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12,700 | In statistics, randomness is commonly used to create simple random samples. This allows surveys of completely random groups of people to provide realistic data that is reflective of the population. Common methods of doing this include drawing names out of a hat or using a random digit chart (a large table of random dig... | Randomness | 0.816745 |
12,701 | For example, if a single unstable atom is placed in a controlled environment, it cannot be predicted how long it will take for the atom to decay—only the probability of decay in a given time. Thus, quantum mechanics does not specify the outcome of individual experiments, but only the probabilities. Hidden variable theo... | Randomness | 0.816745 |
12,702 | The distinction has no serious significance, however, as microscopic hotspots such as those proposed to explain catalyst behaviour in several gas-phase catalytic reactions have been demonstrated by post-mortem methods and in-situ methods. Some theoretical and experimental approaches have been published towards the clar... | MORE synthesis | 0.816744 |
12,703 | Where the rate of heat conduction is high between system domains, hot spots would have no long-term existence as the components rapidly reach thermal equilibrium. In a system where the heat transfer is slow, it would be possible to have the presence of a steady state hot spot that may enhance the rate of the chemical r... | MORE synthesis | 0.816744 |
12,704 | However, due to the design of most microwave ovens and to uneven absorption by the object being heated, the microwave field is usually non-uniform and localized superheating occurs. Microwave volumetric heating (MVH) overcomes the uneven absorption by applying an intense, uniform microwave field. Different compounds co... | MORE synthesis | 0.816744 |
12,705 | This approach employs the statement whose independence is being examined, to prove the existence of a set model of ZFC, in which case Con(ZFC) is true. Since ZFC satisfies the conditions of Gödel's second theorem, the consistency of ZFC is unprovable in ZFC (provided that ZFC is, in fact, consistent). Hence no statemen... | ZFC set theory | 0.816742 |
12,706 | Since forcing preserves choice, we cannot directly produce a model contradicting choice from a model satisfying choice. However, we can use forcing to create a model which contains a suitable submodel, namely one satisfying ZF but not C. Another method of proving independence results, one owing nothing to forcing, is b... | ZFC set theory | 0.816742 |
12,707 | The above proof is an example of a non-constructive proof disallowed by intuitionists: The proof is non-constructive because it doesn't give specific numbers a {\displaystyle a} and b {\displaystyle b} that satisfy the theorem but only two separate possibilities, one of which must work. (Actually a = 2 2 {\displaystyle... | Principle of the excluded middle | 0.816742 |
12,708 | In general, intuitionists allow the use of the law of excluded middle when it is confined to discourse over finite collections (sets), but not when it is used in discourse over infinite sets (e.g. the natural numbers). Thus intuitionists absolutely disallow the blanket assertion: "For all propositions P concerning infi... | Principle of the excluded middle | 0.816742 |
12,709 | It allows the development of generic procedures of information processing and thus a unification of basic methods of computer science, in particular of distributed information processing. Information relates to precise questions, comes from different sources, must be aggregated, and can be focused on questions of inter... | Valuation algebra | 0.816739 |
12,710 | It has furthermore been neglected in classical information theory that one wants to extract those parts out of a piece of information that are relevant to specific questions. A mathematical phrasing of these operations leads to an algebra of information, describing basic modes of information processing. Such an algebra... | Valuation algebra | 0.816739 |
12,711 | The term "information algebra" refers to mathematical techniques of information processing. Classical information theory goes back to Claude Shannon. It is a theory of information transmission, looking at communication and storage. However, it has not been considered so far that information comes from different sources... | Valuation algebra | 0.816739 |
12,712 | The operations are well defined since d ( R ⋈ S ) = d ( R ) ∪ d ( S ) {\displaystyle d(R\bowtie S)=d(R)\cup d(S)} If x ⊆ d ( R ) {\displaystyle x\subseteq d(R)} , then d ( π x ( R ) ) = x {\displaystyle d(\pi _{x}(R))=x} .It is easy to see that relational databases satisfy the axioms of a labeled information algebra: s... | Valuation algebra | 0.816739 |
