Orbit space

We have analyzed the symmetry properties and the ground state of an orbital Hubbard model with two orbital flavors, describing a partly filled spin-polarized eg band on a cubic lattice, as in ferromagnetic manganites. We demonstrate that the off-diagonal hopping responsible for transitions between x 2 -y 2 and 3z 2 -r 2 orbitals, and the absence of SU(2) invariance in orbital space, have important implications. One finds that superexchange contributes in all orbital ordered states, the Nagaoka theorem does not apply, and the kinetic energy is much enhanced as compared with the spin case. Therefore, orbital ordered states are harder to stabilize in the Hartree-Fock approximation (HFA), and the onset of a uniform ferro-orbital polarization and antiferro-orbital instability are similar to each other, unlike in spin case. Next we formulate a cubic (gauge) invariant slave boson approach using the orbitals with complex coefficients. In the mean-field approximation it leads to the renormalization of the kinetic energy, and provides a reliable estimate for the ground state energy of the disordered state. Using this approach one finds that the HFA fails qualitatively in the regime of large Coulomb repulsion U → ∞ -the orbital order is unstable, and instead a strongly correlated orbital liquid with disordered orbitals is realized at any electron filling.

We present here a review of the fundamental topics of Hartree-Fock theory in Quantum Chemistry. From the molecular Hamiltonian, using and discussing the Born-Oppenheimer approximation, we arrive to the Hartree and Hartree-Fock equations for the electronic problem. Special emphasis is placed in the most relevant mathematical aspects of the theoretical derivation of the final equations, as well as in the results regarding the existence and uniqueness of their solutions. All Hartree-Fock versions with different spin restrictions are systematically extracted from the general case, thus providing a unifying framework. Then, the discretization of the one-electron orbitals space is reviewed and the Roothaan-Hall formalism introduced. This leads to a exposition of the basic underlying concepts related to the construction and selection of Gaussian basis sets, focusing in algorithmic efficiency issues. Finally, we close the review with a section in which the most relevant modern developments (specially those related to the design of linear-scaling methods) are commented and linked to the issues discussed. The whole work is intentionally introductory and rather self-contained, so that it may be useful for non experts that aim to use quantum chemical methods in interdisciplinary applications. Moreover, much material that is found scattered in the literature has been put together here to facilitate comprehension and to serve as a handy reference.

Projections via the action of a Hilbert Lie group of a class of semi-martingales (given by Itô fields) defined on Hilbert manifolds are investigated. Using Itô calculus, we show that the drift term arising in the projected process can be interpreted in terms of a regularised trace of the second fundamental form of the orbits. For group actions with finite dimensional orbit space, we introduce a notion of strongly harmonic functions resp. regularised Brownian motion, which project onto harmonic functions resp. onto Brownian motion, whenever the orbits are minimal (in a regularised sense). We relate this projection procedure of semi-martingales to the Faddeev-Popov procedure in gauge field theory.

Increasing research is being done into industrial uses for the microgravity environment aboard orbiting space vehicles. However, there is some concern over the effects of reaction forces produced by moving objects, especially motors, robotic actuators, and astronauts. Reaction forces produced by the movement of these objects may manifest themselves as undesirable accelerations in the space vehicle making the vehicle unusable for microgravity applications. It is desirable to provide compensation for such forces using active means. This paper presents the design and experimental evaluation of the NASA three degree of freedom reaction compensation platform, a system designed to be a testbed for the feasibility of active attenuation of reaction forces caused by moving objects in a microgravity environment. Unique 'linear motors,' which convert electrical current directly into rectilinear force, are used in the platform design. The linear motors induce accelerations of the disp...

