Symmetric boundary values for the Dirichlet problem for harmonic maps from the disc into the 2-sphere

Let Aut(D) denote the group of biholomorphic diffeormorphisms from the unit disc D onto itself and O(3) the group of orthogonal transformations of the unit sphere S ². The existence of multiple solutions to the Dirichlet problem for harmonic maps from D into S ² is related to the symmetries (if any) of the boundary value γ : ∂D → S ², by invariance of the Dirichlet energy under the action of Aut(D) × O(3). In this paper, we classify the stabilizers in Aut(D) × O(3) of boundary values in H ^1/2(S ¹, S ²) and . We give two applications to the Dirichlet problem for harmonic maps. This work was partially supported by the CMLA, Ecole Normale Supérieure de Cachan, Cachan, France.     

Regularization of a Nonstandard Cauchy Problem for a Dynamic Lame System

We consider the Cauchy problem for dynamic Lame systems in the cylinder G_T = D × (0,T) constructed over a domain D in a three-dimensional space, where the initial data are given in some strip in the lateral surface of the cylinder. The strip has the form S × (0,T), where S is an open subset of the boundary surface of the domain D. This problem is ill-posed. Under certain requirements to the configuration of S, we derive an explicit formula for solutions to this problem.

     

Iteration of mapping classes and limits of hyperbolic 3-manifolds

Let ϕ∈Mod(S) be an element of the mapping class group of a surface S. We classify algebraic and geometric limits of sequences {Q(ϕ ⁱ X,Y)} _i=1 ^∞ of quasi-Fuchsian hyperbolic 3-manifolds ranging in a Bers slice. When ϕ has infinite order with finite-order restrictions, there is an essential subsurface D _ϕ⊂S so that the geometric limits have homeomorphism type S×ℝ-D _ϕ×{0}. Typically, ϕ has pseudo-Anosov restrictions, and D _ϕ has components with negative Euler characteristic; these components correspond to new asymptotically periodic simply degenerate ends of the geometric limit. We show there is an s≥1 depending on ϕ and bounded in terms of S so that {Q(ϕ ^si X,Y)} _i=1 ^∞ converges algebraically and geometrically, and we give explicit quasi-isometric models for the limits. Oblatum 4-I-1999 & 19-VII-2000?Published online: 30 October 2000     

Delay moments for FIFO GI/GI/s queues

For stable FIFO GI/GI/s queues, s ≥ 2, we show that finite (k+1)st moment of service time, S, is not in general necessary for finite kth moment of steady-state customer delay, D, thus weakening some classical conditions of Kiefer and Wolfowitz (1956). Further, we demonstrate that the conditions required for E[D ^k]<∞ are closely related to the magnitude of traffic intensity ρ (defined to be the ratio of the expected service time to the expected interarrival time). In particular, if ρ is less than the integer part of s/2, then E[D] < ∞ if E[S^3/2]<∞, and E[D^k]<∞ if E[S^k]<∞, ^k≥ 2. On the other hand, if s-1 < ρ < s, then E[D^k]<∞ if and only if E[S^k+1]<∞, k ≥ 1. Our method of proof involves three key elements: a novel recursion for delay which reduces the problem to that of a reflected random walk with dependent increments, a new theorem for proving the existence of finite moments of the steady-state distribution of reflected random walks with stationary increments, and use of the classic Kiefer and Wolfowitz conditions. This revised version was published online in June 2006 with corrections to the Cover Date.     

Size direction games over the real line. III

We prove, using the continuum hypothesis, thatD (the direction player) has a winning strategy in {ie442-1} for some uncountableX, and that there is an uncountableX which intersects each perfect nowhere-dense set of reals in a countable set such thatD does not win in {ie442-2} for everya. We also give another proof to the fact that Γ^S (X) is a win forD is countable.     

Twisted conjugacy classes in symplectic groups,mapping class groups and braid groups (with an appendix written jointly with Francois Dahmani)

We prove that the symplectic group Sp(2n,\mathbbZ){Sp(2n,\mathbb{Z})} and the mapping class group Mod _S of a compact surface S satisfy the R _∞ property. We also show that B _n (S), the full braid group on n-strings of a surface S, satisfies the R _∞ property in the cases where S is either the compact disk D, or the sphere S ². This means that for any automorphism f{\phi} of G, where G is one of the above groups, the number of twisted f{\phi}-conjugacy classes is infinite.     

Ascending Nets of Prüfer V-Multiplication Domains

Abstract

Let D be an integral domain, S ? D a multiplicative set such that aD _S  ∩ D is a principal ideal for each a ∈ D and let D ^(S) = ? _s∈S D[X/s]. It is known that if D is a Prüfer v-multiplication domain (resp., generalized GCD domain, GCD domain), then so is D ^(S) respectively. When D is a Noetherian domain, we obtain a similar result for the power series analog D ^((S)) = ? _s∈S D[[X/s]] of D ^(S). Our approach takes care simultaneously of both cases D ^(S) and D ^((S)).     

