On the sensitive dynamical system and the transition from the apparently deterministic process to the completely random process

The detailed analysis of the dynamical process of coin tossing is made. Through calculations, it is illustrated how and why the result is extremely sensitive to the initial conditions. It is also shown that, as the initial height of the mass center of the coin increases, the final configuration, i.e. head or tail, becomes more and more sensitive to the initial parameters (the initial velocity angular velocity, and the initial orientation), the coefficient of the air drag, and the energy absorption factor of the surface on which the coin bounces. If we keep the head upward initially but allow a small range for the change of some other initial parameters, the frequency that the final configuration is head, would be 1 if the initial height h of the mass center is sufficiently small, and would be clo to 1/2 if h is sufficiently large. An interesting question is how this frequency changes continuously from 1 to 1/2 as h increases. Detailed calculations show that such a transition is very similar to the transition from laminar to turbulent flows. A basic difference between the transition stage and the completely random stage is indicated: In the completely random stage, the deterministic process of the individual case is extremely sensitive to the initial conditions and the dynamical parameters, out the statistical properties of the ensemble are insensitive to the small changes of the initial conditions and the dynamical parameters. On the contrary, in the transition stage, both the deterministic process of the individual case and the statistical properties of the ensemble are sensitive to the initial conditions and the dynamical parameters. The mechanism for this feature of the transition stage is the existence of the long-train structure in the parameter space. The illuminations of this analysis on some other random phenomena are discussed.     

Whose Thoughts Are They, Anyway? Dimensionally Exploding Bion's “Double-Headed Arrow” into Coadapting, Transitional Space

The physics and biology that found psychoanalysis account for discontinuous experience only in the presence of nonmeasurable, metaphysical operators; these include the ego and its subsystems as well as biological experience inherited through Lamarckian principles. Complex, self-organizing systems, however, can link biology to experience without metaphysics. They can also account for psychoanalytically relevant behaviors without appealing to stable internal representations. These behaviors include what W. R. Bion called transformation in O and its corollary, the appearance of the selected fact. By dimensionally exploding the double-headed arrow that he used to link the states Ps and D in his model for thinking (Ps D), we can generate a space that is, at once, psychoanalytically imaginal and dynamically coadapting. Isomorphic to D. W. Winnicott's transitional space, it is self-organizing. It is describable according to dynamics formulated by W J. Freeman, S. Kauffman and C. Langton and it can generate instantaneous conscious contents by way of a selective process analogous to spatio-temporal binding. As a whole, this model supports a clinical stance advanced by D. W. Winnicott as play, within transitional space.     

On elastic-plastic structures with associated stress-strain relations allowing for work softening

Summary For the elementary (finite or infinitesimal) constituents of the structure are assumed generalized stress-strain relations which satisfy the condition of normality but which may exhibit work-softening, concavity of yield surfaces, variation of elastic coefficients with stress and/or plastic strain.The usual phenomena of geometric instability are excluded. Sufficient conditions are formulated for overall stability in spite of the presence of unstable elements, and for uniqueness of the incremental boundary-values problem. Conditions are discussed with a view to applications and expressed in terms of positive definiteness of appropriate quadratic forms.Finally, yield surfaces and flow laws for the structure are examined, and among other things their necessary association is shown.

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Sommario Per i costituenti elementari (finiti o infinitesimi) della struttura si assumono legami incrementali tra sforzi e deformazioni generalizzati che soddisfano alla condizione di normalità ma che per il resto sono generici, cioè tali da presentare eventualmente incrudimento negativo, concavità del campo elastico, variazione dei coefficienti elastici con gli sforzi e/o con le deformazioni plastiche.Esclusi per il sistema i fenomeni usuali di instabilità geometrica si formulano condizioni sufficienti per la stabilità del complesso nonostante la presenza di parti a funzionamento instabile, e per l'unicità del problema incrementale al contorno. Le condizioni sono discusse in vista delle applicazioni ed espresse in termini di definizione positiva di forme quadratiche opportune.Si esaminano infine le superfici di snervamento e le leggi di scorrimento per l'intera struttura e, tra l'altro, se ne dimostra la necessaria associazione.

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First published in Italian in Rendiconti dell'Istituto Lombardo Classe Scienze e Lettere, A 100, 1966.The present investigation has been promoted and financed by the Consiglio Nazionale delle Ricerche (C.N.R.) at the Istituto di Scienza delle Costruzioni of the Facoltà di Architettura, Politecnico of Milano. Part of it was carried out at Brown University (Providence R. I. - U.S.A.). The author acknowledges with tanks the NATO Fellowship assigned by the C.N.R. in 1964, the encouragement and kind hospitality he received from Professor D.C. Drucker, and the interesting discussions he had with Professor Drucker and Dr. A. C. Palmer.     

