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For an odd prime number $q$, the primitive positive definite formshave discriminant $-8q$ and lie in distinct proper equivalence classes. The first represents one, whereas the second has minimum positive value two.

The Jacobi constant is the conserved energy-like integral of the circular restricted three-body problem in the rotating frame.

A no-signalling box is a conditional probability distribution $P(x,y|a,b)$ whose marginal distribution for either party is independent of the other party's input. It can therefore exhibit correlations stronger than quantum correlations without transmitting information superluminally.

Attaching a new vertex that is a leaf of a graph multiplies the chromatic polynomial by $t-1$: after colouring the old graph, the new vertex may receive any colour except its neighbour's. Starting with one vertex and repeatedly adding a leaf of a graph constructs any finite tree. Thus a tree with $n\ge 1$ vertices hasIn contrast, every pair of vertices in a complete graph must receive different colours, so the multiplication principle gives the falling factorialFor positive integers $t<n$ this product vanishes, expressing the impossibility of assigning distinct colours to all $n$ vertices.

Deleting an edge removes one colour constraint. Edge contraction describes exactly the newly admitted colourings that violate that constraint. If $e$ joins distinct vertices $u,v$ of a simple graph $G$, write $G-e$ for the graph with that edge deleted and $G/e$ for the graph with $u,v$ identified and $e$ removed. Parallel edges produced by the contraction may be reduced to one edge, since they impose the same condition on a proper vertex coloring. ThenTo see why, divide the proper vertex colorings of $G-e$ into two classes. Those with different colours at $u,v$ are exactly the colourings of $G$. Those with equal colours at $u,v$ correspond one-to-one to colourings of $G/e$: the merged vertex receives their common colour. Subtracting the second class leaves the first.

For the triangle, delete $AC$ to get a three-vertex path graph, or contract $AC$ to get two vertices joined by an edge after removing the redundant parallel edge. Consequently,At $t=3$, this says $6=12-6$. Of the twelve path colourings pictured above, six give $A,C$ the same colour; these are precisely the ones ruled out when we restore $AC$.

This recurrence is useful because it reduces a constrained counting problem to smaller graphs. A graph on $n$ vertices with no edges has $P_G(t)=t^n$, since every vertex chooses its colour independently. Repeated edge deletion and edge contraction reduce to this case and show that the counting function is a polynomial. The same recurrence gives its leading coefficients. The method is conceptually simple, although the number of recursive cases can still grow rapidly.

A connected graph of maximum degree $\Delta$ has chromatic number at most $\Delta$ unless it is a complete graph or an odd cycle.

If $r<s$, coupling by extra independent squared normals gives ${\chi }_s^2={\chi }_r^2+W$ with $W\ge 0$, so every fixed upper quantile increases with the degrees of freedom.

On projective space, the Euler sequence is

The total Chern class is $c(E)=1+c_1(E)+c_2(E)+\cdots$.

The chemical potential of photons in thermal equilibrium is zero because their number is not conserved.

The Chebyshev equation has self-adjoint formFor $U_n=T_n^{\prime }$, differentiation givesThe respective orthogonality weights are $(1-x^2{)}^{-1/2}$ and $(1-x^2{)}^{1/2}$.

Every square matrix over an algebraically closed field is similar to an upper-triangular matrix. Choose an eigenvector, extend it to a basis, and apply induction to the induced lower-right block.

For real matrices $A,B$, $p(t)=\det (A+tB)$ is a real polynomial. If $p(i)\ne 0$, it is not the zero polynomial, so some real $\lambda$ satisfies $p(\lambda )\ne 0$.

The expansion $\log (\sin x/x)=-x^2/6+O(x^4)$ converts products of small sinc factors into Gaussian characteristic-function limits.

The method of characteristics solves a first-order partial differential equation by reducing it to ordinary differential equations along characteristic curves.

The characteristic flow map $Z_{s,t}(x)$ is the point reached at time $t$ by the characteristic curve that passes through $x$ at time $s$. Whenever the flow is invertible, transport solutions pull their initial data back by $Z_{0,t}^{-1}$.

The character space ${\Phi }_A$ is the set of all characters on $A$.

Let $G\times G$ act on $\mathbb{C}G$ by $(g,h)x=gx{h}^{-1}$. Its character is zero unless $g$ and $h$ are conjugate, in which case it equals $|C_G(g)|$. Character column orthogonality gives the same character forwhere the $V_i$ run through the irreducible complex representations. Maschke's theorem therefore gives an isomorphism of the two representations.

A forcing order has the countable chain condition when every antichain in a forcing order is countable. Such forcing preserves cardinals and cofinalities at least ${\aleph }_1$.

A degreewise short exact sequence $0\to A\to B\to C\to 0$ of chain complexes induces a long exact sequence of homology groups through the connecting homomorphisms.