Under the hypotheses of Jensen's formula, $f(0)\ne 0$ ensures that the radial zero-counting function vanishes for sufficiently small positive $t$. Integrating each zero's contribution separately givesThe values of $n_f$ at its finitely many jump points do not change the integral. For $0<r<R$, every zero in the smaller open disc supplies at least $\log (R/r)$ per unit of multiplicity of a zero, soThe second inequality bounds the circle log-mean of a holomorphic function by the logarithm of the largest boundary complex modulus, recovering the Jensen zero-count bound. An upper bound on boundary growth and a fixed nonzero centre value therefore constrain interior zeros. The factor $\log (R/r)$ also explains why the same estimate becomes weaker as the smaller radius approaches the boundary.

For constant adiabatic exponent $\gamma >4/3$ and a star whose central mass density is its maximum, the homologous radial fluid displacement ${\xi }_r=r$ makes the derivative term in the weighted stellar pulsation Rayleigh quotient vanish. The remaining quotient is $(3\gamma -4)$ times a positive weighted average of $g/r$. Since $m(r)\le 4\pi {\rho }_c{r}^3/3$, $g/r\le 4\pi G{\rho }_c/3$, proving the bound. For $\gamma <4/3$, the same trial quotient is negative and proves radial instability; the inequality with the central density must not be extended with its sign unchanged.

On $[0,1)$ with Lebesgue measure, the functions $r_n(t)=(-1{)}^{\lfloor 2^{n}t\rfloor }$, $n\ge 1$, are independent symmetric Rademacher random variables. Every prescribed pattern for the first $n$ functions occupies a dyadic interval of measure $2^{-n}$. Their closed span in $L^1$ is isomorphic to the l2 sequence space by the Khintchine inequality.

The canonical projection of a quotient set is the function $q(x)=[x]$. It is surjective: every set element of $X/\sim$ is $[x]$ for some $x\in X$. Two set elements have the same image exactly when they are related by the equivalence relation. Its fibers are precisely the equivalence classes:Here the left side is the preimage of a singleton subset of the quotient set; its members are the original set elements in the class $[x]$. In the diagram, the three arrows arriving at $[1]$ start at $1$, $4$ and $7$.

For $A_T=\mu A+3\hat{s}A|A{|}^2-10A|A{|}^4+4A_{XX}$, a nonzero uniform real state with $B=A_0^2$ obeys $\mu =-3\hat{s}B+10B^2$. Real amplitude sidebands grow at $6\hat{s}B-40B^2-4{\ell }^2$, while phase sidebands grow at $-4{\ell }^2$. The maximum amplitude growth coefficient is $9{\hat{s}}^2/40$, attained at $B=3\hat{s}/40$. Thus nonzero uniform states cannot have modulation bifurcations if the smallest allowed slow wavenumber exceeds $3\hat{s}/(4\sqrt{10})$. For a physical domain of length $L$ and $X={\epsilon }^{2}x$, this gives $L<8\pi \sqrt{10}/(3s)$. The zero state is an exception: it has sideband thresholds $\mu =4{\ell }^2$ for every positive allowed wavenumber, so an assertion including all uniform states would be false.

Build the cumulative universe over an infinite set $A$ of Quine atoms, and extend every permutation of $A$ recursively to its other sets. An object has finite support if every permutation fixing that finite subset of $A$ fixes the object. Retain objects whose members, recursively away from the atomic self-loops, also have finite support. Parameters and domains give finite supports for definable subsets and functional ranges; the collection of all retained subsets of a retained set is again supported. These observations verify separation, replacement and power set, and the pure natural-number hierarchy gives infinity. There is no choice function on the two-element subsets of $A$: swap two atoms outside its supposed finite support, fixing their unordered pair but interchanging either possible selected member.

Start with distinct formal atoms $q\in A$ with membership extensions $\{q\}$. Recursively adjoin a unique representative $S(X)$ of every subset $X$ of a previous stage, except that $S(\{q\})=q$ uses the atom already present. Define membership by these extensions. Each set-sized collection of objects has bounded construction rank, so it has a representative at a later stage. This gives full extensionality, unlike adjoining an ordinary singleton distinct from a Quine atom. Definable subsets and ranges, unions and power sets are set-sized collections in the ambient model and hence have representatives. The pure finite hierarchy supplies infinity. Every permutation of the atoms extends recursively; hereditary finite support restriction gives the hereditarily finite-supported Quine-atom model, whose two-element atom subsets have no choice function.