The density of orbital space debris constitutes an increasing environmental challenge. There are three ways to alleviate the problem: debris mitigation, debris removal and collision avoidance. This paper addresses collision avoidance, by describing a method that contributes to achieving a requisite increase in orbit prediction accuracy. Batch least-squares differential correction is applied to the publicly available two-line element (TLE) catalog of space objects. Using a high-precision numerical propagator, we fit an orbit to state vectors derived from successive TLEs. We then propagate the fitted orbit further forward in time. These predictions are compared to precision ephemeris data derived from the International Laser Ranging Service (ILRS) for several satellites, including objects in the congested sun-synchronous orbital region. The method leads to a predicted range error that increases at a typical rate of 100 meters per day, approximately a 10-fold improvement over TLE's propagated with their associated analytic propagator (SGP4). Corresponding improvements for debris trajectories could potentially provide initial conjunction analysis sufficiently accurate for an operationally viable collision avoidance system. We discuss additional optimization and the computational requirements for applying all-on-all conjunction analysis to the whole TLE catalog, present and near future. Finally, we outline a scheme for debris-debris collision avoidance that may become practicable given these developments.

This overview is intended as a lightweight companion to the long article [20]. One of the main results there is the determination of the rational cohomology of the stable mapping class group, in agreement with the Mumford conjecture [26]. This is part of a recent development in surface theory which was set in motion by Ulrike Tillmann's discovery [34] that Quillen's plus construction turns the classifying space of the stable mapping class group into an infinite loop space. Tillmann's discovery depends heavily on Harer's homological stability theorem [15] for mapping class groups, which can be regarded as one of the high points of geometric surface theory.

Les auteurs rapportent un cas exceptionnel d'occlusion vasculaire rétinienne, suite à un gliome du nerf optique chez une adolescente de 11 ans venue consulter pour perte de la vision et déviation oculaire Mots clés : gliome du nerf optique ; occlusion vasculaire rétinienne ; strabisme

Earth-orbiting objects, large enough to be tracked, have been surveyed for possible systematic debris removal. Based upon the statistical collision studies of others, it was determined that objects in orbits approximately 1000 km above the Earth's surface are at greatest collisional risk. This study has identified Russian C-1B boosters as the most important target of opportunity for debris removal. Currently, more than 100 in tact boosters are orbiting the Earth with apogees between 950 km and 1050 km. Using data provided by Energia USA, specific information on the C-1B booster, in terms of rendezvous and capture strategies, has been discussed.

An architectural approach to the evolutionary growth of an orbiting space station from a small manned satellite to a fully independent, self-sustainable space colony facility is presented. Social and environmental factors, ease of transportation via the space shuttle, and structural design are considered.

When Government drawings, specifications, or other data are used for any purpose other than in connection with a definitely related Government procurement operation, the United States Government thereby incurs no responsibility nor any obligation whatsoever, and the fact that the Government may have formulated, furnished, or in any way supplied the said drawings, specifications, or other data, is not to be regarded by implication or otherwise as in any manner licensing the holder or any other person or corporation, or conveying any rights or permission to manufacture, use, or sell any patented invention that may in any way be related thereto.

There is a Nahm transform for two-dimensional gauge fields which establishes a oneto-one correspondence between the orbit space of U (N) gauge fields with topological charge k defined on a torus and that of U (k) gauge fields with charge N on the dual torus. The main result of this paper is to show that a similar duality transform cannot exist for CP N instantons. This fact establishes a significative difference between 4-D gauge theories and CP N models. The result follows from the global analysis of the moduli space of instantons based on a complete and explicit parametrization of all self-dual solutions on the twodimensional torus. The boundary of the space of regular instantons is shown to coincide with the space of singular instantons. This identification provides a new approach to analyzing the role of overlapping instantons in the infrared sector of CP N sigma models.

We investigate discrete groups G of isometries of a complete connected Riemannian manifold M which are generated by reflections, in particular those generated by disecting reflections. We show that these are Coxeter groups, and that the orbit space M/G is isometric to a Weyl chamber C which is a Riemannian manifold with corners and certain angle conditions along intersections of faces. We can also reconstruct the manifold and its action from the Riemannian chamber and its equipment of isotropy group data along the faces. We also discuss these results from the point of view of Riemannian orbifolds.

