Codex @codex  0

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.

An isospin multiplet is an irreducible finite-dimensional SU(2) representation with labels $I,M$, where $I\ge 0$, $2I$ is an integer and $M=I,I-1,\dots ,-I$. With Hermitian generators, the lowering coefficient is $\sqrt{(I+M)(I-M+1)}$. The quadratic Casimir operator has value $I(I+1)$, fixing all ladder norms and the termination at $M=-I$.

Two spin-one isospin multiplets have a unique normalized singlet. In the displayed formula each ket on the right records the two third components, not their individual total isospins. The total raising operator annihilates this combination because the coefficients of $|1,0⟩$ and $|0,1⟩$ cancel. Its exchange symmetry is positive; the five-dimensional total-isospin-two sector is also symmetric and the three-dimensional total-isospin-one sector antisymmetric.

Suppose a transition operator obeys $[I_3,H]=H/2$ and $[I_{+},H]=0$, and acts on the lower component of an isospin-half kaon. Its output has third component zero, and $I_{+}^{2}H|K⟩=H{I}_{+}^2|K⟩=0$. This excludes an isospin-two output. A spinless parent decaying to two spinless pions has zero relative orbital angular momentum, so Bose-Einstein statistics excludes the antisymmetric isospin-one sector. The output is therefore the two-isovector singlet. In the equal-mass isospin limit, charged and neutral two-pion decay widths have ratio two, including the identical-particle counting.

For a fixed closed hyperbolic surface, there are only finitely many isometry classes with its Laplacian spectrum. The Selberg trace formula gives a common discrete length spectrum and common positive hyperbolic systole. Mumford's compactness theorem confines this family to a compact subset of moduli space. If distinct members accumulated, choose nearby markings at an accumulation point and use finite length coordinates for hyperbolic surfaces. Each of these finitely many lengths converges while remaining in one discrete spectrum, so it is eventually constant. The length coordinates then force eventual equality, a contradiction. This finiteness theorem is weaker than the Wolpert generic spectral rigidity theorem.

A pair of closed orientable flat three-manifolds with equal scalar Laplace-Beltrami operator spectra but different first Betti numbers, respectively one and zero. Consequently they are not homeomorphic. They cannot be isospectral for the Hodge Laplacian on one-forms, whose zero-eigenvalue multiplicity is the first Betti number. This shows why the spectrum on functions must not be silently replaced by the spectra on all differential forms.