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A vacuum mode ${\bar{h}}_{\mu \nu }=H_{\mu \nu }{e}^{ik\cdot x}$ has a null wavevector and a transverse symmetric amplitude. Residual gauge freedom can further reduce it to two physical transverse-traceless polarizations.

An infinitesimal coordinate change generated by ${\xi }_{\mu }$ changesIt preserves Lorenz gauge exactly when $\square {\xi }_{\mu }=0$.

The evaluation map from the based path space gives a fiber sequenceSince $PX$ is contractible, it relates invariants of $X$ to those of its loop space.

With treatment coding, the intercept describes the reference factor level and each displayed factor coefficient is a contrast against that level. Changing the reference level reparametrizes the same fitted model: fitted probabilities and likelihood do not change, but coefficients representing different pairwise contrasts can have different standard errors and p-values.

For independent grouped counts $Y_i\sim \mathrm{Bin}(n_i,p_i)$, grouped-binomial logistic regression specifiesFitting the proportions $Y_i/n_i$ with binomial weights $n_i$ gives the same likelihood.

For a signed margin $u=yh(x)$, the logistic loss isIt is convex and has derivative ${\varphi }^{\prime }(u)=-e^{-u}/(1+e^{-u})$.

For binary independent responses, logistic regression sets

The logarithms of a fundamental system of $r+s-1$ units form a basis of the hyperplaneand their integer span is a full lattice in $H$.

The kernel of $\mathrm{Log}:{\mathcal{O}}_K^{\times }\to {\mathbb{R}}^{r+s}$ is exactly $\mu (K)$. If every conjugate of a unit has modulus one, all its powers lie in a bounded subset of the discrete Minkowski lattice ${\mathcal{O}}_K$, so two powers coincide and the unit is a root of unity.

If, for some real $u$,then partial summation continues $-{\zeta }^{\prime }(s-iu)/\zeta (s-iu)$ meromorphically to $\Re s>1/2$ with no pole except at $1+iu$. The zeta function has no zero to the right of the critical line, and its functional equation then implies the Riemann hypothesis.

The classical zero-free region and a truncated Perron contour givefor some constant $c>0$.

A scheme is a locally ringed space covered by open subsets isomorphic to spectra of commutative rings. It retains both the points defined by prime ideals and the local algebra of functions around them.

The product topology on $X\times Y$ is generated by the seminorms $(x,y)\mapsto p(x)$ and $(x,y)\mapsto q(y)$. Its continuous dual is canonically $X^{\ast }\oplus Y^{\ast }$.

If $C$ is a nonempty open convex subset of a real locally convex space and $x_0\notin C$, a continuous linear functional strictly separates them: after choosing a sign, $f(c)<f(x_0)$ for every $c\in C$. Apply the Hahn-Banach theorem to the Minkowski functional of a translate of $C$.

For an absorbing convex set $U$ containing zero, its Minkowski functional is $p_U(x)=\inf \{t>0:x\in tU\}$. If $U$ is open and convex, then $p_U$ is sublinear and $U=\{x:p_U(x)<1\}$.

If $X$ is locally compact Hausdorff and $A\cap K$ is closed in every compact subset $K$, then $A$ is closed. Around each point outside $A$, use a compact neighbourhood and the relative openness of its complement of $A\cap K$.

If $R$ is a Dedekind domain and $0\notin S$, then $S^{-1}R$ is an integrally closed Noetherian domain of dimension at most one. It is therefore either a Dedekind domain or, when its dimension is zero, a field.

An improper mixed state is the reduced density matrix of a subsystem entangled with unobserved degrees of freedom. A quantum state readout device returns that reduced state until a relevant collapse lies in its past light cone.

A proper mixed state represents classical uncertainty over states selected by an actual preparation or collapse. Once that selecting event lies in the readout point's past light cone, a quantum state readout device returns the selected component rather than merely the ensemble average.

A quantum state readout device at $x$ hypothetically outputs a classical description of the local quantum state under objective collapse without changing that state. Its output can respond to a remote collapse only after the collapse enters the past light cone of $x$, so the device preserves relativistic causality.