On some new criteria for infinite differentiability of periodic functions

We study the set of infinitely differentiable periodic functions in terms of generalized -derivatives defined by a pair of sequences ψ ₁ and ψ ₂. It is shown that every function ƒ from the set has at least one derivative whose parameters ψ ₁ and ψ ₂ decrease faster than any power function and, at the same time, for any function ƒ ∈ different from a trigonometric polynomial, there exists a pair ψ whose parameters ψ ₁ and ψ ₂ have the same rate of decrease and for which the -derivative no longer exists. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 10, pp. 1399–1409, October, 2007.     

A characterization of wavelet families arising from biorthogonal MRA’s of multiplicityd

In this article we give a necessary and sufficient condition for a pair of wavelet families in L²(ℝ ⁿ ), to arise from a pair of biorthogonal MRA’s. The condition is given in terms of simple equations involving the functions ψ^ℓ and . To work in greater generality, we allow multiresolution analyses of arbitrary multiplicity, based on lattice translations and matrix dilations. Our result extends the characterization theorem of G. Gripenberg and X. Wang for dyadic orthonormal wavelets in L²(ℝ),and includes, as particular cases, the sufficient conditions of P. Auscher and P.G. Lemarié in the biorthogonal situation.     

Dual Wavelet Frames and Riesz Bases in Sobolev Spaces

This paper generalizes the mixed extension principle in L ₂(ℝ ^d ) of (Ron and Shen in J. Fourier Anal. Appl. 3:617–637, 1997) to a pair of dual Sobolev spaces H ^s (ℝ ^d ) and H ^−s (ℝ ^d ). In terms of masks for φ,ψ ¹,…,ψ ^L ∈H ^s (ℝ ^d ) and , simple sufficient conditions are given to ensure that (X ^s (φ;ψ ¹,…,ψ ^L ), forms a pair of dual wavelet frames in (H ^s (ℝ ^d ),H ^−s (ℝ ^d )), where

<Emphasis Type="Italic">ψ</Emphasis>-continuous functions

Let be a ψ-density topology for a fixed function ψ. This paper is concerned with the family of ψ-continuous functions, that means continuous functions from (ℝ, ) into (ℝ, ). The family of such functions forms a lattice and is not closed under addition and uniform convergence. There exist functions ψ for which even linear functions are not ψ-continuous.      

A note on the crossed product $$ \mathcal{R}(\mathcal{A},\alpha ) $$ associated with PSL2(ℝ)

Given the hyperbolic measure dxdy/y ² on the upper half plane ℍ, the rational actions of PSL₂(ℝ) on ℍ induces a continuous unitary representation α of this group on the Hilbert space L ²(ℍ, dxdy/y ²). Supposing that = {M _f : f ∈ L ^∞ (ℍ, dxdy/y ²)}, we show that the crossed product is of type I. In fact, the crossed product is *-isomorphic to the von Neumann algebra , where is the abelian group von Neumann algebra generated by the left regular representation of K. This work was supported by the Youth Foundation of Sichuan Education Department of China (Grant No. 2003B017)     

Propagation of oscillations to 2D incompressible Euler equations

The asymptotic expansions are studied for the vorticity to 2D incompressible Euler equations with-initial vorticity , where ϕ₀(x) satisfies |d ϕ₀(x)|≠0 on the support of and is sufficiently smooth and with compact support in ℝ² (resp. ℝ²×T) The limit,v(t,x), of the corresponding velocity fields {v ^ɛ(t,x)} is obtained, which is the unique solution of (E) with initial vorticity ω₀(x). Moreover, (ℤ²)) for all 1≽p∞, where and ϕ(t,x) satisfy some modulation equation and eikonal equation, respectively.     

On the relations among best approximation and Fourier coefficients

We discuss the relations among the best approximation E _n (f) and the Fourier coefficients of a function, under the conditions that and , where MVBVS^* is the class of the so-called Strong Mean Value Bounded Variation Sequences. This work was supported by NSERC RCD grant and AARMS of Canada (Yu D S), by NSERC of Canada (Zhou P), and by the National Natural Science Foundation of China (Grant No. 10471130) and the Open Fund (Grant No. PLN0613) of State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation (Southwest Petroleum University) (Zhou S P)     

