Die Nuklearität der Ultradistributionsräume und der Satz vom Kern II

Continuing the research of part I conditions equivalent to ()- or ()-nuclearity of spaces of ultradifferential functions and their duals as well as some applications are given. To get these results it is shown that tensor products of smooth sequence spaces, power series spaces, and spaces S(M_q) introduced in part I are isomorphic to suitable sequence spaces of the same class, which are stable provided the factors are stable power series spaces. Hence it is possible to establish isomorphisms between different functions spaces, to calculate the nuclearity types of tensor products by the nuclearity types of the factors, and to prove that the class of ()- or ()-nuclear spaces is closed under forming tensor products iff is multiplicatively stable.     

Calculi with monotone deductions and their economic interpretation

Let the possible ways of development of some system from the initial state X₀ be given by the deductive system <;X₀> (X₀ is an axiom, the algorithm defines the relation of deducibility in one step). Let Y₁,..., Y_e be all states directly derivable from X [i.e., (X)={y₁,..., Y_e]. Let be an algorithm assigning for each X transition probabilities p₁,..., p _l , where is the transition probability to the special state STOP. defines a probability measure on the set of all deductions. We define the information in the pair < ;X₀> by the forla: where px is the probability of being in X directly before STOP. We consider , assigning a fixed p for each X and satisfying the condition p₁= ...=p _l . Then the information in < ; X₀> becomes a function <;X₀> of one p. The essential characteristic of the system <;X > is given by the asymptotic behavior of <;X₀> as p 0. This characteristic corresponds well with the intuitive notion of the relative power of calculi. Now we consider <,X>(p) as a function of X. For many types of systems there is a useful strategy for maximizing this function (the strategy of increasing freedom of choice); we consider in this connection the simplest systems of economic character. Let X, Y, Z be n-dimensional vectors with nonnegative components (the components are interpreted as resources and products of a certain economic system, gives the technological possibility of transformations of the resources). Let Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 88, pp. 90–104, 1979.     

Parabolic subgroups of Chevalley groups over a semilocal ring

Let G be the Chevalley group over a commutative semilocal ring R which is associated with a root system . The parabolic subgroups of G are described in the work. A system =() of ideals in R ( runs through all roots of the system ) is called a net of ideals in the commutative ring R if ₊ for all those roots and for which + is also a root. A net is called parabolic if =R for >0. The main theorem: under minor additional assumptions all parabolic subgroups of G are in bijective correspondence with all parabolic nets . The paper is related to two works of K. Suzuki in which the parabolic subgroups of G are described under more stringent conditions.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 75, pp. 43–58, 1978.     

On the exactness of a theorem of G.A. Fomin

, . . [1] - . .     

On convergence and summability of Fourier series

. , , –1<<0. .

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The present work was written on the basis of two earlier works received byAnalysis Mathematica on January 16, 1979, and July 20, 1979.     

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).     

A chord-stretching map of a convex loop is an isometry

Let ⁿ be n-dimensional Euclidean space, and let : [0, L] ⁿ and : [0, L] ⁿ be closed rectifiable arcs in ⁿ of the same total length L which are parametrized via their arc length. is said to be a chord-stretched version of if for each 0s tL, |(t)–(s)| |(t)–(s)|. is said to be convex if is simple and if ([0, L]) is the frontier of some plane convex set. Individual work by Professors G. Choquet and G. T. Sallee demonstrated that if were simple then there existed a convex chord-stretched version of . This result led Professor Yang Lu to conjecture that if were convex and were a chord-stretched version of then and would be congruent, i.e. any chord-stretching map of a convex arc is an isometry. Professor Yang Lu has proved this conjecture in the case where and are C ² curves. In this paper we prove the conjecture in general.     

Eulerian numbers with fractional order parameters

Summary The aim of this paper is to generalize the well-known Eulerian numbers, defined by the recursion relationE(n, k) = (k + 1)E(n – 1, k) + (n – k)E(n – 1, k – 1), to the case thatn is replaced by . It is shown that these Eulerian functionsE(, k), which can also be defined in terms of a generating function, can be represented as a certain sum, as a determinant, or as a fractional Weyl integral. TheE(, k) satisfy recursion formulae, they are monotone ink and, as functions of , are arbitrarily often differentiable. Further, connections with the fractional Stirling numbers of second kind, theS(, k), > 0, introduced by the authors (1989), are discussed. Finally, a certain counterpart of the famous Worpitzky formula is given; it is essentially an approximation ofx in terms of a sum involving theE(, k) and a hypergeometric function.Dedicated to the memory of Alexander M. Ostrowski on the occasion of the 100th anniversary of his birth.     

