{"https://codexery.com/geometry-and-shapes/e/geometry-and-shapes/triangle-inequality/": {"t": "Triangle inequality", "s": "A fundamental theorem on triangle side lengths."}, "/e/algebra/real-valued-function/": {"t": "Real-valued function", "s": "A function that assigns a real number to each domain element."}, "https://codexery.com/mathematics-and-cryptography/e/mathematics-and-cryptography/stefan-banach/": {"t": "Stefan Banach", "s": "Polish mathematician and founder of modern functional analysis."}, "https://codexery.com/geometry-and-topology/e/geometry-and-topology/basis-linear-algebra/": {"t": "Basis (linear algebra)", "s": "A basis is a linearly independent spanning set of a vector space."}, "https://codexery.com/geometry-and-shapes/e/geometry-and-shapes/plane-geometry/": {"t": "Plane (geometry)", "s": "Two-dimensional Euclidean space with distance and angle."}, "/e/algebra/quaternion/": {"t": "Quaternion", "s": "Four-dimensional number system for three-dimensional rotations."}, "https://codexery.com/cities/e/landmark-cities/new-york-city/": {"t": "New York City", "s": "The most populous city in the United States, a global center of finance, culture, and diplomacy."}, "https://codexery.com/probability-statistics/e/probability-and-statistics/probability-theory/": {"t": "Probability theory", "s": "Mathematical study of randomness and uncertainty."}, "https://codexery.com/communication-and-media/e/communication-and-media/signal-processing/": {"t": "Signal processing", "s": "Subfield of electrical engineering for analyzing and modifying signals."}, "/e/algebra/quadratic-form/": {"t": "Quadratic form", "s": "A homogeneous polynomial of degree two in several variables."}, "https://codexery.com/geometry-and-shapes/e/geometry-and-shapes/unit-circle/": {"t": "Unit circle", "s": "A circle of radius 1, central to trigonometry and complex analysis."}, "https://codexery.com/geometry-and-topology/e/geometry-and-topology/convex-set/": {"t": "Convex set", "s": "A set containing all line segments between its points."}, "https://codexery.com/historical-figures/e/notable/variste-galois/": {"t": "Évariste Galois", "s": "Mathematician and revolutionary who solved a 350-year-old problem."}, "https://codexery.com/classical-mechanics-and-dynamics/e/classical-mechanics-and-dynamics/rotation/": {"t": "Rotation", "s": "Rotational motion leaves at least one point unchanged."}, "/e/algebra/rational-function/": {"t": "Rational function", "s": "Ratio of two polynomial functions, fundamental in algebra and analysis."}, "https://codexery.com/economic-concepts/e/economic-concepts/economics/": {"t": "Economics", "s": "Social science studying production, distribution, and consumption of goods and services.", "i": "https://codexery.com/economic-concepts/img/economic-concepts/economics.webp", "c": "Pk0001 · CC BY-SA 4.0"}, "/e/algebra/polynomial-ring/": {"t": "Polynomial ring", "s": "Ring formed from polynomials with coefficients in another ring."}, "https://codexery.com/geometry-and-topology/e/geometry-and-topology/algebraic-geometry/": {"t": "Algebraic geometry", "s": "Branch of mathematics studying zeros of polynomial equations."}, "/e/algebra/ring-mathematics/": {"t": "Ring (mathematics)", "s": "Algebraic structure with two binary operations, generalizing integer arithmetic."}, "/e/algebra/product-mathematics/": {"t": "Product (mathematics)", "s": "The result of multiplication, central to arithmetic and algebra."}, "https://codexery.com/mathematical-logic-and-computation/e/mathematical-logic-and-computation/algorithm/": {"t": "Algorithm", "s": "A finite sequence of logical instructions for solving problems."}, "/e/algebra/quadratic-formula/": {"t": "Quadratic formula", "s": "Closed-form expression for solving quadratic equations."}, "/e/algebra/quadratic-equation/": {"t": "Quadratic equation", "s": "A second-degree polynomial equation with at most two solutions."}, "https://codexery.com/scientists-and-mathematicians/e/scientists-and-mathematicians/niels-henrik-abel/": {"t": "Niels Henrik Abel", "s": "Norwegian mathematician who proved the impossibility of solving the general quintic equation in radicals."