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A family satisfies this principle with speed $a_n\to \infty$ and rate function $I$ when the displayed variational identity holds for every bounded continuous real function $f$. A large deviation principle implies it by the Varadhan lemma. The lower bound restricts the expectation to a neighborhood of a chosen point; the upper bound partitions the bounded range of $f$ into small intervals and applies the closed-set probability bounds.

For nonnegative $X_n$, the identity also givesFor $\mu >0$, truncate $f(s)=-\mu s$ below at $-\mu R$. The resulting bounded-function identity applies, while the difference of the two exponential expectations is at most $e^{-\mu R{a}_n}$. Choosing $R$ sufficiently large makes this error negligible on the claimed exponential scale. The case $\mu =0$ is immediate. This extension identifies moment-decay rates of a passive scalar from fluctuations of its logarithmic stretching.

Let $M$ be a Poisson random measure of intensity $\mu$, let $\mu (A)<\infty$, and let $f\ge 0$ be measurable. Apply the Laplace functional of a Poisson random measure to $f+t{1}_A$ and differentiate from the right at $t=0$. On the random side, differentiation is dominated by the integrable random variable $M(A)$; on the intensity side, by $1_A$. This proves the displayed identity when $\int (1-e^{-f})d\mu$ is finite. When this integral is infinite, $e^{-M(f)}=0$ almost surely, and the same identity holds with both sides zero.

Fix the exceptional-zero threshold $\beta >1-\eta /\log q$, with $\eta$ less than one third of the constant in Landau theorem for two real Dirichlet characters. If distinct primitive conductors satisfy $q_1<q_2\le q_1^2$, then $\log (q_1{q}_2)\le 3\log q_1$; both exceptional zeros lie in the forbidden two-zero interval. Thus the conductors grow at least quadratically. The same argument with equal conductors proves at most one exceptional real primitive character for each modulus. Existence is not asserted, and a consistently sufficiently small threshold is essential.

// Target: number-theory.bigb

For a smooth function $M(q)$, its differential defines a Lagrangian submanifold of the cotangent bundle: pullback of the canonical one-form on a cotangent bundle is $dM$, so pullback of the symplectic form $-d\theta$ is zero. The graph has half the ambient dimension. With $q=(S,V,N)$ and momentum coordinates $(T,-P,\mu )$, the first law of thermodynamics identifies the equilibrium graph with $dM$. This geometric statement does not require extensivity or thermodynamic stability.

A Lagrangian submanifold in a Calabi–Yau geometry is special when the phase of its holomorphic volume form is constant. With orientation chosen so that the real part is positive, it is a calibrated submanifold for $\mathrm{Re}(e^{-i\theta }\Omega )$. In ${\mathbb{C}}^3$, its spatial three-fold can describe a supersymmetric curved D3-brane.

A foliation of a symplectic manifold is Lagrangian when every leaf has half the ambient dimension and the restricted symplectic form vanishes. The tangent distribution is a real polarization in geometric quantization. Restricted prequantum connections are flat on individual leaves, but nontrivial holonomy can obstruct global parallel sections.

The graph of a one-form $\alpha$ on a smooth manifold $Q$ is a Lagrangian submanifold of its cotangent bundle exactly when $d\alpha =0$. Indeed the graph pulls the tautological one-form back to $\alpha$, and therefore pulls the canonical symplectic form back to $-d\alpha$. Locally the Poincaré lemma writes $\alpha =dS$, giving $p_i={\partial }_{i}S$. Globally a closed one-form may have nonzero periods and no single generating function.

Relative to a fixed Lagrangian subspace $L_0$, the Maslov cycle consists of Lagrangian subspaces $L$ with $L\cap L_0\ne 0$. In a two-dimensional symplectic space all lines are Lagrangian, and the cycle is the single line $L_0$ in the projective line. Fibrewise versions yield sections of a projective-line bundle and record passage through a prescribed vertical direction.

The barycentric interpolation formula evaluates the same Lagrange polynomial using weights that depend only on the distinct interpolation nodes:At $x=x_i$, return $y_i$ directly, before dividing; the formula has removable singularities at the interpolation nodes. Multiplying every $w_i$ by the same nonzero constant leaves the answer unchanged. Driscoll and Braun derive this formula as a more efficient, numerically stable way to evaluate polynomial interpolation.^[ref]

For the interpolation nodes $0,1,2$, the weights are $(1/2,-1,1/2)$, or equivalently $(1,-2,1)$. With data $(1,2,5)$, at $x=1/2$ the denominator using the latter weights is $2+4-2/3=16/3$ and the numerator is $2+8-10/3=20/3$, giving $p_2(1/2)=5/4$ again. For eleven interpolation nodes in the Runge phenomenon figure, the generator computes the weights by the direct products above and returns stored values exactly at every interpolation node.

The direct products require $O(n^2)$ arithmetic operations once; each subsequent value uses $O(n)$ arithmetic operations. Here $O$ denotes Big O notation. At large polynomial degree or on poorly scaled intervals, forming these products can exceed the range of floating-point arithmetic; normalization or specialized weight formulas may be needed. Stable evaluation also cannot cure sensitivity intrinsic to the chosen interpolation nodes, measured by the Lebesgue constant of interpolation, or the approximation failure in the Runge phenomenon.

For $n+2$ increasingly ordered nodes $y_j$, let $r_j$ be their Lagrange cardinal polynomials, $t=\sum f(y_j)r_j$, and $r=\sum (-1{)}^j{r}_j$. The leading coefficient of $r_j$ has sign $(-1{)}^{n+2-j}$, so all summands in the leading coefficient of $r$ have the same nonzero sign. There is a unique $\lambda$ canceling degree $n+1$ in $t-\lambda r$. Its errors at the nodes are $\lambda (-1{)}^j$, and a better degree-at-most-$n$ approximation would have a difference polynomial with at least $n+1$ roots. Thus it minimizes the maximum node error. Increasing order matters: for the permutation $(0,1/3,1,2/3)$ the alternating interpolant is already the quadratic $-1+9x(1-x)$, so the leading-coefficient assertion for $n=2$ fails.

