Polar Decomposition

The aim of the paper is to give some results relating to λcommuting operators where pairs of different classes of operators are included. Under considerations we take pairs of λ-commuting operators that include normal, hyponormal, quasihyponormal and isometric operators. We focus on algebraic relations between different operators. In fact, we have worked on operators products and we have presented some results.

For in vivo determination of optically active (chiral) substances in turbid media, like for example glucose in human tissue, the backscattering geometry is particularly convenient. However, recent polarimetric measurements performed in the backscattering geometry have shown that, in this geometry, the relatively small rotation of the polarization vector arising due to the optical activity of the medium is totally swamped by the much larger changes in the orientation angle of the polarization vector due to scattering. We show that the change in the orientation angle of the polarization vector arises due to the combined effect of linear diattenuation and linear retardance of light scattered at large angles and can be decoupled from the pure optical rotation component using polar decomposition of Mueller matrix. For this purpose, the method developed earlier for polar decomposition of Mueller matrix was extended to incorporate optical rotation in the medium. The validity of this approach for accurate determination of the degree of optical rotation using the Mueller matrix measured from the medium in both forward and backscattering geometry was tested by conducting studies on chiral turbid samples prepared using known concentration of scatterers and glucose molecules.

The different orthogonal relationships that exists in the Löwdin orthogonalizations is presented. Other orthogonalization techniques such as polar decomposition (PD), principal component analysis (PCA) and reduced singular value decomposition (SVD) can be derived from Löwdin methods. It is analytically shown that the polar decomposition is presented in the symmetric orthogonalization; principal component analysis and singular value decomposition are in the canonical orthogonalization. The canonical orthogonalization can be brought in into the form of reduced SVD or vice-versa. The analytic relation between symmetric and canonical orthogonalization methods is established. The inter-relationship between symmetric orthogonalization and singular value decomposition is presented.

The Copenhagen Interpretation describes individual systems, using the same Hilbert space formalism as does the statistical ensemble interpretation (SQM). This leads to the well-known paradoxes surrounding the Measurement Problem. We extend this common mathematical structure to encompass certain natural bundles with connections over the Hilbert sphere S. This permits a consistent extension of the statistical interpretation to interacting individual systems, thereby resolving these paradoxes. Suppose V is a physical system in interaction with another system W. The state vector of V+W has a set of polar decompositions with a vector q of complex coefficients. These are parameterized by the right toroid T of amplitudes q, and comprise a singular toroidal bundle over S, which comprises the enlarged state space of V+W. We prove that each T has a unique natural convex partition yielding the correct SQM probabilities. In the extended theory V and W synchronously assume pure spectral states a...

With the help of a useful mathematical tool, the polar decomposition of closed operators, and a simple observation, i.e. the unique relation between tensor-product states and compact operators, we manage to give a compact and coherent account of the various properties of higher-order-Schmidt-representations.

In this note, we discuss measure theoretic characterizations for weighted composition operators in some operator classes on L 2 (F) such as, p-quasihyponormal, p-paranormal, p-hyponormal and weakly hyponormal. Some examples are then presented to illustrate that weighted composition operators lie between these classes.

The polarization properties of any medium are completely described by the sixteen element Mueller matrix that relates the polarization parameters of the light incident on the medium to that emerging from it. Measurement of all the elements of the matrix requires a minimum of sixteen measurements involving both linear and circularly polarized light. However, for many diagnostic applications, it would be useful if the polarization parameters can be quantified with linear polarization measurements alone. In this paper, we present a method based on polar decomposition of Mueller matrix for quantification of the polarization parameters of a scattering medium using the nine element (3 × 3) Mueller matrix that requires linear polarization measurements only. The methodology for decomposition of the 3 × 3 Mueller matrix is based on the previously developed decomposition process for sixteen element (4 × 4) Mueller matrix but with an assumption that the depolarization of linearly polarized light due to scattering is independent of the orientation angle of the incident linear polarization vector. Studies conducted on various scattering samples demonstrated that this assumption is valid for a turbid medium like biological tissue where the depolarization of linearly polarized light primarily arises due to the randomization of the field vector's direction as a result of multiple scattering. For such medium, polar decomposition of 3 × 3 Mueller matrix can be used to quantify the four independent polarization parameters namely, the linear retardance (δ ), the circular retardance (ψ), the linear depolarization coefficient (Δ) and the linear diattenuation (d) with reasonable accuracy. Since this approach requires measurements using linear polarizers only, it considerably simplifies measurement procedure and might find useful applications in tissue diagnosis using the retrieved polarization parameters.

