Kennzeichnungen von Minimalschraubflächen Mittels Vertikalisophoten und Horizontalschattengrenzen

In Euclidean space E³, let be a (regular C-) minimal surface without planar points having locally (without loss of generality) the spherical representation n(u,v)=(cos v/cosh u, sin v/cosh u, tanh u), (u,v)G². The corresponding (isothermal) parametrization : x(u,v), (u,v)G can be expressed using agenerating Function (u,v) which satisfies _{uu + vv} – 2_utanh u + =0; the v-curves (coordinate curves u=u₀) in , along each of which the angle between the normal n(u,v) of and the x₃-axis is constant, are thevertical- isophotes of , the u-curves (v=v₀) being their orthogonal trajectories (theorems 1, 2). Considering u-curves and/or v-curves of having additional geometric properties (curves of constant/steepest slope, curves of constant Gaussian curvature, asymptotic curves, lines of curvature or geodesies of ) we prove many newgeometric characterizations of theright helicoid, thecatenoid andScherk's second surface (theorems 3–7). All of these surfaces areminimal hélicoidal surfaces.     

Regularity of solutions and interfaces of a generalized porous medium equation inR N

Summary We consider the Cauchy problem for the generalized porous medium equation u_t=(u) where u=u(x, t), xRⁿ and t>0, and the initial datum u(x, 0) is assumed to be nonnegative, integrable mid to nave compact support. The nonlinearity (u) is a C¹ function defined for uO which grows like a power of u. Our assumptions generalize the porous medium case, (u)=u^m, m>1, and also include the equation of the Marshak waves. This problem has finite speed of propagation. We estimate the rate of growth of the support of the solution with precise estimates for t 0 and t. Our main result deals with the regularity of the solutions. We show that after a certain time t₀ the pressure, defined by v=(u), with (u)=(u)/u and (0)=0, is a Lipschitz-continuous function of x and t and the interface is a Lipschitz-continuous surface in R^N+1; the solution u is Hölder continuous for all times t> 0.Both authors partially supported by CAICYT, Project 2805-83. The second author also supported by USA-Spain Joint Research Grant CCB-8402023.     

Group Classification of Nonlinear Schrödinger Equations

We propose an approach to problems of group classification. By using this approach, we perform a complete group classification of nonlinear Schrödinger equations of the form i _t + + F(, *) = 0.     

Translation Invariant Positive Definite Hermitian Bilinear Ultradistributions

Every translation invariant positive definite Hermitian bilinear functional on the Gel'fand-Shilov space sMpMp(n×nK) of general type S is of the form B(,) = (x)(x)d(x), , sMpMp (n), where is a positive {M}-tempered measure, i.e., for every > 0 exp[-M(|x|)] d(x) < . To prove this we prove Schwartz kernel theorem for {M}-tempered ultradistributions and need Bochner-Schwartz theorem for {M}-tempered ultradistributions. Our result includes most of the quasianalytic cases. Also, we obtain parallel results for the case of Beurling type (Mp.     

Joint torsion of Toeplitz operators with H∞ symbols

We give a new² index theorem for the basic example of Toeplitz operators on the circle. The joint torsion, a non zero complex valued analytic index, of a pair of Fredholm Toeplitz operatorsT andT withH symbols is computed by residues in the disk, and is determined by a monodromy integral which specifies the isomorphism class of a flat line bundle on the circle. When the symbols and are rational a product of joint torsions identifies the isomorphism class of the bundle inH ¹ (S ¹,C ^*), and the identification extends by rational approximation to the case of smooth symbols defined on the circle.Partially supported by National Science Foundation grants to both authors.     

Stabilization theorem for the Milnor K2-functor

Let be an associative ring. For every natural number n there is a canonical homomorphism _n: K_2,n()K₂(), where K₂ is the Milnor functor and K_2,n() the associated unstable K-group. Dennis and Vasershtein have proved that if n is larger than the stable rank of , _n is an epimorphism. It is proved in the article that if n – 1 is greater than the stable rank of , the homomorphism _n is an isomorphism.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 64, pp. 131–152, 1976.     

Einschliessungsaussagen für Differentialoperatoren vierter Ordnung und ein Verfahren zur Berechnung von Schrankenfunktionen

For a linear fourth order ordinary differential operator M we study Range Domain Implications (RDI). Let C_o [O,1] be positive; we show under what conditions there exists a C_O[O,1] such that the following RDI holds: Mu(x) (x) (0x1) u(x) (0x1). In particular we provide a numerical procedure to calculate .RDI are used to obtain error estimations and to solve related nonlinear problems.The basic idea to prove RDI is to split M into a product of second order differential operators which are easier to handle. For the general case that there exists no global splitting the concept of a local splitting is introduced.

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The author would like to thank the European Research Office of the United States Army for their kind interest.     

Asymptotic estimates of approximation of continuous periodic functions by Fourier sums

Asymptotic estimates, expressed in terms of the value of the modulus of continuity of r-th order (r2) at the point t=/n of a functionf C ₂ or of the (, )-derivative of a functionf C _B C, are established for the deviations of continuous periodic functions from their Fourier sums.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 6, pp. 747–755, June, 1990.     

