The optical path length along a ray is the integral of its refractive index over geometric length. A difference $h$ in optical path length produces a phase difference $2\pi h/\lambda$, where $\lambda$ is the vacuum wavelength.

For $1\le p<\infty$ and probability measures with finite $p$th absolute moments,where $\rho$ is the ground metric and $\pi$ ranges over transport plans. For $p=2$ this is the second Wasserstein distance.

On a Polish space $(X,\rho )$, probability measures with finite first moments satisfyThe supremum runs over functions with Lipschitz constant at most one. Fixing $f(x_0)=0$ ensures integrability from the moment assumption.

In Euclidean space, push an optimal transport plan $\pi$ forward by $(x,y)\mapsto (1-t)x+ty$. The resulting probability measures form a constant speed curve for the p-Wasserstein distance:If the plan is induced by a transport map $T$, this becomes the displayed title formula.

On a $d$-dimensional Hilbert space, the Von Neumann entropy is maximized uniquely by $I/d$. Indeed, nonnegativity of quantum relative entropy giveswith equality exactly when $\rho =I/d$.

A vector field preserves a chosen volume form $\omega$ when its local flow pulls $\omega$ back to itself. Infinitesimally this is ${\mathcal{L}}_X\omega =0$, using the Lie derivative. The condition depends on the chosen volume form.

For thickness $h(r,t)$, radial velocity $u(r,t)$ and external pressure $p_{\mathrm{ext}}$, the leading Newtonian fluid stress tensor and incompressibility giveRadial force balance on an annular sector includes the inward projection of hoop traction and external pressure on the sloping broad faces. Together with conservation of mass it yields

A real vector space is a vector space over the field $\mathbb{R}$ of real numbers.

A vector space over the field of complex numbers. A complex-linear map preserves addition and multiplication by complex scalars.

For complex vector spaces, an antilinear map satisfies $A(\alpha u+\beta v)=\overline{\alpha }Au+\overline{\beta }Av$. It is linear over the real numbers and conjugates complex scalar multiplication. This differs from two conjugate linear operators, which are related by similarity rather than conjugate-linearity.

For $x_{n+1}=a_n{x}_n+b_n$, put $U_0=1$ and $U_n={\prod }_{j=1}^n{a}_j$, with $a_j\ne 0$. Dividing by this accumulated homogeneous growth gives $z_n=x_n/U_{n-1}$ and $z_{n+1}-z_n=b_n/U_n$. A telescoping series then provesThis is the discrete counterpart of the integrating factor for a first-order linear differential equation.

For positive integers $k,s,\ell$, some $N$ ensures that every $k$-color finite coloring of $[N]$ contains a monochromatic setIt follows from the Van der Waerden theorem by mathematical induction on the number of colors. If $M$ is a bound for $k-1$ colors, take a long one-color arithmetic progression of length $(\ell -1)M+1$ and step $d$. Either one of $sd,2sd,\dots ,Msd$ has its color, giving the result immediately, or these multiples use only $k-1$ colors. Pull back their finite coloring to $[M]$ and use mathematical induction, then multiply the resulting configuration by $sd$. Enlarge the ambient finite integer interval to include all these multiples. The cases $k=1$ and $\ell =1$ are immediate. The parameter $s$ makes this useful for positive monochromatic solutions of a partition regular equation with arbitrary integer coefficients.

A scalar-field vacuum is a constant field at a minimum of its potential energy, with the minimum shifted to zero when discussing finite-energy classical field-theory solitons. Distinct isolated vacua allow a scalar-field kink to approach different values at the two ends of space.

At the Gaussian fixed point, the engineering dimension of the coupling of ${\varphi }^{2n}$ is $2n-(n-1)d$. It vanishes at the upper critical dimension $d_c=2n/(n-1)$.

If $\varphi$ maps the unit disc conformally to a proper simply connected domain, the nonvanishing holomorphic quotient $F(w)=(\varphi (w)-\varphi (0))/w$ has harmonic log modulus. Its radial mean equals $\log |{\varphi }^{\prime }(0)|$. Boundary values are understood as radial limits: the standard integral-mean bound ${\sup }_{r<1}\int |\varphi (r{e}^{i\theta }){|}^{p}d\theta <\infty$ for $0<p<1/2$, together with the Koebe distortion theorem lower bound $|F(w)|\ge |{\varphi }^{\prime }(0)|/4$, gives uniform integrability of these logarithms. Hence

The Koebe function $k(z)=z/(1-z{)}^2$ is a univalent function from the unit disc onto $\mathbb{C}\setminus (-\infty ,-1/4]$. To see the image, $(1+z)/(1-z)$ maps the disc onto the right half-plane and $k(z)=(((1+z)/(1-z){)}^2-1)/4$. Its normalization $k(0)=0$, $k^{\prime }(0)=1$ proves sharpness of the Koebe quarter theorem.

For a number field $K$ and a finite set $S$ of finite primes, an S-unit is an element of $K^{\times }$ whose discrete valuation is zero at every prime outside $S$. Its group is the unit group of the ring obtained by allowing denominators at $S$. The Dirichlet unit theorem and the valuation homomorphism to ${\mathbb{Z}}^S$ show that this group is a finitely generated abelian group.

With permeability of a porous medium $k(z)=k_0(1+\beta z)$, the Dupuit approximation givesHere $\varphi$ is porosity and $R$ is recharge. The low-height limit has mobility proportional to $h$, whereas large heights have mobility proportional to $h^2$. This changes both the forced filling similarity for power-law diffusion and the separable draining profile for power-law diffusion.

The Dupuit approximation treats flow in a shallow unconfined aquifer as predominantly horizontal, with hydrostatic pressure $p=\rho g(h-z)$ and horizontal pressure gradient independent of depth. Integrating Darcy's law over the saturated depth then gives a volume flux per unit width depending only on the groundwater height and its slope.

In Type Ia supernova cosmology, the predicted apparent magnitude depends on the common absolute magnitude $M_0$ and the Hubble constant only through $\beta =M_0-5{\log }_{10}(H_0/H_{\mathrm{ref}})$, after choosing a fixed reference $H_{\mathrm{ref}}>0$. A flat improper prior on $M_0$ makes the likelihood integrated over $M_0$ independent of $H_0$, by translation of the integration variable. If the other prior factors are independent and the posterior is proper, the marginal posterior of $H_0$ equals its prior. An external calibration of $M_0$ can break this lack of identifiability.