For azimuthal velocity $v(r)e_{\theta }$, the velocity gradient in the rotating polar basis is $\left(\begin{array}{cc}0& -v/r\\ v^{\prime }& 0\end{array}\right)$. The basis rotates along a particle at angular speed $v/r$. Accounting for that rotation in the material derivative replaces $L$ by $L-(v/r)\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)$ in the upper-convected derivative. Its remaining shear rate is $v^{\prime }-v/r$, so the local viscoelastic normal-stress difference follows the same constitutive algebra as simple shear flow.

On a compact base, finitely many global sections can be selected to span every fiber when evaluations of global sections are surjective. They give a surjective morphism $M\times {\mathbb{R}}^N\to E$. Its kernel is a kernel bundle of a surjective vector bundle morphism, and its fibers give the displayed map $\varphi$. The first isomorphism theorem on each fiber identifies the resulting pullback quotient with $E$, with continuously varying inverses. The finite selection follows by choosing a frame's values at each point, extending each value to a global section, then taking a finite cover of the regions where those sections remain independent.

Over ${\mathrm{Gr}}_1({\mathbb{R}}^2)={\mathbb{RP}}^1$, the universal quotient is the orthogonal-complement line bundle, a Möbius line bundle. Writing a section as $a(\theta )(-\sin \theta ,\cos \theta )$ forces $a(\theta +\pi )=-a(\theta )$. Continuity then forces a zero, so no nowhere-zero section exists.

A unitary connection $A$ on an oriented Riemannian smooth four-manifold is anti-self-dual when $F_A^{+}=(F_A+\ast F_A)/2=0$. Its curvature is covariantly closed by the Bianchi identity and solves the Yang-Mills equations. For a closed oriented base, with anti-Hermitian matrices and the fundamental trace, $k=c_2(E)[X]=(8{\pi }^2{)}^{-1}{\int }_X\mathrm{Tr}(F_A\wedge F_A)=\Vert F_A{\Vert }_2^2/(8{\pi }^2)$ for an $SU(2)$ bundle. Thus its charge is nonnegative.

// Destination: geometry-and-topology.bigb

A unitary bundle gauge transformation of a Hermitian vector bundle is a smooth fiber-linear automorphism covering the identity on the base and preserving the Hermitian metric. In a local frame that is orthonormal, it is represented by a unitary matrix $u(x)$.

For the conventionthe curvature form changes by $F_{A^u}=u^{-1}{F}_{A}u$. Thus the gauge transformation changes the fiber frame while preserving the geometric connection on a vector bundle up to equivalence.

If a determinant trivialization is fixed, require the automorphism to preserve it as well. This gives the special unitary group subgroup of the full unitary group gauge transformations. In particular, an $SU(2)$ vector bundle with its fixed determinant uses determinant-one unitary bundle gauge transformations.

// Destination: geometry-and-topology.bigb

Fix a complex line bundle with a Hermitian metric over a closed oriented Riemannian manifold $X$. For a unitary connection $A$, the real curvature form $i{F}_A$ represents a fixed de Rham cohomology class. The Hodge theorem gives its unique harmonic representative $\eta$. Write $\eta -i{F}_A=d\beta$ with $\beta$ a real one-form. Then $A-i\beta$ is a unitary connection with real curvature form equal to $\eta$.

Two such unitary connections differ by a closed imaginary-valued one-form. A gauge transformation removes exactly those differences whose real periods lie in $2\pi \mathbb{Z}$. Their classes under gauge equivalence of principal connections carry a free transitive action ofwhere the integral cohomology is understood through its image in real cohomology. If the first Betti number is zero, there is just one such equivalence class. The compactness assumption supplies the Hodge theorem used here.^[ref]

// Destination: geometry-and-topology.bigb

For a connection with compact structure group on a four-dimensional ball, sufficiently small $L^2$ curvature permits a gauge with $d^{\ast }a=0$, suitable boundary condition and $\Vert a{\Vert }_{W^{1,2}}\le C\Vert F{\Vert }_2$ after rescaling. For anti-self-dual connections, local elliptic regularity also gives scale-invariant curvature derivative estimates on smaller balls. These are local analytic inputs, not the global bubbling compactness conclusion.

