Players hold finite sets of resources, and their coalition value is the number of distinct resources they collectively cover. The value is generally subadditive rather than superadditive: overlapping resources are counted only once. It is a superadditive coalitional game exactly when the players' resource sets are pairwise disjoint.

A set family $\mathcal{F}$ shatters a set $Z$ when its trace of a set family on $Z$ realizes every subset of $Z$:Each possible include-or-exclude pattern on $Z$ must occur, although different patterns may come from different members of $\mathcal{F}$.

For example, on $X=\{1,2\}$ let $\mathcal{F}=\{\varnothing ,\{1\},\{2\}\}$. It shatters either singleton, but it does not shatter $X$, because the subset $\{1,2\}$ never occurs as a trace of a set family. Its VC dimension is therefore one.

The VC dimension is the largest finite size of a shattered set, or infinity if arbitrarily large finite sets are shattered. The Sauer-Shelah lemma bounds the number of traces when a fixed size cannot be shattered. This turns a local pattern-realization condition into a global bound on the size of a set family.

For a finite square matrix $X$, maximize over unitary operators. Write its singular value decomposition as $X=PD{Q}^{†}$. Then $|\mathrm{Tr}(XU)|=|\mathrm{Tr}(D{Q}^{†}UP)|\le {\sum }_j{D}_{jj}$, since every diagonal entry of a unitary has modulus at most one. Choosing $U=Q{P}^{†}$ attains equality, proving the formula for the trace norm.

A regular tournament has equal outdegree and indegree at every vertex. On $2k+1$ vertices, label by residues modulo $2k+1$ and direct $i\to j$ when $j-i\in \{1,\dots ,k\}$. This gives an explicit regular tournament. It contains no outward-oriented star with more than $k$ leaves.

A median order maximizes the number of forward directed edges in a linear ordering of the vertices of a tournament. Every interval inherits a median order. Moving the first vertex of an interval to its end proves that it has at least as many outgoing as incoming edges in that interval; otherwise the move would increase the number of forward edges. Reversing all edges and the order gives the dual assertion for the last vertex.

For a timelike unit normal with squared norm epsilon and vanishing ambient Ricci tensor, contraction of the Gauss equation for a nonnull hypersurface gives the displayed formula. In signature $(+---)$ the timelike normal has epsilon plus one, so the scalar of the induced negative-definite metric is nonnegative. In signature $(-+++)$ the positive-definite spatial metric instead has scalar $-6{\lambda }^2$. Replacing the induced metric by its negative reverses the scalar, so the inequality must not be transferred between these conventions without adjustment.

// Target: general-relativity.bigb

A CM-field is a totally imaginary number field that is a quadratic extension of a totally real number field. Its nontrivial automorphism over that real subfield acts as complex conjugation in every complex embedding. Imaginary quadratic number fields are examples. An arbitrary totally imaginary number field need not have such a real subfield of index two.

For a càdlàg function, replace the last sampled value at $2^{-n}\lceil 2^{n}t\rceil$ by $X_t$. Right continuity makes this change tend to zero. The corrected sum is bounded by the total variation of a function. Conversely every finite partition of an interval can be approximated by dyadic points from the right, retaining the endpoint $t$. The triangle inequality bounds that partition's limiting sum by the dyadic sums. Taking the supremum proves convergence, including the value $+\infty$.

For $p\ge 1$, the displayed quantity measures the accumulated oscillation of a path in a metric space. For $p=1$ it is total variation. A Hölder continuous path of exponent $\alpha$ has finite $p$-variation on a compact interval if $p\alpha \ge 1$. Applying this construction to the Carnot-Carathéodory distance gives the defining regularity of a weak geometric p-rough path.

An aleph-one-like linear order has size ${\aleph }_1$ and every point has only countably many predecessors. Consequently every proper initial segment is countable, since it is bounded by any point outside it. This differs from requiring every nonempty interval to be uncountable.

A total order is dense when between every two distinct ordered points lies a third. Endpoints are allowed; “without endpoints” is an additional condition. A dense order with two points $a<b$ cannot be a well-order, since $\{x:a<x\}$ is nonempty and has no least member. Its first-order axioms are the total-order axioms plus $\forall x,y(x<y\Rightarrow \exists z(x<z<y))$. Countability and uncountability are not first-order axiomatizable restrictions, by the upward and downward witness constructions of the Löwenheim-Skolem theorem.

