Convergence of biorthogonal series of biharmonic eigenfunctions by the method of titchmarsh

Canonical edge problems for the biharmonic equation can be solved by separating variables. The eigenvalues and eigenvectors arising in this separation are derived from a reduced system of ordinary differential equations along lines suggested in the excellent work of R. C. Smith (1952). We study the reduced system which is governed by a vector ordinary differential equation. A solution of the biharmonic problem, governed by a partial differential equation, can be found only if the prescribed data is restricted to a subspace of the space spanned by the eigenfunctions of the reduced problem. The theory leads to problems in generalized harmonic analysis which seek conditions under which arbitrary vector fields f(y) with values in ² can be represented in terms of eigenvectors of the reduced problem. This paper adds new theorems and conjectures to the theory. We extend Smith's generalization to fourth-order problems of the methods introduced by Titchmarsh (1946) to study eigenfunction expansions associated with second-order problems. We use this method to prove that, if f(y)=[(f ₁(y), f ₂y)], -1y1, f(y) C¹[-1, 1], f L₂[-1, 1], then the series expressing f(y) converges uniformly to f(y) in the open interval (-1, 1), uniformly in [-1, 1] if f ₁(±1)=0 and, in any case, to [0, f ₂(±1)-f ₁(±1)] at y=±1. This is unlike Fourier series, which converge to the mean value of the periodic extension of a function. The series exhibits a Gibbs phenomenon near the end points of discontinuity when f ₁(±1) 0.The Gibbs undershoot and overshoot for the step function vector [1, 0] and ramp function vector [y, 0] are computed numerically. The undershoot and overshoot are much larger than in the case of Fourier series and, unlike Fourier series, the Gibbs oscillations do not appear to be entirely suppressed by Féjer's method of summing Cesaro sums. We show that, when f(y) has interior points of discontinuity, the series for f(y) diverges and we present numerical results which indicate that, in this divergent case, the Cesaro sums converge to f(y) apparently with Gibbs oscillations near the point of discontinuity.     

Solitary waves at the interface of a two-layer fluid

In this paper, we discuss the solitary waves at the interface of a two-layer incompressible inviscid fluid confined by two horizontal rigid walls, taking the effect of surface tension into account. First of all, we establish the basic equations suitable for the model considered, and hence derive the Korteweg-de Vries (KdV) equation satisfied by the first-order elevation of the interface with the aid of the reductive perturbation method under the approximation of weak dispersion. It is found that the KdV solitary waves may be convex upward or downward. It depends on whether the signs of the coefficients and of the KdV equation are the same or not. Then we examine in detail two critical cases, in which the nonlinear effect and the dispersion effect cannot balance under the original approximation. Applying other appropriate approximations, we obtain the modified KdV equation for the critical case of first kind (=0), and conclude that solitary waves cannot exist in the case considered as >0, but may still occur as <0, being in the form other than that of the KdV solitary wave.As for the critical case of second kind (=0), we deduce the generalized KdV equation, for which a kind of oscillatory solitary waves may occur. In addition, we discuss briefly the near-critical cases. The conclusions in this paper are in good agreement with some classical results which are extended considerably.     

An elementary proof of the polar factorization of vector-valued functions

We present an elementary proof of an important result of Y. Brenier [Br1, Br2], namely, that vector fields in ^d satisfying a nondegeneracy condition admit the polar factorization (*) u(x)=(s(x)), where is a convex function and s is a measure-preserving mapping. Brenier solves a minimization problem using Monge-Kantorovich theory; whereas we turn our attention to a dual problem, whose Euler-Lagrange equation turns out to be (*).     

