Torsion and conformally Anosov flows in contact Riemannian geometry

We prove that on a compact (non Sasakian) contact metric 3-manifold with critical metric for the Chern-Hamilton functional, the characteristic vector field ξ is conformally Anosov and there exists a smooth curve in the contact distribution of conformally Anosov flows. As a consequence, we show that negativity of the ξ-sectional curvature is not a necessary condition for conformal Anosovicity of ξ (this completes a result of [4]). Moreover, we study contact metric 3-manifolds with constant ξ-sectional curvature and, in particular, correct a result of [13].     

Rigidity for critical metrics of the volume functional

Geodesic balls in a simply connected space forms , or are distinguished manifolds for comparison in bounded Riemannian geometry. In this paper we show that they have the maximum possible boundary volume among Miao–Tam critical metrics with connected boundary provided that the boundary of the manifold has a lower bound for the Ricci curvature. In the same spirit we also extend a rigidity theorem due to Boucher et al. 7 and Shen 18 to n‐dimensional static metrics with positive constant scalar curvature, which gives us a partial answer to the Cosmic no‐hair conjecture.     

Cugliandolo-Kurchan equations for dynamics of Spin-Glasses

We study the Langevin dynamics for the family of spherical p-spin disordered mean-field models of statistical physics. We prove that in the limit of system size N approaching infinity, the empirical state correlation and integrated response functions for these N-dimensional coupled diffusions converge almost surely and uniformly in time, to the non-random unique strong solution of a pair of explicit non-linear integro-differential equations intensively studied by Cugliandolo and Kurchan. Research partially supported by NSF grants #DMS-0406042, #DMS-FRG-0244323     

A characterization for isometries and conformal mappings of pseudo-Riemannian manifolds

Generalizing theorems of Myers-Steenrod and of Hawking, we obtain characterizations for isometries and conformal mappings of pseudo-Riemannian spaces (M, g): Define a local distance function on convex normal neighbourhoods by (p, q) =g(exp _p ^–1 q, exp _p ^–1 q). Then every homeomorphismf locally preserving these functions is an isometry. If (M, g) has indefinite signature andf locally preserves distance zero, it is a conformal diffeomorphism.     

Drinfeld–Sokolov reduction for quantum groups and deformations of W-algebras

We define deformations of W-algebras associated to comple semisimple Lie algebras by means of quantum Drinfeld-Sokolov reduction procedure for affine quantum groups. We also introduce Wakimoto modules for arbitrary affine quantum groups and construct free field resolutions and screening operators for the deformed W-algebras. We compare our results with earlier definitions of q-W-algebras and of the deformed screening operators due to Awata, Kubo, Odake, Shiraishi [60],[6], [7], Feigin, E. Frenkel [22] and E. Frenkel, Reshetikhin [34]. The screening operator and the free field resolution for the deformed W-algebra associated to the simple Lie algebra sl₂ coincide with those for the deformed Virasoro algebra introduced in [60]. The author is supported by the Swiss National Science Foundation.     

New characterizations of spheres, cylinders and W-curves

We study hypersurfaces (curves, resp.) of Euclidean space of arbitrary dimension such that the chord joining any two points on the hypersurface (curve, resp.) meets it at the same angle.     

On Nonprojective Block Components of Lefschetz Characters for Sporadic Geometries

This work examines the possible projectivity of 2-modular block parts of nonprojective Lefschetz characters over 2-local geometries of several sporadic groups. Previously known results on M ₁₂, J ₂, and HS are mentioned for completeness. The main new results are on the sporadic groups Suz, Co ₃, Ru, O′N, and He. For each group, the Lefschetz character is calculated, and its 2-modular block parts are examined for projectivity. In each case it is confirmed that a nonprincipal block part contains a nonprojective summand. The case of O′N is additionally found to have a nonprojective summand in its principal block part. Nineteen of the sporadic groups (including many previously known cases) are categorized into three classes based on projectivity properties of their Lefschetz characters.     

Characteristic functional equations of polynomials and the morera-carleman theorem

Several characteristic functional equations satisfied by classes of polynomials of bounded degree are examined in connection with certain generalizations of the Morera-Carleman Theorem. Certain functional equations which have nonanalytic polynomial solutions are also considered.     

On the dynamics of a third order Newton's approximation method

We provide an answer to a question raised by S. Amat, S. Busquier, S. Plaza on the qualitative analysis of the dynamics of a certain third order Newton type approximation function , by proving that for functions f twice continuously differentiable and such that both f and its derivative do not have multiple roots, with at least four roots and infinite limits of opposite signs at , has periodic points of any prime period and that the set of points a at which the approximation sequence does not converge is uncountable. In addition, we observe that in their Scaling Theorem analyticity can be replaced with differentiability.     