Exact Solutions of the Hydrodynamic Equations Derived from Partially Invariant Solutions

The paper proposes a heuristic approach to constructing exact solutions of the hydrodynamic equations based on the specificity of these equations. A number of systems of hydrodynamic equations possess the following structure: they contain a reduced system of n equations and an additional equation for an extra function w. In this case, the reduced system, in which w = 0, admits a Lie group G. Taking a certain partially invariant solution of the reduced system with respect to this group as a seed:rdquo; solution, we can find a solution of the entire system, in which the functional dependence of the invariant part of the seed solution on the invariants of the group G has the previous form. Implementation of the algorithm proposed is exemplified by constructing new exact solutions of the equations of rotationally symmetric motion of an ideal incompressible liquid and the equations of concentrational convection in a plane boundary layer and thermal convection in a rotating layer of a viscous liquid.     

An extension of Reiner's “Deborah Number” concept to a Wide field of rheological investigations

Summary Reiner defined a numeric, which he called theDeborah Number to represent the ratio of a relaxation time to a natural (observation) time. This implies aMaxwell model but is readily extended to complete relaxation spectra. Similar Numbers are proposed for retardation times and also for some conditions of coagulation thixotropy and for data from certain psychophysical experiments.     

Shock Waves in a Fluid with Bubbles Containing Evaporating Drops of a Liquefied Gas

The propagation and reflection of one-dimensional plane unsteady waves and pulses in a mixture of a fluid with two-phase bubbles containing evaporating drops is investigated. A significant effect of unsteady evaporation of the drops in the zone ahead of the shock wave on the wave propagation is demonstrated. The evaporation of the drops results in a pressure increase ahead of the wave and the shock wave as it were climbs to increasing pressure level. In contrast to bubbly fluids with single-phase bubbles, in a fluid with two-phase bubbles, at a fixed phase volume fraction, a decrease in bubble size results in an increase rather than a decrease of the oscillation amplitude. The wave reflection from a solid wall is essentially nonlinear and the maximum pressure attained at the wall is several times greater than the incident-wave intensity.     

On universal solutions in initially stressed bodies

A saddle point (or maximum-minimum) principle is set up for the quasi-static boundary-value problem in linear viscoelasticity. The appropriate class of convolution-type functionals for it is taken in terms of bilinear forms with a weight function involving the Fourier transform. The minimax property is shown to hold as a direct consequence of thermodynamic restrictions on the relaxation function. This approach can be extended to further linear evolution problems where initial data are not prescribed.     

Boundary Layer Stabilization by “Artificial Turbulence”

The possibility of boundary layer stabilization by artificial turbulence, i.e. spanwise-periodic transverse flows induced by mass forces, is considered. It is shown that as the result of additional momentum transfer by the transverse flow the velocity profile in the boundary layer becomes fuller. This results in the suppression of unstable disturbances and increases the laminar-flow interval by 3–4 times.     

Conical flows in the neighborhood of breaks in the edges of surfaces separating cocurrent supersonic streams of perfect gas

An investigation is made into the conical flows which occur when a perfect (inviscid and nonheat conducting) gas flows over the terminal edges of surfaces with breaks separating an external and an internal flow with velocity vectors parallel to the line of intersection of the surfaces. Such flows are observed, in particular, in the neighborhood of breaks in the outlet edge of a nozzle of rectangular cross section with a straight or skewed exit plane under conditions of underexpanded flow of a supersonic jet into a cocurrent supersonic stream. By means of a linear analysis flow patterns corresponding to various flow interaction regimes and edge geometries are constructed and a law of similarity is formulated. The validity of the results thus obtained is confirmed by examples of the numerical solution of the complete nonlinear system of Euler equations. In this connection, within the framework of the approach outlined in [1], as a rule, together with the shocks and characteristic surfaces bounding the conical flow in question, the shear discontinuity separating the external and internal streams is constructed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 119–127, January–February, 1935.     

