Spreading of solute across the mean flow direction through diffusion and pore-scale velocity variations. In a reflecting layer of depth $H$, a constant transverse dispersion coefficient gives mixing time of order $H^2/D_T$. Repeated exchange between different longitudinal speeds produces Taylor dispersion on times long compared with this mixing time.

For a real number $x$, define $\Vert x{\Vert }_{\mathbb{R}/\mathbb{Z}}={\min }_{k\in \mathbb{Z}}|x-k|$. It lies in $[0,1/2]$ and obeys $\Vert \ell x\Vert \le |\ell |\Vert x\Vert$ for an integer $\ell$. It is a distance on the circle, rather than a norm on the real vector space.

For a universal constant $C>0$ and $0<\epsilon \le 1$, every real $\alpha$ admits $1\le n\le {\epsilon }^{-C}$ with distance to the nearest integer of $\alpha n^2$ less than $\epsilon$. A Fejér kernel detects failure of recurrence as a large quadratic exponential sum. The Van der Corput inequality for finite scalar sequences then produces a short linear near-return, whose suitable multiple gives the quadratic return. To include $\epsilon >1$, use the bound $\max \{1,{\epsilon }^{-C}\}$.

A vector $\omega \in {\mathbb{R}}^n$ satisfying $|k\cdot \omega |\ge \gamma |k{|}_1^{-\tau }$ for all nonzero integer vectors $k$, with $\gamma >0$. This excludes resonances and controls the small divisors in analytic cohomological equations on a Diophantine torus. Different choices of norm change the constant but not the nature of the condition.

An irrational real number is badly approximable when $\Vert q\alpha \Vert \ge c/q$ for some $c>0$ and every positive integer $q$, where the double bars mean distance to the nearest integer. Equivalently every rational approximation obeys $|\alpha -a/q|\ge c/q^2$. The Diophantine bound for the square root of two gives a concrete example and supplies separated rotations for bilinear cancellation for badly approximable phases.

The space of Taylor-equivalence classes $j_x^{k}f$ of smooth maps $M^m\to N^n$, with derivatives through order $k$ agreeing at the source point. The projection $j_x^{k}f\mapsto (x,f(x))$ has total dimension $m+n\left(\genfrac{}{}{0ex}{}{m+k}{k}\right)$ and fiber dimension $n((\genfrac{}{}{0ex}{}{m+k}{k})-1)$. First-order fibers are canonically $\mathrm{Hom}(T_{x}M,T_{y}N)$; higher-order fibers have coordinate-dependent Taylor descriptions. The truncation to order $k-1$ is an affine bundle modeled on ${\mathrm{Sym}}^k(T_x^{\ast }M)\otimes T_{y}N$.

A differential three-form is a smooth, totally antisymmetric covariant tensor of rank three. For a Killing-Yano two-form, its covariant derivative is such a tensor, because it is antisymmetric in the first two and last two slots.

For a diffeomorphism, pull a type $(r,s)$ tensor field back by the inverse differential on vector slots and the dual differential on covector slots. Invertibility distinguishes this construction from pullback of purely covariant tensors along an arbitrary smooth map. This operation preserves tensor products and tensor contractions, supplying the flow definition of the Lie derivative of a tensor field.

For an even-length Boolean string, one phase query followed by a known unitary extension can produce a zero-pair outcome with amplitude proportional to the zero-minus-one imbalance, or an index pair with amplitude proportional to the difference of their phase signs. A zero-pair observation certifies nonbalance. Every nonzero pair has opposite bits, so deleting it preserves the imbalance and allows recursion on two fewer indices. Ending with an empty list certifies balance. This quantum circuit is exact on all inputs and uses at most half the original length in queries, even without the constant-or-balanced promise.

The maximally mixed density operator on an $N$-dimensional Hilbert space assigns equal weights to every vector of any orthonormal basis. It is invariant under every unitary operator. Its purity is $\mathrm{Tr}({\rho }_{\ast }^2)=1/N$, the minimum possible because the nonnegative eigenvalues sum to one and obey the Cauchy-Schwarz inequality. For $N>1$ it is a mixed state; for $N=1$ it is the sole pure state. A rank-$k$ projector has probability $k/N$ in this state.

