The maximal Fejér operator on the spacesH p (R×···×R)

It is shown that the maximal operator of the Fejér means of a tempered distribution is bounded from thed-dimensional Hardy spaceH _p (R×···×R) toL _p (R ^d ) (1/2<p<∞) and is of weak type (H ₁ ^?i ,L ₁) (i=1,…,d), where the Hardy spaceH ₁ ^?i is defined by a hybrid maximal function. As a consequence, we obtain that the Fejér means of a functionf ∈H ₁ ^?i ?L(logL) ^d?1 converge a.e. to the function in question. Moreover, we prove that the Fejér means are uniformly bounded onH _p (R×···×R) whenever 1/2<p<∞. Thus, in casef ∈H _p (R×···×R) the Fejér means converge tof inH _p (R×···×R) norm. The same results are proved for the conjugate Fejér means, too.     

Topological change in mean convex mean curvature flow

Consider the mean curvature flow of an (n+1)-dimensional compact, mean convex region in Euclidean space (or, if n<7, in a Riemannian manifold). We prove that elements of the mth homotopy group of the complementary region can die only if there is a shrinking S ^k ×R ^n?k singularity for some k≤m. We also prove that for each m with 1≤m≤n, there is a nonempty open set of compact, mean convex regions K in R ⁿ⁺¹ with smooth boundary ?K for which the resulting mean curvature flow has a shrinking S ^m ×R ^n?m singularity.     

Multivariate model with a Kronecker product covariance structure: S. N. Roy method of estimation

In this paper we consider a (p × q)-matrix X = (X ₁, ..., X _q ), where a pq-vector vec (X) = (X ₁ ^T , ...,X _q ^T ) ^T is assumed to be distributed normally with mean vector vec (M) = (M ₁ ^T , ...,M _q ^T ) ^T and a positive definite covariance matrix Λ. Suppose that Λ follows a Kronecker product covariance structure, that is Λ = Φ?Σ, where Φ = (? _ij ) is a (q × q)-matrix and Σ = (σ _ij ) is a (p × p)-matrix and the matrices Φ, Σ are positive definite. Such a model is considered in [4], where the maximum likelihood estimates of the parameters M, Φ, Σ are obtained. Using S. N. Roy’s technique (see, e.g., [3]) of the multivariate statistical analysis, we obtain consistent and unbiased estimates of M, Φ, Σ as in [4], but with less calculations.     

On the expansion with respect to Steklov means of the second differences of functions

We shall prove that every function locally integrable in then-dimensional Euclidean spaceR ⁿ can be expanded into a series whose terms are the Steklov means of the second differences of the given function. In addition, the lengths of the edges of the cubes with respect to which averaging is taken form an infinite decreasing geometric progression. The series obtained in this way converge almost everywhere inR ⁿ . If the function expanded belongs to the Lebesgue spaceL _p on a compact set ofR ⁿ for some 1≤p<∞, then the expansion converges also in the norm of this space.     

Difference Sets in {\mathbb{Z}}_4^m and {\mathbb{F}}_2^{2m}

Let R=GR(4,m) be the Galois ring of cardinality 4^m and let T be the Teichmüller system of R. For every map λ of T into { -1,+1} and for every permutation Π of T, we define a map φ _λ ^Π of Rinto { -1,+1} as follows: if x∈R and if x=a+2b is the 2-adic representation of x with x∈T and b∈T, then φ _λ ^Π (x)=λ(a)+2Tr(Π(a)b), where Tr is the trace function of R . For i=1 or i=-1, define D _i as the set of x in R such thatφ _λ ^Π =i. We prove the following results: 1) D _i is a Hadamard difference set of (R,+). 2) If φ is the Gray map of R into ${\mathbb{F}}_2^{2m}$ , then (D _i) is a difference set of ${\mathbb{F}}_2^{2m}$ . 3) The set of D _i and the set of φ(D _i) obtained for all maps λ and Π, both are one-to-one image of the set of binary Maiorana-McFarland difference sets in a simple way. We also prove that special multiplicative subgroups of R are difference sets of kind D _i in the additive group of R. Examples are given by means of morphisms and norm in R.     

