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The computation of the eigenvalues of a Sturm-Liouville problem is a difficult task, when high-index eigenvalues are computed. In most previous methods, it can be seen that the uncertainty of the results increases as the estimated eigenvalues grow larger. This paper is to present some new methods in which, not only the error of calculating the higher eigenvalues does not grow, but it also vanishes as eigenvalues tend to infinity. Moreover, the proposed method gives good estimates of eigenfunctions corresponding to high eigenvalues.

In this paper, a simplified two-level finite element method with backtracking is proposed for the stream function formulation of the Navier–Stokes equations. This method requires the solution of two small systems on the coarse space and one symmetric, positive linear problem on the refined space. Error analysis for the case of Clough–Tocher or Bogner–Fox–Schmit elements is presented and the optimal asymptotic estimation is obtained in the H 2 - and H 1 -norms. Finally, numerical examples are given to verify the theoretical analysis.

Haar wavelet operational matrix has been widely applied in system analysis, system identification, optimal control and numerical solution of integral and differential equations. In the present paper we derive the Haar wavelet operational matrix of the fractional order integration, and use it to solve the fractional order differential equations including the Bagley-Torvik, Ricatti and composite fractional oscillation equations. The results obtained are in good agreement with the existing ones in open literatures and it is shown that the technique introduced here is robust and easy to apply.

This paper deals with the stability analysis of the Euler–Maclaurin methods for neutral differential equations with piecewise continuous arguments u ′ ( t ) = au ( t ) + ∑ i = 0 N a i u ( i ) ( [ t ] ) . The stability regions of the Euler–Maclaurin methods are determined. The conditions under which the analytic stability region is contained in the numerical stability region are obtained and some numerical experiments are given.

In this paper, it is shown how information about an orbit of a chaotic dynamical system can be recovered given only measurements with “large” error. Assuming that the imprecision of measurements manifests itself as a finite partition of the phase space, general, necessary, and sufficient conditions under which it is possible to reconstruct orbits are presented. Two reconstructing methods and their pseudocodes are presented and analyzed. The applicability of those methods is tested using a series of computer experiments for various one-dimensional mappings on the unit interval. Potential applications are also discussed.

In this paper, we study several overlapping domain decomposition methods for the numerical solutions of some linear and semilinear elliptic stochastic partial differential equations discretized by the finite element methods. In particular, we show that the algorithms converge and the convergence rates are independent of the finite element mesh parameter, as well as the number of subdomains used in the domain decomposition.

This paper presents a mathematical model for heat and moisture transfer through cloth. A two-dimensional mathematical model, which considers complicated heat and mass transfer is developed. The coupled partial differential equations are created based on integrations of porous medium equations and heat, diffusion equations. A non-linearized implicit finite-difference method is presented to find numerical solutions of the two-dimensional simulation model. Results obtained by the present method are found to agree satisfactorily with the experimental data available in the literature.

The aim of this paper is to construct a unified family of meromorphic functions, which is related to many known functions such as a unified family of partial zeta type functions, a unified family of L-functions, and so on. We investigate and derive many properties of this family of meromorphic functions. Moreover, we compute the residues of this family of meromorphic functions at their poles. We also give some applications and remarks involving this family of meromorphic functions. Akdeniz Üniversitesi

In this paper, we introduce a new modified iteration process for approximating common endpoint of a multivalued α-nonexpansive mapping and a multivalued mapping satisfying condition (E′) in uniformly convex hyperbolic spaces. As a by-product, we improve the main results of Panyanak (2018) with a new and faster algorithm for two multivalued mappings. Moreover, we give a numerical example to substantiate our results. Our work is new and holds simultaneously in uniformly convex Banach spaces as well as CAT(0) spaces.

Dynamic structure-based neural networks are being extensively applied in many fields of science and engineering. A novel dynamic structure-based neural network determination approach using orthogonal genetic algorithm with quantization is proposed in this paper. Both the parameter (the threshold of each neuron and the weight between neurons) and the transfer function (the transfer function of each layer and the network training function) of the dynamic structure-based neural network are optimized using this approach. In order to satisfy the dynamic transform of the neural network structure, the population adjustment operation was introduced into orthogonal genetic algorithm with quantization for dynamic modification of the population's dimensionality. A mathematical example was applied to evaluate this approach. The experiment results suggested that this approach is feasible, correct and valid.

