Surgery and the Spectrum of the Dirac Operator

We show that for generic Riemannian metrics on a simply-connected closed spin manifold of dimension at least 5 the dimension of the space of harmonic spinors is no larger than it must be by the index theorem. The same result holds for periodic fundamental groups of odd order. The proof is based on a surgery theorem for the Dirac spectrum which says that if one performs surgery of codimension at least 3 on a closed Riemannian spin manifold, then the Dirac spectrum changes arbitrarily little provided the metric on the manifold after surgery is chosen properly.Comment: 23 pages, 4 figures, to appear in J. Reine Angew. Mat

Generalized Euler-Poincar\'e equations on Lie groups and homogeneous spaces, orbit invariants and applications

We develop the necessary tools, including a notion of logarithmic derivative for curves in homogeneous spaces, for deriving a general class of equations including Euler-Poincar\'e equations on Lie groups and homogeneous spaces. Orbit invariants play an important role in this context and we use these invariants to prove global existence and uniqueness results for a class of PDE. This class includes Euler-Poincar\'e equations that have not yet been considered in the literature as well as integrable equations like Camassa-Holm, Degasperis-Procesi, $\mu$CH and $\mu$DP equations, and the geodesic equations with respect to right invariant Sobolev metrics on the group of diffeomorphisms of the circle

AFRICA. NO. Mapas generales (17). 1:10800000

Escalas gráficas de 80 leguas portuguesas [= 4,6 cm], 80 millas holandesas [= 4,5 cm] y 100 leguas inglesas de 23 1/9 al grado [= 4,9 cm]. Coordenadas referidas al meridiano de Londres (O 28°30'-E 18°50'/N 31°00'-S 111°30'). Red geográfica de 5° en 5°. Orientado con lis en rosa de dieciséis vientosRelieve de perfilSeñala mediante colores las fronteras entre los distintos reinados y territoriosIndica la dirección de las corrientes marinasFigura amplia leyenda sobre las características generales de la costa y breves notas históricas, distribuidas por todo el documentoInserta: A correct chart of the Gold Coast : According to the Sr. Danville on a Larger Scale. Escala [ca. 1:1500000], 20 leguas marinas inglesas y francesas de veinte al grado [= 7,3 cm]. Nudo de rumbos e indicación de los límites entre los reino

A characterization of Dirac morphisms

Relating the Dirac operators on the total space and on the base manifold of a horizontally conformal submersion, we characterize Dirac morphisms, i.e. maps which pull back (local) harmonic spinor fields onto (local) harmonic spinor fields.Comment: 18 pages; restricted to the even-dimensional cas

A model problem for conformal parameterizations of the Einstein constraint equations

We investigate the possibility that the conformal and conformal thin sandwich (CTS) methods can be used to parameterize the set of solutions of the vacuum Einstein constraint equations. To this end we develop a model problem obtained by taking the quotient of certain symmetric data on conformally flat tori. Specializing the model problem to a three-parameter family of conformal data we observe a number of new phenomena for the conformal and CTS methods. Within this family, we obtain a general existence theorem so long as the mean curvature does not change sign. When the mean curvature changes sign, we find that for certain data solutions exist if and only if the transverse-traceless tensor is sufficiently small. When such solutions exist, there are generically more than one. Moreover, the theory for mean curvatures changing sign is shown to be extremely sensitive with respect to the value of a coupling constant in the Einstein constraint equations.Comment: 40 pages, 4 figure

The Dirac operator on generalized Taub-NUT spaces

We find sufficient conditions for the absence of harmonic $L^2$ spinors on spin manifolds constructed as cone bundles over a compact K\"ahler base. These conditions are fulfilled for certain perturbations of the Euclidean metric, and also for the generalized Taub-NUT metrics of Iwai-Katayama, thus proving a conjecture of Vi\csinescu and the second author.Comment: Final version, 16 page

Blowup Criterion for the Compressible Flows with Vacuum States

We prove that the maximum norm of the deformation tensor of velocity gradients controls the possible breakdown of smooth(strong) solutions for the 3-dimensional compressible Navier-Stokes equations, which will happen, for example, if the initial density is compactly supported \cite{X1}. More precisely, if a solution of the compressible Navier-Stokes equations is initially regular and loses its regularity at some later time, then the loss of regularity implies the growth without bound of the deformation tensor as the critical time approaches. Our result is the same as Ponce's criterion for 3-dimensional incompressible Euler equations (\cite{po}). Moreover, our method can be generalized to the full Compressible Navier-Stokes system which improve the previous results. In addition, initial vacuum states are allowed in our cases.Comment: 17 page

Chemical and vibratory signals used in alarm communication in the termite Reticulitermes flavipes (Rhinotermitidae)

Termites have evolved diverse defence strategies to protect themselves against predators, including a complex alarm communication system based on vibroacoustic and/or chemical signals. In reaction to alarm signals, workers and other vulnerable castes flee away while soldiers, the specialized colony defenders, actively move toward the alarm source. In this study, we investigated the nature of alarm communication in the pest Reticulitermes flavipes. We found that workers and soldiers of R. flavipes respond to various danger stimuli using both vibroacoustic and chemical alarm signals. Among the danger stimuli, the blow of air triggered the strongest response, followed by crushed soldier head and light flash. The crushed soldier heads, which implied the alarm pheromone release, had the longest-lasting effect on the group behaviour, while the responses to other stimuli decreased quickly. We also found evidence of a positive feedback, as the release of alarm pheromones increased the vibratory communication among workers and soldiers. Our study demonstrates that alarm modalities are differentially expressed between castes, and that the response varies according to the nature of stimuli