A conformational population is the fraction of otherwise specified molecules occupying a particular conformation. It describes how a conformational ensemble is distributed, rather than a single mean structure. For two equilibrium conformations of the same molecular composition, let $G_A,G_B$ be their effective molar Gibbs free energies at fixed temperature and pressure. Their population ratio isThe effective free energies include each state's internal degeneracy. A higher-free-energy state can remain appreciably populated even though it is not the most common structure.

Populations and exchange rates are different observables. A two-state kinetic model $A\rightleftarrows B$ with rates $k_{AB},k_{BA}$ has equilibrium balance $k_{AB}{p}_A=k_{BA}{p}_B$. Multiplying both rates by ten leaves the populations unchanged but speeds exchange tenfold.

In a primary nuclear magnetic resonance study of cyclophilin A, measurements detected coordinated conformational exchange in the free enzyme and related motions during turnover, illustrating why a static structure is insufficient to describe enzyme dynamics.^[ref] Inferring a catalytic mechanism requires the relevant exchange and reaction rates as well as the conformational populations.

A SET domain is a protein domain found in many methyltransferases that modify lysine residues in histones and other proteins. It contributes to the catalytic machinery that transfers a methyl group to a selected target. The domain's presence suggests a family relationship, but does not by itself specify which residue a particular enzyme modifies.

In 2000, experiments on SUV39H1 and related proteins identified histone H3 lysine 9 methylation activity. Mutational and biochemical analysis localized the catalytic requirement to the SET domain together with adjacent cysteine-rich regions.^[ref] This linked a recognizable sequence region to a concrete chemical reaction on chromatin, while also showing why isolating a domain name from its surrounding structure can oversimplify enzyme function.

A Gla domain is a protein domain containing several gamma-carboxyglutamate residues. These are glutamate residues changed by a Vitamin K-dependent post-translational modification: an additional carboxyl group gives the modified residue extra capacity to bind calcium ions. In proteins such as prothrombin, the resulting calcium ion interactions help organize the domain's protein folding and its association with a phospholipid membrane. The residue chemistry, the folded domain and its membrane interaction are related but distinct levels of explanation.

The unusual residue was identified experimentally in 1974. J. Stenflo and colleagues found that a prothrombin fragment's electrophoretic mobility could not be explained by its ordinary amino acid composition. Nuclear magnetic resonance and mass spectrometry identified gamma-carboxyglutamic acid. The extra chemical group helped explain the fragment's calcium ion-binding behaviour.^[ref]

Later, M. Soriano-Garcia and colleagues used X-ray crystallography to resolve a calcium-bound prothrombin fragment at $2.2\text{A˚}$ resolution. Much of the Gla domain was disordered in the calcium-free crystal, whereas the calcium-bound structure was well defined and contained seven bound calcium ions. Their coordination by carboxylate groups and water connected residue chemistry to a three-dimensional structure. Seven is the count for that reported structure, rather than a universal definition of every Gla domain.^[ref]

The S1 RNA-binding domain is a protein domain found in many RNA-associated proteins, named after its original identification in ribosomal protein S1. It is a reusable RNA-interaction module; its presence does not identify the entire host protein as a ribosomal component or a nuclease.^[ref]

A 1997 structural experiment examined the S1 domain of Escherichia coli polynucleotide phosphorylase using nuclear magnetic resonance. The resulting structure was a five-stranded antiparallel beta barrel. Conserved residues on one face and neighboring loops suggested an RNA-binding surface; that proposal distinguished an observed fold from an inferred contact function.^[ref]

The 2016 RNase R study supplies a functional example. Altering residues in its S1 domain changed interactions with duplex RNA, while separate measurements examined nucleotide binding, conformational changes and degradation. A binding domain can therefore influence catalysis by positioning a substrate or stabilizing a productive state, without itself supplying all catalytic chemistry.^[ref]

For an architectural counting example, a hypothetical protein with two S1 domains has two copies of that module, not automatically two RNA-cleaving active sites. Domain count, binding behavior and enzyme activity are different observations that require different evidence.

A knowledge extractor is the algorithm that turns sufficiently successful proving behavior into a valid cryptographic witness in the definition of a proof of knowledge. The definition specifies its access to the prover and its efficiency guarantee, often in terms of acceptance probability above a knowledge error. The real verifier need not execute the extractor or learn the witness.^[ref]

A concrete algebraic step appears in the graph-isomorphism proof protocol. Suppose two accepting continuations with the same commitment $H$ supply mappingsThe extractor returnsTo check it, take any edge of $G_0$. The first mapping sends it to an edge of $H$, and the inverse second mapping sends that edge to an edge of $G_1$. Both mappings are bijections, so their composition is a graph isomorphism. The same argument handles nonedges.

