SOME SOLUTIONS OF MAXWELL EQUATIONS

This article gives a class of solution of Maxwellequations which implies the solution of arbitrary ho-lomorphic hypercomplex functional with brief discus-sion of their physical meaning.It exhibites somewider nature than classical electromagnetic theory:(l)An electromagnetic field may spread non-periodically and may have the holographic nature andfree shape.(2)The electromagnetic field may spread more slow-ly,equally or more quickly than light speed.(3)The electromagnetic field may spread in arbi-trary curved direction.(4)There is a branch electromagnetic field in in-tergrowth space.An intergrowth world may exist.This article also makes an inference and a descrip-tion about the example of the above-mentioned strangeelectromagnetic field.It relates to some hard-to-explain phenomena through different physical models.     

RESPONSE OF PARAMETRICALLY EXCITED DUFFING-VAN DER POL OSCILLATOR WITH DELAYED FEEDBACK

The dynamical behaviour of a parametrically excited Duffing-van der Pol oscillator under linear-plus-nonlinear state feedback control with a time delay is concerned. By means of the method of averaging together with truncation of Taylor expansions, two slow-flow equations on the amplitude and phase of response were derived for the case of principal parametric resonance. It is shown that the stability condition for the trivial solution is only associated with the linear terms in the original systems besides the amplitude and frequency of parametric excitation. And the trivial solution can be stabilized by appreciate choice of gains and time delay in feedback control. Different from the case of the trivial solution, the stability condition for nontrivial solutions is also associated with nonlinear terms besides linear terms in the original system. It is demonstrated that nontrivial steady state responses may lose their stability by saddle-node (SN) or Hopf bifurcation (HB) as parameters vary. The simulations, obtained by numerically integrating the original system, are in good agreement with the analytical results.     

RECURRENT NEURAL NETWORK MODEL BASED ON PROJECTIVE OPERATOR AND ITS APPLICATION TO OPTIMIZATION PROBLEMS

The recurrent neural network (RNN) model based on projective operator was studied. Different from the former study, the value region of projective operator in the neural network in this paper is a general closed convex subset of n-dimensional Euclidean space and it is not a compact convex set in general, that is, the value region of projective operator is probably unbounded. It was proved that the network has a global solution and its solution trajectory converges to some equilibrium set whenever objective function satisfies some conditions. After that, the model was applied to continuously differentiable optimization and nonlinear or implicit complementarity problems. In addition, simulation experiments confirm the efficiency of the RNN.     

DISCUSSION ON THE SIF FOR POINTS ON BORDER OF ELLIPTICAL FLAT CRACK INSIDE INFINITE SOLID UNDER UNIFORM TENSION

Using the results of crack surface displacement field in Green-Sneddon’s solution and coordinate transformation,this paper hasderived an expression K_1(x_1,z_1,α)for SIF at any point and at anyorientation on the border of elliptical flat crack inside infinitesolid under uniform tension.As a complement of Irwin’s work,it is shown that for any pointed point on the elliptical border theSIF defined on normal plane takes the maximum value.And it shouldbe pointed out that in some works some idea concerning Irwin’s con-tents is open to question.An expression K_I in terms of polarangle which is more intuitional than centrifugal angle is proposedfor SIF at any point on the elliptical border.     

STABILITY OF SYSTEM OF TWO-DIMENSIONAL NON-HYDROSTATIC REVOLVING FLUIDS

Applying the theory of stratification, it is proved that the system of the two-dimensional non-hydrostatic revolving fluids is unstable in the two-order continuous function class. The construction of solution space is given and the solution approach is offered. The sufficient and necessary conditions of the existence of formal solutions are expressed for some typical initial and boundary value problems and the calculating formulae to formal solutions are presented in detail.     

AN ANALYTICAL SOLUTION OF G. I. TAYLOR’S THEORY OF PLASTIC DEFORMATION IN IMPACT OF CYLINDRICAL PROJECTILES AND ITS IMPROVEMENT

The theory of plastic deformation in the impact of cylindrical projectiles on rigid targets was first introduced by G. I. Taylor(1948)^[1]. The importance of this theory lies in the fact that the dynamic yield strength of the materials can be determined from the measurement of the plastic deformation of flat-ended cylindrical projectiles. From the experimental results^[2] we find that the dynamic yield strength is independent of impact velocity, and that it is higher than the static yield strength in general, and several times higher than the static yield strength in certain cases. This gives an important foundation for the study of elastoplastic impact problems in general. However, it is well known that the complexity of differential equations in Taylor’s theory compelled us to use the troublesome numerical solution. In this paper, the analytical solution of all the equations in Taylor’s theory is given in parametrical form and the results are discussed in detail.In the latter part of this paper, the method of calculation of impulse of impact is improved by considering the processes of radial’ movement of materials. The analytical solution of the improved theory shows that it gives better agreement with the experimental results than that of original Taylor’s theory.     

