Integral approximation of the characteristic function of an interval by trigonometric polynomials

We apply the results on the integral approximation of the characteristic function of an interval by the subspace $ \mathcal{T}_{n - 1} $ \mathcal{T}_{n - 1} of trigonometric polynomials of order at most n − 1, which were obtained by the authors earlier, to investigate the Jackson inequality between the best uniform approximation of a continuous periodic function by the subspace $ \mathcal{T}_{n - 1} $ \mathcal{T}_{n - 1} and its modulus of continuity of the second order. The corresponding method of the uniform approximation of continuous periodic functions by trigonometric polynomials is constructed.     

Distortio n theorems o n the Lie ball $$ \mathcal{R}_{IV} $$ ( n) i n ℂ n

In this paper, we introduce the subfamilies H _m ($ \mathcal{R}_{IV} $ \mathcal{R}_{IV} (n)) of holomorphic mappings defined on the Lie ball $ \mathcal{R}_{IV} $ \mathcal{R}_{IV} (n) which reduce to the family of holomorphic mappings and the family of locally biholomorphic mappings when m = 1 and m → +∞, respectively. Various distortion theorems for holomophic mappings H _m ($ \mathcal{R}_{IV} $ \mathcal{R}_{IV} (n)) are established. The distortion theorems coincide with Liu and Minda’s as the special case of the unit disk. When m = 1 and m → +∞, the distortion theorems reduce to the results obtained by Gong for $ \mathcal{R}_{IV} $ \mathcal{R}_{IV} (n), respectively. Moreover, our method is different. As an application, the bounds for Bloch constants of H _m ($ \mathcal{R}_{IV} $ \mathcal{R}_{IV} (n)) are given.     

A <Emphasis Type="Italic">τ</Emphasis>-function solution of the sixth painlevé transcendent

We represent and analyze the general solution of the sixth Painlevé transcendent $ \mathcal{P}_6 $ \mathcal{P}_6 in the Picard-Hitchin-Okamoto class in the Painlevé form as the logarithmic derivative of the ratio of τ-functions. We express these functions explicitly in terms of the elliptic Legendre integrals and Jacobi theta functions, for which we write the general differentiation rules. We also establish a relation between the $ \mathcal{P}_6 $ \mathcal{P}_6 equation and the uniformization of algebraic curves and present examples.     

Fourier-Feynman transforms of unbounded functionals on abstract Wiener space

Huffman, Park and Skoug established several results involving Fourier-Feynman transform and convolution for functionals in a Banach algebra S on the classical Wiener space. Chang, Kim and Yoo extended these results to abstract Wiener space for a more generalized Fresnel class $ \mathcal{F}_{\mathcal{A}_1 ,\mathcal{A}_2 } $ \mathcal{F}_{\mathcal{A}_1 ,\mathcal{A}_2 } _A1,A2 than the Fresnel class $ \mathcal{F} $ \mathcal{F} (B)which corresponds to the Banach algebra S. In this paper we study Fourier-Feynman transform, convolution and first variation of unbounded functionals on abstract Wiener space having the form

On the shift semigroup on the Hardy space of Dirichlet series

We develop a Wold decomposition for the shift semigroup on the Hardy space $ \mathcal{H}^2 $ \mathcal{H}^2 of square summable Dirichlet series convergent in the half-plane $ \Re (s) > 1/2 $ \Re (s) > 1/2 . As an application we have that a shift invariant subspace of $ \mathcal{H}^2 $ \mathcal{H}^2 is unitarily equivalent to $ \mathcal{H}^2 $ \mathcal{H}^2 if and only if it has the form $ \phi \mathcal{H}^2 $ \phi \mathcal{H}^2 for some $ \mathcal{H}^2 $ \mathcal{H}^2 -inner function φ.     

Imaginary powers of a Laguerre differential operator

Imaginary powers associated to the Laguerre differential operator $ L_\alpha = - \Delta + |x|^2 + \sum _{i = 1}^d \frac{1} {{x_i^2 }}(\alpha _i^2 - 1/4) $ L_\alpha = - \Delta + |x|^2 + \sum _{i = 1}^d \frac{1} {{x_i^2 }}(\alpha _i^2 - 1/4) are investigated. It is proved that for every multi-index α = (α₁,...α _d ) such that α _i ≧ −1/2, α _i ∉ (−1/2, 1/2), the imaginary powers $ \mathcal{L}_\alpha ^{ - i\gamma } ,\gamma \in \mathbb{R} $ \mathcal{L}_\alpha ^{ - i\gamma } ,\gamma \in \mathbb{R} , of a self-adjoint extension of L _α, are Calderón-Zygmund operators. Consequently, mapping properties of $ \mathcal{L}_\alpha ^{ - i\gamma } $ \mathcal{L}_\alpha ^{ - i\gamma } follow by the general theory.     