Generalized Besov spaces and Triebel-Lizorkin spaces

In this paper the classical Besov spaces Bsp.q and Triebel-Lizorkin spaces Fsp.q for s ∈R are generalized in an isotropy way with the smoothness weights {|2j|aln}∞j=0. These generalized Besov spaces and Triebel-Lizorkin spaces, denoted by Bap.q and Fap.q for a ∈Irk and k ∈N, respectively, keep many interesting properties, such as embedding theorems (with scales property for all smoothness weights), lifting properties for all parameters a, and duality for index 0 < p < ∞. By constructing an example, it is shown that there are infinitely many generalized Besov spaces and generalized Triebel-Lizorkin spaces lying between Bs,p.q and ∪tsBt,p.q,and between Fsp.q and ∪ts Ftp.q, respectively. Between Bs,p,q and ∪tsBt,p.qq,and between Fsp,qand ∪tsFtp.q,respectively.     

Normality concerning shared values

Let be a family of meromorphic functions in a plane domain D, and a and b be finite non-zero complex values such that . If for and , then is normal. We also construct a non-normal family of meromorphic functions in the unit disk Δ={|z|<1} such that for every and in Δ, where m is a given positive integer. This answers Problem 5.1 in the works of Gu, Pang and Fang. This work was supported by National Natural Science Foundation of China (Grant Nos. 10671093, 10871094) and the Natural Science Foundation of Universities of Jiangsu Province of China (Grant No. 08KJB110001), the Qing Lan Project of Jiangsu, China and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry     

Infinitely many radial solutions to elliptic problems with critical Sobolev and Hardy terms

Let Ω RN be a ball centered at the origin with radius R > 0 and N 7, 2* = 2N/N-2. We obtain the existence of infinitely many radial solutions for the Dirichlet problem -△u = μ |x|2 u |u|2*-2u λu in Ω, u = 0 on аΩ for suitable positive numbers μ and λ. Such solutions are characterized by the number of their nodes.     

<Emphasis Type="Italic">K</Emphasis><Subscript>0</Subscript>-group and approximate similarity invariants of strongly irreducible operators

Let H olenote a complex separable Hilbert space and L(H) denote the collection of bounded linear operators on H. An operator T ∈ L(H) is said to be strongly irreducible if T does not commute with any nontrivial idempotent. Herrero and Jiang showed that the norm-closure of the class of all strongly irreducible operators is the class of all operators with connected spectrum. This result can be considered as an approximate inverse of the Riesz decomposition theorem. In the paper, we give a more precise charact...     

Dominant Families of Pointed Curves in Projective Spaces

Fix non-negative integers r, e, m, g, s such that r ≥ 3, 0 ≤ m < r, e > 0, g + s ≤ er + max{0, m − 1} + 2, g ≤ (e − 1)r + max{0,m − 1} and 0 ≤ s ≤ er + 2. Set d := er + m. Fix any such that and S is in linearly general position. Fix an ordering of the points P ₁, . . . , P _s of S. Here we prove the existence of an irreducible family Γ of smooth, non-degenerate and connected curves with degree d and genus g, all of them containing S and such that the induced map is dominant. Received: September 19, 2006.     

Approximately isometric lifting in quasidiagonal extensions

Let 0 → I → A → A/I → 0 be a short exact sequence of C*-algebras with A unital. Suppose that the extension 0 → I → A → A/I → 0 is quasidiagonal, then it is shown that any positive element (projection, partial isometry, unitary element, respectively) in A/I has a lifting with the same form which commutes with some quasicentral approximate unit of I consisting of projections. Furthermore, it is shown that for any given positive number ε, two positive elements (projections, partial isometries, unitary elements, respectively) in A/I, and a positive element (projection, partial isometry, unitary element, respectively) a which is a lifting of , there is a positive element (projection, partial isometry, unitary element, respectively) b in A which is a lifting of such that ∥a−b∥ < . As an application, it is shown that for any positive numbers ε and in U(A/I) ₀ , there exists u in U(A)₀ which is a lifting of such that cel(u) < cel. This work was supported by National Natural Science Foundation of China (Grant No. 10771161)     

Aluthge Transforms of Complex Symmetric Operators

If denotes the polar decomposition of a bounded linear operator T, then the Aluthge transform of T is defined to be the operator . In this note we study the relationship between the Aluthge transform and the class of complex symmetric operators (T iscomplex symmetric if there exists a conjugate-linear, isometric involution so that T = CT*C). In this note we prove that: (1) the Aluthge transform of a complex symmetric operator is complex symmetric, (2) if T is complex symmetric, then and are unitarily equivalent, (3) if T is complex symmetric, then if and only if T is normal, (4) if and only if T ² = 0, and (5) every operator which satisfies T ² = 0 is necessarily complex symmetric. This work partially supported by National Science Foundation Grant DMS 0638789.     

