In the model of reversible enzyme inhibition, competitive inhibition makes substrate binding and inhibitor binding mutually exclusive: $EI$ is allowed but $ESI$ is absent. Set ${\alpha }^{\prime }=1$, equivalently take $K_i^{\prime }\to \infty$. ThenFor the chosen $I=30\mu \mathrm{M}$, $\alpha =4$: the half-limiting substrate concentration rises from $20$ to $80\mu \mathrm{M}$, while the limiting reaction rate remains $100\mu \mathrm{M}\mathrm{mi}{\mathrm{n}}^{-1}$. At $s=200\mu \mathrm{M}$ the inhibited reaction rate is $500/7\approx 71.43\mu \mathrm{M}\mathrm{mi}{\mathrm{n}}^{-1}$, compared with the uninhibited $1000/11\approx 90.91\mu \mathrm{M}\mathrm{mi}{\mathrm{n}}^{-1}$. “Overcome by substrate” means that their ratio tends to one as $s\to \infty$, not that any finite addition removes the difference.

An inhibition constant sets an inhibitor concentration scale in an enzyme kinetics model. In the rapid-binding construction under reversible enzyme inhibition, $K_i=[E]I/[EI]$ and $K_i^{\prime }=[ES]I/[ESI]$ are dissociation constants. Smaller values favour the corresponding bound population at fixed free inhibitor concentration. More generally, fitted inhibition constants need not be true binding dissociation constants; a mechanism must justify that interpretation.^[ref]

For example, choosing $I=K_i$ makes $[EI]=[E]$ in this model, but does not imply that half of all enzyme is $EI$: $ES$ and $ESI$ also contribute to the total population.

A frequency at which a specified driven response is resonant. The value can depend on the observable and damping: the electric current amplitude of a driven series RLC circuit peaks at $f_0=1/(2\pi \sqrt{LC})$, while a different circuit readout need not peak at the same frequency.

A resonator stores oscillatory energy in one or more normal modes. Driving near a mode's frequency can produce resonance. Losses limit the response and determine its quality factor. After the drive is removed, an isolated weakly damped mode decays approximately exponentially; an observed room response may combine many modes.

For real $\beta >1$, choose a branch of the complex logarithm whose argument interval contains $[0,2\pi /\beta ]$ strictly, and integrate $\mathrm{Log}z/(1+z^{\beta })$ over an annular sector of angle $\alpha =2\pi /\beta$. The one enclosed simple pole is $e^{i\pi /\beta }$, with residue $-i\pi e^{i\pi /\beta }/{\beta }^2$. The circular arcs tend to zero, with bounds $O(ϵ|\log ϵ|)$ and $O(R^{1-\beta }\log R)$.

Writing $I$ for the title integral and $J={\int }_0^{\infty }(1+x^{\beta }{)}^{-1}dx$, the two radial segments and the residue theorem giveDivide by $e^{i\pi /\beta }$ and equate real and imaginary parts. This gives $J=\pi /[\beta \sin (\pi /\beta )]$ and the title formula. When $1<\beta <2$, the sector extends below the negative real axis; the principal complex logarithm therefore cannot be used throughout that sector. A cut along a ray just below the positive axis avoids this issue for every $\beta >1$.

Indirect selection of bacterial mutants recovers a source using a related sample's tested phenotype, rather than directly challenging the retained lineage. A positive example uses a surviving replica biological colony to locate its unexposed source. A negative example uses failure to grow in one condition to locate a source that does grow elsewhere. The second is particularly useful for finding auxotrophy: dependence on an externally supplied nutrient.

Construct a four-by-four teaching layout with rows A–D and columns 1–4. The master has 13 biological colonies; A4, D1 and D4 are empty. A growth-supporting control replica grows at every occupied master site except B2 and C3. A replica lacking an assumed nutrient X grows at those same control sites except A2 and C1. All other requirements are stipulated equal. These are invented readouts, not observations from the Lederberg replica plating experiment.

Let $M$, $C$ and $S$ be the finite sets of sites with growth on the master, control and X-omission replica respectively. In this construction $S\subseteq C\subseteq M$, with cardinalitiesThe set differencecontains the two condition-dependent candidates. Pick from A2 and C1 on the master, then compare fresh samples of each recovered source with and without X. Restoration of growth when X is supplied supports an X-requirement phenotype, subject to the scientific controls; growth alone does not identify a particular gene or mutation.

