20,975 research outputs found
A Kam Theorem for Space-Multidimensional Hamiltonian PDE
We present an abstract KAM theorem, adapted to space-multidimensional
hamiltonian PDEs with smoothing non-linearities. The main novelties of this
theorem are that: the integrable part of the hamiltonian may contain
a hyperbolic part and as a consequence the constructed invariant tori may be
unstable. It applies to singular perturbation problem. In this paper
we state the KAM-theorem and comment on it, give the main ingredients of the
proof, and present three applications of the theorem .Comment: arXiv admin note: text overlap with arXiv:1502.0226
Convergence and adiabatic elimination for a driven dissipative quantum harmonic oscillator
We prove that a harmonic oscillator driven by Lindblad dynamics where the
typical drive and loss channels are two-photon processes instead of
single-photon ones, converges to a protected subspace spanned by two coherent
states of opposite amplitude. We then characterize the slow dynamics induced by
a perturbative single-photon loss on this protected subspace, by performing
adiabatic elimination in the Lindbladian dynamics.Comment: submitted to IEEE-CDC 201
On average control generating families for singularly perturbed optimal control problems with long run average optimality criteria
The paper aims at the development of tools for analysis and construction of
near optimal solutions of singularly perturbed (SP) optimal controls problems
with long run average optimality criteria. The idea that we exploit is to first
asymptotically approximate a given problem of optimal control of the SP system
by a certain averaged optimal control problem, then reformulate this averaged
problem as an infinite-dimensional (ID) linear programming (LP) problem, and
then approximate the latter by semi-infinite LP problems. We show that the
optimal solution of these semi-infinite LP problems and their duals (that can
be found with the help of a modification of an available LP software) allow one
to construct near optimal controls of the SP system. We demonstrate the
construction with a numerical example.Comment: 36 pages, 4 figures. arXiv admin note: substantial text overlap with
arXiv:1309.373
A Kernel Perspective for Regularizing Deep Neural Networks
We propose a new point of view for regularizing deep neural networks by using
the norm of a reproducing kernel Hilbert space (RKHS). Even though this norm
cannot be computed, it admits upper and lower approximations leading to various
practical strategies. Specifically, this perspective (i) provides a common
umbrella for many existing regularization principles, including spectral norm
and gradient penalties, or adversarial training, (ii) leads to new effective
regularization penalties, and (iii) suggests hybrid strategies combining lower
and upper bounds to get better approximations of the RKHS norm. We
experimentally show this approach to be effective when learning on small
datasets, or to obtain adversarially robust models.Comment: ICM
Robust â„‹2 Performance: Guaranteeing Margins for LQG Regulators
This paper shows that ℋ2 (LQG) performance specifications can be combined with structured uncertainty in the system, yielding robustness analysis conditions of the same nature and computational complexity as the corresponding conditions for ℋ∞ performance. These conditions are convex feasibility tests in terms of Linear Matrix Inequalities, and can be proven to be necessary and sufficient under the same conditions as in the ℋ∞ case.
With these results, the tools of robust control can be viewed as coming full circle to treat the problem where it all began: guaranteeing margins for LQG regulators
Measure, Topology and Probabilistic Reasoning in Cosmology
I explain the difficulty of making various concepts of and relating to
probability precise, rigorous and physically significant when attempting to
apply them in reasoning about objects (e.g., spacetimes) living in
infinite-dimensional spaces, working through many examples from cosmology. I
focus on the relation of topological to measure-theoretic notions of and
relating to probability, how they diverge in unpleasant ways in the
infinite-dimensional case, and are difficult to work with on their own as well
in that context. Even in cases where an appropriate family of spacetimes is
finite-dimensional, however, and so admits a measure of the relevant sort, it
is always the case that the family is not a compact topological space, and so
does not admit a physically significant, well behaved probability measure.
Problems of a different but still deeply troubling sort plague arguments about
likelihood in that context, which I also discuss. I conclude that most standard
forms of argument used in cosmology to estimate the likelihood of the
occurrence of various properties or behaviors of spacetimes have serious
mathematical, physical and conceptual problems.Comment: 26 page
Singularity theory study of overdetermination in models for L-H transitions
Two dynamical models that have been proposed to describe transitions between
low and high confinement states (L-H transitions) in confined plasmas are
analysed using singularity theory and stability theory. It is shown that the
stationary-state bifurcation sets have qualitative properties identical to
standard normal forms for the pitchfork and transcritical bifurcations. The
analysis yields the codimension of the highest-order singularities, from which
we find that the unperturbed systems are overdetermined bifurcation problems
and derive appropriate universal unfoldings. Questions of mutual equivalence
and the character of the state transitions are addressed.Comment: Latex (Revtex) source + 13 small postscript figures. Revised versio
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