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Let G G be a finite group and M M be a compact piecewise linear (PL) manifold. Define a PL G G -isotopy to be a level-preserving PL action of G G on M × [ 0 , 1 ] M \times [0,1] . In this paper PL G G -isotopies are studied and PL G G -isotopic actions (which need not be equivalent) are characterized.

A continuum-Mechanical Approach to the Mechanics of Rock Masses A quasihomogeneous region of a discontinuous rock mass is approximated by a generalized Cosserat continuum. Blocks of the rock mass are represented by mass points, and to each point a number of vectors — the directors — is assigned. These directors determine the size and shape of the individual blocks. For this continuum are derived the basic equations of motion and the thermodynamic relations. The stress state is represented by a non-symmetric stress tensor.

A formula for the ignition time, that Bradley obtained empirically, is derived analytically by a rigorous asymptotic analysis of the limit of large activation energy. Two terms in the expansion must be retained. The second term reveals the existence—just prior to ignition—of a reactive-diffusive zone at the surface of the solid and a transient-diffusive zone in the interior. The analysis also exhibits a universal correction factor for Bradley's formula, of order unity.

SummaryThe paper deals with the behaviour of three shock waves meeting at a point in a perfect gas. It is shown that the equations of motion can be reduced to a single polynomial equation of degree 10. The real roots of this equation are studied to determine their physical significance. In addition, the appearance of degenerate shock systems is shown to be associated with the formation of certain multiple roots of the polynomial equation.

Geometrically speaking these conditions express the fact that x=r(w) represents a surface in R 3 of constant mean curvature H, having finite area A(t) and an assigned boundary F*. Under various conditions on F* and H existence proofs for such surfaces have been given in the literature ([1], [5], [6], [8], [9], and [10]; see also [7] for a complete discussion of the non-parametric case). If one considers the totality of all boundary curves F* lying in the unit ball I~[ < 1, the sharpest result hitherto obtained is due to HILDEBRANDT [6], who proved that in this case the class ~ (F* , H) is nonempty, provided that [HI < 1. On the other hand, from geometric considerations it is very plausible that for a given curve F* the surfaces x =z(w) should cease to exist, if [HI exceeds a certain critical number c(F*). The purpose of this note is to substantiate this fact by estimating the quantity c(F*) for a class of boundary curves F*. Introduce the numbers

This paper presents a method of numerical solution of boundary value problems governed by a set of ordinary differential equations. The highest derivative is chosen as the unknown function. Governing equations are transformed into a set of integral equations. The kernels of integral equations turn out to be influence functions for deflection and/or bending moments of a corresponding beam, and can, therefore, be computed using well-known methods of structural analysis. Finally, the unknown function (highest derivative) is approximated in a defined manner and the solution is obtained through numerical integration.

In this paper, we consider the Dirichlet Laplacian operator -{\Delta} on a curved quantum guide in R n (n = 2, 3) with an asymptotically straight reference curve. We give uniqueness results for the inverse problem associated to the reconstruction of the curvature by using either observations of spectral data or a boot-strapping method. keywords: Inverse Problem, Quantum Guide, Curvature

1. Let E be a closed set in the complex plane and f a meromorphic function outside E omitting a set 3". We shall consider the following problem: I f E is thin, under what conditions is F thin, too? In Chapter 2 we consider the case when E and F are of Hausdorff dimension less than one. In Chapter 3 E and F are countable sets with one limit point, and in Chapter 4 E is a countable set whose points converge to infinity, f is entire, and F is allowed to contain at most one finite value.

The method presented in Part I of this paper1 is extended to problems governed by a system of partial differential equations. The basic clarity and simplicity of the method is preserved. The domain of integration is covered by a net of orthogonal (not necessarily equidistant) straight lines. Choosing again highest derivatives with respect to each of the co-ordinates as unknowns, it is possible to reduce the problem to a system of integral equations along the lines of the adopted net. The subsequent procedure is basically the same as in Part I. The results prove to be of remarkable accuracy even for very coarse nets.

