper, we can analyze and characterize the methods for generating maximal period pseudorandom sequences constructed by quadratic functions.

• Communications of the Korean Mathematical Society
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• v.24 no.2
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• pp.171-180
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• 2009

In this paper, we give three criteria for non-polynomial functions interpolated from the set of univariate polynomials of degree less than m over a finite field to be a permutation on the same set.

• Journal of information and communication convergence engineering
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• v.2 no.3
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• pp.193-197
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• 2004

In this paper, a combination problem of controllers which are the same type of $H_{\infty}$ controllers designed with different weighting functions. This approach can remove the difficulty in the selection of the weighting functions. As a sub-controller, the Youla type of $H_{\infty}$ controller is used. In the $H_{\infty}$ controller, Youla parameterization is used to minimize $H_{\infty}$ norm of mixed sensitivity function by using polynomial approach. Computer simulation results show the robustness improvement and the performance improvement.

• Korean Journal of Mathematics
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• v.30 no.4
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• pp.629-642
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• 2022

In a recent paper, Sakar and Güney introduced a new class of bi-close-to-convex functions and determined the estimates for the general Taylor-Maclaurin coefficients of functions therein. The main purpose of this note is to give a generalization of this class. Also we point out the proof by Sakar and Güney is incorrect and present a correct proof.

• Bulletin of the Korean Mathematical Society
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• v.55 no.2
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• pp.625-631
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• 2018

Let R be a commutative ring with identity and let R[x] be the collection of polynomials with coefficients in R. There are a lot of multiplications in R[x] such that together with the usual addition, R[x] becomes a ring that contains R as a subring. These multiplications are from a class of functions ${\lambda}$ from ${\mathbb{N}}_0$ to ${\mathbb{N}}$. The trivial case when ${\lambda}(i)=1$ for all i gives the usual polynomial ring. Among nontrivial cases, there is an important one, namely, the case when ${\lambda}(i)=i!$ for all i. For this case, it gives the well-known Hurwitz polynomial ring $R_H[x]$. In this paper, we completely determine the Krull dimension of $R_H[x]$ when R is a $Pr{\ddot{u}}fer$ domain. Let R be a $Pr{\ddot{u}}fer$ domain. We show that dim $R_H[x]={\dim}\;R+1$ if R has characteristic zero and dim $R_H[x]={\dim}\;R$ otherwise.

• Proceedings of the KIEE Conference
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• 2005.07d
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• pp.2888-2890
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• 2005

In this paper, we introduce a new structure of fuzzy-neural networks Fuzzy Set-based Polynomial Neural Networks (FSPNN). The two underlying design mechanisms of such networks involve genetic optimization and information granulation. The resulting constructs are Fuzzy Polynomial Neural Networks (FPNN) with fuzzy set-based polynomial neurons (FSPNs) regarded as their generic processing elements. First, we introduce a comprehensive design methodology (viz. a genetic optimization using Genetic Algorithms) to determine the optimal structure of the FSPNNs. This methodology hinges on the extended Group Method of Data Handling (GMDH) and fuzzy set-based rules. It concerns FSPNN-related parameters such as the number of input variables, the order of the polynomial, the number of membership functions, and a collection of a specific subset of input variables realized through the mechanism of genetic optimization. Second, the fuzzy rules used in the networks exploit the notion of information granules defined over systems variables and formed through the process of information granulation. This granulation is realized with the aid of the hard C- Means clustering (HCM). The performance of the network is quantified through experimentation in which we use a number of modeling benchmarks already experimented with in the realm of fuzzy or neurofuzzy modeling.

• 제어로봇시스템학회:학술대회논문집
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• 2000.10a
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• pp.372-372
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• 2000

In this paper, we propose Neuro Fuzzy Polynomial Networks(NFPN) based on Polynomial Neural Network(PNN) and Neuro-Fuzzy(NF) for model identification of complex and nonlinear systems. The proposed NFPN is generated from the mutually combined structure of both NF and PNN. The one and the other are considered as the premise part and consequence part of NFPN structure respectively. As the premise part of NFPN, NF uses both the simplified fuzzy inference as fuzzy inference method and error back-propagation algorithm as learning rule. The parameters such as parameters of membership functions, learning rates and momentum coefficients are adjusted using genetic algorithms. As the consequence part of NFPN, PNN is based on Group Method of Data Handling(GMDH) method and its structure is similar to Neural Networks. But the structure of PNN is not fixed like in conventional Neural Networks and self-organizing networks that can be generated. NFPN is available effectively for multi-input variables and high-order polynomial according to the combination of NF with PNN. Accordingly it is possible to consider the nonlinearity characteristics of process and to get better output performance with superb predictive ability. In order to evaluate the performance of proposed models, we use the nonlinear function. The results show that the proposed FPNN can produce the model with higher accuracy and more robustness than any other method presented previously.

• The Journal of Natural Sciences
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• v.4
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• pp.191-220
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• 1991

A new texture segmentation-based image coding technique which performs segmentation based on roughness of textural regions and properties of the human visual system (HVS) is presented. This method solves the problems of a segmentation-based image coding technique with constant segments by proposing a methodology for segmenting an image texturally homogeneous regions with respect to the degree of roughness as perceived by the HVS. The fractal dimension is used to measure the roughness of the textural regions. The segmentation is accomplished by thresholding the fractal dimension so that textural regions are classified into three texture classes; perceived constant intensity, smooth texture, and rough texture. An image coding system with high compression and good image quality is achieved by developing an efficient coding technique for each segment boundary and each texture class. For the boundaries, a binary image representing all the boundaries is created. For regions belonging to perceived constant intensity, only the mean intensity values need to be transmitted. The smooth and rough texture regions are modeled first using polynomial functions, so only the coefficients characterizing the polynomial functions need to be transmitted. The bounda-ries, the means and the polynomial functions are then each encoded using an errorless coding scheme. Good quality reconstructed images are obtained with about 0.08 to 0.3 bit per pixel for three different types of imagery ; a head and shoulder image with little texture variation, a complex image with many edges, and a natural outdoor image with highly textured areas.