For the quaternionic tautological line bundle regarded as a complex rank-two bundle, take $q=c_2^{MU}$. Its ordinary image generates degree-four integral cohomology, and its square generates degree eight. The Atiyah-Hirzebruch spectral sequence collapses because the cells have dimensions $0,4,8$ and the coefficients are even. The classes $1,q,q^2$ lift a free basis of the associated graded module, and $q^3=0$ because the theory is connective and the space has dimension eight.

For $p\equiv 3(\mathrm{mod}4)$ and $z=x+iy\ne 0$ with rational $x,y$, divide $x^2+y^2$ by the square of the common minimum $p$-adic valuation. Its remaining unit cannot vanish modulo $p$, because $-1$ is not a square there. The norm therefore has even valuation. In particular a rational parameter $a$ with odd valuation at such a prime is not a norm, and $(-1,a{)}_{\mathbb{Q}}$ is a division quaternion algebra: its norm $N(z)-aN(w)$ cannot vanish for a nonzero element $z+wj$.

Near a collar edge adjoining a film of thickness $h_f$, local mass conservation gives $h^3{h}^{\prime \prime \prime }=c(h-h_f)$. The scaling $h=h_{f}H$, $x-x_f=h_{f}X/c^{1/3}$ yields the displayed equation. Its linearization at $H=1$ has exponents $1$ and $-1/2\pm i\sqrt{3}/2$. The positive growing rear branch approaching one as $X\to -\infty$ has one amplitude, removable by translation; the front's decaying oscillatory branches have two amplitudes, leaving one parameter after translation. This dimension count describes local branches; global positivity and selection require matching. If their core asymptotes, after removing linear terms by translation, are $H_{-}\sim a_{-}{X}^2+b_{-}$ and $H_{+}\sim a_{+}(\lambda )X^2+b_{+}(\lambda )$, matching to the collar gives $A/2=a_{\pm }{c}^{2/3}/h_{\pm }$ and $B=b_{\pm }{h}_{\pm }$. Consequently $a_{+}{b}_{+}=a_{-}{b}_{-}$; a selected root with $\alpha =a_{+}({\lambda }_0)>a_{-}$ yields $c=(A{h}_{+}/(2\alpha ){)}^{3/2}$, $B=h_{+}{b}_{+}({\lambda }_0)$ and $h_{-}/h_{+}=a_{-}/\alpha$. Local linearization alone does not establish uniqueness of this global matching root.

The primary roll and temperature modes force only a horizontally uniform temperature mode proportional to $\sin (2\pi z)$ at quadratic order. Its amplitude is $c=-\alpha ab/(8\pi )$. Its feedback on the primary temperature mode saturates magnetoconvection, giving the displayed positive squared amplitude above threshold. Thus a simple roll onset is a supercritical pitchfork bifurcation for every $Q\ge 0$. The three-mode Fourier truncation and full perturbation calculation agree through the cubic solvability condition because all forced quadratic modes are retained; this is not an assertion that the truncation gives an exact finite-amplitude PDE solution.

// Destination: dynamical-systems.bigb

With ${\mu }_i=e^{\beta x_i}$, define $U={\sum }_i{x}_i(Y_i-{\mu }_i)$ and $I={\sum }_i{x}_i^2{\mu }_i$. The estimating equation and a Taylor expansion give $\hat{\beta }-{\beta }_0\approx I({\beta }_0{)}^{-1}U({\beta }_0)$. If responses are independent with correct means and $\mathrm{Var}(Y_i)=\varphi {\mu }_i$, then $EU=0$ and $\mathrm{Var}U=\varphi I$. Consequently the first-order estimator variance is $\varphi /I$, even if the working Poisson distribution is wrong. Regular asymptotics require growing information and control of leverage and the Taylor remainder.

