Solutions of the Dirac–Fock Equations and the Energy of the Electron-Positron Field

We consider atoms with closed shells, i.e. the electron number N is 2, 8, 10,..., and weak electron-electron interaction. Then there exists a unique solution γ of the Dirac–Fock equations with the additional property that γ is the orthogonal projector onto the first N positive eigenvalues of the Dirac–Fock operator . Moreover, γ minimizes the energy of the relativistic electron-positron field in Hartree–Fock approximation, if the splitting of into electron and positron subspace is chosen self-consistently, i.e. the projection onto the electron-subspace is given by the positive spectral projection of. For fixed electron-nucleus coupling constant g:=α Z we give quantitative estimates on the maximal value of the fine structure constant α for which the existence can be guaranteed.     

Homogenization of the nonlinear Kelvin–Voigt model of viscoelasticity and of the Prager model of plasticity

Denoting by the stress tensor, by the linearized strain tensor, by A the elasticity tensor, and assuming that is a convex potential, the inclusion accounts for nonlinear viscoelasticity, and encompasses both the linear Kelvin–Voigt model of solid-type viscoelasticity and the Prager model of rigid plasticity with linear kinematic strain-hardening. This relation is assumed to represent the constitutive behavior of a space-distributed system, and is here coupled with the dynamical equation. An initial- and boundary-value problem is formulated, and the existence and uniqueness of the solution are proved via classical techniques based on compactness and monotonicity. A composite material is then considered, in which the function and the tensor A rapidly oscillate in space. A two-scale model is derived via Nguetseng’s notion of two-scale convergence. This provides a detailed account of the mesoscopic state of the system. Any dependence on the fine-scale variable is then eliminated, and the existence of a solution of a new single-scale macroscopic model is proved. The final outcome is at variance with the nonlinear extension of the generalized Kelvin–Voigt model, which is based on an apparently unjustified mean-field-type hypothesis.     

Superfluous Order and the Proper Solution of the Maxwell Equation

In this paper it is proved that Maxwell equation is equivalent to a fourth order equation.Under a certain condition,its general solution is given by     

Study of the inertial effect and the nonlinearities of?the?CRONE suspension based on the hydropneumatic technology

In this paper, we emphasize two main effects involved in the CRONE car suspension technology (CRONE: French acronym for Commande Robuste d??Ordre Non Entier). In a first time, we present the influence of the inductive or inertial effect of the pipes that links the different cells of the hydropneumatic car suspension. These components are mainly resistive and capacitive devices. Then, we analyze the nonlinear relations that link the hydraulic power variables (the flow and the pressure) of the hydraulic resistors and the hydropneumatic accumulators and we study the effect of the nonlinear terms on the car suspension response. Our study is based on the gamma RC arrangement developed in Altet et al. (In: Analysis and design of hybrid systems??proceedings of ADHS03, pp. 63?C68. Elsevier, Amsterdam, 2003) and Serrier et al. (In: Proceedings of IDETC/CIE 2005: ASME 2005 international design engineering technical conferences and computers and information in engineering conference, Long Beach, CA, USA, 24?C28 September 2005). In a second time, we focus only on the gamma RLC arrangement, introduced in Abi Zeid Daou et al. (Int. J. Electron. 96(12):1207?C1223, 2009). We show whether the parasite effect due to the pipes or the nonlinear RC components affect the system??s response. The simulation results show that neither the inertial effect caused by these parasite pipes of one meter length nor the use of the nonlinear resistors or the accumulators modifies the response of the gamma RC arrangement.     

On the Ellipticity of the Middle Surface of a Shell and its Application to the Asymptotic Analysis of ‘Membrane’ Shells

We consider a surface S = (), where ² is a bounded, connected, open set with a smooth boundary and : ³ is a smooth map; let () denote the components of the two-dimensional linearized strain tensor of S and let 0 with length 0 > 0. We assume the the norm ,|| ()||0, in the space V0() = { H¹() × H¹() × L²(); = 0 on 0 } is equivalent to the usual product norm on this space. We then establish that this assumption implies that the surface S is uniformly elliptic and that we necessarily have 0 = .     

Study of the effects of inner pressure and surface tension on the fibre drawing process with the aid of an analytical asymptotic fibre drawing model and the numerical solution of the full N.–St. equations

Micro-structured optical fibres (i.e. fibres that contain holes) have assumed a high profile in recent years and given rise to many novel optical devices. The problem of manufacturing such fibres by heating and then drawing a preform is considered for the case of annular capillaries. A fluid mechanics model suggested in the literature that uses asymptotic analysis based on the small aspect ratio of capillaries has been compared with the full 3D set of the N.–St. equations, for modelling the fabrication of capillaries. The final asymptotic equations, analysed in some asymptotic limits, are solved numerically and then compared with the N.–St. solutions, obtained with a commercial finite elements solver. These asymptotic limits give valuable practical information about the control parameters that influence the drawing process, taking into account the effects of surface tension and inner pressure, since those are of essential importance during drawing. It is shown that the asymptotic method delivers reliable results as long as the inner pressure does not exceed too high values.     

