The ridge function representation of polynomials and an application to neural networks

The first goal of this paper is to establish some properties of the ridge function representation for multivariate polynomials, and the second one is to apply these results to the problem of approximation by neural networks. We find that for continuous functions, the rate of approximation obtained by a neural network with one hidden layer is no slower than that of an algebraic polynomial.     

Approximation of discrete functions and Chebyshev polynomials orthogonal on the uniform grid

Let _N+2m ={−m, −m+1, …, −1, 0, 1, …,N−1,N, …,N−1+m}. The present paper is devoted to the approximation of discrete functions of the formf : _N+2m → ℝ by algebraic polynomials on the grid Ω _N ={0, 1, …,N−1}. On the basis of two systems of Chebyshev polynomials orthogonal on the sets Ω _N+m and Ω _N , respectively, we construct a linear operatorY _{n+2m, N} =Y _{n+2m, N} (f), acting in the space of discrete functions as an algebraic polynomial of degree at mostn+2m for which the following estimate holds (x ε Ω _N ):

Fuzzy polynomial regression with fuzzy neural networks

Recently, fuzzy linear regression is considered by Mosleh et al. [1]. In this paper, a novel hybrid method based on fuzzy neural network for approximate fuzzy coefficients (parameters) of fuzzy polynomial regression models with fuzzy output and crisp inputs, is presented. Here a neural network is considered as a part of a large field called neural computing or soft computing. Moreover, in order to find the approximate parameters, a simple algorithm from the cost function of the fuzzy neural network is proposed. Finally, we illustrate our approach by some numerical examples.     

On the solution of the Abel equation of the second kind by the shifted Chebyshev polynomials

This paper presents a new approximate method of Abel differential equation. By using the shifted Chebyshev expansion of the unknown function, Abel differential equation is approximately transformed to a system of nonlinear equations for the unknown coefficients. A desired solution can be determined by solving the resulting nonlinear system. This method gives a simple and closed form of approximate solution of Abel differential equation. The solution is calculated in the form of a series with easily computable components. The numerical results show the effectiveness of the method for this type of equation. Comparing the methodology with some known techniques shows that the present approach is relatively easy and highly accurate.     

Adjoint polynomials of bridge–path and bridge–cycle graphs and Chebyshev polynomials

The chromatic polynomial of a simple graph G with n>0 vertices is a polynomial of degree n, where α_k(G) is the number of k-independent partitions of G for all k. The adjoint polynomial of G is defined to be , where is the complement of G. We find explicit formulas for the adjoint polynomials of the bridge–path and bridge–cycle graphs. Consequence, we find the zeros of the adjoint polynomials of several families of graphs.     

Discriminants of polynomials related to Chebyshev polynomials: The “Mutt and Jeff ” syndrome

The discriminants of certain polynomials related to Chebyshev polynomials factor into the product of two polynomials, one of which has coefficients that are much larger than the other?s. Remarkably, these polynomials of dissimilar size have “almost” the same roots, and their discriminants involve exactly the same prime factors.     

Memory‐efficient Arnoldi algorithms for linearizations of matrix polynomials in Chebyshev basis

Novel memory‐efficient Arnoldi algorithms for solving matrix polynomial eigenvalue problems are presented. More specifically, we consider the case of matrix polynomials expressed in the Chebyshev basis, which is often numerically more appropriate than the standard monomial basis for a larger degree d. The standard way of solving polynomial eigenvalue problems proceeds by linearization, which increases the problem size by a factor d. Consequently, the memory requirements of Krylov subspace methods applied to the linearization grow by this factor. In this paper, we develop two variants of the Arnoldi method that build the Krylov subspace basis implicitly, in a way that only vectors of length equal to the size of the original problem need to be stored. The proposed variants are generalizations of the so‐called quadratic Arnoldi method and two‐level orthogonal Arnoldi procedure methods, which have been developed for the monomial case. We also show how the typical ingredients of a full implementation of the Arnoldi method, including shift‐and‐invert and restarting, can be incorporated. Numerical experiments are presented for matrix polynomials up to degree 30 arising from the interpolation of nonlinear eigenvalue problems, which stem from boundary element discretizations of PDE eigenvalue problems. Copyright © 2013 John Wiley & Sons, Ltd.     

