Eigenvalues of a one-dimensional Dirac operator pencil

We study the spectrum of a one-dimensional Dirac operator pencil, with a coupling constant in front of the potential considered as the spectral parameter. Motivated by recent investigations of graphene waveguides, we focus on the values of the coupling constant for which the kernel of the Dirac operator contains a square integrable function. In physics literature such a function is called a confined zero mode. Several results on the asymptotic distribution of coupling constants giving rise to zero modes are obtained. In particular, we show that this distribution depends in a subtle way on the sign variation and the presence of gaps in the potential. Surprisingly, it also depends on the arithmetic properties of certain quantities determined by the potential. We further observe that variable sign potentials may produce complex eigenvalues of the operator pencil. Some examples and numerical calculations illustrating these phenomena are presented.

Theoretical insight into diagnosing observation error correlations using observation-minus-background and observation-minus-analysis statistics

To improve the quantity and impact of observations used in data assimilation it is necessary to take into account the full, potentially correlated, observation error statistics. A number of methods for estimating correlated observation errors exist, but a popular method is a diagnostic that makes use of statistical averages of observation-minus-background and observation-minus-analysis residuals. The accuracy of the results it yields is unknown as the diagnostic is sensitive to the difference between the exact background and exact observation error covariances and those that are chosen for use within the assimilation. It has often been stated in the literature that the results using this diagnostic are only valid when the background and observation error correlation length scales are well separated. Here we develop new theory relating to the diagnostic. For observations on a 1D periodic domain we are able to the show the effect of changes in the assumed error statistics used in the assimilation on the estimated observation error covariance matrix. We also provide bounds for the estimated observation error variance and eigenvalues of the estimated observation error correlation matrix. We demonstrate that it is still possible to obtain useful results from the diagnostic when the background and observation error length scales are similar. In general, our results suggest that when correlated observation errors are treated as uncorrelated in the assimilation, the diagnostic will underestimate the correlation length scale. We support our theoretical results with simple illustrative examples. These results have potential use for interpreting the derived covariances estimated using an operational system.

Accumulation of complex eigenvalues of an indefinite Sturm--Liouville operator with a shifted Coulomb potential

For a particular family of long-range potentials V, we prove that the eigenvalues of the indefinite Sturm–Liouville operator A = sign(x)(−Δ+V(x)) accumulate to zero asymptotically along specific curves in the complex plane. Additionally, we relate the asymptotics of complex eigenvalues to the two-term asymptotics of the eigenvalues of associated self-adjoint operators.

Untangling associations between chironomid taxa in Neotropical streams using local and landscape filters

1. Analyses of species association have major implications for selecting indicators for freshwater biomonitoring and conservation, because they allow for the elimination of redundant information and focus on taxa that can be easily handled and identified. These analyses are particularly relevant in the debate about using speciose groups (such as the Chironomidae) as indicators in the tropics, because they require difficult and time-consuming analysis, and their responses to environmental gradients, including anthropogenic stressors, are poorly known. 2. Our objective was to show whether chironomid assemblages in Neotropical streams include clear associations of taxa and, if so, how well these associations could be explained by a set of models containing information from different spatial scales. For this, we formulated a priori models that allowed for the influence of local, landscape and spatial factors on chironomid taxon associations (CTA). These models represented biological hypotheses capable of explaining associations between chironomid taxa. For instance, CTA could be best explained by local variables (e.g. pH, conductivity and water temperature) or by processes acting at wider landscape scales (e.g. percentage of forest cover). 3. Biological data were taken from 61 streams in Southeastern Brazil, 47 of which were in well-preserved regions, and 14 of which drained areas severely affected by anthropogenic activities. We adopted a model selection procedure using Akaike`s information criterion to determine the most parsimonious models for explaining CTA. 4. Applying Kendall`s coefficient of concordance, seven genera (Tanytarsus/Caladomyia, Ablabesmyia, Parametriocnemus, Pentaneura, Nanocladius, Polypedilum and Rheotanytarsus) were identified as associated taxa. The best-supported model explained 42.6% of the total variance in the abundance of associated taxa. This model combined local and landscape environmental filters and spatial variables (which were derived from eigenfunction analysis). However, the model with local filters and spatial variables also had a good chance of being selected as the best model. 5. Standardised partial regression coefficients of local and landscape filters, including spatial variables, derived from model averaging allowed an estimation of which variables were best correlated with the abundance of associated taxa. In general, the abundance of the associated genera tended to be lower in streams characterised by a high percentage of forest cover (landscape scale), lower proportion of muddy substrata and high values of pH and conductivity (local scale). 6. Overall, our main result adds to the increasing number of studies that have indicated the importance of local and landscape variables, as well as the spatial relationships among sampling sites, for explaining aquatic insect community patterns in streams. Furthermore, our findings open new possibilities for the elimination of redundant data in the assessment of anthropogenic impacts on tropical streams.

