The reflexive closure of a binary relation $R\subseteq A\times A$ is $R\cup {\Delta }_A$, where ${\Delta }_A=\{(a,a):a\in A\}$. Every reflexive relation containing $R$ must contain the diagonal, so this is the least such extension.

The inclusion of compact Hausdorff spaces in topological spaces has a left adjoint. Its unit $X\to RX$ is universal among continuous functions from $X$ to compact Hausdorff spaces and has dense image. Completeness follows from products in a category and closed equalizers. For any map $X\to K$, its closed image has a dense subset of cardinality at most $|X|$ and hence cardinality at most $2^{2^{|X|}}$. Representatives of all such compact Hausdorff spaces and maps establish the solution-set condition, so the general adjoint functor theorem gives the reflection. The unit need not be injective or a topological embedding for an arbitrary $X$.

In the symmetric resonant cubic amplitude equation, put $A=\sqrt{r}{e}^{i\Omega t}$, $B=C{e}^{i\Omega t}$, with $d_R>0$, ${\beta }_R<0$. The transverse equation is $\dot{C}=-(d+\beta r)C-\beta r\overline{C}$. Its real matrix has trace $-2(d_R+{\beta }_{R}r)$ and determinant $|d{|}^2+2rQ$, where $Q=\mathrm{Re}(\beta \overline{d})$. A positive steady threshold requires $Q<0$; a simple zero eigenvalue also requires $Q\ne Q_{\ast }={\beta }_R|d{|}^2/(2d_R)$. A Hopf bifurcation requires $Q>Q_{\ast }$, giving a positive determinant at $r_H$. For $Q<Q_{\ast }$ the first instability is steady, producing reflection-related mixed phase-locked periodic branches. For $Q>Q_{\ast }$ it is Hopf, producing modulated oscillations, generically quasiperiodic in the original variables. Equality is a double-zero degeneracy.

A planar rotation about a point on the axis of a Euclidean reflection, composed after that Euclidean reflection, is a Euclidean reflection whose axis has been rotated by half the rotation angle. In coordinates centred at the fixed point, with the original axis horizontal, write $S=\mathrm{diag}(1,-1)$. ThenThis matrix fixes the direction $(\cos (\alpha /2),\sin (\alpha /2))$ and reverses its perpendicular direction. The axis is an unoriented line, so its angle is defined modulo $\pi$.

For $1\le k\le n\le q$, choose distinct evaluation points $a_1,\dots ,a_n$ in a finite field ${\mathbb{F}}_q$ and nonzero coordinate multipliers $v_1,\dots ,v_n$. The generalized Reed-Solomon code isMultiplying coordinates by nonzero field elements preserves Hamming weight, so this is a linear code with parameters $[n,k,n-k+1]$, just as for a Reed-Solomon code. In particular it is a maximum distance separable code.

Its dual code is ${\mathrm{GRS}}_{n-k}(a,u)$, where one may takeA common nonzero multiplier on all $u_i$ gives the same dual code. For $k=n$ the dual code is the zero code, interpreted as having no nonzero evaluation polynomials.

Let $n=q-1$ and let $\omega$ be a primitive root of unity in ${\mathbb{F}}_q$. The length-$n$ cyclic code with consecutive zeros of a cyclic code ${\omega }^b,\dots ,{\omega }^{b+n-k-1}$ has dimension $k$. It equals the generalized Reed-Solomon code of vectors $({\omega }^{j(1-b)}f({\omega }^j){)}_{j=0}^{n-1}$, $\mathrm{deg}f<k$. The choice $b=1$ is the unweighted Reed-Solomon code on these evaluation points; allowing other offsets makes this family closed under dual codes.

For based spaces there is a natural bijection $[\Sigma X,Y]\cong [X,\Omega Y]$. A map on the suspension sends a point $x$ and suspension coordinate $t$ to a loop $t\mapsto f(x,t)$; the collapsed ends and basepoint give based loops. The reverse construction evaluates such loops. The two constructions commute with based homotopies. Applied to spheres it gives ${\pi }_{n+1}Y\cong {\pi }_n\Omega Y$.

An octet operator between baryon octet states has two reduced matrix elements, because the tensor square of the SU(3) adjoint representation contains two octets. The symmetric invariant $d_{ris}$ and antisymmetric invariant $f_{ris}$ give independent component tensors. Equivalently, represent baryons by traceless matrices $B$ and let the operator act as $F[t_i,B]+D(\{t_i,B\}-\frac{2}{3}\mathrm{tr}(t_{i}B)I)$. Normalization choices are absorbed into $F,D$; Hermitian operators permit real coefficients in a Hermitian octet basis.

For positive matter and vacuum fractions with ${\Omega }_m+{\Omega }_{\Lambda }=1$, redshift drift vanishes when $s^2={\Omega }_m{s}^3+{\Omega }_{\Lambda }$, $s=1+z$. Factoring out the trivial observer root $s=1$ leaves ${\Omega }_m{s}^2-{\Omega }_{\Lambda }s-{\Omega }_{\Lambda }=0$, giving the displayed positive root of a polynomial. It lies at a positive redshift exactly when ${\Omega }_m<2/3$. It is independent of the Hubble constant at fixed density fractions and is not the redshift of the acceleration-to-deceleration transition.

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 $V\cdot e_i$.