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An additive quadruple is a tuple $(a,b,c,d)$ satisfying $a+b=c+d$.

An abelian category is an additive category with all kernels and cokernels in which every monomorphism and epimorphism is normal.

The symmetric two-particle statesform the spin-one triplet. On this subspace, ${\mathbf{S}}^2$ has eigenvalue $1(1+1){\hslash }^2=2{\hslash }^2$.

With $J=j_1+j_2$, the maximal state is the product $|J,J⟩=|j_1,j_1⟩|j_2,j_2⟩$. Lowering it once givesThe orthogonal combination is the highest-weight state $|J-1,J-1⟩$.

The tensor product of spin-$j_1$ and spin-$j_2$ irreducible representations decomposes into one copy of each spinThe change-of-basis entries between uncoupled and coupled states are Clebsch--Gordan coefficients.

For labels $Y\in \{-1,1\}$ and a real-valued score $f$, the exponential loss is $\varphi (Yf(X))=e^{-Yf(X)}$. Its population and empirical risks are

The principle of stationary action says that the physical path between fixed endpoints is a stationary point of the action.

For $R<r<2R$, a rigid inner sphere and harmonic pressureproduce a standing radial field. With $k=\omega /c_0$ and $\alpha =kR$,The pressure and radial velocity are in temporal quadrature, so the period-averaged acoustic intensity vanishes.

For $R(t)=a+\mathrm{Re}(ϵe^{i\omega t})$, put $k=\omega /c_0$. To first order in $ϵ/a$, the exterior outgoing velocity-potential amplitude isIt satisfies ${\hat{u}}_r(a)={\hat{\varphi }}^{\prime }(a)=i\omega ϵ$.

The alpha-disk prescription writes turbulent kinematic viscosity as $\nu =\alpha c_{s}H\simeq \alpha c_s^2/\Omega$, with dimensionless $\alpha \lesssim 1$. It reflects eddies no larger than the disk thickness and no faster than the sound speed.

Viscous disk evolution combines mass conservation with radial transport of angular momentum by stress. In a steady Keplerian disk, the sum of advective and stress-carried angular-momentum fluxes is independent of radius.

For a field extension $L/K$, an extension of an absolute value $|\cdot |$ on $K$ is an absolute value on $L$ whose restriction to $K$ is $|\cdot |$.

Two nontrivial absolute values on a field are equivalent when they induce the same topology. Equivalently, one is a positive real power of the other.

A non-Archimedean absolute value satisfies the ultrametric inequality $|x+y|\le \max (|x|,|y|)$.

There are only finitely many algebraic numbers of bounded degree and bounded Absolute multiplicative Weil height. The finite set can be enumerated effectively from the bounded coefficients of their primitive minimal polynomials.

The degree and Absolute multiplicative Weil height bounds are both essential.

If $\gamma \ne 0$ is algebraic of degree at most $D$, then every embedding satisfiesIndeed, $H({\gamma }^{-1})=H(\gamma )$ and the contribution of one Archimedean place to the height bounds $|\gamma {|}^{-1/D}$.

Let $P,Q\in \mathbb{Z}[X_1,\dots ,X_k]$ have degree at most $n_j$ in $X_j$. Thenwhenever the quotient is defined. At each non-Archimedean place, the integral coefficients contribute at most one; at the Archimedean places, the triangle inequality contributes the polynomial length. Multiplying these local estimates and using the product formula gives the claim.

If the primitive minimal polynomial of an algebraic number $\alpha$ has degree $d$, then

For a nonzero vector $\mathbf{a}=(a_0,\dots ,a_n)$ over a number field $K$, its absolute projective height isThe product formula makes this unchanged by multiplying all coordinates by the same nonzero scalar. The projective height of a linear form is the projective height of its coefficient vector.

The absolute logarithmic Weil height is $h(\alpha )=\log H(\alpha )$.