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Starting with the problem of finding a mass-distribution on a sphere, admitting in each point P a density, function of the point P and of the mass contained in a certain domain depending on P, the author proves the existence and the uniqueness of the solution of (5) under assumption (1), (2), (3). As a generalization, he also studies hyperbolic partial differential equations having solutions periodic in one of the variables with period depending on the others.

A systematic procedure is developed for finding analytical solutions of certain ordinary linear differential equations of arbitrary order, describing reflection and conversion of waves in multiwave media having a concentrated region of nonuniformity. This procedure involves transformation to equations solvable by generalized hypergeometric functions which have the form of normal modes, exp(−ikz), in uniform regions. It is applicable when the variable coefficients in the original equation contain only linear combinations of one varying parameter, p(z), and in certain cases its derivatives or ratios thereof, if p(z) is proportional to tanhz. Arbitrary gradients and dissipation are allowed. The applicability of this procedure to solve for the reflection and conversion coefficients is indicated for second‐, fourth‐, and sixth‐order problems in magnetostatic, spin, and elastic‐wave conversion in magnetic insulators such as yttrium iron garnet, and for electromagnetic‐to‐electrostatic wave conversion in hot pla...

We give a method for computing nontrivial solutions to the nonlinear partial differential equation ∇2Ψ + λ2sinh Ψ = 0, with Ψ = 0 on a square boundary. The method consists of a Newton-Raphson iteration, in which successive corrections to Ψ must satisfy a linearized partial differential equation. We give a direct solution algorithm for the linearized equation, which is suitable for small meshes. Using this method, we establish the nonuniqueness of solutions by finding six solutions for the same value of λ. Calculation of these solutions required from 4 to about 40 iterations each, depending upon the accuracy of the initial approximation. These solutions are the lowest-mode members of three classes of solutions possessing (1) rectangular, (2) quasicylindrical, and (3) diagonal symmetry.

The fundamental constant of Grothendieck's inequality, defined below, was shown by Grothendieck to be less than sinh π/2=2.301+. We improve the bound slightly, and show that for the positive definite case π/2 suffices.

It is pointed out that if the 70 percent and 95 percent points are used in fitting, the laboratory curve and the hyperbolic curve will compare vary well in this region but may differ seriously at low stress levels. The best overall fit between laboratory curves and hypergolas is obtained using the 50 percent and 95 percent points but the user should use points appropriate to the stress levels of major interest. Comment is made on the values of R sub f, and compaction conditions, and a linear failure equation. The use is unidacted of the log-log relationship to obtain a single analytic relationship through the stress range from 10 psi-1,000 psi. Both the tangent Poisson ration model and the hyperbolic model work well for samples compocted wet of optimum because they involve essentially constant values of Poissons ratio and thus degenerated into linear models in both cases. It is pointed out that the use of average axial strain and averate radial strain (derived from average values of volumetric and axial strains) is more likely to yield accurate results than the use of an average axial strain with the radial strain measured at a single point.

The main result of our paper is a proof of the uniqueness of the classical solution of a mixed problem for a second-order hyperbolic equation in a normal cylinder.

AN explicit difference scheme for solving the mixed problem for a symmetric first-order hyperbolic system is constructed, on the basis of Galerkin's approximation of the spatial operator and the introduction of an operator of the “artificial viscosity” type. The convergence of the scheme is investigated.

The multivariable approach to the synthesis of networks composed of a finite number of uniform lossless transmission lines, commensurable or incommensurable, and lumped passive elements is verified by showing that the multivariable rational matrix, W(λ0,λ1, .. ,λn) is bounded real in (n + 1) complex variables if and only if W is bounded real in p after substituting, λ0 = α0p + β0 and λi = tanh (αip + βi) for λi (1⩽i⩽n) where all the α's and β's are nonnegative and arbitrary, except for being not simultaneously zero in like indexed pairs. Consequences of this result are discussed.

This paper considers the Cauchy problem for a hyperbolic equation with constant operator coefficients in Hilbert space: where and are selfadjoint operators and is semibounded.As an example we consider ultraparabolic systems.Bibliography: 10 items.

This paper deals with symmetric hyperbolic systems, ∂ u / ∂ t = L u \partial u/\partial t = Lu , where L is equal to the homogeneous, constant coefficient operator L 0 {L_0} for | x | > R |x| > R . Under the hypothesis that L has simple null bicharacteristics and these propagate to infinity, local decay of solutions and completeness of the wave operators relating solutions of ∂ u / ∂ t = L u \partial u/\partial t = Lu and solutions of ∂ u / ∂ t = L 0 u \partial u/\partial t = {L_0}u are established. Results of this type for elliptic L are due to Lax and Phillips. The proof here is based, in part, on a new estimate of the regularity of the L 2 {L^2} -solutions of the equation L u + ( i λ + ε ) u = g Lu + (i\lambda + \varepsilon )u = g for smooth g with support in | x | ≤ R |x| \leq R .