// Destination: geometry-and-topology.bigb

For positive gain and nonlinear-loss coefficients, a constant-density exciton-polariton condensate with physical velocity $v\hat{x}$ has $\psi =\sqrt{\gamma /\Gamma }\exp [i(mvx-\mu t)/\hslash ]$. Its chemical potential obeys the displayed relation. For prescribed real $\mu$, a real velocity requires $\mu \ge g\gamma /\Gamma$. If chemical potential is determined by the solution, any real velocity is allowed. Linear density-phase perturbations in the comoving frame satisfy $\lambda (\lambda +2\gamma /\hslash )+{ϵ}_k({ϵ}_k+2g{n}_{\ast })/{\hslash }^2=0$, where ${ϵ}_k={\hslash }^2{k}^2/(2m)$. Repulsive coupling has no growing modes in this simple model; attractive coupling has a long-wavelength instability even though the plane wave exists.

Let $f(r)=(-1{)}^{s_2(r)}$ be the Thue–Morse sequence, let $e(t)=e^{2\pi it}$, and let $N\ge 2$. For each positive integer $m\le N^{1/100}$, take an integer interval $I_m\subseteq [1,N/m]$. The range condition ensures that every product $mn$ remains at most $N$.

Choose the integer interval $J_m$ from $m\min I_m$ to $m\max I_m$, omitting an empty $I_m$. Character orthogonality givesThe uniform interval cancellation for binary digit parity bounds each inner exponential sum by $C{N}^{\sigma }$, with $\sigma =2-\frac{3}{4}{\log }_{2}3<1$. The average over $a$ does not increase that bound. ThereforeThe exponent is negative, so this normalized Type I sum bound is smaller than every fixed negative power of $\log N$ for large $N$. No cancellation between the outer terms is needed; each inner progression is controlled separately.

Conditional on the pooled number $s$ of successes in $m+n$ exchangeable random variables that are binary, all placements of those successes are equally likely under a common success-chance mixture. The count in a chosen batch of size $n$ therefore has a hypergeometric distribution, with mass $\left(\genfrac{}{}{0ex}{}{s}{y}\right)\left(\genfrac{}{}{0ex}{}{m+n-s}{n-y}\right)/\left(\genfrac{}{}{0ex}{}{m+n}{n}\right)$. Under the uniform-count predictive property, $s$ is uniform on $0,\dots ,m+n$. Dividing the joint pooled-and-batch probability by the first batch's uniform marginal gives the posterior predictive distribution, with factor $(m+1)/(m+n+1)$. Substituting $s=x+y$ in the hypergeometric expression does not give a single normalized hypergeometric mass as $y$ varies, since its population success total then varies too.

A two-way analysis of variance with the full interaction and one observation in each observed factor cell has one independently varying fitted cell mean per observation. A cell-indicator design matrix has full row matrix rank, even when some combinations are missing, so the model is a saturated statistical model. Its residual sum of squares is zero and its residual degrees of freedom are zero. Missing combinations make some coefficients nonidentifiable; they do not restore a pure-error estimate. Standard errors and an ordinary residual-based F-test for interactions therefore cannot be recovered merely by fitting this larger model.

For observed cells with means $a_i+b_j$, form a bipartite graph joining row $i$ to column $j$ when that cell is observed. If all levels occur, the design rank is $r+c-k$, where $k$ is the number of connected components. Indeed a coefficient perturbation in the kernel satisfies $a_i=-b_j$ on every observed edge, and hence has exactly one freely chosen constant on each component. A connected design has rank $r+c-1$ and is identifiable after a single reference constraint. Missing cells destroy the balanced-design orthogonality of row and column effects, so sequential sums of squares can depend on order.

In a fully balanced crossed design, centered indicator columns for the two factors are orthogonal because each pair of levels occurs equally often. Missing cells can change the level-pair counts so that the displayed cross-product becomes nonzero. Then sequential sums of squares depend on term order, although the final additive fit is unchanged. Testing either factor conditional on the other requires its extra sum of squares from the appropriate reduced model.

