Regularization of Nonlinear Ill-Posed Variational Inequalities and Convergence Rates

Let H be a Hilbert space and K be a nonempty closed convex subset of H. For f H, we consider the (ill-posed) problem of finding u K for which 0 for all v K, where A : H H is a monotone (not necessarily linear) operator. We study the approximation of the solutions of the variational inequality by using the following perturbed variational inequality: for f H, f – f , find u, K for which 0 for all v K, where , , and are positive parameters, and K, a perturbation of the set K, is a nonempty closed convex set in H. We establish convergence and a rate O(1 / 3) of convergence of the solutions of the regularized variational inequalities to a solution of the original variational inequality using the Mosco approximation of closed convex sets, where A is a weakly differentiable inverse-strongly-monotone operator.     

On Piecewise-Constant Approximation of Continuous Functions of n Variables in Integral Metrics

We consider the approximation by piecewise-constant functions for classes of functions of many variables defined by moduli of continuity of the form (₁, ..., _n ) = ₁(₁) + ... + _n ( _n ), where _i ( _i ) are ordinary moduli of continuity that depend on one variable. In the case where _i ( _i ) are convex upward, we obtain exact error estimates in the following cases: (i) in the integral metric L ₂ for (₁, ..., _n ) = ₁(₁) + ... + _n ( _n ); (ii) in the integral metric L _p (p 1) for (₁, ..., _n ) = c ₁₁ + ... + c _{n n} ; (iii) in the integral metric L _{(2, ..., 2, 2r)} (r = 2, 3, ...) for (₁, ..., _n ) = ₁(₁) + ... + _{n – 1}( _{n – 1}) + c _{n n} .     

Generalizations of Perfect, Semiperfect, and Semiregular Rings

For a ring R and a right R-module M, a submodule N of M is said to be -small in M if, whenever N + X = M with M/X singular, we have X = M. If there exists an epimorphism p: P M such that P is projective and Ker(p) is -small in P, then we say that P is a projective -cover of M. A ring R is called -perfect (resp., -semiperfect, -semiregular) if every R-module (resp., simple R-module, cyclically presented R-module) has a projective -cover. The class of all -perfect (resp., -semiperfect, -semiregular) rings contains properly the class of all right perfect (resp., semiperfect, semiregular) rings. This paper is devoted to various properties and characterizations of -perfect, -semiperfect, and -semiregular rings. We define (R) by (R)/Soc(R_R) = Jac(R/Soc(R_R)) and show, among others, the following results:

Direct and inverse estimates for a singular Cauchy integral along a closed curve

A new metric characteristic() of rectifiable Jordan curves is introduced. We will find an estimate of the type of the Zygmund estimate for an arbitrary rectifiable closed Jordan curve in its terms. It is shown that the Plemel'-Privalov theorem on the invariance of Holder's spaces is true for the class of curves satisfying the condition(gd), which is much wider than the class of piecewise smooth curves (the presence of cusps is admissible). The Bari-Stechkin theorem on the necessary conditions of action of a singular operator in the spaces H is generalized. It is shown that this theorem is valid for every curve which has a continuous tangent at least at one point and ().Translated from Matematicheskie Zametki, Vol. 19, No. 3, pp. 365–380, March, 1976.The author is thankful to S. B. Stechkin, D. E. Men'shov, and P. L. Ul'yanov in whose seminars his attention was directed to the significance of the inverse estimate.     

Linearly similar models of Toeplitz operators

Let be a simply connected domain with a piecewise smooth boundary and assume that the function F is meromorphic in , does not have poles on , and the index of each point \F() with respect to the curve F() is nonnegative (at the positive traversal of the curve ). Under these assumptions, for a class of Banach spaces (including the Hardy-Smirnov spaces, the analytic spacesLip , the Bergman, Bloch spaces, etc.) one defines the Toeplitz operator T_F and one establishes its similarity to the operator of multiplication f·f by the projection of a specially selected Riemann surface _*=_*(T_F called the ultraspectrum of the operator T_F.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 157, pp. 113–123, 1987.The author is deeply grateful to his scientific advisor N. K. Nikol'skii for the formulation of the problem and for his constant interest in the paper, and also to A. A. Borichev, A. L. Vol'berg, and B. M. Solomyak for useful discussions.     

