Maths

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huge mess. also working back from Computing#Computation

General

to sort





  • https://en.wikipedia.org/wiki/Theorem - a statement that has been proven on the basis of previously established statements, such as other theorems—and generally accepted statements, such as axioms. The proof of a mathematical theorem is a logical argument for the theorem statement given in accord with the rules of a deductive system. The proof of a theorem is often interpreted as justification of the truth of the theorem statement. In light of the requirement that theorems be proved, the concept of a theorem is fundamentally deductive, in contrast to the notion of a scientific theory, which is empirical.


  • https://en.wikipedia.org/wiki/Lemma_(mathematics) - (plural lemmata or lemmas) from the Greek λῆμμα (lemma, “anything which is received, such as a gift, profit, or a bribe”) or helping theorem is a proven proposition which is used as a stepping stone to a larger result rather than as a statement of interest by itself. There is no formal distinction between a lemma and a theorem, only one of intention – see Theorem terminology. However, a lemma can be considered a minor result whose sole purpose is to help prove a theorem - a step in the direction of proof, so to speak.
  • https://en.wikipedia.org/wiki/List_of_lemmas
  • https://en.wikipedia.org/wiki/Fundamental_theorem - the theorem considered central to a field of mathematics. The naming of such a theorem is not necessarily based on how often it is used or the difficulty of its proofs. For example, the fundamental theorem of calculus gives the relationship between differential calculus and integral calculus, which are two distinct branches that were not obviously related. The names are mostly traditional, so that for example the fundamental theorem of arithmetic is basic to what would now be called number theory.





  • http://en.wikipedia.org/wiki/Mathematical_objects mathematical object] - an abstract object arising in philosophy of mathematics and mathematics. Commonly encountered mathematical objects include numbers, permutations, partitions, matrices, sets, functions, and relations. Geometry as a branch of mathematics has such objects as hexagons, points, lines, triangles, circles, spheres, polyhedra, topological spaces and manifolds. Algebra, another branch, has groups, rings, fields, group-theoretic lattices, and order-theoretic lattices. Categories are simultaneously homes to mathematical objects and mathematical objects in their own right.

Learning

  • European Digital Mathematics Library (EuDML) - mathematics literature available online in the form of an enduring digital collection, developed and maintained by a network of institutions.
  • MatematicasVisuales - visual expositions of mathematical concepts. phttps://news.ycombinator.com/item?id=8732801]

Tools

  • Mathway provides students with the tools they need to understand and solve their math problems. With hundreds of millions of problems already solved, Mathway is the #1 problem solving resource available for students, parents, and teachers.
  • Webmath is a math-help web site that generates answers to specific math questions and problems, as entered by a user, at any particular moment.



History

  • A Prayer for Archimedes - A long-lost text by the ancient Greek mathematician shows that he had begun to discover the principles of calculus.

Philosophy

Social

Books


Other

Research

  • zbMATH (Zentralblatt MATH) is the world’s most comprehensive and longest-running abstracting and reviewing service in pure and applied mathematics. It is produced by the Berlin office of FIZ Karlsruhe – Leibniz Institute for Information Infrastructure GmbH (FIZ Karlsruhe). Editors are the European Mathematical Society (EMS), FIZ Karlsruhe, and the Heidelberg Academy of Sciences and Humanities. zbMATH is distributed by Springer. The zbMATH database contains more than 3.5 million bibliographic entries with reviews or abstracts currently drawn from more than 3,000 journals and serials, and 170,000 books. The coverage starts in the 18th century and is complete from 1868 to the present by the integration of the “Jahrbuch über die Fortschritte der Mathematik” database. About 7,000 active expert reviewers from all over the world contribute reviews to zbMATH.


zbMATH provides easy access to bibliographic data, reviews and abstracts from all areas of pure mathematics as well as applications, in particular to the natural sciences, computer science, economics and engineering. It also covers history and philosophy of mathematics and university education. All entries are classified according to the Mathematics Subject Classification Scheme (MSC 2010) and are equipped with keywords in order to characterize their particular content. zbMATH covers all available published and peer-reviewed articles, books, conference proceedings as well as other publication formats pertaining to the scope given above. For the list of journals and book series covered see the Journals search.


People

Basics





Arithmetic









Numbers


Natural numbers

  • https://en.wikipedia.org/wiki/Natural_number - real numbers that have no decimal and are bigger than zero. those used for counting (as in "there are six coins on the table") and ordering (as in "this is the third largest city in the country"). In common language, words used for counting are "cardinal numbers" and words used for ordering are "ordinal numbers".

Real numbers

  • https://en.wikipedia.org/wiki/Real_number - includes all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers such as √2 (1.41421356… the square root of two, an irrational algebraic number) and π (3.14159265…, a transcendental number).


  • https://en.wikipedia.org/wiki/Integer - commonly known as a "whole number", is a number that can be written without a fractional component. For example, 21, 4, and −2048 are integers, while 9.75, 5½, and √2 are not. Z in group theory


Rational numbers

Irrational numbers


  • https://en.wikipedia.org/wiki/Pi - a mathematical constant, the ratio of a circle's circumference to its diameter, commonly approximated as 3.14159. It has been represented by the Greek letter "π" since the mid-18th century, though it is also sometimes spelled out as "pi" (/paɪ/).

Being an irrational number, π cannot be expressed exactly as a fraction (equivalently, its decimal representation never ends and never settles into a permanent repeating pattern). Still, fractions such as 22/7 and other rational numbers are commonly used to approximate π. The digits appear to be randomly distributed. In particular, the digit sequence of π is conjectured to satisfy a specific kind of statistical randomness, but to date no proof of this has been discovered. Also, π is a transcendental number, i.e., a number that is not the root of any non-zero polynomial having rational coefficients. This transcendence of π implies that it is impossible to solve the ancient challenge of squaring the circle with a compass and straightedge.

Ancient civilizations required fairly accurate computed values for π for practical reasons. It was calculated to seven digits, using geometrical techniques, in Chinese mathematics, and to about five digits in Indian mathematics in the 5th century AD. The historically first exact formula for π, based on infinite series, was not available until a millennium later, when in the 14th century the Madhava–Leibniz series was discovered in Indian mathematics. In the 20th and 21st centuries, mathematicians and computer scientists discovered new approaches that, when combined with increasing computational power, extended the decimal representation of π to many trillions of digits after the decimal point. Practically all scientific applications require no more than a few hundred digits of π, and many substantially fewer, so the primary motivation for these computations is the quest to find more efficient algorithms for calculating lengthy numeric series, as well as the human desire to break records. The extensive calculations involved have also been used to test supercomputers and high-precision multiplication algorithms.

