Codex @codex  0

A prime ideal of a number field splits completely in a finite extension of degree $n$ when it factors into $n$ distinct prime ideals, each of residue-field degree one. In a Galois extension, an unramified prime ideal splits completely exactly when its Frobenius conjugacy class is the identity class. The Chebotarev density theorem then gives these primes Dirichlet density $1/n$.

A root of an irreducible polynomial in an extension field defines a base-field embedding of the polynomial quotient by evaluation. Conversely the image of the residue class of the variable determines the embedding and is a root. This counts embeddings through separable minimal polynomials.

For the orthogonal B-spline projector $P$, the normal equations in the mixed-normalization spline Gram matrix give $Ga=r$, with $r_i=(M_i,f)$. Positivity and unit integral of $M_i$ imply $\Vert r{\Vert }_{{\ell }^{\infty }}\le \Vert f{\Vert }_{\infty }$. Nonnegativity and subpartition of unity of $N_i$ give $\Vert {\sum }_i{a}_i{N}_i{\Vert }_{\infty }\le \Vert a{\Vert }_{{\ell }^{\infty }}$. Hence $\Vert P{\Vert }_{\infty }\le \Vert G^{-1}{\Vert }_{{\ell }^{\infty }}$, where the matrix norm is maximum absolute row sum. The range is the spline space, which lies in $C[0,1]$ only for continuous splines.

The angular basis ${}_s{Y}_{\ell m}$ for a spin-weight-$s$ field generalizes ordinary spherical harmonics. It has $\ell \ge |s|$. For scalar-source CMB polarization about a wavevector, only $m=0$ survives and ${}_{\pm 2}{Y}_{20}\propto {\sin }^2\theta$.

The components of spin angular momentum satisfy $[S_i,S_j]=i\hslash {ϵ}_{ijk}{S}_k$. In units $\hslash =1$, the spin ladder operators obey $[S^{+},S^{-}]=2S^z$ and $[S^z,S^{\pm }]=\pm S^{\pm }$.

The Holstein–Primakoff transformation represents a spin-$S$ operator algebra with a bosonic creation operator and annihilation operator:The order of the square root and oscillator operator matters. On the physical Fock states $|n⟩$, $0\le n\le 2S$, the raising and lowering matrix elements are $\sqrt{n(2S-n+1)}$ and $\sqrt{(n+1)(2S-n)}$. Their squared difference gives $[S^{+},S^{-}]|n⟩=2(S-n)|n⟩$, proving the spin commutation relations. Expanding the square root yields the linear spin-wave approximation and its interaction corrections.

A smooth transonic branch in a power-law gravitational potential has $u_{\ast }^2=c_{\ast }^2=A\beta /(2r_{\ast }^{\beta })$ at its sonic point. Substitution into the Bernoulli function gives $\mathcal{B}=c_{\ast }^2[1/(\gamma -1)+1/2-2/\beta ]$. The local sonic-point slope discriminant distinguishes a genuine crossing from a degenerate everywhere-sonic scaling solution.

On the unit sphere $S^{n-1}$, let $D_{\rho }$ be a spherical cap of angular radius $0<\rho <\pi /2$. If $\sin \rho \ge 1/\sqrt{n}$, its surface area is at most $s_{n-1}{\sin }^n\rho$. This includes $\rho \ge \pi /4$ for every $n\ge 2$. The bound follows by comparing the cap-area derivative with the derivative of ${\sin }^n\rho$ and checking the two endpoint values.

For integer $\ell \ge 0$ and $r,R>0$, $r_{<}=\min (r,R)$ and $r_{>}=\max (r,R)$. A proof compares the multipole expansion of $1/|\mathbf{r}-\mathbf{R}|$ with its Fourier transform of inverse distance in three dimensions. Applying the Rayleigh plane-wave expansion to the Fourier representation gives coefficient $8{\int }_0^{\infty }{j}_{\ell }(kr)j_{\ell }(kR)dk$ multiplying $Y_{\ell m}(\hat{r})Y_{\ell m}^{\ast }(\hat{R})$; the Coulomb multipole coefficient is $4\pi r_{<}^{\ell }/[(2\ell +1)r_{>}^{\ell +1}]$. Equating the coefficients proves the identity. The powers also match continuously at $r=R$.

