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  • Articles  (222)
  • Applied Mathematics  (185)
  • CR: 5.15  (37)
  • 2010-2014
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  • Articles  (222)
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  • 1
    Electronic Resource
    Electronic Resource
    Sublinear convergence of the Chord method at singular points (1983)
    Springer
    Numerische Mathematik 42 (1983), S. 147-154 
    Details
    ISSN: 0945-3245
    Keywords: AMS (MOS): 65H10 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We give sufficient conditions for convergence of the chord method for a class of singular problems. The rate of convergence is sublinear.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01395307
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  • 2
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    A method of solving nonlinear variational problems by nonlinear transformation of the objective functional. Part I (1984)
    Springer
    Numerische Mathematik 43 (1984), S. 59-82 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65K10 ; CR: 5.15 ; 5.41
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary A general globally convergent iterative method for solving nonlinear variational problems is introduced. The method is applied to a temperature control problem and to the minimal surface problem. Several aspects of finite element implementation of the method are discussed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01389638
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  • 3
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    Das modifizierte Newton-Verfahren in verallgemeinerten Banach-Räumen (1984)
    Springer
    Numerische Mathematik 43 (1984), S. 91-104 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65J15 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary A Newton-Kantorowitsch-analysis of the modified Newton's method in generalized Banach-spaces is given. The application of generalized norms — mappings from a linear space into an ordered Banach-space — improves convergence- and existence results as well as error estimates compared with the real-norm theory.
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    URL: http://dx.doi.org/10.1007/BF01389640
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  • 4
    Electronic Resource
    Electronic Resource
    A general convergence theorem for the reduced gradient method (1984)
    Springer
    Numerische Mathematik 43 (1984), S. 241-247 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 90C30 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Consider the problem of minimizing a real function subject to linear equality constraints and to nonnegativity bounds on the variables. A convergence theorem is established for a general algorithm model based on the reduced gradient method. The most meaningful assumptions that are made, concern two crucial points: the choice of the independent variables and that of the search direction.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01390125
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  • 5
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    Error bounds of Newton type process on Banach spaces (1982)
    Springer
    Numerische Mathematik 39 (1982), S. 175-193 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS) (1980) ; Primary 65J05, 65J15 ; Secondary 47H10 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary For solving a nonlinear operator equationF(x)=0 in Banach spaces, the Newton's method or Newton type methods are important numerical techniques. We use the properties of real equationt=Φ(t) majorizing an operator equationx=Gx to find a fixed point ofG as a solution of equationF(x)=0. Various type of operatorsG are considered in this paper. For a nonlinear operatorG, we would find a real function Φ majorizing the operatorG and it will be related to a rate of convergence $$\omega (r) = \frac{{r^2 }}{{2(r^2 + d)^{1/2} }}.$$ It follows thatG has a fixed point as a solution ofF(x)=0. Practical limitations of error bounds like as in Potra and Pták [5] are discribed.
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    URL: http://dx.doi.org/10.1007/BF01408692
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  • 6
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    Some convergence results for asynchronous algorithms (1982)
    Springer
    Numerische Mathematik 39 (1982), S. 325-340 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65H10 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Description / Table of Contents: Summary In this paper we present convergence results for the asynchronous algorithms based essentially on the notion of classical contraction. We generalize, in particular, all convergence results for those algorithms which are based on the vectorial norm hypothesis, in wide spread use recently. Certain problems, for which the vectorial norm hypothesis can be difficult or even impossible to verify, can nontheless be tackled within the scope of the classical contraction that we adopte.
    Notes: Resumé Nous présentons dans cet article des résultats de convergence des algorithmes asynchrones basés essentiellement sur la notion classique de contraction. Nous généralisons, en particulier, tous les résultats de convergence de ces algorithmes qui font l'hypothèse de contraction en norme vectorielle qui récemment a été très souvant utilisée. Par ailleurs, l'hypothèse de contraction en norme vectorielle peut se trouver difficile, voire impossible à vérifier pour certains problèmes qui peuvent être cependant abordés dans le cadre de la contraction classique que nous adoptons.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01407866
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  • 7
    Electronic Resource
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    Local convergence analysis for partitioned quasi-Newton updates (1982)
    Springer
    Numerische Mathematik 39 (1982), S. 429-448 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65K05 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary This paper considers local convergence properties of inexact partitioned quasi-Newton algorithms for the solution of certain non-linear equations and, in particular, the optimization of partially separable objective functions. Using the bounded deterioration principle, one obtains local and linear convergence, which impliesQ-superlinear convergence under the usual conditions on the quasi-Newton updates. For the optimization case, these conditions are shown to be satisfied by any sequence of updates within the convex Broyden class, even if some Hessians are singular at the minimizer. Finally, local andQ-superlinear convergence is established for an inexact partitioned variable metric method under mild assumptions on the initial Hessian approximations.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01407874
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  • 8
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    Bisection is optimal (1982)
    Springer
    Numerische Mathematik 40 (1982), S. 111-117 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65H10 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We seek an approximation to a zero of a continuous functionf:[a,b]→ℝ such thatf(a)≦0 andf(b)≧0. It is known that the bisection algorithm makes optimal use ofn function evaluations, i.e., yields the minimal error which is (b−a)/2 n+1, see e.g. Kung [2]. Traub and Wozniakowski [5] proposed using more general information onf by permitting the adaptive evaluations ofn arbitrary linear functionals. They conjectured [5, p. 170] that the bisection algorithm remains optimal even if these general evaluations are permitted. This paper affirmatively proves this conjecture. In fact we prove optimality of the bisection algorithm even assuming thatf is infinitely many times differentiable on [a, b] and has exactly one simple zero.
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    URL: http://dx.doi.org/10.1007/BF01459080
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  • 9
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    On a numerical treatment for the curve-tracing of the homotopy method (1983)
    Springer
    Numerische Mathematik 42 (1983), S. 323-329 
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    ISSN: 0945-3245
    Keywords: AMS (MOS): 65K05 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary A Newton-like iteration scheme is proposed for the tracing of an implicitly defined smooth curve. This scheme originates from the study of the continuous Newton-Raphson method for underdetermined systems and, hence, inherits the characteristic property of orthogonality. Its domain of attraction is formed and makes it possible to trace this curve more efficiently.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01389577
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  • 10
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    A generalized conjugate gradient algorithm for minimization (1980)
    Springer
    Numerische Mathematik 35 (1980), S. 223-230 
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    ISSN: 0945-3245
    Keywords: AMS (MOS): 65K05 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary A generalized conjugate gradient algorithm which is invariant to a nonlinear scaling of a strictly convex quadratic function is described, which terminates after at mostn steps when applied to scaled quadratic functionsf: R n →R1 of the formf(x)=h(F(x)) withF(x) strictly convex quadratic andh∈C 1 (R1) an arbitrary strictly monotone functionh. The algorithm does not suppose the knowledge ofh orF but only off(x) and its gradientg(x).
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    URL: http://dx.doi.org/10.1007/BF01396318
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  • 11
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    On the rate of superlinear convergence of a class of variable metric methods (1980)
    Springer
    Numerische Mathematik 35 (1980), S. 293-313 
    Details
    ISSN: 0945-3245
    Keywords: AMS (MOS) ; 90C30 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary This paper considers a class of variable metric methods for unconstrained minimization. Without requiring exact line searches each algorithm in this class converges globally and superlinearly on convex functions. Various results on the rate of the superlinear convergence are obtained.