We investigate the rudiments of Riemannian geometry on orbit spaces M/G for isometric proper actions of Lie groups on Riemannian manifolds. Minimal geodesic arcs are length minimising curves in the metric space M/G and they can hit strata which are more singular only at the end points. This is phrased as convexity result. The geodesic spray, viewed as a (strata-preserving) vector field on T M/G, leads to the notion of geodesics in M/G which are projections under M → M/G of geodesics which are normal to the orbits. It also leads to 'ballistic curves' which are projections of the other geodesics. In examples (Hermitian and symmetric matrices, and more generally polar representations) we compute their equations by singular symplectic reductions and obtain generalizations of Calogero-Moser systems with spin.

We investigate discrete groups G of isometries of a complete connected Riemannian manifold M which are generated by reflections, in particular those generated by disecting reflections. We show that these are Coxeter groups, and that the orbit space M/G is isometric to a Weyl chamber C which is a Riemannian manifold with corners and certain angle conditions along intersections of faces. We can also reconstruct the manifold and its action from the Riemannian chamber and its equipment of isotropy group data along the faces. We also discuss these results from the point of view of Riemannian orbifolds.

We consider an effective action of a compact pn´1q-torus on a smooth 2nmanifold with isolated fixed points. We prove that under certain conditions the orbit space is a closed topological manifold. In particular, this holds for certain torus actions with disconnected stabilizers. There is a filtration of the orbit manifold by orbit dimensions. The subset of orbits of dimensions less than n´1 has a specific topology which we axiomatize in the notion of a sponge. In many cases the original manifold can be recovered from its orbit manifold, the sponge, and the weights of tangent representations at fixed points.

We consider an effective action of a compact pn´1q-torus on a smooth 2nmanifold with isolated fixed points. We prove that under certain conditions the orbit space is a closed topological manifold. In particular, this holds for certain torus actions with disconnected stabilizers. There is a filtration of the orbit manifold by orbit dimensions. The subset of orbits of dimensions less than n´1 has a specific topology which we axiomatize in the notion of a sponge. In many cases the original manifold can be recovered from its orbit manifold, the sponge, and the weights of tangent representations at fixed points.

A scalable integral direct, distributed-data parallel algorithm for fourindex transformation is presented. The algorithm was implemented in the context of the second-order Moller Plesset (MP2) energy evaluation, yet it is easily adopted for other electron correlation methods, where only MO integrals with two indices in the virtual orbitals space are required. The major computational steps of the MP2 energy are the two-electron integral evaluation (9(N 4) and transformation into the MO basis (9(0N¢), where N is the number of basis functions, and O the number of occupied orbitals, respectively. The associated maximal communication costs scale as (9(n~O2V N), where V and nx denote the number of virtual orbitals, and the number of symmetry-unique shells. The largest local and global memory requirements are (9(N 2) for the MO coefficients and (9(OV N) for the three-quarter transformed integrals, respectively. Several aspects of the implementation such as symmetry-treatment, integral prescreening, and the distribution of data and computational tasks are discussed. The parallel efficiency of the algorithm is demonstrated by calculations on the phenanthrene molecule, with 762 primitive Gaussians, contracted to 412 basis functions. The calculations were performed on an IBM SP2 with 48 nodes. The measured wall clock time on 48 nodes is less than 15 min for this calculation, and the speedup relative to single-node execution is estimated to 527. This superlinear speedup is a result of exploiting both the compute power and the aggregate memory of the parallel computer. The latter reduces the number of passes through the AO integral list, and hence the operation count of the calculation. The test calculations also show that the evaluation of the two-electron integrals dominates the calculation, despite the higher scaling of the transformation step.

