Consider an initially resting layer governed by the linear shallow water equations, with mean depth $h$, constant nonzero Coriolis parameter $f$, and a localized along-ridge-uniform height disturbance ${\zeta }_0(x)$. Define $c=\sqrt{gh}$ and the positive Rossby deformation radius $R_D=c/|f|$. The linearized shallow-water potential-vorticity anomaly is $Q=v_x-u_y-f\zeta /h$, initially $Q_0=-f{\zeta }_0/h$.

The stationary geostrophic balance has $u_s=0$ and $v_s=g{\zeta }_s^{\prime }/f$. Substituting in conserved $Q$ givesThe decaying Green function of this modified Helmholtz equation givesIts kernel is positive, has integral one and has width $R_D$: the balanced surface is a smoothed version of the initial ridge with the same signed excess volume per unit ridge length. For a positive Gaussian function ${\zeta }_0(x)=A{e}^{-x^2/(2{\sigma }^2)}$, the final height is positive, even and lower at its central peak; the balanced along-ridge velocity is odd. Reversing $f$ changes the sign of this velocity without changing the final height or its decay length. Using $c/f$ as a positive kernel width would instead require the explicit assumption $f>0$.

For each spatial Fourier transform component, initial rest fixesThe second term consists of inertia-gravity waves. For localized data on the infinite line they spread away and leave the steady component locally. This local geostrophic adjustment does not remove wave energy globally; a finite reflecting basin can retain oscillations.

For $2<p<3$ and a continuous path $h$ of bounded variation, translate the first level by $h$ and add the mixed integrals $\int X_{s,r}\otimes d{h}_r$, $\int h_{s,r}\otimes d{X}_r$ and $\int h_{s,r}\otimes d{h}_r$ to the second level. Integration by parts defines the second mixed integral. The operation is continuous in the rough path metric and the variation norm of $h$. For canonical lifts it is precisely the enhancement of the sum of the paths.

The geodesic deviation equation makes the relative gravitational acceleration proportional to the separation $L$ and a typical Riemann curvature tensor component $\mathcal{R}$. Thus tidal acceleration vanishes as $L\to 0$ at fixed bounded curvature. A uniform electric field can give different accelerations $qE/m$ at the same point, independently of separation. Relative tidal changes accumulated over a fixed long time need not vanish as a fraction of the initial separation; a local spacetime experiment must control its duration as well as its spatial size.

An external source is a prescribed field coupled to a dynamical field or operator in the action, for example $S_J=S_0+\int J_i(x){\varphi }_i(x)d^{d}x$. Functional derivatives of the generating functional with respect to $J$ insert the corresponding fields into correlation functions. The source is varied to generate these insertions, rather than integrated over as a dynamical degree of freedom. Transforming it in the appropriate representation maintains an invariant source coupling when deriving a Ward identity.

In a finite-dimensional integral, $u_i{e}^{b\cdot u}={\partial }_{b_i}{e}^{b\cdot u}$. Expand $e^{V(u)}$ as a formal perturbation series to obtain $\int du{e}^{-u^{T}Au/2+b\cdot u+V(u)}=e^{V({\partial }_b)}{I}_0(b)$. With a finite-mode regulator, a Minkowski source insertion satisfies $\varphi (x)e^{i\int J\varphi }=(1/i)\delta e^{i\int J\varphi }/\delta J(x)$; this gives the displayed interacting generating functional. The substitution is a perturbative differential-operator identity, not an assertion that the interaction exponential defines a convergent series. Divide the resulting functional by $Z[0]$ for normalized correlation functions.

For a commutative Noetherian base $D$ of finite dimension $d$, the noncommutative Krull dimension of $D(\sigma ,t)$ equals $d$ unless a height-$d$ maximal ideal is periodic under $\sigma$ or contains infinitely many translates ${\sigma }^n(t)$. This is the commutative-base specialization of Theorem 5.3 in Krull dimension of Generalized Weyl Algebras with non-commutative coefficients.

A selective measurement in quantum mechanics conditions on a known outcome. For Kraus operator $A_i$, the Born rule gives $p_i=\mathrm{Tr}(A_i^{†}{A}_i\rho )$ and the conditional density operator is $A_i\rho A_i^{†}/p_i$ when $p_i>0$. Conditioning changes the state and can change the conditional reduced density matrix of a correlated remote system. That conditional state is available as such only to an observer with the outcome information.