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    URL: http://dx.doi.org/10.1007/BF01396414
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  • 12
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    Quotienten-Differenzen-Algorithmus: Beweis der Regeln von Rutishauser (1982)
    Springer
    Numerische Mathematik 40 (1982), S. 93-98 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65D15 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Description / Table of Contents: Zusammenfassung Der Quotienten-Differenzen-Algorithmus nach Rutishauser ist geeignet zur Bestimmung von Polen meromorpher Funktionen, gegeben durch eine Taylorreihe. Sind mehrere Pole betragsgleich, so kann eine Polynomfolge bestimmt werden, deren Grenzpolynom diese Pole als Nullstellen hat. Die Konvergenz wurde von Rutishauser jedoch nicht bewiesen. Ein Beweis wird in der vorliegenden Arbeit präsentiert.
    Notes: Summary The Quotient-Difference Algorithm of Rutishauser can be used for the determination of poles of a meromorphic function given by its power series. If some of the poles have same modulus, a sequence of polynomials can be determined such that the limiting polynomial has exactly these poles as zeros. The convergence has not been proved by Rutishauser, however. A proof is presented in this paper.
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    URL: http://dx.doi.org/10.1007/BF01459078
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  • 13
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    On the steps of convergence of approximate eigenvectors in the Rayleigh-Schrödinger series (1983)
    Springer
    Numerische Mathematik 42 (1983), S. 31-50 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65J10 ; 41A25 ; 41A65 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Error bounds are given for the iterative computation of the eigenvectors in the Rayleigh-Schrödinger series. These bounds remove the discrepancy in the theoretical behaviour and numerical results, noted by Redont, under the assumption of collectively compact convergence. As a particular case, it follows that the eigenvector in the iterative Galerkin method proposed by Sloan improves upon the eigenvector in the Galerkin method. This is illustrated by numerical experiments.
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    URL: http://dx.doi.org/10.1007/BF01400916
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  • 14
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    Imperfect conjugate gradient algorithms for extended quadratic functions (1983)
    Springer
    Numerische Mathematik 42 (1983), S. 97-105 
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    ISSN: 0945-3245
    Keywords: AMS (MOS): 65K05 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Generalized conjugate gradient algorithms which are invariant to a nonlinear scaling of a strictly convex quadratic function are described. The algorithms when applied to scaled quadratic functionsf∶R n →R 1 of the formf(x)=h(F(x)) withF(x) strictly convex quadratic andh∈C 1(R 1) an arbitrary strictly monotone functionh generate the same direction vectors as for the functionF without perfect steps.
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    URL: http://dx.doi.org/10.1007/BF01400920
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  • 15
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    Sharp error bounds for Newton's process (1980)
    Springer
    Numerische Mathematik 34 (1980), S. 63-72 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65H10 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary The method of nondiscrete mathematical induction is applied to the Newton process. The method yields a very simple proof of the convergence and sharp apriori estimates; it also gives aposteriori bounds which are, in general, better than those given in [1].
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01463998
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  • 16
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    Sur la convergence théorique de la méthode du gradient réduit généralisé (1980)
    Springer
    Numerische Mathematik 34 (1980), S. 73-85 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65K05 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary The theoretical convergence of Generalized Reduced Gradient Method (GRG) has not been proved; the purpose of this paper is to propose two theoritical and general variants of the original method and a proof of their convergence.
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    URL: http://dx.doi.org/10.1007/BF01463999
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  • 17
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    The numerical solution of the inverse Stefan problem (1980)
    Springer
    Numerische Mathematik 34 (1980), S. 411-429 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65N10 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary The inverse Stefan problem can be understood as a problem of nonlinear approximation theory which we solved numerically by a generalized Gauss-Newton method introduced by Osborne and Watson [19]. Under some assumptions on the parameter space we prove its quadratic convergence and demonstrate its high efficiency by three numerical examples.
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    URL: http://dx.doi.org/10.1007/BF01403678
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  • 18
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    Starlike domains of convergence for Newton's method at singularities (1980)
    Springer
    Numerische Mathematik 35 (1980), S. 95-111 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65H10 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Given a solutionx * of a system of nonlinear equationsf with singular Jacobian ∇f(x *) we construct an open starlike domainR of initial points, from which Newton's method converges linearly tox *. Under certain conditions the union of those straight lines throughx *, that do not intersect withR is shown to form a closed set of measure zero, which is necessarily disjoint from any starlike domain of convergence. The results apply to first and higher order singularities.
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    URL: http://dx.doi.org/10.1007/BF01396373
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  • 19
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    Algorithmes mixtes asynchrones. Etude de convergence monotone (1984)
    Springer
    Numerische Mathematik 44 (1984), S. 363-369 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65H10 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Description / Table of Contents: Summary In this paper we introduce two classes of iterative algorithms which we call “Asynchronous mixed algorithms” and we study their convergence under partial ordering. These algorithms can be implemented just as well on monoprocessors as on multiprocessors, and, along with their convergence study, constitute a generalization of the mixed classical “Newton-relaxation” algorithms.
    Notes: Résumé Nous introduisons dans cet article deux classes d'algorithmes itératifs que nous appelons «Algorithmes mixtes asynchrones» et nous en étudierons la convergence selon un ordre partiel. Ces algorithmes sont implémentables aussi bien sur les monoprocesseurs que sur les multiprocesseurs, et avec leur étude de convergence constituent une généralisation des algorithmes mixtes classiques «Newton-relaxation».
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    URL: http://dx.doi.org/10.1007/BF01405568
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  • 20
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    A quadratic approximation method for minimizing a class of quasidifferentiable functions (1984)
    Springer
    Numerische Mathematik 45 (1984), S. 411-430 
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    ISSN: 0945-3245
    Keywords: AMS MOS ; 90C30 ; 65K05 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary An unconstrained nonlinear programming problem with nondifferentiabilities is considered. The nondifferentiabilities arise from terms of the form max [f 1(x), ...,f n (x)], which may enter nonlinearly in the objective function. Local convex polyhedral upper approximations to the objective function are introduced. These approximations are used in an iterative method for solving the problem. The algorithm proceeds by solving quadratic programming subproblems to generate search directions. Approximate line searches ensure global convergence of the method to stationary points. The algorithm is conceptually simple and easy to implement. It generalizes efficient variable metric methods for minimax calculations.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01391417
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  • 21
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    Root finding by divided differences (1981)
    Springer
    Numerische Mathematik 37 (1981), S. 93-104 
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    ISSN: 0945-3245
    Keywords: AMS (MOS): 65H05 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary A recursive method is presented for computing a simple zero of an analytic functionf from information contained in a table of divided differences of its reciprocalh=1/f. A good deal of flexibility is permitted in the choice of ordinate and derivative values, and in the choice of the number of previous points upon which to base the next estimate of the required zero. The method is shown to be equivalent to a process of fitting rational functions with linear numerators to data sampled fromf. Asymptotic and regional convergence properties of such a process have already been studied; in particular, asymptotically quadratic convergence is easily obtained, at the expense of only one function evaluation and a moderate amount of “overhead” computation per step. In these respects the method is comparable with the Newton form of iterated polynomial inverse interpolation, while its regional convergence characteristics may be superior in certain circum-stances. It is also shown that the method is not unduly sensitive to round-off errors.
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    URL: http://dx.doi.org/10.1007/BF01396188
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  • 22
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    Superlinear convergence of symmetric Huang's class of methods (1981)
    Springer
    Numerische Mathematik 38 (1981), S. 209-218 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65K05 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary In this paper the problem of minimizing the functionalf:D∈R n →R is considered. Typical assumptions onf are assumed. A class of Quasi-Newton methods, namely Huang's class of methods is used for finding an optimal solution of this problem. A new theorem connected with this class is presented. By means of this theorem some convergence results known up till now only for the methods which satisfy Quasi-Newton condition are extended, that is the results of superlinear convergence of variable metric methods in the cases of exact and asymptotically exact minimization and the so-called direct-prediction case. This theorem allows to interpretate one of the parameters as the scaling parameter.
