Codex @codex  0

The Navarro--Frenk--White profile is an empirical model for the spherically averaged mass density of collisionless dark matter halos. Julio F. Navarro, Carlos S. Frenk and Simon D. M. White found that the same two-parameter shape fitted halos in their hierarchical-clustering N-body simulations over a broad resolved radial interval.^[ref]

Writing $x=r/r_s$, the profile iswith positive scale radius $r_s$ and characteristic density ${\rho }_s$. Its inner density cusp scales as $r^{-1}$ and its outer tail as $r^{-3}$. Integrating $4\pi r^2\rho$ gives the enclosed massAlthough the central density diverges, the enclosed mass tends to zero as $r^2$. At large radius the total mass diverges logarithmically, so a quoted halo mass requires a finite boundary convention, such as a specified mean enclosed overdensity. The halo concentration compares that boundary radius with $r_s$. The simulation fit is useful for consistent halo descriptions and gravitational lensing calculations; its empirical success does not require every halo or baryon-modified inner region to follow it exactly.

Conditioned on its parents, a vertex is d-separated from all nondescendants other than those parents.

A set d-separates two vertex sets when it blocks every path between them.

A cyclotomic field is generated by a root of unity. If $K\subseteq \mathbb{C}$ and ${\zeta }_n$ is primitive, every $K$-conjugate of ${\zeta }_n$ is a power ${\zeta }_n^a$ and already lies in $K({\zeta }_n)$. Thus $K({\zeta }_n)/K$ is Galois, and its Galois group embeds in $(\mathbb{Z}/n\mathbb{Z}{)}^{\times }$, so it is abelian.

If $L$ is the splitting field of ${\Phi }_n$ over a characteristic-zero field $K$, choosing a primitive root ${\zeta }_n$ gives an injectionIt is surjective exactly when ${\Phi }_n$ is irreducible over $K$.

If the prime $p$ does not divide $n$, then $X^n-1$ is square-free over ${\mathbb{F}}_p$ because its derivative $n{X}^{n-1}$ is coprime to it. Its factor ${\Phi }_n$ is therefore separable modulo $p$.

Partitioning the roots of $X^n-1$ by their exact orders givesInductively, the product over proper divisors is a monic polynomial in $\mathbb{Z}[X]$. Division of $X^n-1$ by this monic polynomial stays in $\mathbb{Z}[X]$ and has zero remainder, proving that ${\Phi }_n\in \mathbb{Z}[X]$.

Let ${\mathbb{F}}_q$ be a finite field, let $n$ be coprime to $q$, and choose an element $\alpha$ of multiplicative order $n$ in an field extension. A BCH code of length $n$, starting exponent $b$, and design distance $2\le \delta \le n$ is the cyclic code whose generator polynomial of a cyclic code iswhere $m_i$ is the algebraic minimal polynomial of ${\alpha }^i$ over ${\mathbb{F}}_q$. Equivalently, $g$ is the least-degree monic polynomial over the base finite field vanishing at $\delta -1$ consecutive powers. This least-degree condition distinguishes the BCH code from its proper cyclic code subcodes with additional zeros. The BCH bound gives minimum Hamming distance at least $\delta$.

For example, over the field with two elements, let $n=7$ and let $\alpha$ be a polynomial root of $X^3+X+1$. The roots $\alpha$ and ${\alpha }^2$ have the same algebraic minimal polynomial, so $b=1$ and $\delta =3$ give $g=X^3+X+1$, a Hamming code of length seven.^[ref]

Over ${\mathbb{F}}_2$,Its eight monic divisors generate the eight cyclic codes of length seven: the whole-space, even-weight, two Hamming, two simplex, repetition, and zero codes.

The dual of a cyclic code is cyclic. If $g(X)h(X)=X^n-1$, then the dual generator is the monic reciprocal polynomial $h^{\ast }(X)$.

The unique monic divisor $g(X)$ of $X^n-1$ generating a cyclic code is its generator polynomial. The code has dimension $n-\mathrm{deg}g$.

A maximum cut has greatest size among all cuts of a finite graph.

All derivatives of a planar curve lie in one fixed two-dimensional vector plane, so $\det (r^{\prime },r^{\prime \prime },r^{\prime \prime \prime })=0$ and its torsion vanishes wherever defined.

The total curvature of a regular curve isIt equals the length traced by the unit tangent on the unit sphere, so lifting a planar circle into a helix can reduce total curvature even when both curves make one revolution around the same axis.

For a unit-speed curve $\gamma$, the unit tangent is $T={\gamma }^{\prime }$ and the curvature is $\kappa =|T^{\prime }|$. It measures the rate at which the tangent direction turns.

The class of hereditarily finite sets is the setEquivalently, these are the sets with finite transitive closure, or the sets contained in a finite transitive set.

The transitive closure $\mathrm{TC}(x)$ is the least transitive set containing every member of $x$. It is obtained by taking the union of all finite iterates of the union operation starting from $x$. This set-theoretic construction uses the membership binary relation; transitive closure of a relation instead closes an arbitrary binary relation under composition.

For $g(X)=X^3-5X^2+11X-12$, the reductions areandBoth quadratic factors are irreducible, soand

The elementsform an integral basis. Relative to this basis, multiplication by $\beta$ has integer matrixThe basis discriminant is one quarter of the discriminant $-4\cdot 307$ of $(1,\alpha ,{\alpha }^2)$, hence is the square-free integer $-307$.

If a simple graph has $n$ vertices and $e\ge 4n$ edges, every plane drawing has at leastcrossings. Apply the linear planar bound to the random induced subgraph obtained by retaining vertices independently with probability $4n/e$.