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A New Digital Method for Three-Dimensional Stress Analysis in Elastic Media A new method for three-dimensional stress analysis in elastic media is outlined. According to this approach the problem is formulated purely in terms of conditions prevailing at the bounding surfaces of an elastic body. Stress and displacements on these surfaces, as well as inside the body, are expressed in terms of the superimposed effect of a set of basic solutions having suitable properties. The method is developed in matrix form. Some general properties of the governing matrices are established with a view to developing suitable numerical procedures. The convergence characteristics of two iterative schemes are considered. The paper concludes with observations concerning the generality of the approach.

For a kernel belonging toL p-space,p>1, the rate of convergence of Hoeffding's [5]U-statistic to its expectation is studied; this includes as a special case a similar result on the sample mean previously studied by Chung [3]. Also, anL p-convergence result of Pyke and Root [8] on the sample partial sum is extended toU-statistics.

The general implicit finite-difference approximation of second order devised by von Neumann is applied to the initial boundary value problem for a somewhat generalized wave equation. By means of eigenvalue expansion it is shown that the method is uniform convergent of order $$\log \left( {\Delta x^{ - 1} } \right)O\left( {\Delta t^2 + \Delta x^2 } \right)$$ Δt, Δx mesh widths). Moreover, the convergence on the linet=T reveals to be proportional toT.

This paper investigates some convergence properties of Fiacco and McCormick's SUMT algorithm. A convex programming problem is given for which the SUMT algorithm generates a unique unconstrained minimizing trajectory having an infinite number of accumulation points, each a global minimizer to the original convex programming problem.

ATaylor-series correction to the maximum inplane shear stress was studied as a means of extending the data zone in photoelastic determination of stress-intensity factors beyond the singular region of a two-degree-of-freedom analysis. Convergence properties were obtained by comparing with several complete two-dimensional solutions. Experiments were performed on two kinds of three-dimensional problems, plates containing surface flows in both bending and extension. Results were analyzed by both a two-degree-of-freedom and aTaylor-series correction method (TSCM). Results were compared to theories of F. W. Smith and A. S. Kobayashi and R. C. Shah. It was concluded that

A method is described for minimizing a continuously differentiable functionF(x) ofn variables subject to linear inequality constraints. It can be applied under the same general assumptions as any method of feasible directions. IfF(x) is twice continuously differentiable and the Hessian matrix ofF(x) has certain properties, then the algorithm generates a sequence of points which converges superlinearly to the unique minimizer ofF(x). No computation of secondorder derivatives is required.

The General Fermat Problem asks for the minimum of the weighted sum of distances fromm points inn-space. Dozens of papers have been written on variants of this problem and most of them have merely reproduced known results. This note calls attention to the work of Weiszfeld in 1937, who may have been the first to propose an iterative algorithm. Although the same algorithm has been rediscovered at least three times, there seems to be no completely correct treatment of its properties in the literature. Such a treatment, including a proof of convergence, is the sole object of this note. Other aspects of the problem are given scant attention.

In the theory of optimization, in connection with ordinary and partial differential equations, a number of closure and lower closure theorems have been obtained in different contexts and under a variety of conditions and modes of convergence. In particular "seminormali ty" conditions (property (Q) and its variants) have played different roles. In this paper we first prove closure and lower closure theorems for orientor fields in a rather abstract context, all based on weak convergence and MAZUR'S theorem (w167 and 5). In the context of orientor fields, these theorems can be given the most satisfactory formulation and simplest proofs (see, e.g., th. (4.i), (5.i)). Furthermore, in the present new approach, the interplay of "seminormali ty" conditions and modes of convergence can be easily seen: the stronger the mode of convergence, the weaker are the "seminormali ty" conditions that are needed. From these theorems we then derive, as corollaries, closure and lower closure statements for Mayer and Lagrange problems (~ 6 and 7) and lower semicontinuity statements for free problems (w Under suitable hypotheses, no seminormality condition (or property (Q)) is needed. Further theorems without seminormality conditions, as well as other details, are discussed in [3]. In particular, we show that seminormality conditions can be removed, not only under standard Lipschitz requirements, as expected, but also under much more satisfactory simple growth conditions, as proposed some time ago by E. H. ROTHE for free problems. Applications to multidimensional Lagrange problems are discussed in [8].

