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.     

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 .     

A sharp weightedL 2-estimate for the solution to the time-dependent Schrödinger equation

For Ξ∈R ⁿ ,t∈R andf∈S(R ⁿ ) define $\left( {S^2 f} \right)\left( t \right)\left( \xi \right) = \exp \left( {it\left| \xi \right|^2 } \right)\hat f\left( \xi \right)$ . We determine the optimal regularitys ₀ such that $\int_{R^n } {\left\| {(S^2 f)[x]} \right\|_{L^2 (R)}^2 \frac{{dx}}{{(1 + |x|)^b }} \leqslant C\left\| f \right\|_{H^s (R^n )}^2 ,s > s_0 } ,$ holds whereC is independent off∈S(R ⁿ ) or we show that such optimal regularity does not exist. This problem has been treated earlier, e.g. by Ben-Artzi and Klainerman [2], Kato and Yajima [4], Simon [6], Vega [9] and Wang [11]. Our theorems can be generalized to the case where the exp(it|ξ|²) is replaced by exp(it|ξ|^a),a≠2. The proof uses Parseval's formula onR, orthogonality arguments arising from decomposingL ²(R ⁿ ) using spherical harmonics and a uniform estimate for Bessel functions. Homogeneity arguments are used to show that results are sharp with respect to regularity.     

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.     

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.     

Commutative properties of singularly perturbed operators

Suppose the self-adjoint operatorA in the Hilbert spaceH commutes with the bounded operatorS. Suppose another self-adjoint operatorā is singularly perturbed with respect toA, i.e., it is identical toA on a certain dense set inH. We study the following question: Under what conditions doesā also commute withS? In addition, we consider the case whenS is unbounded and also the case whenS is replaced by a singularly perturbed operator S. As application, we consider the Laplacian inL ₂(R ^q ) that is singularly perturbed by a set of δ functions and commutes with the symmetrization operator inR ^q ,q=2, 3, or with regular representations of arbitrary isometric transformations inR ^q ,q≤3.     

Ombres Convexité, régularité et sous-harmonicité

Letf:V→R be a function defined on a subsetV ofR ⁿ ×R ^d let?:x→inf{f(x t);t such that(x t)∈V} denote theshadow off and letΦ={(x t)∈V; f(x t)=?(x)} This paper deals with the characterization of some properties of ? in terms of the infinitesimal behavior off near points ζ∈Φ proving in particular a conjecture of J M Trépreau concerning the cased=1 Characterizations of this type are provided for the convexity the subharmonicity or theC ^{1 1} regularity of ? in the interior ofI={x∈ R ⁿ;εR ^d (x t)∈V} and in theC ^{1 1} case an expression forD ²? is given To some extent an answer is given to the following question: which convex function ?:I→R I interval ?R (resp which function √:I→R of classC ^{1 1}) is the shadow of aC ² functionf:I×R→R?     

Absolutely continuous cocycles over irrational rotations

For homeomorphisms $$\left( {z,w} \right)\mathop \to \limits^{T\varphi } \left( {z . e^{2xi\alpha } ,\varphi \left( z \right)w} \right)$$ (z, w ∈S ¹,α is irrational,?:S ¹→S ¹) of the torusS ¹×S ¹ it is proved thatT? has countable Lebesgue spectrum in the orthocomplement of the eigenfunctions whenever? is absolutely continuous with nonzero topological degree and the derivative of? is of bounded variation. Some other cocycles with bounded variation are studied and generalizations of the above result to certain distal homeomorphisms on finite dimensional tori are presented.     

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.     

The properties of orthogonal matrix-valued wavelet packets in higher dimensions

In the paper matrix-valued multiresolution analysis and matrix-valued wavelet packets of spaceL ²(R ⁿ ,C ^{s x s}) are introduced. A procedure for constructing a class of matrix-valued wavelet packets in higher dimensions is proposed. The properties for the matrix-valued multivariate wavelet packets are investigated by using integral transform, algebra theory and operator theory. Finally, a new orthonormal basis ofL ²(R ⁿ ,C ^{s x s}) is derived from the orthogonal multivariate matrix-valued wavelet packets.     

On lattices of congruences of relational systems and universal algebras

Let \(\mathfrak{X}\) =〈X;R〉 be a relational system.X is a non-empty set andR is a collection of subsets ofX ^α, α an ordinal. The system of equivalence relations onX having the substitution property with respect to members ofR form a complete latticeC( \(\mathfrak{X}\) ) containing the identity but not necessarilyX×X. It is shown that for any relational system (X;R) there is a groupoid definable onX whose congruence lattice isC( \(\mathfrak{X}\) )U{X×X} . Theorem 2 and Corollary 2 contain some interesting combinatorial pecularities associated with oriented complete graphs and simple groupoids.     

Additive operators preserving rank-additivity on symmetry matrix spaces

We characterize the additive operators preserving rank-additivity on symmetry matrix spaces. LetS _n(F) be the space of alln×n symmetry matrices over a fieldF with 2,3 ∈F ^*, thenT is an additive injective operator preserving rank-additivity onS _n(F) if and only if there exists an invertible matrixU∈M _n(F) and an injective field homomorphism ? ofF to itself such thatT(X)=cUX ^?U^T, ?X=(x_ij)∈S_n(F) wherec∈F ^*,X ^?=(?(x _ij)). As applications, we determine the additive operators preserving minus-order onS _n(F) over the fieldF.     

