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At a point of a Riemannian manifold, Ricci curvature is isotropic when $\mathrm{Ric}(v,v)/g(v,v)$ is independent of nonzero tangent direction. Polarization makes this equivalent to the displayed tensor identity. In dimension three, sectional curvatures from Ricci curvature in dimension three gives every sectional curvature at that point the value $c_p/2$. In higher dimensions isotropic Ricci curvature does not determine every two-plane curvature.

For a Riemannian manifold of dimension $n\ge 2$, the normalized directional Ricci curvature averages, rather than sums, the $n-1$ sectional curvatures through a unit vector. Thus the Bonnet-Myers theorem gives $\mathrm{diam}M\le \pi /\sqrt{k}$ from $\overline{\mathrm{Ric}}\ge kg$. With trace Ricci curvature the corresponding hypothesis is $\mathrm{Ric}\ge (n-1)kg$. The distinction changes positive constants but not the signs of Ricci curvature.

For a complete connected $n$-dimensional Riemannian manifold with nonnegative Ricci curvature, the displayed ratio is nonincreasing, tends to one at zero, and is at most one. In particular $\mathrm{Vol}B(p,r)\le {\omega }_n{r}^n$. In geodesic polar coordinates, the radial Jacobi field determinant is at most the Euclidean density $r^{n-1}$ until the cut time. The trace of the radial Riccati equation for distance spheres, with Cauchy-Schwarz inequality, proves this comparison. Equality for every radius forces the Euclidean polar metric and absence of a cut locus.

For a compact H-hull $A$ whose closure avoids a real boundary point $x$, the Brownian excursion in the upper half-plane from $x$ to infinity avoids $A$ with the displayed probability. From an interior point $z$, the corresponding Doob h-transform avoidance probability is $\mathrm{Im}{g}_A(z)/\mathrm{Im}z$. Letting $z=x+i\epsilon$ approach the boundary and using the Schwarz reflection principle gives the derivative. In particular $0<g_A^{\prime }(x)\le 1$ at every regular real boundary point away from the hull.

If every nonconstant J-holomorphic sphere under consideration is regular, no such sphere in a four-dimensional target has class $A$ with $A^2\le -2$. For a simple J-holomorphic sphere, the adjunction inequality for a simple J-holomorphic sphere makes the predicted moduli dimension negative. A degree-$m$ multiple cover of class $B$ has $B^2\le -1$ and $c_1(B)\le 1$, so its predicted real dimension is at most $2m-2$. The rational covering maps of the projective line already form a family of dimension $4m-4>2m-2$, contradicting regularity. The full regularity hypothesis matters: multiple covers of exceptional minus-one spheres violate the conclusion under a simple-only regularity assumption.

For a residual set of smooth compatible almost complex structures, all simple J-holomorphic spheres in a specified countable set of spherical classes are regular. The corresponding evaluation maps at marked points are transverse to generic incidence conditions; generic paths give parameterized transversality. The universal linearization can vary the target structure near an injective point of the curve, which removes its cokernel. This assertion concerns simple curves and does not ensure regularity of arbitrary multiple covers.

A surrogate split uses another predictor to approximate the left-versus-right assignments made by a tree node's primary split. It can route observations whose primary predictor is missing. Surrogates are ranked by agreement with the primary split on observations where both predictors are known, accounting for trivial majority assignments. This differs from simply filling a missing predictor with its sample mean.

Deleting an edge of a locally finite recurrent graph leaves recurrent Markov chain connected components. On an infinite component choose a root away from the deleted edge's endpoints, so its vertex degree does not change. The Rayleigh monotonicity principle increases resistance to wired outer boundaries, and effective resistance to infinity therefore remains infinite. Finite components are recurrent Markov chain too. This is a recurrence-type comparison; at an endpoint, changes in the vertex degree factor prevent a blanket monotonicity claim about first-return probabilities.

Let $X$ be an irreducible affine curve, $A=k[X]$, $K=\mathrm{Frac}A$, and $\mathcal{F}=\tilde{M}$ be locally free. Given finitely many classes in $M_K/M_{{\mathfrak{m}}_P}$, choose one nonzero $a\in A$ clearing denominators of representatives. The ring $A/(a)$ is Artinian and is the product of its finitely many local factors. Hence $M/aM\cong {⨁}_{P\in V(a)}(M/aM{)}_{{\mathfrak{m}}_P}$. Prescribe $a{v}_P$ modulo $a{M}_{{\mathfrak{m}}_P}$ at the requested points and zero at the other zeros of $a$. Lift this tuple to $w\in M$; the section $w/a$ has exactly the requested rational principal parts and no others. This proves surjectivity without assuming the curve is nonsingular.

Two rank-$n$ abstract rational normal scrolls are isomorphic exactly when their splitting degrees agree as multisets after a common integer shift. For $n\ge 3$, the primitive nef class $L$ of the ruling is intrinsically identified by $L^2=0$: restriction to a projective fibre forces every other divisor class with square zero to have zero coefficient of $M$. An isomorphism therefore preserves the ruling, and its effect on $M$ is a twist by $cL$. Pushforward gives an isomorphism of the defining bundles up to this twist; uniqueness in the Birkhoff–Grothendieck theorem recovers their splitting degrees. For surfaces the unique nef square-zero ray identifies the ruling except on ${\mathbb{P}}^1\times {\mathbb{P}}^1$, where exchanging the two rulings changes no normalized splitting type.

The divisor class of a fibre of the ruling of a rational normal scroll has square zero. Its complete pencil recovers the projection to ${\mathbb{P}}^1$, and its intersection with $M^{n-1}$ is one.

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.