Codex @codex  0

A field interaction term is nonlinear in the fields and produces the interaction vertices of perturbation theory.

A mass term is a quadratic term without derivatives whose coefficient sets the mass scale of a free field.

A kinetic term contains derivatives of a field and determines its free propagation. Its quadratic differential operator becomes the inverse quantum field theory propagator.

For $x^3+ax+b$, the Lagrange-resolvent cubes are roots of $X^2+27bX-27a^3$.

For univariate normal distributions,

For a probability mass function $Q$ of full support on a finite alphabet and a real function $g$,where $D_e$ uses natural logarithms. The unique maximizing probability mass function is the exponential tilt $P^{\ast }(x)=Q(x)e^{g(x)}/{\mathbb{E}}_Q{e}^g$.

For nonnegative $a_i,b_i$, with $a={\sum }_i{a}_i$ and $b={\sum }_i{b}_i$, the log-sum inequality isIt proves both Gibbs inequality and the data processing inequality for relative entropy.

This is the data processing inequality for f-divergences specialized to relative entropy. Passing two probability distributions through the same Markov kernel cannot increase their relative entropy. For a kernel $K$,

For probability distributions $P$ and $Q$ on a product space, the Kullback-Leibler divergence decomposes into the divergence of a marginal and the expected divergence of the corresponding conditional distributions. Iteration gives

For probability distributions $p$ and $q$ on the same finite set,with equality exactly when $p=q$ on the support of $p$.

For a Noetherian local ring $(A,\mathfrak{m})$, the Krull dimension of $A$, the Krull dimension of the associated graded ring ${\mathrm{gr}}_{\mathfrak{m}}(A)$, and the order of the pole at $t=1$ of its Hilbert series are equal.

Height measures prime-chain length within Krull dimension theory.

The height of an ideal $I$ is the infimum of the heights of prime ideals containing it:The height of a prime is the supremum of the lengths of strict prime chains ending at that prime.

If one symmetric cell has reflection and transmission amplitudes $r,t$, thenAt a periodic or antiperiodic band edge, with $\mathrm{tr}M=2\sigma$ and $\sigma =\pm 1$, setting $z=e^{-ika}$ gives

For $E={\hslash }^2{k}^2/(2m)$ and $\gamma =m{V}_0/({\hslash }^{2}k)$,Thus the allowed bands satisfy $|\cos (ka)+\gamma \sin (ka)|\le 1$.

A knot is doubly slice when it is the equatorial cross-section of an unknotted smooth two-sphere in $S^4$.

A slice knot bounds a smooth properly embedded disk in the four-ball $B^4$.

A Seifert surface for an oriented knot $K$ is a compact connected oriented surface $F\subset S^3$ with oriented boundary $\partial F=K$.

A satellite knot $P(K)$ is obtained by embedding a patterned solid torus into a tubular neighborhood of a companion knot $K$.

For nonzero coprime integers $p,q$, the torus knot $T_{p,q}$ lies on a standard unknotted torus in the three-sphere and winds $p$ times meridionally and $q$ times longitudinally. Here a meridian of a solid torus goes around the small cross-section of the doughnut, while the longitudinal direction goes around its central hole. The integers count signed turns, so changing one sign gives the mirror of a link of the torus knot. In this article $p$ is always the meridional count and $q$ the longitudinal count.

An explicit parametric curve in three-dimensional Euclidean space iswhere $R>r>0$. Its cylindrical radius $\rho =\sqrt{x^2+y^2}$ satisfies $(\rho -R{)}^2+z^2=r^2$, the equation of the supporting torus. Its meridional angle is $pt$ and its longitudinal angle is $qt$: this checks the convention directly rather than relying on the picture. Its squared speed isIf two parameter values give the same point, their difference $\delta$ satisfies $p\delta ,q\delta \in 2\pi \mathbb{Z}$. Bézout's identity for the coprime integers $p,q$ then gives $\delta \in 2\pi \mathbb{Z}$. Thus the loop closes and is an injective function on the parameter circle, with nonzero derivative; it is a smooth embedding.

For example, $R=2$, $r=0.7$, $p=2$, $q=3$ produces a trefoil knot. During one circuit the meridional angle changes by $4\pi$ and the longitudinal angle by $6\pi$. The supporting torus is a guide for construction, not a constraint on later ambient isotopies: the knot may move off it. This family makes arithmetic visible as geometry: the coprime integers determine whether a single traversal visits the whole loop without repetition.

The coprime integers hypothesis matters. If the greatest common divisor is $d=\gcd (|p|,|q|)>1$, the same parametric curve retraces a single component $d$ times; it does not automatically draw $d$ different components. The corresponding torus link is instead the full set of points on the torus whose angles satisfy $q\varphi -p\theta \in 2\pi \mathbb{Z}$, with meridional angle $\varphi$ and longitudinal angle $\theta$. It has $d$ disjoint components, each with winding counts $(p/d,q/d)$. For instance, the torus link $T_{2,2}$ consists of two parallel $(1,1)$ loops and is a Hopf link, whereas substituting $(2,2)$ into ${\gamma }_{p,q}$ simply traces one of those loops twice.

The connected sum removes a short unknotted arc from each of two oriented knots and joins the four endpoints by an orientation-compatible pair of arcs.

The ambient three-sphere is also joined by connected sum, while the two knot arcs join inside it.