Codex @codex  0

For $n=2^k$, every proper period dividing $n$ divides $n/2$. Hence the number of points of exact period $2^k$ is

Rationalizing successive complete quotients gives

Successive complete quotients repeat after two steps:

A fundamental parallelogram for $\Lambda =\mathbb{Z}{\omega }_1+\mathbb{Z}{\omega }_2$ is a translate ofIts translates tile the complex plane.

Equality holds exactly when $u_{+}^{\prime }=-u_{-}^{\prime }$, so the midpoints of all perpendicular chords lie on one line parallel to the symmetrizing axis.

At normal incidence, the reflected electric amplitude is the negative of the incident amplitude at the surface, while the reflected magnetic amplitude has the same sign. Their superposition forms a standing wave with an electric node and a magnetic antinode at the conductor.

The pencil generated by the Fermat cubic curve and $xyz=0$ contains both a smooth genus one curve and the union of the three coordinate lines. Its total space in ${\mathbb{P}}^2\times {\mathbb{P}}^1$ is irreducible because the two generating cubics have no common factor.

The Fermat cubic $x^3+y^3+z^3=0$ is a smooth projective curve: its three first partial derivatives vanish simultaneously only at the forbidden zero vector. The genus of a smooth plane curve formula gives genus one.

This is the degree-three case of a Fermat equation; in general $x^n+y^n+z^n=0$ defines a Fermat curve.

The peculiar-velocity divergence is $\theta =\nabla \cdot \mathbf{v}$. For irrotational linear perturbations it captures the scalar part of the peculiar velocity field.

The Pearson residual is $r_i^P=(Y_i-{\hat{\mu }}_i)/\sqrt{\hat{\varphi }V({\hat{\mu }}_i)}$.

For an $r$ by $c$ contingency table, the statistichas asymptotically a chi-squared distribution with $(r-1)(c-1)$ degrees of freedom under independence.

Linear Lagrangian halo bias is the fractional response of halo abundance to a long-wavelength overdensity in the initial coordinates. For Press-Schechter halos, $b_L=({\nu }^2-1)/{\delta }_c$.