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  • Articles  (41)
  • Applied Mathematics  (41)
  • 1975-1979  (41)
  • Mathematics  (41)
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  • Mathematics  (41)
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  • 1
    Electronic Resource
    Electronic Resource
    Analysis of Generalised Galerkin methods in the numerical solution of elliptic equations (1979)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 1 (1979), S. 3-15 
    Details
    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The Petrov-Galerkin projection method is outlined for the solution of the linear elliptic equation Lu = f with homogeneous boundary conditions. By choosing appropriate finite dimensional trial and test spaces, the methods of weighted residuals, collocation, and H1 Galerkin can be interpreted within the Petrov-Galerkin projection method framework.The important question of how best to choose the trial and test functions to suit a particular type of problem is then discussed. Objective criteria associated with the matrix which governs the Petrov-Galerkin numerical process are proposed.
    Additional Material: 1 Tab.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/mma.1670010102
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  • 2
    Electronic Resource
    Electronic Resource
    The construction of solutions to the heat equation backward in timeDedicated to the memory of our “Doctor-Father” Professor Arthur Erdélyi. (1979)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 1 (1979), S. 32-39 
    Details
    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Solutions to the backwards heat equation are approximated by solutions of a pseudo-heat equation. Solutions to this modified equation are constructed by means of a fundamental solution and potential theory, and it is shown that the fundamental solution can be approximated by various expansions in special functions.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/mma.1670010104
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  • 3
    Electronic Resource
    Electronic Resource
    On the treatment of free boundary problems with the heat equation via optimal control (1979)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 1 (1979), S. 40-61 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: In this paper we consider a free boundary problem of general type for the heat equation in one space dimension. We formulate this problem as an optimal control problem and derive necessary conditions for a solution of it. In order to compute a solution of the control problem, we apply the projection-gradient-method. Simple numerical examples illustrate the results.
    Additional Material: 3 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/mma.1670010105
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  • 4
    Electronic Resource
    Electronic Resource
    The convergence of the horizontal line method for Maxwell's equations (1979)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 1 (1979), S. 101-113 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The interior initial-boundary value problem for Maxwell's equations is considered for certain homogeneous boundary conditions and currents which satisfy corresponding compatibility conditions. Under the assumption that a solution exists, it is shown by means of suitable normed spaces that the sequence of discrete solutions obtained from the horizontal line method converges discretely to this solution. A priori estimations are given both for the discrete solutions and the solution of the given problem.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/mma.1670010109
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  • 5
    Electronic Resource
    Electronic Resource
    Masthead (1979)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 1 (1979) 
    Details
    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/mma.1670010201
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  • 6
    Electronic Resource
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    Error estimates for sampling approximation of non-bandlimited functionsThis research was carried out under the DFG „Schwerpunktprogramm Digitale Signalverarbeitung“ (Bu 166/29, 31). (1979)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 1 (1979), S. 127-137 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: A new proof of the sampling theorem for non-bandlimited signals is presented. It leads to better rates of convergence for the envolved approximation process. Furthermore, the order of convergence is given for an extended class of functions.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/mma.1670010202
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  • 7
    Electronic Resource
    Electronic Resource
    Hopf-Bifurkation bei Lasern (1979)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 1 (1979), S. 194-213 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The semiclassical equations describing a ring laser show two successive bifurcations, one stationary and one Hopf bifurcation. This phenomenon is analyzed mathematically. The initial value problem for the laser equations and the stability of the stationary solutions are discussed in detail. The transition to ultrashort laser pulses is shown to be a Hopf bifurcation. The direction of the bifurcation is determined for a numerical example. It turns out that it depends on the parameters of the system.
    Additional Material: 2 Ill.
