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Sobolev Space

The Sobolev space W^k, p(U) consists of all locally summable functions u: U to R such that for each multi-index alpha with |alpha| <= k, the weak derivative D^alphau exists and belongs to L^p(U).

Laplace's Equation

Laplace's equation Delta u = 0 is the prototype elliptic PDE. Every major result in elliptic theory — mean value properties, maximum principles, regularity, Harnack inequalities — appears first and

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Basis and Dimension

Algebra Linear Algebra

Let $V$ be a vector space over a field $F$. A basis for $V$ is a subset $\mathcal{B} \subset V$ such that: 1. $\mathcal{B}$ is linearly independent: for any finite collection $v_1, \ldots, v_k \in \mathcal{B}$ and scalars $c_1, \ldots, c_k \in F$, \begin{align} c_1 v_1 + c_2 v_2 + \cdots + c_k v_k = 0 \implies c_1 = c_2 = \cdots = c_k = 0. \end{align} 2. $\mathcal{B}$ spans $V$: every $v \in V$ can be written as a finite linear combination \begin{align} v = c_1 v_1 + c_2 v_2 + \cdots + c_k v_k \end{align} for some $v_1, \ldots, v_k \in \mathcal{B}$ and $c_1, \ldots, c_k \in F$.

Updated 3 weeks ago

Matrix

Algebra Linear Algebra

Let $\mathbb{F}$ be a field and let $m, n \in \mathbb{N}$. An $m \times n$ matrix over $\mathbb{F}$ is a function \begin{align} A: \{1, \dots, m\} \times \{1, \dots, n\} &\to \mathbb{F} \\ (i, j) &\mapsto A_{ij}, \end{align} where $A_{ij}$ is called the $(i,j)$-entry of $A$. We display $A$ as a rectangular array with $m$ rows and $n$ columns: \begin{align} A &= \begin{pmatrix} A_{11} & A_{12} & \cdots & A_{1n} \\ A_{21} & A_{22} & \cdots & A_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ A_{m1} & A_{m2} & \cdots & A_{mn} \end{pmatrix}. \end{align} The set of all $m \times n$ matrices over $\mathbb{F}$ is denoted $\mathbb{F}^{m \times n}$.

Updated 3 weeks ago

Orthogonality

Algebra Linear Algebra

Let $V$ be a real inner product space. Two vectors $v, w \in V$ are orthogonal, written $v \perp w$, if $\langle v, w \rangle = 0$. More generally, a vector $v \in V$ is orthogonal to a subset $S \subset V$ if $\langle v, s \rangle = 0$ for every $s \in S$.

Updated 3 weeks ago

Linear Independence

Algebra Linear Algebra

Let $V$ be a vector space over a field $F$, and let $v_1, v_2, \ldots, v_k \in V$. The list $(v_1, \ldots, v_k)$ is linearly independent if the only solution to \begin{align} c_1 v_1 + c_2 v_2 + \cdots + c_k v_k &= 0 \end{align} with $c_1, \ldots, c_k \in F$ is $c_1 = c_2 = \cdots = c_k = 0$. The list is linearly dependent if there exist scalars $c_1, \ldots, c_k \in F$, not all zero, satisfying this equation.

Updated 3 weeks ago

Principal Ideal Domain

Algebra Ring Theory

Every student of algebra eventually meets the integers and polynomial rings side by side and notices something striking: both have a division algorithm, both factor elements uniquely, and in both rings every ideal is generated by a single element. This is not a coincidence. These two rings share a s

Updated 3 weeks ago

Unique Factorisation Domain

Algebra Ring Theory

Consider the integers. You learned early on that every integer greater than can be broken into prime factors, and that this decomposition is essentially unique: , not in any fundamentally different sense. This is so familiar it feels like a fact about numbers themselves rather than a structural prop

Updated 3 weeks ago

Sylow Theorems

Group Theory Group Theory

Let $p$ be a prime. A finite group $G$ is a $p$-group if $|G| = p^k$ for some $k \ge 0$. More generally, a subgroup $H \le G$ is a $p$-subgroup of $G$ if $|H|$ is a power of $p$.