Parametric estimation with a class of M-estimators

Consider a class of M-estimators indexed by a criterion function ψ. When the function ψ is taken to be in a class of functions $ \mathcal{F} $ \mathcal{F} , a family of processes indexed by the class $ \mathcal{F} $ \mathcal{F} is obtained and called M-processes. Pooling the M-estimators in such class may be used to define new kind of estimators. In order to get the asymptotic properties of these pooled estimators, the convergence in probability of the corresponding M-process is studied uniformly on $ \mathcal{F} $ \mathcal{F} together with their weak convergence towards a Gaussian process. An application to location estimation is presented and discussed.     

Integral-type operators from a mixed norm space to a bloch-type space on the unit ball

Let $ \mathbb{B} $ \mathbb{B} be the unit ball in ℂ ⁿ and let H($ \mathbb{B} $ \mathbb{B} ) be the space of all holomorphic functions on $ \mathbb{B} $ \mathbb{B} . We introduce the following integral-type operator on H($ \mathbb{B} $ \mathbb{B} ):

Smooth maps of a foliated manifold in a symplectic manifold

Let M be a smooth manifold with a regular foliation $ \mathcal{F} $ \mathcal{F} and a 2-form ω which induces closed forms on the leaves of $ \mathcal{F} $ \mathcal{F} in the leaf topology. A smooth map f: (M, $ \mathcal{F} $ \mathcal{F} ) → (N, σ) in a symplectic manifold (N, σ) is called a foliated symplectic immersion if f restricts to an immersion on each leaf of the foliation and further, the restriction of f*σ is the same as the restriction of ω on each leaf of the foliation. If f is a foliated symplectic immersion then the derivative map Df gives rise to a bundle morphism F: TM → T N which restricts to a monomorphism on T $ \mathcal{F} $ \mathcal{F} ⊆ T M and satisfies the condition F*σ = ω on T $ \mathcal{F} $ \mathcal{F} . A natural question is whether the existence of such a bundle map F ensures the existence of a foliated symplectic immersion f. As we shall see in this paper, the obstruction to the existence of such an f is only topological in nature. The result is proved using the h-principle theory of Gromov.     

A generalization of Mathieu subspaces to modules of associative algebras

We first propose a generalization of the notion of Mathieu subspaces of associative algebras $ \mathcal{A} $ \mathcal{A} , which was introduced recently in [Zhao W., Generalizations of the image conjecture and the Mathieu conjecture, J. Pure Appl. Algebra, 2010, 214(7), 1200–1216] and [Zhao W., Mathieu subspaces of associative algebras], to $ \mathcal{A} $ \mathcal{A} -modules $ \mathcal{M} $ \mathcal{M} . The newly introduced notion in a certain sense also generalizes the notion of submodules. Related with this new notion, we also introduce the sets σ(N) and τ(N) of stable elements and quasi-stable elements, respectively, for all R-subspaces N of $ \mathcal{A} $ \mathcal{A} -modules $ \mathcal{M} $ \mathcal{M} , where R is the base ring of $ \mathcal{A} $ \mathcal{A} . We then prove some general properties of the sets σ(N) and τ(N). Furthermore, examples from certain modules of the quasi-stable algebras [Zhao W., Mathieu subspaces of associative algebras], matrix algebras over fields and polynomial algebras are also studied.     

Approximation and numerical realization of 3D contact problems with given friction and a coefficient of friction depending on the solution

The paper presents the analysis, approximation and numerical realization of 3D contact problems for an elastic body unilaterally supported by a rigid half space taking into account friction on the common surface. Friction obeys the simplest Tresca model (a slip bound is given a priori) but with a coefficient of friction $ \mathcal{F} $ \mathcal{F} which depends on a solution. It is shown that a solution exists for a large class of $ \mathcal{F} $ \mathcal{F} and is unique provided that $ \mathcal{F} $ \mathcal{F} is Lipschitz continuous with a sufficiently small modulus of the Lipschitz continuity. The problem is discretized by finite elements, and convergence of discrete solutions is established. Finally, methods for numerical realization are described and several model examples illustrate the efficiency of the proposed approach.     

Note on $$ \mathcal{I}R $$-closed and $$ A_{\mathcal{I}R} $$-sets

We characterize and discuss the properties of $ \mathcal{I}R $ \mathcal{I}R -closed sets and $ A_{\mathcal{I}R} $ A_{\mathcal{I}R} -sets. Also, we give characterizations of weakly $ \mathcal{I} $ \mathcal{I} -locally closed sets and $ \mathcal{I} $ \mathcal{I} -submaximal spaces. A characterization of codense ideals in terms of $ \mathcal{I}R $ \mathcal{I}R -closed sets is also given.     

