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For sufficiently small nonzero $\epsilon$,is an embedded antipodally invariant curve on the round sphere. It is not a great circle and hence not a geodesic, while the antipodal isometry exchanges its two complementary discs.

A non-equatorial latitude on the round sphere is preserved transitively by rotations about its axis but has nonzero geodesic curvature. Its two complementary spherical caps are not isometric.

If a closed oriented surface is cut along a curve into two isometric surfaces with boundary, their Euler characteristics and total Gaussian curvatures agree. Applying Gauss-Bonnet to both halves, whose induced boundary orientations are opposite, gives

If ambient isometries preserving a connected curve act transitively on it, then the magnitude of its geodesic curvature is constant. Continuity then makes the signed geodesic curvature either identically zero or of one fixed sign.

A coordinate $q_i$ is ignorable, or cyclic, when $\partial L/\partial q_i=0$. The Euler--Lagrange equation then gives ${\dot{p}}_i=0$, so its generalized momentum is conserved.

A generalized linear mixed model adds latent random effects to the linear predictor of a generalized linear model.

IRLS fits a generalized linear model by repeatedly solving a weighted least-squares approximation to its score equations.

Logistic regression models the conditional log odds as an affine function of predictors.

For fixed exposures $x_i\ge 0$, a Poisson exposure model takes independent observationsIt has mean and variance $\theta x_i$ and is therefore heteroscedastic when the exposures differ.

A Poisson regression models independent count responses byIts likelihood is ${\prod }_i{e}^{-{\mu }_i}{\mu }_i^{y_i}/y_i!$.

The canonical link identifies the linear predictor with the exponential family's natural parameter: $g(\mu )=\theta$, where $\mu =b^{\prime }(\theta )$.

For a generalized linear model with variance function $V$, the Pearson estimator is

Binomial regression models independent $Y_i\sim \mathrm{Bin}(m_i,p_i)$ and relates $p_i$ to a linear predictor. Logistic and probit regression use the logit and inverse-normal links respectively.

A link function connects the response mean to the linear predictor through $g({\mu }_i)={\eta }_i$.

The linear predictor of a generalized linear model is the linear combination ${\eta }_i=x_i^T\beta$ of the explanatory variables.

If $F(s)=c{e}^{-\chi s}$ with $\chi >0$, the Gelfand-Levitan-Marchenko equation has a separable kernel. Writing $K(x,y)=-B(x)e^{-\chi y}$ reduces the integral equation to one algebraic equation for $B(x)$ and reconstructs a single soliton.

For an embedded submanifold $M\subseteq N$ of a Riemannian manifold, the vector-valued second fundamental form isHere $\overline{\nabla }$ is the ambient Levi-Civita connection and the superscript denotes projection onto the normal bundle. For a hypersurface in Euclidean space with unit normal $N$, the scalar form is $II(X,Y)=⟨A(X,Y),N⟩=⟨-dN(X),Y⟩$. In a surface parametrization its coefficients are $e=N\cdot X_{uu}$, $f=N\cdot X_{uv}$, and $g=N\cdot X_{vv}$.

Two disjoint simple closed curves on a sphere bound an annulus $A$. If both are geodesics, its boundary geodesic-curvature terms vanish, whereas $\chi (A)=0$. Gauss-Bonnet would givewhich is impossible when $K>0$ everywhere.

The total Gaussian curvature of a closed surface is two pi times its Euler characteristic.

For a standard ring torus, Gaussian curvature is positive outside, negative inside, and zero along the top and bottom circles.