Generalization of Gδ*-Diagonals and ωΔ-Spaces

In this paper we introduce the concepts of a quasi-G*-diagonal and quasi--space as generalizations of the concepts of G*-diagonal and -space respectively. It is shown that a quasi-Moore space may be characterized in terms of these concepts. As a consequence we obtain the following metrization theorems: every paracompact -space with quasi-G-diagonal is metrizable and every collectionwise normal quasi--space is metrizable.     

On certain indestructibility of strong cardinals and a question of Hajnal

A model in which strongness of is indestructible under ⁺ -weakly closed forcing notions satisfying the Prikry condition is constructed. This is applied to solve a question of Hajnal on the number of elements of { |2 <}.     

Point homogeneous flat affine planes

All flat projective planes (P,) whose automorphism group contains a 2-dimensional, connected closed subgroup fixing at most one line 1, are classified, except the following 2 classes: (1) – L₂, and not transitive on P1, and (2) S₁xR.Research supported in part by National Research Council of Canada (NRC)Herrn Professor Dr. Werner Burau zum 70. Geburtstag     

On a Particular Class of Minihypers and Its Applications. I. The Result for General q

This article is the first in a series of three articles that discuss a particular class of minihypers and its applications. Proving that for small and < N, a {v _{+ 1}, v ; N, q}-minihyper consists of a sum of -spaces, we show that the excess points of an s-cover with excess of PG(N, q), (s + 1)|(N + 1), form a sum of s-spaces, and that no maximal partial s-spreads with deficiency of PG(N, q), (s + 1)|(N + 1), exist. The case q square will be studied in greater detail in [7] and further applications of these classification results on this class of minihypers will be published in [8].     

Thermal stability of a reactive slab with time-periodic heating of one surface

Summary The stability of time-periodic thermal oscillations in a slab with chemical reaction is examined. One surface is at a fixed temperature and the temperature of the other varies with amplitude and frequency. The reaction, which is of zero-order, is characterised by the Frank-Kamenetskii parameter(,,), where(=RT ₀/E) is the activation energy parameter. Perturbation series in and are used to calculate the critical values _crit; for _crit the slab can support time-periodic temperature oscillations.

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Zusammenfassung Die Stabilität von zeitperiodischen thermischen Schwingungen in einem Stab mit chemischer Reaktion wird untersucht. Eine Oberfläche hat eine feste Temperatur und auf der anderen verändert sich die Temperatur mit der Amplitude und der Frequenz. Die Reaktion der Ordnung null, wird vom Frank-Kamenetskii Parameter(,,) bestimmt, wobei(=RT ₀/E) der Aktivierungs-Energie Parameter ist. Reihenentwicklungen in und werden für die Berechnung des kritischen Wertes _crit benützt; für _crit können im Stab zeitperiodische Temperaturschwingungen bestehen.

     

On upper bounds of Fourier-Walsh coefficients

An upper bound is established for the upper bounds of the Fourier-Walsh coefficients a_n(f) whose modulus of continuity (,f) does not exceed a given modulus of continuity (). In the case of convex majorants of (), these bounds are attained for individual ordinal numbers n.Translated from Matematicheskie Zametki, Vol. 6, No. 6, pp. 725–736, December, 1969.     

Asymptotic behavior of a functional Gaussian integral

Functional Gaussian integrals relative to a Wiener measure are considered, where both the integrand and the integration measure depend on some parameter . Asymptotic relations are obtained for such integrals for ^–1/2 ( is the deviation), when and tend to 0. Such relations are useful in the investigation of the asymptotic behavior of expressions of the partition function type tr exp (—tH) for t 0.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 184, pp. 26–36, 1990.     

Parabelscharen und grenzkreise in der isotropen ebene

It is known by H. Sachs [5] that the classical curve theorem of ABRAMESCU also holds in isotropic geometry. Generalising an idea due to O. Röschel [2] we regard all inscribed parabolas (s, t) of a triangle (t). This triangle is formed by the tangents of three neighbouring points of a C -curve k(t) in an isotropic plane. Let U((t)) be the circumcircle of (t) and I((t)) the incircle of the triangle (t) whose midpoints of the sides are the vertices of (t). The circle U((t)) is the locus of the isotropic focal points of (s, t) and the incircle I((T)) the envelope of the isotropic axes of (s, t). We prove that the ABRAMESU-circle — lim U((t)) — is identical with the locus of the focal points of lim (s, t) and the circle lim I((t)) with the envelope of the axes of lim (s, t). The characteristic points, different from k(t), of the circles lim U((t)) and lim I((t)) determine the direction of the affine-normal of k(t).Herrn Professor Helmut Mäurer zum 60. Geburtstag gewidmet     

