In a normal linear model with an intercept, $n$ observations and $p$ independent mean coefficients, adjusted $R^2$ isIt compares the residual variance estimate with the unmodelled sample variance, adjusting the raw coefficient of determination for fitted dimension. It can be negative and is not guaranteed to increase when another regressor is added. In penalized regression a related convention uses the fit's effective degrees of freedom instead of $p$; the precise smoothing-software variance convention should be specified.

A coadjoint orbit is the orbit of $\ell \in {\mathfrak{g}}^{\ast }$ under ${\mathrm{Ad}}_g^{\ast }\ell$, defined by $⟨{\mathrm{Ad}}_g^{\ast }\ell ,\xi ⟩=⟨\ell ,{\mathrm{Ad}}_{g^{-1}}\xi ⟩$. For a connected Lie group these orbits are the symplectic leaves of its Lie-Poisson manifold. Their tangent vectors are infinitesimal coadjoint actions, and their natural two-form is the Kirillov–Kostant–Souriau symplectic form.

For an oriented rank-$k$ bundle over $S^n$, with $k\ge 2$, normalize its clutching function to be based at the identity. A nowhere-zero section exists precisely when the displayed homotopy class vanishes. If it vanishes, extend $\gamma e_k$ across one hemisphere and use the constant section $e_k$ on the other. Conversely normalize any nowhere-zero section using the fiber metric. Its two hemisphere maps satisfy ${\sigma }_{-}=\gamma {\sigma }_{+}$ on the boundary. The first hemisphere makes ${\sigma }_{+}$ homotopic to a constant; the second makes $\gamma {\sigma }_{+}$ null-homotopic. Thus $\gamma e_k$ is null-homotopic. If $k>n$, the lower homotopy group of the sphere vanishes and a section always exists. Rank one is already trivial because $SO(1)$ is trivial.

Let $H$ be a closed subgroup of a Lie group $G$, and let nonzero vectors $X_j$ in the ambient Lie algebra $\mathfrak{g}$ tend to zero with $\exp X_j\in H$ and $X_j/\Vert X_j\Vert \to X$. For any real $t$, choose integers $m_j$ with $m_j\Vert X_j\Vert \to t$. Then $(\exp X_j{)}^{m_j}=\exp (m_j{X}_j)\to \exp (tX)$. Closedness gives $\exp (tX)\in H$. Compactness of the unit sphere supplies a limit direction from every sequence approaching the identity through nonidentity subgroup elements.

For a closed subgroup $H$ of a Lie group, this set is a real Lie subalgebra. Scalar closure is immediate; the Lie product formula in a Lie group and closedness of $H$ give addition. It is a closed vector subspace, also directly by continuity of the Exponential map of a Lie group. Conjugating its one-parameter subgroups by elements of $H$ preserves it. Differentiating the Adjoint representation of a Lie group along $\exp (sX)$ gives $[X,Y]\in \mathfrak{h}$ for $X,Y\in \mathfrak{h}$. After the closed-subgroup theorem establishes smoothness, $\mathfrak{h}$ is exactly the tangent space $T_{e}H$.

Choose a vector subspace $\mathfrak{m}$ complementary to the Lie algebra of a closed subgroup $H$. The differential of $F$ at $(0,0)$ is $(U,V)\mapsto U+V$, so the inverse function theorem makes $F$ a local diffeomorphism. Sufficiently small nonzero $U\in \mathfrak{m}$ cannot satisfy $\exp U\in H$: otherwise limit directions of a closed subgroup give a nonzero element of $\mathfrak{m}\cap \mathfrak{h}$. Consequently $H$ is locally exactly the coordinate slice $U=0$, proving that it is an embedded submanifold.

The Chapman estimator is a correction intended to reduce the small-sample bias of the Lincoln-Petersen estimator. Under the fixed-size hypergeometric sampling assumptions it is exactly an unbiased estimator when $M+C\ge N$; it can have nonzero bias otherwise. This qualification and the formula are stated in Barker, §1.1.1, equation (1.3), citing Chapman's work through Seber. For the toy observations,It is an alternative estimator, not a maximum-likelihood estimate for this capture-recapture model. The Chapman estimator remains finite when $R=0$, but a finite output does not supply the missing overlap information or repair unequal catchability, lost marks, or population change.

The Lincoln-Petersen estimator is the ratio estimator obtained by matching the observed overlap to its hypergeometric expected value. It is defined when $R>0$. To distinguish that ratio from the exact integer maximization, consecutive positive likelihoods satisfyThis comparison is valid for $N\ge M+C-R$. For the toy counts, the numerator minus the denominator is $500-5(N+1)$. Thus the likelihood function increases up to ninety-nine, ties at ninety-nine and one hundred, then decreases for every larger integer. This independently establishes the global maximum-likelihood estimates, beyond the plotting window. The adjacent-ratio method is given in Guiteras, §8.

Average a closed one-form over fibre rotations. The average differs from the original by an exact form because each rotation is isotopic to the identity; the Cartan formula for the Lie derivative supplies an explicit homotopy integral. Its vertical value $a$ is constant, since closedness and rotational invariance give $da=0$. Subtracting $a\omega$ leaves a basic form ${\pi }^{\ast }\xi$, and closedness gives $d\xi =aK{\Omega }_a$. For a zero-flux magnetic field, the Liouville mean of $\zeta (X+fV)$ is zero: the directional mean of $\xi (v)$ vanishes, the vertical mean is $2\pi a\int f{\Omega }_a$, and the coboundary has zero mean by volume preservation.

