The Navier-Stokes equations in the weak- LnL^n space with time-dependent external force

Abstract. We consider the Navier-Stokes equations with time-dependent external force, either in the whole time or in positive time with initial data, with domain either the whole space, the half space or an exterior domain of dimension . We give conditions on the external force sufficient for the unique existence of small solutions in the weak- space bounded for all time. In particular, this result gives sufficient conditions for the unique existence and the stability of small time-periodic solutions or almost periodic solutions. This result generalizes the previous result on the unique existence and the stability of small stationary solutions in the weak- space with time-independent external force. Received: 30 March 1999 / Accepted: 21 September 1999 / Published online: 28 June 2000     

Syzygies of Multi-Variable Higher Spin Dirac Operators on $$\mathbb{R}^{3}$$

This paper is a short report on the generalization of some results of our previous paper [12] to the case of spin j/2 Dirac operators in real dimension three for arbitrary odd integer j. We use an explicit formula for the local expression of such operators to study their algebraic properties, construct the compatibility conditions of the overdetermined system associated to the operator in several spatial variables, and we prove that its associated algebraic complex, dual do the BGG sequence coming from representation theory, has substantially the same pattern as the Cauchy-Fueter complex. The author is a member of the Eduard Čech Center and his research is supported by the relative grants.     

Embedding some Riemann surfaces into $${\mathbb {C}^2}$$ with interpolation

We prove that several types of open Riemann surfaces, including the finitely connected planar domains, embed properly into such that the values on any given discrete sequence can be arbitrarily prescribed. Kutzschebauch supported by Schweizerische Nationalfonds grant 200021-107477/1.     

Eigenfunctions of the Laplace operators for a building of type[(A)\tilde]2\tilde A_2

Let Δ be a thick affine building of type and of order q. We prove that each eigenfunction of the Laplace operators of Δ is the Poisson transform of a suitable finitely additive measure on the maximal boundary Ω.     

Examples of scalar-flat hypersurfaces in $${\mathbb{R}^{n+1}}$$

Given a hypersurface M of null scalar curvature in the unit sphere , n ≥ 4, such that its second fundamental form has rank greater than 2, we construct a singular scalar-flat hypersurface in as a normal graph over a truncated cone generated by M. Furthermore, this graph is 1-stable if the cone is strictly 1-stable.     

A Class of Integral Operators on the Unit Ball of \mathbb{C}^{n}

For real parameters a, b, c, and t, where c is not a nonpositive integer, we determine exactly when the integral operator

Wiener-Hopf operators on L_{\omega}^{2}(\mathbb{R}^{+})

Let be a weighted space with weight . In this paper we show that for every Wiener-Hopf operator T on and for every a I, there exists a function such that

Equivariant gluing constructions of contact stationary Legendrian submanifolds in $${\mathbb {S}^{2n+1}}$$

A contact-stationary Legendrian submanifold of is a Legendrian submanifold whose volume is stationary under contact deformations. The simplest contact-stationary Legendrian submanifold (actually minimal Legendrian) is the real, equatorial n-sphere S ₀. This paper develops a method for constructing contact-stationary (but not minimal) Legendrian submanifolds of by gluing together configurations of sufficiently many many U(n + 1)-rotated copies of S ₀. Two examples of the construction, corresponding to finite cyclic subgroups of U(n + 1) are given. The resulting submanifolds are very symmetric; are geometrically akin to a ‘necklace’ of copies of S ₀ attached to each other by narrow necks and winding a large number of times around before closing up on themselves; and are topologically equivalent to .     

$${\mathcal{C}}_{0}$$ ( X)-algebras,stability and strongly self-absorbing $${\mathcal{C}}^{*}$$ -algebras

We study permanence properties of the classes of stable and so-called -stable -algebras, respectively. More precisely, we show that a (X)-algebra A is stable if all its fibres are, provided that the underlying compact metrizable space X has finite covering dimension or that the Cuntz semigroup of A is almost unperforated (a condition which is automatically satisfied for -algebras absorbing the Jiang–Su algebra tensorially). Furthermore, we prove that if is a K ₁-injective strongly self-absorbing -algebra, then A absorbs tensorially if and only if all its fibres do, again provided that X is finite-dimensional. This latter statement generalizes results of Blanchard and Kirchberg. We also show that the condition on the dimension of X cannot be dropped. Along the way, we obtain a useful characterization of when a -algebra with weakly unperforated Cuntz semigroup is stable, which allows us to show that stability passes to extensions of -absorbing -algebras. Research supported by: Deutsche Forschungsgemeinschaft (through the SFB 478), by the EU-Network Quantum Spaces - Noncommutative Geometry (Contract No. HPRN-CT-2002-00280), and by the Center for Advanced Studies in Mathematics at Ben-Gurion University     

