Sum Of Orbits Research Articles

On two geometric means and sum of adjoint orbits

In this paper, we study the metric geometric mean introduced by Pusz and Woronowicz and the spectral geometric mean introduced by Fiedler and Pták, originally for positive definite matrices. The relation between t-metric geometric mean and t-spectral geometric mean is established via log majorization. The result is then extended in the context of symmetric space associated with a noncompact semisimple Lie group. For any Hermitian matrices X and Y, So's matrix exponential formula asserts that there are unitary matrices U and V such thateX/2eYeX/2=eUXU⁎+VYV⁎. In other words, the Hermitian matrix log⁡(eX/2eYeX/2) lies in the sum of the unitary orbits of X and Y. So's result is also extended to a formula for adjoint orbits associated with a noncompact semisimple Lie group.

The polytopes of the H3 group with 60 vertices and their orbit decompositions.

The goal of this article is to compare the geometrical structure of polytopes with 60 vertices, generated by the finite Coxeter group H3, i.e. an icosahedral group in three dimensions. The method of decorating a Coxeter-Dynkin diagram is used to easily read the structure of the reflection-generated polytopes. The decomposition of the vertices of the polytopes into a sum of orbits of subgroups of H3 is given and presented as a `pancake structure'.

Gradient flows for the minimum distance to the sum of adjoint orbits

Let G be a connected semisimple Lie group and its Lie algebra. Let be the Cartan decomposition corresponding to a Cartan involution θ of . The Killing form B induces a positive definite symmetric bilinear form B θ on defined by B θ(X, Y) = − B(X, θY). Given , we consider the optimization problem where the norm ‖ · ‖ is induced by B θ and K is the connected subgroup of the G with the Lie algebra . We obtain the gradient flow of the corresponding smooth function on the manifold K × ··· × K. Our results give unified extensions of several results of Li, Poon and Schulte-Herbrüggen [C.K. Li, Y.T. Poon, and T. Schulte-Herbruggen, Least-squares approximation by elements from matrix orbits achieved by gradient flows on compact Lie groups, Math. Comp. 80 (2011), 1601–1621]. They are also true for reductive Lie groups.

Determinants of sum of orbits under compact Lie group

We study the determinants on the sum of orbits of two elements in the Lie algebra of a compact connected subgroup in the unitary group. As an application, the extremal determinant expressions are obtained for the symplectic group.

Generalized Littlewood–Richardson rule and sum of coadjoint orbits of compact Lie groups

Let G be a compact connected semisimple Lie group. We extend to all irreducible finite-dimensional representations of G a result of Heckman which provides a relation between the generalized Littlewood–Richardson rule and the sum of G-coadjoint orbits. As an application of our result, we describe the eigenvalues of a sum of two real skew-symmetric matrices.

Sums of adjoint orbits and [formula omitted]-singular dichotomy for [formula omitted

Let G be a real compact connected simple Lie group, and g its Lie algebra. We study the problem of determining, from root data, when a sum of adjoint orbits in g , or a product of conjugacy classes in G, contains an open set. Our general methods allow us to determine exactly which sums of adjoint orbits in su ( m ) and products of conjugacy classes in SU ( m ) contain an open set, in terms of the highest multiplicities of eigenvalues. For su ( m ) and SU ( m ) we show L 2 -singular dichotomy: The convolution of invariant measures on adjoint orbits, or conjugacy classes, is either singular to Haar measure or in L 2 .

Determinant and Pfaffian of sum of skew symmetric matrices

We completely describe the determinants of the sum of orbits of two real skew symmetric matrices, under similarity action of orthogonal group and the special orthogonal group respectively. We also study the Pfaffian case and the complex case.

Analysis of multiple scattering iterations for high-frequency scattering problems. II: The three-dimensional scalar case

In this paper, we continue our analysis of the treatment of multiple scattering effects within a recently proposed methodology, based on integral-equations, for the numerical solution of scattering problems at high frequencies. In more detail, here we extend the two-dimensional results in part I of this work to fully three-dimensional geometries. As in the former case, our concern here is the determination of the rate of convergence of the multiple-scattering iterations for a collection of three-dimensional convex obstacles that are inherent in the aforementioned high-frequency schemes. To this end, we follow a similar strategy to that we devised in part I: first, we recast the (iterated, Neumann) multiple-scattering series in the form of a sum of periodic orbits (of increasing period) corresponding to multiple reflections that periodically bounce off a series of scattering sub-structures; then, we proceed to derive a high-frequency recurrence that relates the normal derivatives of the fields induced on these structures as the waves reflect periodically; and, finally, we analyze this recurrence to provide an explicit rate of convergence associated with each orbit. While the procedure is analogous to its two-dimensional counterpart, the actual analysis is significantly more involved and, perhaps more interestingly, it uncovers new phenomena that cannot be distinguished in two-dimensional configurations (e.g. the further dependence of the convergence rate on the relative orientation of interacting structures). As in the two-dimensional case, and beyond their intrinsic interest, we also explain here how the results of our analysis can be used to accelerate the convergence of the multiple-scattering series and, thus, to provide significant savings in computational times.

