C-spaces and simplicial complexes

Given a class

of simplicial complexes G, we introduce the notion of a

-C-space. In the definition of a C-space, open disjoint families v _i refine coverings u _i . The nerves of these families are zero-dimensional complexes. In our definition, the nerve of a family vi must embed in the complex G _i of the class

. We give a complete characterization of bicompact

-C-spaces.

     

On Jordan ideals and submodules: Algebraic and analytic aspects

Let

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be an algebra, and let X be an arbitrary

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-bimodule. A linear space Y ? X is called a Jordan

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-submodule if Ay + yA ∈ Y for all A ∈

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and y ∈ Y. (For X =

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, this coincides with the notion of a Jordan ideal.) We study conditions under which Jordan submodules are subbimodules. General criteria are given in the purely algebraic situation as well as for the case of Banach bimodules over Banach algebras. We also consider symmetrically normed Jordan submodules over C*-algebras. It turns out that there exist C*-algebras in which not all Jordan ideals are ideals.

     

Bernstein <Emphasis Type="Italic">n</Emphasis>-widths of Besov embeddings on Lipschitz domains

In this paper, using an equivalent characterization of the Besov space by its wavelet coefficients and the discretization technique due to Maiorov, we determine the asymptotic degree of the Bernstein n-widths of the compact embeddings Bq0s＋t（Lp0（Ω））→Bq1s（Lp1（Ω））, t〉max{d（1/p0-1/p1）, 0}, 1 ≤ p0, p1, q0, q1 ≤∞,where Bq0s＋t（Lp0（Ω）） is a Besov space defined on the bounded Lipschitz domain Ω ？ Rd. The results we obtained here are just dual to the known results of Kolmogorov widths on the related classes of functions.     

Optimization of the boundary control of string vibrations in the excitation problem with a fixed endpoint in the class <Emphasis Type="Italic">W</Emphasis><Stack><Subscript>2</Subscript><Superscript>2</Superscript></Stack>

We consider the problem of optimal boundary control of string vibrations by a displacement at one endpoint with the other endpoint being fixed; the problem is considered in the space

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and then, more generally, in the space

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, p ≥ 1. The control brings the vibration process from the quiescent state to an arbitrarily prescribed state in a time that is a multiple of the double string length.

     

Boundedness and compactness of an integral operator between <Emphasis Type="Italic">H</Emphasis><Superscript>∞</Superscript> and a mixed norm space on the polydisk

This addendum to [1] completely characterizes the boundedness and compactness of a recently introduced integral type operator from the space of bounded holomorphic functions H ^∞(\(\mathbb{D}^n \)) on the unit polydisk \(\mathbb{D}^n \) to the mixed norm space

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with p, q ∈ [1,∞) and α = (α₁, ..., α _n ) such that α _j > ?1 for every j = 1, ..., n. We show that the operator is bounded if and only if it is compact and if and only if g ∈

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, where \(\vec q\) = (q, ..., q).

     

<Emphasis Type="Italic">G</Emphasis><Subscript>2</Subscript>-holonomy metrics connected with a 3-Sasakian manifold

We construct complete noncompact Riemannian metrics with G ₂-holonomy on noncompact orbifolds that are ?³-bundles with the twistor space Open image in new window as a spherical fiber.     

A note on a result of Skiba

A subgroup H of a group G is called weakly s-permutable in G if there is a subnormal subgroup T of G such that G = HT and H ∩ T ≤ H _sG , where H _sG is the maximal s-permutable subgroup of G contained in H. We improve a nice result of Skiba to get the following

Theorem. Let ? be a saturated formation containing the class of all supersoluble groups

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and let G be a group with E a normal subgroup of G such that G/E ∈ ?. Suppose that each noncyclic Sylow p-subgroup P of F*(E) has a subgroup D such that 1 < |D| < |P| and all subgroups H of P with order |H| = |D| are weakly s-permutable in G for all p ∈ π(F*(E)); moreover, we suppose that every cyclic subgroup of P of order 4 is weakly s-permutable in G if P is a nonabelian 2-group and |D| = 2. Then G ∈ ?.