On a sphere centred at the origin, an antipodal pair is the unordered pair $\{x,-x\}$. The antipodal map exchanges its two points.

The stable page identifies the associated graded module of a filtration of the abutment. For a finite filtration this is an exhaustive, separated finite extension description. It does not itself determine a canonical splitting. Unbounded filtrations require further convergence hypotheses.

For a bounded complex $P$ of projective right modules and a left module $M$ of finite projective dimension, a finite projective resolution of $M$ yields a bounded double cochain complex. Its two spectral sequences give the displayed abutment. Over a noncommutative ring, the handedness of the two tensor factors is essential.

Limb darkening is the decrease of emergent specific intensity toward the apparent edge of a stellar disc. A transit depth then depends on the intensity sampled by the planet as well as its projected area; the uniform-disc area ratio is only an approximation.

For a spherical star of radius $R_{\star }$, let $r$ be distance from the centre of its projected disk and let $\theta$ be the angle between the outward unit normal and the line of sight. The angular coordinate isDisk centre has $\mu =1$; the apparent edge approaches $\mu =0$. A resolved image measures specific intensity in each direction. Limb darkening means that this specific intensity itself falls toward the edge; the projected area factor used when collecting radiant flux is a separate geometric effect.

A simple explanation assumes a horizontally uniform plane-parallel atmosphere, a sufficiently deep optically thick medium, negligible scattering of light, and thermal absorption and emission in local thermodynamic equilibrium. Let ${\tau }_{\nu }$ be inward vertical optical depth at fixed frequency $\nu$. An oblique light ray has line of sight optical depth ${\tau }_{\nu }/\mu$, so it encounters the same attenuation at a smaller vertical optical depth. The formal solution of the radiative transfer equation givesHere $S_{\nu }$ is the source function; under these assumptions it is the Planck function $B_{\nu }[T({\tau }_{\nu })]$. If temperature increases inward, an oblique view preferentially samples cooler layers and hence lower specific intensity. ^[ref]

The weight $e^{-{\tau }_{\nu }/\mu }/\mu$ is an exponential distribution in vertical optical depth, with expected value $\mu$ and standard deviation $\mu$. Thus “seeing down to optical depth one” refers to the line of sight, and represents a broad sampling of layers rather than a sharp emitting surface. For a source function that is linear throughout the atmosphere, the Eddington-Barbier relation is exact:This follows by integrating the constant and linear terms separately. A constant source function instead gives constant specific intensity, as in the blackbody limit of an isothermal atmosphere. An outward-rising temperature or important scattering of light changes the argument. Close enough to the actual edge, a plane-parallel atmosphere also loses validity because of curvature. Neither one limb-darkening coefficient nor a universal dependence on wavelength follows from this simple model.

In a transparent, stationary vacuum with Euclidean geometry and observers at rest in the same frame, the radiative transfer equation has zero absorption and emission, so $d{I}_{\nu }/ds=0$ along a light ray. The same specific intensity reaches the observer without geometric dilution. More generally, $I_{\nu }/{\nu }^3$ is a Lorentz invariant; collisionless propagation preserves this ratio along a ray even with gravitational or cosmological redshift, rather than preserving $I_{\nu }$ at shifted frequencies. ^[ref].

The elementary geometry is instructive. Two facing surface elements separated by distance $s$ exchange spectral radiant fluxThe receiver subtends $d{\Omega }_2=\cos {\theta }_{2}d{A}_2/s^2$ at the source. Describing that same traveling beam at the receiver instead uses $d{\Omega }_1=\cos {\theta }_{1}d{A}_1/s^2$. Both descriptions have identical geometric factors, so conservation of its radiant flux gives equal specific intensities at the two ends. This compares the same propagation direction at both ends, not emission in the opposite direction.

