Related papers: On the close interaction between algorithmic rando…
In this work we introduce the randomness which is truly quantum mechanical in nature arising as an act of measurement. For a composite classical system we have the joint entropy to quantify the randomness present in the total system and…
We generalise the randomness test definitions in the literature for both the Martin-L\"of and Schnorr randomness of a series of binary outcomes, in order to allow for interval-valued rather than merely precise forecasts for these outcomes,…
This article is a brief personal account of the past, present, and future of algorithmic randomness, emphasizing its role in inductive inference and artificial intelligence. It is written for a general audience interested in science and…
Although some information-theoretic measures of uncertainty or granularity have been proposed in rough set theory, these measures are only dependent on the underlying partition and the cardinality of the universe, independent of the lower…
This book is an introductory course to basic commutative algebra with a particular emphasis on finitely generated projective modules. We adopt the constructive point of view, with which all existence theorems have an explicit algorithmic…
While Kolmogorov complexity is the accepted absolute measure of information content of an individual finite object, a similarly absolute notion is needed for the relation between an individual data sample and an individual model summarizing…
In this paper, we relate the framework of mod-$\phi$ convergence to the construction of approximation schemes for lattice-distributed random variables. The point of view taken here is that of Fourier analysis in the Wiener algebra, allowing…
A coarse description of a subset A of omega is a subset D of omega such that the symmetric difference of A and D has asymptotic density 0. We study the extent to which noncomputable information can be effectively recovered from all coarse…
Classification is an important goal in many branches of mathematics. The idea is to describe the members of some class of mathematical objects, up to isomorphism or other important equivalence in terms of relatively simple invariants. Where…
Random matrix theory is a well-developed area of probability theory that has numerous connections with other areas of mathematics and its applications. Much of the literature in this area is concerned with matrices that possess many exact…
We define a measure of competitive performance for distributed algorithms based on throughput, the number of tasks that an algorithm can carry out in a fixed amount of work. This new measure complements the latency measure of Ajtai et al.,…
With the increasing interplay between experimental and computational approaches at multiple length scales, new research directions are emerging in materials science and computational mechanics. Such cooperative interactions find many…
The influence theorem for product measures on the discrete space {0,1}^N may be extended to probability measures with the property of monotonicity (which is equivalent to `strong positive-association'). Corresponding results are valid for…
We show from a categorical point of view that probability measures on certain measurable or topological spaces arise canonically as the extension of probability distributions on countable sets. We do this by constructing probability monads…
Nies and Scholz introduced the notion of a state to describe an infinite sequence of qubits and defined quantum-Martin-Lof randomness for states, analogously to the well known concept of Martin-L\"of randomness for elements of Cantor space…
Classical designs of randomized experiments, going back to Fisher and Neyman in the 1930s still dominate practice even in online experimentation. However, such designs are of limited value for answering standard questions in settings,…
Let $(Z_k)_{k\geq 1}$ be a sequence of independent and identically distributed complex random variables with common distribution $\mu$ and let $P_n(X):=\prod_{k=1}^n (X-Z_k)$ the associated random polynomial in $\mathbb C[X]$. In [Kab15],…
We present two theorems concerned with algorithmic randomness and differentiability of functions of several variables. Firstly, we prove an effective form of the Rademacher's Theorem: we show that computable randomness implies…
This paper studies projections of uniform random elements of (co)adjoint orbits of compact Lie groups. Such projections generalize several widely studied ensembles in random matrix theory, including the randomized Horn's problem, the…
This work is meant to be a step towards the formal definition of the notion of algorithm, in the sense of an equivalence class of programs working "in a similar way". But instead of defining equivalence transformations directly on programs,…