A mass analyzer is the part of a mass spectrometer that makes ions distinguishable or selectively transmits them according to mass-to-charge ratio. The measured coordinate can be position, arrival time or transmission under a specified field setting. A separate mass spectrometer detector records the resulting signal.

A quadrupole mass analyzer selects trajectories using combined static and time-varying electric fields: chosen mass-to-charge values remain on stable paths while others are lost from the transmitted beam.^[ref] A time-of-flight mass spectrometer instead encodes the ratio in arrival time after acceleration. Different physical designs can therefore provide the same kind of ionic-mass information through different observables.

An original magnetic example exposes an important assumption. For speed $v$ perpendicular to a uniform magnetic field $B$, balancing the Lorentz force and circular acceleration givesIf the incoming ions have the same selected speed and charge, mass values in the ratio $56/54$ give bending radii in that ratio, about $1.0370$. But if ions instead start negligibly slowly and gain energy from the same electric potential difference $U$, substituting $qU=m{v}^2/2$ gives $r=\sqrt{2mU/q}/B$. Their radius ratio is then $\sqrt{56/54}\approx 1.0184$. The initial-condition choice changes the mapping.

Analysis requires this mapping and its assumptions, not just separated peaks. Mass resolving power describes the ability to distinguish nearby peak coordinates under a stated width criterion; accurate assignment of those coordinates additionally needs instrument calibration and the correct charge-state interpretation.

A mass spectrometer detector turns arriving ions into a recorded electrical signal. It is distinct from the mass analyzer that makes mass-to-charge ratio information distinguishable. The measured response can depend on collection, conversion and amplification as well as on how many ions arrived.

A Faraday cup collects electric charge directly. Ideally, a flux $\Phi$ of ions with known charge $q$ gives $I=q\Phi$. For $10^6$ singly charged positive ions per second, the current is $1.602176634\times 10^{-13}\mathrm{A}$, about $0.1602\mathrm{pA}$. Escaping secondary electrons and background currents must be accounted for when inferring particle delivery.^[ref]

An electron multiplier instead uses secondary emission to amplify the initial conversion signal. The manufacturer's detector description shows discrete dynodes, a continuous-dynode alternative, and a conversion dynode before the multiplier.^[ref] In an illustrative equal-stage model, ten stages with mean yield $\delta =4$ give mean gain $G=4^{10}=1048576$. This is a model calculation, not a measured gain for a named instrument.

Detector gain and detection efficiency answer different questions: the size of an output event and the probability of registering an arriving ion. Increasing gain cannot recover missed ions. At high signals, mass spectrometer detector saturation can also invalidate proportionality between ion delivery and recorded response. Quantitative peak comparisons therefore need a tested response range and appropriate instrument calibration, not merely a large signal.

A computer virus is self-replicating code that propagates by becoming part of host computer programs or suitable files. Its replication depends on the host's execution context. This distinguishes it from a self-contained computer worm, although both are conventionally grouped as malware.^[ref]

Replication and any additional harmful effect are separate features. A computer virus can make unauthorized changes simply by incorporating itself into another host; a dramatic data-destruction effect is not necessary to recognize that replication mechanism.

For an original counting example, imagine ten archived copies of the same infected host file on an isolated storage medium. That is ten stored artifacts, not evidence of ten running processes or ten infected computers. Conversely, one executing host can affect more than one stored artifact. Counts of detected files, affected machines and active executions measure different things.

The biological analogy is useful only in a limited sense: a computer virus depends on a host context for reproduction, but its mechanism is computation and file modification. Digital evidence about copies and execution should support the particular incident claim being made, rather than treating every occurrence of a byte pattern as a separate observed infection event.

A MALDI matrix is material combined with an analyte to assist its release and ionization during MALDI. In ultraviolet MALDI, the MALDI matrix absorbs light at the laser wavelength, coupling the pulse to a sample that need not absorb efficiently on its own. Its chemical and physical properties influence the resulting mass spectrum.

An early experiment compared amino acids at different laser wavelengths. In a mixture of alanine and tryptophan irradiated at $266\mathrm{nm}$, an alanine molecular-ion signal appeared at about one tenth of the irradiance required for alanine alone. Absorbing tryptophan therefore supplied a concrete matrix-assisted effect; the useful observation was the changed ion-production threshold.^[ref]

An idealized optical model illustrates why absorption matters without predicting ion yield. If a uniform layer has absorption coefficient $\alpha$ and thickness $d$, the Beer-Lambert law gives absorbed fraction $1-e^{-\alpha d}$, ignoring reflection and scattering. Optical depths $\alpha d=1$ and $3$ give approximately $63\%$ and $95\%$. These are absorbed-light fractions, not the fractions of analyte molecules that become detected ions; chemistry and subsequent transport still intervene.

The MALDI matrix can also affect spatial interpretation. A tissue-imaging study at $20\mu \mathrm{m}$ sampling found matrix-crystal clusters larger than the sampling scale; their distribution contributed granular structure to the ion image.^[ref] Matrix preparation is consequently part of the measurement, not an invisible supporting surface. Mass spectrometry image normalization may reduce some common intensity variation, but it cannot make every matrix-dependent response an absolute concentration measurement.