Application of the method of integral diffusion to investigation of turbulent transfer

A number of authors have critically examined semiempirical mixing length theories [1]. A defect of these theories is connected with the fact that the magnitude of the mixing length, which is assumed to be small in constructing the theory, turns out in experiments to be comparable with the characteristic dimensions of the flow region. Thus, the concept of volume convection [2–4] or integral diffusion [5], which is understood to be a transfer mechanism in which the friction stress is not expressed in terms of the velocity gradient, is introduced along with the concept of gradient diffusion. In addition, there are a number of experimental papers [6] in which it is shown that the turbulent friction stress cannot be equal to zero at the place in the flow where the derivative of the velocity is equal to zero. Mixing length theory does not describe this effect.It is possible to generalize mixing length theory [7–9] in a way which eliminates these defects. Flow of an incompressible fluid is considered.     

About the “normal growth” approximation in the dynamical theory of phase transitions

Nonequilibrium phase transitions can often be modeled by a surface of discontinuity propagating into a metastable region. The physical hypothesis of normal growth presumes a linear relation between the velocity of the phase boundary and the degree of metastability. The phenomenological coefficient, which measures the mobility of the phase boundary, can either be taken from experiment or obtained from an appropriate physical model. This linear approximation is equivalent to assuming the surface entropy production (caused by the kinetic dissipation in a transition layer) to be quadratic in a mass flux.In this paper we investigate the possibility of deducing the normal growth approximation from the viscosity-capillarity model which incorporates both strain rates and strain gradients into constitutive functions. Since this model is capable of describing fine structure of a thick advancing phase boundary, one can derive, rather than postulate, a kinetic relation governing the mobility of the phase boundary and check the validity of the normal growth approximation.We show that this approximation is always justified for sufficiently slow phase boundaries and calculate explicitly the mobility coefficient. By using two exact solutions of the structure problem we obtained unrestricted kinetic equations for the cases of piecewise linear and cubic stress-strain relations. As we show, the domain of applicability of the normal growth approximation can be infinitely small when the effective viscosity is close to zero or the internal capillary length scale tends to infinity. This singular behavior is related to the existence of two regimes for the propagation of the phase boundary — dissipation dominated and inertia dominated.     

Die Reduktion von nichteuklidischen geometrischen Objekten in eine euklidische Form und physikalische Deutung der Reduktion durch Eigenspannungszustände in Kristallen

Summary Self-stresses in crystals are closely related to tearing processes. These processes have a meaning only in connection with geometrical objects. The present paper deals with tearing processes connected with the geometrical objects metric and connexion. A tearing process is commonly associated with either one of these processes. It is therefore of interest to investigate the question whether a tearing process which is connected with both objects exists. The investigation of this problem leads to the subdivision of the self-stresses according to the metric properties of the connexion: metrical stresses caused by dislocations and disclinations and nonmetrical stresses caused by non-mechanical sources and by extra-matter.It should be mentioned that the mathematical part of this paper has an independent interest for the understanding of the structures associated with higher differential geometry.

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Vorgelegt von T. Meixner

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Herrn Professor Dr. A. Seeger danke ich für die verständnisvolle Förderung dieser Arbeit. Herrn Dr. C. Teodosiu bin ich für zahlreiche Diskussionen sehr dankbar.

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The problems of nonlinear bending for orthotropic rectangular plate with four clamped edges

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Direct derivation of the two-shaft system balance conditions for the second order reciprocal inertia forces on the V-type eight cylinders internal combustion engines with a plane crankshaft

By employing the four shafts balance concept paper [1] has reported a balance regime for the second order reciprocal inertia forces on the V-type eight cylinder internal combustion engines with a plane crankshaft. Thereafter, paper [2] has acquired a two-shafts balance regime, but through a rather tedious roudabout degenerating manipulation. The present article has, but starting out directly from the two-shafts balance concept, successfully acquired the same results as those in paper [2]. In addition, we propose, herein, a third balance system which might be, in general, called the slipper balance regime.     

An application of non-singularity BEM to plate bending, buckling and natural vibration problems

In this paper the fundamental solution of the singular governing equation of plate static bending is taken as the Green's function, which can satisfy the governing equation precisely in the plate region. Based on the principle of superposition, let the function values on the plate boundary, induced by a set of the Green's function sources (including the known sources in the plate region and the unknown sources in the fictitious region), satisfy the prescribed conditions on specially chosen boundary matching points, and the corresponding semi-analytical and semi-numerical solution can be obtained, which is free from the restraint of boundary forms and boundary conditions. The more matching points there are on the boundary, the better the accuracy of results is. Finally, in static bending problems a set of linear algebraic equations has to be computed; in buckling problems the minimum value of buckling eigenvalue equation has to be found; in natural vibration problems the eigenvalues of the frequency equation have to be calculated. Numerical examples are given and the results are compared with those by the analytical method and other methods. It can be seen that they are very close to each other.     

