Overview

The course is an experiment to go beyond the confines of technology to a deeper understanding that links computation, mathematics, and the innate human ability to think. This linking is vital for computational problem solving. We will present mathematics as a critical thinking tool with wide applicability. Someone with critical thinking skills can be trusted to make decisions independently and does not need constant handholding. The independence gained by critical thinking can get you into interesting careers or help you embark on your own ventures.

We will learn mathematics not as a collection of formulae, theorems, and proofs but as a way to invent new concepts needed to solve challenging computational problems. We will discuss how to perform computational experiments to explore ways to solve problems, and through experiments invent new concepts. We will make every effort to invoke curiosity and creativity. Through experiments, we will get to see the sublime beauty and the super computational power emanating from mathematical concepts that simplify complex computations.

Mathematics involves the highest order of creativity of the human mind. The mathematics that we take for granted today was non-existent at one point and it was created by amazing thinkers, often in pursuit of solving important problems that had remained unsolved for centuries. By going through the amazing progression of mathematical thinking, we will see that mathematics is not about memorization but about learning to think critically and creatively.

Critical thinking is all the more important now when humans have to compete with computers. Critical thinking invariably entails:

• Getting the big picture surrounding a problem
• Acquiring the ability to ask questions to eliminate ambiguity and noise from the problem description
• Recognizing and managing the inherent complexity of the problem
• Connecting the dots to go from the known to the unknown

As an example, we will present the great invention of Newton’s Calculus. Newton had a big picture of computing missed by modern offerings of Calculus, which present it as limits and proofs that came much later. Newton was inspired by the first algebra book in England by John Wallis — a mathematical giant who invented the algebra with infinity. Newton was amazed by the method Wallis invented to compute π. Building on Wallis’s algebra with infinity, Newton made the quantum jump to his invention — the universal computing method to reduce complex computations to simple arithmetic operations. We will see that Newton’s calculus and discrete mathematics are united as the pursuit of inventing new ways to perform complex computations simply and precisely.

Euler, the great successor of Newton, invented graphs to solve a famous puzzle. The challenges of modern computing technologies invariably translate to graph problems. Solving them is critically important because of the direct link they have to our existence and sustenance. With all the power of supercomputers in foreseeable future, we cannot even enumerate the combinatorial possibilities in these problems. New mathematical inventions are necessary. We will discuss a new mathematical invention to address challenging problems of modern computing technologies.

This course page is alive — we will continue to add new material and update existing material.

Course Modules

• Binomials
• Newton’s Calculus with Binomial Theorem
• Euler’s Sequencing Gadget and Applications
• Path Algebra
• Linear Algebra for Computing
• Mathematics by AI vs. Mathematics by Humans

A Course on Mathematical Thinking — Introduction was originally published in ENSOFT on Medium, where people are continuing the conversation by highlighting and responding to this story.

Preview

Binomial numbers have fascinated humans since antiquity. These numbers appear mysteriously in solving a very broad range of challenging problems. Binomial numbers are the backbone of Newton’s Calculus and the cornerstone of probability theory and statistics. Using binomials Gauss invented the binomial distribution to approximate the bell curve (normal distribution). This module presents the fundamentals of binomial numbers and gives a glimpse their mysterious appearance is applications. Specifically, we discuss:

• Binomial expansion
• Binomial coefficients
• Binomials in power sets
• Binomials in Venn diagrams
• Binomials in probability and statistics
• A glimpse of the big binomial computation problem and its astonishing solution
• Counting and other problems using Binomials

Newton’s invention of Calculus was inspired by a mysterious binomial formula he found in the first Algebra book in England written by John Wallis. Binomial expansion is so central to Newton’s Calculus that, by his own wish, it is engraved on his tomb. Binomials as the backbone of Newton’s Calculus is a fascinating topic. We cover it in a separate module.

Lecture Notes

A. Binomials and Counting Problems

B. Binomial Identities

C. A Systematic Method for Solving Counting Problems

Experimental AI Generated Audio Notes

We are including AI generated audio notes. These audios are generated using the podcast feature of Google’s NotebookLM. We think you will find these podcasts intriguing. Beware that AI may misinterpret the material or may add erroneous information. Always keep in mind that these audios are supplementary to the lecture notes. This is an experiment, let us know if you find it helpful.

