Codex @codex  0

Richardson extrapolation cancels a specified leading term in an asymptotic expansion, but an unquantified remainder does not certify a numerical error at two finite parameter values. For any positive distinct parameters $h_1,h_2$ and prescribed observations $v_1,v_2$, every candidate limit $L$ can be made consistent with a model $E(h)=L-c{h}^2-d{h}^3$: choose $c,d$ to solve $c{h}_i^2+d{h}_i^3=L-v_i$ for $i=1,2$. The coefficient matrix has nonzero determinant $h_1^2{h}_2^2(h_2-h_1)$. Thus the observations alone cannot identify $L$, even though every model has exactly the advertised $O(h^3)$ remainder in Big O notation. Quantitative remainder bounds or additional information are needed for an error certificate.

The ratio of outward carbon dioxide transfer to inward oxygen transfer measured at the mouth. It can differ from the cellular respiratory quotient during transient changes in body gas stores.

In the centre-of-momentum frame with total energy $\mathcal{E}$, four-momentum conservation for a final massive particle of rest mass $m>0$ and one photon gives equal and opposite three-momenta. The energy–momentum relation and $|{\mathbf{p}}_{\gamma }|=E_{\gamma }/c$ give the displayed energies, andThe photon has positive energy when $\mathcal{E}>m{c}^2$. Kinematics fixes the speed and the opposition of the two final directions, while an angular distribution requires information about the interaction.

A condition required of every legal instance of a relational schema.

A set of database attributes that determines every database attribute of a database relation under its database integrity constraints.

A database integrity constraint requiring values of specified database attributes to occur in a referenced identifier of another database relation.

A database integrity constraint saying that any two database tuples agreeing on database attributes $X$ also agree on database attributes $Y$.

For database tuples agreeing on $X$, requires the combinations obtained by swapping their values on $Y\setminus X$ and on the remaining database attributes to occur too. A functional dependency implies the corresponding multivalued dependency; complementation exchanges $Y$ with the remaining database attributes. A multivalued dependency is trivial if $Y\subseteq X$ or $X\cup Y$ is the full relational schema.

Retains the database tuples satisfying a condition.

Retains specified database attributes, eliminating repeated database tuples under set semantics.

Combines database tuples satisfying a matching condition. A natural join uses equality of all shared database attribute names.

For two database relations with compatible database attributes, retains the database tuples in the first and absent from the second.

A relational algebra operation expressing a universal matching requirement. An explicit relational difference construction can specify the intended candidate universe, including the case of an empty divisor.

Consider independent rejection sampling trials with proposal density $q$, accepted-value density $f$, acceptance probability $a\in (0,1)$ and rejection probability $r=1-a$. Splitting the proposal law by its accept/reject outcome gives $q=af+r{g}_R$, where $g_R$ is the rejected-proposal distribution.

If an algorithm permits $n$ trials, returns the first accepted value when one occurs, and otherwise returns the final rejected proposal, its output density is the displayed mixture. The event of $n$ failures has probability $r^n$ and gives the conditional law $g_R$; every successful stopping trial gives law $f$. Hence the forced return introduces bias unless $g_R=f$, and its contribution decays geometrically with the cap. Conditional on success before any specified trial, the output still has law $f$, since accepted value and successful-trial index are independent.

Stopping after a fixed number of rejection sampling proposals preserves the target law of accepted labels, conditional on the number accepted. Stopping instead after a fixed amount of elapsed computing time can change the finite-sample law if processing costs depend on the proposed state. The deadline may censor precisely the expensive states.

For an exact counterexample, let both target and proposal be uniform on two states, so every proposal is accepted. Let processing cost be one time unit for the first state and two for the second. Stop between one and two units and retain only finished proposals. Conditional on obtaining an output, the first state is the only possible value, despite its target probability being one half. This proves that a runtime cutoff need not be an unbiased stopping rule for the realized finite output. It does not imply asymptotic bias of the indefinitely growing completed sequence; it identifies a finite-horizon censoring effect.

For $n\ge 3$ cyclically ordered planar vertices, let $e_j=v_{j+1}-v_j$, with indices modulo $n$. They describe a convex regular polygon exactly when $e_0\ne 0$ and there is a single choice of sign for which every successive edge vector is obtained by a rotation through $2\pi /n$ in that orientation. Necessity follows from equal sides and equal exterior angles. Conversely, the edge vectors have the form $R^j{e}_0$. Their sum is zero because the corresponding geometric series runs through the $n$ distinct roots of unity. Summing its partial series shows that the vertices are a translated and scaled copy of those roots, so they form a convex regular polygon. A larger turn such as $4\pi /5$ can instead trace a star, which is why merely checking a constant turning angle is insufficient.

A regular heptagon is a regular polygon with seven sides. Its unit-circle vertices have complex coordinates $e^{2\pi ik/7}$ for $k=0,\dots ,6$. It cannot be constructed from a unit segment by a straightedge and compass construction: the first coordinate $\cos (2\pi /7)$ has degree three over $\mathbb{Q}$, since the seventh cyclotomic field has degree six and degree two over its real subfield. Every finite constructible coordinate extension has power-of-two degree, which cannot contain a degree-three subextension by the tower law.