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A weak large deviation principle has the usual lower bound on every open set, but requires the upper bound only on compact sets. It does not by itself control probabilities escaping every compact set. Exponential tightness provides that missing control.

Let $A_i$ be independent random variables with normal distribution $N(\mu ,{\sigma }^2)$, ${\sigma }^2>0$, and let $B$ be an independent exponential random variable of rate $\lambda >0$. At speed $L$, $(B+{\sum }_{i=1}^L{A}_i)/L$ has the good rate functionThe product large-deviation principle gives the joint cost $\lambda b+(z-\mu {)}^2/(2{\sigma }^2)$ for $b\ge 0$, and the contraction principle for large deviations gives its infimal convolution under $b+z=x$. The minimum occurs at $b=(x-\mu -\lambda {\sigma }^2{)}^{+}$. The fixed exponential perturbation changes the far upper tail even though it vanishes in probability after division by $L$; a fixed Gaussian random variable divided by $L$ instead has exponential equivalence with zero at speed $L$.

Let $X$ have the positive-support geometric distribution $\mathbb{P}(X=k)=p(1-p{)}^{k-1}$ with $0<p<1$. As $L\to \infty$, $X/L$ satisfies a large deviation principle of speed $L$ and good rate functionFor every real threshold $x$, the exact upper-tail formula isThe maximum with zero matters at $x=0$ and at thresholds below the first lattice point; the uncapped exponent would otherwise give a Probability greater than one.

For positive $x$, dividing the logarithm by $L$ gives the limit $-ax$. These tails bound closed sets through their leftmost nonnegative point. For an open neighborhood of a positive $x$, choose integers $k_L$ with $k_L/L\to x$; their individual masses have the same logarithmic limit. Neighborhoods of zero have Probability tending to one, supplying rate zero there. Negative-only sets have Probability zero, and the rate sublevel sets are compact intervals.

For example, with $p=1/2$, $L=100$ and $x=1/5$, the event is $X\ge 20$ and has Probability $2^{-19}$. Meanwhile $X/L$ converges in Probability to zero. The nontrivial rate describes the rare excursions away from that degenerate weak limit.

For a fixed finite number of nonnegative terms and $a_L\to \infty$, use ${\max }_j{p}_{j,L}\le {\sum }_j{p}_{j,L}\le r{\max }_j{p}_{j,L}$. The logarithm of $r$, divided by the large-deviation speed, tends to zero, and a finite maximum commutes with the limit superior. If all individual logarithmic limits exist, the limit of the sum is their maximum.

Let $X_i$ be independent and identically distributed with mean $\mu$, variance ${\sigma }^2\in (0,\infty )$, and a moment-generating function finite on an open interval about zero. Choose $b_n>0$ with $b_n\to \infty$ and $b_n/\sqrt{n}\to 0$. The moderate deviation principle foris a large deviation principle of speed $b_n^2$ and good rate function $I(x)=x^2/(2{\sigma }^2)$.

To see the Gaussian quadratic rate, center one summand and expand its logarithmic moment-generating function at zero. For every fixed real $\theta$,The limiting function is finite and differentiable everywhere, so the Gärtner–Ellis theorem applies; its Legendre-Fenchel transform is the displayed rate. This proof states an actual sufficient exponential-moment hypothesis rather than leaving the assumptions implicit.

For $b_n=n^{1/4}$, a fixed positive $Z_n=x$ corresponds to a sample-mean displacement of order $n^{-1/4}$, between the ordinary $n^{-1/2}$ central-limit scale and a fixed order-one displacement. Its tail has logarithmic Probability of order $-\sqrt{n}{x}^2/(2{\sigma }^2)$. The scaling is intermediate, but the rate is still quadratic under these hypotheses.

If a large deviation principle has a finite continuous rate function, then the infima on any open set and its closure agree. Approximate every closure point by points of the set and use continuity. The open-set lower and closed-set upper bounds consequently match, giving the displayed logarithmic limit, including the empty-set convention. A single-point event need not obey such an equality.

A cardinal $\kappa$ is supercompact if for every cardinal $\lambda \ge \kappa$ there is an elementary embedding $j:V\to M$ into a transitive class with critical point of an elementary embedding $\kappa$, $j(\kappa )>\lambda$, and $M^{\lambda }\subseteq M$. This closure supplies every sufficiently small rank segment needed to transfer existential witnesses into $M$.

A scalar $\lambda$ for which a nonzero Laplacian eigenfunction satisfies this equation, with specified boundary conditions or on a closed manifold, is an eigenvalue of the positive Laplacian. A Dirichlet Laplacian eigenvalue imposes zero boundary values; other boundary conditions generally change the spectrum.

The imaginary part of the entire function $e^{z^2}$ is a harmonic function. On the upper half-plane it vanishes at $y=0$ and tends to zero as $y\to \infty$ for every fixed $x$, but is nonzero and grows rapidly in the horizontal direction. Adding it to a Poisson integral preserves those boundary and vertical-decay conditions. A boundedness or suitable transform-growth condition is therefore needed to make the half-plane Dirichlet problem unique.

