Global embedding of the Kerr black hole event horizon into hyperbolic 3-space

An explicit global and unique isometric embedding into hyperbolic 3-space, H^3, of an axi-symmetric 2-surface with Gaussian curvature bounded below is given. In particular, this allows the embedding into H^3 of surfaces of revolution having negative, but finite, Gaussian curvature at smooth fixed points of the U(1) isometry. As an example, we exhibit the global embedding of the Kerr-Newman event horizon into H^3, for arbitrary values of the angular momentum. For this example, considering a quotient of H^3 by the Picard group, we show that the hyperbolic embedding fits in a fundamental domain of the group up to a slightly larger value of the angular momentum than the limit for which a global embedding into Euclidean 3-space is possible. An embedding of the double-Kerr event horizon is also presented, as an example of an embedding which cannot be made global.Comment: 16 pages, 13 figure

Holographic Uniformization

We derive and study supergravity BPS flow equations for M5 or D3 branes wrapping a Riemann surface. They take the form of novel geometric flows intrinsically defined on the surface. Their dual field-theoretic interpretation suggests the existence of solutions interpolating between an arbitrary metric in the UV and the constant-curvature metric in the IR. We confirm this conjecture with a rigorous global existence proof.Comment: 52 pages, 3 figure

Long time dynamics and coherent states in nonlinear wave equations

We discuss recent progress in finding all coherent states supported by nonlinear wave equations, their stability and the long time behavior of nearby solutions.Comment: bases on the authors presentation at 2015 AMMCS-CAIMS Congress, to appear in Fields Institute Communications: Advances in Applied Mathematics, Modeling, and Computational Science 201

Nonlinear Instability for the Critically Dissipative Quasi-Geostrophic Equation

We prove that linear instability implies non-linear instability in the energy norm for the critically dissipative quasi-geostrophic equation.Comment: 16 pages, corrected typos, a global bound that was obtained for the unforced equation by Kiselev-Nazarov-Volberg obtained for the forced equation and utilized in the paper

How Gibbs distributions may naturally arise from synaptic adaptation mechanisms. A model-based argumentation

This paper addresses two questions in the context of neuronal networks dynamics, using methods from dynamical systems theory and statistical physics: (i) How to characterize the statistical properties of sequences of action potentials ("spike trains") produced by neuronal networks ? and; (ii) what are the effects of synaptic plasticity on these statistics ? We introduce a framework in which spike trains are associated to a coding of membrane potential trajectories, and actually, constitute a symbolic coding in important explicit examples (the so-called gIF models). On this basis, we use the thermodynamic formalism from ergodic theory to show how Gibbs distributions are natural probability measures to describe the statistics of spike trains, given the empirical averages of prescribed quantities. As a second result, we show that Gibbs distributions naturally arise when considering "slow" synaptic plasticity rules where the characteristic time for synapse adaptation is quite longer than the characteristic time for neurons dynamics.Comment: 39 pages, 3 figure

Prescribing the Jacobian in critical spaces

International audienceWe consider the Sobolev space $X=W^{s,p}({\mathbb S}^m ; {\mathbb S}^{k-1})$. We prove the existence of a robust distributional Jacobian $Ju$ for $u\in X$ provided $sp\ge k-1$. This generalizes a result of Bourgain, Brezis and the second author (Comm. Pure Appl. Math. 2005), where the case $m=k$ is considered. In the critical case where $sp=k-1$, we identify the image of the map $X\ni u\mapsto Ju$. This extends a result of Alberti, Baldo and Orlandi (J. Eur. Math. Soc. 2003) for $s=1$ and $p=k-1$. We also present a new, analytical, dipole construction method