}, "/e/algebra/polynomial/": {"t": "Polynomial", "s": "A mathematical expression of variables and coefficients using addition and multiplication."}, "/e/algebra/product-category-theory/": {"t": "Product (category theory)", "s": "Universal construction generalizing Cartesian product across categories."}, "https://codexery.com/sets-and-logic/e/sets-and-logic/set-mathematics/": {"t": "Set (mathematics)", "s": "A set is a collection of different things, typically mathematical objects."}, "/e/algebra/ring-theory/": {"t": "Ring theory", "s": "Study of algebraic structures with addition and multiplication."}, "https://codexery.com/combinatorics/e/combinatorics/bijection/": {"t": "Bijection", "s": "A function pairing each element of two sets exactly once.", "i": "https://codexery.com/combinatorics/img/combinatorics/bijection.webp", "c": "Schapel · Public domain"}, "https://codexery.com/geometry-and-topology/e/geometry-and-topology/affine-transformation/": {"t": "Affine transformation", "s": "A transformation preserving lines and parallelism in affine spaces."}, "https://codexery.com/historical-figures/e/notable/david-hilbert/": {"t": "David Hilbert", "s": "German mathematician and philosopher of mathematics"}, "/e/algebra/norm-mathematics/": {"t": "Norm (mathematics)", "s": "A function measuring vector length in a normed space."}, "/e/algebra/projection-linear-algebra/": {"t": "Projection (linear algebra)", "s": "A linear operator that is idempotent, generalizing graphical projection."}, "https://codexery.com/geometry-and-topology/e/geometry-and-topology/cartesian-product/": {"t": "Cartesian product", "s": "Set of all ordered pairs from two sets."}, "https://codexery.com/geometry-and-topology/e/geometry-and-topology/topological-space/": {"t": "Topological space", "s": "A set with a structure defining closeness without distance."}, "https://codexery.com/set-theory-logic/e/set-theory-and-logic/ordered-pair/": {"t": "Ordered pair", "s": "Ordered pair: a fundamental mathematical object with order significance."}, "https://codexery.com/sets-and-logic/e/sets-and-logic/empty-set/": {"t": "Empty set", "s": "The unique set containing no elements."}, "https://codexery.com/probability-statistics/e/probability-and-statistics/parameter/": {"t": "Parameter", "s": "A characteristic that helps define or classify a system.", "i": "https://codexery.com/probability-statistics/img/probability-and-statistics/parameter.webp", "c": "Yomomo · CC BY-SA 4.0"}, "https://codexery.com/set-theory-and-foundations/e/set-theory-and-foundations/class-set-theory/": {"t": "Class (set theory)", "s": "A class is a collection of mathematical objects defined by a property, differing from sets to avoid paradoxes."}, "https://codexery.com/sets-and-logic/e/sets-and-logic/set-theory/": {"t": "Set theory", "s": "The branch of mathematical logic that studies sets, founded by Georg Cantor in the 1870s."}, "https://codexery.com/set-theory-logic/e/set-theory-and-logic/operation-mathematics/": {"t": "Operation (mathematics)", "s": "A function combining elements of a set into another element."}, "https://codexery.com/computing-digital/e/computing-and-digital/computer-programming/": {"t": "Computer programming", "s": "The composition of instructions for computers to follow."}, "/e/algebra/rank-linear-algebra/": {"t": "Rank (linear algebra)", "s": "Rank measures the dimension of a matrix's column or row space."}, "https://codexery.com/probability-statistics/e/probability-and-statistics/ordinary-least-squares/": {"t": "Ordinary least squares", "s": "A method for estimating linear regression parameters by minimizing squared errors.", "i": "https://codexery.com/probability-statistics/img/probability-and-statistics/ordinary-least-squares.webp", "c": "Eteri byazrova (GraphMath) · CC BY-SA 4.0"}, "https://codexery.com/science-measurement/e/science-and-measurement/machine-learning/": {"t": "Machine learning", "s": "Statistical algorithms that learn from data without explicit programming."