15,531 research outputs found
KdV-type equations linked via Baecklund transformations: remarks and perspectives
Third order nonlinear evolution equations, that is the Korteweg-deVries
(KdV), modified Korteweg-deVries (mKdV) equation and other ones are considered:
they all are connected via Baecklund transformations. These links can be
depicted in a wide Baecklund Chart} which further extends the previous one
constructed in [22]. In particular, the Baecklund transformation which links
the mKdV equation to the KdV singularity manifold equation is reconsidered and
the nonlinear equation for the KdV eigenfunction is shown to be linked to all
the equations in the previously constructed Baecklund Chart. That is, such a
Baecklund Chart is expanded to encompass the nonlinear equation for the KdV
eigenfunctions [30], which finds its origin in the early days of the study of
Inverse scattering Transform method, when the Lax pair for the KdV equation was
constructed. The nonlinear equation for the KdV eigenfunctions is proved to
enjoy a nontrivial invariance property. Furthermore, the hereditary recursion
operator it admits [30 is recovered via a different method. Then, the results
are extended to the whole hierarchy of nonlinear evolution equations it
generates. Notably, the established links allow to show that also the nonlinear
equation for the KdV eigenfunction is connected to the Dym equation since both
such equations appear in the same Baecklund chart.Comment: 18 page
Group invariance principles for causal generative models
The postulate of independence of cause and mechanism (ICM) has recently led
to several new causal discovery algorithms. The interpretation of independence
and the way it is utilized, however, varies across these methods. Our aim in
this paper is to propose a group theoretic framework for ICM to unify and
generalize these approaches. In our setting, the cause-mechanism relationship
is assessed by comparing it against a null hypothesis through the application
of random generic group transformations. We show that the group theoretic view
provides a very general tool to study the structure of data generating
mechanisms with direct applications to machine learning.Comment: 16 pages, 6 figure
Range unit root tests
Since the seminal paper by Dickey and Fuller in 1979, unit-root tests have conditioned the standard approaches to analyse time series with strong serial dependence, the focus being placed in the detection of eventual unit roots in an autorregresive model fitted to the series. In this paper we propose a completely different method to test for the type of "long-wave" patterns observed not only in unit root time series but also in series following more complex data generating mechanism. To this end, our testing device analyses the trend exhibit by the data, without imposing any constraint on the generating mechanism. We call our device the Range Unit Root (RUR) Test since it is constructed from running ranges of the series. These statistics allow a more general characterization of a strong serial dependence in the mean behavior, thus endowing our test with a number of desirable properties. Among these properties are the invariance to nonlinear monotonic transformations of the series and the robustness to the presence of level shifts and additive outliers. In addition, the RUR test outperforms the power of standard unit root tests on near-unit-root stationary time series
A range unit root test
Since the seminal paper by Dickey and Fuller in 1979, unit-root tests have conditioned the standard approaches to analyse time series with strong serial dependence, the focus being placed in the detection of eventual unit roots in an autorregresive model fitted to the series. In this paper we propose a completely different method to test for the type of long-wave patterns observed not only in unit root time series but also in series following more complex data generating mechanisms. To this end, our testing device analyses the trend exhibit by the data, without imposing any constraint on the generating mechanism. We call our device the Range Unit Root (RUR) Test since it is constructed from running ranges of the series. These statistics allow a more general characterization of a strong serial dependence in the mean behavior, thus endowing our test with a number of desirable properties, among which its error-model-free asymptotic distribution, the invariance to nonlinear monotonic transformations of the series and the robustness to the presence of level shifts and additive outliers. In addition, the RUR test outperforms the power of standard unit root tests on near-unit-root stationary time series and is asymptotically immune to noise
Uniqueness of Simultaneity
We consider the problem of uniqueness of certain simultaneity structures in
flat spacetime. Absolute simultaneity is specified to be a non-trivial
equivalence relation which is invariant under the automorphism group Aut of
spacetime. Aut is taken to be the identity-component of either the
inhomogeneous Galilei group or the inhomogeneous Lorentz group. Uniqueness of
standard simultaneity in the first, and absence of any absolute simultaneity in
the second case are demonstrated and related to certain group theoretic
properties. Relative simultaneity with respect to an additional structure X on
spacetime is specified to be a non-trivial equivalence relation which is
invariant under the subgroup in Aut that stabilises X. Uniqueness of standard
Einstein simultaneity is proven in the Lorentzian case when X is an inertial
frame. We end by discussing the relation to previous work of others.Comment: LeTeX-2e, 18 pages, no figure
Symmetry and History Quantum Theory: An analogue of Wigner's Theorem
The basic ingredients of the `consistent histories' approach to quantum
theory are a space \UP of `history propositions' and a space \D of
`decoherence functionals'. In this article we consider such history quantum
theories in the case where \UP is given by the set of projectors \P(\V) on
some Hilbert space \V. We define the notion of a `physical symmetry of a
history quantum theory' (PSHQT) and specify such objects exhaustively with the
aid of an analogue of Wigner's theorem. In order to prove this theorem we
investigate the structure of \D, define the notion of an `elementary
decoherence functional' and show that each decoherence functional can be
expanded as a certain combination of these functionals. We call two history
quantum theories that are related by a PSHQT `physically equivalent' and show
explicitly, in the case of history quantum mechanics, how this notion is
compatible with one that has appeared previously.Comment: To appear in Jour.Math.Phys.; 25 pages; Latex-documen
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