When baryons are a small fraction of the gravitating matter, their perturbation is forced by the cold dark matter potential. Comparing pressure frequency $c_s^2{k}^2$ with $4\pi G{a}^2{\bar{\rho }}_C$ yields the scale ${\lambda }_J=c_s\sqrt{\pi /(G{\bar{\rho }}_C)}$ for following the growing matter mode. It is not an absolute statement that every sub-Jeans baryon perturbation is zero or cannot be forced: a slowly varying response is ${\delta }_B\simeq (k_J^2/k^2){\delta }_C$. In the pressureless limit the difference from CDM is a constant plus a decaying mode during matter domination, so baryons catch up in relative amplitude.

NIRSpec provides multiple spectroscopic modes within $0.6$–$5.3\mu \mathrm{m}$. Its time-series modes can measure exoplanet transmission spectra and exoplanet secondary eclipses; individual settings have narrower coverage. See NASA's instrument overview.

For a low-mass singular isothermal sphere satellite on a circular orbit of radius $D$ in a singular isothermal sphere host, ${\Omega }^2=2{\sigma }_h^2/D^2$ and ${\Phi }_h^{\prime \prime }=-2{\sigma }_h^2/D^2$. The radial differential effective acceleration is $({\Omega }^2-{\Phi }_h^{\prime \prime })r$. Balancing it with satellite gravity $2{\sigma }_d^2/r$ gives the displayed Jacobi tidal radius and bound mass $m_t=\sqrt{2}D{\sigma }_d^3/(G{\sigma }_h)$. Equal mean density gives a different order-one approximation, $r_t=D{\sigma }_d/{\sigma }_h$; eccentric orbits require an orbital-history prescription.

The Jacobi abstruse identity is an identity of theta constants, with spatial argument zero:It does not assert the same relation at a general nonzero spatial argument. With $q=e^{2\pi i\tau }$, the Jacobi triple product converts it into the oscillator-product identitywhere $\mathbb{N}=\{1,2,\dots \}$ and $|q|<1$. In a critical RNS string, dividing by the eight-boson oscillator product proves equality of GSO projection-retained Neveu–Schwarz sector and Ramond sector state multiplicities at every mass level.

For real characteristics $\alpha ,\beta$ and $\mathrm{Im}\tau >0$, Gaussian decay gives normal convergence, an entire function of $z$ and holomorphic dependence on $\tau$. Integer shifts of alpha leave the function unchanged; a unit shift of beta multiplies it by $e^{2\pi i\alpha }$. Its spatial quasi-periods are $\theta (z+1)=e^{2\pi i\alpha }\theta (z)$ and $\theta (z+\tau )=e^{-\pi i\tau -2\pi i(z+\beta )}\theta (z)$. Binary characteristic labels often mean alpha and beta divided by two, so the convention must be stated.

Use period-one spatial normalization and $Q=e^{\pi i\tau }$. The conventional odd function is ${\theta }_1=-\theta [1/2,1/2]$ with the characteristic convention of theta function with characteristics. Differentiating its Jacobi triple product gives ${\theta }_1^{\prime }(0)=2\pi Q^{1/4}{\prod }_{m\ge 1}(1-Q^{2m}{)}^3$. The three even constant products have product $2Q^{1/4}{\prod }_{m\ge 1}(1-Q^{2m}{)}^3$, proving the identity. The sign changes if one instead calls the characteristic series itself theta1.

For $A={\mu }^{-1}I+U-U^T$, with $U$ the four-dimensional unit superdiagonal, the Jacobi method has eigenvalues $2i\mu \cos (j\pi /5)$, $j=1,\dots ,4$. The Gauss-Seidel method has two zero eigenvalues and $-{\mu }^2(3\pm \sqrt{5})/2$. Therefore both methods converge for every starting vector exactly when $0<|\mu |<(\sqrt{5}-1)/2$. This equality of convergence ranges follows from their spectra, not from positive-definiteness of the nonsymmetric coefficient matrix.

Assume the invariant and coinvariant groups are finite and $f_M$ generates the characteristic ideal over ${\mathbb{Z}}_p[[S]]$, $S=\gamma -1$. The structure theorem and multiplicativity of finite Euler characteristics prove the formula; finite module errors have Euler characteristic one. If $f_M\mid f_N$ and the two Euler characteristics agree, the quotient has p-adic-unit constant term and is therefore a unit power series. Thus divisibility plus this finite arithmetic comparison upgrades to equality of characteristic ideals.

For a continuous semimartingale $X$,where $L^0(X)$ is the local time of a semimartingale at zero.

The one-dimensional Itô formula for polynomials follows by induction on powers using the stochastic product rule. If it holds for $X^k$, its martingale part gives $[X^k,X]=k{X}^{k-1}\cdot [X]$. The product rule for $X^{k}X$ then yields the coefficient $k(k+1)/2$ of the bracket term for $X^{k+1}$.

// Target: probability-and-statistics.bigb

For the uniform set family of vertex sets of $r$-cliques, its $(r-2)$-fold lower shadow consists of graph edges. Repeated use of the Kruskal-Katona theorem bounds clique count from edge count. Sixteen four-cliques have at least twenty-three triples in their shadow and at least eighteen edges in their second shadow, by the expansions $16=\left(\genfrac{}{}{0ex}{}{6}{4}\right)+\left(\genfrac{}{}{0ex}{}{3}{3}\right)$ and $23=\left(\genfrac{}{}{0ex}{}{6}{3}\right)+\left(\genfrac{}{}{0ex}{}{3}{2}\right)$. Hence a graph with fifteen edges has at most fifteen four-cliques, attained by the complete graph on six vertices.

