For critical site percolation on the triangular lattice, scaling-limit crossing probabilities transform naturally under conformal maps between planar domains. This rigorously established property supports descriptions of interfaces by SLE and calculations of planar percolation critical exponents. Transferring it to a different lattice or to bond percolation requires a separate universality argument.

The identity follows because the identity matrix and the three Pauli matrices form an orthogonal basis of two-by-two matrices under the trace inner product. It rewrites a spin-vector bilinear square as a scalar four-fermion expression, with the Grassmann signs supplied when fields are reordered. This is the algebra behind a spin-vector Hubbard–Stratonovich decoupling.

Three light quarks have a six-dimensional single-quark spin-flavour space ${\mathbb{C}}^3\otimes {\mathbb{C}}^2$. A symmetric spatial ground state and the antisymmetric three-quark colour singlet require the symmetric cube of that space. Its dimension is $\left(\genfrac{}{}{0ex}{}{8}{3}\right)=56$, and restriction to flavour $SU(3)$ and spin $SU(2)$ gives a baryon decuplet of spin $3/2$ and a baryon octet of spin $1/2$. The two mixed-permutation multiplicity spaces pair once in the symmetric sector, explaining why the two flavour-octet copies in the unconstrained tensor cube do not represent two distinct ground-state octets.

A Hamiltonian path is a path in a graph that visits every graph vertex exactly once. Removing any edge from a Hamilton cycle produces a Hamiltonian path. Conversely, a Hamiltonian path closes to a Hamilton cycle exactly when its endpoints are joined by an edge.

For any positive integer $c$, a homogeneous rational matrix is partition regular over $c\mathbb{N}$ if and only if it is partition regular over $\mathbb{N}$. Pull a colouring of $c\mathbb{N}$ back along $n\mapsto cn$ and scale the resulting solution. Conversely, restrict a colouring of all positive integers to $c\mathbb{N}$. The argument depends on homogeneity and does not automatically extend to inhomogeneous equations or prescribed distinctness conditions.

An exponential damping factor makes many otherwise conditionally convergent or divergent oscillatory integrals absolutely convergent. Removing the damping after integration defines an Abel limit when it exists. For the Bessel function $J_1$, ${\int }_0^{\infty }{e}^{-ak}{J}_1(kR)dk=R^{-1}(1-a/\sqrt{a^2+R^2})$, giving the Mestel disk radial field as $a\downarrow 0$. This does not remove an unrelated additive divergence of the potential at low wavenumber.

A finite undirected graph can be oriented with outdegree at least an integer $k\ge 0$ at every vertex if and only if every vertex subset $U$ is incident to at least $k|U|$ distinct edges. The max-flow min-cut theorem and integral max-flow theorem prove sufficiency by assigning distinct edges to vertices through an incidence flow network; each assigned edge is oriented away from that vertex. Necessity follows by counting outgoing edges from $U$.

For a nonzero constant-coefficient polynomial differential operator in one variable, every homogeneous distribution solution is a classical solution, indeed analytic. Factor the polynomial into powers of distinct characteristic roots of a constant-coefficient differential equation. The kernel decomposition for coprime polynomials reduces the equation to $(D-\lambda {)}^{m}v=0$, and multiplication of a distribution by a smooth function reduces this to $D^m(e^{-\lambda x}v)=0$. The fact that a distribution with zero derivative is constant, iterated, makes the latter distribution a polynomial of degree less than $m$.

If a continuous nondecreasing modulus $\mu$ satisfies this divergence condition, an estimate $|b(t,x)-b(t,y)|\le A(t)\mu (|x-y|)$ with locally integrable $A$ gives uniqueness of characteristic curves. Apply a regularized integral comparison to their separation. For the log-Lipschitz modulus, ${\delta }^{\prime }(t)\le A(t)\delta (1-\log \delta )$ gives $\delta (t)\le \exp (1-(1-\log \delta (0))\exp (-{\int }_0^{t}A))$ while $\delta \le 1$. Zero initial separation stays zero. Unlike the usual linear Gronwall inequality, this criterion allows non-Lipschitz vector fields.

Differentiating a smooth characteristic flow map in its initial point gives this linear equation, with $Y(s)=I$. The Liouville formula for a fundamental matrix then gives $\det Y(t)=\exp ({\int }_s^t\mathrm{div}b(r,X(r))dr)$. Incompressible Hamiltonian flows preserve Lebesgue measure because this divergence vanishes.

Independence factors the events that both variables exceed a threshold or both lie below it. This gives the two cumulative distribution functions above, including for distributions with atoms. The minimum formula is most naturally expressed through survival probabilities, while the maximum formula uses the CDFs directly.