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    URL: http://dx.doi.org/10.1007/BF01397090
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  • 23
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    Quasi-Newton-Verfahren vom Rang-Eins-Typ zur Lösung unrestringierter Minimierungsprobleme (1981)
    Springer
    Numerische Mathematik 38 (1981), S. 219-228 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65K05 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary In this paper new Quasi-Newton methods of rank-one type for solving the unconstrained minimization problem are developed. The methods belong to the Oren-Luenberger class (for negative parametersΘ k ) and they generate always positive definite updating matrices. Moreover it is shown that these methods are invariant by scaling of the objective function.
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    URL: http://dx.doi.org/10.1007/BF01397091
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  • 24
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    Quasi-Newton-Verfahren vom Rang-Eins-Typ zur Lösung unrestringierter Minimierungsprobleme (1981)
    Springer
    Numerische Mathematik 38 (1981), S. 229-244 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65 K05 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary It is shown that the methods developed in part I of this paper [6] converge at leastn-step quadratically to the global minimum point of the objective functionf, if the Hessian off is uniformly positive definite, uniformly bounded and Lipschitz continuous and if the method is used with restart and asymptotically exact line search.
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    URL: http://dx.doi.org/10.1007/BF01397092
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  • 25
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    Some supplementary results on the 1+ $$\sqrt 2 $$ order method for the solution of nonlinear equations (1982)
    Springer
    Numerische Mathematik 38 (1982), S. 383-392 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65H10 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Recently an iterative method for the solution of systems of nonlinear equations having at leastR-order 1+ $$\sqrt 2 $$ for simple roots has been investigated by the author [7]; this method uses as many function evaluations per step as the classical Newton method. In the present note we deal with several properties of the method such as monotone convergence, asymptotic inclusion of the solution and convergence in the case of multiple roots.
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    URL: http://dx.doi.org/10.1007/BF01396439
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  • 26
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    Approximation of optimal distributed control problems governed by variational inequalities (1982)
    Springer
    Numerische Mathematik 38 (1982), S. 393-416 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 49D05, 49D07, 65K10 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary A method for approximating the optimal control and the optimal state for a class of distributed control problems governed by variational inequalities is given. It uses a Rayleigh-Ritz-Galerkin scheme, regularising techniques and a gradient algorithm. A numerical example is given.
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    URL: http://dx.doi.org/10.1007/BF01396440
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  • 27
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    An error analysis for the secant method (1982)
    Springer
    Numerische Mathematik 38 (1982), S. 427-445 
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    ISSN: 0945-3245
    Keywords: AMS(MOS) 65J15, 65G99 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Sharp a priori and a posteriori error bounds are given for the secant method for solving non-linear equations in Banach spaces. The numerical stability of this method is also investigated. The stability results are analogous to those obtained by Lancaster for Newton's method.
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    URL: http://dx.doi.org/10.1007/BF01396443
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  • 28
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    Ein lokal überlinear konvergentes Verfahren zur Bestimmung von Rückkehrpunkten implizit definierter Raumkurven (1982)
    Springer
    Numerische Mathematik 38 (1982), S. 455-465 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65H10, 47H17 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary A numerical technique is described for computing turning points of a space curveL implicitly defined by a nonlinear system ofn equations inn+1 variables. The basic idea is a local parametrization ofL where the parameter that gives the next iterate is determined by applying one step of the well-known method for minimizing a real function using cubic Hermite interpolation with two nodes. The method is shown to convergeQ-super-linearly and withR-order of at least two. A numerical example concerning the analysis of nonlinear resistive circuits shows the algorithm to work effectively on real life problems.
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    Optimal solution of nonlinear equations satisfying a Lipschitz condition (1984)
    Springer
    Numerische Mathematik 43 (1984), S. 225-240 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65 H 10 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary For a given nonnegative ε we seek a pointx * such that |f(x *)|≦ε wheref is a nonlinear transformation of the cubeB=[0,1] m into ℝ (or ℝ p ,p〉1) satisfying a Lipschitz condition with the constantK and having a zero inB. The information operator onf consists ofn values of arbitrary linear functionals which are computed adaptively. The pointx * is constructed by means of an algorithm which is a mapping depending on the information operator. We find an optimal algorithm, i.e., algorithm with the smallest error, which usesn function evaluations computed adaptively. We also exhibit nearly optimal information operators, i.e., the linear functionals for which the error of an optimal algorithm that uses them is almost minimal. Nearly optimal information operators consists ofn nonadaptive function evaluations at equispaced pointsx j in the cubeB. This result exhibits the superiority of the T. Aird and J. Rice procedure ZSRCH (IMSL library [1]) over Sobol's approach [7] for solving nonlinear equations in our class of functions. We also prove that the simple search algorithm which yields a pointx *=x k such that $$\left| {f(x_k )} \right| = \mathop {\min }\limits_{1 \leqq j \leqq n} \left| {f(x_j )} \right|$$ is nearly optimal. The complexity, i.e., the minimal cost of solving our problem is roughly equal to (K/ε) m .
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    Solving systems of polynomial equations by bounded and real homotopy (1984)
    Springer
    Numerische Mathematik 43 (1984), S. 397-418 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65H10, 58C99, 55M25 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary The homotopy method for solving systems of polynomial equations proposed by Chow, Mallet-Paret and Yorke is modified in two ways. The first modification allows to keep the homotopy solution curves bounded, the second one to work with real polynomials when solving a system of real equations. For the first method numerical results are presented.
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    On the convergence order of accelerated root iterations (1984)
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    Numerische Mathematik 44 (1984), S. 463-476 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65H05 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary A Gauss-Seidel procedure for accelerating the convergence of the generalized method of the root iterations type of the (k+2)-th order (k∈N) for finding polynomial complex zeros, given in [7], is considered in this paper. It is shown that theR-order of convergence of the accelerated method is at leastk+1+σ n (k), where σ n (k)〉1 is the unique positive root of the equation σ n -σ-k-1 = 0 andn is the degree of the polynomial. The examples of algebraic equations in ordinary and circular arithmetic are given.
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    Une méthode numérique pour le calcul des points de retournement. Application à un problème aux limites non-linéaire (1981)
    Springer
    Numerische Mathematik 37 (1981), S. 433-444 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65H10 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary An iterative method is presented for determining a turning point in a branch of solutions of an algebraic system of equations depending on a parameter. One shows that the convergence is superlinear and that this method may constitute a complement for the Newton method. With a finite element method, numerical results are given testing the process on the discrete version of a mildly nonlinear boundary value problem to determine the turning point in the positive solution branch.
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    Nonlinear successive over-relaxation (1984)
    Springer
    Numerische Mathematik 44 (1984), S. 309-315 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65H10 ; CR: 5.15
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    Topics: Mathematics
    Notes: Summary We study the convergence of Gauss-Seidel and nonlinear successive overrelaxation methods for finding the minimum of a strictly convex functional defined onR n .
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    A generalization of regula falsi (1980)
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    Numerische Mathematik 36 (1980), S. 333-346 
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    ISSN: 0945-3245
    Keywords: AMS (MOS) ; 65H10 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary The method of nondiscrete mathematical induction is applied to a multistep variant of the secant method. Optimal conditions for convergence as well as error estimates, sharp in every step, are obtained.
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    Global approximate Newton methods (1981)
    Springer
    Numerische Mathematik 37 (1981), S. 279-295 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65H 10 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We derive a class of globally and quadratically converging algorithms for a system of nonlinear equations,g(u)=0, whereg is a sufficiently smooth homeomorphism. Particular attention is directed to key parameters which control the iteration. Several examples are given that have been successful in solving the coupled nonlinear PDEs which arise in semiconductor device modelling.