Higher Order Asymptotic Theory for Discriminant Analysis in Exponential Families of Distributions

This paper deals with the problem of classifying a multivariate observation X into one of two populations Π₁: p(x; w⁽¹⁾) S and Π₂: p(x; w⁽²⁾) S, where S is an exponential family of distributions and w⁽¹⁾ and w⁽²⁾ are unknown parameters. Let ; be a class of appropriate estimators ( ⁽¹⁾, ⁽²⁾) of (w⁽¹⁾, w⁽²⁾ based on training samples. Then we develop the higher order asymptotic theory for a class of classification statistics D = [ | = log{p(X; ⁽¹⁾)/p(X; ⁽²⁾)}, ( ⁽¹⁾, ⁽²⁾) ;]. The associated probabilities of misclassification of both kinds M( ) are evaluated up to second order of the reciprocal of the sample sizes. A classification statistic is said to be second order asymptotically best in D if it minimizes M( ) up to second order. A sufficient condition for to be second order asymptotically best in D is given. Our results are very general and give us a unified view in discriminant analysis. As special results, the Anderson W, the Cochran and Bliss classification statistic, and the quadratic classification statistic are shown to be second order asymptotically best in D in each suitable classification problem. Also, discriminant analysis in a curved exponential family is discussed.     

All Vacuum Near Horizon Geometries in D-dimensions with (D − 3) Commuting Rotational Symmetries

We explicitly construct all stationary, non-static, extremal near horizon geometries in D dimensions that satisfy the vacuum Einstein equations, and that have D−3 commuting rotational symmetries. Our work generalizes [arXiv:0806.2051] by Kunduri and Lucietti, where such a classification had been given in D = 4,5. But our method is different from theirs and relies on a matrix formulation of the Einstein equations. Unlike their method, this matrix formulation works for any dimension. The metrics that we find come in three families, with horizon topology S ² × T ^D-4, or S ³ × T ^D-5, or quotients thereof. Our metrics depend on two discrete parameters specifying the topology type, as well as (D − 2)(D − 3)/2 continuous parameters. Not all of our metrics in D ≥ 6 seem to arise as the near-horizon limits of known black hole solutions.     

On the sum of the total domination numbers of a digraph and its converse

A vertex subset S of a digraph D is called a dominating set of D if every vertex not in S has an in-neighbor in S. A dominating set S of D is called a total dominating set of D if the subdigraph induced by S has no isolated vertices. The total domination number of D, denoted by γt(D), is the minimum cardinality of a total dominating set of D. We show that for any connected digraph D of order n≥3, γt(D)+γt(D^? )≤5n/3, where D? is the converse of D. Furthermore, we characterize the oriented trees for which the equality holds.     

Large bipartite graphs with given degree and diameter

We give constructions of bipartite graphs with maximum Δ, diameter D on B vertices, such that for every D ≥ 2 the lim inf_Δ→∞B. Δ^1-D = b_D > 0. We also improve similar results on ordinary graphs, for example, we prove that lim_Δ→∞N · Δ^?D = 1 if D is 3 or 5. This is a partial answer to a problem of Bollobás.     

New Examples of Willmore Surfaces in S n

A surface x: M S ⁿ is called a Willmore surface if it is a criticalsurface of the Willmore functional _M (S – 2H ²)dv, where H isthe mean curvature and S is the square of the length of the secondfundamental form. It is well known that any minimal surface is aWillmore surface. The first nonminimal example of a flat Willmoresurface in higher codimension was obtained by Ejiri. This example whichcan be viewed as a tensor product immersion of S ¹(1) and a particularsmall circle in S ²(1), and therefore is contained in S ⁵(1) gives anegative answer to a question by Weiner. In this paper we generalize theabove mentioned example by investigating Willmore surfaces in S ⁿ (1)which can be obtained as a tensor product immersion of two curves. We inparticular show that in this case too, one of the curves has to beS ¹(1), whereas the other one is contained either in S ²(1) or in S ³(1). In the first case, we explicitly determine the immersion interms of elliptic functions, thus constructing infinetely many newnonminimal flat Willmore surfaces in S ⁵. Also in the latter casewe explicitly include examples.     

Dilatation estimates for quasiconformal extensions

LetD andD′ be ring domains inB ⁿ , withS ⁿ⁻¹ as one boundary component, and let be a homeomorphism which isK-quasiconformal inD and withf(S ⁿ⁻¹)=S ⁿ⁻¹. According to a result of Gehringf÷S ⁿ⁻¹ admits an extension which is quasiconformal inB ⁿ . We find here an upper bound for the dilatation ofg in terms ofn, K, and modD. This work was started during a visit to Université de Paris, financed by a cultural exchange program between France and Finland.     

S1-Valued Harmonic Maps with High Topological Degree: Asymptotic Behavior of the Singular Set

We prove that the singularities of harmonic maps from a domain D in the plane to S ¹ minimizing a renomalized energy tend to go to the boundary when their number becomes large.     

