Related papers: Exploring Progressions: A Collection of Problems
This book is devoted to the problem of sequential probability forecasting, that is, predicting the probabilities of the next outcome of a growing sequence of observations given the past. This problem is considered in a very general setting…
Curriculum Learning for Reinforcement Learning is an increasingly popular technique that involves training an agent on a sequence of intermediate tasks, called a Curriculum, to increase the agent's performance and learning speed. This paper…
In this paper, we study $k$-term arithmetic progressions $N, N+d, ..., N+(k-1)d$ of powerful numbers. Under the $abc$-conjecture, we obtain $d \gg_\epsilon N^{1/2 - \epsilon}$. On the other hand, there exist infinitely many $3$-term…
This monograph presents a class of algorithms called coordinate descent algorithms for mathematicians, statisticians, and engineers outside the field of optimization. This particular class of algorithms has recently gained popularity due to…
Using standard methods (due to Janson, Stein-Chen, and Talagrand) from probabilistic combinatorics, we explore the following general theme: As one progresses from each member of a family of objects ${\cal A}$ being "covered" by at most one…
Solving a system of polynomial equations is a ubiquitous problem in the applications of mathematics. Until recently, it has been hopeless to find explicit solutions to such systems, and mathematics has instead developed deep and powerful…
In this series of three articles, we give an exposition of various results and open problems in three areas of algebraic and geometric combinatorics: totally non-negative matrices, representations of the symmetric group, and hyperplane…
One of the central problems in additive combinatorics is to determine how large a subset of the first $N$ integers can be before it is forced to contain $k$ elements forming an arithmetic progression. Around 25 years ago, Gowers proved the…
The article provides a pedagogical review aimed at graduate students in materials science, physics, and applied mathematics, focusing on recent developments in the subject. Following a brief summary of concepts from complex analysis, the…
There are many extremely challenging problems about existence of monochromatic arithmetic progressions in colorings of groups. Many theorems hold only for abelian groups as results on non-abelian groups are often much more difficult to…
This is a (long) survey about applications of percolation theory in equilibrium statistical mechanics. The chapters are as follows: 1. Introduction 2. Equilibrium phases 3. Some models 4. Coupling and stochastic domination 5. Percolation 6.…
Sequence transformations accomplish an acceleration of convergence or a summation in the case of divergence by detecting and utilizing regularities of the elements of the sequence to be transformed. For sufficiently large indices, certain…
This very preliminary text is related to ``Algorithms on Texts'', also called ``Algorithmic Stringology''. It is an extension of the book ``125 Problems in Text Algorithms'' providing, in the same compact style, more problems with…
This text contains over three hundred specific open questions on various topics in additive combinatorics, each placed in context by reviewing all relevant results. While the primary purpose is to provide an ample supply of problems for…
Let $\mathcal{T}$ be a collection of 3-element subsets $S$ of $\{1, \ldots,n\}$ with the property that if $i<j<k$ and $a<b<c$ are two 3-element subsets in $S$, then there exists an integer sequence $x_1 < x_2 < \cdots < x_n$ such that $x_i,…
This is a survey on algorithmic questions about combinatorial and geometric properties of convex polytopes. We give a list of 35 problems; for each the current state of knowledege on its theoretical complexity status is reported. The…
This work describes numerical methods that are useful in many areas: examples include statistical modelling (bioinformatics, computational biology), theoretical physics, and even pure mathematics. The methods are primarily useful for the…
Throughout this book, we discuss some open problems in various branches of science, including mathematics, theoretical physics, astro-physics, geophysics etc. It is of our hope that some of the problems discussed in this book will find…
Enumeration of tilings is the mathematical study concerning the total number of coverings of regions by similar pieces without gaps or overlaps. Enumeration of tilings has become a vibrant subfield of combinatorics with connections and…
This document is an exposition of an assortment of open problems arising from the exact enumeration of (perfect) matchings of finite graphs. Roughly half have been solved at the time of this writing; see the document "Twenty Open Problems…