At distinct nodes $a_0,\dots ,a_n$, the Lagrange interpolation polynomial has basis ${\ell }_j(x)={\prod }_{i\ne j}(x-a_i)/(a_j-a_i)$. Differentiating its exact expression for a polynomial of degree at most $n$ gives the displayed weights. Their uniqueness follows because the nodal evaluations form a basis of the dual space of that polynomial space.

This lacunary power series has radius of convergence one. For $|z|<1$, the inequality $n!\ge n$ for $n\ge 1$ gives absolute convergence by a geometric series comparison. For $|z|>1$ its terms do not tend to zero. At every point of the unit circle the terms have modulus one, so the term test for divergence proves divergence there too. The two initial terms both equal $z$, since $0!=1!=1$; this finite repetition does not affect any convergence conclusion.

For $n$ independent normal responses of common variance at $g$ distinct predictor settings, a rank-$p$ normal linear model has residual degrees $n-p$. The saturated setting-mean model has $g$ parameters and pure error degrees $n-g$. Splitting orthogonal residual spaces gives lack of fit degrees $g-p$. Under the fitted mean model, the two squared lengths divided by the common variance are independent chi-squared variables; their ratio of mean squares has the F-distribution shown. This test is unavailable without repeated settings or positive lack-of-fit degrees.

The exponentially weighted Hilbert space consists of measurable functions with ${\int }_{\mathbb{R}}|u(x){|}^2{e}^{2\gamma x}dx<\infty$. Multiplication by $e^{\gamma x}$ is a unitary map onto L2 space. Requiring unweighted square integrability as well produces an intersection that is incomplete under the weighted norm alone. For example, $1_{[-N,0]}$ is a weighted Cauchy sequence for $\gamma >0$, but its weighted limit $1_{(-\infty ,0]}$ is not in unweighted L2 space.

The cell-average projection replaces a function by its average on each grid cell. It is independent of the representative of an L2 function, unlike point sampling. The Cauchy-Schwarz inequality gives $h|U_m{|}^2\le {\int }_{\text{cell }m}|u{|}^{2}dx$, hence $h{\sum }_m|U_m{|}^2\le \Vert u{\Vert }_{L^2}^2$. Viewed as a piecewise constant function, the projection is the orthogonal projection onto the grid's piecewise constant linear subspace. Density of continuous compactly supported functions and the contraction estimate give convergence in the L2 norm as the mesh tends to zero.

For a continuous local martingale starting at zero, finite expected terminal quadratic variation is equivalent to being an L2-bounded continuous martingale. Stop at bounded path and bracket levels and use $\mathbb{E}(M_t^{\tau }{)}^2=\mathbb{E}[M{]}_{t\wedge \tau }$. Uniform integrability removes the localization; Doob L2 maximal inequality justifies convergence of squares. The terminal identity is $\mathbb{E}{M}_{\infty }^2=\mathbb{E}[M{]}_{\infty }$. For nonzero initial values one must additionally assume $M_0\in L^2$.

A generalized limit on real bounded sequences is a positive linear functional of operator norm one that extends the ordinary limit of a sequence. Such a functional can be chosen by the Hahn-Banach theorem: extend the limit on the convergent sequence space while dominating it by the sublinear functional $p(x)={\mathrm{limsup}}_n{x}_n$. Applying the same domination to $-x$ gives the displayed lower bound. A Banach limit additionally has invariance under the left shift on bounded sequences.

The usual zero-over-zero L'Hopital rule requires the denominator's derivative to be nonzero throughout a sufficiently small deleted neighborhood. A limit taken only where that derivative is nonzero is insufficient. An explicit counterexample uses $x_n=2^{-n}$, $h_n=4^{-n}$, $S(u)=3u^2-2u^3$ and $q(u)=16u^2(1-u{)}^2$. On $[3x_n/4,x_n]$ put $g=h_n$ and $f=h_{n}q((x-3x_n/4)/(x_n/4))$. On $[x_n/2,3x_n/4]$ put $g=h_{n+1}+(h_n-h_{n+1})S((x-x_n/2)/(x_n/4))$ and $f=0$. Extend $g=1,f=0$ beyond one. Values and first derivatives agree at the joins, both functions tend to zero, and $g$ is positive. Where $g^{\prime }\ne 0$, it lies inside a transition interval and $f^{\prime }=0$. Thus $f^{\prime }/g^{\prime }$ has natural-domain limit zero, but $f/g$ equals zero at $x_n$ and one at $7x_n/8$, so has no limit.

For uniform flow $\mathbf{U}$ plus a line vortex of counterclockwise circulation $\Gamma$, the outward advective momentum flux around a large circle is $\rho \mathbf{\Gamma }\times \mathbf{U}/2$. The Bernoulli equation supplies an equal pressure contribution. Their sum is the force on the fluid; the force on an obstacle per unit span is its negative, $\rho \mathbf{U}\times \mathbf{\Gamma }$. Retaining only the advective term loses half the lift and obscures its sign.

On $G={\mathbb{Z}}_p^{\times }/\{\pm 1\}$, this element of the total quotient ring of the Iwasawa algebra becomes an integral measure after multiplication by every difference $[a]-[1]$. Its even moments interpolate the displayed Euler-corrected zeta values. Its trivial-character component has a pole, so the pseudomeasure itself must not be treated as an element of the integral group algebra. Multiplication by the augmentation ideal removes the pole.