Formal Sciences

Statistics Building Blocks of Data Science Statistics word cloud collection samples survey

Statistics

Statistics are present in almost every aspect of life. As a primary form of mathematical analysis, statistical data influences everything from insurance rates to consumer economics. Learn more about why statistics careers are fun and highly coveted among professionals ...
systems science chaos theory written on small chalkboard

Systems Science

Systems science is a field of study that organizes independent parts within a larger whole. The process helps to identify places where improvements might make every element of a network improve. Read about how the study of systems has significant implications for the natural and human-made world ...
Chinese Remainder Theorem (Examples Included) Fresh chicken eggs in a basket

Chinese Remainder Theorem

This definitive guide covers proofs, examples, algorithms, applications, and history of the Chinese Remainder Theorem. It also includes links to additional resources such as online articles, courses, books, and tutors to help students learn from a variety of sources. Professionals can also use these resources to increase their knowledge of the field or help structure courses for their students ...
Differentiation Related Rates Applying Implicit Differentiation to rippled drop io n a pond waves

Related Rates (Applying Implicit Differentiation)

In this topic we show how implicit differentiation and the chain rule can be used to calculate the rate of change of one variable in terms of the rate of change of another variable (which may be more easily measured). Introduction to Related Rates The procedure of solving a related rates problem is to find an equation that relates two ...
Data Services Focused analyst showing annual financial report on laptop screen

Data Services for Self-Publishing (Everything to Know)

The data world is hard to break into if you're not literate in numbers and statistics. Luckily, professionals are available to offer data services. This way, your book is ready for distribution while ensuring your content is informative and accessible. Find out what data services can do for your next self-published project ...
Linear Regression Using R Developing programming and coding technologies working in a software engineers developing applications together in office

Linear Regression Using R: An Introduction

Stuck writing linear regressions? Fret not. Linear Regression Using R: An Introduction to Data Modeling will take you back to basics. It is a textbook for beginners that explains how to work in this particular computer language. It takes a keen mind and a wealth of knowledge to data mine. The ability to extract data and analyze it has become ...
Notes on Diffy Qs Happy young university students studying with books and laptops in library

Notes on Diffy Qs: Differential Equations for Engineers

Do you see differential equations and think dreadful equations? This doesn't have to be. Notes on Diffy Qs is an engineer's guide to differential equations. The book is suited for engineering students taking their first course on "diffy Qs." Differential equations are the language of many sciences, including engineering. Therefore, they are very important to understanding the work that engineers ...
Mathematics Scientist Adding Microscopy To Brain Study App

Mathematics

Mathematics uses numbers as a language to explore some of the world's most complex theories and problems. Children in school associate math with difficulty or confusion. With proper study, the subject can become an exciting way to view the world. Read all about the topic and its many branches ...
Computer Science Scientist Adding Microscopy To Brain Study App

Computer Science

Computer science is the study of what computer applications can do for society. A degree in the field offers several opportunities for jobs and areas of study. The growing influence of digital technology makes computer science one of the most important subjects today. Learn about the history and importance of this field ...
Current Events in Mathematics Close-up Shot of a Hand Holding Chalk and Writing Complex and Sophisticated Mathematical Formula Equation on the Blackboard

Current Events in Mathematics

Current events in mathematics point to a growing field with a diversifying range of utilities and practices. The history of mathematics is an interesting dive into the past, present and future. It's long been one of the most studied disciplines. In fact, it's older than most other fields of study in general. To be fair, it does date back to ...
Confluent Relations using Reduction Relations arrow all pointing together to a single point

Confluent Relations (using Reduction Relations)

We discuss confluent relations; in particular, we prove Newman's Lemma: that local confluence, confluence, the Church-Rosser property, and the unique normal forms property are all equivalent for a well-founded relation. We also give a generalization of Newman's lemma based on the Buchberger-Winkler's Property. Reduction Relations Let $\longrightarrow$ be a relation on $X.$ If there exists $c\in X$ such that $a\stackrel{*}{\longrightarrow} ...
Well Founded Relations Well Founded Induction in a swirling cloud of numbers down to the ground Flying numbers and math symbols

Well-Founded Relations (and Well-Founded Induction)

Well-Founded induction is a generalization of mathematical induction. Well-Founded Induction Definition. Let $\longrightarrow$ be a relation on $X.$ 1) If $A\subseteq X$ and $a\in A,$ then $a$ is called a $\longrightarrow$-minimal element of $A$ if there does not exist $b\in A$ such that $a\longrightarrow b.$ 2) If each nonempty subset of $X$ has a $\longrightarrow$-minimal element, then $\longrightarrow$ is called ...
Partial Order Relations Mappings on Ordered Sets

Partial Order Relations (Mappings on Ordered Sets)

We discuss many properties of ordered sets including Noetherian ordered sets and order ideals. We also detail monotone mappings and isomorphisms between ordered sets. Ordered Sets Throughout we assume $(X,\geq)$ is an ordered set. By this we mean that $X$ is a set and that $\geq$ is binary relation on $X$ that is reflexive, antisymmetric, and transitive. Definition. A subset ...
Equivalence Relations Properties and Closures a binary network graph being handed off

