Molecular diffusion is the component of material transport caused by the random thermal motion of molecules, measured relative to bulk motion of their surroundings. A diffusion coefficient is a property of a particular substance in a particular medium under specified conditions, not a universal speed of spreading. Its dependence on the solute, solvent and temperature is precisely what makes a spreading experiment informative. A random walk idealizes this motion; the Brownian motion description provides a continuous approximation over sufficiently long times. ^[ref]

A motivating experiment: watch dye enter a gel. Samprovalaki, Robbins and Fryer studied rhodamine 6G and methylene blue entering agar gel from two side reservoirs. A covered slab, $10\times 5\times 1\mathrm{mm}$, sat on a controlled thermal stage; the dye was brought to the same temperature. They photographed the gel every ten minutes, averaged image intensity across a strip, and fitted the changing concentration to a diffusion equation model. The cover limited evaporation and stopped dye travelling over the gel surface. Runs at $30$, $50$ and $70{}^{\circ }\mathrm{C}$ were repeated in triplicate. The measured spatial colour change therefore supplied a quantitative transport test, rather than merely showing that dye eventually spreads. Changing lighting also changes image intensity, so the optical response had to be checked. Importantly for interpretation, their constant-diffusion coefficient fit was imperfect and the gel coefficient is an effective transport coefficient, not automatically the value in free water. ^[ref]

A finite-pulse version to test width growth. The following is a proposed teaching variant, not that paper's apparatus or a report of measurements. Make a long, narrow, transparent gel channel of uniform cross-section. Join a thin dyed gel band between two undyed lengths of the same gel, with good contact and no air gaps; the dye should initially fill the cross-section so the dominant concentration variation is along the channel coordinate $x$. After assembly, keep the covered channel horizontal at uniform temperature, away from vibration. The dye is a finite initial pulse: no reservoir supplies more dye. Choose a dye whose image response can be calibrated and whose interaction with the gel is small enough for an approximately constant diffusion coefficient to be plausible.

Photograph the whole pulse at recorded times with fixed lighting, focus and exposure, including a length scale. Use uniformly dyed standards of the same gel thickness and a blank gel to convert image intensity to concentration $C(x,t)$; do not assume raw brightness is proportional to concentration. Average across the channel width and examine several parallel strips to check the one-dimensional approximation. Measure the initial profile instead of treating the band as infinitely thin. For a finite pulse with negligible end effects, define its spatial integral $Q$, centre $\bar{x}$ and width $\sigma$ byHere ${\sigma }^2$ is the variance of the normalized profile $C/Q$, and $\sigma$ is its standard deviation. If $C$ is measured in $\mathrm{mol}{\mathrm{m}}^{-3}$, then $Q$ has units $\mathrm{mol}{\mathrm{m}}^{-2}$; multiplying by the channel cross-section gives the total amount of substance. For the unbounded one-dimensional diffusion equation with constant $D$, conserved $Q$, finite variance and sufficiently decaying tails,Thus plot width squared against time: its slope determines the diffusion coefficient, while its intercept includes the finite starting width. This moment law holds even for a non-Gaussian distribution initially; it follows by applying integration by parts twice to the diffusion equation. ^[ref]

For the particularly readable case of an initially Gaussian distribution centred at zero, uniform advection at speed $u$ and constant diffusion coefficient $D$ give the one-dimensional concentration profileIts centre moves as $\bar{x}=ut$ and its variance grows as ${\sigma }_0^2+2Dt$. Our illustrative choice ${\sigma }_0=0.50\mathrm{mm}$ and $D=0.20\mathrm{m}{\mathrm{m}}^2{\mathrm{h}}^{-1}$ gives widths $0.50$, $0.81$ and $1.36\mathrm{mm}$ at zero, one and four hours. These are calculated values, not a measured dye coefficient. Once $2Dt$ dominates the initial variance, width grows as $\sqrt{2Dt}$: four times the elapsed time gives approximately twice the width. A rectangular starting band will not immediately have a Gaussian distribution, although the same width-squared law still applies under the stated assumptions.

Controls and limits of the interpretation. In the proposed experiment, check that $Q$ stays constant and that the centre remains stationary while the width increases. Include a blank and a uniformly dyed gel photographed over the same period: an apparent signal change there exposes lighting drift, dye fading or evaporation. Repeat with a lower starting dye concentration and a second initial width; agreement of the slopes is a check on constant $D$ and on preparation effects. Keep the pulse far from the channel ends and record its faint tails: truncation and background errors can strongly bias a variance. Poor contact, dye binding, chemical reactions or transport in a liquid film can invalidate the simple model. A gel suppresses bulk liquid motion, but its inferred diffusivity need not equal that in the pure solvent. A stationary centre alone does not exclude symmetric flows, and linear width-squared growth alone does not prove a molecular mechanism: turbulence can also produce an effective diffusive description. These are reasons to combine the profile test with the controlled setup.

