Fractional Modified Dyadic Integral and Derivative on ℝ<Subscript>+</Subscript>

For functions in the Lebesgue space L(ℝ₊), a modified strong dyadic integral J _α and a modified strong dyadic derivative D ^(α) of fractional order α > 0 are introduced. For a given function f ∈ L(ℝ₊), criteria for the existence of these integrals and derivatives are obtained. A countable set of eigenfunctions for the operators J _α and D ^(α) is indicated. The formulas D ^(α)(J ^α(f)) = f and J _α(D ^(α)(f)) = f are proved for each α > 0 under the condition that . We prove that the linear operator is unbounded, where is the natural domain of J _α. A similar statement for the operator is proved. A modified dyadic derivative d ^(α)(f)(x) and a modified dyadic integral j _α(f)(x) are also defined for a function f ∈ L(ℝ₊) and a given point x ∈ ℝ₊. The formulas d ^(α)(J _α(f))(x) = f(x) and j _α(D ^(α)(f)) = f(x) are shown to be valid at each dyadic Lebesgue point x ∈ ℝ₊ of f.__________Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 39, No. 2, pp. 64–70, 2005Original Russian Text Copyright © by B. I. GolubovSupported by the Russian Foundation for Basic Research (grant no. 05-01-00206).     

A continuity property for the inverse of mañé's projection

Let X be a compact subset of a separable Hilbert space H with finite fractal dimension d _F(X), and P ₀an orthogonal projection in H of rank greater than or equal to 2 d _F (X) + 1. For every > 0, there exists an orthogonal projection P in H of the same rank as P ₀, which is injective when restricted to X and such that P – P ₀ < This result follows from Mañé's paper. Thus the inverse (P|_X)^–1 of the restricted mapping P|_X: X PX is well defined. It is natural to ask whether there exists a universal modulus of continuity for the inverse of Mañé's projection (P| _X )^–1. It is known that when H is finite dimensional then (P| _X )^–1 is Hölder continuous. In this paper we shall prove that if X is a global attractor of an infinite dimensional dissipative evolutionary equation then in some cases (e.g. two-dimensional Navier-Stokes equations with homogeneous Dirichlet boundary conditions) for every x, y X such that , where is a positive constant.     

Berger-Shaw's theorem forp-hyponormal operators

For an-multicyclicp-hyponormal operatorT, we shall show that |T|^2p –|T ^*|^2p belongs to the Schatten and that tr Area ((T)).     

Interpolation of Intersections Generated by a Linear Functional

Let (X ₀, X ₁) be a Banach couple such that X ₀ ∩ X ₁ is dense in X ₀ and X ₁. By (X ₀, X ₁)_θ,q , 0 < θ < 1, 1 ⩽ q < ∞, we denote the spaces of the real interpolation method. Let ψ be a nonzero linear functional defined on some linear space M ⊂ X ₀ + X ₁ and such that ψ ∈ (X ₀ ∩ X ₁)*, and let N = Ker ψ. We examine conditions under which the natural formula

On quasi degree quadrature rules

Summary In this paper we examine quadrature rules for the integral which are exact for all with +d. We specify three distinct families of solutions which have properties not unlike the standard Gauss and Radau quadrature rules. For each integerd the abscissas of the quadrature rules lie within the closed integration interval and are expressed in terms of the zeros of a polynomialq _d(y). These polynomialsq _d(y), (d=0, 1, ...), which are not orthogonal, satisfy a three term recurrence relation of the type Q_d+1(y)=(y+_d+1)q_d(y)–_d+1yq_d–1(y) and have zeros with the standard interlacing property.This work was supported by the Applied Mathematical Sciences Research Program (KC-04-02) of the Office of Energy Research of the U.S. Department of Energy under Contract W-31-109-Eng-38     

On Dirichlet Processes Associated with Second Order Divergence Form Operators

Let be a d - dimensional Markov family corresponding to a uniformly elliptic second order divergence form operator. We show that for any quasi continuous in the Sobolev space the process (X) admits under P ^x a decomposition into a martingale additive functional (AF) M and a continuous AF A of zero quadratic variation for almost every starting point x if q=2, for quasi every x if q>2 and for every if is continuous, d=1 and or d>1 and q>d. Our decomposition enables us to show that in the case of symmetric operator the energy of A equals zero if q=2 and that the decomposition of (X) into the martingale AF M and the AF of zero energy A is strict if for some q>d. Moreover, our decomposition provides a probabilistic representation of A .     

On the Self-Similar Jordan Arcs Admitting Structure Parametrization

We study the attractors γ of a finite system of contraction similarities S _j (j = 1,..., m) in ℝ^d which are Jordan arcs. We prove that if a system possesses a structure parametrization (ℐ,ϕ) and ℱ(ℐ) is the associated family of ℐ then we have one of the following cases:1. The identity mapping Id does not belong to the closure of ℱ(ℐ). Then (if properly rearranged) is a Jordan zipper.2. The identity mapping Id is a limit point of ℱ(ℐ). Then the arc γ is a straight line segment.3. The identity mapping Id is an isolated point of .We construct an example of a self-similar Jordan curve which implements the third case.Original Russian Text Copyright © 2005 Aseev V. V. and Tetenov A. V.The authors were supported by the Program “Universities of Russia” (Grant UR.04.01.456).__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 4, pp. 733–748, July–August, 2005.     

