3,770 research outputs found
Classical and Quantum Parts of the Quantum Dynamics: the Discrete-Time Case
In the study of open quantum systems modeled by a unitary evolution of a
bipartite Hilbert space, we address the question of which parts of the
environment can be said to have a "classical action" on the system, in the
sense of acting as a classical stochastic process. Our method relies on the
definition of the Environment Algebra, a relevant von Neumann algebra of the
environment. With this algebra we define the classical parts of the environment
and prove a decomposition between a maximal classical part and a quantum part.
Then we investigate what other information can be obtained via this algebra,
which leads us to define a more pertinent algebra: the Environment Action
Algebra. This second algebra is linked to the minimal Stinespring
representations induced by the unitary evolution on the system. Finally in
finite dimension we give a characterization of both algebras in terms of the
spectrum of a certain completely positive map acting on the states of the
environment
Matrix-F5 algorithms and tropical Gr\"obner bases computation
Let be a field equipped with a valuation. Tropical varieties over can
be defined with a theory of Gr\"obner bases taking into account the valuation
of . Because of the use of the valuation, this theory is promising for
stable computations over polynomial rings over a -adic fields.We design a
strategy to compute such tropical Gr\"obner bases by adapting the Matrix-F5
algorithm. Two variants of the Matrix-F5 algorithm, depending on how the
Macaulay matrices are built, are available to tropical computation with
respective modifications. The former is more numerically stable while the
latter is faster.Our study is performed both over any exact field with
valuation and some inexact fields like or In the latter case, we track the loss in precision,
and show that the numerical stability can compare very favorably to the case of
classical Gr\"obner bases when the valuation is non-trivial. Numerical examples
are provided
Nonparametric estimation of the local Hurst function of multifractional Gaussian processes
A new nonparametric estimator of the local Hurst function of a
multifractional Gaussian process based on the increment ratio (IR) statistic is
defined. In a general frame, the point-wise and uniform weak and strong
consistency and a multidimensional central limit theorem for this estimator are
established. Similar results are obtained for a refinement of the generalized
quadratic variations (QV) estimator. The example of the multifractional
Brownian motion is studied in detail. A simulation study is included showing
that the IR-estimator is more accurate than the QV-estimator
A wavelet analysis of the Rosenblatt process: chaos expansion and estimation of the self-similarity parameter
By using chaos expansion into multiple stochastic integrals, we make a
wavelet analysis of two self-similar stochastic processes: the fractional
Brownian motion and the Rosenblatt process. We study the asymptotic behavior of
the statistic based on the wavelet coefficients of these processes. Basically,
when applied to a non-Gaussian process (such as the Rosenblatt process) this
statistic satisfies a non-central limit theorem even when we increase the
number of vanishing moments of the wavelet function. We apply our limit
theorems to construct estimators for the self-similarity index and we
illustrate our results by simulations
On the degree of the polynomial defining a planar algebraic curves of constant width
In this paper, we consider a family of closed planar algebraic curves
which are given in parametrization form via a trigonometric
polynomial . When is the boundary of a compact convex set, the
polynomial represents the support function of this set. Our aim is to
examine properties of the degree of the defining polynomial of this family of
curves in terms of the degree of . Thanks to the theory of elimination, we
compute the total degree and the partial degrees of this polynomial, and we
solve in addition a question raised by Rabinowitz in \cite{Rabi} on the lowest
degree polynomial whose graph is a non-circular curve of constant width.
Computations of partial degrees of the defining polynomial of algebraic
surfaces of constant width are also provided in the same way.Comment: 13 page
Moment bounds and central limit theorems for Gaussian subordinated arrays
A general moment bound for sums of products of Gaussian vector's functions
extending the moment bound in Taqqu (1977, Lemma 4.5) is established. A general
central limit theorem for triangular arrays of nonlinear functionals of
multidimensional non-stationary Gaussian sequences is proved. This theorem
extends the previous results of Breuer and Major (1981), Arcones (1994) and
others. A Berry-Esseen-type bound in the above-mentioned central limit theorem
is derived following Nourdin, Peccati and Podolskij (2011). Two applications of
the above results are discussed. The first one refers to the asymptotic
behavior of a roughness statistic for continuous-time Gaussian processes and
the second one is a central limit theorem satisfied by long memory locally
stationary process
Non-parametric estimation of time varying AR(1)--processes with local stationarity and periodicity
Extending the ideas of [7], this paper aims at providing a kernel based
non-parametric estimation of a new class of time varying AR(1) processes (Xt),
with local stationarity and periodic features (with a known period T), inducing
the definition Xt = at(t/nT)X t--1 + t for t N and with a t+T
at. Central limit theorems are established for kernel estima-tors
as(u) reaching classical minimax rates and only requiring low order moment
conditions of the white noise (t)t up to the second order
Detecting changes in the fluctuations of a Gaussian process and an application to heartbeat time series
The aim of this paper is first the detection of multiple abrupt changes of
the long-range dependence (respectively self-similarity, local fractality)
parameters from a sample of a Gaussian stationary times series (respectively
time series, continuous-time process having stationary increments). The
estimator of the change instants (the number is supposed to be known)
is proved to satisfied a limit theorem with an explicit convergence rate.
Moreover, a central limit theorem is established for an estimator of each
long-range dependence (respectively self-similarity, local fractality)
parameter. Finally, a goodness-of-fit test is also built in each time domain
without change and proved to asymptotically follow a Khi-square distribution.
Such statistics are applied to heart rate data of marathon's runners and lead
to interesting conclusions
- …