Analysis of multiple scattering iterations for high-frequency scattering problems. I: the two-dimensional case

We present an analysis of a recently proposed integral-equation method for the solution of high-frequency electromagnetic and acoustic scattering problems that delivers error-controllable solutions in frequency-independent computational times. Within single scattering configurations the method is based on the use of an appropriate ansatz for the unknown surface densities and on suitable extensions of the method of stationary phase. The extension to multiple-scattering configurations, in turn, is attained through consideration of an iterative (Neumann) series that successively accounts for further geometrical wave reflections. As we show, for a collection of two-dimensional (cylindrical) convex obstacles, this series can be rearranged into a sum of periodic orbits (of increasing period), each corresponding to contributions arising from waves that reflect off a fixed subset of scatterers when these are transversed sequentially in a periodic manner. Here, we analyze the properties of these periodic orbits in the high-frequency regime, by deriving precise asymptotic expansions for the “currents” (i.e. the normal derivative of the fields) that they induce on the surface of the obstacles. As we demonstrate these expansions can be used to provide accurate estimates of the rate at which their magnitude decreases with increasing number of reflections, which defines the overall rate of convergence of the multiple-scattering series. Moreover, we show that the detailed asymptotic knowledge of these currents can be used to accelerate this convergence and, thus, to reduce the number of iterations necessary to attain a prescribed accuracy.

The sum of unitary similarity orbits containing only special operators

Let B ( H ) be the algebra of bounded linear operator acting on a Hilbert space H (over the complex or real field). Characterization is given to A 1 , … , A k ∈ B ( H ) such that for any unitary operators U 1 , … , U k , ∑ j = 1 k U j ∗ A j U j is always in a special class S of operators such as normal operators, self-adjoint operators, unitary operators. As corollaries, characterizations are given to A ∈ B ( H ) such that complex, real or nonnegative linear combinations of operators in its unitary orbit U ( A ) = { U ∗ AU : U unitary } always lie in S .

A quadratic inequality for sum of co-adjoint orbits

We obtain an e¤ective lower bound on the distance of sum of coadjoint orbits from the origin. Even when the distance is zero, thus the symplectic quotient is well-de ned, our result give a nontrivial constraint on these co-adjoint orbits. In the particular case of unitary groups, we recover the quadratic inequality for eigenvalues of Hermitian matrices satisfying A+B = C: This quadratic inequality was obtained earlier by the authors using completely di¤erent means, namely Klyachko’s theory of toric stable re‡exive sheaves and the Chern number inequality for Hermitian Yang-Mills connection. 1 Introduction Given any rank r Hermitian matrix A, we may order its eigenvalues in such a way that 1 (A) 2 (A) ::: r (A) ; and denote (A) := ( 1 (A) ; ; r (A)) 2 R as its spectrum. In [K2] (Also see [Fu] for an excellent account of the subject), Klyachko discovered following series of linear inequalities for Hermitian matrices A;B;C satisfying A+B = C : X k2K k (C) X i2I i (A) + X j2J j (B) for some triple of subsets I; J;K f1; 2; :::; rg of the same cardinality and such that the associated Schubert cycle sK is a component of sI sJ . This result can be interpreted as describing the linear inequalities which determine the intersection of the sum of co-adjoint orbits O (A) + O (B) + O ( C) with the positive Weyl chamber for the unitary group, which is a convex polytope by Kirwan’s convexity theorem. Klyachko’s result was generalized to any compact Lie group by Berenstein and Sjamaar in the beautiful paper [BeS], they are 1 able to relate above eigenvalue inequalities to the convexity of the image of the moment map. All above inequalities are linear on eigenvalues of matrices. In [LW], we nd a natural quadratic inequality on these eigenvalues by relating the Hermitian matrices to stable re‡exive sheaves over the projective spaces. More precisely, we have shown Theorem 1 Suppose A1; ; AN are rank r Hermitian matrices satisfying N X