}, "/e/algebra/quadratic-function/": {"t": "Quadratic function", "s": "A polynomial function of degree two, graphed as a parabola."}, "https://codexery.com/geometry-and-shapes/e/geometry-and-shapes/tangent/": {"t": "Tangent", "s": "The line that just touches a curve at a point."}, "https://codexery.com/computing-software/e/computing-and-software/programming-language/": {"t": "Programming language", "s": "Engineered languages for expressing computer programs."}, "https://codexery.com/classical-mechanics-and-dynamics/e/classical-mechanics-and-dynamics/motion/": {"t": "Motion", "s": "Motion is the continuous change of position in space."}, "https://codexery.com/mechanics-and-fluid-dynamics/e/mechanics-and-fluid-dynamics/displacement-geometry/": {"t": "Displacement (geometry)", "s": "Vector representing shortest distance and direction of motion."}, "https://codexery.com/philosophical-concepts/e/philosophical-concepts/time/": {"t": "Time", "s": "Time is the fourth dimension, measured by clocks and calendars."}, "https://codexery.com/classical-mechanics/e/classical-mechanics/velocity/": {"t": "Velocity", "s": "A vector measure of speed with direction."}, "https://codexery.com/acids-and-bases/e/acids-and-bases/acid-strength/": {"t": "Acid strength", "s": "Acid strength measures proton donation tendency via dissociation constants.", "i": "https://codexery.com/acids-and-bases/img/acids-and-bases/acid-strength.webp", "c": "Linsley, Clyde Maurice; Bauer, Frederick Charles · Public domain"}, "https://codexery.com/acids-and-bases/e/acids-and-bases/acid-dissociation-constant/": {"t": "Acid dissociation constant", "s": "Quantitative measure of acid strength in solution.", "i": "https://codexery.com/acids-and-bases/img/acids-and-bases/acid-dissociation-constant.webp", "c": "Д.Ильин own work, vectorization, based on File:Citric acid speciation.png by Pet · CC BY-SA 4.0"}, "https://codexery.com/historical-figures/e/notable/euclid/": {"t": "Euclid", "s": "Ancient Greek mathematician, author of the Elements."}, "https://codexery.com/scientists-and-mathematicians/e/scientists-and-mathematicians/brahmagupta/": {"t": "Brahmagupta", "s": "Indian mathematician and astronomer who formalized zero and the quadratic formula"}, "https://codexery.com/historical-figures/e/scientists/ren-descartes/": {"t": "René Descartes", "s": "Father of modern philosophy and analytic geometry"}, "https://codexery.com/geometry-and-topology/e/geometry-and-topology/differential-geometry/": {"t": "Differential geometry", "s": "Study of smooth shapes and spaces using calculus and algebra."}, "https://codexery.com/geometry-and-topology/e/geometry-and-topology/manifold/": {"t": "Manifold", "s": "A topological space locally resembling Euclidean space near each point."}, "https://codexery.com/probability-statistics/e/probability-and-statistics/multivariate-normal-distribution/": {"t": "Multivariate normal distribution", "s": "Generalization of normal distribution to multiple dimensions.", "i": "https://codexery.com/probability-statistics/img/probability-and-statistics/multivariate-normal-distribution.webp", "c": "MartinThoma · CC0"}, "https://codexery.com/geometry-and-topology/e/geometry-and-topology/analytic-geometry/": {"t": "Analytic geometry", "s": "Study of geometry using a coordinate system."}, "https://codexery.com/geometry-and-topology/e/geometry-and-topology/algebraic-topology/": {"t": "Algebraic topology", "s": "Branch of mathematics using algebra to study topological spaces."}, "https://codexery.com/scientists-and-mathematicians/e/scientists-and-mathematicians/leonhard-euler/": {"t": "Leonhard Euler", "s": "Swiss polymath who founded graph theory and topology, introduced key mathematical notation, and was the most prolific mathematician of the 18th century."}, "https://codexery.com/historical-figures/e/notable/carl-friedrich-gauss/": {"t": "Carl Friedrich Gauss", "s": "German mathematician, astronomer, geodesist, and physicist"}, "https://codexery.com/scientists-and-mathematicians/e/scientists-and-mathematicians/carl-gustav-jacob-jacobi/": {"t": "Carl Gustav Jacob Jacobi", "s": "German mathematician who revolutionized elliptic functions and dynamics."