Continuity and Invariance of the Sacker–Sell Spectrum

The Sacker–Sell (also called dichotomy or dynamical) spectrum \(\varSigma \) is a fundamental concept in the geometric, as well as for a developing bifurcation theory of nonautonomous dynamical systems. In general, it behaves merely upper-semicontinuously and a perturbation theory is therefore delicate. This paper explores an operator-theoretical approach to obtain invariance and continuity conditions for both \(\varSigma \) and its dynamically relevant subsets. Our criteria allow to avoid nonautonomous bifurcations due to collapsing spectral intervals and justify numerical approximation schemes for \(\varSigma \).     

The moiré method and the evaluation of principal-moment and stress directions

This paper discusses, in relation to the moiré method as used for the solution of plate bending and two-dimensional stress problems, two graphical techniques for the determination of the directions of principal moments and stresses.The so-called isoclinic method and the point method are described.The application of these new techniques on three different models—a circular disk under diametrically opposite loads and two different circular plates subjected to a lateral load—are fully discussed.The graphically determined principal-stress and moment directions show excellent agreement with analytically determined comparable values.Paper was presented at 1963 SESA Annual Meeting held in Boston, Mass., on November 6–8.     

Local Minimizers and Quasiconvexity – the Impact of Topology

The aim of this paper is to discuss the question of existence and multiplicity of strong local minimizers for a relatively large class of functionals : from a purely topological point of view. The basic assumptions on are sequential lower semicontinuity with respect to W^1,p-weak convergence and W^1,p-weak coercivity, and the target is a multiplicity bound on the number of such minimizers in terms of convenient topological invariants of the manifolds and .In the first part of the paper, we focus on the case where is non-contractible and proceed by establishing a link between the latter problem and the question of enumeration of homotopy classes of continuous maps from various skeleta of into . As this in turn can be tackled by the so-called obstruction method, it is evident that our results in this direction are of a cohomological nature.The second part is devoted to the case where =^N and is a bounded smooth domain. In particular we consider integralswhere the above assumptions on can be verified when the integrand F is quasiconvex and pointwise p-coercive with respect to the gradient argument. We introduce and exploit the notion of a topologically non-trivial domain and under this establish the first existence and multiplicity result for strong local minimizers of that in turn settles a longstanding open problem in the multi-dimensional calculus of variations as described in [6].     

Where is the rudder of a fish？： the mechanism of swimming and control of self-propelled fish school

Numerical simulation and control of self- propelled swimming of two- and three-dimensional biomimetic fish school in a viscous flow are investigated. With a parallel computational fluid dynamics package for the two- and three-dimensional moving boundary problem, which combines the adaptive multi-grid finite volume method and the methods of immersed boundary and volume of fluid, it is found that due to the interactions of vortices in the wakes, without proper control, a fish school swim with a given flap- ping rule can not keep the fixed shape of a queue. In order to understand the secret of fish swimming, a new feedback con- trol strategy of fish motion is proposed for the first time, i,e., the locomotion speed is adjusted by the flapping frequency of the caudal, and the direction of swimming is controlled by the swinging of the head of a fish. Results show that with this feedback control strategy, a fish school can keep the good order of a queue in cruising, turning or swimming around circles. This new control strategy, which separates the speed control and direction control, is important in the construction of biomimetic robot fish, with which it greatly simplifies the control devices of a biomimetic robot fish.     

The Mechanical Properties of the Slurry and the Resistance Force of a Sphere Moving With Uniform Velocity in the Slurry

It is commonly considered that the mechanical properties of the slurryare different from that of ordinary Newtonian fluid,and can be describedby that of Bingham fluid.Hence its shearing stress should be described bythe formula of the shearing stress of Bingham fluid.But the author holdsthe contrary opinion and firmly believes that the slurry is a highly viscousfluid with very long relaxation time such as those of asphalt,glass,etc.In this article,we have discussed the mechanical properties of the slurryand the reslstance of a sphere moving with uniform veloclty in the slurry.In the process of discussion.we use Stokes solution of the vlscous fluidpassing around a sphere.When the sphere is in equilibrium under theaction of gravitational force,the force of buoyancy and the resistance,we get the velocity of sedimentation.When the velocity of sedimentationis equal to zero,we get the relation between the yield stress of Binghamfluid and the diameter of the particles which will not sink.The theoreticalresults     