This verifies the output once the two continuations have been obtained. A full extraction proof must also justify obtaining them, for example through permitted rewinding in cryptographic proofs, within the promised expected extraction time.

A zero-knowledge simulator solves a different task: it reproduces a verifier-view distribution without the witness. An extractor produces a witness from successful proving behavior. Confusing those output requirements would obscure both the privacy and knowledge guarantees.

The knowledge error $\kappa (x)$ of a proof of knowledge is a threshold in its extraction guarantee. If a prover's acceptance probability $p(x)$ exceeds that threshold, the definition requires a knowledge extractor with an appropriate expected extraction time. It is not simply the probability that the extractor outputs an incorrect witness.^[ref]

In one standard formulation, on an input in the witness relation's domain,where $C$ is an appropriate polynomial. The extraction guarantee includes the relation and oracle-access conditions as well as this bound.^[ref]

For an original numerical comparison, set $\kappa =1/2$. Acceptance probability $p=0.6$ leaves a gap of0.1, giving a factor10 multiplying $C(|x|)$ in this bound. At $p=0.51$, the factor is100. As success approaches the threshold, the permitted expected running time worsens. At $p=\kappa$, this displayed guarantee does not apply.

The single-challenge graph-isomorphism proof protocol illustrates why a nonzero threshold can be meaningful. A strategy can prepare a relabeling of one public graph and answer only its corresponding challenge, succeeding with probability $1/2$ without using an isomorphism between the two public graphs. Success beyond that baseline is what the extraction analysis exploits.^[ref]

Rewinding in cryptographic proofs is a simulation or extraction technique that returns an algorithmic participant to an earlier state and explores another continuation. The saved state includes the relevant earlier messages and randomness. The technique is justified by the simulator's or extractor's access in the mathematical model; it is not an extra power of the ordinary verifier during a live exchange.^[ref]

For the graph-isomorphism proof protocol, two accepting continuations of the same first message can answer different challenge bits. A knowledge extractor can combine their response mappings to obtain a cryptographic witness. If the prover instead starts again with a different first message, the responses need not combine in that way.

An original retry calculation explains an efficiency condition. Suppose independent simulation attempts succeed with probability $q$ and each failed attempt returns to the required earlier state. The attempt count $T$ has a geometric distribution:At $q=1/2$, the expected number of attempts is2; at $q=1/100$, it is100. Constant or inverse-polynomial success can therefore support expected polynomial time, whereas exponentially rare success generally cannot.

A valid argument must also account for the cost of each attempt and show that the resulting output distribution is correct. Restoring a state, conditioning on success and restarting an entire protocol are distinct operations.

A propositional proof system specifies efficiently checkable certificates for logical tautologies. In the functional formulation, it is a polynomial-time computable surjection from finite binary strings onto the encoded tautologies: the input is a proof and the output is the formula proved. Soundness means no false formula is output; completeness means every logical tautology has some proof.

For example, a truth table can certify $p\vee \neg p$ by checking its two assignments. The same method for $n$ variables may require $2^n$ rows, so efficient verification in the proof's length does not imply short proofs in the formula's length. Proof complexity asks which formulas require long certificates in particular systems. Searching for a proof and checking a supplied proof are different computational tasks.^[ref]

Classical Laguerre geometry studies oriented hyperplanes, oriented spheres and their oriented tangency in Euclidean space. Its transformations preserve that contact relation. In the plane the objects are oriented straight lines and oriented circles; points can be represented as radius-zero circles. The orientation and signed radius are part of the data, so ordinary unoriented tangency alone loses information.^[ref]

For example, fix a unit normal $u$ and an oriented line $u\cdot x=h$. Under a consistent signed-radius convention, a circle with centre $c$ and signed radius $r$ is in oriented contact when $u\cdot c-h=r$. The absolute-distance equation $|u\cdot c-h|=|r|$ forgets which side and orientation participate. This linear signed-contact relation motivates coordinate models for tangency and the use of Laguerre geometry in studying incircular nets.

An assigned value in proficiency testing $x_{\mathrm{pt}}$ is the benchmark attributed to the proficiency-test item for evaluation of participant results. Its origin, measurement uncertainty and relationship to the specified measurand are part of its meaning. It is not automatically a true value known without uncertainty.

