Completing the minus chain complex in powers of $U$ replaces $\mathbb{Z}[U]$ by the formal power series ring $\mathbb{Z}[[U]]$; its homology is written ${\mathbf{HF}}^{-}$. This permits infinite sums of Heegaard Floer cobordism maps whose exponents tend to infinity, as occur in surgery formulas. A series with constant coefficient $\pm 1$ is a unit: its inverse is found recursively, or by the geometric series for its positive-order part. Such a series need not have a polynomial inverse, which is why exact triangles involving sums over all Spin-c extensions require a completion in the minus theory. Individual homogeneous cobordism maps are still defined on the ordinary minus groups.

The reduced minus and plus groups areThe connecting homomorphism of the fundamental long exact sequence induces an isomorphism $\delta :H{F}_{\mathrm{red}}^{+}\to H{F}_{\mathrm{red}}^{-}$ of degree $-1$. This follows immediately from exactness: its image is the indicated kernel, and its kernel is precisely the image quotiented out on the plus side. These groups isolate the information lost by inverting $U$.

Let ${\mathfrak{s}}_k$ on $S^1\times S^2$ have $⟨c_1({\mathfrak{s}}_k),[S^2]⟩=2k\ne 0$. The ordinary groups areA sufficiently wound genus-one diagram gives a two-generator summand whose chain differential is multiplication by $1-U^{|k|}$: the two relevant disks have opposite coherent signs and basepoint multiplicities differing by $|k|$. The map is injective over $\mathbb{Z}[U]$, so its homology is the displayed quotient. Multiplication by $U$ is invertible in the quotient, showing that minus-to-infinity is an isomorphism and that the plus and hat groups vanish. In the U-adic completion of Heegaard Floer homology, $1-U^{|k|}$ is a unit, so both completed minus and infinity groups vanish. This example shows why genus bounds for the completed-minus or plus groups must not be transferred unchanged to the ordinary minus theory.

For $x^{\prime \prime }+ϵc(1-x^2)x^{\prime }+x=k$, use $T=ϵt$ and $x=k-r(T)\cos (t+\varphi (T))$. Removal of secular terms gives ${\varphi }_T=0$ and the displayed amplitude equation. With $R=r^2$, $R_T=c(k^2-1)R+(c/4)R^2$, which is explicitly integrable. If $c$ is replaced by a slowly varying coefficient $C(T)$, the same solution uses the accumulated time ${\int }_0^{T}C(s)ds$. The method of multiple scales is uniform on bounded slow-time intervals only while its amplitude and correction estimates remain bounded.

If $V$ is a value in a lattice-ordered group of $t>1$, every convex lattice subgroup strictly containing $V$ contains $t$. Thus the convex lattice subgroup generated by $V$ and $t$ is the unique least subgroup properly containing $V$. It is called the cover of $V$. There is no intermediate convex lattice subgroup. A normal-valued lattice-ordered group requires $V$ to be a normal subgroup of this cover, not necessarily of the whole group. Values are prime convex lattice subgroups: maximality omitting $t$ makes them meet-irreducible in the lattice of convex lattice subgroups, so their ordered right cosets form a chain.

Let $p,q,x\ge 1$ in a lattice-ordered group. Under the displayed hypothesis, suppose $d=(x{q}^{-2}{p}^{-2})\vee 1>1$ and choose a value $P$ of $d$. If $x\in P$, then $1\le d\le x$ contradicts its omission. Otherwise extend $P$ to a value $V$ of $x$. The right-coset test is $Vu\le Vv⟺(u{v}^{-1})\vee 1\in V$. If $p\notin V$, put $c=(x{q}^{-2}{p}^{-1})\vee 1\in V$. Then $(p{c}^{-1})\vee 1\notin V$, while its inverse positive part is $(c{p}^{-1})\vee 1=d$. These parts have meet one, so primeness of $P$ forces $d\in P$. If $p\in V$, put $c=(x{q}^{-1})\vee 1\in V$ and $t=p^{2}q\notin V$; the disjoint pair $(t{c}^{-1})\vee 1$ and $(c{t}^{-1})\vee 1=d$ gives the same contradiction. No normality assumption is used in this separation argument.

Differentiate the Kirchhoff stress tensor relation $\mathbf{\tau }=FS{F}^T$, using $\dot{F}=LF$ for the velocity gradient. Subtracting the two deformation terms gives the displayed upper-convected derivative. This removes the transport generated solely by deformation of a fixed material stress tensor.

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 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. 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 determinant-preserving unitary bundle automorphism covering the identity on the base. For an $SU(2)$ bundle its values are in the fiberwise special unitary group. We use $A^u=u^{-1}Au+u^{-1}du$. Its curvature transforms by $F_{A^u}=u^{-1}{F}_{A}u$. The automorphism is a change of unitary frame, not a change of the base point.

// Destination: geometry-and-topology.bigb

The real curvature $i{F}_A$ of a unitary line connection has a fixed de Rham class. Let $\eta$ be its harmonic representative and write $\eta -i{F}_A=d\beta$. Then $A-i\beta$ has curvature $\eta$. Any two such connections differ by a closed imaginary one-form. If $b_1=0$, that form is exact and comes from a circle-valued gauge transformation. In general their gauge classes form a torsor for $H^1(X;\mathbb{R})/(2\pi H^1(X;\mathbb{Z}))$.

// 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.

The additive character orthogonality filter $1_{m\mid r}=m^{-1}{\sum }_{a=0}^{m-1}e(ar/m)$ restricts the uniform interval cancellation for binary digit parity to multiples of $m$, without increasing its bound. Each inner Type I sum is therefore $O(N^{\sigma })$. Summing the at most $N^{1/100}$ outer terms proves the displayed estimate, which is smaller than every fixed negative power of $\log N$ for large $N$.

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.