A function is Hölder continuous with exponent $0<\alpha \le 1$ when $|f(x)-f(y)|\le C|x-y{|}^{\alpha }$ for some constant $C$.

On a bounded Lipschitz domain $\Omega \subset {\mathbb{R}}^d$, the Sobolev embedding $W^{1,p}(\Omega )\hookrightarrow L^q(\Omega )$ is compact whenever $q$ lies strictly below the critical Sobolev exponent.

The sobrification has one point for every nonempty irreducible closed subset $A$ of $X$, with opens $\hat{U}=\{A:A\cap U\ne \varnothing \}$. Irreducibility gives $\hat{U}\cap \hat{V}=\widehat{U\cap V}$, and the map from original opens to these opens is bijective, detected by singleton closures. An irreducible closed subset of the new space has complement $\hat{W}$; $F=X\setminus W$ is an irreducible closed subset of $X$ and uniquely generates it. Thus the new space is a sober space.

Let $Y_t=Y_0{\prod }_{j=1}^t{Z}_j$, where $Y_0>0$ is deterministic and the positive factors are independent and integrable. Use the natural filtration. Its finite-horizon Snell envelope isIndeed, backward induction and independence of random variables give $\mathbb{E}[U_{t+1}\mid {\mathcal{F}}_t]=Y_t{c}_{t+1}\mathbb{E}{Z}_{t+1}$; take the maximum with immediate exercise $Y_t$. The constants are finite and at least one. Equivalently, $c_t$ is the maximum of $1$ and the successive products ${\prod }_{j=t+1}^u\mathbb{E}{Z}_j$ for $t<u\le T$. If the factors have a common mean $\lambda >1$, waiting until maturity is optimal and $c_t={\lambda }^{T-t}$. Integrability and independence of random variables from the current information are essential to the finite deterministic coefficient conclusion.

The fixed-vector space is $V^K=\{v:\pi (k)v=v\text{ for all }k\in K\}$. Conjugation gives an isomorphism $V^K\to V^{gK{g}^{-1}}$, $v\mapsto \pi (g)v$. If $K\subseteq L$, then $V^L\subseteq V^K$.

An admissible representation is a smooth representation for which every fixed-vector space $V^K$ for a compact-open subgroup has finite dimension. Admissibility can hold for an infinite-dimensional representation; it bounds each local fixed level, not their union.

The right action on compactly supported locally constant functions is a smooth representation. For a compact-open subgroup $K$, its fixed-vector space has basis the characteristic functions of cosets in $G/K$, since compact support on that discrete quotient is finite support. Consequently this representation is admissible exactly when $G$ is compact.

The unnormalized Jacquet module is the space of coinvariants of a unipotent subgroup $U$. If a subgroup $T$ normalizes $U$, it acts on this quotient. Every linear functional transforming by a linear character trivial on $U$ factors through it. Normalized Jacquet functors include an additional modulus twist; that convention must be specified.

A smooth linear character is a one-dimensional smooth representation, equivalently a homomorphism whose kernel is open. On a local multiplicative group, each smooth linear character is trivial on some sufficiently deep principal-unit subgroup. Its inverse has the same kernel.

A smooth vector in any abstract group representation of a locally profinite group has an open stabilizer, equivalently is fixed by some compact-open subgroup. Smooth vectors form an invariant linear subspace: use intersections for addition and conjugation for translation. A smooth representation is one in which every vector has this property.

A smooth map $f:M\to N$ is transverse to an embedded submanifold $S\subset N$ whenThis is a sum of vector subspaces, not necessarily a direct sum. Equivalently the composition of $d{f}_p$ with the quotient to $T_{f(p)}N/T_{f(p)}S$ is surjective. In a slice chart for an embedded submanifold of codimension $k$, the $k$ normal coordinate functions composed with $f$ have surjective derivative on their common zero set. If the preimage is empty the condition holds vacuously.

A continuous map from a smooth manifold to ${\mathbb{R}}^d$ can be approximated by a smooth map with any prescribed positive continuous pointwise error bound. In coordinate charts, smoothing by local convolution and then combining with a smooth partition of unity gives the approximation. On a compact subset of an open chart domain, a smooth cutoff allows the resulting straight-line homotopy to be supported in that domain. This permits the Sard theorem to be used after approximation; the original continuous map can be space-filling and need not omit a point.

For smooth maps from a compact manifold with a fixed Riemannian metric to Euclidean space, the displayed metric measures uniform convergence of maps and their first derivatives. For a manifold embedded in Euclidean space use its induced metric. Different fixed metrics give equivalent topologies on a compact domain.

A smooth embedding of a compact manifold remains an embedding under sufficiently small C1 metric on a smooth mapping space perturbations. A positive uniform lower bound on its differential preserves immersion. Finite coordinate neighborhoods give uniform local injectivity for nearby maps, and compactness separates the remaining pairs of points. A uniform second-derivative bound gives an explicit local Taylor estimate, although it is not necessary for the C1 openness statement.

Choose finitely many coordinate charts on a compact manifold and smooth functions supported in them, positive on a cover. Extend the displayed chart products by zero. A positive first coordinate and its matching chart coordinates separate points, while their derivatives detect every nonzero tangent vector. Thus the map is an injective immersion; compactness makes it a smooth embedding. Extra zero coordinates make its codimension positive if required.

The internal function spectrum is characterized by $[C,F(A,E){]}_{\mathrm{st}}=[C\wedge A,E{]}_{\mathrm{st}}$. For a based space $K$, notation $F(K,E)$ abbreviates $F({\Sigma }^{\infty }K,E)$.

For a tension-dominated membrane with internal dimension $D$, the massless Gaussian height covariance has spectrum $k_{B}T/(\sigma q^2)$. Removing the uniform translation mode leaves the height difference well-defined. Its variance grows linearly with distance for $D=1$, logarithmically for $D=2$, and approaches a cutoff-dependent constant for $D>2$. A microscopic cutoff is necessary when the short-distance integral diverges. This distinguishes rough positional order from a failure of the local small-slope approximation.

A slip line is a characteristic curve of a yielded plane-strain plastic equilibrium system. The two orthogonal families carry stress and velocity compatibility relations. Their directions need not agree with principal stress directions.

Parameterize a fixed yield surface by stress-space arc length $l$ with ${\xi }_{,l}=-\cos 2\varphi$ and ${\tau }_{,l}=-\sin 2\varphi$. For mean normal stress $\sigma$, force balance gives $\nabla \sigma =M\nabla l$, where $M=\left(\begin{array}{cc}\cos 2\varphi & \sin 2\varphi \\ \sin 2\varphi & -\cos 2\varphi \end{array}\right)$. Its eigenvectors along angles $\varphi$ and $\varphi +\pi /2$ have eigenvalues $+1$ and $-1$, proving the displayed slip line invariants.

For an associated flow rule in the reduced plane-strain yield coordinates, the rate-of-strain tensor has zero extension along both slip line directions. Write velocity as $u{e}_{\alpha }+v{e}_{\beta }$, with the local orthonormal axes turning through angle $\varphi$. Differentiating along an $\alpha$ curve gives longitudinal extension $du-vd\varphi$; along a $\beta$ curve it gives $dv+ud\varphi$. Setting these extensions to zero proves the relations for the corresponding anisotropic yield surface parameterization.