Generalization of the Hardy-Littlewood theorem on functions with derivatives in the space H1

Suppose f is a function that is analytic in the disk D = {z:¦z1¦ < 1} and belongs to the Hardy space H₁. Then, by the Hardy-Littlewood theorem, the following conditions are equivalent: (a) f H₁; (b) f coincides with some function of bounded variation almost everywhere on D; (c) almost everywhere on D, the function f coincides with some absolutely continuous function; (d) for an integral modulus of continuity (f, ) for the function f, we have (f, ) = O(). This article presents a generalization of this theorem to higher derivatives in the space H_p. The notions of generalized absolute continuity, generalized variation, and higher-order moduli of smoothness are used for this purpose.Translated from Matematicheskie Zametki, Vol. 52, No. 1, pp. 87–93, July, 1992.     

Sharp Error Bounds for Asymptotic Expansions of the Distribution Functions for Scale Mixtures

Let Z be a random variable with the distribution function G(x) and let s be a positive random variable independent of Z. The distribution function F(x) of the scale mixture X = sZ is expanded around G(x) and the difference between F(x) and its expansion is evaluated in terms of a quantity depending only on G and the moments of the powers of the variable of the form s/gr - 1, where (> 0) and (= ±1) are parameters indicating the types of expansion. For = -1, the bound is sharp under some extra conditions. Sharp bounds for errors of the approximations of the scale mixture of the standard normal and some gamma distributions are given either by analysis ( = -1) or by numerical computation ( = 1).     

Weak solutions of the Stokes problem in weighted Sobolev spaces

For n2 we consider the Stokes problem in ⁿ, -u + p=f, -divu=g, in weighted Soboiev spaces H ₆ ^m,r , where the weights are proportional to (1+|x|). We prove the existence of weak solutions for any K, whereK is a discrete set of critical values. Furthermore, we characterize the solutions of the homogeneous problem.This research was supported by the DFG research group Equations of Hydrodynamics, Universities of Bayreuth and Paderborn.     

Uniformities for the convergence in law and in probability

The convergence (X _n, Y_n)0 is investigated and characterized for probability metrics which metrize convergence in distribution or in probability. Some related metrics are also considered.     

Rarefied gas flow through long slots

The transition flow of a gas between two reservoirs through a two-dimensional slot is studied. The width of the slot is much smaller than the length but finite. The ratio of the equilibrium number densities in the reservoirs is near 1 and the equilibrium temperatures are the same. Galerkin's method of solution is applied to the linearized BGK equation and the total mass flow is evaluated as a function of the inverse Knudsen number and the length-to-half width ratioL. Limits 1, 1 andL 1 are investigated and good agreement with other authors is found. Simple analytical expressions are obtained for these cases.

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Riassunto Si studia la corrente gassosa tra due serbatoi uniti da un canale bi-dimensionale nel regime di transizione. La larghezza del canale è molto più piccola della sua lunghezza ma è finita. II rapporto tra le densità di molecole nei serbatoi è prossimo a 1 e le temperature d'equilibrio sono le stesse. II metodo di soluzione di Galerkin viene applicato all'equazione di Bhatnagar, Gross e Krook linearizzata e si valuta la protata in funzione del numero inverso di Knudsen e del rapportoL tra lunghezza e semi-larghezza. Si studiano i limiti 1, 1 eL1 e si trova un boun accordo con precedenti lavori. Per questi casi si trovano semplici espressioni analitiche.

     

Some remarks on factoriality of certain hypersurfaces in \mathbb{P}^4

Let V be a reduced and irreducible hypersurface of degree k 3. In this paper we prove that if the singular locus of V consists of ₂ ordinary double points, ₃ ordinary triple points and if ₂ + 4₃ < (k – 1)², then any smooth surface contained in V is a complete intersection on V.Received: 7 January 2004     

The order of magnitude of the arithmetic means of double orthogonal series

, - , (C, 1,1), (C, 1,0) (C, 0,1)- . , , . - .

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Dedicated to Academician S. M. Nikol'skii on his 80th birthday