Certain classes of generating functions for the Jacobi and related hypergeometric polynomials

For a certain class of generalized hypergeometric polynomials, the authors first derive a general theorem on bilinear, bilateral, and mixed multilateral generating functions and then apply these generating functions in order to deduce the corresponding results for the classical Jacobi and Laguerre polynomials. They also consider several linear generating functions for these polynomials as well as for some multivariable Jacobi and multivariable Laguerre polynomials which were investigated in recent years. Some of the linear generating functions, presented in this paper, are associated with the Stirling numbers of the second kind.     

A class of new transforms tailored for the hypergeometric series

A new class of transforms, tailored for the hypergeometric series, is proposed and tested on a number of series. From a knowledge of a given number of terms of a sequence, these transforms reproduce the functions with a better accuracy than the Levin-like transforms. Though there exists a correlation between the approximative power of a rational approximant and its ability to predict the leading terms of a series, there are exceptions to this, especially in the case of divergent series. The new transforms can, in most cases, predict a number of extra terms not used in the construction of the approximants.     

A new approach to feature selection based on the Karhunen-Loeve expansion

After surveying existing feature selection procedures based upon the Karhunen-Loeve (K-L) expansion, the paper describes a new K-L technique that overcomes some of the limitations of the earlier procedures. The new method takes into account information on both the class variances and means, but lays particular emphasis on the classification potential of the latter. The results of a series of experiments concerned with the classification of real vector-electrocardiogram and artificially generated data demonstrate the advantages of the new method. They suggest that it is particularly useful for pattern recognition when combined with classification procedures based upon discriminant functions obtained by recursive least squares analysis.     

Some Integrals of the products of laguerre polynomials

The evaluation of an integral of the product of Laguerre polynomials was discussed recently in this Journal by Mavromatis [12] (1990) and Lee [9] (1997) [see also Ong and Lee [14] (2000)]. The main object of the present sequel to these earlier works is to consider a family of such integrals of the products of Laguerre, Hermite, and other classical orthogonal polynomials in a systematic and unified manner. Relevant connections of some of these integral formulas with various known integrals, as well as the computational and numerical aspects of the results presented here, are also pointed out.     

Granular type-2 membership functions: A new approach to formation of footprint of uncertainty in type-2 fuzzy sets

In this study, a new approach for the formation of type-2 membership functions is introduced. The footprint of uncertainty is formed by using rectangular type-2 fuzzy granules and the resulting membership function is named as granular type-2 membership function. This new approach provides more degrees of freedom and design flexibility in type-2 fuzzy logic systems. Uncertainties on the grades of membership functions can be represented independently for any region in the universe of discourse and free of any functional form. So, the designer could produce nonlinear, discontinuous or hybrid membership functions in granular formation and therefore could model any desired discontinuity and nonlinearity. The effectiveness of the proposed granular type-2 membership functions is firstly demonstrated by simulations done on noise corrupted Mackey–Glass time series prediction. Secondly, flexible design feature of granular type-2 membership functions is illustrated by modeling a nonlinear system having dead zone with uncertain system parameters. The simulation results show that type-2 fuzzy logic systems formed by granular type-2 membership functions have more modeling capabilities than the systems using conventional type-2 membership functions and they are more robust to system parameter changes and noisy inputs.     

A note on the optimal expansion of Volterra models using Laguerre functions

This work tackles the problem of expanding Volterra models using Laguerre functions. A strict global optimal solution is derived when each multidimensional kernel of the model is decomposed into a set of independent orthonormal bases, each of which parameterized by an individual Laguerre pole intended for representing the dominant dynamic of the kernel along a particular dimension. It is proved that the solution derived minimizes the upper bound of the squared norm of the error resulting from the practical truncation of the Laguerre series expansion into a finite number of functions. This is an extension of the results in Campello, Favier and Amaral [(2004). Optimal expansions of discrete-time Volterra models using Laguerre functions. Automatica, 40, 815-822.], where an optimal solution was obtained for the usual yet particular case in which a single Laguerre pole is used for expanding a given kernel along all its dimensions. It is also proved that the particular and extended solutions are equivalent to each other when the Volterra kernels are symmetric.     

Advances on the theory of generalized Bessel functions and applications to multiphoton processes

The generalized Bessel functions (GBF) are presented within the context of a more comprehensive formalism. We also discuss the partial differential equations defining GBF and their modified versions of integer order (MGBF). Numerical results are given for the first-kind MGBF as well as for the GBF with the imaginary parameter, whose importance in multiphoton processes is extensively discussed.     

