In $⟨G,t\mid ta{t}^{-1}=\varphi (a),a\in A⟩$, a pinch is a subword $ta{t}^{-1}$ with $a\in A$, or $t^{-1}bt$ with $b\in B=\varphi (A)$. It can be replaced by a base-group element, removing two stable letters. Britton's lemma asserts that every identity word with stable letters must have a pinch after base-group multiplication is performed.

A reduced sequence is a word $g_0{t}^{{ϵ}_1}{g}_1\cdots t^{{ϵ}_k}{g}_k$ with no pinch in an HNN extension. If $k>0$, Britton's lemma makes it nontrivial and prevents it from representing a base-group element. Reduced sequences need not be unique; uniqueness requires the fixed coset representatives in the normal form theorem for an HNN extension.

For a Normal element of a C-star algebra, the C-star identity gives $\Vert a^2\Vert =\Vert a{\Vert }^2$, and iteration gives $\Vert a^{2^n}\Vert =\Vert a{\Vert }^{2^n}$. The spectral radius formula then gives $r(a)=\Vert a\Vert$. This proves the isometry of the Gelfand transform for a commutative C-star algebra directly from its defining norm identity.

In an orthonormal basis, rank-one dephasing is the quantum channel $\Delta (\rho )={\sum }_i|{\psi }_i⟩⟨{\psi }_i|\rho |{\psi }_i⟩⟨{\psi }_i|$. It removes off-diagonal entries while preserving the basis probabilities. Its output Von Neumann entropy is their Shannon entropy, and the relative-entropy identity for rank-one dephasing quantifies the entropy increase.

The bright region where the stream from Roche-lobe overflow strikes an accretion disk. Dissipation of relative kinetic energy produces a localized source of light whose visibility changes around the orbital period.

With real $k,U$ and $\mu >k^2$, substitution into the nonlinear Ginzburg-Landau equation cancels the material derivative and gives $Q^2=\mu -k^2$. A complex $k$ makes the modulus spatially varying, so it generally does not admit this constant-amplitude nonlinear Ginzburg-Landau equation ansatz.

For $\delta A_{\mu }={\partial }_{\mu }\omega$, the real quadratic gauge functional varies by $(\square +2A\cdot \partial )\omega$. Its Faddeev-Popov operator depends on the gauge field, producing a ghost-gauge vertex even in an Abelian gauge theory. With $\Psi =\int \bar{c}(F[A]+\xi h/2)$, a consistent left BRST symmetry convention gives $s\Psi =hF+\xi h^2/2-\bar{c}(\square +2A\cdot \partial )c$. The auxiliary-field integral at $\xi =0$ imposes the exact condition; finite $\xi$ gives a weighted gauge condition. This real functional is distinct from a complex quadratic Abelian gauge condition.

For the series over all nonzero integer pairs, the constant Fourier coefficient is $2\zeta (2s)y^s+2\sqrt{\pi }\Gamma (s-1/2)\zeta (2s-1)y^{1-s}/\Gamma (s)$. Integrating over one real period unfolds the translated intervals and then uses the Gaussian integral. Multiplication by ${\pi }^{-s}\Gamma (s)$ expresses it as $2\Lambda (2s)y^s+2\Lambda (2s-1)y^{1-s}$ in terms of the completed Riemann zeta function.

For independent samples from an exponential distribution, a test of rate at most ${\lambda }_0$ against rate greater than ${\lambda }_0$ rejects for small sample sums. All simple-alternative likelihood ratios against ${\lambda }_0$ decrease with that sum; its gamma distribution calibrates the size at ${\lambda }_0$. Monotonicity of the rejection probability in the rate proves that this is a uniformly most powerful test for the composite hypotheses.

On a horizontal bed, the shallow-ice approximation with a linear basal drag law gives $u=-(\rho g/\mu )h_x(h/\beta +hz-z^2/2)$ and $q=-(\rho g/\mu )(h^3/3+h^2/\beta )h_x$. With no ice accumulation or ice ablation, mass conservation is $h_t+q_x=0$. Internal shear supplies the cubic mobility and basal sliding the quadratic mobility.

Under Newton's law of cooling with fixed ambient temperature $T_0$ and $k>0$, an instantaneous jump $\beta$ at $t_{\ast }$ sets $T(t_{\ast }+)=T(t_{\ast }-)+\beta$. The subsequent temperature is $T_0+[T(t_{\ast }-)+\beta -T_0]e^{-k(t-t_{\ast })}$. Equivalently, for an initial temperature $T_i$, the full solution isThis piecewise model has the distributional derivative equation $T^{\prime }+k(T-T_0)=\beta \delta (t-t_{\ast })$, with the Dirac delta distribution representing the added temperature.

