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 $v_1,\dots ,v_n$ expresses each vector of the toric lattice uniquely as ${\sum }_i{a}_i{v}_i$ with $a_i\in \mathbb{Z}$. Relative to any fixed integral basis, vectors form another lattice basis exactly when their determinant is $\pm 1$. A primitive vector is one whose integer span is saturated; it can be completed to such a basis by the Euclidean algorithm.

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 $L$.

For a geometric random variable $X$ with $\mathbb{P}(X=k)=p(1-p{)}^{k-1}$, $0<p<1$, the family $X/L$ has large-deviation speed $L$ and good rate function $I(x)=ax$ for $x\ge 0$, infinite otherwise, where $a=-\log (1-p)$. The exact tail $\mathbb{P}(X/L\ge x)=(1-p{)}^{\lceil Lx\rceil -1}$ gives the upper bound. A lattice point tending to any positive $x$ supplies the open-neighborhood lower bound; neighborhoods of zero have probability tending to one. Finite sublevel sets are compact intervals.