E Φ andh Φ fail to beM-ideals inL Φ andl Φ in the case of the Orlicz norm

It is well known that in the case of the Luxemburg normE ^Φ (resp.h ^Φ) is anM ideal inL ^Φ (resp.l ^Φ), see [1], [9], [15] and [6]; [17] and [18]. It is proved in this paper that in the case of the Orlicz normE ^Φ (resph ^Φ) is anM-ideal inL ^Φ (resp.l ^Φ) iff Φ satisfies the suitable Δ₂ or Φ^*(a(Φ^*)), where a(Φ^*) is linear on the interval [0,u]} and Φ^* denotes the function complementary to Φ in the sense of Young. It is also proved that any linear continuous regular (i.e. order continuous) functional ξ_υ overE ^Φ (resp.h ^Φ) generated byv∈ L(h ^Φ*) (resp.v∈ L(h ^Φ*)) which attains its norm on the unit sphereS(E ^Φ) (resp.S(h ^Φ)), has a unique norm-preserving extension toL ^Φ (resp.l ^Φ). Finally, it is proved thatL ^Φ (resp.l ^Φ) has the property that any linear continuous regular functional ξ_υ overE ^Φ (resp.h ^Φ) has a unique norm-preserving extension toL ^Φ (resp.l ^Φ) iff Φ orE ^Φ satisfies the suitable Δ₂ and in the second case Φ^* attains the value 1.     

On the subspace L((x ∧ y)m) of Sm(∧2?4)

We take the exterior power ℝ⁴ ∧ ℝ⁴ of the space ℝ⁴, its mth symmetric power V = S ^m (∧²ℝ⁴) = (ℝ⁴ ∧ ℝ⁴) ∨ (ℝ⁴ ∧ ℝ⁴) ∨ ... ∨(ℝ⁴ ∧ ℝ⁴), and put V ₀ = L((x ∧ y)∨ ... ∨(x ∧ y): x, y ∈ ℝ⁴). We find the dimension of V ₀ and an algorithm for distinguishing a basis for V ₀ efficiently. This problem arose in vector tomography for the purpose of reconstructing the solenoidal part of a symmetric tensor field. Original Russian Text Copyright ? 2009 Gubarev V. Yu. The author was supported by the State Maintenance Program for the Leading Scientific Schools of the Russian Federation (Grant NSh-344.2008.1). __________ Novosibirsk. Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 50, No. 3, pp. 503–514, May–June, 2009.     

Estimates for parametric Marcinkiewicz integrals in BMO and Campanato spaces

In this paper, the authors consider the behaviors of a class of parametric Marcinkiewicz integrals μ _Ω ^ρ , μ _Ω,λ ^*,ρ and μ _Ω,S ^ρ on BMO(ℝ ⁿ ) and Campanato spaces with complex parameter ρ and the kernel Ω in Llog⁺ L(S ⁿ⁻¹). Here μ _Ω,λ ^*,ρ and μ _Ω,S ^ρ are parametric Marcinkiewicz functions corresponding to the Littlewood-Paley g _λ *-function and the Lusin area function S, respectively. Under certain weak regularity condition on Ω, the authors prove that if f belongs to BMO(ℝ ⁿ ) or to a certain Campanato space, then [μ _Ω,λ ^*,ρ (f)]², [μ _Ω,S ^ρ (f)]² and [μ _Ω ^ρ (f)]² are either infinite everywhere or finite almost everywhere, and in the latter case, some kind of boundedness are also established.     

A bound for the representability of large numbers by ternary quadratic forms and nonhomogeneous waring equations

The solvability of the equation n = x ² + y ² + 6pz ² (p is a fixed large prime) is proved under some natural congruential conditions and the assumption nm ¹² > p ²¹. As an implication, the solvability of the equation n = x ² + y ² + u ³ + v ³ + z ⁴ + w ¹⁶ + t ^4k+1 for all sufficiently large n is established. Bibliography: 13 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 357, 2008, pp. 5–21.     

On the chromatic polynomial of a graph

Let P(G,λ) be the chromatic polynomial of a graph G with n vertices, independence number α and clique number ω. We show that for every λ≥n, ()^α≤≤ () ^{n −ω}. We characterize the graphs that yield the lower bound or the upper bound.?These results give new bounds on the mean colour number μ(G) of G: n− (n−ω)() ^{n −ω}≤μ(G)≤n−α() ^α. Received: December 12, 2000 / Accepted: October 18, 2001?Published online February 14, 2002     