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    Une méthode numérique pour le calcul des points de retournement. Application à un problème aux limites non-linéaire (1981)
    Springer
    Numerische Mathematik 37 (1981), S. 445-452 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65H10 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary A finite element method (P1) with numerical integration for approximating the boundary value problem −Δu=λe u is considered. It is shown that the discrete problem has a solution branch (with turning point) which converges uniformely to a solution branch of the continuous problem. Error estimates are given; for example it is found that $$\left| {\lambda _0 - \lambda _h^0 } \right| = 0(h^{2 - \varepsilon } )$$ , ɛ〉0, whereλ 0 andλ h 0 are critical values of the parameter λ for continuous and discrete problems.
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    Partitioned variable metric updates for large structured optimization problems (1982)
    Springer
    Numerische Mathematik 39 (1982), S. 119-137 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65H10 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary This paper presents a minimization method based on the idea of partitioned updating of the Hessian matrix in the case where the objective function can be decomposed in a sum of convex “element” functions. This situation occurs in a large class of practical problems including nonlinear finite elements calculations. Some theoretical and algorithmic properties of the update are discussed and encouraging numerical results are presented.
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    The effect of a thin inclusion of high rigidity in an elastic body (1980)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 2 (1980), S. 251-270 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: We study an elastic body comprising an inclusion whose thickness is 2∊ and for which Lamé's constants are λ∊ and μ. We are interested in the limit behaviour of the inclusion, when ∊ → 0 and μ∊ → ∊; that is when the inclusion becomes thiner and more rigid. Different behaviours are possible, according to the rate at which μ∊ converges to infinity; the limit inclusion may “vanish” if it is not very rigid, but also it may be entirely solid.
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    Nonselfadjoint operators in diffraction and scattering (1980)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 2 (1980), S. 327-346 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
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    Partial asymptotic stability via limiting equations (1981)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 3 (1981), S. 516-522 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: In this paper the limiting equations approach is applied to study the stability properties with respect to a part of the state variables for nonautonomous dynamical systems. Sufficient conditions are given for uniform asymptotic eventual strongly partial stability and for uniform asymptotic partial stability. An application of the results is given.
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    On spherical spline interpolation and approximation (1981)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 3 (1981), S. 551-575 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: Spherical spline functions are introduced by use of Green's surface functions with respect to the (Laplace-)Beltrami operator of the (unit) sphere. Natural (spherical) spline functions are used to interpolate data discretely given on the sphere. A method is presented that allows the smoothing of irregularities in measured values or experimental data. Extensions of Peano's theorem and Sard's theory of best approximation to the spherical case are given by integral formulas. Schoenberg's theorem is transcribed into spherical nomenclature.
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    Local solvability of a nonstationary leakage problem for an ideal incompressible fluid, 3 (1982)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 4 (1982), S. 1-14 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: In this paper we prove the existence and uniqueness of solutions of the leakage problem for the Euler equations in bounded domain Ω C R3 with corners π/n, n = 2, 3… We consider the case where the tangent components of the vorticity vector are given on the part S1 of the boundary where the fluid enters the domain. We prove the existence of an unique solution in the Sobolev space Wpl(Ω), for arbitrary natural l and p 〉 1. The proof is divided on three parts: (1) the existence of solutions of the elliptic problem in the domain with corners \documentclass{article}\pagestyle{empty}\begin{document}$$ {\rm rot }\upsilon {\rm = }\omega {\rm, div }\upsilon = 0,\upsilon \cdot \bar n||_{\partial \Omega } = 6 $$\end{document} where v - velocity vector, ω - vorticity vector and n is an unit outward vector normal to the boundary,(2) the existence of solutions of the following evolution problem for given velocity vector \documentclass{article}\pagestyle{empty}\begin{document}$$ \begin{array}{l} \omega _t + \upsilon ^\kappa \omega _x \kappa - \omega ^\kappa \upsilon _x \kappa = F \equiv {\rm rot }f \\ \omega |_{t = 0} = \omega _0,\omega |_{s1} = \eta \\ \end{array} $$\end{document}(3) the method of successive approximations, using solvability of problems (1) and (2).
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    Galerkin approximations for the equation Lu + Nu = f (1982)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 4 (1982), S. 286-290 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Topics: Mathematics
    Notes: Introducing the concept of a supercontinuous operator, we obtain a general convergence theorem for Galerkin approximations. Under the stronger assumption that N is a monotone operator with N(0) = 0, we show norm convergence of the unique Galerkin approximations to the unique solution.
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    Biorthogonal sequences of solutions of the generalized feller equation (1982)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 4 (1982), S. 317-353 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The generalized Feller equation is a linear, autonomous, parabolic equation of a positive space variable and a time variable. Its coefficients are power functions of the space variable, and they depend on four parameters. In general, the equation is singular at the origin and at infinity. It contains as special cases the special Feller equation, the Kepinski equation, and the standard heat equation. The main objective of the present paper is to establish series expansions of solutions of the generalized Feller equation in terms of the elements of two sequences of particular solutions. The elements of one of these sequences are particular initial condition solutions. The two sequences are biorthogonal. The main result is that a solution does have the desired expansion property if and only if it has the Huygens property in some neighborhood of the origin of the time variable.
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    Ein gemischtes randwertproblem der kreiszylinderschale mit beliebigen ausschnitten (1982)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 4 (1982), S. 354-381 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The problem of stress determination in the area of cut-outs in circular cylindrical shells at given loads is of great interest in industrial practice. This work deals with a mixed boundary value problem of a differential equation derived according to the theory of shallow shells. On part Ct1 of the boundary, the displacements are given, whereas the stresses are specified on the remaining part Ct2. Starting from the Betti-Maxwell principle and with the aid of the fundamental solutions for unit loads and unit displacements, integral representations can be derived for the displacement functions as well as the stress functions.The problem is then transformed into an equivalent system of Fredholm integral equations of the first kind with logarithmic kernels as the main part. As the integral equations together with the auxiliary conditions form a strongly elliptical system of pseudo-differential operators, the Galerkin method converges. Assuming that curves Ct1 and Ct1 do not have points of intersection and that the data are sufficiently regular, the required functions are approximated by cubic splines and, for simplicity's sake, the integral equation system is solved by approximation with a collocation method. In view of the complicated terms of the kernel functions, the kernels are split into a regular and a singular part, the regular part being in turn replaced by cubic splines. The remaining integrations are done numerically by means of Gaussian quadrature formulae. The applicability of the method is demonstrated with the example of a cylinder under internal pressure.
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    Zum 80. Geburtstag von Wolfgang Haack (1982)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 4 (1982), S. 451-453 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
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    Additional Material: 1 Ill.
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    Zur theorie rotierender und schwingender schaufelkränze in einer unterschallstromung durch einen ringkanalDie vorliegende Arbeit wurde von der Deutschen Forschungsgemeinschaft unter der Numraer Me 261/3 unterstützt. (1982)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 4 (1982), S. 454-479 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
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    Notes: In this paper the three-dimensional perturbation flow induced by a rotating and oscillating blade row which operates in a subsonic flow in axial direction of an annular channel is studied. The velocity potential is reduced to the infinite Hilbert space vector of Fourier coefficients of an eigen-function expansion with respect to vanishing normal derivatives on both cylinder walls. These coefficients satisfy an infinite set of ordinary differential equations of second order after an application of a one-dimensional Fourier transform in axial direction.Several canonical two-part mixed boundary value problems are then investigated by reduction to “infinite two-by-two-Wiener-Hopf functional systems”. In case of strong factorizability of certain matrix-operator-valued functions on the line these systems may be solved explicitely. Criteria for the factorization are not given here.