     

Sur les fonctions arithmétiques non entières

A theorem of Baker says that a function F entire on ? ^d such that F(? ^d ) ? ? and increasing slower (in a precise sense) than \(2^{z_1 + \cdots + z_d } \) is necessarily a polynomial. This is a multivariate generalisation of the celebrated theorem of Pólya (case d = 1). Using the theory of analytic functionals with non-compact carrier, Yoshino proved a general theorem dealing with the growth of arithmetic analytic functions, which implies that the conclusion of Baker’s theorem holds if F is only assumed to be holomorphic on the domain

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, j = 1,..., d}.

The case d = 1 was also treated in a different way by Gel’fond and Pólya by means of the characteristic function of Carlson-Nörlund. This function was introduced to bound in a nearly optimal way the growth of holomorphic functions of one variable that can be expanded in a Newton interpolation series in the half-plane

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.

In this article, we show how this characteristic function can also be used to bound the growth of multivariate functions defined on

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that can be expanded in multiple Newton series. These considerations enable us to improve Gel’fond-Pólya’s and Yoshino’s theorems, in particular, to remove or to weaken certain of their technical conditions.

     

<Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-boundedness of a boundary integral operator on a contour with a peak

Results on the solvability of boundary integral equations on a plane contour with a peak obtained in collaboration with V.G. Maz’ya are developed. Earlier, it was proved that, on a contour Γ with an outward peak, the operator of the boundary equation of the Dirichlet boundary value problem maps the space ? _{p, β + 1} (Γ) continuously onto \(\mathcal{N}_{p,\beta } (\Gamma )\). The norm of a function in ? _{p, β} (Γ) is defined as

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, provided that the peak is at the origin. In this case, the norms on the spaces \(\mathcal{N}_{p,\beta }^ \mp (\Gamma )\) are defined by

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, where q _± are the intersection points of Γ with the circle {z: |z| = |q|} and δ > 0 is a fixed small number. On a contour with an inward peak, the operator of the boundary equation of the Dirichlet problem continuously maps ? _{p, β + 1} (Γ) onto ? _{p, β}(Γ), where ? _{p, β}(Γ) is the direct sum of \(\mathcal{N}_{p,\beta }^ + (\Gamma )\) (Γ) and the space

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(Γ) of functions on Γ of the form p(z) = Σ _{k = 0} ^m t ^(k)Rez ^k with the parameter m = [μ ? β ? p ^?1]. The operator I ? 2W of the boundary integral equation of plane elasticity theory, where W is the elastic double-layer potential, is considered. The main result is that the operator I ? 2W continuously maps the space ? _{p, β + 1} × ? _{p, β + 1}(Γ) to the space \(\mathcal{N}_{p,\beta }^ - \times \mathcal{N}_{p,\beta }^ - (\Gamma )\).

On a contour with an inward peak, the obtained representation of the operator I ? 2W and theorems on the boundedness of auxiliary integral operators imply that the images of vector-valued functions from ? _{p, β + 1} × ? _{p, β + 1}(Γ) have components representable as sums of functions from the spaces \(\mathcal{N}_{p,\beta }^ - (\Gamma )\)(Γ) and ? _{p, β}(Γ).     

Weakly infinite-dimensional spaces modulo simplicial complexes

The classes of spaces

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-wid and ?-wid are introduced for the class

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of finite simplicial complexes and the class ? of compact polyhedra. If

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, then

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-wid = wid, ?-wid = S-wid. It is proved that S-wid ? ?-wid and ?-wid = S-? _τ -wid for any triangulation τ of the class ?.

     

Strongly Gauduchon Spaces

The authors define strongly Gauduchon spaces and the class■■, which are generalization of strongly Gauduchon manifolds in complex spaces. Comparing with the case of Kahlerian, the strongly Gauduchon space and the class■are similar to the Kahler space and the Fujiki class■■ respectively. Some properties about these complex spaces are obtained. Moreover, the relations between the strongly Gauduchon spaces and the class■■ are studied.     

Abelian groups small with respect to reduced groups

In this paperwe study

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-small abelian groups, where

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is the class of all reduced groups.