Remarks on the bifurcation of solutions of a nonlinear eigenvalue problem

Conclusions We have investigated solutions of equation (3) when ² is an eigenvalue of the linearized operator (13) and when it is not. In Section 4 we have shown that for 0 and ² = _i ² we have exactly two nontrivial solutions which bifurcate to the right of _i ² ; these solutions are shown to exist in an interval ( _i ² , _i ² + ₀). The method of Section 3 may then be used to extend these two solutions to the right of _i ² + ₀ providing that ²= _i ² + ₀ is not an eigenvalue of the linear operator (13) evaluated at = ± ₁. Either a solution can be uniquely extended, or there exists a value of ²where the bifurcation method must be applied again³.While the method used here gives the exact number of solutions bifurcating from _i ² , other problems remain open; for example, it is still not proven that the two bifurcating branches have i zeros, as is the case for Hammerstein operators with oscillation kernels [4]. The conjecture of Odeh and Tadjbakhsh that there are exactly 2(i+1) nontrivial solutions in the interval _i ² < _i ^+1/2 remains un-answered, although it would be proven if one could show that there is no secondary bifurcation as in the cases of Kolodner [7] and Coffman [8].     

A unified intrinsic functional expansion theory for solitary waves

A new theory is developed here for evaluating solitary waves on water, with results of high accuracy uniformly valid for waves of all heights, from the highest wave with a corner crest of 120&#176;down to very low ones of diminishing height. Solutions are sought for the Euler model by employing a unified expansion of the logarithmic hodograph in terms of a set of intrinsic component functions analytically determined to represent all the intrinsic properties of the wave entity from the wave crest to its outskirts. The unknown coefficients in the expansion are determined by minimization of the mean-square error of the solution, with the minimization optimized so as to take as few terms as needed to attain results as high in accuracy as attainable. In this regard, Stokes‘s formula, F^2μπ= tanμπ, relating the wave speed (the Froude number F) and the logarithmic decremen t# of its wave field in the outskirt, is generalized to establish a new criterion requiring (for minimizing solution error) the functional expansion to contain a finite power series in M terms of Stokes‘s basic term (singular in #), such that 2Mμ is just somewhat beyond unity, i.e. 2Mμ≌ 1. This fundamental criterion is fully validated by solutions for waves of various amplitude-to-water depth ratio α= a/h, especially about α≌0.01, at which M = 10by the criterion.In this pursuit, the class of dwarf solitary waves, defined for waves with ≌≤ 0.01, is discovered as a group of problems more challenging than even the highest wave. For the highest wave, a new solution is determined here to give the maximum height αhst = 0.8331990, and speed Fhst = 1.290890, accurate to the last significant figure, which seems to be a new record.     

Some problems of nonisothermal steady flow of a viscous fluid

The paper presents solutions to the problems of plane Couette flow, axial flow in an annulus between two infinite cylinders, and flow between two rotating cylinders. Taking into account energy dissipation and the temperature dependence of viscosity, as given by Reynolds's relation =₀ exp (–T) (₀, =const). Two types of boundary conditions are considered: a) the two surfaces are held at constant (but in general not equal) temperatures; b) one surface is held at a constant temperature, the other surface is insulated.Nonisothermal steady flow in simple conduits with dissipation of energy and temperature-dependent viscosity has been studied by several authors [1–11]. In most of these papers [1–6] viscosity was assumed to be a hyperbolic function of temperature, viz. =_m 1/1+²(T–T_m.Under this assumption the energy equation is linear in temperature and can he easily integrated. Couette flow with an exponential viscosity-temperature relation. =₀ e ^–T (₀, =const), (0.1) was studied in [7, 8]. Couette flow with a general (T) relation was studied in (9).Forced flow in a plane conduit and in a circular tube with a general (T) relation was studied in [10]. In particular, it has been shown in [10] that in the case of sufficiently strong dependence of viscosity on temperature there can exist a critical value of the pressure gradient, such that a steady flow is possible only for pressure gradients below this critical value.In a previous work [11] the authors studied Polseuille flow in a circular tube with an exponential (T) relation. This thermohydrodynamic problem was reduced to the problem of a thermal explosion in a cylindrical domain, which led to the existence of a critical regime. The critical conditions for the hydrodynamic thermal explosion and the temperature and velocity profiles were calculated.In this paper we treat the problems of Couette flow, pressureless axial flow in an annulus, and flow between two rotating cylinders taking into account dissipation and the variation of viscosity with temperature according to Reynolds's law (0.1). The treatment of the Couette flow problem differs from that given in [8] in that the constants of integration are found by elementary methods, whereas in [8] this step involved considerable difficulties. The solution to the two other problems is then based on the Couette problem.     