Effect of the Stone–Wales defect on the structure and mechanical properties of single-wall carbon nanotubes in axial stretch and twist

The effect of the Stone–Wales defect due to the rotation of a pair of neighboring atoms on the equilibrium structure and mechanical properties of single-wall carbon nanotubes in axial stretch and twist is considered. The position of carbon atoms in a test section consisting of a number of repeated units hosting a solitary Stone–Wales defect is computed by minimizing the Tersoff–Brenner potential. The energy invested in the defect is found to decrease as the radius of the nanotube becomes smaller. Numerical computations for nanotubes with zigzag and armchair chiralities show that inclined, axial, and circumferential defect orientations have a strong influence on the mechanical response in axial stretch and twist. Stretching may cause the defect energy to become negative, revealing the possibility of spontaneous defect formation leading to failure. In some cases, stretching may eliminate the defect and purify the nanotube. When the tube is twisted around its axis, a neck develops at the location of the defect, signaling possible disintegration.     

Estimate of the Latent Free Energy and Damage at the Tip of an Opening–Mode Crack

A method of estimating the latent elastic energy associated with the microinhomogeneity of the stress and plastic–strain fields inside the plastic zone localized near the tip of an opening–mode crack (Dugdale zone) under conditions of plane stresses is proposed. The microinhomogeneity of plastic flow upon small strain hardening is taken into account only in the form of considerable distortion of the geometry of the free surfaces of the plastic zone. The damage that developes because of release of the latent free energy is estimated depending on the magnitude of the crack opening.     

The Symplectic Evans Matrix,¶and the Instability of Solitary Waves and Fronts

Hamiltonian evolution equations which are equivariant with respect to the action of a Lie group are models for physical phenomena such as oceanographic flows, optical fibres and atmospheric flows, and such systems often have a wide variety of solitary-wave or front solutions. In this paper, we present a new symplectic framework for analysing the spectral problem associated with the linearization about such solitary waves and fronts. At the heart of the analysis is a multi-symplectic formulation of Hamiltonian partial differential equations where a distinct symplectic structure is assigned for the time and space directions, with a third symplectic structure – with two-form denoted by Ω– associated with a coordinate frame moving at the speed of the wave. This leads to a geometric decomposition and symplectification of the Evans function formulation for the linearization about solitary waves and fronts. We introduce the concept of the symplectic Evans matrix, a matrix consisting of restricted Ω-symplectic forms. By applying Hodge duality to the exterior algebra formulation of the Evans function, we find that the zeros of the Evans function correspond to zeros of the determinant of the symplectic Evans matrix. Based on this formulation, we prove several new properties of the Evans function. Restricting the spectral parameter λ to the real axis, we obtain rigorous results on the derivatives of the Evans function near the origin, based solely on the abstract geometry of the equations, and results for the large |λ| behaviour which use primarily the symplectic structure, but also extend to the non-symplectic case. The Lie group symmetry affects the Evans function by generating zero eigenvalues of large multiplicity in the so-called systems at infinity. We present a new geometric theory which describes precisely how these zero eigenvalues behave under perturbation. By combining all these results, a new rigorous sufficient condition for instability of solitary waves and fronts is obtained. The theory applies to a large class of solitary waves and fronts including waves which are bi-asymptotic to a nonconstant manifold of states as $|x|$ tends to infinity. To illustrate the theory, it is applied to three examples: a Boussinesq model from oceanography, a class of nonlinear Schr?dinger equations from optics and a nonlinear Klein-Gordon equation from atmospheric dynamics. Accepted August 7, 2000 ?Published online January 22, 2001     

“Non-free” interaction of plane shock waves and the boundary layer in the neighborhood of the leading edge of a plate with slip

The interaction between a normally impinging shock wave and the boundary layer on a plate with slip is studied in the neighborhood of the leading edge using various experimental methods, including special laser technology, to visualize the supersonic conical gas flows. It is found that in the “non-free” interaction, when the leading edge impedes the propagation of the boundary layer separation line upstream, the structure of the disturbed flow is largely identical to that in the developed “free” interaction, but with higher parameter values and gradients in the leading part of the separation zone. The fundamental property of developed separation flows, namely, coincidence of the values of the pressure “plateau” in the separation zone and the pressure behind the oblique shock above the separation zone of the turbulent boundary layer, is conserved. Moscow. e-mail: ostap@inmech.msu.su. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 57–69, May–June, 2000. The work was carried out with financial support from the Russian Foundation for Basic Research (project No. 97-01-00099).     