Certain invariants of extension algebras of C({\mathbb{T}}^{2})

In this paper, we compute certain invariants of extension algebras of the torus algebra by , where is the C*-algebra of compact operators on an infinite dimensional separable Hilbert space H. These extension algebras are also constructed up to isomorphism. Received: 5 July 2007, Revised: 14 February 2008     

On the long time behaviour of solutions to dissipative wave equations in \mathbb{R}^{2}

In this paper we present a parabolic approach to studying the diffusive long time behaviour of solutions to the Cauchy problem:

Nondicritical $${\mathbb {C}^*}$$-actions on two-dimensional Stein manifolds

We study and classify actions of the complex multiplicative group on a nonsingular Stein surface with an isolated nondicritical singularity. We prove that the corresponding foliation exhibits a holomorphic first integral of a type F = f ⁿ g ^m where f and g are global holomorphic functions and . Under some additional conditions on the functions f and g we prove analytic linearization for the action. Our results can be viewed as extension of the original work of Masakazu Suzuki.     

Laguerre geometry of hypersurfaces in $$\mathbb{R}^{n}$$

Laguerre geometry of surfaces in is given in the book of Blaschke [Vorlesungen über Differentialgeometrie, Springer, Berlin Heidelberg New York (1929)], and has been studied by Musso and Nicolodi [Trans. Am. Math. soc. 348, 4321–4337 (1996); Abh. Math. Sem. Univ. Hamburg 69, 123–138 (1999); Int. J. Math. 11(7), 911–924 (2000)], Palmer [Remarks on a variation problem in Laguerre geometry. Rendiconti di Mathematica, Serie VII, Roma, vol. 19, pp. 281–293 (1999)] and other authors. In this paper we study Laguerre differential geometry of hypersurfaces in . For any umbilical free hypersurface with non-zero principal curvatures we define a Laguerre invariant metric g on M and a Laguerre invariant self-adjoint operator : TM → TM, and show that is a complete Laguerre invariant system for hypersurfaces in with n≥ 4. We calculate the Euler–Lagrange equation for the Laguerre volume functional of Laguerre metric by using Laguerre invariants. Using the Euclidean space , the semi-Euclidean space and the degenerate space we define three Laguerre space forms , and and define the Laguerre embeddings and , analogously to what happens in the Moebius geometry where we have Moebius space forms S ⁿ , and (spaces of constant curvature) and conformal embeddings and [cf. Liu et al. in Tohoku Math. J. 53, 553–569 (2001) and Wang in Manuscr. Math. 96, 517–534 (1998)]. Using these Laguerre embeddings we can unify the Laguerre geometry of hypersurfaces in , and . As an example we show that minimal surfaces in or are Laguerre minimal in .C. Wang Partially supported by RFDP and Chuang-Xin-Qun-Ti of NSFC.     

On nonatomic submeasures on {\mathbb{N}}

A submeasure μ defined on the subsets of is nonatomic if for every ℓ ≥ 1 there exists a partition of into a finite number of parts on which μ is bounded from above by 1/ℓ. In this paper we answer several natural questions concerning nonatomic submeasures d _F that are determined (like the standard density) by a family F of finite subsets of . We first show that if the number of n-element sets in F grows at most exponentially with n, then d _F is nonatomic; but if this growth condition fails, then d _F need not be nonatomic in general. We next prove that, for a nonatomic submeasure d _F , the minimal number of sets in a 1/ℓ-small partition of can grow arbitrarily fast with ℓ. We also give a simple example of a nonatomic submeasure that is not equivalent to a submeasure of type d _F . The second author acknowledges a generous support of the Foundation for Polish Science.     

