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A physical-strip pole of a diagonal S-matrix can come from exchange of an existing particle in the crossed channel. For external masses $m_a,m_b$, use the momentum difference, rather than the sum, to identify the exchanged mass through the displayed invariant. Crossing symmetry turns the pole at $i\alpha$ into a direct pole at $i(\pi -\alpha )$ of the particle-antiparticle channel. For $m_B=2m\cos (u/2)$, the pole of $S_{BA}$ at $iu/2$ has $t=m^2$ and exchanges the existing charge-one particle; it is not a new charge-three bound state.

For $J_h^{\mu }={\sum }_f{Q}_f{\bar{q}}_f{\gamma }^{\mu }{q}_f$ without a factor of $e$, use $i\int e^{iqx}⟨T{J}_h^{\mu }(x)J_h^{\nu }(0)⟩d^{4}x=(q^{\mu }{q}^{\nu }-q^2{g}^{\mu \nu }){\Pi }_J(q^2)$. The inclusive tensor is $W_J^{\mu \nu }=2\pi {\rho }_J(s)(q^{\mu }{q}^{\nu }-s{g}^{\mu \nu })$. A massless free-quark calculation gives ${\rho }_J=N_c{\sum }_f{Q}_f^2/(12{\pi }^2)$; strong interactions alter this function.

For $q=(q^0,\mathbf{q})$ with $\mathbf{q}\ne 0$, the Coulomb gauge propagator hasThe temporal component is instantaneous; the spatial part projects onto transverse photon polarizations. Although these components are not individually Lorentz covariant, their contraction with conserved external currents equals the covariant answer. Zero-spatial-momentum limits should be taken after combining the current contractions.

Define the inclusive photon-field tensor for hadronic final states by $W_A^{\mu \nu }=2\pi {\rho }_h(s)(-g^{\mu \nu }+q^{\mu }{q}^{\nu }/s)$ for timelike $q$ with $s=q^2$ and $q^0>0$. Here each final state has relativistic normalization and the sum includes the four-momentum delta function. This hadronic photon spectral density has mass dimension $-2$. At leading electromagnetic order, ${\rho }_h=e^2{\rho }_J/s$, with ${\rho }_J$ the dimensionless hadronic electromagnetic-current spectral density. Stating the tensor convention prevents mismatched powers of $s$ or $e$ in an inclusive cross-section.

The interaction vertex $-i\lambda$ and scalar propagator $-i/(p^2-iϵ)$ give one four-point bubble diagram for each of three momentum channels, each with Feynman-diagram symmetry factor $1/2$. The two-point tadpole diagram is zero as a scaleless integral in dimensional regularization. If full proper vertices include the free quadratic kernel, ${\hat{\tau }}_2^{(0)}=-p^2$; the interaction self-energy convention instead starts at zero.

The excess free energy per area of a solid-liquid phase boundary. Curvature changes the equilibrium temperature through the Gibbs--Thomson relation. Crystal anisotropy can make this energy and its associated surface stiffness depend on orientation.

Premelting is the persistence of a liquid layer at a surface or interface below the bulk melting temperature. Molecular disjoining pressure and phase equilibrium can stabilize the film. Its hydraulic connection to a reservoir permits frost heave when water freezes onto the base of loaded ice.

A crystal's interfacial free energy and attachment kinetics depend on the interface normal $\mathbf{n}$. The orientation-dependent energy $\gamma$ and mobility $M$ select directions even when bulk diffusion is isotropic. This supplements the transport mechanism of crystal dendrites; isotropic transport alone does not specify the number of preferred arms.

An equivalent large-scale permeability of a porous medium relating imposed pressure drop to total Darcy flux through a heterogeneous sample. Parallel flow paths add conductances, whereas successive resistive segments add pressure drops. It depends on direction and boundary conditions; it need not be the arithmetic mean of the local permeabilities.

A dimensionless reduction of a porous formation's intrinsic permeability of a porous medium for one fluid phase in multiphase flow. Its effective phase permeability is $K{k}_{ri}$, and its phase mobility is $k_{ri}/{\mu }_i$. It depends on saturation and pore-scale phase configuration; it is not the magnetic use of relative permeability.

For fixed positive integer period $n$, integer-valued periodic functions on $\mathbb{Z}$ are in bijection with ${\mathbb{Z}}^n$ and hence form a countable set. Their union over periods is a countable union of countable sets. In contrast, period-one integer-valued functions on $\mathbb{Q}$ can prescribe arbitrary binary values on its infinitely many distinct classes modulo $\mathbb{Z}$. The power set of a countably infinite set injects into them, proving uncountability.

A one-factorization partitions every edge of a graph into perfect matchings. Each matching uses every vertex exactly once. A total of synthemes is precisely a one-factorization of the complete graph on six points.

A minimum-weight perfect matching pairs every vertex exactly once while minimizing total edge weight. General weighted perfect matching has polynomial-time algorithms. In the Christofides algorithm, it corrects precisely the odd-degree vertices of a minimum spanning tree. Alternating edges of a cyclic order on an even number of vertices give two perfect matchings; the cheaper one costs at most half that cycle's cost.

If an infinite percolation cluster exists almost surely, the probability that a large square box intersects one tends to one. The four rotationally symmetric exterior-side arm events are increasing. By the Harris-FKG inequality, their common probability is at least $1-(1-P(D_N){)}^{1/4}$, where $D_N$ is their union. Thus a pair of specified opposite sides both has an infinite exterior arm with probability at least $1-2(1-P(D_N){)}^{1/4}$, tending to one. The rays start at a boundary graph vertex and immediately leave the graph vertex box.

The leading power $\beta$ of the percolation probability approached from above the percolation critical probability. It quantifies how the density of the infinite percolation cluster first becomes positive.

The exponent $\delta$ describes the tail of the critical cluster seen from a specified vertex of a graph. If the expected number of size-$s$ clusters per site scales as $s^{-\tau }$, weighting by cluster size gives $1/\delta =\tau -2$ when these scaling laws hold.

The critical cluster mass-radius scaling exponent $D_f$ describes clusters of mass roughly $r^{D_f}$ at radius $r$. The displayed relation is the below-upper-critical-dimension scaling prediction, not an identity from the definition. In the planar scaling description $D_f=91/48$.

The mean-field predictions model large clusters by branching processes. Rigorous results apply in specified high-dimensional or sufficiently spread-out regimes, using tools such as lace expansion and the percolation triangle condition. The full list involves different observables; the triangle condition alone should not be presented as proving every spatial exponent. With these exponents $2\beta +\gamma =3$, whereas $d\nu =d/2$, explaining the failure of naive hyperscaling relation above six.

A Pauli gate applies a Pauli matrix as a unitary operator to a qubit. The Pauli X gate, Pauli Y gate and Pauli Z gate are common correction operations in quantum teleportation.

An influence-function representer is a centered square-integrable function representing derivatives of a statistical functional along all admissible score functions. Representers may differ by a function orthogonal to the statistical tangent space. This pathwise definition is distinct from defining an influence function solely by point-mass contamination paths.