The higher rank rigidity theorem for manifolds with no focal points

We say that a Riemannian manifold M has rank M ≥ k if every geodesic in M admits at least k parallel Jacobi fields. The Rank Rigidity Theorem of Ballmann and Burns–Spatzier, later generalized by Eberlein–Heber, states that a complete, irreducible, simply connected Riemannian manifold M of rank k ≥ 2 (the “higher rank” assumption) whose isometry group Γ satisfies the condition that the Γ-recurrent vectors are dense in SM is a symmetric space of noncompact type. This includes, for example, higher rank M which admit a finite volume quotient. We adapt the method of Ballmann and Eberlein–Heber to prove a generalization of this theorem where the manifold M is assumed only to have no focal points. We then use this theorem to generalize to no focal points a result of Ballmann–Eberlein stating that for compact manifolds of nonpositive curvature, rank is an invariant of the fundamental group.     

Search for exclusive multibody non- decays at the resonance

Using data collected at the psi(3770) resonance with the CLEO-c detector at the Cornell e+e- storage ring, we present searches for 25 charmless decay modes of the psi(3770), mostly multibody final states. No evidence for charmless decays is found.     

Femtosecond laser ablation characteristics of nickel-based superalloy C263

Femtosecond laser (180 fs, 775 nm, 1 kHz) ablation characteristics of the nickel-based superalloy C263 are investigated. The single pulse ablation threshold is measured to be 0.26±0.03 J/cm² and the incubation parameter ξ=0.72±0.03 by also measuring the dependence of ablation threshold on the number of laser pulses. The ablation rate exhibits two logarithmic dependencies on fluence corresponding to ablation determined by the optical penetration depth at fluences below ∼5 J/cm² (for single pulse) and by the electron thermal diffusion length above that fluence. The central surface morphology of ablated craters (dimples) with laser fluence and number of laser pulses shows the development of several kinds of periodic structures (ripples) with different periodicities as well as the formation of resolidified material and holes at the centre of the ablated crater at high fluences. The debris produced during ablation consists of crystalline C263 oxidized nanoparticles with diameters of ∼2–20 nm (for F=9.6 J/cm²). The mechanisms involved in femtosecond laser microprocessing of the superalloy C263 as well as in the synthesis of C263 nanoparticles are elucidated and discussed in terms of the properties of the material.     

Implicit QR algorithms for palindromic and even eigenvalue problems

In the spirit of the Hamiltonian QR algorithm and other bidirectional chasing algorithms, a structure-preserving variant of the implicit QR algorithm for palindromic eigenvalue problems is proposed. This new palindromic QR algorithm is strongly backward stable and requires less operations than the standard QZ algorithm, but is restricted to matrix classes where a preliminary reduction to structured Hessenberg form can be performed. By an extension of the implicit Q theorem, the palindromic QR algorithm is shown to be equivalent to a previously developed explicit version. Also, the classical convergence theory for the QR algorithm can be extended to prove local quadratic convergence. We briefly demonstrate how even eigenvalue problems can be addressed by similar techniques. D. S. Watkins partly supported by Deutsche Forschungsgemeinschaft through Matheon, the DFG Research Center Mathematics for key technologies in Berlin.     

Reversed-phase poly(styrene-divinylbenzene) materials optimised for large scale preparative and process purification of synthetic peptides and recombinant proteins

Rigid macroporous copolymers of styrene and divinylbenzene have been designed for large-scale preparative and process-scale purification of synthetic peptides and recombinant proteins. The polymeric particles are mechanically stable and hence able to operate at the required high linear velocities. The pore size and pore morphology has been optimised to enable unhindered solute diffusion whilst providing maximum available surface area to enhance loading capacity. A 100 A pore size has been developed for synthetic peptides and a 300 A pore size for recombinant proteins. Precise control of particle size, within the range 10 to 20 microm, is possible which together with the very narrow particle size distribution enables maximum resolution/loading to be obtained within the pressure limits of the instrumentation being used. The chemical stability of the polymer enables cleaning in place with 1 M sodium hydroxide without particle dissolution or a deterioration in selectivity. These materials can be packed into compression hardware and are manufactured as single lots up to 100 kg (300 l) batch size.     

The crystal and molecular structure of 3,3,6,6-tetramethyl-s-tetrathiane, ((CH3)2CS2]2,

The X-ray crystal structure of 3,3,6,6-tetramethyl-s-tetrathiane (“duplodithioacetone”) confirms earlier nmr studies predicting the molecule to adopt a twist-boat conformation.     

Invariants of linear maps on matrix algebras

Let L be a linear map on the space of n×n matrices over a field. We determine the structure of the maps L that preserve commutativity. We also determine the structure of all linear maps on complex matrices that preserve the higher numerical range. The main tool is the Fundamental Theorem of Projective Geometry.     

Class numbers of imaginary quadratic fields

The classical class number problem of Gauss asks for a classification of all imaginary quadratic fields with a given class number . The first complete results were for by Heegner, Baker, and Stark. After the work of Goldfeld and Gross-Zagier, the task was a finite decision problem for any . Indeed, after Oesterlé handled , in 1985 Serre wrote, ``No doubt the same method will work for other small class numbers, up to 100, say.' However, more than ten years later, after doing , Wagner remarked that the case seemed impregnable. We complete the classification for all , an improvement of four powers of 2 (arguably the most difficult case) over the previous best results. The main theoretical technique is a modification of the Goldfeld-Oesterlé work, which used an elliptic curve -function with an order 3 zero at the central critical point, to instead consider Dirichlet -functions with low-height zeros near the real line (though the former is still required in our proof). This is numerically much superior to the previous method, which relied on work of Montgomery-Weinberger. Our method is still quite computer-intensive, but we are able to keep the time needed for the computation down to about seven months. In all cases, we find that there is no abnormally large ``exceptional modulus' of small class number, which agrees with the prediction of the Generalised Riemann Hypothesis.

     

Hydrogenic Rydberg states of molecular van der Waals complexes: resolved Rydberg spectroscopy of DABCO-N2

The complementary threshold ionization techniques of MATI and ZEKE spectroscopy have been used to reveal well-resolved, long-lived (>10 micros) hydrogenic Rydberg series (50< or =n< or =98) in a van der Waals complex formed between a polyatomic molecule and a diatomic molecule for the first time. The series are observed within 50 cm(-1) of the adiabatic ionization threshold as well as two core-excited thresholds corresponding to excitation of up to two quanta in the van der Waals vibrational mode.