11,218 research outputs found
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
Local and global geometry of Prony systems and Fourier reconstruction of piecewise-smooth functions
Many reconstruction problems in signal processing require solution of a
certain kind of nonlinear systems of algebraic equations, which we call Prony
systems. We study these systems from a general perspective, addressing
questions of global solvability and stable inversion. Of special interest are
the so-called "near-singular" situations, such as a collision of two closely
spaced nodes.
We also discuss the problem of reconstructing piecewise-smooth functions from
their Fourier coefficients, which is easily reduced by a well-known method of
K.Eckhoff to solving a particular Prony system. As we show in the paper, it
turns out that a modification of this highly nonlinear method can reconstruct
the jump locations and magnitudes of such functions, as well as the pointwise
values between the jumps, with the maximal possible accuracy.Comment: arXiv admin note: text overlap with arXiv:1211.068
Complete Solutions for a Combinatorial Puzzle in Linear Time
In this paper we study a single player game consisting of black checkers
and white checkers, called shifting the checkers. We have proved that the
minimum number of steps needed to play the game for general and is . We have also presented an optimal algorithm to generate an optimal
move sequence of the game consisting of black checkers and white
checkers, and finally, we present an explicit solution for the general game
Joint Image Reconstruction and Segmentation Using the Potts Model
We propose a new algorithmic approach to the non-smooth and non-convex Potts
problem (also called piecewise-constant Mumford-Shah problem) for inverse
imaging problems. We derive a suitable splitting into specific subproblems that
can all be solved efficiently. Our method does not require a priori knowledge
on the gray levels nor on the number of segments of the reconstruction.
Further, it avoids anisotropic artifacts such as geometric staircasing. We
demonstrate the suitability of our method for joint image reconstruction and
segmentation. We focus on Radon data, where we in particular consider limited
data situations. For instance, our method is able to recover all segments of
the Shepp-Logan phantom from angular views only. We illustrate the
practical applicability on a real PET dataset. As further applications, we
consider spherical Radon data as well as blurred data
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