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The Euclidean norm of $x=(x_1,\dots ,x_n)\in {\mathbb{R}}^n$ is

The trace norm of a finite-dimensional operator $A$ is the sum of its singular values:For a Hermitian operator $A=A_{+}-A_{-}$, it equals $\mathrm{Tr}{A}_{+}+\mathrm{Tr}{A}_{-}$.

For $x\in {\mathbb{R}}^d$, the L1 norm is

This applies normalization of an algebraic curve to the two local branches of a node.

Normalization separates a node's two branches; a rational nodal cubic has normalization ${\mathbb{P}}^1$ and two points above its node.

The geometric genus of an irreducible curve is the genus of its smooth projective normalization.

A unit normal is a normal vector of Euclidean norm one. Given a nonzero normal vector $n_0$, normalization gives $n=n_0/\Vert n_0\Vert$. This makes a dot product $w\cdot n$ measure the signed length of the normal component, without an extra factor from the length of $n_0$. On a hypersurface, the two opposite choices $n$ and $-n$ correspond to opposite normal orientations; a continuous choice on an orientable surface fixes its orientation of a surface.

A closed subset $A$ of a normal space is a countable intersection of open sets exactly when it is the zero set of some continuous function $f:X\to [0,1]$.

Under a surjective homomorphism, the image of a normal subgroup is normal.

The preimage of a normal subgroup under any group homomorphism is normal.

A weak shock has a small relative pressure jump. Its density jump agrees with the reversible adiabatic prediction through second order in the pressure jump, while entropy production first appears at third order.

In the rest frame of a one-dimensional shock, states with velocities $w_0,w_1$ satisfyand

These are the Rankine-Hugoniot conditions specialized to the mass, momentum and energy fluxes of a perfect gas.

The shock frame is an inertial reference frame in which the shock is stationary. A travelling shock then becomes a steady flow through a fixed discontinuity.

A one-way normal model assigns each factor level its own mean while assuming independent Gaussian errors with a common variance.

This Gaussian model is the setting for a one-way instance of analysis of variance.

Under finite moments and a nonsingular limiting design moment, laws of large numbers make least-squares estimators converge to the true coefficient.

Least-squares differentiation gives $X^{T}X\hat{\beta }=X^{T}Y$, or its precision-weighted analogue.

Weighted least squares minimizes ${\sum }_i{w}_i(Y_i-X_i^T\beta {)}^2$; inverse-variance weights give the efficient estimator under known heteroscedasticity.

Among linear unbiased estimators in a homoscedastic linear model, least squares has minimum variance.

For a linear regression design matrix $X$ with full column rank, the ordinary least squares coefficient vector is $\hat{\beta }=(X^{T}X{)}^{-1}{X}^{T}Y$. In a normal linear model with error covariance matrix ${\sigma }^{2}I$, it is an unbiased estimator with covariance matrix ${\sigma }^2(X^{T}X{)}^{-1}$. Replacing ${\sigma }^2$ by the residual variance estimate gives the reported coefficient standard errors.

In simple linear regression with a nonconstant predictor,

Multicollinearity means that predictor columns are strongly linearly related. It inflates coefficient variances and can make individual effects imprecise even when a joint test strongly rejects that all corresponding coefficients vanish.

With ${\hat{\sigma }}^2=\Vert Y-X\hat{\beta }{\Vert }^2/(n-p)$, a simultaneous $(1-\alpha )$ confidence region for $\beta$ is