We report multiconfigurational linear response calculations of the one-bond and geminal indirect nuclear spin-spin couplings constants in SiH,, GeH,, and SnH,. Results of a basis set study on SnH, using 13 different basis sets on Sn are given, showing the effects on the individual contributions to the coupling constant. In addition to the more conventional choice of active orbital space (one correlating orbital for each strongly occupied valence orbital), calculations employing extended active spaces were carried out. The results are compared with available experimental data showing that correlation effects are very important. For the one-bond couplings good agreement with experiment, within l-5%, is obtained, whereas the geminal proton-proton couplings show larger deviations. The calculations reported here are, to our knowledge, the first that take into consideration correlation effects in all four contributions to the coupling constants for molecules as large as GeH, and SnH4. All X-H couplings are dominated by the Fermi contact term while the picture is more diverse for the geminal couplings.

We report multiconfigurational linear response calculations of the one-bond and geminal indirect nuclear spin-spin couplings constants in SiH,, GeH,, and SnH,. Results of a basis set study on SnH, using 13 different basis sets on Sn are given, showing the effects on the individual contributions to the coupling constant. In addition to the more conventional choice of active orbital space (one correlating orbital for each strongly occupied valence orbital), calculations employing extended active spaces were carried out. The results are compared with available experimental data showing that correlation effects are very important. For the one-bond couplings good agreement with experiment, within l-5%, is obtained, whereas the geminal proton-proton couplings show larger deviations. The calculations reported here are, to our knowledge, the first that take into consideration correlation effects in all four contributions to the coupling constants for molecules as large as GeH, and SnH4. All X-H couplings are dominated by the Fermi contact term while the picture is more diverse for the geminal couplings.

We prove that the group of Hamiltonian automorphisms of a symplectic 4-manifold contains only finitely many conjugacy classes of maximal compact tori with respect to the action of the full symplectomorphism group. We also extend to rational and ruled manifolds a result of Kedra which asserts that, if M is a simply connected symplectic 4-manifold with b 2 ≥ 3, and if f M δ denotes a blow-up of M of small enough capacity δ, then the rational cohomology algebra of the Hamiltonian group Ham(f M δ) is not finitely generated. Both results are based on the fact that in a symplectic 4-manifold endowed with any tamed almost structure J, exceptional classes of minimal symplectic area are J-indecomposable. Some applications and examples are given.

Given an orientable complete hyperbolic 3-manifold of finite volume M we construct a canonical class α(M) in H 3 (B(SL 2 (C), T)) with B(SL 2 (C), T) the SL 2 (C)-orbit space of the classifying space for a certain family of isotropy subgroups. We prove that α(M) coincides with the Bloch invariant β(M) of M defined by Neumann and Yang in [13], giving with this a simpler proof that the Bloch invariant is independent of an ideal triangulation of M. We also give a new proof of the fact that the Bloch invariant lies in the Bloch group B(C).

We present here a group theoretical analysis of the structure of the space ~ of orbits in the classical (plane) Kepler problem, and relate it to the description of the Kepler orbits as curves in configuration and in velocity spaces. A Minkowskian parametrization in f~ is introduced which allows us a clear description of many aspects of this problem. In particular, this parametrization suggests us the introduction in f~ of a Lorentzian metric, whose conformal group SO(3, 2) contains a seven-dimensional subgroup which is induced by point transformations in the configuration space P(. A SO(2, 1) subgroup of this group still acts transitively on X, which is thus identified as a homogeneous space for SO(2, 1); each regular Kepler orbit is the trace of a one-dimensional subgroup whose canonical parameter automatically equals to the classical anomalies. These results are somehow a configuration space analogous of the geometrical structure of the Kepler problem in the velocity space previously known.

This paper addresses the computation of the required trajectory correction maneuvers (TCM) for a halo orbit space mission to compensate for the launch velocity errors introduced by inaccuracies of the launch vehicle. By combining dynamical systems theory with optimal control techniques, we are able to provide a compelling portrait of the complex landscape of the trajectory design space. This approach enables automation of the analysis to perform parametric studies that simply were not available to mission designers a few years ago, such as how the magnitude of the errors and the timing of the first trajectory correction maneuver affects the correction ∆V. The impetus for combining dynamical systems theory and optimal control in this problem arises from design issues for the Genesis Discovery mission being developed for NASA by the Jet Propulsion Laboratory.

