Algorithms for Solvents of Matrix Polynomials

In an earlier paper we developed the algebraic theory of matrix polynomials. Here we introduce two algorithms for computing "dominant" solvents. Global convergence of the algorithms under certain conditions is established

The Algebraic Theory of Matrix Polynomials

A matrix S is a solvent of the matrix polynomial M(X)=A₀Xᵐ +...+ Am if M(S)=O where A, X, and S are square matrices. In this paper we develop the algebraic theory of matrix polynomials and solvents. We define division and interpolation, investigate the properties of block Vandermonde matrices, and define and study the existence of a complete set of solvents. We study the relation between the matrix polynomial problem and the lambda-matrix problem, which is to find a scalar A₀λᵐ + A₁λᵐ⁻¹ +...+ Am is singular. In a future paper we extend Traub’s algorithm for calculating zeros of scalar polynomials to matrix polynomials and establish global convergence properties of this algorithm for a class of matrix polynomials

Polychromatic flow cytometry is more sensitive than microscopy in detecting small monoclonal plasma cell populations

Background There is an emerging role for flow cytometry (FC) in the assessment of small populations of plasma cells (PC). However, FC's utility has been questioned due to consistent underestimation of the percentage of PC compared to microscopy. Methods A retrospective study was performed on bone marrow samples analysed by 8-colour FC. Plasma cell populations were classified as polyclonal or monoclonal based on FC analysis. FC findings were compared with microscopy of aspirates, histology and immunohistochemistry of trephine biopsies, and immunofixation (IFX) of serum and/or urine. Results FC underestimated PC compared to aspirate and trephine microscopy. The 10% diagnostic cutoff for MM on aspirate microscopy corresponded to a 3.5% cutoff on FC. Abnormal plasma cell morphology by aspirate microscopy and clonality by FC correlated in 229 of 294 cases (78%). However, in 50 cases, FC demonstrated a monoclonal population but microscopy reported no abnormality. In 15 cases, abnormalities were reported by microscopy but not by FC. Clonality assessment by trephine microscopy and FC agreed in 251/280 cases (90%), but all 29 discordant cases were monoclonal by FC and not monoclonal by microscopy. These cases had fewer PC and proportionally more polyclonal PC, and when IFX detected a paraprotein, it had the same light chain as in the PC determined by FC. Conclusions FC was more sensitive in detecting monoclonal populations that were small or accompanied by polyclonal PC. This study supports the inclusion of FC in the evaluation of PC, especially in the assessment of small population

Renormalization of the Lattice HQET Isgur-Wise Function

We compute the perturbative renormalization factors required to match to the continuum Isgur-Wise function, calculated using lattice Heavy Quark Effective Theory. The velocity, mass, wavefunction and current renormalizations are calculated for both the forward difference and backward difference actions for a variety of velocities. Subtleties are clarified regarding tadpole improvement, regulating divergences, and variations of techniques used in these renormalizations.Comment: 28 pages, 0 figures, LaTeX. Final version accepted for publication in Phys. Rev. D. (Minor changes.

Charge Deficiency, Charge Transport and Comparison of Dimensions

We study the relative index of two orthogonal infinite dimensional projections which, in the finite dimensional case, is the difference in their dimensions. We relate the relative index to the Fredholm index of appropriate operators, discuss its basic properties, and obtain various formulas for it. We apply the relative index to counting the change in the number of electrons below the Fermi energy of certain quantum systems and interpret it as the charge deficiency. We study the relation of the charge deficiency with the notion of adiabatic charge transport that arises from the consideration of the adiabatic curvature. It is shown that, under a certain covariance, (homogeneity), condition the two are related. The relative index is related to Bellissard's theory of the Integer Hall effect. For Landau Hamiltonians the relative index is computed explicitly for all Landau levels.Comment: 23 pages, no figure

Modular Lie algebras and the Gelfand-Kirillov conjecture

Let g be a finite dimensional simple Lie algebra over an algebraically closed field of characteristic zero. We show that if the Gelfand-Kirillov conjecture holds for g, then g has type A_n, C_n or G_2.Comment: 20 page

The Principal Element of a Frobenius Lie Algebra

We introduce the notion of the \textit{principal element} of a Frobenius Lie algebra \f. The principal element corresponds to a choice of F\in \f^* such that $F[-,-]$ non-degenerate. In many natural instances, the principal element is shown to be semisimple, and when associated to \sl_n, its eigenvalues are integers and are independent of $F$. For certain ``small'' functionals $F$, a simple construction is given which readily yields the principal element. When applied to the first maximal parabolic subalgebra of \sl_n, the principal element coincides with semisimple element of the principal three-dimensional subalgebra. We also show that Frobenius algebras are stable under deformation.Comment: 10 page

Of Bounces, Branes and Bounds

Some recent studies have considered a Randall-Sundrum-like brane world evolving in the background of an anti-de Sitter Reissner-Nordstrom black hole. For this scenario, it has been shown that, when the bulk charge is non-vanishing, a singularity-free ``bounce'' universe will always be obtained. However, for the physically relevant case of a de Sitter brane world, we have recently argued that, from a holographic (c-theorem) perspective, such brane worlds may not be physically viable. In the current paper, we reconsider the validity of such models by appealing to the so-called ``causal entropy bound''. In this framework, a paradoxical outcome is obtained: these brane worlds are indeed holographically viable, provided that the bulk charge is not too small. We go on to argue that this new finding is likely the more reliable one.Comment: 15 pages, Revtex; references added and very minor change