The integrability of conjugate functions

For any functionf of L(0, 2), we prove that there is a function L(0, 2) such that ¦(x)¦ = ¦f(x)¦ almost everywhere and L(0, 2), where is the conjugate of.Translated from Matematicheskie Zametki, Vol. 4, No. 4, pp. 461–465, October, 1968.     

The arithmetic of the characteristic Pólya functions

In the framework of the theory of D. Kendall's delphic semigroups are considered problems of divisibility in the semigroup of convex characteristic functions on the semiaxis (0,). Letn ()={:₁¦₁1 or ₁=}, and I_o()={: ₁¦ ₁ N()}. The following results are proved: 1) The semigroup is almost delphic in the sense of R. Davidson. 2) N() is a set of the type G which is dense in (in the topology of uniform convergence on compacta). 3) The class I_o() contains only the function identically equal to one.Translated from Matematicheskie Zametki, Vol. 21, No. 5, pp. 717–725, May, 1977.The author thanks I. V. Ostrovskii for the formulation of the problem and valuable remarks.     

A relation between Chung's and Strassen's laws of the iterated logarithm

Summary Let W(t) be a standard Wiener process and let f(x) be a function from the compact class in Strassen's law of the iterated logarithm. We investigate the lim inf behavior of the variable sup ¦W(xT)(2T loglog T)^–1/2–f(x)¦, 0x1 suitably normalized as T.This extends Chung's result valid for f(x)0, stating that lim inf.[ sup ¦(2T loglogT)^–1/2 W(xT)¦(loglog T)^–1]=/4 a.s. T 0x1     

Best approximation by splines on classes of periodic functions in the metric of L

We have obtained the exact value of the upper bound on the best approximations in the metric of L on the classes W^rH of functionsf C ₂ ^r for which ¦f ^(r) (x)-f ^(r) (x)) ¦ <(¦ x-xf) [ (t) is the upwards-convex modulus of continuity] by subspaces of r-th order polynomial splines of defect 1 with respect to the partitioning k/n.Translated from Matematicheskie Zametki, Vol. 20, No. 5, pp. 655–664, November, 1976.     

Some more general bounds on the residual term of the Lagrange interpolation formula

Error bounds are derived for the Lagrange interpolation formula and for the k-th derivative of the residual term of this formula in terms of the Lipschitz constant of the n-th derivative for the case with (n+1) nodes and also for the case when the functions satisfy a special condition: G x, y, z [a, b]: ¦ (x)(z–y) +(y)(x–z)+(z)(y–x)¦G¦(x–y)(y–z)(z–x)¦.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 73, pp. 27–32, 1992.     

Classes caractéristiques d''une π-algèbre et suite spectrale en K-théorie bivariante

Résumé Pour tout groupe discret et pour toute -algèbre D, la C ^*-algèbre D(E) (dont la définition exacte est donnée dans la section 4) est la version équivariante de la C ^*-algèbre C(B, D) des fonctions continues sur B, le classifiant du groupe, à valeurs dans D et qui s'annulent à l'infini. Si D désigne une autre -algèbre, nous définissons une suite spectrale en K-théorie bivariante dont les premiers termes sont donnés par les groupes H ^p (B, KK(D, D)) et qui converge (lorsque B est de dimension finie) vers KK(B; D(E), D(E)). Cette suite spectrale généralise celle de Kasparov mais est obtenue de manière différente: en étendant la définition des quasihomomorphismes aux C(X)-algèbres (X est une espace topologique localement compact), on a recours à des méthodes homotopiques telles les décompositions de Postnikov et le calcul des groupes d'homotopie des espaces d'équivalences d'homotopie. Sous certaines hypothèses, ces mÊmes constructions nous permettent de définir, pour toute -algèbre D, une obstruction, appelée classe secondaire de la -algèbre D, qui détermine la différentielle d ₂ de la suite spectrale de Kasparov.

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For all discrete group and all -algebra D, the C ⁺-algebra D(E) (whose exact definition is given in Section 4) is the equivariant version of the C ^*-algebra C(B, D) of continuous functions from B (the classifiant of the group) to D, vanishing at infinity. If D is another -algebra, we define a spectral sequence in bivariant K-theory whose first terms are given by the groups H ^p (B, KK(D, D)) and which converges (if B of finite dimension) to KK(B; D(E), D(E)). This spectral sequence generalises the spectral sequence given by Kasparov but it is obtained in a quite different way: by extending the definition of quasihomomorphisms to the C(X)-algebras (where X is a locally compact topological space), we use homotopical methods, like Postnikov decompositions and the calculus of homotopy groups of spaces of homotopy equivalences. Furthermore, under certain hypotheses, with these constructions, we define an obstruction, called the secondary class of the -algebra D, which determines the differential d ₂ of the Kasparov spectral sequence.

     

Some ratio limit Theorems for a general state space Markov Process

Summary Consider a discrete time parameter Markov Process with stationary probability functions, a general state spaceX and the Harris recurrence condition. This then implies the existence and essential uniqueness of a sigma-finite stationary measure. It is also assumed that the class of measurable sets contains single point sets. LetP ^(m)(x, S) denote them-step transition probability fromx toS andp ^(m)(x, ·), the component ofP ^(m)(x, ·) which is absolutely continuous with respect to. Let =C: C, for some and }. The paper here presented contains theorems of which the following is typical:Theorem: LetS with(S)>0, measurableBS, (B)>0 andqB with uniformly iny, yB for all non-negative measurable f. Then for all measurableAS with(A)>0,k=0,±1, ±2,... in measure onS. If the g.c.d. ()=1 and with(X)< then the above limit holds in measure onX.This research was partially supported by National Science Foundation Grant Gp-3906.The author would like to convey his appreciation to Professor Steven Orey for his invaluable guidance, advice and encouragement.     

