Related papers: On Kummer's test of convergence and its relation t…
Given observations from a stationary time series, permutation tests allow one to construct exactly level $\alpha$ tests under the null hypothesis of an i.i.d. (or, more generally, exchangeable) distribution. On the other hand, when the null…
In this work we study the phenomenon of self-testing from the first principles, aiming to place this versatile concept on a rigorous mathematical footing. Self-testing allows a classical verifier to infer a quantum mechanical description of…
The goal of property testing is to quickly distinguish between objects which satisfy a property and objects that are $\epsilon$-far from satisfying the property. There are now several general results in this area which show that natural…
The concept of measurement is discussed. It is argued that counting process in mathematics is also measurement which requires a basic unit. The idea of scale is put forward. The basic unit itself, which are composed of the infinitesimal of…
Cauchy's sum theorem is a prototype of what is today a basic result on the convergence of a series of functions in undergraduate analysis. We seek to interpret Cauchy's proof, and discuss the related epistemological questions involved in…
Taking several statistical examples, in particular one involving a choice of experiment, as points of departure, and making symmetry assumptions, the link towards quantum theory developed in Helland (2005a,b) is surveyed and clarified. The…
The main result is a generalization of Keller's recursion equation for finding a prime number given the previous primes. We also examine the convergence of the limit in Keller's equation and the convergence of the limit in the general…
We primarily investigate congruences modulo $p$ for finite sums of the form $\sum_k\binom{rk}{k}x^k/k$ over the ranges $0<k<p$ and $0<k<p/r$, where $p$ is a prime larger than the positive integer $r$. Here $x$ is an indeterminate, thus…
We introduce equivalence testing procedures for linear regression analyses. Such tests can be very useful for confirming the lack of a meaningful association between a continuous outcome and a continuous or binary predictor. Specifically,…
We study quantum algorithms for verifying properties of the output probability distribution of a classical or quantum circuit, given access to the source code that generates the distribution. We consider the basic task of uniformity…
We propose a novel modular debiasing technique applicable to any discrete random source, addressing the fundamental challenge of reliably extracting high-quality randomness from inherently imperfect physical processes. The method involves…
This paper presents the Quanta Prime Sequence (QPS) and its foundational theorem, showcasing a unique class of polynomials with substantial implications. The study uncovers profound connections between Quanta Prime numbers and essential…
In this paper, we investigate the stabilizers of certain multisets $\mod p^k$ with respect to their natural multiplicative action, completely describing them for a certain family of polynomials whenever $p$ is an odd prime. This elucidates…
Quantum hypothesis testing concerns the discrimination between quantum states. This paper introduces a novel lower bound for asymmetric quantum hypothesis testing that is based on the Nussbaum-Szko{\l}a mapping. The lower bound provides a…
Time irreversibility, defined as the lack of invariance of the statistical properties of a system or time series under the operation of time reversal, has received an increasing attention during the last decades, thanks to the information…
The considered problem is uniform convergence of sequences of hypergeometric series. We give necessary and sufficient conditions for uniformly dominated convergence of infinite sums of proper bivariate hypergeometric terms. These conditions…
Outlier hypothesis testing is studied in a universal setting. Multiple sequences of observations are collected, a small subset of which are outliers. A sequence is considered an outlier if the observations in that sequence are distributed…
Convergent infinite products, indexed by all natural numbers, in which each factor is a rational function of the index, can always be evaluated in terms of finite products of gamma functions. This goes back to Euler. A purpose of this note…
The measurement process in quantum mechanics is usually described by the von Neumann projection postulate, which forms a basic constituent of the laws of quantum mechanics. Since this postulate requires the outside observer of the system,…
We study the singular series associated to a cubic form with integer coefficients. If the number of variables is at least $10$, we prove the absolute convergence (and hence positivity) under the assumption of Davenport's Geometric…