First-line indentation shifts the beginning of the first formatted line of a text paragraph relative to its normal starting edge. In a left-to-right layout, a positive indent reserves space on the left; the corresponding starting edge is on the right in a right-to-left layout. In CSS, text-indent controls this placement.^[ref]

In a simple left-to-right column of width $W=40$ with indentation $d=10$, the first line begins at $x=10$ and has available typesetting line width $W-d=30$. Later lines can begin at $x=0$ and use all 40 units. A negative indent $d=-5$ instead places the start at $x=-5$, extending into the margin if no other layout constraint intervenes.

This changes the first line's geometry without adding printable glyphs. Replacing the indent by several stored spaces would couple the result to word spacing, white-space processing and the chosen computer font. A line breaking model can represent the change explicitly through its sequence of typesetting line widths.

Basic research seeks understanding of underlying phenomena without being directed at a particular practical application. Applied research instead starts from a specific practical objective. The distinction concerns the aim of the investigation: either kind can use experiments, observations and mathematical models.^[ref]

For example, investigating how an unfamiliar material responds to light can seek a general explanation of its behaviour; optimizing a particular photodetector for a specified use has an application-driven goal. The first investigation can later enable useful technology without having started as a design project. Conversely, the practical aim of the second does not remove its need for careful measurement and a testable physical explanation. These categories describe research purposes, rather than a ranking of scientific quality.

Applied research investigates a question with a particular practical objective. It still seeks new knowledge; the distinction from basic research concerns the aim of the investigation rather than whether it uses experiments, observations or mathematical models.^[ref]

For example, investigating how a material converts light into an electrical signal can support the design of a photodetector for a specified use. A controlled study can vary the material or illumination and compare the resulting sensitivity and noise. The intended detector supplies a practical objective, while the measured relation provides reusable knowledge.

This differs from simply assembling a known design or carrying out routine production. Equally, a practical aim does not guarantee a useful result: a sound experiment can establish that a proposed mechanism fails. Applied research and basic research can therefore share methods and inform each other without being distinguished by whether the outcome is successful.

Cognitive science investigates the mind and intelligence by combining methods from disciplines including psychology, neuroscience, linguistics, computer science and artificial intelligence. Its scientific questions concern mechanisms of perceiving, learning, reasoning and acting, rather than just naming mental abilities.^[ref]

A computational model of cognition makes an account precise enough to compare with observations. For example, consider two hypotheses about people performing synchronous speech. One predicts that a designated follower always trails a leader by a constant interval $d$. Another allows both people to adjust their timing. Recording corresponding speech landmarks produces a sequence of signed timing differences; this can test whether the proposed constant-follower account fits the task.

Fred Cummins's paired-reading experiments supply a concrete connection between linguistics and quantitative behavioural measurement: corresponding speech events can be closely aligned, and both speakers adjust their timing relative to individual readings.^[ref]

A successful behavioural prediction does not uniquely identify a neural mechanism. Combining task performance, explicit models and independent physiological measurements helps distinguish explanations that agree on one observable. The field's different methods constrain one another without requiring every question to be answered at the same descriptive level.

A physical constant is a quantity whose value is independent of the particular experiment within a stated physical theory and regime. Examples include the speed of light in vacuum and Planck constant. A numerical value depends on the units of measurement; a dimensionless constant avoids that dependence.

Some constants have exact numerical values by the definition of the International System of Units. Others, such as the fine-structure constant, remain measured quantities with experimental uncertainty. Exact unit-defining values do not imply that every realization of those units is exact.^[ref]

Transport phenomena are processes that redistribute quantities such as mass, momentum and energy. Bulk motion produces advection, while microscopic motion and interactions can produce diffusion, heat conduction and viscosity.

Constitutive laws relate a flux to a driving gradient: Fourier's law gives heat flux $\mathbf{q}=-k\nabla T$, and Fick's first law gives diffusive flux $\mathbf{j}=-D\nabla c$. Combining these relations with conservation balances gives evolution equations. The coefficients and the appropriate transported quantity depend on the medium; a formal similarity between two diffusion equations does not make their physical diffusivities identical.

A physical theory is an organized set of principles and mathematical relations that predicts measurable behavior within a stated domain. It connects idealized quantities to experimental procedures, allowing observations to test more than an isolated fitted formula.

For example, Newtonian mechanics relates force to acceleration and predicts orbital motion, while general relativity changes the description of gravity when spacetime curvature becomes relevant. A theory can remain useful within its approximation regime even after a broader theory is established. Its usefulness depends on explicit assumptions, reproducible predictions and known accuracy, rather than on treating every successful approximation as exact at all scales.

Computational physics uses numerical algorithms to obtain and test predictions of physical models. It includes solving differential equations, simulating stochastic processes and extracting physical quantities from data. The mathematical model, its numerical approximation and its agreement with an experiment are separate questions: an accurately executed algorithm can still solve an unsuitable model.

A simple example is the harmonic oscillator $\ddot{x}=-{\omega }^{2}x$. The explicit Euler method advances it byFor a unit mass, define $E_n=(v_n^2+{\omega }^2{x}_n^2)/2$. Direct substitution givesThe physical oscillator conserves energy, but this algorithm creates it on every step. The discrepancy is a numerical effect, not a discovery about the oscillator. Changing the time step and comparing with a known solution exposes the problem; for long integrations, the choice of numerical method matters as well as the size of the step.

