12,206 research outputs found
Moment inversion problem for piecewise D-finite functions
We consider the problem of exact reconstruction of univariate functions with
jump discontinuities at unknown positions from their moments. These functions
are assumed to satisfy an a priori unknown linear homogeneous differential
equation with polynomial coefficients on each continuity interval. Therefore,
they may be specified by a finite amount of information. This reconstruction
problem has practical importance in Signal Processing and other applications.
It is somewhat of a ``folklore'' that the sequence of the moments of such
``piecewise D-finite''functions satisfies a linear recurrence relation of
bounded order and degree. We derive this recurrence relation explicitly. It
turns out that the coefficients of the differential operator which annihilates
every piece of the function, as well as the locations of the discontinuities,
appear in this recurrence in a precisely controlled manner. This leads to the
formulation of a generic algorithm for reconstructing a piecewise D-finite
function from its moments. We investigate the conditions for solvability of the
resulting linear systems in the general case, as well as analyze a few
particular examples. We provide results of numerical simulations for several
types of signals, which test the sensitivity of the proposed algorithm to
noise
Two refreshing views of Fluctuation Theorems through Kinematics Elements and Exponential Martingale
In the context of Markov evolution, we present two original approaches to
obtain Generalized Fluctuation-Dissipation Theorems (GFDT), by using the
language of stochastic derivatives and by using a family of exponential
martingales functionals. We show that GFDT are perturbative versions of
relations verified by these exponential martingales. Along the way, we prove
GFDT and Fluctuation Relations (FR) for general Markov processes, beyond the
usual proof for diffusion and pure jump processes. Finally, we relate the FR to
a family of backward and forward exponential martingales.Comment: 41 pages, 7 figures; version2: 45 pages, 7 figures, minor revisions,
new results in Section
A bounded jump for the bounded Turing degrees
We define the bounded jump of A by A^b = {x | Exists i <= x [phi_i (x)
converges and Phi_x^[A|phi_i(x)](x) converges} and let A^[nb] denote the n-th
bounded jump. We demonstrate several properties of the bounded jump, including
that it is strictly increasing and order preserving on the bounded Turing (bT)
degrees (also known as the weak truth-table degrees). We show that the bounded
jump is related to the Ershov hierarchy. Indeed, for n > 1 we have X <=_[bT]
0^[nb] iff X is omega^n-c.e. iff X <=_1 0^[nb], extending the classical result
that X <=_[bT] 0' iff X is omega-c.e. Finally, we prove that the analogue of
Shoenfield inversion holds for the bounded jump on the bounded Turing degrees.
That is, for every X such that 0^b <=_[bT] X <=_[bT] 0^[2b], there is a Y
<=_[bT] 0^b such that Y^b =_[bT] X.Comment: 22 pages. Minor changes for publicatio
A Galois connection between Turing jumps and limits
Limit computable functions can be characterized by Turing jumps on the input
side or limits on the output side. As a monad of this pair of adjoint
operations we obtain a problem that characterizes the low functions and dually
to this another problem that characterizes the functions that are computable
relative to the halting problem. Correspondingly, these two classes are the
largest classes of functions that can be pre or post composed to limit
computable functions without leaving the class of limit computable functions.
We transfer these observations to the lattice of represented spaces where it
leads to a formal Galois connection. We also formulate a version of this result
for computable metric spaces. Limit computability and computability relative to
the halting problem are notions that coincide for points and sequences, but
even restricted to continuous functions the former class is strictly larger
than the latter. On computable metric spaces we can characterize the functions
that are computable relative to the halting problem as those functions that are
limit computable with a modulus of continuity that is computable relative to
the halting problem. As a consequence of this result we obtain, for instance,
that Lipschitz continuous functions that are limit computable are automatically
computable relative to the halting problem. We also discuss 1-generic points as
the canonical points of continuity of limit computable functions, and we prove
that restricted to these points limit computable functions are computable
relative to the halting problem. Finally, we demonstrate how these results can
be applied in computable analysis
Stability estimates for the fault inverse problem
We study in this paper stability estimates for the fault inverse problem. In
this problem, faults are assumed to be planar open surfaces in a half space
elastic medium with known Lam\'e coefficients. A traction free condition is
imposed on the boundary of the half space. Displacement fields present jumps
across faults, called slips, while traction derivatives are continuous. It was
proved in \cite{volkov2017reconstruction} that if the displacement field is
known on an open set on the boundary of the half space, then the fault and the
slip are uniquely determined. In this present paper, we study the stability of
this uniqueness result with regard to the coefficients of the equation of the
plane containing the fault. If the slip field is known we state and prove a
Lipschitz stability result. In the more interesting case where the slip field
is unknown, we state and prove another Lipschitz stability result under the
additional assumption, which is still physically relevant, that the slip field
is one directional
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