Nematode movement along a chemical gradient in a structurally heterogeneous environment : 2. Theory

L'influence de l'hétérogénéité sur la diffusion chimique et le déplacement des nématodes est étudiée par le biais d'un modèle théorique. Ce modèle prend en compte trois facteurs influant sur le déplacement des nématodes : la structure du sol, la stratégie de recherche de nourriture et la chémotaxie. Utilisant un modèle continu, nous avons mis au point un système discret permettant de simuler les traces des nématodes dans chacune des quatre situations définies par Anderson et al. (1997). Nous avons montré que l'hétérogénéité structurale provoque aussi bien des taux variables de concentrations du composé attractif dans des aires réduites que la reconnaissance de ce composé. L'hétérogénéité structurale du sol limite également la stratégie de recherche de nourriture du nématode lequel adopte alors une stratégie permettant d'éviter les pièges structuraux. Il est démontré que des augmentations localisées de la densité structurale accroissent significativement la reconnaissance du composé attractif. (Résumé d'auteur

Nematode movement along a chemical gradient in a structurally heterogeneous environment : 1 . Experiment

L'interaction entre l'hétérogénéité structurale et les gradients chimiques, ainsi que leur influence sur le déplacement des nématodes, ont été étudiées. Trois dispositifs expérimentaux ont été utilisés qui comprennent un nématode (#Caenorhabditis elegans) placé sur une couche homogène de milieu nutritif gélosé dans une boîte de Petri avec ou sans présence d'une source bactérienne de nourriture (#Escherichia coli) utilisée comme attractif. L'hétérogénéité structurale est réalisée en ajoutant des grains de sable en une seule épaisseur dans chacun des traitements homologues. Toutes les traces ont été relevées à l'aide d'un dispositif de vidéo à séquences temporelles et les données digitalisées avant analyse. Les répartitions des angles de changement de direction et les dimensions fractales des traces sont calculées pour chaque traitement. Il se révèle un effet statistiquement significatif (P inférieur ou égal à 0,01) de tous les traitements sur le déplacement des nématodes. En présence d'un produit attractif, le déplacement du nématode est plus linéaire et dirigé vers la source bactérienne. L'hétérogénéité structurale provoque un déplacement plus linéaire que dans le cas d'un milieu homogène. La dimension fractale des traces du nématode est significativement (P inférieur ou égal à 0,01) plus élevée pour les traitements sans sable ni bactéries que pour les autres traitements. Ces résultats permettent, pour la première fois, de quantifier le degré auquel les nématodes utilisent un comportement de recherche de nourriture au hasard dans un milieu homogène et adoptent un déplacement mieux orienté en présence d'un produit attractif. Finalement, lorsqu'une hétérogénéité est présente, la stratégie de recherche de nourriture devient plutôt une stratégie d'évitement permettant au nématode d'échapper aux "pièges" structuraux, tels les pores en cul-de-sac, et de pouvoir ainsi continuer à réagir à l'attraction. (Résumé d'auteur

Topological representations of matroid maps

The Topological Representation Theorem for (oriented) matroids states that every (oriented) matroid can be realized as the intersection lattice of an arrangement of codimension one homotopy spheres on a homotopy sphere. In this paper, we use a construction of Engstr\"om to show that structure-preserving maps between matroids induce topological mappings between their representations; a result previously known only in the oriented case. Specifically, we show that weak maps induce continuous maps and that the process is a functor from the category of matroids with weak maps to the homotopy category of topological spaces. We also give a new and conceptual proof of a result regarding the Whitney numbers of the first kind of a matroid.Comment: Final version, 21 pages, 8 figures; Journal of Algebraic Combinatorics, 201

Anderson localization as a parametric instability of the linear kicked oscillator

We rigorously analyse the correspondence between the one-dimensional standard Anderson model and a related classical system, the `kicked oscillator' with noisy frequency. We show that the Anderson localization corresponds to a parametric instability of the oscillator, with the localization length determined by an increment of the exponential growth of the energy. Analytical expression for a weak disorder is obtained, which is valid both inside the energy band and at the band edge.Comment: 7 pages, Revtex, no figures, submitted to Phys. Rev.

Instabilities in a Two-Component, Species Conserving Condensate

We consider a system of two species of bosons of equal mass, with interactions $U^{a}(|x|)$ and $U^{x}(|x|)$ for bosons of the same and different species respectively. We present a rigorous proof -- valid when the Hamiltonian does not include a species switching term -- showing that, when $U^{x}(|x|)>U^{a}(|x|)$, the ground state is fully "polarized" (consists of atoms of one kind only). In the unpolarized phase the low energy excitation spectrum corresponds to two linearly dispersing modes that are even a nd odd under species exchange. The polarization instability is signaled by the vani shing of the velocity of the odd modes.Comment: To appear in Phys. Rev.

