For a non-elementary discrete subgroup generated by $A,B\in \mathrm{SL}(2,\mathbb{C})$, the displayed inequality is a necessary trace constraint. Failure of it proves nondiscreteness when non-elementarity has independently been checked. It is not a sufficient condition for discreteness and must not be applied to an elementary pair without its hypothesis.

The Gaussian Bianchi group is the Kleinian group of determinant-one two-by-two matrices over the Gaussian integers, modulo $\{\pm I\}$. Its matrix entries lie in a discrete subring of $\mathbb{C}$, so it is a discrete subgroup of $PS{L}_2(\mathbb{C})$. The Borromean link group is a torsion-free subgroup of index 24.

The valence formula for the modular group shows that $\Delta =(E_4^3-E_6^2)/1728$ has no zeros in the half-plane. For every $a\in \mathbb{C}$, the weight-twelve form $E_4^3-a\Delta$ has total weighted zero order one. At $a=0$ its sole zero is the order-three zero at the cubic elliptic orbit; at $a=1728$ its sole zero is the order-two zero at the quadratic elliptic orbit. Otherwise it has one simple ordinary zero orbit. Thus each value of the Klein j-invariant occurs on exactly one modular orbit and classifies one homothety of complex lattices.

For hindered settling with normalized particle volume fraction $s$, upward coordinate and downward isolated settling velocity magnitude $V_0$, the scalar conservation law has conservation law flux $f(s)=-V_{0}s(1-s{)}^{\alpha }$. For $\alpha >0$, its convexity changes at $s=2/(\alpha +1)$ when this lies in the physical interval. This change distinguishes spreading rarefaction waves from compressing characteristic curves; a flux with an inflection can produce both a shock wave and a rarefaction wave in one transition. At $\alpha =0$ the flux is linear and the profile simply translates.

Choose a real even smooth compactly supported Fourier transform $\eta$ equal to one near zero and let $K$ be its inverse Fourier transform. Repeated integration by parts makes $K$ a Schwartz function. It is real, integrable and square-integrable, with unit integral. Moment differentiation of the Fourier transform gives $\int x^{j}K(x)dx=i^j{\eta }^{(j)}(0)=0$ for every positive integer $j$. Such a kernel for density estimation therefore satisfies arbitrary finite-order moment cancellation, and necessarily takes negative values. If the transform plateau contains the frequency support of a bandlimited function $f$, the convolution theorem proves $K\ast f=f$.

The Epanechnikov kernel for density estimation is a nonnegative symmetric truncated parabola with integral one, ${\mu }_2(K_E)=1/5$, and $R(K_E)=3/5$. It has the smallest optimized asymptotic mean integrated squared error among nonnegative second-order density kernels, up to rescaling. Its canonical kernel has scale $15^{1/5}$ and support $[-15^{1/5},15^{1/5}]$.

The Gaussian density kernel is the standard normal probability density used as a unit-integral kernel for density estimation. Its second moment is one and $R(K_N)=1/(2\sqrt{\pi })$. Its canonical kernel has scale $(2\sqrt{\pi }{)}^{-1/5}$. It is a function of one smoothing displacement, distinct from the two-argument Gaussian kernel used as a positive-semidefinite kernel.

The triweight kernel is a compactly supported symmetric probability density with variance $1/9$. Its value and first two derivatives vanish at the support endpoints. Its integrated squared second derivative is $35$, and it minimizes that curvature among twice continuously differentiable densities with zero mean and the same variance, by the triweight density minimizes integrated squared curvature inequality.

For a quasistatic cantilever of Kelvin-Voigt material, the bending moment is $M=EI{y}_{xx}+{\eta }_{s}I{y}_{xxt}$. A tip load gives $M=F(t)(\ell -x)$, so the free-tip displacement obeys the displayed equation with $\tau ={\eta }_s/E$. A step load gives a finite asymptotic deflection. If inertia is retained, the interior equation is $\rho A{y}_{tt}+EI{y}_{xxxx}+{\eta }_{s}I{y}_{xxxxt}=0$, with the corresponding moment and shear conditions at the tip.

A state with $k$ B smoothings has at most $s_A+k$ circles, so its largest A exponent is at most $c(D)+2s_A-2$. The dual argument bounds the smallest exponent below by $-c(D)-2s_B+2$. For a connected link diagram, the Turaev surface inequality bounds the resulting breadth by $4c(D)$. For a connected reduced alternating diagram, adequacy prevents cancellation at both extremes and planar Euler characteristic gives equality. A disconnected projection needs the additional term in the Jones polynomial breadth bound for a disconnected diagram.

For an oriented 3-manifold, take the linear span of isotopy classes of framed links, including the empty link, and impose the local bracket smoothing relation and the trivial-circle relation with factor $-A^2-A^{-2}$. Fixing framed boundary endpoints gives its relative version. Gluing manifolds or stacking cylinders induces maps between skein modules; framed isotopy, rather than unframed Reidemeister-I invariance, is built into this construction.

