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K-fold cross-validation partitions observations into $K$ folds, trains $K$ times while holding out one fold at a time, and combines the held-out losses.

Leave-one-out cross-validation performs one fit with each observation held out in turn and averages the held-out losses.

A Hecke algebra is a deformation of the group algebra of a Coxeter group, obtained by deforming the quadratic relations for its simple generators while retaining the braid relations.

A BN-pair in a group $G$ consists of subgroups $B,N$ such that $G=⟨B,N⟩$, $H=B\cap N$ is normal in $N$, and $W=N/H$ has distinguished involutory generators satisfying the Bruhat multiplication axioms. The quotient $W$ is its Weyl group.

A hyperbolic Coxeter group has a nondegenerate Coxeter Gram matrix of Lorentzian signature in its standard geometric realization.

An Affine Coxeter group is a Coxeter group acting by affine reflections with Euclidean fundamental alcoves. An irreducible affine Coxeter group has a positive-semidefinite Coxeter Gram matrix with a one-dimensional radical.

A finite Coxeter group is a Coxeter group with finitely many elements. Its Coxeter Gram matrix is positive definite.

A Coxeter system $(W,I,M)$ records a Coxeter group $W$, its set $I$ of simple generators, and its Coxeter matrix $M=(m_{ij})$.

On ${\mathbb{R}}^2\setminus \{0\}$, the powers of $T(x,y)=(2x,y/2)$ act freely with locally disjoint translates. Nevertheless $(2^{-n},1)\to (0,1)$ while $T^n(2^{-n},1)\to (1,0)$, so the two distinct limiting orbits cannot be separated in the quotient.

A standardized mean difference divides a group mean difference by a pooled standard deviation.

This gives a dimensionless effect size, so mean differences can be compared across measurement scales.

When $Y_p=0$, $\mathcal{F}=X_{H^{+}}$, and

The flux of the local Couette-Poiseuille profile is

For $U_0=0$, $U_1=U$, and constant $G=p_x$, zero shear at $y=0$ requiresThe velocity becomes $u=U{y}^2/h^2$, the flux per unit width is $Uh/3$, and the fluid exerts shear $-2\mu U/h$ on the moving upper wall.

A beta plane retains the leading meridional variation $f=f_0+\beta y$ of the Coriolis parameter.

An f-plane approximates the Coriolis parameter by a constant $f_0$ over a limited horizontal region.

The dual sequence consists of the coordinate functionals $(e_n^{\ast })$ in $X^{\ast }$. It is always a basic sequence, though it need not span the entire dual space.

For even length $N=2m$, split the input of a discrete Fourier transform into its even and odd samples, $x_{2r}$ and $x_{2r+1}$. Let $E_k,O_k$ be their length-$m$ discrete Fourier transforms, with negative-exponent convention. Splitting the original sum by the parity of its index gives, for $0\le k<m$,The sign change follows from ${\omega }_N^m=-1$. Thus the same two short transforms supply both halves of the result through an FFT butterfly. For $N=2^p$, recursive splitting reaches length one after $p$ levels. Each level needs $O(N)$ arithmetic operations, so the time complexity is $O(N\log N)$. For example, a length-four transform uses the two length-two transforms of $(x_0,x_2)$ and $(x_1,x_3)$, then twiddle factors $1$ and $-i$.^[ref]

For $F(x)=e^{-x}{1}_{x\ge 0}$, induction in the convolution integral givesIts Fourier transform in the angular-frequency convention is $(1+ik{)}^{-n}$.

For nonzero terms, ifthen $\sum a_n$ converges absolutely. If the ratio has a limit greater than one, the terms do not tend to zero and the series diverges.

For positive terms, if the ratio of two sequences tends to a finite positive number, their series either both converge or both diverge.