TY - JOUR A1 - Hermann, Andreas T1 - Zero sets of eigenspinors for generic metrics JF - Communications in analysis and geometry N2 - Let M be a closed connected spin manifold of dimension 2 or 3 with a fixed orientation and a fixed spin structure. We prove that for a generic Riemannian metric on M the non-harmonic eigenspinors of the Dirac operator are nowhere zero. The proof is based on a transversality theorem and the unique continuation property of the Dirac operator. Y1 - 2014 SN - 1019-8385 SN - 1944-9992 VL - 22 IS - 2 SP - 177 EP - 218 PB - International Press of Boston CY - Somerville ER - TY - JOUR A1 - Rungrottheera, Wannarut A1 - Schulze, Bert-Wolfgang T1 - Weighted spaces on corner manifolds JF - Complex variables and elliptic equations N2 - We study spaces on manifolds with double weights and iterated discrete and continuous asymptotics, and their relationship with corner pseudo-differential operators. KW - manifolds with corners KW - iterated asymptotics KW - operators with corner symbols KW - 35J70 KW - 47G30 KW - 58J40 Y1 - 2014 U6 - https://doi.org/10.1080/17476933.2013.876416 SN - 1747-6933 SN - 1747-6941 VL - 59 IS - 12 SP - 1706 EP - 1738 PB - Routledge, Taylor & Francis Group CY - Abingdon ER - TY - JOUR A1 - Ludewig, Matthias T1 - Vector fields with a non-degenerate source JF - Journal of geometry and physics N2 - We discuss the solution theory of operators of the form del(x) + A, acting on smooth sections of a vector bundle with connection del over a manifold M, where X is a vector field having a critical point with positive linearization at some point p is an element of M. As an operator on a suitable space of smooth sections Gamma(infinity)(U, nu), it fulfills a Fredholm alternative, and the same is true for the adjoint operator. Furthermore, we show that the solutions depend smoothly on the data del, X and A. KW - Recursive transport equations KW - First order PDE KW - Fredholm alternative KW - Heat kernel coefficients KW - WKB expansion KW - Semiclassical analysis Y1 - 2014 U6 - https://doi.org/10.1016/j.geomphys.2014.01.014 SN - 0393-0440 SN - 1879-1662 VL - 79 SP - 59 EP - 76 PB - Elsevier CY - Amsterdam ER - TY - JOUR A1 - Klein, Markus A1 - Rama, Juliane T1 - Time asymptotics of e(-ith(kappa)) for analytic matrices and analytic perturbation theory JF - Asymptotic analysis N2 - In quantum mechanics the temporal decay of certain resonance states is associated with an effective time evolution e(-ith(kappa)), where h(.) is an analytic family of non-self-adjoint matrices. In general the corresponding resonance states do not decay exponentially in time. Using analytic perturbation theory, we derive asymptotic expansions for e(-ith(kappa)), simultaneously in the limits kappa -> 0 and t -> infinity, where the corrections with respect to pure exponential decay have uniform bounds in one complex variable kappa(2)t. In the Appendix we briefly review analytic perturbation theory, replacing the classical reference to the 1920 book of Knopp [Funktionentheorie II, Anwendungen und Weiterfuhrung der allgemeinen Theorie, Sammlung Goschen, Vereinigung wissenschaftlicher Verleger Walter de Gruyter, 1920] and its terminology by standard modern references. This might be of independent interest. KW - resonances KW - exponential decay KW - long-time corrections KW - Fermi golden rule KW - analytic perturbation theory Y1 - 2014 U6 - https://doi.org/10.3233/ASY-141226 SN - 0921-7134 SN - 1875-8576 VL - 89 IS - 3-4 SP - 189 EP - 233 PB - IOS Press CY - Amsterdam ER - TY - JOUR A1 - Schulze, Bert-Wolfgang A1 - Wei, Y. T1 - The Mellin-edge quantisation for corner operators JF - Complex analysis and operator theory N2 - We establish a quantisation of corner-degenerate symbols, here called Mellin-edge quantisation, on a manifold with second order singularities. The typical ingredients come from the "most singular" stratum of which is a second order edge where the infinite transversal cone has a base that is itself a manifold with smooth edge. The resulting operator-valued amplitude functions on the second order edge are formulated purely in terms of Mellin symbols taking values in the edge algebra over . In this respect our result is formally analogous to a quantisation rule of (Osaka J. Math. 37:221-260, 2000) for the simpler case of edge-degenerate symbols that corresponds to the singularity order 1. However, from the singularity order 2 on there appear new substantial difficulties for the first time, partly caused by the edge singularities of the cone over that tend to infinity. Y1 - 2014 U6 - https://doi.org/10.1007/s11785-013-0289-3 SN - 1661-8254 SN - 1661-8262 VL - 8 IS - 4 SP - 803 EP - 841 PB - Springer CY - Basel ER - TY - JOUR A1 - Hoegele, Michael A1 - Pavlyukevich, Ilya T1 - The exit problem from a neighborhood of the global attractor for dynamical systems perturbed by heavy-tailed levy processes JF - Stochastic analysis and applications N2 - We consider a finite-dimensional deterministic dynamical system with the global attractor ? which supports a unique ergodic probability measure P. The measure P can be considered as the uniform long-term mean of the trajectories staying in a bounded domain D containing ?. We perturb the dynamical system by a multiplicative heavy