Einbettung von topologischen Raumgeometrien auf R3 in den reellen affinen Raum

Betten [1] had defined topological spatial geometries on R ³: In R ³ a system L of closed subsets homeomorphic to R (the lines) and a system ? of closed subsets homeomorphic to R ² (the planes) are given such that through any two different points passes exactly one line and through any three non-collinear points passes exactly one plane. Furthermore, ? and ? carry topologies such that the operations of joining and intersection are continuous. It is proved that any topological spatial geometry on R ³ can be imbedded into R ³ as an open convex subset K such that the lines in ? (planes in ?) are mapped onto intersections of lines (planes) of R ³ with K. The collineation group of the geometry is isomorphic to the subgroup of the colineation group of real projective space consisting of the automorphisms that map K into itself. In particular, it is a Lie group of dimension ?12.     

Average width and optimal recovery of the anisotropic classesW r H α (R d ) in the metricC k (R d )

This paper concerns the problems of the average widths and the optimal recovery of the anisotropic Hölder classesW ^r H ^α (R ^d ) of smooth functions defined on the Euclidean spaceR ^d in the metricC ^k (R ^d ). The weak asymptotic behavior is established for the corresponding quantities.     

Embeddings into power series rings

Let T be an ordered ring without divisors of zero, and letA be the set of archimedean subgroups of T generated by a Banaschewski functionτ. Let_XΠ_Δ R be the power series ring of the real numbers ? over the totally ordered semigroup Δ of archimedean classes of T, and letχ be the usual Banaschewski function on_XΠ_Δ R. The following are equivalent:

1. τ satisfies the additional condition; for convex subgroups P,Q of T, where
2. There exists a one-to-one homomorphism Γ:T→_XΠ_Δ R of ordered rings such that for every convex subgroup Q of_XΠ_Δ R, there exists a convex subgroup P of T such that \(\Gamma (P) \subseteq Q\) and \(\Gamma (\tau (P)) \subseteq \chi (Q)\) .

     

A note on invariant sets of iterated function systems

We prove that the family of all invariant sets of iterated systems of contractions R ^N → R ^N is a nowhere dense F _σ type subset in the space of the nonempty compact subsets of R ^N equipped with the Hausdorff metric.     

Sets of vectors with many orthogonal paris:

What is the most number of vectors inR ^d such that anyk+1 contain an orthogonal pair? The 24 positive roots of the root systemF ₄ inR ⁴ show that this number could exceeddk.     

On a J. C. C. Nitsche’s Type Inequality for the Hyperbolic Space H3

Let H ³ be the hyperbolic space identified with the unit ball B ³ = {x ∈ R ³ : |x| < 1} with the Poincaré metric d _h and let ??(x ₀, p, q) : = {x : p < d _h (x, x ₀) < q} ? H ³ be a hyperbolic annulus with the inner and outer radii 0 < p < q < ∞. We prove that if there exists a hyperbolic harmonic diffeomorphism between annuli ??(x ₀, a, b) and ?? (y ₀, α, β) in the hyperbolic space H ³, then β/α>1+ψ(a,b), where ψ is a positive function. In addition, for given two annuli in the hyperbolic space H ³, satisfying certain conditions, we construct radial harmonic mappings between them.     

Contraction mappings in interval analysis

Under certain conditions, the contraction mapping fixed point theorem guarantees the convergence of the iterationx _i+1=f(x _i ) toward a fixed point of the functionf:R ⁿ →R ⁿ. When an interval extensionF off is used in a similar iteration scheme to obtain a sequence of interval vectors these conditions need not provide convergence to a degenerate interval vector representing the fixed point, even if the width of the initial interval vector is chosen arbitrarily small. We give a sufficient condition on the extensionF in order that the convergence is guaranteed. The centered form of Moore satisfies this condition.     

Operators in Krein Space

We study self-adjoint operators in Krein space. Our goal is to show that there is a relationship between the following classes of operators: operators with a compact “corner,” definitizable operators, operators of classes (H) and K(H), and operators of class D _κ ⁺.     

Duals of Hardy Spaces of B-valued Martingales for 0<r≤1

The dual of B-valued martingale Hardy space Hs(p)r(B) with small index 0 r ≤ 1,which is associated with the conditional p-variation of B-valued martingale,is characterized.In order to obtain the results,a new type of Campanato spaces for B-valued martingales is introduced and the classical technique of atomic decompositions is improved.Some results obtained here are connected closely with the p-uniform smoothness and q-uniform convexity of the underlying Banach space.     