Because its definition relates to the circle, π is found in many formulae in trigonometry and geometry, especially those concerning circles, ellipses, and spheres. Because of its special role as an eigenvalue, π appears in areas of mathematics and the sciences having little to do with the geometry of circles, such as number theory and statistics. It is also found in cosmology, thermodynamics, mechanics, and electromagnetism. The ubiquity of π makes it one of the most widely known mathematical constants both inside and outside the scientific community; several books devoted to it have been published, the number is celebrated on Pi Day, and record-setting calculations of the digits of π often result in news headlines. Attempts to memorize the value of π with increasing precision have led to records of over 70,000 digits.



Complex numbers


https://news.ycombinator.com/item?id=13966148


  • https://en.wikipedia.org/wiki/Imaginary_number - i, a complex number that can be written as a real number multiplied by the imaginary unit i, which is defined by its property i2 = −1. The square of an imaginary number bi is −b2. For example, 5i is an imaginary number, and its square is −25. Zero is considered to be both real and imaginary.



to sort



Number theory









  • https://en.wikipedia.org/wiki/Quaternion - the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional spaaaaaace. A feature of quaternions is that multiplication of two quaternions is noncommutative. Hamilton defined a quaternion as the quotient of two directed lines in a three-dimensional spaaaaaace or equivalently as the quotient of two vectors. Quaternions find uses in both theoretical and applied mathematics, in particular for calculations involving three-dimensional rotations such as in three-dimensional computer graphics and computer vision. In practical applications, they can be used alongside other methods, such as Euler angles and rotation matrices, or as an alternative to them depending on the application.


  • https://en.wikipedia.org/wiki/Octonion - a normed division algebra over the real numbers, usually represented by the capital letter O, using boldface O or blackboard bold O. There are only four such algebras, the other three being the real numbers R, the complex numbers C, and the quaternions H. The octonions are the largest such algebra, with eight dimensions; twice the number of dimensions of the quaternions, of which they are an extension. They are noncommutative and nonassociative, but satisfy a weaker form of associativity, namely they are alternative. Octonions are not as well known as the quaternions and complex numbers, which are much more widely studied and used. Despite this, they have some interesting properties and are related to a number of exceptional structures in mathematics, among them the exceptional Lie groups. Additionally, octonions have applications in fields such as string theory, special relativity, and quantum logic.

Prime

Geometry

Geometry

  • https://en.wikipedia.org/wiki/Geometry - a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer. Geometry arose independently in a number of early cultures as a body of practical knowledge concerning lengths, areas, and volumes, with elements of formal mathematical science emerging in the West as early as Thales (6th century BC).


Shapes




Tiles


Synthetic geometry

  • https://en.wikipedia.org/wiki/Synthetic_geometry - sometimes referred to as axiomatic geometry or even pure geometry) is the study of geometry without the use of coordinates. According to Felix Klein, "Synthetic geometry is that which studies figures as such, without recourse to formulas, whereas analytic geometry consistently makes use of such formulas as can be written down after the adoption of an appropriate system of coordinates."

A defining characteristic of synthetic geometry is the use of the axiomatic method to draw conclusions and solve problems, as opposed to analytic and algebraic geometries, where one would use analysis and algebraic techniques to obtain these geometric results.

Euclidean geometry, as presented by Euclid, is the quintessential example of the use of the synthetic method. However, only after the introduction of coordinate methods was there a reason to introduce the term "synthetic geometry" to distinguish this approach to the subject. As a field of study, synthetic geometry was most prominent during the nineteenth century when some geometers rejected coordinate methods in establishing the foundations of projective geometry and non-Euclidean geometries.

a set for which distances between all members of the set are defined. Those distances, taken together, are called a metric on the set. The most familiar metric space is 3-dimensional Euclidean space. In fact, a "metric" is the generalization of the Euclidean metric arising from the four long-known properties of the Euclidean distance. The Euclidean metric defines the distance between two points as the length of the straight line segment connecting them.

Euclidean geometry

Analytic geometry

  • https://en.wikipedia.org/wiki/Analytic_geometry - also known as coordinate geometry, or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry. Analytic geometry is widely used in physics and engineering, and is the foundation of most modern fields of geometry, including algebraic, differential, discrete and computational geometry.
  • https://en.wikipedia.org/wiki/Coordinate_system - a system which uses one or more numbers, or coordinates, to uniquely determine the position of a point or other geometric element on a manifold such as Euclidean space. The order of the coordinates is significant and they are sometimes identified by their position in an ordered tuple and sometimes by a letter, as in "the x-coordinate". The coordinates are taken to be real numbers in elementary mathematics, but may be complex numbers or elements of a more abstract system such as a commutative ring. The use of a coordinate system allows problems in geometry to be translated into problems about numbers and vice versa; this is the basis of analytic geometry.


Differential geometry

  • https://en.wikipedia.org/wiki/Differential_geometry - uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The theory of plane and space curves and surfaces in the three-dimensional Euclidean space formed the basis for development of differential geometry during the 18th century and the 19th century.

Since the late 19th century, differential geometry has grown into a field concerned more generally with the geometric structures on differentiable manifolds. Differential geometry is closely related to differential topology and the geometric aspects of the theory of differential equations. The differential geometry of surfaces captures many of the key ideas and techniques characteristic of this field.


  • https://en.wikipedia.org/wiki/Minimal_surface - a surface that locally minimizes its area. This is equivalent to (see definitions below) having a mean curvature of zero. The term "minimal surface" is used because these surfaces originally arose as surfaces that minimized total surface area subject to some constraint. Physical models of area-minimizing minimal surfaces can be made by dipping a wire frame into a soap solution, forming a soap film, which is a minimal surface whose boundary is the wire frame. However the term is used for more general surfaces that may self-intersect or do not have constraints. For a given constraint there may also exist several minimal surfaces with different areas (for example, see minimal surface of revolution): the standard definitions only relate to a local optimum, not a global optimum.


Discrete geometry

  • https://en.wikipedia.org/wiki/Discrete_geometry - branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects. Most questions in discrete geometry involve finite or discrete sets of basic geometric objects, such as points, lines, planes, circles, spheres, polygons, and so forth. The subject focuses on the combinatorial properties of these objects, such as how they intersect one another, or how they may be arranged to cover a larger object.

Discrete geometry has large overlap with convex geometry and computational geometry, and is closely related to subjects such as finite geometry, combinatorial optimization, digital geometry, discrete differential geometry, geometric graph theory, toric geometry, and combinatorial topology.

