Related papers: If Archimedes would have known functions
Almost all problems in applied mathematics, including the analysis of dynamical systems, deal with spaces of real-valued functions on Euclidean domains in their formulation and solution. In this paper, we describe the the tool Ariadne,…
Working with letters that represent unknown constants, i.e., parameters, has been historically challenging for students. This is an important skill for their success in many future quantitative settings, and yet it appears this topic is…
The paper suggests a short survey of integration algorithms which evolved since 1982. These theorems and algorithms form discrete versions of the calculus theorems.
I am presenting a first-ever scientific collection of short sayings on probability and statistics expressed by most various men of science, many classics included, from antiquity to Kepler to our time. Quite understandably, the reader will…
Scientists have demonstrated that quantum computing has presented novel approaches to address computational challenges, each varying in complexity. Adapting problem-solving strategies is crucial to harness the full potential of quantum…
Finite simple graphs are a playground for classical areas of mathematics. We illustrate this by looking at some theorems. These are slightly enhanced preparation notes for a talk given at the joint AMS meeting of January 16, 2014 in…
Just as the $\lambda$-calculus uses three primitives (abstraction, application, variable) as the foundation of functional programming, inheritance-calculus uses three primitives (record, definition, inheritance) as the foundation of…
A brief review of the history of the conic sections would not be complete without an exhaustively tolerable account of all the things related to the subject that can be found in the extensive work of the wise Archimedes. There is no strong…
We are convinced of the usefulness of sketches and diagrams during mathematical work but the observation is made in our practices that they are not spontaneously used by students. In order to study the understanding and use of sketches by…
This work explores a possible course of evolution of mathematics in ancient times in India when there was no script, no place-value system, and no zero. Reviewing examples of time-reckoning, large numbers, sacrificial altar-making, and…
Technology is currently ubiquitous and is also part of the educational system at all levels. It started with communication technology systems, and later continued with digital competence. Nowadays, although these previous concepts are still…
This paper aims to give a brief account of the mathematical work of the 7th-century Armenian polymath and natural philosopher Anania Shirakatsi. The three sections of Anania's ``Book of Arithmetic'' -- tables of arithmetic operations, a…
This work exploits the logical foundation of session types to determine what kind of type discipline for the pi-calculus can exactly capture, and is captured by, lambda-calculus behaviours. Leveraging the proof theoretic content of the…
In this paper, we trace the development of the theory of the calculus of variations. From its roots in the work of Greek thinkers and continuing through to the Renaissance, we see that advances in physics serve as a catalyst for…
The lambda-calculus is a peculiar computational model whose definition does not come with a notion of machine. Unsurprisingly, implementations of the lambda-calculus have been studied for decades. Abstract machines are implementations…
In this paper we use theoretical frameworks from mathematics education and cognitive psychology to analyse Cauchy's ideas of function, continuity, limit and infinitesimal expressed in his Cours D'Analyse. Our analysis focuses on the…
Passive documents and active programs now widely comingle. Document languages include Turing-complete programming elements, and programming languages include sophisticated document notations. However, there are no formal foundations that…
Cost functions provide a framework for constructions of sets Turing below the halting problem that are close to computable. We carry out a systematic study of cost functions. We relate their algebraic properties to their expressive…
This Note describe my own recollection of the first 30 years of Category Theory, it is not the result of any historical investigation. The choice of concepts and its evaluation is my own, necessarily subjective. It follows a chronological…
There have been several modifications of how basic calculus has been taught, but very few of these modifications have considered the computational tools available at our disposal. Here, we present a few tools that are easy to develop and…