Recently R.E. Kalaba reduced the solution of a system of integral equations to the solution of a Cauchy system. In this paper we derive the same Cauchy system via the Wiener-Hopt procedure.

In recent years the use of invariant imbedding in the solution of a variety of problems has been increasing. In this paper, application of this method to problems in heat conduction and radiation is demonstrated. No previous knowledge of invariant imbedding is assumed. The method is applied to several relatively simple problems. The initial value problem obtained by the method is numerically stable. Sample calculations are presented which demonstrate the accuracy of the algorithm.

An initial value method for the integral equation of the column is presented for determining the buckling load of columns. The differential equation of the column is reduced to a Fredholm integral equation. An initial value problem is derived for this integral equation, which is reduced to a set of ordinary differential equations with prescribed initial conditions in order to find the Fredholm resolvent. The singularities of the resolvent occur at the eigenvalues. Integration of the equations proceeds until the integrals become excessively large, indicating that a critical load has been reached. To check this method, numerical results are given for two examples, for which the critical load is well known. One is the Euler load of a simply supported beam, and the other case is the buckling load of a cantilever beam under its own weight. The advantage of this initial value method is that it can be applied easily to solve other nonlinear problems for which the critical loads are unknown. This approach will be illustrated in future papers.

In this paper, it is shown that the method introduced recently by Kalaba and Scott for solving nonlinear characteristic value problems for integral operators is equivalent to a nonlinear characteristic value problem for a matrix. This is demonstrated by finding the explicit solution to the set of differential equations used by them. The results generalize those proved recently by the author for the linear problem.

This paper compares the relative efficiencies of the invariant imbedding method with the traditional solution techniques of successive approximations (Picard method), linear algebraic equations, and Sokolov's method of averaging functional corrections in solving numerically two representatives of a class of Fredholm integral equations. The criterion of efficiency is the amount of computing time necessary to obtain the solution to a specified degree of accuracy. The results of this computational investigation indicate that invariant imbedding has definite numerical advantages; more information was obtained in the same length of time as with the other methods, or even in less time. The conclusion emphasized is that a routine application of invariant imbedding may be expected to be computationally competitive with, if not superior to, a routine application of other methods for the solution of some classes of Fredholm integral equations.

A novel general direct iteration algorithm for the evaluation of branching points of nonlinear boundary-value problems is suggested. The method proposed takes advantage of the GPM concept. The analysis presented in the paper encompasses also the determination of branching points in nonlinear algebraic equations. The method can be applied to problems arising in a number of physical and mathematical applications. The technique is tailored to problems of diffusion and heat conduction accompanied by a chemical reaction.

The purpose of this paper is to present an iterative scheme for solving the radiative-transfer equation in a scattering, absorbing and emitting slab with specularly reflecting boundary. The medium under consideration is anisotropic, nonhomogeneous and azimuthally unsymmetric, and the boundary surface can be either transparent or reflecting. A novelty of this scheme is that it leads to an explicit recursion formula for the determination of the solution as well as an error estimate for the iterations. The recursion formula involves straightforward integrations of a well-behaved function and can be used to calculate numerical results by using a computer. It is shown that the sequence of iterations from the recursion formula converges uniformly to a unique positive solution of the problem, so that this method also gives an existence and uniqueness theorem.

In this paper we consider the stabilization of constant linear systems by linear feedback controls. In particular, the problem of the number of components of the state which must be measured to achieve a prescribed location of the closed-loop poles is studied. A simple, computable necessary and sufficient condition is given for the omission of state variables from the measurement process. The theoretical results are illustrated by examples and a discussion of related topics for future research is given.

The matrix sign function has several interesting properties which form the basis of new solution algorithms for problems which occur frequently in systems and control theory applications. Presented in this paper are new algorithms, based on the matrix sign function, for the solution of algebraic matrix Riccati equations, Lyapunov equations, coupled Riccati equations, spectral factorization, matrix square roots, pole assignment, and the algebraic eigenvalue-eigenvector problem. Examples of the application of each algorithm are also presented.