This paper deals with two-dimensional stresses due to an important class of body forces. Gravity stresses and temperature stresses are produced by body forces of the type investigated. In the case of gravity stresses it is proved that these stresses may be calculated by forgetting about the action of gravity on the body and applying instead only external normal loads that are identical with a hydrostatic pressure. This opens the possibility of applying photoelastic tests to the measurement of gravity stresses; a model loaded by these external pressures would show an isochromatic fringe pattern identical with the one that would appear in a model submitted to an intense gravity field. In Fig. 5 is shown the type of external loading to be applied to find the gravity stresses in a dam. In the case of two-dimensional temperature stress in cylindrical bodies where the temperature distribution is stationary, it is shown that a distinction is to be made between solid and hollow cylinders. For a solid cylinder the only thermal stress component appearing is an axial stress acting perpendicularly to cross-sectional planes. In other words, the solid cylinder expands freely along the cross-sectional planes. For a hollow cylinder a general stress condition may arise which is identical with the stresses produced by cutting a longitudinal slit of a certain small width in the cylinder wall and sticking together the two sides of the slit (dislocation).

Investigation of Soret and Hall effects on unsteady MHD free convection heat and mass transfer flow of a viscous, incompressible, electrically conducting and optically thick radiating fluid past an impulsively moving infinite vertical plate with ramped temperature through a uniform porous medium in a rotating system in the presence of first order chemical reaction is carried out. Exact analytical solution of the governing equations for fluid velocity, fluid temperature and species concentration subject to appropriate initial and boundary conditions is obtained using Laplace transform technique. Expressions for shear stress, rate of heat transfer and rate of mass transfer at the plate are derived for both ramped temperature and isothermal plates. The numerical values of primary and secondary fluid velocities, fluid temperature and species concentration are displayed graphically whereas those of shear stress and rate of mass transfer at the plate are presented in tabular form for various values of pertinent flow parameters.Keywords: Magnetohydrodynamics (MHD), Hall current, chemical reaction, Soret effect, Ramped wall temperature.

It is both an honour and a responsibility to deliver a lecture commemorating the names of two scientists who have made unique contributions to fluid dynamics, and whose names perhaps above all others will remain immortal in our subject. The range of topics that would be suitable for such a lecture is very wide, because there is scarcely a branch of fluid dynamics in which the work of Reynolds and Prandtl does not continue to play a significant part. In a lecture of current interest there is, therefore, no problem of relating the subject matter to the names to be commemorated; the influence of Reynolds and Prandtl remains as strong as when they were alive, and many of the problems that engaged their attention are even today only partially solved. The lecturer must, therefore, use other considerations for guidance in the selection of his subject from the limited range on which he is qualified to speak, for few could hope to work over the wide field covered by Reynolds and Prandtl. He must if possible select a, topic in which major contributions have originated from Germany and the United Kingdom, on which research is actively in progress in both countries, and for which progress has relied on the combination of theoretical and experimental work to provide acceptable simplifying assumptions that characterised many of the investigations of Reynolds and Prandtl.

The air ejector, in its various forms, is a device which has many applications in engineering practice, and several attempts have been made to analyse its mode of action, some of these having been supported by experimental work. Most of the experimental results available are related to ejectors in which relatively high-pressure steam is utilized as the driving fluid, but even in these cases the information provided is restricted to a narrow field.The investigation described relates to an air ejector employing as the driving fluid air at a relatively low pressure, not exceeding 40 lb. per sq. in. (abs.), and covering a wide range of operating conditions by means of interchangeable nozzles. Two distinct experimental arrangements were built—one for the set of conditions in which the ejector draws in a relatively small quantity of suction fluid and pumps it through a relatively high pressure-ratio, and the other covering conditions in which the quantity of suction fluid is much larger, but the pressure ratio is quite small. For a given initial pressure and quantity of driving fluid, the rate of mass flow of suction fluid depends chiefly on the diameter of the combining tube, in which the driving and suction fluids mix; in the experiments, the ratio of com-bining-tube area to driving-nozzle area was varied in twelve steps, covering a range of area ratios from 1·44 to 1,110·0, and compression ratios ranging from about 3 to about 1·001.Efforts were made to find the best proportions of those parts of the ejector which exert a major influence on performance, and certain conclusions are drawn from the results of the experiments. Theoretical aspects of the problem are briefly discussed.