Matériels et méthodes: Etude rétrospective de 15 cas de contexte écho-clinique incertain sur une période de 2 ans. Des séquences axiales et coronales pondérées en TI, T2 et Tl Fatsat avec injection de Gadolinium toutes réalisées avec antenne de surface ont constitué le protocole de routine de l'IRM scrotale. La confrontation chirurgicale a été réalisée pour 8 cas et un suivi évolutif pour 7 cas. Résultats: Sur des arguments morphologiques et de prise de contraste, l'IRM a permis une bonne différenciation entre les lésions bénignes et les masses tissulaires testiculaires et extra-testiculaires ainsi qu'une meilleure analyse du canal inguinal. Conclusion: Le recours à l'IRM testiculaire est de plus en plus rapporté en littérature. En dépit de son coût et de son accessibilité, l'IRM scrotale reste une excellente méthode d'imagerie indiquée pour les cas douteux en échographie. Mots clés: Testicules et annexes

We algebraically characterize the Davis hyperbolic 4-manifold as the orbit space of the unique torsion-free normal subgroup of index 14,400 of the (5, 3, 3, 5) Coxeter simplex reflection group acting on hyperbolic 4-space. We determine the homology, injectivity radius, and the group of isometries of the Davis manifold. We show that the Davis manifold is a spin manifold.

This paper is a continuation of our previous paper [9] in which we developed the theory for classifying geometric fibrations of compact, connected, flat n-orbifolds, over a 1-orbifold, up to affine equivalence. In this paper, we apply our theory to classify all the geometric fibrations of compact, connected, flat 3-orbifolds, over a 1-orbifold, up to affine equivalence. 2 JOHN G. RATCLIFFE AND STEVEN T. TSCHANTZ both a Seifert fibration and its dual co-Seifert fibration of a compact, connected, flat 3-orbifold via Theorem 4 and Corollary 1 of [9] (see also [7]). We will organize our description into 17 tables, one for each 2-space group type of the generic fiber of the co-Seifert fibration. We will go through the 2-space groups in reverse order because the complexity of the description generally increases in reverse order of the IT number of a 2-space group. That the computer generated affine classification of the co-Seifert geometric fibrations described in Table 1 of [5] is correct and complete follows from Theorems 7-12 of [9] and Lemmas 1-33 below. (1) In each of these 17 tables, the first column lists the IT (international tables) number of the 3-space group Γ. (2) The second column lists the generic fibers (V /N, V ⊥ /K) of the co-Seifert and Seifert fibrations corresponding to a 2-dimensional, complete, normal subgroup N of Γ with V = Span(N) and K = N ⊥. We will denote a circle by O and a closed interval by I. (3) The third column lists the isomorphism type of the structure group Γ/NK, with C n indicating a cyclic group of order n, and D n a dihedral group of order 2n. (4) The fourth column lists the quotients (V /(Γ/K), V ⊥ /(Γ/N)) under the action of the structure group. Note that (V /(Γ/K), V ⊥ /(Γ/N)) are the bases of the Seifert and co-Seifert fibrations. (5) The fifth column indicates how generators of the structure group act diagonally on the Cartesian product V /N × V ⊥ /K. The structure group action was derived from the standard affine representation of Γ in Table 1B of [1]. We denote a rotation of O of (360/n) • by n-rot., and a reflection of O or I by ref. We denote the identity map by idt. (6) The sixth column lists the classifying pairs for the co-Seifert fibrations. These are pairs {α, β} of affinities of V /N that classify co-Seifert fibrations via Theorems 7-10 of [9]. For C n actions, α and β are inverse affinities of order n. For D 1 actions, α = idt., and β has order 2. For D n actions, with n > 1, α and β are affinities of order 2 whose product has order n. In particular, for D n actions given by (α, ref.), (γ, n-rot.), the classifying pair is {α, β} with γ = αβ, except for the cases n = 3, 6 at the end of Tables 16 and 17, which require an affine deformation. We indicate a classifying pair that falls into the case E 1 ∩ E 2 = {0} of Theorem 10 of [9] by an asterisk. Tables 1-17 were first computed by hand, and then they were double checked by a computer calculation. Let C ∞ be the standard infinite cyclic 1-space group e 1 + I , and let D ∞ be the standard infinite dihedral 1-space group e 1 + I, −I. If M is a 2-space group, we define the symmetry group of the flat orbifold E 2 /M by Sym(M) = Isom(E 2 /M). We will identify V /N with E 2 /M and the group of affinities Aff(V /N) of V /N with Aff(M) = Aff(E 2 /M). 3. Generic Fiber * 632 (30 • − 60 • right triangle) with IT number 17 The 2-space group M with IT number 17 is * 632 in Conway's notation [2] or p6m in IT notation. See space group 4/4/1/1 in Table 1A of [1] for the standard affine representation of M. The flat orbifold E 2 /M is a 30 • − 60 • right triangle.

