Packing-type Measures of the Sample Paths of Fractional Brownian Motion

Abstract   Let Λ = {λ _k } be an infinite increasing sequence of positive integers with λ _k →∞. Let X = {X(t), t ∈? R ^N } be a multi-parameter fractional Brownian motion of index α(0 < α < 1) in R ^d . Subject to certain hypotheses, we prove that if N < αd, then there exist positive finite constants K ₁ and K ₂ such that, with unit probability,

A superadditive property of Hadamard’s gamma function

Hadamard’s gamma function is defined by

On the fourth moment in the Rankin-Selberg problem

If

Logarithmic Estimates for Submartingales and Their Differential Subordinates

In the paper we determine, for any K>0 and α∈[0,1], the optimal constant L(K,α)∈(0,∞] for which the following holds: If X is a nonnegative submartingale and Y is α-strongly differentially subordinate to X, then

Solutions to Some Open Problems in Fluid Dynamics

Let u=u（x,t,uo）represent the global solution of the initial value problem for the one-dimensional fluid dynamics equation ut-εuxxt＋δux＋γHuxx＋βuxxx＋f（u）x=αuxx,u（x,0）=uo（x）, whereα〉0,β〉0,γ〉0,δ〉0 andε〉0 are constants.This equation may be viewed as a one-dimensional reduction of n-dimensional incompressible Navier-Stokes equations. The nonlinear function satisfies the conditions f（0）=0,｜f（u）｜→∞as ｜u｜→∞,and f∈C^1（R）,and there exist the following limits Lo=lim sup/u→o f（u）/u^3 and L∞=lim sup/u→∞ f（u）/u^5 Suppose that the initial function u0∈L^I（R）∩H^2（R）.By using energy estimates,Fourier transform,Plancherel＇s identity,upper limit estimate,lower limit estimate and the results of the linear problem vt-εv（xxt）＋δvx＋γHv（xx）＋βv（xxx）=αv（xx）,v（x,0）=vo（x）, the author justifies the following limits（with sharp rates of decay） lim t→∞[（1＋t）^（m＋1/2）∫｜uxm（x,t）｜^2dx]=1/2π（π/2α）^（1/2）m!!/（4α）^m[∫R uo（x）dx]^2, if∫R uo（x）dx≠0, where 0!!=1,1!!=1 and m!!=1·3…（2m-3）…（2m-1）.Moreover lim t→∞[（1＋t）^（m＋3/2）∫R｜uxm（x,t）｜^2dx]=1/2π（x/2α）^（1/2）（m＋1）!!/（4α）^（m＋1）[∫Rρo（x）dx]^2, if the initial function uo（x）=ρo′（x）,for some functionρo∈C^1（R）∩L^1（R）and∫Rρo（x）dx≠0.     

Global behavior of the branch of positive solutions for nonlinear Sturm–Liouville problems

We consider the nonlinear Sturm–Liouville problem

On the number of divisors which are values of a polynomial

Let τ(n) be the number of positive divisors of an integer n, and for a polynomial P(X)∈ℤ[X], let

Properties of Riesz fractional derivatives for Korteweg–de Vries solitons

Riesz fractional derivatives are defined as fractional powers of the Laplacian, D ^α  = (?Δ) ^α/2 for ${\alpha \in \mathbb{R}}Riesz fractional derivatives are defined as fractional powers of the Laplacian, D ^α  = (−Δ) ^α/2 for a ? \mathbbR{\alpha \in \mathbb{R}}. For the soliton solution of the Korteweg–de Vries equation, u ₀(X) with X = x − 4t, these derivatives, u _α (X) = D ^α u ₀(X), and their Hilbert transforms, v _α (X) = −HD ^α u ₀(X), can be expressed in terms of the full range Hurwitz Zeta functions ζ₊(s, a) and ζ₋(s, a), respectively. New properties are established for u _α (X) and v _α (X). It is proved that the functions w _α (X) = u _α (X) + iv _α (X) with α > −1 are solutions of the differential equation