Inhomogeneous Vortex Patterns in Rotating Bose-Einstein Condensates

We consider a 2D rotating Bose gas described by the Gross-Pitaevskii (GP) theory and investigate the properties of the ground state of the theory for rotational speeds close to the critical speed for vortex nucleation. While one could expect that the vortex distribution should be homogeneous within the condensate we prove by means of an asymptotic analysis in the strongly interacting (Thomas-Fermi) regime that it is not. More precisely we rigorously derive a formula due to Sheehy and Radzihovsky [Phys. Rev. A 70, 063620(R) (2004)] for the vortex distribution, a consequence of which is that the vortex distribution is strongly inhomogeneous close to the critical speed and gradually homogenizes when the rotation speed is increased. From the mathematical point of view, a novelty of our approach is that we do not use any compactness argument in the proof, but instead provide explicit estimates on the difference between the vorticity measure of the GP ground state and the minimizer of a certain renormalized energy functional.Comment: 41 pages, journal ref: Communications in Mathematical Physics: Volume 321, Issue 3 (2013), Page 817-860, DOI : 10.1007/s00220-013-1697-

A discrete time neural network model with spiking neurons II. Dynamics with noise

We provide rigorous and exact results characterizing the statistics of spike trains in a network of leaky integrate and fire neurons, where time is discrete and where neurons are submitted to noise, without restriction on the synaptic weights. We show the existence and uniqueness of an invariant measure of Gibbs type and discuss its properties. We also discuss Markovian approximations and relate them to the approaches currently used in computational neuroscience to analyse experimental spike trains statistics.Comment: 43 pages - revised version - to appear il Journal of Mathematical Biolog

Neuroarchitecture of Aminergic Systems in the Larval Ventral Ganglion of Drosophila melanogaster

Biogenic amines are important signaling molecules in the central nervous system of both vertebrates and invertebrates. In the fruit fly Drosophila melanogaster, biogenic amines take part in the regulation of various vital physiological processes such as feeding, learning/memory, locomotion, sexual behavior, and sleep/arousal. Consequently, several morphological studies have analyzed the distribution of aminergic neurons in the CNS. Previous descriptions, however, did not determine the exact spatial location of aminergic neurite arborizations within the neuropil. The release sites and pre-/postsynaptic compartments of aminergic neurons also remained largely unidentified. We here used gal4-driven marker gene expression and immunocytochemistry to map presumed serotonergic (5-HT), dopaminergic, and tyraminergic/octopaminergic neurons in the thoracic and abdominal neuromeres of the Drosophila larval ventral ganglion relying on Fasciclin2-immunoreactive tracts as three-dimensional landmarks. With tyrosine hydroxylase- (TH) or tyrosine decarboxylase 2 (TDC2)-specific gal4-drivers, we also analyzed the distribution of ectopically expressed neuronal compartment markers in presumptive dopaminergic TH and tyraminergic/octopaminergic TDC2 neurons, respectively. Our results suggest that thoracic and abdominal 5-HT and TH neurons are exclusively interneurons whereas most TDC2 neurons are efferent. 5-HT and TH neurons are ideally positioned to integrate sensory information and to modulate neuronal transmission within the ventral ganglion, while most TDC2 neurons appear to act peripherally. In contrast to 5-HT neurons, TH and TDC2 neurons each comprise morphologically different neuron subsets with separated in- and output compartments in specific neuropil regions. The three-dimensional mapping of aminergic neurons now facilitates the identification of neuronal network contacts and co-localized signaling molecules, as exemplified for DOPA decarboxylase-synthesizing neurons that co-express crustacean cardioactive peptide and myoinhibiting peptides

Tempo and Mode in Evolution of Transcriptional Regulation

Perennial questions of evolutionary biology can be applied to gene regulatory systems using the abundance of experimental data addressing gene regulation in a comparative context. What is the tempo (frequency, rate) and mode (way, mechanism) of transcriptional regulatory evolution? Here we synthesize the results of 230 experiments performed on insects and nematodes in which regulatory DNA from one species was used to drive gene expression in another species. General principles of regulatory evolution emerge. Gene regulatory evolution is widespread and accumulates with genetic divergence in both insects and nematodes. Divergence in cis is more common than divergence in trans. Coevolution between cis and trans shows a particular increase over greater evolutionary timespans, especially in sex-specific gene regulation. Despite these generalities, the evolution of gene regulation is gene- and taxon-specific. The congruence of these conclusions with evidence from other types of experiments suggests that general principles are discoverable, and a unified view of the tempo and mode of regulatory evolution may be achievable