Plane entry flows and energy estimates for the navier-stokes equations

One of the classic problems of laminar flow theory is the development of velocity profiles in the inlet regions of channels or pipes. Such entry flow problems have been investigated extensively, usually by approximate techniques. In a recent paper [4], Horgan & Wheeler have provided an alternative approach, based on an energy method for the stationary Navier-Stokes equations. In [4], concerned with laminar flow in a cylindrical pipe of arbitrary cross-section, an analogy is drawn between the end effect issue of concern here, called the end effect, and the celebrated Saint-Venant's Principle of the theory of elasticity.In this paper, I consider the two-dimensional analog of the problem treated in [4] with a view to providing a more explicit formulation of the energy approach to entry flow problems. The flow development in a semi-infinite channel with parallel-plates is analyzed within the framework of the stationary Navier-Stokes equations. Introduction of a stream function leads to a formulation in terms of a boundary-value problem for a single fourth order nonlinear elliptic equation. In the case of Stokes flow, this problem is formally equivalent to a boundary-value problem for the biharmonic equation considered by Knowles [5] in the analysis of Saint-Venant's Principle in plane elasticity. The main result is an explicit estimate which establishes the exponential spatial flow development and leads to an upper bound for an appropriately defined entrance length. These results are obtained using differential inequality techniques analogous to those developed in investigation of Saint-Venant's Principle.     

Stress analysis of plates with a circular hole reinforced by flange reinforcing member

This paper deals with the problem of stress analysis of plates with a circular hole reinforced by flange reinforcing member. The so called flange reinforcing member here means that the reinforcing member is built up by setting shapes or bars with any section shape on both sides of the plates along the edge of the hole. Two cases of external loads are considered. In one case the external loads are stressesσ_X^(∞),σ_Y^(∞),and τ_XY^(∞) acting at infinite point of the plate, and in the other the external loads are linear distributed normal stresses. The procedure of solving the problems mentioned above consists of three steps. Firstly, the reinforcing member is taken out from the plates and considered to be a circular bar being solved to determine its deformation under the action of radial force q₀(θ) and tangential force t₀(θ) which are forces acting upon each other between reinforcing member and plate. Secondly, the displacements of plate with a circular hole under the action of q₀(θ) and t₀(θ) and external loads are determined. Finally, forces q₀(θ) and t₀(θ) are obtained by the compatibility of deformations between reinforcing member and plate. Then the internal forces and displacements of reinforcing member and plate are deduced from q₀(θ) and t₀(θ) obtained.     

A simple variational approach to a converse of the Lagrange-Dirichlet theorem

Communicated by K. Kirchgässner     

Berechnung des dynamischen Verhaltens von instationären Temperaturfeldern in zylindrischen Bauteilen

Zusammenfassung Im ersten Teil dieser Untersuchung wird zur Betrachtung des dynamischen Verhaltens instationärer Temperaturfelder in den Wandungen zylindrischer Rohre ein mathematisches Modell erstellt und mit Hilfe der Laplace-Transformation ausgewertet. Im einzelnen werden dabei die Übertragungsfunktionen der Rohrwandtemperaturen hergeleitet und für den Fall der Abweichung vom stationären Zustand unter dem Einfluß äußerer Störungen explizit dargestellt.Im zweiten Teil der Untersuchung wird das sich daraus ergebende dynamische Verhalten der Wandtemperatur fluiddurchströmter Rohre für einige Beispiele in Form von Ortskurven dargestellt.