A. Why Newton chose Binomials over Gravity?

B. Binomial Identities and Committee Problems

A Course on Mathematical Thinking — Binomials was originally published in ENSOFT on Medium, where people are continuing the conversation by highlighting and responding to this story.

This article has a new home.

Thanks for reading us here on Medium. We’ve moved our articles to our own site, where we keep our SimDiff automation guides up to date and add new ones as SimDiff grows. You’ll find the latest version of this article at SimDiff and GitHub Workflows on ensoftcorp.com.

We hope to see you there, and thanks for following along all these years.

The ENSOFT team

Integrating SimDiff in GitHub workflows was originally published in ENSOFT on Medium, where people are continuing the conversation by highlighting and responding to this story.

Thanks for reading us here on Medium. We’ve moved our articles to our own site, where we keep our SimDiff automation guides up to date and add new ones as SimDiff grows. You’ll find the latest version of this article at SimDiff and Azure DevOps on ensoftcorp.com.

We hope to see you there, and thanks for following along all these years.

The ENSOFT team

SimDiff and Azure DevOps: Our Guide Has Moved was originally published in ENSOFT on Medium, where people are continuing the conversation by highlighting and responding to this story.

Thanks for reading us here on Medium. We’ve moved our articles to our own site, where we keep our SimDiff automation guides up to date and add new ones as SimDiff grows. You’ll find the latest version of this article at SimDiff and GitLab Workflows on ensoftcorp.com.

We hope to see you there, and thanks for following along all these years.

The ENSOFT team

SimDiff and GitLab Workflows: Our Guide Has Moved was originally published in ENSOFT on Medium, where people are continuing the conversation by highlighting and responding to this story.

Thanks for reading us here on Medium. We’ve moved our articles to our own site, where we keep our research in one place and add new work as it comes out. You’ll find the latest version of this article at A High School Algebra Solution for the Shortest Path Problem on ensoftcorp.com.

We hope to see you there, and thanks for following along all these years.

The ENSOFT team

A High School Algebra Solution for the Shortest Path Problem: Our Article Has Moved was originally published in ENSOFT on Medium, where people are continuing the conversation by highlighting and responding to this story.

This article has a new home.

Thanks for reading us here on Medium. We’ve moved our articles to our own site, where we keep our SimDiff compatibility details up to date and add new ones as SimDiff grows. You’ll find the latest version of this article at SimDiff and MATLAB Releases on ensoftcorp.com.

We hope to see you there, and thanks for following along all these years.

The ENSOFT team

SimDiff — MathWorks Compatibility was originally published in ENSOFT on Medium, where people are continuing the conversation by highlighting and responding to this story.

This article has a new home.

Thanks for reading us here on Medium. We’ve moved our articles to our own site, where we keep our XCSG reference pages up to date and add new ones as our tools grow. You’ll find the latest version of this article at XCSG Interprocedural Flows on ensoftcorp.com.

We hope to see you there, and thanks for following along all these years.

The ENSOFT team

XCSG Compendium — Interprocedural Flows was originally published in ENSOFT on Medium, where people are continuing the conversation by highlighting and responding to this story.

This article has a new home.

Thanks for reading us here on Medium. We’ve moved our articles to our own site, where we keep our XCSG reference pages up to date and add new ones as our tools grow. You’ll find the latest version of this article at XCSG Data Flow on ensoftcorp.com.

We hope to see you there, and thanks for following along all these years.

The ENSOFT team

XCSG Compendium — Data Flow was originally published in ENSOFT on Medium, where people are continuing the conversation by highlighting and responding to this story.

This article has a new home.

Thanks for reading us here on Medium. We’ve moved our articles to our own site, where we keep our XCSG reference pages up to date and add new ones as our tools grow. You’ll find the latest version of this article at XCSG Control Flow on ensoftcorp.com.

We hope to see you there, and thanks for following along all these years.

The ENSOFT team

XCSG Compendium — Control Flow was originally published in ENSOFT on Medium, where people are continuing the conversation by highlighting and responding to this story.