     

<Emphasis Type="Italic">K</Emphasis>-Radical classes and product radical classes of <Emphasis Type="Italic">MV</Emphasis>-algebras

For an MV-algebra let J ₀( ) be the system of all closed ideals of ; this system is partially ordered by the set-theoretical inclusion. A radical class X of MV-algebras will be called a K-radical class iff, whenever ∈ X and is an MV-algebra with J ₀( ) ≅ J ₀( ), then ∈ X. An analogous notation for lattice ordered groups was introduced and studied by Conrad. In the present paper we show that there is a one-to-one correspondence between K-radical classes of MV-algebras and K-radical classes of abelian lattice ordered groups. We also prove an analogous result for product radical classes of MV-algebras; product radical classes of lattice ordered groups were studied by Ton. This work has been partially supported by the Slovak Academy of Sciences via the project Center of Excellence-Physics of Information, Grant I/2/2005.     

Commuting variety of Witt algebra

Let \(\mathfrak{g} = W_1 \) be the Witt algebra over an algebraically closed field k of characteristic p > 3; and let Open image in new window be the commuting variety of g. In contrast with the case of classical Lie algebras of P. Levy [J. Algebra, 2002, 250: 473–484], we show that the variety Open image in new window is reducible, and not equidimensional. Irreducible components of Open image in new window and their dimensions are precisely given. As a consequence, the variety Open image in new window is not normal.     

On stability of CR-mappings between nilpotent lie groups of step two

A quasi-CR-mapping from a nilpotent Lie group

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of step two to another such group satisfies a Beltrami-type system of partial differential equations which is usually not elliptic but subelliptic when the group

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is strongly 2-pseudoconcave. We derive an integral representation formula for CR-mappings from a strongly 2-pseudoconcave nilpotent Lie group of step two to another such group and establish the Hölder continuity of ε-quasi-CR-mappings and the stability of CR-mappings between such groups.

     

Approximation by local <Emphasis Type="Italic">L</Emphasis>-splines corresponding to a linear differential operator of the second order

For the class of functions

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, where \(\mathcal{L}_2 (D)\) is a linear differential operator of the second order whose characteristic polynomial has only real roots, we construct a noninterpolating linear positive method of exponential spline approximation possessing extremal and smoothing properties and locally inheriting the monotonicity of the initial data (the values of a function

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at the points of a uniform grid). The approximation error is calculated exactly for this class of functions in the uniform metric.

     

On <Emphasis Type="Italic">S</Emphasis>-quasinormal and <Emphasis Type="Italic">c</Emphasis>-normal subgroups of a finite group

Let be a saturated formation containing the class of supersolvable groups and let G be a finite group. The following theorems are presented: (1) G ∈ if and only if there is a normal subgroup H such that G/H ∈ and every maximal subgroup of all Sylow subgroups of H is either c-normal or S-quasinormally embedded in G. (2) G ∈ if and only if there is a normal subgroup H such that G/H ∈ and every maximal subgroup of all Sylow subgroups of F*(H), the generalized Fitting subgroup of H, is either c-normal or S-quasinormally embedded in G. (3) G ∈ if and only if there is a normal subgroup H such that G/H ∈ and every cyclic subgroup of F*(H) of prime order or order 4 is either c-normal or S-quasinormally embedded in G. Supported by the Natural Science Foundation of China and the Natural Science Foundation of Guangxi Autonomous Region (No. 0249001). Corresponding author. Supported in part by the Natural Science Foundation of China (10571181), NSF of Guangdong Province (06023728) and ARF(GDEI).     

Geodesic mapping onto Kählerian spaces of the first kind

In the present paper a generalized Kählerian space Open image in new window of the first kind is considered as a generalized Riemannian space \(\mathbb{G}\mathbb{R}_N \) with almost complex structure F _i ^h that is covariantly constant with respect to the first kind of covariant derivative.Using a non-symmetric metric tensor we find necessary and sufficient conditions for geodesic mappings f: Open image in new window with respect to the four kinds of covariant derivatives. These conditions have the form of a closed system of partial differential equations in covariant derivatives with respect to unknown components of the metric tensor and the complex structure of the Kählerian space Open image in new window .