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A shortest conjugacy representative is a word having the fewest written letters among all words representing elements of a fixed conjugacy class. This minimum equals the least group word length of an element in the class; a longer spelling of that element is not itself a shortest representative. Such a word exists because lengths are nonnegative integers. It is freely and cyclically reduced: free cancellation shortens the same representative, while removing mutually inverse first and last letters shortens a conjugate. Every cyclic rotation has the same minimal length.

For a random vector indexed by vertices $V$, its conditional independence graph is the undirected graph with $u-v$ absent exactly when $Z_u\perp Z_v\mid Z_{V\setminus \{u,v\}}$. In a nonsingular multivariate normal distribution, its edges are the nonzero off-diagonal entries of the precision matrix.

On a smooth manifold, the sequence of sheaves of abelian groupsis exact. The additive smooth-function sheaf is a fine sheaf, so the connecting map identifies $H^1(X,{\mathcal{C}}_{\mathbb{C}}^{\infty ,\ast })$ with $H^2(X,\mathbb{Z})$. The former classifies smooth complex line bundles and the connecting class is their First Chern class.

Every open set in the Ellentuck topology is a completely Ramsey set. Fix a finite stem $s$. An infinite tail accepts a finite extension $a$ if its entire basic neighborhood $[s\cup a,A]$ lies in the open set; it rejects $a$ if no infinite subtail accepts it. Decisions persist under thinning. A fusion chooses an infinite set deciding every finite extension. For any rejected extension, only finitely many possible next elements can yield accepted extensions, since infinitely many would themselves provide an accepting tail. A second thinning therefore makes all finite extensions rejected. Openness then rules out any point of the set in the resulting neighborhood.

A compatible triple consists of a symplectic form, a compatible almost complex structure, and the Riemannian metric $g(u,v)=\omega (u,Jv)$. Equivalently $g(Ju,v)=\omega (u,v)$.

Before cosmological recombination, photon-baryon sound speed controls the baryonic comoving Jeans length. The tightly coupled photon-baryon fluid has $c_s^2=c^2/[3(1+R_b)]$, where $R_b=3{\bar{\rho }}_b/(4{\bar{\rho }}_{\gamma })\propto a$. Using total background density in the instantaneous dynamical Jeans estimate, the scale grows as $a$ during radiation domination, approaches a plateau in strongly baryon-loaded matter domination, and drops when photon decoupling removes photon pressure support. Subsequently, adiabatic monatomic-gas cooling gives $T_b\propto a^{-2}$ and a comoving Jeans length proportional to $a^{-1/2}$. Residual Compton scattering can delay that cooling law.

Adding $d\varphi$ to a free-energy density adds a constant at fixed total compositional order parameter. In the common-tangent construction for phase coexistence, it shifts the tangent slope by $d$ while preserving its points of contact. Thus coexistence compositions are unchanged, whereas their chemical potential shifts. This does not assert unchanged equilibrium composition at a fixed externally imposed chemical potential.

A Lobatto IIIA method collocates at the Lobatto quadrature nodes, including both endpoints of the time interval. Its three-stage version hasThe Butcher order conditions give order four. Its stability function iswhich is A-stable but not L-stable. It is not algebraically stable: the first diagonal entry of the defining matrix is $2b_1{a}_{11}-b_1^2=-1/36$. This distinguishes linear A-stability from nonlinear algebraic stability.

For a hypersurface $N$ of a symplectic manifold, the restriction of the symplectic form has a one-dimensional kernel, its characteristic line field. On a regular value level $N=H^{-1}(c)$, this line is generated by the Hamiltonian vector field of $H$.

For a parabolic wave equation driven by longitudinal Gaussian white noise with transverse covariance kernel $B$, the mean envelope obeys ${\partial }_{x}m=i{\Delta }_{\perp }m/(2k_0)-\gamma m$. This closure requires independent increments in the longitudinal direction; statistical homogeneity and one-point Gaussian distributions alone are insufficient.

A single crossing-symmetric scattering amplitude has equivalent pole expansions in different exchange channels. Its scattering-amplitude factorization reproduces the string spectrum and couplings in each channel without adding the channel expansions again.

At genus zero the residual Möbius transformation group fixes three marked insertion points, leaving one complex position to integrate. The corresponding worldsheet ghost fields give $|z_{ij}{z}_{jk}{z}_{ki}{|}^2$ for the three fixed points. For on-shell weight-$(1,1)$ string vertex operators, these ghost factors cancel the coordinate dependence of the fixed-point normalization, including the limit at infinity. The remaining Koba-Nielsen factor for four tachyons is the integrand of the complex beta integral, producing the Virasoro–Shapiro amplitude.

Classification in statistical learning with two possible labels. Given the conditional probability $\eta (x)=\mathbb{P}(Y=1\mid X=x)$, a Bayes classifier under zero-one loss predicts one exactly when $\eta (x)\ge 1/2$, with arbitrary decisions at ties.

With the Church Booleans $\mathsf{T}=\lambda ab.a$ and $\mathsf{F}=\lambda ab.b$, encode a pair byThen $\mathsf{Fst}(\mathsf{Pair}AB){\to }_{\beta }^{\ast }A$ and the second projection similarly returns $B$. Pairing an iteration counter with a computed value implements primitive recursion using only Church numeral iteration.

The lambda termreturns the Church Boolean $\mathsf{T}$ on $c_0$ and $\mathsf{F}$ on every positive Church numeral. Zero iterations leave the initial true value, while at least one iteration replaces it by false. Normal-order beta reduction evaluates a numeral conditional without evaluating an unchosen branch.

With normalized Choi state $C_{\Lambda }$ and the output factor first,Expand $C_{\Lambda }=d^{-1}{\sum }_{i,j}\Lambda (|i⟩⟨j|)\otimes |i⟩⟨j|$. The partial trace multiplies each coefficient by $\mathrm{Tr}(|i⟩⟨j|X^T)=X_{ij}$, reconstructing $\Lambda (X)$ by linearity.

A channel's Choi state is its normalized Choi matrix,It is a density operator and satisfies ${\mathrm{Tr}}_{\mathrm{out}}{C}_{\Lambda }=I_{\mathrm{in}}/d$. Some authors use an unnormalized Choi matrix of trace $d$, so reconstruction factors depend on the convention.

A chiral gauge spectrum has independent left-handed Weyl spinors in complex gauge group representations without an obligatory conjugate pair of left-handed Weyl spinors. The right-handed antiparticles required by the CPT theorem do not themselves make a spectrum vectorlike.

A vectorlike gauge spectrum pairs complex gauge group representations $R$ and $\bar{R}$ among independent left-handed Weyl spinors. Such a pair permits a gauge-invariant fermion mass and occurs in a full charged hypermultiplet of four-dimensional $\mathcal{N}=2$ extended supersymmetry.

For $X=x+\sqrt{2}\theta G+{\theta }^{2}F$, the nilpotency condition gives $x^2=xG=0$ and $2xF=GG$. On the branch with invertible commuting part of $F$, $x=GG/(2F)$, while the goldstino $G$ and auxiliary field $F$ are independent. Products of three identical two-component odd spinor entries vanish, verifying the remaining constraints. The scalar is therefore composite. The branch hypothesis matters: $X=0$ also obeys the constraint and need not break supersymmetry.