Existence and uniqueness of complete constant mean curvature surfaces at infinity ofH 3

A set of conditions are given, each equivalent to the constancy of mean curvature of a surface in H ³.It is shown that analogs of these equivalences exist for surfaces in S _∞ ² ,the bounding ideal sphere of H ³,leading to a notion of constant mean curvature at infinity of H ³.A parametrization of all complete constant mean curvature surfaces at infinity of H ³ is given by holomorphic quadratic differentials on Ĉ,C, and D.     

A characterization of compact convex polyhedra in hyperbolic 3-space

Summary In this paper we study the extrinsic geometry of convex polyhedral surfaces in three-dimensional hyperbolic spaceH ³. We obtain a number of new uniqueness results, and also obtain a characterization of the shapes of convex polyhedra inH ³ in terms of a generalized Gauss map. This characterization greatly generalizes Andre'ev's Theorem.Oblatum 12-XI-1991 & 29-V-1992     

A Theorem on Discrete,Torsion Free Subgroups of Isom H <Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

Let H ⁿ be the hyperbolic n-space with n⩾ 2. Suppose that Γ H ⁿ is a discrete, torsion free subgroup and a is a point in the domain of discontinuity Ω (Γ). Let p be the projection map from H ⁿ to the quotient manifold M=H ⁿ/Γ. In this paper we prove that there exists an open neighborhood U of a in H ⁿ∪ Ω (Γ) such that p is an isometry on U∩ H ⁿ.Mathematics Subject Classification (2000). primary: 51M10; secondary: 22E40, 57M50.     

Coding of a translation of the two-dimensional torus

Let α be in the two-dimensional torus T ² = R ²/Z ². Assume that the translation map T : x → x + α acts ergodically. We present a symbolic coding of the map T which shares several properties with the Sturmian coding of a one-dimensional translation. The symbolic dynamical system is metrically isomorphic to the geometric dynamical system (T ², T). The coding is of quadratic growth complexity and 2-balanced. Moreover, there is a geometric underpinning, the coding is related to a fundamental domain for the action of Z ² on R ² and also to bounded remainder sets.      

Coding of a translation of the two-dimensional torus

Let α be in the two-dimensional torus T ² = R ²/Z ². Assume that the translation map T : x → x + α acts ergodically. We present a symbolic coding of the map T which shares several properties with the Sturmian coding of a one-dimensional translation. The symbolic dynamical system is metrically isomorphic to the geometric dynamical system (T ², T). The coding is of quadratic growth complexity and 2-balanced. Moreover, there is a geometric underpinning, the coding is related to a fundamental domain for the action of Z ² on R ² and also to bounded remainder sets.     

Formes antihermitiennes devenant hyperboliques sur un corps de déploiement

Let H=(a,b)_F be a division quaternion algebra over a field F of characteristic not 2. Denote by τ the canonical involution on H and by K a splitting field of H. If h is a skew-hermitian form over (H,τ) then, by extension of scalars to K and by Morita equivalence, we obtain a quadratic form h_K over K. This gives a map of Witt groups ρ:W⁻¹(H,τ)→W(K) induced by ρ(h)=h_K. When K is a generic splitting field of H we prove in this note that the map ρ is injective.     

ON MULTIOSCULATING SPACES

The aim of this paper is to study varieties with second Gauss map not birational. In particular we classify such varieties in dimension two. We prove that there are two types of surfaces S of P ⁿ (C), with n > 5, not satisfying Laplace equations, with second Gauss map t ² not birational: i. surfaces such that the image of the second Gauss map is one-dimensional and containing a one-dimensional family of curves. Each curve of the family is contained in some P ³ ? P ⁿ .

ii. surfaces such that the second Gauss map is generically finite of degree at least two. In this case the image of the second Gauss map is two-dimensional, locally embedded in a Laplace congruence and meeting the general generatrix in more than one point.