Stochastic programming with incomplete information:a surrey of results on postoptimization and sensitivity analysis

The possibility of successful applications of stochastic programming decision models has been limited by the assumed complete knowledge of the distribution Fof the random parameters as well as by the limited scope of the existing numerical procedures.

We shall give a survey of selected methods which can be used to deal with the incomplete knowledge of the distribution F, namely to study robustness of the optimal solution and the optimal value of the objective function relative to small changes of the underlying distribution and to get error bounds in approximation schemes.     

Exponential stability, exponential expansiveness, and exponential dichotomy of evolution equations on the half-line

LetU=(U(t, s)) _tsO be an evolution family on the half-line of bounded linear operators on a Banach spaceX. We introduce operatorsG _O,G _X andI _X on certain spaces ofX-valued continuous functions connected with the integral equation , and we characterize exponential stability, exponential expansiveness and exponential dichotomy ofU by properties ofG _O,G _X andI _X , respectively. This extends related results known for finite dimensional spaces and for evolution families on the whole line, respectively.This work was done while the first author was visiting the Department of Mathematics of the University of Tübingen. The support of the Alexander von Humboldt Foundation is gratefully acknowledged. The author wishes to thank R. Nagel and the Functional Analysis group in Tübingen for their kind hospitality and constant encouragement.Support by Deutsche Forschungsgemeinschaft DFG is gratefully acknowledged.     

A product formula for semigroups of Lipschitz operators associated with abstract quasilinear evolution equations

The notion of semigroups of Lipschitz operators associated with abstract quasilinear evolution equations is introduced and a product formula for such semigroups is established. The product formula obtained in the paper is applied to the solvability of the Cauchy problem for a first order quasilinear system through a finite difference scheme of the Lax‐Friedrichs type.     

Existance and Uniqueness for Boussinesq type equations on a circle

We establish local and global existence results for Boussinesq type equations on a circle, employing Fourier series and a fixed point argument.     

Transitivity and Dense Periodicity for Graph Maps

In 1984, Blokh proved [A. M. Blockh, On transitive mappings of one-dimensional branched manifolds, Differential-Difference Equations and Problems of Mathematical physics (Russian), Akad. Nauk Ukrain. SSR Inst. Mat., Kiev, 131, pp. 3–9, 1984] that any topologically transitive continuous map from a graph into itself which has periodic points has a dense set of periodic points and has positive topological entropy (in this proof a crucial role is played by the specification property, which implies these two statements). Also, he characterized the topologically transitive continuous graph maps without periodic points. Unfortunately, this clever paper is only available in Russian (except for a translation to English of the statements of the theorems without proofs—see [A. M. Blockh, The connection between entropy and transitivity for one-dimensional mappings, Uspekhi Mat. Nauk, 42(5(257)) (1987), pp. 209–210]).     

Stability and robustness of collocation methods for Abel-type integral equations

Summary We study stability aspects of collocation methods for Abel-type integral equations of the first kind using piecewise polynomials. These collocation methods may be formulated as projection methods. Stability is defined as the boundedness of the sequence of projectors in their natural setting. Robustness is essentially the optimal asymptotic insensitivity to perturbations in the data. We show that stability and robustness are equivalent for the above collocation methods. This allows us to obtain optimal error estimates for some methods that are well-known to be robust. We also present numerical results for some methods which appear to be robust.Research supported in part by the United States Army under Contract No. DAAG 29-83-K-0109     

Summation of series and Gaussian quadratures,II

Continuing previous work, we discuss applications of our summation/integration procedure to some classes of complex slowly convergent series. Especially, we consider the series of the form , where 0<v1 andR(s) is a rational function. Such cases were recently studied by Gautschi, using the Laplace transform method. Also, we give an appropriate method for calculating values of the Riemann zeta function , which can be transformed to a weighted integral on (0,+)of the functiont exp (–z/2)log(1- _m ² t ²))cos(z arctan( _m t, _m >=2/((2m+1)),m₀, involving the hyperbolic weightw(t)=1/cosh² t. Numerical results are included to illustrate the method.Dedicated to Luigi Gatteschi on the occasion of his 70th birthday     

On automorphisms of Enriques surfaces and their entropy

Consider an arbitrary automorphism of an Enriques surface with its lift to the covering K3 surface. We prove a bound of the order of the lift acting on the anti‐invariant cohomology sublattice of the Enriques involution. We use it to obtain some mod 2 constraint on the original automorphism. As an application, we give a necessary condition for Salem numbers to be dynamical degrees on Enriques surfaces and obtain a new lower bound on the minimal value. In the Appendix, we give a complete list of Salem numbers that potentially could be the minimal dynamical degree on Enriques surfaces and for which the existence of geometric automorphisms is unknown.