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    URL: http://dx.doi.org/10.1002/mma.1670010206
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  • 8
    Electronic Resource
    Electronic Resource
    Masthead (1979)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 1 (1979) 
    Details
    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/mma.1670010301
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  • 9
    Electronic Resource
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    On the integral equation method for the plane mixed boundary value problem of the Laplacian (1979)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 1 (1979), S. 265-321 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Although the plane boundary value problem for the Laplacian with given Dirichlet data on one part Γ2 and given Neumann data on the remaining part Γ2 of the boundary is the simplest case of mixed boundary value problems, we present several applications in classical mathematical physics. Using Green's formula the problem is converted into a system of Fredholm integral equations for the yet unknown values of the solution u on Γ2 and the also desired values of the normal derivatie on Γ1. One of these equations has principal part of the second kind, whereas that one of the other is of the first kind. Since any improvement of constructive methods requires higher regularity of u but, on the other hand, grad u possesses singularities at the collision points Γ1 ∩ Γ2 even for C∞ data, u is decomposed into special singular terms and a regular rest. This is incorporated into the integral equations and the modified system is solved in appropriate Sobolev spaces. The solution of the system requires to solve a Fredholm equation of the first kind on the arc Γ2 providing an improvement of regularity for the smooth part of u. Since the integral equations form a strongly elliptic system of pseudodifferential operators, the Galerkin procedure converges. Using regular finite element functions on Γ1 and Γ2 augmented by the special singular functions we obtain optimal order of asymptotic convergence in the norm corresponding to the energy norm of u and also superconvergence as well as high orders in smoother norms if the given data are smooth (and not the solution).
    Additional Material: 8 Ill.
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    URL: http://dx.doi.org/10.1002/mma.1670010302
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  • 10
    Electronic Resource
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    On a class of elliptic singular perturbations with applications in population genetics (1979)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 1 (1979), S. 432-441 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: With the maximum principle for differential equations asymptotic estimates are made for a class of linear elliptic singular perturbation problems with resonant turning point behaviour in some of the independent variables. The method is applied to the stationary solution of the Kolmogorov backward equation in population genetics.
    Additional Material: 1 Ill.
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    URL: http://dx.doi.org/10.1002/mma.1670010308
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  • 11
    Electronic Resource
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    Zum 70. Geburtstag von Henry Görtler (1979)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 1 (1979), S. 443-444 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/mma.1670010402
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  • 12
    Electronic Resource
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    On the time-dependent Nernst-Planck equationsProf. Henry Görtler zum 70. Geburtstag (1979)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 1 (1979), S. 480-490 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: For the one-dimensional Nernst-Planck equations of the theory of ion-transport in thin layers or membranes which mathematically represent a nonlinear coupled parabolic integro-differential system some global existence and uniqueness theorems for inner and outer problems are given using a priori estimates of the maximum principle type.
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    URL: http://dx.doi.org/10.1002/mma.1670010406
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  • 13
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    The direction of the secondary wave motion in the Ekman boundary layerDedicated to Professor Henry Görtler on the occasion of his 70th birthday (1979)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 1 (1979), S. 491-497 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The secondary wave motion in the free Ekman layer, on which a constant shear stress acts, is examined by an analytic method. The results show, that the waves move away from the center of rotation.
    Additional Material: 2 Ill.
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    URL: http://dx.doi.org/10.1002/mma.1670010407
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  • 14
    Electronic Resource
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    An existence theorem for the unmodified Vlasov Equation (1979)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 1 (1979), S. 530-544 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The initial value problem for the modified Vlasov equation with a mollification parameter δ 〉 0, as introduced by Batt, has a unique global solution in the weak sense whenever f0 ε L1 and f0 ≧ 0 λ-a.e. Assuming boundedness of f0 and boundedness of the kinetic energy, it is shown that, as δ → 0, there are subsequences δn → 0 such that the corresponding solutions converge weakly in the measure-theoretical sense. The limits are shown to be global weak solutions of the initial value problem for Vlasov's equation, and these solutions are seen to be weakly continuous with respect to t. For the plasma physical case, boundedness of the kinetic energy is a consequence of energy conservation.