The guide distinguishes assignment by formulation, a certified reference material, results from one laboratory, an expert-laboratory consensus or participant consensus. These methods provide different evidence and dependencies; a simple interlaboratory consensus value is only one possibility.^[ref]

For an original formulation example, suppose the relevant mass concentration is analyte mass divided by final solution volume:The assigned value in this stipulated complete-transfer model is $c=2.00mg/L$. If $u(m)=0.02\mathrm{mg}$ and $u(V)=0.005\mathrm{L}$ are independent standard uncertainties, first-order propagation givesso $u(c)\approx 0.0224mg/L$. Purity, incomplete transfer, stability or nonuniform mixing would require additional model terms. Formulation alone may not reproduce extraction or preparation difficulties encountered in routine samples.^[ref]

Participant consensus introduces a different dependency. If four laboratories report 100 and a fifth reports 110, their arithmetic mean is 102. The fifth laboratory's deviation from this benchmark is 8, whereas its deviation from the other four laboratories' mean is 10. In general,where ${\bar{x}}_{-i}$ is the arithmetic mean excluding participant $i$. This original identity shows participant influence on a consensus assigned value; the same result being assessed helps determine its comparator. It does not imply that every scheme should use a leave-one-out rule, but the dependence must not be mistaken for an independent reference comparison.

The provider should record how the assigned value was obtained and how its uncertainty was evaluated. A precise participant consensus can retain shared bias in an interlaboratory consensus, while an independent reference can itself have substantial uncertainty. Those are distinct limitations requiring different evidence.

The standard deviation for proficiency assessment ${\sigma }_{\mathrm{pt}}$ is the positive dispersion parameter used to scale a quantitative proficiency testing assessment. In a proficiency-test z-score, it specifies how large a deviation is relative to the scheme's performance criterion. It is not necessarily the observed standard deviation of participant results, a standard error of their mean, or one participant's standard uncertainty.^[ref]

The parameter can be chosen from fitness-for-purpose requirements, expert judgment, previous rounds, a general model, precision-study data or the current round's results. The choice must match the intended assessment. The guide favors fitness-for-purpose criteria where possible because a denominator determined by each round's participant spread can change the apparent performance scale.^[ref]

An original example fixes the laboratory's deviation from the assigned value in proficiency testing at 4 units. Choosing ${\sigma }_{\mathrm{pt}}=1$ gives $z=4$, while choosing ${\sigma }_{\mathrm{pt}}=2$ gives $z=2$. The measurement did not improve when the denominator changed. This is why the criterion must be justified independently of a desire to obtain a favorable score.

The assessment scale can also depend on the quantity level. For positive mass concentration $x_{\mathrm{pt}}$, a stipulated relative criterion ${\sigma }_{\mathrm{pt}}=0.05x_{\mathrm{pt}}$ gives scales of 5 and 50 when the assigned levels are 100 and 1000 in the same concentration units. Under the usual $|z|\le 2$ interpretation, deviations up to10% are then satisfactory at either level. A constant absolute scale of 5 would instead impose a much tighter relative criterion at the higher level. Neither choice is universal; the intended measurement use decides which is meaningful.

Uncertainty of the assigned value is a separate input. The guide's criterion $u(x_{\mathrm{pt}})<0.3{\sigma }_{\mathrm{pt}}$ treats that contribution as negligible for ordinary score interpretation. Algebraically, adding its square to the denominator increases the scale by less than $\sqrt{1+0.3^2}\approx 1.044$, illustrating the size of the approximation. If the contribution is not negligible, the provider must account for it through an appropriate evaluation rather than relabel it as participant variation.^[ref]

Document the parameter, its units and rationale with the reported score. A useful assessment criterion remains interpretable across rounds and reflects the performance needed for the task, rather than merely rewarding agreement within a poorly performing group.

A proficiency-test z-score expresses a laboratory result's deviation from the assigned value in proficiency testing in units of the scheme's standard deviation for proficiency assessment:The score is dimensionless because numerator and denominator describe the same kind of quantity in the same units. Its sign gives the direction of the deviation. The denominator is an assessment scale, not automatically the participant's own measurement uncertainty.^[ref]

For an original resistance example, take $x_{\mathrm{pt}}=100\Omega$, ${\sigma }_{\mathrm{pt}}=2\Omega$ and a reported value $105\Omega$. The score is $2.5$. Under the guide's commonly used interpretation, $|z|\le 2$ is satisfactory, $2<|z|<3$ raises a warning, and $|z|\ge 3$ raises an action signal. Thus this example is questionable; a result of $96\Omega$ gives $z=-2$ and lies exactly at the satisfactory boundary, whereas $106\Omega$ gives $z=3$. The scheme must state its interpretation.^[ref]

The word z does not by itself make this a hypothesis test with a universally fixed error probability. If scores actually followed a standard normal distribution, the stated thresholds would have the corresponding normal-tail probabilities. That distribution is an additional model assumption; the numerical scoring rule alone does not establish it.