On characterization of multivariate stable distributions via random linear statistics

LetX ₁,X ₂, ...,X _n be independent and identically distributed random vectors inR ^d , and letY=(Y ₁,Y ₂, ...,Y _n )′ be a random coefficient vector inR ⁿ , independent ofX _j ^/′ . We characterize the multivariate stable distributions by considering the independence of the random linear statistic $$U = Y_1 X_1 + Y_2 X_2 + \cdot \cdot \cdot + Y_n X_n $$ and the random coefficient vectorY.     

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.     

Canonical Extensions and Profinite Completions of Semilattices and Lattices

Canonical extensions of (bounded) lattices have been extensively studied, and the basic existence and uniqueness theorems for these have been extended to general posets. This paper focuses on the intermediate class \({{\boldsymbol{\mathcal{S}}}}_{\wedge}\) of (unital) meet semilattices. Any \({\mathbf S}\in {{\boldsymbol{\mathcal{S}}}}_{\wedge}\) embeds into the algebraic closure system Filt(Filt(S)). This iterated filter completion, denoted Filt²(S), is a compact and \({\textstyle{\bigvee}\,}{\textstyle{\bigwedge}\,}\) -dense extension of S. The complete meet-subsemilattice S ^δ of Filt²(S) consisting of those elements which satisfy the condition of \({\textstyle{\bigwedge}\,}{\textstyle{\bigvee}\,}\) -density is shown to provide a realisation of the canonical extension of S. The easy validation of the construction is independent of the theory of Galois connections. Canonical extensions of bounded lattices are brought within this framework by considering semilattice reducts. Any S in \({{\boldsymbol{\mathcal{S}}}}_{\wedge}\) has a profinite completion, \({\rm Pro}_{{{\boldsymbol{\mathcal{S}}}}_{\wedge}}({\mathbf S})\) . Via the duality theory available for semilattices, \({\rm Pro}_{{{\boldsymbol{\mathcal{S}}}}_{\wedge}}({\mathbf S})\) can be identified with Filt²(S), or, if an abstract approach is adopted, with \({\mathbb F_{\sqcup}}({\mathbb F_{\sqcap}}({\mathbf S}))\) , the free join completion of the free meet completion of S. Lifting of semilattice morphisms can be considered in any of these settings. This leads, inter alia, to a very transparent proof that a homomorphism between bounded lattices lifts to a complete lattice homomorphism between the canonical extensions. Finally, we demonstrate, with examples, that the profinite completion of S, for \({\mathbf S} \in {{\boldsymbol{\mathcal{S}}}}_{\wedge}\) , need not be a canonical extension. This contrasts with the situation for the variety of bounded distributive lattices, within which profinite completion and canonical extension coincide.     

On the smoothness of the symbol of multidimensional singular integral operators

Пустьf — характерист ика, аΦ — соответству ющий символ n-мерного сингу лярного интегрального опера тора, которые являютс я однородными функция ми нулевой степени вR ⁿ,n≧2, с нулевыми средни ми значениями на един ичной сфереS ^n?1 пространств аR ⁿ . В статье установлива ются связи между хара ктеристикойf и символом? в термин ах наилучших равномерн ых приближений полин омами по сферическим гармони кам и сферических равномерных модулей непрерывности, котор ые позволяют установить связь так же в терминах пространст вW ^r H ^w (S ^n?1). Доказывается, ч то установленные резул ьтаты в терминах пространствW ^r H ^w (S ^n?1) неу лучшаемые.     

Critical sets in 3-space

Given a non-empty compact set C ?R ³, is C the set of critical points for some smooth proper functionf :R ³ →R ₊? In this paper we prove that the answer is “yes” for Antoine’s Necklace and most but not all tame links.     

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.     

Existence and non-existence of positive solutions of the scalar field system in R n,I

Letn > 3 andΩ be either the entire spaceR ⁿ or a Euclidean ball in R ⁿ . Consider the following boundary value problem (I) $$\{ _{\Delta v - u + u^q = 0,}^{\Delta u - v + v^p = 0,} u,v > 0, x \in \Omega $$ with homogeneous Dirichlet boundary data (replaced byu, v → 0 as ¦x¦ → ∞ when Ω=R ⁿ ), where p > 1 and q > 1. In this paper, we investigate the question of existence and non-existence of solutions of (I) and prove that (I) admits a solution if and only if $$\frac{1}{{p + 1}} + \frac{1}{{q + 1}} > \frac{{n - 2}}{n}$$ . The existence on a ball and onR ⁿ are established by a variational approach and an approximation argument respectively. The Pohozaev identity is used to show non-existence onR ⁿ .     

Stable Stationary Harmonic Maps to Spheres

For k ≥ 3, we establish new estimate on Hausdorff dimensions of the singular set of stable-stationary harmonic maps to the sphere S^k. We show that the singular set of stable-stationary harmonic maps from B5 to 83 is the union of finitely many isolated singular points and finitely many HSlder continuous curves. We also discuss the minimization problem among continuous maps from B^n to S^2.