Related papers: Bernstein polynomials: a bibliometric data analysi…
Special functions are essential in theoretical and applied mathematics and have various applications in the applied sciences. Mathematicians have studied them for centuries, but there is still no bibliometric analysis that summarises the…
AI is transforming scientific research by revealing new ways to understand complex physical systems, but its impact remains constrained by the lack of large, high-quality domain-specific datasets. A rich, largely untapped resource lies in…
A basic experiment in probability theory is drawing without replacement from an urn filled with multiple balls of different colours. Clearly, it is physically impossible to overdraw, that is, to draw more balls from the urn than it…
Methods of high-dimensional probability play a central role in applications for statistics, signal processing theoretical computer science and related fields. These lectures present a sample of particularly useful tools of high-dimensional…
Many of the current scientific advances in the life sciences have their origin in the intensive use of data for knowledge discovery. In no area this is so clear as in bioinformatics, led by technological breakthroughs in data acquisition…
A comprehensive artificial intelligence system needs to not only perceive the environment with different `senses' (e.g., seeing and hearing) but also infer the world's conditional (or even causal) relations and corresponding uncertainty.…
We prove various theorems on approximation using polynomials with integer coefficients in the Bernstein basis of any given order. In the extreme, we draw the coefficients from $\{ \pm 1\}$ only. A basic case of our results states that for…
We consider a basic problem in unsupervised learning: learning an unknown \emph{Poisson Binomial Distribution}. A Poisson Binomial Distribution (PBD) over $\{0,1,\dots,n\}$ is the distribution of a sum of $n$ independent Bernoulli random…
Scientific endeavors such as large astronomical surveys generate databases on the terabyte scale. These, usually multidimensional databases must be visualized and mined in order to find interesting objects or to extract meaningful and…
In 1910 Einstein published a crucial aspect of his understanding of Boltzmann entropy. He essentially argued that the likelihood function of any system composed by two probabilistically independent subsystems {\it ought} to be factorizable…
Grouped data are commonly encountered in applications. The Bernstein polynomial model is proposed as an approximate model in this paper for estimating a univariate density function based on grouped data. The coefficients of the Bernstein…
Proteins are sequences of amino acids that serve as the basic building blocks of living organisms. Despite rapidly growing databases documenting structural and functional information for various protein sequences, our understanding of…
Though it goes without saying that linear algebra is fundamental to mathematical biology, polynomial algebra is less visible. In this article, we will give a brief tour of four diverse biological problems where multivariate polynomials play…
We establish the existence of the Bernstein polynomial in one indeterminate $t$, and provide a method for its explicit computation. The Bernstein polynomial is associated with finitely generated modules over the Weyl algebra, known as…
The field of astronomy has arrived at a turning point in terms of size and complexity of both datasets and scientific collaboration. Commensurately, algorithms and statistical models have begun to adapt --- e.g., via the onset of artificial…
This paper shows how bibliometric assessment can be implemented at individual level. This has been successfully done at the University of Vienna carried out by the Bibliometrics and Publication Strategies Department of the Vienna University…
Mathematics education, a crucial and basic field, significantly influences students' learning in related subjects and their future careers. Utilizing artificial intelligence to interpret and comprehend math problems in education is not yet…
Classical statistics deals with determined and precise data analysis. But in reality, there are many cases where the information is not accurate and a degree of impreciseness, uncertainty, incompleteness, and vagueness is observed. In these…
Scientific datasets play a crucial role in contemporary data-driven research, as they allow for the progress of science by facilitating the discovery of new patterns and phenomena. This mounting demand for empirical research raises…
Modern statistical software and machine learning libraries are enabling semi-automated statistical inference. Within this context, it appears easier and easier to try and fit many models to the data at hand, reversing thereby the Fisherian…