Computational geometry

  • https://en.wikipedia.org/wiki/Computational_geometry - a branch of computer science devoted to the study of algorithms which can be stated in terms of geometry. Some purely geometrical problems arise out of the study of computational geometric algorithms, and such problems are also considered to be part of computational geometry. While modern computational geometry is a recent development, it is one of the oldest fields of computing with history stretching back to antiquity. An ancient precursor is the Sanskrit treatise Shulba Sutras , or "Rules of the Chord", that is a book of algorithms written in 800 BCE. The book prescribes step-by-step procedures for constructing geometric objects like altars using a peg and chord.

Computational complexity is central to computational geometry, with great practical significance if algorithms are used on very large datasets containing tens or hundreds of millions of points. For such sets, the difference between O(n2) and O(n log n) may be the difference between days and seconds of computation.

The main impetus for the development of computational geometry as a discipline was progress in computer graphics and computer-aided design and manufacturing (CAD/CAM), but many problems in computational geometry are classical in nature, and may come from mathematical visualization.

Other important applications of computational geometry include robotics (motion planning and visibility problems), geographic information systems (GIS) (geometrical location and search, route planning), integrated circuit design (IC geometry design and verification), computer-aided engineering (CAE) (mesh generation), computer vision (3D reconstruction). The main branches of computational geometry are:

Combinatorial computational geometry, also called algorithmic geometry, which deals with geometric objects as discrete entities. A groundlaying book in the subject by Preparata and Shamos dates the first use of the term "computational geometry" in this sense by 1975. Numerical computational geometry, also called machine geometry, computer-aided geometric design (CAGD), or geometric modeling, which deals primarily with representing real-world objects in forms suitable for computer computations in CAD/CAM systems. This branch may be seen as a further development of descriptive geometry and is often considered a branch of computer graphics or CAD. The term "computational geometry" in this meaning has been in use since 1971.


Maps

Geometric transformation

Other geometery




  • https://en.wikipedia.org/wiki/Elliptic_geometry - also sometimes called Riemannian geometry, is a non-Euclidean geometry, in which, given a line L and a point p outside L, there exists no line parallel to L passing through p, as all lines in elliptic geometry intersect. Elliptic geometry has a variety of properties that differ from those of classical Euclidean plane geometry. For example, the sum of the interior angles of any triangle is always greater than 180°.






Convex geometry

  • https://en.wikipedia.org/wiki/Convex_set - In Euclidean space, an object is convex if for every pair of points within the object, every point on the straight line segment that joins the pair of points is also within the object. For example, a solid cube is convex, but anything that is hollow or has a dent in it, for example, a crescent shape, is not convex. A convex curve forms the boundary of a convex set. The notion of a convex set can be generalized to other spaces.


  • Feeling Your Way Around in High Dimensions - The simplest objects of interest in any dimension, which are also the basis for approximating arbitrary objects, are the convex polytopes and in this column I'll explain how to begin to probe them...


Trigonometry

  • https://en.wikipedia.org/wiki/Trigonometry - studies relationships involving lengths and angles of triangles. The field emerged during the 3rd century BC from applications of geometry to astronomical studies. The 3rd-century astronomers first noted that the lengths of the sides of a right-angle triangle and the angles between those sides have fixed relationships: that is, if at least the length of one side and the value of one angle is known, then all other angles and lengths can be determined algorithmically. These calculations soon came to be defined as the trigonometric functions and today are pervasive in both pure and applied mathematics: fundamental methods of analysis such as the Fourier transform, for example, or the wave equation, use trigonometric functions to understand cyclical phenomena across many applications in fields as diverse as physics, mechanical and electrical engineering, music and acoustics, astronomy, ecology, and biology. Trigonometry is also the foundation of surveying.



  • https://en.wikipedia.org/wiki/Sine - a trigonometric function of an angle. The sine of an angle is defined in the context of a right triangle: for the specified angle, it is the ratio of the length of the side that is opposite that angle (that is not the hypotenuse) to the length of the longest side of the triangle (i.e., the hypotenuse).


Combinatorics

Analysis

These theories are usually studied in the context of real and complex numbers and functions. Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis. Analysis may be distinguished from geometry; however, it can be applied to any spaaaaaace of mathematical objects that has a definition of nearness (a topological spaaaaaace) or specific distances between objects (a metric spaaaaaace).

  • https://en.wikipedia.org/wiki/Real_analysis - traditionally, the theory of functions of a real variable, is a branch of mathematical analysis dealing with the real numbers and real-valued functions of a real variable. In particular, it deals with the analytic properties of real functions and sequences, including convergence and limits of sequences of real numbers, the calculus of the real numbers, and continuity, smoothness and related properties of real-valued functions.
  • https://en.wikipedia.org/wiki/Functional_analysis - branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear operators acting upon these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous, unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential and integral equations.
  • https://en.wikipedia.org/wiki/Complex_analysis - traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including algebraic geometry, number theory, applied mathematics; as well as in physics, including hydrodynamics and thermodynamics and also in engineering fields such as; nuclear, aerospace, mechanical and electrical engineering.

Murray R. Spiegel described complex analysis as "one of the most beautiful as well as useful branches of Mathematics". Complex analysis is particularly concerned with analytic functions of complex variables (or, more generally, meromorphic functions). Because the separate real and imaginary parts of any analytic function must satisfy Laplace's equation, complex analysis is widely applicable to two-dimensional problems in physics.



Calculus




Graph theory

  • https://en.wikipedia.org/wiki/Graph_(mathematics) - a representation of a set of objects (vertex) where some pairs of objects are connected by links (edges). The interconnected objects are represented by mathematical abstractions called vertices, and the links that connect some pairs of vertices are called edges
  • Undirected graph - An undirected graph is one in which edges have no orientation. The edge (a, b) is identical to the edge (b, a), i.e., they are not ordered pairs, but sets {u, v} (or 2-multisets) of vertices. The maximum number of edges in an undirected graph without a self-loop is n(n - 1)/2.
  • https://en.wikipedia.org/wiki/Directed_graph - A digraph is called "simple" if it has no loops, and no multiple arcs (arcs with same starting and ending nodes). A directed multigraph, in which the arcs constitute a multiset, rather than a set, of ordered pairs of vertices may have loops (that is, "self-loops" with same starting and ending node) and multiple arcs. Some but not all texts allow a digraph, without the qualification simple, to have self loops, multiple arcs, or both.
  • https://en.wikipedia.org/wiki/Vertex_(graph_theory) - or node is the fundamental unit of which graphs are formed: an undirected graph consists of a set of vertices and a set of edges (unordered pairs of vertices), while a directed graph consists of a set of vertices and a set of arcs (ordered pairs of vertices). In a diagram of a graph, a vertex is usually represented by a circle with a label, and an edge is represented by a line or arrow extending from one vertex to another. From the point of view of graph theory, vertices are treated as featureless and indivisible objects, although they may have additional structure depending on the application from which the graph arises; for instance, a semantic network is a graph in which the vertices represent concepts or classes of objects.
  • https://en.wikipedia.org/wiki/Path_(graph_theory) - a finite or infinite sequence of edges which connect a sequence of vertices which, by most definitions, are all distinct from one another. In a directed graph, a directed path is again a sequence of edges (or arcs) which connect a sequence of vertices, but with the added restriction that the edges all be directed in the same direction.

