If $\iota$ is the least inaccessible limit of inaccessibles, $V_{\iota }$ models ZFC and internally has unboundedly many inaccessible cardinals, while none of its ordinals is an inaccessible limit of inaccessibles. Rank agreement makes inaccessibility below $\iota$ absolute. If a model of unboundedly many inaccessibles has no such limit in the first place, no cutoff is needed. This proves the corresponding relative-consistency nonimplication.

The norm-one idele group is the kernel of the idelic modulus. Its quotient by the diagonal $K^{\times }$ is compact for a number field and surjects onto the ideal class group.

For an axisymmetric vertical mode of a shearing sheet with a vertical magnetic field, radial buoyancy frequency $N_r$, and $A=k^2{v}_A^2>0$, the nonstationary amplitudes obey ${\sigma }^4+(2A+N_r^2+{\Omega }^2){\sigma }^2+A(A+N_r^2-3{\Omega }^2)=0$. The two roots for ${\sigma }^2$ are real. A growing mode exists exactly when $A+N_r^2<3{\Omega }^2$. If the advected buoyancy displacement variable is retained as an independent amplitude, the full characteristic polynomial also has a stationary factor $\sigma$.

In units $G=M=a=1$, velocity integration of $F=C{L}^{p-2}{E}^{(3p+1)/2}$ gives $C=(p+1)\Gamma (2p+2)/[2^{(p+5)/2}{\pi }^{5/2}\Gamma (p/2)\Gamma ((3p+3)/2)]$. The beta function separates the radial-speed and angular integrals. This constant is positive for every $p>0$.

For a non-null Killing vector field $V$ of a Levi-Civita connection, hypersurface orthogonality implies that $\eta =V^{♭}/F$, with $F=g(V,V)$, is a closed differential form. The Killing equation and $V^{♭}\wedge d{V}^{♭}=0$ give $2F{\nabla }_{\mu }{V}_{\nu }=V_{\nu }{\partial }_{\mu }F-V_{\mu }{\partial }_{\nu }F$, whence $d\eta =0$. The Poincaré lemma gives $\eta =d\varphi$ locally. For a timelike $V$, $\varphi$ supplies local static time coordinates. This does not assert global exactness.

The hyperbolic-sine infinite product gives, by taking its logarithmic derivative,The identity holds away from the poles, with locally convergent sums. For real $z>0$, it makes $(z\coth z-1)/z^2$ positive and strictly decreasing. It is useful for Hartmann flow flux monotonicity and small-argument expansions of hyperbolic functions.

Suppose an emitted spectral luminosity is $L_{\nu }=A{\nu }^{-s}$ for every $\nu \ge {\nu }_0$, where $A>0$ and $h{\nu }_0$ is an ionization threshold. The total emitted ionizing luminosity and photon rate use different weights:Thus a finite emitted mean photon energy for this unbounded power law requires the stronger condition $s>1$ and equalsFor $s=2$, the mean is twice the threshold energy. As $s\downarrow 1$, the high-frequency tail makes the mean diverge even though the photon count stays finite. For $0<s\le 1$, substituting into the finite-mean formula would therefore be invalid; for $s\le 0$ even the unbounded photon-rate integral diverges.

A finite high-frequency cutoff changes both integrals and removes these upper-tail divergences. The mean energy deposited in a nebula also need not equal the emitted mean: absorption weights the spectrum with its photoionization cross section, and energy above threshold can be distributed among heat, excitation and secondary ionizations. The excess $h\nu -h{\nu }_0$, rather than the whole photon energy, is the initial kinetic energy of the photoelectron in the simplest single-ionization event.

At fixed disk radius, balancing local heating and cooling gives thermal equilibria parameterized by surface density and temperature. In the hydrogen-ionization disk instability, this locus has cold and hot stable branches joined by unstable equilibria. Crossing the turning points drives the state changes of a dwarf nova. Thermal stability of an accretion disk tests the local heating and cooling response, while viscous evolution of an accretion disk changes the surface density.

In a frame of a vector bundle, write a vector bundle covariant derivative as $\nabla (ev)=e(dv+\Gamma v)$. The associated horizontal subspace at $(x,v)$ consists of tangent pairs $(\xi ,-{\Gamma }_x(\xi )v)$. Under a frame change $e^{\prime }=eg$, the connection one-form changes by ${\Gamma }^{\prime }=g^{-1}\Gamma g+g^{-1}dg$, and this horizontal equation changes to the corresponding one in the new frame. Thus the subspaces glue to a unique smooth linear connection on a vector bundle. Its vertical projection gives back the original derivative. The Leibniz rule ensures locality, allowing this construction from an operator initially defined on global sections.

