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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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Holder Space

Analysis Functional Analysis

Let $n\in \mathbb{N}_0$ and let $U\subseteq \mathbb{R}^n$ be an open, bounded set. Fix an integer $k\in \mathbb{N}_0$ and an exponent $\alpha\in (0,1]$. A function in this space is a map \begin{align} u:\overline U &\to \mathbb{R}\\ x &\mapsto u(x). \end{align} Define the Hölder seminorm map \begin{align} [\cdot]_{C^{0,\alpha}(\overline U)}: C^0(\overline U) &\to [0,\infty]\\ f &\mapsto \sup_{\substack{x,y\in \overline U\\ x\neq y}} \frac{|f(x)-f(y)|}{|x-y|^\alpha}. \end{align} For a multi-index $\beta=(\beta_1,\dots,\beta_n)\in \mathbb{N}_0^n$ with order $|\beta|=\beta_1+\cdots+\beta_n$, define the partial derivative operator \begin{align} D^\beta u := \partial_1^{\beta_1}\cdots \partial_n^{\beta_n}u \quad \text{on }U. \end{align} The Hölder space $C^{k,\alpha}(\overline U)$ consists of all functions $u\in C^k(\overline U)$ such that the norm \begin{align} \|u\|_{C^{k,\alpha}(\overline U)} := \sum_{|\beta|\le k}\|D^\beta u\|_{L^\infty(U)} + \sup_{|\beta|=k}\,[D^\beta u]_{C^{0,\alpha}(\overline U)} \end{align} is finite, where \begin{align} \|D^\beta u\|_{L^\infty(U)} := \sup_{x\in U}|D^\beta u(x)|. \end{align}

Updated Jan 20

Calculus of Variations (PDEs)

Analysis Partial Differential Equations

Let $n \in \mathbb{N}$ and let $U \subset \mathbb{R}^n$ be a bounded open set. Let $L: \mathbb{R}^n \times \mathbb{R} \times \bar{U} \to \mathbb{R}$ be a smooth function, called the Lagrangian. We denote the arguments of $L$ by $L(p, z, x)$, where $p \in \mathbb{R}^n$ corresponds to the gradient $\nabla u$, $z \in \mathbb{R}$ corresponds to the value $u(x)$, and $x \in \bar{U}$ is the position. The energy functional $I$ is the map: \begin{align} I: \mathcal{A} &\to \mathbb{R} \\ w &\mapsto \int_U L(\nabla w(x), w(x), x) \, dx \end{align} where $\mathcal{A}$ is the admissible class of functions, typically defined by boundary conditions: \begin{align} \mathcal{A} = \{ w \in C^2(\bar{U}) : w = g \text{ on } \partial U \} \end{align} for a given boundary function $g$.

Updated Jan 19

Dynamics of ODEs

Calculus Differential Equations

Let $n \in \mathbb{N}$ and let $U \subseteq \mathbb{R}^n$ be an open set. Let $f: U \to \mathbb{R}^n$ be a continuous function. An autonomous ODE system is defined by the equation: \begin{align} \frac{d}{dt} X(t) = f(X(t)) \end{align} where the solution is a differentiable map: \begin{align} X: I &\to U \\ t &\mapsto X(t) \end{align} defined on an open interval $I \subseteq \mathbb{R}$. We also study parameter-dependent families of solutions. Let $\epsilon \in \mathbb{R}$ be a parameter. We consider the map: \begin{align} X: I \times \mathbb{R} &\to U \\ (t, \epsilon) &\mapsto X(t, \epsilon) \end{align} which satisfies the differential equation for each fixed $\epsilon$: \begin{align} \frac{\partial}{\partial t} X(t, \epsilon) = f(X(t, \epsilon)). \end{align} In this context, we analyze how the trajectory $X(\cdot, \epsilon)$ varies as the parameter $\epsilon$ changes, typically manifesting through variations in the initial condition $X(0, \epsilon)$.

Updated Jan 19

Partition of Unity

Geometry Topology

Let $U \subset \mathbb{R}^n$ be an open set, and let $\mathcal{V} = \{V_i\}_{i \in I}$ be an open cover of $U$ (i.e., $U = \bigcup_{i \in I} V_i$). A partition of unity subordinate to the cover $\mathcal{V}$ is a collection of smooth functions $\{\zeta_i\}_{i \in I}$ defined on $U$ satisfying the following properties: 1. Smoothness: $\zeta_i \in C^\infty(U)$ for all $i \in I$. 2. Range: $0 \le \zeta_i(x) \le 1$ for all $x \in U$. 3. Support: For each $i$, the support of $\zeta_i$ is contained in the cover set $V_i$: \begin{align} \operatorname{supp}(\zeta_i) \subset V_i. \end{align} 4. Locally Finite: For any compact subset $K \subset U$, $\operatorname{supp}(\zeta_i) \cap K \neq \emptyset$ for only finitely many indices $i$. 5. Unity: For every $x \in U$: \begin{align} \sum_{i \in I} \zeta_i(x) = 1. \end{align}

