Unsteady motion of a spheroid in an elastico-viscous liquid

In this paper we have considered the unsteady motion of a spheroid moving from rest in viscoelastic fluid. The paper has been divided into two sections. In Section A a general formula as well as approximate formula for the drag has been calculated. In Section B the expression for the velocity of the spheroid started from rest under the action of constant force has been investigated.

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Zusammenfassung In dieser Arbeit wird die nicht-stationäre Bewegung eines Spheroids von der Ruhe in einem viskoelastischen Fluid betrachtet. Die Arbeit ist in zwei Abschnitte geteilt. Im ersten Teil werden allgemeine Formeln sowie Näherungen für den Widerstand entwickelt. Im zweiten Abschnitt werden Ausdrücke für die Geschwindigkeit eines Speroids hergeleitet, welches von der Ruhe aus unter dem Einfluss einer konstanten Kraft sich bewegt.

     

Fuzzy power sets and fuzzy implication operators

The theory of fuzzy power sets, which has hitherto been insufficiently developed, is shown very naturally to require the use of a fuzzy implication operator (Section 1). Six such operators are gathered from the literature on multiple-valued logic (Section 2), and their effects on fuzzy power-set theory are compared throughout the rest of the paper. After certain fundamental definitions of set characteristics (Section 3), the six operators are carried in parallel while working out basic aspects of power-set theory. Among these are the properties of the set-inclusion relation and the set-equivalence relation (Section 4), two distinct concepts of disjointness (Section 5), questions of consistency in the relations between a set and its complement (Section 6), and a very concrete theorem on a difference among the operators with regard to the derivation of crisp conclusions from fuzzy premises (Section 7). Finally (Section 8), emphasis is placed on the dependence of the choice of operators upon the purposes the user has in hand.     

NATURAL BOUNDARY ELEMENT METHOD FOR THREE DIMENSIONAL EXTERIOR HARMONIC PROBLEM WITH AN INNER PROLATE SPHEROID BOUNDARY

In this paper, we study natural boundary reduction for Laplace equation with Dirichletor Neumann boundary condition in a three-dimensional unbounded domain, which is theoutside domain of a prolate spheroid. We express the Poisson integral formula and naturalintegral operator in a series form explicitly. Thus the original problem is reduced to aboundary integral equation on a prolate spheroid. The variational formula for the reducedproblem and its well-posedness are discussed. Boundary element approximation for thevariational problem and its error estimates, which have relation to the mesh size andthe terms after the series is truncated, are also presented. Two numerical examples arepresented to demonstrate the effectiveness and error estimates of this method.     

On the origin of convection

The method of investigating bifurcations developed in [1 and 2] is applicable to many hydrodynamic problems. In the present paper it is applied to investigate the origin of convection in a horizontal fluid layer heated from below.

Secondary stationary flows are of particular interest in the convection problem since the loss of stability is associated with these flows: “the principle of the change in stability” is not only valid here but has been proved rigorously [3]. It has also been proved that secondary stationary flows are generated by branching off from the state of rest [4 and 5].

The problem under consideration is invariant relative to the group of motions of a horizontal plane.

The single solution invariant relative to this whole group is the rest solution. When this solution is unstable, it is natural to expect the occurrence of solutions invariant relative to some subgroup of the group of motions. If the mentioned subgroup is generated by a pair of translations (in perpendicular directions), we arrive at doubly-periodic solutions (Section 1), and if invariance relative to rotation through a certain angle is required in addition, we arrive at solutions of hexagonal type (Section 2). As is known, precisely these latter are realized in convection experiments [6]. Deductions on the existence of doubly-pertodic convection flows are elucidated in Theorem 1.1, and the existence of solutions of hexagonal convection type is asserted in Theorem 1.2. The applied method has slight connection with the boundary conditions. Only for definiteness is it assumed that the boundaries of the layer are solid walls on each of which the temperature is specified.     

Barely Transitive Locally Nilpotent p-Groups

My paper [2], which appeared in the special issue dedicatedto the late Professor Brian Hartley, contained an error in Lemma2.2 as a result of overlooking the case that the exponents k_ion p. 359 may be divisible by arbitrary powers of p, which waspointed out by Professor H. Smith. In order to overcome thissituation all of Section 2, except Lemma 2.1, has been rewritten.However this does not cause any change in the rest of the paper.In particular the original assertions remain correct.     