Combined perturbation bounds: Ⅱ. Polar decompositions

In this paper, we study the perturbation bounds for the polar decomposition A= QH where Q is unitary and H is Hermitian. The optimal (asymptotic) bounds obtained in previous works for the unitary factor, the Hermitian factor and singular values of A are σ2r||△Q||2F ≤ ||△A||2F,1/2||△H||2F ≤ ||△A||2F and ||△∑||2F ≤ ||△A||2F, respectively, where ∑ = diag(σ1, σ2,..., σr, 0,..., 0) is the singular value matrix of A and σr denotes the smallest nonzero singular value. Here we present some new combined (asymptotic)perturbation bounds σ2r ||△Q||2F 1/2||△H||2F≤ ||△A||2F and σ2r||△Q||2F ||△∑ ||2F ≤||△A||2F which are optimal for each factor. Some corresponding absolute perturbation bounds are also given.     

Binomial Thue equations,ternary equations and power values of polynomials

We explicitly solve the equation Ax ⁿ   − By ⁿ  = ±1 and, along the way, we obtain new results for a collection of equations Ax ⁿ   − By ⁿ  = z ^m with m ∈ {3, n}, where x, y, z, A, B, and n are unknown nonzero integers such that n ≥ 3, AB = p ^α q ^β with nonnegative integers α and β and with primes 2 ≤ p < q < 30. The proofs depend on a combination of several powerful methods, including the modular approach, recent lower bounds for linear forms in logarithms, somewhat involved local considerations, and computational techniques for solving Thue equations of low degree.     

Multiple points of trajectories of multiparameter fractional Brownian motion

Consider 0<α<1 and the Gaussian process Y(t) on ℝ ^N with covariance E(Y(s)Y(t))=|t|^2α+|s|^2α−|t−s|^2α, where |t| is the Euclidean norm of t. Consider independent copies X ¹,…,X ^d of Y and␣the process X(t)=(X ¹(t),…,X ^d (t)) valued in ℝ ^d . When kN≤␣(k−1)αd, we show that the trajectories of X do not have k-multiple points. If N<αd and kN>(k−1)αd, the set of k-multiple points of the trajectories X is a countable union of sets of finite Hausdorff measure associated with the function ϕ(ɛ)=ɛ ^{k N /α−( k −1) d} (loglog(1/ɛ)) ^k . If N=αd, we show that the set of k-multiple points of the trajectories of X is a countable union of sets of finite Hausdorff measure associated with the function ϕ(ɛ)=ɛ ^d (log(1/ɛ) logloglog 1/ɛ) ^k . (This includes the case k=1.) Received: 20 May 1997 / Revised version: 15 May 1998     

Characterizations of BMO associated with Gauss measures via commutators of local fractional integrals

Let dγ(x) ≡ π ^−n/2 e ^{−|x| 2} dx for all x ∈ ℝ ⁿ be the Gauss measure on ℝ ⁿ . In this paper, the authors establish the characterizations of the space BMO(γ) of Mauceri and Meda via commutators of either local fractional integral operators or local fractional maximal operators. To this end, the authors first prove that such a local fractional integral operator of order β is bounded from L ^p (γ) to L ^p/(1−pβ)(γ), or from the Hardy space H ¹(γ) of Mauceri and Meda to L ^1/(1−β)(γ) or from L ^1/β (γ) to BMO(γ), where β ∈ (0, 1) and p ∈ (1, 1/β).     

On the Cantor-Bendixson derivative, resolvable ranks, and perfect set theorems of A. H. Stone

A Borel derivative on the hyperspace 2 ^X of a compactumX is a Borel monotone mapD: 2 ^X →2 ^X . The derivative determines a Cantor-Bendixson type rank δ:2^X → ω₁ ∪ {∞} . We show that ifA⊂2 ^X is analytic andZ⊂A intersects stationary many layers δ₋₁({ξ}), then for almost all σ,F∩δ₋₁({ξ}) cannot be separated fromZ ∩∪ _a<ξ ^δ−1({a}) (and also fromZ ∩∪ _a>ξ ^δ−1({a}) by anyF _σ-set. Another main result involves a natural partial order on 2 ^X related to the derivative. The results are obtained in a general framework of “resolvable ranks” introduced in the paper. During our work on this paper the second author was a Visiting Professor at the Miami University, Ohio. This author would like to express his gratitude to the Department of Mathematics and Statistics for the hospitality.     

Algebraic relations for reciprocal sums of odd terms in Fibonacci numbers

In this paper, we prove the algebraic independence of the reciprocal sums of odd terms in Fibonacci numbers ∑ _n=1^∞ F _2n−1⁻¹, ∑ _n=1^∞ F _2n−1⁻², ∑ _n=1^∞ F _2n−1⁻³ and write each ∑ _n=1^∞ F _2n−1^−s (s≥4) as an explicit rational function of these three numbers over ℚ. Similar results are obtained for various series including the reciprocal sums of odd terms in Lucas numbers.      