A totally ordered set is a set equipped with a total order. Internally in a topos, weak totality does not imply decidable equality. The empty ordered set is allowed unless an inhabitation axiom is explicitly imposed.

A maximum of a subset $S$ of a total order is an element $m\in S$ with $s\le m$ for every $s\in S$. If it exists it is unique. Every nonempty finite subset has a maximum, by repeated pairwise comparison. Unlike a supremum, a maximum must belong to the subset. The completion time of jobs processed on finitely many parallel machines is the maximum of the machine completion times.

A minimum of $A$ is a member of $A$ less than or equal to every member of $A$. It equals the infimum when that lower bound is attained. A nonempty finite subset of a totally ordered set has a minimum; a nonempty open real interval does not.

Every closed regular space curve has total curvature at least $2\pi$. Equality holds exactly for convex plane curves traversed once. Here curvature is unsigned and integration is with respect to arc length.

For coprime p,q with absolute values at least two, the knot group is $⟨a,b\mid a^p=b^q⟩$. Its nontrivial central element $z=a^p=b^q$ maps nontrivially to the abelianization. A meridian $a^u{b}^v$, where $qu+pv=1$, has infinite order in the quotient $C_{|p|}\ast C_{|q|}$. A composite knot group would be an amalgamated free product of proper factors over its meridian subgroup, whose center lies in that subgroup. Then z would be a meridian power, contradicting the infinite order of the meridian modulo z.

// Target: geometric-group-theory.bigb

Collapsing an essential meridian circle of a torus to one point gives the pinched torus. In the torus's CW structure, the two-cell attaches to two circles by the commutator. Collapsing one circle leaves one one-cell and a two-cell attached by a null-homotopic word. Thus the quotient is homotopy equivalent to $S^1\vee S^2$. Its cellular chain groups are $\mathbb{Z}$ in degrees zero, one and two, with zero differentials, so its integral homology groups are $H_0=H_1=H_2=\mathbb{Z}$ and vanish in all higher degrees. This illustrates that collapsing a circle can preserve a top-dimensional homology class even while reducing the first homology rank.

Take $u,v$ modulo $\pi$; shifting either coordinate by $\pi$ leaves $h$ unchanged. Its partial derivatives are $h_u=\sin v\sin (2u-v)$ and $h_v=\sin u\sin (u-2v)$. The only critical points are $(0,0)$, $(\pi /3,2\pi /3)$ and $(2\pi /3,\pi /3)$. The latter are a nondegenerate minimum and maximum, with values $\mp 3\sqrt{3}/8$. At the origin the leading cubic is $uv(u-v)$, linearly equivalent to a monkey saddle; its critical point is isolated but degenerate. Thus three smooth critical points coexist with the Morse lower bound of four for the torus.

For an abelian group $G$ and an integer $m\ge 1$, its $m$-torsion subgroup is $G[m]=\{x\in G:mx=0\}$, the kernel of multiplication by $m$. In a finite abelian group, every nonempty fiber of this homomorphism has cardinality $|G[m]|$, so $|G:mG|=|G[m]|$. This follows directly: if $mx=my$, then $x-y\in G[m]$, and conversely translating by an element of $G[m]$ leaves the image unchanged. On the cyclic group ${\mathbb{Z}}_N$, the solutions of $mx=0$ number $\gcd (m,N)$; divide both integers by their greatest common divisor to see that $x$ must be a multiple of $N/\gcd (m,N)$.

Enumerate all words $w_i$ on two generators in a free pro-p group. Choose powers $p^{e_i}$ so large that ${\sum }_i{\tau }^{p^{e_i}}<2\tau -1$ for some $1/2<\tau <1$, and impose the closed normal relations $w_i^{p^{e_i}}=1$. The Golod-Shafarevich infinitude criterion for pro-p groups makes the resulting pro-$p$ group infinite. Its abstract subgroup generated by the two generator images is dense and infinite; each element is represented by an enumerated word and has finite order. The ambient finite quotients separate its elements, so it is also a residually finite group. The abstract subgroup, rather than the whole topological group, is the desired finitely generated torsion group.