A map between simply connected spaces is a rational homotopy equivalence when it induces isomorphisms ${\pi }_n(X)\otimes \mathbb{Q}\to {\pi }_n(Y)\otimes \mathbb{Q}$ in every degree. Equivalently it induces an equivalence of their rationalizations, or a rational homology isomorphism by the rational Whitehead theorem. Spaces have the same rational homotopy type when they are connected by a zigzag of such maps. This is distinct from rational homotopy equivalence of algebraic cycles.

On an irreducible algebraic curve, let $\mathcal{R}(\mathcal{F})$ be the constant rational-section sheaf and put $\mathcal{P}(\mathcal{F})(U)={⨁}_{P\in U}\mathrm{Rat}(\mathcal{F})/{\mathcal{F}}_P$. Every open subset is quasi-compact, so compatible finite-support families glue with finite support. Its stalk at $P$ is the displayed rational-principal-part quotient. A rational section is regular outside finitely many points, making its principal-part map well defined. The resulting sequence is stalkwise exact. Both right-hand sheaves are flasque, and consequently $H^1(X,\mathcal{F})$ is the cokernel of the global principal-part map and $H^q(X,\mathcal{F})=0$ for $q\ge 2$. The curve and Zariski-topology hypotheses distinguish finite support here from locally finite pole sets in complex analysis.

A rational maximal torus is elliptic when it is contained in no proper rational parabolic subgroup. This definition allows the split central torus of the ambient group; an absolute absence of every split subtorus is not the right condition when the centre is nontrivial.

For a nonzero discrete Liouville period group, this is its positive generator. In Euclidean symplectic space, the same group is obtained from signed symplectic areas of disks with boundary on $L$, by the Stokes theorem and contractibility of the ambient space. A product of circles of common radius $r$ is a rational Lagrangian submanifold with $\gamma (L)=\pi r^2$. Translation changes a primitive only by an exact form and hence leaves the period group unchanged.

At a smooth point with local parameter $u$, expand a rational differential as ${\sum }_j{a}_j{u}^{j}du$ in the completed local field; its residue is $a_{-1}$. Formal substitution shows this coefficient is independent of the parameter, in arbitrary characteristic. For a finite separable extension of these local fields, residues commute with the differential trace, with a sum over branches. The monogenic trace formula in the trace-dual module of a finite curve map proves this compatibility by coefficient extraction.

For tangential data with ${\beta }_1={\beta }_2=0$, the harmonic function $q=xy$ has zero prescribed data on both axes but a growing nonzero derivative. For normal data with ${\beta }_1={\beta }_2=\pi /2$, $q=\log |z|$ also has zero prescribed data, and $q_z=1/(2z)$ decays at infinity but is singular at the corner. Thus decay of the data, and even decay of the solution gradient, does not alone determine the quadrant representation. Requiring the reflected gradient to decay at infinity and satisfy $|z|H(z)\to 0$ at the corner removes both examples. Without such conditions, add homogeneous holomorphic terms obeying the reflection parity.

For real harmonic $q$, prescribe $-\sin {\beta }_1{q}_x+\cos {\beta }_1{q}_y=h_1$ on the vertical axis and $\cos {\beta }_2{q}_x-\sin {\beta }_2{q}_y=h_2$ on the horizontal axis. Their orthogonal derivatives $j_1,j_2$ determine the missing gradient components. If $H_1,J_1$ use the kernel $e^{ky}$ and $H_2,J_2$ use $e^{-ikx}$, the clockwise spectra ${\hat{q}}_1={\int }_0^{i\infty }{e}^{-ikz}{q}_{z}dz$, ${\hat{q}}_2=-{\int }_0^{\infty }{e}^{-ikz}{q}_{z}dz$ have the displayed expressions. Their global relation is ${\hat{q}}_1+{\hat{q}}_2=0$ in the third spectral quadrant.

