Critical fluid limit of a gated processor sharing queue

We consider a sequence of single-server queueing models operating under a service policy that incorporates batches into processor sharing: arriving jobs build up behind a gate while waiting to begin service, while jobs in front of the gate are served according to processor sharing. When they have been completed, the waiting jobs move in front of the gate and the cycle repeats. We model this system with a pair of measure valued processes describing the jobs in front of and behind the gate. Under mild asymptotically critical conditions and a law-of-large-numbers scaling, we prove that the pair of measure-valued processes converges in distribution to an easily described limit, which has an interesting periodic dynamics

Clifford algebras and new singular Riemannian foliations in spheres

Using representations of Clifford algebras we construct indecomposable singular Riemannian foliations on round spheres, most of which are non-homogeneous. This generalizes the construction of non-homogeneous isoparametric hypersurfaces due to by Ferus, Karcher and Munzner.Comment: 21 pages. Construction of foliations in the Cayley plane added. Proofs simplified and presentation improved, according to referee's suggestions. To appear in Geom. Funct. Ana

Pseudo-Riemannian geodesic foliations by circles

We investigate under which assumptions an orientable pseudo-Riemannian geodesic foliations by circles is generated by an $S^1$-action. We construct examples showing that, contrary to the Riemannian case, it is not always true. However, we prove that such an action always exists when the foliation does not contain lightlike leaves, i.e. a pseudo-Riemannian Wadsley's Theorem. As an application, we show that every Lorentzian surface all of whose spacelike/timelike geodesics are closed, is finitely covered by $S^1\times \R$. It follows that every Lorentzian surface contains a non-closed geodesic.Comment: 14 page

Fluid Models of Many-server Queues with Abandonment

We study many-server queues with abandonment in which customers have general service and patience time distributions. The dynamics of the system are modeled using measure- valued processes, to keep track of the residual service and patience times of each customer. Deterministic fluid models are established to provide first-order approximation for this model. The fluid model solution, which is proved to uniquely exists, serves as the fluid limit of the many-server queue, as the number of servers becomes large. Based on the fluid model solution, first-order approximations for various performance quantities are proposed

A microRNA cluster in the Fragile-X region expressed during spermatogenesis targets FMR1.

Testis-expressed X-linked genes typically evolve rapidly. Here, we report on a testis-expressed X-linked microRNA (miRNA) cluster that despite rapid alterations in sequence has retained its position in the Fragile-X region of the X chromosome in placental mammals. Surprisingly, the miRNAs encoded by this cluster (Fx-mir) have a predilection for targeting the immediately adjacent gene, Fmr1, an unexpected finding given that miRNAs usually act in trans, not in cis Robust repression of Fmr1 is conferred by combinations of Fx-mir miRNAs induced in Sertoli cells (SCs) during postnatal development when they terminate proliferation. Physiological significance is suggested by the finding that FMRP, the protein product of Fmr1, is downregulated when Fx-mir miRNAs are induced, and that FMRP loss causes SC hyperproliferation and spermatogenic defects. Fx-mir miRNAs not only regulate the expression of FMRP, but also regulate the expression of eIF4E and CYFIP1, which together with FMRP form a translational regulatory complex. Our results support a model in which Fx-mir family members act cooperatively to regulate the translation of batteries of mRNAs in a developmentally regulated manner in SCs

Sub-Riemannian geodesics on nested principal bundles

We study the interplay between geodesics on two non-holono\-mic systems that are related by the action of a Lie group on them. After some geometric preliminaries, we use the Hamiltonian formalism to write the parametric form of geodesics. We present several geometric examples, including a non-holonomic structure on the Gromoll-Meyer exotic sphere and twistor space.Comment: 10 page

Exotic Spaces in Quantum Gravity I: Euclidean Quantum Gravity in Seven Dimensions

It is well known that in four or more dimensions, there exist exotic manifolds; manifolds that are homeomorphic but not diffeomorphic to each other. More precisely, exotic manifolds are the same topological manifold but have inequivalent differentiable structures. This situation is in contrast to the uniqueness of the differentiable structure on topological manifolds in one, two and three dimensions. As exotic manifolds are not diffeomorphic, one can argue that quantum amplitudes for gravity formulated as functional integrals should include a sum over not only physically distinct geometries and topologies but also inequivalent differentiable structures. But can the inclusion of exotic manifolds in such sums make a significant contribution to these quantum amplitudes? This paper will demonstrate that it will. Simply connected exotic Einstein manifolds with positive curvature exist in seven dimensions. Their metrics are found numerically; they are shown to have volumes of the same order of magnitude. Their contribution to the semiclassical evaluation of the partition function for Euclidean quantum gravity in seven dimensions is evaluated and found to be nontrivial. Consequently, inequivalent differentiable structures should be included in the formulation of sums over histories for quantum gravity.Comment: AmsTex, 23 pages 5 eps figures; replaced figures with ones which are hopefully viewable in pdf forma

