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For a T0 space $X$, evaluation against all continuous maps $g:X\to S$ is a homeomorphism onto its image under the mapinto a power of the Sierpiński space. The $T_0$ axiom makes it injective, and the coordinate inverse images of $\{1\}$ recover every open subset of $X$.

Between parallel boundaries, a guided shear-horizontal mode has a discrete transverse wavenumber $q_n$ and dispersion relation ${\omega }_n^2=c_S^2(k^2+q_n^2)$. Its cutoff frequency is $c_S{q}_n$.

A sheaf of modules over the sheaf of smooth functions is fine when every locally finite open cover admits a subordinate family of sheaf endomorphisms summing to the identity. Fine sheaves on a paracompact space are acyclic, so their positive-degree sheaf cohomology vanishes.

The Chebyshev estimate isIt gives upper and lower bounds of the correct linear order without asserting the Prime number theorem asymptotic $\psi (x)\sim x$.

A scheme $X$ has characteristic $p$ when $p\cdot 1=0$ in its structure sheaf of a scheme. Equivalently, its structure morphism $X\to \mathrm{Spec}\mathbb{Z}$ factors through $\mathrm{Spec}{\mathbb{F}}_p$.

The structure morphism of a scheme $X$ is its unique morphism $X\to \mathrm{Spec}\mathbb{Z}$. More generally, an $S$-scheme comes equipped with a specified morphism to the base scheme $S$.

For a payoff $h(S_T,M_T)$ with $M_t={\max }_{s\le t}{S}_s$, the pair $(S_t,M_t)$ is a sufficient Markov state for valuation. If an independent restarted stock path has initial value $y$, terminal value $z$, and running maximum $w$, then the continued payoff is

If $\chi$ is an irreducible character of a finite group $G$ and $H\le G$, thenEquality holds exactly when $\chi$ vanishes on $G\setminus H$.

D. Higman's criterion says that an $RG$-module $M$ is relative projective module for $H$ exactly when there is $\alpha \in {\mathrm{End}}_{RH}(M)$ satisfying

Let a finite group $G$ act on an algebra $A$ by conjugation and let $H\le G$. The transfer ideal from $H$ isThe notation records both the source subgroup and the ambient fixed-point algebra.

If $L$ is a simplicial subcomplex of $K$, thenIts homology is the simplicial relative homology $H_k(K,L)$.

If ${\rho }_1,\dots ,{\rho }_k$ are the complex irreducible representations of a finite group $G$ and $n_j=\dim {\rho }_j$, decomposition of the regular representation gives

Let $X=X_1\bigsqcup \cdots \bigsqcup X_r$ and $Y=Y_1\bigsqcup \cdots \bigsqcup Y_s$ be set partitions of the two parts of a bipartite graph. The pair of partitions is $\epsilon$-regular whenThus a uniformly random pair in $X\times Y$ lies in an irregular cell pair with probability at most $\epsilon$.

For the zero-frequency dispersion relationwrite $\mathbf{k}=\kappa (\cos \varphi ,\sin \varphi )$. Thenwhere $\psi$ is the direction of the group velocity. Maximizing the angular slope gives a downstream wake wedge of semi-angle

A finite extension of number fields is unramified everywhere when every finite prime ideal is unramified. If infinite places are included in the convention, one also requires every real place to remain real.

A quasar light curve records its measured brightness as a function of time. Intrinsic accretion variability, measurement noise, and gravitational lensing can all contribute structure.

For a full-column-rank $n\times k$ matrix $V$, extend the thin QR decomposition $V=Q_1{R}_1$ to an orthogonal matrix $Q=(Q_1{Q}_2)$. Thenso $Q^T$ maps the column space of $V$ onto the first $k$ coordinate directions. Successive Householder transformations construct the same reduction without first forming a full orthogonal basis.

Let $\theta =[\overline{a_0,\dots ,a_n}]$, and let ${\theta }^{\prime }$ be its algebraic conjugate, the other fixed point of the Möbius transformation represented by $M(a_0)\cdots M(a_n)$. ThenThe identity follows by transposing the continued-fraction matrix and substituting $X=-1/Y$ in its fixed-point equation.

The derivative at the identity of a smooth curve in $\mathrm{SE}(3)$ is a vector fieldAfter shifting the origin perpendicular to $a\ne 0$, it becomes rotation about the axis parallel to $a$ plus translation along that axis.

For $u^{\prime }=F(x,u)$, a point-symmetry generator $V=\xi {\partial }_x+\eta {\partial }_u$ must satisfyIt follows by applying the first prolongation to $u^{\prime }-F=0$ and restricting to its solution manifold.