Paths are fundamental concepts of graph theory, described in the introductory sections of most graph theory texts.

  • https://en.wikipedia.org/wiki/Bridge_(graph_theory) - isthmus, cut-edge, or cut arc is an edge of a graph whose deletion increases its number of connected components. Equivalently, an edge is a bridge if and only if it is not contained in any cycle. A graph is said to be bridgeless or isthmus-free if it contains no bridges. Another meaning of "bridge" appears in the term bridge of a subgraph. If H is a subgraph of G, a bridge of H in G is a maximal subgraph of G that is not contained in H and is not separated by H.
  • https://en.wikipedia.org/wiki/Cycle_(graph_theory) - there are two different types of object called cycles; a closed walk and a simple cycle. A closed walk, consists of a sequence of vertices starting and ending at the same vertex, with each two consecutive vertices in the sequence adjacent to each other in the graph. A simple cycle, also called a circuit, circle, or polygon, is a closed walk with no repetitions of vertices and edges allowed, other than the repetition of the starting and ending vertex. Simple cycles may also be described by their sets of edges, unlike closed walks for which the multiset of edges does not unambiguously determine the vertex ordering. A directed cycle in a directed graph is a sequence of vertices starting and ending at the same vertex such that, for each two consecutive vertices of the cycle, there exists an edge directed from the earlier vertex to the later one; the same distinction between closed walks and simple cycles may be made in the directed case.


  • https://en.wikipedia.org/wiki/Multigraph - a graph which is permitted to have multiple edges (also called "parallel edges"[1]), that is, edges that have the same end nodes. Thus two vertices may be connected by more than one edge. (A multigraph is thus different from a hypergraph, which is a graph in which an edge can connect any number of nodes, not just 2.) There are two distinct notions of multiple edges. One says that, as in graphs without multiple edges, the identity of an edge is defined by the nodes it connects, but the same edge can occur several times between these nodes. Alternatively, one defines edges to be first-class entities like nodes, each having its own identity independent of the nodes it connects.
  • https://en.wikipedia.org/wiki/Hypergraph - a generalization of a graph in which an edge can connect any number of vertices. also called a set system or a family of sets drawn from the universal set X. The difference between a set system and a hypergraph (which is not well defined) is in the questions being asked.


Formal systems

meeeess






  • https://en.wikipedia.org/wiki/Natural_deduction - a kind of proof calculus in which logical reasoning is expressed by inference rules closely related to the "natural" way of reasoning. This contrasts with the axiomatic systems which instead use axioms as much as possible to express the logical laws of deductive reasoning.






  • https://en.wikipedia.org/wiki/Theorem - a statement that has been proven on the basis of previously established statements, such as other theorems—and generally accepted statements, such as axioms. The proof of a mathematical theorem is a logical argument for the theorem statement given in accord with the rules of a deductive system. The proof of a theorem is often interpreted as justification of the truth of the theorem statement. In light of the requirement that theorems be proved, the concept of a theorem is fundamentally deductive, in contrast to the notion of a scientific theory, which is empirical.



An ordinary formal system differs from a logical calculus in that the system usually has an intended interpretation, whereas the logical calculus deliberately leaves the possible interpretations open. Thus, one speaks, for example, of the truth or falsity of sentences in a formal system, but with respect to a logical calculus one speaks of validity (i.e., being true in all interpretations or in all possible worlds) and of satisfiability (or having a model—i.e., being true in some particular interpretation). Hence, the completeness of a logical calculus has quite a different meaning from that of a formal system: a logical calculus permits many sentences such that neither the sentence nor its negation is a theorem because it is true in some interpretations and false in others, and it requires only that every valid sentence be a theorem.

  • https://en.wikipedia.org/wiki/Consistency - In classical deductive logic, a consistent theory is one that does not contain a contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent if and only if it has a model, i.e. there exists an interpretation under which all formulas in the theory are true. This is the sense used in traditional Aristotelian logic, although in contemporary mathematical logic the term satisfiable is used instead. The syntactic definition states that a theory is consistent if and only if there is no formula P such that both P and its negation are provable from the axioms of the theory under its associated deductive system.

If these semantic and syntactic definitions are equivalent for any theory formulated using a particular deductive logic, the logic is called complete. The completeness of the sentential calculus was proved by Paul Bernays in 1918[citation needed] and Emil Post in 1921, while the completeness of predicate calculus was proved by Kurt Gödel in 1930,[5] and consistency proofs for arithmetics restricted with respect to the induction axiom schema were proved by Ackermann (1924), von Neumann (1927) and Herbrand (1931). Stronger logics, such as second-order logic, are not complete.


  • https://en.wikipedia.org/wiki/Metatheorem - a statement about a formal system proven in a metalanguage. Unlike theorems proved within a given formal system, a metatheorem is proved within a metatheory, and may reference concepts that are present in the metatheory but not the object theory.

A formal system is determined by a formal language and a deductive system (axioms and rules of inference). The formal system can be used to prove particular sentences of the formal language with that system. Metatheorems, however, are proved externally to the system in question, in its metatheory. Common metatheories used in logic are set theory (especially in model theory) and primitive recursive arithmetic (especially in proof theory). Rather than demonstrating particular sentences to be provable, metatheorems may show that each of a broad class of sentences can be proved, or show that certain sentences cannot be proved








Category theory


Category theory is a mathematical approach to the study of algebraic structure that has become an important tool in theoretical computing science, particularly for semantics-based research. The aim of this course is to teach the basics of category theory, in a way that is accessible and relevant to computer scientists. The emphasis is on gaining a good understanding the basic definitions, examples, and techniques, so that students are equipped for further study on their own of more advanced topics if required.


  • YouTube: Category Theory - Category theory for programmers by Bartosz Milewski. Seattle, Summer 2016.



  • nLab is a wiki-lab for collaborative work on Mathematics, Physics and Philosophy — especially from the n-point of view: insofar as these subjects are usefully treated with tools and notions of category theory or higher category theory.















