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An affine quadrupole mode has axis perturbations whose fractional sum vanishes. It changes the ellipsoidal shape without changing volume to first order.

The affine breathing mode changes all three principal axes by the same fraction and therefore changes the stellar volume and density.

If the spectral radius of $A$ is below one, $x_{n+1}=A{x}_n+b$ converges to the unique fixed point $(I-A{)}^{-1}b$.

The affine surfaceis birational to ${\mathbb{A}}^2$ and has singular locus $\{(1,0,z):z\in k\}$, an irreducible affine line.

An irreducible affine plane cubic has at most one singular point. If two existed, restriction of its cubic polynomial to their joining line would have a zero of multiplicity at least two at each point. A polynomial of degree at most three cannot have those four zeros unless it vanishes identically, making the line a component and contradicting irreducibility.

At a smooth point of a hypersurface, restriction to the tangent hyperplane has zero linear term. The natural hyperplane intersection therefore has tangent dimension one larger than its expected dimension and is singular at that point.

The holographic dictionary identifies leading near-boundary bulk-field coefficients with CFT sources and subleading normalizable coefficients with operator expectation values.

In a large-rank gauge theory, a single-trace operator is dual at leading order to a single bulk field, while multi-trace operators correspond to multiparticle bulk states.

A Witten diagram is a position-space perturbative diagram in anti-de Sitter spacetime whose external bulk-to-boundary propagators end at boundary operator insertions.

Three scalar primaries have a conformally fixed position dependence, leaving one theory-dependent coefficient $C_{123}$.

Conformal invariance fixes the two-point function of identical scalar primaries of dimension $\Delta$ to $C_2/|x_{12}{|}^{2\Delta }$.

A Euclidean conformal transformation pulls the flat metric back to a position-dependent positive multiple of itself. Its infinitesimal generator obeys the conformal Killing equation.

For every nonnegative integer $k$, the entry $(A^k{)}_{uv}$ equals the number of length-$k$ walks from $u$ to $v$. This follows by induction, since matrix multiplication appends one adjacent vertex to each walk.

If a finite set $A$ has at least $\eta |A{|}^3$ additive quadruples, then it contains $A^{\prime }$ with $|A^{\prime }|\ge {\eta }^C|A|$ and $|A^{\prime }+A^{\prime }|\le {\eta }^{-C}|A^{\prime }|$ for an absolute constant $C$.

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