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On a compact smooth manifold without boundary of dimension at least two, a residual set of smooth Riemannian metrics consists of nowhere locally homogeneous metrics. Thus such metrics are dense in the smooth topology by the Baire category theorem. In dimension one, arclength coordinates always provide nontrivial local isometries.

For $M>0$ with integral one, the Cauchy-Schwarz inequality makes the mass functional bounded on the weighted Hilbert space. The vector $M$ has norm one, so the velocity-reset projection is the orthogonal projection onto its span. Consequently $L=\Pi -I$ is bounded and self-adjoint. This Hilbert geometry differs from the norm-one positive reset projection on phase-space $L^1$.

Completing the square in $S[q]=\frac{1}{2}qKq+Jq$ with $K=-{\partial }_t^2-m^2+iϵ$ gives the displayed source factor, where $G=K^{-1}$. In the frequency convention $e^{i\omega t}$, $\tilde{G}=({\omega }^2-m^2+iϵ{)}^{-1}$ and $G(t)=-i{e}^{-im|t|}/(2m)$ for $m>0$. The physical time-ordered oscillator quantum field theory propagator is $iG$. Sources are first taken as test functions so the pairings are defined.

For a minimally coupled scalar field, put $q^2=h^{ij}{D}_i\varphi D_j\varphi$ using the spatial covariant derivative. Spatial and normal projections of its stress-energy tensor give $\rho ={\Pi }^2/2+q^2/2+V$, ${\mathcal{J}}_i=-\Pi D_i\varphi$, and $S_{ij}=D_i\varphi D_j\varphi +h_{ij}({\Pi }^2/2-q^2/2-V)$. The trace-free stress is $D_i\varphi D_j\varphi -h_{ij}{q}^2/3$, with coefficient one. This makes the scalar anisotropic stress second order in a spatial gradient expansion.

In the usual critical Lorentz-invariant bosonic string, level-one transverse vector polarizations fit a massless representation, whereas a massive vector would require an extra polarization. This fixes the intercept to one. Polarization counting alone does not establish that conclusion in every dimension: in three dimensions the one transverse polarization transforms trivially under the transverse rotation group and can be interpreted as a scalar. The quantum consistency assumptions must accompany an argument that the first level must be massless.

For two normal matrices of the same size, a pairing of their eigenvalues has squared Euclidean discrepancy at most the squared Frobenius norm of their difference. For Hermitian matrices, the increasing real eigenvalue order gives such a pairing, because it minimizes squared discrepancy. For real symmetric matrices, the norm square is $\mathrm{Tr}(A-B{)}^2$.

Here take $X,Y$ iid, centered with variance one, and with finite moment-generating functions at all real arguments. Independence of their sum and difference gives $M(2t)=M(t{)}^{3}M(-t)$. Dividing the positive and negative argument identities shows $M(t)/M(-t)$ satisfies the symmetry step of dyadic rigidity of a moment-generating function, hence $M$ is even. The identity becomes $M(2t)=M(t{)}^4$, and the unit-variance expansion then forces $M(t)=e^{t^2/2}$. The uniqueness theorem for moment-generating functions identifies the standard normal distribution. This is a precise sufficient-hypothesis version; the displayed implication is understood with those stated assumptions.

The predictor values are random, and the regression function is the conditional mean of the response. For independent pairs and uniformly bounded conditional error variances, kernel-window occupancies govern the stochastic part of a local polynomial regression risk bound. This differs from fixed-design nonparametric regression, where the predictor values are deterministic.

Reachability within $2^k$ steps can be tested by enumerating a midpoint and recursively testing two half-paths. Configurations have $O(s)$ bits and recursion depth $O(s)$, yielding deterministic space $O(s^2)$. The space-bound discovery by exit reachability permits this simulation without assuming that the supplied function $s$ is constructible.

The function $1/\cos (1/z)$ has simple poles at $z=1/[\pi (n+1/2)]$, accumulating at zero. Its origin is therefore a non-isolated singularity, despite the underlying cosine of $1/z$ having an isolated essential singularity.

An additive nontrivial valuation on the rational field is equivalent to exactly one P-adic valuation. Integers have nonnegative values; some prime must have positive value, and Bézout's identity prevents two distinct primes from doing so. Unique factorization then gives the formula, with $c>0$ in the real-valued case. The trivial valuation is a separate possibility if admitted by the definition.

