A lattice basis $v_1,\dots ,v_n$ of a toric lattice $N$ expresses each element uniquely as ${\sum }_i{a}_i{v}_i$ with $a_i\in \mathbb{Z}$. Relative to a fixed integral basis, the new vectors form a lattice basis exactly when the matrix of their coordinates has determinant $\pm 1$.

Being a real basis is weaker. The vectors $(1,0),(1,2)$ are linearly independent over the real numbers, but their integer combinations have even second coordinate and do not fill ${\mathbb{Z}}^2$. Their determinant is two.

A primitive lattice vector is nonzero and generates a saturated sublattice. In coordinates its entries have greatest common divisor one. The Euclidean algorithm completes it to a lattice basis. The nonzero condition excludes the zero vector, whose zero subgroup is saturated but which cannot be a basis vector.^[ref]

Potassium chloride is $\mathrm{KCl}$, a one-to-one ionic compound of potassium and chloride. Its solid lattice and the hydration of its dissolved ions contribute opposing terms to its enthalpy of solution.

Dissolution calorimetry therefore probes the whole transformation from a specified solid to a specified solution state. The observed heat is not simply a direct measurement of the energy of one potassium-chloride pair.

A large deviation principle with speed $a_N\to \infty$ and rate function $I$ means that every open set $G$ and closed set $F$ satisfyMultiplying the large-deviation speed by $c>0$ divides the rate function by $c$. The speed must match the rare-event scale: a fixed exponential random variable divided by $N^p$ has speed $N^p$, whereas a sum of $N$ independent random variables with a normal distribution, divided by $N^p$, has speed $N^{2p-1}$ when $p>1$.

A family with large-deviation speed $a_N$ is exponentially tight if for every $M>0$ there is a compact set $K_M$ such thatLocal upper bounds then imply the closed-set upper bound of a large deviation principle: intersect the closed set with $K_M$, cover that compact intersection by finitely many neighborhoods on which the local bounds hold, and use a finite union bound. The remaining Probability has exponent at most $-M$; let $M\to \infty$.

Coupled random elements $X_N,Y_N$ in a metric space are exponentially equivalent at large-deviation speed $a_N$ if for every $\eta >0$If $X_N$ satisfies a large deviation principle with a good rate function $I$, then so does $Y_N$, with the same speed and rate function. For the upper bound, $\{Y_N\in F\}$ is contained in $\{X_N\in F^{\eta }\}\cup \{d(X_N,Y_N)>\eta \}$, where $F^{\eta }$ is the closed $\eta$-neighborhood. Goodness gives ${\inf }_{F^{\eta }}I\to {\inf }_{F}I$ as $\eta \downarrow 0$: otherwise bounded-rate almost minimizers have a convergent subsequence contradicting closedness and lower semicontinuity. For an open-set lower bound, choose a ball about any $x\in G$ with a slightly larger ball still in $G$, subtract the superexponentially small error from the probability that $X_N$ belongs to the smaller ball, and shrink the ball. No independence assumption is required.

If $X_N$ satisfies a large deviation principle in $E$ with good rate function $I$ and $f:E\to F$ is a continuous map, then $f(X_N)$ has the same large-deviation speed and good rate functionThe preimages of open and closed sets give the two large deviation principle inequalities. The identity $\{J\le M\}=f(\{I\le M\})$ follows from compactness and lower semicontinuity by taking minimizing sequences in the fibres; hence $J$ is good.

Suppose $X_N$ and $Y_N$ are independent random variables in Polish spaces, each satisfying a large deviation principle at speed $a_N$, with good rate functions $I$ and $K$. Their pair has good rate function $I(x)+K(y)$. On open rectangles, independence makes the logarithmic lower bounds add. For the upper bound, the marginal families are exponentially tight, and so is their pair. Cover a closed set intersected with the product of the two compact containment sets by finitely many small rectangles and add the corresponding local upper bounds. The finite union bound and then increasing the containment level give the closed-set bound. Goodness follows because $\{I+K\le M\}$ is closed inside $\{I\le M\}\times \{K\le M\}$.

Let $X_N\in {\mathbb{R}}^d$ and $a_N\to \infty$. Define the scaled cumulant-generating functionSuppose its pointwise limit $\Lambda$ exists, is lower semicontinuous, has $0$ in the interior of its effective domain, and is essentially smooth: it is differentiable on that interior, which is nonempty, and its gradient norm diverges at every finite boundary point approached from the interior. Then $X_N$ satisfies a large deviation principle with good rate function given by the Legendre-Fenchel transformThe Chernoff bound supplies the upper estimate. The lower estimate uses exponential tilting at a parameter whose gradient selects the required mean; essential smoothness supplies the boundary approximation needed for the full lower bound. A finite boundary slope can leave an affine branch of ${\Lambda }^{\ast }$ outside this argument, so a separate lower-bound proof is then necessary.

The exponential integral of a fixed measurable function is controlled, on its logarithmic scale, by the function's essential infimum. Suppose $0<{\int }_A{e}^{-\varphi }dx<\infty$ and $\alpha ={\mathrm{essinf}}_A\varphi$ is finite. For $\theta \ge 1$, the integral is at most $e^{-(\theta -1)\alpha }{\int }_A{e}^{-\varphi }dx$. For every $\epsilon >0$, the set $A_{\epsilon }=\{x\in A:\varphi (x)<\alpha +\epsilon \}$ has positive finite Lebesgue measure, and the integral is at least $|A_{\epsilon }|e^{-\theta (\alpha +\epsilon )}$. Taking logarithms, dividing by $\theta$, and then letting $\epsilon \downarrow 0$ proves the displayed limit. The argument needs no isolated or nondegenerate minimizer. The Laplace principle for probability measures treats families whose distributions also change with the scaling parameter.

If random elements satisfy a large deviation principle at speed $a_n\to \infty$ with rate function I, then a bounded continuous real function f satisfies the displayed exponential-integral identity. For the upper bound, partition its bounded range into finitely many closed intervals of width epsilon and apply the closed-set upper bound to their preimages. The largest exponential term controls the finite sum. For the lower bound, restrict to an open neighborhood where f is within epsilon of its value at a chosen point and use the open-set lower bound. Let epsilon tend to zero and optimize over the point. Boundedness avoids a separate exponential tail assumption.

Suppose a large deviation principle at speed $a_n$ has a good rate function with unique zero mu. At any slower speed $b_n\to \infty$ with $a_n/b_n\to \infty$, its rate is zero at mu and infinity elsewhere. On a closed set excluding mu, the positive original rate gives decay faster than every exponential at speed $b_n$. Neighborhoods of mu have Probability tending to one, giving the slower-speed open-set lower bound. The new rate is good because its finite sublevel set is the singleton mu.

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.