Ribbon Schur Functions

We present a new determinantal expression for Schur functions. Previous expressions were due to Jacobi, Trudi, Giambelli and others (see [7]) and involved elementary symmetric functions or hook functions. We give, in Theorem 1.1, a decomposition of a Schur function into ribbon functions (also called skew hook functions, new functions by MacMahon, and MacMahon functions by others). We provide two different proofs of this result in Sections 2 and 3. In Section 2, we use Bazin's formula for the minors of a general matrix, as we already did in [6], to decompose a skew Schur function into hooks. In Section 3, we show how to pass from hooks to ribbons and conversely. In Section 4, we generalize to skew Schur functions. In Section 5, we give some applications, and show how such constructions, in the case of staircase partitions, generalize the classical continued fraction for the tangent function due to Euler.     

Periodic solutions for coupled systems of differential-difference and difference equations

Summary The objective of this paper is to give necessary and sufficient conditions for the existence of periodic solutions of coupled systems of differential-difference and difference equations. By differentiating the difference equation, we obtain a system of neutral differential-difference equations and we get the original problem by putting a side condition on the neutral equation; that is, by restricting the initial data to lie on certain manifold in the space of all initial data. This allows us to treat the problem using the methods of neutral functional differential equations. In[8], Hale and Martinez-Amore exploited a certain change of variables to obtain some results on the stability of this systems. In Section 2, we summarize those ideas. The effect of the side condition is reflected in the variation of constants formula in Section 3. In this section, the variation of constants formula is decomposed via eigenspaces. In Section 4, we give a theorem on the Fredholm alternative for periodic solutions which is basic to the application of the usual theory to perturbed linear problems. I want to express my most deep gratitude to Professor J. K. Hale for his advice and suggestions which led to considerable improvements of this paper. Entrata in Redazione l'8 marzo 1978. Research was supported in the form of Grant from the Program of Cultural Exchange between the United States and Spain.     

On the fast solution of Toeplitz-block linear systems arising in multivariate approximation theory

When constructing multivariate Padé approximants, highly structured linear systems arise in almost all existing definitions [10]. Until now little or no attention has been paid to fast algorithms for the computation of multivariate Padé approximants, with the exception of [17]. In this paper we show that a suitable arrangement of the unknowns and equations, for the multivariate definitions of Padé approximant under consideration, leads to a Toeplitz-block linear system with coefficient matrix of low displacement rank. Moreover, the matrix is very sparse, especially in higher dimensions. In Section 2 we discuss this for the so-called equation lattice definition and in Section 3 for the homogeneous definition of the multivariate Padé approximant. We do not discuss definitions based on multivariate generalizations of continued fractions [12, 25], or approaches that require some symbolic computations [6, 18]. In Section 4 we present an explicit formula for the factorization of the matrix that results from applying the displacement operator to the Toeplitz-block coefficient matrix. We then generalize the well-known fast Gaussian elimination procedure with partial pivoting developed in [14, 19], to deal with a rectangular block structure where the number and size of the blocks vary. We do not aim for a superfast solver because of the higher risk for instability. Instead we show how the developed technique can be combined with an easy interval arithmetic verification step. Numerical results illustrate the technique in Section 5.Research partly funded by FWO-Vlaanderen.     

Dissecting orthoschemes into orthoschemes

We exhibit a dissection, with one degree of freedom, of an arbitrary orthoscheme in Euclidean, spherical or hyperbolic d-space into d+1 orthoschemes (Section 2); this can be interpreted as a set of relations in the scissors congruence group or, weaker, as a set of functional equations for the volume. Besides special cases where the dissection is into mutually congruent parts, we obtain, in the spherical case and for a special value of the parameter, scissors congruence formulae similar to Schläfli's period formulae for the spherical orthoscheme volume (see Section 5). In Section 6 we use the dissection to explain the structure of the volume formula for asymptotic hyperbolic 3-orthoschemes due to Lobachevsky. Finally, in Section 7, by exploiting symmetries, we show that two systems of special volume relations of Schläfli (in spherical d-space) and Coxeter (for all three geometries in dimension 3) hold even on the level of dissection. In particular, it seems that all the presently known exact values for the volume of special spherical 3-simplexes hold, independently of Schläfli's differential formula, as consequences of scissors congruence relations.     

Differential Forms and the Change of Variable Formula for Multiple Integrals

Recently P. Lax has produced a novel approach to the proof of the change of variable formula for multiple integrals. In Section 1 we give a variant of Lax's proof, using the language of differential forms. In Sections 2 and 3 we discuss extensions involving more singular maps and integrands.     

