At each point there is at most one supported Haar wavelet per scale. If a detail coefficient is below $\delta$, its value there is at most $\delta 2^{j/2}/\sqrt{L}$. Summing these bounds up to level $K$ gives the displayed geometric estimate. Selecting $K$ so the final term is of the order of the modulus of continuity controls all omitted fine details, even though arbitrary deletions from a uniformly convergent series need not preserve convergence.

In a shallow-water potential-vorticity inversion with the background gradient explicitly compensated, uniform PV gives $({\nabla }^2-L_R^{-2}){\psi }^{\prime }=-fb/H$. The displayed Gaussian function solves it for $b=ϵ[4(1-r^2/a^2)+a^2/L_R^2]e^{-r^2/a^2}$. This is not, without that compensation, a steady solution of the standard finite-depth constant-f current problem: its residual is $U{\psi }_x^{\prime }/L_R^2$. Closed streamlines can occur while Rossby number and surface displacement remain small.

A Euclidean body has an axis-parallel box of the same volume with every coordinate-projection volume no larger than its own. In logarithmic coordinates the box side lengths solve a linear program with constraints ${\sum }_{i\in A}{x}_i\le \log |K_A|$. Its dual uses nonnegative exact fractional covers; the uniform cover inequality bounds every dual objective below by $\log |K|$, proving existence by strong duality.

An alternative proof minimizes an array of candidate projection volumes subject to the finitely many inequalities from irreducible uniform covers. Tight constraints force the array to factor into its singleton coordinates, which become the side lengths of the box.

Let $g_D(x)={\mathbb{E}}_x{T}_D$, possibly infinite, and suppose the closed ball $\overline{B(x,r)}$ lies in the open set $D$. A Strong Markov property decomposition at an independent random-radius exit time, using uniform ball sampling by Brownian stopping, gives the displayed identity in the nonnegative extended reals. Indeed the expected exit time from a radius-$s$ ball started at its centre is $s^2/n$, and averaging this against $n{s}^{n-1}/r^n$ gives $r^2/(n+2)$. The identity gives local integrability whenever $g_D(x)$ is finite.

The extraordinary wave's electric displacement field lies in that plane and is perpendicular to the wavevector; its electric field need not be. For angle $\theta$ between the wavevector and optic axis,Thus $n_{\mathrm{ex}}(0)=n_o$ and $n_{\mathrm{ex}}(\pi /2)=n_e$.^[ref]

A short derivation starts from the existing plane wave in an anisotropic dielectric equation. Put $\mathbf{k}=(k_x,0,k_z)$ and $k_0=\omega /c$, with angular frequency $\omega$ and vacuum speed of light $c$. The $y$ component gives the ordinary wave relation $k_x^2+k_z^2=n_o^2{k}_0^2$. The determinant of the $x,z$ system givesSubstituting $k_x=n_{\mathrm{ex}}{k}_0\sin \theta$ and $k_z=n_{\mathrm{ex}}{k}_0\cos \theta$ yields the displayed directional refractive index.

The ordinary wave's electric field is perpendicular to the plane containing the wavevector and optic axis. Its refractive index $n_o$ is direction independent at fixed frequency.

The symmetry direction of a uniaxial optical medium. For a wavevector along it, the two refractive indices coincide: there is no birefringence for that propagation direction, even though $n_e\ne n_o$.

A free particle following Langevin dynamics initialized at zero position and velocity has relaxation time $\tau =m/\zeta$ and thermal diffusivity $D=k_{B}T/\zeta$. Its position is the convolution of the random force with $[1-e^{-t/\tau }]/\zeta$. The fluctuation-dissipation relation for a Langevin particle gives force covariance amplitude $2\zeta k_{B}T$, and integrating the squared kernel yields the displayed variance. At short times it grows as $2D{t}^3/(3{\tau }^2)$ because the initial velocity is cold; at long times it grows as $2Dt$. A Maxwellian initial velocity produces a different short-time transient.

For two Bernoulli random variables with joint probabilities $a,b,c,d$ in the order $(0,0),(0,1),(1,0),(1,1)$, their covariance is $d-(c+d)(b+d)=ad-bc$. Zero covariance fixes the $(1,1)$ probability to the product of its marginals; subtracting this from each marginal fixes all three remaining joint probabilities to their products. Thus uncorrelated random variables taking only zero and one are independent, including degenerate marginals. The implication is special to this two-point setting and fails for general distributions.

There are ${\aleph }_1$ pairwise disjoint stationary subsets of the nonzero countable limit ordinals. In a Ulam matrix on omega-one, each row partitions a stationary tail into countably many cells, so some cell is stationary. One column contains stationary cells in uncountably many rows, and the column cells are disjoint. Unions indexed by different subsets of ${\omega }_1$ then differ on a stationary set.

The displayed sum is the first-order projection of a centred U-statistic onto sums of functions of individual observations. Its summands are independent and have mean zero. Subtracting it leaves a remainder with conditional expectation zero in each individual observation, a useful decomposition for variance and normal-limit calculations.