A review of some studies of skill acquisition indicates that a simple hyperbolic function fits data on individual performance well. The function yields an explicit value of prior experience related to the task. The particular function for an individual can be adjusted to a “hypothetical first” trial which is common to other members of the sample population. The individual constants in the adjusted functions may then be averaged arithmetically to give a mean function, which is also an hyperbola. Standard deviations of the values of the constants may also be computed. These too will be hyperbolic functions.

A method valid at all temperatures belowT c is presented for the self-consistent solution of the Bogoliubov-de Gennes equations. Computation is reduced to manageable proportions by using an Ansatz for the pair potential, and determining the parameters in the Ansatz by requiring self-consistency. For the simple geometry of a semiinfinite slab of superconductor in contact with an insulator (or vacuum), the Ansatz Δ=Δ∞ tanh αz is found to be good close to the transition temperature and correctly recovers the Ginzburg-Landau theory asT →T c.

Reference-stress parameters, based on the Norton constitutive equation, are developed for creep bending of circular and elliptical tubes. The parameters are presented in the form of a design chart. The reference stress methods apply also to other simple constitutive equations of the power law, exponential and hyperbolic sine forms, and also to two complex equations describing creep behaviour over the entire range from primary to tertiary. In these equations the functions of stress and time are not separable and reference stress techniques provide a convenient method of evaluating stationary stress distributions and investigating the variation of these stresses with time. Aluminium alloy tubes were creep tested with cyclic uniform bending moments. Creep strains to fracture were measured by an end-rotation method. The creep behaviour of the tubes was well predicted by reference-stress cyclic tensile creep tests.

A staple draftometer is described using a "Hall Effect" wattmeter as the transducer for measuring dynamic forces in staple slivers during drafting. The procedure for calibration and use of the instrument is also given. A novel feature of the apparatus is the use of a decade resistance box to calibrate the desired full-scale force range. Drafting force was a hyperbolic function of draft ratio (beyond 1.5× draft), and the relationship between the two variables can be expressed in terms of a constant as shown by the following equation: (Drafting Force) X (Draft Ratio) = K Thus the drafting force curve in terms of draft ratios can be defined by a single numher. An increase in crimp frequency (0-15 crimps per inch range) gave an exponential increase in sliver drafting force. Addition of luhricants such as Ucon® or silicone oils to clean T-54 Dacron® polyester fiber slivers produced a significant reduction in drafting force, most of which occurred during the addition of the first 0.1% lubricant. The absolute drafting force is governed by specific properties of the lubricant.

The "curve of autocatalytic growth" (and also the generalized logistic) is essentially the hyperbolic tangent. Use of this fact simplifies the theoretical and practical handling of the equation. Demonstration of a good fit of the autocatalytic curve to a given growth-curve is no proof that the growth is autocatalytic, since the autocatalytic curve can be made to fit curves of similar appearance but very different nature. The variety of forms which the autocatalytic curve can assume is evidence of the futility of such an attempted proof.

The natural frequency of roof-shaped panels is evaluated on the basis of Reissner's simplified equations for thin shallow shells, which are approximately solved with the aid of Galerkin's method. The flexural vibration mode is assumed in a form of Fourier series and a homogeneous solution of stress function is expressed in terms of trigonometric and hyperbolic functions in order to satisfy several sets of the in-plane boundary conditions exactly. The modal equations are generally coupled and hence every natural mode is a general combination of the assumed modes. Numerical results in two-mode approximation indicate that the conditions at straight edges are more influential than the conditions at curved edges for a panel of λ ≳ 1, where λ is defined as the ratio of the length of straight edges to that of curved edges. In particular, the restriction on the normal displacement at straight edges raises the natural frequency of the lower frequency mode to a great extent. For λ ≲ 1 the conditions at curved edges are very influential. At curved edges the condition of zero tangential displacement gives more restrictions to a panel of λ < 1.83 rather than that of zero normal displacement. As for the higher frequency mode the influence of the in-plane boundary conditions is diminished.