12,713 | Uncertain information Random variables with values in information algebras represent probabilistic argumentation systems (Haenni, Kohlas & Lehmann 2000). Semantic information Information algebras introduce semantics by relating information to questions through focusing and combination (Groenendijk & Stokhof 1984);(Flor... | Valuation algebra | 0.816739 |
12,714 | Valuation algebras Dropping the idempotency axiom leads to valuation algebras. These axioms have been introduced by (Shenoy & Shafer 1990) to generalize local computation schemes (Lauritzen & Spiegelhalter 1988) from Bayesian networks to more general formalisms, including belief function, possibility potentials, etc. (... | Valuation algebra | 0.816739 |
12,715 | The pairs ( ϕ , x ) {\displaystyle (\phi ,x)\ } , where ϕ ∈ Φ {\displaystyle \phi \in \Phi } and x ∈ D {\displaystyle x\in D} such that ϕ ⇒ x = ϕ {\displaystyle \phi ^{\Rightarrow x}=\phi } form a labeled Information Algebra. More precisely, in the two-sorted algebra ( Φ , D ) {\displaystyle (\Phi ,D)\ } , the followin... | Valuation algebra | 0.816739 |
12,716 | In representation theory, the Ferrers diagrams correspond to the irreducible representations of the symmetric group of permutations, and the Young tableaux with a given shape form a basis of the irreducible representation with that shape. Therefore, the telephone numbers give the sum of the degrees of the irreducible r... | Telephone number (mathematics) | 0.816736 |
12,717 | Therefore, the telephone numbers also count involutions. The problem of counting involutions was the original combinatorial enumeration problem studied by Rothe in 1800 and these numbers have also been called involution numbers.In graph theory, a subset of the edges of a graph that touches each vertex at most once is c... | Telephone number (mathematics) | 0.816736 |
12,718 | Simple Viterbi-decoded convolutional codes are now giving way to turbo codes, a new class of iterated short convolutional codes that closely approach the theoretical limits imposed by Shannon's theorem with much less decoding complexity than the Viterbi algorithm on the long convolutional codes that would be required f... | Convolutional code | 0.816736 |
12,719 | Also, use of a thermostable polymerase eliminates the need to add new enzyme to each round of thermocycling. A single closed tube in a relatively simple machine can be used to carry out the entire process. Thus, the use of Taq polymerase was the key idea that made PCR applicable to a large variety of molecular biology ... | Taq DNA polymerase | 0.816733 |
12,720 | In the early 1980s, Kary Mullis was working at Cetus Corporation on the application of synthetic DNAs to biotechnology. He was familiar with the use of DNA oligonucleotides as probes for binding to target DNA strands, as well as their use as primers for DNA sequencing and cDNA synthesis. In 1983, he began using two pri... | Taq DNA polymerase | 0.816733 |
12,721 | In molecular biology, alanine scanning is a site-directed mutagenesis technique used to determine the contribution of a specific residue to the stability or function of a given protein. Alanine is used because of its non-bulky, chemically inert, methyl functional group that nevertheless mimics the secondary structure p... | Alanine scanning | 0.816732 |
12,722 | The Plant Molecular Biology is a peer-reviewed scientific journal covering all aspects of plant molecular biology. It was established in 1981 and is published by Springer Science+Business Media. The editor-in-chief is Motoaki Seki.According to the Journal Citation Reports, the journal has a 2020 impact factor of 4.076. | Plant Molecular Biology | 0.81673 |
12,723 | In particle physics, an interaction point (IP) is the place where particles collide in an accelerator experiment. The nominal interaction point is the design position, which may differ from the real or physics interaction point, where the particles actually collide. A related, but distinct, concept is the primary verte... | Interaction point | 0.816728 |
12,724 | "A designation "flow graph" that includes both the Mason graph and the Coates graph, and a variety of other forms of such graphs appears useful, and agrees with Abrahams and Coverley's and with Henley and Williams' approach.A directed network – also known as a flow network – is a particular type of flow graph. A networ... | Flow graph (mathematics) | 0.816725 |
12,725 | A flow graph is a form of digraph associated with a set of linear algebraic or differential equations: "A signal flow graph is a network of nodes (or points) interconnected by directed branches, representing a set of linear algebraic equations. The nodes in a flow graph are used to represent the variables, or parameter... | Flow graph (mathematics) | 0.816725 |
