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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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Fourier Transform

Analysis Harmonic Analysis

Let $f \in L^1(\mathbb{R}^n; \mathbb{C})$. The Fourier Transform is the mapping $\mathcal{F}$ defined as follows: \begin{align} \mathcal{F}: L^1(\mathbb{R}^n) &\to L^\infty(\mathbb{R}^n) \\ f &\mapsto \hat{f} \end{align} where the function $\hat{f}$ is given by the integral: \begin{align} \hat{f}: \mathbb{R}^n &\to \mathbb{C} \\ \xi &\mapsto \frac{1}{(2\pi)^{n/2}} \int_{\mathbb{R}^n} f(x) e^{-i x \cdot \xi} \, dx. \end{align} Here, the factor $(2\pi)^{-n/2}$ is the scaling normalization chosen to ensure the transform is unitary on $L^2(\mathbb{R}^n)$.

Updated Feb 28

Hilbert Space

Analysis Functional Analysis

Let $H$ be a real vector space. An inner product on $H$ is a map $(\cdot\,,\cdot)_H: H \times H \to \mathbb{R}$ satisfying, for all $x, y, z \in H$ and $\alpha \in \mathbb{R}$: 1. Symmetry: $(x, y)_H = (y, x)_H$. 2. Linearity in the first argument: $(\alpha x + z, y)_H = \alpha(x, y)_H + (z, y)_H$. 3. Positive definiteness: $(x, x)_H \ge 0$, with equality if and only if $x = 0$.

Updated Feb 23

Hyperbolic Systems of First-Order Equations

Analysis Partial Differential Equations

The second-order hyperbolic theory treats a single scalar equation $u_{tt} + Lu = f$ for one unknown function $u$. Many important physical systems — the equations of gas dynamics, electromagnetism, elasticity — naturally involve several coupled unknowns evolving simultaneously. These are described by first-order hyperbolic systems, where the…

Updated Feb 22

Second-Order Hyperbolic Equations

Analysis Partial Differential Equations

The parabolic theory models diffusion — processes that smooth out irregularities and dissipate energy over time. Many physical systems, however, are governed by wave propagation: vibrating membranes, acoustic pressure, electromagnetic fields. These are described by hyperbolic equations, where the second-order time derivative $u_{tt}$ replaces the…

Updated Feb 22

Hopf Bifurcation

Calculus Differential Equations

Let $A(\mu) := D_X F(0, \mu)$ be the Jacobian matrix of the system at the origin. The system satisfies the Hopf Spectral Hypothesis at $\mu=0$ if the spectrum $\sigma(A(0))$ satisfies: 1. Imaginary Crossing: There exists a pair of simple, purely imaginary eigenvalues: \begin{align} \lambda_{1,2}(0) = \pm i\omega_0, \quad \omega_0 > 0 \end{align} 2. Spectral Gap: All other eigenvalues $\lambda_j$ satisfy the hyperbolic condition: \begin{align} \text{Re}(\lambda_j) \neq 0, \quad \forall j > 2 \end{align}

Updated Feb 21

Entropy Condition (Scalar Conservation Laws)

Analysis Partial Differential Equations

The conservation of a physical quantity — mass, momentum, energy — in the absence of sources takes the form $u_t + f(u)_x = 0$, where $u(x,t)$ is the conserved density and $f(u)$ is the flux. This equation is deceptively simple. For smooth initial data it has a smooth solution locally in time via the method of characteristics. But because…

Updated Feb 21

Distributional Derivative

Analysis Partial Differential Equations

Let $T \in \mathcal{D}'(\Omega)$ be a distribution. Let $\alpha \in \mathbb{N}_0^n$ be a multi-index. The distributional derivative of order $\alpha$, denoted $\partial^\alpha T$, is the functional defined by the mapping: \begin{align} \partial^\alpha T: \mathcal{D}(\Omega) &\to \mathbb{R} \\ \varphi &\mapsto (-1)^{|\alpha|} \langle T, \partial^\alpha \varphi \rangle. \end{align} Here, $\langle T, \psi \rangle$ denotes the action of the distribution $T$ on the test function $\psi$.

Updated Feb 21

Algebraic Geometry Overview

Algebra Abstract Algebra

Algebraic geometry is a branch of mathematics studying zeros of multivariate polynomials. An algebraic variety is the set of solutions of a system of polynomial equations.

Updated Feb 10

Brouwer Fixed Point Theorem

Geometry Topology

The Brouwer fixed point theorem is a foundational result in topology and nonlinear analysis. It addresses the fundamental question of when a continuous function mapping a set into itself admits a "fixed point"—a point $\bar{x}$ such that $f(\bar{x}) = \bar{x}$. Unlike the contraction mapping principle, which relies on strict metric properties to…

Updated Jan 23

Contraction Mapping Principle

Analysis Functional Analysis

Let $(X, d)$ be a metric space. A mapping $T: X \to X$ is called a contraction mapping (or simply a contraction) if there exists a constant $k \in \mathbb{R}$ with $0 \le k < 1$ such that for all $x, y \in X$: \begin{align} d(T(x), T(y)) \le k d(x, y). \end{align} The constant $k$ is called the Lipschitz constant or contraction factor.

Updated Jan 23

Rademacher's Theorem

Analysis Real Analysis

Let $U \subseteq \mathbb{R}^n$ be an open set. A function $f: U \to \mathbb{R}^m$ is called Lipschitz continuous (or simply Lipschitz) if there exists a constant $L \ge 0$ such that $|f(x) - f(y)| \le L |x - y|$ for all $x, y \in U$. The smallest such $L$ is denoted by $\text{Lip}(f)$.

Updated Jan 21

Spherical Integrals and Radial Functions

Analysis Harmonic Analysis

Let $n \in \mathbb{N}$ with $n \ge 2$. The open ball of radius $R > 0$ centered at the origin is the set $B(0,R) \subseteq \mathbb{R}^n$ defined by: \begin{align} B(0,R) &:= \{ x \in \mathbb{R}^n : |x| < R \}. \end{align} A function $g: B(0,R) \to \mathbb{R}$ is called radial if there exists a profile function $f: [0, R) \to \mathbb{R}$ such that: \begin{align} g(x) &= f(|x|) \quad \text{for all } x \in B(0,R). \end{align} We assume $f$ is Borel measurable and integrable with respect to the relevant measures.

Updated Jan 21

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Fourier Transform

Hilbert Space

Hyperbolic Systems of First-Order Equations

Second-Order Hyperbolic Equations

Hopf Bifurcation

Entropy Condition (Scalar Conservation Laws)

Distributional Derivative

Algebraic Geometry Overview

Brouwer Fixed Point Theorem

Contraction Mapping Principle

Rademacher's Theorem

Spherical Integrals and Radial Functions

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Explore Mathematics

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Browse by Area

Trending This Week

Sobolev Space

Cambridge IA Numbers and Sets

Continuity (Real Analysis)

Recently Updated

All Pages

Fourier Transform

Hilbert Space

Hyperbolic Systems of First-Order Equations

Second-Order Hyperbolic Equations

Hopf Bifurcation

Entropy Condition (Scalar Conservation Laws)

Distributional Derivative

Algebraic Geometry Overview

Brouwer Fixed Point Theorem

Contraction Mapping Principle

Rademacher's Theorem

Spherical Integrals and Radial Functions