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For a transition with fixed rest wavelength, ${\lambda }_{\mathrm{obs}}=(1+z){\lambda }_{\mathrm{rest}}$. The apparent spectroscopic velocity shift is $dv=cd{\lambda }_{\mathrm{obs}}/{\lambda }_{\mathrm{obs}}=cdz/(1+z)$. It is a convenient wavelength-shift unit, not necessarily a change in a source's peculiar velocity. Omitting the denominator $1+z$ overestimates the signal at high redshift.

The Sandage-Loeb test seeks the spectroscopic velocity drift of stable cosmological absorption features over an observer-time baseline. Many Lyman-alpha forest lines in bright distant quasars can be combined, but wavelength calibration, instrumental stability and peculiar accelerations must be controlled below the tiny cosmological signal. The absorption-template proposal is developed in the original 1998 redshift-drift study.

In a thin-shell giant model with a nonrelativistic degenerate core, $R_c\propto M_c^{-1/3}$. For a gas-pressure radiative layer with $\kappa \propto \rho T^{-3}$ and specific energy generation $ϵ\propto \rho T^{10}$, the deep profile has $P\propto T^4$, $\rho \propto T^3$ and $T\propto M_c/r$. Integrating the strongly concentrated shell power gives $L\propto C^2{M}_c^{16}{R}_c^{-13}$, while radiative transport gives $C^2\propto M_c/L$. Therefore $L^2\propto M_c^{17}{R}_c^{-13}$, yielding the displayed exponent. This is a model-specific scaling, not a universal core-mass law; it fails when envelope stripping removes the shell fuel or the confining structure.

Use a Kramers radiative-zero envelope around a stellar core and specific heating $ϵ={ϵ}_0\rho T^{67/4}$. Since $\rho \propto C{T}^{13/4}$ and $T\propto M_c/r$, $ϵ\propto C{r}^{-20}$ and $ϵ(1.05R_c)/ϵ(R_c)=1.05^{-20}\simeq 0.377$. The volume heating is $\rho ϵ\propto C^2{T}^{93/4}$. Integrating $4\pi r^2\rho ϵ$ from $R_c$ outwards gives $L\propto C^2{M}_c^{93/4}{R}_c^{-81/4}$, while radiative transport gives $C^2\propto M_c/L$. Therefore $L^2\propto M_c^{97/4}{R}_c^{-81/4}$, or $L\propto M_c^{97/8}{R}_c^{-81/8}$. For a core sequence with fixed $R_c$, the mass exponent is $97/8$. The steep integrand confines most heating to a thin shell; using the exterior constant-luminosity profile throughout it estimates the coefficient rather than resolving the shell structure.

A finitely generated group embeds in a finitely presented group if and only if it has a recursive presentation of a group. The necessity follows by enumerating proofs that substituted generator words are trivial in the larger finite group presentation. The sufficiency is the embedding theorem; it also applies to effectively countably generated recursive presentations after an effective two-generator embedding. This is the theorem of Subgroups of finitely presented groups.

If the increments of $W$ have mean $\mu$ and centered increments satisfy a causal autoregressive process of order one with $|\varphi |<1$, sum their recursive forecasts to obtain the displayed expression. For fixed forecast origin, ${\hat{W}}_{T,k}/k\to \mu$. One must observe the last increment, requiring $T\ge 2$ or a known initial level $W_0$; a single observed level without an initial-level model does not identify that increment.

For $G_{xx}=R$, zero bending moment and transverse force at the leading end give $G(x)={\int }_0^x(x-s)R(s)ds$. The two trailing-end free-end bending boundary conditions hold exactly when the displayed two integrals vanish. They express total transverse force and torque balance. Adding unknown rigid translation and yaw to a prescribed deformation supplies the two degrees of freedom of the swimming recoil correction. Its principal inertial matrix is the weighted moment matrix of $1$ and $x-x_c$ with weight $m_b+m$, hence is positive definite for a nondegenerate positive mass distribution.