Uncertainty in the benchmark can change the comparison. The proficiency-test z-prime score usesWith the same $5\Omega$ deviation and ${\sigma }_{\mathrm{pt}}=2\Omega$, an assigned-value standard uncertainty $1.5\Omega$ changes the denominator to $2.5\Omega$ and the modified score to 2. A smaller modified score may reflect a less certain benchmark, not improved laboratory measurement.^[ref]

A score should therefore travel with the assigned-value method, assessment scale and relevant uncertainty information. Comparing scores from schemes with different criteria can confuse a change in the benchmark with a change in performance.

A proficiency-test item is the material, artefact, instrument or information supplied for assessment in a proficiency testing scheme. It is the object to which a specified task and result refer; it is not itself the participant's reported result or the assigned value in proficiency testing. The primary guide's definition covers samples and physical standards as well as data or other information used for assessment.^[ref]

For a material example, the provider can prepare a batch, divide it into sealed units, and distribute one coded unit to each laboratory. The specified measurand, amount to be sampled, preparation instructions, reporting units and handling conditions belong to the comparison design. The material must support the particular measurement procedures being assessed, rather than merely have a convenient nominal concentration.

Reference material homogeneity and reference material stability protect the interpretation. An original counterexample gives two laboratories bottles whose actual mass concentrations are $8$ and $12mg/L$, while both are scored against $10mg/L$ with assessment scale $0.5mg/L$. Perfect measurement of each bottle produces proficiency-test z-scores of $-4$ and $+4$. Those discrepancies arise from unequal items, not from incorrect laboratory measurement. A perfectly stable material can still be nonuniform, and a uniform batch can still change during transport.

A simple item heterogeneity contribution to proficiency-test scores can be written in a measurement model,where ${\delta }_{j(i)}$ describes the supplied unit's deviation, $b_i$ the laboratory effect and ${\epsilon }_i$ residual variation. Observing $x_i$ alone cannot assign all of its difference from $\theta$ to $b_i$. Replicate portions from selected units and stability checks across the relevant interval supply additional evidence.

The required degree of item uniformity depends on the assessment criterion. Narrowing the standard deviation for proficiency assessment can make previously tolerable item differences significant. The guide consequently requires checking whether item homogeneity and stability remain appropriate when choosing a stricter assessment scale.^[ref]

When a single artefact is circulated sequentially, changes between visits can similarly be confounded with laboratory differences. Record the item identifier, its preparation or transport history and the actual observation conditions so the reported value refers to a reproducible task.

The compactly generated product of compactly generated spaces $X,Y$ iswhere the right-hand product has the ordinary product topology and $k$ denotes kification. Kification retains the same points but declares a subset closed when its inverse image is closed under every continuous map from a compact Hausdorff space. Its topology can be finer than the ordinary product topology.

For a compactly generated source $Z$, a set map $Z\to X{\times }_{k}Y$ is continuous exactly when both coordinate maps are continuous. Thus this is the categorical product in the compactly generated category, even when the ordinary topological product fails to be compactly generated. If one factor is locally compact Hausdorff, the ordinary product is already suitable.

For example, the compactly generated product of finite CW complexes has the usual product topology. The modification matters in general function-space and homotopy constructions, where products and mapping spaces must interact correctly.^[ref]

Dempster-Shafer theory represents evidence that may support a set of possibilities without assigning its support to individual possibilities. On a finite sample space $\Theta$, a basic probability assignment is a function $m:2^{\Theta }\to [0,1]$ with $m(\varnothing )=0$ and ${\sum }_{B\subseteq \Theta }m(B)=1$. The associated belief function and plausibility function areThe belief function counts support committed wholly to $A$; the plausibility function also counts support not excluding $A$. These are generally not additive probability measures.^[ref]

For $\Theta =\{r,b\}$, let $m(\{r\})=0.6$ and $m(\Theta )=0.4$. Then $\mathrm{Bel}(\{r\})=0.6$ and $\mathrm{Pl}(\{r\})=1$, while the corresponding values for $\{b\}$ are $0$ and $0.4$. The uncommitted $0.4$ has not been asserted to be evidence specifically for $b$. Rules for combining separate bodies of evidence need assumptions about how the evidence was obtained; they are not automatic consequences of these definitions.

Reliability theory uses probability theory and statistical inference to describe whether a component or system continues to perform its specified function. For a nonnegative failure time $T$, its reliability at time $t$ is the survival function $R(t)=\mathbb{P}(T>t)$. If $T$ has probability density function $f$, the hazard function $h(t)=f(t)/R(t)$ describes the instantaneous failure rate conditional on survival, where $R(t)>0$.