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    Über die existenz und eindeutigkeit einer lösung für ein randwertproblem für eine gleichung vom gemischten typ in einem rechteck (1982)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 4 (1982), S. 510-528 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
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    Notes: In the paper a boundary value problem is studied for the equation of mixed typek(y)uxx + uyy + r(x, y)u = f(x, y) in the rectangular domain {(x, y)| -1 〈 x 〈 +1, yc 〈 y 〈 yH} with yc 〈 0, yH 〉 0, k(y) = sign y|y|m, m 〉 0 (and more generally for a function k = k(y) with k(O) = 0, k(y)y 〉 O for y ≠ O). Specific for the stated problem is that no data are prescribed on the line {(x, yc), -1 〈 x 〈 +1}. It is proved that the formulated problem is well-posed in the sense that there is at most one quasi-regular solution and that a generalized solution exists. The energy-integral-(abc-)method is used to show uniqueness and to obtain an apriori estimate for the solution of the adjoint problem whence the existence statement follows.
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    Rounding errors in numerical solutions of two linear equations in two unknownsDedicated to Professor Wolfgang Haack on the occasion of his eightieth birthday. (1982)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 4 (1982), S. 549-571 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
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    Notes: New condition numbers and stability constants for the numerical behaviour of Cramer's rule and Gaussian elimination for solving two linear equations in two unknowns under data perturbations and rounding errors of floating-point arithmetic are established. By these means fundamental error estimates and stability theorems are proved. The error estimates are illustrated by a series of numerical examples.
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    The sommerfeld half-plane problem revisited, II the factoring of a matrix of analytic functionsThese results were announced at the XXth General Assembly of the U.R.S.I. held in Washington, D.C., August 1981. (1983)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 5 (1983), S. 14-21 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
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    Notes: A half-plane under plane wave excitation obeys a Dirichlet boundary condition on one side and a Neumann boundary condition on the other. These boundary conditions contrast the ones used by A. Sommerfeld in his classical paper. The present problem leads to a system of integral equations of the Wiener-Hopf type which may be solved by a matrix factoring method suggested by A. E. Heins in 1950.
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    Asymptotic equipartition of energy for equations of elasticity (1983)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 5 (1983), S. 186-194 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: In this paper we give a review of the equipartition of energy results of Goldstein and Sandefur [3], [4], [5] as well as proving a new result in the case of a particular fourth order equation. These results are then applied to the equations of elasticity to give a weak asymptotic orthogonality for the shear and pressure waves. In the case of boundary value problems in the interior of a bounded domain we get weak asymptotic orthogonality in the average.
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    Variational derivatives and p-gradients of functionals on spaces of continuously differentiable functions (1983)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 5 (1983), S. 530-543 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The purpose of this paper is to introduce and study a new type of derivative - the variational gradient - for a functional on Cn[a, b]. Local and global versions of this concept are analyzed. This notion provides a natural approach to variational derivatives on Cn[a, b] under rather mild smoothness assumptions on the functional. When applied in the context of the Calculus of Variations, the notion of the variational gradient captures the natural boundary conditions (as well as the Euler-Lagrange equations) under weaker smoothness assumptions than those usually required using Gǎteaux variations.Conditions are established for the existence of the variational derivative and an integral representation for the Gǎteaux variation in terms of the variational derivative is derived. Conditions for the variational derivative to be differentiable are also established.
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    Asymptotic study of the macroscopic behaviour of a solid-fluid mixture (1980)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 2 (1980), S. 1-11 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: In the framework of homogenization theory we study a mixture of an elastic solid and a viscous compressible fluid with periodic structure and its limit behaviour as the period tends to zero Existence, uniqueness and convergence theorems are given. The limit behaviour is viscoelastic.
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    Zwei Klassen vollständiger Funktionensysteme zur Behandlung der Randwertaufgaben der Schwingungsgleichung ΔU + k2U = 0 (1980)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 2 (1980), S. 48-67 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: In this paper we present new methods to solve the classical Dirichlet and Neumann problems for ΔU + k2U = 0. We prove that the solutions of this equation for a region S containing G restricted to G are dense in L2(∂G). Introducing a basis in the space of solutions for S we find a complete orthogonal system in L2(∂G) which can be used to solve the boundary value problems by means of approximation in the Hilbertspace norm. Regularity estimates lead to series expansions in G.The well-known basis systems obtained by separation of variables thus may be used for every regular region without the very special geometric restrictions. Another class of basis systems may be obtained in analogy to the Runge. theorems by considering types of singularity functions.
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    URL: http://dx.doi.org/10.1002/mma.1670020106
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    A singular perturbation approach for a reaction diffusion system arising in nuclear reactor theory (1980)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 2 (1980), S. 91-107 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
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    Notes: The qualitative behavior of the solutions of a reaction-diffusion system arising in the theory of nuclear reactors is investigated by means of singular perturbation techniques, the small parameter ∊ representing the inverse of neutron velocity. At the lowest approximation for small ∊ the solutions exhibit oscillations about the unique strictly positive equilibrium, much as in the case of the associated lumped parameter system. A two-times representation for solutions of small amplitudes is also given.
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    On uniqueness of Neumann-Tricomi problem in R2 (1980)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 2 (1980), S. 168-177 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
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    Notes: We consider the equation of mixed type (k(y) ⋛ 0 whenever y ⋛ 0) in a region G which is bounded by the curves: A piecewise smooth curve Γ lying in the half-plane y 〉 0 which intersects the line y = 0 at the points A(-1, 0) and B(0, 0). For y 〈 0 by a piecewise smooth curve Γ through A which meets the characteristic of (1) issued from B at the point P and the curve Γ which consists of the portion PB of the characteristic through B. We obtain sufficient conditions for the uniqueness of the solution of the problem L[u] = f, dnu: = k(y)uxdy - uydx|γ0 = = Ψ(s) for a “general” function k(y), when r(x, y) is not necessarily zero and Γ1 is of a more general form then in the papers of V. P. Egorov [6], [7].
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    URL: http://dx.doi.org/10.1002/mma.1670020204
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    Numerical solution of Hopf bifurcation problems (1980)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 2 (1980), S. 178-190 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
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    Notes: We present two numerical methods for the solution of Hopf bifurcation problems involving ordinary differential equations. The first one consists in a discretization of the continuous problem by means of shooting or multiple shooting methods. Thus a finite-dimensional bifurcation problem of special structure is obtained. It may be treated by appropriate iterative algorithms. The second approach transforms the Hopf bifurcation problem into a regular nonlinear boundary value problem of higher dimension which depends on a perturbation parameter ∊. It has isolated solutions in the ∊-domain of interest, so that conventional discretization methods can be applied. We also consider a concrete Hopf bifurcation problem, a biological feedback inhibition control system. Both methods are applied to it successfully.
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    Zur L∞-Konvergenz finiter Elemente bei parabolischen Differentialgleichungen (1980)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 2 (1980), S. 221-234 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: For a parabolic model problem it is shown that the convergence rate of higher order finite element approximation is quasioptimal in L∞. Moreover, the estimate does not depend on the interval [0, T]. The essential tool of the proof is a modified weighted norm technique.
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    Scattering of elastic waves through a heterogeneous medium (1980)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 2 (1980), S. 271-287 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: In this paper we study the elastic wave diffraction in R3 through a heterogeneous medium, with periodic structure, which occupies a bounded domain. We show that, as the period tends to zero, the solution tends, in some sense, to the solution corresponding to the diffraction by an obstacle made of the classical “homogenized medium”. An analogous result is also proved for the scattering frequencies and the associated scattering functions.
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    Masthead (1980)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 2 (1980) 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
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    Über die Gaußsche Methode zur angenäherten Berechnung von IntegralenMeinem verehrten Lehrer, Herrn Prof. Dr. Cl. Müller, in Dankbarkeit zum 60. Geburtstag gewidmet. (1980)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 2 (1980), S. 397-409 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: The purpose of this paper is the application of Green's theory and Green-Lagrange integral formulas relative to Legendre's differential operator to obtain integral expressions of remainder terms in Gaussian mechanical quadratures.
