For a simple polygon with cyclically ordered vertices of a polygon $(x_i,y_i)$, its signed area is $\frac{1}{2}{\sum }_i(x_i{y}_{i+1}-x_{i+1}{y}_i)$, with indices read cyclically. Reversing the traversal changes the sign; taking the absolute value gives the area. Each summand is a two-dimensional determinant.

A cyclic polygon has all its vertices of a polygon on a single circle, its circumcircle. Every nondegenerate triangle is a cyclic polygon, but a general quadrilateral is not: a convex quadrilateral is cyclic exactly when opposite interior angles sum to $\pi$. Every regular polygon is cyclic. For a side subtending central angle $\theta$, the chord length is $2R\sin (\theta /2)$, where $R$ is the circumradius; this converts placement on a circle into side-length calculations.

A regular star polygon $\{p/q\}$ joins every $q$-th vertex of a regular $p$-gon. When $\gcd (p,q)=1$, this traces one closed polygonal circuit.

A dodecagon is a polygon with twelve sides. A regular dodecagon has twelve equal sides and twelve equal interior angles, each $10\pi /12=5\pi /6$, or $150^{\circ }$. Joining its centre to its vertices divides it into twelve congruent triangles with central angle $\pi /6$. If its circumradius is $R$, these triangles give area $12R^2\sin (\pi /6)/2=3R^2$.

The weight-average molar mass divided by the number-average molar mass. Its conventional written symbol is Đ. With number weighting it equals $1+\mathrm{Var}(M)/⟨M{⟩}^2$, so it is at least one and equals one only for a uniform molar mass. It measures one aspect of distribution breadth, not the full shape of a molar mass distribution.

Formation of Si–O–Si linkages from silanol groups, with elimination of water or another small molecule according to the reacting functional groups.

Use of identified pollen in sediments or other material to reconstruct vegetation and environmental change. Production, transport and preservation affect the relationship between counts and vegetation.

The Brent variant of Pollard rho finds factors of a composite integer $N$ by iterating a map such as $x\mapsto x^2+c(\mathrm{mod}N)$. A collision modulo an unknown prime factor can be detected through $\gcd (|x-y|,N)$. Brent's variant organizes comparisons in blocks whose lengths double, reducing the work of cycle detection compared with comparing two iterates at every step.^[ref]

It can also batch differences: compute their product modulo $N$ and take one greatest common divisor per block. If that gcd is $N$, retry the saved differences individually to recover a proper factor, or restart with a different orbit. The commonly used square-root-of-factor runtime estimate is heuristic and depends on treating the orbit modulo that factor like a random map.

Pyroelectricity is a change in spontaneous electric polarization with temperature. When electrodes collect the corresponding change of surface electric charge, a changing temperature can produce an electric current even without an applied voltage.

For electrode area $A$, coefficient $p$ and sufficiently uniform sample temperature, the ideal short-circuit signal is $I=pAdT/dt$. Heating and cooling reverse its sign. A constant elevated temperature does not sustain that ideal current once the temperature stops changing. Thermal gradients, leakage and mechanical strain can add signals, motivating the distinction between primary pyroelectric effect and secondary pyroelectric effect.

electric polarization present in a polar material without an externally applied electric field.

A phase without spontaneous switchable electric polarization, although an applied electric field can induce electric polarization. A material can be paraelectric above a transition to a ferroelectric phase.

A tensor derivative of polarization with respect to a sublattice displacement at specified macroscopic electrical conditions. It includes electronic rearrangement and need not equal a formal ionic charge.

For integer angular frequency $m\ge 0$, the Laplacian in polar coordinates gives the displayed identity on $r>0$ by direct differentiation. Consequently a source $\beta r^q\cos (m\theta )$ has particular solution $\beta r^{q+2}\cos (m\theta )/[(q+2{)}^2-m^2]$ when the denominator is nonzero. The sine case is identical. If $s=q+2=\pm m\ne 0$, differentiating the identity with respect to $s$ gives the resonant particular solution $\beta r^s\ln r\cos (m\theta )/(2s)$. For $s=m=0$, the radial source $\beta r^{-2}$ instead has particular solution $\beta (\ln r{)}^2/2$. These are local statements on the punctured plane; boundedness, smoothness at the origin and boundary data must be imposed separately by adding suitable harmonic functions.

