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If sample transformations preserve the model and both hypotheses, the invariance principle restricts attention to tests whose decisions are unchanged by those transformations. Such tests are functions of a maximal invariant. In estimation, the analogous requirement is equivariance, accompanied by an invariant loss function so that risks transform compatibly. The principle respects symmetries; it does not by itself prove optimality among all unrestricted procedures.

A statistic is complete for a parameter family if the displayed implication holds for every measurable $g$ integrable under all parameter values. This property does not itself assert sufficiency; a complete sufficient statistic has both properties. A nonconstant unbiased estimator of a known constant provides a direct obstruction to completeness.

A timelike Killing vector with hypersurface orthogonality admits local coordinates with $k={\partial }_t$ and $d{s}^2=N(\mathbf{x}{)}^{2}d{t}^2-h_{ij}(\mathbf{x})d{x}^{i}d{x}^j$. The only connection coefficients with time indices that can be nonzero are ${\Gamma }_{0i}^0={\partial }_i\log N$ and ${\Gamma }_{00}^i=N{h}^{ij}{\partial }_{j}N$. Substitution into the Ricci tensor gives $R_{0i}=0$. Thus $R_{ab}{k}^b$ has only its time component and is proportional to $k_a$. Equivalently $k_d{R}^d{}_{[a}{k}_{b]}=0$. This local statement does not require the Vacuum Einstein equations and is independent of the overall curvature-sign convention.

In units $G=\hslash =c=k_B=1$, a smooth Wick rotation of the four-dimensional static patch of de Sitter spacetime has Euclidean time period $2\pi \sqrt{3/\Lambda }$. Its compact geometry is a round four-sphere of radius $\sqrt{3/\Lambda }$. The Einstein field equations give $R=4\Lambda$, and direct integration gives volume $24{\pi }^2/{\Lambda }^2$, hence gravitational Euclidean action $I_E=-3\pi /\Lambda$. With zero assigned internal energy, $I_E=\beta E-S$ gives Bekenstein-Hawking entropy $S=3\pi /\Lambda =A/4$.

In a linear Gaussian state-space model, let the predicted state mean and covariance be $m_t^{-},P_t^{-}$. The observation innovation process has conditional mean zero and covariance ${\Sigma }_t=A{P}_t^{-}{A}^{\top }+R$, with innovation ${ϵ}_t=y_t-A{m}_t^{-}$. The conditional multivariate normal distribution gives the predictive observation density. Multiplying these conditional densities by the chain rule gives the displayed likelihood function; equivalently the innovations are independent Gaussian vectors. The Kalman filter supplies $m_t^{-},P_t^{-}$ recursively, so one evaluates the exact observed-data likelihood function without integrating all latent states jointly.

The latent state follows a random walk, while observations add independent measurement noise. If the two noises have variances $W,V$, the differences have autocovariances $W+2V$ at lag zero, $-V$ at lag one, and zero at larger lags. For $V,W>0$, write $q=(W+2V+\sqrt{W^2+4WV})/2$ and $\vartheta =-V/q$. The differenced process is an invertible moving-average model with coefficient $\vartheta$ and innovation variance $q$. Its level has a unit root.

For an autoregressive moving-average model of order $(1,1)$, this state has observation row $F=(\varphi ,1,\theta )$ and transition matrixThe state noise is $(0,{\epsilon }_t,0{)}^{\top }$. Its first transition row reconstructs $X_{t-1}$ and its third row shifts the previous white noise value.

A gas is a fluid that expands to occupy its available container and is readily compressed compared with an ordinary liquid. Its pressure reflects microscopic momentum transfer to boundaries. In a sufficiently dilute classical gas, molecular separation is large enough that the ideal gas law $pV=N{k}_{B}T$ can approximate its equilibrium response.

A real gas can depart from that law because intermolecular interactions and finite molecular size become significant. Compressing a gas or changing its temperature therefore tests an equation of state, rather than a universally exact proportionality. A vapor is the gaseous form of a substance under conditions where a condensed phase can be reached by suitable compression or cooling. The same substance can pass continuously between liquid-like and gas-like states around its liquid-gas critical point, so a complete phase description requires a phase diagram as well as the broad word gas.

A liquid flows under a sustained shear stress while usually resisting a change of volume much more strongly than a gas. It adopts the shape of a container but, when it coexists with a gas and gravity permits, can form a free surface rather than expanding to fill all available space. These are mechanical observations, not a requirement that every liquid be exactly incompressible.

At rest in a uniform gravitational field, a liquid develops the hydrostatic pressure variation needed to support its weight. During flow, viscosity relates rate of deformation to viscous stress for a Newtonian fluid. Releasing a small surface disturbance probes its dynamical response through gravity waves and surface tension. Thus its static pressure, resistance to rapid compression and resistance to shear rate are distinct measurable properties. Liquid and gas can also connect continuously around a liquid-gas critical point, so the broad labels do not imply a first-order boundary along every thermodynamic path.

A solid resists sustained changes of shape on the timescale being considered. It can support a static shear stress, whereas an ordinary liquid continues to deform under sustained shear. Small loads often produce recoverable elastic deformation; larger loads can cause plastic deformation or fracture.