On three differential equations associated with Chebyshev polynomials of degrees 3, 4 and 6

We shall study the differential equation y'~2= T_n(y)-(1-2μ~2);where μ~2 is a constant, T_n(x) are the Chebyshev polynomials with n = 3, 4, 6.The solutions of the differential equations will be expressed explicitly in terms of the Weierstrass elliptic function which can be used to construct theories of elliptic functions based on _2F_1(1/4, 3/4; 1; z),_2F_1(1/3, 2/3; 1; z), _2F_1(1/6, 5/6; 1; z) and provide a unified approach to a set of identities of Ramanujan involving these hypergeometric functions.     

Independence polynomials of circulants with an application to music

The independence polynomial of a graph G is the generating function I(G,x)=∑_k≥0i_kx^k, where i_k is the number of independent sets of cardinality k in G. We show that the problem of evaluating the independence polynomial of a graph at any fixed non-zero number is intractable, even when restricted to circulants. We provide a formula for the independence polynomial of a certain family of circulants, and its complement. As an application, we derive a formula for the number of chords in an n-tet musical system (one where the ratio of frequencies in a semitone is 2^1/n) without ‘close’ pitch classes.     

Some properties and an application of multivariate exponential polynomials

In the paper, the authors introduce a notion “multivariate exponential polynomials” which generalize exponential numbers and polynomials, establish explicit formulas, inversion formulas, and recurrence relations for multivariate exponential polynomials in terms of the Stirling numbers of the first and second kinds with the help of the Faà di Bruno formula, two identities for the Bell polynomials of the second kind, and the inversion theorem for the Stirling numbers of the first and second kinds, construct some determinantal inequalities and product inequalities for multivariate exponential polynomials with the aid of some properties of completely monotonic functions and other known results, derive the logarithmic convexity and logarithmic concavity for multivariate exponential polynomials, and finally find an application of multivariate exponential polynomials to white noise distribution theory by confirming that multivariate exponential polynomials satisfy conditions for sequences required in white noise distribution theory.     

An application of Liapunov''s method for the analysis of neural networks

The stability is studied of a class of nonlinear dynamical systems which possess many nonlinearities and many equilibrium states. As a special case, the analyzed class of systems includes analog neural networks. Sufficient conditions for the nonoscillatory behaviour of these systems, in the form of frequency domain criteria, are presented. The main result is proved relying on a suitable Liapunov function which is subsequently used for the simultaneous computation of regions of attraction for each stable equilibrium.     

Cash demand forecasting in ATMs by clustering and neural networks

To improve ATMs’ cash demand forecasts, this paper advocates the prediction of cash demand for groups of ATMs with similar day-of-the week cash demand patterns. We first clustered ATM centers into ATM clusters having similar day-of-the week withdrawal patterns. To retrieve “day-of-the-week” withdrawal seasonality parameters (effect of a Monday, etc.) we built a time series model for each ATMs. For clustering, the succession of seven continuous daily withdrawal seasonality parameters of ATMs is discretized. Next, the similarity between the different ATMs’ discretized daily withdrawal seasonality sequence is measured by the Sequence Alignment Method (SAM). For each cluster of ATMs, four neural networks viz., general regression neural network (GRNN), multi layer feed forward neural network (MLFF), group method of data handling (GMDH) and wavelet neural network (WNN) are built to predict an ATM center’s cash demand. The proposed methodology is applied on the NN5 competition dataset. We observed that GRNN yielded the best result of 18.44% symmetric mean absolute percentage error (SMAPE), which is better than the result of Andrawis, Atiya, and El-Shishiny (2011). This is due to clustering followed by a forecasting phase. Further, the proposed approach yielded much smaller SMAPE values than the approach of direct prediction on the entire sample without clustering. From a managerial perspective, the clusterwise cash demand forecast helps the bank’s top management to design similar cash replenishment plans for all the ATMs in the same cluster. This cluster-level replenishment plans could result in saving huge operational costs for ATMs operating in a similar geographical region.     