The General Mathematical Theory of Plasticity and the Il’yushin Postulates of Macroscopic Definability and Isotropy

The physical laws characterizing the relation between stresses and strains are considered and analyzed in the general modern theory of elastoplastic deformations and in its postulates of macroscopic definability and isotropy for initially isotropic continuous media. The fundamentals of this theory in continuum mechanics were developed by A.A. Il’yushin in the mid-twentieth century. His theory of small elastoplastic deformations under simple loading became a generalization of Hencky’s deformation theory of flow, whereas his theory of elastoplastic processes which are close to simple loading became a generalization of the Saint-Venant–Mises flow theory to the case of hardening media. In these theories, the concepts of simple arid complex loading processes arid the concept of directing form change tensors are introduced; the Bridgman law of volume elastic change and the universal Roche–Eichinger laws of a single hardening curve under simple loading are adopted; and the Odquist hardening for plastic deformations is generalized to the case of elastoplastic hardening media for the processes of almost simple loading without consideration of a specific history of deformations for the trajectories with small arid mean curvatures. In this paper we discuss the possibility of using the isotropy postulate to estimate the effect of forming parameters in the stress-strain state appeared due to the strain-induced anisotropy during the change of the internal structures of materials. We also discuss the possibility of representing the second-rank symmetric stress and strain tensors in the form of vectors in the linear coordinate six-dimensional Euclidean space. An identity principle is proposed for tensors and vectors.     

Complete Characterization and Synthesis of the Response Function of Elastodynamic Networks

The response function of a network of springs and masses, an elastodynamic network, is the matrix valued function W(ω), depending on the frequency ω, mapping the displacements of some accessible or terminal nodes to the net forces at the terminals. We give necessary and sufficient conditions for a given function W(ω) to be the response function of an elastodynamic network, assuming there is no damping. In particular we construct an elastodynamic network that can mimic a suitable response in the frequency or time domain. Our characterization is valid for networks in three dimensions and also for planar networks, which are networks where all the elements, displacements and forces are in a plane. The network we design can fit within an arbitrarily small neighborhood of the convex hull of the terminal nodes, provided the springs and masses occupy an arbitrarily small volume. Additionally, we prove stability of the network response to small changes in the spring constants and/or addition of springs with small spring constants.     

Investigation of the influence of inner-shell photoionization and photoexcitation on the Heα spectrum in photoionized plasmas

The present paper investigates the influence of inner shell photoionization and photoexcitation on the He_α, i.e., the 1s² to 1s2p transition in He-like ions, and the associated satellite spectra in photoionized plasmas. A comparison of the importance of these processes is made relative to other atomic processes as a function of the electron temperature and irradiation conditions. For the formation of the He_α and the satellite spectra, the K-shell photoionization is found to have significant contribution under low radiation temperature and/or intensity, when lithium- and beryllium-like ions have high abundance, but highly ionized H-like ions are rare.     

Results of an experimental and numerical study of the aerodynamic heating of the undersurface of delta wings with sharp leading edges at mach numbers M∞=6.1 and 8

The heat transfer between a supersonic flow and the undersurface of delta wings with leading-edge sweep angles x=65 and 70° is investigated in a shock tunnel at angles of attack 15°. The supersonic inviscid flow over these wings in regimes in which the bow shock is attached to the sharp leading edges is calculated numerically. The compressible boundary layer problem is solved for the calculated inviscid flow fields in the laminar, transition and turbulent flow zones. The calculations and experimental values of the heat flux on the surface of the wings are compared. The calculations are in satisfactory agreement with the experimental data in the laminar and transition zones, but diverge significantly (by up to 20%) in the turbulent zone.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 183–188, July–August, 1991.The authors wish to thank A. A. Golubinskii for assisting with the solution of the problem of supersonic inviscid gas flow over a wing.     

A Physical Model of the Structure and Attenuation of Shock Waves in Metals

In this paper,a physical model of the structure and attenuation of shockwaves in metals is presented.In order to establish the constitutive equa-tions of materials under high velocity deformation and to study the struc-ture of transition zone of shock wave.two independent approaches are in-volved.Firstly,the specific internal energy is decomposed into the elasticcompression energy and elastic deformation energy,and the later is represent-ed by an expansion to third-order terms in elastic strain and entropy.includ-ing the coupling effect of heat and mechanical energy.Secondly,a plasticrelaxation function describing the behaviour of plastic flow under high tem-perature and high pressure is suggested from the viewpoint of dislocationdynamics.In addition.a group of ordinary differential equations has beenbuilt to determine the thermo-mechanical state variables in the transitionzone of a steady shock wave and the thickness of the high pressure shockwave.and an analytical solution of the equations can be foun