Si considerano dapprima le soluzioni generalizzateF dell’equazione Fxx − Fyy= ϕ(x, y, F). Tali soluzioniF risultano rispettivamente limitate o quasi-periodiche iny, uniformemente perx ∈I, in una strisciaI ×R del pianoxy, secondo che siano rispettivamente limitate o quasi-periodiche per ogni fissatox ∈ Ω, Ω essendo un insieme di misura esterna positiva. Mediante l’equazione Fxx−Fyy = q(x)·F vengono poi introdotte ed esaminate nuove classi di funzioni quasi-periodiche sull’asse reale. Queste funzioni «q-quasi-periodiche» vengono definite come quelle funzioni ƒ definite su un intervalloI reale, in corrispondenza delle quali esiste inI ×R una soluzione generalizzataF dell’equazione Fxx−Fyy= q(x)·F soddisfacente la condizione F(x,0) = ƒ(x), oveF e una funzione quasi-periodica iny secondo Bohr. First generalized solutionsF of Fxx−Fyy= ϕ(x,y,F) are treated. SuchF are bounded resp. almost-periodic iny, uniformly inx ∈I, on a stripI ×R of thexy-plane, if they are only bounded resp. a.p. for each fixedx ∈ Ω, Ω being a set of positive outer measure. Then with the aid of Fxx−Fyy = q(x)·F new classes of a.p. functions on the real line are introduced and discussed. These “q-a.p.” functions are defined as functions ƒ on a real intervalI, to which there exists a generalized solutionF of Fxx−Fyy= q(x)·Fon I×R withF(x, 0)= ƒ(x),F being Bohr-a.p. iny.

In this paper the Lax-Wendroff procedure is extended to the scalar case of a two-dimensional hyperbolic conservation law. Explicit third and fourth order accuracy finite-difference operators are constructed for solving quasi-linear initial value problems. Stability conditions are obtained and are used in numerical computations. The computational results which are presented demonstrate that large amounts of computing time and memory space are saved without loss of accuracy. (Author)

The initial deflections exist between frames for the stiffend cylinders fabricated by welding. In this paper the influence of the initial deflection to the elastic behavior of a cylinder subjected to external hydrostatic pressure was studied using the mathematical expressions for the intial deflections as follows(1) wi1=δ (1-4ξ2) (2) wi2=δ/wξ=0-wξ=1/2 (w-wξ=1/2) (3) wi3=A sinh κ2ξ sin κ1ξ+D cosh κ2ξ cos κ1ξ+kA theoretical analysis indicates that the influences of initial deflections are different by its forms.The pressures at which the total stress on the surface of the cylinder with initial deflection wi3 equal to the yielding stress of material were calculated for the cylinders having diffrent dimentions. Results of these calculations are shown Fig. 4, which are nondimentional plots of ψ=qa/Yh versus ν2=b2/ah.

Two difference methods for approximating some first-order nonlinear hyperbolic differential equations are considered. The methods apply to problems arising in a number of physical applications. Each of the methods is explicit and can be implemented on a computer easily. It is proved that the methods are first-order convergent in the maximum norm. For one of the methods in order to obtain convergence it is necessary to monitor, and perhaps change, the size of the time step as the computation proceeds. The other method is unconditionally convergent.

It is well known that the well posed Cauchy problem admits the consistent stable finite difference scheme. By the Lax's equivalence theorem the solutions of that scheme converge to the true solutions of the Cauchy problem. In the case of mixed problem these general theory is not yet established. Especially in the case of m-space variable we can find no example of consistent stable finite difference scheme. But we can say in the case of well posed mixed problem consistency and stability assures L convergence of approximate solutions to the true solution. Many authors obtained the condition for the stability of finite difference scheme with boundary condition (Strang [1], Kreiss [23, Osher [3]). But they did not find consistent stable scheme which approximate well posed mixed problem for hyperbolic system. We consider in a half space first order symmetric hyperbolic system. On the plane boundary we set dissipative boundary condition. In the interior of the region we must take the consistent scheme which is stable for the Cauchy problem. Here we consider a modified Friedrichs scheme as a most simple explicit scheme which is stable for the Cauchy problem. We can find consistent boundary scheme which assures the stability of the whole scheme.