Updated 3 weeks ago

Group Action

Group Theory Group Theory

Let $(G, \cdot)$ be a group and let $X$ be a set. A left group action of $G$ on $X$ is a function \begin{align} G \times X &\to X \\ (g, x) &\mapsto g \cdot x \end{align} satisfying the following two axioms: 1. Identity: $e \cdot x = x$ for all $x \in X$, where $e \in G$ is the identity element. 2. Compatibility: $(gh) \cdot x = g \cdot (h \cdot x)$ for all $g, h \in G$ and all $x \in X$. When these axioms hold, we say $G$ acts on $X$ (on the left), and write $G \curvearrowright X$. The set $X$ is called a $G$-set.

Updated 3 weeks ago

Euclidean Domain

Algebra Ring Theory

An integral domain $R$ is a Euclidean domain if there exists a function \begin{align} N: R \setminus \{0\} &\to \mathbb{Z}_{\ge 0} \end{align} called a Euclidean norm (or Euclidean valuation) on $R$, satisfying: for every $a \in R$ and every $b \in R \setminus \{0\}$, there exist $q, r \in R$ such that \begin{align} a &= bq + r \end{align} and either $r = 0$ or $N(r) < N(b)$.

Updated 3 weeks ago

Polynomial Ring

Algebra Ring Theory

Let $R$ be a commutative ring with identity $1_R$. The polynomial ring $R[x]$ is the set of all formal expressions \begin{align} f = a_0 + a_1 x + a_2 x^2 + \cdots + a_n x^n \end{align} where $n \ge 0$ and $a_0, a_1, \ldots, a_n \in R$. Two such expressions are equal if and only if they have identical coefficients in each degree. Addition is defined coefficient-wise: \begin{align} \left(\sum_{i=0}^{m} a_i x^i\right) + \left(\sum_{j=0}^{n} b_j x^j\right) &= \sum_{k=0}^{\max(m,n)} (a_k + b_k) x^k \end{align} (with $a_k = 0$ for $k > m$ and $b_k = 0$ for $k > n$), and multiplication is defined by the Cauchy product: \begin{align} \left(\sum_{i=0}^{m} a_i x^i\right) \cdot \left(\sum_{j=0}^{n} b_j x^j\right) &= \sum_{k=0}^{m+n} \left(\sum_{i+j=k} a_i b_j\right) x^k. \end{align}

Updated 3 weeks ago

Module

Algebra Abstract Algebra

Let $R$ be a ring with multiplicative identity $1_R$. A left $R$-module is an abelian group $(M, +)$ together with a scalar multiplication map \begin{align} R \times M &\to M \\ (r, m) &\mapsto r \cdot m \end{align} satisfying, for all $r, s \in R$ and $m, n \in M$: 1. $r \cdot (m + n) = r \cdot m + r \cdot n$ (distributivity over module addition), 2. $(r + s) \cdot m = r \cdot m + s \cdot m$ (distributivity over ring addition), 3. $(rs) \cdot m = r \cdot (s \cdot m)$ (compatibility with ring multiplication), 4. $1_R \cdot m = m$ (unital axiom).

Updated 3 weeks ago

Real Numbers

Analysis Real Analysis

Take any introductory calculus course and you will encounter the following fact: every bounded increasing sequence converges. This is stated as though it were obvious, but it is not. The statement is false for the rational numbers . Consider the sequence of rational truncations of : \begin{align*}

Updated 3 weeks ago

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Explore Mathematics

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Sobolev Space

Laplace's Equation

Table of content in different sections

Recently Updated

All Pages

Basis and Dimension

Matrix

Orthogonality

Linear Independence

Principal Ideal Domain

Unique Factorisation Domain

Sylow Theorems

Group Action

Euclidean Domain

Polynomial Ring

Module

Real Numbers