In a complex representation of a finite group $N$, the $\theta$-isotypic component is the sum of all irreducible submodules with character $\theta$. The character idempotent $e_{\theta }=\theta (1)|N{|}^{-1}{\sum }_{n\in N}\theta (n^{-1})n$ projects onto it. Every $N$-submodule is preserved by these projections and decomposes into its intersections with the isotypic components. If $N◃G$, conjugation permutes them according to ${\theta }^g(n)=\theta (g^{-1}ng)$.

This ternary quadratic form has a nonzero zero over ${\mathbb{Q}}_p$ exactly when $p\ne 7$. Away from $2,5,7,13$, the finite-field square-set intersection proof supplies a nonsingular residue zero, and Hensel lemma lifts it. At 5 use $(0,1,1)$ modulo 5; at 13 use $(3,1,0)$. At 2 hold $x=2,z=1$ and apply the strong form of Hensel lemma to $7Y^2+33$ at $Y=1$: the function and derivative have valuations 3 and 1. At 7, the equation modulo 7 forces $x=z=0$ because 5 is not a square; reduction after dividing by 7 then forces $y=0$. Any normalized integral nonzero solution would therefore have all coordinates divisible by 7, a contradiction.

A rank-three Cartesian tensor invariant under every proper rotation matrix in three dimensions is a multiple of the Levi-Civita symbol. Half-turns about coordinate axes force every nonzero component to contain each coordinate label once, and quarter-turns make these components alternating. Conversely, the determinant formula $R_{ia}{R}_{jb}{R}_{kc}{ϵ}_{abc}=(\det R){ϵ}_{ijk}$ proves invariance when $\det R=1$. Requiring invariance under reflections as an ordinary tensor forces $b=0$; invariance under proper rotations alone permits nonzero $b$.

Contract an isotropic tensor with the Kronecker delta or the Levi-Civita symbol, using invariance under proper rotations. The three displayed contractions have ranks one, two and three. The only isotropic vector is zero; an isotropic second-rank tensor is a scalar delta, and an isotropic third-rank tensor is a scalar epsilon. For $T_{ijkl}=\alpha {\delta }_{ij}{\delta }_{kl}+\beta {\delta }_{ik}{\delta }_{jl}+\gamma {\delta }_{il}{\delta }_{jk}$, direct contraction gives $\mu =3\alpha +\beta +\gamma$ and $\nu =\beta -\gamma$.

The rank-four law $c_{ijkl}=\alpha {\delta }_{ij}{\delta }_{kl}+\beta {\delta }_{ik}{\delta }_{jl}+\gamma {\delta }_{il}{\delta }_{jk}$ sends a strain array to $P_{ij}=\alpha {\delta }_{ij}{T}_{kk}+\beta T_{ij}+\gamma T_{ji}$. Applying the scalar, symmetric-traceless and axial tensor decomposition gives the displayed independent actions on the three components. The inverse requires all relevant coefficients nonzero. Restricting to a symmetric second-rank tensor eliminates the axial part and gives $P_{ij}=\alpha {\delta }_{ij}{T}_{kk}+(\beta +\gamma )T_{ij}$. Without that symmetry restriction the two-parameter form is not generally valid.

When only the bulk modulus differs from an isotropic matrix, the elastic polarization is spherical: $\tau =(\kappa -{\kappa }^0)\mathrm{tr}(\epsilon )I$. The double trace of the isotropic elastic strain Green kernel makes the trace equation local, giving the displayed formula. Individual strain components still depend on a nonlocal potential of the spherical polarization, so the complete infinitesimal strain tensor is not generally a local multiple of its reference value.

In an inviscid locally isothermal Keplerian disk, a radial normal mode of wavenumber $k$ has pressure variable $\xi$ proportional to a Probabilists' Hermite polynomial in $x=z/H$. Gaussian-weighted finite energy excludes the exponentially growing vertical branch. Polynomial termination gives $n=0,1,\dots$ and the local dispersion relation $({\hat{\omega }}^2-n{\Omega }_K^2)({\hat{\omega }}^2-{\Omega }_K^2)=k^2{c}_s^2{\hat{\omega }}^2$, where $\hat{\omega }=\omega -m{\Omega }_K$. At singular values introduced by eliminating velocity components, the original equations must also be checked. The $n=0$ propagating branch is an inertial-acoustic wave; $n=1$ has vertically uniform vertical velocity and odd pressure perturbation, permitting bending disturbances.

A locally isothermal disk in vertically harmonic external gravity has hydrostatic equilibrium $c_s^2{\rho }_z=-{\Omega }_z^{2}z\rho$, hence $\rho ={\rho }_0{e}^{-z^2/(2H^2)}$ with $H=c_s/{\Omega }_z$. Because $c_s$ is independent of height, integrating $p=c_s^2\rho$ gives the displayed pressure. The harmonic-gravity approximation has the accuracy of the thin disk expansion.

For mass density $\rho ={\rho }_0{\mathrm{sech}}^2(z/H)$ with $H^2=c_s^2/(2\pi G{\rho }_0)$, a zero-frequency gravitational mode with $k\ne 0$ obeys $W=0$. Its potential satisfies ${\varphi }_{zz}+[2{\mathrm{sech}}^2(z/H)/H^2-k^2]\varphi =0$. The substitution $\tau =\tanh (z/H)$ gives the associated Legendre differential equation of degree one and order $|k|H$. Solutions decaying at both infinities exist only at $|k|H=1$, when $\varphi \propto \mathrm{sech}(z/H)$. Longer horizontal wavelengths are unstable. Neutral mass-preserving relabeling displacements and the uniform vertical translation at $k=0$ are distinct from this fragmentation threshold.