Liquid scintillation counting methods are described for the determination, with a precision of & ZOJO, of the absolute activities of 04 stock solutions to be used in marinc productivity studies. The efficiency of a windowless Geiger counter was found to bc 58% by counting very thin films of labcllcd plastic whose absolute activities were then dctcrmined by liquid scintillation counting. The zero-thickness Gcigcr activities of several Cl4 stock solutions wcrc determined with a precision of t 3%, and an accuracy of k 5%. This method was compared with the method of extrapolating self-absorption curves of BaC1403 planchets of varying thicknesses. It was found that the self-absorption curve between 0.5 ancl 6.0 mg/cm2 conformed equally well to an cxponcntial as to a hyperbolic function. Extrapolation of thcsc functions to 0 mg/cm2 gave zero-thickness activities 24% and 17% rcspcctively lower than those obtained by the new method described, showing that the self-absorption curve variccl considerably from both the exponential and hyperbolic function at thickncsscs below 0.5 mg/cm2. This is attributed largely to back-scattering. Empirical self-absorption curves were found to bc unsatisfactory due to the impossibility of preparing planchcts of BaCOs in the critical range of 0 to 0.5 mg/cm2.

and p is the number of dimensions. The numerical algorithm consists of the conversion of a three-point triangular operator, which reduces to the successive application of known formulae. The number of operations to determine a solution for a new time layer is proportional to the number of nodes of the space network and is a quantity of the same order as the number of operations for a purely explicit scheme. Thus the schemes put forward below are economic.

In measuring the distance at which we can observe the fine details of the three stimuli shown in Fig. 3 (striped squares, whose areas are 25, 5 and 1 cm2), we obtain that visual acuity is inversely proportional to the area of the stimulus. Furthermore, visual acuity increases (decreases) in the same way as decreases (increases) the area of the stimulus. This relation should be expressed by an hyperbolic function.

1. When a keyway is cut in a cylindrical shaft, subject to torsion only, and of circular cross section, the maximum stress in the keyed part will considerably exceed that in the full part. To form an estimate of this increase, Filont has investigated the torsion in a shaft the cross section of which is composed of confocal ellipses and hyperbolas. His results are expressed in infinite series of trigonometric and hyperbolic functions, and their numerical computation is necessarily somewhat laborious. Moreover, his numerical results (1. c., ? 22-23) extend only up to a ratio of minor and major axes in the ellipse equal to tanh (2r/2) = 0.917, and this section differs too much from a circle to allow a sufficiently close numerical estimate of the stress ratio in the latter case. In the present paper, a cross section will be considered which frequently occurs in practice: the shaft section is taken as a circle, and the keyway is formed by a circular arc intersecting the former orthogonally. We shall first determine the exact formulas for the stress distribution, and then derive from these an approximate formula, adapted to practical use, in which the ratio of the maximum stresses in the keyed and full parts of the shaft is represented by the expression

In the last ten years, a general theory of mixed problems for linear hyperbolic systems of first order has been developed. In this paper, we treat the L2 well-posed mixed problems for strongly hyperbolic systems of first order with constant coefficients. It is well known that Cauchy problem for hyperbolic systems of first order with constant coefficients is L2 well-posed if and only if it is strongly hyperbolic DL2j. The family of strongly hyperbolic systems contains the strictly and symmetric hyperbolic systems. But, Strang's condition for strong hyper bolicity seems to be useless except for Cauchy problem. So, in Sec. 3, we prove another condition for strong hyperbolicity which is useful for not only Cauchy problem, but also mixed problems, the lacunas of Riemann's matrix of strongly hyperbolic systems [1], the propagation of singularities Q8] and others. The L2 well-posed mixed problems for the strictly and symmetric hyperbolic systems have been fully investigated. But, even if hyperbolic systems have constant coefficients, nothing is known in general for ones with multiple characteristics except for systems with constant multiple characteristics [[KT] and 2x2 systems [JL3]. We consider the mixed problem for hyperbolic systems of first order with constant coefficients

Hyperbolic differential operators with constant coefficients introduced and studied systematically by Gårding (4), were characterized by the existence of the fundamental solution with some cone condition, according to Hörmander (6). Recently Ehrenpreis, extending the notion of hyperbolicity due to Gårding, has defined hyperbolic operators for distributions with compact support in the convolution sense. Under the hypothesis that the operator is invertible as a distribution, he has established a theorem analogous to the theorem of Hörmander mentioned above (3).