For a two-qubit density operator, the real three-by-three matrix $T$ encodes the spin-product expectation values: unit measurement axes $\mathbf{a},\mathbf{b}$ have $E(\mathbf{a},\mathbf{b})={\mathbf{a}}^{T}T\mathbf{b}$. The tensor does not include local Bloch-vector data and alone need not specify the complete density operator. The CHSH optimum from the correlation tensor depends on the two largest eigenvalues of $T^{T}T$.

For radiation in cosmology with zero cosmological constant, $p=\rho /3$. The Friedmann acceleration equation gives $\ddot{a}/a=-8\pi G\rho /3$, so the deceleration parameter equals the cosmological density parameter, regardless of spatial curvature. This is a relation for a pure radiation fluid, not a general identity for a mixture.

This region $0\le r<R$ of de Sitter spacetime is static for the central observer, with a cosmological Killing horizon at $r=R$. The central proper time fixes the normalization of ${\partial }_t$. Static observers have acceleration $r/(R^2\sqrt{1-r^2/R^2})$. Horizon-crossing null coordinates remove the coordinate singularity, while the global extension has compact spherical spatial slices and spacelike past/future conformal infinity.

For $L=D_t^2+2k{D}_t+k^2+{\omega }^2$, with $k>0$ and $\omega >0$, the causal Green function is $G(t)=H(t)e^{-kt}\sin (\omega t)/\omega$, where $H$ is the Heaviside step function. It satisfies $LG=\delta (t)$: it is continuous at zero, with a unit jump in its first derivative. Its smooth positive-time kernel $g$ obeys $g(0)=0,g^{\prime }(0)=1$. The zero-frequency limit is $H(t)t{e}^{-kt}$. Shifting $G$ gives the zero-initial-data response to a delayed Dirac delta function.

With the Fourier transform convention $\hat{f}(\xi )=\int f(t)e^{-i\xi t}dt$, the transformed equation gives $\hat{G}(\xi )=[k^2+{\omega }^2-{\xi }^2+2ik\xi {]}^{-1}$. Its poles $\xi =ik\pm \omega$ lie in the upper half of the complex plane. Inverting by contour integration therefore gives zero for negative time and an exponentially damped oscillation for positive time.

A massless open-string vector polarization tangential to a planar D-brane becomes a worldvolume gauge field. Transverse polarizations become scalar fields describing fluctuations of its position. This supports the interpretation of a Dirichlet plane as a dynamical extended object. The unprojected bosonic open-string tachyon additionally signals that its brane ground state is unstable; the massless field interpretation does not remove that instability.

At an ordinary thermodynamic critical point, two coexisting phases become indistinguishable: their order parameter difference vanishes, and the correlation length and appropriate response functions typically diverge. A continuous phase transition can be approached there by tuning the thermal and conjugate-field variables. A renormalization-group fixed point controls its singular long-distance behavior. A triple point with three separated minima is not an ordinary thermodynamic critical point, and a tricritical point requires an additional tuning and has a different scaling theory.

Two simple symmetric random walks on the planar integer lattice starting at sites of the same parity can be coupled to agree after a finite time almost surely. Run them independently until they meet, then give them identical increments. Their difference walk has return probability $p_{2n}(0,0)$ at time $n$; the divergent return sum proves Markov-chain recurrence and makes the meeting time finite.

In hydrogen-only cosmology, charge neutrality gives $n_e=n_p$ and baryon conservation gives $n_b=n_p+n_H$. The fraction $X_e$ lies between zero and one, with $n_H=(1-X_e)n_b$. With helium present, electron abundance per hydrogen nucleus and per total baryon are different conventions and must be specified. The hydrogen-only Saha equation gives the equilibrium fraction.

When the effective recombination rate per electron becomes smaller than the expansion rate, the small remaining ionized fraction can no longer track chemical equilibrium. A nonzero electron fraction remains, exceeding the exponentially small Saha equation prediction. The effective rate includes atomic radiative bottlenecks; chemical freeze-out is distinct from photon decoupling by scattering.

Salinity measures dissolved salt content. A transport model must specify its units; salt mass fraction is one convenient convention. A dilute solution's ice liquidus is often approximated by $T_L=T_m-mC$. Salt-free ice has zero salinity, and the moving interface's salt balance distinguishes salt rejection during freezing from dilution during melting.