For speed of light $c(y)=c_0/\sqrt{1-ky}$, where $c_0>0$ and $1-ky>0$, the travel-time integrand in Fermat's principle is $c_0^{-1}\sqrt{(1-ky)(1+y^{\prime 2})}$. For nonzero $k$, the Beltrami identity giveswhere $y_0$ is the turning height. The corresponding stationary rays are the parabolasFor equal-height endpoints $(\pm x_0,0)$ with $x_0>0$, symmetry gives $x_c=0$ and $(k{x}_0{)}^2=4k{y}_0(1-k{y}_0)$. Real turning heights therefore require $|k|x_0\le 1$, not merely a one-sided condition on signed $k$. When the inequality is strict, the two algebraic values are $y_0=[1\pm \sqrt{1-k^2{x}_0^2}]/(2k)$; only rays remaining inside the specified optical layer are admissible. Equality merges the two branches. At $k=0$, the medium has constant speed of light and the ray joining these endpoints is the straight horizontal segment; the formulas containing $1/k$ must be replaced by that limit.

Online text: mathcs.clarku.edu/~djoyce/elements/bookI/defI7.html

Definition from Euclid's Elements by Euclid. The linked English adaptation and guide are by David E. Joyce; Clark University is the web host. No publication year is established from this page.

Cited passages: Book I, Definition 7; Joyce's guide distinguishing bounded figures from the infinite plane surface.

Molecular beam epitaxy grows crystalline films by directing controlled beams of material toward a crystalline substrate in a vacuum system. A typical apparatus has source cells, shutters, a heated substrate holder and diagnostic tools such as electron diffraction. Opening or closing shutters changes which species reach the surface, enabling layered heterostructures. Growth quality depends on surface diffusion, incorporation and desorption as well as the arriving flux. The method's value is precise control of composition and interfaces; depositing material in vacuum alone does not ensure epitaxial crystal ordering.

Epitaxial growth using volatile metal-organic precursors delivered in the gas phase. Precursor decomposition and surface incorporation grow a crystalline compound layer or nanostructure; temperature and mass transport influence composition.

The protocol used in the trypsin colorimetric assay requires $40$–$60\%$ inhibition and rejects hazy or yellow BAPNA solution.^[ref]

The trypsin colorimetric assay explains blank subtraction and conversion to a reaction rate using the Beer-Lambert law. This protocol is an activity comparison at one substrate level, not evidence that aprotinin obeys any particular curve above. To test the model, an experimental extension would vary both free inhibitor and free substrate concentrations, estimate early linear reaction rates before appreciable depletion, and fit the whole collection by nonlinear regression. Include a no-inhibitor series and inhibitor-matched blanks: inhibitor-dependent absorbance drift must not be mistaken for changed catalysis.

There is a mathematical reason to collect more than one reaction rate. At $s=20\mu \mathrm{M}$ with the chosen uninhibited parameters, measuring $v=25\mu \mathrm{M}\mathrm{mi}{\mathrm{n}}^{-1}$ would imply only $\alpha +{\alpha }^{\prime }=4$. Both $(\alpha ,{\alpha }^{\prime })=(2,2)$ and $(3,1)$ satisfy that observation, although their high-$s$ limits differ. A broader substrate range tests which model fits. Even a fitted curve leaves binding geometry unresolved. Check activity recovery after reducing inhibitor concentration separately; the simple equation also ceases to apply when free inhibitor is substantially depleted by binding or inhibitor exchange has not equilibrated during the measured interval.

Construct a minimal model with one substrate and no appreciable accumulated reaction product. Only the enzyme-substrate complex $ES$ produces product, with catalytic rate constant $k_{\mathrm{cat}}$; $EI$ and $ESI$ are inactive. Write $s=[S]$ and $I=[\mathrm{I}]$ for free molar concentrations, and $E_T$ for total enzyme molar concentration. Assume a suitable quasi-steady-state approximation, rapid inhibitor binding and release relative to turnover, and negligible depletion of free inhibitor. In this construction the population relations areHere $K_M$ is the uninhibited Michaelis constant, and $K_i,K_i^{\prime }$ are the inhibition constants for the two indicated binding branches. The symbol $I$ denotes a free concentration, not an amount added to the tube. Define $V_{\max }=k_{\mathrm{cat}}{E}_T$. Conservation of enzyme gives an original short derivation:Thus fitting a Michaelis-Menten equation at a fixed $I$ would return the apparent parameters“Apparent” means conditional on that inhibitor concentration. The uninhibited Michaelis constant has not been redefined. Dimensional analysis checks the denominator: both $\alpha K_M$ and ${\alpha }^{\prime }s$ have units of molar concentration; the resulting $v$ has units of reaction rate.