For the shearing-wave energy profile, stationary points satisfy $-2\mathrm{Re}T=(1+T^2{)}^2$. At large Reynolds number, the initial minimum is at $T_{-}\sim -(2\mathrm{Re}{)}^{1/3}$ and the subsequent maximum is at $T_{+}\sim -1/(2\mathrm{Re})$. Their energy ratio is $G_{\max }\sim (2\mathrm{Re}{)}^{2/3}{e}^{-2/3}$. The Reynolds number here uses $A$, half the background shear rate; using $2A$ instead changes the coefficient convention.

// Target: astrophysics.bigb

The spinor covariant derivative commutator is a differential two-form in its derivative indices, so the two-component spinor calculus decomposition defines symmetric curvature operators ${\Delta }_{AB}$ and ${\Delta }_{A^{\prime }{B}^{\prime }}$. With ${ϵ}^{AB}{ϵ}_{AB}=2$, the unprimed projection is ${\Delta }_{AB}={ϵ}^{A^{\prime }{B}^{\prime }}[{\nabla }_{A{A}^{\prime }},{\nabla }_{B{B}^{\prime }}]/2$, and the primed projection is analogous. For a torsion-free connection these are algebraic curvature derivations: they annihilate scalar functions and obey the Leibniz rule on tensor products.

The totally symmetric unprimed four-index spinor represents one chiral part of the Weyl tensor. The other part is represented by a primed spinor, conjugate to the unprimed one on a Lorentzian real slice. The trace-free symmetry separates conformal curvature from the Ricci tensor and scalar-curvature terms.

A two-component two-component spinor is an element of a two-dimensional complex spin space with invariant antisymmetric form ${ϵ}_{AB}$. In complexified four-dimensional geometry there are independent unprimed and primed spin spaces, and vectors are represented by $x^{A{A}^{\prime }}$. Raising an index uses the antisymmetric form, so ${\pi }^{A^{\prime }}{\pi }_{A^{\prime }}=0$. This identity makes the incidence-plane directions ${\pi }^{A^{\prime }}{\partial }_{A{A}^{\prime }}$ totally null.

In a homogeneous overlapping two-component Bose-Einstein condensate, the long-wavelength energy change at fixed species numbers is one half the integral of the density perturbation quadratic form with matrix $\left(\begin{array}{cc}{U}_{11}& U_{12}\\ U_{12}& U_{22}\end{array}\right)$. Strict positive definiteness gives the displayed criterion. Strong positive cross-repulsion favors separation; sufficiently strong cross-attraction instead produces an unstable joint density mode. Equality is marginal at this order, not strict stability.

In the conformal frame for $g={\Omega }^2\eta$, identify the spinor fibres using the chosen vierbein. If ${ϵ}_0$ is a flat constant spinor field, the spin connection of a conformally flat metric gives ${\nabla }_aϵ={\hat{\gamma }}_a{\hat{\gamma }}^c({\partial }_c\log \Omega )ϵ/2$. Contracting with curved ${\gamma }^a={\Omega }^{-1}{\hat{\gamma }}^a$ yields ${\gamma }^a{\nabla }_aϵ=n{\Omega }^{-1}{\hat{\gamma }}^c({\partial }_c\log \Omega )ϵ/2$. Thus ${\nabla }_aϵ={\gamma }_a{\gamma }^b{\nabla }_bϵ/n$: the rescaled parallel spinor is a twistor spinor.

The local frame $X_i^m$ on $X_i\ne 0$ identifies a section with a homogeneous rational function of degree $m$ on the punctured affine cone. Homogenizing a local rational expression and cancelling common factors writes every nonzero section as a quotient of coprime homogeneous polynomials $F/G$ with $\mathrm{deg}F-\mathrm{deg}G=m$. Conversely, such a rational function regular on the cone gives local coefficients $r/X_i^m$, which glue with the twisting sheaf's transition functions of a vector bundle.