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For $s>0$ in the finite moment-generating function domain and integrable $X$, Jensen inequality gives the lower bound and $X\le \mathrm{esssup}X$ almost surely gives the upper bound. The Holder inequality makes $K(s)=\log \mathbb{E}{e}^{sX}$ convex with $K(0)=0$. Thus $K(t)\le (t/s)K(s)$ for $0<t<s$, and the effective bandwidth is nondecreasing. It need not be strictly increasing: deterministic traffic gives a constant bandwidth. Infinite peaks or exponential moments require extended-value conventions.

For a nondecreasing process $A$ with stationary increments and independent increments, suppose $\mathbb{E}{e}^{\theta A(t)}=e^{t\Lambda (\theta )}$. If $\Lambda (\theta )\le C\theta$, then $e^{\theta (A(t)-Ct)}$ is a nonnegative supermartingale. Applying the maximal inequality for a nonnegative supermartingale up to time $T$ and then letting $T\to \infty$ givesThe stationary workload of a stable queue fed by $A$ has this supremum distribution, using the past input and stationary increments and independent increments. For independent flows, the constraint is ${\sum }_j{a}_j(\theta )\le C$, where $a_j(\theta )={\Lambda }_j(\theta )/\theta$ is the long-time effective bandwidth. The result bounds a continuous-time supremum directly; a one-time Chernoff bound by itself does not do that.

Let a slotted queue have stationary input, constant service $c$, and finite stationary queue workload $Q={\sup }_{n\ge 0}(A_n-cn)$. Write $K_n(\theta )=\log \mathbb{E}{e}^{\theta A_n}$ for its past cumulative arrivals. The union bound and Chernoff bound give the displayed inequality for $B>0$. It requires no temporal independence. If each $K_n(\theta )$ is finite and $K_n(\theta )/n\to \kappa (\theta )<c\theta$, the series is finite: its tail is bounded by a geometric series. Consequently ${\mathrm{limsup}}_{L\to \infty }{L}^{-1}\log \mathbb{P}(Q>Lb)\le -\theta b$ for $b>0$. The condition is that the asymptotic effective bandwidth $\kappa (\theta )/\theta$ be less than service capacity.

Independent slots carry either zero work or $r>0$ units, the latter with probability $p$. Their effective bandwidth is the displayed expression, with small-parameter limit $pr$ and large-parameter limit $r$ when $0<p<1$. For $pr<c<r$, the positive root of $\log (1-p+p{e}^{\theta r})=c\theta$ determines the optimized exponential overflow bound. In the special case $r=2,c=1,p<1/2$, the stationary queue workload is geometric with ratio $p/(1-p)$, and the root is ${\theta }_{\ast }=\log ((1-p)/p)$, giving exactly $\mathbb{P}(Q\ge k)=e^{-{\theta }_{\ast }k}$ for integers $k\ge 0$.

For independent Poisson arrival counts of mean $\nu$ per slot, with one unit of work per arrival, the cumulant-generating function is $\nu (e^{\theta }-1)$. Its effective bandwidth is the displayed expression. If $c>\nu$, the equation $\nu (e^{\theta }-1)=c\theta$ has a unique positive root: strict convexity, negative initial derivative after subtracting $c\theta$, and divergence to infinity prove existence and uniqueness. Independent Poisson streams combine by replacing $\nu$ with the sum of their intensities.

Let $X$ count $k$-cliques in a binomial random graph on $m$ vertices and $Y$ count distinct ordered pairs sharing an edge. The clique conflict graph gives $Z_k\ge X^2/(X+Y)$: order its vertices randomly and keep each one preceding all its neighbours, then apply the Cauchy-Schwarz inequality. Another application gives $\mathbb{E}{Z}_k\ge (\mathbb{E}X{)}^2/(\mathbb{E}X+\mathbb{E}Y)$. If $\mathbb{E}Y/(\mathbb{E}X{)}^2=O(k^4/m^2)$ and $1/\mathbb{E}X$ is no larger, then $\mathbb{E}{Z}_k=\Omega (m^2/k^4)$. Changing one edge changes $Z_k$ by at most one, so the Azuma-Hoeffding inequality for an edge-exposure martingale yields the displayed exponential absence bound. This supports a union bound over exponentially many adaptive candidate subsets.

For the upper plane-wave branch of the quintic real Ginzburg-Landau equation, put $\Lambda =2R^2(2R^2-\alpha )>0$. A modulation of wavenumber $p$ has growth rates $-p^2-\Lambda /2\pm \sqrt{{\Lambda }^2/4+4Q^2{p}^2}$. Long-wave phase diffusion is positive when $4Q^2<\Lambda$, giving the displayed stability boundary. At equality the leading phase growth is $-p^4/\Lambda$, so the boundary is marginal in the diffusion coefficient but damped at the next spatial order. In the $(\mu ,Q)$ plane the boundary is $\mu =2Q^2-\alpha [\alpha +\sqrt{{\alpha }^2+16Q^2}]/8$.