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    Zur numerischen Lösung gewöhnlicher Differential-gleichungen mit Splines in einem SonderfallHerrn Claus Müller zum 60. Geburtstag am 20. 2. 1980 gewidmet (1980)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 2 (1980), S. 419-428 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: In an earlier paper [1] a general procedure has been presented to obtain polynomial spline approximations for the solution of the initial value problem for ordinary differential equations. In this paper the general procedure is described by an equivalent one step method. Furthermore two convergence theorems are proved for a special case which is not included in the general convergence or divergence theory given in [1].
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    Some remarks on the horizontal line methodHerrn Claus Müller zum 60. Geburtstag am 20. 2. 1980 gewidmet (1980)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 2 (1980), S. 471-479 
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    Notes: In the following paper a methodological survey with respect to applications of the horizontal line method (Rothe's method) to a class of initial boundary value problems is given. By means of results from abstract perturbation theory, convergence results and error estimates are established for several special initial boundary value problems of mathematical physics.
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    Elektrodynamik mit SpinHerrn Prof. Dr. Claus Müller zu seinem 60. Geburtstag am 20. 2. 1980 gewidmet (1980)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 2 (1980), S. 457-470 
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    Notes: Atomistic equations of the electromagnetic field for a particle with spin are derived from a Lagrangian. These equations are consistent with the equations of motion for such a particle. The resulting phenomenological equations are the well-known equations of Maxwell for the electromagnetic field in matter. The atomistic field equations for a particle with spin and magnetic moment give a dipole field. This result and the corresponding quantum mechanics for a particle with spin are applied to compute the hyperfine structure of the hydrogen atom by perturbation theory.
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    On the boundary value problem of the biharmonic operator on domains with angular corners (1980)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 2 (1980), S. 556-581 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: The paper is concerned with boundary singularities of weak solutions of boundary value problems governed by the biharmonic operator. The presence of angular corner points or points at which the type of boundary condition changes in general causes local singularities in the solution. For that case the general theory of V. A. Kondrat'ev provides a priori estimates in weighted Sobolev norms and asymptotic singular representations for the solution which essentially depend on the zeros of certain transcendental functions. The distribution of these zeros will be analysed in detail for the biharmonic operator under several boundary conditions. This leads to sharp a priori estimates in weighted Sobolev norms where the weight function is characterized by the inner angle of the boundary corner. Such estimates for “negative” Sobolev norms are used to analyse also weakly nonlinear perturbations of the biharmonic operator as, for instance, the von Kármán model in plate bending theory and the stream function formulation of the steady state Navier-Stokes problem. It turns out that here the structure of the corner singularities is essentially the same as in the corresponding linear problem.
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    Bifurcation results for a class of periodic quasi-linear parabolic equations (1981)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 3 (1981), S. 11-20 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: We consider the problem where a and f are 1-periodic in t, a is positive, f satisfies appropriate decreasing conditions; smoothness of a, f, ∂Ω is also assumed. Denote by λ0 the principal eigenvalue of Δ with zero Dirichlet boundary conditions, and define . We prove: (a) if ε ≤ 0, then no non-negative periodic solution exists but zero, and any solution with continuous non-negative initial datum converges to zero uniformly as t → ∞; (b) if ε 〉 0, then a unique non trivial non-negative 1-periodic solution u* exists, and any solution with continuous, non-negative not identically zero initial datum approaches uniformly u* as t → ∞.
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    Free boundary problems for the laplace equation: A priori estimates, existence theorems, asymptotic behaviours (1981)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 3 (1981), S. 38-69 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: This work is devoted to Free Boundary Problems with Laplace equation Δu = f in the domain and the two conditions on the Free Boundary u = 1 and |grad u| = λ = const. In the model problems which we study, three cases arise: 1) f = 0, λ is given, 2) f = 0, λ is unknown and the length of the Free Boundary is given, 3) f ≠ 0, λ = 0 (Obstacle Problem).
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    A linear Uzawa-algorithm for the steady-state Navier-Stokes problem (1981)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 3 (1981), S. 115-120 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: An Uzawa algorithm of Temam is improved so that the Navier-Stokes boundary value problem is reduced to a sequence of linear Poisson problems. A simple finite element method is shown to give a convergent sequence of approximate solutions. Finally a penalty variant is discussed.
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    Eine Integralgleichungsmethode für akustische Reflexionsprobleme an lokal gestörten Ebenen (1981)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 3 (1981), S. 128-144 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: In this paper we consider the reflection of acoustic waves at an unbounded surface which coincides with a plane outside a sufficiently large sphere. We prove uniqueness and existence theorems for the corresponding boundary value problems for the reduced wave equation with Dirichlet and Neumann data by employing integral equation methods.
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    A singular perturbation problem for linear operators with an application to electrostatic and magnetostatic boundary and transmission problemsPart of this work was carried out while both authors were at the Department of Mathematical Sciences, University of Delaware, Newark, DE 19711, USA. (1981)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 3 (1981), S. 249-274 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: The treatment of certain electro- and magnetostatic boundary and transmission problems by boundary integral equations leads to parameter-dependent integral equations of the second kind. The integral operators involved have the property that the dimension of their nullspaces changes between two nonzero values (depending on the geometry of the problem) as the parameter tends to zero. We investigate the continuous dependence of solutions to these equations on the parameter. To this end, we treat the problem of continuous dependence of solutions to parameter-dependent linear operator equations of the second kind in a Banach space in the framework of generalized inverses.
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    Über das asymptotische Verhalten thermoelastischer Wellen im R3Diese Arbeit entstand mit Unterstützung des durch die Deutsche Forschungsgemeinschaft getragenen Sonderforschungsbereiches 72 an der Universität Bonn. (1981)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 3 (1981), S. 312-317 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: In [L] the author provided existence theorems for initial-boundary-value problems in thermoelasticity for unbounded domains. Continuing this work the asymptotic behaviour of solutions in R3 with respect to t → ∞ will be described in this paper.
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    Stable solutions of some ill-posed problems (1981)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 3 (1981), S. 336-363 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: Stable solutions of equations of the first kind and equations of the second kind at a characteristic value are given. Iterative processes for solutions are constructed. Extension of operators is used to turn an ill-posed problem into a well-posed one.
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    Mathematische Behandlung ernes Modells der Haemodialyse (1981)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 3 (1981), S. 393-404 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
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    Notes: In this paper a simple mathematical model for the process of hemodialysis is presented. This model is based on a system of two linear differential equations of first order with partly discontinuous coefficients that describe the time-development of the concentrations of a certain toxin (like urea) in the intra- and extracellular part of the human body. The main result is the existence of periodic positive solutions of this system under the natural assumption that the generation of the toxin and its removal by hemodialysis are periodic processes. These periodic positive solutions are also computed numerically for a realistic choice of the coefficients of the modelling differential equations.
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    On the convexity of equilibrium plasma configurations (1981)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 3 (1981), S. 435-443 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: We investigate the problem of plasma confinement by a magnetic field in an infinite cylinder. We show that if the cylinder has convex cross-section, then there exists an equilibrium plasma configuration with convex cross-section.
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    The impedance boundary value problem for the time harmonic Maxwell equations (1981)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 3 (1981), S. 475-487 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: The existence of a unique solution to Maxwell's equations defined in an exterior domain with impedance boundary condition is established for all frequencies. This is accomplished by reducing this problem to that of solving a system of singular integral equations and then regularizing this system such that the Riesz theory is applicable. We also consider the inverse problem in which it is desired to determine the impedance from a knowledge of the far field pattern. By restricting the impedance to lie a priori in a compact set results are obtained on the existence, uniqueness, and stability of the solution to this inverse scattering problem.