Generalization of istratescu's theorem on contractive mappings in metric spaces

In this paper we consider mappings T:XX of metric spaces, satisfying the condition: , where is some right semicontinuous function. We prove that if is a nondecreasing function, ()< for >0, –() as ,, then the map T has a fixed point and for any pointxX. Interesting examples are given.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol, 83, pp. 73–82, 1979.     

Waterman classes and triangular sums of double Fourier series

Let ^a ={nln^a (n+1)}, where a R. The following results are established: For every &fnof ^a BV ((- ]²), the triangular partial sums of its Fourier series are uniformly bounded if a = -1, and converge everywhere if a < -1.For every a>0, there exists &fnof ^a BV ((- ]²) such that the triangular partial sums of its Fourier series are unbounded at the point (0;0).     

High-Energy Relations Between Elastic Scattering Amplitudes of Particles and Antiparticles

We find conditions under which the ratio of particle and antiparticle total scattering cross sections as s. If the forward elastic scattering amplitudes become purely real asymptotically, then their ratio tends to –1. We prove that /_–(s)1 for ₊(s)/m ², where m is the -meson mass. We show that the asymptotic relations obtained have finite-energy analogues for some processes.     

On Strongly π-Regular Group Rings

Let R be an associative ring with unit. An element x R is said to be left (right) -regular if there exist y R and a positive integer n such that . If x is both left and right -regular, then it is said to be strongly -regular. R is said to be a strongly -regular ring if all its elements are strongly -regular. In this paper we determine some conditions which are necessary or sufficient for a group ring to be strongly -regular.AMS Subject Classification (1991): 16E50, 16U99     

A generalization of a theorem of M. Riesz to the case of functions of several variables

The following theorem was proved by M. Riesz: Iff(x) L(–,),f(x) 0 and the conjugate functionf (x) is also integrable on [-, ], thenf(x) L log⁺L. The analog of this theorem for functions of several variables is established.Translated from Matematicheskie Zametki, Vol. 4, No. 3, pp. 269–280, November, 1968.     

The Harish-Chandra theorem for the quantum algebrau q (sl (3))

A basis of a quantum universal enveloping algebraU is constructed; the following theorem is proved with the help of this basis: For any nonzero element U, there exists a finite-dimensional representation such that(u) 0.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45 No. 3, pp. 436–439, March, 1993.     

Combinatorics and flexure

Combinatorial identities, trigonometric formulas, together with complex variable techniques are used to derive exact and closed expressions for the six flexure functions of certain isotropic cylinders under flexure. The cross sections are bounded either by the closed curvesr=a cosⁿ (/n) (–<) or the closed curvesr=asin(/n)ⁿ(–<), wheren isa positive integer (n>1).

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Résumé Des identiteés combinatoires et des formules trigonométriques avec des techniques de variables complexes sont utilisées pour dériver des expressions exactes et simples pour les six fonctions de flexion de quelques cylindres isotropiques. Les sections sont limitées par les courbes ferméesr=a cosⁿ /n(–) et les courbesr=asin/nⁿ(–) où est un entier positif (n>1).

     

The best one-sided approximation of the classes WrHω

In this paper we calculate the upper bounds of the best one-sided approximations, by trigonometric polynomials and splines of minimal defect in the metric of the space L, of the classes W^rH (r = 2, 4, 6, ...) of all 2-periodic functions f(x) that are continuous together with their r-th derivative f^r(x) and such that for any points x and x we have ¦f ^r (x) f^r (x) ¦ (x–x¦), where (t) is a modulus of continuity that is convex upwards.Translated from Matematicheskie Zametki, Vol. 21, No. 3, 313–327, March, 1977.     

On the best estimate for perimeters of plane sets with the angle property

We investigate the problem of finding the maximum length of perimeters of plane sets with fixed diameter d, such that every point of the boundary of the set is a vertex of an open angle of opening which does not intersect the set. First we consider plane curves which satisfy such angle property in a finite number of directions, and among them we find the one of maximum length. Then we prove that the perimeter of any plane set with the angle property is less than or equal to d(sin /2)^-2; this is the best estimate when /2.     

Langevin-Type Models II: Self-Targeting Candidates for MCMC Algorithms*

The Metropolis-Hastings algorithm for estimating a distribution is based on choosing a candidate Markov chain and then accepting or rejecting moves of the candidate to produce a chain known to have as the invariant measure. The traditional methods use candidates essentially unconnected to . We show that the class of candidate distributions, developed in Part I (Stramer and Tweedie 1999), which self-target towards the high density areas of , produce Metropolis-Hastings algorithms with convergence rates that appear to be considerably better than those known for the traditional candidate choices, such as random walk. We illustrate this behavior for examples with exponential and polynomial tails, and for a logistic regression model using a Gibbs sampling algorithm. The detailed results are given in one dimension but we indicate how they may extend successfully to higher dimensions.     

227-01227-01227-01-property and normal subgroups

Summary D-property (=set of primes) in finite groups is not in general inherited by subgroups. In this paper, as evidence in favor of the following conjecture (F. Gross): (o) If a finite group G satisfies D then its normal subgroups satisfy D-property as well. the Author shows that if the D and the D-properties (=set of the primes not in ) hold together in a finite group G, then both are inherited by the normal subgroups of G. As a corollary, the characterization of the groups satisfying both the properties D and D is given in terms of the composition factors.     

Formal Dimension for Semisimple Symmetric Spaces

If G is a semisimple Lie group and (, ) an irreducible unitary representation of G with square integrable matrix coefficients, then there exists a number d() such that