A population principal component explains the displayed fraction of total variance. The first $q$ components explain ${\sum }_{j\le q}{\lambda }_j/{\sum }_j{\lambda }_j$. Comparing covariance and correlation analyses requires different denominators: the original variance sum for the former and the number of nonconstant standardized variables for the latter.

A principal component is the projection of centered data onto a unit eigenvector of its covariance matrix. Ordering the components by their eigenvalues orders their variances. Repeated eigenvalues allow any orthonormal basis of the corresponding eigenspace.

Two conventions occur for principal component loadings: the unit eigenvector ${\ell }_j$ defining the direction, and the scaled vector $r_j=\sqrt{{\lambda }_j}{\ell }_j$. For principal component analysis on a correlation matrix and a positive eigenvalue ${\lambda }_j>0$, entries of $r_j$ are the correlation coefficients between each standardized variable and the component scaled to variance one. Indeed, $\mathrm{Cov}(Z,{\ell }_j^{T}Z)={\lambda }_j{\ell }_j$ and $\mathrm{Var}({\ell }_j^{T}Z)={\lambda }_j$.

Thus $\Vert r_j{\Vert }^2={\lambda }_j$, whereas $\Vert {\ell }_j\Vert =1$. At a zero eigenvalue, the scaled vector is zero but the component has zero variance, so correlations with a unit-variance version are undefined. Reporting both the loading convention and the eigenvalue avoids conflating directions, correlations and degenerate components.

A scree plot graphs ordered covariance or correlation eigenvalues against component number. A transition from a steep decline to a flatter tail suggests a dimension, but this is a diagnostic rather than a proof of the number of latent factors. In factor analysis, it should be combined with model fit, substantive interpretation and uncertainty assessment.

Regress a response on selected principal component scores of centered predictors. If $X=UD{V}^T$ is a singular value decomposition and the first $q$ positive-singular-value components are retained, the retained score matrix is $U_q{D}_q$ and the fitted coefficient in predictor coordinates is $V_q{D}_q^{-1}{U}_q^{T}y$. The fitted centered response is $U_q{U}_q^{T}y$: this follows by applying the least-squares normal equations to the orthogonal score columns. Truncating small singular directions limits variance amplification, but directions of large predictor variance need not be those best predicting the response; select $q$ using prediction assessment rather than explained predictor variance alone.

For $n=q-1$ and $a_j={\omega }^j$, the two evaluation forms are $a_j^{1-b}f(a_j)$ and $a_j^{b}h(a_j)$, with $\mathrm{deg}f<k$ and $\mathrm{deg}h<n-k$. Their dot product is ${\sum }_j{a}_{j}f(a_j)h(a_j)=0$, since every appearing exponent is between one and $n-1$ and the sums of those powers of $a_j$ vanish. Comparing dimensions proves equality with the dual code. In particular, the dual of the unweighted evaluation code has multipliers $a_j$; it need not be an unweighted evaluation code on the same ordered points.

With syndromes $S_{\ell }={\sum }_i{E}_i{X}_i^{b+\ell }$ and distinct error-location values $X_i$, form $S(z)={\sum }_{\ell =0}^{n-k-1}{S}_{\ell }{z}^{\ell }$ and the error locator polynomial $\Lambda (z)={\prod }_i(1-X_{i}z)$. There is an error evaluator polynomial $\Omega$ of degree less than the number of errors satisfying the key equation. When the number of errors is at most $\lfloor (n-k)/2\rfloor$, the Berlekamp-Massey algorithm or extended Euclidean algorithm recovers these polynomials, normalized by $\Lambda (0)=1$.

In an orientable three-manifold, an embedded nonseparating two-sphere has a product neighborhood. Join its two boundary spheres by an arc outside that neighborhood and take a regular neighborhood of the union. This region is $S^2\times S^1$ with an open ball removed. Its boundary sphere exhibits the displayed connected sum. Consequently a prime but nonirreducible orientable three-manifold is $S^2\times S^1$.

Every closed connected orientable three-manifold is a finite connected sum of prime three-manifolds, unique up to order and the insertion of three-sphere factors. The corresponding fundamental group is the free product of the summand groups. A prime orientable three-manifold is either irreducible or $S^2\times S^1$; this separates aspherical infinite-group cases from the infinite cyclic case.

If all nontrivial zeta zeros satisfy $\Re \rho \le 1-c$ with fixed $0<c<1/2$, the Riemann–von Mangoldt formula gives ${\sum }_{|\rho |<T}|\rho {|}^{-1}\ll {\log }^{2}T$. The truncated explicit formula for the second Chebyshev function then bounds the error by $O(x^{1-c}{\log }^{2}T+x{\log }^{2}x/T)$. Choosing $T=x^c$ gives the displayed bound and, for every positive $\epsilon$, $O_{\epsilon }(x^{1-c+\epsilon })$.