Multivariate analysis using ‘and’ and ‘or’

The connectives ‘and’ and ‘or’ are potentially useful in multivariate analysis and theory construction. They are simple, logical ways to connect two or more variables together. However, until recently there has been no framework for operationalizing these connectives for continuous variables, and this lack has severely limited their use. Using fuzzy set theory as a basis for such a framework, this paper lays out the necessary tools and models to permit the use of ‘and’ and ‘or’ in multivariate analysis.Section 1 introduces conventional operators for ‘and’ and ‘or’, and Section 2 provides suitable extensions and generalizations of them. Section 3 sets out the required least-squares techniques for fitting these generalized operators to data, first in the context of ANOVA problems and then in regression contexts, for single-connective (three-variable) models. The theoretical developments and examples from real data-sets demonstrate the utility of ‘and’ and ‘or’ as a means to cell-specific interpretations of interaction effects which can also readily be translated into English. Section 4 extends these developments to multivariate, multiple-connective models and discusses issues of generalizability. The paper concludes (Section 5) with a brief discussion of remaining unsolved problems, future prospects for more sophisticated models, and computer programs.     

Multiplicative partial integration and the Trotter Product Formula

For a long time it has been apparent that the Trotter Product Formula, a simple version of which reads $\text{exp}\text{{(A + B)t} =}\text{lim}{}_{\mathrm{n\to \infty }}\text{exp {(}\text{At}\text{n}\text{) · (}\text{Bt}\text{n}\text{)}}{}^{\mathrm{n}}$, where A, B are (non-commuting) operators and t is real, is related to the formula for the multipicative integral of the sum of two functions. But the precise connection between the two has not been exposed in the literature. Our purpose is to fill this lacuna. The theory of the multiplicative integral is outlined in Section 1, and the formula for the integral of the sum of two functions and the Trotter Product Formula are discussed in Section 2. Possible extensions are alluded to in Section 3.     

Time-dependent operator matrices and inhomogeneous Cauchy problems

The theory of operator matrices has been applied recently in various fields (cf. [4], [9], [10]). In particular, it is possible to solve inhomogeneous abstract Cauchy problems using the theory of operator matrices on appropriate product spaces (see [9]). For nonautonomous Cauchy problems, however, it seems that there is still lacking a systematic theory of operator matrices. As a first step towards such a theory, in this paper we will show that nonautonomous inhomogeneous Cauchy problems can be solved by using operator matrices. First we will give simplified proofs of known wellposedness results for these problems (Section 2). In Section 3 we use the results of Section 2 for a discussion of nonautonomous Cauchy problems in the context of abstract operator matrices. This paper is part of a research project supported by the Deutsche Forschungsgemeinschaft DFG. The support of DAAD is gratefully acknowledged. The authors wish to thank Rainer Nagel and Klaus Jochen Engel for helpful comments.     

Formal methods in pattern recognition: A review

There is lot of excitement with Pattern Recognition methods with high precision, since this problem area is a well-established field of Operations Research (O.R.). Recent work of some researchers has shown that O.R. methods in general and Optimisation methods in particular, can be applied to give some very good results. Thus this research area has been won back from the Artificial Intelligence community and is quickly becoming once more a fast growing field in O.R. The aim of this review is to examine the early success of classification methods and Pattern Recognition methods, consider their downfall and examine the new techniques that have been applied to make it like a resurgent Phoenix. It will be shown that optimisation methods, if carried out properly, through a formal analysis of their structure and their requirements can achieve correct classification with probability one. Many researchers make it more difficult for themselves by not considering the formalisation of the task concerned and so adapt heuristics to the problem. Computational methods taken from the Irvine Repository database on recognition instances will be placed in evidence. The outline of the paper is as follows. After the introduction a historical sketch of the field will be presented. Then in Section 3, the need for formal methods will be argued and various results on formal requirements as convergence etc. will be derived. Many of these formal requirements are of course related to the best-unbiased estimate (b.u.e) requirements in Statistics. In Section 4 some popular algorithms for Pattern Recognition will be presented and their degree of satisfaction of the formal requirements stressed, allowing in Section 5 to present many applications, so that conclusions can be reached in Section 6. It will be found that the satisfaction of the formal requirements is a necessary and sufficient condition to reach recognition with probability one.     