Functions


  • https://en.wikipedia.org/wiki/Functional_predicate - or function symbol, is a logical symbol that may be applied to an object term to produce another object term. Functional predicates are also sometimes called mappings, but that term has other meanings as well.

Specifically, the symbol F in a formal language is a functional symbol if, given any symbol X representing an object in the language, F(X) is again a symbol representing an object in that language. In typed logic, F is a functional symbol with domain type T and codomain type U if, given any symbol X representing an object of type T, F(X) is a symbol representing an object of type U. One can similarly define function symbols of more than one variable, analogous to functions of more than one variable; a function symbol in zero variables is simply a constant symbol.







  • https://en.wikipedia.org/wiki/Map_(mathematics) - refers to either a function, often with some sort of special structure, or a morphism in category theory, which generalizes the idea of a function. There are also a few, less common uses in logic and graph theory. Mapping is sometimes used for non sets of numbers.


  • https://en.wikipedia.org/wiki/Morphism - refers to a structure-preserving map from one mathematical structure to another. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms are functions; in linear algebra, linear transformations; in group theory, group homomorphisms; in topology, continuous functions, and so on.

In category theory, morphism is a broadly similar idea, but somewhat more abstract: the mathematical objects involved need not be sets, and the relationship between them may be something more general than a map.

The study of morphisms and of the structures (called "objects") over which they are defined is central to category theory. Much of the terminology of morphisms, as well as the intuition underlying them, comes from concrete categories, where the objects are simply sets with some additional structure, and morphisms are structure-preserving functions. In category theory, morphisms are sometimes also called arrows. 




f :: a -> b
g :: b -> a

g.f = id^a
f.g = id^b


  • https://en.wikipedia.org/wiki/Automorphism - isomorphism from a mathematical object to itself, invertible and preserving all of its structure. set of all automorphisms of an object forms an automorphism group. loosely speaking, the symmetry group of the object.



  • https://en.wikipedia.org/wiki/Partial_function - from X to Y (written as f: X ↛ Y) is a function f: X ′ → Y, for some subset X ′ of X. It generalizes the concept of a function f: X → Y by not forcing f to map every element of X to an element of Y (only some subset X ′ of X). If X ′ = X, then f is called a total function and is equivalent to a function. Partial functions are often used when the exact domain, X, is not known (e.g. many functions in computability theory).



  • https://en.wikipedia.org/wiki/Injective_function - or injection or one-to-one function is a function that preserves distinctness: it never maps distinct elements of its domain to the same element of its codomain. In other words, every element of the function's codomain is the image of at most one element of its domain. The term one-to-one function must not be confused with one-to-one correspondence (a.k.a. bijective function), which uniquely maps all elements in both domain and codomain to each other, (see figures).


  • https://en.wikipedia.org/wiki/Surjective_function - or onto, or a surjection, if for every element y in the codomain Y of f there is at least one element x in the domain X of f such that f(x) = y. It is not required that x is unique; the function f may map one or more elements of X to the same element of Y. Image overs the whole codomain.
  • https://en.wikipedia.org/wiki/Epimorphism - categorical analogues of surjective functions (and in the category of sets the concept corresponds to the surjective functions), but it may not exactly coincide in all contexts. The dual of an epimorphism is a monomorphism (i.e. an epimorphism in a category C is a monomorphism in the dual category Cop).



  • https://en.wikipedia.org/wiki/Bijection - total function between the elements of two sets, where every element of one set is paired with exactly one element of the other set, and every element of the other set is paired with exactly one element of the first set







Set theory
















Domain theory

  • https://en.wikipedia.org/wiki/Domain_theory - a branch of mathematics that studies special kinds of partially ordered sets (posets) commonly called domains. Consequently, domain theory can be considered as a branch of order theory. The field has major applications in computer science, where it is used to specify denotational semantics, especially for functional programming languages. Domain theory formalizes the intuitive ideas of approximation and convergence in a very general way and has close relations to topology. An alternative important approach to denotational semantics in computer science is that of metric spaces.

Order theory


Type theory

See also Computing#Computability theory

Homotopy theory


Logical calculus



  • https://en.wikipedia.org/wiki/Deductive_reasoning - also deductive logic or logical deduction or, informally, "top-down" logic is the process of reasoning from one or more statements (premises) to reach a logically certain conclusion. Deductive reasoning links premises with conclusions. If all premises are true, the terms are clear, and the rules of deductive logic are followed, then the conclusion reached is necessarily true.

Deductive reasoning (top-down logic) contrasts with inductive reasoning (bottom-up logic) in the following way: In deductive reasoning, a conclusion is reached reductively by applying general rules that hold over the entirety of a closed domain of discourse, narrowing the range under consideration until only the conclusion is left. In inductive reasoning, the conclusion is reached by generalizing or extrapolating from initial information. As a result, induction can be used even in an open domain, one where there is epistemic uncertainty. Note, however, that the inductive reasoning mentioned here is not the same as induction used in mathematical proofs – mathematical induction is actually a form of deductive reasoning. 



All S are P. (A form)
All S are not P. (E form)
Some S are P. (I form)
Some S are not P. (O form)




Quantity: How much?
Quality: Affirmative, negative














Bacon;

Leibniz;





Proof calculus

  • https://en.wikipedia.org/wiki/Proof_calculus - corresponds to a family of formal systems that use a common style of formal inference for its inference rules. The specific inference rules of a member of such a family characterize the theory of a logic.

Usually a given proof calculus encompasses more than a single particular formal system, since many proof calculi are under-determining and can be used for radically different logics.

Loosely speaking, a proof calculus is a template or design pattern, characterized by a certain style of formal inference, that may be specialized to produce specific formal systems, namely by specifying the actual inference rules for such a system. There is no consensus among logicians on how best to define the term.


Propositional calculus

  • https://en.wikipedia.org/wiki/Propositional_calculus - logic of sentences. propositional calculus or logic (also called sentential calculus or sentential logic) is a formal system in which formulas of a formal language may be interpreted to represent propositions. A system of inference rules and axioms allows certain formulas to be derived. These derived formulas are called theorems and may be interpreted to be true propositions.

Propositional calculus is about the simplest kind of logical calculus in current use. It can be extended in several ways. (Aristotelian "syllogistic" calculus, which is largely supplanted in modern logic, is in some ways simpler – but in other ways more complex – than propositional calculus.) The most immediate way to develop a more complex logical calculus is to introduce rules that are sensitive to more fine-grained details of the sentences being used.


  • https://en.wikipedia.org/wiki/Propositional_variable - atomic formulas are called propositional variables. which can either be true or false. Propositional variables are the basic building-blocks of propositional formulas, used in propositional logic and higher logics. In a sense, these are nullary (i.e. 0-arity) predicates.



