Related papers: An uncertainty principle for arithmetic sequences
Parametric estimation for diffusion processes is considered for high frequency observations over a fixed time interval. The processes solve stochastic differential equations with an unknown parameter in the diffusion coefficient. We find…
Fourier analysis and representation of circular distributions in terms of their Fourier coefficients, is quite commonly discussed and used for model-free inference such as testing uniformity and symmetry etc. in dealing with 2-dimensional…
These are lecture notes from a course in arithmetic dynamics given in Grenoble in June 2017. The main purpose of this text is to explain how arithmetic equidistribution theory can be used in the dynamics of rational maps on P^1. We first…
Heisenberg's uncertainty principle is usually taken to express a limitation of operational possibilities imposed by quantum mechanics. Here we demonstrate that the full content of this principle also includes its positive role as a…
In this paper, we generalise Hardy's uncertainty principle to vector-valued functions, and hence to operators. The principle for operators can be formulated loosely by saying that the kernel of an operator cannot be localised near the…
The aim of this paper is to prove some new uncertainty principles for the windowed Hankel transform. They include uncertainty principle for orthonormal sequence, local uncertainty principle, logarithmic uncertainty principle and…
The concept of complexity appears in virtually all areas of knowledge. Its intuitive meaning shares similarities across fields, but disagreements between its details hinders a general definition, leading to a plethora of proposed…
We study the process of dispersion of low-regularity solutions to the Schr\"odinger equation using fractional weights (observables). We give another proof of the uncertainty principle for fractional weights and use it to get a lower bound…
In this note we consider spectral cut-off estimators to solve a statistical linear inverse problem under arbitrary white noise. The truncation level is determined with a recently introduced adaptive method based on the classical discrepancy…
Modelling uncertainty in Machine Learning models is essential for achieving safe and reliable predictions. Most research on uncertainty focuses on output uncertainty (predictions), but minimal attention is paid to uncertainty at inputs. We…
Fair predictive algorithms hinge on both equality and trust, yet inherent uncertainty in real-world data challenges our ability to make consistent, fair, and calibrated decisions. While fairly managing predictive error has been extensively…
A simple proof of the convergence of the variational regularization, with the regularization parameter, chosen by the discrepancy principle, is given for linear operators under suitable assumptions. It is shown that the discrepancy…
Supervised machine learning and predictive models have achieved an impressive standard today, enabling us to answer questions that were inconceivable a few years ago. Besides these successes, it becomes clear, that beyond pure prediction,…
This paper introduces the order-theoretic concept of lattices along with the concept of consistent quantification where lattice elements are mapped to real numbers in such a way that preserves some aspect of the order-theoretic structure.…
We give a necessary and sufficient condition to achieve the most uncertain exponent in the fractal uncertainty principle of discrete Cantor sets. The condition will be described as distributed spectral pairs, which is a generalization of…
The spectral theory of graphs provides a bridge between classical signal processing and the nascent field of graph signal processing. In this paper, a spectral graph analogy to Heisenberg's celebrated uncertainty principle is developed.…
We describe a new framework of a sublinear expectation space and the related notions and results of distributions, independence. A new notion of G-distributions is introduced which generalizes our G-normal-distribution in the sense that…
Neural networks make accurate predictions but often fail to provide reliable uncertainty estimates, especially under covariate distribution shifts between training and testing. To address this problem, we propose a Bayesian framework for…
A mathematical framework for information-theoretic analysis is established, with a new viewpoint of describing transmitted messages and communication channels by the nonlinear expectation theory, beyond the framework of classical…
It is well-known that if a subset A of a finite Abelian group G satisfies a quasirandomness property called uniformity of degree k, then it contains roughly the expected number of arithmetic progressions of length k, that is, the number of…