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Postmating reproductive isolation acts after mating. Specify whether the test concerns fertilization, offspring survival, or the offspring's own reproduction; those outcomes are not interchangeable.

Premating isolation reduces mating between biological populations. The dietary divergence and mating isolation in Drosophila pseudoobscura assay measures this stage using observed pairings.

An intertwiner between finite-dimensional irreducible complex representations is either zero or an isomorphism; an endomorphism of one irreducible is scalar.

For a finite group $G$ acting on a finite set $X$, the number of orbits isIt follows by counting pairs $(g,x)$ with $gx=x$ first by $g$ and then by $x$.

Finite-dimensional complex representations of $S{U}_2$ are completely reducible, and their irreducibles are indexed by nonnegative integers.

The representation ring is the Grothendieck group of finite-dimensional complex unitary representations of a compact group, with addition induced by direct sum and multiplication by tensor product. As an abelian group it is freely generated by the irreducible unitary representations. Taking dimension gives a ring homomorphism $R(G)\to \mathbb{Z}$.

A kinematic redshift is the relativistic Doppler effect from relative motion of emitter and receiver. With positive radial recession speed $v$, its small-velocity limit is $z_{\mathrm{kin}}=v/c+O(v^2/c^2)$. A peculiar velocity gives such a factor in addition to cosmological redshift.

Superluminal signalling would transmit controllable information outside the source event's future light cone and would violate relativistic causality.

For identical point vortices on the plane, use $\omega ={\sum }_{a}d{x}_a\wedge d{y}_a$, ${\iota }_{X_H}\omega =dH$, and $L=\frac{1}{2}{\sum }_a(x_a^2+y_a^2)$. The generator $Y$ of simultaneous counterclockwise rotations is $-X_L$, and $L$ is its moment map with the convention ${\iota }_Y\omega =-dL$. In coordinates rotating with angular velocity $\Omega$, the Hamiltonian vector field is $X_H-\Omega Y=X_{H+\Omega L}$. Hence a collision-free configuration rigidly rotating at that velocity is exactly a critical point of the augmented Hamiltonian function $H+\Omega L$. For $H=-{\sum }_{a<b}\log |{\mathbf{r}}_a-{\mathbf{r}}_b{|}^2$, scaling all positions shows ${\sum }_a{\mathbf{r}}_a\cdot {\nabla }_{a}H=-N(N-1)$; a rotating relative equilibrium therefore obeys $2\Omega L=N(N-1)$. For $N\ge 2$ this implies positive $\Omega$ in the stated circulation convention. Reversing the contraction convention requires consistent changes to the Hamiltonian and rotation signs.

Osmolality counts dissolved particles per mass of solvent; it characterizes a chemical solution, rather than the total amount of water in an organism.

Blood pressure is force per area exerted by blood on its surroundings, measured relative to a specified reference. Arterial pressure varies during the cardiac cycle, giving systolic and diastolic values. Pressure, flow and vascular resistance are related but are different physical quantities.

For an orthogonal projection, $P^k=P$ for $k\ge 1$. Summing the exponential series proves the displayed identity. At $\theta =\pi$, the operator is $I-2P$, which changes the sign on the range and fixes its orthogonal complement. In Grover search algorithm, the marked-state reflection and the uniform-state reflection compose to the search iterate up to an irrelevant global phase.

A proposed modern teaching measurement puts a pipette-measured portion of an iron ${\mathrm{Fe}}^{2+}$ chemical solution in a conical flask, acidifies with excess dilute sulfuric acid, and adds standardized potassium permanganate from a clamped burette while swirling. The documented endpoint principle is that added purple permanganate is consumed until the first small excess leaves a pale purple colour; no separate colour indicator is required. Insufficient acidity can produce brown manganese dioxide, invalidating the assumed five-electron half-reaction.^[ref]

Record initial and final burette readings; their difference is the delivered volume $V$. The following controls are proposed additions to test the inference. Run a reagent blank with the same water, sulfuric acid and endpoint criterion but no sample, obtaining $V_b$. Use a separately established potassium permanganate concentration $c$, clear the burette tip of air, and repeat sample portions. A known added amount of substance of ${\mathrm{Fe}}^{2+}$ should increase consumption by one fifth of that amount. A blank estimates reagent demand and the small endpoint excess; it cannot remove an interfering sample reactant.

Assuming complete, selective reaction according to the acidic permanganate-iron balance, the corrected inference isWith $c$ in $\mathrm{mol}{\mathrm{L}}^{-1}$, use $V$ and $V_b$ in litres. As an original stipulated calculation, not an experimental result, take $c=0.02000\mathrm{mol}{\mathrm{L}}^{-1}$, $V=12.60\mathrm{mL}$ and $V_b=0.10\mathrm{mL}$. Then the permanganate consumption is $0.2500\mathrm{mmol}$ and the inferred ${\mathrm{Fe}}^{2+}$ amount is $1.250\mathrm{mmol}$. In a $25.00\mathrm{mL}$ sample portion this corresponds to $0.05000\mathrm{mol}{\mathrm{L}}^{-1}$.

In this example the corrected volume is $12.50\mathrm{mL}=0.01250\mathrm{L}$. The acidic permanganate-iron balance gives $z_A=1$ for ${\mathrm{Fe}}^{2+}$ and $z_T=5$ for permanganate. Thus $0.2500\mathrm{mmol}$ of permanganate accepts $1.250\mathrm{mmol}$ of formal electrons, exactly what $1.250\mathrm{mmol}$ of ${\mathrm{Fe}}^{2+}$ supplies. The amount of substance ratio is five to one, despite equality of the exchanged electron amounts.