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    URL: http://dx.doi.org/10.1002/mma.1670010410
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  • 15
    Electronic Resource
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    A finite element method for the solution of a potential theory integral equation (1979)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 1 (1979), S. 581-587 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: This paper discusses a finite element approximation for an integral equation of the second kind deduced from a potential theory boundary value problem in two variables. The equation is shown to admit a unique solution, to be variational and coercive in the Hilbert space of functions σ ε H1/2(Γ), frd γ = 0. The Galerkin method with finite elements as trial functions is shown to lead to an optimal rate of convergence.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/mma.1670010415
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  • 16
    Electronic Resource
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    Finite elements for incompressible flow (1979)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 1 (1979), S. 16-31 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: We describe in this paper a finite element method for the solution of viscous incompressible flow problems which incorporates an approximate form of the incompressibility condition automatically into the finite element basis. Several examples of such finite elements are presented and applied to a simple test problem.
    Additional Material: 4 Ill.
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    URL: http://dx.doi.org/10.1002/mma.1670010103
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  • 17
    Electronic Resource
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    On the limiting behaviour of solutions to boundary integral equations associated with time harmonic wave equations for small frequencies (1979)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 1 (1979), S. 89-100 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The treatment of boundary value problems for Helmholtz equation and for the time harmonic Maxwell's equations by boundary integral equations leads to integral equations of the second kind which are uniquely solvable for small positive frequencies λ. However, the integral equations obtained in the limiting case λ = 0 which are related to boundary value problems of potential theory in general are not uniquely solvable since the corresponding boundary value problems are not. By first considering in a general setting of a Banach space X the limiting behaviour of solutions φλ to the equation φλ - K λ φ λ = fλ as λ → 0 where {Kλ: X → X, λ ∊ (0,α)}, α 〉 0, denotes a family of compact linear operators such that I - Kλ (I identity) is bijective for λ∊(0,α) whilst I - K0 is not and ‖ Kλ - K0‖ →, 0, ‖fλ - f0‖ → 0, λ → 0, and then applying the results to the boundary integral operators, the limiting behaviour of the integral equations is considered. Thus, the results obtained by Mac Camey for the Helmholtz equation are extended to the case of non-connected boundaries and Werner's results on the integral equations for the Maxwell's equations are extended to the case of multiply connected boundaries.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/mma.1670010108
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  • 18
    Electronic Resource
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    Zur Theorie der Integraltransformationen (1979)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 1 (1979), S. 114-125 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Let A be a selfadjoint uniformly elliptic differential operator. Let the underlying domain be bounded. Eigenvalue problems can be solved then, and an arbitrary square integrable function may be developed in a Fourier series relative to the eigenfunctions. In general elliptic differential operators have a continuous spectrum, if the underlying domain is unbounded. In this case the spectral theorem provides a representation of a given function by an integral transformation. The spectral projector can be calculated, if the outgoing and incoming solutions are known (radiation condition). Thus integral transformations may be derived very easily. Four examples will be given: the Fourier sine transform, the Lebedev transform, the transformation belonging to the Dirichlet problem of the plate equation and, finally, the Fourier transformation.
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    URL: http://dx.doi.org/10.1002/mma.1670010110
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  • 19
    Electronic Resource
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    Ein algebraisches Verfahren zur Lösung von gemischten Anfangsrandwertproblemen aus der Theorie instationärer GitterströmungenHerrn Prof. Dr. H. Söhngen zu seinem 70. Geburtstage am 30. November 1978 gewidmet. (1979)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 1 (1979), S. 158-186 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Mixed boundary value problems for the wave-equation in the x-y-plane, with boundaries parallel to the x-axis, are usually treated by Laplacetransformation. To use this method numerically we present it here by means of the Mikusiński-calculus. In this paper our main interest lies in the treatment of mixed initial-boundary value problems arising in unsteady subsonic cascade flow.