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Computation of the dynamic behaviour of unsteady-state temperature fields in cylindrical structures

In the first part of this paper a mathematical model is developed allowing the investigation of unsteady-state temperature fields in the walls of cylindrical pipes. Evaluation is done by means of Laplace-transformation. In particular the transfer function of the pipe wall temperature is derived, explicitly shown for the case of deviations from steady-state influenced by external disturbances. In the second part of this paper the resulting dynamical behaviour of the wall temperature of heat pipes containing a fluid is shown by means of Nyquist plots for several examples.

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Formelzeichen a Temperaturleitzahl m²/sec - A, B, A^*, B^*, , ¯B Integrationskonstanten °C - ber, bei, ker, kei Kelvin-Funktionen - ber₁, bei₁, ker₁, kei₁ kelvin-Funklionerion - Bi Biot-Zahl - c spezifische Wärme kJ/kg K - F Übertragungsfunktion - i –1 (imaginäre Einheit) - I₀, K₀, I₁, K₁ modifizierte Bessel-Funktionen - N Nenner (Gl. (39)) - r Rohrradius m - R normierter Abstand von der Innenwand % - s (komplexe) Laplace-Variable 1/sec - t Zeitvariable sec - T Zeitkonstante sec - u Integrationsvariable (Gl. (15)) - Y₀₀, Y₁₀, Y₁₁ Hilfsfunktionen (Gl. (35)-(37)) - Wärmeübergangszahl kW/K m² - kleine Änderung - Laplace-Operator 1/m² - Umgebungstemperatur °C - Rohrwandtemperatur °C - Wärmeleitfähigkeit kW/K m - Dichte kg/m³ - (komplexe) Kennvariable (Gl. (11)) - Frequenz 1/sec - Variable (Gl. (45)) Indizes a Rohraußenwand - FDS Frischdampfsammelrohr - F Fluid - H Heizgas - i Rohrinnenwand - m Mittel - VD Verdampferrohr - W Rohrwand - 0 zum Zeitpunkt t=t₀ - -(Überstreichung) stationärer Zustand Herrn Prof. Dr.-Ing. R. Quack zum 65. Geburtstag gewidmet.     

Reflection and diffraction of SH waves due to a line source in a heterogeneous medium

The field due to a line source of harmonic SH waves embedded in a semi infinite medium whose density and rigidity vary exponentially with depth is derived in the integral form. The displacement due to diffraction at any point in the shadow zone is obtained and, by the saddle point method of evaluation of the integral, the field at any point in the illuminated region is also found. Finally, geometrical interpretation is given to the different rays arriving in the illuminated as well as in the shadow zone.Nomenclature b shear wave velocity on the free surface - C wave velocity - H _v ⁽¹⁾ (p), H _v ⁽²⁾ (p) Hankel's function of the first and second kind respectively - k Fourier transform parameter with respect to x - v the displacement - fourier transform of v with respect to x - X grazing angle - , small positive constants - positive constant, –/2 - ₀ coefficient of rigidity at the free surface - coefficient of rigidity - _is values of _i at the saddle point (i=1, 2, 3, 4) - the density of the medium - ₀ the density of the medium at the free surface - /2 frequency of vibration     

Finite difference analysis of plane Poiseuille and Couette flow developments

Summary The problem of flow development from an initially flat velocity profile in the plane Poiseuille and Couette flow geometry is investigated for a viscous fluid. The basic governing momentum and continuity equations are expressed in finite difference form and solved numerically on a high speed digital computer for a mesh network superimposed on the flow field. Results are obtained for the variations of velocity, pressure and resistance coefficient throughout the development region. A characteristic development length is defined and evaluated for both types of flow.Nomenclature h width of channel - L ratio of development length to channel width - p fluid pressure - p ₀ pressure at channel mouth - P dimensionless pressure, p/ ² - P ₀ dimensionless pressure at channel mouth - P pressure defect, P ₀–P - (P)₀ pressure defect neglecting inertia - Re Reynolds number, uh/ - u fluid velocity in x-direction - mean u velocity across channel - u ₀ wall velocity - U dimensionles u velocity u/ - U _c dimensionless centreline velocity - U ₀ dimensionless wall velocity - v fluid velocity in y-direction - V dimensionless v velocity, hv/ - x coordinate along channel - X dimensionless x-coordinate, x/h ² - y coordinate across channel - Y dimensionless y-coordinate, y/h - resistance coefficient, - ₀ resistance coefficient neglecting inertia - fluid density - fluid viscosity     

