Peripheral separability and cusps of arithmetic hyperbolic orbifolds

For X = R, C, or H it is well known that cusp cross-sections of finite volume X-hyperbolic (n+1)-orbifolds are flat n-orbifolds or almost flat orbifolds modelled on the (2n+1)-dimensional Heisenberg group N_{2n+1} or the (4n+3)-dimensional quaternionic Heisenberg group N_{4n+3}(H). We give a necessary and sufficient condition for such manifolds to be diffeomorphic to a cusp cross-section of an arithmetic X-hyperbolic (n+1)-orbifold. A principal tool in the proof of this classification theorem is a subgroup separability result which may be of independent interest.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-32.abs.htm

Arbitrarily large families of spaces of the same volume

In any connected non-compact semi-simple Lie group without factors locally isomorphic to SL_2(R), there can be only finitely many lattices (up to isomorphism) of a given covolume. We show that there exist arbitrarily large families of pairwise non-isomorphic arithmetic lattices of the same covolume. We construct these lattices with the help of Bruhat-Tits theory, using Prasad's volume formula to control their covolumes.Comment: 9 pages. Syntax corrected; one reference adde

Optical Study of GaAs quantum dots embedded into AlGaAs nanowires

We report on the photoluminescence characterization of GaAs quantum dots embedded into AlGaAs nano-wires. Time integrated and time resolved photoluminescence measurements from both an array and a single quantum dot/nano-wire are reported. The influence of the diameter sizes distribution is evidenced in the optical spectroscopy data together with the presence of various crystalline phases in the AlGaAs nanowires.Comment: 5 page, 5 figure

Self-consistent calculations of quadrupole moments of the first 2+ states in Sn and Pb isotopes

A method of calculating static moments of excited states and transitions between excited states is formulated for non-magic nuclei within the Green function formalism. For these characteristics, it leads to a noticeable difference from the standard QRPA approach. Quadrupole moments of the first 2+ states in Sn and Pb isotopes are calculated using the self-consistent TFFS based on the Energy Density Functional by Fayans et al. with the set of parameters DF3-a fixed previously. A reasonable agreement with available experimental data is obtained.Comment: 5 pages, 6 figure

Temperature-dependent magnetospectroscopy of HgTe quantum wells

We report on magnetospectroscopy of HgTe quantum wells in magnetic fields up to 45 T in temperature range from 4.2 K up to 185 K. We observe intra- and inter-band transitions from zero-mode Landau levels, which split from the bottom conduction and upper valence subbands, and merge under the applied magnetic field. To describe experimental results, realistic temperature-dependent calculations of Landau levels have been performed. We show that although our samples are topological insulators at low temperatures only, the signature of such phase persists in optical transitions at high temperatures and high magnetic fields. Our results demonstrate that temperature-dependent magnetospectroscopy is a powerful tool to discriminate trivial and topological insulator phases in HgTe quantum wells

Fixed points and amenability in non-positive curvature

Consider a proper cocompact CAT(0) space X. We give a complete algebraic characterisation of amenable groups of isometries of X. For amenable discrete subgroups, an even narrower description is derived, implying Q-linearity in the torsion-free case. We establish Levi decompositions for stabilisers of points at infinity of X, generalising the case of linear algebraic groups to Is(X). A geometric counterpart of this sheds light on the refined bordification of X (\`a la Karpelevich) and leads to a converse to the Adams-Ballmann theorem. It is further deduced that unimodular cocompact groups cannot fix any point at infinity except in the Euclidean factor; this fact is needed for the study of CAT(0) lattices. Various fixed point results are derived as illustrations.Comment: 33 page

There is no "Theory of Everything" inside E8

We analyze certain subgroups of real and complex forms of the Lie group E8, and deduce that any "Theory of Everything" obtained by embedding the gauge groups of gravity and the Standard Model into a real or complex form of E8 lacks certain representation-theoretic properties required by physical reality. The arguments themselves amount to representation theory of Lie algebras in the spirit of Dynkin's classic papers and are written for mathematicians.Comment: Final version, to appear in Communications in Mathematical Physics. The main difference, from the previous version, is the creation of a new section, containing a response to Lisi's objections to our wor