Convergence of vector bundles with metrics of Sasaki-type

If a sequence of Riemannian manifolds, $X_i$, converges in the pointed Gromov-Hausdorff sense to a limit space, $X_\infty$, and if $E_i$ are vector bundles over $X_i$ endowed with metrics of Sasaki-type with a uniform upper bound on rank, then a subsequence of the $E_i$ converges in the pointed Gromov-Hausdorff sense to a metric space, $E_\infty$. The projection maps $\pi_i$ converge to a limit submetry $\pi_\infty$ and the fibers converge to its fibers; the latter may no longer be vector spaces but are homeomorphic to $\R^k/G$, where $G$ is a closed subgroup of $O(k)$ ---called the {\em wane group}--- that depends on the basepoint and that is defined using the holonomy groups on the vector bundles. The norms $\mu_i=\|\cdot\|_i$ converges to a map $\mu_{\infty}$ compatible with the re-scaling in $\R^k/G$ and the $\R$-action on $E_i$ converges to an $\R-$action on $E_{\infty}$ compatible with the limiting norm. In the special case when the sequence of vector bundles has a uniform lower bound on holonomy radius (as in a sequence of collapsing flat tori to a circle), the limit fibers are vector spaces. Under the opposite extreme, e.g. when a single compact $n$-dimensional manifold is re-scaled to a point, the limit fiber is $\R^n/H$ where $H$ is the closure of the holonomy group of the compact manifold considered. An appropriate notion of parallelism is given to the limiting spaces by considering curves whose length is unchanged under the projection. The class of such curves is invariant under the $\R$-action and each such curve preserves norms. The existence of parallel translation along rectifiable curves with arbitrary initial conditions is also exhibited. Uniqueness is not true in general, but a necessary condition is given in terms of the aforementioned wane groups $G$.Comment: 44 pages, 1 figure, in V.2 added Theorem E and Section 4 on parallelism in the limit space

Exotic Differentiable Structures and General Relativity

We review recent developments in differential topology with special concern for their possible significance to physical theories, especially general relativity. In particular we are concerned here with the discovery of the existence of non-standard (``fake'' or ``exotic'') differentiable structures on topologically simple manifolds such as $S^7$, \R and $S^3\times {\bf R^1}.$ Because of the technical difficulties involved in the smooth case, we begin with an easily understood toy example looking at the role which the choice of complex structures plays in the formulation of two-dimensional vacuum electrostatics. We then briefly review the mathematical formalisms involved with differentiable structures on topological manifolds, diffeomorphisms and their significance for physics. We summarize the important work of Milnor, Freedman, Donaldson, and others in developing exotic differentiable structures on well known topological manifolds. Finally, we discuss some of the geometric implications of these results and propose some conjectures on possible physical implications of these new manifolds which have never before been considered as physical models.Comment: 11 pages, LaTe

A Markovian event-based framework for stochastic spiking neural networks

In spiking neural networks, the information is conveyed by the spike times, that depend on the intrinsic dynamics of each neuron, the input they receive and on the connections between neurons. In this article we study the Markovian nature of the sequence of spike times in stochastic neural networks, and in particular the ability to deduce from a spike train the next spike time, and therefore produce a description of the network activity only based on the spike times regardless of the membrane potential process. To study this question in a rigorous manner, we introduce and study an event-based description of networks of noisy integrate-and-fire neurons, i.e. that is based on the computation of the spike times. We show that the firing times of the neurons in the networks constitute a Markov chain, whose transition probability is related to the probability distribution of the interspike interval of the neurons in the network. In the cases where the Markovian model can be developed, the transition probability is explicitly derived in such classical cases of neural networks as the linear integrate-and-fire neuron models with excitatory and inhibitory interactions, for different types of synapses, possibly featuring noisy synaptic integration, transmission delays and absolute and relative refractory period. This covers most of the cases that have been investigated in the event-based description of spiking deterministic neural networks