// Target: analysis.bigb

For ${\gamma }^{\mu \nu \rho }{\partial }_{\nu }{\psi }_{\rho }-m{\gamma }^{\mu \rho }{\psi }_{\rho }=0$ with $m\ne 0$, divergence gives ${\gamma }^{\mu \rho }{\partial }_{\mu }{\psi }_{\rho }=0$. Gamma contraction then forces $\gamma \cdot \psi =0$, hence $\partial \cdot \psi =0$. The remaining equation is $(/\partial +m){\psi }_{\mu }=0$ in a mostly-plus convention. In the rest frame the temporal component vanishes, leaving a spatial vector times an on-shell two-component spinor; its gamma trace removes the spin-one-half sector. The remaining spin-three-halves representation has four physical states, including longitudinal helicities $\pm 1/2$.

Gauge equivalence ${\psi }_{\mu }\sim {\psi }_{\mu }+{\partial }_{\mu }ϵ$ permits gamma-trace gauge. The Rarita-Schwinger field equation then gives transversality and a massless Dirac equation for each component. For momentum along the third axis, residual gauge removes the temporal and longitudinal components. The remaining constraint is ${\psi }_2={\gamma }^1{\gamma }^2{\psi }_1$, with ${\psi }_1$ in the two-dimensional kernel of $/p$. Spinor helicity $\pm 1/2$ is paired with vector helicity of the same sign, leaving only $\pm 3/2$. A Majorana spinor reality condition identifies antiparticles; a Dirac vector-spinor has separate antiparticle modes.

The gamma trace contracts the vector index of a Rarita-Schwinger field with a gamma matrix, producing an ordinary spinor. In the massive free theory it vanishes by the field equations. In the massless theory a gauge transformation changes it by ${\gamma }^{\mu }{\partial }_{\mu }ϵ$, allowing a gamma-traceless gauge for propagating modes. Together with transversality this removes the unwanted spin-one-half sector.

Let $M$ be a membership structure satisfying extensionality and every pure-membership separation instance, with an external automorphism $j$ and domains $D_i=j^i(D_0)$ represented by objects of $M$. Suppose every $M$-subset of $D_i$ is an element of $D_{i+1}$. Interpret NFU on the objects of $D_0$ using the displayed set predicate and membership, with all subset statements evaluated in $M$. Atoms have no members, and extensionality applied to $j(y),j(z)\subseteq D_0$ gives extensionality for sets. A stratified formula becomes an ordinary pure-membership formula by sending each variable of type $i$ to $j^i(x)$ and restricting its quantifier to $D_i$. A set predicate on sort $i$ means containment in $D_{i-1}$; adjacent-sort membership is guarded by containment in the lower domain. Separation yields the extension $X\subseteq D_i$, the closure hypothesis places it in $D_{i+1}$, and $j^{-(i+1)}(X)\in D_0$ is the required untyped set. This proves every comprehension instance. The guard is essential: an object not contained in a lower domain can still have some ordinary members there, which must not become members of a typed atom. Internal ranks $V_{j^i(\alpha )}$ with $j(\alpha )>\alpha$ give one instance of these hypotheses; shifted rank objects in an elementary model of an expanded $V_{\omega +\omega }$ give a consistency proof without assuming Con(ZFC).

This hierarchy measures the complexity of successive finite selections. Start with ${\mathcal{M}}_0=\{\varnothing \}$. At a successor stage, retain the previous family and allow one smaller integer to be prepended. At a countable limit $\lambda$, choose positive ordinals ${\lambda }_j\uparrow \lambda$ and admit sets in ${\mathcal{M}}_{{\lambda }_j}$ whose minimum is at least $j$. Each family is hereditary and spreading, has no infinite branch, and its extension-tree rank on every infinite subset is at least $\alpha$. These properties make the hierarchy useful for transfinite constructions of unconditional basic sequences.