The objective of this paper is the rigorous derivation of an invariant imbedding algorithm for the solution of the integral equation o(z)@?g(z)+@c@?^~"0K(|z-z'|)o(z')dz' for z>=0, under suitable restrictions on g, K, and @c. First a set of conditions is determined under which Eq. (1) has a unique solution. The function o(z) is shown to be approximated almost uniformly for Y=0 and as x->~ by the solution of

Any attempt to establish models is meaningless unless it leads to a precise and well-founded model. To achieve this, the search for a model must be governed by an adequate theoretical framework. Delattre's theory of transformation systems has this aim. The identification of a model evidently becomes more difficult with increasing number of parameters to be evaluated, and this number in turn is governed by the quantity of information gained from the experiments. The identification may also be hampered by incompleteness of the experimental results, if these do not relate to all the experimental parameters concerned in the dynamics of the system. The identification algorithms given in this paper are based on modern methods of numerical analysis, and allow the unrestricted use of all available experimental measurements. The principles of application of the method are described with reference to the interpretation of the survival curves of irradiated micro-organisms.

This paper presents a new method to implement Taylor series in floating point arithmetic to generate trigonometric functions. With this method the number of terms that need to be computed is determined a priori from the exponent in the floating point representation of the argument, thereby minimizing execution time while maintaining a specified level of significance.

In the case of a multinomial distribution @P"1([email protected]"1)[email protected]"2([email protected]"2)+...+ @P"k([email protected]"k) is at times referred to as Gini's index of diversity. In this paper, we present the distributional properties of the statistic , based on samples of size n for a homogeneous multinomial distribution. For [email protected][email protected]? 12 and [email protected][email protected]?12, we give a short table of the pdf, cdf, and the moments of D. For large values of n, we mention some results for the asymptotic distribution of D for the general multinomial distribution.

Approximating a solution to the Fredholm integral equation o(x)[email protected](x) + @! ^b"aK(x, y)o(y) dy by the Nystrom method involves some numerical quadrature for approximating the integral, producing a linear system satisfied by approximate function values of o. This paper discusses the use of generalized product-interpolatory formulas which model o as one mth-degree polynomial on each subinterval and model K as a (possibly large) sequence of nth-degree polynomials. In cases where K is varying much more rapidly than o this allows for o to be sampled much less often than K. Since K is modeled as a sequence of polynomials, its frequent sampling does not require a prohibitive increase in the degree of the interpolating polynomials. Coefficient formulas and examples are given for the (m,n) cases (1,1), (1,2), (2,1) and (2,2).

In this paper the authors consider the problem of the existence, and iteration to a fixed point or a zero, of an operator on a closed subset of an abstract space. The results generalize the construction mapping principle. A generalized or cone norm is used.

The general problem treated in this paper is the derivation of the distribution of intervals between neighboring points or ''events'' on a line. For the case in which the events are independently and uniformly distributed the distribution of intervals is well known. However, from the applications point of view, the nonuniform case is highly important, a situation motivating the derivation of the results in this paper.

The intent of this paper is to show that the Nordsieck-Gear methods with maximum polynomial degree k+1, first described in [1], admit of matched starting methods which are exact for all polynomials of degree =

A theory of the scattering of electromagnetic waves by homogeneous spheres, the so-called Mie theory, is presented in a unique and coherent manner in this paper. We begin with Maxwell's equations, from which the vector wave equations are derived and solved by means of the two orthogonal solutions to the scalar wave equation. The transverse incident electric field is mapped in spherical coordinates and expanded in known mathematical functions satisfying the scalar wave equation. Determination of the unknown coefficients in the scattered and internal fields is achieved by matching the electromagnetic boundary conditions on the surface of a sphere. Far-field solutions for the electric field are then given in terms of the scattering functions. Transformation of the electric field to the reference plane containing incident and scattered waves is carried out. Extinction parameters and the phase matrix are derived from the electric field perpendicular and parallel to the reference plane. On the basis of the independent-scattering assumption, the theory is extended to cases involving a sample of homogeneous spheres.

This paper discusses some properties of matrix polynomials and a computational procedure for finding the matrix roots of such polynomials and their relationship to spectral factorization. Polynomials of order n with square matrix coefficients of order N are considered. The computational procedure is of interest in the analysis and design of multivariable control systems.

A new approach to adaptive control is proposed. The principal distinguishing feature is the direct estimation and updating of the criterion function by means of a filtering operation on a vector of transitional pseudo-return functions. The data storage and computational problems often associated with explicit probability distribution updating via Bayes' rule are thus avoided. Convergence properties are established for a simple linear criterion function filter designed for a class of adaptive control problems typified by a well-known two-armed bandit problem. Optimality properties are established for the filter in a companion paper.

