A steady electrical circuit described by Ohm's law, Kirchhoff's current law, Kirchhoff's voltage law, and ideal independent electrical sources or linear dependent electrical sources. Fixed sources make the terminal response an affine function when the circuit equations admit a unique voltage for each prescribed terminal electric current.

Frequency modulation carries a message by changing the instantaneous frequency of a carrier wave. In the ideal model the wave amplitude stays constant: information changes the rate at which the wave phase advances. Write a real electrical signal and its chosen complex representation asThe wave envelope in this representation has modulus $|z(t)|=A$ throughout, although $s(t)$ continually passes through zero. A smaller instantaneous value of $s(t)$ therefore does not mean a smaller wave amplitude. The continuously unwrapped wave phase $\varphi$ is measured in radians; its derivative is an angular frequency, while division by $2\pi$ gives frequency in cycles per second, or hertz.

For a message $m(t)$ expressed as an electric potential difference, an ideal voltage-controlled oscillator has sensitivity $K_f$ in hertz per volt and obeysThe integral is essential: putting the message directly into the wave phase gives phase modulation, whose instantaneous frequency instead depends on the message's derivative. Amplitude modulation changes the multiplying wave amplitude. These are different ways to encode the same kind of message signal.

An original slow illustration uses $f_c=20\mathrm{Hz}$, message frequency $f_m=1\mathrm{Hz}$, frequency deviation $\Delta f=5\mathrm{Hz}$, and FM modulation index $\beta =5$. With zero initial wave phase,Here $t$ is in seconds. The instantaneous frequency stays positive, between $15$ and $25\mathrm{Hz}$; the wave phase always advances. The oscillations bunch up where $f_i$ is large and spread out where it is small, while the wave envelope remains at $A$.

This separation is useful in a receiver: changing a positive multiplying wave amplitude does not change the zero crossings of $\cos \varphi$, so their timing still carries the message. Noise that moves those crossings, a vanishing wave amplitude, and frequency response distortion can still damage it. The surprising beauty of FM is that a constant wave envelope can conceal a rich frequency spectrum: the components interfere to change timing instead of the envelope's height.

An oscillator whose frequency is adjusted by an input electric potential difference. An approximately linear control characteristic provides a model for frequency modulation.

Amplitude modulation encodes a message by varying the multiplying wave amplitude of a carrier wave. Conventional AM adds a positive constant before multiplication, allowing a simple envelope detector to recover the message within its limits. Multiplication translates the message's frequency spectrum into a band around the carrier, adapting a slow message to a faster channel. This is an application of the modulation property of the Fourier transform.^[ref]

For an original single-message-frequency voltage model, chooseHere $t$ is time, $A$ is the peak voltage amplitude of the unmodulated carrier wave, ${\omega }_c$ is its angular frequency, $\Omega$ is the message angular frequency, and $m$ is the dimensionless AM modulation index. The message shape is $\cos (\Omega t)$; $m$ measures how deeply it changes the multiplying factor. Define the signed factor $q(t)=1+m\cos (\Omega t)$. The nonnegative wave envelope of the representation $z(t)=Aq(t)e^{\mathrm{i}{\omega }_{c}t}$ isThe instantaneous voltage $s(t)$ crosses zero every carrier cycle, even when $E(t)$ is large. Its value at one time, its wave envelope, and its carrier-component amplitude are three different quantities. The narrow-envelope intuition additionally needs $\Omega \ll {\omega }_c$; the algebraic model only assumes $\Omega <{\omega }_c$.

The beauty is that two views reveal different parts of the same operation. In time, the message appears to ride on the oscillations. In the frequency spectrum, amplitude modulation sidebands show it as a pair of translated components. An envelope detector reads their combined shape without needing the carrier's wave phase; coherent demodulation instead uses a synchronized reference and preserves the signed factor.

An electrical engineer designs or analyzes systems involving electrical energy, electromagnetic fields and information-carrying signals. The work includes circuits, power conversion, communication, instrumentation and control. A sensor readout, for example, needs an amplifier that preserves the signal while accounting for noise, bandwidth, loading and supply limitations. The mathematical circuit model must be compared with component tolerances and measured behaviour, because an ideal amplifier equation does not capture every physical limit.

An antenna couples electrical circuits to propagating electromagnetic radiation for transmission or reception.

A computer scientist studies computation, information representation and algorithms. The work ranges from proving limits in computability theory and computational complexity theory to designing systems and evaluating them empirically. An analysis of binary search asks which ordering assumptions make elimination valid and how many comparisons are needed; a systems experiment asks how representation and hardware affect measured execution. Mathematics, formal models and reproducible measurement answer complementary questions.

A fixed-size unit of stored information in a computational cost model. Counting machine words separately from bits makes assumptions about identifier and value sizes explicit: for example, the storage bound for an adjacency list assumes each vertex identifier fits into a constant number of machine words.

Software consists of instructions and associated data used by a computer. A spreadsheet is software that stores spreadsheet cells and evaluates their spreadsheet formulas.

