On the Margulis constant for Kleinian groups, I curvature

The Margulis constant for Kleinian groups is the smallest constant $c$ such that for each discrete group $G$ and each point $x$ in the upper half space ${\bold H}^3$, the group generated by the elements in $G$ which move $x$ less than distance c is elementary. We take a first step towards determining this constant by proving that if $\langle f,g \rangle$ is nonelementary and discrete with $f$ parabolic or elliptic of order $n \geq 3$, then every point $x$ in ${\bold H}^3$ is moved at least distance $c$ by $f$ or $g$ where $c=.1829\ldots$. This bound is sharp

Investigation of the reaction of the lunar surface to the impact of a lunar probe final report

Flash phenomena associated with hypervelocity impact for estimating flash from impact of lunar prob

Neutron Diffuse Scattering from Polar Nanoregions in the Relaxor Pb(Mg1/3Nb2/3)O3

We have studied the neutron diffuse scattering in the relaxor PMN. The diffuse scattering appears around the Burns temperature (~620K), indicating its origin from the polar nanoregions (PNR). While the relative diffuse intensities are consistent with previous reports, they are entirely different from those of the lowest-energy TO phonon. Because of that, it has been considered that this TO mode could not be the ferroelectric soft mode. Recently, a neutron scattering study has unambiguously shown that the TO mode does soften on cooling. If the diffuse scattering in PMN originates from the soft mode condensation, then the atomic displacements must satisfy the center of mass condition. But, the atomic displacements determined from diffuse scattering intensities do not fulfill this condition. To resolve this contradiction, we propose a simple model in which the total atomic displacement consists of two components: $\delta_{CM}$ is created by the soft mode condensation, satisfying the center of mass condition, and, $\delta_{shift}$ represents a uniform displacement of the PNR along their polar direction relative to the surrounding (unpolarized) cubic matrix. Within this framework, we can successfully describe the neutron diffuse scattering intensities observed in PMN.Comment: 7 pages, 7 figures (Revised: 11-16-2001

New Perspectives on Eye Development and the Evolution of Eyes and Photoreceptors

Recent experiments on the genetic control of eye development have opened up a completely new perspective on eye evolution. The demonstration that targeted expression of one and the same master control gene, that is, Pax6 can induce the formation of ectopic eyes in both insects and vertebrates, necessitates a reconsideration of the dogma of a polyphyletic origin of the various eye types in all the animal phyla. The involvement of Pax6 and six1 and six3 genes, which encode highly conserved transcription factors, in the genetic control of eye development in organisms ranging from planarians to humans argues strongly for a monophyletic origin of the eye. Because transcription factors can control the expression of any target gene provided it contains the appropriate gene regulatory elements, the conservation of the genetic control of eye development by Pax6 among all bilaterian animals is not due to functional constraints but a consequence of its evolutionary history. The prototypic eyes postulated by Darwin to consist of two cells only, a photoreceptor and a pigment cell, were accidentally controlled by Pax6 and the subsequent evolution of the various eye types occurred by building onto this original genetic program. A hypothesis of intercalary evolution is proposed that assumes that the eye morphogenetic pathway is progressively modified by intercalation of genes between the master control genes on the top of the hierarchy and the structural genes like rhodopsin at the bottom. The recruitment of novel genes into the eye morphogenetic pathway can be due to at least two different genetic mechanisms, gene duplication and enhancer fusion. In tracing back the evolution of eyes beyond bilaterians, we find highly developed eyes in some box-jellyfish as well as in some Hydrozoans. In Hydrozoans the same orthologous six genes (six1 and six3) are required for eye regeneration as in planarians, and in the box jellyfish Tripedalia a pax B gene, which may be a precursor of Pax6, was found to be expressed in the eyes. In contrast to the adults, which have highly evolved eyes, the Planula larva of Tripedalia has single- celled photoreceptors similar to some unicellular protists. For the origin of photoreceptor cells in metazoa, I propose two hypotheses, one based on cellular differentiation and a more speculative one based on symbiosis. The former assumes that photoreceptor cells originated from a colonial protist in which all the cells were photosensitive and subsequent cellular differentiation to give rise to photoreceptor cells. The symbiont hypothesis, which I call the Russian doll model, assumes that photosensitivity arose first in photosynthetic cyanobacteria that were subsequently taken up into red algae as primary chloroplasts. The red algae in turn were taken up by dinoflagellates as secondary chloroplasts and in some species evolved into the most sophisticated eye organelles, as found, for example, in some dinoflagellates like Erythropsis and Warnovia, which lack chloroplasts. Because dinoflagellates are commonly found as symbionts in cnidarians, the dinoflagellates may have transferred their photoreceptor genes to cnidarians. In cnidarians such as Tripedalia the step from photoreceptor organelles to multicellular eyes has occurred. These two hypotheses, the cellular differentiation and the symbiont hypothesis, are not mutually exclusive and are the subject of further investigation

Soft Mode Dynamics Above and Below the Burns Temperature in the Relaxor Pb(Mg_1/3Nb_2/3)O_3

We report neutron inelastic scattering measurements of the lowest-energy transverse optic (TO) phonon branch in the relaxor Pb(Mg_1/3Nb_2/3)O_3 from 400 to 1100 K. Far above the Burns temperature T_d ~ 620 K we observe well-defined propagating TO modes at all wave vectors q, and a zone center TO mode that softens in a manner consistent with that of a ferroelectric soft mode. Below T_d the zone center TO mode is overdamped. This damping extends up to, but not above, the waterfall wave vector q_wf, which is a measure of the average size of the PNR.Comment: 4 pages, 4 figures; modified discussion of Fig. 3, shortened captions, added reference, corrected typos, accepted by Phys. Rev. Let

Theory of magnetism with temporal disorder applied to magnetically doped ZnO

A dynamic model of the asymmetric Ising glass is presented: an Ising model with antiferromagnet bonds with probabilities q arranged at random in a ferromagnetic matrix. The dynamics is introduced by changing the arrangement of the antiferromagnetic bonds after n Monte Carlo steps but keeping the same value of q and spin configuration. In the region where there is a second order transition between the ferromagnetic and paramagnetic states the dynamic behaviour follows that expected for motional narrowing and reverts to the static behaviour only for large n. There is a different dynamic behaviour where there is a first order transition between the ferromagnetic and spin glass states where it shows no effects of motional narrowing. The implications of this are discussed. This model is devised to explain the properties of doped ZnO where the magnetisation is reduced when the exchange interactions change with time.Comment: Paper was presented at MMM 2008 and is accepted for publication in J.A.

Density matrix renormalisation group study of the correlation function of the bilinear-biquadratic spin-1 chain

Using the recently developed density matrix renormalization group approach, we study the correlation function of the spin-1 chain with quadratic and biquadratic interactions. This allows us to define and calculate the periodicity of the ground state which differs markedly from that in the classical analogue. Combining our results with other studies, we predict three phases in the region where the quadratic and biquadratic terms are both positive.Comment: 13 pages, Standard Latex File + 5 PostScript figures in separate (New version with SUBSTANTIAL REVISIONS to appear in J Phys A