-\fracddX(Pa(X)\fracdwdX)+Qa(X)w = lra(X)w,       X ? \mathbbR,-\frac{\rm d}{{\rm d}X}\left(P_{\alpha}(X)\frac{{\rm d}w}{{\rm d}X}\right)+Q_{\alpha}(X)w = \lambda\rho_{\alpha}(X)w,\qquad X \in \mathbb{R},      11. Further properties of Azimi-Hagler banach spaces      Parviz Azimi  H. Khodabakhshian 《Czechoslovak Mathematical Journal》2009,59(4):871-878 For the Azimi-Hagler spaces more geometric and topological properties are investigated. Any constructed space is denoted by X α,p . We show (i)  The subspace [(e nk )] generated by a subsequence (e nk ) of (e n ) is complemented. (ii)  The identity operator from X α,p to X α,p when p > q is unbounded. (iii)  Every bounded linear operator on some subspace of X α,p is compact. It is known that if any X α,p is a dual space, then (iv)  duals of X α,1 spaces contain isometric copies of ℓ ∞ and their preduals contain asymptotically isometric copies of c 0. (v)  We investigate the properties of the operators from X α,p spaces to their predual.       12. On<Emphasis Type="Italic">M</Emphasis>-structure,the asymptotic-norming property and locally uniformly rotund renormings      Eduardo Nieto  Migdalia Rivas 《Arkiv f?r Matematik》2002,40(2):323-333 Letr, s ∈ [0, 1], and letX be a Banach space satisfying theM(r, s)-inequality, that is, where π X is the canonical projection fromX *** ontoX *. We show some examples of Banach spaces not containingc 0, having the point of continuity property and satisfying the above inequality forr not necessarily equal to one. On the other hand, we prove that a Banach spaceX satisfying the above inequality fors=1 admits an equivalent locally uniformly rotund norm whose dual norm is also locally uniformly rotund. If, in addition,X satisfies wheneveru *,v * ∈X * with ‖u *‖≤‖v *‖ and (x α * ) is a bounded weak* null net inX *, thenX can be renormed to satisfy the,M(r, 1) and theM(1, s)-inequality such thatX * has the weak* asymptotic-norming property I with respect toB X .      13. A necessary and sufficient condition for singular nonlinear second-order boundary value problems to have positive solutions      Zhao Zengqin 《高校应用数学学报(英文版)》1998,13(1):15-24 In this paper the following result is obtained: Suppose f(g,u,v) is nonnegative, continuous in (a, 6) ×R+ ×R + ; f may be singular at κ = a(and/or κ = b) and υ = 0; f is nondecreasing on u for each κ,υ,nonincreasing on υ for each κ,u; there exists a constant q ε (0,1) such that . Then a necessary and sufficient condition for the equation u′’+f(κ,u,u) = 0 on the boundary condition au(.a)-βu′ (a) = 0, γ(b)+δu′(b) = 0 to have C1(I) nonzero solutions is that where α,β,γ,δ are nonnegative real numbers, Δ= (b-a)αγ + αγ+βδ+βγ>0, e(κ) =G(κ,κ), G(κ,y) is Green’s function of above mentioned boundary value problem (when f(κ,u,υ)≡0). Project supported by the Natural Science Foundation of Shandong Province.      14. The Besov subspace consisting of most non-smooth functions      E. I. Berezhnoi 《Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences)》2009,44(3):163-171 Under the assumptions that Δ(f, h)(t) = |f(t + h) − f(t)|, X is a symmetric space of functions in [0, 1], α ∈ (0, 1) and p ∈ [1, ∞) are any fixed number, by the triple (X, α, p) a Besov type space Λ X,p α is constructed, where the norm is given by the equality For any α 0 ∈ (0, 1), it is shown that there exists an infinite-dimensional, closed subspace of Λ X,p α0, such that any non-identically zero function does not belong to the subspace Λ X,p α with α > α 0. The work is done under the financial support of RFFI, Project Cod 08-01-00669      15. Spectral Exponent of Finite Sums of Weighted Positive Operators in <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-Spaces      Urszula Ostaszewska  Krzysztof Zajkowski 《Positivity》2007,11(4):549-562 In this paper we investigate the spectral exponent, i.e. logarithm of the spectral radius of operators having the form and acting in spaces Lp(X, μ), where X is a compact topological space, φk∈C(X), φ = (φk)k=1N∈C(X)N, and are linear positive operators (Ukf≥ 0 for f≥ 0). We consider the spectral exponent ln r(Aφ) as a functional depending on vector-function φ. We prove that ln r(Aφ) is continuous and on a certain subspace of C(X)N is also convex. This yields that the spectral exponent is the Fenchel-Legendre transform of a convex functional defined on a set of continuous linear positive and normalized functionals on the subspace of coefficients φ that is       16. Spectra of Bipartite P- and Q-Polynomial Association Schemes      John S. Caughman 《Graphs and Combinatorics》1998,14(4):321-343  Let Y=(X,{R i }0≤i≤D) denote a symmetric association scheme with D≥3, and assume Y is not an ordinary cycle. Suppose Y is bipartite P-polynomial with respect to the given ordering A 0, A 1,…, A D of the associate matrices, and Q-polynomial with respect to the ordering E 0, E 1,…,E D of the primitive idempotents. Then the eigenvalues and dual eigenvalues satisfy exactly one of (i)–(iv). (i) (ii) D is even, and (iii) θ* 0>θ0, and (iv) θ* 0>θ0, D is odd, and Received: February 13, 1996 / Revised: October 16, 1996      17. Independent Sets In Association Schemes      C. D. Godsil  M. W. Newman 《Combinatorica》2006,26(4):431-443 Let X be k-regular graph on v vertices and let τ denote the least eigenvalue of its adjacency matrix A(X). If α(X) denotes the maximum size of an independent set in X, we have the following well known bound: . It is less well known that if equality holds here and S is a maximum independent set in X with characteristic vector x, then the vector is an eigenvector for A(X) with eigenvalue τ . In this paper we show how this can be used to characterise the maximal independent sets in certain classes of graphs. As a corollary we show that a graph defined on the partitions of {1, . . . ,9} with three cells of size three is a core. * Researchs upported by NSERC.      18. New asymptotic formula for eigenvalues of nonlinear Sturm-Liouville problems      Tetsutaro Shibata 《Journal d'Analyse Mathématique》2007,102(1):347-358 We study the nonlinear Sturm-Liouville problem where λ > 0 is an eigenvalue parameter and f(u) is a rapidly increasing function. For better understanding of the global behavior of the bifurcation branch in R+ × L 2(I), we establish precise asymptotic formulas up to the third term for the eigenvalue λ(α) associated with the eigenfunction u α with ‖u α‖2 = α, as α → ∞. We show that there exists a new type of asymptotic formula for λ (α) as α → ∞.      19. Precise rates in the law of the iterated logarithm under dependence assumptions      Wel Dong LIU Zheng Yan LIN 《数学学报(英文版)》2008,24(1):59-74 Let {X, X1, X2,...} be a strictly stationaryφ-mixing sequence which satisfies EX = 0,EX^2（log2{X}）^2〈∞and φ（n）=O（1/log n）^Tfor some T〉2.Let Sn=∑k=1^nXk and an=O（√n/（log2n）^γ for some γ〉1/2.We prove that limε→√2√ε^2-2∑n=3^∞1/nP（｜Sn｜≥ε√ESn^2log2n＋an）=√2.The results of Gut and Spataru （2000） are special cases of ours.      20. A Wiener–Wintner theorem for the Hilbert transform      Michael Lacey  Erin Terwilleger 《Arkiv f?r Matematik》2008,46(2):315-336 We prove the following extension of the Wiener–Wintner theorem and the Carleson theorem on pointwise convergence of Fourier series: For all measure-preserving flows (X,μ,T t ) and f∈L p (X,μ), there is a set X f ⊂X of probability one, so that for all x∈X f , The proof is by way of establishing an appropriate oscillation inequality which is itself an extension of Carleson’s theorem.

