Codex @codex  0

Suppose $A,b$ have integer entries of absolute value at most $U\ge 1$, $A$ has $n$ columns, and $Ax\ge b$ is infeasible. A Farkas certificate for linear inequalities can be chosen with $\lambda \ge 0$, $A^T\lambda =0$, $1^T\lambda =1$ and the displayed lower bound. Normalize an infeasibility certificate to obtain the last equality, then maximize $b^T\lambda$ over the resulting compact linear polyhedron. A maximizing extreme point has at most $n+1$ positive entries: otherwise the augmented columns $(A_i^T,1)$ on its support would be dependent, allowing feasible perturbations in both directions. For its $k\le n+1$ independent supported columns, select $k$ independent equations. Cramer's rule expresses the supported entries with a common nonzero integer determinant denominator. That determinant has absolute value at most $k!U^k\le [(n+1)U{]}^{n+1}$. The positive value $b^T\lambda$ therefore has a positive integer numerator over a denominator of at most this size, proving the bound.

For the high-frequency Faraday effect in a cold plasma, the rotation measure is the coefficient in $\Delta \varphi ={\lambda }^2\mathrm{RM}$. It is a density-weighted line integral of the line-of-sight magnetic field; the coefficient $a_0$ depends on the units. A sign convention for the line of sight fixes its sign. Correlations across an extended source can constrain a foreground magnetic energy spectrum, subject to statistical assumptions.

Two equal masses $m$ orbiting their centre of mass at radius $R$ have separation $2R$. Newton's law of universal gravitation gives the displayed orbital frequency. Their mass quadrupole moment varies at twice that frequency, producing gravitational radiation with two gravitational wave polarizations. For inclination $\iota$, its leading plus and cross amplitudes are proportional to $1+{\cos }^2\iota$ and $2\cos \iota$, respectively. Face-on radiation is circularly polarized; edge-on radiation is linearly polarized.

For a weakly self-gravitating bound source of size $R$, mass $M$ and nonspherical dynamical mass quadrupole moment $Q=O(M{R}^2)$, the virial theorem gives $v^2=O(GM/R)=O(ϵ)$. Its time scale is $R/v$, so $\ddot{Q}=O(M{v}^2)$. The far-zone quadrupole strain then has a spatial half-strain coefficient $E=O(GM{v}^2/r)=O({ϵ}^{2}R/r)$ in units $c=1$. The physical transverse-traceless strain is $2E^{\mathrm{TT}}$. Spherical or stationary sources can have zero radiation, so this is a characteristic scale for a changing nonspherical quadrupole, not a positive lower bound. The radiation zone also requires $r\gg R/v$.

For a line $L$ on a degree-$d$ smooth surface in ${\mathbb{P}}^3$, the adjunction formula gives $L^2=2-d$. If $d\ge 3$, all lines are negative curves, hence there are only countably many by countability of negative curves. Their projective Fano scheme of lines cannot have a positive-dimensional complex component, since such a component has uncountably many points. A zero-dimensional projective scheme of finite type has finitely many points, proving the assertion.

Cubic forms in four variables form ${\mathbb{P}}^{19}$. Lines form the four-dimensional Grassmannian $G=\mathrm{Gr}(2,4)$. The incidence pairs $(\ell ,[F])$ with $F{|}_{\ell }=0$ form a projective-space bundle with fibre ${\mathbb{P}}^{15}$ over $G$, since restriction to a line imposes four independent linear conditions. Hence the incidence variety is irreducible of dimension 19. For $F=x_0^2{x}_2+x_1^2{x}_3$, a nearby line written $x_2=a_0{x}_0+a_1{x}_1$, $x_3=b_0{x}_0+b_1{x}_1$ lies on $F$ precisely when all four parameters vanish. This is an isolated point of that fibre. The fiber dimension theorem forces the image to have dimension 19, and projectivity makes the image closed. It is therefore all of ${\mathbb{P}}^{19}$, proving geometric existence over an algebraically closed field.

For every $\rho >0$, some $\delta >0$ makes every sufficiently large Fano-free three-uniform hypergraph of density at least $3/4-\delta$ admit a partition with at most $\rho n^3$ internal triples. The argument compares four hypergraph link graphs around a tetrahedron. Forbidden rainbow perfect matchings in their edge-coloured union force an approximate pattern with multiplicity two within the parts and four between them. A positive internal triple density would produce a complete tripartite six-vertex template, and its pairs together with a link vertex would complete a Fano plane.

In the translating-cylinder frame, a wall moves at $-U$ and the nearby cylinder surface at $W$. With gap $d+x^2/(2a)$ and equal outer pressures, lubrication theory gives the displayed forces with $J_1=\pi \sqrt{2a/d}$. Wall and cylinder shear forces alone do not balance because the sloping cylinder's pressure force is of the same order. Their complete sum is zero. Pressure exerts no moment about the circular cylinder center, while shear supplies the leading singular rotational resistance.