Phase-contrast magnetic resonance imaging estimates motion from differences in the phase of nuclear magnetic resonance signals acquired with controlled magnetic field gradients. The encoding measures a component of velocity, with its sign set by the gradient and reconstruction convention. A 1982 paper described an early flow-sensitive phase-encoding approach.^[ref]

For a constant-velocity teaching model, write position along the encoding direction as $x(t)=x_0+vt$. If $G(t)$ is the effective gradient, including refocusing signs, thenHere $\gamma$ is the gyromagnetic ratio in radians per second per tesla, and $m_1$ is the gradient first moment. With balanced zeroth moments, two encodings giveThese equations are an explicit idealized derivation, not a claim that every measured voxel contains one uniform velocity.

Define the velocity encoding limit by $v_{\mathrm{enc}}=\pi /(\gamma |\Delta m_1|)$. If it is $10cm/s$, a corrected phase difference of $+\pi /2$ corresponds to $+5cm/s$ under the chosen sign convention. Phase wrapping modulo $2\pi$ causes an ambiguity beyond the encoding range, related to aliasing.

A 1991 gated study measured differently timed cerebrospinal fluid motion in ventricular and subarachnoid compartments over the cardiac cycle.^[ref] Opposite velocities during equal parts of a cycle can average to zero despite substantial oscillatory motion. A volumetric flow rate additionally requires integrating the normal velocity across a defined cross-section; peak speed alone does not supply transported volume.

Magnetic resonance velocity encoding makes a component of motion measurable through the phase of a nuclear magnetic resonance signal. Controlled magnetic field gradients give different phase shifts to spins with different trajectories. Phase-contrast magnetic resonance imaging compares encodings to extract a velocity component, rather than interpreting ordinary image brightness as speed.^[ref]

In an original idealized model, assume constant velocity $v$ along one encoding axis and effective gradients with equal gradient zeroth moments. Then the encoding difference obeyswhere $\gamma$ is the angular gyromagnetic ratio and $\Delta m_1$ is the gradient first moment difference. For $\gamma =2\pi \cdot 42.58\times 10^6\mathrm{rad}{\mathrm{s}}^{-1}{\mathrm{T}}^{-1}$, $\Delta m_1=2\times 10^{-8}\mathrm{T}{\mathrm{s}}^2/m$ and $v=0.10m/s$, this gives $\Delta \varphi \simeq 0.535\mathrm{rad}$. These stipulated values illustrate the conversion, rather than a particular scanner experiment.

Reversing the moment difference reverses the phase sign under this convention. Reducing its magnitude decreases sensitivity to a given velocity while increasing the velocity encoding limit. A measured voxel can contain a range of velocities, and background phase or phase aliasing can complicate reconstruction.

The encoded component also differs from volumetric flow rate: estimating transported volume requires combining the appropriate normal velocity with area and time, not merely recording one peak speed.

Computational learning theory studies what an algorithm can learn under an explicit observation model, accuracy requirement and resource bound. Sample complexity measures how much information is needed; time complexity measures the computation used to turn it into a predictor. A class can admit statistically sufficient samples while presenting a difficult computational Search problem.

In realizable probably approximately correct learning, labelled examples are sampled independently from a fixed but unknown distribution and some hypothesis in the class labels them perfectly. The learner must return a hypothesis whose true error is at most $\epsilon$ with probability at least $1-\delta$. For a finite class $H$, any consistent learner satisfies this guarantee whenIndeed, a fixed hypothesis with error greater than $\epsilon$ matches all $m$ examples with probability at most $(1-\epsilon {)}^m\le e^{-m\epsilon }$; a union bound over $H$ proves the claim. The argument bounds information requirements, but does not supply an efficient method for finding a consistent hypothesis.

In online learning, a learner chooses $a_t$ before observing the round's loss function ${\ell }_t$. Its external cumulative regret against a fixed action set $A$ isNo-regret learning seeks a guarantee $R_T=o(T)$, often in expectation for a randomized algorithm, under stated feedback and adversary assumptions. The average excess loss then tends to zero. The comparator is the best single action in hindsight; it does not choose the best action separately on every round.

For example, against two prediction rules, an exponential-weights learner starts with equal weights and multiplies rule $i$'s weight by $\exp (-\eta {\ell }_t(i))$ after seeing all rule losses. Sampling in proportion to the weights gradually shifts probability toward better rules. For losses in $[0,1]$, an appropriate learning rate gives an order-$\sqrt{T}$ expected bound for this fixed two-rule setting. This compares performance even without independent observations from one fixed distribution; it does not guarantee small absolute loss when every rule predicts badly.

A word embedding represents a word by a vector, usually learned so that relationships in the data become useful relationships between vectors. A vocabulary of $V$ words can be stored as a matrix with $V$ rows and $d$ columns: selecting a word selects one row. Unlike a separate indicator for each word, a dense representation can share predictive information across words.

For example, training a neural language model can place words occurring in similar contexts near one another, making evidence about “cat” useful when predicting contexts involving “dog”. This is statistical similarity, not a guarantee of synonymy: opposite words can occur in similar contexts too. The 2003 neural probabilistic model learned these vectors jointly with next-word probabilities.^[ref]

A neural language model uses a neural network to assign a probability distribution to language sequences. An autoregressive model factorizes the probability of words $w_1,\dots ,w_T$ asTraining by maximum likelihood estimation rewards the observed next words; generation repeatedly samples from the conditional distribution. If a two-word sequence has first-word probability $0.2$ and conditional second-word probability $0.3$, its joint probability is $0.06$.

Yoshua Bengio and coauthors' 2003 model jointly learned a distributed representation of words and a neural probability predictor. Sharing representation parameters allowed similar contexts to inform one another, rather than treating every observed word sequence as an unrelated event.^[ref]