Richard Feynman's lecture on planetary motion gives a concrete historical teaching example: forces determine accelerations, which update velocities and positions through successive arithmetic steps. A computer extends the same approach to systems whose coupled motion has no convenient closed expression.^[ref] Conservation checks, comparison against solvable limits, and convergence under finer numerical resolution help distinguish a reliable calculation from an attractive plot. Agreement with measurements then tests the physical model itself.

Medical physics applies physics to quantitative problems in medical imaging, nuclear medicine and radiation therapy. It connects an instrument's signal to a physical quantity, tests the model behind that inference, and establishes its measurement uncertainty. Examples include reconstructing an image from radiation or acoustic measurements and determining the absorbed dose delivered by an ionizing radiation beam.

The sealed-water calorimeter of Steve R. Domen gives a concrete measurement example. Domen's 1994 study placed sealed water in a glass cylinder, 110 mm long and 33 mm in diameter, inside a 30 cm water-filled acrylic cube. Two calibrated thermistors formed opposite arms of a Wheatstone bridge; their change in electrical resistance measured the small temperature rise. The glass also limited convection.^[ref]

In the ideal water calorimetry balance, deposited energy becomes heat, givingwhere $D$ is absorbed dose and $c_p$ specific heat capacity. Using a rounded illustrative $c_p=4180\mathrm{J}\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{K}}^{-1}$, depositing $1\mathrm{J}$ in $1\mathrm{kg}$ corresponds to one gray and onlyThis small signal makes thermal control and measurement calibration central to the experiment.

Domen also investigated the heat defect: radiation-induced chemistry can release or store energy, changing the heat readout. Approximately 3700 measurements varied dissolved-gas conditions and compared the response with other calorimeters. The test examined whether a measured temperature rise represented the intended absorbed dose, rather than assuming the equivalence.^[ref]

Metrological traceability connects such measurements through documented calibrations and their measurement uncertainties. In the IAEA/WHO network, national laboratories and secondary-standard laboratories link reference dosimeters to the instruments used in practice. A primary measurement standard supplies the reference realization; the calibration chain transfers it to other instruments.^[ref]

A mathematical theorem is a statement established by a mathematical proof from specified assumptions and previously justified results. Its hypotheses belong to the claim: proving a conclusion for one class of objects does not automatically prove it for a larger class.

For every nonnegative integer $n$, the sum of the first $n$ odd positive integers is $n^2$. A proof by mathematical induction starts with the empty sum at $n=0$; adding the next term gives $n^2+(2n+1)=(n+1{)}^2$. This argument establishes all cases from one implication. Checking many numerical cases can suggest the result, but does not replace the inductive step unless the full stated domain is finite and exhaustively covered.

A special function is a named family of functions with a developed theory of identities, transformations, asymptotic behavior and numerical evaluation, often arising repeatedly as solutions of differential equations or integral representations. The boundary with elementary functions is conventional rather than a strict mathematical classification.

For example, the regular solution of $x^2{y}^{\prime \prime }+x{y}^{\prime }+x^{2}y=0$ normalized by $y(0)=1$ is the order-zero Bessel function $J_0$. Such a function packages a recurring radial-wave calculation into a reusable object. The Gamma function similarly extends factorial values through $\Gamma (n+1)=n!$ for nonnegative integers, allowing identities involving integer parameters to be studied continuously.

A rational univariate representation expresses a finite algebraic solution set using roots of one univariate polynomial and rational coordinate functions. For distinct solutions, one may use a separating parameter $t$, a square-free polynomial $p(t)$, and formulas $x_i=q_i(t)/q_0(t)$ with $q_0$ nonzero at the relevant roots. Recovering multiplicities requires additional care beyond merely listing the distinct points.

For $x^2-2=0$ and $(x-1)y-1=0$, take $p(t)=t^2-2$, $x=t$ and $y=1/(t-1)$. Neither root of $p$ equals one, so both coordinate formulas are defined. Solving one polynomial therefore recovers both solutions of the original two-equation system.^[ref]

A computation applies specified rules to represented information to obtain a result. The rules may operate on exact integers, symbolic expressions or approximate numerical values; the representation determines which operations and errors must be considered.

The Euclidean algorithm computes a greatest common divisor by replacing $(a,b)$ with $(b,a\mathrm{mod}b)$ until the second entry is zero. For $(30,18)$ the pairs become $(18,12)$, $(12,6)$ and $(6,0)$, yielding six. The identity $\gcd (a,b)=\gcd (b,a\mathrm{mod}b)$ proves correctness, while decreasing positive remainders prove termination. Computability theory asks whether a suitable algorithm exists; computational complexity asks what resources it requires.

Scientific computing turns scientific models into reproducible calculations through numerical and symbolic algorithms, with explicit attention to error and computational cost. The task includes selecting a model representation, solving the resulting equations and checking which features survive controlled changes in resolution or precision.

For a one-dimensional heat equation solved by explicit finite differences over a fixed physical time, halving the spatial grid spacing doubles the number of unknowns and typically requires four times as many time steps for stability. With linear work per step, the cost grows by about a factor of eight. This connects a physical resolution choice with an algorithmic resource requirement rather than treating finer resolution as free.

The Abel Prize is an international award for outstanding work in mathematics, named after Niels Henrik Abel. It was first awarded in 2003 to Jean-Pierre Serre. The Norwegian Academy of Science and Letters administers the selection.^[ref]

The mathematical work behind an award is more informative than the honour alone. The 2009 award recognized Mikhail Gromov's contributions to geometry and topology, including global geometric questions whose behaviour cannot be read from local coordinates alone.^[ref] The 2021 award to László Lovász and Avi Wigderson recognized contributions connecting discrete mathematics and theoretical computer science.^[ref] The prize's scope therefore spans several mathematical traditions.