For a single-valued harmonic function tending to zero both at the origin and at infinity, match an angular Fourier series across the unit circle. A nonzero mode with continuous interface value $K_n\cos (n\theta )$ has interior field $K_n{r}^n\cos (n\theta )$ and exterior field $K_n{r}^{-n}\cos (n\theta )$. Its exterior-minus-interior radial derivative is $-2n{K}_n\cos (n\theta )$. Thus a prescribed jump coefficient $j_n$ determines $K_n=-j_n/(2n)$. Sine modes obey the same relation. The constant mode must have zero jump under these decay conditions.

For the Laplace equation on $0<y<1$, prescribe bottom data $g$ and zero top data. A horizontal Fourier transform reduces the equation to ${\hat{u}}_{yy}-s^2\hat{u}=0$, giving the displayed multiplier and its limit $1-y$ at $s=0$. Integrable transformed data give a bounded inverse-transform solution. The bounded solution is unique by the maximum principle for harmonic functions, using barriers $\epsilon \cosh (\delta x)\cos (\delta (y-1/2))$ for $0<\delta <\pi$ on expanding rectangles. Without a growth condition, one can add $e^{\pi x}\sin (\pi y)$, so the two boundary traces alone do not determine an arbitrary harmonic function on an infinite strip.

For a regular interior and a decaying exterior harmonic potential with continuous values on the unit sphere, the same coefficient multiplies the two radial branches of each Legendre polynomial mode. Its exterior-minus-interior radial derivative is $-(2n+1)a_n{P}_n$. Thus an axisymmetric prescribed jump $j_n{P}_n(\cos \theta )$ produces coefficient $a_n=-j_n/(2n+1)$. This diagonal matching rule reduces a spherical interface problem to its angular-mode coefficients.

For $0<x<L$, $0<y<H$, choose $k=m\pi /H>0$ and seek a harmonic function with zero values on the horizontal edges, boundary value $\sin (ky)$ at $x=L$, and $\varphi +\alpha {\partial }_n\varphi =0$ at $x=0$, with constant $\alpha \ge 0$. Separation of variables gives $\varphi =[\alpha k\cosh (kx)+\sinh (kx)]\sin (ky)/[\alpha k\cosh (kL)+\sinh (kL)]$. The outward derivative at the left edge is $-{\partial }_x$, giving $F(0)-\alpha F^{\prime }(0)=0$. The denominator is positive and Poisson uniqueness with a nonnegative Robin normal coefficient proves uniqueness in the regular solution class.

A single-valued harmonic function in a planar annulus has radial Fourier modes $1,\log r$ and $r^n,r^{-n}$ multiplying $\cos (n\theta ),\sin (n\theta )$ for integers $n\ge 1$. Substitute $R(r)\Theta (\theta )$ in Laplace's equation: periodicity gives ${\Theta }^{\prime \prime }+n^2\Theta =0$, while $r^2{R}^{\prime \prime }+r{R}^{\prime }-n^{2}R=0$. Regularity at the origin excludes $\log r,r^{-n}$; boundedness at infinity excludes $\log r,r^n$. Coefficients must give locally convergent series, with the boundary regularity required by the particular Dirichlet problem. Mere finiteness at each finite exterior point does not imply boundedness at infinity.

A Generalized Lane-Emden equation is a semilinear partial differential equation whose highest-order term is the Laplacian:For a radial function in $d$ dimensions it becomes $u^{\prime \prime }+(d-1)u^{\prime }/r+u^p=0$. The case $d=3$ gives the classical Lane-Emden equation.

The spinodal is the boundary where a metastable phase loses local stability. In a mean-field free-energy description the relevant curvature vanishes there, separating local minima from unstable configurations.

Fix the spatial mean of a scalar order parameter to $M$ and sum the Boltzmann weights of all configurations with that mean. The resulting constrained partition function $Z_M$ defines this free-energy density, with an additive normalization fixed consistently. For an applied conjugate field $h$, $Z(h)=\int dM{e}^{-{\beta }_{T}V[F_c(M)-hM]}$, up to the chosen measure normalization. The thermodynamic limit gives $f(h)={\inf }_M[F_c(M)-hM]$. A homogeneous Landau free energy is an approximation to this constrained potential; phase-separated configurations can convexify it in the full thermodynamic limit.

An isotropic linear elasticity material with equal positive Lamé parameters has Poisson's ratio $1/4$ and the displayed compressional/shear speed ratio. This simplifying model fixes the relative amplitudes in the tensile-crack radiation pattern; it does not describe every stable isotropic solid.

Application $MN$ applies the lambda term $M$ to the lambda term $N$. It associates to the left: $MNP=(MN)P$. Together with variables and lambda abstraction, it generates all lambda terms.

A pointed complete partial order $D$ is reflexive when there are Scott continuous maps $e:[D\to D]\to D$ and $p:D\to [D\to D]$ with $p\circ e=\mathrm{id}$. Define $a\cdot b=p(a)(b)$ and interpret lambda abstraction using $e$. The inverse law gives beta soundness; $e\circ p=\mathrm{id}$ additionally gives eta soundness.