For a small steady lower-boundary height $b(x,y)$, impermeability gives $w={\mathbf{u}}_g\cdot \nabla b$ at the reference plane. Substitution in the buoyancy equation $D_g(f_0{\psi }_z)+N^{2}w=0$ yields the displayed boundary condition. For nonzero Fourier modes in a uniform crossing current, this fixes ${\psi }_z(0)=-N^{2}b/f_0$. An upper radiation condition or decay condition is still needed to select the physical response.

A stationary boundary corrugation can resonate with a Rossby wave whose intrinsic propagation cancels the background current. For constant $N$, rigid upper height $H$, and $m=\pi /H$, define ${\kappa }^2=k^2+{\ell }^2+f_0^2{m}^2/N^2$. When $U=\beta /{\kappa }^2$, a lower corrugation proportional to $e^{ikx}\sin (\ell y)$ admits a disturbance proportional to $e^{ikx}\sin (\ell y)[Bt\cos (mz)+C(z-H)\sin (mz)]$, with $B=2ik{f}_0{U}^2/\beta$ and $C=N^2/(f_{0}m)$. Its wave activity grows quadratically with time while the upward Eliassen–Palm flux grows linearly. The linear prediction applies only while the disturbance remains small.

Choose a finite cover of an affine scheme by principal opens on which the quasi-coherent sheaf is associated with a module. The sheaf gluing axiom expresses global sections as the kernel of the difference map from the finite product of chart sections to the finite product of overlap sections. Localization commutes with kernels and finite products. Localizing at $a$ therefore gives the same kernel for the cover intersected with $D(a)$, proving the formula. The principal-open formulas identify the sheaf with that associated with its global-section module; no finite-generation hypothesis on the module is needed.

In $0\to \mathcal{F}\to \mathcal{G}\to \mathcal{E}\to 0$, assume the outer sheaves are quasi-coherent. On an affine open, any section of $\mathcal{E}$ has local lifts to $\mathcal{G}$ on a finite principal cover. Their differences are a Čech cocycle in the module defining $\mathcal{F}$. Exactness of the unit-ideal localization Čech complex makes this cocycle a coboundary, so adjusted lifts glue. Thus sections are exact on the affine open and on every principal open in it. Comparing the resulting sequences after localization, the outer isomorphisms force the middle localization map to be an isomorphism. Hence $\mathcal{G}$ is associated with its global-section module and is quasi-coherent.

For an affine variety with coordinate ring $A$ and an $A$-module $M$, localization defines sections on principal open subsets by the displayed formula. Restrictions are the natural localization maps. Equivalently, a section is a family of stalk values which locally come from one fraction $m/f^r$. These data form a quasi-coherent sheaf, with global sections $M$ and stalk $M_{{\mathfrak{m}}_P}$ at $P$. Sheaf tensor products correspond to tensor products over $A$; pullback of a sheaf of modules corresponds to extension of scalars, and affine direct image to restriction of scalars.

On an affine scheme, a morphism between quasi-coherent sheaves is determined by the induced map on global sections, because its maps on principal open subsets are their localizations. Exactness of localization gives the displayed identity. Thus the kernel sheaf of a morphism between quasi-coherent sheaves is quasi-coherent on an arbitrary scheme, by applying the affine argument locally. No finite generation is needed.

For a negligible tensor contribution, the historical COBE horizon amplitude ${\delta }_H\sim 2\times 10^{-5}$ corresponds to curvature power ${\mathcal{P}}_{\mathcal{R}}\sim 25{\delta }_H^2/4$. In quartic hilltop inflation, $\alpha ={\pi }^2{\mathcal{P}}_{\mathcal{R}}/(8N^3)$ is of order $10^{-14}$ at $N=50$, independent of the small symmetry-breaking scale at this order. The exact inferred amplitude depends on the late cosmology and tilt used in the COBE transfer calculation.

// Target: cosmic-inflation.bigb

Using the approximate large-angle CMB relation $T/S\simeq -7n_T$ gives $T/S\simeq 7{\sigma }^4/(96M^4{N}^3)$ in quartic hilltop inflation. It is suppressed by the fourth power of the small-field scale. This large-angle temperature ratio is not the primordial tensor-to-scalar ratio $r=16{ϵ}_V$; their transfer functions and normalization differ.

// Target: cosmology.bigb