We characterize the spectral behavior of a primal Schur-complement-based block diagonal preconditioner for saddle point systems, subject to low-rank modifications. This is motivated by a desire to reduce as much as possible the computational cost of matrix-vector products with the (1,1) block, while keeping the eigenvalues of the preconditioned matrix reasonably clustered. The formulation leads to a perturbed hyperbolic quadratic eigenvalue problem. We derive interlacing results, highlighting the differences between this problem and perturbed linear eigenvalue problems. As an example, we consider primal-dual interior point methods for semidefinite programs, and express the eigenvalues of the preconditioned matrix in terms of the centering parameter.

Let ∆ n (T ) denote the n-times iterated Aluthge transform of T , i.e. ∆ 0 (T ) = T and ∆ n (T ) = ∆(∆ n-1 (T )), n ∈ N. We prove that the sequence {∆ n (T )} n∈N converges for every r × r diagonalizable matrix T . We show that the limit ∆ ∞ (•) is a map of class C ∞ on the similarity orbit of a diagonalizable matrix, and on the (open and dense) set of r × r matrices with r different eigenvalues.

Let λ ∈ (0, 1) and let T be a r × r complex matrix with polar decomposition T = U |T |. Then, the λ-Aluthge transform is defined by * * Partially supported by CONICET (PIP 4463/96), Universidad de La Plata (UNLP 11 X472) and ANPCYT (PICT03-09521).

We prove several eigenvalue inequalities for the differences of various means of two positive invertible operators A and B on a separable Hilbert space, under the assumption that A -B is compact. Equality conditions of these inequalities are also obtained.

We have successfully applied the polar decomposition (PD) to the scattering matrix of coupling metallic nanospheres. The Discrete Dipole Approximation method (DDA) has been used as an intermediate tool to calculate these matrices. We also present a simple model based on the interference to justify the presence of a minimum in the scattered intensity. It is also shown how PD provides us with a frame in which the scattering systems can be characterized by independent parameters representing magnitudes of simple (virtual) elements constituting an alternative to conventional Mueller Matrix analysis methods.

It is well known that the Schatten p-norm defined on the space of matrices is useful and possesses nice properties. In this paper, we explore the concept of Schatten p-norm on R via the structure of Euclidean Jordan algebra. Two types of Schatten p-norm on R are defined and the relationship between these two norms is also investigated.

Large, 3D ice formations such as icicles exhibit a high degree of geometric and optical complexity. Modeling these features by hand can be a daunting task, so we present a novel physically-based algorithm for simulating this phenomenon. Solidification is usually posed as a so-called 'Stefan problem', but the problem in its classic form is inappropriate for simulating the ice typically found in a winter scene. We instead use the 'thin-film' variant of the Stefan problem to derive velocity equations for a level set simulation. However, due to the scales involved in the problem, even an adaptive grid level set solver is still insufficient to track the tip of an icicle. Therefore, we derive an analytical solution for the icicle tip and use it to correct the level set simulation. The results appear to be in agreement with experimental data. We also present a physically-based technique for modeling ripples along the ice surface that alleviates the need to explicitly track small-scale geometry. To our knowledge, our approach is the most complete model available, and produces complex visual phenomena that no previous method has been able to capture.

Zebrafish are powerful animal models for understanding biological processes and the molecular mechanisms involved in different human diseases. Advanced optical techniques based on fluorescence microscopy have become the main imaging method to characterize the development of these organisms at the microscopic level. However, the need for fluorescence probes and the consequent high light doses required to excite fluorophores can affect the biological process under observation including modification of metabolic function or phototoxicity. Here, without using any labels, we propose an implementation of a Mueller-matrix polarimeter into a commercial optical scanning microscope to characterize the polarimetric transformation of zebrafish preserved at different embryonic developmental stages. By combining the full polarimetric measurements with statistical analysis of the Lu and Chipman mathematical decomposition, we demonstrate that it is possible to quantify the structural changes of the...