If a design matrix $X$ has a nonzero null vector $z$, the normal linear model has identical observation distributions at $\beta$ and $\beta +z$. Any estimator has the same expectation at those two parameters, and therefore cannot be an unbiased estimator of both coefficient vectors. Full column rank is required for unbiased estimation of the entire unconstrained coefficient vector; identifiable linear combinations can still be estimated without it.

A non-backtracking random walk remembers its preceding vertex and excludes that neighbor from the next choice. At vertices of degree at least two, it is a Markov chain on directed edges: $(a,b)$ moves to $(b,c)$ with probability $1/(\mathrm{deg}b-1)$ for each neighbor $c\ne a$. Dead ends require an additional convention. An initial vertex without a preceding edge uses the ordinary first-step neighbor distribution.

A small proposal scale increases acceptance while making successive states almost equal. A large scale often proposes low-density states and creates long rejection runs. A pilot comparison can optimize effective sample size of a Markov chain per unit computing time, or expected accepted squared displacement, rather than acceptance alone. Tune on representative functions and then fix the scale for production; an arbitrarily history-dependent adaptation is not automatically a chain with the desired invariant distribution.

To update a positive parameter, a symmetric additive proposal distribution can be generated on the whole real line. Reject nonpositive proposals; accept positive ones using the target density ratio. The off-diagonal proposal remains symmetric. Repeatedly redrawing until the proposal is positive instead truncates and renormalizes the proposal differently at different current states, so a Hastings correction is then required. A symmetric walk in $\log v$ is another valid method, but its transformed target includes the Jacobian determinant factor $v$.

For a conditional count mean $\mathbb{E}(Y\mid x,b)=\exp (b+{\beta }_0+{\beta }^{\mathsf{T}}x)$, suppose the distribution of $b$ is independent of the covariates and has finite $\mathbb{E}{e}^b$. Integrating over $b$ gives the displayed logarithmic link function mean: every slope stays unchanged, and only the intercept shifts. This is an exact property of the log link; it does not hold in this form for the logit.

For $q\ge 1$, the random-cluster measure lies between the two Bernoulli product measures displayed above. The upper Holley condition ratio is $q^{k(\omega \wedge \eta )-k(\eta )}\ge 1$. The lower ratio is $q^{k(\omega \vee \eta )-k(\omega )+|\eta \setminus \omega |}\ge 1$, because adding an edge reduces the component count by at most one. The comparisons persist under wired limits and give $p_c(1)\le p_c(q)\le q{p}_c(1)/(1+(q-1)p_c(1))$.

Choose any vertex $v$ of a red-blue colored complete graph $K_6$. Its five incident edges go to the other five vertices. By the pigeonhole principle, at least three have the same color. Call their other endpoints $a,b,c$ and, after swapping color names if needed, suppose $va,vb,vc$ are red.

If any of $ab,ac,bc$ is red, it completes a red triangle in a graph with $v$. If none is red, all three are blue, so $a,b,c$ themselves form a blue triangle in a graph. Both possibilities force a monochromatic triangle in a graph. This proves the upper bound for the Ramsey number for triangles using only five incident edges and the three pairs among the selected neighbors; the colors of the other edges cannot defeat the argument.

Both panels show all 15 edges of a complete graph on six vertices. Faint colored edges are example choices outside the decisive argument; bold edges exhibit the forced triangle in a graph. The selected neighbor color could instead be blue, giving the same argument with the color names exchanged.

Place five labeled vertices around a pentagon. In the complete graph $K_5$, color the five perimeter edges red and the five diagonals blue. Writing $ij$ for the unordered pair $\{i,j\}$, the red pairs are $12,23,34,45,15$, and the blue pairs are $13,14,24,25,35$. This assigns a color to every pair, including every diagonal.

The red subgraph is the cycle graph $1,2,3,4,5,1$; the blue subgraph is the cycle graph $1,3,5,2,4,1$. Neither contains a triangle in a graph. Consequently, none of the ten triples has all three edges red or all three blue, proving the lower bound for the Ramsey number for triangles. Crossings in the drawing are not additional vertices.