Existence for the Cahn-Hilliard phase separation model with a nondifferentiable energy

Summary The Cahn-Hilliard model for phase separation in a binary alloy leads to the equations (I) u_t=w, (II) w= (u)– u with an associated energy functional F(u)=f [(u)+ +¦u¦²/2] dx. In this paper we discuss the existence theory for initial bounday value problems arising from modifications to the Cahn-Hilliard model due to the addition of the non-differentiable term ¦u¦dx to the energy F(u).     

Asymptotic normality of p-norm estimators in multiple regression

Summary The consistency and asymptotic normality of p-norm estimators (1<p<2) is established by applying some of the ideas of Huber (1973), where asymptotic normality of the so-called M-estimators for regression parameters is shown. A central role is played by a weight function . Huber assumed that , and are bounded. This is, however, not the case for p-norm estimators with 1<p<2, but some of his ideas can still be applied.     

Büschel linearer Abbildungen

If , , are linear mappings out of a projective space (P,G) into a projective space (P', G') and , then is said to belong to the pencil <,<> of linear mappings spanned by and if in the main (x), (x), (x) are collinear for all x P. We give some sufficient conditions for x P and , , such that (x) is uniquely determined by giving, and (z), z P.

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Herrn Prof. Dr.Helmut Karzel zum 60. Geburtstag gewidmet     

Minimization of the maximal penalty in the case of preemption of jobs

For a class of structural sets of penalty functions={_i} _i=1 ⁿ with lower quasiconvex functions _i defined for sets of jobs={_i} _i=1 ⁿ , one gives an algorithm for solving the problem n /1/ preemp ¦ max, having order 0(np), where n is the number of jobs _i and p is the total length of the completion of all jobs of the set.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 102, pp. 61–67, 1980.In conclusion, the author expresses her gratitude to K. V. Shakhbazyan for his interest in this paper.     

Sur les équations fonctionnelles auxq-différences

Summary In this paper, we study the convergence of formal power series solutions of functional equations of the formP _k(x)(^[k](x))=(x), where ^[k] (x) denotes thek-th iterate of the function.We obtain results similar to the results of Malgrange and Ramis for formal solutions of differential equations: if(0) = 0, and(0) =q is a nonzero complex number with absolute value less than one then, if(x)=a(n)x ⁿ is a divergent solution, there exists a positive real numbers such that the power seriesa(n)q ^sn(n+1)2 x ⁿ has a finite and nonzero radius of convergence.

     

On the parallel factorization of matrices over principal ideal rings

We describe all the factorizations A=BC (up to associates) of a matrix A over a commutative principal ideal domain parallel to the factorization D^A= of its canonical diagonal form D^A ( and are diagonal matrices), that is, the factorizations such that the matrices B and C are equivalent to the matrices and respectively.Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 4, 1997, pp. 96–100.     

Asymptotic behavior for t→∞ of the solutions of the equation ψxx+u(x,t)ψ+(λ/4)ψ=0 with a potential u,satisfying the Korteweg-de Vries equation. II

This is the third paper in a series of papers of the authors, devoted to a rigorous investigation of the asymptotic behavior of the solutions of the KdV equation as t. The immediate purpose of the paper is the investigation of the solution of the Schrödinger equation in the neighborhood of the singular point x=3t for a special class of potentials, introduced in the previous papers. As it will be proved, in the final analysis this class of potentials describes the asymptotic behavior of the solutions, decreasing for x, of the KdV equation as t. The solution , far from the singular point, has been investigated earlier. In the paper we investigate a series which gives the solution of the Schrödinger equation and we consider the asymptotic properties of this series.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 138, pp. 8–32, 1984.     

Asymptotic Characteristics of Compact Sets and Stepanets Classes

We obtain estimates for the -entropy and -capacity of sets of periodic functions with mean value zero that have a (, )-derivative belonging to the space L ²(0, 2).     

Integral averages and oscillation of second order sublinear differential equations

New oscillation criteria are given for the second order sublinear differential equation

Elliptic boundary-value problems with discontinuous nonlinearities

For a class of elliptic boundary value problems with discontinuous nonlinearities, the existence of strong solutions is established. Two applications are then developed. In particular, one of them is devoted to implicit elliptic equations of the form (–u)=(u), where is a continuous function and has a set of discontinuity points of Lebesgue measure zero. The abstract framework where these problems are studied is that of set-valued analysis.     

Inclusion statements for operator equations by a continuity principle

The paper is concerned with Range-Domain Implications MvCvK, where M is a given operator and C,K denote given sets. Sufficient conditions are derived by a very general continuity principle. Various special cases are considered such as inverse-positivity, MvMwvw, and a generalization H(,[,])MvH(,[,]) v, where Mu=H(u,u) and [,] denotes an order interval. These results are applied to differential operators related to boundary or initial value problems. The goal is to furnish a simple uniform approach, to explain its application, and to provide a kind of survey on what problems have been treated in this way.