This paper aims at computing the rotation matrix and angles of rotations using Newton and Halley's methods in the generalized polar decomposition. The method extends the techniques of Newton's and Halley's methods for iteratively finding the zeros of polynomial equation of single variable to matrix rotation valued problems. It calculates and estimates the eigenvalues using Chevbyshev's iterative method while computing the rotation matrix. The sample problems were tested on a randomly generated matrix of order from the family of matrix market. Numerical examples are given to demonstrate the validity of this work.

This paper examines a condition for the existence and uniqueness of a finite deformation field whenever a Gram-Schmidt (QR) factorization of the deformation gradient F is used. First, a compatibility condition is derived, provided that a right Cauchy-Green tensor C = F T F is prescribed. It is well-known that under this condition a vanishing of the Riemann curvature tensor R ensures compatibility of a finite deformation field. We derive a restriction imposed on Laplace stretch U , arising from a QR decomposition of the deformation gradient, through this compatibility condition. The derived condition on Laplace stretch is unambiguous, because a Cholesky factorization of the right Cauchy-Green tensor ensures the existence of a unique Laplace stretch. Although a vanishing of the Riemann curvature tensor provides a necessary and sufficient compatibility condition from a purely geometric point of view, this condition lacks a direct physical interpretation in a sense that one cannot identify the restrictions imposed by this condition on a quantity that can be readily measured from experiments. On the other hand, our compatibility condition restricts dependence of components of a Laplace stretch on certain spatial variables in a reference configuration. Unlike the symmetric right Cauchy-Green stretch tensor U obtained from a traditional polar decomposition of F, the components of Laplace stretch can be measured from experiments. Thus, this newly derived compatibility condition provides a physical meaning to the somewhat abstract idea of the traditionally used compatibility condition, viz., a vanishing of the Riemann curvature tensor. Couplings between certain components of the Laplace stretch representing shear and elongation play a crucial role in deriving this condition. Finally, implications of this compatibility condition are discussed.

A laboratory simulation experiment-of the interaction of the solar wind and the earth's magnetic field is being conducted at Lewis Research Center of NASA. The plasma flow is produced by an MPD a r c and a magnetic dipole is used to simulate the earth's magnetic field. The initial investigations of the stagnation region of the flow show a rapid rise in the plasma potential. The magnetic field in this region is also altered from the dipole type behavior it exhibits when no flow exists. This region of change in plasma potential and magnetic field has a thickness characterized by the electron cyclotron radius. It appears to be a merging together of a collisionless deceleration and stagnation of the flow. However, whether o r not a collisionless shock wave exists has not been conclusively demonstrated.

