A constructive approach to the quantum (cosh ϕ)2 model. I. The method of the Gel'fand-Levitan-Marchenko equations

In a Hubert , with the aid of the postulated Gel'fand-Levitan-Marchenko quantum equations, one introduces the fields ₁(x) and ₂(x), which are the quantum analogues of the classical fields cosh (x) and sinh (x) in the sinh-Gordon model. It is shown that the fields _j(x) satisfy the Wightman axioms, including the invariance relative to reflections of space-time and mutual local commutativity. In addition, one proves the asymptotic completeness of the theory and one computes explicitly the scattering operator. In the developed approach, no cut-offs are used and, therefore, there are no renormalization effects.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 146, pp. 147–190, 1985.     

Asymptotic Analysis and Modeling of the Jointing of a Massive Body with Thin Rods

Asymptotic formulas are obtained for solutions of the anisotropic elasticity problem for a body with cavities into which thin rods are inserted, the outer ends of the rods being rigidly fixed. The surface of the body and the lateral surface of the rods are assumed load-free, but the entire elastic junction is subject to mass forces. The elastic materials are inhomogeneous and the stiffness of the rods may differ greatly from that of the body, their ratio being of the order h with an arbitrary exponent ; for = 0, the junction is homogeneous. Together with the asymptotic formulas, we construct and justify an asymptotic model of the junction. This model is applicable for a wide range of the exponent and preserves the parameter h in the conjugation conditions but is represented by a regularly perturbed problem. Since the leading asymptotic term involves fields with strong singularities, we have to give correct statements of the limit problem for a body with one-dimensional rods. For this purpose, we use the theory of self-adjoint extensions of operators or the technique of weighted spaces with separated asymptotics. The justification of our asymptotic expansions utilizes weighted anisotropic Korn inequalities, which take into account the mutual position of the rods and provide the best possible a priori estimates of the solutions. In contrast to other investigations, we describe, in explicit terms, the dependence of the bounds in the error estimates on the right-hand sides of the original problem. We also discuss the relationship between the asymptotic ansatz formulas and the weighted norms in the asymptotically precise Korn inequality.__________Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 24, pp. 95–214, 2004.     

The Wiener-Hopf method for systems of pseudodifferential equations with an application to crack problems

The aim of this paper is to develop the Wiener-Hopf method for systems of pseudo-differential equations with non-constant coefficients and to apply it to the describtion of the asymptotic behaviour of solutions to boundary integral equations for crack problems when a crack occurs in a linear anisotropic elastic medium. The method was suggested in [15] for scalar pseudo-differential equations with constant coefficients and applied in [7] to the crack problems in the isotropic case. The existence and a-priori smoothness of solutions for the anisotropic case has been proved in [11, 12], while the isotropic case has been treated earlier in [7, 25, 41, 50]. Our results improve even those for the isotropic case obtained in [7, 50]. Asymptotic estimates for the behaviour of solutions in the anisotropic case have been obtained in [28] by a different method.In memoriam, dedicated to Professor Dr. V.D. Kupradze on the occasion of the 90th anniversary of his birthThis work was carried out during the first author's visit in Stuttgart in 1992 and supported by the DFG priority research programme Boundary Element Methods within the guest-programme We-659/19-2.     

Tritronquée Solutions of Perturbed First Painlevé Equations

We consider solutions of the class of ODEs y=6y ²–x , which contains the first Painlevé equation (PI) for =1. It is well known that PI has a unique real solution (called a tritronquée solution) asymptotic to and decaying monotonically on the positive real line. We prove the existence and uniqueness of a corresponding solution for each real nonnegative 1.     

Asymptotic Density and the Asymptotics of Partition Functions

Let A be a set of positive integers with gcd (A) = 1, and let p _A (n) be the partition function of A. Let c ₀ = 2/3. If A has lower asymptotic density and upper asymptotic density , then lim inf log p _A (n)/c ₀ n and lim sup log p _A (n)/c ₀ n . In particular, if A has asymptotic density > 0, then log p _A (n) c₀n. Conversely, if > 0 and log p _A (n) c ₀ n, then the set A has asymptotic density .     

Small Ball Probabilities Around Random Centers of Gaussian Measures and Applications to Quantization

Let be a centered Gaussian measure on a separable Hilbert space (E, ). We are concerned with the logarithmic small ball probabilities around a -distributed center X. It turns out that the asymptotic behavior of –log (B(X,)) is a.s. equivalent to that of a deterministic function _R (). These new insights will be used to derive the precise asymptotics of a random quantization problem which was introduced in a former article by Dereich, Fehringer, Matoussi, and Scheutzow.⁽⁸⁾     

A simplicial approximation of a solution for the nonlinear complementarity problem

For a given map f from the n-dimensional Euclidean space En into itself, we consider the complementary problem of finding a nonnegative vector x in En whose imagef(x) is also nonnegative and such that the two vectors are orthogonal. It is the unifying mathematical form for several problems arising in different fields such as mathematical programming, game theory and economics.In this paper a new algorithm is developed based on the adjacent simplex technique , which was used by Garcia, Lemke and Lüthi for approximating an equilibrium point of a noncooperative n-person game. An almost-complementary path leads to a complementary simplex, which approximates a stationary point. Because most of the existence proofs for the nonlinear complementarity use the relationship between stationary points and complementarity, the algorithm gives constructive proofs for many existence theorems. If a better approximation is desired, the algorithm may be restarted from any point. The dimension of the simplices on the path is varying, which computationally should result in some savings.     