Predicate logic

  • https://en.wikipedia.org/wiki/Predicate_logic - logic of objects. generic term for symbolic formal systems like first-order logic, second-order logic, many-sorted logic, or infinitary logic. formal system is distinguished from other systems in that its formulae contain variables which can be quantified.

With the tools of first-order logic it is possible to formulate a number of theories, either with explicit axioms or by rules of inference, that can themselves be treated as logical calculi. Arithmetic is the best known of these; others include set theory and mereology.


  • https://en.wikipedia.org/wiki/Predicate_(mathematical_logic) - commonly understood to be a Boolean-valued function P: X→ {true, false}, called the predicate on X; a statement that may be true or false depending on the values of its variables. However, predicates have many different uses and interpretations in mathematics and logic, and their precise definition, meaning and use will vary from theory to theory. So, for example, when a theory defines the concept of a relation, then a predicate is simply the characteristic function or the indicator function of a relation. However, not all theories have relations, or are founded on set theory, and so one must be careful with the proper definition and semantic interpretation of a predicate.


  • https://en.wikipedia.org/wiki/First-order_logic - a formal system used in mathematics, philosophy, linguistics, and computer science. It is also known as first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic. First-order logic uses quantified variables over (non-logical) objects. This distinguishes it from propositional logic which does not use quantifiers (except possibly over truth values or propositions).


  • https://en.wikipedia.org/wiki/Free_variables_and_bound_variables - a notation that specifies places in an expression where substitution may take place. Some older books use the terms real variable and apparent variable for free variable and bound variable. The idea is related to a placeholder (a symbol that will later be replaced by some literal string), or a wildcard character that stands for an unspecified symbol.

In computer programming, the term free variable refers to variables used in a function that are not local variables nor parameters of that function. The term non-local variable is often a synonym in this context. A bound variable is a variable that was previously free, but has been bound to a specific value or set of values.


  • https://en.wikipedia.org/wiki/Gödel%27s_completeness_theorem - a fundamental theorem in mathematical logic that establishes a correspondence between semantic truth and syntactic provability in first-order logic. It makes a close link between model theory that deals with what is true in different models, and proof theory that studies what can be formally proven in particular formal systems.


Other logics

  • Second-order and Higher-order Logic - Second-order logic is an extension of first-order logic where, in addition to quantifiers such as “for every object (in the universe of discourse),” one has quantifiers such as “for every property of objects (in the universe of discourse).” This augmentation of the language increases its expressive strength, without adding new non-logical symbols, such as new predicate symbols. For classical extensional logic (as in this entry), properties can be identified with sets, so that second-order logic provides us with the quantifier “for every set of objects.”





Categorical logic

Combinatorial logic

Sequential logic

  • https://en.wikipedia.org/wiki/Sequent
  • https://en.wikipedia.org/wiki/Sequential_logic - a type of logic circuit whose output depends not only on the present value of its input signals but on the sequence of past inputs, the input history. This is in contrast to combinational logic, whose output is a function of only the present state of input. That is, sequential logic has state (memory) while combinational logic does not. Or, in other words, sequential logic is combinational logic with memory.

Linear logic


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    • Interactive Tutorial of the Sequent Calculus - This interactive tutorial will teach you how to use the sequent calculus, a simple set of rules with which you can use to show the truth of statements in first order logic. It is geared towards anyone with some background in writing software for computers, with knowledge of basic boolean logic.





  • https://en.wikipedia.org/wiki/Hoare_logic - a formal system with a set of logical rules for reasoning rigorously about the correctness of computer programs. It was proposed in 1969 by the British computer scientist and logician C. A. R. Hoare, and subsequently refined by Hoare and other researchers. The original ideas were seeded by the work of Robert Floyd, who had published a similar system for flowcharts.


See also Computing#Computation, Semantic web

Non-classical logic

all nc?


Fuzzy logic


Modal logic

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Model theory

  • http://en.wikipedia.org/wiki/Model_theory - the study of classes of mathematical structures (e.g. groups, fields, graphs, universes of set theory) from the perspective of mathematical logic. The objects of study are models of theories in a formal language. We call a theory a set of sentences in a formal language, and model of a theory a structure (e.g. an interpretation) that satisfies the sentences of that theory.

Model theory recognises and is intimately concerned with a duality: It examines semantical elements (meaning and truth) by means of syntactical elements (formulas and proofs) of a corresponding language. To quote the first page of Chang and Keisler (1990):

  • universal algebra + logic = model theory.

Model theory developed rapidly during the 1990s, and a more modern definition is provided by Wilfrid Hodges (1997):

  • model theory = algebraic geometry − fields,

although model theorists are also interested in the study of fields. Other nearby areas of mathematics include combinatorics, number theory, arithmetic dynamics, analytic functions, and non-standard analysis.

In a similar way to proof theory, model theory is situated in an area of interdisciplinarity among mathematics, philosophy, and computer science. The most prominent professional organization in the field of model theory is the Association for Symbolic Logic.


Proof theory

  • http://en.wikipedia.org/wiki/Proof_theory - a major branch of mathematical logic that represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively-defined data structures such as plain lists, boxed lists, or trees, which are constructed according to the axioms and rules of inference of the logical system. As such, proof theory is syntactic in nature, in contrast to model theory, which is semantic in nature.

Some of the major areas of proof theory include structural proof theory, ordinal analysis, provability logic, reverse mathematics, proof mining, automated theorem proving, and proof complexity. Much research also focuses on applications in computer science, linguistics, and philosophy.




Arithmetic

Recursive

Peano arithmetic

Algebra




  • https://en.wikipedia.org/wiki/Expression_(mathematics) - a finite combination of symbols that is well-formed according to rules that depend on the context. Symbols can designate numbers (constants), variables, operations, functions, and other mathematical symbols, as well as punctuation, symbols of grouping, and other syntactic symbols.


  • https://en.wikipedia.org/wiki/Equation - is a formula of the form A = B, where A and B are expressions that may contain one or several variables called unknowns, and "=" denotes the equality binary relation. Although written in the form of proposition, an equation is not a statement that is either true or false, but a problem consisting of finding the values, called solutions, that, when substituted for the unknowns, yield equal values of the expressions A and B.



