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Derivation (differential algebra)
In differential algebra, a derivation is a linear map on an algebra obeying the Leibniz product rule, the algebraic abstraction of differentiation begun by Joseph Fels Ritt in the 1930s.
Derivative
The derivative of a function is the limit of its average rate of change over shrinking intervals, giving the instantaneous rate of change and tangent slope at a point.
Derivative-free optimization
Derivative-free optimization (DFO) is a class of numerical algorithms that minimize or maximize a function using only its values, when the function is a black-box simulation with no derivative information.
Derived algebraic geometry
Derived algebraic geometry, or DAG, is a branch of mathematics studying derived schemes and stacks, classical schemes enhanced with homotopical, higher categorical data, with foundations laid in the 2000s by Toën, Vezzosi, and Lurie.
Derived category
A derived category is a mathematical construction that replaces an abelian category by a category of chain complexes, invented by Grothendieck and Verdier in the early 1960s.
Derived functor
A derived functor, in left and right variants, is a homological algebra construction measuring how far a functor fails to be exact, computed by applying it to a projective or injective resolution and taking homology.
Descartes' rule of signs
Descartes' rule of signs, also called Descartes' sign rule, is an algebra theorem stating that a polynomial's positive real roots number at most its coefficient sign changes.
Descent theory
Descent theory is a set of techniques in algebraic geometry and category theory for recovering an object from its pullback along a covering morphism, presented in general form by Alexander Grothendieck in 1960.
Descriptive complexity theory
Descriptive complexity theory is a branch of finite model theory that characterizes complexity classes by the logic needed to define problems, notably NP via Fagin's 1974 theorem.
Descriptive geometry
Descriptive geometry is a field of geometry and technical drawing, created by Gaspard Monge around 1765, that represents 3D objects on 2D paper so true lengths and angles can be found.
Descriptive set theory
Descriptive set theory is the branch of mathematical logic that studies definable sets of real numbers and Polish spaces, including Borel, analytic, and coanalytic sets.
Descriptive statistics
Descriptive statistics is the branch of statistics that summarizes observed data set values using measures like the mean, median, standard deviation, histograms, and box plots.
Design effect
The design effect (deff or DEFF) is a survey sampling measure comparing a complex design's variance to that of simple random sampling, coined by Leslie Kish in 1965.
Design matrix
A design matrix is the matrix of independent-variable values used to fit a statistical model, with one row per observation and one column per model term.
Design space
A design space, in pharmaceutical manufacturing, is the combination of input variables and process parameters demonstrated to provide assurance of quality, codified in the ICH Q8 guideline of 2005.
Design-based inference
Design-based inference is a framework for statistical inference in which uncertainty is quantified through the randomization used to select samples, not through assumed models, tracing to Jerzy Neyman's 1934 paper.
Dessin d'enfant
A dessin d'enfant, French for "child's drawing," is a finite bicolored graph on a surface that determines a Riemann surface; the absolute Galois group acts on these objects.
Detailed balance
Detailed balance is a condition on stochastic systems, such as Markov chains, requiring that probability flow between each pair of states equals the reverse flow, implying reversibility.
Determinacy (set theory)
In set theory, determinacy is the property of an infinite two-player game that one player has a winning strategy; the full Axiom of Determinacy contradicts the Axiom of Choice.
Determinacy and large cardinals
Determinacy and large cardinals refers to results linking determinacy axioms, which say certain infinite games have winning strategies, with large cardinal axioms, beginning with work from the 1960s onward.
Determinant
The determinant of a square matrix is a scalar that measures how the matrix scales oriented volume and is zero exactly when the matrix is not invertible.
Determinantal point process
A determinantal point process is a random point process whose points repel each other, used in machine learning for selecting diverse subsets and in spatial statistics.
Deterministic finite automaton
A deterministic finite automaton, also called a deterministic finite state machine, is a finite-state machine that reads input one symbol at a time and recognizes exactly the regular languages.
Detrended correspondence analysis
Detrended correspondence analysis (DCA) is an ordination method used in ecology and archaeology, introduced by Hill and Gauch in 1980 to fix distortions in correspondence analysis.
Detrended fluctuation analysis
Detrended fluctuation analysis (DFA) is a statistical method that detects long-range correlations in nonstationary time series, introduced by C.-K. Peng and colleagues in 1994 to analyze DNA sequences.
Developable surface
A developable surface, also called a singly curved surface, is a surface in three-dimensional space that can be flattened onto a plane without stretching or tearing, such as cones and cylinders.
Deviance information criterion
The Deviance Information Criterion (DIC), developed by David J. Spiegelhalter and colleagues, is a Bayesian model selection criterion that compares candidate models using deviance plus a parameter penalty.
Deviation (statistics)
In statistics, a deviation, also known as a statistical deviation, is the difference between an observation and a central value such as the mean or median, and deviations underlie measures like variance and standard deviation.
DFT matrix
The DFT matrix is a complex square matrix whose entries are powers of an N-th root of unity, so multiplying it by a vector computes the discrete Fourier transform.
Diagonal
A diagonal is a line segment joining two non-adjacent vertices of a polygon or polyhedron; a polygon with n vertices has n(n−3)/2 diagonals, and a triangle has none.