Foliations and polynomial diffeomorphisms of R(3)

Let Y = (f, g, h): R(3) -> R(3) be a C(2) map and let Spec(Y) denote the set of eigenvalues of the derivative DY(p), when p varies in R(3). We begin proving that if, for some epsilon > 0, Spec(Y) boolean AND (-epsilon, epsilon) = empty set, then the foliation F(k), with k is an element of {f, g, h}, made up by the level surfaces {k = constant}, consists just of planes. As a consequence, we prove a bijectivity result related to the three-dimensional case of Jelonek`s Jacobian Conjecture for polynomial maps of R(n).

A note on the eigenvalues of a special class of matrices

In the analysis of stability of a variant of the Crank-Nicolson (C-N) method for the heat equation on a staggered grid a class of non-symmetric matrices appear that have an interesting property: their eigenvalues are all real and lie within the unit circle. In this note we shall show how this class of matrices is derived from the C-N method and prove that their eigenvalues are inside [-1, 1] for all values of m (the order of the matrix) and all values of a positive parameter a, the stability parameter sigma. As the order of the matrix is general, and the parameter sigma lies on the positive real line this class of matrices turns out to be quite general and could be of interest as a test set for eigenvalue solvers, especially as examples of very large matrices. (C) 2010 Elsevier B.V. All rights reserved.

Large time behaviour for functional differential equations with dominant eigenvalues of arbitrary order

The spectral theory for linear autonomous neutral functional differential equations (FDE) yields explicit formulas for the large time behaviour of solutions. Our results are based on resolvent computations and Dunford calculus, applied to establish explicit formulas for the large time behaviour of solutions of FDE. We investigate in detail a class of two-dimensional systems of FDE. (C) 2009 Elsevier Inc. All rights reserved.

One-dimensional trapped atoms: critical coupling parameter and critical number of particles

In this paper we consider the case of a Bose gas in low dimension in order to illustrate the applicability of a method that allows us to construct analytical relations, valid for a broad range of coupling parameters, for a function which asymptotic expansions are known. The method is well suitable to investigate the problem of stability of a collection of Bose particles trapped in one- dimensional configuration for the case where the scattering length presents a negative value. The eigenvalues for this interacting quantum one-dimensional many particle system become negative when the interactions overcome the trapping energy and, in this case, the system becomes unstable. Here we calculate the critical coupling parameter and apply for the case of Lithium atoms obtaining the critical number of particles for the limit of stability.

The Yang-Lee edge singularity in spin models on connected and non-connected rings

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Generalized partition function zeros of 1D spin models and their critical behavior at edge singularities

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Population dynamics of Musca domestica (Diptera: Muscidae): experimental and theoretical studies at different temperatures

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

EIGENVALUES of FIBONACCI STOCHASTIC ADDING MACHINE

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Virasoro constraints and flavor-topology duality in QCD

We derive Virasoro constraints for the zero momentum part of the QCD-like partition functions in the sector of topological charge v. The constraints depend on the topological charge only through the combination N-f +betav/2 where the value of the Dyson index beta is determined by the reality type of the fermions. This duality between flavor and topology is inherited by the small-mass expansion of the partition function and all spectral sum rules of inverse powers of the eigenvalues of the Dirac operator. For the special case beta =2 but arbitrary topological charge the Virasoro constraints are solved uniquely by a generalized Kontsevich model with the potential V(X) = 1/X.

SUPERSYMMETRY, VARIATIONAL METHOD AND HULTHEN POTENTIAL

The formalism of supersymmetric quantum mechanics provides us with the eigenfunctions to be used in the variational method to obtain the eigenvalues for the Hulthen potential.

Blumenthal's theorem for Laurent orthogonal polynomials

We investigate polynomials satisfying a three-term recurrence relation of the form B-n(x) = (x - beta(n))beta(n-1)(x) - alpha(n)xB(n-2)(x), with positive recurrence coefficients alpha(n+1),beta(n) (n = 1, 2,...). We show that the zeros are eigenvalues of a structured Hessenberg matrix and give the left and right eigenvectors of this matrix, from which we deduce Laurent orthogonality and the Gaussian quadrature formula. We analyse in more detail the case where alpha(n) --> alpha and beta(n) --> beta and show that the zeros of beta(n) are dense on an interval and that the support of the Laurent orthogonality measure is equal to this interval and a set which is at most denumerable with accumulation points (if any) at the endpoints of the interval. This result is the Laurent version of Blumenthal's theorem for orthogonal polynomials. (C) 2002 Elsevier B.V. (USA).