A bulk cell expansion generally leaves an oscillatory Dirichlet mismatch. A boundary corrector can solve the homogeneous oscillatory conductivity equation with boundary data cancelling that mismatch. It restores the prescribed physical boundary trace while leaving the leading bulk cell formula unchanged. Estimates depend on boundary regularity and on the coefficient assumptions.

For conditionally independent paired counts with means ${\lambda }_i$ and ${\lambda }_i{e}^{\beta }$, condition on every pair total $N_i=Y_{i0}+Y_{i1}$. The joint conditional likelihood removes all pair-specific rates, leaving the displayed independent binomial distributions. A zero-total pair contributes no information. With unrestricted fixed baseline rates, this conditional likelihood has the same parameter-dependent part as the ordinary profile likelihood; a specified Gamma–Poisson hierarchical model can retain additional information by imposing structure on those rates.

With $\Theta =T/T_{\min }>1$, the two-dimensional self-gravitating rod equilibrium has the displayed energy per unit length for the potential reference $\psi (R_e)=-GM\log (V/V_0)$. Integration of $-\rho \psi /2$ gives its gravitational energy; its two-dimensional Maxwell-Boltzmann velocity distribution has kinetic energy $NkT$. At fixed area, differentiation gives $C=Nk[2+2\Theta \log (1-{\Theta }^{-1})+\Theta /(\Theta -1)]$. Expanding the logarithm shows $C/(Nk)=1+{\sum }_{j\ge 3}(j-2)/(j{\Theta }^{j-1})>1$. The energy increases continuously from $-\infty$ to $+\infty$, so there is no finite-energy endpoint or turning point of the regular sequence. This excludes the classical gravothermal catastrophe in this model, while leaving a distinct canonical collapse at or below its minimum temperature.

When the birefringence separation of optical propagation constants $\Delta k=2\pi \Delta n/\lambda$ is large compared with the spatial rate of rotation of the optic axes, light with linear polarization in a local eigenpolarization follows the twisting nematic director. Thickness alone is insufficient if birefringence vanishes. Strong field alignment along the propagation direction suppresses the transverse optical anisotropy and hence the quarter-turn polarization rotation.

For isometry families $f_x:U\to U\prime$ and $g_y:U\prime \to U\prime \prime$, the complement of $g_y{f}_x(V)$ decomposes orthogonally as the complement of $g_y(W)$ plus the image under $g_y$ of the complement of $f_x(V)$ in $W$. Their fiberwise one-point compactifications therefore give the same Thom space as the external direct sum. The resulting stage isomorphisms respect structure maps and give $Y{⋉}_g(X{⋉}_{f}E)\cong (Y\times X){⋉}_{g\circ f}E$ in the stable homotopy category.

For an isolated uniform sphere with fixed positive cosmological constant, energy conservation and the virial theorem give $2\eta x^3-(2+\eta )x+1=0$, $x=R_v/R_m$. The physical root starts at $x=1/2$ when $\eta =0$ and decreases for $0<\eta <1$. At the limiting value $\eta =1$, the cubic factors as $(x-1)(2x^2+2x-1)$; the collapsing branch approaches $(\sqrt{3}-1)/2$. Finite turnaround therefore gives a strict lower bound, whose infimum is approached rather than attained. Compared with zero vacuum energy at the same turnaround data, the negative vacuum potential and its virial contribution require stronger final gravitational binding.

For the Brusselator, the stationary spatial threshold is $b_T=(1+a\sqrt{D_u/D_v}{)}^2$ and the homogeneous oscillatory threshold is $b_H=1+a^2$. A genuine Turing instability appears first only if $b_T<b_H$, which rearranges to the displayed diffusivity-ratio bound. Equality gives simultaneous stationary spatial and homogeneous oscillatory neutral modes. Otherwise homogeneous loss of stability occurs first.

On a specified finite domain, only its boundary-compatible wavenumbers can grow. Solve the Brusselator mode determinant for its neutral value of $b$ and minimize over those nonzero modes. Minimizing over all continuous $k$ instead gives the usual threshold and $k_c^2=a/\sqrt{D_u{D}_v}$. The finite-mode minimum must still precede the homogeneous threshold to describe a Turing instability.

For $u_t=D{u}_{xx}-u(u-r)(u-1)-\rho (v-u)$ and $0=v_{xx}-v+u$, a Fourier mode perturbation about zero has $v=u/(1+k^2)$ and growth rate $\sigma (k)=-r-D{k}^2+\rho k^2/(1+k^2)$. If $\rho >D$, its maximum is $-r+(\sqrt{\rho }-\sqrt{D}{)}^2$ at $k^2=\sqrt{\rho /D}-1$; if $\rho \le D$, its maximum is $-r$ at zero. Replacing $(u,v,r)$ by $(1-u,1-v,1-r)$ gives the corresponding criterion at the other homogeneous equilibrium.