We prove the Farrell-Jones Isomorphism Conjecture for groups acting properly discontinuously via isometries on (real) hyperbolic n-space H n with finite volume orbit space. We then apply this result to show that, for any Bianchi group Γ, W h(Γ),K 0 (ZΓ), and K i (ZΓ) vanish for i ≤ −1.

In this paper it is provided a globalization theorem which strengthens a theorem of Michael for local properties of the spaces and maps. Applications include a globalization theorem for numerability for spaces, metrizability and some new proofs for known globalization theorems for SEP, including the Uniformization Theorem of Hurewicz.

The Schottky Problem is to characterize the Jacobians in the space of principally polarized abelian varieties (PPAVs). Besides its intrinsic interest, it is related to deep questions in PDE and Physics. The perspective of Information Theory leads to the numerical investigation of properties of the distribution of the periods, especially in cases of relatively large genus. In turn, numerical results lead to conjectures that the periods of hyperelliptic curves are band-limited, and that the squared moduli of the periods of a general surface are Zipfian.

The Schottky Problem is to characterize the Jacobians in the space of principally polarized abelian varieties (PPAVs). Besides its intrinsic interest, it is related to deep questions in PDE and Physics. The perspective of Information Theory leads to the numerical investigation of properties of the distribution of the periods, especially in cases of relatively large genus. In turn, numerical results lead to conjectures that the periods of hyperelliptic curves are band-limited, and that the squared moduli of the periods of a general surface are Zipfian.

Spin–orbit coupling constants between singlet and triplet states are evaluated as residues of multiconfiguration linear response functions. In this approach, the spin–orbit coupling constants are automatically determined between orthogonal and noninteracting states. Sample calculations are presented for the X 3Σ−g–b 1Σ+g transition in O2 and the 1A1–3B1 transition in CH2. The convergence of the coupling constants is examined as a function of basis set and level of correlation. An exotic behavior is observed in the correlation of the 1A1 state for CH2 when increasing the active space, demonstrating an intricate coupling between the dynamic and static correlation. In general, the results indicate that reliable spin–orbit coupling constants between valence states may be obtained with a 4s3p2d1f basis set for first row atoms and a modest active orbital space.

The problem of classifying, up to isometry, the orientable 3-manifolds that arise by identifying the faces of a Platonic solid was completely solved in a nice paper of Everitt [B. Everitt, 3-manifolds from Platonic solids, Topology Appl. 138 (2004) 253-263]. His work completes the classification begun by Best [L.A. Best, On torsion-free discrete subgroups of PSL 2 (C) with compact orbit space, Canad.

We present here a group theoretical analysis of the structure of the space ~ of orbits in the classical (plane) Kepler problem, and relate it to the description of the Kepler orbits as curves in configuration and in velocity spaces. A Minkowskian parametrization in f~ is introduced which allows us a clear description of many aspects of this problem. In particular, this parametrization suggests us the introduction in f~ of a Lorentzian metric, whose conformal group SO(3, 2) contains a seven-dimensional subgroup which is induced by point transformations in the configuration space P(. A SO(2, 1) subgroup of this group still acts transitively on X, which is thus identified as a homogeneous space for SO(2, 1); each regular Kepler orbit is the trace of a one-dimensional subgroup whose canonical parameter automatically equals to the classical anomalies. These results are somehow a configuration space analogous of the geometrical structure of the Kepler problem in the velocity space previously known.

