A degree-$m$ rational map of the complex projective line is represented by two degree-at-most-$m$ polynomials without a common root, modulo common scalar multiplication. This is an open set of ${\mathbb{CP}}^{2m+1}$. Quotienting by domain Möbius transformations subtracts three complex dimensions, giving real dimension $4m-4$ for degree $m\ge 2$. These are the local covering parameters of a multiple-covered J-holomorphic sphere.

The rational homotopy type of a simply connected space is its equivalence class under zigzags of rational homotopy equivalences, or equivalently the homotopy type of its rationalization of a topological space. For the usual finite-type spaces, it is encoded contravariantly by the isomorphism class of the Sullivan minimal model. Isomorphic cohomology rings determine this type only when additional information, such as formality of a topological space, makes the cochain algebra equivalent to the cohomology algebra.

A Padé approximant is a rational function with numerator degree at most $m$ and denominator degree at most $n$, normalized by $Q_n(0)=1$, such that $Q_n(z)f(z)-P_m(z)=O(z^{m+n+1})$ near zero, when such an approximant exists. For the exponential function, its $[1/1]$ approximant is $(1+z/2)/(1-z/2)$. Applying this rational function to a matrix gives the Crank-Nicolson method. Diagonal and subdiagonal approximants also give stability functions of Gauss collocation methods and Radau IIA methods respectively.

For two weight-$k$ level-one cusp forms, use the primitive real-analytic Eisenstein series and $d\mu =dxdy/y^2$. The Rankin–Selberg unfolding identity isInitially this follows by absolute convergence, unfolding to $0\le x<1$, and integrating the Fourier series in $x$. Exponential cusp decay and polynomial growth of the primitive real-analytic Eisenstein series make the integral holomorphic for $\mathrm{Re}w>1$, continuing the Dirichlet series to $\mathrm{Re}s>k$.

NFU has a unary predicate $S$ for sets, atoms have no members, extensionality applies to sets, and comprehension applies to every stratified formula. A stratification assigns integer types to variables so that equality uses equal types and $x\in y$ requires $\mathrm{type}(y)=\mathrm{type}(x)+1$. A unary set predicate adds no type difference.

Start with the actual set structure $B=(V_{\omega +\omega },\in )$. This structure satisfies extensionality and every pure-membership separation instance: the subset defined in a set $a$ of rank below $\omega +\omega$ still has rank below $\omega +\omega$. We do not assert that $B$ satisfies replacement or all of ZFC. Expand it by Skolem functions. In it use the sequence of rank objects $V_{\omega +n+6}$, $n<\omega$. For two such objects $d<e$ in sequence order, we havewith quantifiers in $B$. The second statement holds because every subset of $V_{\alpha }$ belongs to $V_{\alpha +1}$, and the later rank is at least $\alpha +1$.

Introduce constants $d_i$, $i\in \mathbb{Z}$, and require them to be order indiscernibles in the Skolem language, satisfying these two assertions whenever $i<j$. Also require $V_{\omega +5}\subseteq d_i$ for every $i$; this fixed rank object is first-order definable in $B$. Every finite part of these requirements together with $\mathrm{Th}(B^{\ast })$ is satisfiable: color finite increasing tuples of the original rank sequence by the truth values of the finitely many formulas mentioned, and use Ramsey's theorem to get a homogeneous subsequence. The rank assertions hold for every increasing tuple there. The compactness theorem gives a model of the full theory. In its Skolem hull $N$ generated by the $d_i$, the integer shift extends to an automorphism $j$ with $j(d_i)=d_{i+1}$, by the term-transport proof of the Ehrenfeucht-Mostowski theorem. Thus $N$ still satisfies extensionality and all pure-membership separation instances, andThis is a set-sized first-order model; no satisfaction predicate for a proper-class universe is being assumed.