In a previous paper [1] on hypernumbers it was suggested that there were significant ramifications to be explored and exemplified. In particular, the following important areas were only mentioned in passing, but are now more fully examined: the hypernumber use of idempotents, zero divisors, and nilpotents; the concept of bimatrices; and a survey, necessarily brief, of hypernumbers beyond @e and i, that is, beyond the proper square and fourth roots of unity respectively. The term hypernumber was introduced into mathematics by the present author in 1966 [2] to denote domains of number including or beyond the arithmetic of ordinary numbers, complex numbers, quaternions, matrices, or octaves (Cayley-Graves numbers); and the term has since been used by Kline [3] and by Spencer and Moon [4], but unnecessarily restricted to forms of i and @e, and too often only to matrix forms.

In this paper we describe the solution of a rather intractable differential eigenvalue problem arising from a stability problem in fluid dynamics. Techniques used for solving the problem include finite differences, a variety of shooting methods and Riccati transformations (invariant imbedding). A description is given of the difficulties encountered. The invariant imbedding technique was found to be the most efficient method for obtaining the solution accurately. Though our results are for one difficult problem, we anticipate that our conclusions will be widely applicable to problems of similar type.

This paper discusses the fitting of differential equation models to economic data. In particular, it treats the problem of describing the growth of capital in terms of a differential equation containing several parameters. The parameters are to be estimated on the basis of data. This estimation problem is formulated as a nonlinear boundary value problem. The rapidly convergent successive approximation method of quasilinearization is described and applied. Representative results of numerical experiments are presented, showing the effectiveness of the approach. Suggestions for additional studies are made.

Invariant imbedding, or the Riccati transformation, has been used to solve unstable ordinary differential equations for a few years. This paper compares the above method with parallel or multiple shooting and a method using Chebyshev series. Parallel shooting gives a solution as accurate as that obtained using the Riccati transformation, in a comparable time.

The solution of linear equations by iterative methods requires for convergence that the absolute magnitudes of all the eigenvalues of the iteration matrix should be less than unity. The test for convergence however is often difficult to apply because of the computation required. In this paper a method for determining the convergence of the Gauss-Seidel iteration is proposed. The method involves the numerical integration of initial value differential equations in the complex plane around the unit circle. The Gauss-Seidel method converges if the number of roots inside the unit circle is equal to the order of the iteration matrix.

In this paper the method of stochastic linearization is employed to develop new approximate algorithms for nonlinear smoothing. Both fixed-point and fixed-interval smoothing cases are considered. An example is included to illustrate the use of the algorithms.

Lg-splines find application in many areas of mathematics, physics, and engineering. Especially in the last, there is need for recursive algorithms suitable for real-time application. In this paper we investigate a dynamical structure of the Hilbert space underlying the spline interpolation problem. We use these insights to develop a recursive algorithm for computing Lg-splines interpolating extended Hermite- Birkhoff data. We also investigate the relationship of our algorithm to a basic theorem due to Jerome and Schumaker regarding the smoothness properties of such splines and to algorithms based on their theorem.

Ordinary N-term integral quadratures require the evaluation of the entire integrand at N points. However, m-by-n product type quadratures involve the evaluation of one factor of the integrand at m points and the reamaining factor at n points. The principal results of this paper include the generalization of the product-type quadrature concept to arbitrary weight functions and to infinite as well as finite intervals, the calculation of the mn coefficients of this product quadrature formula from the LU decoposition of one n-byn, and the extension of the precision of the formula. Nuerical examples are included to illustrate the application of Gaussian product-type quadratures and to compare them with the ordinary Gaussian quadratures.

This paper presents an algorithm for computing the product of any two polynomials in the U-algebra of the split three dimensional simple Lie algebra L over an algebraically closed field of prime characteristic K, and expressing the product as a linear combination of standard monomials. All coefficients are taken from the prime field.

Team decision theory studies the problem of how a group of decision makers should use information to coordinate their actions. Mathematically, the task is to find functions that maximize an objective functional. The Euler equations take the form of a system of integral equations. In this paper, it will be shown that a class of such integral equations has solutions that are identical to the solutions of a system of initial-valued integrodifferential equations. This Cauchy system describes the sensitivity of the solutions to underlying parameters and provides an efficient technique for solving difficult team decision problems. An analysis of a profit maximizing firm demonstrates the usefulness of the Cauchy system.