In Ref. 2, four algorithms of dual matrices for function minimization were introduced. These algorithms are characterized by the simultaneous use of two matrices and by the property that the one-dimensional search for the optimal stepsize is not needed for convergence. For a quadratic function, these algorithms lead to the solution in at mostn+1 iterations, wheren is the number of variables in the function. Since the one-dimensional search is not needed, the total number of gradient evaluations for convergence is at mostn+2.

A method of Regula Falsi type for finding a simple root of a non-linear equation is presented. It is similar to the Pegasus procedure, but exhibits a higher asymptotic convergence efficiency. This is accomplished by eliminating one or two of the slower Pegasus substeps.

The general Fermat problem is to find the minimum of the weighted sum of distances fromm destination points in Euclideann-space. Kuhn recently proved that a classical iterative algorithm converges to the unique minimizing point , for any choice of the initial point except for a denumerable set. In this note, it is shown that although convergence is global, the rapidity of convergence depends strongly upon whether or not  is a destination.

It is shown that in the LCAO-MO-SCF problem, if the molecular orbital orthonormality constraints are introduced in the manner first suggested by Fletcher, then the Hessian of the problem is singular. It is suggested that this singularity may well account for the slow convergence observed using direct energy minimization methods to solve the SCF problem. Ways of avoiding the consequences of this singularity are discussed.

The problem under consideration consists in maximizing a separable concave objective functional on a class of non-negative Lebesgue integrable functions satisfying a system of linear constraints. The problem is approximated by two sequences of concave separable programming problems with linear constraints. The convergence of the sequences of optimum values of these problems is investigated in the general case and the convergence of the sequences of optimum solutions in a special case. A numerical example is given.

A coupled iteration scheme for approximating the solution of nonlinear boundary value problems framed in the context of alternative problems (i.e. auxiliary equation and bifurcation equation) is discussed. Under suitable hypotheses, convergence of the scheme proves existence of a solution, and in some cases the convergence to the bifurcation point is accelerated.

This paper establishes quantitative bounds for the variation of an isolated local minimizer for a general nonlinear program under perturbations in the objective function and constraints. These bounds are then applied to establish rates of convergence for a class of recursive nonlinear-programming algorithms.

A simple two-layer model of finite-amplitude frontal waves[4] was modified to obtain a frequency equation describing stable, rapidly propagating very short waves. A schematic parametrisation of convective activity is permissible in the model. The low-level convergence is thus introduced in a parametric way.

The analysis of regions of convergence of expansions of the right-hand sides of the differential equations of motion in the vicinity of the collinear libration points in the circular restricted three-body problem in powers of coordinates of the infinitesimal body due to F. R. Moulton (Moulton, 1920) is shown to be erroneous and his results are corrected. The generalisation of Moulton's results to analogous expansions of the equations in the elliptic problem of three bodies made by R. W. Farquhar (Farquhar, 1968) is shown to be groundless.

Convergence properties of descent methods are investigated for the case where the usual requirement that an exact line search be performed at each iteration is relaxed. The error at each iteration is measured by the relative decrease in the directional derivative in the search direction. The objective function is assumed to have continuous second derivatives and the eigenvalues of the Hessian are assumed to be bounded above and below by positive constants. Sufficient conditions are given for establishing that a method converges, or that a method converges at a linear rate. These results are used to prove that the order of convergence for a specific conjugate gradient method is linear, provided the error at each iteration is suitably restricted.

Shooting methods for nonlinear boundary value problems are examined. It is shown that the methods converge whenever the problem is well posed in the sense that the solution to be computed is isolated.