Explanation of the “anomalous” dispersal of the discrete particles in a two-phase turbulent jet with allowance for their inertia

The anomalous dispersal of the discrete particles in a two-phase turbulent jet (growth of their concentration in the region of the axis, rapid damping of the concentration in the initial sections of the jet, and wavelike axial distribution) is studied with allowance for the initial conditions and the Magnus effect.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 36–41, January–February, 1984.I thank G. N. Abramovich and A. N. Sekundov for discussing the results.     

Electrode layers in flows of an inviscid plasma with good conductivity

The structure of the electromagnetic electrode layers that are produced in flows across a magnetic field by a completely ionized and inviscid plasma with good conductivity and a high magnetic Reynolds number is examined in a linear approximation. Flow past a corrugated wall and flow in a plane channel of slowly varying cross section with segmented electrodes are taken as specific examples. The possibility is demonstrated of the formation of nondissipative electrode layers with thicknesses on the order of the Debye distance or electron Larmor radius and of dissipative layers with thicknesses on the order of the skin thickness, as calculated from the diffusion rate in a magnetic field [2].In plasma flow in a transverse magnetic field, near the walls, along with the gasdynamie boundary layers, which owe their formation to viscosity, thermal conductivity, etc. (because of the presence of electromagnetic fields, their structures may vary considerably from that of ordinary gasdynamic layers), proper electromagnetic boundary layers may also be produced. An example of such layers is the Debye layer in which the quasi-neutrality of the plasma is upset. No less important, in a number of cases, is the quasi-neutral electromagnetic boundary layer, in which there is an abrupt change in the frozen-in parameter k=B/p (B is the magnetic field and p is the density of the medium). This layer plays a special role when we must explicitly allow for the Hall effect and the related formation of a longitudinal electric field (in the direction of the veloeiryv of the medium). We will call this the magnetic layer. The magnetic boundary layer can be dissipative as well as noudissipative (see below). The dissipative magnetic layer has been examined in a number of papers: for an incompressible medium with a given motion law in [1], for a compressible medium with good conductivity in [2], and with poor conductivity in [3]. In the present paper, particular attention will be devoted to nondissipative magnetic boundary layers.     

Das erweiterte Korrespondenzgesetz für die dynamische Idealviskosität und die Idealwärmeleitfähigkeit reiner Stoffe

Zusammenfassung Zur Berechnung der dynamischen Idealviskosität Ideal (T) und der Idealwärmeleitfähigkeit ideal (T) benötigt man die kritische TemperaturT _kr, das kritische spezifische Volum _kr, die MolmasseM, den kritischen Parameter _kr und die molare isochore WärmekapazitätC _v(T). Sowohl das theoretisch, als auch das empirisch abgeleitete erweiterte Korrespondenzgesetz ergeben eine für praktische Zwecke ausreichende Genauigkeit für die Meßwertwiedergabe, die bei den assoziierenden Stoffen und den Quantenstoffen jedoch geringer ist als bei den Normalstoffen.

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The extended correspondence law for the ideal dynamic viscosity and the ideal thermal conductivity of pure substances

For the calculation of the ideal dynamic viscosity Ideal (T) and the ideal thermal conductivity ideal (T) the critical temperatureT _kr, the critical specific volumev _kr, the molecular massM, the critical parameter _kr, and the molar isochoric heat capacityC _v(T) is needed. Not only the theoretically determined but also the empirically determined extended correspondence law gives for practical use a good representation of the measured data, which for the associating substances and the quantum substances is not so good as for the normal substances.

     

On the existence of limit cycles of Liénard equation

In this paper, we have proved several theorems which guarantee that the Liénard equation has at least one or n limit cycles without using the traditional assmuption G(±) =+. Thus some results in [3–5] are extended. The limit cycles can he located by our theorems. Theorems 3 and 4 give sufficient conditions for the existence of n limit cycles having no need of the conditions that the function F(x) is odd or nth order compatible with each other or nth order contained in each other.     

Some aspects of dilute suspension theory

A theory proposed by the author as representative of the flow of a general suspension contains three interaction forces, f, S and N. For a quasi-concentrated suspension and for a dilute suspension, N and S, N are omitted, respectively. For the latter special case, we treat diffusion of a fluid through an elastic solid. For a quasi-concentrated suspension, we show that F and S depend on the gradient of the motion gradient. We demonstrate the existence of interesting phenomena: non-simple behavior, dissipative effects, generalized lift and drag forces.Presented at the second conference Recent Developments in Structured Continua, May 23 – 25, 1990, in Sherbrooke, Québec, Canada.