For a body lying above an obstacle with height $f$, its normal displacement graph $w$ must obey the displayed inequality. Equality allows contact and strict inequality means separation. This is a unilateral mechanical constraint: motion away from the obstacle is allowed, but motion through it is forbidden. In nonadhesive frictionless contact the associated normal contact force can push the body outwards but cannot pull it into the obstacle.

For a nonadhesive obstacle, $g$ is the signed separation and $r$ its outward normal contact force. Nonpenetration gives $g\ge 0$, absence of adhesion gives $r\ge 0$, and reaction is possible only at contact, giving $gr=0$. In a small-slope taut-string model, $g=w-f$ and force balance gives $r=p-T{w}^{\prime \prime }$ for downward applied load $p$. The three conditions then become $(w-f)(T{w}^{\prime \prime }-p)=0$, $w\ge f$, $T{w}^{\prime \prime }-p\le 0$. Multiplication by admissible displacement differences and integration gives the associated variational inequality, connecting a local contact law to constrained energy minimization.

A toric contact lens has different optical powers along different principal meridians, allowing correction of astigmatism. Its orientation matters because rotating those meridians changes the direction of correction. Optical power and rotational stability are therefore separate design requirements.

For smooth equality constraints $h_j=0$ and inequality constraints $g_i\ge 0$, this condition requires linear independence of all equality gradients and active inequality gradients at the feasible point. It ensures existence of Lagrange multipliers at a local optimum and uniqueness of their values for a fixed objective gradient. It implies the Mangasarian–Fromovitz constraint qualification by solving the independent gradient equations with zero equality derivatives and positive active inequality derivatives.

At a feasible point for smooth $h=0$, $g\ge 0$, the equality gradients must be independent and there must be a direction $v$ with $Dhv=0$ and $D{g}_{i}v>0$ for every active inequality. It supplies a first-order direction strictly into all active inequality constraints while respecting the equality tangent space. This constraint qualification ensures the usual Karush-Kuhn-Tucker conditions are necessary for local optimality. For a strictly convex quadratic inequality restricted to an affine equality space, it holds at every feasible point exactly when a strictly feasible point exists; moving from a boundary point toward a strictly feasible point supplies the required direction.

Retention of a mass of faeces, commonly in the rectum. An enema is one possible treatment following assessment. ^[ref]

Every real interval is connected. Otherwise a separation into two nonempty relatively open sets defines a continuous real function equal to zero on one side and one on the other. Choose one point of each set. The intermediate value theorem on the closed segment between them requires the value $1/2$, a contradiction. The converse also holds: a connected subset of the real line contains every point between two of its points, since omitting an intermediate point separates it into its intersections with the two open half-lines.

Suppose $X,Y$ are nonempty connected spaces, and $A⊊X$, $B⊊Y$ are proper subsets. Then $C=(X\times Y)\setminus (A\times B)$, with its subspace topology from the product topology, is a connected space. No closed set or open set hypothesis on $A,B$ is needed.

Choose $x_0\in X\setminus A$, $y_0\in Y\setminus B$. The two topological subspaces $\{x_0\}\times Y$ and $X\times \{y_0\}$ are homeomorphic to the factors, intersect at $(x_0,y_0)$, and lie in $C$. Their union is a connected subset $H$. Every $(x,y)\in C$ has $x\notin A$ or $y\notin B$. In the first case its entire vertical copy $\{x\}\times Y$ lies in $C$ and meets $H$; in the second case the horizontal copy $X\times \{y\}$ does so. Thus $C$ is a connected union with a connected hub.

Both properness assumptions are needed for this general result. Taking $X=Y=\mathbb{R}$, $A=X$ and $B=\{0\}$ gives the two disjoint open half-planes $y<0$ and $y>0$, a separation of a topological space of the complement.

Let $X$ be a compact Hausdorff space and let $K_i$ be compact subsets such that all finite intersections are nonempty and connected. Their total intersection $K$ is nonempty by the finite intersection characterization of compactness, and compact because it is closed in $X$.

If $K$ were not connected, its separation of a topological space would divide it into two nonempty compact subsets. Separation of disjoint compact subsets gives disjoint ambient open sets $U,V$ containing the two pieces. The family consisting of $X\setminus (U\cup V)$ and all $K_i$ has empty total intersection, so some finite subfamily has empty intersection. Hence a finite intersection $K_J$ of the $K_i$ lies in $U\cup V$. Since $K\subseteq K_J$ meets both open sets, they disconnect $K_J$, a contradiction. The hypothesis on all finite intersections is stronger than requiring each individual $K_i$ to be connected.

A subset of a topological space is a connected set when its subspace topology is connected. Thus connectedness is tested using relatively open sets, not only subsets open in the ambient space.

A topological space is disconnected space if it can be written as the union of two disjoint nonempty open subsets. Such a pair is a separation of a topological space. Each of the subsets is also closed, and conversely a nontrivial subset that is both open and closed gives a separation of a topological space with its complement.

