503 research outputs found
On the stability of critical chemotactic aggregation
We consider the two dimensional parabolic-elliptic Patlak-Keller-Segel model
of chemotactic aggregation for radially symmetric initial data. We show the
existence of a stable mechanism of singularity formation and obtain a complete
description of the associated aggregation process.Comment: 80 page
Well-posedness and convergence of the Lindblad master equation for a quantum harmonic oscillator with multi-photon drive and damping
We consider the model of a quantum harmonic oscillator governed by a Lindblad
master equation where the typical drive and loss channels are multi-photon
processes instead of single-photon ones; this implies a dissipation operator of
order 2k with integer k>1 for a k-photon process. We prove that the
corresponding PDE makes the state converge, for large time, to an invariant
subspace spanned by a set of k selected basis vectors; the latter physically
correspond to so-called coherent states with the same amplitude and uniformly
distributed phases. We also show that this convergence features a finite set of
bounded invariant functionals of the state (physical observables), such that
the final state in the invariant subspace can be directly predicted from the
initial state. The proof includes the full arguments towards the well-posedness
of the corresponding dynamics in proper Banach spaces of Hermitian trace-class
operators equipped with adapted nuclear norms. It relies on the Hille-Yosida
theorem and Lyapunov convergence analysis.Comment: 20 pages, submitte
Adiabatic elimination for open quantum systems with effective Lindblad master equations
We consider an open quantum system described by a Lindblad-type master
equation with two times-scales. The fast time-scale is strongly dissipative and
drives the system towards a low-dimensional decoherence-free space. To perform
the adiabatic elimination of this fast relaxation, we propose a geometric
asymptotic expansion based on the small positive parameter describing the
time-scale separation. This expansion exploits geometric singular perturbation
theory and center-manifold techniques. We conjecture that, at any order, it
provides an effective slow Lindblad master equation and a completely positive
parameterization of the slow invariant sub-manifold associated to the
low-dimensional decoherence-free space. By preserving complete positivity and
trace, two important structural properties attached to open quantum dynamics,
we obtain a reduced-order model that directly conveys a physical interpretation
since it relies on effective Lindbladian descriptions of the slow evolution. At
the first order, we derive simple formulae for the effective Lindblad master
equation. For a specific type of fast dissipation, we show how any Hamiltonian
perturbation yields Lindbladian second-order corrections to the first-order
slow evolution governed by the Zeno-Hamiltonian. These results are illustrated
on a composite system made of a strongly dissipative harmonic oscillator, the
ancilla, weakly coupled to another quantum system.Comment: 9 pages, one figur
Accelerated Spectral Clustering Using Graph Filtering Of Random Signals
We build upon recent advances in graph signal processing to propose a faster
spectral clustering algorithm. Indeed, classical spectral clustering is based
on the computation of the first k eigenvectors of the similarity matrix'
Laplacian, whose computation cost, even for sparse matrices, becomes
prohibitive for large datasets. We show that we can estimate the spectral
clustering distance matrix without computing these eigenvectors: by graph
filtering random signals. Also, we take advantage of the stochasticity of these
random vectors to estimate the number of clusters k. We compare our method to
classical spectral clustering on synthetic data, and show that it reaches equal
performance while being faster by a factor at least two for large datasets
Compressive Spectral Clustering
Spectral clustering has become a popular technique due to its high
performance in many contexts. It comprises three main steps: create a
similarity graph between N objects to cluster, compute the first k eigenvectors
of its Laplacian matrix to define a feature vector for each object, and run
k-means on these features to separate objects into k classes. Each of these
three steps becomes computationally intensive for large N and/or k. We propose
to speed up the last two steps based on recent results in the emerging field of
graph signal processing: graph filtering of random signals, and random sampling
of bandlimited graph signals. We prove that our method, with a gain in
computation time that can reach several orders of magnitude, is in fact an
approximation of spectral clustering, for which we are able to control the
error. We test the performance of our method on artificial and real-world
network data.Comment: 12 pages, 2 figure
Quantitative Robustness Analysis of Flat Timed Automata
Whereas formal verification of timed systems has become a very active field of research, the idealized mathematical semantics of timed automata cannot be faithfully implemented. Recently, several works have studied a parametric semantics of timed automata related to implementability: if the specification is met for some positive value of the parameter, then there exists a correct implementation. In addition, the value of the parameter gives lower bounds on sufficient resources for the implementation. In this work, we present a symbolic algorithm for the computation of the parametric reachability set under this semantics for flat timed automata. As a consequence, we can compute the largest value of the parameter for a timed automaton to be safe
The Authorization Policy Existence Problem
International audienceConstraints such as separation-of-duty are widely used to specify requirements that supplement basic authorization policies. However, the existence of constraints (and authorization policies) may mean that a user is unable to fulïŹll her/his organizational duties because access to resources is denied. In short, there is a tension between the need to protect resources (using policies and constraints) and the availability of resources. Recent work on workïŹow satisïŹability and resiliency in access control asks whether this tension compromises the ability of an organization to achieve its objectives. In this paper, we develop a new method of specifying constraints which subsumes much related work and allows a wider range of constraints to be speciïŹed. The use of such constraints leads naturally to a range of questions related toâpolicy existenceâ, where a positive answer means that an organizationâs objectives can be realized. We provide an overview of our results establishing that some policy existence questions, notably for those instances that are restricted to user-independent constraints, are ïŹxed-parameter tractable
Fluid-Structure Energy Transfer of a Tensioned Beam Subject to Vortex-Induced Vibrations in Shear Flow
The fluid-structure energy transfer of a tensioned beam of length to diameter ratio 200, subject to vortex-induced vibrations in linear shear flow, is investigated by means of direct numerical simulation at three Reynolds numbers, from 110 to 1,100. In both the in-line and cross-flow directions, the high-wavenumber structural responses are characterized by mixed standing-traveling wave patterns. The spanwise zones where the flow provides energy to excite the structural vibrations are located mainly within the region of high current where the lock-in condition is established, i.e. where vortex shedding and cross-flow vibration frequencies coincide. However, the energy input is not uniform across the entire lockin region. This can be related to observed changes from counterclockwise to clockwise structural orbits. The energy transfer is also impacted by the possible occurrence of multi-frequency vibrations. Topics: Energy transformation, Fluids, Shear flow, Vortex-induced vibrationBP America Production CompanyBP-MIT Major Projects Progra
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