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Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach.

Francis Nier (2004)

Journées Équations aux dérivées partielles

We present here a simplified version of recent results obtained with B. Helffer and M. Klein. They are concerned with the exponentally small eigenvalues of the Witten Laplacian on 0 -forms. We show how the Witten complex structure is better taken into account by working with singular values. This provides a convenient framework to derive accurate approximations of the first eigenvalues of Δ f , h ( 0 ) and solves efficiently the question of weakly resonant wells.

Quantitative concentration inequalities on sample path space for mean field interaction

François Bolley (2010)

ESAIM: Probability and Statistics

We consider the approximation of a mean field stochastic process by a large interacting particle system. We derive non-asymptotic large deviation bounds measuring the concentration of the empirical measure of the paths of the particles around the law of the process. The method is based on a coupling argument, strong integrability estimates on the paths in Hölder norm, and a general concentration result for the empirical measure of identically distributed independent paths.

Quantitative stability for sumsets in n

Alessio Figalli, David Jerison (2015)

Journal of the European Mathematical Society

Given a measurable set A n of positive measure, it is not difficult to show that | A + A | = | 2 A | if and only if A is equal to its convex hull minus a set of measure zero. We investigate the stability of this statement: If ( | A + A | - | 2 A | ) / | A | is small, is A close to its convex hull? Our main result is an explicit control, in arbitrary dimension, on the measure of the difference between A and its convex hull in terms of ( | A + A | - | 2 A | ) / | A | .

Quantized distributed output regulation of multi-agent systems

Xiaoli Wang, Yumin Chen (2016)

Kybernetika

Motivated by digital communication channel, we consider the distributed output regulation problem for linear multi-agent systems with quantized state measurements. Quantizers take finitely many values and have an adjustable "zoom" parameter. Quantized distributed output regulation concerns designing distributed feedback by employing quantized technique for multi-agent systems such that all agents can track an active leader, and/or distributed disturbance rejection. With the solvability conditions...

Quantum diffusion and generalized Rényi dimensions of spectral measures

Jean-Marie Barbaroux, François Germinet, Serguei Tcheremchantsev (2000)

Journées équations aux dérivées partielles

We estimate the spreading of the solution of the Schrödinger equation asymptotically in time, in term of the fractal properties of the associated spectral measures. For this, we exhibit a lower bound for the moments of order p at time T for the state ψ defined by [ 1 T 0 T | X | p / 2 e - i t H ψ 2 d t ] . We show that this lower bound can be expressed in term of the generalized Rényi dimension of the spectral measure μ ψ associated to the hamiltonian H and the state ψ . We especially concentrate on continuous models.

Quantum Euler-Poisson systems: Existence of stationary states

Ansgar Jüngel, Hailiang Li (2004)

Archivum Mathematicum

A one-dimensional quantum Euler-Poisson system for semiconductors for the electron density and the electrostatic potential in bounded intervals is considered. The existence and uniqueness of strong solutions with positive electron density is shown for quite general (possibly non-convex or non-monotone) pressure-density functions under a “subsonic” condition, i.e. assuming sufficiently small current densities. The proof is based on a reformulation of the dispersive third-order equation for the electron...

Quantum graph spectra of a graphyne structure

Ngoc T. Do, Peter Kuchment (2013)

Nanoscale Systems: Mathematical Modeling, Theory and Applications

We study the dispersion relations and spectra of invariant Schrödinger operators on a graphyne structure (lithographite). In particular, description of different parts of the spectrum, band-gap structure, and Dirac points are provided.

Quantum optimal control using the adjoint method

Alfio Borzì (2012)

Nanoscale Systems: Mathematical Modeling, Theory and Applications

Control of quantum systems is central in a variety of present and perspective applications ranging from quantum optics and quantum chemistry to semiconductor nanostructures, including the emerging fields of quantum computation and quantum communication. In this paper, a review of recent developments in the field of optimal control of quantum systems is given with a focus on adjoint methods and their numerical implementation. In addition, the issues of exact controllability and optimal control are...

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