In a cyclic group with generator $g$, $n$ participants with secret exponents $a_j$ can exchange $n$ tokens around a ring. Token $j$ starts at $g^{a_j}$ and is successively exponentiated by its next $n-2$ recipients before being sent to the final recipient, whose exponent is the only one missing. That recipient completes the exponentiation locally. Each token makes $n-1$ transmissions, and everyone obtains $g^{{\prod }_j{a}_j}$. Thus $n(n-1)$ transmitted group elements suffice. As with two-party Diffie-Hellman key exchange, authentication is needed.

For a differential three-form $A$ with $F=dA$ on an oriented eleven-dimensional manifold, varying $\int (\frac{1}{2}F\wedge \star F+F\wedge F\wedge A)$ with fixed metric gives $\delta S=\int \delta A\wedge (d\star F+3F\wedge F)$ up to a boundary term. Indeed $d(\delta A\wedge F\wedge A)=d\delta A\wedge F\wedge A-\delta A\wedge F\wedge F$, so the two differentiated $F$ factors and the differentiated $A$ factor contribute equally. The shift $A\mapsto A+d\Lambda$ changes the density by $d(F\wedge F\wedge \Lambda )$, preserving the field equation.

For $n=1$, differentiability at $a$ means that the difference quotient has a finite two-sided limit of a function. For $n\ge 2$, require derivatives through order $n-1$ throughout some open neighbourhood of $a$, and require the final difference quotientto exist as a finite real number. The last derivative need not exist elsewhere or be continuous at $a$. In particular, existence of $f^{\prime }(a)$ by itself does not justify taking a second derivative: the function $f^{\prime }$ must first be defined in a neighbourhood of $a$.

At a point with $f(x)\ne 0$, factor the difference of squares and divide by $f(x+h)+f(x)\to 2f(x)$ to show that differentiability of $f^2$ is equivalent to that of $f$. At a zero of $f$, its square has a local minimum; if differentiable, its derivative must vanish. The difference quotient then vanishes exactly when $f(x+h{)}^2/|h|\to 0$, equivalently $f(x+h)/\sqrt{|h|}\to 0$. For example, $f(t)=|t{|}^{\alpha }$ with $1/2<\alpha <1$ is not differentiable at zero, but its square is.

If all partial derivatives exist near a point and are continuous there, successive coordinate increments and the one-variable mean value theorem express $f(a+h)-f(a)$ as ${\sum }_j{h}_j{\partial }_{j}f(a+{\eta }_j(h))$ with ${\eta }_j(h)\to 0$. Subtracting the displayed linear map leaves $o(\Vert h\Vert )$, proving Fréchet differentiability. In two dimensions the absolute remainder is at most $ϵ(|h_1|+|h_2|)\le \sqrt{2}ϵ\Vert h\Vert$ for small enough increments.

The function $f(x)=x^2\sin (1/x^2)$ for $x\ne 0$, extended by $f(0)=0$, is differentiable everywhere. At zero its derivative is zero by the difference quotient, while away from zero $f^{\prime }(x)=2x\sin (1/x^2)-2x^{-1}\cos (1/x^2)$ is unbounded near zero. Differentiability of a function is therefore weaker than continuity of its derivative.

Set $f(0)=0$. The difference quotient has Euclidean norm $|t|$, so $f^{\prime }(0)=0$. For $t\ne 0$, differentiation gives $\Vert f^{\prime }(t){\Vert }^2=1+4t^2$. Thus this differentiable curve has a derivative discontinuous at zero, despite each scalar component satisfying the usual one-dimensional derivative properties.

Put $m=\left(\genfrac{}{}{0ex}{}{k}{2}\right)$, take odd $q\equiv 1(\mathrm{mod}2m)$, and let $H$ be the index-$m$ multiplicative subgroup. If a $k$-set $A$ has its unordered pair differences in distinct $H$-cosets, choose $S\subset H$ containing exactly one element from each pair $\{h,-h\}$. Then $\{sA:s\in S\}$ is a difference family for a Steiner 2-design. Indeed one unordered pair supplies each coset, and its two difference orientations multiplied by $S$ cover that coset once. Translations yield a $2$-$(q,k,1)$ Steiner system. The cyclotomic pattern extension by a second-moment bound supplies suitable $A$ for all sufficiently large $q$ in this progression.

Characterize the instrument and fixture before inferring permittivity: use the prescribed open and short measurements to compensate residual parallel response and series electrical impedance, and check a known reference capacitor or the required load standard. Keep cables and electrode geometry reproducible. An open fixture measurement is distinct from the empty-gap reference capacitor; blindly subtracting its entire capacitance would remove the desired reference signal. Compensation cannot determine an incorrectly measured specimen thickness or eliminate all specimen-dependent fringing field of a capacitor errors. ^[ref].

