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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).

Cambridge IA Numbers and Sets

A proof is a finite sequence of statements, each of which is either an axiom, a previously established result, or follows from earlier statements in the sequence by a valid logical rule, such that the

Discrete Mathematics 1

Continuity (Real Analysis)

Consider the polynomial p(x) = x^3 - 2. Since p(1) = -1 0, the function changes sign on [1, 2], so there must be a root somewhere in between — namely sqrt[3]2. This reasoning is the Intermediate Value

Analysis 1

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Cambridge IB Groups, Rings and Modules

A group is a triple $(G, \cdot, e)$ where $G$ is a set, $\cdot : G \times G \to G$ is a binary operation, and $e \in G$ is an element, satisfying: \begin{align} &\text{(Associativity)} \quad (a \cdot b) \cdot c = a \cdot (b \cdot c) \quad \text{for all } a, b, c \in G, \\ &\text{(Identity)} \quad a \cdot e = e \cdot a = a \quad \text{for all } a \in G, \\ &\text{(Inverses)} \quad \text{for all } a \in G \text{ there exists } a^{-1} \in G \text{ with } a \cdot a^{-1} = a^{-1} \cdot a = e. \end{align}

Updated Mar 30

Cambridge II Linear Analysis

Analysis Functional Analysis

A normed vector space is a pair $(V, \|\cdot\|)$ where $V$ is a vector space over a field $\mathbb{F}$ (either $\mathbb{R}$ or $\mathbb{C}$) and $\|\cdot\|: V \to \mathbb{R}$ is a norm, i.e. a function satisfying: (i) $\|v\| \ge 0$ for all $v \in V$, with equality if and only if $v = 0$. (ii) $\|\lambda v\| = |\lambda| \cdot \|v\|$ for all $\lambda \in \mathbb{F}$, $v \in V$. (iii) $\|v + w\| \le \|v\| + \|w\|$ for all $v, w \in V$ (the triangle inequality).

Updated Mar 30

Cambridge II Statistical Physics

Probability & Statistics Probability Theory

Statistical physics bridges the gap between the microscopic laws governing individual particles and the macroscopic phenomena we observe in everyday matter. A gas in a box contains on the order of $10^{23}$ particles, each obeying Newton's laws (or quantum mechanics), yet we describe the gas with just a handful of quantities — pressure,…

Updated Mar 29

Cambridge IA Groups

Group Theory Group Theory

Suppose you hold a square card and are asked: in how many ways can you put it back so that it occupies exactly the same region of the table? You can rotate it by $0°$, $90°$, $180°$, or $270°$; you can also flip it about any of its four axes of symmetry. These eight operations can be composed (do one, then another) and undone (every flip and…

Updated Mar 27

Wave Equation

Analysis Partial Differential Equations

A vibrating string, a drumhead, a sound wave in air — each of these systems transmits disturbances at a finite speed. If we pluck a guitar string at one end, the displacement travels along the string as a pulse; it does not appear instantaneously everywhere. This is the defining physical characteristic of wave propagation, and the mathematical…

Updated Mar 27

Cambridge IB Complex Analysis

Analysis Complex Analysis

A domain is a non-empty, open, path-connected subset $U \subseteq \mathbb{C}$.

Updated Mar 26

Cambridge IA Differential Equations

Calculus Differential Equations

Differential equations sit at the centre of applied mathematics: Newton's second law, the heat equation, population dynamics, electrical circuits — each reduces to a relationship between an unknown function and its derivatives. Solving such equations requires not only ingenuity specific to each problem type, but a robust toolkit of calculus…

Updated Mar 25

Cambridge IA Numbers and Sets

Discrete Mathematics Set Theory

Updated Mar 24

Tempered Distributions

Analysis Harmonic Analysis

The space of Tempered Distributions, denoted by $\mathcal{S}'(\mathbb{R}^n)$, is the topological dual of the Schwartz space. Explicitly, it is the set of all continuous linear functionals $u: \mathcal{S}(\mathbb{R}^n) \to \mathbb{C}$. Linearity means $u \circ (\alpha \phi + \beta \psi) = \alpha (u \circ \phi) + \beta (u \circ \psi)$ for all $\phi, \psi \in \mathcal{S}(\mathbb{R}^n)$ and $\alpha, \beta \in \mathbb{C}$. Continuity implies that if a sequence of test functions $\{\phi_k\}_{k=1}^\infty$ converges to zero in $\mathcal{S}(\mathbb{R}^n)$ (meaning $\|\phi_k\|_{\alpha, \beta} \to 0$ for all multi-indices $\alpha, \beta$), then the sequence of complex numbers $\{u \circ \phi_k\}_{k=1}^\infty$ converges to zero. Here, $u \circ \phi$ denotes the action of the distribution $u$ on the test function $\phi$.

Updated Mar 20

Distribution

Analysis Functional Analysis

Let $\Omega \subseteq \mathbb{R}^n$ be a non-empty open set. The space of test functions on $\Omega$, denoted $\mathcal{D}(\Omega)$, is the vector space of all smooth functions with compact support contained in $\Omega$: \begin{align} \mathcal{D}(\Omega) &:= \{ \varphi \in C^\infty(\Omega) \mid \mathrm{supp}(\varphi) \subset \Omega \text{ is compact} \}. \end{align}

Updated Mar 20

Derivative

Calculus Differential Calculus

Let $E \subseteq \mathbb{R}$, let $a \in E$ be a limit point of $E$, and let $f : E \to \mathbb{R}$. The derivative of $f$ at $a$ is \begin{align} f'(a) &:= \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}, \end{align} provided this limit exists and is finite. Equivalently, \begin{align} f'(a) &= \lim_{x \to a} \frac{f(x) - f(a)}{x - a}. \end{align} If $f'(a)$ exists, $f$ is said to be differentiable at $a$. If $f$ is differentiable at every point of an open set $U \subseteq E$, $f$ is differentiable on $U$, and the function \begin{align} f' : U &\to \mathbb{R} \\ a &\mapsto f'(a) \end{align} is called the derivative of $f$.

Updated Mar 19

Strict Inductive Limit Topology

Analysis Geometry Functional Analysis Topology

The strict inductive limit topology is the canonical construction for building a locally convex topological vector space from a countable ascending chain of locally convex subspaces, each already equipped with its own topology. Rather than seeking a single family of seminorms controlling the entire space, the construction declares a set open if…

Updated Mar 19

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Explore Mathematics

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Sobolev Space

Cambridge IA Numbers and Sets

Continuity (Real Analysis)

Recently Updated

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Cambridge IB Groups, Rings and Modules

Cambridge II Linear Analysis

Cambridge II Statistical Physics

Cambridge IA Groups

Wave Equation

Cambridge IB Complex Analysis

Cambridge IA Differential Equations

Cambridge IA Numbers and Sets

Tempered Distributions

Distribution

Derivative

Strict Inductive Limit Topology