The subject of this thesis is Galois correspondence for von Neumann algebras and its interplay with non-commutative probability theory. After a brief introduction to representation theory for compact groups, in particular to Peter-Weyl theorem, and to operator algebras, including von Neumann algebras, automorphism groups, crossed products and decomposition theory, we formulate first steps of a non-commutative version of probability theory and introduce nonabelian analogues of stochastic processes and martingales. The central objects are a von Neumann algebra M and a compact group G acting on M, for which we give in three consecutive steps, i.e. for inner, spatial and general automorphism groups one-to-one correspondences between subgroups of G and von Neumann subalgebras of M. Furthermore, we identify non-abelian martingales in our approach and prove for them a convergence theorem. Zusammenfassung: Gegenstand dieser Dissertation ist Galois-Korrespondenz für von Neumann Algebren und deren Zusammenspiel mit nichtkommutativer Wahrscheinlichkeitstheorie. Nach einer kurzen Einführung in die Darstellungstheorie für kompakte Gruppen, insbesondere in das Theorem von Peter-Weyl, und in Operatoralgebren, einschließlich von Neumann Algebren, Gruppen von Automorphismen, semidirekte Produkte und Dekompositionstheorie, formulieren wir die ersten Schritte einer nichtkommutativen Version der Wahrscheinlichkeitstheorie und präsentieren nichtabelsche Analoga für stochastische Prozesse und Martingale. Die zentralen Objekte dieser Arbeit sind eine von Neumann Algebra M und eine kompakte Gruppe G, die auf M wirkt, für die wir in drei aufeinander aufbauenden Fällen, d.h. für innere, äußere und allgemeine Gruppen von Automorphismen, bijektive Korrespondenzen zwischen Untergruppen von G und von Neumann Unteralgebren von M angeben. Darüberhinaus identifizieren wir in unserem Formalismus nichtabelsche Martingale und beweisen für diese ein Konvergenztheorem. Our formulation of non-commutative martingales is based on the ansatz of Coja-Oghlan and Michaliček . A real valued random variable on classical probability space (Ω, A, P ) from the functional analytic viewpoint is understood as (1.3) spatial and finally general automorphism groups. The investigation of Izumi et al. is localised in the framework of our ansatz. At the end, we apply our results to non-commutative probability, namely we identify non-commutative martingales and prove a convergence theorem for them. We conclude this thesis we a summary and outlook. Lemma 2.10 The set D(G) of all coefficients constitutes an involutive algebra over C, the so-called coefficient algebra of G.

We study the spectrum of the almost Mathieu hamiltonian : where θ is an irrational number and x is in the circle ΊΓ. For a small enough coupling constant μ and any x there is a closed energy set of non-zero measure in the absolutely continuous spectrum of H. For sufficiently high μ and almost all x we prove the existence of a set of eigenvalues whose closure has positive measure. The two results are obtained for a subset of irrational numbers θ of full Lebesgue measure.

In this masters thesis we prove by contradiction the irrationality of the numbers e, π 2 , and √n m, where m, n ∈ N and √n m ∈/ N. Alongside we also prove the transcendence of the numbers e and π, which have distinctly different proofs, as the proof for the transcendence of the number π makes use of the theory of elementary symmetric polynomials. In the last chapter, we also prove the main theorem of this work, which is the Gelfond-Schneider theorem on the transcendence of numbers of the form α β , where α 6= 0, 1 and β is irrational. This work is one of the rare translations of this theorem in Slovene; the only other translation appears in an undergraduate thesis from 1978 [6]. We conclude the thesis by listing several consequences of the theorem and a number of authors who advanced the work in transcendental number theory

In this masters thesis we prove by contradiction the irrationality of the numbers e, π 2 , and √n m, where m, n ∈ N and √n m ∈/ N. Alongside we also prove the transcendence of the numbers e and π, which have distinctly different proofs, as the proof for the transcendence of the number π makes use of the theory of elementary symmetric polynomials. In the last chapter, we also prove the main theorem of this work, which is the Gelfond-Schneider theorem on the transcendence of numbers of the form α β , where α 6= 0, 1 and β is irrational. This work is one of the rare translations of this theorem in Slovene; the only other translation appears in an undergraduate thesis from 1978 [6]. We conclude the thesis by listing several consequences of the theorem and a number of authors who advanced the work in transcendental number theory

We study the star order on the algebra L(H) of bounded operators on a Hilbert space H. We present a new interpretation of this order which allows to generalize to this setting many known results for matrices: functional calculus, semi lattice properties, shorted operators and orthogonal decompositions. We also show several properties for general Hilbert spaces regarding the star order and its relationship with the functional calculus and the polar decomposition, which were unknown even in the finite dimensional setting. We also study of the existence of strong limits for star-monotone sequences and nets.

Given an arbitrary finite sequence of vectors in a finite-dimensional Hilbert space, we describe an algorithm, which computes a Parseval frame for the subspace generated by the input vectors while preserving redundancy exactly. We further investigate several of its properties. Finally, we apply the algorithm to several numerical examples.

In this paper we present a model for the realistic simulation of the mechanical behavior of cloth based on the Finite Elements Method. The use of this method in a material with the elastic properties of cloth has some problems of convergence to which we propose a solution in this work.