Black Hole Entropy is Noether Charge

We consider a general, classical theory of gravity in $n$ dimensions, arising from a diffeomorphism invariant Lagrangian. In any such theory, to each vector field, $\xi^a$, on spacetime one can associate a local symmetry and, hence, a Noether current $(n-1)$-form, ${\bf j}$, and (for solutions to the field equations) a Noether charge $(n-2)$-form, ${\bf Q}$. Assuming only that the theory admits stationary black hole solutions with a bifurcate Killing horizon, and that the canonical mass and angular momentum of solutions are well defined at infinity, we show that the first law of black hole mechanics always holds for perturbations to nearby stationary black hole solutions. The quantity playing the role of black hole entropy in this formula is simply $2 \pi$ times the integral over $\Sigma$ of the Noether charge $(n-2)$-form associated with the horizon Killing field, normalized so as to have unit surface gravity. Furthermore, we show that this black hole entropy always is given by a local geometrical expression on the horizon of the black hole. We thereby obtain a natural candidate for the entropy of a dynamical black hole in a general theory of gravity. Our results show that the validity of the ``second law" of black hole mechanics in dynamical evolution from an initially stationary black hole to a final stationary state is equivalent to the positivity of a total Noether flux, and thus may be intimately related to the positive energy properties of the theory. The relationship between the derivation of our formula for black hole entropy and the derivation via ``Euclidean methods" also is explained.Comment: 16 pages, EFI 93-4

Gap Fluctuations in Inhomogeneous Superconductors

Spatial fluctuations of the effective pairing interaction between electrons in a superconductor induce variations of the order parameter which in turn lead to significant changes in the density of states. In addition to an overall reduction of the quasi-particle energy gap, theory suggests that mesoscopic fluctuations of the impurity potential induce localised tail states below the mean-field gap edge. Using a field theoretic approach, we elucidate the nature of the states in the `sub-gap' region. Specifically, we show that these states are associated with replica symmetry broken instanton solutions of the mean-field equations.Comment: 11 pages, 3 figures included. To be published in PRB (Sept. 2001

A Note on Conserved Charges of Asymptotically Flat and Anti-de Sitter Spaces in Arbitrary Dimensions

The calculation of conserved charges of black holes is a rich problem, for which many methods are known. Until recently, there was some controversy on the proper definition of conserved charges in asymptotically anti-de Sitter (AdS) spaces in arbitrary dimensions. This paper provides a systematic and explicit Hamiltonian derivation of the energy and the angular momenta of both asymptotically flat and asymptotically AdS spacetimes in any dimension D bigger or equal to 4. This requires as a first step a precise determination of the asymptotic conditions of the metric and of its conjugate momentum. These conditions happen to be achieved in ellipsoidal coordinates adapted to the rotating solutions.The asymptotic symmetry algebra is found to be isomorphic either to the Poincare algebra or to the so(D-1, 2) algebra, as expected. In the asymptotically flat case, the boundary conditions involve a generalization of the parity conditions, introduced by Regge and Teitelboim, which are necessary to make the angular momenta finite. The charges are explicitly computed for Kerr and Kerr-AdS black holes for arbitrary D and they are shown to be in agreement with thermodynamical arguments.Comment: 27 pages; v2 : references added, minor corrections; v3 : replaced to match published version forthcoming in General Relativity and Gravitatio

Some comments on "The Mathematical Universe"

I discuss some problems related to extreme mathematical realism, focusing on a recently proposed "shut-up-and-calculate" approach to physics (arXiv:0704.0646, arXiv:0709.4024). I offer arguments for a moderate alternative, the essence of which lies in the acceptance that mathematics is (at least in part) a human construction, and discuss concrete consequences of this--at first sight purely philosophical--difference in point of view.Comment: 11 page

Sample-size dependence of the ground-state energy in a one-dimensional localization problem

We study the sample-size dependence of the ground-state energy in a one-dimensional localization problem, based on a supersymmetric quantum mechanical Hamiltonian with random Gaussian potential. We determine, in the form of bounds, the precise form of this dependence and show that the disorder-average ground-state energy decreases with an increase of the size $R$ of the sample as a stretched-exponential function, $\exp( - R^{z})$, where the characteristic exponent $z$ depends merely on the nature of correlations in the random potential. In the particular case where the potential is distributed as a Gaussian white noise we prove that $z = 1/3$. We also predict the value of $z$ in the general case of Gaussian random potentials with correlations.Comment: 30 pages and 4 figures (not included). The figures are available upon reques