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A nonidentity orientation-preserving isometry of the hyperbolic plane is parabolic when it has one fixed point on the ideal boundary and none in the plane. In the upper half-plane model, every parabolic isometry is conjugate to a nonzero horizontal translation.

This is one type of isometry of the hyperbolic plane, classified by its fixed points.

The Gauss equation has singularities at $0,1,\infty$ and normalized exponent-zero solution $F(A,B;C;u)$ at the origin.

A Papperitz or P-symbol lists the three regular singular points of a second-order equation and the two characteristic exponents at each.

This symbol records the local exponents of Riemann's differential equation.

A singular point of a linear differential equation is irregular when it fails the regular-singular criterion. For $y^{\prime \prime }+P(x)y^{\prime }+Q(x)y=0$, a pole of order greater than one in $P$ or greater than two in $Q$ makes the point irregular.

For$z_0$ is a regular singular point when $(z-z_0)P(z)$ and $(z-z_0{)}^{2}Q(z)$ are analytic at $z_0$.

Substitution of a Frobenius behavior $(z-z_0{)}^{\rho }$ into the leading singular terms gives the indicial equation and its characteristic exponents.

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.