A computer is a physical system that represents information and carries out defined operations on it. In a stored-program design, instructions and data are held in computer memory, and a processor interprets instructions to update state. A simple program can read two represented numbers, add them and store the result; the represented bit patterns acquire meaning through the chosen encoding. Digital computers use discrete states, while analogue computers represent quantities through continuously varying physical variables. A mathematical model of computation abstracts these mechanisms to study capability and resource use.

Chromatography separates components of a mixture through different migration rates in contact with a chromatographic stationary phase while a chromatographic mobile phase transports them. Liquid chromatography describes the physical state of the transporting phase; size-exclusion chromatography describes a separation mechanism. Classification by phase and classification by mechanism answer different questions. In ideal size-exclusion chromatography, differential access to liquid-filled pores replaces binding as the source of differential migration.

In many other forms of chromatography, a solute repeatedly leaves the chromatographic mobile phase and returns to it. Reversible adsorption holds it at a surface; partition distributes it between the chromatographic mobile phase and an immobilized liquid chromatographic stationary phase. Spending more time stationary means less progress with the moving fluid. Two initially overlapping components can therefore emerge as separate bands without either being permanently trapped. Chromatography is useful both for comparing a sample with reference substances and for collecting separated components; a visible band need not contain just one substance.

Relative retention belongs to the sample together with the chosen chromatographic stationary phase, chromatographic mobile phase, eluent composition, and temperature. Changing these conditions can change the separation order. Chemical polarity helps explain interactions in some systems but is not a universal ranking rule. In thin-layer chromatography, a spot's TLC retardation factor is consequently conditional evidence about identity, rather than a fixed identifier intrinsic to a molecule.

The branch of chemistry concerned with identifying substances in a sample and measuring their quantities or concentrations. A bicinchoninic acid assay, for example, estimates protein content through an optical measurement.

A laboratory beaker is a broad cylindrical vessel, usually with a pouring lip, used for holding, mixing, or heating samples. Its shape supports access and stirring rather than precise volume delivery.

Graduation marks give approximate volume indications. Accurate volumetric work instead uses apparatus designed and calibrated for the relevant measurement, so an observed liquid level in a beaker should not be treated as an exact delivered volume.

Biochemistry studies the chemistry of living systems: what their molecules are, how they recognize one another, and how their chemical reactions support the organization of a biological cell. It connects biology to chemical kinetics, stoichiometry and thermodynamics. A useful biochemical explanation follows both material and energy: which substrate becomes which reaction product, what sets the reaction rate, and how the transformation is driven.

Consider three connected examples. The stereochemistry of a sugar helps determine how it interacts with an enzyme's active site; the molecular formula alone cannot describe that recognition. Selective enzymes then direct material through metabolism, while ATP can participate in energy coupling that drives otherwise unfavourable transformations. These are different questions about the same working system, rather than three names for its energy supply.^[ref]

This viewpoint is useful because it separates possible explanations for a stopped process. Missing substrate, inactive enzyme, depleted ATP and accumulated reaction product call for different interventions. Studying a cell-free extract can isolate some of those causes while leaving a substantial chemical reaction network intact.

The use of physical principles and quantitative measurements to investigate living systems. A force measurement on a molecular motor, for example, connects mechanical output with a proposed chemical cycle rather than treating the motor as an unexplained black box.

A member of a chemical family including dopamine, noradrenaline and adrenaline, characterized by a catechol ring and an amine-containing side chain. Their signaling effects depend on receptor type and location rather than chemical-family membership alone.

A scientist who studies life and living systems. A biologist may investigate molecules, biological cells, whole organisms or interactions among organisms, using observation, experiments and quantitative models to connect biological patterns with explanations.

Photography records a spatial distribution of light as an image. An optical lens or optical pinhole establishes a correspondence between scene directions and positions on a receiving surface; a photographic emulsion or photodetector preserves the resulting signal. Image formation and recording are separate parts of the process.

This makes photography useful for scientific measurement: a photograph can preserve the geometry of a nuclear emulsion track or the changing appearance of an astronomical source. Interpreting the record still requires calibration. Image brightness depends on the instrument and the duration of illumination as well as on the source, and a visible feature's width can reflect the point spread function rather than the object's physical diameter.

Engineering applies scientific knowledge, mathematical models and systematic design to create systems meeting practical requirements. A design problem combines objectives with constraints: a bridge must carry specified loads while satisfying limits on deformation, materials and cost. Analysis predicts behaviour, experiments test the predictions, and manufacturing choices determine what can be built consistently. Mechanical engineering, electrical engineering, civil engineering and chemical engineering emphasize different systems, but all connect an idealized model to measured performance and uncertainty in real components.

A nonidentity Möbius transformation has one or two distinct fixed points on the Riemann sphere. If there are two, send them to zero and infinity; the conjugated map fixes both and is a nonzero scaling. If there is one, send it to infinity; the resulting affine map cannot have slope different from one, since that would give another fixed point. It is a nonzero translation, which a scaling conjugates to $z+1$. The identity is the scaling $\mu =1$. Fixed-point sets are carried bijectively under conjugacy, so a translation is not conjugate to a scaling.