    Additional Material: 4 Ill.
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    URL: http://dx.doi.org/10.1002/mma.1670010204
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  • 20
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    Asymptotic expansions for central limit theorems for general linear stochastic processes. I: General theorems on rates of convergence (1979)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 1 (1979), S. 241-264 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: This paper deals with so-called general linear stochastic processes (GLSP), defined by T. Kawata in 1972 in generalization of work of R. Lugannani and J. B. Thomas of 1967/71. These second order processes (which are not necessarily stationary nor have independent increments) are described by rather weak requirements, so that several processes such as some random noise and pulse train processes are specific models of these GLSP. Part I is concerned with two general theorems giving asymptotic expansions (including those for the density function) in the central limit theorem for such GLSP, together with error rates. The assumptions for the corresponding θ- and o-error estimates seem rather natural: in the former, apart from assumptions on the inherent structure of such GLSP, the existence of certain moments of higher order as well as a Cramer-type condition are assumed, in the latter in addition a Lindeberg-type condition of higher order. Fourier analytic machinery is used for the proofs.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/mma.1670010208
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  • 21
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    Asymptotic expansions for central limit theorems for general linear stochastic processes. II: Models of the general random noise and pulse train processes (1979)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 1 (1979), S. 346-353 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: This part is concerned with the applications of the theorems on error estimates for asymptotic expansions in the central limit theorem for general linear stochastic processes of Part I to two important typical models. These are the general random noise process and a class of pulse train processes that occur in stochastic signal theory and in a variety of digital communication schemes. The assumptions on the inherent structure of the general theorems turn out to be simple here.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/mma.1670010304
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  • 22
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    On boundary controllability of one-dimensional vibrating systems (1979)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 1 (1979), S. 322-345 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: In this paper boundary controllability of one-dimensional vibrating system such as the vibrating string or the vibrating beam is studied. In particular we are concerned with the question whether it is possible to transfer a given initial state of vibration into rest within a given time such that the system stays in rest when the control is turned off.This problem is rephrased as a typical trigonometric moment problem which is solved within the framework of an abstract moment problem in a Hilbert space. The results of null-controllability which are obtained are substantially based on classical results of Ingham and Redheffer concerning trigonometric inequalities and incompleteness of certain sequences of trigonometric functions, respectively. The representation of the general statements follows closely the lines of a paper of Russell. Besides a special case is treated where explicit representations of boundary controls can be given that transfer the system to a permanent rest position. This special case includes amplitude boundary control of the vibrating string and the freely supported beam.
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    URL: http://dx.doi.org/10.1002/mma.1670010303
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    Stationary solutions of reaction-diffusion equations (1979)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 1 (1979), S. 418-431 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Given a semilinear reaction-diffusion equation. If the corresponding ordinary differential equation admits a convex compact positively invariant set and the boundary data assume their values in this set then the first and third boundary value problem have stationary solutions. The proofs are based on Weinberger's strong invariance principle, some related tools and the Leray-Schauder degree. The theorem is applied to several equations from theoretical biology, also in the case of distinct diffusion rates.
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    URL: http://dx.doi.org/10.1002/mma.1670010307
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  • 24
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    Masthead (1979)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 1 (1979) 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/mma.1670010401
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  • 25
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    Separation of variables and commuting operatorsDedicated to Professor Henry Görtler on the occasion of his 70th birthday (1979)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 1 (1979), S. 468-479 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: For any differential operator L, separable in some coordinate system, we construct a set of commuting operators Si such that 1) each Si maps solutions of Lu = 0 into solutions, and 2) the separated solutions of Lu = 0 are simultaneous eigenfunctions of Si. Moreover, a certain description of the separable coordinate system by Si is possible. The paper generalizes results obtained by W. Miller and others on many equations of mathematical physics.