With $r=x-\bar{x}$, lateral relative velocity $w=({\partial }_t+U{\partial }_x)h$, $Q=\int mwdx$ and $P=\int rmwdx$, the Lighthill elongated-body theory force on the fluid is $F_z=({\partial }_t+U{\partial }_x)(mw)$. Integration by parts gives $\int F_{z}dx=\dot{Q}+U[mw{]}_0^L$ and $\int r{F}_{z}dx=\dot{P}+U[rmw{]}_0^L-UQ$. The endpoint terms are momentum fluxes and must not be discarded when a tail has nonzero added mass. If both endpoint added masses vanish, total lateral body-plus-fluid momentum is conserved; a finite-mass body cannot then finish with zero relative crossflow and a new nonzero transverse velocity.

Put $B_j=[m(x)(x-\bar{x}{)}^j{]}_0^L$, $M_j=\int m(x)(x-\bar{x}{)}^{j}dx$, $A=\int \alpha (t)dt$, and $C=B_0(B_2-M_1)-B_1(B_1-M_0)$. For quadratic bending $h_0=-\alpha (t)(x-\bar{x}{)}^2$, the linear recoil correction in elongated-body theory with a stable post-stroke relaxation gives the displayed angle. It follows by setting time derivatives to zero in the integrated lateral and angular momentum equations. The simpler $2UA$ is exact in the zero-body-mass limit, or when $B_1=M_0$, but is not a generic finite-body-mass identity. For $L=1$, $\bar{x}=1/2$, and $m(x)=m_{\ast }{x}^2$, it becomes $2UA/(1+2M_b/m_{\ast })$. Body deformation inertia affects derivative forcing during the stroke, not this final integral relation when the deformation and its time derivative return to zero.

For a real basis matrix $E$ with basis vectors as rows, the reciprocal basis has row matrix $E^{-T}$. If either basis is orthonormal, both are, and $E^{-T}=E$. Thus $E$ is an orthogonal matrix, with determinant either $1$ or $-1$. It is a rotation matrix precisely when the original ordered basis is positively oriented relative to the reference coordinates. For example, $\mathrm{diag}(1,1,-1)$ is its own orthonormal reciprocal basis matrix but is an improper orthogonal transformation.

For a three-dimensional basis with nonzero scalar triple product $\Delta$, its reciprocal basis satisfies ${\hat{e}}_2\times {\hat{e}}_3=e_1/\Delta$ and the cyclic analogues. Consequently its scalar triple product is $1/\Delta$, and taking the reciprocal basis again recovers the original basis. The coordinates of $V$ in the original basis are $V\cdot {\hat{e}}_i$; in the reciprocal basis they are $V\cdot e_i$.

For independent affected sib pairs with IBD counts $n_0,n_1,n_2$, the likelihood function of the recombination fraction is proportional to $s^{2n_0+n_1}(1-s{)}^{2n_2+n_1}$, with $s=2\theta (1-\theta )$. At fixed sample size, the total shared-copy count $2n_2+n_1$ is a sufficient statistic by the Fisher-Neyman factorization theorem. Thus a one-sided likelihood-ratio test has the same ordering as a test for excess mean IBD sharing. This equivalence depends on the stated fully penetrant recessive model and does not extend to arbitrary disease models.

// Target: biology.bigb

A motivating experiment used isolated membranes from A-431 biological cells, which supplied abundant EGFR. Separate membrane aliquots were incubated at $0{}^{\circ }\mathrm{C}$ with ATP carrying radioactive phosphorus in its terminal phosphate group, plus magnesium or manganese ions. Radioactivity in material recovered by chemical acid-induced precipitation supplied the readout. Adding EGF beforehand enhanced incorporation relative to the unstimulated preparation; this placed a ligand-dependent biochemical response in a cell-free membrane system.^[ref]

The follow-up comparisons separated increased addition from suppressed removal: phosphate removal was rapid but unaffected by EGF, and anti-EGF antibody reversed activation by removing the ligand. These observations support reversible regulation of membrane-associated protein kinase activity.^[ref] Total incorporation alone cannot identify which protein or amino acid residue received the label, prove intrinsic receptor catalysis, or predict an intact biological cell's response.