A GASEOUS SHOCK WAVE THEORY OF THE GALACTIC SPIRAL STRUCTURE

In this paper, the galactic spiral structure is studied bythe galactic shock wave of interstellar gas with self-gravita-tion. The perturbed gravitation of stars is not a necessarycondition for the existence of such Shock.It is proved firstof all that there exists solution of local Shock wave even ifthe perturbed gravitation is absent.The condition │ω_(no)│>a isrequired for such solution. The spiral structure can onlg beexplained by the shock solution when the difference of densitybetween the regions of arm and interarm is larger. The granddesign of shock wave with self-gravitation is obtained by theiterative method.The features of shock wave can be analyzedqualitatively in the velocity plane for a special perturbedgravitation Which is used to simulate the self-gravitation.ofinterstellar gas.As the mass distribution in proto-galacticdisk is irregular initially.the grand design of the galacticShock wave was developed by the processes of Winding,growth ofinstability and overlapping of waves.Hence.it giv     

NEW THEORY FOR EQUATIONS OF NON-FUCHSIAN TYPE——REPRESENTATION THEOREM OF TREE SERIES SOLUTION (Ⅱ)

Our main result consists in proving the representation theorem. Irregular integral is a new type of analytic function, represented by a compound Taylor-Fourier tree series, in which each coefficient of the Fourier series is a Taylor series, while the Taylor coefficients are tree series in terms of equations parameters, higher order correction terms to each coefficient having tree structure with inexhaustible proliferation.The solution obtained is proved to be convergent absolutely and uniformly in the region defined by coefficient functions of the original equation, provided the structure parameter is less than unity. Direct substitution shows that our tree series solution satisfies the equation explicity generation by generation.As compared with classical theory our method not only furnishes explicit expression of irregular integral, leading to the solution of Poincare problem, but also provides possibility of extending the scope of investigation for analytic theory to equations with various kinds of singularities in a unifying way.Exact explicit analytic expression for irregular integrals can be obtained by means of correspondence principle.It is not difficult to prove the convergence of the tree series solution obtained. Direct substitution shows it satisfies the equation.The tree series is automorphic, which agrees completely with Poincaré’s conjecture.     

THE SCHRDINGER EQUATION OF THIN SHELL THEORIES

This work is the continuation of the discussions of[50]and[51].In this paper:(A)The Love-Kirchhoff equation of small deflection problem for elastic thin shellwith constant curvature are classified as the same several solutions of Schr(?)dingerequation,and we show clearly that its form in axisymmetric problem;(B)For example for the small deflection problem,we extract the general solution ofthe vibration problem of thin spherical shell with equal thickness by the force in centralsurface and axisymmetric external field,that this is distinct from ref.[50]in variable.Today the variable is a space-place,and is not time;(C)The von Kármán-Vlasov equation of large deflection problem for shallow shellare classified as the solutions of AKNS equations and in it the one-dimensional problem isclassified as the solution of simple Schr(?)dinger equation for eigenvalues problem,and wetransform the large deflection of shallow shell from nonlinear problem into soluble linearproblem.     

A note on general solution of Stokes equations

Li et al. (2015) claim that it is sufficient to use two harmonic functions to express the general solution of Stokes equations. In this paper, we demonstrate that this is not true in a general case and that we in fact need three scalar harmonic functions to represent the general solution of Stokes equations (Venkatalaxmi et al., 2004).     

Existence of the minimal positive solution of some nonlinear elliptic systems when the nonlinearity is the sum of a sublinear and a superlinear term

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The value of the critical exponent for reaction-diffusion equations in cones

Let D R ^N be a cone with vertex at the origin i.e., D = (0, )x where S ^N–1 and x D if and only if x = (r, ) with r=¦x¦, . We consider the initial boundary value problem: u _t = u+u ^p in D×(0, T), u=0 on Dx(0, T) with u(x, 0)=u ₀(x) 0. Let ₁ denote the smallest Dirichlet eigenvalue for the Laplace-Beltrami operator on and let ₊ denote the positive root of (+N–2) = ₁. Let p ^* = 1 + 2/(N + ₊). If 1 < p < p ^*, no positive global solution exists. If p>p ^*, positive global solutions do exist. Extensions are given to the same problem for u _t=+¦x¦ u ^p .This research was supported in part by the Air Force Office of Scientific Research under Grant # AFOSR 88-0031 and in part by NSF Grant DMS-8 822 788. The United States Government is authorized to reproduce and distribute reprints for governmental purposes not withstanding any copyright notation therein.     

