Mathematics Articles

Mathematics Scientist Adding Microscopy To Brain Study App

Mathematics (the Ultimate Guide to Becoming a Mathematician)

Mathematics Articles Surprisingly, mathematics has no widely accepted definition. It has been defined in various ways for centuries, such as the science of quantity, logic, or intuition. A few main schools of thought exist today about how to formally define it, but let’s leave that for more in-depth mathematics articles. In a nutshell, you can …

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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 (5 Stories You Need to Know)

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, …

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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  …

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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 …

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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, …

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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 $R$ …

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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, …

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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  $$ f|_A(x_1)=f|_A(x_2)  \Longrightarrow f(x_1)=f(x_2) \Longrightarrow x_1=x_2 $$ …

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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 …

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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= …

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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 …

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Diagonalization of a Matrix fluid dynamics amplitude reaches a critical level start to occur that causes large amounts of wave energy to be transformed into turbulent kinetic energy

Diagonalization of a Matrix (with Examples)

An $n\times n$ matrix $A$ is called diagonalizable if $A$ is similar to some diagonal matrix $D.$ If the matrix of a linear transformation $T$ with respect to some basis is diagonal then we call $T$ diagonalizable. Diagonalization Theorem Theorem. An $n\times n$ matrix $A$ is diagonalizable if and only if it has $n$ linearly …

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Divergence and Curl of a Vector Field Young woman, physics teacher draws a diagram of the electric field

Divergence and Curl of a Vector Field

The Divergence and Curl Definition. Let $\mathbf{V}$ be a given vector field. The divergence of $\mathbf{V}$ is defined by div $\mathbf{V}=\nabla \cdot \mathbf{V}$ and the curl of $\mathbf{V}$ is defined by curl $\mathbf{V}=\nabla \times \mathbf{V}$ where \begin{equation} \nabla =\frac{\partial }{\partial x}\mathbf{i}+\frac{\partial }{\partial y}\mathbf{j}+\frac{\partial }{\partial z}k\end{equation} is the del operator. For a discussion on the physical …

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Determinant of a Matrix Example of calculating of the determinant of a given two by two matrix

Determinant of a Matrix (Theory and Examples)

Determinants and Trace The determinant function can be defined by essentially two different methods. The advantage of the first definition, one which uses permutations, is that it provides an actual formula for $\det(A)$, a fact of theoretical importance. The disadvantage is that, quite frankly, computing a determinant by this method can be cumbersome. A pattern …

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Vector Fields and Gradient Fields

Vector Fields and Gradient Fields

Introduction to Vector Fields Definition. A vector field in $\mathbb{R}^n$ is a function $\mathbf{V}$ that assigns a vector from each point in its domain. A vector field with domain $D$ in $\mathbb{R}^n$ has the form \begin{equation} {V}(x_1, \ldots, x_n) = \langle u_1 (x_1, \ldots,x_n), \ldots, u_n (x_1, \ldots, x_n) \rangle \end{equation} where the scalar functions …

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Inner Products and Orthonormal Bases Face detection and recognition of man Computer vision and machine learning concept

Inner Products and Orthonormal Bases

Recall that the norm of $x\in \mathbb{R}^n$ defined by $\left|\left |x\right|\right | = \sqrt{x_1^2+x_2^2}$ is not linear. To injective linearity into the discussion we introduce the dot product: for $x,y\in \mathbb{R}^n$ the dot product of $x$ and $y$ is defined as $x \cdot y=x_1 y_1+\cdots +x_n y_n.$ Obviously $x \cdot x=\left|\left |x\right|\right |^2$, and with …

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