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A dense porous gravity current of height $h$ places excess hydrostatic pressure $\Delta \rho gh$ on a basal seal. For seal permeability of a porous medium $\lambda$ and thickness $b$, Darcy law gives downward flux $\lambda \Delta \rho gh/(\mu b)$. Dividing the stored height-volume balance by porosity $\varphi$ gives a linear drainage term $-\Omega h$. The coefficient has inverse-time units and does not contain an extra factor of the aquifer permeability.

Sea-ice lead-width distributions depend on observational thresholds, resolution and whether leads are counted individually or weighted by area. Observations commonly exhibit a broad, approximately power law range, with many narrow leads and a relatively heavy tail.

// Target: geophysics.bigb

The pullback of the truth point along a local operator $j$ is the subobject $J\hookrightarrow \Omega$. A monomorphism with characteristic map $\chi$ is j-dense exactly when $\chi$ factors through $J$. Its internal order is inherited from the Heyting algebra of truth values. Having a least global element is an internal condition over all parameter objects, not just a condition on global truth values.

The extrapolation height at which a logarithmic wall profile vanishes. For a fully rough wall it is set mainly by the geometry; for a smooth wall it scales with $\nu /u_{\ast }$. It is not the thickness of the molecular viscous sublayer or the literal location of a no-slip surface.

The dimensionless coefficient in the logarithmic wall law and its mixing-length form $\ell =\kappa z$. Its numerical value is determined from the turbulence closure and measurements, rather than from dimensional analysis alone.

Fix $T>0$, let $X$ be Brownian motion, and let $\tau =\inf \{u:X_u=-1\}$. Set $q(t)=t/(T-t)$ for $t<T$ andThe hitting time is finite almost surely, so each price path reaches one before $T$ and stays there. Stopping additionally on hitting $n+2$ makes the stock price a bounded martingale; these stopping times increase to $T$. Hence $S$ is a positive local martingale, and the original probability measure is an equivalent local martingale measure, excluding admissible arbitrage. Nevertheless $S_T=B_T$ while $S_0=2\ne 1=B_0$. The zero-cost short-stock/long-two-cash portfolio has terminal gain one but wealth $2-S_t$, which has no deterministic lower bound. It is not admissible, resolving the apparent contradiction.

A finite extension of the complete discretely valued field $\mathbb{C}((t))$ has residue field $\mathbb{C}$, hence is totally ramified. If $\pi$ is its uniformizer, write $t=u{\pi }^n$. A root of the residue of $u$ lifts by Hensel lemma to an $n$th root $v$ of $u$, since the residue characteristic is zero. Then $s=v\pi$ satisfies $s^n=t$. Distinct valuations modulo $n$ make $1,s,\dots ,s^{n-1}$ linearly independent over the base, so $L=\mathbb{C}((t))(s)$. Expanding in $s$ with constant coefficients identifies it with $\mathbb{C}((s))$. Its automorphisms are $s\mapsto {\zeta }_{n}s$, giving a cyclic Galois group.

For arbitrary positive integers $e,f$, the displayed extension has degree $ef$, ramification index $e$, and residue degree $f$. If $|t|=p^{-1}$, its unique extended absolute value on a field is $|\sum a_n{u}^n|=p^{-\min \{n:a_n\ne 0\}/e}$. A basis consists of ${\gamma }_i{u}^j$, where the ${\gamma }_i$ form an basis of ${\mathbb{F}}_{p^f}$ and $0\le j<e$. Regrouping a Laurent series by its exponent modulo $e$ proves the basis assertion. When $p$ divides $e$ the extension can be inseparable; the valuation calculation still applies.

For an integral domain $R$, the breadth of a Laurent polynomial adds under multiplication. If two Laurent polynomials multiply to one, both therefore have breadth zero: they are monomials, and their coefficients must be units of $R$. Conversely every displayed monomial is invertible. For $R=\mathbb{Z}$ the units are precisely $\pm t^k$, explaining the normalization ambiguity of an Alexander polynomial.

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.