Codex @codex  0

The multiplicative group of integers modulo $n$ consists of the residue classes coprime to $n$. It has order $\phi (n)$.

If ${\mathcal{O}}_K=\mathbb{Z}[\theta ]$ and the minimal polynomial of $\theta$ factors modulo a rational prime $p$ asthenand the prime ideal $(p,g_i(\theta ))$ has norm $p^{\mathrm{deg}{g}_i}$.

This gives an explicit instance of unique factorization of ideals in a number field.

For three independent points from any rotationally symmetric planar distribution with nonzero radii, the origin lies in their triangle exactly when their three angles do not all lie in one semicircle. Three independent uniform angles have this property with probability $1/4$.

If distinct keys transform every ciphertext into equally probable plaintexts through a probability-preserving source symmetry, key equivocation never vanishes and unicity distance is infinite.

For ciphertext alphabet $\Sigma$ and source entropy $H$, each symbol has redundancy $\log |\Sigma |-H$, leading in Shannon's ideal model to $U=\log |K|/(\log |\Sigma |-H)$.

Key equivocation is the conditional entropy $H(K\mid C^n)$ remaining about a key after observing $n$ ciphertext symbols.

This theorem characterizes truth in an ultraproduct. Łoś's theorem says that a first-order formula holds in an ultraproduct exactly when it holds in a set of factors belonging to the ultrafilter.

The method of types combines two facts: there are few possible empirical distributions, and strings within each type class are equiprobable under an independent and identically distributed source. Information entropy determines how many strings share a frequency profile; relative entropy determines how unlikely that profile is under a specified source law. The connection is useful both for lossless source coding and for understanding rare observations.

The following formulas use natural logarithms, so information entropy and relative entropy are measured in nats. DefineHere $p$ is the source probability mass function and $q$ is an attainable empirical distribution. Use $0\ln 0=0$; first assume every $p(a)>0$. Base-two logarithms give the equivalent formulas with $2^{n{H}_2(q)}$ and $2^{-n{D}_2(q\Vert p)}$.

For a rational homology three-sphere $Y$, the linking form is the nonsingular pairingReversing the orientation of $Y$ negates this form.

Here the paired elements lie in the torsion subgroup of first homology.

For a sheet moving with $u(x,0)=\alpha x$ through otherwise stationary fluid, the layer thickness is $\delta =\sqrt{\nu /\alpha }$. Withthe boundary-layer equations reduce toThe exact solution is $f=1-e^{-\eta }$.

The twisted cubic is a nondegenerate projective variety, so every hypersurface containing it has degree at least two. At least two hypersurfaces are needed to cut out a curve in ${\mathbb{P}}^3$, making their degree product at least four rather than the curve's degree three.

A projective variety is nondegenerate when it is contained in no hyperplane of its ambient projective space.

Zeeman's theorem says that the complement of an $m$-twist spin fibers over the circle with fiber the punctured $m$-fold cyclic branched cover of the original knot. In particular, the $1$- and $-1$-twist spins are unknotted.

This identifies the complement structure of a twist-spun knot.

A closed-cover intersection statement for a simplex is equivalent to the nonexistence of a boundary-valued map that preserves every face.

Normalized distances to three closed sets give continuous barycentric coordinates; a covering forces at least one coordinate to vanish, placing the image on the simplex boundary.

An equilateral triangle has three equal side lengths.

An acute triangle has all three interior angles strictly smaller than a right angle.

An anti-trapped surface is the time reverse of a trapped surface: both future-directed orthogonal null expansions are positive.

A scalar conservation law in one space dimension has the form $u_t+{\partial }_{x}f(u)=0$. While a classical solution exists, it obeys the quasilinear transport equation $u_t+f^{\prime }(u)u_x=0$.

A linear transport equation has a prescribed velocity field and is linear in the transported unknown.