Logistic evolutionary product-unit neural networks: Innovation capacity of poor Guatemalan households

A new logistic regression algorithm based on evolutionary product-unit (PU) neural networks is used in this paper to determine the assets that influence the decision of poor households with respect to the cultivation of non-traditional crops (NTC) in the Guatemalan Highlands. In order to evaluate high-order covariate interactions, PUs were considered to be independent variables in product-unit neural networks (PUNN) analysing two different models either including the initial covariates (logistic regression by the product-unit and initial covariate model) or not (logistic regression by the product-unit model). Our results were compared with those obtained using a standard logistic regression model and allow us to interpret the most relevant household assets and their complex interactions when adopting NTC, in order to aid in the design of rural policies.     

Global analysis of planar neural networks

In this paper, the global qualitative analysis of cubic dynamical systems is established. These systems are used as learning models of planar neural networks.     

The use of Chebyshev cardinal functions for the solution of a partial differential equation with an unknown time-dependent coefficient subject to an extra measurement

A numerical technique is presented for the solution of a parabolic partial differential equation with a time-dependent coefficient subject to an extra measurement. The method is derived by expanding the required approximate solution as the elements of Chebyshev cardinal functions. Using the operational matrix of derivative, the problem can be reduced to a set of algebraic equations. From the computational point of view, the solution obtained by this method is in excellent agreement with those obtained by previous works and also it is efficient to use.     

Laurent-Padé approximants to four kinds of Chebyshev polynomial expansions. Part II: Clenshaw-Lord type approximants

Laurent-Padé (Chebyshev) rational approximantsP _m (w, w ⁻¹)/Q _n (w, w ⁻¹) of Clenshaw-Lord type [2,1] are defined, such that the Laurent series ofP _m /Q _n matches that of a given functionf(w, w ⁻¹) up to terms of orderw ^±(m+n) , based only on knowledge of the Laurent series coefficients off up to terms inw ^±(m+n) . This contrasts with the Maehly-type approximants [4,5] defined and computed in part I of this paper [6], where the Laurent series ofP _m matches that ofQ _n f up to terms of orderw ^±(m+n ), but based on knowledge of the series coefficients off up to terms inw ^±(m+2n). The Clenshaw-Lord method is here extended to be applicable to Chebyshev polynomials of the 1st, 2nd, 3rd and 4th kinds and corresponding rational approximants and Laurent series, and efficient systems of linear equations for the determination of the Padé-Chebyshev coefficients are obtained in each case. Using the Laurent approach of Gragg and Johnson [4], approximations are obtainable for allm≥0,n≥0. Numerical results are obtained for all four kinds of Chebyshev polynomials and Padé-Chebyshev approximants. Remarkably similar results of formidable accuracy are obtained by both Maehly-type and Clenshaw-Lord type methods, thus validating the use of either.     

A new formula for fractional integrals of Chebyshev polynomials: Application for solving multi-term fractional differential equations

A new explicit formula for the integrals of shifted Chebyshev polynomials of any degree for any fractional-order in terms of shifted Chebyshev polynomials themselves is derived. A fast and accurate algorithm is developed for the solution of linear multi-order fractional differential equations (FDEs) by considering their integrated forms. The shifted Chebyshev spectral tau (SCT) method based on the integrals of shifted Chebyshev polynomials is applied to construct the numerical solution for such problems. The method is then tested on examples. It is shown that the SCT yields better results.