}, "https://codexery.com/combinatorics/e/combinatorics/permutation/": {"t": "Permutation", "s": "A bijection rearranging a set's elements into a new order."}, "https://codexery.com/mathematicians/e/mathematicians/james-joseph-sylvester/": {"t": "James Joseph Sylvester", "s": "English mathematician who shaped matrix theory and American mathematics.", "i": "https://codexery.com/mathematicians/img/mathematicians/james-joseph-sylvester.webp", "c": "Alfred Edward Emslie · Public domain"}, "https://codexery.com/historical-eras/e/historical-eras/middle-kingdom-of-egypt/": {"t": "Middle Kingdom of Egypt", "s": "Period of reunification and stability after the First Intermediate Period.", "i": "https://codexery.com/historical-eras/img/historical-eras/middle-kingdom-of-egypt.webp", "c": "Gert7 · via Wikipedia: Middle Kingdom of Egypt · CC BY-SA 4.0"}, "https://codexery.com/historical-figures/e/notable/diophantus/": {"t": "Diophantus", "s": "Greek mathematician, author of Arithmetica, often called the father of algebra."}, "https://codexery.com/languages/e/languages/latin/": {"t": "Latin", "s": "A classical language of the Italic branch, Latin evolved into the Romance languages and remained a key scholarly tongue into the early 19th century."}, "https://codexery.com/geometry-and-shapes/e/geometry-and-shapes/surface-mathematics/": {"t": "Surface (mathematics)", "s": "A two-dimensional mathematical model generalizing the plane."}, "https://codexery.com/geometry-and-shapes/e/geometry-and-shapes/three-dimensional-space/": {"t": "Three-dimensional space", "s": "Three coordinates define a point in space."}, "https://codexery.com/design-digital-arts/e/design-and-digital-arts/3d-computer-graphics/": {"t": "3D computer graphics", "s": "3D computer graphics use geometric data to render digital images."}, "https://codexery.com/science-measurement/e/science-and-measurement/magnetic-resonance-imaging/": {"t": "Magnetic resonance imaging", "s": "Medical imaging technique using magnetic fields and radio waves.", "i": "https://codexery.com/science-measurement/img/science-and-measurement/magnetic-resonance-imaging.webp", "c": "UCSF Health · CC BY-SA 4.0"}, "https://codexery.com/cities/e/capitals/dublin/": {"t": "Dublin", "s": "Capital of Ireland, city of the black pool and the hurdled ford."}, "https://codexery.com/world-heritage-tentative-list/e/world-heritage-tentative-list/dunsink-observatory/": {"t": "Dunsink Observatory", "s": "Historic Irish observatory, home of Hamilton's quaternion discovery."}, "https://codexery.com/classical-mechanics-and-dynamics/e/classical-mechanics-and-dynamics/kinematics/": {"t": "Kinematics", "s": "Subfield of physics and geometry describing motion without forces."}, "https://codexery.com/electromagnetism/e/electromagnetism/maxwell-s-equations/": {"t": "Maxwell's equations", "s": "Foundation of classical electromagnetism and optics."}, "https://codexery.com/chemists-and-biochemists/e/chemists-and-biochemists/josiah-willard-gibbs/": {"t": "Josiah Willard Gibbs", "s": "American scientist who created statistical mechanics and modern vector calculus."}, "https://codexery.com/design-digital-arts/e/design-and-digital-arts/computer-graphics/": {"t": "Computer graphics", "s": "Computer graphics generates and manipulates images using computers."}, "https://codexery.com/computing-digital/e/computing-and-digital/computer-simulation/": {"t": "Computer simulation", "s": "Running mathematical models on computers to represent real-world systems."}, "https://codexery.com/languages-and-dialects/e/languages-and-dialects/german-language/": {"t": "German language", "s": "German, known natively as Deutsch (pronounced [dɔɪ̯tʃ]), belongs to the West Germanic branch of the Indo-European language family and is spoken primarily across"}, "/e/algebra/rank-nullity-theorem/": {"t": "Rank–nullity theorem", "s": "Domain dimension equals rank plus nullity."