Further properties of Azimi-Hagler banach spaces

For the Azimi-Hagler spaces more geometric and topological properties are investigated. Any constructed space is denoted by X _α,p . We show

On<Emphasis Type="Italic">M</Emphasis>-structure,the asymptotic-norming property and locally uniformly rotund renormings

Letr, s ∈ [0, 1], and letX be a Banach space satisfying theM(r, s)-inequality, that is,

A necessary and sufficient condition for singular nonlinear second-order boundary value problems to have positive solutions

In this paper the following result is obtained: Suppose f(g,u,v) is nonnegative, continuous in (a, 6) ×R⁺ ×R ⁺ ; f may be singular at κ = a(and/or κ = b) and υ = 0; f is nondecreasing on u for each κ,υ,nonincreasing on υ for each κ,u; there exists a constant q ε (0,1) such that

The Besov subspace consisting of most non-smooth functions

Under the assumptions that Δ(f, h)(t) = |f(t + h) − f(t)|, X is a symmetric space of functions in [0, 1], α ∈ (0, 1) and p ∈ [1, ∞) are any fixed number, by the triple (X, α, p) a Besov type space Λ _X,p ^α is constructed, where the norm is given by the equality

Spectral Exponent of Finite Sums of Weighted Positive Operators in <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-Spaces

In this paper we investigate the spectral exponent, i.e. logarithm of the spectral radius of operators having the form

Spectra of Bipartite P- and Q-Polynomial Association Schemes

 Let Y=(X,{R _i }_0≤i≤D) denote a symmetric association scheme with D≥3, and assume Y is not an ordinary cycle. Suppose Y is bipartite P-polynomial with respect to the given ordering A ₀, A ₁,…, A _D of the associate matrices, and Q-polynomial with respect to the ordering E ₀, E ₁,…,E _D of the primitive idempotents. Then the eigenvalues and dual eigenvalues satisfy exactly one of (i)–(iv). (i)

Independent Sets In Association Schemes

Let X be k-regular graph on v vertices and let τ denote the least eigenvalue of its adjacency matrix A(X). If α(X) denotes the maximum size of an independent set in X, we have the following well known bound:

New asymptotic formula for eigenvalues of nonlinear Sturm-Liouville problems

We study the nonlinear Sturm-Liouville problem

Precise rates in the law of the iterated logarithm under dependence assumptions

Let {X, X1, X2,...} be a strictly stationaryφ-mixing sequence which satisfies EX = 0,EX^2（log2{X}）^2〈∞and φ（n）=O（1/log n）^Tfor some T〉2.Let Sn=∑k=1^nXk and an=O（√n/（log2n）^γ for some γ〉1/2.We prove that limε→√2√ε^2-2∑n=3^∞1/nP（｜Sn｜≥ε√ESn^2log2n＋an）=√2.The results of Gut and Spataru （2000） are special cases of ours.     

A Wiener–Wintner theorem for the Hilbert transform

We prove the following extension of the Wiener–Wintner theorem and the Carleson theorem on pointwise convergence of Fourier series: For all measure-preserving flows (X,μ,T _t ) and f∈L ^p (X,μ), there is a set X _f ⊂X of probability one, so that for all x∈X _f ,