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A commutative Artinian ring has finite composition length as a module over itself. It has finitely many maximal ideals, and its Jacobson radical $J$ is nilpotent. The Chinese remainder theorem makes $R/J$ a finite product of fields; each Artinian module $J^i/J^{i+1}$ has finite-dimensional components over those fields. Adding their lengths along the finite radical filtration proves the assertion, and in particular the ascending chain condition.

A nonzero element of a number field is totally positive when its image is positive under every real field embedding. In a totally real number field, the principal fractional ideals having a totally positive generator form the subgroup used to define the narrow ideal class group. Signs of different generators are related by the unit signature map.

An associated prime of an $R$-module $M$ is a prime ideal equal to the annihilator ${\mathrm{Ann}}_R(m)$ of some nonzero $m\in M$. The set of these primes is denoted ${\mathrm{Ass}}_R(M)$. Over a Noetherian ring, any maximal member among the annihilators of nonzero elements is prime: if $abm=0$ and $bm\ne 0$, maximality forces ${\mathrm{Ann}}_R(bm)={\mathrm{Ann}}_R(m)$, so $am=0$. Hence every nonzero module has an associated prime.

The effective dimension for a specified anisotropic rescaling is the total scaling weight of its coordinate measure. With $D_{\perp }$ directions of weight 1 and $D_{\parallel }$ directions of weight $\theta$, $d_{\mathrm{eff}}=D_{\perp }+\theta D_{\parallel }$. The uniaxial Gaussian Lifshitz point has $\theta =1/2$, so $d_{\mathrm{eff}}=D-1/2$. This is not a topological or Hausdorff dimension.

A closed conic enlargement of an oscillatory integral amplitude support is obtained by taking the closure of its frequency-direction projection in $X\times S^{k-1}$ and lifting that set radially to nonzero frequencies. It records limiting spatial locations and frequency directions, including limits reached only at arbitrarily large frequency. Ordinary support in $X\times {\mathbb{R}}^k$ need not record those limits. This distinction is necessary in singular support bounds for an oscillatory integral.

The alternating conjugacy class splitting criterion gives seven conjugacy classes in $A_6$. Their cycle types and sizes are $1^6:1$, $2^2{1}^2:45$, $31^3:40$, $3^2:40$, $42:90$, and two $51$ classes of size 72. Representatives of the split classes are a 5-cycle and its square. Their sizes sum to 360. A normal subgroup must be a union of these classes containing the identity, so these explicit sizes can prove simplicity using Lagrange's theorem.

A zenith blind spot is a region in which an altazimuth mount cannot follow a target because the required axis motion exceeds a drive constraint. A finite slew across an exact zenith crossing gives a lost-track interval; a full forbidden footprint additionally depends on the target trajectory, acceleration and the control strategy. It need not be circular.

In the exact constant-opacity, pure-radiation pressure diffusion model, dynamic viscosity is height-independent. Imposing an additional exact pointwise relation $\mu \propto p(z)$ forces constant pressure, incompatible with nonzero vertical hydrostatic equilibrium away from the midplane. The characteristic-pressure alpha disk prescription can still be used for order-of-magnitude scalings; it is not an exact simultaneous vertical solution.

An almost complex structure $J$ gives the $+i$ and $-i$ eigenbundles of its complex-linear extension. Their projections are $(I-iJ)/2$ and $(I+iJ)/2$. For an integrable structure the first is the holomorphic tangent bundle.

A rational parametrization expresses the coordinates of an algebraic curve as rational functions of a parameter, with a rational inverse on a dense open subset. It identifies the function field with $K(t)$ and the smooth projective normalization of an algebraic curve with the projective line. A nonconstant parametrizing map without an inverse is a weaker notion; even this cannot exist for an elliptic curve in characteristic zero.

For an algebra for a monad $(X,a)$, its action $a:TX\to X$ satisfies $a{\eta }_X=1_X$.

