Four mutually tangent circles with signed curvatures $k_i$ satisfy $(k_1+k_2+k_3+k_4{)}^2=2(k_1^2+k_2^2+k_3^2+k_4^2)$. Curvature is the reciprocal of the radius, with an enclosing circle assigned negative curvature. Given three curvatures, solving the quadratic equation gives $k_4=k_1+k_2+k_3\pm 2\sqrt{k_1{k}_2+k_2{k}_3+k_3{k}_1}$, providing the two completions of the configuration.

A circular segment is the part of a disk bounded by a chord and one of the two circular arcs joining its endpoints. For radius $R$ and the chosen arc's central angle $0\le \varphi \le 2\pi$ in radians, its area is $R^2(\varphi -\sin \varphi )/2$. Subtract the signed central triangle of area $R^2\sin \varphi /2$ from the circular sector of area $R^2\varphi /2$. Angles below $\pi$ give the smaller segment and angles above $\pi$ the larger one.

At a point $T$ on a circle with centre $O$, the tangent line is perpendicular to the radius $OT$. For example, differentiating $|\mathbf{r}(s)-O{|}^2=R^2$ along a parametric curve on the circle gives $(\mathbf{r}-O)\cdot {\mathbf{r}}^{\prime }=0$, showing that the radial vector and tangent direction are orthogonal vectors. Conversely, the perpendicular straight line through $T$ has only that point in common with the circle: for a nonzero displacement $v$ along it, $|T+v-O{|}^2=R^2+|v{|}^2>R^2$.

Two distinct circles are tangent when they meet at exactly one point. For external tangency the distance between centers is the sum of the radii; for internal tangency it is their absolute difference. These relations turn many circle packing problems into exact distance equations.

The power of a point $P$ relative to a circle with center $O$ and radius $r$ is $O{P}^2-r^2$. If a line through $P$ meets the circle at $A$ and $B$, the product of the directed distances $PA$ and $PB$ equals this quantity.

Given three mutually externally tangent circles with disjoint interiors, a Soddy circle is a circle tangent to all three. The inner solution lies in the curvilinear triangular gap. In signed curvatures $b_i=\pm 1/r_i$, the two possible fourth curvatures satisfyA containing circle uses negative curvature. For three unit circles, the inner curvature is $3+2\sqrt{3}$, while the outer curvature $3-2\sqrt{3}<0$ describes a containing circle. The formula is the quadratic solution of the Descartes' circle theorem; zero curvature is interpreted as a tangent line.

Astatotilapia burtoni is a cichlid fish used in research on social status, reproduction and neuroendocrine signalling. Changes in social circumstances can be related to measurable changes in behavior and physiological state within the same individual.

CheW is a coupling protein linking chemotaxis receptor arrays to CheA-associated signalling complexes. Removing it from a reconstituted system can test its functional requirement, while a physical-interface assay addresses direct contacts.

Unconstrained thermal, mechanical and particle-exchange contacts require equality of temperature, pressure and the chemical potential of each exchangeable species. Walls, external fields, interfaces and chemical reactions change which variations are allowed.

The derivative of Gibbs energy with respect to reaction extent at fixed temperature and pressure: ${\Delta }_{r}G={\sum }_i{\nu }_i{\mu }_i$. For a reversible electrochemical cell reaction, ${\Delta }_{r}G=-nFE$.

The stoichiometric sum of standard molar entropies of products minus those of reactants: ${\Delta }_r{S}^{\circ }={\sum }_i{\nu }_i{S}_i^{\circ }$.

The temperature relation $\mathrm{d}\ln K/\mathrm{d}T={\Delta }_r{H}^{\circ }/(R{T}^2)$ for a dimensionless thermodynamic equilibrium constant. With constant reaction enthalpy, integration gives the difference of reciprocal temperatures.