For two independent rate-$\lambda >0$ exponential distributions, set $U=\min (X,Y)$ and $V=\max (X,Y)$. Summing the two ordering branches gives joint probability density $2{\lambda }^2{e}^{-\lambda (2u+d)}$ for $(U,d=V-U)$ with $u,d>0$. It factors into independent exponential probability density functions. Thus $V$ is distributed as $X+Y/2$, with expected value $3/(2\lambda )$ and variance $5/(4{\lambda }^2)$. This spacing result is the order-statistic manifestation of the memorylessness of the exponential distribution.

A continuous surjective linear map between Fréchet spaces is an open map. The Baire category theorem first shows that the closure of the image of every convex balanced zero-neighbourhood is a zero-neighbourhood: the target is the countable union of scalar multiples of this image, and an interior point of its closure can be translated to zero using convexity and symmetry.

To remove the closure, fix a domain zero-neighbourhood $U$ and choose convex balanced zero-neighbourhoods $U_n$ so that every series ${\sum }_n{u}_n$, $u_n\in U_n$, converges to a point of $U$. With an increasing defining sequence of seminorms $p_j$, one can ensure $p_j(u_n)<2^{-n}$ for $j\le n$ and make the finitely many seminorms defining $U$ summable within its bounds. Completeness gives convergence. Choose target zero-neighbourhoods $V_n\subset \overline{A(U_n)}$ shrinking to zero. For $y\in V_1$, choose $u_n\in U_n$ successively so that $y-A{\sum }_{j\le n}{u}_j\in V_{n+1}$. Density permits each correction. Continuity then gives $y=A{\sum }_n{u}_n\in A(U)$, proving openness.

When the continuous Fourier transform $F$ of an integrable function vanishes outside $[-\pi ,\pi ]$, its periodic Fourier coefficients are the integer samples of the continuous representative of $f$, with the sign of the index reversed. Projecting $F$ in $L^2[-\pi ,\pi ]$ and then applying the Fourier inversion theorem gives the displayed reconstruction using the normalized sinc function, with each quotient defined as one when $t=n$. The Cauchy-Schwarz inequality controls the reconstruction error uniformly in $t$. Bessel's inequality bounds the vector of shifted sinc functions, giving absolute uniform tails when the sample sequence belongs to ${\ell }^2$.

A number field is totally real when every embedding into the complex numbers has image in the real numbers. The maximal real subfield of a cyclotomic field is a basic example. Ordinary Hilbert class fields split at real places and therefore retain this property.

In a planar warped spacetime, contraction of the Ricci tensor with a null vector in an orthonormal frame gives $R_{ab}{k}^a{k}^b=-2(\log A{)}^{\prime \prime }(k^3{)}^2$. The Einstein field equations identify this with $8\pi T_{ab}{k}^a{k}^b$; metric terms, including a cosmological constant, vanish in a null contraction. Thus the null energy condition is equivalent to concavity of $\log A$ in the proper transverse coordinate $z$. Null directions confined to a planar slice saturate the condition.

A minimal normal subgroup is a nontrivial normal subgroup containing no smaller nontrivial normal subgroup of the ambient group. It need not be a minimal nontrivial subgroup. In a finite soluble group, it is elementary abelian: its derived subgroup must be trivial, a characteristic Sylow subgroup selects one prime, and the characteristic subgroup of elements killed by that prime gives exponent $p$.

Here $\Delta \psi =\psi (v^{\prime })+\psi (v_{\ast }^{\prime })-\psi (v)-\psi (v_{\ast })$ is the pairwise collision change, not a Laplace operator. Average over the two particle labels and use collision reversal to obtain the formula. Every collision invariant has $\Delta \psi =0$, giving mass, momentum and kinetic energy conservation. A second symmetrization replaces $f{f}_{\ast }\Delta \psi /2$ by $(f^{\prime }{f}_{\ast }^{\prime }-f{f}_{\ast })(-\Delta \psi )/4$.

The angular collision rate is integrable at each nonzero relative velocity, with any needed speed-growth bounds stated separately. This makes the separate gain and loss integrals available under suitable velocity integrability. Non-cutoff kernels can have infinite angular collision rate and require cancellation rather than the same naive gain-loss split.

Write $f=M(1+h)$ around a fixed Maxwellian and expand the nonlinear Boltzmann collision operator to first order. The weak Boltzmann collision identity gives the nonnegative dissipation form in $L^2(Mdv)$, and polarization proves symmetry on an appropriate common domain. Its zero modes are the collision invariants when the scattering kernel is nondegenerate. A spectral gap, and self-adjointness of an unbounded realization, require additional domain and kernel hypotheses.