For a real Ginzburg–Landau equation plane wave with amplitude squared $b/g>0$, perturbing its amplitude and phase couples Fourier modes on either side of its carrier wavenumber. The displayed eigenvalues follow from the real-linear two-component perturbation equation. Their small-$p$ phase branch is $-\xi (\mu -3\xi q^2)p^2/(\mu -\xi q^2)+O(p^4)$, so the robust infinite-domain stable band is $q^2<\mu /(3\xi )$. On a finite periodic envelope domain only discrete $p$ are allowed; an instability requires an allowed $p$ with $p^2<6q^2-2\mu /\xi$. Uniform amplitude perturbations alone cannot determine this Eckhaus instability.

With uniform thermal wind $U=\Lambda z$ above a plane boundary, write the zero-interior-potential vorticity response as ${\psi }^{\prime }=\mathrm{Re}\{A{e}^{-\mu z+i(kx+ly)}\}$, where $\mu =N\sqrt{k^2+l^2}/f$ and $f>0$. Boundary vertical velocity $ϵ\cos (kx+ly-{\omega }_{0}t)$ gives $C=N^2ϵ/(f\mu )$ and natural angular frequency $\omega =\Lambda k/\mu$. An initially zero response has $A=C(e^{-i{\omega }_{0}t}-e^{-i\omega t})/[i(\omega -{\omega }_0)]$. Its continuous resonant limit is $Ct{e}^{-i\omega t}$: forcing in phase with the boundary mode supplies energy every cycle, and linear undamped theory predicts secular growth.

Start at one and apply $n$ updates, each multiplying the current value by $1+1/n$. The final value is $(1+1/n{)}^n$: one update gives $2$, two give $2.25$, ten give approximately $2.59374$, and one hundred give approximately $2.70481$. Increasing the number of steps, while reducing each increment proportionally, approaches Euler's number rather than producing unbounded accumulation.

The natural logarithm makes the limit transparent. For $u>0$, its integral definition gives $u/(1+u)\le \ln (1+u)\le u$, by bounding $1/t$ on $[1,1+u]$. With $u=1/n$,The squeeze theorem gives a logarithmic limit of one; continuity of the exponential function gives $(1+1/n{)}^n\to e$. This links a discrete multiplicative rule with continuous exponential growth, without assuming that a finite number of steps equals the limiting value.

The linear incompressible flow $\mathbf{u}=(-\omega x,-\omega y,2\omega z)$ compresses the transverse plane and stretches the axial direction. An axial magnetic field obeys $B_t-\omega r{B}_r=2\omega B+\eta {\Delta }_{\perp }B$. In ideal magnetohydrodynamics, its amplitude grows as $e^{2\omega t}$ while its transverse width decreases as $e^{-\omega t}$. Thus its magnetic energy per unit axial length grows as $e^{2\omega t}$, even when opposite-polarity angular sectors give zero signed magnetic flux. Finite magnetic diffusivity limits the amplification of localized finite-flux Gaussian fields.

An anti-dynamo theorem gives a geometric or analytic restriction excluding self-sustained dynamo action under stated magnetic diffusivity, domain and boundary conditions. It excludes regeneration against resistive decay, not all transient amplification or field induced by an external source. Important restrictions include Cowling anti-dynamo theorem, the toroidal-velocity anti-dynamo theorem, and the planar anti-dynamo theorem. Backus' necessary condition for dynamo action instead bounds the stretching required for possible regeneration.

An axisymmetric magnetic mode is unchanged by axial rotations, while an equatorial magnetic mode and a corresponding velocity mode transform as $A,V\mapsto e^{i\theta }(A,V)$. Magnetic reversal sends $(B,A,V)$ to $(-B,-A,V)$. For magnetic evolution linear in magnetic amplitudes, rotational equivariance requires $\dot{B}=Bf(|V{|}^2)+2\mathrm{Re}[A^{\ast }Vg(|V{|}^2)]$ and $\dot{A}=Ah(|V{|}^2)+BVj(|V{|}^2)+A^{\ast }{V}^2\ell (|V{|}^2)$, with $f$ real and the other functions complex. Velocity monomials $B^p{A}^q(A^{\ast }{)}^r{V}^s(V^{\ast }{)}^t$ must have even $p+q+r$ and weight $q-r+s-t=1$. These conditions enumerate the cubic normal form without assuming reflection symmetry or nonlinear magnetic self-interactions.