Counting and effective rigidity in algebra and geometry

The purpose of this article is to produce effective versions of some rigidity results in algebra and geometry. On the geometric side, we focus on the spectrum of primitive geodesic lengths (resp., complex lengths) for arithmetic hyperbolic 2-manifolds (resp., 3-manifolds). By work of Reid, this spectrum determines the commensurability class of the 2-manifold (resp., 3-manifold). We establish effective versions of these rigidity results by ensuring that, for two incommensurable arithmetic manifolds of bounded volume, the length sets (resp., the complex length sets) must disagree for a length that can be explicitly bounded as a function of volume. We also prove an effective version of a similar rigidity result established by the second author with Reid on a surface analog of the length spectrum for hyperbolic 3-manifolds. These effective results have corresponding algebraic analogs involving maximal subfields and quaternion subalgebras of quaternion algebras. To prove these effective rigidity results, we establish results on the asymptotic behavior of certain algebraic and geometric counting functions which are of independent interest.Comment: v.2, 39 pages. To appear in Invent. Mat

Development of Grazing Incidence Multilayer Mirrors for Hard X-ray Focusing Telescopes

We are developing depth-graded, multilayer-coated mirrors for astrophysical hard X-ray focusing telescopes. In this paper, we discuss the primary technical challenges associated with the multilayer coatings, and report on progress to date. We have sputtered constant cl-spacing and depth-graded W / Si multilayers onto 0.3- 0.5 mm thick DURAN glass (AF45 and D263) and 0.4 mm thick epoxy replicated aluminum foils (ERAFs) , both of which are potential mirror substrates. We have characterized the interfacial roughness, uniformity, and stress of the coatings. The average interfacial roughness of each multilayer was measured from specular reflectivity scans (Bi = Br) using Cu K0 X-rays. The thin film stress was calculated from the change in curvature induced by the coating on flat glass substrates. Thickness and roughness uniformity were measured by taking specular reflectivity scans of a multilayer deposited on the inside surface of a quarter cylinder section. We found that interfacial roughness (a) in the multilayers was typically between 3.5 and 4.0 A on DESAG glass, and between 4.5 and 5.0 A on the ERAFs. Also, we found that coatings deposited on glass that has been thermally formed into a cylindrical shape performed as well as flat glass. The film stress, calculated from Stoney's equation, for a 200 layer graded multilayer was approximately 200 MPa. Our uniformity measurements show that with no baffles to alter the deposition profile on a curved optic, the layer thickness differs by "'203 between the center and the edge of the optic. Interfacial roughness, however, remained constant, around 3.6 A, throughout the curved piece, even as the layer spacing dropped off

Consequences of local gauge symmetry in empirical tight-binding theory

A method for incorporating electromagnetic fields into empirical tight-binding theory is derived from the principle of local gauge symmetry. Gauge invariance is shown to be incompatible with empirical tight-binding theory unless a representation exists in which the coordinate operator is diagonal. The present approach takes this basis as fundamental and uses group theory to construct symmetrized linear combinations of discrete coordinate eigenkets. This produces orthogonal atomic-like "orbitals" that may be used as a tight-binding basis. The coordinate matrix in the latter basis includes intra-atomic matrix elements between different orbitals on the same atom. Lattice gauge theory is then used to define discrete electromagnetic fields and their interaction with electrons. Local gauge symmetry is shown to impose strong restrictions limiting the range of the Hamiltonian in the coordinate basis. The theory is applied to the semiconductors Ge and Si, for which it is shown that a basis of 15 orbitals per atom provides a satisfactory description of the valence bands and the lowest conduction bands. Calculations of the dielectric function demonstrate that this model yields an accurate joint density of states, but underestimates the oscillator strength by about 20% in comparison to a nonlocal empirical pseudopotential calculation.Comment: 23 pages, 7 figures, RevTeX4; submitted to Phys. Rev.