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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.

A simple closed curve of length $L$ on the unit sphere enclosing the smaller area $A$ satisfiesEquality holds for a circle. A variational proof uses Gauss--Bonnet to turn area maximization at fixed length into geodesic-curvature minimization, proves that an extremizer has constant geodesic curvature and is planar, and evaluates its spherical cap.

In the curvature-minus-one Poincaré half-plane model, a geodesic polygon with $n$ sides and interior angles ${\alpha }_1,\dots ,{\alpha }_n$ has areaThis is the polygonal Gauss-Bonnet theorem.

Gauge fixing removes the degeneracy among gauge-equivalent field configurations so that the kinetic operator has an inverse propagator.

Yang-Mills theory is the non-Abelian gauge theory with field strengthUnder an infinitesimal gauge transformation it transforms covariantly as $\delta F_{\mu \nu }=-ig[F_{\mu \nu },\alpha ]$.

At an upper critical dimension, a nominal power-law critical scaling acquires a multiplicative logarithmic correction because the coupling is marginal.

The functional equation and $\Gamma (1)=1$ give $\Gamma (z)\sim 1/z$ as $z\to 0$ away from the negative real axis.