April 17, 2025 |
Section: Notes
Simple Types Before ░░░░ dependent types with ░░░░ ░░░░ ░░░░ ░░░░ we’ll take ░░░░ ░░░░ ░░░░ something ░░░░ ░░░░ ░░░░ simple types░░░░ starting ░░░░ ░░░░ ░░░░ ░░░░ type of all, the functon type $\_\to\_$. ░░░░ ░░░░ ░░░░ ░░░░ types is that ░░░░ ░░░░ not ░░░░ what to ░░░░ ░░░░ We’ll ░░░░ this with ░░░░ which work ░░░░ any ░░░░ called $\textsf{Set}$ in Agda. That is, ░░░░ will ░░░░ polymorphic ░░░░ without introducing a ░░░░ ░░░░ ░░░░ ░░░░ which ░░░░ ░░░░ do later once we have dependent ░░░░ in our arsenal. ░░░░ of polymorphic constructions ░░░░ ░░░░ $\textsf{id}$ and composition $\_ \circ \_$ ░░░░ ░░░░ ░░░░ a first ░░░░ of ░░░░ ░░░░
March 17, 2025 |
Section: Notes
Part I: Rings The Integers The Well Ordering Principle and Induction ░░░░ general ░░░░ ░░░░ the ░░░░ ░░░░ ░░░░ algebraic structures; we will try to ░░░░ their ░░░░ by ░░░░ ░░░░ which ░░░░ ░░░░ ░░░░ ░░░░ properties. ░░░░ ░░░░ ░░░░ the ░░░░ of ░░░░ $\mathbb{Z}$░░░░ ░░░░ ░░░░ with the ░░░░ ░░░░ of addition $(+)$ ░░░░ multiplication $(\cdot)$. We ░░░░ spend ░░░░ ░░░░ trying ░░░░ ░░░░ how $\mathbb{Z}$ is ░░░░ ░░░░ ░░░░ ░░░░ to these two operations, ░░░░ ░░░░ ░░░░ ░░░░ several ░░░░ ░░░░ ░░░░ will ░░░░ take a ░░░░ ░░░░ those ░░░░ ░░░░ ░░░░ ring axioms, and eventually aim at studying all structures that are defined by requiring a set $A$ along with two operations (which will be called $+$ and $\cdot$ even if they may have nothing to do with the conventional $+$ and $\cdot$) to satisfy the ring axioms. These structures will be called rings: from this perspective, $\mathbb{Z}$ is a particular example of a ring. Other examples being $\mathbb{Q}$ (rational numbers), $\mathbb{R}$ (real numbers), $\mathbb{C}$ (complex numbers); but many more exist, and most of them have nothing to do with numbers░░░░
August 1, 2024 |
Section: Notes
Part I: Essentials Newton’s Laws of Motion Space and Time ░░░░ three ░░░░ of motion are ░░░░ in ░░░░ ░░░░ four ░░░░ underlying ░░░░ ░░░░ ░░░░ ░░░░ space░░░░ time░░░░ mass░░░░ ░░░░ force░░░░ ░░░░ ░░░░ by ░░░░ ░░░░ ░░░░ ░░░░
Space ░░░░ ░░░░ $P$ of the three-dimensional ░░░░ we live ░░░░ can ░░░░ labeled ░░░░ ░░░░ position ░░░░ $\bf{r}$ ░░░░ ░░░░ the distance ░░░░ direction ░░░░ $P$ ░░░░ ░░░░ ░░░░ ░░░░ $O$. ░░░░ most ░░░░ way to ░░░░ a ░░░░ ░░░░ in ░░░░ ░░░░ its ░░░░ ░░░░ ░░░░ ░░░░ the directions ░░░░ by ░░░░ orthonormal basis. ░░░░ ░░░░ ░░░░ ░░░░ ░░░░ introducing unit vectors for each of ░░░░ coordinate ░░░░
August 1, 2023 |
Section: Notes
Introduction Part 1: Historical Introduction The Two Basic Concepts of Calculus The remarkable progress ░░░░ ░░░░ ░░░░ made in science ░░░░ ░░░░ ░░░░ the last ░░░░ is ░░░░ ░░░░ large ░░░░ ░░░░ the ░░░░ of mathematics. That branch of mathematics known as ░░░░ and ░░░░ calculus serves ░░░░ a ░░░░ ░░░░ powerful tool ░░░░ tackling a ░░░░ of problems that arise ░░░░
░░░░ Physics ░░░░ Astronomy
Engineering Chemistry Geology ░░░░ Biology Social Sciences Calculus ░░░░ more ░░░░ a technical ░░░░ it ░░░░ ░░░░ collection of ░░░░ ░░░░ that ░░░░ interesting thinking ░░░░ for ░░░░ ░░░░ ideas ░░░░ ░░░░ ░░░░ with speed, area, volume, rate of growth, continuity, tangent line, and ░░░░ ░░░░ ░░░░ ░░░░ ░░░░ ░░░░ ░░░░ ░░░░ subject ░░░░ its ░░░░ ░░░░ Most ░░░░ these ideas ░░░░ be ░░░░ ░░░░ ░░░░ they revolve ░░░░ two rather ░░░░ problems of a ░░░░ nature. We proceed ░░░░ ░░░░ brief description of these problems.