     

Lie theory and the Chern-Weil homomorphism

Let P→B be a principal G-bundle. For any connection θ on P, the Chern-Weil construction of characteristic classes defines an algebra homomorphism from the Weil algebra Wg=Sg^∗⊗∧g^∗ into the algebra of differential forms A=Ω(P). Invariant polynomials inv(Sg^∗)⊂Wg map to cocycles, and the induced map in cohomology inv(Sg^∗)→H(Abasic) is independent of the choice of θ. The algebra Ω(P) is an example of a commutativeg-differential algebra with connection, as introduced by H. Cartan in 1950. As observed by Cartan, the Chern-Weil construction generalizes to all such algebras.In this paper, we introduce a canonical Chern-Weil map Wg→A for possibly non-commutativeg-differential algebras with connection. Our main observation is that the generalized Chern-Weil map is an algebra homomorphism “up to g-homotopy”. Hence, the induced map inv(Sg^∗)→Hbasic(A) is an algebra homomorphism. As in the standard Chern-Weil theory, this map is independent of the choice of connection.Applications of our results include: a conceptually easy proof of the Duflo theorem for quadratic Lie algebras, a short proof of a conjecture of Vogan on Dirac cohomology, generalized Harish-Chandra projections for quadratic Lie algebras, an extension of Rouvière's theorem for symmetric pairs, and a new construction of universal characteristic forms in the Bott-Shulman complex.     

Torsion in the full orbifold <Emphasis Type="Italic">K</Emphasis>-theory of abelian symplectic quotients

Let (M, ω, Φ) be a Hamiltonian T-space and let H í T{H\subseteq T} be a closed Lie subtorus. Under some technical hypotheses on the moment map Φ, we prove that there is no additive torsion in the integral full orbifold K-theory of the orbifold symplectic quotient [M//H]. Our main technical tool is an extension to the case of moment map level sets the well-known result that components of the moment map of a Hamiltonian T-space M are Morse-Bott functions on M. As first applications, we conclude that a large class of symplectic toric orbifolds, as well as certain S ¹-quotients of GKM spaces, have integral full orbifold K-theory that is free of additive torsion. Finally, we introduce the notion of semilocally Delzant which allows us to formulate sufficient conditions under which the hypotheses of the main theorem hold. We illustrate our results using low-rank coadjoint orbits of type A and B.     

Zéros d’applications holomorphes deC n dansC n

It is known that, unlike the one dimensional case it is not possible to find an upper bound for the zeros of an entire map fromC ⁿ toC ⁿ ,n≥2, in terms of the growth of the map. However, if we only consider the “non-degenerate” zeros, that is, the zeros where the jacobian is not “too small”, it becomes possible. We give a new proof of this fact.      

SOME FUNCTIONAL ANALYTIC PROPERTIES OF COMPOSITION OPERATORS

Abstract

This is a report on a number of recent results on composition operators which map, for 0 < p ? q ∞, the Hardy space H^p (on the unit disk in the complex plane) into H ^q. Attention is focused on questions of boundedness (existence), compactness, order boundedness and, in connection with the latter, on relating the absolutely summing and nuclearity character as well as special factorization properties of the operator to function theoretic properties of the defining symbol. Moreover, tools are provided to show that certain classes of operators can well be distinguished already on the level of composition operators.     

A uniformly differentiable approximation scheme for delay systems using splines

A new spline-based scheme is developed for linear retarded functional differential equations within the framework of semigroups on the Hilbert spaceR ⁿ ×L ². The approximating semigroups inherit in a uniform way the characterization for differentiable semigroups from the solution semigroup of the delay system (e.g., among other things the logarithmic sectorial property for the spectrum). We prove convergence of the scheme in the state spacesR ⁿ ×L ² andH ¹. The uniform differentiability of the approximating semigroups enables us to establish error estimates including quadratic convergence for certain classes of initial data. We also apply the scheme for computing the feedback solutions to linear quadratic optimal control problems.Work done by K. Ito was supported by AFOSR under Contract No. F-49620-86-C-0111, by NASA under Grant No. NAG-1-517, and by NSF under Grant No. UINT-8521208. Work done by F. Kappel was supported by AFOSR under Grant No. 84-0398 and by FWF(Austria) under Grants S3206 and P6005.     

Tensor products and discriminants of unital quadratic forms over commutative rings

A tensor product for unital quadratic forms is introduced which extends the product of separable quadratic algebras and is naturally associative and commutative. It admits a multiplicative functor vdis, the vector discriminant, with values in symmetric bilinear forms. We also compute the usual (signed) discriminant of the tensor product in terms of the discriminants of the factors. The orthogonal group scheme of a nonsingular unital quadratic formQ of even rank is isomorphic toZ ₂×SO(Q ⁰) whereQ ⁰ is the restriction of –Q to the space of trace zero elements. We use cohomology to interpret the action of separable quadratic algebras on unital quadratic forms, and to determine which forms of odd rank can be realized asQ ⁰.     

