Let $P=p_2/p_1$ and $D={\rho }_2/{\rho }_1$ be the pressure and mass density ratios across a normal shock wave in a perfect gas with specific-heat ratio $\gamma$. Eliminating the velocities from the Rankine-Hugoniot conditions for a perfect gas givesAs $P\to \infty$, the ratio tends to the shock compression ratio $(\gamma +1)/(\gamma -1)$.

For a finite CW complex, every element in the kernel of the rank map in topological K-theory is a nilpotent element. One proof adjoins one positive-dimensional cell at a time: the restriction kernel is square-zero by the relative product in topological K-theory. For connected spaces the rank-zero kernel is Reduced topological K-theory; for disconnected spaces the kernel of restriction to just one basepoint can contain nonzero idempotents.

A set $E\subseteq [\mathbb{N}{]}^{\omega }$ is completely Ramsey if every basic Ellentuck topology neighborhood $[s,A]$ admits an infinite $B\subseteq A$ with $[s,B]\subseteq E$ or $[s,B]\cap E=\varnothing$. Keeping the finite stem $s$ makes this stronger than being a Ramsey set of infinite subsets.

Using the axiom of choice, enumerate all infinite subsets $A_{\alpha }$ of the positive integers in order type $2^{{\aleph }_0}$. By transfinite recursion, choose two previously unused infinite subsets $X_{\alpha },Y_{\alpha }$ of each $A_{\alpha }$. This is possible because each $[A_{\alpha }{]}^{\omega }$ has cardinality $2^{{\aleph }_0}$ and fewer choices have been made at stage $\alpha$. The set $\{X_{\alpha }\}$ is not a Ramsey set of infinite subsets, since every $[A_{\alpha }{]}^{\omega }$ meets both it and its complement.

For all real $s\ge 0$, this definition uses the normalized integral valuation on $L$. Equivalently $G_s=G_{\lceil s\rceil }$. The group $G_0$ is the inertia group and $G_1$ is the wild inertia group.

Spectral flux density is radiant power received per unit area and per unit wavelength or frequency. The two conventions satisfy $F_{\lambda }d\lambda =-F_{\nu }d\nu$ under $\nu =c/\lambda$, hence $F_{\lambda }=(c/{\lambda }^2)F_{\nu }$ when the arguments refer to the same photons. Integrating over a photometric passband gives its measured flux.

If release imparts no velocity kick, a grain's initial Kepler orbit under reduced stellar attraction has $q=r_b$, $Q=r_b/(1-2\beta )$, and $e=\beta /(1-\beta )$ for $0\le \beta <1/2$. Its pericentre distance remains at the dust birth ring, while its apocentre distance grows with the radiation-pressure coefficient.

Inside any three-dimensional domain containing a ball, let $u_{\epsilon }(x)={\epsilon }^{1/2}\varphi ((x-x_0)/\epsilon )$ for a fixed smooth compactly supported nonconstant function. The right-hand side $f_{\epsilon }=\Delta u_{\epsilon }$ has constant $L^2$ norm, whereas $\Vert \nabla u_{\epsilon }{\Vert }_{\infty }={\epsilon }^{-1/2}\Vert \nabla \varphi {\Vert }_{\infty }$ diverges. The functions have zero boundary trace, giving a scaling counterexample to a uniform $L^2$-forcing gradient estimate.

For an orientation-preserving quasiconformal map, the almost-everywhere coefficient ${\mu }_f=f_{\bar{z}}/f_z$. It is a coordinate-dependent representative of a tensor; the modulus and the resulting maximal dilatation are coordinate-independent.

The essential supremum of the ratio of the largest to the smallest infinitesimal singular value of a quasiconformal map. It equals $(1+\Vert {\mu }_f{\Vert }_{\infty })/(1-\Vert {\mu }_f{\Vert }_{\infty })$ in terms of the Beltrami coefficient.

An antiquark is the antiparticle of a quark, with opposite electric charge, baryon number and additive flavor numbers, the same mass and spin, and the conjugate color representation. Thus an anti-up quark has charge $-2e/3$, whereas an anti-down quark has charge $+e/3$.

A quark–antiquark pair can have zero net baryon number and combine into a color singlet, supplying the minimal valence content of a meson. Flavor cancellation is separate from electric-charge cancellation: $u\bar{d}$ has baryon number zero but electric charge $+e$. Antiquarks also occur in the nonvalence partonic content of hadrons, so their presence does not imply that the entire hadron is an antimatter counterpart of another one.

