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Let $X$ be a finite combinatorial 2-complex satisfying $C^{\prime }(1/6)$, and let $Y$ be a connected CW subcomplex of its universal cover with no missing shells. If the largest cell perimeter is $L$, then $Y^{(1)}$ is $(L/2)$-quasiconvex in the universal cover's unit-edge 1-skeleton. Bound an ambient metric geodesic and an intrinsic metric geodesic of $Y^{(1)}$ by a reduced disc diagram. A shell on the ambient side would shorten a metric geodesic. A shell on the intrinsic side belongs to $Y$ and would shorten its intrinsic metric geodesic. Spurs are likewise excluded from side interiors. The Greendlinger ladder theorem forces a ladder between the two marked corners. Each cell gives a path of length at most half its perimeter from either side to the other.

Ageostrophic flow is the part of velocity beyond leading geostrophic flow. Its small horizontal divergence supplies vertical motion and the stretching term in the quasi-geostrophic potential-vorticity equation.

Define $J(a,b)=a_x{b}_y-a_y{b}_x$ and $\zeta ={\nabla }_h^2\psi$. Combining quasi-geostrophic potential vorticity dynamics with buoyancy evolution eliminates the pressure tendency and givesThe beta term remains when the nonlinear terms are omitted. For a steady weak response to heating about rest, $w=R/N^2$ solves this equation using $\beta {\psi }_x=f_0{R}_z/N^2$ and suitable no-through-flow and decay conditions.

Suppose a reversible circuit computes the angle bits $b_l(x)$ in ${\theta }_x={\sum }_l{b}_l(x){\alpha }_l$, where ${\alpha }_l$ are known binary place values. Apply a controlled unitary gate $R_y(2{\alpha }_l)$ to the target for each set angle bit. The rotations about the y-axis commute and their angles add, giving $R_y(2{\theta }_x)$. Finally perform uncomputation of the angle and arithmetic workspace. An $L$-bit angle needs $L$ controlled rotations and twice the reversible angle-computation cost. This preserves coherent superpositions of inputs and underlies the HHL controlled reciprocal rotation.

In an $N$-dimensional computational basis, the uniform superposition state isMeasurement in that basis gives the uniform distribution on a finite set. For $N=2^n$, Hadamard gates prepare it from $|0^n⟩$. Its reflection $2|s_N⟩⟨s_N|-I$ is a Grover diffusion operator.

A qutrit is a three-dimensional quantum system, with pure states $\alpha |0⟩+\beta |1⟩+\gamma |2⟩$ satisfying $|\alpha {|}^2+|\beta {|}^2+|\gamma {|}^2=1$. A qubit-qutrit state is a bipartite state on ${\mathbb{C}}^2\otimes {\mathbb{C}}^3$.

The two-dimensional quantum torus is the algebra with invertible generators $X,Y$ and relation $YX=qXY$, for $q\in k^{\times }$. Its basis consists of $X^i{Y}^j$ with $i,j\in \mathbb{Z}$, with the same multiplication rule as the quantum plane. It is a right Noetherian ring: successive adjunction of $X^{-1}$, $Y$, and $Y^{-1}$ satisfies the hypotheses of the noncommutative Hilbert basis theorem.

If $M$ is a positive contraction and $p=⟨\psi |M^2|\psi ⟩>0$, the normalized successful post-measurement state is $M|\psi ⟩/\sqrt{p}$. Since $M^2\le M$,A likely outcome therefore preserves a pure input well in quantum fidelity when its measurement operator is positive. Positivity excludes a hidden unitary rotation that could otherwise change the state even for an outcome of probability one.

At tree level, the Feynman diagram has an electron-positron current and a muon-antimuon current joined by one virtual photon. In the massless limit, the spin average and final-state spin sum give $\overline{|\mathcal{M}{|}^2}=e^4(1+{\cos }^2\theta )$, and the integrated relativistic scattering cross-section is $4\pi {\alpha }^2/(3s)$.