This paper gives several sets of sufficient conditions that alocal solutionx k exists of the problem $$\min _{R^k } f^k (x)$$ ,k=1, 2,..., such that {x k } has cluster points that arelocal solutions of a problem of the form min R f(x). The results are based on a well-known concept of topological, orpoint-wise convergence of the sets {R k } toR. Such results have been used to construct and validate large classes of mathematical programming methods based on successive approximations of the problem functions. They are also directly applicable to parametric and sensitivity analysis and provide additional characterizations of optimality for large classes of nonlinear programming problems.

The construction of model atmospheres by means of differential corrections is discussed. Either flux or flux gradients may be minimized, with arbitrary weighting with depth. For the simple atmospheres studied, no convergence problems were encountered even when very poor first approximations were used; and flux constancy in the radiative case was generally attained in three or four iterations. Quantities computed in one iteration may be re-used, not only in subsequent iterations within the model, but also in the construction of other models with differentT eff org; and so the method is particularly suited for grid computations.

Two semidiscrete collocation approximations using smooth cubic splines are developed as approximations to the solution of two-point linear parabolic boundary value problems.L∞-convergence results are presented for these two approximations as well as the piecewise linear Galerkin approximation. Several computational examples are given to illustrate the convergence results and demonstrate the applicability of the method.

In the present study we consider a normal separable stochastic continuous field, and we prove the convergence of a Karhunen series with probability 1 for all parameter values. This leads in particular, to the nonrandomness of points of the discontinuity and values of the discontinuity. A criterion is presented for the convergence of the canonical expansion in a uniform norm.

The article discusses the problem of the depletion of a round stratum with a gasified liquid, in the case of the work of a round battery of boreholes with constant end-face pressures. The nonlinear differential equations describing the filtration process are replaced by finite-difference equations, which are solved numerically using a computer program. The condition for the convergence of the finite-difference scheme used is given. The proposed algorithm is used to solve a concrete example. By making analogous calculations for different variants of a given number of boreholes, of the end-face pressures, of the radius of the battery, valuable information can be obtained with respect to their optimal values, the time of the start of secondary methods of exploitation, the volume of gas dissolved in the petroleum, as well as the output of petroleum, which is easily calculated from the known values of the pressure and the petroleum saturation at any given moment of time.

Browne [1967] has given a method of solving the problem (originally stated by Mosier, [1939]) of finding a least squares fit to a specified factor structure. The problem is one of minimizing the sum of squared residuals of φ —FT with Diag (T'T)=I. Browne's solution involves the eigenvectors and values ofF'F and leads to an iterative solution. This paper gives a form of the solution which does not involve solution of an eigenvalue problem but does require an iteration similar to Browne's. It suggests the possible existence of a singularity, and a simple modification of Browne's computational procedure is proposed which deals with this case. A better starting value for the iteration is also proposed for which convergence is guaranteed using the ordinary Newton iteration.

In this work we are concerned with the stationary neutron transport Boltzmann equation (in its integral form) in plane and spherical symmetry; both the study of the properties of the exact and approximate solution are based on functional analysis. We have taken advantage of the fact that the kernel of the integral equation is even, to semplify the general discussion; among the results, we quote that relating to the continuity of the solution with respect to the optical thickness. Moreover we emphasized that the Ritz method seems the most convenient one for the approximate evaluation of the solution; in this connexion, some observations have been made about the convergence of the approximate solutions to the exact one.

We conclude by mentioning two counter-examples. An example may be constructed in which the initial configuration has an average γ and the velocity distribution although symmetric is such that if it is truncated there will be convergence to P(γ) but without truncation there will not be convergence. It is also possible to show that if Z ∋ W there may be convergence to P(γ) without the initial configuration having an average.

For a certain nonconservative system with complicated dynamic boundary conditions, Ritz's method, or an extended version of Galerkin's method, respectively are applied to the calculation of the eigenvalue usingadmissible coordinate functions which satisfy the geometric boundary conditions but do not satisfy all of the dynamic boundary conditions. By showing the analogy of Ritz or the extended Galerkin method with the approximation of a certain integro-differential equation, means are obtained for proving the convergence of Ritz or the extended Galerkin method indirectly by proving the convergence of the analoguous approximate integro-differential equation.