The Problem of the Pointwise Fourier Inversion for Piecewise Smooth Functions of Several Variables

We study the pointwise convergence problem for the inverse Fourier transform of piecewise smooth functions, i.e., whether S_rD f (\bx) ? f (\bx)S_{\rho D} f (\bx) \to f (\bx) as r? ￥\rho \to \infty . r? ￥\rho \to \infty . Here for \bx,\bxi ? \Rn\bx,\bxi \in \Rn S_rDf(\bmx)=\dsf1(2p)^n/2\intli_rD [^(f)](\bxi) e^{\dst iá\bmx,\bxi?} d\bxi . S_{\rho D}f(\bm{x})=\dsf1{(2\pi)^{n/2}}\intli_{\rho D} \widehat{f}(\bxi) e^{\dst i\langle\bm{x},\bxi\rangle} d\bxi~. is the partial sum operator using a convex and open set DD containing the origin, and rD={ r\bxi:\bxi ? D }\rho D=\left\{ \rho \bxi:\bxi\in D \right\}.     

Green Integrals on Manifolds with Cracks

We prove the existence of a limit in H ^m (D)of iterations of a double layer potential constructed from the Hodge parametrix on a smooth compact manifold with boundary, X, and a crack S D, D being a domain in X. Using this result we obtain formulas for Sobolev solutions to the Cauchyproblem in D with data on S, for an elliptic operator A of order m 1, whenever these solutions exist.This representation involves the sum of a series whose terms are iterationsof the double layer potential. A similar regularisation is constructed also for a mixed problem in D.     

Minimal entropy and collapsing with curvature bounded from below

We show that if a closed manifold M admits an ℱ-structure (not necessarily polarized, possibly of rank zero) then its minimal entropy vanishes. In particular, this is the case if M admits a non-trivial S ¹-action. As a corollary we obtain that the simplicial volume of a manifold admitting an ℱ-structure is zero.?We also show that if M admits an ℱ-structure then it collapses with curvature bounded from below. This in turn implies that M collapses with bounded scalar curvature or, equivalently, its Yamabe invariant is non-negative.?We show that ℱ-structures of rank zero appear rather frequently: every compact complex elliptic surface admits one as well as any simply connected closed 5-manifold.?We use these results to study the minimal entropy problem. We show the following two theorems: suppose that M is a closed manifold obtained by taking connected sums of copies of S ⁴, ℂP ², ²,S ²×S ²and the K3 surface. Then M has zero minimal entropy. Moreover, M admits a smooth Riemannian metric with zero topological entropy if and only if M is diffeomorphic to S ⁴,ℂP ²,S ²×S ²,ℂP ²#  ² or ℂP ²# ℂP ². Finally, suppose that M is a closed simply connected 5-manifold. Then M has zero minimal entropy. Moreover, M admits a smooth Riemannian metric with zero topological entropy if and only if M is diffeomorphic to S ⁵,S ³×S ², then on trivial S ³-bundle over S ² or the Wu-manifold SU(3)/SO(3). Oblatum 13-III-2002 & 12-VIII-2002?Published online: 8 November 2002 G.P. Paternain was partially supported by CIMAT, Guanajuato, México.?J. Petean is supported by grant 37558-E of CONACYT.     

On the restricted arc-connectivity of s-geodetic digraphs

For a strongly connected digraph D the minimum ,cardinality of an arc-cut over all arc-cuts restricted arc-connectivity λ′（D） is defined as the S satisfying that D - S has a non-trivial strong component D1 such that D - V（D1） contains an arc. Let S be a subset of vertices of D. We denote by w＋（S） the set of arcs uv with u ∈ S and v S, and by w-（S） the set of arcs uv with u S and v ∈ S. A digraph D = （V, A） is said to be λ′-optimal if λ′（D） =ξ′（D）, where ξ′（D） is the minimum arc-degree of D defined as ξ（D） = min {ξ′（xy） ： xy ∈ A}, and ξ′（xy） = min（｜ω＋（{x,y}）｜, ｜w-（{x,y}）｜, ｜w＋（x） ∪ w- （y） ｜, ｜w- （x） ∪ω＋ （y）｜}. In this paper a sufficient condition for a s-geodetic strongly connected digraph D to be λ′-optimal is given in terms of its diameter. Furthermore we see that the h-iterated line digraph Lh（D） of a s-geodetic digraph is λ′-optimal for certain iteration h.     

Strange bundles onP 2 and elementary transformations

Summary Here we begin the study of moduli of vector bundles on a surfaceS with a fixed restriction to a divisorD. Here we stress the caseD≊P ¹. In this way we construct many families of stable rank-2 bundles onP ² with unbalanced general splitting type (in characteristicp>0).

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Riassunto Si comincia qui lo studio dei moduli di fibrati vettoriali su una superficieS con una assegnata restrizione ad un divisoreD (quasi sempre qui conD≊P ¹). In caratteristicap si ottengono così molte famiglie di fibrati stabili suP ² con ?strana? restrizione ad una retta generica.