A method for the calculation of eigenfunction expansions

In this paper, a method is presented for the calculation of the coefficients of the series expansion of a function f(t), in the base orthonormal set made up by the eigenfunctions of the self-adjoint operator $\text{L: L(x(t)) = (}\text{d}\text{dt}\text{)( p(t)(}\text{dx(t)}\text{dt}\text{))?q(t)x(t)}$. We show that the values of the numbers $\text{t}\text{}$x_k> can be obtained by solving the differential equation L + λ) y(t) = Kf(t), in the interval of definition, for each of the eigenvalues λ of L and by using as initial conditions those which determine one of its associated orthonormal functions. This makes the method specially interesting for its implementation on a hybrid computer: One advantage of the proposed method is that the analysis of f(t) does not require the simultaneous presence of the functions of the base set and the problem signal, thus eliminating both the problems of the synchronized generation of signals and the need for storing it in memory.     

Some remarks on a conjecture in parameter adaptive control

A.S. Morse has raised the following question: Do there exist differentiable functions

$\text{f:R}{}^2\text{→ R}\text{and}\text{g:R}{}^2\text{→ R}$

with the property that for every nonzero real number λ and every (x₀, y₀) ∈ $\text{R}$² the solution (x(t),y(t)) of

$\text{x}\text{?}\text{(t) = x(t) + λf(x(t),y(t))}$

,

$\text{y}\text{?}\text{(t) = g(x(t),y(t))}$

,

$\text{x(0) = x}{}_0\text{, y(0) = y}{}_0$

, is defined for all t ? 0 and satisfies

$\text{lim}\text{t → + ∞}$

and y(t) is bounded on [0,∞)? We prove that the answer is yes, and we give explicit real analytic functions f and g which work. However, we prove that if f and g are restricted to be rational functions, the answer is no.     

A recursive and a grammatical characterization of the exponential-time languages

We characterize the class of all languages which are acceptable in exponential time by means of recursive and grammatical methods. (i) The class of all languages which are acceptable in exponential time is uniquely characterized by the class of all (0-1)-functions which can be generated, starting with the initial functions of the Grzegorczyk-class $\text{E}$², by means of subtitution and limited recursion of the form f(x, y + 1) = h(x, y), f(x, y), f(x, l(x, y))), l(x, y) ? y. (ii) The class of all languages which are acceptable in exponential time is equal to the class of all languages generated by context-sensitive grammars with context-free control sets.     

One step integration methods of third-fourth order accuracy with large hyperbolic stability limits

One step integration methods of third and fourth order accuracy that use K function evaluations to solve the system of differential equations $\text{d}\text{y}\text{d}\text{t}\text{= A · y}$ are proposed. These methods are shown to have a hyperbolic stability limit of y (K ? 1)² ? 1 which approaches the theoretical maximum limit of K ? 1 at large K obtained for methods of lower order accuracy.     

Solution errors in finite element analysis

The development of the finite element method so far indicates that it is a discretization technique especially suited for positive definite, self-adjoint, elliptic systems, or systems with such components. The application of the method leads to the discretized equations in the form of $\text{u}\text{?}\text{=}\text{f(u)}$, where u lists the response of the discretized system at n preselected points called nodes. Instead of explicit expressions, vector function f and its Jacobian f,_u are available only numerically for a numerically given u. The solution of $\text{u}\text{?}\text{=}\text{f(u)}$ is usually a digital computer. Due to finiteness of the computer wordlength, the numerical solution $\text{u}{}_{\mathrm{c}}$ is in general different from u. Let $\text{u}\text{(}\text{x}\text{, t)}$ denote the actual response of the system in continuum at points corresponding to those of u. In the literature. $\text{u}\text{(}\text{x}\text{, t)-}\text{u}$ is called the discretization errors, $\text{u}\text{-}\text{u}{}_{\mathrm{c}}$ the round-off errors, and the s is. $\text{u}\text{(}\text{x}\text{, t)-}\text{u}{}_{\mathrm{c}}$ is called the solution errors. In this paper, a state-of-the-art survey is given on the identification, growth, relative magnitudes, estimation, and control of the components of the solution errors.     

