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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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Convolution

Analysis Harmonic Analysis

Convolution is the operation that "smears" one function against another, producing a weighted average that inherits the best properties of both factors. It is the algebraic operation underlying three pillars of analysis: $L^p$ theory (where Young's inequality controls its integrability), Fourier analysis (where the Convolution Theorem converts it…

Updated 2 days ago

Heat Equation

Analysis Partial Differential Equations

The homogeneous Heat Equation for a function $u : \Omega \times (0, \infty) \to \mathbb{R}$ is the second-order linear PDE: \begin{align} \frac{\partial u}{\partial t} - \Delta u = 0 \end{align} where $\Omega \subseteq \mathbb{R}^n$ is an open set, $t > 0$ denotes time, and $\Delta = \sum_{i=1}^n \frac{\partial^2}{\partial x_i^2}$ is the spatial Laplacian.

Updated 2 days ago

Dirichlet Kernel

Analysis Harmonic Analysis

The Dirichlet kernel is the convolution kernel that represents the Fourier partial sums: $S_N f = f D_N$. Every question about convergence of Fourier series — pointwise, uniform, or in $L^p$ — reduces to a question about $D_N$. The fundamental difficulty of the theory is that $D_N$ is not a positive kernel: it oscillates, its $L^1$ norm grows…

Updated 2 days ago

Fourier Series (Trigonometric)

Analysis Harmonic Analysis

The Orthogonal System page establishes the abstract theory: given a complete orthonormal system in a Hilbert space, every element has a convergent Fourier expansion and Parseval's identity holds. But it leaves open the concrete question: which orthonormal system should we use, and is it complete? This page answers both questions for the…

Updated 2 days ago

Group

Group Theory Group Theory

A group is a pair $(G, \cdot)$ where $G$ is a non-empty set and $\cdot : G \times G \to G$ is a binary operation satisfying: 1. Associativity. For all $a, b, c \in G$, $(a \cdot b) \cdot c = a \cdot (b \cdot c)$. 2. Identity. There exists an element $e \in G$ such that $e \cdot g = g \cdot e = g$ for all $g \in G$. 3. Inverses. For each $g \in G$, there exists $g^{-1} \in G$ such that $g \cdot g^{-1} = g^{-1} \cdot g = e$.

Updated 2 days ago

Fourier Series

Analysis Harmonic Analysis

A Fourier series is the expansion of a function in terms of a complete orthonormal system, expressing the function as an infinite sum of basis elements weighted by its Fourier coefficients. The theory splits into two layers: the abstract Hilbert space framework (which works for any ONS) and the concrete analysis of specific systems (where the…

Updated 2 days ago

Brownian Motion

A standard Brownian motion (or Wiener process) is a stochastic process $(W_t)_{t \ge 0}$ on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ satisfying: 1. Initial condition: $W_0 = 0$ almost surely. 2. Independent increments: For any $0 \le s < t$, the increment $W_t - W_s$ is independent of $\sigma(W_r : r \le s)$. 3. Gaussian increments: For any $0 \le s < t$, $W_t - W_s \sim \mathcal{N}(0, t - s)$. 4. Continuous paths: Almost surely, the map $t \mapsto W_t(\omega)$ is continuous.

Updated 2 days ago

Androma Several Complex Variables I: Domains and Holomorphy

Analysis Complex Analysis

Let $z_j = x_j + i y_j$ be coordinates on $\mathbb{C}^n$. The Wirtinger derivatives with respect to $z_j$ and $\bar{z}_j$ are the first-order differential operators \begin{align} \partial_{z_j} &= \frac{1}{2}\left(\frac{\partial}{\partial x_j} - i\frac{\partial}{\partial y_j}\right), \qquad \partial_{\bar{z}_j} = \frac{1}{2}\left(\frac{\partial}{\partial x_j} + i\frac{\partial}{\partial y_j}\right). \end{align}

Updated 2 days ago

Set Theory I (see Axiomatic Set Theory entry above).

Discrete Mathematics Logic notes Androma Tree: Discrete Mathematics, Logic

A class is a collection described by a first-order formula $\varphi(x, p_1, \dots, p_n)$ with set parameters $p_1, \dots, p_n$.

Updated 2 days ago

Modular Forms I: Classical Theory

notes

The complex upper half-plane is \begin{align} \mathbb H := \{z \in \mathbb C : \operatorname{Im}(z) > 0\}. \end{align}

Updated 2 days ago

Martingale

Probability & Statistics Probability Theory

Let $(\Omega, \mathcal F, \mathbb P)$ be a probability space with filtration $(\mathcal F_t)_{t \in T}$, where $T$ is a linearly ordered time set. A real-valued stochastic process $(X_t)_{t \in T}$ is a martingale with respect to $(\mathcal F_t)_{t \in T}$ if: 1. $(X_t)_{t \in T}$ is adapted to $(\mathcal F_t)_{t \in T}$; 2. $(X_t)_{t \in T}$ is integrable; 3. for all $s, t \in T$ with $s \le t$, \begin{align} \mathbb E[X_t \mid \mathcal F_s] &= X_s \end{align} almost surely.

Updated 2 days ago

Differential Forms and de Rham Cohomology

Let $U \subset \mathbb R^n$ be open. A differential $k$-form on $U$ is a smooth assignment $\omega$ sending each point $x \in U$ to an alternating $k$-linear map \begin{align} \omega_x : (\mathbb R^n)^k \to \mathbb R. \end{align} The vector space of smooth $k$-forms on $U$ is denoted $\Omega^k(U)$. For $k=0$, set $\Omega^0(U)=C^\infty(U)$.

Updated 2 days ago

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Convolution

Heat Equation

Dirichlet Kernel

Fourier Series (Trigonometric)

Group

Fourier Series

Brownian Motion

Androma Several Complex Variables I: Domains and Holomorphy

Set Theory I (see Axiomatic Set Theory entry above).

Modular Forms I: Classical Theory

Martingale

Differential Forms and de Rham Cohomology

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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

Convolution

Heat Equation

Dirichlet Kernel

Fourier Series (Trigonometric)

Group

Fourier Series

Brownian Motion

Androma Several Complex Variables I: Domains and Holomorphy

Set Theory I (see Axiomatic Set Theory entry above).

Modular Forms I: Classical Theory

Martingale

Differential Forms and de Rham Cohomology