A solid need not be a crystal. A crystalline specimen has long-range structural order, while an amorphous solid such as ordinary glass lacks the same repeating lattice arrangement. The mechanical classification and the structural classification answer different questions. Applying a small lateral load to the top of a block while holding its base fixed tests shear resistance; a finite shear displacement at static balance is a solid-like response. At long times some solids also creep, so the observation interval and applied stress matter when comparing a material's response with an ideal elastic model.

A protostar is a forming star whose energy comes chiefly from gravitational contraction and accretion, before sustained core hydrogen burning supplies its luminosity. A fixed-mass contraction model approximates the stage after most accretion has ended; an actively accreting protostar need not have fixed mass or the same Hayashi track.

In a normal linear model with residual degrees of freedom $\nu \ge 2$, an internally studentized residual satisfies $|{\eta }_i|\le \sqrt{\nu }$ and ${\eta }_i^2/\nu \sim \mathrm{Beta}(1/2,(\nu -1)/2)$, provided its leverage is less than one. Choose an orthonormal basis of the residual space with its first direction proportional to the projection of the $i$th coordinate vector. The Gaussian coordinates in this basis are independent standard normal random variables $Z_1,\dots ,Z_{\nu }$, and ${\eta }_i=\sqrt{\nu }{Z}_1/({\sum }_j{Z}_j^2{)}^{1/2}$. The beta distribution follows from the independent chi-squared distributions of $Z_1^2$ and ${\sum }_{j=2}^{\nu }{Z}_j^2$. The bounded residual is not exactly distributed according to Student's t-distribution; it approaches the standard normal distribution as $\nu \to \infty$.

An upstream-facing probe that senses stagnation pressure. Determining speed also requires a separate static pressure and mass density.

A sequential model of a topological spectrum consists of based spaces $E_j$ and structure maps $S^1\wedge E_j\to E_{j+1}$. Indexed models use finite-dimensional subspaces of a universe for spectra and structure maps $S^{W\ominus V}\wedge E(V)\to E(W)$. Different standard models present the same stable homotopy category.

The stable homotopy category is obtained from topological spectra by inverting stable equivalences of spectra. Its morphisms are abelian groups; for a cellular source and fibrant target they are spectrum homotopy classes of point-set maps of topological spectra.

The derived smash product of spectra is the symmetric monoidal product on the stable homotopy category. It has unit sphere spectrum, preserves cofiber sequences of spectra and homotopy colimits of spectra in each variable, and satisfies ${\Sigma }^{\infty }X\wedge {\Sigma }^{\infty }Y\simeq {\Sigma }^{\infty }(X\wedge Y)$.

The homotopy cofiber of spectra of $u:A\to B$ is the derived pushout $B{\cup }_{A}CA$. It fits into a cofiber sequence of spectra.

A homotopy colimit of spectra is the derived colimit of a diagram of topological spectra. For a sequential diagram $T_0\to T_1\to \cdots$, it is the cofiber of spectra of $1-s:{\bigvee }_j{T}_j\to {\bigvee }_j{T}_j$, where $s$ uses the bonding maps into the next summands.

A universe for spectra is a countably infinite-dimensional real inner product space $U$, used to index finite-dimensional suspension directions. A standard model is the algebraic direct sum $U={⨁}_{i\ge 1}\mathbb{R}{e}_i$ with an orthonormal basis, carrying its direct-limit topology. Its vectors have finite support; this universe is not the completed Hilbert space ${\ell }^2$. Indexed topological spectra assign spaces to finite-dimensional vector subspaces $V\subseteq U$ and use the corresponding orthogonal differences for structure maps.^[ref]

A family of indexing vector subspaces is cofinal when every finite-dimensional $W\subseteq U$ is contained in some member of the family. The flag $V_n=\mathrm{span}(e_1,\dots ,e_n)$ is cofinal: a finite basis of $W$ uses only finitely many of the $e_i$. In contrast, $W_n=\mathrm{span}(e_2,e_4,\dots ,e_{2n})$ has unbounded dimension but never contains $e_1$, so it is not cofinal. Cofinality ensures that restricting the index system does not discard any finite collection of suspension directions.

For binary linear codes $C_2\subseteq C_1\subseteq {\mathbb{F}}_2^N$, the CSS code has basis $|x+C_2⟩=|C_2{|}^{-1/2}{\sum }_{y\in C_2}|x+y⟩$ for $x\in C_1/C_2$. Its X stabilizers come from $C_2$ and its Z stabilizers from the dual code $C_1^{\perp }$; their commutation follows from orthogonality. For positive logical dimension, $d_X=\min \{\mathrm{wt}(x):x\in C_1\setminus C_2\}$ and $d_Z=\min \{\mathrm{wt}(z):z\in C_2^{\perp }\setminus C_1^{\perp }\}$, with quantum code distance $d=\min (d_X,d_Z)$. Thus arbitrary errors on at most $\lfloor (d-1)/2\rfloor$ qubits are correctable. Stabilizer errors are harmless even if their weight is small; the distances exclude them.