In the Mott insulating phase of the transition metal oxides, the effective orbital-orbital interaction is directional both in orbital space and in real space. We discuss a classical realization of directional coupling in two dimensions. Despite extensive degeneracy of the ground state, the model exhibits partial orbital ordering in the form of directional ordering of fluctuations at low temperatures stabilized by an entropy gap. Transition to the disordered phase is shown to be in the Ising universality class through exact mapping and multicanonical Monte Carlo simulations.

We investigate discrete groups G of isometries of a complete connected Riemannian manifold M which are generated by reflections, in particular those generated by disecting reflections. We show that these are Coxeter groups, and that the orbit space M/G is isometric to a Weyl chamber C which is a Riemannian manifold with corners and certain angle conditions along intersections of faces. We can also reconstruct the manifold and its action from the Riemannian chamber and its equipment of isotropy group data along the faces. We also discuss these results from the point of view of Riemannian orbifolds.

We compute the period matrices of the Riemann surfaces given by the equations w 2 = z 2g+2 − 1 (Accola-Maclachlan surfaces) and w 2g+2 = z(z − 1) g−1 (z + 1) g+2 (Kulkarni surfaces). Furthermore, we obtain the triples of matrices associated to the real forms for these Riemann surfaces.

The High Resolution Dynamics Limb Sounder (HIRDLS) experiment was designed to provide global temperature and composition data on the region from the upper troposphere to the mesopause with vertical and horizontal resolution not previously available. The science objectives are the study of small-scale dynamics and transports, including stratosphere-troposphere exchange, upper troposphere/lower stratosphere chemistry, aerosol, cirrus and PSC distributions, and gravity waves. The instrument features 21 channels, low noise levels, high vertical resolution, and a mechanical cooler for long life. During launch most of the optical aperture became obscured, so that only a potion of an optical beam width at a large azimuth from the orbital plane on the side away from the Sun can see the atmosphere. Irrecoverable loss of capabilities include limitation of coverage to the region 65°S-82°N and inability to obtain longitudinal resolution finer than an orbital spacing. While this optical blockage also impacted radiometric performance, extensive effort has gone into developing corrections for the several effects of the obstruction, so that radiances from some of the channels can be put into retrievals for temperature. Changes were also necessary for the retrieval algorithm. The validation of the resulting temperature retrievals is presented to demonstrate the effectiveness of these corrections. The random errors range from $0.5 K at 20 km to $1.0 at 60 km, close to those predicted. Comparisons with high-resolution radiosondes, lidars, ACE-FTS, and ECMWF analyses give a consistent picture of HIRDLS temperatures being 1-2 K warm from 200 to 10 hPa and within ±2 K of standards from 200 to 2 hPa (but warmer in the region of the tropical tropopause), above which HIRDLS appears to be cold. Comparisons show that both COSMIC and HIRDLS can see small vertical features down to about 2 km

The heating of coexisting gas and liquid phases of pure fluid through its critical point makes the fluid extremely compressible, expandable, slows the diffusive transport, and decreases the contact angle to zero (perfect wetting by the liquid phase). We have performed experiments on near-critical fluids in a variable volume cell in the weightlessness of an orbiting space vehicle, to suppress buoyancy-driven flows and gravitational constraints on the liquid-gas interface. The high compressibility, high thermal expansion, and low thermal diffusivity lead to a pronounced adiabatic heating called the piston effect. We have directly visualized the near-critical fluid's boundary layer response to a volume quench when the external temperature is held constant. We have found that when the system's temperature T is increased at a constant rate past the critical temperature T c, the interior of the fluid gains a higher temperature than the hot wall (overheating). This extends previous results in temperature quenching experiments in a similarly prepared system when the gas is clearly isolated from the wall. Large elliptical wetting film distortions are also seen during these ramps. By ray tracing through the elliptically shaped wetting film, we find very thick wetting film on the walls. This wetting film is at least one order of magnitude thicker than films that form in the Earth's gravity. The thick wetting film isolates the gas bubble from the wall allowing gas overheating to occur due to the difference in the piston effect response between gas and liquid. Remarkably, this overheating continues and actually increases when the fluid is ramped into the single-phase supercritical phase.