Krümmungskennzeichnung der Ketten einer allgemeinen Geometrie Σ(ℜ,\mathfrak{L})

Lie theory renders a concept of curvature for the geometries (,) introduced by W. Benz [1]. Among regular curves in the chains of (, ) are characterized by =0.     

Large transversals to small families of unit disks

Summary It is proved that if the nonempty intersection of bounded closed convex sets AnB is contained in (A + F)U(B+F) and one of the following holds true: (i) the space X is less-than-three dimensional, (ii) AUB is convex, (iii) F is a one-point set, then AnBCA+F or AnBCB+F (Theorems 2 and 3). Moreover, under some hypotheses the characterization of A and B such that AnB is a summand of AUB is given (Theorem 3).     

Stationary Cardinals

Summary This paper will define a new cardinal called aStationary Cardinal. We will show that every weakly ₁ ¹ -indescribable cardinal is a stationary cardinal, every stationary cardinal is a greatly Mahlo cardinal and every stationary set of a stationary cardinal reflects. On the other hand, the existence of such a cardinal is independent of that of a ₁ ¹ -indescribable cardinal and the existence of a cardinal such that every stationary set reflects is also independent of that of a stationary cardinal. As applications, we will show thatV=L implies ¹ holds if is ₁ ¹ -indescribable and so forth.     

Moduli of smoothness and best approximation in the spacesL p , 0<p<1

L ^p , 0<<1, .     

Martingale Hardy spaces,BMO and VMO spaces with nonlinearly ordered stochastic basis

VMO. , ⁺, «» VMO . « » .     

Some functional inequalities and their Baire category properties

Summary In the paper we consider, from a topological point of view, the set of all continuous functionsf:I I for which the unique continuous solution:I – [0, ) of(f(x)) (x, (x)) and(x, (x)) (f(x)) (x, (x)), respectively, is the zero function. We obtain also some corollaries on the qualitative theory of the functional equation(f(x)) = g(x, (x)). No assumption on the iterative behaviour off is imposed.     

On the solution of a generalized Schrödinger-type equation

Lu(t)+(u,F)g(t)=f(t), tS. , ( F, g). .

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The authors wish to thank Professor Yu. A. Rozanov for his help and discussions.     

Realisierung endlicher Gruppen als Galoisgruppen

A talk at the 1984 Oberwolfach meeting on Algebraic Number Theory will be summarized. It surveyed some new results on the realization of finite groups as Galois groups over the fields and ^ab, where ^ab is the maximal abelian extension field of .     

On the discrete mean square of Dirichlet <Emphasis Type="Italic">L</Emphasis>-functions at 1

There have been many results obtained so far for the mean square of the (absolute) value of the Dirichlet L-function L(s,) in the critical strip 0<<1, especially on the critical line , but relatively few results were known for discrete mean value of |L(1,)|² till W. Zhang had published papers improving the error term step by step, which have recently been superseded by M. Katsurada and K.Matsumoto in which they succeeded in deriving an asymptotic formula for ₀|L(1,)|². The object of our paper is to point out a structural property contained in the formation of the mean square, to find out the niryana–the true body of the above sum.Dedicated to Professor Jean Louis Nicolás on his sixtieth birthdayin final form: 7 October 2003     

Discrete Spectrum of Nonstationary Stochastic Processes on Groups

Vector-valued, asymptotically stationary stochastic processes on -compact locally compact abelian groups are studied. For such processes, we introduce a stationary spectral measure and show that it is discrete if and only if the asymptotically stationary covariance function is almost periodic. Using an almost periodic Fourier transform we recover the discrete part of the spectral measure and construct a natural, consistent estimator for the latter from samples of the process.     

Eine Bemerkung zu den höheren Pythagoraszahlen reeller Körper

Becker has shown in [1] that for the 4-th Pythagoras number of the field (X) the inequality P₄ ((X)) 36 holds. In this paper we will show P₄ ((X)) 24 and P₄ (K) 3 for all real pythagorean fields K.     

New classes of self-similar symmetric stable random fields

We construct two new classes of symmetric stable self-similar random fields with stationary increments, one of the moving average type, the other of the harmonizable type. The fields are defined through an integral representation whose kernel involves a norm on ⁿ . We examine how the choice of the norm affects the finite-dimensional distributions. We also study the processes which are obtained by projecting the random fields on a one-dimensional subspace. We compare these projection processes with each other and with other well-known self-similar processes and we characterize their asymptotic dependence structure.The research was done at Boston University while the first author was on leave from the Hugo Steinhaus Center, Poland. The second author was partially supported by the ONR Grant N00014-90-J-1287 at Boston University and by a grant of the United States-Israel Binational Science Foundation.     

Über eine Verallgemeinerung der Nicoletti-Aufgabe für Funktional-Differentialgleichungen mit voreilendem Argument

We consider a functional differential equation (1) (t)=F(t,) fort[0,+) together with a generalized Nicoletti condition (2)H()=. The functionF: [0,+)×C ⁰[0,+)B is given (whereB denotes the Banach space) and the value ofF (t, ) may depend on the values of (t) fort[0,+);H: C ⁰[0,+)B is a given linear operator and B. Under suitable assumptions we show that when the solution :[0,+)B satisfies a certain growth condition, then there exists exactly one solution of the problem (1), (2).