Compose the lens-to-quadrant Möbius map with the logarithm having argument in $(0,\pi /2)$ on the first quadrant. Polar coordinates show that the latter maps bijectively to the horizontal strip $0<\mathrm{Im}\zeta <\pi /2$. Its inverse is the exponential function, so the combined inverse is $(e^{\zeta }-(1+i))/(e^{\zeta }+(1+i))$. Both derivatives are nonzero, proving conformality. The logarithm's branch fixes the strip's vertical position; adding $2\pi i$ shifts that position without changing the exponential inverse.

The map $z\mapsto z^2$ is a bijective conformal map from the open first quadrant to the complex upper half-plane. In polar coordinates it doubles arguments from $(0,\pi /2)$ to $(0,\pi )$, and its derivative $2z$ never vanishes inside the quadrant. Its inverse is the branch of the square root with argument in $(0,\pi /2)$. By conformal invariance of harmonicity, a harmonic function $f$ on the half-plane pulls back to $g(x,y)=f(x^2-y^2,2xy)$. The two boundary rays map to the positive and negative real axes, respectively.

For $a>0$, this conformal map takes $\mathrm{Re}z>0$, $0<\mathrm{Im}z<a$ bijectively onto the upper half-plane. Its imaginary part is $\sinh (\pi x/a)\sin (\pi y/a)>0$, and its derivative is nonzero in the interior. The end wall maps to $[-1,1]$ and the two long walls to the complementary real rays. To prove the map is onto and injective, first put $s=e^{\pi z/a}$, which maps onto $|s|>1$, $\mathrm{Im}s>0$. The equation $s+s^{-1}=2\zeta$ has exactly one root outside the unit circle when $\mathrm{Im}\zeta >0$: the roots have product one and neither has unit modulus. The identity $\mathrm{Im}\zeta =(1-|s{|}^{-2})\mathrm{Im}s/2$ puts that root in the correct half-plane. This permits planar potential flow problems in a closed-end channel to be solved by the method of images.

On $S_h=\{z:0<\mathrm{Im}z<h\}$, let $\beta \in S_h$ and write $e^{\pi \beta /h}=A+iB$ with $B>0$. The exponential function $E(z)=e^{\pi z/h}$ is a conformal map from the strip bijectively onto the upper half-plane, sending $ih/2$ to $i$. The affine map $w\mapsto A+Bw$ is an automorphism of the half-plane sending $i$ to $E(\beta )$. Conjugate it by $E$, using the principal complex logarithm as the inverse. The displayed formula is a conformal automorphism of the strip and satisfies $g_{\beta }(ih/2)=\beta$. The branch range $0<\mathrm{Im}\mathrm{Log}w<\pi$ is what prevents an unwanted vertical period in this inverse.

If a degree-$N$ polynomial $f(\zeta )={\sum }_{j=1}^N{a}_j{\zeta }^j$ maps the unit disc conformally onto $D$, Green's theorem expresses $M_n={\iint }_D{z}^{n}dxdy$ as the residue of $f^n{f}^{\prime }{\sum }_j{\bar{a}}_j{\zeta }^{-j}$ times $\pi$. Its lowest possible power is $n-N$, so no residue occurs when $n\ge N$. For a quadratic map, $M_0=\pi (|a_1{|}^2+2|a_2{|}^2)$ and $M_1=\pi a_1^2{\bar{a}}_2$. The first moment can be complex; the zeroth is area.

A researcher who studied how nacre's layered structure produces structural coloration. In a study with Chunhui Zhou and Elina Myagkaya, Fan performed the theoretical analysis and computational simulation and designed the experiments. Modeling interference between reflected light from successive layers made the color pattern a probe of layer thickness.^[ref]

A researcher who coauthored a study of structural coloration in nacre. Zhou investigated the origin of pearl iridescence and arranged and supervised the experiments. The study compared the spatial color pattern with a layered optical model, using interference to infer structural information that ordinary color matching alone can obscure.^[ref]

A coauthor of a study connecting nacre's structural coloration with its layered internal structure. Myagkaya performed the scanning electron microscopy analysis, providing a direct structural measurement to accompany the optical model. The work illustrates how measurements of reflected light can constrain layer thickness, while microscopy checks the structure responsible for the color.^[ref]

Mass concentration is the mass of a specified constituent divided by the final sample volume, ${\rho }_i=m_i/V$. Its SI unit is $\mathrm{kg}{\mathrm{m}}^{-3}$; laboratory data also commonly use mass per litre.^[ref]

For example, $10\mathrm{g}$ of a solute in a final solution volume of $1\mathrm{L}$ gives $10\mathrm{g}{\mathrm{L}}^{-1}$. The denominator is the solution volume after mixing, not necessarily the volume of solvent initially added.

If $w_i$ is mass fraction and $\rho$ the mixture mass density, then ${\rho }_i=w_i\rho$. Heating or compressing the sample can change the mass concentration through a change of volume while leaving the mass fraction unchanged.