Every algebra for a monad $(A,a)$ is the coequalizer, in the Eilenberg-Moore category, of $F(TA)\rightrightarrows F(A)$ with underlying arrows ${\mu }_A$ and $Ta$, followed by $a:F(A)\to (A,a)$. The pair has common section $F{\eta }_A$. If an algebra morphism $h:F(A)\to (B,b)$ equalizes the pair, the unique induced algebra morphism is $h{\eta }_A:(A,a)\to (B,b)$. This explicit proof does not require general colimits of algebras.

Suppose nonnegative numbers satisfy $a_0{b}_0\le c_0{d}_0$, $a_1{b}_1\le c_1{d}_1$, and $a_0{b}_1,a_1{b}_0\le c_1{d}_0$. Set $x=a_0{b}_1$, $y=a_1{b}_0$, $M=c_1{d}_0$, $N=c_0{d}_1$. Then $x,y\le M$ and $xy\le MN$. For $M>0$, $(M-x)(M-y)\ge 0$ gives $x+y\le M+xy/M\le M+N$; for $M=0$, $x=y=0$. Adding the two diagonal bounds proves $(a_0+a_1)(b_0+b_1)\le (c_0+c_1)(d_0+d_1)$. This lets the four functions theorem sum out one Boolean coordinate while preserving its pointwise hypothesis.

In the ABL rule, probabilities depend on the full intermediate quantum measurement instrument. A binary Lüders rule quantum measurement preserves coherence within its unresolved complement, whereas a fully resolved projective measurement removes it. Summing fine-grained probabilities after quantum measurement does not generally reproduce the coarse-grained experiment. Different postselection success rates are part of the distinction.

Among retained payouts $g(S)$ with $0\le g(x)\le x$ and the same expected value as $\min (S,M)$, aggregate stop loss reinsurance minimizes the variance. Pointwise $(g(x)-M{)}^2\ge (\min (x,M)-M{)}^2$, and subtracting the identical squared distance of their common expected value from $M$ proves the claim. Equality requires equal payouts almost surely.

If $S$ has exponential distribution with expected value $\mu$, the tail integral formula for moments gives $\mathbb{E}\min (S,M)=\mu (1-e^{-m})$ and $\mathrm{Var}(\min (S,M))={\mu }^2(1-2m{e}^{-m}-e^{-2m})$, where $m=M/\mu$. At matching retained expected value, the excess variance under quota share reinsurance is $2{\mu }^2{e}^{-m}(m-1+e^{-m})\ge 0$.

In a vector space over a field $k$, an affine line is a translate of a one-dimensional vector subspace. Its direction is that vector subspace, so multiplying $v$ by a nonzero scalar preserves the direction. Over a finite field of size $q$ it contains exactly $q$ distinct points. This usage concerns affine subsets of vector spaces, rather than the affine line as an affine scheme.

A map between affine varieties is a morphism when its coordinate functions are restrictions of ambient polynomials. Pullback identifies these morphisms contravariantly with homomorphisms of their coordinate rings.

A classical variety is affine if finitely many global regular functions generate the unit ideal and their nonvanishing opens are affine. Localization of global sections on a principal open identifies their coordinate rings with $A_{f_i}$, where $A=\Gamma (X,{\mathcal{O}}_X)$. Choose a finite-type subalgebra of $A$ containing the functions, their unit-ideal coefficients and numerators of generators of every $A_{f_i}$. Its affine variety has the same principal affine charts, whose isomorphisms glue globally. This avoids assuming finite generation of $A$ before affineness is known.

Writing $u=e^{-\beta x/2}v$ transforms $u_t=u_{xx}+\beta u_x$ into $v_t=v_{xx}-{\beta }^{2}v/4$. A prescribed coordinate derivative $u_x=g$ at an endpoint becomes $v_x-(\beta /2)v=e^{\beta x/2}g$ there. Thus a Neumann boundary condition becomes a Robin boundary condition. The signs are coordinate-derivative signs at both endpoints; outward normal derivatives require a sign change at the left endpoint.