For one massless neutrino flavor with one helicity per member and $|{\mu }_{\nu }|\gg T_{\nu }$, the Fermi-Dirac distribution fills momentum states up to the positive chemical potential. If ${\mu }_{\nu }>0$, the neutrino density is ${\mu }_{\nu }^4/(8{\pi }^2)$ and the antineutrino contribution is approximately $3T_{\nu }^4{e}^{-{\mu }_{\nu }/T_{\nu }}/{\pi }^2$. For negative chemical potential the roles reverse. Thus the leading total energy density is even in ${\mu }_{\nu }$ and has no factor of two from the two species. Extra populated internal degrees of freedom do multiply it. This distinction prevents using a positive-degeneracy approximation for an exponentially dilute negative-chemical-potential distribution.

Combining the degenerate neutrino and antineutrino energy density with a critical density bound constrains the neutrino degeneracy parameter through the fourth root of the available energy budget. The numerical bound depends on $T_{\nu ,0}/T_{\gamma ,0}$ and the number of degenerate flavors. The standard ratio $(4/11{)}^{1/3}$ assumes neutrino decoupling before electron-positron annihilation in cosmology; it should be stated as a thermal-history assumption if the degeneracy is very large. The massless formula is inappropriate for a species that is nonrelativistic today.

Initial data whose normal derivative is nonzero at a wall cannot satisfy homogeneous Neumann boundary conditions continuously all the way to the initial time. For example, $-\mathrm{erf}(ay)$ has wall derivative $-2a{e}^{-a^2}/\sqrt{\pi }$ at $y=\pm 1$, which is nonzero for finite $a>0$. A parabolic mild solution of an abstract Cauchy problem can still take those initial data in an integral or square-integrable sense and satisfy the boundary condition for every positive time. An exactly compatible smooth profile is needed for a classical solution on the closed time interval.

With the tail-positive incidence convention, a network edge's reduced cost is $r_{ij}=c_{ij}-{\pi }_i+{\pi }_j$. For uncapacitated minimum-cost flow, dual feasibility requires $r_{ij}\ge 0$ and complementary slackness requires $f_{ij}{r}_{ij}=0$. A negative non-tree reduced cost identifies a candidate cost-decreasing graph cycle direction in the network simplex algorithm. This sign convention differs from the row-column potentials of a transportation problem.

A feasible positive step $\theta$ along that direction changes the objective by the corresponding negative network reduced cost times $\theta$. If a basic flow already lies at a blocking bound, the allowable step can instead be zero: that is a degenerate network simplex algorithm pivot, with no immediate objective decrease.

For known size $\theta >0$, the negative binomial distribution is an exponential family with natural parameter $\eta =\log [\mu /(\theta +\mu )]<0$ and cumulant $b_{\theta }(\eta )=-\theta \log (1-e^{\eta })$. A generalized linear model can use a logarithmic link function for its mean, even though that link is not canonical. Jointly estimating the size changes the family and variance function, requiring additional updates beyond a single fixed-family GLM fit; this is what MASS::glm.nb does.

A natural transformation all of whose components are isomorphisms. Their inverses form the inverse natural transformation.

The infinitesimal generator of a semigroup for $P_{t}f=e^{-tk}f$ is multiplication by $-k$ on its maximal domain. The bound $|(1-e^{-tk})/t|\le k$ proves the norm derivative when $kf$ is integrable; in $C_0$ use a compact-small-tail argument on $kf$. Conversely norm convergence identifies the pointwise derivative as $-kf$. Limits of graph pairs satisfy the same multiplication identity, proving it is a closed operator.

For a nonempty finite index set $I$, this hypothesis asserts that every member $H_i$ is true simultaneously. A valid local test must have its stated size under every parameter satisfying the entire intersection. The Bonferroni correction within $I$ always supplies such a test from valid elementary p-values.

Place $n$ labelled balls independently into $m$ labelled bins, each with probability $1/m$. The vector of bin counts has a multinomial distribution, and each count separately has the binomial distribution $\mathrm{Bin}(n,1/m)$. Distinct bin counts have covariance $-n/m^2$, so they are dependent for $n>0$ and $m>1$. Counts of bins of a specified size are sums of indicator variables; their expectations follow from the linearity of expectation, without any independence between bins.