By contrast,is unresolved sampling or viability in this first comparison. Neither site should be counted as an X-requirement candidate. The three sites outside $M$ have no master source to recover at all. The short logical argument is membership testing: a candidate must belong to $C$, establishing control growth, and not belong to $S$, establishing the contrasting readout. Being outside $S$ alone supplies only the second condition. Among the seven empty X-omission sites, two are candidates, two are unresolved and three have no master source. The beauty of the method is that a shared location turns this logical distinction into a practical retrieval route.

Replica plating controls distinguish a growth phenotype from failure of the copying process. A replica on a cell culture medium that supports the starting biological colonies is a touch-transfer scientific control: growth at a site demonstrates that viable biological cells reached that control site. If a source is visible on the master but absent on both control and restrictive replicas, the restrictive readout is unresolved. If it grows on the control but fails on the restrictive replica, the difference is a candidate condition-dependent phenotype.

The touch-transfer scientific control still cannot guarantee transfer to every other replica. A known growing comparison on the restrictive cell culture medium, repeat sampling, and retesting a recovered source address different possible failures. Likewise, an orientation mark checks which source a site represents; it cannot check whether any biological cells arrived. Gentle, aligned contact helps preserve the map. In Investigations in Molecular Cell Biology by Clare M. O’Connor, section 12.6: Exercise 2 - Replica plating and complementation, the instructions align master and carrier marks and call for light, even contact.^[ref]

For indirect selection of bacterial mutants, keep the recovery route on the unexposed master. Returning a carrier that has touched a challenged replica to that master would introduce a possible exposure route. The original experiment's stray-bacteriophage check addresses contamination of the retained lineage, while the orientation and touch-transfer scientific controls address spatial identity and successful sampling. These questions must be checked separately.

Joshua Lederberg and Esther Lederberg reported replica plating in 1952. They autoclave-sterilized velveteen and secured it pile-up on a wood or cork cylinder with a metal hoop. Inverted agar plates touched the fabric lightly. Pencil marks or pin indentations fixed orientation. Complete-medium and minimal-medium replicas compared growth of biological colonies.^[ref]

Escherichia coli W-1 grew on plain agar; serial T-1 bacteriophage replicas showed resistant biological colonies at matching sites. Corresponding unexposed master regions seeded repeated broth-and-plain-agar enrichment, yielding resistant isolates. Stray-bacteriophage checks were negative; resistance persisted without challenge. This supported preexisting heritable resistance. The authors noted variable transfer and moisture-dependent resolution.^[ref]

The Luria-Delbrück experiment uses variation between independently grown cell cultures to investigate the timing of resistant mutations. The Lederberg replica plating experiment adds a spatial route to a recoverable source: the tested samples identify an unexposed lineage. These are complementary arguments about mutation and natural selection. Growth readouts establish a tested phenotype; identifying the responsible nucleotide sequence requires other evidence, such as DNA sequencing.

Replica plating spatial correspondence is the preserved association between a master site and its sampled sites on replicas. Label a master site by an ordered pair $(r,c)$, specifying its row and column. Keeping the same orientation and viewing convention makes that label refer to the same source on every plate. The biological colonies need not have identical outlines, because the transferred samples can contain different numbers of biological cells.

A simple orientation check shows why a circular outline is insufficient. In a constructed four-by-four layout, a half-turn sends $(r,c)$ to $(5-r,5-c)$. Thus B2, or $(2,2)$, would be confused with C3, or $(3,3)$. Applying the half-turn twice returns the original ordered pair, but applying it once generally selects the wrong source. An asymmetric orientation mark fixes that ambiguity. Two contact transfers also preserve the handedness when all plates are viewed from the same side: any reflection in going from plate to fabric is undone in going from fabric to plate. A fabric-side view must not be silently substituted for a plate-side view.

