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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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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

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

Expectation

Probability & Statistics Probability Theory

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space and let $X: \Omega \to \mathbb{R}$ be a random variable. Write $X^+ = \max(X, 0)$ and $X^- = \max(-X, 0)$, so that $X = X^+ - X^-$ and $|X| = X^+ + X^-$. If $\min(\mathbb{E}[X^+], \mathbb{E}[X^-]) < \infty$, the expectation of $X$ is \begin{align} \mathbb{E}[X] = \mathbb{E}[X^+] - \mathbb{E}[X^-] \in [-\infty, \infty]. \end{align} The random variable $X$ is called integrable if $\mathbb{E}[|X|] = \mathbb{E}[X^+] + \mathbb{E}[X^-] < \infty$, in which case $\mathbb{E}[X] \in \mathbb{R}$.

Updated 2 days ago

Sobolev Space

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

Updated Apr 16

Analytic Continuation

Analysis Complex Analysis

Let $\Omega_1, \Omega_2 \subset \mathbb{C}$ be connected open sets with $\Omega_1 \cap \Omega_2 \neq \varnothing$, and let $f_j: \Omega_j \to \mathbb{C}$ be holomorphic for $j = 1, 2$. We say that $f_2$ is a direct analytic continuation of $f_1$ to $\Omega_2$ if \begin{align} f_1(z) &= f_2(z) \quad \text{for all } z \in \Omega_1 \cap \Omega_2. \end{align}

Updated 2 days ago

Meromorphic Function

Analysis Complex Analysis

Let $\Omega \subset \mathbb{C}$ be a connected open set. A function $f: \Omega \to \mathbb{C} \cup \{\infty\}$ is meromorphic on $\Omega$ if there exists a discrete closed subset $P \subset \Omega$ such that: 1. $f$ is holomorphic on $\Omega \setminus P$, 2. every point of $P$ is a pole of $f$. The set $P$ is called the pole set of $f$. We write $f \in \mathcal{M}(\Omega)$ for the class of meromorphic functions on $\Omega$.

Updated 2 days ago

Winding Number

Analysis Complex Analysis

Let $\gamma: [a, b] \to \mathbb{C}$ be a piecewise $C^1$ closed curve, so that $\gamma(a) = \gamma(b)$. Let $z_0 \in \mathbb{C} \setminus \operatorname{im}(\gamma)$. The winding number of $\gamma$ around $z_0$ is the integer \begin{align} n(\gamma, z_0) &= \frac{1}{2\pi i} \oint_\gamma \frac{dz}{z - z_0}. \end{align}

Updated 2 days ago

Androma Notation

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Laplace's Equation

Analysis Partial Differential Equations

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Lebesgue Measure

Analysis Tree: Analysis

Let $\mathcal{M}_{\mathcal{L}}(\mathbb{R}^n)$ denote the class of Lebesgue measurable subsets of $\mathbb{R}^n$. Lebesgue measure on $\mathbb{R}^n$ is the function \begin{align} \mathcal{L}^n: \mathcal{M}_{\mathcal{L}}(\mathbb{R}^n) &\to [0,\infty] \\ E &\mapsto \mathcal{L}^{n,}(E). \end{align*}

Updated 1 day ago

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Brownian Motion

Martingale

Expectation

Sobolev Space

Analytic Continuation

Meromorphic Function

Winding Number

Androma Notation

Laplace's Equation

Lebesgue Measure