We study the structured condition number of differentiable maps between smooth matrix manifolds, extending previous results to maps that are only \BbbR-differentiable for complex manifolds. We present algorithms to compute the structured condition number. As special cases of smooth manifolds, we analyze automorphism groups, and Lie and Jordan algebras associated with a scalar product. For such manifolds, we derive a lower bound on the structured condition number that is cheaper to compute than the structured condition number. We provide numerical comparisons between the structured and unstructured condition numbers for the principal matrix logarithm and principal matrix square root of matrices in automorphism groups as well as for the map between matrices in automorphism groups and their polar decomposition. We show that our lower bound can be used as a good estimate for the structured condition number when the matrix argument is well conditioned. We show that the structured and unstructured condition numbers can differ by many orders of magnitude, thus motivating the development of algorithms preserving structure.

It has been argued by Valatin in 1951 that the exterior algebra is the natural mathematical framework for the N-body problem of identical Fermionic particles [1]. However, the tools developed in this mathematical field [2] have remained little exploited up to now in quantum physics. It is the purpose of this talk to review some of the key exterior algebra concepts and techniques that we find particularly relevant for physicists and chemists. We will focus on the concept of p-internal space and the derived one of porthogonality [3], which generalizes that of strong-orthogonality and can be viewed as a graded indistinguishability measure for electronic states. p-orthogonality has been applied in the past to constrain geminal models [4], and work in progress shows that computational cost can be drastically reduced by using new geminal ansätze based on such algebraic constraints. Time permitting, we will show the connection between the concept of “cancelator space” and Configuration Int...

In 1992, C. Vallée showed that the metric tensor field C = ∇Θ T ∇Θ associated with a smooth enough immersion Θ : Ω → R 3 defined over an open set Ω ⊂ R 3 necessarily satisfies the compatibility relation CURL Λ + COF Λ = 0 in Ω, where the matrix field Λ is defined in terms of the field U = C 1/2 by Key words and phrases. Classical differential geometry, fundamental theorem of Riemannian geometry, Pfaff systems, polar factorization, nonlinear three-dimensional elasticity.

In this article, we generalize some norms inequalities for sums, differences, and products of absolute value operators. Our results based on Minkowski type inequalities and generalized forms of the Cauchy-Schwarz inequality. Some other related inequalities are also discussed.

Some singular value inequalities for matrices are given. Amongother inequalities it is proved that if f and g be nonnegative functions on[0, ∞) which are continuous and satisfying the relation f (t)g(t) = t, for allt ∈ [0, ∞), thensj ∗XB ∗XB ) 1≤ sj((A∗f(| X2(| X∗|)A+ A∗f2(| X∗|)A) ⊕ (B∗g2(| X |)B+ B∗g2(| X |)B)),2(| X 1(| X 2(| X 1+ Af(| X|)A2) ⊕ (Bg(| X |)B1+ Bg(| X |)B2)),for j = 1, 2, ..., n, where A1, A, B, B2, X are square matrices. Our results inthis article generalize some existing singular value inequalities of matrices

In [F. Uhlig, Explicit polar decomposition and a near-characteristic polynomial: The 2 × 2 case, Linear Algebra Appl., 38:239-249, 1981], the author gives a representation for the factors of the polar decomposition of a nonsingular real square matrix of order 2. Uhlig's formulae are generalized to encompass all nonzero complex matrices of order 2 as well as all order n complex matrices with rank at least n − 1.

In (F. Uhlig, Explicit polar decomposition and a near-characteristic polynomial: The 2 × 2c ase,Linear Algebra Appl., 38:239-249, 1981), the author gives a representation for the factors of the polar decomposition of a nonsingular real square matrix of order 2. Uhlig's formulae are generalized to encompass all nonzero complex matrices of order 2 as well as all order n complex matrices with rank at least n � 1.

Some singular value inequalities for matrices are given. Among other inequalities it is proved that if f and g be nonnegative functions on [0, ∞) which are continuous and satisfying the relation f (t)g(t) = t, for all t ∈ [0, ∞), then s j (A

In this article, we generalize some norms inequalities for sums, differences, and products of absolute value operators. Our results based on Minkowski type inequalities and generalized forms of the Cauchy-Schwarz inequality. Some other related inequalities are also discussed.

