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For $y^{\prime \prime }+P(z)y^{\prime }+Q(z)y=0$, the finite point $z_0$ is ordinary when $P$ and $Q$ are holomorphic at $z_0$.

In a flat radiation-plus-$\Lambda$ universe,The expansion changes from deceleration to acceleration when the radiation and cosmological-constant terms are equal, at

With zero cosmological constant, the condition $\rho +3P\ge 0$ implies $\ddot{a}\le 0$ andwherever $H\ne 0$.

For positive spatial curvature, zero cosmological constant, and $\rho ,P\ge 0$, continuity makes $\rho a^3$ nonincreasing during expansion. The negative curvature term then prevents unbounded growth of $a$, while the acceleration equation forces $\dot{a}$ to reach zero in finite time.

For a homogeneous pressureless fluid in comoving coordinates, the Euler equation giveswhile the Poisson equation givesTaking the divergence of the first relation and comparing them yieldsThis local fluid derivation describes a homogeneous infinite universe without choosing a physical centre of expansion.

For a unit-speed curve on the unit sphere,under the compatible signed-torsion and geodesic-curvature conventions.

A scalar assignment $Q(\gamma ,s)$ is a pointwise Euclidean invariant in the weak covariance sense when it is unchanged by proper Euclidean motions and satisfiesunder translation of the arc-length parameter.

The unit-speed circular helixhas constant curvature $a/(a^2+b^2)$ and constant torsion $b/(a^2+b^2)$.

A Euclidean motion has the form $E(x)=Ax+b$ with $A\in O(3)$. It preserves Euclidean distance, angles, and all extrinsic geometric quantities transformed with their orientations.

This is the dimension-three case $E(3)=O(3)⋉{\mathbb{R}}^3$ of the Euclidean group.

For a sufficiently smooth unit-speed curve with $\kappa >0$, the Frenet frame is the positively oriented orthonormal frame

Writing $\mathcal{F}=n_e/n_B$ couples the two equilibrium equations:

For a frame $L$, an $L$-valued matrix $f:A⇝B$ is a function $A\times B\to L$. Composition is matrix multiplication with join as addition and meet as multiplication.

An integral ideal is a fractional ideal of an integral domain $A$ that is contained in $A$. For a number field, take $A$ to be its ring of integers ${\mathcal{O}}_K$.

The cutoffs ${\theta }_n(\xi )=(1-|\xi |/n{)}_{+}$ increase pointwise to one, while their Fourier transforms are nonnegative and have integral $2\pi$ under the angular-frequency convention.

The squared sinc function is nonnegative and integrable; it occurs as the Fourier transform of a triangular function.

It is the pointwise square of the sinc function, with its normalization inherited from that function.

The triangular functionhas angular-frequency Fourier transform $4{\sin }^2(\xi /2)/{\xi }^2$.

For $y^{\prime }=i\omega y$ with real nonzero $\omega$, explicit Euler has amplification factor $1+i\omega h$, whose modulus $\sqrt{1+{\omega }^2{h}^2}$ exceeds one for every $h>0$.

If $C$ is Hermitian positive definite and $D$ is Hermitian, every eigenvalue of $CD$ is real. Equivalently, $CD$ is similar to the Hermitian matrix $C^{1/2}D{C}^{1/2}$.

If a real Fourier polynomial $c(x)$ is strictly positive, then $C_{nm}={\hat{c}}_{n-m}$ is Hermitian positive definite becausefor every nonzero $z$.

The degree-$d$ moment curve is $\gamma (t)=(t,t^2,\dots ,t^d)$. Tangent or radial directions from separated parameter intervals are quantitatively transverse by the Vandermonde determinant.