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    URL: http://dx.doi.org/10.1002/mma.1670010405
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    Generalized Hopf bifurcation in Hilbert spaceDedicated to Prof. Dr. H. Görtler on the occasion of his 70 th birthday (1979)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 1 (1979), S. 498-513 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: We consider a family of semilinear evolution equations in Hilbert space of the form with, in general, unbounded operators *A(λ), F(λ·) depending analytically on a real parameter λ. We assume that the origin is a stationary solution, i.e. F(λ,0) = 0, for all λ ε R and that the linearization (with respect to u) at the origin is given by du/dt + A(λ)u = 0. Our essential assumption is the following: A(λ) possesses one pair of simple complex conjugate eigenvalues μ(λ) = Re μ(λ) ± i Im μ(λ) such that Im μ(0) 〉 0 and for some m ε N or If m = 1 the curves of eigenvalues μ(λ) cross the imaginary axis transversally at ±i Im μ(0). In this case a unique branch of periodic solutions emanates from the origin at λ = 0 which is commonly called Hopf bifurcation. If μ(λ) and the imaginary axis are no longer transversal, i.e. m 〉 1, we call a bifurcation of periodic solutions, if it occurs, a generalized Hopf bifurcation. It is remarkable that up to m such branches may exist. Our approach gives the number of bifurcating solutions, their direction of bifurcation, and its asymptotic expansion. We regain the results of D. Flockerzi who established them in a completely different way for ordinary differential equations.
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    URL: http://dx.doi.org/10.1002/mma.1670010408
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  • 27
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    Masthead (1979)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 1 (1979) 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/mma.1670010101
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    Sonar echo analysis (1979)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 1 (1979), S. 70-88 
    Details
    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Pulse mode sonar operation is analyzed under the assumption that the scattering object Γ lies in the far fields of both the transmitter and the receiver. It is shown that, in this approximation, the sonar signal is a plane wave s(x · θ0 - t) near Γ, where θ0 is a unit vector directed from the transmitter toward Γ, and similarly the echo is a plane wave e(x · θ - t) near the receiver, where θ is a unit vector directed from Γ toward the receiver. Moreover, if Γ is stationary with respect to the sonar system then it is shown that where š(ω) is the Fourier transform of S(τ) and T+(ωθ,ωθ0) is the scattering amplitude in the direction θ due to the scattering by Γ of a time-harmonic plane wave with frequence ω and propagation direction θ0. A generalization of this relation is derived for moving scatterers.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/mma.1670010107
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    Periodic solutions of ẋ(t) = -f(x(t), x(t - 1)) (1979)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 1 (1979), S. 62-69 
    Details
    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The existence of periodic solutions of general first order nonlinear difference-differential equations is shown using an appropriate cone of initial data and an asymptotic fixed point theorem.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/mma.1670010106
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    Lösung von Randwertproblemen der Schwingungsgleichung für ein Modell eines TriebwerkeinlaufsHerrn Prof. Dr. H. Söhngen zu seinem 70. Geburtstag am 30. November 1978 gewidmet (1979)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 1 (1979), S. 138-157 
    Details
    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: We consider a simple model of a jet engine inlet consisting of two coaxial cylinders. The inner cylinder extends in both directions to infinity and represents the hub of a jet engine. The outer cylinder is assumed to be semiinfinite representing the engine's jacket. This arrangement is approached by a compressible and invisced gas moving at a velocity U less than the speed of sound. Two problems are formulated. In problem A a normal speed distribution is given in the compressor-inlet plane, in problem B we are given a pressure distribution. We look for the induced pressure and velocity fields. Using the acoustical approximation of the fundamental equations the Wiener-Hopf-method is applied then leading, after a multiplicative and additive decomposition procedure, to an infinite system of linear equations. These involve the expansion coefficients of the solution with respect to the eigenfunctions of the concentric cylindrical duct. Applying arguements from perturbation theory it is possible to extract the essential information from this system of equations. Apart from the possible exception of a denumerable set of values of the distance L of the blade-row from the inlet the problem has a unique solution.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/mma.1670010203
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    Global existence for two-velocity models of the Boltzmann equation (1979)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 1 (1979), S. 187-193 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: If A is a symmetric 2 × 2-matrix, then the initial value problem describes the evolution in time of a fictive gas whose particles can move only with the velocities u1 and v2. It is proved that, for continuous initial values vanishing at infinity, (1) has a global solution if an H-Theorem holds for the gas described by (1). The validity of an H-Theorem is expressed by the properties of A.
