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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.

The antiparticle of a quark.

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.

For density operators, Klein's inequality gives $D(\rho \Vert \sigma )\ge 0$, with equality exactly when $\rho =\sigma$. If the support of a positive operator $\rho$ is not contained in that of $\sigma$, the quantum relative entropy is $+\infty$.

For a rigid body rotating around its third principal axis, the anisotropic part of its second mass moment tensor oscillates at angular frequency $2\Omega$. The quadrupole formula gives the displayed luminosity. Equal transverse principal moments remove this rotating quadrupole and its leading gravitational wave emission.

The iterative soft-thresholding algorithm applies the proximal gradient method to $\Vert x{\Vert }_1+\frac{1}{2}\Vert Ax-y{\Vert }_2^2$:Here $S_{\tau }$ is coordinatewise soft thresholding. The objective is convex and coercive, so iterates converge to a minimizer in finite dimensions; uniqueness requires additional hypotheses.