Nevanlinna-Pick Interpolation for {\mathbb{C}} + BH^\infty

We study the Nevanlinna-Pick problem for a class of subalgebras of H ^∞. This class includes algebras of analytic functions on embedded disks, the algebras of finite codimension in H ^∞ and the algebra of bounded analytic functions on a multiply connected domain. Our approach uses a distance formula that generalizes Sarason’s [23] work. We also investigate the difference between scalar-valued and matrix-valued interpolation through the use of C ^*-envelopes. This research was partially supported by the NSF grant DMS 0300128. This research was completed as part of my Ph.D. dissertation at the University of Houston.     

Finiteness properties for subgroups of GL (n,\mathbb Z)(n,\mathbb Z)

Abstract. We construct finitely presented subgroups of GL that have infinitely many conjugacy classes of finite subgroups. This answers a question of Grunewald and Platonov. We suggest a variation on their question. Received: 26 August 1999 / Revised: 28 September 1999 / Published online: 8 May 2000     

Amenability,tubularity, and embeddings into $$\mathcal{R}^{\omega}$$

Suppose M is a tracial von Neumann algebra embeddable into (the ultraproduct of the hyperfinite II₁-factor) and X is an n-tuple of selfadjoint generators for M. Denote by Γ(X; m, k, γ) the microstate space of X of order (m, k ,γ). We say that X is tubular if for any ε >  0 there exist and γ > 0 such that if then there exists a k × k unitary u satisfying for each 1 ≤  i ≤  n. We show that the following conditions are equivalent:

Covering and packing in \Bbb Zn{\Bbb Z}^n and \Bbb Rn{\Bbb R}^n, (I)

Given a finite subset A{\cal A} of an additive group \Bbb G{\Bbb G} such as \Bbb Zⁿ{\Bbb Z}^n or \Bbb Rⁿ{\Bbb R}^n , we are interested in efficient covering of \Bbb G{\Bbb G} by translates of A{\cal A} , and efficient packing of translates of A{\cal A} in \Bbb G{\Bbb G} . A set S ì \Bbb G{\cal S} \subset {\Bbb G} provides a covering if the translates A + s{\cal A} + s with s ? Ss \in {\cal S} cover \Bbb G{\Bbb G} (i.e., their union is \Bbb G{\Bbb G} ), and the covering will be efficient if S{\cal S} has small density in \Bbb G{\Bbb G} . On the other hand, a set S ì \Bbb G{\cal S} \subset {\Bbb G} will provide a packing if the translated sets A + s{\cal A} + s with s ? Ss \in {\cal S} are mutually disjoint, and the packing is efficient if S{\cal S} has large density. In the present part (I) we will derive some facts on these concepts when \Bbb G = \Bbb Zⁿ{\Bbb G} = {\Bbb Z}^n , and give estimates for the minimal covering densities and maximal packing densities of finite sets A ì \Bbb Zⁿ{\cal A} \subset {\Bbb Z}^n . In part (II) we will again deal with \Bbb G = \Bbb Zⁿ{\Bbb G} = {\Bbb Z}^n , and study the behaviour of such densities under linear transformations. In part (III) we will turn to \Bbb G = \Bbb Rⁿ{\Bbb G} = {\Bbb R}^n .     

Factoriality condition of some nodal threefolds in {\mathbb{P}^4}

We prove that for n = 8, 9, 10, 11, a nodal hypersurface of degree n in is factorial if it has at most (n − 1)² − 1 nodes. The author is grateful to Ivan Cheltsov for valuable comments and suggestions.     

An asymptotic theorem for minimal surfaces and existence results for minimal graphs in $${\mathbb H^2 \times \mathbb R}$$

In this paper we prove a general and sharp Asymptotic Theorem for minimal surfaces in . As a consequence, we prove that there is no properly immersed minimal surface whose asymptotic boundary Γ_∞ is a Jordan curve homologous to zero in such that Γ_∞ is contained in a slab between two horizontal circles of with width equal to π. We construct vertical minimal graphs in over certain unbounded admissible domains taking certain prescribed finite boundary data and certain prescribed asymptotic boundary data. Our admissible unbounded domains Ω in are non necessarily convex and non necessarily bounded by convex arcs; each component of its boundary is properly embedded with zero, one or two points on its asymptotic boundary, satisfying a further geometric condition. The first author wish to thank Laboratoire Géométrie et Dynamique de l’Institut de Mathématiques de Jussieu for the kind hospitality and support. The authors would like to thank CNPq, PRONEX of Brazil and Accord Brasil-France, for partial financial support.