// Target: number-theory.bigb

Fix an irreducible polynomial f of degree d with nonzero constant term. In $G{L}_{dm}(q)$, the number of matrices whose minimal polynomial is a power of f is $|G{L}_{dm}(q)|q^{d(m^2-m)}/|G{L}_m(q^d)|$. The primary matrix centralizer formula is identical to the unipotent centralizer formula over $F_{q^d}$, and summing reciprocal centralizer orders reduces to counting nilpotent matrices by the Fitting decomposition.

In the Noetherian local ring $A=k[[u,v]]$ with maximal ideal $\mathfrak{m}=(u,v)$, the embedding dimension is two but ${\mathfrak{m}}^r$ needs $r+1$ generators for every $r\ge 1$. The degree-$r$ monomials $u^r,u^{r-1}v,\dots ,v^r$ generate ${\mathfrak{m}}^r$, and their images form a basis of ${\mathfrak{m}}^r/{\mathfrak{m}}^{r+1}$, so no smaller generating list exists. These ideals are $\mathfrak{m}$-primary: if $ab\in {\mathfrak{m}}^r$ and $a\notin {\mathfrak{m}}^r$, then $b$ cannot be a unit; hence $b\in \mathfrak{m}$ and $b^r\in {\mathfrak{m}}^r$. Consequently embedding dimension bounds the minimum over all maximal-primary ideals, because the maximal ideal itself is eligible, but does not bound the number of generators of each such ideal.

For free embedding bosons, the operator product expansion of the stress tensor with ${\zeta }_{\mu }:\partial X^{\mu }{e}^{ipX}:$ has a cubic pole $-i{\alpha }^{\prime }(p\cdot \zeta ):e^{ipX}:/2$. Its vanishing is precisely the primary condition, with weight $1+{\alpha }^{\prime }{p}^2/4$. The two-derivative closed-string vertex has the two displayed transversality conditions from its holomorphic and antiholomorphic cubic poles. Being primary does not require $p^2=0$; that extra condition follows when a physical integrated vertex must have conformal weights $(1,1)$.

A primary state is a simultaneous eigenvector of $L_0,{\tilde{L}}_0$ with eigenvalues $h,\tilde{h}$ that is annihilated by all positive modes of both Virasoro algebras. Negative modes generate descendants. The state–operator correspondence identifies it with a primary operator inserted at the origin.

Let a sufficiently smooth $V$ satisfy the pricing equation for a local volatility model, and assume its proposed holdings are stochastically integrable. Hold $h_t^S=V_S(t,S_t)$ stocks and $h_t^B=[V(t,S_t)-S_t{V}_S(t,S_t)]/B_t$ units of the bank account. Their value is $V(t,S_t)$. The Itô formula and the pricing equation giveThus the holdings form a self-financing strategy with terminal value $g(S_T)$, rather than merely a decomposition of today's value. This is a delta hedge. If $V\ge 0$, the replicating strategy has nonnegative wealth and satisfies the standard admissible trading strategy lower-bound condition. The stock and bank wealth amounts are $S_t{V}_S$ and $V-S_t{V}_S$; they differ from the numbers of asset units held.

With $x=\log S$ and $u=e^{r(T-t)}V(e^x,t)$, a local volatility pricing equation becomes the displayed backward equation. Central finite differences and the explicit Euler method give $u_i^{j-1}=u_i^j+k[a_i^j{D}_{xx}{u}_i^j+b_i^j{D}_x{u}_i^j]$. Nonnegative weights require $k{\sigma }_i^2/h^2\le 1$ and $|b_i|h\le {\sigma }_i^2$. Under these conditions the update is a convex combination, yielding maximum-norm stability. With smooth coefficients, positive diffusion, fixed positive remaining maturity and suitable consistent boundary treatment, its ordinary error is $O(k+h^2)$. Volatility bounds alone do not imply all this regularity or a Greek error expansion.

A prequantum line bundle is a Hermitian line bundle over a symplectic manifold with unitary connection on a vector bundle of the displayed curvature. With $\omega =-d\theta$, a local connection is $\nabla =d-i\theta /\hslash$. Reversing the Hamiltonian-vector-field convention or passing to the dual line bundle reverses the corresponding curvature convention. Integrality of $\omega /(2\pi \hslash )$ on closed two-cycles is the topological existence condition.

For ${\iota }_{X_f}\omega =df$ and $F_{\nabla }=i\omega /\hslash$, the curvature commutator and $[X_f,X_g]=-X_{\{f,g\}}$ give $[\hat{f},\hat{g}]=i\hslash \widehat{\{f,g\}}$. In cotangent coordinates, $\theta =pdq$ gives $\hat{f}=-i\hslash X_f+f-\theta (X_f)$. Restriction to a vertical polarization yields $\hat{q}=q$ and $\hat{p}=-i\hslash {\partial }_q$.

A premeasure on an algebra of sets extends to a measure on the generated sigma-algebra. The extension is unique when the premeasure is sigma-finite.

For a commutative monoid object $(M,\ast ,e)$ in preframes, the indicated subset is a frame. Its bottom is $e$, its binary join is $\ast$, and its directed joins and finite meets are inherited. Every homomorphism from a frame regarded as a join monoid factors uniquely through this subset. This gives a coreflective subcategory.