<Emphasis Type="Italic">g</Emphasis>-closed sets in ideal topological spaces

Characterizations and properties of $ \mathcal{I}_g $ \mathcal{I}_g -closed sets and $ \mathcal{I}_g $ \mathcal{I}_g -open sets are given. A characterization of normal spaces is given in terms of $ \mathcal{I}_g $ \mathcal{I}_g -open sets. Also, it is established that an $ \mathcal{I}_g $ \mathcal{I}_g -closed subset of an $ \mathcal{I} $ \mathcal{I} -compact space is $ \mathcal{I} $ \mathcal{I} -compact.     

Weak convergence in the dual of weak L<Superscript>p</Superscript>

We consider a Dedekind σ-complete Banach lattice E whose dual is weakly sequentially complete. Suppose that E has a positive element u and a family of positive operators $ \mathcal{G} $ \mathcal{G} such that

The Hasse-Witt invariant in some towers of function fields over finite fields

In this article we investigate the p-rank of function fields in several good towers. To do this we first recall and establish some properties of the behaviour of the p-rank under extensions. Then we compute the p-ranks of function fields in several optimal towers over a quadratic field $ \mathbb{F}_{q^2 } $ \mathbb{F}_{q^2 } , as well as for a specific good tower over a cubic field $ \mathbb{F}_{q^3 } $ \mathbb{F}_{q^3 } , which was introduced by Bassa, Garcia and Stichtenoth.     

Analogues of Horn’s theorem for finite unions of starshaped sets in ℝ<Superscript><Emphasis Type="Italic">d</Emphasis></Superscript>

Fix k, d, 1 ≤ k ≤ d + 1. Let $ \mathcal{F} $ \mathcal{F} be a nonempty, finite family of closed sets in ℝ ^d , and let L be a (d − k + 1)-dimensional flat in ℝ ^d . The following results hold for the set T ≡ ∪{F: F in $ \mathcal{F} $ \mathcal{F} }. Assume that, for every k (not necessarily distinct) members F ₁, …, F _k of $ \mathcal{F} $ \mathcal{F} ,∪{F _i : 1 ≤ i ≤ k} is starshaped and the corresponding kernel contains a translate of L. Then T is starshaped, and its kernel also contains a translate of L.     

Real moments of the restrictive factor

Let λ be a real number such that 0 < λ < 1. We establish asymptotic formulas for the weighted real moments Σ _n≤x R ^λ (n)(1 − n/x), where R(n) =$ \prod\nolimits_{\nu = 1}^k {p_\nu ^{\alpha _\nu - 1} } $ \prod\nolimits_{\nu = 1}^k {p_\nu ^{\alpha _\nu - 1} } is the Atanassov strong restrictive factor function and n =$ \prod\nolimits_{\nu = 1}^k {p_\nu ^{\alpha _\nu } } $ \prod\nolimits_{\nu = 1}^k {p_\nu ^{\alpha _\nu } } is the prime factorization of n.     

A$\mathcal{T}$-algebras and extensions of A$\mathbb{T}$-algebras

Lin and Su classified A$ \mathcal{T} $ \mathcal{T} -algebras of real rank zero. This class includes all A$ \mathbb{T} $ \mathbb{T} -algebras of real rank zero as well as many C*-algebras which are not stably finite. An A$ \mathcal{T} $ \mathcal{T} -algebra often becomes an extension of an A$ \mathbb{T} $ \mathbb{T} -algebra by an AF-algebra. In this paper, we show that there is an essential extension of an A$ \mathbb{T} $ \mathbb{T} -algebra by an AF-algebra which is not an A$ \mathcal{T} $ \mathcal{T} -algebra. We describe a characterization of an extension E of an A$ \mathbb{T} $ \mathbb{T} -algebra by an AF-algebra if E is an A$ \mathcal{T} $ \mathcal{T} -algebra.     

The generalized Roper-Suffridge extension operator in Banach spaces (III)

Suppose that X is a complex Banach space with the norm ‖·‖ and n is a positive integer with dim X ⩾ n ⩾ 2. In this paper, we consider the generalized Roper-Suffridge extension operator $ \Phi _{n,\beta _2 ,\gamma _2 , \ldots ,\beta _{n + 1} ,\gamma _{n + 1} } (f) $ \Phi _{n,\beta _2 ,\gamma _2 , \ldots ,\beta _{n + 1} ,\gamma _{n + 1} } (f) on the domain $ \Omega _{p_1 ,p_2 , \ldots ,p_{n + 1} } $ \Omega _{p_1 ,p_2 , \ldots ,p_{n + 1} } defined by