Parametric excitation of waves at an interface

Experiments on the parametric excitation of waves at a fluid interface show a strong disagreement with theoretical results [1–3], since the latter do not take into account the influence of the second medium. This proves to be especially important at low frequencies. Thus, for a water-air interface with an excitation frequency = 60 sec^–1 the contribution amounts to 10%,and with = 30 sec^–1, even 20%. In this paper the stability of the interface of two viscous, incompressible fluids of finite depth in a variable gravity field is considered. The problem is put in the linear form by making an expansion with respect to the small viscosity and is solved by taking the Laplace transform with respect to time. A second-order integrodifferential equation with periodic coefficients is obtained for the deviation of the interface from the equilibrium position; its solution is sought by the method of averaging [4]. It is shown that the presence of the second fluid significantly raises the threshold of instability.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 167–170, March–April, 1977.     

Performance prediction of dual cylindrical wave laser devices for flow measurements

The dual cylindrical wave system is a variant of laser Doppler velocimetry, in which two cylindrical waves of laser light are used to illuminate a moving particle. This instrument is being used for local measurement of the unsteady skin friction in turbulent boundary layers, as well as droplet sizing in spray flows. In the present work, performance of these new devices is examined using the electromagnetic theory of light. Various requirements for the design and operation of these instruments have been further elaborated and extended. The accuracy of the previous experimental results has also been considered. The optics-related errors are shown to be negligible in the measurements of streamwise as well as spanwise wall velocity gradients. However, rigorous simulations appear to be essential for proper calibration of the particle sizing device.List of symbols A, B, C three particle positions - a half-width of an optical slit - a _gm amplitude of a plane wave in the spectrum of a cylindrical wave - d _f fringe spacing - d _p particle diameter - E amplitude of the electric oscillation in the optical field - E _c combined electric field of two cylindrical waves - E _o maximum strength of the electric field at the source of a cylindrical wave - E _s electric field of a scattered wave - E _y time-dependent electric field in the case of electric polarization - f characteristic length for the phase of the scattering amplitude - f _a anisotropic frequency - f _D Doppler frequency - F _DCW transfer function of DCW system for particle sizing - F _pDA Phase Doppler transfer function - g wall velocity gradient - g _m measured wall velocity gradient - I ₀, I₂ integrals in the asymptotic expansion of the scattering amplitude - I _s intensity of the scattered light - k wave number of laser light in the fluid medium - m refractive index of the particle relative to the surrounding medium - N ₀ nominal number of fringes resulting from interference of two cylindrical waves - P phase of a plane wave - P ₁, P₂ phases of plane waves from downstream and upstream cylindrical waves respectively - P _s scattered light power at a receiving aperture - r unit vector in the direction of light scattering - r _D distance of the signal detector from the particle center - S scattering amplitude of a cylindrical wave - S ₁, S₂ Scattering amplitudes of the cylindrical waves emanating from S₁ and S₂ respectively - magnitude of the scattering amplitude for a plane wave - S _c combined scattering amplitude of two cylindrical waves - S₁, S₂ downstream and upstream sources of cylindrical waves, respectively - S scattering amplitude of a plane wave - s half-spacing between sources of the cylindrical waves - t time - u velocity along x-axis - w ₀ 1/e half-width of the field distribution at the waist of a laser sheet - X ₀ nominal width of the fringe volume along the particle path - X particle position in the measuring volume - x, y, z Cartesian coordinates Greek symbols angle of the direction of wave propagation from x-axis - coefficient of the second-order term in the phase function of a cylindrical wave - angular size of the signal receiving aperture - incremental used for numerical differentiation the ratio of to _p - — ₀ integration parameter for I₀ and I₂ - half-angle between the directions of propagation of two waves - wavelength of laser in the fluid medium - ₀ wavelength of laser in vacuum - parameter defining the direction of propagation of a plane wave - 1/e 1/e half-width of the function A - ₀ direction of propagation of the dominant plane wave in the spectrum - _0s the direction of propagation of the plane wave that contributes predominantly to scattering in a particular direction - _p the value of . corresponding to one cycle of P - _s change in corresponding to a lobe of the scattering amplitude - a dimensional form of that determines lobes in the scattered field - signal phase, ₀ + _a - _a anisotropic phase shift - ₀ phase difference between two indicent waves - off-axis angle - elevation angle - circular frequency of laser light     