System structure matters. For independent components with reliabilities $R_i(t)$, a series reliability system fails at the first component failure, so $R_{\mathrm{series}}(t)={\prod }_i{R}_i(t)$. A parallel reliability system works while at least one component works, so $R_{\mathrm{parallel}}(t)=1-{\prod }_i(1-R_i(t))$. Two independent components each having reliability $0.9$ therefore give $0.81$ in series and $0.99$ in parallel. These product formulas depend on independence; a common cause of failure can invalidate them.^[ref]

Stochastic analysis develops calculus and analytic tools for stochastic processes, especially when their sample paths are too irregular for ordinary differentiation. Brownian motion is the central example: its increments have size of order $\sqrt{\Delta t}$ rather than $\Delta t$, and its paths are almost surely nowhere differentiable.

The Itô integral is defined through an appropriate limit of sums whose integrands use information available before each increment. Its Itô isometry gives $\mathbb{E}({\int }_0^t{H}_{s}d{B}_s{)}^2=\mathbb{E}{\int }_0^t{H}_s^{2}ds$ for square-integrable adapted integrands. The resulting Itô formula includes a second-derivative correction:Taking expectations recovers $\mathbb{E}{B}_t^2=t$. Ordinary calculus would miss the extra term and predict the wrong variance. This is why stochastic calculus is needed for stochastic differential equations and their stochastic partial differential equation counterparts, rather than treating random trajectories as differentiable curves.^[ref]

The detection probability $P_D$ is the conditional probability that a specified procedure declares a detection when the target or signal condition is present. With $D$ the detection event and $H_1$ the signal-present condition,It depends on the signal, noise model, measurement conditions and decision rule. In a statistical hypothesis test, it is the statistical power at the specified alternative. Its complement is the probability of missing that signal; the false positive rate instead conditions on signal absence.^[ref]

High $P_D$ does not by itself imply that most detections are real. Suppose, in an invented collection of10,000 independent trial conditions,100 contain a target. A procedure detecting90% of targets and falsely alarming in5% of the9,900 target-absent conditions has expected counts90 real detections and495 false alarms. Only $90/(90+495)\approx 15.4\%$ of its alarms then correspond to targets. These are expected counts under the example model, not measured detector performance.

More generally, if $\pi =P(H_1)$ and $P_{FA}=P(D\mid H_0)$ with $H_0$ meaning absence, Bayes' theorem givesThe order of conditioning is therefore essential. Changing a decision threshold commonly trades detection against false alarms; a receiver operating characteristic displays that tradeoff.

For an odd prime $p$, integers $x,y$ with $p\mid x-y$ and $p\nmid xy$, and a positive integer $n$ with $x^n\ne y^n$, the lifting-the-exponent lemma givesHere $v_p$ is the prime valuation. The divisibility and unit hypotheses cannot be dropped.

Write $x=y+a$ with $v_p(a)\ge 1$. In the binomial expansion of $(y+a{)}^p-y^p$, the first term $p{y}^{p-1}a$ has valuation $v_p(a)+1$, while every later term has greater valuation. Thus raising to the $p$th power increases the difference valuation by exactly one. An exponent coprime to $p$ leaves it unchanged; factoring $n=p^{r}m$ proves the formula.

For example, $v_3(4^9-1)=v_3(3)+v_3(9)=3$. At $p=2$ a separate rule is needed: for odd $x,y$ and even $n$, $v_2(x^n-y^n)=v_2(x-y)+v_2(x+y)+v_2(n)-1$. It gives $v_2(3^4-1)=1+2+2-1=4$, showing why the odd-prime formula alone is insufficient.

A probable prime is an integer that passes a specified necessary test for primality without thereby receiving a proof of primality. The qualification must identify the test and bases. For example, an odd $n$ passing a Fermat test to base $a$ satisfies $a^{n-1}\equiv 1(\mathrm{mod}n)$, but a composite can satisfy that congruence.

The Miller-Rabin primality test imposes a stronger condition on a chain of modular squares. Yet $2047=23\cdot 89$ passes even that test to base 2: writing $2046=2\cdot 1023$, one has $2^{1023}=(2^{11}{)}^{93}\equiv 1(\mathrm{mod}2047)$. A pass therefore means “not disproved by this test”, whereas a failing witness proves compositeness. With independent uniformly sampled bases, the standard worst-case error bound decreases exponentially with the number of rounds; it is a conditional test-error bound, not automatically the posterior probability that the particular input is prime.^[ref]