38,572 research outputs found
A strong operator topology adiabatic theorem
We prove an adiabatic theorem for the evolution of spectral data under a weak
additive perturbation in the context of a system without an intrinsic time
scale. For continuous functions of the unperturbed Hamiltonian the convergence
is in norm while for a larger class functions, including the spectral
projections associated to embedded eigenvalues, the convergence is in the
strong operator topology.Comment: 15 pages, no figure
A randomized and fully discrete Galerkin finite element method for semilinear stochastic evolution equations
In this paper the numerical solution of non-autonomous semilinear stochastic
evolution equations driven by an additive Wiener noise is investigated. We
introduce a novel fully discrete numerical approximation that combines a
standard Galerkin finite element method with a randomized Runge-Kutta scheme.
Convergence of the method to the mild solution is proven with respect to the
-norm, . We obtain the same temporal order of
convergence as for Milstein-Galerkin finite element methods but without
imposing any differentiability condition on the nonlinearity. The results are
extended to also incorporate a spectral approximation of the driving Wiener
process. An application to a stochastic partial differential equation is
discussed and illustrated through a numerical experiment.Comment: 31 pages, 1 figur
A Markov growth process for Macdonald's distribution on reduced words
We give an algorithmic-bijective proof of Macdonald's reduced word identity
in the theory of Schubert polynomials, in the special case where the
permutation is dominant. Our bijection uses a novel application of David
Little's generalized bumping algorithm. We also describe a Markov growth
process for an associated probability distribution on reduced words. Our growth
process can be implemented efficiently on a computer and allows for fast
sampling of reduced words. We also discuss various partial generalizations and
links to Little's work on the RSK algorithm.Comment: 16 pages, 5 figure
Scaling asymptotics for quantized Hamiltonian flows
In recent years, the near diagonal asymptotics of the equivariant components
of the Szeg\"{o} kernel of a positive line bundle on a compact symplectic
manifold have been studied extensively by many authors. As a natural
generalization of this theme, here we consider the local scaling asymptotics of
the Toeplitz quantization of a Hamiltonian symplectomorphism, and specifically
how they concentrate on the graph of the underlying classical map
Delayed loss of stability in singularly perturbed finite-dimensional gradient flows
In this paper we study the singular vanishing-viscosity limit of a gradient
flow in a finite dimensional Hilbert space, focusing on the so-called delayed
loss of stability of stationary solutions. We find a class of time-dependent
energy functionals and initial conditions for which we can explicitly calculate
the first discontinuity time of the limit. For our class of functionals,
coincides with the blow-up time of the solutions of the linearized system
around the equilibrium, and is in particular strictly greater than the time
where strict local minimality with respect to the driving energy gets
lost. Moreover, we show that, in a right neighborhood of , rescaled
solutions of the singularly perturbed problem converge to heteroclinic
solutions of the gradient flow. Our results complement the previous ones by
Zanini, where the situation we consider was excluded by assuming the so-called
transversality conditions, and the limit evolution consisted of strict local
minimizers of the energy up to a negligible set of times
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