A new approach of goodness-of-fit testing for exponentiated laws applied to the generalized Rayleigh distribution

Goodness-of-fit statistics are considered which are appropriate for generalized families of distributions, resulting from exponentiation. The tests employ a variation of the data determined by the cumulative distribution function of the corresponding non-generalized distribution. The resulting test, which makes use of the Mellin transform of the transformed data, is shown to be consistent. Simulation results for the case of the generalized Rayleigh distribution show that the proposed test compares well with standard methods based on the empirical distribution function.     

A new approach to the decomposition of Boolean functions by the method of <Emphasis Type="Italic">q</Emphasis>-partitions. III. Joint decomposition of a function system

The paper considers the set-theoretical approach to the joint decomposition of systems of Boolean functions of variables specified in different representation forms. The approach is based on the method of q-partitions of conjuncterms and concept of decomposition clones. Theorems on joint decomposition of a system of full and partial functions are formulated. The approach is illustrated by examples. Parts I and II of this article are published in No. 5 (2001) and No. 1 (2002). __________ Translated from Kibernetika i Sistemnyi Analiz, No. 2, pp. 39–58, March–April 2007.     

Fractional calculus — A new approach to the analysis of generalized fourth-order diffusion–wave equations

The homotopy perturbation method is applied to the generalized fourth-order fractional diffusion–wave equations. The problem is formulated in the Caputo sense. Moreover, a reliable scheme for calculating nonlinear operators is proposed. The results reveal that the present method is very effective and convenient.     

A Fuzzy set based approach to generalized landscape theory of aggregation

In this paper, we firstly reformulate the landscape theory of aggregation (Axelrod and Bennett, 1993) in terms of an optimization problem, and then straightforwardly propose a fuzzy-set-theoretic based extension for it. To illustrate efficiency of the proposal, we make a simulation with the proposed framework for the international alignment of the Second World War in Europe. It is shown that the obtained results are essentially comparable to those given by the original theory. Consequently, the fuzzy-set-theoretic based extension of landscape theory can allow us to analyze a wide variety of aggregation processes in politics, economics, and society in a more flexible manner. Shigemasa Suganuma: He received the M.S. degree in knowledge science from Japan Advanced Institute of Science and Technology,, Ishikawa, Japan in 2000. He currently takes a doctor's course in School of Knowledge Science, Japan Advanced Institute of Science and Technology (JAIST). His research interest includes agent based simulation and its application to social and political concerns, industry and environmental behavior. Van-Nam Huynh, Ph.D.: He received the B.S. in Mathematics (1990) and Ph.D. (1999) from University of Quinhon, Vietnam and Institute of Information Technology, Vietnam Academy of Science and Technology, respectively. From April 2001 to March 2002, he was a postdoctoral fellow awarded by INOUE Foundation for Science at JAIST. He is currently a Research Associate in School of Knowledge Science, JAIST, Japan. His current research interests include fuzzy logic and approximate reasoning, uncertainty formalisms in knowledge-based systems, decision making. Yoshiteru Nakamori, Ph.D.: He received the B.S., M.S., and Ph.D. degrees all in applied mathematics and physics from Kyoto University, Kyoto, Japan. He is currently a Professor in School of Knowledge Science, JAIST. His research interests include development of modeling methodology based on hard as well as soft data, and support systems for soft thinking around hard data. Shouyang Wang, Ph.D.: He received the Ph.D. degree in Operations Research from Chinsese Academy of Sciences (CAS), Beijing in 1986. He is currently a Bairen distinguished professor of Management Science at Academy of Mathematics and Systems Sciences of CAS and a Lotus chair professor of Hunan University in Changsha. He is the editor-in-chief or a co-editor of 12 journals. He has published 120 journal articles. His current research interest includes decision analysis, system engineering and knowledge management.     

A collocation approach to solve the Riccati-type differential equation systems

In this paper, a collocation method is presented for the solutions of the system of the Riccati-type differential equations with variable coefficients. The proposed approach consists of reducing the problem to a nonlinear algebraic equation system by expanding the approximate solutions in terms of the Bessel polynomials with unknown coefficients. The unknown coefficients of the Bessel polynomials are found by using the matrix operations of derivatives together with the collocation method. The proposed method gives the analytic solutions when the exact solutions are polynomials. Also, an error analysis technique based on the residual function is introduced for the suggested method. The reliability and efficiency of the proposed scheme are demonstrated by some numerical examples. Comparing the methodology with some known techniques shows that the presented approach is relatively easy and highly accurate. All of the numerical calculations have been done by using a program written in Maple.     

