Splitting stationary sets from weak forms of Choice

Working in the context of restricted forms of the Axiom of Choice, we consider the problem of splitting the ordinals below $\lambda$ of cofinality $\theta$ into $\lambda$ many stationary sets, where $\theta < \lambda$ are regular cardinals. This is a continuation of \cite{Sh835}

Scaling of distributions of sums of positions for chaotic dynamics at band-splitting points

The stationary distributions of sums of positions of trajectories generated by the logistic map have been found to follow a basic renormalization group (RG) structure: a nontrivial fixed-point multi-scale distribution at the period-doubling onset of chaos and a Gaussian trivial fixed-point distribution for all chaotic attractors. Here we describe in detail the crossover distributions that can be generated at chaotic band-splitting points that mediate between the aforementioned fixed-point distributions. Self affinity in the chaotic region imprints scaling features to the crossover distributions along the sequence of band splitting points. The trajectories that give rise to these distributions are governed first by the sequential formation of phase-space gaps when, initially uniformly-distributed, sets of trajectories evolve towards the chaotic band attractors. Subsequently, the summation of positions of trajectories already within the chaotic bands closes those gaps. The possible shapes of the resultant distributions depend crucially on the disposal of sets of early positions in the sums and the stoppage of the number of terms retained in them

Stationary local random countable sets over the Wiener noise

The times of Brownian local minima, maxima and their union are three distinct examples of local, stationary, dense, random countable sets associated with classical Wiener noise. Being local means, roughly, determined by the local behavior of the sample paths of the Brownian motion, and stationary means invariant relative to the L\'evy shifts of the sample paths. We answer to the affirmative Tsirelson's question, whether or not there are any others, and develop some general theory for such sets. An extra ingredient to their structure, that of an honest indexation, leads to a splitting result that is akin to the Wiener-Hopf factorization of the Brownian motion at the minimum (or maximum) and has the latter as a special case. Sets admitting an honest indexation are moreover shown to have the property that no stopping time belongs to them with positive probability. They are also minimal: they do not have any non-empty proper local stationary subsets. Random sets, of the kind studied in this paper, honestly indexed or otherwise, give rise to nonclassical one-dimensional noises, generalizing the noise of splitting. Some properties of these noises and the inter-relations between them are investigated. In particular, subsets are connected to subnoises

Electron acceleration with improved Stochastic Differential Equation method: cutoff shape of electron distribution in test-particle limit

We develop a method of stochastic differential equation to simulate electron acceleration at astrophysical shocks. Our method is based on It\^{o}'s stochastic differential equations coupled with a particle splitting, employing a skew Brownian motion where an asymmetric shock crossing probability is considered. Using this code, we perform simulations of electron acceleration at stationary plane parallel shock with various parameter sets, and studied how the cutoff shape, which is characterized by cutoff shape parameter $a$, changes with the momentum dependence of the diffusion coefficient $\beta$. In the age-limited cases, we reproduce previous results of other authors, $a\approx2\beta$. In the cooling-limited cases, the analytical expectation $a\approx\beta+1$ is roughly reproduced although we recognize deviations to some extent. In the case of escape-limited acceleration, numerical result fits analytical stationary solution well, but deviates from the previous asymptotic analytical formula $a\approx\beta$.Comment: corrected typos, 10 pages, 4 figures, 2 tables, JHEAp in pres

Proximal Iteratively Reweighted Algorithm with Multiple Splitting for Nonconvex Sparsity Optimization

This paper proposes the Proximal Iteratively REweighted (PIRE) algorithm for solving a general problem, which involves a large body of nonconvex sparse and structured sparse related problems. Comparing with previous iterative solvers for nonconvex sparse problem, PIRE is much more general and efficient. The computational cost of PIRE in each iteration is usually as low as the state-of-the-art convex solvers. We further propose the PIRE algorithm with Parallel Splitting (PIRE-PS) and PIRE algorithm with Alternative Updating (PIRE-AU) to handle the multi-variable problems. In theory, we prove that our proposed methods converge and any limit solution is a stationary point. Extensive experiments on both synthesis and real data sets demonstrate that our methods achieve comparative learning performance, but are much more efficient, by comparing with previous nonconvex solvers