Dynamo quenching is the reduction or modification of magnetic-field generation by magnetic feedback on the generating flow and its correlations. The Lorentz force can change shear or turbulence, while magnetic helicity affects mean-field regeneration. A phenomenological scalar example is $\alpha (B)={\alpha }_0/(1+B^2/B_{\ast }^2)$. In the illustrative amplitude equation $\dot{B}=[c\alpha (B)-d]B$ with $c,d>0$, a growing field saturates at $B^2=B_{\ast }^2(c{\alpha }_0/d-1)$ when $c{\alpha }_0>d$. This demonstrates the negative feedback mechanism without claiming that algebraic quenching is a complete solar closure.

Let a solenoidal flow be confined to a sphere $V$, with the insulating boundary condition for the radial magnetic scalar. Define $N={\int }_V{P}^2$, $D={\int }_{{\mathbb{R}}^3}|\nabla P{|}^2$, $M={\int }_{{\mathbb{R}}^3}|\mathbf{B}{|}^2$ and $q={\max }_V|\mathbf{u}\cdot \mathbf{x}|$. Integration by parts inside and outside the sphere givesThe Cauchy-Schwarz inequality therefore makes $q^2\ge {\eta }^{2}D/M$ necessary whenever $N^{\prime }\ge 0$ and $D>0$. A uniform strict violation with a positive gap forces decay, using a Sobolev inequality to bound $N$ by a constant times $D$. A bounded statistically steady field instead requires $q_{\ast }^2\ge {\eta }^2⟨D⟩/⟨M⟩$, with $q_{\ast }={\sup }_{t}q(t)$. This is a necessary condition for dynamo action, not a sufficient criterion or a pointwise assertion at every instant of an arbitrary time-dependent solution.

// Target: analysis.bigb

For a smooth heat equation solution $w_t=\Delta w$ with $w_t+\alpha {\partial }_{n}w=0$ on the boundary and nonnegative $\alpha$, Green's first identity gives the displayed dissipation identity for $E={\int }_V|\nabla w{|}^2$. The outward normal derivative is used. Writing the boundary loss as $\alpha ({\partial }_{n}w{)}^2$ avoids division by $\alpha$ and remains valid where $\alpha =0$. A stationary solution is harmonic with $\alpha {\partial }_{n}w=0$; it need not be a particular fixed-data Dirichlet problem solution.

Assume $a=c/r$ decreases outward, tends to zero at $R$, and define $h(a)=-d\ln r/da$. Differentiation of the Duvall law gives $F^{\prime }(w)={\int }_0^{w}h(b)b/[w^2\sqrt{w^2-b^2}]db$. Interchanging integrals and using ${\int }_b^{a}dw/[w\sqrt{w^2-b^2}\sqrt{a^2-w^2}]=\pi /(2ab)$ proves the displayed Abel transform inversion. For a finite outer reference value $w_s$, both lower limits become $w_s$, after removing the known outer-layer contribution to the travel time. Inversion requires a monotone branch and amplifies errors in differentiated frequency data.

For saturated Darcy flow through a long dam occupying a channel of width $\alpha z^2$, let hydraulic conductivity be $K=k\rho g/\mu$ and free-surface height $h(x)$. The Dupuit approximation gives $Q=-K\alpha h^3{h}_x/3$. Fixed heads $h(0)=H$ and $h(L)=\eta$ give $h^4=H^4-(H^4-{\eta }^4)x/L$ and the displayed steady discharge. Porosity relates pore velocity to superficial Darcy velocity but does not independently multiply this discharge when $k$ is intrinsic permeability. For depth-dependent permeability, integrate the horizontal conductivity over the saturated cross-section first.

For $n\ge 1$, any distinct node set in $[-1,1]$ different from the $n+1$ Chebyshev polynomial extrema admits one degree-$n$ polynomial $q$, bounded by one at all those nodes, whose every nonconstant derivative violates the corresponding sharp endpoint bound. In decreasing order set $q(t_i)=(-1{)}^i$. The endpoint derivative signs of Lagrange cardinal polynomials give $q^{(k)}(1)={\sum }_i|L_i^{(k)}(1)|$. Interpolating $T_n$ on the same nodes showsEach term is nonnegative and at least one is positive, because the other node set includes a nonextremal point. Uniqueness is a claim about node sets: permuting the same nodes does not change their norming property.

Start with cash one, switch to $X/a$ when a nonnegative supermartingale first reaches at most $a$, and switch to cash $q=b/a$ when it next reaches at least $b$. Repeat with coefficients multiplied by $q$ after every completed upcrossing. At each switch the new value is at most the old one, so pasting supermartingales with downward jumps gives a supermartingale when stopped after any fixed number of switches. Its initial value is $\min (X_0/a,1)$, and its value at the $k$th completed crossing is exactly $q^k$. Applying its expected value bound proves the Dubins upcrossing inequality.