Related papers: Exploring Progressions: A Collection of Problems
We show that essentially the Fibonacci sequence is the unique binary recurrence which contains infinitely many three-term arithmetic progressions. A criterion for general linear recurrences having infinitely many three-term arithmetic…
With this work we aim to show how Mathematica can be a useful tool to investigate properties of combinatorial structures. Specifically, we will face enumeration problems on independent subsets of powers of paths and cycles, trying to…
Combinatorics is a powerful tool for dealing with relations among objectives mushroomed in the past century. However, an more important work for mathematician is to apply combinatorics to other mathematics and other sciences not merely to…
This review is focused on the borderline region of theoretical physics and mathematics. First, we describe numerical methods for the acceleration of the convergence of series. These provide a useful toolbox for theoretical physics which has…
Large language models (LLMs) have shown remarkable progress in mathematical problem-solving, but evaluation has largely focused on problems that have exact analytical solutions or involve formal proofs, often overlooking approximation-based…
We present a mathematical and algorithmic scheme for learning the principal geometric elements in an image or 3D object. We build on recent work that convexifies the basic problem of finding a combination of a small number shapes that…
We study multivariate problems like function approximation, numerical integration, global optimization and dispersion. We obtain new results on the information complexity $n(\varepsilon,d)$ of these problems. The information complexity is…
In this book, there are five chapters: Systems of Linear Equations, Vector Spaces, Homogeneous Systems, Characteristic Equation of Matrix, and Matrix Dot Product. It has also exercises at the end of each chapter above to let students…
We investigate upper-division student difficulties with direct integration in multiple contexts involving the calculation of a potential from a continuous distribution (e.g., mass, charge, or current). Integration is a tool that has been…
This work connects two mathematical fields - computational complexity and interval linear algebra. It introduces the basic topics of interval linear algebra - regularity and singularity, full column rank, solving a linear system, deciding…
This paper surveys the machine learning literature and presents in an optimization framework several commonly used machine learning approaches. Particularly, mathematical optimization models are presented for regression, classification,…
Here practical aspects of conducting research via computer simulations are discussed. The following issues are addressed: software engineering, object-oriented software development, programming style, macros, make files, scripts, libraries,…
In this volume, we encouraged a broad array of submissions that highlight issues of Diversity, Equity, and Inclusion (DEI) in introductory mathematics programs with special attention to precalculus, differential calculus, and integral…
There are ten chapters in this dissertation, which focuses on nine contents: growth estimates for a class of subharmonic functions in the half plane; growth estimates for a class of subharmonic functions in the half space; a generalization…
We study arithmetic inequalities for multiplicative, sub(super)-multiplicative, sub(super)-homogeneous functions. Applications for the classical arithmetic functions are pointed out.
What kind of problem-solving instruction can help students apply what they have learned to solve the new and unfamiliar problems they will encounter in the future? We propose that mathematical sensemaking, the practice of seeking coherence…
The following work is an exploration into certain topics in the broad world of integrable models, both classical and quantum, and consists of two main parts of roughly equal length. The first part, consisting of chapters 1-3, concerns…
This book contains a large number of exercises related to different stochastic disciplines. Difficulty of the problems varies from the basic level in the first chapter up to the analysis of articles in Probability, Statistics and Computer…
We study the problem of generating monomials of a polynomial in the context of enumeration complexity. In this setting, the complexity measure is the delay between two solutions and the total time. We present two new algorithms for…
Mathematica offers, by way of the package Combinatorics, many useful functions to work on graphs and ordered structures, but none of these functions was specific enough to meet the needs of our research group. Moreover, the existing…