tailed Levy noise of small intensity E>0 and solve the asymptotic first exit time and location problem from D in the limit of E?0. In contrast to the case of Gaussian perturbations, the exit time has an algebraic exit rate as a function of E, just as in the case when ? is a stable fixed point studied earlier in [9, 14, 19, 26]. As an example, we study the first exit problem from a neighborhood of the stable limit cycle for the Van der Pol oscillator perturbed by multiplicative -stable Levy noise. KW - alpha-stable Levy process KW - Canonical (Marcus) SDE KW - First exit location KW - First exit time KW - Global attractor KW - Ito SDE KW - Multiplicative noise KW - Regular variation KW - Stratonovich SDE KW - Van der Pol oscillator Y1 - 2014 U6 - https://doi.org/10.1080/07362994.2014.858554 SN - 0736-2994 SN - 1532-9356 VL - 32 IS - 1 SP - 163 EP - 190 PB - Taylor & Francis Group CY - Philadelphia ER - TY - JOUR A1 - Chang, Der-Chen A1 - Habal, Nadia A1 - Schulze, Bert-Wolfgang T1 - The edge algebra structure of the Zaremba problem JF - Journal of pseudo-differential operators and applications N2 - We study mixed boundary value problems, here mainly of Zaremba type for the Laplacian within an edge algebra of boundary value problems. The edge here is the interface of the jump from the Dirichlet to the Neumann condition. In contrast to earlier descriptions of mixed problems within such an edge calculus, cf. (Harutjunjan and Schulze, Elliptic mixed, transmission and singular crack problems, 2008), we focus on new Mellin edge quantisations of the Dirichlet-to-Neumann operator on the Neumann side of the boundary and employ a pseudo-differential calculus of corresponding boundary value problems without the transmission property at the interface. This allows us to construct parametrices for the original mixed problem in a new and transparent way. Y1 - 2014 U6 - https://doi.org/10.1007/s11868-013-0088-7 SN - 1662-9981 SN - 1662-999X VL - 5 IS - 1 SP - 69 EP - 155 PB - Springer CY - Basel ER - TY - JOUR A1 - Kiselev, Oleg M. A1 - Tarkhanov, Nikolai Nikolaevich T1 - The capture of a particle into resonance at potential hole with dissipative perturbation JF - Chaos, solitons & fractals : applications in science and engineering ; an interdisciplinary journal of nonlinear science N2 - We study the capture of a particle into resonance at a potential hole with dissipative perturbation and external periodic excitation. The measure of resonance solutions is evaluated. We also derive an asymptotic formula for the parameter range of those solutions which are captured into resonance. Y1 - 2014 U6 - https://doi.org/10.1016/j.chaos.2013.11.003 SN - 0960-0779 SN - 1873-2887 VL - 58 SP - 27 EP - 39 PB - Elsevier CY - Oxford ER - TY - JOUR A1 - Blanchard, Gilles A1 - Delattre, Sylvain A1 - Roquain, Etienne T1 - Testing over a continuum of null hypotheses with False Discovery Rate control JF - Bernoulli : official journal of the Bernoulli Society for Mathematical Statistics and Probability N2 - We consider statistical hypothesis testing simultaneously over a fairly general, possibly uncountably infinite, set of null hypotheses, under the assumption that a suitable single test (and corresponding p-value) is known for each individual hypothesis. We extend to this setting the notion of false discovery rate (FDR) as a measure of type I error. Our main result studies specific procedures based on the observation of the p-value process. Control of the FDR at a nominal level is ensured either under arbitrary dependence of p-values, or under the assumption that the finite dimensional distributions of the p-value process have positive correlations of a specific type (weak PRDS). Both cases generalize existing results established in the finite setting. Its interest is demonstrated in several non-parametric examples: testing the mean/signal in a Gaussian white noise model, testing the intensity of a Poisson process and testing the c.d.f. of i.i.d. random variables. KW - continuous testing KW - false discovery rate KW - multiple testing KW - positive correlation KW - step-up KW - stochastic process Y1 - 2014 U6 - https://doi.org/10.3150/12-BEJ488 SN - 1350-7265 SN - 1573-9759 VL - 20 IS - 1 SP - 304 EP - 333 PB - International Statistical Institute CY - Voorburg ER - TY - JOUR A1 - Kiselev, Oleg A1 - Tarkhanov, Nikolai Nikolaevich T1 - Scattering of trajectories at a separatrix under autoresonance JF - Journal of mathematical physics N2 - The subject of this paper is solutions of an autoresonance equation. We look for a connection between the parameters of the solution bounded as t -> -infinity, and the parameters of two two-parameter families of solutions as t -> infinity. One family consists of the solutions which are not captured into resonance, and another of those increasing solutions which are captured into resonance. In this way we describe the transition through the separatrix for equations with slowly varying parameters and get an estimate for parameters before the resonance of those solutions which may be captured into autoresonance. (C) 2014 AIP Publishing LLC. Y1 - 2014 U6 - https://doi.org/10.1063/1.4875105 SN - 0022-2488 SN - 1089-7658 VL - 55 IS - 6 PB - American Institute of Physics CY - Melville ER -