Updated Jan 16

Envelope of Shocks

Analysis Partial Differential Equations

Let $(V, \Phi)$ be a local coordinate chart for the boundary $\Gamma$ around a point $\tilde{c}$, where $V' = \Phi(\Gamma \cap V) \subset \mathbb{R}^{n-1}$ is the flattened parameter space with coordinates $y' = (y_1, \dots, y_{n-1})$. Let $\Psi: I \times V' \to \Omega$ be the Local Characteristic Flow Map defined by $\Psi(s, y') = X(s, y')$, where $X(s, y')$ is the spatial component of the solution to the characteristic ODE system. The Jacobian Matrix $D\Psi(s, y')$ is the $n \times n$ matrix identified by the column vectors corresponding to the characteristic velocity and the wavefront tangent vectors: \begin{align} D\Psi(s, y') = \begin{pmatrix} \vline & \vline & & \vline \\ \frac{\partial X}{\partial s} & \frac{\partial X}{\partial y_1} & \cdots & \frac{\partial X}{\partial y_{n-1}} \\ \vline & \vline & & \vline \end{pmatrix} \end{align} The Envelope of Shocks (relative to this chart) is the set of points $x \in \Omega$ where this matrix is singular: \begin{align} \det(D\Psi(s, y')) = 0 \end{align} The full envelope is the union of such sets over an atlas covering $\Gamma$.

Updated Jan 15

Implicit Function Theorem

Calculus Multivariable Calculus

A point $p \in \mathbb{R}^k$ is called a regular point of a map $F: \mathbb{R}^k \to \mathbb{R}^m$ (where $k > m$) if the total derivative $D F(p)$ has maximal rank (rank $m$). The IFT implies that near any regular point of the zero level set, the set looks like the graph of a function (possibly after permuting coordinates).

Updated Jan 14

Inverse Function Theorem

Calculus Multivariable Calculus

The Inverse Function Theorem provides sufficient conditions for a function to be locally invertible and describes the derivative of that inverse. It is a cornerstone of differential geometry and multivariable calculus, bridging linear approximations with local non-linear behavior. In single-variable calculus, a function is invertible near a…

Updated Jan 14

Boundary

Geometry Topology

Let $U \subset \mathbb{R}^n$ be a subset. The boundary of $U$, denoted by $\partial U$, is the set of points in the closure of $U$ that do not belong to the interior of $U$: \begin{align} \partial U := \bar{U} \setminus U^\circ. \end{align}

Updated Jan 13

Derivative of a radon measure

Analysis Measure Theory

For each point $x\in\mathbb{R}^n$, define the upper and lower derivatives of a Radon measure $\nu$ with respect to another Radon measure $\mu$ by \begin{align} \overline D_{\mu}\nu(x) &:=\limsup_{r\to 0} \frac{\nu\bigl(B(x,r)\bigr)}{\mu\bigl(B(x,r)\bigr)} \quad\text{if }\mu\bigl(B(x,r)\bigr)>0\text{ for all }r>0, \\ &:=+\infty\quad\text{if }\mu\bigl(B(x,r)\bigr)=0\text{ for some }r>0, \end{align} \begin{align} \underline D_{\mu}\nu(x) &:=\liminf_{r\to 0} \frac{\nu\bigl(B(x,r)\bigr)}{\mu\bigl(B(x,r)\bigr)} \quad\text{if }\mu\bigl(B(x,r)\bigr)>0\text{ for all }r>0, \\ &:=+\infty\quad\text{if }\mu\bigl(B(x,r)\bigr)=0\text{ for some }r>0. \end{align}

Updated Oct 2

Mutually Singular Measures

Analysis Measure Theory

In the study of measures, besides understanding when one measure is controlled by another (absolute continuity), it's equally important to have a notion for when two measures are entirely disjoint. This is captured by the concept of mutual singularity. Two measures, $\nu$ and $\mu$, are mutually singular if they "live" on separate, disjoint sets.…

Updated Aug 5

Absolutely Continuous Measures

Analysis Measure Theory

In measure theory, we often want to compare two different ways of assigning "size" to sets, represented by measures $\mu$ and $\nu$. A natural question arises: if one measure $\nu$ is "controlled" by another measure $\mu$, can we express $\nu$ in terms of $\mu$? The notion of control is captured by absolute continuity, denoted $\nu \ll \mu$, which…

Updated Aug 5

Hausdorff Measure

Analysis Measure Theory

Lebesgue measure quantifies the $n$-dimensional “volume’’ of subsets of $\mathbb R^{n}$. Whenever a set has topological dimension strictly less than $n$, its Lebesgue measure equals $0$. For example, a circle sitting in $\mathbb R^{3}$ has $3$-dimensional Lebesgue measure $0$ because a one-dimensional curve occupies no volume. The natural…

Updated Aug 3

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Holder Space

Calculus of Variations (PDEs)

Dynamics of ODEs

Partition of Unity

Envelope of Shocks

Implicit Function Theorem

Inverse Function Theorem

Boundary

Derivative of a radon measure

Mutually Singular Measures

Absolutely Continuous Measures

Hausdorff Measure

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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

Holder Space

Calculus of Variations (PDEs)

Dynamics of ODEs

Partition of Unity

Envelope of Shocks

Implicit Function Theorem

Inverse Function Theorem

Boundary

Derivative of a radon measure

Mutually Singular Measures

Absolutely Continuous Measures

Hausdorff Measure