Асимптоический Анализ Распределения Статистики Диксона

. u 60-u u     

Converse theorems for multidimensional Kantorovich operators

L _p [0, l]. . . - .

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Supported by National Science Foundation, Zhejiang Provincial Science Foundation of China, and Alexander von Humboldt Foundation of Germany.     

Interpolation of generalized analytical functions of complex variable

A general approach is proposed to the interpolation of x -analytical functions of a complex variable with an arbitrary –,+[Basis x -analytical functions whose imaginary pan is a polynomial in x, and y are obtained in explicit form.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 58, pp. 3–9, 1986.     

Modified (0, 2)-Interpolation on the Roots of Jacobi Polynomials. II (Convergence)

In [4] A. M. Chak, A. Sharma and J. Szabados characterized the Jacobi matrices P(,), (, > –1) for which the (0,2)-interpolation problem is regular. It follows from their result, that if n is odd and = , or if , are both odd integers and n > 1 + ( + )/2, then the (0,2)-interpolation problem is not regular. Recently, the author proved that for , both odd integers, the (0,2)-interpolation problem augmented with boundary (Hermite-type) conditions at the endpoints of the interval [–1,1] is regular. In this paper the convergence of this modified (0,2)-interpolation procedure is studied, if the inner nodal points are the roots of the ultraspherical polynomials with odd integer parameter.     

Gibbs states of the Hopfield model in the regime of perfect memory

Summary We study the thermodynamic properties of the Hopfield model of an autoassociative memory. IfN denotes the number of neurons andM (N) the number of stored patterns, we prove the following results: IfM/N 0 asN , then there exists an infinite number of infinite volume Gibbs measures for all temperaturesT<1 concentrated on spin configurations that have overlap with exactly one specific pattern. Moreover, the measures induced on the overlap parameters are Dirac measures concentrated on a single point and the Gibbs measures on spin configurations are products of Bernoulli measures. IfM/N , asN for small enough, we show that for temperaturesT smaller than someT()<1, the induced measures can have support only on a disjoint union of balls around the previous points, but we cannot construct the infinite volume measures through convergent sequences of measures.Work partially supported by the Commission of the European Communities under contract No. SC1-CT91-0695     

A generalized integral with applications to trigonometric series

, . . .

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The authors wish to thank the referee whose comments improved the presentation of the paper. In fact, the present form of Lemma 2, which was originally very long, is due to the referee.     

On the uniform (C, 1)-summability of Fourier series and integrability of series with respect to multiplicative orthonormal systems

H (G), f(g)H (G) , (, 1)- OHMC G. , OHMC, A. H. . , . , OHMC, lim supp _n=, , ,n .. . , 117 234 . . -      

A note on rational approximation rate for Müntz system {x λn } with λ n ↘ 0

[Zho2] {x ⁿ } , _n 0 n .

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Supported in part by an NSERC Postdoctoral Fellowship and a CRF grant of University of Alberta.     

Convolution semigroups and central limit theorem associated with a dual convolution structure

We consider hypergroups associated with Jacobi functions ⁽⁾ (x), (–1/2). We prove the existence of a dual convolution structure on [0,+[i(]0,s ₀]{{) =++1,s ₀=min(,–+1). Next we establish a Lévy-Khintchine type formula which permits to characterize the semigroup and the infinitely divisible probabilities associated with this dual convolution, finally we prove a central limit theorem.     

Singular perturbations of variational problems arising from a two-phase transition model

Given that, are two Lipschitz continuous functions of to ₊ and thatf(x, u, p) is a continuous function of × × ^N to [0, + [ such that, for everyx, f(x,·, 0) reaches its minimum value 0 at exactly two points(x) and(x), we prove the convergence ofF (u) = (1/) f (x, u, Du) dx when the perturbation parameter goes to zero. A formula is given for the limit functional and a general minimal interface criterium is deduced for a wide class of two-phase transition models. Earlier results of [19], [21], and [22] are extended with new proofs.