On representation of the solution of the Neumann problem in a domain with peak by the harmonic simple layer potential

The problem of finding a solution of the Neumann problem for the Laplacian in the form of a simple layer potential Vρ with unknown density ρ is known to be reducible to a boundary integral equation of the second kind to be solved for density. The Neumann problem is examined in a bounded n-dimensional domain Ω⁺ (n > 2) with a cusp of an outward isolated peak either on its boundary or in its complement Ω⁻ = R ⁿ \Ω⁺. Let Γ be the common boundary of the domains Ω^±, Tr(Γ) be the space of traces on Γ of functions with finite Dirichlet integral over R ⁿ , and Tr(Γ)* be the dual space to Tr(Γ). We show that the solution of the Neumann problem for a domain Ω⁻ with a cusp of an inward peak may be represented as Vρ⁻, where ρ⁻ ∈ Tr(Γ)* is uniquely determined for all Ψ⁻ ∈ Tr(Γ)*. If Ω⁺ is a domain with an inward peak and if Ψ⁺ ∈ Tr(Γ)*, Ψ⁺ ⊥ 1, then the solution of the Neumann problem for Ω⁺ has the representation u ⁺ = Vρ⁺ for some ρ⁺ ∈ Tr(Γ)* which is unique up to an additive constant ρ₀, ρ₀ = V ⁻¹(1). These results do not hold for domains with outward peak.     

Asymptotic expansions in limit theorems for stochastic processes. I

Summary. For some families of locally infinitely divisible Markov processes η ^ɛ (t), 0≦ t≦ T, with frequent small jumps, limit theorems for expectations of functionals F(η ^ɛ [0,T]) are proved of the form | E ^ɛ F(η ^ɛ [0,T])−E ⁰ F(η ⁰ [0,T])|≦ const ⋅ k(ɛ) , E ^ɛ F(η ^ɛ [0,T])=E ⁰ [F(η ⁰ [0,T])+ k(ɛ) ⋅ A ₁ F(η ⁰ [0,T])]+o(k(ɛ))  (ɛ↓ 0) , where A ₁ is a linear differential operator acting on functionals, and the constant is expressed in terms of the local characteristics of the processes η ^ɛ (t) and the norms of the derivatives of the functional F. Received: 1 April 1994 / In revised form: 30 September 1995     

Error bounds for asymptotic expansion of the scale mixtures of the normal distribution

Summary LetX be a standard normal random variable and let σ be a positive random variable independent ofX. The distribution of η=σX is expanded around that ofN(0, 1) and its error bounds are obtained. Bounds are given in terms of E(σ ²V−σ ²−1) ^k whereσ ²Vσ ⁻² denotes the maximum of the two quantitiesσ ² andσ ⁻², andk is a positive integer, and of E(σ ²−1) ^k , ifk is even. The Institute of Statistical Mathematics     

Correlation structure of intermittency in the parabolic Anderson model

Consider the Cauchy problem ∂u(x, t)/∂t = ℋu(x, t) (x∈ℤ^d, t≥ 0) with initial condition u(x, 0) ≡ 1 and with ℋ the Anderson Hamiltonian ℋ = κΔ + ξ. Here Δ is the discrete Laplacian, κ∈ (0, ∞) is a diffusion constant, and ξ = {ξ(x): x∈ℤ ^d } is an i.i.d.random field taking values in ℝ. G?rtner and Molchanov (1990) have shown that if the law of ξ(0) is nondegenerate, then the solution u is asymptotically intermittent. In the present paper we study the structure of the intermittent peaks for the special case where the law of ξ(0) is (in the vicinity of) the double exponential Prob(ξ(0) > s) = exp[−e ^{s /θ}] (s∈ℝ). Here θ∈ (0, ∞) is a parameter that can be thought of as measuring the degree of disorder in the ξ-field. Our main result is that, for fixed x, y∈ℤ ^d and t→∈, the correlation coefficient of u(x, t) and u(y, t) converges to ∥w _ρ∥⁻² _ℓ2Σ_{z ∈ℤd} w _ρ(x+z)w _ρ(y+z). In this expression, ρ = θ/κ while w _ρ:ℤ^d→ℝ⁺ is given by w _ρ = (v _ρ)^{⊗ d} with v _ρ: ℤ→ℝ⁺ the unique centered ground state (i.e., the solution in ℓ²(ℤ) with minimal l ²-norm) of the 1-dimensional nonlinear equation Δv + 2ρv log v = 0. The uniqueness of the ground state is actually proved only for large ρ, but is conjectured to hold for any ρ∈ (0, ∞). empty It turns out that if the right tail of the law of ξ(0) is thicker (or thinner) than the double exponential, then the correlation coefficient of u(x, t) and u(y, t) converges to δ _{x, y} (resp.the constant function 1). Thus, the double exponential family is the critical class exhibiting a nondegenerate correlation structure. Received: 5 March 1997 / Revised version: 21 September 1998     