The High Resolution Dynamics Limb Sounder (HIRDLS) experiment was designed to provide global temperature and composition data on the region from the upper troposphere to the mesopause with vertical and horizontal resolution not previously available. The science objectives are the study of small-scale dynamics and transports, including stratosphere-troposphere exchange, upper troposphere/lower stratosphere chemistry, aerosol, cirrus and PSC distributions, and gravity waves. The instrument features 21 channels, low noise levels, high vertical resolution, and a mechanical cooler for long life. During launch most of the optical aperture became obscured, so that only a potion of an optical beam width at a large azimuth from the orbital plane on the side away from the Sun can see the atmosphere. Irrecoverable loss of capabilities include limitation of coverage to the region 65°S-82°N and inability to obtain longitudinal resolution finer than an orbital spacing. While this optical blockage also impacted radiometric performance, extensive effort has gone into developing corrections for the several effects of the obstruction, so that radiances from some of the channels can be put into retrievals for temperature. Changes were also necessary for the retrieval algorithm. The validation of the resulting temperature retrievals is presented to demonstrate the effectiveness of these corrections. The random errors range from $0.5 K at 20 km to $1.0 at 60 km, close to those predicted. Comparisons with high-resolution radiosondes, lidars, ACE-FTS, and ECMWF analyses give a consistent picture of HIRDLS temperatures being 1-2 K warm from 200 to 10 hPa and within ±2 K of standards from 200 to 2 hPa (but warmer in the region of the tropical tropopause), above which HIRDLS appears to be cold. Comparisons show that both COSMIC and HIRDLS can see small vertical features down to about 2 km

The pure Yang-Mills theory defined in the four-dimensional Euclidean space has a rich and interesting structure even at the classical level. The discovery of regular solutions to the Yang-Mills field equations, which correspond to absolute minima of the action (Belavin et al. , 1975},has led to an intensive study ot such a

A large database of anomalies, registered by 220 satellites in different orbits over the period 1971-1994 has been compiled. For the first time, data from 49 Russian Kosmos satellites have been included in a statistical analysis. The database also contains a large set of daily and hourly space weather parameters. A series of statistical analyses made it possible to quantify, for different satellite orbits, space weather conditions on the days characterized by anomaly occurrences. In particular, very intense fluxes (>1000 pfu at energy >10 MeV) of solar protons are linked to anomalies registered by satellites in high-altitude (>15 000 km), near-polar (inclination >55 •) orbits typical for navigation satellites, such as those used in the GPS network, NAVSTAR, etc. (the rate of anomalies increases by a factor ∼20), and to a much smaller extent to anomalies in geostationary orbits, (they increase by a factor ∼4). Direct and indirect connections between anomaly occurrence and geomagnetic perturbations are also discussed.

We investigate the rudiments of Riemannian geometry on orbit spaces M/G for isometric proper actions of Lie groups on Riemannian manifolds. Minimal geodesic arcs are length minimising curves in the metric space M/G and they can hit strata which are more singular only at the end points. This is phrased as convexity result. The geodesic spray, viewed as a (strata-preserving) vector field on T M/G, leads to the notion of geodesics in M/G which are projections under M → M/G of geodesics which are normal to the orbits. It also leads to 'ballistic curves' which are projections of the other geodesics. In examples (Hermitian and symmetric matrices, and more generally polar representations) we compute their equations by singular symplectic reductions and obtain generalizations of Calogero-Moser systems with spin.