}, "https://codexery.com/combinatorics/e/combinatorics/recurrence-relation/": {"t": "Recurrence relation", "s": "Equation defining sequence terms from previous ones."}, "https://codexery.com/classical-mechanics/e/classical-mechanics/force/": {"t": "Force", "s": "A vector action causing change in motion, shape, or pressure.", "i": "https://codexery.com/classical-mechanics/img/classical-mechanics/force.webp", "c": "SpaceX · CC0"}, "https://codexery.com/separation-and-purification-techniques/e/separation-and-purification-techniques/spectroscopy/": {"t": "Spectroscopy", "s": "Spectroscopy measures and interprets electromagnetic spectra interacting with matter."}, "https://codexery.com/probability-statistics/e/probability-and-statistics/sample-space/": {"t": "Sample space", "s": "Set of all possible outcomes of an experiment or random trial."}, "https://codexery.com/probability-and-stochastic-processes/e/probability-and-stochastic-processes/random-variable/": {"t": "Random variable", "s": "A measurable function from sample space to measurable space."}, "https://codexery.com/geometry-and-topology/e/geometry-and-topology/compact-space/": {"t": "Compact space", "s": "Compactness makes a space behave like a finite set."}, "https://codexery.com/geometry-and-topology/e/geometry-and-topology/metric-space/": {"t": "Metric space", "s": "A set with a distance function between its points."}, "https://codexery.com/geometry-and-topology/e/geometry-and-topology/affine-space/": {"t": "Affine space", "s": "Geometric structure preserving parallelism and ratios, not distances."}, "https://codexery.com/combinatorics/e/combinatorics/algebraic-combinatorics/": {"t": "Algebraic combinatorics", "s": "Area of mathematics combining abstract algebra and combinatorics."}, "https://codexery.com/topology-and-abstract-spaces/e/topology-and-abstract-spaces/topology/": {"t": "Topology", "s": "Study of properties preserved under continuous deformations."}, "https://codexery.com/mathematicians/e/mathematicians/jean-pierre-serre/": {"t": "Jean-Pierre Serre", "s": "French mathematician, Fields Medalist, and first Abel Prize laureate."}, "https://codexery.com/combinatorics/e/combinatorics/combinatorics/": {"t": "Combinatorics", "s": "Mathematics of counting, arrangement, and optimization in finite systems.", "i": "https://codexery.com/combinatorics/img/combinatorics/combinatorics.webp", "c": "Dom walden · CC BY-SA 4.0"}, "https://codexery.com/mathematicians/e/mathematicians/hermann-weyl/": {"t": "Hermann Weyl", "s": "Universal mathematician who bridged pure mathematics and theoretical physics.", "i": "https://codexery.com/mathematicians/img/mathematicians/hermann-weyl.webp", "c": "via Wikipedia: Hermann Weyl · see source"}, "https://codexery.com/historical-figures/e/notable/alexander-grothendieck/": {"t": "Alexander Grothendieck", "s": "German-born French mathematician and leading figure in modern algebraic geometry"}, "https://codexery.com/complexity-theory/e/complexity-theory/time-complexity/": {"t": "Time complexity", "s": "A measure of algorithm running time as input size grows."}, "/e/algebra/representation-theory/": {"t": "Representation theory", "s": "Studies algebraic structures via linear transformations."}, "https://codexery.com/european-scientific-pioneers/e/european-scientific-pioneers/richard-dedekind/": {"t": "Richard Dedekind", "s": "German mathematician who defined real numbers via Dedekind cuts."}, "https://codexery.com/scientists-and-mathematicians/e/scientists-and-mathematicians/emmy-noether/": {"t": "Emmy Noether", "s": "Groundbreaking mathematician who transformed abstract algebra and physics."}, "https://codexery.com/mathematicians/e/mathematicians/ernst-kummer/": {"t": "Ernst Kummer", "s": "German mathematician known for work in hypergeometric series, Kummer surfaces, and contributions to Fermat's Last Theorem.", "i": "https://codexery.com/mathematicians/img/mathematicians/ernst-kummer.webp", "c": "via Wikipedia: Ernst Kummer · see source"}}