Reynolds number dependence of the drag coefficient for laminar flow through electrically-heated photoetched screens

Experiments are performed to measure the drag coefficient of electrically-heated screens. Square-pattern 80 mesh and 100 mesh screens of 50.8 m-wide wires photoetched from 50.8 m thick Inconel sheets are examined. Ambient air is passed through these screens at upstream velocities yielding wire-width Reynolds numbers from 2 to 35, and electrical current is passed through the screens to generate heat fluxes from o to 0.17 MW/m², based on the total screen area. The dependence of the drag coefficient on Reynolds number and heat flux is determined for these two screens by measuring pressure drops across the screens for a variety of conditions in these ranges. In all cases, heating is found to increase the drag coefficient above the unheated value. A correlation relating the heated drag coefficient to the unheated drag coefficient is developed based on the idea that the main effect of heating at these levels is to modify the Reynolds number through modifying the viscosity. This correlation is seen to reproduce the experimental results closely.List of Symbols A total screen cross sectional area - C fitting coefficient, near unity - c _D heated drag coefficient - c _{D, 0} unheated drag coefficient - C _p air specific heat at constant pressure - D photoetched wire width, sheet thickness - h _s stagnation point heat-transfer coefficient - k air thermal conductivity - M distance between adjacent wires - O open area fraction - p air pressure - p air pressure drop across screen - Pr Prandtl number for air, c _p/k - Q total electrical power to screen - R radius of curvature at stagnation point - Re _D wire width Reynolds number, UD/ - T air temperature - U air speed upstream of screen - air specific heat ratio - air density - air viscosity - exponent in temperature power law for viscosity - () quantity () evaluated at heated screen temperature The authors thank John Lewin and Bob Meyer for their assistance in the design and fabrication of the heated screen test facility and Tom Grasser for his help in performing the experiments. This work was performed at Sandia National laboratories, supported by the U.S. Department of Energy under contract number DE-AC04-94AL85000.     

On natural convection from a vertical plate with a prescribed surface heat flux in porous media

This paper presents a theoretical and numerical investigation of the natural convection boundary-layer along a vertical surface, which is embedded in a porous medium, when the surface heat flux varies as (1 +x ²)), where is a constant andx is the distance along the surface. It is shown that for > -1/2 the solution develops from a similarity solution which is valid for small values ofx to one which is valid for large values ofx. However, when -1/2 no similarity solutions exist for large values ofx and it is found that there are two cases to consider, namely < -1/2 and = -1/2. The wall temperature and the velocity at large distances along the plate are determined for a range of values of .Notation g Gravitational acceleration - k Thermal conductivity of the saturated porous medium - K Permeability of the porous medium - l Typical streamwise length - q _w Uniform heat flux on the wall - Ra Rayleigh number, =gK(q _w /k)l/(v) - T Temperature - Too Temperature far from the plate - u, v Components of seepage velocity in the x and y directions - x, y Cartesian coordinates - Thermal diffusivity of the fluid saturated porous medium - The coefficient of thermal expansion - An undetermined constant - Porosity of the porous medium - Similarity variable, =y(1+x ) ^/3/x ^1/3 - A preassigned constant - Kinematic viscosity - Nondimensional temperature, =(T – T )Ra^1/3 k/qw - Similarity variable, = =y(log_e x)^1/3/x ^2/3 - Similarity variable, =y/x ^2/3 - Stream function     

