Related papers: Ernest Michael and theory of continuous selections
Monotonicity and convex analysis arise naturally in the framework of multi-marginal optimal transport theory. However, a comprehensive multi-marginal monotonicity and convex analysis theory is still missing. To this end we study extensions…
Topological Machine Learning (TML) is an emerging field that leverages techniques from algebraic topology to analyze complex data structures in ways that traditional machine learning methods may not capture. This tutorial provides a…
In the year 2014, the field of selection principles found its way into several additional, fascinating mathematical realms. The field enters the consensus as a mainstream part of set theory and topology, and as a promising direction for…
We provide a formal introduction into the classic theorems of general topology and its axiomatic foundations in set theory. In this second part we introduce the fundamental concepts of topological spaces, convergence, and continuity, as…
For a compact and convex window, Mecke described a process of tessellations which arise from cell divisions in discrete time. At each time step, one of the existing cells is selected according to an equally-likely law. Independently, a line…
In the article we give an appreciation of Edward Nelson's multifaceted contribution to mathematics, and particularly to foundational theories of infinitesimals.
An elementary biostatistical theory based on a selectivity-variability principle is proposed to address a question raised by Charles Darwin, namely, how one sex of a sexually dimorphic species might tend to evolve with greater variability…
The subject of stochastic approximation was founded by Robbins and Monro [Ann. Math. Statist. 22 (1951) 400--407]. After five decades of continual development, it has developed into an important area in systems control and optimization, and…
Natural selection and random drift are competing phenomena for explaining the evolution of populations. Combining a highly fit mutant with a population structure that improves the odds that the mutant spreads through the whole population…
Statistical models with latent structure have a history going back to the 1950s and have seen widespread use in the social sciences and, more recently, in computational biology and in machine learning. Here we study the basic latent class…
Yule's 1925 paper introducing the branching model that bears his name was a landmark contribution to the biodiversity sciences. In his paper, Yule developed stochastic models to explain the observed distribution of species across genera and…
Turing's theory of pattern formation is a universal model for self-organization, applicable to many systems in physics, chemistry and biology. Essential properties of a Turing system, such as the conditions for the existence of patterns and…
Factor analysis, a classical multivariate statistical technique is popularly used as a fundamental tool for dimensionality reduction in statistics, econometrics and data science. Estimation is often carried out via the Maximum Likelihood…
Evolutionary multi-objective clustering (EMOC), a modern clustering technique, has been widely applied to extract patterns, allowing us to analyze different aspects of complex data by considering multiple criteria. In this article, we…
Multivariate statistical analysis is concerned with observations on several variables which are thought to possess some degree of inter-dependence. Driven by problems in genetics and the social sciences, it first flowered in the earlier…
The design of deep neural networks remains somewhat of an art rather than precise science. By tentatively adopting ergodic theory considerations on top of viewing the network as the time evolution of a dynamical system, with each layer…
We introduces a general linear framework that unifies the study of multi-winner voting rules and proportionality axioms, demonstrating that many prominent multi-winner voting rules-including Thiele methods, their sequential variants, and…
An algebraization of the notion of topology has been proposed more than seventy years ago in a classical paper by McKinsey and Tarski. However, in McKinsey and Tarski's setting the model theoretical notion of homomorphism does not…
Modern biological tools have made it possible to unequivocally demonstrate the deep relationship among species in terms of genes and basic molecular mechanisms. In addition, results from genetic, physical and physiological approaches…
Maximum entropy principle (MEP) offers an effective and unbiased approach to inferring unknown probability distributions when faced with incomplete information, while neural networks provide the flexibility to learn complex distributions…