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The simplicial mapping space is defined byIf $A\hookrightarrow B$ is a monomorphism and $X$ is a Kan complex, restriction $\underline{\mathrm{Hom}}(B,X)\to \underline{\mathrm{Hom}}(A,X)$ is a Kan fibration.

A Kan complex is a simplicial set in which every simplicial horn, including every outer horn, has a filler.

A quasicategory is a simplicial set in which every inner horn has a filler.

The standard simplicial $n$-simplex is the representable simplicial set

For a simplicial abelian group $A$, one convention for normalized chains isEquivalently, it is the quotient of the unnormalized chain complex by its degenerate subcomplex.

For $d$ independent symmetric walks on $\mathbb{Z}$, the probability that all are at the origin at time $2n$ isThe Borel-Cantelli lemmas therefore show that simultaneous returns occur only finitely often almost surely when $d\ge 3$.

For the symmetric walk and $V_{2n}=|\{0\le j\le 2n:S_j=0\}|$,Since the central binomial coefficient satisfies $\left(\genfrac{}{}{0ex}{}{2k}{k}\right)/4^k\asymp k^{-1/2}$, this expectation is of order $\sqrt{n}$.

For unit left eigenvector $u$ and right eigenvector $v$ of a real matrix at a simple eigenvalue $\lambda$, its normwise sensitivity isIt measures the first-order amplification of a matrix perturbation into an eigenvalue perturbation.

Every tree $T$ satisfies the Sidorenko conjecture. If a bipartite host graph has edge density $\alpha$, then a uniformly random bipartition-respecting map from a $k$-vertex tree into the host is a graph homomorphism with probability at least ${\alpha }^{k-1}$.

If an ordinal is partitioned into two copies of type $\beta$, its type is below $\beta +\beta +\beta$; two copies do not always give a strict bound.

The connecting map sends a homology class in the quotient complex to the class obtained by lifting a representative, applying the middle differential, and identifying the result in the subcomplex.

A cosmological caustic is the image of a singular point of the Lagrangian map. The ideal collisionless density diverges there, although velocity dispersion and other small-scale physics regularize the singularity.

For $X=U\cup V$, the short exact sequence of sheaves obtained by restricting to $U$, $V$, and $U\cap V$ induces a long exact sequence

For an open cover $\mathcal{U}=(U_i)$, the Čech cochain group iswith the alternating sum of restrictions as differential. Its cohomology is the Čech cohomology of $\mathcal{F}$ with respect to $\mathcal{U}$.

A sheaf is flasque when every restriction map is surjective. Flasque sheaves are acyclic for global sections, so $H^i(X,\mathcal{F})=0$ for $i>0$.

For a proper Euclidean motion of Euclidean three-space $E(x)=Ax+b$, transport the unit normal by $\tilde{N}(E(p))=AN(p)$. The corresponding shape operators are orthogonally conjugate:Their determinant and trace, hence Gaussian curvature and mean curvature, are unchanged.

At a point of a smooth surface, a normal section of a surface can bend by different amounts depending on its tangent vector. The principal curvatures are the largest and smallest signed normal curvatures among unit tangent vectors. Equivalently they are the two eigenvalues $k_1,k_2$ of the shape operator, with an orthonormal basis $e_1,e_2$ of corresponding principal directions. Here the labels follow the selected directions, without requiring $k_1\le k_2$. ^[ref]

For a unit tangent vector making angle $\theta$ with $e_1$, Euler formula for normal curvature givesThe normal curvature is a weighted average of the principal curvatures; the extremes occur in the principal directions. If the principal curvatures coincide, every direction in the tangent plane is a principal direction: this is an umbilical point. The Gaussian curvature and mean curvature summarize the pair by

For an outward-oriented sphere of radius $R$, both principal curvatures are $-1/R$, so the normal curvature is $-1/R$ in every direction. For an outward-oriented circular cylinder, choose $e_1$ around the circumference and $e_2$ along the axis. The principal curvatures are $-1/R,0$, giving $k_n(\theta )=-{\cos }^2\theta /R$; halfway between the principal directions, $k_n(\pi /4)=-1/(2R)$.

At the origin of the upward-oriented hyperbolic paraboloid $z=(x^2-y^2)/2$, the principal directions are the coordinate axes and the principal curvatures are $1,-1$ in the chosen length unit. Thus $k_n(\theta )=\cos (2\theta )$, with zero normal curvature at $\theta =\pi /4,3\pi /4$. Zero normal curvature in these directions coexists with Gaussian curvature $-1$: the shape operator is not zero. The mean curvature is zero at this point; this does not say that the whole hyperbolic paraboloid is a minimal surface.

Each graph uses angle $\theta$ in the tangent plane, measured from the indicated first principal direction. The sphere and circular cylinder have $R=1$; the saddle graph refers only to the origin. Reversing the unit normal reflects each normal curvature graph through zero, negates both principal curvatures and the mean curvature, and preserves their product, the Gaussian curvature.

The unique cubic subfield is the maximal real subfieldandIt is the fixed field of complex conjugation.

The unique quadratic subfield of $\mathbb{Q}({\zeta }_7)$ isThe conjugate period is $-1-\eta$, and

If finitely many open convex subsets $K_1,\dots ,K_n$ of a locally convex space have empty intersection, some continuous linear map $T:X\to {\mathbb{R}}^{n-1}$ preserves that fact: ${\bigcap }_{i}T(K_i)=\varnothing$.