Global Well-Posedness of mKdV in L 2 (𝕋, ℝ)

ABSTRACT

We show that the Miura map L ²(𝕋) → H ^?1(𝕋), r?r _x  + r ² is a global fold and then apply our results on global well-posedness of KdV in H ^?1(𝕋) to show that mKdV is globally well-posed in L ²(𝕋).     

Walsh Multipliers for Dyadic Hardy Spaces

In this article we are interested in conditions on the coefficients of a Walsh multiplier operator that imply the operator is bounded on certain dyadic Hardy spaces H ^p , 0 < p < ∞. In particular, we consider two classical coefficient conditions, originally introduced for the trigonometric case, the Marcinkiewicz and the Hörmander–Mihlin conditions. They are known to be sufficient in the spaces L ^p , 1 < p < ∞. Here we study the corresponding problem on dyadic Hardy spaces, and find the values of p for which these conditions are sufficient. Then, we consider the cases of H ¹ and L ¹ which are of special interest. Finally, based on a recent integrability condition for Walsh series, a new condition is provided that implies that the multiplier operator is bounded from L ¹ to L ¹, and from H ¹ to H ¹. We note that existing multiplier theorems for Hardy spaces give growth conditions on the dyadic blocks of the Walsh series of the kernel, but these growth are not computable directly in terms of the coefficients.     

A Note on the Double Affine Hecke Algebra of Type GL2

We express the double affine Hecke algebra ? associated to the general linear group GL₂(k) (here, k is a field with char(k) ≠ 2) as an amalgamated free product of quadratic extensions over the three-dimensional quantum torus 𝒪_q((k^×)³). With an eye towards proving ring-theoretic results pertaining to ?, a general treatment of amalgamated products of Ore and quadratic extensions is given. We prove an analogue of the Hilbert Basis Theorem for an amalgamated product Q of quadratic extensions and determine conditions for when the one-sided ideals of Q are principal or doubly-generated. Furthermore, we determine sufficient conditions which imply Q is a principal ideal ring. Finally, we construct an explicit isomorphism from ? to the amalgamated free product ring of quadratic extensions over 𝒪_q((k^×)³), a ring known to be noetherian. Therefore, it follows that ? is noetherian.     

Three results on the regularity of the curves that are invariant by an exact symplectic twist map

A theorem due to G. D. Birkhoff states that every essential curve which is invariant under a symplectic twist map of the annulus is the graph of a Lipschitz map. We prove: if the graph of a Lipschitz map h:T→R is invariant under a symplectic twist map, then h is a little bit more regular than simply Lipschitz (Theorem 1); we deduce that there exists a Lipschitz map h:T→R whose graph is invariant under no symplectic twist map (Corollary 2). Assuming that the dynamic of a twist map restricted to a Lipschitz graph is bi-Lipschitz conjugate to a rotation, we obtain that the graph is even C ¹ (Theorem 3). Then we consider the case of the C ⁰ integrable symplectic twist maps and we prove that for such a map, there exists a dense G _δ subset of the set of its invariant curves such that every curve of this G _δ subset is C ¹ (Theorem 4).     

The cyclicity of the elliptic segment loops of the reversible quadratic Hamiltonian systems under quadratic perturbations

Denote by Q^H and Q^R the Hamiltonian class and reversible class of quadratic integrable systems. There are several topological types for systems belong to Q^H∩Q^R. One of them is the case that the corresponding system has two heteroclinic loops, sharing one saddle-connection, which is a line segment, and the other part of the loops is an ellipse. In this paper we prove that the maximal number of limit cycles, which bifurcate from the loops with respect to quadratic perturbations in a conic neighborhood of the direction transversal to reversible systems (called in reversible direction), is two. We also give the corresponding bifurcation diagram.     

Optimally conditioned optimization algorithms without line searches

New variable metric algorithms are presented with three distinguishing features:

1. They make no line searches and allow quite arbitrary step directions while maintaining quadratic termination and positive updates for the matrixH, whose inverse is the hessian matrix of second derivatives for a quadratic approximation to the objective function.
2. The updates fromH toH ₊ are optimally conditioned in the sense that they minimize the ratio of the largest to smallest eigenvalue ofH ^?1 H ₊.
3. Instead of working with the matrixH directly, these algorithms represent it asJJ ^T, and only store and update the Jacobian matrix J. A theoretical basis is laid for this family of algorithms and an example is given along with encouraging numerical results obtained with several standard test functions.