An oscillation theory for second-order integral differential equations

The primary purpose of this paper is to give an oscillation theory for second-order integral differential equations. It is shown that this theory follows in a natural way as “a corollary” from the more abstract approximation theory of quadratic forms given previously by the author. Thus, our ideas are primarily constructive and quantitative as opposed to the usual qualitative methods. We also note that the usual oscillation theory for second-order differential equations follows directly by our methods. Furthermore, our methods provide a unified theory for eigenvalue problems, optimization problems, and numerical approximation problems within this setting.In Section 1 we give the preliminaries for the remainder of the paper. In Section 2 we define the basic quadratic form and integral differential equation and give the relationships between them. These relationships are used (in Section 3) to give a theory of oscillation in our setting and some basic oscillation results. Finally, in Section 4 we give some deeper oscillation results.To emphasize the unifying methods of our ideas, this paper is presented as a companion paper to “A Numerical Approximation Theory for Second Order Integral Differential Equations.”     

Bayesian graduation of FHA/HUD single-family home mortgage insurance contracts — Section 203

In this paper we will describe a procedure used to construct the survivorship (or double-decrement) table for FHA Section 203 mortgage insurance contracts on single-family home mortgages having a 30-year term. The methodological approach is based on a procedure known in the actuarial literature as Bayesian graduation. This procedure has been described by Kimeldorf and Jones (1967) and Hickman and Miller (1977).     

A study on the structure of planning behaviour

Although there is an abundance of literature on planning, it is largely either about how we actually plan, about how we should plan, or about how we should organize our planning, and little effort seems to have been contributed to investigate how we actually organize our planning, which should become the basis for the above three kinds of research. In this paper, we consider first the relationship between planning and decision-making with a simple example (Section 2), and then the benefits and costs of planning more specifically (Section 3). After these preparatory considerations, we postulate a hypothesis which describes planning behaviour of individuals and organizations, with necessary definitions and assumptions (Section 4). Some propositions attained from the hypothesis are also included. For testing the validity of the hypothesis, an organism model of organization which deals with an organization as an open system in the environment is introduced (Section 5.1). We make additional assumptions to let the model plan (Section 5.2), and the behaviour of the model is generally supported by the practitioners and researcher interviewed (Section 5.3). Finally we discussed briefly the methodology employed and the possible applications of the hypothesis (Section 6).     

The small dispersion limit of the Korteweg-de Vries equation. III

In Parts I and II we have derived explicit formulas for the distribution limit u of the solution of the KdV equation as the coefficient of u_xxx tends to zero. This formula contains n parameters β_1, …, β_n whose values, as well as whose number, depends on x and t. In Section 4 we have shown that for t<tb, n=1, and the value of β, was determined. In Section 5 we have shown that the parameters β_i satisfy a nonlinear system of partial differential equations. In Part III, Section 6 we show that for t large, n=3, and we determine the asymptotic behavior of β_1, β_2, β_3, and of u and u ², for t large. The explicit formulas show that u and u ² are O(t^?1) and O(t^-2) respectively (see formulas (6.2) and (6.24)). In Section 7 we study initial data whose value tends to zero as x→+∞, and to -1 as x→?∞. If one accepts some plausible guesses about the behavior of solutions with such initial data, we derive an explicit formula for the solution and determine the large scale asymptotic behavior of the solution: . The function s(ζ) is expressible in terms of complete elliptic integrals; a similar formula is derived for U ². In Section 8 we indicate how to extend the treatment of this series of papers to multihumped (but still negative) initial data.     

On nonlinear differential equations in Hilbert spaces

In a recent paper of the same title, L. A. Medeiros has considered the question of uniqueness for the Cauchy problem for ordinary differential equations in a complex Hilbert space. Section 1 contains a discussion of Medeiros' results, and provides a motivation for the improved uniqueness results in Section 2     

Radon-Nikodym derivatives of a class of weak distributions on Hilbert spaces

The problem of derivatives of weak distributions is studied in the context of likelihood ratios of signals in noise, the independent case. We show that the derivative is defined in that case and obtain a formula for it. The main result is in Section 2; the necessary introductory material is in Section 1. The application to the linear case is given in Section 3, and in Section 4, a non-linear example, in which we show for the first time that the correction term in the white noise version of the Girsanov formula is a random variable whose expected value is the mean square estimation error.Research supported in part under AFOSR Grant No. 73-2492, Applied Math Division, USAF.