The up quark $u$ is the first-generation up-type quark, with electric charge $+2e/3$, spin $1/2$ and baryon number $1/3$. It is a light quark, and the up quark and down quark together account for the net valence flavor content of ordinary nuclei.^[ref]

The minimal proton content $uud$ has total charge $2(2e/3)-e/3=e$ and baryon number one. The proton's mass is not simply the sum of three free-quark masses: its interacting quark and gluon fields carry most of its energy. The quark model organizes the flavor content, while quantum chromodynamics describes the dynamics needed to compute hadron properties.

The down quark $d$ is the first-generation down-type quark, with electric charge $-e/3$, spin $1/2$ and baryon number $1/3$. Together with the up quark, it supplies the net valence flavor content of ordinary protons and neutrons.^[ref]

The minimal neutron content $udd$ has total electric charge zero and baryon number one. In beta decay, a down-to-up weak charged current can turn a neutron into a proton, accompanied by the required leptons. This is a flavor change, rather than an electron previously stored inside the neutron escaping. Additional quark pairs and gluons are part of the full hadron state.

The strange quark $s$ is the second-generation down-type quark, with electric charge $-e/3$, spin $1/2$, baryon number $1/3$ and strangeness $-1$. Its strange antiquark has the opposite additive quantum numbers.^[ref]

The strong interaction can create a strange–antistrange pair without changing total strangeness. A later weak interaction can change net strangeness, allowing a strange hadron to decay into nonstrange products. This separation of production and decay mechanisms makes the strangeness label useful in reconstructing particle reactions. Its sign is a convention for the flavor quantum number, rather than the sign of an additional physical electric charge.

The charm quark $c$ is the second-generation up-type quark, with electric charge $+2e/3$, spin $1/2$ and baryon number $1/3$. It is heavier than the strange quark and is commonly treated as a heavy quark.^[ref]

A charm–charm antiquark pair supplies the minimal valence content of charmonium, while a charm quark bound with a light antiquark gives an open-charm meson. These systems probe different aspects of quantum chromodynamics: internal heavy-pair motion in the former and heavy–light dynamics in the latter. The approximate flavor classification that includes charm does not make charm and the light quarks equal in mass.

The bottom quark $b$ is the third-generation down-type quark, with electric charge $-e/3$, spin $1/2$ and baryon number $1/3$. It is a heavy quark; its weak interaction decays make bottom-containing hadrons unstable. The charge assignment is listed alongside the other flavors in ^[ref].

A bottom quark combined with a light antiquark forms the minimal valence content of a bottom meson. Measuring the decay products and displaced decay point of such a hadron helps identify bottom production in a particle detector. The observed particles are hadrons and their descendants, rather than an isolated bottom quark propagating freely through the detector.

A constituent quark that does not participate in the elementary weak vertex of a meson decay. It combines with a produced quark or antiquark in the final hadron.

For $V=-A{e}^{-\mu r}/r$, the normalized exponential with scale $\alpha$ gives $E(\alpha )={\hslash }^2{\alpha }^2/(2m)-4A{\alpha }^3/(\mu +2\alpha {)}^2$. Choosing $\alpha =\mu /2$ proves binding if $\mu <Am/{\hslash }^2$. Failure to obtain a negative trial expectation outside this range does not prove absence of a bound state. At equality a small positive admixture of the scale-$\mu$ exponential has a negative cross matrix element and proves binding too.

For an ensemble of density operators with average $\overline{\omega }={\sum }_j{p}_j{\omega }_j$,Expanding the trace definitions cancels the $\log {\omega }_j$ terms, and the remaining sum combines to a trace against $\overline{\omega }$. Positive-weight ensemble states have their support contained in that of the average, so the first sum on the right is finite in finite dimension. If the reference-state support condition fails, both sides are $+\infty$. The identity is associated with Donald's work on relative entropy.

In bits, the quantum Pinsker inequality isThe trace-distance-preserving binary measurement converts the trace distance into classical total variation distance without loss. Apply the data-processing inequality for quantum relative entropy to that measurement, then classical Pinsker's inequality. The factor $\ln 2$ is omitted when relative entropy uses natural logarithms.