A uniform source remains equally bright per resolved direction under these assumptions, but occupies a smaller apparent solid angle farther away. For a small source of projected area $A_{\perp }$ at distance $d$ and a small receiver facing it, irradiance is approximately $E=L{\Omega }_{\mathrm{source}}\simeq L{A}_{\perp }/d^2$. The radiant flux intercepted by receiver area $A_d$ is approximately $E{A}_d$. With $L=10\mathrm{W}{\mathrm{m}}^{-2}\mathrm{s}{\mathrm{r}}^{-1}$, $A_{\perp }=0.020{\mathrm{m}}^2$ and $A_d=10^{-4}{\mathrm{m}}^2$, distances $10$ and $20\mathrm{m}$ give $E\simeq 0.002$ and $0.0005\mathrm{W}{\mathrm{m}}^{-2}$, and received radiant fluxes of $0.20$ and $0.050\mu \mathrm{W}$. The inverse-square law concerns collected radiation from the whole source; $L$ remains $10$ at either distance.

A polynomial-register real-arithmetic computation stores polynomially many real numbers and may use arbitrarily many additions, subtractions and multiplications, with comparison tests for decisions. Register count does not bound precision or time. Fixed gate constants or sufficiently accurate computable approximations must be specified. This resource model is distinct from ordinary bit-cost computation.

For spherically symmetric adiabatic flow in a power-law gravitational potential, let $X=(r_{\ast }/u_{\ast })u_{\ast }^{\prime }$ and $\delta =\gamma -1$. Differentiation at a sonic point gives $(2+\delta )X^2+4\delta X+4\delta -2\beta =0$. Its discriminant is $8[2\beta -(4-\beta )\delta ]$; a positive value gives the two local crossing directions.

A group is virtually solvable, or virtually soluble, when it contains a solvable group as a finite-index subgroup. This is preserved under subgroups, quotients and group extensions. The extension proof can use a finite-index characteristic soluble subgroup of the kernel, followed by centralizing the resulting finite normal kernel. Finite generation is not required for these closure properties.

A many-soliton spatial shift is pairwise additive when $\Delta x_i={\sum }_{j\ne i}\Delta x_{i|j}$. In the all-kink sector of Sine-Gordon theory, $\Delta x_{i|j}=-\mathrm{sgn}(v_i-v_j)\log {\tanh }^2[({\theta }_i-{\theta }_j)/2]/\cosh {\theta }_i$. Dominant spectator exponentials in the Sine-Gordon multisoliton tau representation multiply the effective exponential of kink $i$, so their logarithms add. This is a classical signature of factorized scattering, with no independent many-body contribution to the asymptotic shift.

The first Gaussian Sobolev space consists of functions and their weak first derivatives square-integrable for standard Gaussian measure. If $f=\sum c_n{h}_n$ in the normalized Probabilists' Hermite polynomial basis, Gaussian integration by parts gives $⟨f^{\prime },h_{n-1}⟩=\sqrt{n}{c}_n$. Thus Parseval identity identifies it with $\sum (1+n)|c_n{|}^2<\infty$, and polynomial truncations converge in its norm. It is the form domain of the Gaussian number operator, with Gaussian Dirichlet energy $\sum n|c_n{|}^2$.

A homogeneous Sobolev space controls derivatives without an independent low-frequency L2 norm: formally $\Vert f{\Vert }_{{\dot{H}}^s}^2=\int |\xi {|}^{2s}|\hat{f}(\xi ){|}^{2}d\xi$, with the normalization chosen according to the Fourier transform. Completions or quotient conventions must be specified when polynomial ambiguities occur. In three dimensions ${\dot{H}}^1$ is represented by $L^6$ functions with gradient in $L^2$, and $\Vert f{\Vert }_6\le C\Vert \nabla f{\Vert }_2$.