In Euclidean signature, for a canonically normalized real scalar field with potential $V$, the one-loop contribution to the quantum effective action is ${\Gamma }_1[\varphi ]=\frac{1}{2}\mathrm{Tr}\log [-{\partial }^2+V^{\prime \prime }(\varphi )]$, with a suitable regularization in quantum field theory. Expanding this functional determinant generates one-particle-irreducible Feynman diagrams with arbitrarily many external fields.

A finite higher-point scalar vertex is a term in the quantum effective action with more external scalar fields than the original interaction, whose coefficient remains finite after renormalization. It can survive removal of an ultraviolet cutoff even when no corresponding independent counterterm is needed. Its coefficient is then predicted by the renormalized parameters and external momenta.

Scattering that resolves the constituents of a hadron at large spacelike momentum transfer, with an inclusive sum over final hadron states.

In electron scattering from a proton, write the reaction as $e^{-}p\to e^{-}X$: the outgoing electron is measured, while $X$ denotes all allowed final hadron states. Large momentum transfer supplies a short-distance probe; a large final invariant mass separates the deep continuum from elastic scattering and individual particle resonances. The useful surprise is that an extended proton can have a response characteristic of much smaller constituents. Deep inelastic scattering turns measurements outside the proton into constraints on its internal parton distribution functions.

The SLAC-MIT deep inelastic electron scattering experiment connects this definition to an actual measurement. Its evidence came from distributions of outgoing electron energies and scattering angles, rather than a photograph of a quark. Bjorken scaling concerns deep-inelastic structure functions after known scattering factors have been accounted for; it does not say that the raw differential scattering cross-section is constant.

Color charge is the gauge group charge of Quantum chromodynamics, associated with its $SU(3)$ symmetry. A quark transforms in the fundamental three-dimensional representation, an antiquark in its conjugate, and a lepton is a singlet. The word color is a label for this charge, unrelated to visible light.

The strong coupling constant parametrizes the strength of Quantum chromodynamics interactions. Its dependence on the renormalization scale is governed by the renormalization-group beta function. It is dimensionless in four dimensions.

A Pythagorean triple consists of three positive integers satisfying $x^2+y^2=z^2$. It is primitive when their greatest common divisor is one. Reducing their square numbers modulo $3$ shows that at least one leg is divisible by $3$; reducing modulo $4$ shows that both legs cannot be odd.

For a compact metric space with a continuous map $T$, suppose $y$ is a minimal point and $x,y$ are proximal. Every neighborhood $U$ of $y$ admits arbitrarily large $n$ with $T^{n}x,T^{n}y\in U$. First shrink $U$ to an open neighborhood, then choose an open $V$ with $y\in V$ and $\overline{V}\subseteq U$. Minimality and compactness give a finite cover of the orbit closure of $y$ by $T^{-j}V$, $0\le j\le J$. Uniform continuity of these finitely many iterates turns a sufficiently close proximal encounter into a common visit to $U$ after at most $J$ more steps. If the proximal encounters occur only at bounded times, a zero distance at some time makes the two future orbits coincide, and minimality then gives arbitrarily late common visits. Thus injectivity is not needed for this general lemma.

If distinct points $x,y$ of a dynamical system on a compact metric space satisfy $d(T^{n}x,T^{n}y)\to 0$ in one compatible metric, no compatible metric can make $T$ an isometry. On a compact space, all compatible metrics give the same asymptotic-pair property by uniform continuity, whereas an isometry preserves the strictly positive distance between distinct points. In a full shift, a constant sequence and a sequence differing at just one coordinate converge to each other under forward shifts, proving this obstruction directly in the product topology.

In a proportional hazards model $h_i(t)=h_0(t){\varphi }_i$, the hazard multiplier ${\varphi }_i>0$ is the relative hazard against the reference multiplier one. In a Cox proportional-hazards model it is $e^{x_i^T\beta }$. Ratios of two time-constant multipliers give constant hazard ratios; time-dependent multipliers require a corresponding extension.

The utility ${\sum }_r{n}_r\log x_r$ sums logarithmic utilities over individual flows. Its first-order condition gives proportional fairness. Its negative is a strictly convex function on active flow rates, giving uniqueness of those rates on a convex set of feasible allocations.