Computing in hypernumber arithmetics is discussed, and specifically that of M-algebra, which includes the operations of complex, quaternion, and Cayley numbers (octaves or octonions) as subsets of itself. It is shown that modern quantum gravitation theory requires minimally the 16-dimensional space of the author's M-arithmetic (announced in Appl. Math. and Comput., 1976, p. 211 f. and 1978, p. 45 f.), which has 4320 units (including positive and negative) rather than the mere 2 units of ordinary or ''real'' arithmetic, the 4 units of complex arithmetic, the 24 units of quaternion arithmetic, or the 240 units of octonion or octave arithmetic. Thus computer programming is the natural tool for computations in advanced quantum physics. It turns out that more than three kinds of i-type hypernumbers and more than three kinds of the \Ge-type are needed to ensure the necessary nilpotent and noncommutative algebra required in unified field theory. It is also shown that ''more than three'' here means ''at least seven''; and it turns out that a 16-dimensional arithmetic is needed for such computations. The following paper contextualizes, characterizes, and specifies that arithmetic as the apex of a hierarchy susceptible of clear geometric definition. And the hypernumbers needed in quantized unified field theory are specified.

This paper concerns the computation of a surface area given by a parametric representation on an arbitrary domain D. The basic approach is to construct an analytic mapping with nonzero Jacobian from a simply connected domain; say a unit square, to D. Then the integration used to determine the area can be easily performed on the unit square. For computational purposes the mapping is estimated by a standard relaxation method and an algorithm is developed to estimate the desired area by a finite sum. With sufficiently large sums the error of this estimation can be made arbitrarily small.

In this paper we prove an existence theorem of Peano type for Sobolev differential equations. Also included are theorems on the continuation of solutions, differential inequalities and comparison of solutions of Sobolev differential equations.

This paper deals with the following problems: (1) the development of a general theory which incorporates several methods as special cases; (2) the applicability of moment methods to a class of nonlinear problems, (3) the specification of the class of admissible weighting functions; (4) the estimation of the number of parameters; (5) the elimination of the cutoff error in the analysis of fluorescence decay data.

In a previous paper the use of the Hadamard product was introduced for the study of a steady-state air-pollution problem. In the present paper it is solved in the time dependent case. The solution of the partial differential equation leads to rather unusual integrations.

This paper gives some relationships between stability properties of general processes and the same properties of their limiting processes. The obtained results concern asymptotic stability and total stability.

A basic problem in linear algebra is the determination of the largest eigenvalue (Perron root) of a positive matrix. In the present paper a new differential equation method for finding the Perron root is given. The method utilizes the initial value differential system developed in a companion paper for individually tracking the eigenvalue and corresponding right eigenvector of a parametrized matrix.

In this paper we show the relation between the eigenvalue consistency index and cycles of length k in a complete directed graph. Cardinal consistency, which requires the introduction of a scale to represent the degree of preferences among different alternatives, is discussed, and a new index to measure this concept is derived.

The overall Kaplan-Meier estimate of the survival curve underlying a study population does not individually account for the ages at which patients were first seen and thence followed up in the study. Since the number of patients in these age-specific subgroups varies quite arbitrarily for a given study, the overall Kaplan-Meier curve which assumes that all patients were seen from the beginning of the study usually either overestimates or underestimates the survival probability, depending on the varying sample sizes. In this paper, we therefore propose an estimate of the overall survival curve underlying a study population based on the individual age-specific Kaplan-Meier estimates and the probability distribution of the age at which patients were first seen in the study. In the particular case when there is no censoring, it is shown that the usual overall Kaplan-Meier survival curve which assumes that all patients irrespective of their age at entry into the study were followed from the beginning of the study is identical with the proposed estimate. This result does not seem to necessarily hold good in the presence of arbitrary censoring. Further, the estimate is unbiased and consistent, a property which the Kaplan-Meier estimate also enjoys. The efficiency of the proposed estimate relative to the Kaplan-Meier estimate is being investigated and will be reported along with results of its application in a separate communication.

This paper presents a special purpose linear programming algorithm to solve a least absolute value regression problem with upper and lower bounds on the parameters. The algorithm exploits the problem's special structure by maintaining a compact representation of the basis inverse and by allowing for the capability to combine several simplex iterations into one. Computational results with a computer code implementation of the algorithm are given.