Approximation from shift-invariant spaces

We give sufficient conditions on a single function ? so that the principal shift-invariant space generated by ? provides a prescribed order of approximation inL _p (R ^d ), 1<p<∞, and inH _p (R ^d ), 0<p≤1. In particular, our conditions are given in terms of $\hat \varphi$ and are satisfied even when ? does not decay quickly at infinity.     

On diversity and stability of unit bases for the Euclidean metric

We prove the existence of a family Ω(n) of 2 ^c (where c is the cardinality of the continuum) subgraphs of the unit distance graph (E ⁿ , 1) of the Euclidean space E ⁿ , n ≥ 2, such that (a) for each graph G ? Ω(n), any homomorphism of G to (E ⁿ , 1) is an isometry of E ⁿ ; moreover, for each subgraph G ₀ of the graph G obtained from G by deleting less than c vertices, less than c stars, and less than c edges (we call such a subgraph reduced), any homomorphism of G ₀ to (E ⁿ , 1) is an isometry (of the set of the vertices of G ₀); (b) each graph G ? Ω(n) cannot be homomorphically mapped to any other graph of the family Ω(n), and the same is true for each reduced subgraph of G.     

On rings whose number of centralizers of ideals is finite

LetR be a ring. For the setF of all nonzero ideals ofR, we introduce an equivalence relation inF as follows: For idealsI andJ, I～J if and only ifV _R (I)=V _R(J), whereV _R() is the centralizer inR. LetI _R=F/～. Then we can see thatn(I _R), the cardinality ofI _R, is 1 if and only ifR is either a prime ring or a commutative ring (Theorem 1.1). An idealI ofR is said to be a commutator ideal ifI is generated by{st?ts; s∈S, t∈T} for subsetS andT ofR, andR is said to be a ring with (N) if any commutator ideal contains no nonzero nilpotent ideals. Then we have the following main theorem: LetR be a ring with (N). Thenn(I _R) is finite if and only ifR is isomorphic to an irredundant subdirect sum ofS⊕Z whereS is a finite direct sum of non commutative prime rings andZ is a commutative ring (Theorem 2.1). Finally, we show that the existence of a ringR such thatn(I _R)=m for any given natural numberm.     

Torsion and Abelianization in Equivariant Cohomology

Let X be a topological space upon which a compact connected Lie group G acts. It is well known that the equivariant cohomology H ^* _G (X; Q) is isomorphic to the subalgebra of Weyl group invariants of the equivariant cohomology H ^* _T (X; Q), where T is a maximal torus of G. This relationship breaks down for coefficient rings k other than Q. Instead, we prove that under a mild condition on k the algebra H ^* _G (X; k) is isomorphic to the subalgebra of H ^* _T (X; k) annihilated by the divided difference operators.     

CAT(0) 4-manifolds possessing a single tame point are Euclidean

The concepts of geometric and topological tame point are introduced for a space of nonpositive curvature. These concepts are applied to the characterization problem forCAT(0) 4-manifolds. It is shown that everyCAT(0)M ⁴ having a single (geometric or topological) tame point is homeomorphic toR ⁴. Davis and Januszkiewicz have recently constructedCAT(0)n-manifolds,M ⁿ withn ≥ 5 such that the set of tame points form a dense open subset ofM ⁿ , butM ⁿ ≠R ⁿ .     

Fractional Covers for Convolution Products

A non-zero vector-valued sequence u ∈ ?^q(X′) is a cover for a subset M of ?_P(X) if, for some 0 < α ≤ 1, ∥u * h∥ _∞ ≥ α ∥u∥_q ∥h∥_p for all h ∈ M. Covers of ?¹ = ?¹(R) are important in worst case system identification in ?¹ and in the reconstruction of elements in a normed space from corrupted functional values. We investigate the existence of covers for certain naturally occurring subspaces of ?^p(X). We show that there exist finitely supported covers for some subspaces, and obtain lower bounds for their ’lengths’. We also obtain similar results for covers associated with convolution products for spaces of measurable vector-valued functions defined on the positive real axis.