Some singular value inequalities for matrices are given. Amongother inequalities it is proved that if f and g be nonnegative functions on[0, ∞) which are continuous and satisfying the relation f (t)g(t) = t, for allt ∈ [0, ∞), thensj ∗XB ∗XB ) 1≤ sj((A∗f(| X2(| X∗|)A+ A∗f2(| X∗|)A) ⊕ (B∗g2(| X |)B+ B∗g2(| X |)B)),2(| X 1(| X 2(| X 1+ Af(| X|)A2) ⊕ (Bg(| X |)B1+ Bg(| X |)B2)),for j = 1, 2, ..., n, where A1, A, B, B2, X are square matrices. Our results inthis article generalize some existing singular value inequalities of matrices

The infinite Brownian loop {B 0 t , t ≥ 0} on a Riemannian manifold M is the limit in distribution of the Brownian bridge of length T around a fixed origin 0, when T → +∞. It has no spectral gap. When M has nonnegative Ricci curvature, B 0 is the Brownian motion itself. When M = G/K is a noncompact symmetric space, B 0 is the relativized Φ 0-process of the Brownian motion, where Φ 0 denotes the basic spherical function of Harish-Chandra, i.e. the K-invariant ground state of the Laplacian. In this case, we consider the polar decomposition B 0 t = (K t , X t), where K t ∈ K/M and X t ∈ā + , the positive Weyl chamber. Then, as t → +∞, K t converges and d(0, X t)/t → 0 almost surely. Moreover the processes {X tT / √ T , t ≥ 0} converge in distribution, as T → +∞, to the intrinsic Brownian motion of the Weyl chamber. This implies in particular that d(0, X tT)/ √ T converges to a Bessel process of dimension D = rank M + 2j, where j denotes the number of positive indivisible roots. An ingredient of the proof is a new estimate on Φ 0 .

The infinite Brownian loop {B 0 t , t ≥ 0} on a Riemannian manifold M is the limit in distribution of the Brownian bridge of length T around a fixed origin 0, when T → +∞. It has no spectral gap. When M has nonnegative Ricci curvature, B 0 is the Brownian motion itself. When M = G/K is a noncompact symmetric space, B 0 is the relativized Φ 0-process of the Brownian motion, where Φ 0 denotes the basic spherical function of Harish-Chandra, i.e. the K-invariant ground state of the Laplacian. In this case, we consider the polar decomposition B 0 t = (K t , X t), where K t ∈ K/M and X t ∈ā + , the positive Weyl chamber. Then, as t → +∞, K t converges and d(0, X t)/t → 0 almost surely. Moreover the processes {X tT / √ T , t ≥ 0} converge in distribution, as T → +∞, to the intrinsic Brownian motion of the Weyl chamber. This implies in particular that d(0, X tT)/ √ T converges to a Bessel process of dimension D = rank M + 2j, where j denotes the number of positive indivisible roots. An ingredient of the proof is a new estimate on Φ 0 .

We introduce a new framework for the development of thin plate finite elements, the "twist-Kirchhoff theory." A family of quadrilateral plate elements is derived that takes advantage of the special structure of this new theory. Particular attention is focused on the lowest-order member of the family, an eight degree-of-freedom, four-node quadrilateral element with mid-side rotations whose stiffness matrix is exactly computed with one-point Gaussian quadrature. We prove a convergence theorem for it and various error estimates for the rectangular configuration. These are also generalized to the higherorder elements in the family. Numerical tests corroborate the theoretical results.

The well-know representation theorem for the elasticity tensor C of an isotropic body shows that C[E] = 2#E + ~ tr(E)I (1) for all symmetric tensors E, where tr(E) denotes the trace of E and I is the identity tensor. This theorem is actually a special case of a classical result (cf. e.g. [Je 31, Chapter 7]) on linear, tensor-valued mappings that are isotropic, i.e. C[QHQ T] = QC[HjQ T for all tensors H in the domain of C and all orthogonal tensors Q, where Q~ denotes the transpose of Q.