Nonlinear Eigenvalues for a Quasilinear Elliptic System in Orlicz–Sobolev Spaces

Using an Orlicz–Sobolev Space setting, we consider an eigenvalue problem for a system of the form

Wärmeübergang bei erzwungener turbulenter Strömung in Ringspalten und örtlich beliebig veränderlichen Randbedingungen zweiter Art

Zusammenfassung Das Problem des Wärmeübergangs bei turbulenter Strömung in konzentrischen Ringspalten wird für den dreidimensionalen Fall theoretisch gelöst, wobei die Wandwärmestromdichte sowohl in azimutaler als auch in axialer Richtung beliebig variiert. Die Lösung der Energiegleichung erfolgt mit der klassischen Methode der Superposition und Trennung der Variablen, wobei das dabei auftretende Sturm-Liouvillesche Eigenwertproblem numerisch gelöst wird. Zur Lösung werden Verteilungen für die Geschwindigkeit und anisotropen turbulenten Austauschgrößen verwendet, die mit dem phänomenlogischen Turbulenzmodell von Ramm berechnet wurden. Ergebnisse werden über einen weiten Bereich der Reynolds-Zahl (10⁴ Re 10⁶), der Prandtl-Zahl (0 Pr 100) und für verschiedene Radienverhältnisse diskutiert.

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Turbulent forced convection heat transfer in annuli with arbitrarily varying boundary conditions of second kind

The problem of turbulent flow heat transfer in concentric annuli is analysed for the general threedimensional case in which the wall heat flux varies arbitrarily in both the circumferential and axial directions. The energy equation is solved using the classical method of superposition and separating variables, where the resulting Sturm-Liouville problem are evaluated numerically. The solution is based on velocity profiles and anisotropic thermal turbulent transport properties evaluated by Ramm's phenomenological turbulence model. Results are discussed over a wide range of Reynolds number (10⁴ Re 10⁶), Prandtl number (0 Pr 100) and radius ratio.

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Bezeichnungen a,b Fourierkoeffizienten - B geometrische Funktion, [s(1-r) + r]/(1–s) - C Koeffizienten - D hydraulischer Durchmesser, 2(r₂ – r₁) - E Energietransportfunktion - f axiale Wärmestromdichteverteilung - F azimutale Wärmestromdichteverteilung - g radiale Temperaturfunktion - l Kanallänge - L dimensionslose Kanallänge, 1/D - M axialer Temperaturgradient im thermisch ausgebildeten Bereich - n harmonischer Parameter - Nu Nusselt-Zahl - Pe Péclet-Zahl - Pr Prandtl-Zahl - q Wärmestromdichte - Q dimensionslose Wärmestromdichte, q/q₀ - r dimensionslose radiale Koordinate, (R-r₁)/(r₂-r₁) - r₁,r₂ innerer und äußerer Ringspaltradius - R radiale Koordinate - Re Reynolds-Zahl - s Ringspaltverhältnis, r₁/r₂ - T dimensionslose Temperatur, 2· · (-_E/(D· q₀ - u dimensionslose Geschwindigkeit, U/U_m - U Geschwindigkeit - x dimensionslose axiale Koordinate, X/D - X axiale Koordinate - Wärmeübergangskoeffizient - _un modifizierter Eigenwert - halber Segmentwinkel - turbulente Austauschgröe - Temperatur - dimensionslose Temperaturdifferenz, T - T_m - Wärmeleitfähigkeit - _un Eigenwerte - kinematische Viskosität - azimutale Koordinate - Eigenfunktionen Indizes e thermischer Einlauf - E Eintritt bei x=0 - H Wärme - i Bedingung an der i-ten benetzten Oberfläche (i=1 – Innenrohr, i=2 - Außenrohr) - j Bedingung, wenn nur an der j-ten Oberfläche des Ringspaltes die Wärme übertragen wird (j=1,2) - ij Bedingung an der i-ten Oberfläche, wenn nur an der j-ten Oberfläche des Ringspaltes die Wärme übertragen wird (ij=11, 12, 22, 21) - m mittel - n Ordnung der Harmonischen - r radiale Richtung - u Ordnung des Eigenwertproblems - azimutale Richtung - 0 umfangskonstant - thermisch ausgebildet     

Existence and Multiplicity of Positive Solutions for a Fourth-Order P-Laplace Equations

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The Preston tube in adiabatic compressible flow

A new procedure for the reduction of Preston tube data is introduced, based on the van Driest transformation. It appears to give results agreeing with the better calibration experiments, although a significant assumption in its derivation is violated.List of Symbols M _s Mach number sensed by Preston tube - M Friction Mach number (=u/wall sound speed) - R Gas constant - T _w Wall temperature - d Diameter of Preston tube - h Height of effective centre of Preston tube - p Preston tube pressure difference reading - p _i Equivalent incompressible Preston tube reading - p _w Wall pressure - r Recovery factor (=0.896) - u Friction velocity (=[_w/wall density]^1/2) - Empirical constant allowing for departure from Crocco temperature-velocity correlation (=0.975) - Specific heat ratio - Fluid kinematic viscosity - _w Wall shear-stress     