The linked equilibria among dissolved carbon dioxide, bicarbonate, carbonate and hydrogen ions in seawater. At a fixed alkalinity, added carbon changes speciation and acidity, limiting how much extra carbon can be held at a given atmospheric partial pressure. Exchange and chemical partitioning therefore do not behave like irreversible destruction by a single first-order sink.

Reversible association of two molecular units to form a dimer.

The ionic product of water describes the balance between hydronium and hydroxide produced by self-ionization of water:For a dilute water-based chemical solution whose water activity is approximately one,The ionic thermodynamic activities are dimensionless. In a dilute molal convention, $a_i\simeq b_i/b^{\circ }$, where $b_i$ are molalities and $b^{\circ }=1\mathrm{mol}\mathrm{k}{\mathrm{g}}^{-1}$. The full equilibrium constant for the reaction includes water activity squared in the denominator; concentrated solutions need that solvent contribution as well.

Near ambient pressure at $25{}^{\circ }\mathrm{C}$, $K_w$ is approximately $10^{-14}$. Equal ionic activities in neutral dilute water are therefore approximately $10^{-7}$, giving pH near seven. The value changes with temperature and pressure, so neutrality is not universally the same as pH seven. ^[ref]

For $A\rightleftharpoons 2B$, the conserved amount is $M=N_A+N_B/2$. The total particle count $N_A+N_B$ changes during reaction. Defining a dissociated fraction by $\alpha =N_B/(2M)$ gives $N_A=M(1-\alpha )$.

For ideal-gas single-particle functions $Z_1^{(i)}=C_{i}V{T}^{3/2}$ and reaction $A\rightleftharpoons 2B$, equilibrium gives $K_N=N_B^2/N_A=(C_B^2/C_A)V{T}^{3/2}$. This particle-number ratio contains a volume factor; the concentration ratio $n_B^2/n_A=K_N/V$ does not.

With the weighted inner product ${\int }_{-1}^1\overline{f}gdx/\sqrt{1-x^2}$, the Chebyshev differential equation has the Sturm-Liouville form with coefficients $p=\sqrt{1-x^2}$, $w=(1-x^2{)}^{-1/2}$ and $q=0$. The substitution $x=\cos \theta$ converts its operator to $-d^2/d{\theta }^2$ on $(0,\pi )$ and its inner product to the ordinary one in $d\theta$. The natural endpoint conditions $p{y}^{\prime }\to 0$ become zero derivative conditions at $\theta =0,\pi$.

Consequently the eigenfunctions are the Chebyshev polynomials $T_n(x)=\cos (n\arccos x)$ with eigenvalues $n^2$, $n=0,1,2,\dots$. Their squared norms are $\pi$ for $n=0$ and $\pi /2$ otherwise. In particular $T_1=x$ and $T_3=4x^3-3x$ are orthogonal. The endpoints of the original differential expression are singular; specifying these transformed Neumann boundary conditions defines its self-adjoint realization. The Dirichlet eigenfunctions of the Chebyshev differential operator use a different endpoint condition and have sine rather than cosine profiles.

The densitycontains the term linear in the potential from minimal coupling, but omits $e^2{A}_{\mu }{A}^{\mu }|\varphi {|}^2$. It has exact global phase symmetry, not exact arbitrary local phase symmetry. For a prescribed background its field equation is $(\square +m^2)\varphi +ie(\partial \cdot A)\varphi +2ieA\cdot \partial \varphi =0$, together with its complex conjugate. The omitted quadratic interaction must not be added silently when solving this specified theory.

The vector-minus-axial-vector charged weak current contains the left-chiral projector $P_L=(1-{\gamma }^5)/2$. It violates parity because parity interchanges left and right chiral fields. For a massless particle chirality fixes helicity, but a massive charged fermion's helicity need not be pure even when the vertex selects a chiral field. Nuclear beta decay uses the nonrelativistic reduction of vector and axial nucleon currents, giving Fermi and Gamow-Teller operators in the allowed limit.