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    Study of motions of a vibrating string subject to solid friction (1981)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 3 (1981), S. 287-300 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: In this paper we consider a string moving in its plane and subject to solid friction (Coulomb's law). It is known that when the time increases indefinitely the string reaches an equilibrium position and by analogy with the case of a mass point we ask if the equilibrium is reached after a finite time. We prove that this is the case when the string is initially at rest and 1. when the initial shape possesses a second derivative bounded by certain limits, or 2. when the initial shape is formed by two straight line segments. In the last section we obtain some partial results when the string is initially at rest in the shape of a polygonal line. The case of an arbitrary initial position is still an open problem.
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    On complete recovery of geophysical data (1982)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 4 (1982), S. 33-73 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: One considers an elastic halfspace with depth (x1) dependent density Q and Lamé moduli (λ,m̈). Impulsive stresses τli = δ(x2, x3)δ(t) for x1 = 0 are applied with displacement responses ui = gi(t, x2, x3) at x1 = 0 (i = 1, 2, 3). Let vi(t, x1) = ∫∫uidx2dx3 (i = 1, 2 is enough) and set w(t, x1) = ∫∫x2u1dx2dx3. One obtains a system of 3 differential equations for v1, v2, and w to which the spectral techniques of inverse scattering theory are applied as in [25]. The inverse problems for the uncoupled vi can then be solved to produce 2 functions A1 and A2 involving (Q, λ, μ) as functions of “bound” variables y1 and y2 containing (Q, λ, μ), between which a relation then is determined. Analysis of the coupled equation for w then leads to a Fredholm integral equation whose solution provides an additional relation between (Q, λ, μ) from which (Q, λ, μ) can be determined as functions of x1. The integral equation is reduced to a Volterra type equation by results and techniques of transmutation and then solved by a modification of standard techniques. A number of features and results of independent mathematical interest arise from the transmutation theory.
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    Crank-nicolsen-Galerkin approximation for Maxwell's equations (1982)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 4 (1982), S. 123-130 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
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    Notes: The Maxwell equations are formulated as an evolution equation in a suitable chosen Hilbert space involving a densely defined closed skew-Hermitian operator which generates a unitary group. A Crank-Nicolsen-Galerkin approximation is then established and convergence is shown by arguments from the theory of approximation of groups of operators.
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    On the transition from supercritical to subcritical Hopf bifurcations (1982)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 4 (1982), S. 143-163 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
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    Notes: By using the method of averaging, we give a complete Hopf bifurcation diagram when the differential equation has more than one parameter. The method is applied to a system of parabolic equations with nonlinear coupling on the boundary. This set of equations is a mathematical model which describes the normalized concentrations of substances in a solution produced by two enzymes.
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    Bifurcation problems for discrete variational inequalities (1982)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 4 (1982), S. 243-258 
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    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The buckling of a beam or a plate which is subject to obstacles is typical for the variational inequalities that are considered here. Birfurcation is known to occur from the first eigenvalue of the linearized problem. For a discretization the bifurcation point and the bifurcating branches may be obtained by solving a constrained optimization problem. An algorithm is proposed and its convergence is proved. The buckling of a clamped beam subject to point obstacles is considered in the continuous case and some numerical results for this problem are presented.
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    Monotone schemes for semilinear elliptic systems related to ecology (1982)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 4 (1982), S. 272-285 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Semilinear elliptic systems of partial differential equations related to ecology are studied, with Dirichlet boundary conditions. Monotone sequences of functions which satisfy scalar equations are constructed so that they will converge to upper and lower bounds for the solutions of the systems. In case a related system has a unique positive solution, then these sequences will converge to the solution of the original system. Applications of the monotone sequences to uniqueness and stability are also given.
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    URL: http://dx.doi.org/10.1002/mma.1670040118
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    Spectral properties of certain stiff problems in elasticity and acoustics (1982)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 4 (1982), S. 291-306 
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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 two vibration problems containing a small parameter → 0: a) Vibration of an elastic, slightly compressible body, and b) acoustic vibration of a slightly viscous compressible barotropic fluid in a vessel.The asymptotics of eigenvalues for problem a) is studied by using a uniformly convergent expansion of the stiff type. After a re-scaling of the spectral parameter, the problem b) reduces to an analogous problem, and we prove that, as ε → 0, infinitely many eigenvalues converge to 0 (which is an eigenvalue of infinite multiplicity of the corresponding inviscid acoustic problem).
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/mma.1670040120
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    Spline methods in geodetic approximation problems (1982)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 4 (1982), S. 382-396 
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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 is concerned with spline methods in a reproducing kernel Hilbert space consisting of functions defined and harmonic in the outer space of a regular surface (e.g. sphere, ellipsoid, telluroid, geoid, (regularized) earth's surface). Spline methods are used to solve interpolation and smoothing problems with respect to a (fundamental) system of linear functional giving information about earth's gravity field. Best approximations to linear functionals are discussed. The spline of interpolation is characterized as the spline of best approximation in the sense of an appropriate (energy) norm.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/mma.1670040124
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    A minimax principle for nonlinear eigenvalue problems with applications to nonoverdamped systems (1982)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 4 (1982), S. 415-424 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The theory of Rayleigh functionals for non-linear eigenvalue problems T(λ) u = 0 is extended to cases where the functional is defined only on a proper subset. The theory applies to problems which do not satisfy an overdamping condition and yields a minimax characterization of eigenvalues. Applications to damped free vibrations of an elastic body are discussed.
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    URL: http://dx.doi.org/10.1002/mma.1670040126
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    Applications of bloch expansion to periodic elastic and viscoelastic media (1982)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 4 (1982), S. 433-449 
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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 an unbounded, elastic non homogeneous material with periodic structure. It is shown that the solution of the equations of motion can be expanded in eigenfunctions of periodic operators. The Bloch expansion is used to prove that when the wave-length is long compared to the period of the structure (related to a small parameter ε), the first term of the exact solution expansion, in powers of ε, is the solution of the equations obtained in homogenization theory. The case of a long memory linear viscoelastic material is then studied.
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    URL: http://dx.doi.org/10.1002/mma.1670040128
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    Ein nichtlineares Glättungsverfahren zur rekonstruktion von flugbahnen aus radardaten (1982)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 4 (1982), S. 497-509 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Data processing is an important tool in airspace surveillance and air traffic control today. In this paper the problem is treated how to reconstruct a flown trajectory from its correlated radar plots subject to a certain knowledge of the aircraft manoeuverability and the radar measurement statistics. A variational approach leads to a generalized smoothing spline method. Simulation results are presented.
    Additional Material: 2 Ill.
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    URL: http://dx.doi.org/10.1002/mma.1670040132
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    Masthead (1983)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 5 (1983) 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
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    URL: http://dx.doi.org/10.1002/mma.1670050101
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    Half-stable solution branches for ordinary bifurcation problemsThis paper is an extended version of a part of the author's work [2]. (1983)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 5 (1983), S. 1-13 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: It is shown that second order bifurcation problems with a positive, autonomous nonlinearity have a smooth branch of positive solutions which tends to infinity. Moreover, this branch satisfies a stability rule saying that the solutions are stable if the branch turns to the right and unstable if it turns to the left.
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    A critical case for the solvability of stefan-like problemsWork performed under the auspices of the Italian C.N.R., G.N.F.M. (1983)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 5 (1983), S. 84-96 
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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 one-phase one-dimensional Stefan problem with general data with the aim to investigate some open questions on existence of classical solutions. We show how existence and nonexistence are discriminated by the behavior of the initial datum in the neighborhood of the starting point of the free boundary.
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    URL: http://dx.doi.org/10.1002/mma.1670050107
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    Periodic solutions of u'(t) = -f(u(t - 1)): Multiplicity results and small period solutions (1983)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 5 (1983), S. 162-175 
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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 are concerned with the differential delay equationFor odd and continuous nonlinearities f: R → R. We want to prove existence and multiplicity criteria for so called “slowly oscillating” periodic solutions of (A).The oddness condition is of course a rather restrictive one, but due to results of Nussbaum [9], [10] and Peters [13],[14] and numerical studies of Jürgens, Peitgen and Saupe [7] and Hadeler [6] we know that even for odd nonlinearities f (A) may display a complicated dynamical behaviour. On the other hand the oddness condition allows a classification of symmetry properties of periodic solutions of (A).
