Related papers: Universal Convergence of Semimeasures on Individua…
A measure $\mu$ on the unit circle $\mathbb{T}$ belongs to Steklov class $\mathcal{S}$ if its density $w$ with respect to the Lebesgue measure on $\mathbb{T}$ is strictly positive: $\inf_{\mathbb{T}} w > 0$. Let $\mu$, $\mu_{-1}$ be…
We show that the symmetric portion of correlated coherence is always a valid quantifier of entanglement, and that this property is independent of the particular choice of coherence measure. This leads to an infinitely large class of…
The dynamic time warping (dtw) distance fails to satisfy the triangle inequality and the identity of indiscernibles. As a consequence, the dtw-distance is not warping-invariant, which in turn results in peculiarities in data mining…
We consider sequences of metrics, $g_j$, on a Riemannian manifold, $M$, which converge smoothly on compact sets away from a singular set $S\subset M$, to a metric, $g_\infty$, on $M\setminus S$. We prove theorems which describe when…
For given Boolean algebras $\mathbb{A}$ and $\mathbb{B}$ we endow the space $\mathcal{H}(\mathbb{A},\mathbb{B})$ of all Boolean homomorphisms from $\mathbb{A}$ to $\mathbb{B}$ with various topologies and study convergence properties of…
We study sequential prediction of real-valued, arbitrary and unknown sequences under the squared error loss as well as the best parametric predictor out of a large, continuous class of predictors. Inspired by recent results from…
A universal formulation of uncertainty relations for quantum measurements is presented with additional focus on the representability of quantum observables by classical observables over a given state. Owing to the simplicity and operational…
We detail a simple procedure (easily convertible to an algorithm) for constructing from quasi-uniform samples of $f$ a sequence of linear spline functions converging to the monotone rearrangement of $f$, in the case where $f$ is an almost…
We explore the asymptotic convergence and nonasymptotic maximal inequalities of supermartingales and backward submartingales in the space of positive semidefinite matrices. These are natural matrix analogs of scalar nonnegative…
We study algorithmic randomness notions via effective versions of almost-everywhere theorems from analysis and ergodic theory. The effectivization is in terms of objects described by a computably enumerable set, such as lower semicomputable…
The probability distribution P from which the history of our universe is sampled represents a theory of everything or TOE. We assume P is formally describable. Since most (uncountably many) distributions are not, this imposes a strong…
We show that a spectrum of frequencies obtained by a random perturbation of the integers allows one to represent any measurable function on R by an almost everywhere converging sum of harmonics almost surely.
In this work we prove the undecidability (and $\Sigma^0_1$-completeness) of several theories of semirings with fixed points. The generality of our results stems from recursion theoretic methods, namely the technique of effective…
We study Martin-L\"{o}f random (ML-random) points on computable probability measures on sample and parameter spaces (Bayes models). We consider variants of conditional randomness defined by ML-randomness on Bayes models and those of…
Given the widespread use of lossless compression algorithms to approximate algorithmic (Kolmogorov-Chaitin) complexity, and that lossless compression algorithms fall short at characterizing patterns other than statistical ones not different…
We characterize measure spaces such that the canonical map $L_\infty \to L_1^*$ is surjective. In case of $d$ dimensional Hausdorff measure of a complete separable metric space $X$ we give two equivalent conditions. One is in terms of the…
We consider ensembles of real symmetric band matrices with entries drawn from an infinite sequence of exchangeable random variables, as far as the symmetry of the matrices permits. In general the entries of the upper triangular parts of…
Many statistical estimands of interest (e.g., in regression or causality) are functions of the joint distribution of multiple random variables. But in some applications, data is not available that measures all random variables on each…
Although algorithmic randomness with respect to various non-uniform computable measures is well-studied, little attention has been paid to algorithmic randomness with respect to computable \emph{trivial} measures, where a measure $\mu$ on…
We consider the task of forecasting an infinite sequence of future observations based on some number of past observations, where the probability measure generating the observations is "suspected" to satisfy one or more of a set of…