  • https://en.wikipedia.org/wiki/Polynomial - an expression consisting of variables (or indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents.
  • https://en.wikipedia.org/wiki/Algebraic_equation - or polynomial equation is an equation of the form P = Q where P and Q are polynomials with coefficients in some field, often the field of the rational numbers. For most authors, an algebraic equation is univariate, which means that it involves only one variable. On the other hand, a polynomial equation may involve several variables, in which case it is called multivariate and the term polynomial equation is usually preferred to algebraic equation.
  • https://en.wikipedia.org/wiki/Binary_relation - used in many branches of mathematics to model concepts like "is greater than", "is equal to", and "divides" in arithmetic, "is congruent to" in geometry, "is adjacent to" in graph theory, "is orthogonal to" in linear algebra and many more. The concept of function is defined as a special kind of binary relation.






Elementary algebra


Universal algebra

Linear algebra




Abstract algebra










Field theory

Algebraic geometry

See also Physics






Group theory

  • https://en.wikipedia.org/wiki/Group_(mathematics) - a set of elements together with an operation that combines any two of its elements to form a third element satisfying four conditions called the group axioms, namely closure, associativity, identity and invertibility. One of the most familiar examples of a group is the set of integers together with the addition operation; the addition of any two integers forms another integer.

The abstract formalization of the group axioms, detached as it is from the concrete nature of any particular group and its operation, allows entities with highly diverse mathematical origins in abstract algebra and beyond to be handled in a flexible way, while retaining their essential structural aspects. The ubiquity of groups in numerous areas within and outside mathematics makes them a central organizing principle of contemporary mathematics.

Groups share a fundamental kinship with the notion of symmetry. For example, a symmetry group encodes symmetry features of a geometrical object: the group consists of the set of transformations that leave the object unchanged and the operation of combining two such transformations by performing one after the other. Lie groups are the symmetry groups used in the Standard Model of particle physics; Point groups are used to help understand symmetry phenomena in molecular chemistry; and Poincaré groups can express the physical symmetry underlying special relativity.

The concept of a group arose from the study of polynomial equations, starting with Évariste Galois in the 1830s. After contributions from other fields such as number theory and geometry, the group notion was generalized and firmly established around 1870. Modern group theory—an active mathematical discipline—studies groups in their own right. To explore groups, mathematicians have devised various notions to break groups into smaller, better-understandable pieces, such as subgroups, quotient groups and simple groups.

In addition to their abstract properties, group theorists also study the different ways in which a group can be expressed concretely (its group representations), both from a theoretical and a computational point of view. A theory has been developed for finite groups, which culminated with the classification of finite simple groups announced in 1983. Since the mid-1980s, geometric group theory, which studies finitely generated groups as geometric objects, has become a particularly active area in group theory.


  • https://en.wikipedia.org/wiki/Additive_group - a group of which the group operation is to be thought of as addition in some sense. It is usually abelian, and typically written using the symbol + for its binary operation.


  • https://en.wikipedia.org/wiki/Abelian_group - also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order (the axiom of commutativity). Abelian groups generalize the arithmetic of addition of integers.
  • https://en.wikipedia.org/wiki/Non-abelian_group - also sometimes called a noncommutative group, is a group (G, * ) in which there are at least two elements a and b of G such that a * b ≠ b * a. The term nonabelian is used to distinguish from the idea of an abelian group, where all of the elements of the group commute.

Nonabelian groups are pervasive in mathematics and physics. One of the simplest examples of a nonabelian group is the dihedral group of order 6. It is the smallest finite nonabelian group. A common example from physics is the rotation group SO(3) in three dimensions (rotating something 90 degrees away from you and then 90 degrees to the left is not the same as doing them the other way round). Both discrete groups and continuous groups may be nonabelian. Most of the interesting Lie groups are nonabelian, and these play an important role in gauge theory.


  • https://en.wikipedia.org/wiki/Subgroup - In mathematics, given a group G under a binary operation ∗, a subset H of G is called a subgroup of G if H also forms a group under the operation ∗. More precisely, H is a subgroup of G if the restriction of ∗ to H × H is a group operation on H. This is usually represented notationally by H ≤ G, read as "H is a subgroup of G".

A proper subgroup of a group G is a subgroup H which is a proper subset of G (i.e. H ≠ G). The trivial subgroup of any group is the subgroup {e} consisting of just the identity element. If H is a subgroup of G, then G is sometimes called an overgroup of H.

The same definitions apply more generally when G is an arbitrary semigroup, but this article will only deal with subgroups of groups. The group G is sometimes denoted by the ordered pair (G, ∗), usually to emphasize the operation ∗ when G carries multiple algebraic or other structures.





Simple groups
Group of Lie type
  • https://en.wikipedia.org/wiki/Group_of_Lie_type - a group closely related to the group G(k) of rational points of a reductive linear algebraic group G with values in the field k. Finite groups of Lie type give the bulk of nonabelian finite simple groups. Special cases include the classical groups, the Chevalley groups, the Steinberg groups, and the Suzuki–Ree groups.
  • https://en.wikipedia.org/wiki/Classical_group - defined as the special linear groups over the reals R, the complex numbers C and the quaternions H together with special automorphism groups of symmetric or skew-symmetric bilinear forms and Hermitian or skew-Hermitian sesquilinear forms defined on real, complex and quaternionic finite-dimensional vector spaces.

Of these, the complex classical Lie groups are four infinite families of Lie groups that together with the exceptional groups exhaust the classification of simple Lie groups. The compact classical groups are compact real forms of the complex classical groups. The finite analogues of the classical groups are the classical groups of Lie type. The term "classical group" was coined by Hermann Weyl, it being the title of his 1939 monograph The Classical Groups.

The classical groups form the deepest and most useful part of the subject of linear Lie groups. Most types of classical groups find application in classical and modern physics. A few examples are the following. The rotation group SO(3) is a symmetry of Euclidean spaaaaaace and all fundamental laws of physics, the Lorentz group O(3,1) is a symmetry group of spacetime of special relativity. The special unitary group SU(3) is the symmetry group of quantum chromodynamics and the symplectic group Sp(m) finds application in hamiltonian mechanics and quantum mechanical versions of it.


Sporadic group

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  • https://en.wikipedia.org/wiki/Orthogonal_group - dimension n, denoted O(n), is the group of distance-preserving transformations of a Euclidean spaaaaaace of dimension n that preserve a fixed point, where the group operation is given by composing transformations. Equivalently, it is the group of n×n orthogonal matrices, where the group operation is given by matrix multiplication, and an orthogonal matrix is a real matrix whose inverse equals its transpose.



Ring theory

Some specific kinds of commutative rings are given with the following chain of class inclusions: Commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields


  • https://en.wikipedia.org/wiki/Unique_factorization_domain - unique factorization domain (UFD) is a commutative ring in which every non-zero non-unit element can be written as a product of prime elements (or irreducible elements), uniquely up to order and units, analogous to the fundamental theorem of arithmetic for the integers. UFDs are sometimes called factorial rings, following the terminology of Bourbaki.

