Coupled inductors are inductors whose magnetic flux linkages depend on each other's electric currents. A two-circuit linear model uses two self-inductances and a mutual inductance.

An integrated circuit technology using complementary MOSFETs. CMOS names a circuit technology; active-pixel sensor names a detector architecture.

An ideal insulating gate blocks particle exchange between the gate and substrate of a MOSFET. Their electrochemical potentials, represented by their band-diagram Fermi levels, therefore need not coincide. An applied gate voltage fixes their difference; with $e>0$ and voltage defined by an electron reservoir, ${\tilde{\mu }}_g-{\tilde{\mu }}_s=-e{V}_g$. Contact work-function differences also affect band bending, so equal Fermi levels do not imply flat bands. Each conducting region can separately have a spatially constant equilibrium electrochemical potential.

A ketose is a monosaccharide whose open-chain form contains a ketone group. Fructose is a familiar example, with a carbonyl group at carbon 2 in its open-chain structure.

A ketose can form cyclic structures by reaction of its carbonyl group with one of its own hydroxyl groups. The distinction from an aldose concerns carbonyl connectivity, rather than simply the number of carbon atoms.^[ref]

An aldose is a monosaccharide whose open-chain form contains an aldehyde group. Glucose is an example, even though much of the glucose in aqueous chemical solution has a cyclic structure.

Classification uses the open-chain carbonyl position rather than requiring every molecule to remain open-chain. This connects aldose identity with ring formation and with mutarotation.^[ref]

In water, D-glucose mainly occupies cyclic forms. The C5 hydroxy group adds to the C1 aldehyde, forming a cyclic hemiacetal; reversible opening restores the aldehyde. Reclosure can give either anomer. Their changing proportions cause mutarotation, the approach of optical rotation to an equilibrium value. This pathway involves changing chemical bonds, rather than simply rotating a hydroxy group about a single chemical bond.^[ref]

In this conventional Haworth projection, C5's $\mathrm{C}{\mathrm{H}}_2\mathrm{OH}$ group points up. The alpha anomer's C1 hydroxy group points down, on the opposite face; the beta anomer's points up, on the same face. C2/C3/C4 retain down/up/down hydroxy groups. These are above/below relationships to a reference ring plane, not a claim that the ring is physically flat.^[ref]

The anomeric carbon is C1, the existing carbonyl group carbon that becomes a new stereocenter; no extra carbon chemical atom appears. The two anomers differ at that centre while retaining the other configurations.^[ref]

Trace the ring as C1–C2–C3–C4–C5–O5–C1, where O5 denotes the oxygen originally in the C5 hydroxy group. It contains five carbon chemical atoms and one oxygen; C6 remains attached outside the ring. The C1 carbonyl group oxygen becomes the C1 hydroxy group oxygen. This accounts for every oxygen: O5 in the ring and five hydroxy groups at C1, C2, C3, C4 and C6. Ring closure preserves ${\mathrm{C}}_6{\mathrm{H}}_{12}{\mathrm{O}}_6$. A six-membered ring and a six-carbon sugar are thus different counts.

Recoil is the motion acquired by a body when it exchanges momentum with an emitted, absorbed or scattered particle. For isolated emission from a body initially at rest, momentum conservation makes the remaining body acquire the opposite momentum. If its final mass is $M$ and the recoil is nonrelativistic, its kinetic energy is $E_R\simeq p^2/(2M)$. A photon of energy $E_{\gamma }$ has momentum magnitude $E_{\gamma }/c$, so $E_R\simeq E_{\gamma }^2/(2M{c}^2)$. The transition energy is shared between the photon and the recoiling body; nuclear recoil is an important example.

A uniform square of side $s$ has the displayed moment of inertia about the perpendicular axis through its centre. Integrating $x^2+y^2$ over $[-s/2,s/2{]}^2$ gives $s^4/6$; multiplying by the surface mass density $M/s^2$ gives the result. A square prism of any uniform thickness has the same axial formula when expressed in terms of its total mass. If the side is denoted $2a$ instead, the result is $2M{a}^2/3$, so the side-length convention must be specified before substituting numerical data into a normal mode calculation.

For three mutually perpendicular axes through one point of a nonnegative mass distribution, $I_x+I_y-I_z=2\int z^{2}dm\ge 0$, with cyclic counterparts. Equality holds precisely when the mass distribution is supported in the plane $z=0$, giving the perpendicular axis theorem. The point need not be the centre of mass, and the axes need not be principal axes.