New explicit filters and smoothers for diffusions with nonlinear drift and measurements

The optimal least-squares filtering of a diffusion x(t) from its noisy measurements {y(τ); 0 τ t} is given by the conditional mean E[x(t)|y(τ); 0 τ t]. When x(t) satisfies the stochastic diffusion equation dx(t) = f(x(t)) dt + dw(t) and y(t) = ∫₀^tx(s) ds + b(t), where f(·) is a global solution of the Riccati equation /xf(x) + f(x)² = f(x)² = αx² + βx + γ, for some , and w(·), b(·) are independent Brownian motions, Benes gave an explicit formula for computing the conditional mean. This paper extends Benes results to measurements y(t) = ∫₀^tx(s) ds + ∫₀^t dx(s) + b(t) (and its multidimensional version) without imposing additional conditions on f(·). Analogous results are also derived for the optimal least-squares smoothed estimate E[x(s)|y(τ); 0 τ t], s < t. The methodology relies on Girsanov's measure transformations, gauge transformations, function space integrations, Lie algebras, and the Duncan-Mortensen-Zakai equation.     

Errors in reduction methods

A mathematical basis is given for comparing the relative merits of various techniques used to reduce the order of large linear and non-linear dynamics problems during their numerical integration. In such techniques as Guyan-Irons, path derivatives, selected eigenvectors, Ritz vectors, etc., the nth order initial value problem of $\text{[}\text{/.y}\text{=}\text{f(y)}$ for t > 0, y(0) given] is typically reduced to the mth order (m ? n) problem of $\text{z}\text{?}\text{= g(z)}$ for t > 0, z(0) given] by the transformation $\text{y}\text{=}\text{Pz}$ where P changes from technique to technique. This paper gives an explicit approximate expression for the reduction error e_i in terms of P and the Jacobian of f. It is shown that: (a) reduction techniques are more accurate when the time rate of change of the response y is relatively small; (b) the change in response between two successive stations contributes to the errors at future stations after the change in response is transformed by a filtering matrix H, defined in terms of P; (c) the error committed at a station propagates to future stations by a mixing and scaling matrix G, defined in terms of P, Jacobian of f, and time increment h. The paper discusses the conditions under which the reduction errors may be minimized and gives guidelines for selecting the reduction basis vector, i.e. the columns of P.     

Cylindrical decomposition for systems transcendental in the first variable

We present a generalization of the Cylindrical Algebraic Decomposition (CAD) algorithm to systems of equations and inequalities in functions of the form p(x,f₁(x),…,f_m(x),y₁,…,y_n), where p∈Q[x,t₁,…,t_m,y₁,…,y_n] and f₁(x),…,f_m(x) are real univariate functions such that there exists a real root isolation algorithm for functions from the algebra Q[x,f₁(x),…,f_m(x)]. In particular, the algorithm applies when f₁(x),…,f_m(x) are real exp-log functions or tame elementary functions.     

Two methods for the calculation of the degree of an approximate greatest common divisor of two inexact polynomials

The calculation of the degree d of an approximate greatest common divisor of two inexact polynomials f(y) and g(y) reduces to the determination of the rank loss of a resultant matrix, the entries of which are functions of the coefficients of f(y) and g(y). This paper considers this issue by describing two methods to calculate d, such that knowledge of the noise level imposed on the coefficients of f(y) and g(y) is not assumed. One method uses the residual of a linear algebraic equation whose coefficient matrix and right hand side vector are derived from the Sylvester resultant matrix S(f,g), and the other method uses the first principal angle between a line and a hyperplane, the equations of which are calculated from S(f,g). Computational results on inexact polynomials whose exact forms have multiple roots of high degree are shown and very good results are obtained. These results are compared with the rank loss of S(f,g) for the calculation of d, and it is shown that this method yields incorrect results for these examples.     

Connective stability of competitive equilibrium

The purpose of this paper is to derive necessary and sufficient conditions for connective stability of non-linear matrix systems described by the equation $\text{x}\text{?}\text{= A(t, x) x}$, where the matrix A(t, x) has time-varying nonlinear elements. The obtained results can be used to study stability of competitive equilibrium in as diverse fields as economics and engineering, model ecosystems and arms races.     

Stochastic stabilisation of functional differential equations

In this paper we investigate the problem of stochastic stabilisation for a general nonlinear functional differential equation. Given an unstable functional differential equation dx(t)/dt=f(t,x_t), we stochastically perturb it into a stochastic functional differential equation , where Σ is a matrix and B(t) a Brownian motion while X_t={X(t+θ):-τθ0}. Under the condition that f satisfies the local Lipschitz condition and obeys the one-side linear bound, we show that if the time lag τ is sufficiently small, there are many matrices Σ for which the stochastic functional differential equation is almost surely exponentially stable while the corresponding functional differential equation dx(t)/dt=f(t,x_t) may be unstable.     