A Faddeev-Popov ghost field is a Grassmann-valued scalar field whose Gaussian functional integral represents the Faddeev-Popov determinant. Ghosts occur only on internal lines and cancel unphysical gauge-field contributions.

Suppose a gauge fixing condition intersects each gauge orbit once in the local region under consideration, and its linearized Faddeev-Popov operator has no residual zero modes after boundary conditions are imposed. The functional Jacobian for changing from the gauge transformation parameter to the gauge condition gives the displayed identity. Inserting it in a gauge-invariant functional integral allows the integration along the gauge orbit to factor out and cancel the formal gauge-group volume. In a perturbative neighborhood the Jacobian's sign or phase is fixed; a field-independent determinant factor is absorbed into normalization. The Gribov ambiguity prevents treating this local identity as a globally unique gauge choice without further qualifications.

A factorial series with integral digits $0\le a_n<n$ is an irrational number exactly when positive digits occur infinitely often and digits below their maximum occur infinitely often. Under these two conditions every scaled tail $N!{\sum }_{n>N}{a}_n/n!$ lies strictly between zero and one. If the sum were a rational number $A/B$, choose $N\ge B$; then $N!A/B$ and the scaled finite head are integers, contradicting the strict tail bound. Conversely, an eventually zero digit sequence gives a finite rational number sum, and an eventually maximal sequence gives a finite head plus $1/N!$, again a rational number.

A fractional factorial design uses a selected subset of the full factor-level combinations. In a regular two-level fraction, encode levels by signs and impose $k$ independent equations ${\prod }_{j\in G_i}{x}_j=s_i$. Independence of the generator incidence vectors over ${\mathbb{F}}_2$ leaves $2^{m-k}$ runs. Reduced run count creates aliasing in a fractional factorial design, which must be evaluated against the effects scientifically required.

In a two-level factorial design, a factor subset defines a sign column. The coefficient of this column is its response contrast divided by the number of runs when the columns are orthogonal. A singleton describes a main effect and a pair describes a two-factor interaction. On the full cube, distinct columns have zero inner product: summing over any coordinate in their symmetric difference cancels the terms.

A main effect compares the mean response at different levels of one factor after averaging over the other factors according to the design. With sign coding, its two-level difference is twice the singleton factorial contrast coefficient. In a fractional factorial design, that coefficient can include aliased higher-order effects.

The latent factor vector has mean zero and covariance $I_q$, the specific errors have mean zero and diagonal covariance $\Psi$, and the two are uncorrelated. Thus the off-diagonal covariance is explained by shared factors. Normality additionally gives a likelihood model, but the covariance identity needs only these second-moment assumptions. Positive diagonal entries of $\Psi$ describe variation not explained by the factors. Unlike principal component analysis, the model separates shared and specific variation rather than only maximizing total projected variance.

A latent factor is an unobserved coordinate introduced to explain shared variation among measurements. In an orthogonal factor model, latent factors have variance one and are mutually uncorrelated, with coefficients specified by the factor loadings. They differ from a regression factor, which is a categorical predictor. A fitted covariance alone generally leaves the latent factor orientation undetermined; factor rotation gives equivalent representations.

For a finite connected graph embedded as a plane graph, follow the edges around a face of a plane graph to obtain a closed walk. Count traversals, not distinct edges: a bridge in a graph occurs twice along one facial boundary walk. Each edge contributes two sides in total.^[ref]

For the three-vertex graph path $A-B-C$, the only facial boundary walk is $A,B,C,B,A$, of length four. Both edges are bridges in a graph; neither is a graph cycle. Thus a facial boundary walk need not be a graph cycle.

The outer face is the unique unbounded face of a plane graph for a finite plane graph. It counts once even though it surrounds the whole graph drawing. A tree has only this face of a plane graph.

For the threading convention $d{x}^i-N^{i}dt$ and $K_{ij}=-({\dot{\gamma }}_{ij}+D_i{N}_j+D_j{N}_i)/(2N)$, contract with ${\gamma }^{ij}$. Metric compatibility gives ${\gamma }^{ij}{D}_i{N}_j=D_i{N}^i$, and the Jacobi determinant derivative formula gives ${\gamma }^{ij}{\dot{\gamma }}_{ij}={\partial }_t\log \det \gamma$. With zero shift, $-K/3=(3N{)}^{-1}{\partial }_t\log \sqrt{\gamma }$ is the local volume Hubble parameter. Reversing the shift convention reverses both the normal's spatial component and the shift terms; the two conventions must not be mixed.

For a compact regular locale $X$, put $B=\mathcal{O}(X{)}_{\neg \neg }$ and $\mathcal{O}(\gamma X)=\mathrm{Idl}(B)$. This locale is extremally disconnected. Its natural surjection to $X$ has inverse imageThe Joyal extension lemma gives the frame homomorphism; its injectivity follows by recovering $a$ as the join of the regular opens in this ideal.