A uniform solid with circular cross-sectional radius $r(z)$ can be sliced into thin discs. A slice of thickness $dz$ has mass $dm=\rho \pi r(z{)}^{2}dz$ and axial moment of inertia $d{I}_z=\frac{1}{2}r(z{)}^{2}dm$. ThusAll slice axes coincide, so no parallel axis theorem correction is needed. For a linearly tapered section of length $l$ with end radii $r_0,r_1$, direct integration gives $M=\rho \pi l(r_0^2+r_0{r}_1+r_1^2)/3$ and $I_z=\rho \pi l(r_0^4+r_0^3{r}_1+r_0^2{r}_1^2+r_0{r}_1^3+r_1^4)/10$.

For a cylinder of uniform mass density $\rho$, length $L$, inner radius $R_i$ and outer radius $R_o$, an annular element has $dm=2\pi \rho Lrdr$. Thus the axial moment of inertia is $I=\int r^{2}dm=\pi \rho L(R_o^4-R_i^4)/2$. Dividing by $M=\pi \rho L(R_o^2-R_i^2)$ yields the displayed result. At fixed mass, the solid-cylinder limit is $M{R}_o^2/2$ and the thin-shell limit is $M{R}^2$.

Set $K_i=K_i^{\prime }$, so that $\alpha ={\alpha }^{\prime }$. The reversible enzyme inhibition equation reduces toAt the chosen $I=30\mu \mathrm{M}$, every reaction rate is one quarter of its uninhibited value. The plateau falls to $25\mu \mathrm{M}\mathrm{mi}{\mathrm{n}}^{-1}$, but half that plateau still occurs at $s=20\mu \mathrm{M}$. Increasing substrate cannot restore the uninhibited plateau. An unchanged apparent Michaelis constant here expresses equality of the two binding effects, rather than an absence of inhibitor binding to the enzyme-substrate complex.

Cytochrome c oxidase is respiratory complex IV. It receives electrons from cytochrome c and reduces molecular oxygen to water at metal-containing catalytic centers.

This reaction consumes substrate protons and is coupled to additional proton translocation across the membrane. Distinguishing chemical proton consumption from pumped protons is essential in counting the complex's contribution to the proton-motive force.

Respiratory complex III, the cytochrome bc1 complex, transfers electrons from ubiquinol to cytochrome c. Its Q-cycle mechanism separates the two electrons from a quinol and couples their transfer to vectorial proton movement.

Cytochrome c carries one electron at a time, whereas ubiquinol carries two. The mechanism therefore connects carriers with different electron stoichiometries while contributing to the membrane electrochemical gradient.

Succinate dehydrogenase oxidizes succinate to fumarate and transfers the electrons through bound cofactors to ubiquinone. In mitochondria it is respiratory complex II, linking the citric acid cycle with the respiratory chain.

Unlike complexes I, III, and IV, it does not itself pump protons across the membrane. This distinction explains why electrons entering through succinate contribute differently to the proton-motive output than electrons entering through NADH.

Respiratory complex I, or NADH:ubiquinone oxidoreductase, transfers electrons from NADH to ubiquinone and couples the process to proton translocation across the inner mitochondrial membrane. Its flavin and iron-sulfur centers form part of the electron-transfer pathway.

The resulting contribution to the proton gradient can support ATP synthesis. Electron transfer and proton movement are coupled molecular processes; the complex does not directly phosphorylate ADP. Its physiological direction and behavior depend on redox and membrane-potential conditions.

For height $x$ and horizontal position $y$, Fermat principle in a positive index profile $n(x)=1+\beta x$, $\beta >0$, conserves $n(x)y^{\prime }/\sqrt{1+(y^{\prime }{)}^2}$ on a graph $y(x)$. For positive magnitude $c$ of the horizontal optical momentum, integration gives the displayed catenary. Its minimum height is $x_0=(c-1)/\beta$, at $y=y_0$. Thus $c>1$ produces a smooth turning ray wholly above a road at $x=0$. The two graph branches have opposite signs of $dy/dx$; $x(y)$ continues smoothly through their common horizontal tangent. For a signed negative graph constant, use $|c|$ in the prefactor; $c=0$ gives a vertical ray separately.

The Galactic Center is the central region of the Milky Way, where the compact source Sagittarius A* and nearby stellar motions provide a test of a concentrated gravitational potential. It is a region, rather than another name for the entire black hole or one emitting point.

Infrared observations reduce the obscuration that limits ordinary visible-light measurements. Combining stellar astrometry with line-of-sight velocities constrains an orbit and the central mass; precise departures from a Kepler orbit can test general relativity.^[ref] A resolved apparent trajectory alone does not supply every orbital component or remove reference-frame uncertainty.