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    URL: http://dx.doi.org/10.1002/mma.1670050112
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    Lösungen der poissongleichung und harmonische vektorfelder in unbeschränkten gebieten (1983)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 5 (1983), S. 233-255 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: In the first part of this paper we consider generalised solutions of the Poisson equation Δ U = F in open subsets of Rn(n ≥ 3) with Dirichlet or Neumann boundary data. We prove existence and uniqueness theorems, not only for the corresponding interior and exterior problems, but also for domains with boundaries extending to infinity. In the second part we discuss generalised harmonic fields in open subsets of R3 with vanishing Dirichlet or Neumann boundary condition.
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    URL: http://dx.doi.org/10.1002/mma.1670050117
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    Gruppentheoretische aspekte der signalübertragung und der kardinalen interpolationssplines I (1983)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 5 (1983), S. 195-215 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: As is well known the real Heisenberg nilpotent group A(R) constitutes the group-theoretic embodiment of the Heisenberg canonical commutation relations (CCR) of classical quantum mechanics. In this connection, quantum mechanics stands for the quantum-mechanical description, at a given instant of time, of a non-relativistic microparticle moving in the one-dimensional configuration space R and having the plane R2 as its (flat) phase space. In fact, the (nilpotent) Lie algebra n of Ã(R) reflects the Weyl equations which are global versions of the Heisenberg CCR. Unfortunately, the subject of Heisenberg nilpotent groups is outside quantum mechanics and all the more outside mathematical physics not as commonly known as it should be considering its wide range of applications in a variety of different fields. The present paper which has two parts aims to develop a central topic of nilpotent harmonic analysis, to wit, the microparticle model, the lattice model which will be realized on the Heisenberg compact nilmanifold, and the complex wave model (or Bargmann-Fock-Segal model) of the linear Schrödinger representation of Ã(R) in order to examine geometrically several applications which are governed by the real Heisenberg nilpotent group Ã(R). These applications are in Part I the classical Whittaker-Shannon sampling theorem which is of basic importance in signal processing, to wit, for the transmission of digital signals as well as analog signals, and in Part II the Subbotin-Schoenberg existence and uniqueness theorem of cardinal spline interpolation. Moreover, Part II indicates briefly some connections of the aforementioned models to the Wigner phase-space quasiprobability density function of quantum statistical mechanics via the Schwartz kernels theory on unimodular Lie groups, an approach to the cross- and autoambiguity functions of radar synthesis, and to the Zak transform of solid state physics. The second part also points out some relations of harmonic analysis of the finite nilpotent group A(Z/NZ) to periodic spline interpolants admitting N equidistant knots on the one-dimensional compact torus group T. These last examples should serve mainly as hints for some further lines of investigations in the field of applications of nilpotent harmonic analysis.
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    Das asymptotische verhalten der greenschen funktion N-irregulaärer eigenwertprobleme mit zerfallenden randbedingungen (1983)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 5 (1983), S. 331-345 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: We prove asymptotic estimates for the Green's function of N-irregular eigenvalue problems My = λNγ with splitting boundary conditions.In contrast to the N-regular case the Green's function G(x,ζ,λ) grows exponentially for |λ| → ∞ if x 〉 ζ.These estimates are fundamental for the expansion of functions into a series of eigenfunctions of N-irregular eigenvalue problems. In a subsequent paper it will be shown that this irregular behavior of G(x,ζ,λ) implies that only a very small class of functions can be expanded into a series of eigenfunctions of such problems.
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    URL: http://dx.doi.org/10.1002/mma.1670050122
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    Approximation of the signorini problem with friction, obeying the coulomb law (1983)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 5 (1983), S. 422-437 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The approximation of the Signorini problem with friction by mixed finite element method is studied. The relation between the continuous case and its finite dimensional discretization is analyzed.
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    URL: http://dx.doi.org/10.1002/mma.1670050127
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    Free boundary problem for a two-layer inviscid incompressible fluid (1983)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 5 (1983), S. 476-490 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The existence of a local (in time) classical solution of a free boundary problem for a two-layer inviscid incompressible fluid is shown. The method of successive approximations and the novel approach to Lagrangian coordinates of Solonnikov are used.
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    URL: http://dx.doi.org/10.1002/mma.1670050129
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    Masthead (1984)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 6 (1984) 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
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    URL: http://dx.doi.org/10.1002/mma.1670060101
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    Solvability of an initial boundary value problem for the euler equations in twodimensional domain with corners (1984)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 6 (1984), S. 1-22 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The aim of this paper is to prove the existence and uniqueness of local solutions of some initial boundary value problems for the Euler equations of an incompressible fluid in a bounded domain Ω ⊂ R2 with corners. We consider two cases of a nonvanishing normal component of velocity on the boundary. In three-dimensional case such problems have been considered in papers [12], [13], [14]. Similar problems in domains without corners have been considered in [2]-[6], [11]. In this paper the relation between the maximal corner angle of the boundary and the smoothness of the solutions is shown. The paper consists of four sections. In section 1 two initial boundary value problems for the Euler equations are formulated. In section 2 the existence and uniqueness of solutions of the Laplace equation in twodimensional domain with corners for the Dirichlet and Neumann problems is proved in the Sobolev spaces. In sections 3 and 4 we prove the existence and uniqueness of solutions of problems formulated in section 1, using the method of successive approximations.
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    Faedo-galerkin approximations in equations of evolution (1984)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 6 (1984), S. 41-54 
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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 the evolution equation u' = Au + Nu u(0) = φu is a function on a real interval [0, T] with values in a Hilbert space H, A is a linear operator in H generating a continuous semigroup eAt and N is a nonlinear operator in H.We show that 1Existence of the exact solution implies existence of the Faedo-Galerkin Approximations.2Existence of the Faedo-Galerkin Approximations implies existence of the exact solution.3Uniform convergence of the Faedo-Galerkin Approximations to the exact solution.The Paper consists of two parts. In the first five sections we require that A possesses a complete orthonormal system of eigenfunctions, in section 6 we drop this requirement.
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    URL: http://dx.doi.org/10.1002/mma.1670060104
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    Functional differential equations with certain monotonicity properties (1984)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 6 (1984), S. 97-103 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The present paper is concerned with the study of retarded differential equations in n-space, which satisfy some monotonicity properties. Sufficient conditions are given guaranteeing that solutions are contained in the probability (n - 1)-simplex. Existence and uniqueness of constant solutions are proved. It is also shown that every solution converges to a constant as t → ∞.
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    URL: http://dx.doi.org/10.1002/mma.1670060108
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    On the transmission problem of the helmholtz equation for quadrantsHerrn Prof. Dr. K. Nickel zum 60. Geburtstag am 9. 2. 1984 gewidmet (1984)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 6 (1984), S. 129-157 
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    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The plane transmission problem of the Helmholtz equation for quadrants is characterized by a one-dimensional singular integral equation, which refers to the Fourier transform of the normal derivative of the solution along the x-axis. It is derived by solving the transmission problem for the upper and the lower half-plane involving a Neumann condition at y = 0. This is done by a two-dimensional Laplace transform technique. The inverse Laplace transform with respect to the second cartesian coordinate and the restriction of this one to y = 0 then lead to the integral equation. Thereby the transmission conditions of the original problem at y = 0 have to be taken into account. The resulting integral equation is of generalized Wiener-Hopf-type. It is solved via the contraction theorem imposing restricting conditions on the wave numbers.
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    URL: http://dx.doi.org/10.1002/mma.1670060110
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