Fitting orbits

Nowadays, more than one hundred extra-solar planets are known, and about a dozen of multi-planetary systems have been discovered. Most of them have been detected by the radial velocity (RV) method. The recovery of orbital parameters from RV data leads to several problems. Usually RV data cover irregularly a short time interval which is frequently shorter than the orbital period of the most distant planet. Moreover, observations contain a noise due to the instabilities of the star. The distribution of this noise is unknown. A precise determination of the dynamical state of a multi-planetary system is important for understanding its stability and evolution. In most cases observers determine the orbital parameters for multi-planetary systems just fitting a sum of Keplerian orbits. The parameters obtained in such a way are in most cases the only accessible data about an extra-solar system because the observes very rarely publish their observations. However, the parameters from a multi-Keplerian fit as it has already been observed by many authors, cannot be interpreted as the osculating elements for actual planetary orbits. Moreover, these parameters can be considered as Keplerian elements of: relative, barycentric or Jacobi orbits. One can find arguments that the interpretation of parameters from a multi-Keplerian fit as elements of Keplerian orbits in the Jacobi coordinates is the most proper one, see [Lee and Peale, 2002; Goździewski et al. 2003].To search for other articles by the author(s) go to: http://adsabs.harvard.edu/abstract_service.html

EIGENVALUE SPECTRUM FOR A SINGLE PARTICLE IN A SPHEROIDAL CAVITY: A SEMICLASSICAL APPROACH

Following the semiclassical formalism of Strutinsky et al.,11 we have obtained the complete eigenvalue spectrum for a particle enclosed in an infinitely high spheroidal cavity. Our spheroidal trace formula also reproduces the results of a spherical billiard in the limit η → 1.0. Inclusion of repetition of each family of the orbits with reference to the largest one significantly improves the eigenvalues of the sphere and an exact comparison with the quantum mechanical results is observed up to the second decimal place for kR0 ≥ 7. The contributions of the equatorial, planar (in the axis of symmetry plane) and non-planar (three-dimensional) orbits are obtained from the same trace formula by using the appropriate conditions. The resulting eigenvalues compare very well with the quantum mechanical eigenvalues at normal deformation. It is interesting that the partial sum of equatorial orbits leads to eigenvalues with maximum angular momentum projection, while the summing of planar orbits leads to eigenvalues with Lz = 0 except for L = 1. The remaining quantum mechanical eigenvalues are observed to arise from the three-dimensional (3D) orbits. Very few spurious eigenvalues arise in these partial sums. This result establishes the important role of 3D orbits even at normal deformations.

Armchair cartography - a map of the galactic halo based on observations of local, metal-poor stars

A method is developed by which one can reconstruct a global model of that part of the galactic stellar halo which consists of stars on orbits taking them near the Sun. The model is constructed from observed positions and three-dimensional velocities of local, metal-poor halo stars in terms of a discrete sum of orbits. The method is applied to a sample of 118 local stars, selected without kinematic bias, with |$\text{[Fe/H]} \leqslant - 1.5\,\text{and}\,\langle\text{[Fe/H]}\rangle \simeq - 2.0.$| A Stäckel model of the galactic potential is developed in the paper, much facilitating the implementation of the above method. The density distribution of the reconstructed halo model is found to be well fitted by a power law |$\rho\text{(R)}\propto {R}^{\,-\,3.3\,\pm\,0.2}$| in the intermediate halo |$(8\lt\,R\,\lt20\,\text{kpc}),$| indicating that more than 90 per cent of all stars in the intermediate halo are on orbits taking them near the Sun. The model is therefore a nearly complete model of the galactic stellar halo in this region. The intermediate halo is found to consist of two components: a main, nearly spherical component with axial ratio |$q=0.85\pm0.12$| comprising the large majority of the mass in the halo and a highly flattened component, contributing about 40 per cent of the density at the Sun. The model predicts a typical value of the line-of-sight velocity dispersion towards the galactic poles of |$120-130\,\text{km s}^{-1}$| out to 10 kpc distance for very metal-poor stars |$(\langle\text{[Fe/H]}\rangle \simeq - 2.0).$| This is somewhat more than the |$100-110\,\text{km s}^{-1}$| found for non-local RR Lyrae and blue horizontal branch stars from observations in situ and significantly more than the |$7 4 \pm 10 \,\text{km s}^{-1}$| similarly found for K-giants at the SGP with |$\langle\text{[Fe/H]}\rangle \simeq - 1.3.$| This result would indicate that the halo subsystems become ‘hotter’ (and presumably somewhat rounder) with decreasing metal content, implying a radial metallicity gradient in the halo and suggesting a dissipative formation scenario for the halo. As none of the above stellar subsystems is characterized by dynamically important rotational mean motions, the halo may have formed through an isotropic, dissipative collapse.