On the problem of preventing blowing-up and quenching for semilinear heat equation

In this paper,the global existence of solutions to the IVP=Δu+g(t)f(u) (t>0),u|_(t=0)=u_0(x)and the (?)PVPu_t=Δu-g(t,x)f(u)(t>0,x∈Ω),u|_(t-0)=u|_(?)(?)is investigated. As has been done in [6]the (?)duction of factor g(t) or g(t.x) innonlinear term is to prevent(?) occurrance of blowing-up or quenching of solutions.It isshown in this paper that most of the restrictions on f,g and u_0 in the theorems of[6] maybe cancelled or relaxed,that the smallness of g is required only for t large,and thatunder certain conditions controlling initial state can avoid blowing-up.     

Oscillating convection modes in the surroundings of an air bubble under a horizontal heated wall

The development of different oscillatory modes and their transition into a non-periodic state of convection, initiated by the thermal Marangoni-effect in the vicinity of an air bubble under a horizontal, heated wall, was investigated. In the further surroundings of the air bubble a stably stratified thermal field was maintained. The flow phenomena in the vicinity of the bubble were studied using light sheet and shearing interferometer flow visualization techniques. The observed modes are described with regard to their kinematics. The influence of the Marangoni number and of the bubble geometry on the mode selection is discussed. The boundaries of the different modes and of the non-periodic state are indicated.List of symbols a thermal diffusivity - Bo Bond number, Eq. (4) - c phase velocity, Eq. (6) - g acceleration due to gravity - l characteristic length - Mg Marangoni number, Eq. (1) - n wavenumber - Pr Prandtl number ( = v/a) - r radial coordinate - r _B bubble radius - Ra Rayleigh number ( = ga¦T/r¦l ⁴/va) - Re Reynolds number ( = u _mg l/) - t _p oscillation period - T temperature - T _w wall temperature - u _mg characteristical Marangoni velocity, Eq. (2) - z axial coordinate normal to the heated wall - z _B bubble height Greek letters surface tension - kinematic viscosity - dynamic viscosity Dedicated to Professor Dr.-Ing. Julius Siekmann on the occasion of his 65th birthday     

Numerical studies of slow viscous rotating flow past a sphere—1

The Navier-Stokes equations for a steady, viscous rotating fluid, rotating about the z-axis with angular velocity ω are linearized using the Stokes approximation. The linearized Navier-Stokes equations governing the axisymmetric flow can be written as three coupled partial differential equations for the stream function, vorticity and rotational velocity components. One parameter, R_eω = 2ωa²/v, enters the resulting equations. For R_eω « 1, the coupled equations are solved by the Peaceman-Rachford A.D.I. (Alternating Direction Implicit) method and the resulting algebraic equations are solved by the ‘method of sweeps’. Stream lines for ψ = 0·05, 0·2, 0·5 and magnitude of the vorticity vector z = 0·2 are plotted for R_eω = 0·1, 0·3, 0·5. Correction to the Stokes drag due to the rotation of fluid is calculated.     

Lp-estimates for the nonlinear spatially homogeneous Boltzmann equation

This paper studies L^p-estimates for solutions of the nonlinear, spatially homogeneous Boltzmann equation. The molecular forces considered include inverse k^th-power forces with k > 5 and angular cut-off.The main conclusions are the following. Let f be the unique solution of the Boltzmann equation with f(v,t)(1 + ¦v²¦)^(s ₁ ^{+ /p)/2} L¹, when the initial value f ₀ satisfies f ₀(v) 0, f ₀(v) (1 + ¦v¦²)^(s ₁ ^{+ /p)/2} L¹, for some s₁ 2 + /p, and f ₀(v) (1 + ¦v¦²)^s/2 L^p. If s 2/p and 1 < p < , then f(v, t)(1 + ¦v¦²)^{(s s} ₁)^/2 L^p, t > 0. If s >2 and 3/(1+ ) < p < , thenf(v,t) (1 + ¦v¦²)^(s(s ₁ ^{+ 3/p))/2} L^p, t > 0. If s >2 + 2C₀/C₁ and 3/(l + ) < p < , then f(v,t)(1 + ¦v¦²)^s/2 L^p, t > 0. Here 1/p + 1/p = 1, x y = min (x, y), and C₀, C₁, 0 < 1, are positive constants related to the molecular forces under consideration; = (k – 5)/ (k – 1) for k^th-power forces.Some weaker conclusions follow when 1 < p 3/ (1 + ).In the proofs some previously known L-estimates are extended. The results for L^p, 1 < p < , are based on these L-estimates coupled with nonlinear interpolation.