Use natural units $c=1$ and assume open phase space, $M>m_1+m_2+m_3$. In the parent's rest frame, define $s_{23}=(p_2+p_3{)}^2$. Four-momentum conservation givesThe upper recoil-mass limit follows from $E_1\ge m_1$. For positive $m_2,m_3$, the lower recoil-mass limit follows in their pair rest frame and is attained when their relative momentum vanishes. If both recoil particles are massless, $s_{23}=2E_2{E}_3(1-\cos {\theta }_{23})$ reaches zero with nonzero parallel momenta. If exactly one is massless, $s_{23}$ approaches the nonzero daughter's squared mass only when the massless particle becomes soft. Thus that upper $E_1$ bound is a supremum for a genuine three-body event with all massless particles of positive energy. If $m_1=0$, the lower endpoint is likewise a soft limit. Multiplying the mass expressions for $E_1$ by $c^2$ restores energy units.

A boundary of phase space approached as a massless outgoing particle's energy tends to zero. A zero four-vector is not a physical positive-energy one-particle state. An energy bound requiring such a particle is a supremum or infimum, even when the closed kinematic interval is conventionally described using minimum and maximum. This distinction can matter when applying three-body formulas at mixed massive-massless recoil thresholds.

An antisymmetric tensor combines the electric polarization and magnetization of a medium. For Minkowski metric $(-,+,+,+)$ and units with light speed one, choose $M^{i0}=P_i$ and $M^{ij}={ϵ}_{ijk}{M}_k$, together with $F_{i0}=E_i$ and $F_{ij}={ϵ}_{ijk}{B}_k$. Then the Lorentz scalar interaction is $M^{\mu \nu }{F}_{\mu \nu }/2=\mathbf{P}\cdot \mathbf{E}+\mathbf{M}\cdot \mathbf{B}$. The bound charge and bound current have four-current $j_b^{\mu }={\partial }_{\nu }{M}^{\mu \nu }$, giving ${\rho }_b=-\nabla \cdot \mathbf{P}$ and ${\mathbf{J}}_b={\partial }_t\mathbf{P}+\nabla \times \mathbf{M}$. This current is identically conserved because partial derivatives commute and $M$ is antisymmetric.

A mirror receding at velocity $u=\beta c$ from normally incident light produces two applications of the relativistic Doppler effect, one into the mirror rest frame and one back to the observer. In that rest frame, reflection preserves frequency. Consequently ${\omega }_r={\gamma }^2(1-\beta {)}^2{\omega }_i={\omega }_i(1-\beta )/(1+\beta )$. Equivalently, continuity of wave phase on the moving surface $z=ut$ gives ${\omega }_i-k_{i}u={\omega }_r+k_{r}u$, with $k_i={\omega }_i/c$ and $k_r={\omega }_r/c$. For $|u|\ll c$, the fractional frequency change is $-2u/c$ to first order.

Let $\theta$ be the angle between the source velocity and the emitted photon's propagation direction in the observer's frame. The relativistic Doppler effect gives $\nu ={\nu }_0/[\gamma (1-\beta \cos \theta )]$. A ray transverse in that frame has $\lambda =\gamma {\lambda }_0$, a redshift. A ray transverse in the source frame instead has $\cos \theta =\beta$ by aberration of light, and $\lambda ={\lambda }_0/\gamma$, a blueshift.

The time at which a passing source is closest must also be specified. Put the observer at the origin and the source at $(v{t}_e,d)$ at emission, with finite transverse offset $d>0$. Reception occurs atEmission at closest approach means $t_e=0$, a transverse observer-frame ray received at $t_r=d/c$, hence a redshift. Reception at closest approach means $t_r=0$, so $t_e=-\gamma d/c$ and the ray has $\cos \theta =\beta$, hence a blueshift. Sending $d$ to zero does not remove the distinction between these two limiting event prescriptions. ^[ref]

With incident light travelling to the right, use positive object distance $u$ to the left, image distance $d$ to the right, and curvature radius $R$ toward a right-hand centre. For a ray hitting at small height $Y$, its incoming inclination, outgoing downward inclination and inward-normal downward inclination are $\alpha \simeq Y/u$, $\beta \simeq Y/d$ and $\theta \simeq Y/R$. The paraxial approximation to Snell's law gives $n_1(\alpha +\theta )=n_2(\theta -\beta )$. Cancelling $Y$ proves the title equation. Virtual conjugates and other centre orientations require corresponding signed distances. Applying the relation at both faces of a thin lens proves the lensmaker's equation.