The denominator is the original $25.00\mathrm{mL}$ aliquot, not its final volume after adding sulfuric acid, water and titrant. Such dilution changes concentrations in the flask without changing the initial sampled amount of substance of ${\mathrm{Fe}}^{2+}$. Omitting the $0.10\mathrm{mL}$ titration blank would instead give $1.260\mathrm{mmol}$ and $0.05040\mathrm{mol}{\mathrm{L}}^{-1}$, an overestimate of $0.80\%$ relative to the corrected constructed value. The displayed digits express the stipulated calculation, rather than establish its measurement uncertainty.

The endpoint detects residual permanganate, not the iron identity directly. Other sample reactants that consume permanganate raise the apparent ${\mathrm{Fe}}^{2+}$ amount; prior oxidation of sample ${\mathrm{Fe}}^{2+}$ to ${\mathrm{Fe}}^{3+}$ lowers it. A recovery check probes response to the added iron but does not prove absence of every interference. The readout determines the reacting ${\mathrm{Fe}}^{2+}$ inventory under the stated conditions, rather than total iron in an arbitrary mixture. No measured titres are asserted here.

Approximating decoupling by a single time and retaining only a photon monopole and Doppler dipole omits the finite cosmological visibility function width, diffusion or Silk damping, quadrupole anisotropy and polarization generated as tight coupling fails. The photon-baryon acoustic oscillator also make the emission density and velocity strongly scale dependent. They can be included in the initial monopole and dipole, so their existence alone does not invalidate the subsequent collisionless line-of-sight solution for free-streaming photons. Smaller angles correspond to larger angular multipole $\ell$.

The extended real numbers adjoin $-\infty$ and $+\infty$ to $\mathbb{R}$. They preserve the total order but leave some arithmetic expressions, such as $+\infty -\infty$, undefined.

A number line represents real numbers as points on a line, with zero as origin and a chosen positive direction and unit scale. Addition becomes a shift and multiplication becomes a rescaling about zero.

For nondecreasing real $\beta$, convergence of the displayed improper integral forces $\beta (x)/x\to 1$. For $\lambda >1$, the integral over $[X,\lambda X]$ tends to zero and is at least $(1-{\lambda }^{-1})\beta (X)/X-\log \lambda$. The integral over $[X/\lambda ,X]$ tends to zero and is at most $(\lambda -1)\beta (X)/X-\log \lambda$. These bound the upper and lower limits by $\log \lambda /(1-{\lambda }^{-1})$ and $\log \lambda /(\lambda -1)$. Letting $\lambda \downarrow 1$ squeezes both to one. The limit of the improper integral must be finite.

Asymptotic analysis describes what remains dominant as an argument approaches a specified limit. It compares functions or sequences by scale: how much of a complicated expression can be replaced by a simpler one, and how large is the omitted part? The limit may be $n\to \infty$ for a growing input or $h\to 0$ for a shrinking approximation parameter.

For example, $n^2+10^{5}n$ is eventually dominated by its quadratic term, since division by $n^2$ gives $1+10^5/n\to 1$. But at $n=100$, the linear term contributes $10^7$, compared with only $10^4$ from the quadratic term. A statement about the limiting scale and a good approximation at a particular input are different claims.

Three useful comparisons keep their strengths distinct. Big O notation supplies an eventual upper bound up to a constant multiplier; big theta notation supplies both an upper bound and a lower bound up to constants; asymptotic equivalence makes the ratio tend to one. Little o notation says that a term becomes negligible relative to a specified scale. These distinctions make approximations checkable instead of merely calling one expression small.

The same language connects time complexity with Taylor remainders. In an operation-count model, one asks how a cost scales with input size. In a Taylor expansion, one asks how the approximation error scales with the increment. Neither use requires interpreting the unspecified constant as one. The asymptotic comparison of exponential functions gives another example: dividing by the largest contributing exponential reveals the terms that disappear.

For a subset $A$ of a partially ordered set, its infimum is its greatest lower bound and its supremum is its least upper bound. An upper bound only has to lie above every member; the supremum is the smallest such bound. A lower bound lies below every member; the infimum is the largest such bound. These definitions identify the exact edges of a set even when neither edge belongs to it.

For example, $(0,1)$ and $[0,1]$ both have infimum zero and supremum one. Only $[0,1]$ has a minimum and a maximum: those must be actual members of the set. The set $C=\{1/n:n\in {\mathbb{N}}_{>0}\}$ has infimum zero without a minimum, but its supremum one is also its maximum. Such examples distinguish a bound from an attained extremal value. ^[ref].

For a nonempty subset of the real numbers, completeness of the real numbers guarantees a real supremum when it is bounded above, and a real infimum when it is bounded below. The bound and the ambient partially ordered set both matter: $\{q\in \mathbb{Q}:q\ge 0,q^2<2\}$ has supremum $\sqrt{2}$ in the real numbers, but no supremum in the rational numbers. The detailed explanation belongs to completeness of the real numbers.

Why are unattained bounds useful? They specify a precise limiting target. In mathematical optimization, $f(x)=x$ on $(0,1)$ has infimum zero but no global minimizer. The inputs $x_n=1/n$ for $n\ge 2$ give a minimizing sequence: their objective values approach zero without ever reaching it. Bounds therefore describe what approximation can achieve even when no feasible input is best. The approximation property of an infimum and approximation property of a supremum express this idea quantitatively.

Hardy's inequality bounds averages or weighted functions by their original size or derivatives. For $p>1$ and $f\ge 0$, its integral form is ${\int }_0^{\infty }(x^{-1}{\int }_0^{x}f(t)dt{)}^{p}dx\le (p/(p-1){)}^p{\int }_0^{\infty }f(x{)}^{p}dx$. Spatial weighted forms include the Hardy inequality in Euclidean space.