RELATIVE WIDTH OF SMOOTH CLASSES OF MULTIVARIATE PERIODIC FUNCTIONS WITH RESTRICTIONS ON ITERATED LAPLACE DERIVATIVES IN THE L2-METRIC

For two subsets W and V of a Banach space X, let Kn(W, V, X) denote the relative Kolmogorov n-width of W relative to V defined by Kn (W, V, X) := inf sup Ln f∈W g∈V∩Ln inf ‖f-g‖x,where the infimum is taken over all n-dimensional linear subspaces Ln of X. Let W2(△r) denote the class of 2w-periodic functions f with d-variables satisfying ∫[-π,π]d |△rf(x)|2dx ≤ 1,while △r is the r-iterate of Laplace operator △. This article discusses the relative Kolmogorov n-width of W2(△r) relative to W2(△r) in Lq([-r, πr]d) (1 ≤ q ≤∞), and obtain its weak asymptotic result.     

Two limit theorems on ARIMA models

Let the time series {X(t), t=1, 2, ...} satisfy (B)(1–B) ^d X(t)=(B)e(t), whereB is a backward shift operator, defined byBX(t)=X(t–1), and (z)=1+₁ z+...+ _p z ^p , (z)=1+₁ z+...+ _q z ^q , and all the roots of (z) lie outside the unit circle; {e(t)} is a sequence of iid random variables with mean zero andE|e(t)|^4+r < (r>0). In this paper, the limit properties of , where the integerd1, have been considered.     

Nonuniqueness Sequences for Weighted Algebras of Holomorphic Functions in the Unit Circle

Suppose that is a system of continuous subharmonic functions in the unit disk and is the class of holomorphic functions f in such that log|f(z)| ≤ B _f p _f (z) + C _f , z ∈ , where B _f and C _f are constants and p _f ∈ . We obtain sufficient conditions for a given number sequence Λ = { λ_n} ⊂ to be a subsequence of zeros of some nonzero holomorphic function from , i.e., Λ is a nonuniqueness sequence for .__________Translated from Matematicheskie Zametki, vol. 77, no. 5, 2005, pp. 775–787.Original Russian Text Copyright ©2005 by L. Yu. Cherednikova.     

Homomorphisms between Poisson JC*-Algebras

It is shown that every almost linear mapping of a unital Poisson JC*-algebra to a unital Poisson JC*-algebra is a Poisson JC*-algebra homomorphism when h(2 ⁿ uy) = h(2 ⁿ u) h(y), h(3 ⁿ u y) = h(3 ⁿ u) h(y) or h(q ⁿ u y) = h(q ⁿ u) h(y) for all , all unitary elements and n = 0, 1, 2, · · · , and that every almost linear almost multiplicative mapping is a Poisson JC*-algebra homomorphism when h(2x) = 2h(x), h(3x) = 3h(x) or h(qx) = qh(x) for all . Here the numbers 2, 3, q depend on the functional equations given in the almost linear mappings or in the almost linear almost multiplicative mappings.Moreover, we prove the Cauchy–Rassias stability of Poisson JC*-algebra homomorphisms in Poisson JC*-algebras.*This work was supported by grant No. R05-2003-000-10006-0 from the Basic Research Program of the Korea Science & Engineering Foundation.     

On <Emphasis Type="Bold">(c,p)</Emphasis>-pseudostable Random Variables

In (Oleszkiewicz, Lecture Notes in Math. 1807), K. Oleszkiewicz defined a p-pseudostable random variable X as a symmetric random variable for which the following equation holds:

A strong law for weighted sums of i.i.d. random variables

A strong law is proved for weighted sumsS _n=a _in X _i whereX _i are i.i.d. and {a _in} is an array of constants. When sup(n ^–1|a _in | ^q )^1/q <, 1<q andX _i are mean zero, we showE|X| ^p <,p ^l+q ^–1=1 impliesS _n /n 0. Whenq= this reduces to a result of Choi and Sung who showed that when the {a _in} are uniformly bounded,EX=0 andE|X|< impliesS _n /n 0. The result is also true whenq=1 under the additional assumption that lim sup |a _in |n ^–1 logn=0. Extensions to more general normalizing sequences are also given. In particular we show that when the {a _in} are uniformly bounded,E|X|^1/< impliesS _n /n 0 for >1, but this is not true in general for 1/2<<1, even when theX _i are symmetric. In that case the additional assumption that (x ^1/ log^1/–1 x)P(|X|x)0 asx provides necessary and sufficient conditions for this to hold for all (fixed) uniformly bounded arrays {a _in}.     