In the phase convention $CP|K^0⟩=|{\bar{K}}^0⟩$, the symmetric flavor combination is CP even and the antisymmetric one CP odd. For an equal-diagonal effective matrix with entries b,c off diagonal, choose correlated roots r,s with $r^2=b$, $s^2=c$ and eigenvalues $a\pm rs$. Its eigenvectors are $(r,\pm s{)}^T$. Writing $ϵ=(r-s)/(r+s)$ gives the displayed CP-basis mixtures, up to an overall phase. Their deviation is small only if $|ϵ|\ll 1$; noninvariance alone does not guarantee that. This convention must not be mixed with one in which flavor exchange carries a minus sign.

The Källén–Lehmann spectral density is the nonnegative invariant-mass measure in a scalar vacuum two-point function. With complete intermediate states it satisfies $2\pi \theta (p^0)\rho (p^2)=(2\pi {)}^4{\sum }_{\alpha }|⟨\alpha |\varphi (0)|0⟩{|}^2{\delta }^4(p-p_{\alpha })$, with continuous state measures included in the sum. A stable one-particle state of mass $m$, with covariant normalization and squared field overlap $Z$, contributes $Z\delta (\sigma -m^2)$. Other particles and multiparticle states contribute additional nonnegative weight. The isolated pole of the scalar propagator then has residue $Z$.

An infinite profinite group is hereditarily just infinite if every open subgroup is a just-infinite profinite group. To prove this, it suffices to show that every nonidentity element of every open subgroup has open closed normal closure: any nontrivial closed normal subgroup then contains that normal closure. The Nottingham group is an example. In odd characteristic its depth-commutator formula constructs nonzero elements of each sufficiently high layer inside any such normal subgroup, and successive coefficient correction fills the entire tail.

An irreducible recurrent discrete-time Markov chain with at least two states has $P_{ii}<1$ everywhere. Removing repeated consecutive states from a positive-probability path proves irreducibility of its discrete-time jump chain. Each sojourn in a state is almost surely finite; infinitely many visits to a recurrent state therefore correspond to infinitely many distinct returning sojourns. The jump chain remains recurrent, although deleting holding times may change periodicity.

The Young–Jucys–Murphy elements commute, and symmetric polynomials in them are central. The full center theorem holds integrally as well as over fields, so it survives modular reduction. On an ordinary Specht module, the scalar of such a polynomial is its value on the cell-content multiset. The integral generation step is stronger than knowing this description only over a characteristic-zero field and is essential in the modular block argument.

A Joule-Thomson inversion temperature is a temperature at which the Joule-Thomson coefficient is zero at a specified pressure. There can be more than one such temperature at that pressure; distinguish the upper and lower branches from a maximum or a dilute-gas limiting value.

For stress $-pI+\alpha G_{0}A+2{\mu }_{0}E$ and slip derivative $D_{t}A+\Omega A-A\Omega -\alpha (EA+AE)=-(A-I)/\tau$, the steady viscometric functions are $\mu ={\mu }_0+{\alpha }^2{G}_0\tau /D$, ${\psi }_1=2{\alpha }^2{G}_0{\tau }^2/D$ and ${\psi }_2=-(1-\alpha ){\psi }_1/2$. Solving the three symmetric xy conformation equations gives $A_{xy}=\alpha \tau \dot{\gamma }/D$, $A_{xx}=1+(1+\alpha )\tau \dot{\gamma }{A}_{xy}$, $A_{yy}=1-(1-\alpha )\tau \dot{\gamma }{A}_{xy}$ and $A_{zz}=1$.

In uniaxial extensional flow, the conformation equations give $A_{xx}=1/(1-2\alpha \tau \dot{ϵ})$ and $A_{yy}=A_{zz}=1/(1+\alpha \tau \dot{ϵ})$. Their stress difference gives the displayed extensional viscosity. A stable positive steady conformation requires both denominators positive; a first pole marks loss of a bounded steady response. At zero extension rate the Trouton ratio is three relative to the zero-shear viscosity.

A nonzero entire function vanishing at every nonzero point of the integer lattice has at least a constant times $R^2$ zeros in the disc of radius $R/2$. Each contributes at least $\log 2$ to Jensen's formula at radius $R$. Therefore the circle mean of $\log |F|$, and hence the logarithm of its maximum modulus, is bounded below by $c{R}^2$ for all sufficiently large boundary-zero-free radii. Choosing maximizing points yields $|F(z_j)|>e^{c^{\prime }|z_j{|}^2}$ for some $c^{\prime }>0$ and $|z_j|\to \infty$. If zero is also a zero, first factor out its finite multiplicity; the additional $m\log R$ term does not weaken the quadratic lower bound.