Externally let $D_i=\{x\in N:N\models x\in d_i\}$. The automorphism bijects $D_i$ onto $D_{i+1}$. Interpret a typed structure with sort $i$ equal to $D_i$. At sort $i+1$, declare $y$ a set exactly when $N\models y\subseteq d_i$; otherwise it is an atom. Define adjacent-sort membership byThe guard is necessary: a typed atom might have some ordinary $N$-members in $D_i$, and those must not become its typed members. Set-extensionality follows from extensionality in $N$, because the ordinary members of a typed set all lie in $D_i$. For any typed formula on sort $i$, its quantifiers can be restricted to the corresponding $d_l$, its set predicates replaced by subset assertions, and its memberships replaced by the guarded relation. This is an ordinary pure-membership formula of $N$ with finitely many parameters. Separation gives its extension $X\subseteq d_i$, and the displayed closure assertion gives $X\in d_{i+1}$. It is therefore a typed set witnessing comprehension. The automorphism shifts these sorted domains and preserves the set flags and guarded memberships.

Collapse the types onto $D_0$. DefineObjects not satisfying $S$ have no $E$-members. If two sets have the same $E$-members, their images under $j$ are subsets of $d_0$ with the same ordinary members, so are equal by extensionality of $N$; injectivity of $j$ gives equality of the original objects.

To verify every stratified comprehension instance, assign a variable $v$ its stratification type $t(v)$ and translate its value $a\in D_0$ to $j^{t(v)}(a)\in D_{t(v)}$. Equality is preserved. An atomic $E(x,y)$ translates to the guarded adjacent-sort membership because $t(y)=t(x)+1$; the unary predicate $S(y)$ translates to $j^{t(y)}(y)\subseteq d_{t(y)-1}$. Induction on formulas, using the domain bijections for quantifiers, preserves truth. If the free variable $x$ has type $t$, typed comprehension produces its extension $X\subseteq d_t$, with $X\in d_{t+1}$. Put $b=j^{-(t+1)}(X)\in D_0$. Then $S(b)$ holds, and for every $a\in D_0$,This is exactly the required NFU set. No formula containing the external automorphism $j$ has been used in a separation or comprehension scheme; $j$ only transports values after the pure formula is formed.

The construction also permits the usual infinity requirement. The ordinary $\omega$ and its successor graph $f$ are definable in $N$, hence fixed by $j$, and lie below $V_{\omega +5}$, so belong to all the required domains. Under $E$, the members of $\omega$ are still its ordinary $N$-members. The new Kuratowski pair satisfiesSince $j(f)=f$, membership of this pair in $f$ under $E$ is equivalent to $⟨x,y{⟩}_N\in f$. Thus $f$ is an internal injection of $\omega$ into itself omitting zero, giving a Dedekind-infinite set.

We have built a model of NFU, with infinity if it is included in the formulation, in ordinary ZFC metatheory:In particular $\mathrm{Con}(\mathrm{ZFC})\Rightarrow \mathrm{Con}(\mathrm{NFU})$. This proof does not assume $\mathrm{Con}(\mathrm{ZFC})$ as an additional internal axiom: its starting structure is the set $V_{\omega +\omega }$, not a model of ZFC.

The Fourier expansion of a normalized Eisenstein series and the one-dimensional weight-twelve cusp form space give $691E_{12}=691E_4^3-432000\Delta$. The coefficient of $q^n$, for $n\ge 1$, reduces modulo $691$ to $566{\sigma }_{11}(n)=566\tau (n)$. Since $566$ is invertible modulo $691$, this proves the congruence. Integrality follows from $E_4=1+240\sum {\sigma }_3(n)q^n$ and the product $\Delta =q{\prod }_{n\ge 1}(1-q^n{)}^{24}$.

The modular discriminant satisfies $1728\Delta =E_4^3-E_6^2$, while the Serre derivative gives $2\Theta E_6=E_2{E}_6-E_4^2$. Their integral Fourier expansions have $E_4\equiv 1$ and $E_2\equiv E_6(\mathrm{mod}5)$, because $d^5\equiv d(\mathrm{mod}5)$. Reducing the two identities gives $\Delta \equiv \Theta E_6(\mathrm{mod}5)$. The coefficient of $q^n$ in $\Theta E_6$ is $-504n{\sigma }_5(n)\equiv n{\sigma }_5(n)$.

