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A rotlet dipole is the derivative of a rotlet along the separation of two opposed torques. A torque-free bacterium can have this far field because its rotating flagella and counterrotating body exert spatially separated internal torque reactions. The displayed axial formula uses the observation-coordinate derivative convention and signed torque moment $D$.

For contact with $R_d/R_{\odot }=M_d/M_{\odot }$ and $R_L/a=f(M_d/M{)}^{1/3}$, Kepler's third law gives $P_0=2\pi \sqrt{R_{\odot }^3/(f^{3}G{M}_{\odot })}$. The total binary mass cancels. The linear period relation depends on that particular equilibrium radius sequence; a donor with a different stellar radius response exponent follows a different mass-period relation.

Differentiate the genus-one Hadamard factorization. Its complex difference terms converge as $O(|\rho {|}^{-2})$, and the individual real sums converge by the absolute convergence of the real xi logarithmic derivative. The remaining constant is $\Re B+\sum \Re (1/\rho )$. On the critical line the xi logarithmic derivative has zero real part, while the kernel terms cancel in pairs reflected across that line. Hence this constant vanishes and gives the displayed formula away from zeros.

Riemann hypothesis is equivalent to monotonicity of $\sigma \mapsto |\xi (\sigma +it)|$ for every fixed $t$ on $\sigma \ge 1/2$. Under Riemann hypothesis the real logarithmic derivative of Riemann xi is positive for $\sigma >1/2$. Conversely a zero right of that line would force the nonnegative monotone modulus to vanish on an interval, contrary to the identity theorem. Functional symmetry rules out zeros to the left.

Under Riemann hypothesis, integrate a smoothed explicit formula from $1/2+\delta$ to two, taking $x=\log |t|$. The prime contribution is $O_{\delta }(x^{1-2\delta })$, while the local zero count bounds the zero contribution by $O_{\delta }(x^{-\delta }\log |t|/\log x)$. Both are $o(\log |t|)$ for fixed $0<\delta <1/4$, yielding the displayed bound; other fixed positive shifts follow by standard strip bounds and the Euler product.

The subpower zeta bound to the right of the critical line gives an arbitrarily small exponent at $1/2+\delta$. The functional equation gives exponent $\delta +\epsilon$ at $1/2-\delta$. The Phragmén–Lindelöf principle gives exponent $\delta /2+\epsilon$ at the midpoint. Letting these fixed positive parameters be arbitrarily small proves the Lindelöf hypothesis.

Under a stationary initial law, reversal leaves the law of a finite reversible-chain path unchanged. The first time an ordered occurrence of state $a$ then state $b$ appears therefore has the same distribution as the reverse ordered occurrence. Decomposition at the first entrance relates this symmetry to differences of mean hitting times. If positive diagonal return times are used instead of zero diagonal entrance times, their stationary-weighted terms cancel by Kac's lemma.

For unnormalized model posterior densities $h_j$, model-selection probabilities $s_{jk}$ and parameter-proposal densities $q_{jk}$, accept a proposal $(j,\theta )\to (k,{\theta }^{\prime })$ with the minimum of one and $h_k({\theta }^{\prime })s_{kj}({\theta }^{\prime })q_{kj}(\theta \mid {\theta }^{\prime })/[h_j(\theta )s_{jk}(\theta )q_{jk}({\theta }^{\prime }\mid \theta )]$. A deterministic matching-map formulation instead includes its explicit Jacobian determinant; it must not be counted again when already absorbed into a direct proposal density.

A pseudo-prior is a proper density assigned to a parameter that is inactive under a particular model. Integrating it out leaves that model's likelihood and marginal evidence unchanged. Its choice affects movement between model states, but not the intended marginal posterior over models.

A replicate is an independently allocated instance of a treatment on an experimental unit. Independent replicates permit estimation of response variation; subsamples within one unit do not supply additional independent treatment replication.

For $\beta =-b{g}^3$, ${\gamma }_{\varphi }=c{g}^2$, and $b>0$, one has ${\bar{g}}^2=g^2/(1+b{g}^2\log z)$. Integrating the field-renormalization anomalous dimension gives the displayed logarithmic factor. At large $z$, a nonzero free matching factor yields $d\sim C(\log z{)}^{c/b}$ in the printed RG sign convention.

Relative permeability divided by the phase's dynamic viscosity. In the phase Darcy law, $q_i=-K{\lambda }_i\nabla p_i$ without body forces. Ratios of mobilities determine fractional flow when the phases have a common pressure gradient.

The regular syntactic category has regular formulas-in-context as objects and provably functional regular relations as arrows. It has finite limits and stable image factorizations, with existential quantification supplying images. The regular coverage on this category presents the classifying topos of the theory.

If a regular theory proves an outer coherent sequent $\varphi ⊢{\bigvee }_{i=1}^n{\psi }_i$ with all formulas regular, it proves $\varphi ⊢{\psi }_i$ for some $i$. In the classifying topos, the corresponding subobjects cover the representable associated with $\varphi$. Irreducibility of that representable and full faithfulness of the syntactic Yoneda embedding produce the desired derivation.

For a subfunctor $F\prime \subseteq F$ of a regular-coverage sheaf, its closure consists of sections whose restriction belongs to $F\prime$ along one covering regular epimorphism. Pullbacks prove that local membership is a subfunctor; composition of witnessing covers proves the sheaf condition. It is the smallest subsheaf containing $F\prime$.

If a family of subsheaves covers $yA$, local membership of $1_A$ gives a regular epimorphism $\alpha :B\to A$ belonging to one subsheaf. The sheaf condition descends that section to $1_A$ in the same subsheaf. All maps into $A$ are restrictions of $1_A$, so that subsheaf is $yA$. Thus representable sheaves for the regular coverage are irreducible even for arbitrary unions.

The displayed metric on antipodal classes of the unit sphere induces its quotient topology. Triangle inequality follows by choosing the signs that realize each minimum and applying the Euclidean triangle inequality. Distinct classes have disjoint saturated unions of small balls, proving the quotient is Hausdorff; compactness follows from the continuous quotient map.

For $k>0$, horizontal Fourier transform gives $({\partial }_z^2-k^2)\tilde{\Phi }=4\pi G\tilde{\Sigma }\delta (z)$. Decay and continuity give an exponential on each side; its derivative jump fixes the coefficient. The uniform mode needs a separate linear vertical solution and a potential reference.

The matrix coefficients of a rational $G{L}_m$ action belong to its coordinate ring $\mathbb{C}[g_{ab},\det (g{)}^{-1}]$. Multiplying the action by a sufficiently large determinant twist clears all determinant denominators and produces a polynomial representation. Irreducible rational actions are indexed by decreasing integer highest weights, with negative coordinates allowed.

For finite-dimensional rational representations of complex $G{L}_m$ or $S{L}_m$, average a Hermitian inner product over the compact group $U(m)$ or $SU(m)$ using normalized Haar measure. Orthogonal complements become invariant under the compact group and its complexified Lie algebra. The latter generates the complex group, giving invariant complements and hence complete reducibility. This conclusion does not hold for arbitrary affine algebraic groups, such as the additive group.