Remove the north pole and south pole separately. Each resulting chart is diffeomorphic to the plane, so the Poincaré lemma supplies local primitives of a closed one-form. Their difference has zero derivative on the connected overlap and is constant. Adjust one primitive by that constant and glue. This proves exactness without assuming a prior computation of de Rham cohomology.

A buoyantly rising inviscid bubble in otherwise quiescent unbounded liquid has an exterior Stokeslet with $F=4\pi \mu aU=4\pi \rho g{a}^3/3$. Its tangential traction vanishes on the sphere. The dynamic normal traction is $-3F\cos \theta /(4\pi a^2)$, while the hydrostatic normal-traction variation is $\rho ga\cos \theta$. Their dipoles cancel at the terminal velocity, leaving a constant total normal stress compatible with uniform gas pressure even when surface tension is zero. This is an exact spherical solution in the stated Stokes model, not a proof of shape stability under arbitrary perturbations.

For a clean inviscid bubble a distance $d\gg a$ below a gravity-restored approximately flat free surface, the surface displacement scales as $F/(\rho g{d}^2)=O(a^3/d^2)$. Zero normal velocity and zero tangential traction are imposed to leading order by an opposite normal image Stokeslet at distance $2d$ from the bubble. Its incident velocity is $-F/(8\pi \mu d)$, which gives the displayed correction at fixed buoyancy. The image can be represented by a same-radius inviscid drop of density $2\rho$ in liquid of density $\rho$. Its deformation-producing strain is $O(F/(\mu d^2))$, so the leading fractional capillary deformation is $O(\mathrm{Bo}(a/d{)}^2)$; small Bond number is the conventional stronger sufficient criterion.

Maximizing a node's multinomial likelihood gives class probabilities ${\hat{p}}_{tk}=n_{tk}/n_t$. Twice the negative maximized log-likelihood relative to perfect classification is the displayed deviance, with $0\log 0=0$. It equals twice the node size times its Shannon entropy. A split is chosen by maximizing $D(t)-D(t_L)-D(t_R)$, and tree deviance is the sum over terminal nodes. The training misclassification count is instead ${\sum }_t(n_t-{\max }_k{n}_{tk})$; the two criteria are different.

Pruning balances a tree's empirical loss $R(T)$ against the number of leaves. Increasing $\alpha$ favors a nested sequence of smaller subtrees, obtained by removing splits with the smallest effective cost per removed leaf. Choose complexity using cross-validation or a validation set, and assess the selected tree on a separate test set. The loss can be deviance or misclassification loss; the chosen convention must be specified.

For $p\ge 2$ and $p^{\prime }=p/(p-1)$, interpolate the two-coordinate map $(x,y)\mapsto (x+y,x-y)$ between ${\ell }^1\to {\ell }^{\infty }$ norm one and ${\ell }^2\to {\ell }^2$ norm $\sqrt{2}$. This proves the scalar inequality with factor $2^{1/p}$. Integrating that estimate and applying Minkowski inequality to $|f{|}^{p^{\prime }}+|g{|}^{p^{\prime }}$ in $L^{p/p^{\prime }}$ gives the displayed function inequality.

A planar circle other than the unit circle lifts to a great circle exactly when it meets the unit circle at two opposite points. A great circle other than the equator intersects the equator in an antipodal pair. Conversely the plane containing a lifted circle and an antipodal pair contains their midpoint, the sphere's centre. Algebraically, adding the equations $|z-c{|}^2=r^2$ and $|-z-c{|}^2=r^2$ with $|z|=1$ gives the displayed condition.

For a smooth complex projective algebraic variety, the cycle class map sends an algebraic cycle of codimension $r$ to the Poincare dual of its homology class. Rational equivalence gives the same cohomology class, so the map descends to the Chow group. It respects intersection products, pullback and proper pushforward; a cohomological obstruction therefore obstructs the corresponding proposed identity of Chow classes.

The equivalence class of an algebraic cycle modulo rational equivalence. The Chow group consists of these classes with addition induced by adding cycles.

A chordal filling is the closed connected set obtained from a proper boundary-to-boundary path by adding complementary pockets disconnected from both boundary arcs between its marked endpoints. In $(\mathbb{H},0,\infty )$ it meets the boundary only at zero and infinity and leaves two complementary simply connected sides, adjacent to the negative and positive real axes. Equivalently it is full with respect to the two marked boundary arcs. This definition uses the prime ends of a general simply connected domain and is preserved by conformal maps.

A conformally invariant random chordal filling has exponent $\alpha$ when its avoidance probability for every admissible hull $A$ is the displayed power, where ${\Phi }_A$ fixes zero and has derivative one at infinity. Composition of two hull-removing maps and the chain rule make conditional image avoidance probabilities the same as unconditional ones. The avoidance probabilities determine a chordal filling law principle therefore implies the chordal restriction property. Independent unions followed by filling add exponents, since avoidance probabilities multiply.