Randomness testing plays an important role in cryptography. Randomness is typically examined by batteries of statistical tests. One of the most frequently used test batteries is the NIST Statistical Test Suite. The tests of randomness should be rather fast since they usually process large volumes of data. Unfortunately, this is not the case for the NIST STS, where a complete test can take hours...

We establish a generalization of the Expander Mixing Lemma for arbitrary (finite) simplicial complexes. The original lemma states that concentration of the Laplace spectrum of a graph implies combinatorial expansion (which is also referred to as mixing, or pseudorandomness). Recently, an analogue of this Lemma was proved for simplicial complexes of arbitrary dimension, provided that the skeleto...

The explore-exploit dilemma occurs when we must choose between exploring options that yield information (potentially useful for the future) and exploiting options that yield known reward (certain to be useful right now). We have previously shown that humans use two distinct strategies for resolving this dilemma: the optimal-but-complex ‘directed exploration‘ in which choices are biased towards ...

We show new lower bounds for collision-free transmissions in Radio Networks. Our main result is a tight lower bound of Ω(log n log(1/ )) on the time required by a uniform randomized protocol to achieve a clear transmission with success probability 1 − in a one-hop setting. This result is extended to non-uniform protocols as well. A new lower bound is proved for the important multi-hop setting o...

We investigate the uniqueness of the plane-wave decomposition of temporally deterministic, spatially random fields. Specifically, we consider the decomposition of spatially ergodic and, thus, statistically homogeneous fields. We show that when the spatial power spectrum is injective, the plane waves are the only possible coherent modes. Furthermore, the randomness of such fields originates in t...

This short text summarizes the work in biology proposed in our book, Perspectives on Organisms, where we analyse the unity proper to organisms by looking at it from different viewpoints. We discuss the theoretical roles of biological time, complexity, theoretical symmetries, singularities and critical transitions. We explicitly borrow from the conclusions in some key chapters and introduce them...

Three major puzzles, described later in this section, have captured the attention of macroeconomic finance: the equity-premium, riskfreerate and equity-volatility puzzles. A common strand of these three asset-return puzzles is that markets are behaving as if investors fear some unknown hidden randomness that isn’t obvious from the data. People are acting in the aggregate like there is much more...

We compare various notions of algorithmic randomness. First we consider relativized randomness. A set is ?-random if it is Martin-L?f random relative to 0_1). We show that a set is 2-random if and only if there is a constant c such that infinitely many initial segments x of the set are c-incompressible: C(x) > |x| ? c. The 'only if direction was obtained independently by Joseph Miller. This cha...

Using possibly infinite computations on universal monotone Turing machines, we prove Martin-Löf randomness in ∅′ of the probability that the output be in some set O ⊆ 2≤ω under complexity assumptions about O. §1. Randomness in the spirit of Rice’s theorem for computability. Let 2∗ be the set of all finite strings in the binary alphabet 2 = {0, 1}. Let 2ω be the set of all infinite binary sequen...

Discontinuous structure in landscapes may result in discontinuous, aggregated species bodymass patterns, reflecting the scales of structure available to animal communities within a landscape. The edges of these body-mass aggregations reflect transitions between available scales of landscape structure. Such transitions, or scale breaks, are theoretically associated with increased biological vari...