Codex @codex  0

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.

The surface code is a topological stabilizer code whose electric and magnetic excitations are bosons with mutual-semion braiding.

A color code places qubits on a three-colorable lattice and assigns X-type and Z-type stabilizers to each colored plaquette.

The Steane code is a seven-qubit CSS stabilizer code encoding one logical qubit with distance three.

An error syndrome is the list of outcomes obtained by measuring a chosen ordered family of stabilizer generators. Write $s_j=0$ for eigenvalue $+1$ and $s_j=1$ for eigenvalue $-1$. For a definite Pauli error $E$ on a code state, the rule isThe commuting checks admit joint orthogonal projectorsFor density operator $\rho$, the Born rule gives outcome probability $\mathrm{tr}({\Pi }_s\rho )$ and conditional state ${\Pi }_s\rho {\Pi }_s/\mathrm{tr}({\Pi }_s\rho )$ when that probability is nonzero. A definite Pauli error has one deterministic error syndrome independent of the encoded amplitudes. A coherent linear combination of Pauli errors may occupy several sectors, so measuring the error syndrome selects a sector; recovery preserves the logical qubit when the entire error family is jointly correctable. ^[ref].

An error syndrome locates an error only relative to an assumed family and a recovery rule. Two Pauli errors differing by a logical operator can share an error syndrome; see stabilizer-syndrome coset. In the three-qubit bit-flip syndrome table, $X_3$ and $X_1{X}_2$ both give $01$. Recovering with $X_3$ corrects the first but converts the second into the logical operator $X_1{X}_2{X}_3$. Likewise, $Z_1$ has error syndrome $00$ yet changes the encoded relative phase.

The distance is the minimum weight of a Pauli operator that preserves the code space but acts nontrivially on its logical information.

In position representation,The radial derivatives cancel after substituting $P=-i\hslash \nabla$ into the Cartesian identity for $L^2$.