A Lower Bound for the Sectional Genus of Quasi-Polarized Surfaces

Let (X,L) be a quasi-polarized variety, i.e. X is a smooth projective variety over the complex numbers and L is a nef and big divisor on X. Then we conjecture that g(L) = q(X), whereg(L) is the sectional genus of L and . In this paper, we treat the case . First we prove that this conjecture is true for , and we classify (X,L) withg(L)=q(X), where is the Kodaira dimension of X. Next we study some special cases of .     

Cyclic Designs with Block Size 4 and Related Optimal Optical Orthogonal Codes

We prove the existence of a cyclic (4p, 4, 1)-BIBD—and hence, equivalently, that of a cyclic (4, 1)-GDD of type 4 ^p —for any prime such that (p–1)/6 has a prime factor q not greater than 19. This was known only for q=2, i.e., for . In this case an explicit construction was given for . Here, such an explicit construction is also realized for .We also give a strong indication about the existence of a cyclic (4p 4, 1)-BIBD for any prime , p>7. The existence is guaranteed for p>(2q ³–3q ²+1)²+3q ² where q is the least prime factor of (p–1)/6.Finally, we prove, giving explicit constructions, the existence of a cyclic (4, 1)-GDD of type 6 ^p for any prime p>5 and the existence of a cyclic (4, 1)-GDD of type 8 ^p for any prime . The result on GDD's with group size 6 was already known but our proof is new and very easy.All the above results may be translated in terms of optimal optical orthogonal codes of weight four with =1.     

Zeros of H ∞-Functions on Hyperplanes in \mathbb{B} n

Let ⁿ be the unit ball in ℂⁿ, n ≥ 2. Let T_α = {z ∈ ⁿ : (z, a) = |a|²} for a ∈ ⁿ and denote for a discrete set A in ⁿ. We find a sharp necessary condition for a set A to be a part of the zero-set for a function in H^∞( ⁿ). Bibliography 4 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 303, 2003, pp. 272–278.     

Limit theorems for compact two-point homogeneous spaces of large dimensions

Let be the field , , or of real dimension . For each dimensiond2, we study isotropic random walks(Y ₁)₁0 on the projective space with natural metricD where the random walk starts at some with jumps at each step of a size depending ond. Then the random variablesX ₁ ^d :=cosD(Y ₁ ^d ,x ₀ ^d ) form a Markov chain on [–1, 1] whose transition probabilities are related to Jacobi convolutions on [–1, 1]. We prove that, ford, the random variables (vd/2)(X _l(d) ^d +1) tend in distribution to a noncentral ²-distribution where the noncentrality parameter depends on relations between the numbers of steps and the jump sizes. We also derive another limit theorem for as well as thed-spheresS ^d ford.     

Szegö theorem,Carathéodory domains,and boundedness of calculating functionals

Convex Chebyshev sets M in a linear space X with norm or nonsymmetric norm ‖ · ‖ which are contained in a subspace H of X are considered. It is proved that if | { |_{H, φ} is the nonsymmetric norm on H determined by the Minkowski functional of , where B is the unit ball of X and ‖φ‖, with respect to 0, then M is a Chebyshev set in for any φ. From this result sufficient and necessary conditions for the convexity of Chebyshev sets and bounded Chebyshev sets contained in a subspace H of X are derived.__________Translated from Matematicheskie Zametki, vol. 78, no. 1, 2005, pp. 3–15.Original Russian Text Copyright © 2005 by A. R. Alimov.     

On a generalization of the Gallai-Sylvester theorem

It is shown that ifS ^d , affS=aff ^d , and every hyperplane spanned by (a subset of)S misses fewer thank points ofS(k2), then (a) #Skm ifd=2m–1 is odd and (b) #Skm+1 ifd=2m is even. We also fully describe the extreme sets for which equality holds in (a) or in (b). For oddd the proofs are later modified to purely algebraic ones, and carry over to , where is an arbitrary field. For evend, (b) is generally not true when , but we prove some weaker inequalities that do hold over arbitrary fields.This is part of a Ph.D. thesis, supervised by Professor Micha A. Perles at the Hebrew University of Jerusalem. This research was supported in part by the Landau Center for Mathematical Research.     

TWO-DISTANCE PRESERVING FUNCTIONS FROM EUCLIDEAN SPACE

Let 0 < c < s be fixed real numbers such that , and let f : E² → E ^d for d ≥ 2 be a function such that for every p, q ∈ E ² if |p − q| = c, then |f(p) − f(q)| ≤ c, and if |p − q| = s, then |f(p) − f(q)| ≥ s. Then f is a congruence. This result depends on and expands a result of Rádo et. al. [9], where a similar result holds, but for replacing . We also present a further extensions where E² is replaced by E ⁿ for n > 2 and where the range of c/s is enlarged. This revised version was published online in August 2006 with corrections to the Cover Date.