A nonselective quantum measurement averages over outcomes that are not read or not communicated. Its quantum channel is $\rho \mapsto {\sum }_i{A}_i\rho A_i^{†}$, with ${\sum }_i{A}_i^{†}{A}_i=I$. The completeness condition makes the channel trace preserving. The average equals ${\sum }_i{p}_i{\rho }_i$, where ${\rho }_i$ are the conditional states from a selective quantum measurement. A nonselective projective measurement is the special case in which the operators are orthogonal projectors.

For the Hamiltonian $H=(p^2-x^2)/2$, a Gaussian wave packet $A{e}^{-B{x}^2}$ satisfies $\dot{A}=-i\hslash AB$ and $\dot{B}=-i/(2\hslash )-2i\hslash B^2$. The displayed solution is normalizable for real $\varphi$ exactly when $\sin (2\varphi )>0$; its second moments of a complex Gaussian wave packet areBoth grow as $\hslash e^{2t}/[4\sin (2\varphi )]$, while their difference and hence the mean energy remain constant.

For the weight $e^{-x}$ on $[0,\infty )$, use the zeros of a degree-$n$ Laguerre polynomial as quadrature nodes. Divide a polynomial of degree at most $2n-1$ by that polynomial. The quotient term integrates to zero by orthogonality and vanishes at every node; interpolation of the remainder proves exactness. The two-node rule has nodes $2\pm \sqrt{2}$ and corresponding weights $(2\pm \sqrt{2})/4$, with the larger weight at the smaller node.

The first-kind rule approximates ${\int }_{-1}^{1}f(x)(1-x^2{)}^{-1/2}dx$ at the zeros of the Chebyshev polynomial $T_n$. It integrates all polynomials of degree at most $2n-1$ exactly. The weights follow by integrating the Lagrange interpolation polynomial at those nodes. Orthogonality kills the quotient when a degree-$2n-1$ polynomial is divided by $T_n$; its remainder is recovered by interpolation. No $n$-node rule can integrate every degree-$2n$ polynomial, since the squared nodal polynomial has zero quadrature sum and strictly positive weighted integral.

// Target: probability-and-statistics.bigb

The even weight $1-x^2$ has moments $m_0=4/3$ and $m_2=4/15$. Its monic quadratic orthogonal polynomial is $x^2-1/5$, so the two Gaussian quadrature nodes are $\pm 1/\sqrt{5}$. Symmetry and constant exactness give weights $2/3$ each. These nodes and weights integrate every polynomial of degree at most three exactly. Degree four fails: the true integral is $4/35$, whereas the rule gives $4/75$.

An $(n+1)$-node quadrature exact on degree $n$ is exact through degree $2n+1$ precisely when its node polynomial $p={\prod }_i(x-x_i)$ is orthogonal to every degree-$n$ polynomial for the integration weight. Necessity applies the rule to $pr$. Sufficiency divides an arbitrary degree-$2n+1$ polynomial as $pq+r$, with both $q,r$ of degree at most $n$; the integral and quadrature of $pq$ vanish while both agree on $r$.

For $S[\varphi ]=\frac{1}{2}\int (\partial \varphi {)}^2$ and a source appearing as $-\int J\varphi$ in the exponent, the positive quadratic operator is $A=-\Delta$. Completing the square gives $Z[J]=\mathcal{N}\exp [\frac{1}{2}J{A}^{-1}J]$. If one writes it with a negative quadratic source exponent, its kernel must be $G=-A^{-1}={\Delta }^{-1}$. In the plane $G(z,w)=(2\pi {)}^{-1}\log (|z-w|/L)$, up to harmonic additions and infrared conventions. The physical covariance is its negative. Zero modes are regulated or separated; oscillatory sources in a string path integral produce momentum conservation from the constant mode.

For independent means $\theta \sim N({\mu }_{\theta },t_{\theta })$ and $\varphi \sim N({\mu }_{\varphi },t_{\varphi })$, the first mean's conditional precision is $m/v+1/t_{\theta }$ and conditional precision-weighted mean is ${\sum }_{i\le m}{x}_i/v+{\mu }_{\theta }/t_{\theta }$; replace $m$ by $n-m$ for the second mean. With $v\sim \mathrm{IG}(a_0,b_0)$ its conditional law is $\mathrm{IG}(a_0+n/2,b_0+S_m/2)$. The discrete split has conditional probabilities proportional to $e^{-S_m/(2v)}$. These full conditional distributions give a Gibbs sampler including $m=n$. Averaging the sampled indicators of that configuration estimates its probability; averaging its conditional probabilities gives an estimate by Rao-Blackwellization.

