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We consider the problem of solving a large-scale system of linear equations in a distributed or federated manner by a taskmaster and a set of machines, each possessing a subset of the equations. We provide a comprehensive comparison of two…
In the era of big data, the advancement, improvement, and application of algorithms in academic research have played an important role in promoting the development of different disciplines. Academic papers in various disciplines, especially…
This paper aims first at a simultaneous axiomatic presentation of the proof of optimal convergence rates for adaptive finite element methods and second at some refinements of particular questions like the avoidance of (discrete) lower…
At the University of Colorado Boulder, as part of our broader efforts to transform middle- and upper-division physics courses, we research students' difficulties with particular concepts, methods, and tools in classical mechanics,…
In this paper, we study adaptive online convex optimization, and aim to design a universal algorithm that achieves optimal regret bounds for multiple common types of loss functions. Existing universal methods are limited in the sense that…
Since its creation in 1991, arXiv has become central to the diffusion of research in a number of fields. Combining data from the entirety of arXiv and the Web of Science (WoS), this paper investigates (a) the proportion of papers across all…
This paper is an excerpt of an early version of Chapter 2 of the book "Validity, Reliability, and Significance. Empirical Methods for NLP and Data Science", by Stefan Riezler and Michael Hagmann, published in December 2021 by Morgan &…
The volume of academic paper submissions and publications is growing at an ever increasing rate. While this flood of research promises progress in various fields, the sheer volume of output inherently increases the amount of noise. We…
Datasets are mathematical objects (e.g., point clouds, matrices, graphs, images, fields/functions) that have shape. This shape encodes important knowledge about the system under study. Topology is an area of mathematics that provides…
Given the increase of publications, search for relevant papers becomes tedious. In particular, search across disciplines or schools of thinking is not supported. This is mainly due to the retrieval with keyword queries: technical terms…
In this study, machine learning models were constructed to predict whether judgments made by the European Court of Human Rights (ECHR) would lead to a violation of an Article in the Convention on Human Rights. The problem is framed as a…
Structured documents--tables paired with captions, figures with explanations, equations with the paragraphs that interpret them--are routinely fragmented when indexed for retrieval. Element-level indexing treats every parsed element as an…
The uncertainty measurement of classifiers' predictions is especially important in applications such as medical diagnoses that need to ensure limited human resources can focus on the most uncertain predictions returned by machine learning…
This paper reviews a line of work on decaying turbulence that began with loop equations and culminated in the Euler ensemble as a candidate statistical attractor. Most observable predictions discussed here-including the decay law, velocity…
Discrepancies between scientific papers and their code undermine reproducibility, a concern that grows as automated research agents scale scientific output beyond human review capacity. Whether LLMs can reliably detect such discrepancies…
Accurate yet efficient computation of the Voigt and complex error function is a challenge since decades in astrophysics and other areas of physics. Rational approximations have attracted considerable attention and are used in many codes,…
Large Language Models (LLMs) have demonstrated significant potential in generating mathematical proofs. However, a persistent challenge is that LLMs occasionally make mistakes, while even a minor mistake can invalidate an entire proof.…
Universal approximation theorems are the foundations of classical neural networks, providing theoretical guarantees that the latter are able to approximate maps of interest. Recent results have shown that this can also be achieved in a…
Deep learning algorithms have been applied very successfully in recent years to a range of problems out of reach for classical solution paradigms. Nevertheless, there is no completely rigorous mathematical error and convergence analysis…
The current paper focuses on fundamental groups and Euler characteristics of various digital models of the 2-dimensional sphere. For all models that we consider, we show that the fundamental groups are trivial, and compute the Euler…