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    URL: http://dx.doi.org/10.1002/mma.1670010205
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    Ein schiefes Randwertproblem zu einer elliptischen Differentialgleichung zweiter Ordnung mit einer expliziten L2-Abschätzung für die zweiten Ableitungen der Lösung (1979)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 1 (1979), S. 214-240 
    Details
    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: In connection with the free boundary value problem of determining the earth's surface from measurements of gravitational potential and force-field (“the geodetic boundary problem”), an oblique derivative problem arises, where D0 is some bounded domain, star shaped with respect to the origin. In order to prove a uniquencess theorem for the geodetic boundary problem, it is essential to give estimates for (weighted) L2-norms of the second derivatives of the solutions so that their bounds can be estimated numerically if bounds for the function describing the boundary are known. In this paper a Fredholm inverse for the above problem is constructed and the second derivatives of the solutions are estimated in the desired form.
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    URL: http://dx.doi.org/10.1002/mma.1670010207
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    Conform and mixed finite element schemes for the Dirichlet problem for the Bilaplacian in plane domains with corners (1979)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 1 (1979), S. 354-382 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: In plane domains with corners for the Bilaplacian a uniquely solvable conform variational principle is studied on weighted Sobolev spaces which is equivalent to the standard Dirichlet problem in the weak form. Clamped plates under point forces near corners are handled by this approach. With weighted Hsieh-Clough-Tocher elements on regular triangulations as conform C1-finite elements a new error analysis is performed without higher regularity assumptions on the exact solution than given by the data and the boundary. The rate of convergence of the error depends on the eigenvalue with smallest imaginary part of a clamped infinite wedge since this eigenvalue describes the singularity of the exact solution in a sector with same angle. Using different spaces of trial and test functions in the standard Galerkin procedure it is shown that the error in the weighted energy norm does not pollute. For convex corners asymptotic error estimates, are proved yielding convergence for a mixed method in hydrodynamics where the solution of a system of 2nd order and its Laplacian are approximated simultaneously by C0-finite elements being piecewise polynomials.
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    URL: http://dx.doi.org/10.1002/mma.1670010305
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    Über eine Verallgemeinerung des Begriffs „Charakteristik“ bei partiellen Differentialgleichungen erster Ordnung (1979)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 1 (1979), S. 383-417 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: As, in general, the projections of characteristics into the x-space intersect for finite values of t, the global solution of a conservation law cannot be determined from the characteristic system of the equation, is considered. Only in the linear case, this equation coincides with the equation of the projections of characteristics. For convex h and all x0 this equation has a solution almost everywhere, and the properties of this solution permit to construct a global solution of the conservation law using strips, in the same way as this is done for linear problems by the method of characteristics.
    Additional Material: 1 Ill.
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    URL: http://dx.doi.org/10.1002/mma.1670010306
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    The Crocco transformation for the three-dimensional Prandtl boundary layer equationsDedicated to the 70th birthday of my honored senior collegue and pre-decessor in office professor Dr. Dr. h. c. Henry Görtler (1979)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 1 (1979), S. 445-452 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The three-dimensional fluid flow of a steady or unsteady moving incompressible continuous medium is considered. Several Crocco-type transformations are applied which reduce the system of three equations of the Prandtl boundary layer to a system of two strongly coupled parabolic equations.
