A nonzero covector $\ell$ is null when its squared dual Lorentzian inner product is zero: $g^{\alpha \beta }{\ell }_{\alpha }{\ell }_{\beta }=0$. Raising its index gives a null vector. Null covectors describe normals to characteristic light cones and are the test directions in the classical null condition for wave equations.

For two distinct massless final particles in their centre-of-momentum frame, integrating the energy-momentum delta function gives $d{\Phi }_2=d\Omega /(32{\pi }^2)$. A massless two-particle initial state has flux $2s$, hence $d\sigma /d\cos \theta =\overline{|\mathcal{M}{|}^2}/(32\pi s)$. Average only over the initial spin states physically present.

Loop-space homology is the homology of the based loop space. It also carries the Pontryagin product induced by concatenation of loops. For $n>2$, a Morse theory cell structure on $\Omega S^n$ has one cell in each dimension $j(n-1)$, so its integer homology is $\mathbb{Z}$ in those dimensions and zero elsewhere.

Finite words in a based space, with occurrences of its basepoint deleted. Word concatenation defines a multiplication. For connected based CW spaces, the natural map to $\Omega \Sigma X$ is a weak homotopy equivalence. With free integral homology, the Bott–Samelson theorem identifies its homology algebra with the tensor algebra on reduced homology.

The homology ring formed from loop concatenation and the homology cross product. Homotopy associativity is enough for associativity in homology, and the constant loop is the unit. More generally the same construction applies to a homotopy-associative H-space.

For $n\ge 1$, there is one integral cohomology generator $a_k$ in every degree $2nk$, and its products are $a_i{a}_j=\left(\genfrac{}{}{0ex}{}{i+j}{i}\right)a_{i+j}$. The Bott–Samelson theorem supplies free polynomial homology; its primitive generator gives the binomial diagonal dual to this divided power algebra.

For $I={\int }_{\epsilon }^{\infty }{e}^{-x-\epsilon /x}dx/x$, the omitted interval becomes $J-I={\int }_1^{\infty }{e}^{-t-\epsilon /t}dt/t$. A uniform Taylor expansion gives $J-I=E_1(1)-\epsilon [e^{-1}-E_1(1)]+O({\epsilon }^2)$. Subtracting this correction from the logarithmic expansion of a symmetric exponential integral preserves both the constant and first-order coefficient.

If $f$ is holomorphic near the closed radius-$R$ disc, $f(z_0)\ne 0$, and $|f|\le M$, the displayed estimate holds away from zeros for $|z-z_0|\le R/3$, counting zeros with multiplicity. Factoring local zeros isolates their poles; the remaining logarithm is controlled by disc estimates. If all zeros lie to the left of a point $z$ in real part, their real contributions are nonnegative. This is the disc estimate used by the Landau zero-free-region theorem; a version with general radius ratios is Lemma 24.17 in Montgomery and Vaughan.

For a Keller--Segel model with $c_t=-kb$, a positive-speed travelling wave $B(z),C(z)$, $z=x-vt$, satisfies $v{C}^{\prime }=kB$ and $D{B}^{\prime }+vB-\alpha B{C}^{\prime }/C=0$ when the integrated bacterial flux constant is zero. Writing $\mu =\alpha /D>1$ and $\xi =v(z-z_0)/D$, its positive band solution isTo derive it, integrate $B^{\prime }/B=\mu C^{\prime }/C-v/D$ to obtain $B=K{C}^{\mu }{e}^{-vz/D}$, then separate $v{C}^{\prime }=kB$. The positive integration constant is a translation of the wave. Concentration rises monotonically from zero behind to $C_{\infty }$ ahead; $B$ vanishes at both ends and has a unique maximum at $\xi =-\log (\mu -1)$. The condition $\mu >1$ describes this exponential-tail branch; it is not asserted to exclude every limiting or weak travelling wave at other parameter values.

In a locally small category, the morphisms from $A$ to $B$ form the set $\mathcal{C}(A,B)$. Postcomposition and precomposition give the corresponding covariant and contravariant representable functors.

A morphism is a continuous map $f:X\to Y$ together with a homomorphism ${\mathcal{O}}_Y\to f_{\ast }{\mathcal{O}}_X$ of sheaves of rings whose induced map on every stalk is a local homomorphism. Applied to schemes, this is exactly a morphism of schemes.

An exhaustion of a countable graph is an increasing sequence of finite connected graph vertex sets whose union is the whole graph vertex set. In a locally finite graph, every prescribed finite graph path together with all graph neighbours of its visited graph vertices lies in a sufficiently large exhaustion set. This permits coupling finite-volume and infinite-volume simple random walks up to a finite stopping time.

Here the set is a nonempty compact metrizable convex subset of a Hausdorff locally convex space. The ambient space can be infinite-dimensional. Continuous affine functions separate its points, and weak-star compact dual balls with separable preduals provide examples. This general setting is needed for Choquet theorem.

The fraction relation defining localization of a module gives this criterion. For localization at powers of a single element $f$, a fraction vanishes exactly when some power of $f$ annihilates its numerator. Finitely many such vanishing relations can be cleared with one common exponent. This is distinct from detecting global zero by localizing at every prime ideal.

For a square-integrable discounted claim martingale $dM=hdW$ and discounted stock $d\tilde{S}=\tilde{S}\sigma dW$, choose stock holdings $h/(\tilde{S}\sigma )$. The remaining wealth lies in the bond. A nonnegative claim makes this replication admissible; discounted admissible wealth is a supermartingale, establishing the minimal initial cost.

For a non-dividend-paying stock and constant interest rate $r$, discounted call prices satisfy $C_T=\frac{1}{2}{K}^2\sigma (T,K{)}^2{C}_{KK}-rK{C}_K$. Strike differentiation recovers the discounted terminal density as $C_{KK}$; differentiating the discounted payoff identity supplies the maturity derivative. Put-call parity implies the same equation for put prices.

The exact centered quadratic Gaussian transformation has a connected three-quadratic-vertex contribution $8{\alpha }^3\int d^{3}qP(q)P(|{\mathbf{k}}_1-\mathbf{q}|)P(|{\mathbf{k}}_2+\mathbf{q}|)/(2\pi {)}^3$. It is higher order than the usual tree-level local primordial bispectrum. Infrared or ultraviolet regularization may be needed.

Uniformly sample one positive local term and measure the effect $h_j$. The averaged energy-flag effect on a witness is $H/M$, where $M$ is the number of terms. Repetition with a threshold halfway between the promised flag probabilities uses $O(M^2/(b-a{)}^2)$ witness registers for constant error. A spectral tensor-product analysis, as in QMA parallel repetition with entangled witnesses, proves soundness against entangled witnesses. Constant-locality positive operator-valued measures are efficiently implementable using one ancilla qubit.

A smooth vector field is complete if its maximal integral curves of a vector field exist for every real time. Its local flow is then a global one-parameter group of diffeomorphisms; uniqueness of ordinary differential equations proves the group law and the inverse-time identity. On a compact manifold without boundary, every smooth vector field is complete: compactness gives a uniform positive local existence interval through every point, so a curve can be successively extended past any proposed finite endpoint. On $\mathbb{R}$, $x^2{\partial }_x$ is not complete because its solution $x(t)=x_0/(1-t{x}_0)$ has finite-time blowup for $x_0>0$.

An infinite field with the trivial absolute value on a field is discrete, hence locally compact, but its residue field is the original infinite field. Thus local compactness forces a finite residue field and a discrete nontrivial value group only when the valuation is assumed nontrivial.