1. Single unit impulses were recorded from the ulnar and median nerves of awake human subjects with tungsten electrodes inserted percutaneously in the upper arm. 2. Forty-nine rapidly adapting mechanoreceptors with receptive fields in the glabrous skin area were studied. Thirty-nine units had small receptive fields with distinct borders (RA-receptors) while ten units had large fields with indistinct borders (PC-receptors). 3. The afferent response to stimuli of varying indentation amplitude and velocity of indentation, was analysed. 4. Amplitude thresholds varied from 0·05 to 1·65 mm for the RA-receptors. For the PC-receptors amplitude thresholds ranged from less than 0·05 to 1·95 mm. 5. Velocity thresholds varied for the RA-receptors from 0·4 to 39·3 mm/sec, and for the PC-receptors from 0·5 to 19·6 mm/sec. 6. The conduction velocities of the afferents were all in the A α—β range. For the RA-receptors the conduction velocities ranged from 26 to 91 m/sec (mean = 55·3 ± 3·4), and for PC-receptors the range was from 34 to 61 m/sec (mean = 46·9 ± 3·6). 7. The nerve impulse frequency as a function of indentation velocity was analysed for nineteen RA-receptors and four PC-receptors. A hyperbolic log tangent function of the type first introduced by Naka & Rushton (1966) in studies on S-potentials in the fish retina was found to be the best description of the stimulus—response function for sixteen RA-receptors and two PC-receptors. For the remaining units a pure logarithmic function was the best description. However, the logarithmic function may be, as found in the present study, a special case of the more general log tanh function.

In 1949 I published in the Acta Mathematica (vol. 81) a rather long paper: “L'intégrale de Riemann-Liouville et le problème de Cauchy.” This work will be quoted in the sequel as Acta paper. Only minor local references to this paper will be made here, and knowledge of it is not required for the reading of the present article. The notations used here are slightly different from those used in my former paper.In the Acta paper I introduce multiple integrals and of the Riemann- Liouville type depending on a parameter α and converging for sufficiently large values of α. I give the solution of the Cauchy problem for the wave equation in a unique formula, the same for space-time of odd or even dimensions, implying an analytic continuation with respect to the parameter α.

A perturbation method is used to analyze wave propagation in a class of variable‐section bars that can be treated as a perturbation of a uniform bar. The hyperbolic equation of motion is shown to constitute a singular perturbation problem. An existing perturbation method is modified and extended to obtain a uniformly valid asymptotic expansion of the solution.

MOST THEORETICAL PROCEDURES NOW AVAILABLE FOR DESIGN AND ANALYSIS INVOLVE ONLY LINEARLY ELASTIC SUBGRADE RESISTANCE; IN MANY STRUCTURES SUCH A REPRESENTATION IS QUITE SATISFACTORY, AND THE MOST RATIONAL THEORETICAL APPROACH AT PRESENT IS THE ELASTIC HALF-SPACE REPRESENTATION. TO STUDY THE EFFECT OF NONLINEAR SUBGRADE RESISTANCE ON THE MOTIONS OF A FOUNDATION UNDERGOING HARMONIC VIBRATION, A SINGLE- DEGREE-OF-FREEDOM MODEL WAS DEVELOPED WHICH IN THE ELASTIC DOMAIN EXHIBITED THE SAME TYPE OF MOTION AS THE ELASTIC HALF-SPACE APPROACH. TWO TYPES OF NONLINEAR SUBGRADE RESISTANCE WERE STUDIED. SINCE THE ANALYSIS OF THE NONLINEAR VIBRATION SYSTEM IS NOT A STRAIGHTFORWARD PROCEDURE, THE RESULTS HAVE BEEN PRESENTED IN A DIMENSIONLESS FORM TO MAKE THEM OF PRACTICAL USE. COMPARISONS WITH FIELD DATA ARE PRESENTED, AND THE RESPONSE WAS MORE CLOSELY REPRESENTED IN TERMS OF SHIFT OF RESONANT FREQUENCY AND SHAPE OF THE RESPONSE CURVES THROUGH USE OF THE NONLINEAR HYPERBOLIC TANGENT RESISTANCE THAN BY A LINEARLY ELASTIC RESISTANCE. USE OF THE MODEL FOR PRACTICAL SITUATIONS INVOLVING LINEAR OR NONLINEAR SUBGRADE RESISTANCE, AND FOR SCALING BETWEEN DIFFERENT SIZE FOOTINGS IS PRESENTED. /ASCE/

A technique is presented for accurate approximation of transfer functions whose impulse responses are finite time duration sine and cosine pulses. By paying attention to the effects of errors in the approximation of hyperbolic functions coth Ts and tanh s , an economical synthesis is obtained. The same synthesis can be used for low-error time-delay functions and magnitude functions for bandpasses.

The transient temperature distribution in bonded end closures of annular fuel elements of nuclear reactors is presented analytically. The solution for the temperature distribution is expressed by products of the Bessel functions, hyperbolic function, exponential time function and coupling relations between the end closures and the fuel material of the fuel elements. From a numerical example for the zirconium-alloy end closures and ceramic fuel material in a central nuclear superheating fuel element, the calculated results indicate that the temperature variation in the end closures decreases gradually with increasing time. Fourier number and, to a certain extent, enclosure length.