A numerical or graphical method is given for computing, in strip or plate rolling, the distribution of roll pressure over the arc of contact and the quantities derived from this (e.g. the vertical roll force, the torque, and the power consumption). The method avoids all mathematical approximations previously used in the theoretical treatment of rolling, and permits any given variation of the yield stress and of the coefficient of friction along the arc of contact to be taken into account. It can be used, therefore, in both hot and cold rolling, provided that the basic physical quantities (yield stress and coefficient of friction) are known. The usual assumption that the deformation could be regarded as a locally homogeneous compression has not been made, and the inhomogeneity of stress distribution has been taken into account approximately by using results derived by Prandtl and Nádai from the Hencky treatment of two-dimensional plastic deformation. It is found that the discrepancy between the roll pressure distribution curves calculated from the Kármán theory and those measured by Siebel and Lueg is due to the assumption in the theory that the frictional drag between the rolls and the rolled stock is equal to the product of the roll pressure and the coefficient of friction. If frictional effects are dominant, as in hot rolling, this product may easily exceed the yield stress in shear which is the natural upper limit to the frictional drag, and then static friction, instead of slipping, occurs. This has been taken into account in the present method, and the calculated curves of roll pressure distribution show good agreement with the curves measured by Siebel and Lueg.

Publisher Summary This chapter explores the isomorphisms θ of G for which the boundary function φ exists and is quasisymmetric, that is, there is a ρ ≥1 such that the inequality 1/ρ ≤ [φ(x+ t) – φ (x)] / [φ (x) – φ (x – t)] ≤ ρ holds for every real x and t > 0. Then, φ admits a quasiconformal extension to U. For finitely generated groups G, there is always a quasiconformal extension that induces φ by a theorem of Bers and Greenberg. The chapter presents an implicit characterization for the solvability of the boundary value problem. A quasiconformal mapping inducing θ exists if inequality holds for some .

La technique des équations intégrales est utilisée pour déterminer le champ des tensions et des déplacements dans une éprouvette C. T. 15 en acier 35 NCD 16. Le calcul de l'intégrale J de Rice permet de déterminer dans le cadre de la mécanique linéaire de la rupture le facteur d'intensité de contraintes pour plusieurs longueurs de fissure et ces valeurs sont comparées à celles obtenues à partir d'essais ou de calcul par éléments finis.

This chapter discusses the singular integral equations of the case where the symbol is independent of the pole and the case where the symbol is dependent on the pole. The general equation, the case where the symbol is independent of the pole, can be reduced to the equivalent equation of the Riesz–Schauder type. The chapter also discusses the regularization and domains of constancy of the index, and index theorem. Moreover, the chapter reviews the case where a singular integral is taken over not a Euclidean space but any closed m-dimensional Liapounov manifold that one shall make subject to the conditions of Giraud; namely, the manifold Γ can be orientated; it can be covered by a finite number of partly overlapping parts Γ j , each of which permits of a smooth one-to-one mapping onto a finite region of an m -dimensional Euclidean space E m . In addition, it is proved that the index of a singular equation with a symbol and positively bounded below absolutely was equal to zero.

SUMMARY Based on the variational formulation of linear thermodynamics as developed previously by the writer, thermomechanical reciprocity relations are discussed which lead to new methods of analysis of thermal stresses. These reciprocity relations are quite different from the usual ones derived from the analogy of thermal loading with a combination of surface and body-force distribution. The results are applicable to stationary and transient temperatures in elastic and viscoelastic structures. The methods are entirely variational and do not require the evaluation of the temperature field. The stresses at one point are expressed directly in terms of any arbitrary distribution temperatures applied externally, including the effect of surface heat-transfer layer. The concepts and procedures are illustrated on a simple example. The relation is pointed out between the reciprocity property and the generalization of Castigliano’s principle to thermomechanics.

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