Let $Y_{\ell }$ be a spherical harmonic of degree $\ell$ on the unit sphere. The Laplacian in spherical coordinates separates its angular part from a radial power, giving the displayed identity for $r>0$. Thus a Poisson equation $\Delta \Phi =r^q{Y}_{\ell }$ has the particular solution $\Phi =r^{q+2}{Y}_{\ell }/[(q+2)(q+3)-\ell (\ell +1)]$ when the denominator is nonzero. Add $A{r}^{\ell }{Y}_{\ell }+B{r}^{-\ell -1}{Y}_{\ell }$ to impose the radial boundary conditions. If the denominator vanishes at $p=q+2$, differentiating the displayed identity with respect to $p$ gives instead $\Delta (r^p\log r{Y}_{\ell })=(2p+1)r^{p-2}{Y}_{\ell }$. Regularity at the origin and decay at infinity must be checked separately; the radial identity alone is local away from zero.

For a symmetric electrolyte with mobile charges $\pm q$, bulk density $n_0$ of each sign, and constant permittivity $\epsilon$, the Poisson-Boltzmann equation gives $n_{\pm }=n_0{e}^{\mp q\psi /(k_{B}T)}$. In planar geometry let $E=-{\psi }^{\prime }$ and $\rho =q(n_{+}-n_{-})$. Then $k_{B}T(n_{+}+n_{-}{)}^{\prime }=\rho E$ and $\epsilon E{E}^{\prime }=\rho E$ by Gauss's law, proving that the displayed bulk-subtracted normal stress is constant. At the midpoint of identical equally charged plates, symmetry gives $E=0$ and $\Pi =2n_0{k}_{B}T[\cosh (q{\psi }_{\mathrm{mid}}/(k_{B}T))-1]$. Its ionic osmotic pressure and electrostatic contributions together transmit the force; bare surface-charge repulsion alone omits essential terms.

For point ions of charge $\pm e$ in potential $\psi$, equilibrium electrochemical potentials give $n_{\pm }=n_0\exp [\mp e\psi /(k_{B}T)]$, with $n_0$ the reservoir density of each species. The charge density is $-2e{n}_0\sinh [e\psi /(k_{B}T)]$. A negative wall enriches positive counterions and depletes negative co-ions.

For $|e\psi |\ll k_{B}T$, $\psi \prime \prime ={\kappa }^2\psi$ with ${\kappa }^2=2e^2{n}_0/(\epsilon k_{B}T)$ for a symmetric monovalent salt with per-species reservoir density $n_0$. The half-space solution with $\psi (0)=\zeta$ and vanishing bulk potential is $\zeta e^{-\kappa x}$. If density denotes the sum of species, the written factor of two changes.

A strong negative surface potential creates exponentially enhanced positive counterion density and exponentially suppressed negative co-ion density. The co-ion population remains positive, unlike an extrapolated linear approximation. Finite ion size and correlations eventually limit the applicability of point-ion mean-field theory.

For a symmetric monovalent point-ion electrolyte next to a plane, define $u=e\psi /(k_{B}T)$ and ${\kappa }_D^2=2e^2{n}_{\infty }/(\epsilon k_{B}T)$, with $n_{\infty }$ the density of each sign. The Poisson-Boltzmann equation becomes $u^{\prime \prime }={\kappa }_D^2\sinh u$. Multiplying by $u^{\prime }$ and matching zero field and potential in the bulk gives $(u^{\prime }{)}^2=2{\kappa }_D^2(\cosh u-1)$. The decaying branch integrates to the displayed profile, reducing to $u_0{e}^{-{\kappa }_{D}z}$ at weak potential. This solution assumes a salt reservoir and is distinct from a counterion-only diffuse layer.

A negatively charged plane with monovalent positive counterions in $z>0$ and no salt reservoir has the mean-field profile displayed above, where $b=e/(2\pi {\ell }_B|\sigma |)$ and Bjerrum length ${\ell }_B=e^2/(4\pi \epsilon k_{B}T)$. Its potential relative to the wall is $2k_{B}T\log (1+z/b)/e$. The positive-ion charge density integrates to $|\sigma |$, neutralizing the plane; the field decays algebraically. The potential has no finite zero-at-infinity reference, unlike the screened symmetric-reservoir problem.