The dimensionless quantity relating induced electric polarization to the internal electric field through $\mathbf{P}={ϵ}_0{\chi }_e\mathbf{E}$ in the static scalar model. It describes the material contribution; relative permittivity includes the vacuum term as well, ${ϵ}_r=1+{\chi }_e$. Zero electric susceptibility gives ${ϵ}_r=1$, not zero permittivity.

The scalar ${\chi }_e$ assumes a local response with linearity and isotropy, settled electric polarization, fixed material conditions, and no residual electric polarization at zero electric field. It relates $\mathbf{P}$ to the averaged total internal electric field, which already includes the field of the bound electric charges. It does not relate $\mathbf{P}$ to the field the external electrodes would produce in vacuum. The distinction is essential for a self-consistent calculation.^[ref]

Substitute the scalar relation into the definition of electric displacement field:Comparison with $\mathbf{D}=ϵ\mathbf{E}$ proves $ϵ={ϵ}_0(1+{\chi }_e)$ and ${ϵ}_r=1+{\chi }_e$. Both $\mathbf{D}$ and $\mathbf{P}$ have units $C/{\mathrm{m}}^2$, so ${\chi }_e$ is a dimensionless quantity. In a stipulated static model with ${ϵ}_r=3$, ${\chi }_e=2$: at a prescribed internal electric field, the material contribution $P$ is twice ${ϵ}_{0}E$. This is a hypothetical response, not a measured material constant. The electric susceptibility is a convenient bridge from microscopic dielectric polarization to a macroscopic capacitance measurement, but it is not a measure of electrical conductivity.

The image of the diagonal map in $X\times X$. For a manifold, the diagonal subset is an embedded submanifold diffeomorphic to $X$.

If the input qubit of the Deutsch algorithm acquires conditional environment states with overlap $⟨e_0|e_1⟩=v{e}^{i\alpha }$ between its two Hadamard gates, its fixed zero-for-constant, one-for-balanced rule succeeds with probability $(1+v\cos \alpha )/2$. A known phase can be compensated before the last Hadamard gate, raising the success probability to $(1+v)/2$. An orthogonal pair of environment states gives $v=0$ and destroys all information in the input qubit about the answer; unit overlap modulus preserves coherence up to a known phase.

For smooth fields into ${\mathbb{R}}^l$ with nonnegative differentiable potential and finite energy, Derrick scaling in two spatial dimensions forces the potential integral to vanish. Thus the field takes values where $U=0$ and $\nabla U=0$, so its static field equation makes every component harmonic. Each first derivative is an entire harmonic function in L2 space; the mean-value inequality on arbitrarily large balls forces it to vanish. Hence the field is constant. The scale identity alone does not exclude a pure-gradient model; this additional argument uses the flat unconstrained target. Curved-target harmonic maps, for example the O3 nonlinear sigma model, can instead have finite-energy nonconstant solutions.

The category $D^{+}(X)$ is obtained from bounded below cochain complexes of sheaves by passing to cochain-homotopy classes of maps and then inverting quasi-isomorphisms. It can equally be modeled by injective resolutions. The cohomology-sheaf functors descend to this localization, and an object is zero exactly when all its cohomology sheaves vanish.

A bounded complex is constructible for a given stratification when each of its cohomology sheaves restricts to a finite-rank local system on every stratum. Constructibility concerns cohomology sheaves, not an arbitrary choice of resolution.

On bounded constructible complexes over locally compact stratified spaces, Verdier duality is $D_{X}E=R\mathcal{H}om(E,{\omega }_X^{\bullet })$, where ${\omega }_X^{\bullet }$ is the dualizing complex. For an oriented real $N$-manifold, $D_X{\mathbb{Q}}_X\cong {\mathbb{Q}}_X[N]$. Proper direct image commutes with this duality, and $i_x^{!}{D}_{X}E\cong D_{\{x\}}{i}_x^{\ast }E$. These properties exchange stalk support conditions with costalk conditions.

A ghost is a morphism inducing zero on every cohomology sheaf, but it need not itself be zero. For the equator $j:S^1\hookrightarrow S^2$ and complement $i:U\hookrightarrow S^2$, the connecting morphism $j_{\ast }{\mathbb{Q}}_{S^1}\to i_{!}{\mathbb{Q}}_U[1]$ is a ghost. Its source and target have cohomology sheaves in different degrees, but its induced map $H^1(S^1)\to H_c^2(U)$ is nonzero, proving that the derived morphism is nonzero.

For a perfect fluid with $P=w\rho$, in Newtonian gauge in cosmology the density perturbation on fluid-comoving slices is the displayed combination of the fractional density perturbation and velocity potential. The Einstein energy and momentum constraints combine into ${\nabla }^2\varphi =4\pi G{a}^2\bar{\rho }\Delta$ when scalar anisotropic stress vanishes. This gives a relativistic Poisson constraint valid beyond the subhorizon limit.

Revealed preference infers rankings from choices: a chosen bundle is directly revealed at least as desirable as any bundle affordable in the same budget. Indirect comparisons use chains of such observations.