Quantile regression models a conditional quantile of a response, rather than its conditional mean. Fitting by the check loss uses an asymmetric absolute-error penalty, so different values of $\tau$ describe different parts of the response distribution. Nonlinear models may use any identifiable continuous regression function, not only a linear predictor.

A restricted cubic spline represents a smooth covariate effect by joined cubic pieces, with linear tails beyond the boundary knots. It allows nonlinear regression effects while controlling extrapolation. In a Cox proportional-hazards model, test the nonlinear basis coefficients jointly and report the fitted log-hazard or hazard-ratio curve with uncertainty.

A fractional polynomial uses selected real powers of a positive covariate to model a smooth regression effect. Power zero denotes a logarithm, and a repeated power can generate a term $x^p\log x$. Such terms permit a wider family than ordinary integer-power polynomials, but their selection adds modeling uncertainty and requires validation.

Design of experiments chooses experimental units, treatments, replication, blocks in experimental design and randomization so that scientifically useful treatment contrasts can be estimated with meaningful standard errors. Allocation determines which measurements supply independent treatment information.

A response variable is the outcome whose variation is described by a statistical model or whose change is compared across treatments. Its unit, measurement procedure and observation time must be specified. An outcome consisting of a feature count is one response per sketch, not one independent experimental response per feature.

A statistical functional is pathwise differentiable relative to chosen statistical paths if its derivative along every path depends only on that path's score function and defines a bounded linear functional on their statistical tangent space. The Riesz representation theorem expresses this derivative as an L2 inner product with a unique element of the statistical tangent space, the canonical gradient. A path family and the associated derivative remainder conditions must both be specified; a formal derivative along one convenient family does not establish differentiability along all paths.

Along bounded density tilts, this functional has derivative $4\int f^{4}g=P_f(4f^{3}g)$. At a bounded baseline density its canonical gradient, relative to these regular paths, is $4(f^3-\int f^4)$. A common local bound on nearby densities suffices to obtain the same derivative along arbitrary differentiable-in-quadratic-mean paths in that bounded neighborhood. A bounded baseline alone is insufficient, because of the spike obstruction to density-power differentiability.

Rescaling the spatial metric by the squared static lapse function gives a useful three-dimensional quotient metric $\gamma$. In Vacuum Einstein equations, its Ricci tensor obeys $R_{ij}(\gamma )=2{\partial }_{i}U{\partial }_{j}U$, while $U$ is harmonic. The contracted Bianchi identity recovers the harmonic equation without dividing at critical points.

In units $\hslash =c=k_B=1$, the central inertial observer has temperature $1/(2\pi R)$. The Tolman temperature law redshifts it to the displayed static-observer value. Its acceleration satisfies $4{\pi }^2{T}^2=a^2+R^{-2}$. Near the horizon the temperature approaches $a/(2\pi )$, as predicted by the Unruh effect; at the center temperature remains nonzero despite vanishing acceleration.

A refractory period makes a patient ineligible for another event for a fixed time $r$ after a recurrence. Exclude this time from exposure and risk sets by delaying the next interval's entry to $t_k+r$. In a gap-time recurrent-event model measured from the previous event, the next interval enters at gap age $r$, not zero; the clock is not reset at the end of recovery.

Gravitational fragmentation divides a self-gravitating gas distribution into condensations that can undergo separate collapse. Pressure support, cooling and rotation influence the scales and outcome; it is not equivalent to an instantaneous equal-mass fission of a rigidly rotating sphere. Isothermal fragmentation is a useful limiting case in which thermal adjustment keeps the gas temperature approximately fixed.

Spline interpolation matches prescribed function values at distinct sites by an element of a finite-dimensional spline space. In a B-spline basis its existence and uniqueness for all data are equivalent to invertibility of the B-spline collocation matrix. Distinct ordered sites satisfying the Schoenberg–Whitney theorem give this condition.

In units $\hslash =1$, spin one-half is the two-dimensional irreducible spin angular momentum representation with components $S_i={\sigma }_i/2$. The Pauli matrices give eigenvalues $\pm 1/2$ along every unit axis. The normalized $x$-axis states are $(|{\uparrow }_z⟩\pm |{\downarrow }_z⟩)/\sqrt{2}$.

Use dimensionless spin components, equivalently units $\hslash =1$, in the following formulas.

The Jordan–Wigner transformation represents a one-dimensional spin one-half chain by fermions with a parity string preceding each site. With $P_n={\prod }_{m<n}(1-2n_m)$, one has $S_n^{+}=P_n{c}_n$ and $S_n^z=1/2-n_n$. The string converts the off-site canonical anticommutation relations into commuting spin operators. Adjacent exchange terms become quadratic fermion expressions, while a periodic end bond depends on global fermion parity.

Surface area is the two-dimensional measure induced on the unit sphere in three dimensions. It is invariant under orthogonal rotations. It is not the restriction of ambient three-dimensional Lebesgue measure, under which the sphere has measure zero. Normalizing by the sphere area gives a probability measure.

The Cartesian product $S^p\times S^q$ has one cell in each of dimensions $0,p,q,p+q$, counting the two middle cells separately when $p=q$. For positive $p,q$, its integral cohomology ring has generators $a,b$ of degrees $p,q$, with $a^2=b^2=0$ and $ba=(-1{)}^{pq}ab$. The class $ab$ is a product orientation class. This follows from the Künneth theorem.

Every $g\in S{L}_2(\mathbb{R})$ factors into an upper triangular determinant-one matrix and a rotation matrix. This follows because the triangular subgroup acts transitively on the complex upper half-plane, a group orbit of the full group, while $SO(2)$ is the stabilizer subgroup of $i$. The intersection of the two subgroups is $\{\pm I\}$, giving exactly two matrix pairs $(h,k)$ and $(-h,-k)$. Requiring the diagonal of $h$ to be positive gives uniqueness. For $g=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$ and $q=\sqrt{c^2+d^2}$, the positive branch is $h=\left(\begin{array}{cc}1/q& (ac+bd)/q\\ 0& q\end{array}\right)$ and $k=q^{-1}\left(\begin{array}{cc}d& -c\\ c& d\end{array}\right)$.

Complex two-by-two matrices of determinant one form a complex three-dimensional Lie group, or a real six-dimensional Lie group. Their congruence action on Hermitian two-by-two matrices gives the Lorentz spinor double cover with kernel $\{I,-I\}$. The two Weyl spinor representations are $A$ and $(A^{†}{)}^{-1}$. Treating the group as real is essential for the second, antiholomorphic representation.

A minimum spanning tree of a finite connected undirected weighted graph minimizes the sum of edge weights over all spanning trees. Kruskal's algorithm repeatedly accepts a least-weight edge joining different current connected components. The cut-exchange argument preserves an optimum containing every accepted edge, including when weights tie.

For nonnegative edge costs, an MST gives a lower bound on a travelling salesman tour: removing one tour edge leaves a spanning path whose cost is no greater than the tour cost, and the minimum spanning tree costs no more than that path.

Nonnegativity is needed for this comparison. A triangle with all three edge weights equal to minus one has tour cost $-3$ and minimum spanning tree cost $-2$, so the latter is not a lower bound. The MST optimization problem itself permits negative edge weights.

This topology has basic open sets $[A{]}^{\omega }$ for infinite ground sets $A$, without fixed finite stems. It is coarser than the Ellentuck topology and differs from the ordinary topology on infinite subsets, whose basic cylinders fix an initial segment. The symbol $\tau$ is sometimes used for either coarser convention, so it should be accompanied by a definition.