Maths

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huge mess. also working back from Computing#Computation

General

to sort a



  • 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/List_of_theorems
  • https://en.wikipedia.org/wiki/Lemma_(mathematics) - (plural lemmata or lemmas[1]) 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

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

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

People

Basics

Arithmetic



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

Complex numbers


to sort



  • 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 space. 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 space 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/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.


Shapes

Tiles

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.

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.












Universal algebra

Linear algebra

Abstract algebra

Algebraic geometry

Group theory

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

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.[1] 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.

Combinatorics

Analysis

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.


Calculus


Logic

meeeess 

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)


  • 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.




Bacon;

Leibniz; 





Quantity: How much?
Quality: Affirmative, negative





  • 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.





  • 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.

See also Computing#Computation, Semantic web

Non-classical

all nc?





Software



Set theory




Type theory

Homotopy theory

Field theory

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.


  • 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.



Model theory

The study of interpretation of formal systems is the branch of mathematical logic that is known as model theory.

  • universal algebra + logic = model theory
  • model theory = algebraic geometry − fields


Order theory

Domain theory

Proof theory



Statistics

Information theory

See also Computing

Cellular automaton

Software

  • 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.



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

to sort

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