Uniform computational complexity of the derivatives of C∞-functions

We discuss the uniform computational complexity of the derivatives of C^∞-functions in the model of Ko and Friedman (Ko, Complexity Theory of Real Functions, Birkhäuser, Basel, 1991; Ko, Friedman, Theor. Comput. Sci. 20 (1982) 323–352). We construct a polynomial time computable real function g∈C^∞[−1,1] such that the sequence $\text{{|g}{}^{\mathrm{(n)}}\text{(0)|}}{}_{\mathrm{<mtext>n\in </mtext><mtext>N</mtext>}}$ is not bounded by any recursive function. On the other hand, we show that if f∈C^∞[−1,1] is polynomial time computable and the sequence of the derivatives of f is uniformly polynomially bounded, i.e., |f⁽ⁿ⁾(x)| is bounded by 2^p(n) for all x∈[−1,1] for some polynomial p, then the sequence $\text{{f}{}^{\mathrm{(n)}}\text{}}{}_{\mathrm{<mtext>n\in </mtext><mtext>N</mtext>}}$ is uniformly polynomial time computable.     

A sixth-order extrapolation method for special nonlinear fourth-order boundary value problems

A sixth-order convergent finite difference method is developed for the numerical solution of the special nonlinear fourth-order boundary value problem y^(iv)(x) = f(x, y), a < x < b, y(a) = A₀, y″(a) = B₀, y(b) = A₁ y′(b) = B₁, the simple-simple beam problem.The method is based on a second-order convergent method which is used on three grids, sixth-order convergence being obtained by taking a linear combination of the (second-order) numerical results calculated using the three individual grids.Special formulas are proposed for application to points of the discretization adjacent to the boundaries x = a and x= b, the first two terms of the local truncation errors of these formulas being the same as those of the second-order method used at the other points of each grid.Modifications to these two formulas are obtained for problems with boundary conditions of the form y(a) = A₀, y′(a) = C₀, y(b) = A₁, y′(b) = C₁, the clamped-clamped beam problem.The general boundary value problem, for which the differential equation is y^(iv)(x) = f(x, y, y′, y″, y‴), is also considered.     

Fixed point theorem of generalized quasi-contractive mapping in cone metric space

In this paper, we introduce the generalized quasi-contractive mapping f in a cone metric space (X,d). f is called a generalized quasi-contractive if there is a real λ∈[0,1) such that for all x,y∈X,

d(fx,fy)≤λs     

An upper bound on the mean-square error in the sampling expansion for non-band-limited processes

An upper bound is obtained on the mean-square error involved when a real-valued non-band-limited nonstationary random process x(t) is approximated by the sampling expansion

$\text{∑}\text{n=?∞}\text{∞}\text{x(nT)}\text{sin}\text{π(t?nT)/T}\text{π(t?nT)/T}$

for some T > 0. When the process x(t) is band-limited over [$\text{?}\text{1}\text{2T}\text{,}\text{1}\text{2T}$], this error bound reduces to zero.     

Torsion problem for elastic cylinder with inserts and holes

An integral equation method to solve the classical torsion problem for an elastic cylinder with inserts and holes is treated. The bounded region outside the inserts and the holes will be termed a matrix. As is well-known the solution depends on finding plane harmonic functions in the matrix and inserts such that (a) on the outer boundary of the matrix and the boundaries of the holes the harmonic function in the matrix takes the values $\text{1}\text{2}\text{(x}{}^2\text{+y}{}^2\text{)+c}{}_{\mathrm{j}}$, and (b) on the interfaces of the matrix and the inserts relations exist between the harmonic functions and between their normal derivatives. Here (x, y) are the coordinates of the point on the boundary and c_j, are unknown constants. The usual methods are cumbersome and lengthy. In this paper a straightforward method is presented which is easily programmable. The numerical solution is obtained by evaluating a few integrals either analytically or numerically and solving a system of linear simultaneous equations. An example of a cylinder with an eccentric insert is given which substantiates the theory developed in this paper and is found to agree with known results. However, the method is general and may be applied to a variety of problems.     

Exponential stability in mean square of neutral stochastic differential functional equations

The aim of this paper is to investigate the exponential stability in mean square for a neutral stochastic differential functional equation of the form d[x(t) − G(x_t)] = [f(t,x(t)) + g(t, x_t)]dt + σ(t, x_t)dw(t), where x_t = {x(t + s): − τ s 0}, with τ > 0, is the past history of the solution. Several interesting examples are a given for illustration.