A new approach to spatiotemporal estimation of the river state

The paper proposes a spatiotemporal method of analysis of the river state on the basis of the measurement of pollution level indicators in the context of its application in modern control systems. For the examined objects and measurements of their characteristic parameters described by differential equations, the issue of river state estimation was formulated on the basis of discrete measurements of water pollution carried out in control stations located along the river. Using such determined estimates of all control signals describing the level of pollution in the rivers, higher accuracy of water quality control was obtained in comparison with a case based solely on measurements of some easily accessible parameters. While designing the presented solution, a mathematical model of a physical object was used which was a section of a river understood as a certain mathematical abstraction binding together variables characterizing the state of the object, interaction of external signals and its reaction. It has been assumed that the dissolved oxygen (DO) concentration is the starting point at the stage of water quality status modeling as many other indicators of pollution level are associated with it. Such an indicator may be biochemical oxygen demand (BOD) which also reflects the DO consumption in rivers and represents the degradation of water quality. Having a mathematical model of pollution distribution in the river and real measurements, numerical simulations created the possibility of influencing the changes in pollution indicators (BOD and DO) in specific conditions and in a specific time. The paper verifies the compliance of the proposed mathematical model of the object with the actual representation of the local river Wislok in the Subcarpathian province in Poland.     

A robust method for the computation of new closed‐form Green's functions for microstrip structures

A robust approach for the computation of new closed‐form Green's functions is considered to calculate the symmetrical microstrip Green's functions. In this method, the surface‐wave poles are first extracted using a recursive contour integration method. Then, the remainder is approximated by a series of complex exponentials using the Prony's method or the generalized pencil‐of‐function method (GPOF) along the extended rooftop shaped path in k_ρ‐plane. Subsequently, an analytical identity is employed to obtain the new spatial‐domain Green's functions. It is observed that this method can evaluate the spatial‐domain Green's functions accurately and efficiently for both near and far fields. In addition, there is no erroneous results in the near‐field region when z ≠ z′ and ρ → 0. © 2011 Wiley Periodicals, Inc. Int J RF and Microwave CAE, 2011.     

A New combinatorial approach to optimal embeddings of rectangles

An important problem in graph embeddings and parallel computing is to embed a rectangular grid into other graphs. We present a novel, general, combinatorial approach to (one-to-one) embedding rectangular grids into their ideal rectangular grids and optimal hypercubes. In contrast to earlier approaches of Aleliunas and Rosenberg, and Ellis, our approach is based on a special kind of doubly stochastic matrix. We prove that any rectangular grid can be embedded into its ideal rectangular grid with dilation equal to the ceiling of the compression ratio, which is bothoptimal up to a multiplicative constant and a substantial generalization of previous work. We also show that any rectangular grid can be embedded into its nearly ideal square grid with dilation at most 3. Finally, we show that any rectangular grid can be embedded into itsoptimal hypercube withoptimal dilation 2, a result previously obtained, after much research, through anad hoc approach. Our results also imply optimal simulations of two-dimensional mesh-connected parallel machines by hypercubes and mesh-connected machines, where each processor in the guest machine is simulated by exactly one processor in the host.A preliminary version of this paper appeared at the IEEE International Parallel Processing Symposium, 1994. The research of the third author was supported in part by NSF Grants CCR-9010366 and CCR-9303011.     

A new approach to the design of amperometric gas sensors

Preliminary experiments are described in order to demonstrate the principle of a new design of amperometric gas sensors. In this design, non-porous electrodes are all fabricated onto the gas side of a proton conducting medium so that the oxidation/reduction of the gas to be sensed occurs at the three phase interface between the atmosphere, the electrode and the proton conducting medium, i.e. the system approximates to a line electrode. Such a cell is described which gives good quality responses for the oxidation of SO₂ from the gas phase.     

A transfer-function approach to the interpretation of relaxation spectra of second-order cross-effects in materials science

The interpretation of relaxation spectra of second-order cross-effects is a problem that arises in some branches of materials science when coupling between thermal, mechanical and dielectric quantities is investigated. In this article, a transfer-function approach is combined with thermodynamics to treat this problem in a systematic way. Using quadratic electromechanical interaction as an example, an illustration is given of how an approximated expression for the relaxation strength can be obtained. The method presented is an extension of results on the thermodynamics of first-order coupling given in the literature.     

A new recurslve method for the analysis of linear time invariant dynamic systems via double-term triangular functions （DTTF） in state space environment

This paper presents a new recursive method for system analysis via double-term triangular functions (DTTF) in state space environment. The proposed method uses orthogonal triangular function sets and proves to be more accurate as compared to single term Walsh series (STWS) method with respect to mean integral square error (MISE). This has been established theoretically and comparison of error with respect to MISE is presented for clarity. A numerical example is treated to establish the proposed method. Relevant curves for the solutions of states of the dynamic system are also presented with plots of percentage error for DTTF-based analysis.