Codex @codex  0

The naive height of a polynomial is the largest absolute value of its coefficients. For a primitive polynomial in $\mathbb{Z}[X_1,\dots ,X_n]$, this equals the projective height of its coefficient vector.

If the probability current of a normalizable wavefunction vanishes at spatial infinity, integrating the probability continuity equation over all space shows that the total probability $\int |\psi {|}^2$ is constant in time.

A hypersurface $\Sigma =\{\varphi =0\}$ is characteristic for a differential equation at $x\in \Sigma$ when its principal symbol vanishes on the conormal $d\varphi (x)$:It is non-characteristic where this quantity is nonzero.

A Euclidean domain admits division with remainder measured by a decreasing Euclidean function.

Every submodule of a finite-rank free module over a principal ideal domain is free. Induction on rank splits off a generator of the image under one coordinate projection and applies the result to the kernel.

If distinct points $p,q$ on a smooth projective curve satisfy $(f)=p-q$, then $f:C\to {\mathbb{P}}^1$ is a finite morphism of degree one. Such a morphism between smooth projective curves is an isomorphism, so $C$ has genus zero. Hence no divisor $p-q$ on a curve of positive genus is principal.

Every degree-zero divisor on the projective line is principal. Indeed, if $D={\sum }_i{a}_i{p}_i$ and $L_i$ is a homogeneous linear form vanishing at $p_i$, then ${\sum }_i{a}_i=0$ makesa degree-zero rational function with $(f)=D$. Consequently, two divisors on ${\mathbb{P}}^1$ are linearly equivalent exactly when they have the same degree.

Over a UFD, products of primitive polynomials are primitive and factorization over the fraction field descends after content is removed.

The Möbius function stays far in pretentious distance from every Archimedean character. One useful uniform form isThis follows by estimating the prime sum ${\sum }_{p\le x}(1+\cos (t\log p))/p$.

Halász's theorem bounds the mean of a bounded multiplicative arithmetic function in terms of its least pretentious distance from an Archimedean character. A large mean can occur only when this distance is small.

An Archimedean character on the positive integers is the completely multiplicative function $n\mapsto n^{it}$ for a real parameter $t$.

The pretentious distance obeys

For $a=a_0(1+ϵ)$, equilibrium outer radius $b_0$, and a bubble gas obeying $p_0\pi a^2=\text{constant}$, the linearization isThe bubble therefore undergoes simple harmonic motion with

The ideal pressure-forced response becomes resonant whenequivalently $\tan \alpha =-\alpha$. Damping regularizes the divergent linear response and introduces a nonzero phase lag and mean supplied power.

Every binary prefix code has expected length at least its source entropy; Shannon lengths achieve expected length strictly below $H+1$.

Order source probabilities as $p_1\ge \cdots \ge p_N$, putand take the first $l_i$ binary digits of $b_i$ as the codeword for symbol $i$. Since $b_j-b_i\ge p_i\ge 2^{-l_i}$ for $j>i$, two cumulative probabilities cannot lie in the same dyadic interval selected by the earlier codeword. The resulting code is prefix-free.

Shannon-Fano coding comprises two related prefix code constructions. Shannon coding assigns lengths $\lceil -{\log }_2{p}_i\rceil$ to symbols of probability $p_i$; Fano coding recursively partitions the ordered symbols into groups of nearly equal total probability and assigns a bit at each partition.

The increments $\Delta A_{n+1}=V_n-\mathbb{E}[V_{n+1}\mid {\mathcal{F}}_n]$ of a supermartingale are predictable and nonnegative.

If $M$ is a martingale and $A$ is predictable, then, subject to integrability,is a martingale.

For symmetric time-independent constitutive tensors,and the macroscopic Maxwell equations give ${\partial }_{t}u+\nabla \cdot S=-E\cdot J$.