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An operator product expansion rewrites nearby local local field operators as a sum of local local field operators with separation-dependent coefficient functions: ${\mathcal{O}}_i(x){\mathcal{O}}_j(0)={\sum }_k{C}_{ij}{}^k(x){\mathcal{O}}_k(0)$, in the appropriate short-distance or convergent correlation function sense. The coefficients can contain singular terms, but regular terms and descendants also carry information.

For example, the stress-tensor operator-product expansion in a two-dimensional conformal field theory contains the identity contribution fixed by the central charge, followed by the stress-energy tensor and its derivative. Substituting an operator product expansion into a correlation function reduces a nearby pair to a sum of simpler insertions. Its domain and convergence properties depend on the theory and operator configuration; a schematic short-distance equality is not an unrestricted multiplication rule at every separation.

A graph-theoretic rhombus consists of two triangles in a graph sharing an edge. Every rhombus-free simple graph on $n\ge 4$ vertices has at mostedges, attained by the balanced complete bipartite graph. A triangle-free graph satisfies this by Mantel theorem. If a triangle is present, each vertex outside it has at most one neighbour in it: two neighbours would form another triangle sharing an edge.

For $n=4,5$ this gives at most $4,6$ edges. For $n=6$, the remaining three vertices either have at most two edges, giving at most $3+3+2=8$, or form a second triangle. In the latter case the edges between the triangles form a matching, so there are at most nine edges. For $n\ge 7$, remove a triangle and apply induction to the remaining $n-3\ge 4$ vertices. The edge count is at most $\lfloor (n-3{)}^2/4\rfloor +n\le \lfloor n^2/4\rfloor$. Checking the small bases avoids applying the asserted bound to a three-vertex triangle, where it is false.

Every $n$-vertex graph containing no $K_{r+1}$ satisfies

If $f\in L^1\cap L^{\infty }$ and $\hat{f}\ge 0$, testing against triangular frequency cutoffs and using monotone convergence gives $\Vert \hat{f}{\Vert }_1\le 2\pi \Vert f{\Vert }_{\infty }$.

A triangle-free graph contains no triangle.

For nonnegative integer supplies $s_i$ and demands $d_j$ with equal totals, the complete transportation problem has an integral optimal shipment matrix for every finite real cost array. A feasible matrix exists, and its nonnegative entries are bounded by their row supplies, so the transportation polytope is compact and a linear objective attains its minimum.

Its constraints are the vertex-edge incidence equations of a bipartite graph. Negating the demand rows gives a directed incidence matrix, which is a totally unimodular matrix. After deleting one redundant equality, any independent square basis has determinant $+1$ or $-1$. An integer right side therefore gives integer basic coordinates by Cramer's rule. The Fundamental theorem of linear programming supplies an optimal basic feasible solution, proving integrality.

There is also an algorithmic view. The northwest corner method starts with integer shipments; a cycle pivot adds and subtracts an integer minimum along its alternating cycle, preserving integrality. Zero allocations may be needed to complete a degenerate basis. With forbidden routes, equality of total supply and demand is no longer enough for feasibility, though an existing transportation feasible polytope retains the same incidence-matrix integrality property.

The transportation simplex computes row and column potentials, enters a negative-reduced-cost cell, and pivots around its induced alternating cycle.

The transportation polytope is the set of nonnegative matrices with fixed row and column sums.

If $p$ has degree $n$ and leading coefficient $a_n$, then for nonzero steps $r_1,\dots ,r_n$,Consequently a degree-$n$ polynomial cannot be the sum of $n$ functions each periodic with some nonzero period.

A set is hereditarily countable when its transitive closure is countable. These sets form the transitive set $H_{{\omega }_1}$ and all have rank below ${\omega }_1$.

The relativization ${\phi }^M$ of a formula $\phi$ is obtained recursively by restricting every quantifier to $M$:For parameters in $M$, the ambient statement ${\phi }^M$ holds exactly when the structure $(M,\in )$ satisfies $\phi$.

If a transitive class $M$ contains the empty set and is closed under pairing and union, then $(M,\in )$ satisfies extensionality, empty set, pairing, and union. Transitivity makes all members of each $x\in M$ visible inside $M$, so the ambient witnesses have the same required membership relations internally.

For nonzero $x\in K$, choose $y=x^{-1}$. Thenso the trace form is nondegenerate.

The positive semidefinite trace ball of radius $s\ge 0$ isIt is closed and convex, and $\Vert Z{\Vert }_F\le \mathrm{tr}Z\le s$ for every $Z\in {\mathcal{S}}_s$.

An initial segment contains every element below each of its elements.

An element is greatest when every element of the ordered set is less than or equal to it.

The empty order is the unique total order on the empty set.

A well-order is a total order in which every nonempty subset has a least element.

For the standard longitude $a$ and meridian $b$,

Let $X_{\Sigma }$ be a normal complex toric variety whose toric rays span $N_{\mathbb{R}}$, so it has no torus factor. Form the Cox ring $S=\mathbb{C}[x_{\rho }]$, its toric irrelevant ideal $B_{\Sigma }$, and the diagonalizable groupThe divisor class group grading defines the action of $G$. The Cox construction recovers $X_{\Sigma }$ as the categorical quotient of $\mathrm{Spec}S\setminus V(B_{\Sigma })$ by $G$. It is a geometric quotient exactly when the toric fan is simplicial.

For the projective line, $S=\mathbb{C}[x_0,x_1]$, $B=(x_0,x_1)$ and $G={\mathbb{C}}^{\ast }$ acts by simultaneous scaling, giving $({\mathbb{A}}^2\setminus \{0\})/{\mathbb{G}}_m={\mathbb{P}}^1$. The ray-span condition matters: a positive-dimensional algebraic torus has a zero toric fan and no toric ray variables, so its construction needs the omitted torus factor as well.^[ref]