The same account records an instructive identification error: receptor phosphorylation was initially assigned to the wrong amino acid residue because released modified amino acids migrated together in the electrophoretic electric field. Reanalysis identified phosphotyrosine residues. Retained radioactivity is evidence about the assay's separation and detection, not unique chemical identity.^[ref]

The negative part of a real-valued function $f$ is the nonnegative function $f_{-}(x)=\max (0,-f(x))$. It records the magnitude of the negative values, rather than retaining their sign: if $f(x)=-3$, then $f_{-}(x)=3$. Together with the positive part, it gives $f=f_{+}-f_{-}$ and $|f|=f_{+}+f_{-}$. This decomposition defines the Lebesgue integral of a signed function and makes the integrability condition explicit.

The upper and lower unipotent matrices generate SL2R. For $a\ne 0$, let $w(a)=U(a)L(-a^{-1})U(a)$; multiplication gives $w(a)=\left(\begin{array}{cc}0& a\\ -a^{-1}& 0\end{array}\right)$ and $w(a)w(-1)=\mathrm{diag}(a,a^{-1})$. A determinant-one matrix with upper-left entry $a\ne 0$ factors as $L(c/a)\mathrm{diag}(a,a^{-1})U(b/a)$. If that entry vanishes, first add the second row to the first by $U(1)$. Lower unipotents are conjugates of upper unipotents, so the latter also generate the whole group normally.

For an even-dimensional antisymmetric matrix $M$, put $\theta =(u+iv)/\sqrt{2}$ and $\bar{\theta }=(u-iv)/\sqrt{2}$, with independent Grassmann variables $u,v$. Anticommutation cancels the mixed terms, giving ${\bar{\theta }}^{T}M\theta =(u^{T}Mu+v^{T}Mv)/2$. The constant Grassmann change-of-variables formula changes only the normalization. The two real Pfaffians multiply to $\mathrm{Pf}(M{)}^2=\det M$, agreeing with the complex Grassmann Gaussian integral. In odd dimension $\det M=0$ and both top-degree integrals vanish.

A cubic destabilizing term and quintic saturation describe a subcritical stationary pattern bifurcation. A plane wave $A=R{e}^{iQX}$ has squared amplitude $x=R^2$ satisfying $\mu -Q^2+\alpha x-x^2=0$. The nonzero branches meet at $x=\alpha /2$, $\mu =Q^2-{\alpha }^2/4$. The lower branch is unstable to amplitude perturbations; the upper branch can also undergo an Eckhaus instability of its phase.

A Turing pattern is a spatially structured state produced by interactions between local reactions and diffusion. A Turing instability explains linear pattern onset when a homogeneous equilibrium is stable without diffusion but unstable to a nonzero spatial mode.

For positive concentrations and positive parameters, this reaction–diffusion system has homogeneous equilibrium $(u_{\ast },v_{\ast })=(1/b,1/b^2)$ and reaction Jacobian matrix $\left(\begin{array}{cc}b& -b^2\\ 2/b& -1\end{array}\right)$. Its trace is $b-1$ and determinant is $b$, making the equilibrium reaction-stable exactly when $0<b<1$. Stable complex eigenvalues permit damped oscillations, which must be distinguished from sustained periodic oscillations.

An activator in an activator-inhibitor reaction–diffusion system is a species whose local feedback enhances activation. Near a homogeneous equilibrium, positive self-feedback is reflected in the corresponding diagonal entry of the reaction Jacobian matrix. Cross-coupling to an inhibitor and differing diffusion rates can generate a Turing instability.