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Furstenberg's $\times 2\times 3$ theorem asserts that the double sequence $(2^{m}3^{n}\unicode[STIX]{x1D6FC})_{m,n\geq 1}$ is dense modulo one for every irrational $\unicode[STIX]{x1D6FC}$. The same holds with $2$ and $3$ replaced by any two multiplicatively independent integers. Here we obtain the same result for the sequences $((\begin{smallmatrix}m+n\\ d\end{smallmatrix})a^{m}b^{n}\unicode[STIX]{x1D6FC})_{m,n\geq 1}$ for any non-negative integer $d$ and irrational $\unicode[STIX]{x1D6FC}$ , and for the sequence $(P(m)a^{m}b^{n})_{m,n\geq 1}$ , where $P$ is any polynomial with at least one irrational coefficient. Similarly to Furstenberg's theorem, both results are obtained by considering appropriate dynamical systems. [ABSTRACT FROM AUTHOR]

We consider dynamical systems, consisting of $\mathbb{Z}^{2}$ -actions by continuous automorphisms on shift-invariant subgroups of $\mathbb{F}_{p}^{\mathbb{Z}^{2}}$ , where $\mathbb{F}_{p}$ is the field of order $p$. These systems provide natural generalizations of Ledrappier's system, which was the first example of a 2-mixing $\mathbb{Z}^{2}$ -action that is not 3-mixing. Extending the results from our previous work on Ledrappier's example, we show that, under quite mild conditions (namely, 2-mixing and that the subgroup defining the system is a principal Markov subgroup), these systems are almost strongly mixing of every order in the following sense: for each order, one just needs to avoid certain effectively computable logarithmically small sets of times at which there is a substantial deviation from mixing of this order. [ABSTRACT FROM AUTHOR]

The holographic conceptual approach to cognitive processes in the human brain suggests that, in some parts of the brain, each part of the memory (a neuron or a group of neurons) contains some information regarding the entire data. In Dolev and Frenkel (2010, 2012) we demonstrated how to encode data in a holographic manner using the Walsh-Hadamard transform. The encoding is performed on randomized information, that is then represented by a set of Walsh-Hadamard coefficients. These coefficients turn out to have holographic properties. Namely, any portion of the set of coefficients defines a "blurry image" of the original data. In this work, we describe a built-in error correction technique--enlarging the width of the matrix used in the Walsh-Hadamard transform to produce a rectangular Hadamard matrix. By adding this redundancy, the data can bear more errors, resulting in a system that is not affected by missing coefficients up to a certain threshold. Above this threshold, the loss of data is reflected by getting a "blurry image" rather than a concentrated damage. We provide a heuristic analysis of the ability of the technique to correct errors, as well as an example of an image saved using the system. Finally, we give an example of a simple implementation of our approach using neural networks as a proof of concept., (Copyright © 2016 Elsevier Ltd. All rights reserved.)

We analyze the effect of network topology on the pattern stability of the Hopfield neural network in the case of general graphs. The patterns are randomly selected from a uniform distribution. We start the Hopfield procedure from some pattern v. An error in an entry e of v is the situation where, if the procedure is started at e, the value of e flips. Such an entry is an instability point. Note that we disregard the value at e by the end of the procedure, as well as what happens if we start the procedure from another pattern v' or another entry e' of v. We measure the instability of the system by the expected total number of instability points of all the patterns. Our main result is that the instability of the system does not depend on the exact topology of the underlying graph, but rather only on its degree sequence. Moreover, for a large number of nodes, the instability can be approximated by mΣni=1(1 − Φ(√δi/m−1)), where is the standard normal distribution function and δ1, . . . , δn are the degrees of the nodes.