The well-know representation theorem for the elasticity tensor C of an isotropic body shows that C[E] = 2#E + ~ tr(E)I (1) for all symmetric tensors E, where tr(E) denotes the trace of E and I is the identity tensor. This theorem is actually a special case of a classical result (cf. e.g. [Je 31, Chapter 7]) on linear, tensor-valued mappings that are isotropic, i.e. C[QHQ T] = QC[HjQ T for all tensors H in the domain of C and all orthogonal tensors Q, where Q~ denotes the transpose of Q.

In this paper, we review and investigate some properties of a linear pencil in the frame of the point spectrum and an isolated point in its spectrum. We next give relationships among the spectrum of the linear pencil, the Taylor spectrums of a commuting pair and its spherical Aluthge transform through projection maps and the spectral mapping theorem.

Space Station Freedom must be designed to operate for 30 years in the harsh environment of Low Earth Olbi. The Space Station Freedom Program (SSFP) has established a series of Natural Environment models and databases for use in design and operations planning activities. This paper will describe the suite of models and databases which have either been seiected from among internationally recognized standards or developed specitically for spacecraft design applications. The models indude neutral thermosphere, ionosphere, geomagnetic field, and meteoroid/orbital debris codes as well as databases of predicted solar flux (F10.7) and geomagnetic index (Ap). The models have been integrated with an orbit propagator and used to compute environmental conditions for planned operations altitudes of Space Station Freedom. I II INPUT DATA All of the models require altitude, latitude, longitude, Member AlAA date, and time as input. Some also require solar copyrlgMQ1993 by the Amerkan lnstmne of AemnauUcs and flux (F10.7) or sunspot number (Rz) and Astronautics, Inc. No copydghl Is aueIted In lhe united states under TMe 17. US. code. The US. Qovernmern has a W a bfree license to exerclse all r ! g k under the copy@M claimed herein for Qovemmental purposes. All O I h w @ha am rewwfJ by Ihe ~Qpyr!gM owner.

We consider here the question of lifting the decomposition A s = B 8 ~H ~ bo the exponential sets. Concretely, is every element of G + the product of elements of B + and C, respectively, just as any selfadjoint element of A is the sum of selfadjoint ~lements of B and H? The answer is yes in the following sense: Each a E G + is the positive part of a product bc of elements b E B + and c E C, and both b and c are nniquely determined and depend analytically on a. This can be rephrased as follows: The map (b, c)-+ (bc) + is an analytic diffeomorphism from B + x C onto G +, where for any invertible x E A we denote with x + the positive square root of xx*. This result can be expressed equivalently as: The map (b, c)-+ bcb is a diffeomorphism between the same spaces. Notice that combining tile polar decomposition with these results we can write every invertible g E A as g = bcu, where b E B +, c E C, and u is unitary. This decomposition is unique and the factors b, c, u depend analytically of g. In the case of matrix algebras with = trace/dimension, the factorization corresponds to g = ] det(g)[cu with c > 0, det(c) = 1, and u unitary. This paper extends some results proved by G. Corach and the authors in [2]. Also, Theorem 2 states that the reductive homogeneous space resulting from a conditional expectation satisfies the regularity hypothesis introduced by L. Mata-Lorenzo and L. Recht in [5], Definition 11.1. The situation considered here is the "general context" for regularity indicated in the introduction of the last mentioned paper.

In this paper, we survey various results concerning n-involution operators and k-potent operators in Hilbert spaces. We gain insight by studying the operator equation n T I = , with , 1 k T I k n ≠ ≤ − where , n k ∈ ℕ. We study the structure of such operators and attempt to gain information about the structure of closely related operators, associated operators and the attendant spectral geometry. The paper concludes with some applications in integral equations.

In this paper we obtain a description of all solutions of the truncated matrix Hausdorff moment problem in a general case (no conditions besides solvability are assumed). We use the basic results of Krein and Ovcharenko about generalized sc-resolvents of Hermitian contractions. Necessary and sufficient conditions for the determinateness of the moment problem are obtained, as well. Several numerical examples are provided.