Equivalence Relations (Properties and Closures)

Equivalence Relations We discuss the reflexive, symmetric, and transitive properties and their closures. The relationship between a partition of a set and an equivalence relation on a set is detailed. We then give the two most important examples of equivalence relations. Reflexive, Symmetric, and Transitive Relations Definition. Let $X$ be a set. A relation $R$ on $X$ is called reflexive ...
binary relations graph on a blue background

Binary Relations (Types and Properties)

Let $X$ be a set and let $X\times X=\{(a,b): a,b \in X\}.$ A (binary) relation $R$ is a subset of $X\times X$. If $(a,b)\in R$, then we say $a$ is related to $b$ by $R$. It is possible to have both $(a,b)\in R$ and $(a,b')\in R$ where $b'\neq b$; that is any element in $X$ could be related to any ...
Composition of Functions to do recipes with various soups ingredients and space for text on wooden background

Composition of Functions and Inverse Functions

Composition Theorem. Let $f:X\to Y$ be a function. If $g:Y\to Z$ and $g\circ f$ is injective, then $f$ is injective. Proof. Let $x_1, x_2\in X$. Then $$ f(x_1)=f(x_2) \Longrightarrow (g\circ f)(x_1)=(g\circ f)(x_2) \Longrightarrow x_1=x_2 $$ shows that $f$ is injective. Theorem. Let $f:X\to Y$ be a function. If $g:Y\to Z$ and $g\circ f$ is surjective, then $g$ is surjective. Proof ...
One to One Functions and Onto Functions using pencil and paper

One-to-One Functions and Onto Functions

Injective Functions Definition. Let $X$ and $Y$ be sets. A function $f:X\to Y$ is called injective if $$ \forall x_1,x_2\in X, f(x_1)=f(x_2)\implies x_1=x_2. $$ Theorem. Let $f:X\to Y$ be a function. If $f$ is injective and $A\subseteq X$, then $f|_A$ is injective. Proof. Let $x_1, x_2\in A$. Then $$ f|_A(x_1)=f|_A(x_2) \Longrightarrow f(x_1)=f(x_2) \Longrightarrow x_1=x_2 $$ shows that $f|_A$ is injective ...
Functions Their Properties and Importance busy life with all the daily functions

Functions (Their Properties and Importance)

Let $X$ and $Y$ be sets, we say $f$ is a function from $X$ to $Y$ if $f$ is a subset of $X\times Y$ such that the domain of $f$ is $X$ and $f$ has the property: if $(x,y)\in f$ and $(x,z)\in f$ then $x=z$. For each $x\in X$, the unique $y\in Y$ such that $(x,y)\in f$ is denoted by ...
Families of Sets Finite and Arbitrarily Indexed venn diagrams

Families of Sets (Finite and Arbitrarily Indexed)

Families of Sets Finite Unions and Intersections Given any $n$ sets $A_1, A_2, \ldots, A_n$, we define their union to be the set $$A_1 \cup A_2 \cup \cdots \cup A_n =\{x\mid x\in A_i \text{ for some } i, 1\leq i \leq n\}. $$ and their intersection to the the set $$ A_1 \cap A_2 \cap \cdots \cap A_n =\{x\mid x\in ...
Set Theory Basic Theorems with Many Examples

Set Theory (Basic Theorems with Many Examples)

We discuss the basics of elementary set theory including set operations such as unions, intersections, complements, and Cartesian products. We also demonstrate how to work with families of sets. For a brief discussion of the reviews of (elementary) Halmos' Naive Set Theory read this. A solid understanding of propositional and predicate logic is strongly recommended. To get the most from ...
Canonical Forms and Jordan Blocks Stack of Wooden Blocks with binary code on White Backgroun

Canonical Forms and Jordan Blocks

We discuss Jordan bases and the fact that an operator can be put into Jordan canonical form if its characteristic and minimal polynomials factor into linear polynomials. We demonstrate this with an example and provide several exercises. Jordan Basis A basis of $V$ is called a Jordan basis for $T$ if with respect to this basis $T$ has block diagonal ...
Greens Theorem by Example

Green’s Theorem (by Example)

Green’s Theorem for Simply Connected Regions Green's Theorem is named after the mathematician George Green. Theorem. (Green's Theorem) Let $R$ be a simply connected region with a piecewise smooth boundary curve $C$ oriented counterclockwise and let ${F}= M {i}+ N {j}+0 {k}$ be a continuously differentiable vector field on $R,$ then \begin{align*} \oint_C M dx+Ndy= \iint_R \left( \frac{\partial N}{\partial x}-\frac{\partial ...
Probability Density Functions Famous Gauss curve representing the distribution of probability

Probability Density Functions (Applications of Integrals)

Applications of Integrals We will consider the following applications: average value of a function over a region, mass of a lamina, electric charge, moments and center of mass, moments of inertia, and probability density functions. Average Value Recall the average value of an integrable function of one variable on a closed interval is the integral of the function over the ...

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