A continuum Gaussian chain has separation characteristic function $e^{-a|s-t|}$, $a=k^2{b}^2/6$. Integrating over a contour of length $N$ gives the polymer scattering functionwhere $R_g^2=N{b}^2/6$ in this continuum convention and $\mathcal{D}(x)=2(e^{-x}-1+x)/x^2$, with continuous value $\mathcal{D}(0)=1$. Its limits are $\mathcal{D}(x)=1-x/3+x^2/12+\cdots$ and $\mathcal{D}(x)\sim 2/x$ for large $x$.

For a discrete chain with $N$ sites, retain the self terms and let $z=e^{-k^2{b}^2/6}$. The exact finite expression is $g_N=1+(2/N){\sum }_{\ell =1}^{N-1}(N-\ell )z^{\ell }=(1+z)/(1-z)-2z(1-z^N)/[N(1-z{)}^2]$, with $g_N(0)=N$. Taking $N\to \infty$ while $x=N{k}^2{b}^2/6$ stays fixed gives $g_N/N\to \mathcal{D}(x)$. The continuum expression describes the coil scaling range, whereas the discrete expression tends to one as $kb\to \infty$.

The quotient $[n{]}_q[n-1{]}_q\cdots [n-k+1{]}_q/([k{]}_q\cdots [1{]}_q)$ is the displayed shifted Gaussian binomial coefficient. It is the formal character of ${\Lambda }^k{L}_{n-1}$, where $L_{n-1}$ is the $n$-dimensional irreducible sl2 Lie algebra representation. Indeed its exterior-power weights are sums of $k$ distinct terms of $n-1,n-3,\dots ,1-n$. Parity unimodality of an sl2 character proves unimodality after removing the intervening zero coefficients; equivalently, the ordinary Gaussian polynomial has unimodal coefficients.

For $0\le k\le n$, let $V$ be the $n$-dimensional irreducible sl2 Lie algebra representation. Its weights are $n-1,n-3,\dots ,1-n$. A wedge basis shows $\mathrm{ch}{\Lambda }^{k}V=q^{-k(n-k)}{G}_{n,k}(q^2)$, where $G_{n,k}(t)$ is the Gaussian binomial coefficient. By the Weyl complete reducibility theorem, this character is a nonnegative sum of irreducible strings $q^m+q^{m-2}+\cdots +q^{-m}$. All its weights have parity $k(n-k)$, so their coefficients increase toward zero and decrease afterward. Consequently the coefficients of $G_{n,k}(t)$ are symmetric and unimodal. The symmetric quantum binomial coefficient is this character, a Laurent polynomial; its coefficients are unimodal on the occupied parity. The full integer-exponent sequence need not be unimodal, as $q+q^{-1}$ demonstrates.

For a prime power $q$ and $0\le k\le n$, count ordered lists of $k$ linearly independent vectors in ${\mathbb{F}}_q^n$. The first has $q^n-1$ choices. After $i$ independent choices, their linear span contains $q^i$ vectors, so the next has $q^n-q^i$ choices. Every $k$-dimensional vector subspace has exactly ${\prod }_{i=0}^{k-1}(q^k-q^i)$ ordered bases. Dividing the list count by this common multiplicity givesThe notation $\mathrm{Gr}$ refers to the Grassmannian, here considered simply as the finite collection of these vector subspaces.^[ref]

For $n=4,k=2,q=2$, there are $15\cdot 14=210$ ordered independent pairs. Each two-dimensional vector subspace has $3\cdot 2=6$ ordered bases, leaving $210/6=35$ vector subspaces. There are only $\left(\genfrac{}{}{0ex}{}{4}{2}\right)=6$ coordinate-selected vector subspaces; most of the 35 also contain vectors combining coordinates.

A unique reduced row echelon form gives another count. Its six possible pivot-column patterns have $4,3,2,2,1,0$ freely chosen entries. Each free entry lies in ${\mathbb{F}}_2$, giving $16,8,4,4,2,1$ vector subspaces, respectively. The diagram fixes the zero entries required to make each pivot the first nonzero entry in its row and the only nonzero entry in its column. Different choices describe different row spaces.

The Young diagrams of partitions in a rectangle with at most $k$ rows and at most $n-k$ cells in each row correspond to the lattice paths in the lattice-path interpretation of Gaussian binomial coefficients. Read the shaded row lengths from top to bottom; the result is a nonincreasing list of nonnegative integers. This bijection preserves the number $|\lambda |$ of cells.

In a two-by-two rectangle, the possibilities are the empty integer partition, $(1)$, $(2)$, $(1,1)$, $(2,1)$, and $(2,2)$. Their sizes are $0,1,2,2,3,4$, giving $G_{4,2}(t)$. In particular, there are two allowed diagrams of size two, explaining the middle coefficient.^[ref]