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    URL: http://dx.doi.org/10.1002/mma.1670010403
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    Exotische Lösungen des Bénardschen ProblemsMeinem verehrten Lehrer, Herrn Professor Dr. Dr. h. c. Henry Hörtler zum 70. Geburtstag gewidmet (1979)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 1 (1979), S. 453-467 
    Details
    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: A bifurcation analysis is given for the generalized Bénard problem near the critical value of the Grashoff number. In contrast to earlier investigations where symmetry arguments were used to reduce the boundary-value-problem to a simple-eigenvalue-bifurcation, we only require double periodicity and treat high multiplicity bifurcation. It is shown that solutions of a yet unknown type exist.
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    URL: http://dx.doi.org/10.1002/mma.1670010404
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    Newton iterations without inverting the DerivativeDedicated to Prof. Dr. H. Görtler on the occasion of his 70th birthday (1979)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 1 (1979), S. 514-529 
    Details
    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: We describe an iteration procedure in a Banach space which is quadratically convergent like Newton's method. In fact, it is a modification of the latter. The inverted derivatives are replaced by so-called contractors which are constructed recursively. Moreover, this method is extended to a scale of Banach spaces. It turns out that the rate of convergence remains quadratic, even if the norms of the contractors are increasing exponentially. A hard implicit function theorem results. In particular, this theorem can be applied to prove existence of quasiperiodic solutions for the Lorenz model of stationary convection.
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    URL: http://dx.doi.org/10.1002/mma.1670010409
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    Solutions for systems of nonlinear elliptic equations with nonlinear boundary conditions (1979)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 1 (1979), S. 545-553 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: We look for solutions of systems of nonlinear elliptic equations with nonlinear boundary conditions and values in some compact convex set M. If the nonlinear terms satisfy a sign condition on the boundary of M and the inhomogeneous terms assume their values in this set existence of solutions is proved.The proof is based on the homotopy invariance of the Leray-Schauder degree and Weinberger's strong maximum principle.
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    URL: http://dx.doi.org/10.1002/mma.1670010411
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    On the bifurcation diagram of discrete analogues for ordinary bifurcation problems (1979)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 1 (1979), S. 566-571 
    Details
    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Ordinary bifurcation problems of the form (1) typically have at most one nontrivial, nonnegative solution for λ 〉 0. The paper shows that this is in general not true for discrete analogues to (1) no matter how small the step width h 〉 0 is chosen.
    Additional Material: 1 Ill.
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    URL: http://dx.doi.org/10.1002/mma.1670010413
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    Constructive existence theorems for problems of Thomas-Fermi Type (1979)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 1 (1979), S. 554-565 
    Details
    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: A minimal positive solution of the Thomas-Fermi problem ẅ = λt-1/2 w3/2, w(0) = 1, w(1) = w(1) is shown to exist for each λ 〉 0. It is proved that all positive solutions, for a given value of λ, are strictly ordered and that the minimal positive solution wλ is a decreasing function of λ. Upper and lower analytic bounds for wλ are given and these bounds are shown to initiate sequences of Picard and Newton iterates which converge monotonically to wλ. A comparative analysis of the efficiency of the iteration schemes is presented. The methods used are of a general nature and can be applied to a variety of nonlinear boundary value problems of convex type [14].
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/mma.1670010412
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    On minimal nonlinearities which permit bifurcation from the continuous spectrum (1979)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 1 (1979), S. 572-580 
    Details
    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Bifurcation from the continuous spectrum of a linearized operator is of interest in many physical problems. For example it occurs in the nonlinear Klein-Gordon equation and in nonlinear integrodifferential equations as the Choquard problem; it further appears in nonlinear integral equations of the convolution type.A general theory enclosing all these problems is not yet known. To understand the basic phenomena, we therefore consider monotone differential operators whose linearisations have a purely continuous spectrum. It is shown that in fact the lowest point of the continuous spectrum is a bifurcation point, if the nonlinearity grows sufficiently strong.
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    URL: http://dx.doi.org/10.1002/mma.1670010414
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