It is shown that a numerical treatment of two particles in the continuum is possible in applying the Green's function method to the (d, p)-reaction. A basis set for shell model states including the continuum in an approximate way is established and applied to the expansion of the incoming deuteron wave function into these basis states. As a test for the convergence of this expansion the sum of the squares of the expansion coefficients is calculated and found to be 0.965 instead of 1. The main contribution comes from the bound-bound and the bound-continuum configurations. The continuum-continuum contributions can practically be neglected.

For nonlinear programming problems with equality constraints, Hestenes and Powell have independently proposed a dual method of solution in which squares of the constraint functions are added as penalties to the Lagrangian, and a certain simple rule is used for updating the Lagrange multipliers after each cycle. Powell has essentially shown that the rate of convergence is linear if one starts with a sufficiently high penalty factor and sufficiently near to a local solution satisfying the usual second-order sufficient conditions for optimality. This paper furnishes the corresponding method for inequality-constrained problems. Global convergence to an optimal solution is established in the convex case for an arbitrary penalty factor and without the requirement that an exact minimum be calculated at each cycle. Furthermore, the Lagrange multipliers are shown to converge, even though the optimal multipliers may not be unique.

This paper deals with a confluent form of the topological ?-algorithm which is a method to accelerate the convergence of a sequence of elements of a topological vector space. After giving the rules of the algorithm it is related to some generalizations of the functional Hankel determinants. Some properties and some results about it are proved. An interpretation of the algorithm is given. The last paragraph is devoted with convergence results about the confluent form of the topological ?-algorithm. A parameter is introduced in the algorithm to accelerate the convergence. The optimal value of this parameter is caracterized. By estimating this optimal value, the confluent form of the ?-algorithm is obtained. The paper ends with a remark about the confluent form of the topological ?-algorithm.

Ss lifted pairs of weights simultaneously, one in each hand, and judged either the difference, ratio, or average heaviness of the two weights. Data for the difference and ratio tasks were in general agreement with subtractive and ratio models, but the averaging data showed discrepancies from the constant-weight averaging model similar to those reported in previous psychophysical research. Rescaling was ruled out for the averaging data, because responses to pairs of equal weight were a linear function of subtractive model scale values derived from the difference task data. Scale values for the ratio and difference task data were related exponentially, as were the responses to the pairs, consistent with Torgerson’s conjecture that Ss do not distinguish “differences” from “ratios.” They appear to use the same composition rule but different output functions, depending on the procedures for responding. The scale convergence criterion can thus prevent inappropriate rescaling when a model fails and can dictate rescaling even when a model fits.

The comparisons carried out of the reference and original measurement means of high-frequency ac voltage of the member countries of the COMECON showed their convergence to be within the limits of accuracy estimated by the national metrological organizations. This opens the possibility for unification of methods and means of checking intruments for measuring voltage and, consequently also, for establishing unity of voltage measurements at high frequencies in the member countries of the COMECON.

Linear explosive stochastic difference equations have been studied by White [6], Anderson [1], Rao [3] and the author [4], [5], leading to the derivation of limit distributions of certain sample functions bearing on structural estimation and goodness of fit tests. In this paper the author discusses certain results of similar interest on the structural parametric estimators of a linear explosive model generating a pair of related stochastic processes.

Consider a sequence of extensionsIn of a real-valued functionI. Sufficient conditions for the convergence of the infimum ofIn to the infimum ofI are obtained. These conditions are applied to an eigenvalue problem.

In this paper we consider a class of estimates of a bivariate density function f based on an independent sample of size n. Under the assumption that f is uniformly continuous, the uniform strong consistency of such estimates was first proved by Nadaraya (1970) for a large class of kernel functions. In this note we show that the assumption of the uniform continuity of f is necessary for this type of convergence.

This paper presents a new globally convergent secant method for unconstrained optimization. The root rate of convergence of this algorithm is superlinear, between 1 and 2, decreasing with dimension of the problem. It is shown that on a class of problems, it is substantially more efficient than a number of other algorithms.