Asymptotic behavior of the irrational factor

We study the irrational factor function I(n) introduced by Atanassov and defined by , where is the prime factorization of n. We show that the sequence {G(n)/n} _n≧1, where G(n) = Π _ν=1 ⁿ I(ν)^1/n , is convergent; this answers a question of Panaitopol. We also establish asymptotic formulas for averages of the function I(n). Research of the third author is supported in part by NSF grant number DMS-0456615.     

Classification of infinitely differentiable periodic functions

The set of infinitely differentiable periodic functions is studied in terms of generalized -derivatives defined by a pair of sequences ψ ₁ and ψ ₂. In particular, we establish that every function f from the set has at least one derivative whose parameters ψ ₁ and ψ ₂ decrease faster than any power function. At the same time, for an arbitrary function f ∈ different from a trigonometric polynomial, there exists a pair ψ whose parameters ψ ₁ and ψ ₂ have the same rate of decrease and for which the -derivative no longer exists. We also obtain new criteria for 2π-periodic functions real-valued on the real axis to belong to the set of functions analytic on the axis and to the set of entire functions. Deceased. (A. I. Stepanets) Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 12, pp. 1686–1708, December, 2008.     

A formula for the Bloch norm of a C 1-function on the unit ball of ℂ n

For a C ¹-function f on the unit ball ⊂ ℂ ⁿ we define the Bloch norm by , where is the invariant derivative of f, and then show that . Supported by MNZŽS Serbia, Project No. 144010.     

On some problems of weak consistency of quasi-maximum likelihood estimates in generalized linear models

In this paper,we explore some weakly consistent properties of quasi-maximum likelihood estimates(QMLE) concerning the quasi-likelihood equation in=1 Xi(yi-μ(Xiβ)) = 0 for univariate generalized linear model E(y |X) = μ(X'β).Given uncorrelated residuals {ei = Yi-μ(Xiβ0),1 i n} and other conditions,we prove that βn-β0 = Op(λn-1/2) holds,where βn is a root of the above equation,β0 is the true value of parameter β and λn denotes the smallest eigenvalue of the matrix Sn = ni=1 XiXi.We also show that the convergence rate above is sharp,provided independent non-asymptotically degenerate residual sequence and other conditions.Moreover,paralleling to the elegant result of Drygas(1976) for classical linear regression models,we point out that the necessary condition guaranteeing the weak consistency of QMLE is Sn-1→ 0,as the sample size n →∞.     

Lie algebras determined by finite valued Auslander-Reiten quivers

Let г denote a connected valued Auslander-Reiten quiver, let ℒ(γ) denote the free abelian group generated by the vertex setγ ₀ and let ℒ(Γ) be the universal cover ofг with fundamental groupG. It is proved that whenγ is a finite connected valued Auslander-Reiten quiver,ℒ(γ) is a Lie subalgebra ofℋ(г), and is just the “orbit” Lie algebra ℒ( )/G, where ℋ (г)₁ is the degenerate Hall algebra ofг and ℒ( )/G is the “orbit” Lie algebra induced by .     

On a special basis of approximate eigenvectors with local supports for an isolated narrow cluster of eigenvalues of a symmetric tridiagonal matrix

Let $ \tilde \lambda Let be an approximate eigenvalue of multiplicity m _c = n − r of an n × n real symmetric tridiagonal matrix T having nonzero off-diagonal entries. A fast algorithm is proposed (and numerically tested) for deleting m _c rows of T− I so that the condition number of the r × n matrix B formed of the remaining r rows is as small as possible. A special basis of m _c vectors with local supports is constructed for the subspace kerB. These vectors are approximate eigenvectors of T corresponding to . Another method for deleting m _c rows of T− I is also proposed. This method uses a rank-revealing QR decomposition; however, it requires a considerably larger number of arithmetic operations. For the latter algorithm, the condition number of B is estimated, and orthogonality estimates for vectors of the special basis of kerB are derived. Original Russian Text ? S.K. Godunov, A.N. Malyshev, 2008, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2008, Vol. 48, No. 7, pp. 1156–1166.