Global existence and time decay of small solutions to the landau-ginzburg type equations

We study the Cauchy problem for the nonlinear dissipative equations (0.1) uo∂u-αδu + Β|u|^2/n u = 0,x ∃ Rⁿ,t } 0,u(0,x) = u0(x),x ∃ Rⁿ, where α,Β ∃ C, ℜα 0. We are interested in the dissipative case ℜα 0, and ℜδ(α,Β)≥ 0, θ = |∫ u0(x)dx| ⊋ 0, where δ(α, Β) = ##|α|^n-1n^n/2 / ((n + 1)|α|² + α^{2 n/2}. Furthermore, we assume that the initial data u₀ ∃ L^p are such that (1 + |x|)^αu0 ∃ L¹, with sufficiently small norm ∃ = (1 + |x|)^α u₀ 1 + u₀ p, wherep 1, α ∃ (0,1). Then there exists a unique solution of the Cauchy problem (0.1)u(t, x) ∃ C ((0, ∞); L^∞) ∩ C ([0, ∞); L¹ ∩ L^p) satisfying the time decay estimates for allt0 u(t)||_∞ Cɛt^-n/2(1 + η log 〈t〉)^-n/2, if hg = θ^2/n 2_π ℜδ(α, Β) 0; u(t)||_∞ Cɛt^-n/2(1 + Μ log 〈t〉)^-n/4, if η = 0 and Μ = θ^4/n 4π)² (ℑδ(α, Β))² ℜ((1 + 1/n) υ1-1 υ2) 0; and u(t)||_∞ Cɛt^-n/2(1 + κ log 〈t〉)^-n/6, if η = 0, Μ = 0, κ 0, where υl,l = 1,2 are defined in (1.2), κ is a positive constant defined in (2.31).     

Onl p-complemented copies in Orlicz spaces

Let 1<α≦β<∞ andF be an arbitrary closed subset of the interval [α,β]. An Orlicz sequence spacel ^φ (resp. an Orlicz function spaceL ^φ(μ)) with associated indices α and β is found in such a way that the set of valuesp for which thel ^p-space is isomorphic to a complemented subspace ofl ^φ (resp.L ^φ(μ)) is precisely the given setF (resp.F ∪ {2}). Also, a recent result of Hernández and Peirats [1] is extended showing that, even for the case in which the indices satisfy α_φ ^∞<2<β_φ ^∞, there exist minimal Orlicz function spacesL ^φ(μ) with no complemented copy ofl ^p for anyp ≠ 2. Supported in part by CAICYT grant 0338-84.     

BOUNDEDNESS OF VECTOR-VALUED CALDERON-ZYGMUND OPERATORS ON HERZ SPACES WITH NON-DOUBLING MEASURES

In the paper we obtain vector-valued inequalities for Calderon-Zygmund operator,simply CZO On Herz space and weak Herz space.In particular,we obtain vector-valued inequalities for CZO on Lq(Rd,│x│αdμ)space,with 1＜q＜∞,-n＜α＜n(q-1),and on L1,∞(Rd,│x│αdμ)space,with -n＜α＜0.     

On Estimates of Factorial Quotients

We consider the factorial quotients (2n − 1)!!/(2n)!! in connection with the Wallis formula n ⁻¹(2n)!!²/(2n − 1)!!² → π. We improve the Wallis inequalities (n + 1/2)⁻¹(2n)!!²/(2n − 1)!!² < π < n ⁻¹(2n)!!²/(2n − 1)!!² for π and obtain new estimates of factorial quotients with error order not worse than 1/n ². __________ Translated from Lietuvos Matematikos Rinkinys, Vol. 45, No. 3, pp. 349–358, July–September, 2005.     

Uniqueness of Nelsons diffusions

Let ℒ≔Δ/2+(∇φ/φ) ·∇ be a generalized Schr?dinger operator or generator of Nelsons diffusion, defined on C ^∞ ₀(D) where φ is a continuous and strictly positive function on an open domain D⊂ℝ ^d such that ∇φ∈L _loc ²(D). Some results are given about the two questions below: (i) Whether does ℒ generate a unique semigroup in L ¹(D, φ² dx)? (ii) Whether the semigroup determined by ℒ is strong Feller? Received: 21 October 1997 / Revised version: 3 September 1998