On the holomorphic projectivization by lines, take the line itself as the fibre of $S$. The evaluation inclusion makes it a holomorphic line subbundle of $p^{\ast }E$. Its fibre restriction is $\mathcal{O}(-1)$; the dual is the relative hyperplane line bundle.

A state's holding time in a time-homogeneous continuous-time Markov chain is exponentially distributed with rate $-q_{rr}>0$. Its mean is $-1/q_{rr}$; the probability that its next jump is to state $s$ is $q_{rs}/(-q_{rr})$. An absorbing state has infinite holding time.

Localized wavelets of support diameter $O(2^{-j})$ and $L^1$ norm $O(2^{-j/2})$, with enough vanishing moments, satisfySubtract a Taylor polynomial annihilated by the wavelet, apply the Hölder-Taylor remainder bound on its support of a function, and multiply by its $L^1$ norm. The same argument applies to localized boundary wavelets with the same cancellation; finitely many coarse scaling functions are handled separately.

If $f_j\to f$ uniformly and the Hölder seminorms are uniformly bounded, each fixed difference quotient converges. Taking the supremum over point pairs gives $[f{]}_{\mu }\le {\mathrm{liminf}}_j[f_j{]}_{\mu }$. The supremum norms converge, so the full Hölder norm has the same lower semicontinuity of a Hölder seminorm.

For fixed $q>1$, the coefficient in the Hölder factorization of stochastic exponentials exceeds $\sqrt{q}/(\sqrt{q}+1)>1/2$. Taking $q=1+\epsilon$ and $p=1+{\epsilon }^2$ approaches one-half. Thus a uniform stopped exponential moment at coefficient one-half bounds some $p$th moment of each strict scaling $\mathcal{E}(aM)$, $0<a<1$.

For $f\in L^2(0,1)$ and $0<\alpha <1/2$, the integral $g(x)={\int }_0^{1}f(y)|x-y{|}^{-\alpha }dy$ is well defined for every real $x$. Scaling the squared kernel difference over the real line and using the Cauchy-Schwarz inequality give $|g(x+h)-g(x)|\le C_{\alpha }\Vert f{\Vert }_2|h{|}^{1/2-\alpha }$.

If a real function on an interval obeys $|f(x)-f(y)|\le C|x-y{|}^{\alpha }$ for finite $C$ and $\alpha >1$, its difference quotients tend to zero everywhere. The mean value theorem makes it constant. Equivalently, divide a segment into $N$ pieces to obtain $|f(x)-f(y)|\le C|x-y{|}^{\alpha }{N}^{1-\alpha }\to 0$.

For a homogeneous element $x$ of degree $e>0$ acting on a finite graded module, set $N=(0{:}_{M}x)$ and $C=M/xM$. The multiplication exact sequence $0\to N(-e)\to M(-e)\to M\to C\to 0$ and additivity of component lengths give $(1-t^e)P_M=P_C-t^e{P}_N$. This is the inductive step in the Hilbert-Serre theorem; the annihilator term must be kept unless multiplication by $x$ is injective.

For a Noetherian local ring $(R,\mathfrak{m})$, let $H_R(t)=\mathrm{length}(R/{\mathfrak{m}}^t)$. Its eventual Hilbert-Samuel polynomial has degree $d(R)$, equivalently the pole order at $1$ of the Hilbert series of ${\mathrm{gr}}_{\mathfrak{m}}R$. This is the local length-growth invariant, not the embedding dimension. The prime-chain lower bound for local length proves $\dim R\le d(R)$; the full local dimension theorem gives equality.

A group presentation with a cyclic list of generators $a_i$ and relations $a_i{a}_{i+1}{a}_i^{-1}=a_{i+1}^2$ has no nontrivial finite quotient of a group. In a finite image, all generator orders are odd. Choose the least prime $p$ dividing any nontrivial generator order and let $x$ be the predecessor of a generator $y$ whose order is divisible by $p$. Conjugation by $x$ acts as squaring on $⟨y⟩$, so the multiplicative order of $2$ modulo $p$ divides the order of $x$. That multiplicative order is greater than one and divides $p-1$, giving a prime divisor smaller than $p$ in the order of $x$, a contradiction. This argument applies to any cycle length; it asserts absence of finite quotients, not infinitude of the presented group.

The A2 root system representation $R(2,0)={\mathrm{Sym}}^2{\mathbb{C}}^3$ has dimension six and weights $(2,0),(0,1),(1,-1),(-2,2),(-1,0),(0,-2)$ in Dynkin label coordinates, each with multiplicity one.