Related papers: Frobenius Pseudoprimes
Using an idea of Doug Lind, we give a lower bound for the Perron-Frobenius degree of a Perron number that is not totally-real. As an application, we prove that there are cubic Perron numbers whose Perron-Frobenius degrees are arbitrary…
In 1977, Hugh Williams studied Lucas pseudoprimes to all Lucas sequences of a fixed discriminant. These are composite numbers analogous to Carmichael numbers and they satisfy a Korselt-like criterion: $n$ must be a product of distinct…
We consider the problem of hypothesis testing in the situation where the first hypothesis is simple and the second one is local one-sided composite. We describe the choice of the thresholds and the power functions of different tests when…
We define a simple rule that allows to describe sequences of projective measurements for a broad class of generalized probabilistic models. This class embraces quantum mechanics and classical probability theory, but, for example, also the…
Suppose that R is a two-dimensional normal standard-graded domain over a finite field. We prove that there exists a uniform Frobenius test exponent b for the class of homogeneous ideals in R generated by at most n elements. This means that…
We study a family of ultraproducts of finite fields with the Frobenius automorphism in this paper. Their theories have the strict order property and TP2. But the coarse pseudofinite dimension of the definable sets is definable and…
In this note we generalise a method of Perott to give new proofs that there are infinitely many prime numbers.
Finite frames can be viewed as mass points distributed in $N$-dimensional Euclidean space. As such they form a subclass of a larger and rich class of probability measures that we call probabilistic frames. We derive the basic properties of…
These series of notes serve as an introduction to some of both the classical and modern techniques in Reifenberg theory. At its heart, Reifenberg theory is about studying general sets or measures which can be, in one sense or another,…
Despite significant developments in Proof Theory, surprisingly little attention has been devoted to the concept of proof verifier. In particular, the mathematical community may be interested in studying different types of proof verifiers…
In the modern Bayesian view classical probability theory is simply an extension of conventional logic, i.e., a quantitative tool that allows for consistent reasoning in the presence of uncertainty. Classical theory presupposes, however,…
Let $1<c<d$ be two relatively prime integers, $g_{c,d}=cd-c-d$ and $\mathbb{P}$ is the set of primes. For any given integer $k \geq 1$, we prove that $$\#\left\{p^k\le g_{c,d}:p\in \mathbb{P}, ~p^k=cx+dy,~x,y\in \mathbb{Z}_{\geqslant0}…
Probabilistic concurrent systems are foundational models for modern mobile computing. In this paper, a unifying approach to probabilistic testing equivalences is proposed. With the help of a new distribution-based semantics for…
Uniform proofs are sequent calculus proofs with the following characteristic: the last step in the derivation of a complex formula at any stage in the proof is always the introduction of the top-level logical symbol of that formula. We…
The study of Fourier transforms of probability measures on fractal sets plays an important role in recent research. Faster decay rates are known to yield enhanced results in areas such as metric number theory. This paper focuses on…
With large language models surpassing human performance on an increasing number of benchmarks, we must take a principled approach for targeted evaluation of model capabilities. Inspired by pseudorandomness, we propose pseudointelligence,…
Hypothesis tests under order restrictions arise in a wide range of scientific applications. By exploiting inequality constraints, such tests can achieve substantial gains in power and interpretability. However, these gains come at a cost:…
Two procedures for checking Bayesian models are compared using a simple test problem based on the local Hubble expansion. Over four orders of magnitude, p-values derived from a global goodness-of-fit criterion for posterior probability…
Parameter estimates in misspecified models converge to pseudo-true parameter values, which minimize a population objective function. Pseudo-true values often differ from quantities of economic interest, raising questions of how, if at all,…
A novel type of permutation tests for dendrogram data is studied with respect to two types of metrics for measuring the difference between dendrograms. First, the Frobenius norm is used, and we prove the consistency and efficiency of the…