In a negligible-mass, non-burning stellar envelope with $P=K{T}^{5/2}$ and Kramers' opacity law $\kappa ={\kappa }_1\rho T^{-7/2}$, the stellar radiative temperature gradient is ${\nabla }_{\mathrm{rad}}=3{\kappa }_1\mu K^{2}L/(16\pi ac\mathcal{R}GM{T}^{7/2})$. It decreases inward as the temperature rises, so the inner edge of efficient convection is where it equals $2/5$. If outer matching gives $K\propto g^{1/2}{T}_e^{-2\beta -5/2}$, then $T_b^{7/2}\propto K^{2}L/M\propto T_e^{-4\beta -1}$, since $g=GM/R^2$ and $L=4\pi R^2\sigma T_e^4$. Hence $T_b\propto T_e^{-(8\beta +2)/7}$.

For radial contraction on a star-shaped domain, the homotopy formula is the identity in positive form degrees because pullback by the constant map vanishes there. On functions it is $h_{1}df=f-f(0)$. A nonzero constant is a counterexample to the uncorrected degree-zero identity. The Poincaré lemma is therefore an exactness assertion in positive degrees, while closed degree-zero forms are locally constant.

The unit radial vector field is $E=\nabla r$ on the punctured normal neighbourhood, where $r$ is the radial distance in a normal neighbourhood. Along ${\gamma }_u(t)={\exp }_p(tu)$ with $|u|=1$, it satisfies $E({\gamma }_u(t))={\dot{\gamma }}_u(t)$. The scaled radial field $W=rE=\nabla (r^2/2)$ is smooth at the centre and vanishes there. These conventions have different associated radial functions.

// Target: riemannian-geometry.bigb

Put $z=a_1/\sqrt{a_2}$ using the root with positive real part. The additional resolvent singularity at $s+a_3=z^2/4$ lies on the physical square-root sheet only when $\mathrm{Re}z>0$; otherwise the apparent saddle is on the other sheet. The singularity at $s=-a_3$ remains, including a quartic degeneracy when $a_1=0$. The rightmost physical singularity determines the exponential fixed-position growth rate. Compare it with the temporal maximum $-\mathrm{Re}{a}_3+\max (\mathrm{Re}{a}_1,0{)}^2/(4\mathrm{Re}{a}_2)$ to distinguish absolute wave-packet instability from convective wave-packet instability. Equality cases require their algebraic prefactors.

Reality of the boundary derivatives gives $H_1(\bar{k})=\overline{H_1(k)}$ and $H_2(-\bar{k})=\overline{H_2(k)}$, and the same identities for $J_1,J_2$. The global relation and its conjugate express both quarter-plane oblique boundary spectral functions in terms of the known $H_1,H_2$ and the single bounded holomorphic function $\Phi$ in the upper half-plane. For $\gamma ={\beta }_1+{\beta }_2\in \{0,\pi /2,\pi \}$, its coefficients on the positive real and positive imaginary reconstruction rays are opposite, because $e^{4i\gamma }=1$. Closing the first-quadrant contour removes $\Phi$ by the Cauchy integral theorem.

// Target: analysis.bigb

With only up/down valence quarks and no antiquarks, the quark number sum rule and leading parton model expression give the charge-squared number moment $\int F_2(x)dx/x={\sum }_f{Q}_f^2{N}_f$. The squared charges are $4/9$ for up and $1/9$ for down, so the proton has $8/9+1/9=1$ and the neutron has $4/9+2/9=2/3$. Restricting the flavour list to up/down does not itself remove antiquarks; the no-sea assumption is essential for these individual moments.

