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For observed cell counts $O_j$ and null expected counts $E_j$, the statistic isWith $k$ specified null cell probabilities and sufficiently large expected counts, its null limit is ${\chi }_{k-1}^2$.

A level-$\alpha$ test is uniformly most powerful when its power is at least that of every other level-$\alpha$ test at every parameter value in the alternative.

For a test with rejection region $R$, its power function isIts supremum over the null parameter space is the size of the test.

A joint hypothesis test imposes several parameter restrictions simultaneously and accounts for dependence among their estimators; separate one-parameter confidence intervals do not generally determine its result.

A p-value is the probability, computed under a null hypothesis, of obtaining a test statistic at least as unfavorable to that hypothesis as the observed value.

The significance level is the chosen upper bound on a test's probability of rejecting the null hypothesis when it is true.

A result is statistically significant when its calibrated p-value falls below this chosen threshold.

The null hypothesis is the set of parameter values or probability distributions against which a statistical test controls its probability of rejection.

A minimax rule minimizes the worst-case risk ${\sup }_{\theta \in \Theta }R(\theta ,\delta )$.

For a prior $\pi$, the integrated risk is $r(\pi ,\delta )=\int R(\theta ,\delta )\pi (d\theta )$. Its infimum over decision rules is the Bayes risk, attained by a Bayes rule when one exists.

For a model $\{P_{\theta }:\theta \in \Theta \}$, loss $L$, and decision rule $\delta$, the risk is

Under zero-one loss, a Bayes classifier assigns an observation to a class of greatest posterior probability. For two classes with prior probabilities ${\pi }_0,{\pi }_1$ and densities $f_0,f_1$, it chooses class one exactly when ${\pi }_1{f}_1(x)\ge {\pi }_0{f}_0(x)$.

In static coordinates on anti-de Sitter spacetime, write $f=1+r^2/L^2$, conserved energy per unit rest mass $E=f\dot{t}$, and conserved specific angular momentum $h=r^2\dot{\varphi }$. Timelike normalization gives ${\dot{r}}^2=E^2-V_{\mathrm{eff}}$ with $V_{\mathrm{eff}}=f(1+h^2/r^2)$. A circular geodesic at $r_0>0$ obeys $V_{\mathrm{eff}}^{\prime }=0$, so $h^2=r_0^4/L^2$ and $E=f(r_0)$. Thus $\dot{t}=1$ and $\dot{\varphi }=\pm 1/L$. Both its static-coordinate period and its own proper time period are $2\pi L$, independent of radius. Stability follows from $V_{\mathrm{eff}}^{\prime \prime }(r_0)=8/L^2>0$. A static clock at the orbital radius measures the different period $2\pi L\sqrt{f(r_0)}$, which does depend on radius.

In static coordinates on anti-de Sitter spacetime with curvature radius $L$ and $c=1$, put $f=1+r^2/L^2$. The conserved energy per unit rest mass is $E=fdt/d\tau$. Timelike normalization for a radial geodesic gives $(dr/d\tau {)}^2=E^2-1-r^2/L^2$, hence $d^{2}r/d{\tau }^2=-r/L^2$. An outward launch from the origin at proper time ${\tau }_0$ gives the displayed harmonic solution until its first return, after proper time $\pi L$, independently of $E>1$. Beyond that return, use a signed diameter coordinate or change the radial direction to keep the areal radius nonnegative. This is a coordinate realization of Anti-de Sitter timelike geodesic refocusing.

For Gaussian linear perturbations, the one-loop matter power spectrum iswhere $P_{22}$ correlates two second-order fields and $P_{13}$ correlates a linear field with a third-order field.

The symmetrized kernel $F_n$ convolves $n$ linear density fields to produce the $n$th-order density perturbation. In an Einstein-de Sitter universe,

The Euler-equation kernel is the symmetric function

The continuity-equation kernel isIts symmetrization is ${\alpha }_s=1+({\mathbf{k}}_1\cdot {\mathbf{k}}_2)(k_1^{-2}+k_2^{-2})/2$.

A Yukawa interaction couples left- and right-handed fermions through a scalar field. Replacing the Higgs field by its vacuum expectation value turns a gauge-invariant Standard Model Yukawa interaction into a fermion mass term.

A right-handed neutrino transforms as $(1,1{)}_0$ under $SU(3{)}_c\times SU(2{)}_L\times U(1{)}_Y$, so it is neutral under every Standard Model gauge interaction.

A surface code is a family of topological stabilizer codes on a two-dimensional cellulation. In a standard square-lattice convention, qubits lie on edges, an X-type check multiplies X on edges meeting a vertex, and a Z-type check multiplies Z around a face.

Vertex and face checks commute because their overlaps contain zero or two edges. A local Pauli operator error changes nearby check signs, producing an error syndrome. Chains of errors have syndromes at their endpoints; a chain connecting appropriate boundaries or winding around a nontrivial cycle can instead implement a logical operation. Locality makes repeated syndrome extraction useful, but a sufficiently long undetected chain can still corrupt the encoded state. The associated surface-code anyon model describes the electric and magnetic check excitations and their mutual braiding.