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Given a smooth function $\kappa (s)$, setThen $\gamma$ is unit speed and has signed curvature $\kappa$. This gives the existence part of the fundamental theorem of plane curves.

The cohomology ring is the graded ring whose multiplication is the cup product. It records intersection information that the graded cohomology groups alone do not detect.

If distribution functions $F_n$ converge pointwise to a continuous distribution function $F$, then the convergence is uniform. Choose finitely many points whose successive $F$-increments and two tails are small, then use monotonicity to squeeze $F_n-F$ between its values at adjacent points.

The mean parameter corresponding to the natural parameter $\theta$ is

Differentiating the normalization identity gives

Cardano's formula solves a depressed cubic by taking square roots in its quadratic resolvent and then compatible cube roots.

For elastic scattering with $k-k^{\prime }=q\in {\Lambda }^{\ast }$ and $|k|=|k^{\prime }|=k$,The smallest nonzero diffraction angle is determined by the shortest reciprocal-lattice vector.

This reciprocal-lattice condition is equivalent to Bragg's law, expressed in terms of the lattice-plane spacing and the wavelength.

For a finite set $S$ of crystal sites, the lattice structure factor at momentum transfer $Q$ isIt multiplies the single-atom scattering amplitude.

A three-dimensional Bravais lattice iswhere the primitive vectors $a_1,a_2,a_3$ are linearly independent.

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