The shared numerical choices make the mechanisms comparable. At $I=0$ and $s=20\mu \mathrm{M}$, all three models give $v=50\mu \mathrm{M}\mathrm{mi}{\mathrm{n}}^{-1}$. At $I=10\mu \mathrm{M}$, the respective reaction rates are $100/3$, $25$, and $30\mu \mathrm{M}\mathrm{mi}{\mathrm{n}}^{-1}$; at $I=30\mu \mathrm{M}$ they are $20$, $12.5$, and $50/3\mu \mathrm{M}\mathrm{mi}{\mathrm{n}}^{-1}$. These are independently chosen teaching models, not measured inhibition constants for a named enzyme.

An endocrinologist is a physician or researcher specializing in hormones and endocrine disorders. The field relates glandular secretion and feedback to metabolic, developmental and reproductive function. Assessment of a control pathway requires more than naming one hormone.

The electrophoretic mobility $\mu$ is the proportionality coefficient between drift velocity and a weak applied electric field. Its sign specifies motion relative to the field. When only travel distances are used, the mobility magnitude is often intended. For long linear DNA in a weak-field reptation regime, $|\mu |$ can scale inversely with chain length, whereas free-solution mobility is approximately length-independent. Neither law supplies a universal gel electrophoresis size calibration for arbitrary gel concentration, field strength or molecular topology.

A narrow population band undergoing uniform electrophoresis and ordinary diffusion has mean position $x_0+\mu Et$ and variance ${\sigma }_0^2+2Dt$. This follows from the advection-diffusion equation by changing to the coordinate $x-\mu Et$. For two species with mobility difference $\Delta \mu$, their centre separation is $|\Delta \mu |Et$, whereas diffusion widths grow only as $\sqrt{t}$. Starting from point-like bands with common diffusion coefficient $D$, a required separation of $q$ standard deviations needsA prescribed distance between centres alone is a resolution convention; finite loading widths and other dispersion mechanisms must be included when interpreting actual electrophoresis data.

An electron occupying an atomic orbital without a partner of opposite spin in that atomic orbital.

Friedrich Hund's rules describe the usual ground-term ordering within an atomic electron configuration in the LS coupling regime. First maximize the total spin $S$ subject to the Pauli exclusion principle. Among the terms with that spin, maximize total orbital angular momentum $L$. Within the resulting term, the lowest level has $J=|L-S|$ for a less-than-half-filled open subshell and $J=L+S$ for a more-than-half-filled one. These are ordering rules, not universal replacements for diagonalizing an arbitrary atomic Hamiltonian operator.

For two equivalent $p$ electrons, parallel spins give $S=1$. Their spatial wavefunction must be antisymmetric; the highest allowed orbital projection with parallel spins is $1+0=1$, giving $L=1$. A $p^2$ subshell is less than half filled, so its lowest term symbol is ${}^3{P}_0$, followed within that term by ${}^3{P}_1$ and ${}^3{P}_2$. Closed subshells contribute zero total $L$ and $S$.

A beam of electrons in a partially evacuated tube, studied in the Thomson charge-to-mass experiment.

Release of electrons from a heated material; one source of electrons in an X-ray tube.

Heating a metal gives some electrons enough energy to escape its surface. A heated diode cathode can therefore supply an electric current across a vacuum gap to a collecting diode anode. This makes temperature a control for the supply of electrons. Conversely, measuring that supply at several temperatures lets us estimate the surface work function: the energy barrier measured from the Fermi level to the nearby vacuum level. A Richardson plot turns the strong exponential function dependence into a slope.

Three questions must be kept separate: how many electrons cross the surface, how many traverse the gap, and how many the collecting diode anode records. The Richardson-Dushman law answers the first under specified surface assumptions. Space charge can obstruct the second, while incomplete collection can alter the third.

An electron gun creates, accelerates and shapes a directed electron beam. A source such as a heated cathode supplies electrons, an applied electric potential difference accelerates them toward an anode, and electrodes control the emitted current and beam geometry.

For an electron accelerated from rest through potential difference $V$, the ideal gained kinetic energy is $eV$, with the voltage sign chosen to accelerate it. Focusing optics then control beam position and convergence. Source size, energy spread and space charge affect the useful beam; increasing accelerating voltage alone does not guarantee a smaller probe or higher image resolution.