In units $\hslash =1$, a particle of mass $m$ with potential $V(q)=-Fq$ has classical path $q_c(t)=q_0+[(q_1-q_0)/T-FT/(2m)]t+F{t}^2/(2m)$. Its action and exact quantum-mechanical propagator areThe fluctuation operator is the free one because the potential is linear. This is a special case of classical-path factorization of a quadratic path integral.

For input dimension $D$, an input of entropy $S(\rho )$ gives output entropy $h_2(p)+(1-p)S(\rho )$ and reference-output entropy $h_2(p)+pS(\rho )$. These follow from entropy of an orthogonal quantum mixture and purity of the input purification. Thus the quantum mutual information is $2(1-p)S(\rho )$, maximized at $\rho =I_D/D$. The entanglement-assisted classical capacity is consequently $2(1-p){\log }_{2}D$ bits per use; for a binary input it is $2(1-p)$.

Three Fourier modes form a spatial resonance when their wavevectors sum to zero. Products of the conjugate second and third modes then force the first mode, with cyclic counterparts. Such terms couple both amplitudes and phases, unlike self-saturation terms proportional only to squared magnitudes. For three equal wavenumbers, the vectors are separated by $120$ degrees and support hexagonal pattern interactions.

In $L={\mathbb{Q}}_2(\sqrt{2})$, set $\pi =\sqrt{2}$ and normalize $v_L(\pi )=1$, so $v_L(2)=2$. The two terms of $a+b\pi$ have valuations of opposite parity, giving ${\mathcal{O}}_L={\mathbb{Z}}_2\oplus {\mathbb{Z}}_2\pi$. The nonidentity automorphism sends $\pi$ to $-\pi$, and $v_L(-2\pi )=3$. The lower ramification numbering therefore gives the displayed groups. The different exponent is three.

For traceless generators, the tensor product of Lie algebra representations has generators $t_a\otimes I+I\otimes s_a$. The trace of their summed squares is the sum of the two factor Casimir traces; the cross term vanishes because $\mathrm{tr}{t}_a=\mathrm{tr}{s}_a=0$. Decomposition into irreducible representations, with multiplicities counted, gives the displayed identity. Without tracelessness an extra term $2{\sum }_a\mathrm{tr}{t}_a\mathrm{tr}{s}_a$ is present. A one-dimensional abelian representation of charge one tensored with itself has Casimir four, not the sum two; semisimple Lie algebras avoid this problem because every generator is a sum of commutators.

For generators normalized by ${\mathrm{tr}}_3(t_a{t}_b)={\delta }_{ab}/2$, the quadratic Casimir operator on the irreducible SU(3) representation of Dynkin labels $(a,b)$ has the displayed scalar value. In SU(3) highest-weight coordinates $(p,q)$ it is $p^2+q^2+p+\sqrt{3}q$.

To derive this, write the operator as $H_1^2+H_2^2+{\sum }_{i=1}^3(E_{+}^i{E}_{-}^i+E_{-}^i{E}_{+}^i)$. On a highest-weight vector, the raising operators vanish, and the three root commutators sum to $H_1+\sqrt{3}{H}_2$. Centrality and the Schur lemma extend this value to the representation. The singlet, fundamental, adjoint and decuplet values are $0,4/3,3,6$, respectively. Tensor labels and multiplet dimensions are summarized in the Particle Data Group's SU(n) multiplet guide.

For $e_3=[e_1,e_2]$, $f_3=-[f_1,f_2]$, the operator $C=h_1^2+h_2^2+h_1{h}_2+3(h_1+h_2)+3{\sum }_{i=1}^3{f}_i{e}_i$ commutes with all generators. In its commutator with $e_1$, the Cartan polynomial contributes $3h_1{e}_1$, while the last sum contributes $-3h_1{e}_1$; the two mixed terms involving $f_2{e}_3$ cancel. On highest weight $(p,q)$ its eigenvalue is $p^2+q^2+pq+3p+3q$. This normalization is three times the conventional SU(3) quadratic Casimir eigenvalue.