For $\delta A_{\mu }={\partial }_{\mu }\alpha$, $\delta \varphi =-ie\alpha \varphi$ and $\delta {\varphi }^{\ast }=ie\alpha {\varphi }^{\ast }$, Noether second theorem givesThis holds without imposing the field equations. On the matter equations it becomes a divergence identity for the electromagnetic equation, expressing compatibility with charge conservation. The Hamiltonian description carries the corresponding Gauss law constraint in gauge theory.

To normalize $k[a_1,\dots ,a_n]$ over an arbitrary field, choose a nonzero polynomial relation and a base $M$ larger than all its exponents. The substitutions $b_i=a_i-a_n^{M^i}$ give distinct top weights to its monomials, so its highest power of $a_n$ has a nonzero scalar coefficient. It becomes a monic polynomial equation, making the algebra finite over $k[b_1,\dots ,b_{n-1}]$. Induction proves the Noether normalization lemma, including over finite fields.

For a diffeomorphism-invariant pure metric action, write $\delta S=-\int \sqrt{-g}{E}^{ab}\delta g_{ab}$. Substitute $\delta g_{ab}={\nabla }_a{\xi }_b+{\nabla }_b{\xi }_a$ for an arbitrary compactly supported vector field and integrate by parts. Symmetry of $E^{ab}$ gives ${\nabla }_a{E}^{ab}=0$ without using field equations. This Noether identity concerns the gravitational variational tensor alone.

An axisymmetric logarithmic Newtonian gravitational potential has spheroidal equipotential surfaces and a flat galaxy rotation curve: $R{\Phi }_R(R,0)=V^2$. The singular model is defined away from the origin; a fixed reference length makes the logarithm dimensionless. Its central $r^{-2}$ mass density cusp is locally integrable, but its total mass diverges at large radius.

Rescale the perturbed cosmological metric by $a^{-2}(1+2\varphi )$. To first order, its spatial part is Euclidean and only its lapse is perturbed. The unparameterized null geodesics are unchanged. The nonzero linear connection coefficients are ${\hat{\Gamma }}_{00}^0=2{\partial }_{\eta }\Psi$, ${\hat{\Gamma }}_{0i}^0=2{\partial }_i\Psi$ and ${\hat{\Gamma }}_{00}^i=2{\partial }^i\Psi$. The affine parameter is that of the rescaled metric, not necessarily that of the original spacetime.

The displayed kernel has residue one at zero and vanishes at the endpoints of a radius-$R$ imaginary-axis segment. On either radius-$R$ semicircle, $|1+z^2/R^2|=2|\mathrm{Re}z|/R$, cancelling the inverse real-part bound on a bounded function's Laplace transform tail. Use the full transform only on a right semicircle and a thin analytic leftward path; deform the finite-interval entire transform to the large left semicircle. This proves the Newman Tauberian theorem without assuming analytic continuation across an entire left half-disk.

A Standard Model charged-current box diagram converts $d\bar{s}$ into $s\bar{d}$. Internal quarks run over $u,c,t$, and the flavor sum contains ${\lambda }_i{\lambda }_j$ with ${\lambda }_i=V_{is}^{\ast }{V}_{id}$. The loop function depends on internal masses; CKM matrix unitarity gives ${\sum }_i{\lambda }_i=0$, producing the GIM mechanism cancellation of flavor-independent terms. The amplitude is of order $g^4$ and second order in the weak interaction.

With CPT symmetry, a Hermitian kaon mass matrix is $M=\left(\begin{array}{cc}{r}_0& z\\ z^{\ast }& r_0\end{array}\right)$. Its eigenvalues are $r_0\pm |z|$. For correlated roots $a^2=z$, $b^2=z^{\ast }$ and $ab=|z|$, the normalized lower and upper eigenvectors are $(a,-b)/\sqrt{2|z|}$ and $(a,b)/\sqrt{2|z|}$. They are orthogonal. This is a mass-only result; including decay requires the generally non-Hermitian effective matrix $M-i\Gamma /2$.

In the flavor convention $K_1=(K^0-{\bar{K}}^0)/\sqrt{2}$ and $K_2=(K^0+{\bar{K}}^0)/\sqrt{2}$, the parameter $ϵ=(a-b)/(a+b)$ writes normalized mass states as $(K_1+ϵK_2)/\sqrt{1+|ϵ{|}^2}$ and $(K_2+ϵK_1)/\sqrt{1+|ϵ{|}^2}$. In the Hermitian mass-only approximation with correlated roots, $ϵ=i\tan (\phi /2)$ and the states are orthogonal. For the full non-Hermitian decay matrix the real part need not vanish. The parameter depends on the flavor-state phase convention; observable CP violation involves convention-invariant combinations with decay amplitudes.