States of classical statistical mechanical systems of infinitely many particles. I

We study a general mathematical model of a classical system of infinitely many point particles. The space X of infinite particle configurations is equipped with a natural topology as well as a measurable structure related to it. It is also connected with a family {X _A } of local spaces of finite configurations indexed by bounded open sets A in the one-particle space E. A theorem analogous to Kolmogoroff's fundamental theorem for stochastic processes is proved, according to which a consistent family { _A } of local probability measures _A defined on the X _A gives rise to a unique probability measure on X. We also study the problem of integral representation for positive linear forms defined over some linear space of real functions on X. We prove that a positive linear form F(f), defined for functions f in the class C+P, admits a uniquely determined integral representation F(f)= f () d, where is a probability measure over X.     

Codimension two bifurcations of symmetric cycles in Hamiltonian systems with an antisymplectic involution

Consider a Hamiltonian system with parameters, such that there exists an involution which reverses this Hamiltonian system. Let us assume the linear part L at =0 has only nonzero purely imaginary eigen-values ±ib₁,..., ±ib_n. In this paper, we classify the typical bifurcations of families of symmetric periodic solutions of this system at resonance if b_i/b_j=±1, ±2, or ±1/2 and the number of parameters needed is one or two. First, one puts the Hamiltonians into a convenient normal form. Next, applying a Lyapunov-Schmidt reduction and making further manipulations, one can geta reduced bifurcation equation which can possess certain symmetry. Finally, by using elementary methods from singularity theory or isotopy methods, one obtains the desired bifurcation diagrams.     

Global solvability of one degenerate semilinear differential operator equation

We consider the Cauchy problem for the abstract semilinear differential equation where A and B are linear closed, generally speaking, degenerate operators acting from a Banach space X into a Banach space Y and f(t, u) is a continuously differentiable function. We assume that the resolvent (A + B)^–1 has a pole of at most second order at the point = 0. Global conditions for the existence and uniqueness of a solution of the Cauchy problem are obtained. The results are applied to a nonlinear degenerate initial boundary-value problem with partial derivatives and to a system of differential-algebraic equations of a nonlinear electric circuit.Translated from Neliniini Kolyvannya, Vol. 7, No. 3, pp. 414–429, July–September, 2004.     

On the propagation of elastic waves through composite media

Summary With the aid of an ultrasonic pulse technique, the propagation of elastic waves (longitudinal as well as transverse) through polyurethane rubbers filled with different amounts of sodium chloride particles was studied. The velocity of both longitudinal and transverse waves was found to increase with filler content. From the measured wave velocities, the effective modulus for longitudinal waves,L, bulk modulus,K, and shear modulus,G, were calculated according to the relations for a homogeneous isotropic material. All three moduli appear to be monotonously increasing functions of the filler content over the whole experimentally accessible temperature range (–70 °C to + 70 °C forL andK;}-70 °C to about –20 °C forG) and they, moreover, reflect the glassrubber transition of the binder.Poisson's ratio,, was found to decrease with increasing filler content and show a rise at the high temperature side of the experimentally accessible temperature range (about –20 °C) as a result of the approach of the glass-rubber transition.In addition to the velocities, the attenuation of both longitudinal and transverse waves was measured in the temperature ranges mentioned. It was found that in the hard region tan _L as well as tan _G are independent of the filler content within the accuracy of the measurements. In the rubbery region, however, tan _L, increases with increasing filler content.Finally, the experimental data are compared with a simple macroscopic theory on the elastic properties of composite media.