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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

Analysis 0

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Model Theory I: Foundations

notes Androma

A first-order signature $L$ consists of three specified collections of non-logical symbols: 1. function symbols $f$, each assigned an arity $n_f \in \mathbb N \cup \{0\}$; 2. relation symbols $R$, each assigned an arity $n_R \in \mathbb N$; 3. constant symbols, regarded as function symbols of arity $0$.

Updated 3 days ago

Convex Geometry

notes Androma

Let $x,y \in \mathbb R^n$. The line segment joining $x$ and $y$ is \begin{align} [x,y] := \{(1-t)x + ty : 0 \le t \le 1\}. \end{align}

Updated 3 days ago

Category Theory II: Adjunctions, Limits, and Abelian Categories

notes Androma

Let $F: \mathcal C \to \mathcal D$ and $G: \mathcal D \to \mathcal C$ be functors. An adjunction $F \dashv G$ is a family of bijections \begin{align} \Phi_{c,d}: \mathcal D(Fc,d) \longrightarrow \mathcal C(c,Gd) \end{align} for all objects $c \in \mathcal C$ and $d \in \mathcal D$, natural in both $c$ and $d$.

Updated 3 days ago

Multivariate Statistical Analysis

notes Androma

Let $(\Omega,\mathcal F,\mathbb P)$ be a probability space. A random vector in $\mathbb R^p$ is a measurable map $X:(\Omega,\mathcal F)\to (\mathbb R^p,\mathcal B(\mathbb R^p))$.

Updated 3 days ago

Androma Notation

Character sets — $\mathbb R$, $\mathbb C$, $\mathbb Z$, $\mathbb Q$, $\mathbb N$ (natural numbers start at 1). $\mathbb R^{n}_{+}:=\{x\in \mathbb R^{n}\mid x_i>0\}$; analogously $\mathbb R^{n}_{-}$. $\mathbb R^{n}_{0}:=\mathbb R^{n}\setminus\{0\}$ (the punctured Euclidean space). Points — a generic element of an open set $U\subset \mathbb…

Updated 2 weeks ago

Several Complex Variables II: Sheaves and Stein Theory

Analysis Complex Analysis

A presheaf of abelian groups on $X$ is a contravariant functor $\mathcal{F}$ from the category of open sets of $X$ (with morphisms given by inclusions) to the category of abelian groups. Concretely, $\mathcal{F}$ assigns: - to each open set $U \subseteq X$ an abelian group $\mathcal{F}(U)$, whose elements are called sections of $\mathcal{F}$ over $U$; - to each inclusion $V \subseteq U$ of open sets a group homomorphism $\rho_{UV}: \mathcal{F}(U) \to \mathcal{F}(V)$, called the restriction map, subject to the conditions: $\rho_{UU} = \mathrm{id}_{\mathcal{F}(U)}$ for all $U$, and $\rho_{VW} \circ \rho_{UV} = \rho_{UW}$ whenever $W \subseteq V \subseteq U$. The restriction of a section $s \in \mathcal{F}(U)$ to $V \subseteq U$ is written $s|_V := \rho_{UV}(s)$.

Updated 3 weeks ago

Ergodic Theory I: Foundations

Analysis Measure Theory

Ergodic theory studies measure-preserving dynamical systems and the long-term statistical behaviour they exhibit. When a transformation acts repeatedly on a probability space while preserving the measure, the central questions become: do time averages of observables converge to a limit, and is that

Updated 3 weeks ago

Holomorphic Function

Analysis Complex Analysis

Let $\Omega \subset \mathbb{C}$ be an open set, and let $f: \Omega \to \mathbb{C}$ be a function. Write $f(x + iy) = u(x, y) + iv(x, y)$ where $u, v: \Omega \to \mathbb{R}$ are real-valued. We say $f$ is holomorphic on $\Omega$ if the partial derivatives $\partial_x u$, $\partial_y u$, $\partial_x v$, $\partial_y v$ exist and are continuous on $\Omega$, and satisfy the Cauchy–Riemann equations: \begin{align} \partial_x u &= \partial_y v, \\ \partial_y u &= -\partial_x v. \end{align}

Updated 3 weeks ago

Solvability by Radicals

Algebra Abstract Algebra

Let $K / k$ be a field extension. We say $K$ is a radical extension of $k$ if $K = k(\alpha)$ for some $\alpha \in K$ with $\alpha^n \in k$ for some $n \in \mathbb{N}$.

Updated 3 weeks ago

Separable Extension

Algebra Abstract Algebra

An algebraic extension $K/k$ is a separable extension if every element $\alpha \in K$ is separable over $k$.

Updated 3 weeks ago

Galois Correspondence

Algebra Abstract Algebra

Let $K/k$ be a field extension. The Galois group of $K$ over $k$ is \begin{align} \operatorname{Gal}(K/k) := \{\sigma: K \xrightarrow{\sim} K \mid \sigma \text{ is a field automorphism and } \sigma(a) = a \text{ for all } a \in k\}. \end{align} The group operation is composition of maps.

Updated 3 weeks ago

Galois Group

Group Theory Group Theory

Let $K/k$ be a Galois extension. The Galois group of $K/k$, denoted $\operatorname{Gal}(K/k)$, is the group of all field automorphisms of $K$ that fix $k$ pointwise: \begin{align} \operatorname{Gal}(K/k) := \{ \sigma \in \operatorname{Aut}(K) : \sigma(\alpha) = \alpha \text{ for all } \alpha \in k \}. \end{align} The group operation is composition of automorphisms.

Updated 3 weeks ago

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Model Theory I: Foundations

Convex Geometry

Category Theory II: Adjunctions, Limits, and Abelian Categories

Multivariate Statistical Analysis

Androma Notation

Several Complex Variables II: Sheaves and Stein Theory

Ergodic Theory I: Foundations

Holomorphic Function

Solvability by Radicals

Separable Extension

Galois Correspondence

Galois Group

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Explore Mathematics

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Trending This Week

Sobolev Space

Laplace's Equation

Table of content in different sections

Recently Updated

All Pages

Model Theory I: Foundations

Convex Geometry

Category Theory II: Adjunctions, Limits, and Abelian Categories

Multivariate Statistical Analysis

Androma Notation

Several Complex Variables II: Sheaves and Stein Theory

Ergodic Theory I: Foundations

Holomorphic Function

Solvability by Radicals

Separable Extension

Galois Correspondence

Galois Group