Given a set X of sequences over a finite alphabet, we investigate the following three quantities. (i) The feasible predictability of X is the highest success ratio that a polynomialtime randomized predictor can achieve on all sequences in X. (ii) The deterministic feasible predictability of X is the highest success ratio that a polynomial-time deterministic predictor can achieve on all sequence...

We address the problem of distinguishing regular, chaotic, and random behavior using an electronic circuit modeling of a ball bouncing on an oscillating table. We calculate the correlation dimension of the system from time series data taken from the circuit, and show that this system seems amenable to correlation dimension analysis. In particular we find dimensions of 1.07 for regular data and ...

We present a dimension analysis of a set of solar type I storms and type IV events with different kind of fine structures, recorded at the Trieste Astronomical Observatory. The signature of such types of solar radio events is highly structured in time. However, periodicities are rather seldom, and linear mode theory can provide only limited interpretation of the data. Therefore, we performed an...

We study algorithmic problems on subsets of Euclidean space of low fractal dimension. These spaces are the subject of intensive study in various branches of mathematics, including geometry, topology, and measure theory. There are several well-studied notions of fractal dimension for sets and measures in Euclidean space. We consider a definition of fractal dimension for finite metric spaces whic...

Learning joint probability distributions on n random variables requires exponential sample size in the generic case. Here we consider the case that a temporal (or causal) order of the variables is known and that the (unknown) graph of causal dependencies has bounded in-degree ∆. Then the joint measure is uniquely determined by the probabilities of all (2∆ + 1)-tuples. Upper bounds on the sample...

The Shattering dimension of a class is a real-valued version of the Vapnik-Chervonenkis dimension. We will present a solution to Talagrand’s entropy problem, showing that the L2-covering numbers of every uniformly bounded class of functions are exponential in the shattering dimension of the class. Formally we prove that there are absolute constants K and c such that for every 0 < t ≤ 1 and any ...

We study geometric hypergrahs in a kinetic setting. That is, the set of vertices of the hypergraph is a set of moving points in R with coordinates that are polynomials in time. The hyperedges are all subsets that can be realized by intersecting the set of points at some fixed time with some “simple” geometric shape, such as, say, a halfspace. We show that for many of the static cases where the ...

introduction: water and soil retention curve is one of the most important properties of porous media to obtain in a laboratory retention curve and time associated with errors. for this reason, researchers have proposed techniques that help them to more easily acquired characteristic curve. one of these methods is the use of fractal geometry. determining the relationship between particle size di...