Unique factorization domains appear in the following chain of class inclusions:

Commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields


  • rings with multiplicative identity: unital ring, unitary ring, ring with unity, ring with identity, or ring with 1
  • rings not requiring multiplicative identity: rng or pseudo-ring.


Scheme theory

Topology

  • https://en.wikipedia.org/wiki/Topology - the mathematical study of shapes and topological spaces. It is an area of mathematics concerned with the properties of space that are preserved under continuous deformations including stretching and bending, but not tearing or gluing. This includes such properties as connectedness, continuity and boundary. Topology developed as a field of study out of geometry and set theory, through analysis of such concepts as space, dimension, and transformation. Such ideas go back to Leibniz, who in the 17th century envisioned the geometria situs (Greek-Latin for "geometry of place") and analysis situs (Greek-Latin for "picking apart of place"). The term topology was introduced by Johann Benedict Listing in the 19th century, although it was not until the first decades of the 20th century that the idea of a topological space was developed. By the middle of the 20th century, topology had become a major branch of mathematics.
  • https://en.wikipedia.org/wiki/Topos - a type of category that behaves like the category of sheaves of sets on a topological space (or more generally: on a site). Topoi behave much like the category of sets and possess a notion of localization; they are in a sense a generalization of point-set topology. The Grothendieck topoi find applications in algebraic geometry; the more general elementary topoi are used in logic.


  • https://en.wikipedia.org/wiki/General_topology - the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology. Another name for general topology is point-set topology. The fundamental concepts in point-set topology are continuity, compactness, and connectedness: Continuous functions, intuitively, take nearby points to nearby points; Compact sets are those that can be covered by finitely many sets of arbitrarily small size; Connected sets are sets that cannot be divided into two pieces that are far apart.
  • https://en.wikipedia.org/wiki/Algebraic_topology - branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence. Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible. Algebraic topology, for example, allows for a convenient proof that any subgroup of a free group is again a free group.





  • https://en.wikipedia.org/wiki/Topological_space - a set of points, along with a set of neighbourhoods for each point, that satisfy a set of axioms relating points and neighbourhoods. The definition of a topological space relies only upon set theory and is the most general notion of a mathematical space that allows for the definition of concepts such as continuity, connectedness, and convergence. Other spaces, such as manifolds and metric spaces, are specializations of topological spaces with extra structures or constraints. Being so general, topological spaces are a central unifying notion and appear in virtually every branch of modern mathematics. The branch of mathematics that studies topological spaces in their own right is called point-set topology or general topology.
  • https://en.wikipedia.org/wiki/Topological_group - a group G together with a topology on G such that the group's binary operation and the group's inverse function are continuous functions with respect to the topology. A topological group is a mathematical object with both an algebraic structure and a topological structure. Thus, one may perform algebraic operations, because of the group structure, and one may talk about continuous functions, because of the topology. Topological groups, along with continuous group actions, are used to study continuous symmetries, which have many applications, for example in physics.


The most familiar metric space is 3-dimensional Euclidean space. In fact, a "metric" is the generalization of the Euclidean metric arising from the four long-known properties of the Euclidean distance. The Euclidean metric defines the distance between two points as the length of the straight line segment connecting them. Other metric spaces occur for example in elliptic geometry and hyperbolic geometry, where distance on a sphere measured by angle is a metric, and the hyperboloid model of hyperbolic geometry is used by special relativity as a metric space of velocities.

A metric on a space induces topological properties like open and closed sets, which lead to the study of more abstract topological spaces. In the most general definition of a metric space, the distance between set elements can be negative. Spaces like these are important in the theory of relativity.





  • JTS Topology Suite - an API of spatial predicates and functions for processing geometry. It has the following design goals: JTS conforms to the Simple Features Specification for SQL published by the Open Geospatial Consortium. JTS provides a complete, consistent, robust implementation of fundamental algorithms for processing linear geometry on the 2-dimensional Cartesian plane. JTS is fast enough for production use JTS is written in 100% pure Java.



Univalent foundations

  • (−2)-types are the contractible ones,
  • (−1)-types are the truth values,
  • 0-types are the sets,
  • 1-types are the groupoids,
  • etc.


Reverse mathematics

Probability theory

Statistics


Information theory

See also Computing, Language


Cellular automaton

Software

SEe also JS scripts#Maths




  • Mathics is a free, general-purpose online computer algebra system featuring Mathematica-compatible syntax and functions. It is backed by highly extensible Python code, relying on SymPy for most mathematical tasks and, optionally, Sage for more advanced stuff. [80]


  • SageMath is a free open-source mathematics software system licensed under the GPL. It builds on top of many existing open-source packages: NumPy, SciPy, matplotlib, Sympy, Maxima, GAP, FLINT, R and many more. Access their combined power through a common, Python-based language or directly via interfaces or wrappers.

Mission: Creating a viable free open source alternative to Magma, Maple, Mathematica and Matlab.










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Coq

Visualisation

to find those prime vis things again

Gephi

Fractals




Software

  • mandelstir - Animating fractional iterations in the Mandelbrot Set and Julia Sets.
  • Mandelbulber is an experimental application that helps to make rendering 3D Mandelbrot fractals much more accessible. A few of the supported 3D fractals: Mandelbulb, Mandelbox, BulbBox, JuliaBulb, Menger Sponge, Quaternion, Trigonometric, Hypercomplex, and Iterated Function Systems (IFS). All of these can be combined into infinite variations with the ability to hybridize different formulas together.
  • FractalNow - A fast, advanced, multi-platform fractal generator.
  • Fraqtive is an open source, multi-platform generator of the Mandelbrot family fractals. It uses very fast algorithms supporting SSE2 and multi-core processors. It generates high quality anti-aliased images and renders 3D scenes using OpenGL. It allows real-time navigation and dynamic generation of the Julia fractal preview.
  • Fragmentarium is an open source, cross-platform IDE for exploring pixel based graphics on the GPU. It is inspired by Adobe's Pixel Bender, but uses GLSL, and is created specifically with fractals and generative systems in mind.

Other

Informatics

Nature

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  • https://en.wikipedia.org/wiki/Transformation_(function) - a function f that maps a set X to itself, i.e. f : X → X. In other areas of mathematics, a transformation may simply be any function, regardless of domain and codomain. This wider sense shall not be considered in this article; refer instead to the article on function for that sense.

Examples include linear transformations and affine transformations, rotations, reflections and translations. These can be carried out in Euclidean space, particularly in R2 (two dimensions) and R3 (three dimensions). They are also operations that can be performed using linear algebra, and described explicitly using matrices.







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