12,726 | In mathematics, and especially in category theory, a commutative diagram is a diagram such that all directed paths in the diagram with the same start and endpoints lead to the same result. It is said that commutative diagrams play the role in category theory that equations play in algebra. | Commutative diagrams | 0.816722 |
12,727 | The title of "the father of algebra" is frequently credited to the Persian mathematician Al-Khwarizmi, supported by historians of mathematics, such as Carl Benjamin Boyer, Solomon Gandz and Bartel Leendert van der Waerden. However, the point is debatable and the title is sometimes credited to the Hellenistic mathematic... | Syncopated algebra | 0.81672 |
12,728 | The idea of a function began emerging with Sharaf al-Dīn al-Tūsī, but algebra did not decisively move to the dynamic function stage until Gottfried Leibniz. Abstract stage, where mathematical structure plays a central role. Abstract algebra is largely a product of the 19th and 20th centuries. | Syncopated algebra | 0.81672 |
12,729 | Static equation-solving stage, where the objective is to find numbers satisfying certain relationships. The move away from the geometric stage dates back to Diophantus and Brahmagupta, but algebra didn't decisively move to the static equation-solving stage until Al-Khwarizmi introduced generalized algorithmic processes... | Syncopated algebra | 0.81672 |
12,730 | In addition to the three stages of expressing algebraic ideas, some authors recognized four conceptual stages in the development of algebra that occurred alongside the changes in expression. These four stages were as follows: Geometric stage, where the concepts of algebra are largely geometric. This dates back to the B... | Syncopated algebra | 0.81672 |
12,731 | The ability to do algebra is a skill cultivated in mathematics education. As explained by Andrew Warwick, Cambridge University students in the early 19th century practiced "mixed mathematics", doing exercises based on physical variables such as space, time, and weight. Over time the association of variables with physic... | Syncopated algebra | 0.81672 |
12,732 | Another key event in the further development of algebra was the general algebraic solution of the cubic and quartic equations, developed in the mid-16th century. The idea of a determinant was developed by Japanese mathematician Kowa Seki in the 17th century, followed by Gottfried Leibniz ten years later, for the purpos... | Syncopated algebra | 0.81672 |
12,733 | Modern notation for arithmetic operations was introduced between the end of the 15th century and the beginning of the 16th century by Johannes Widmann and Michael Stifel. At the end of 16th century, François Viète introduced symbols, now called variables, for representing indeterminate or unknown numbers. This created ... | Syncopated algebra | 0.81672 |
12,734 | They also point to his treatment of an equation for its own sake and "in a generic manner, insofar as it does not simply emerge in the course of solving a problem, but is specifically called on to define an infinite class of problems". Victor J. Katz regards Al-Jabr as the first true algebra text that is still extant.A... | Syncopated algebra | 0.81672 |
12,735 | Al-Khwarizmi also introduced the fundamental concept of "reduction" and "balancing" (which he originally used the term al-jabr to refer to), referring to the transposition of subtracted terms to the other side of an equation, that is, the cancellation of like terms on opposite sides of the equation. Other supporters of... | Syncopated algebra | 0.81672 |
12,736 | There are three theories about the origins of Arabic Algebra. The first emphasizes Hindu influence, the second emphasizes Mesopotamian or Persian-Syriac influence and the third emphasizes Greek influence. Many scholars believe that it is the result of a combination of all three sources.Throughout their time in power, t... | Syncopated algebra | 0.81672 |
12,737 | Many of these Greek works were translated by Thabit ibn Qurra (826–901), who translated books written by Euclid, Archimedes, Apollonius, Ptolemy, and Eutocius.Arabic mathematicians established algebra as an independent discipline, and gave it the name "algebra" (al-jabr). They were the first to teach algebra in an elem... | Syncopated algebra | 0.81672 |
12,738 | He was the first to give a general solution to the linear Diophantine equation a x + b y = c , {\displaystyle ax+by=c,} where a , b , {\displaystyle a,b,} and c {\displaystyle c} are integers. Unlike Diophantus who only gave one solution to an indeterminate equation, Brahmagupta gave all integer solutions; but that Bra... | Syncopated algebra | 0.81672 |
12,739 | Al-Karkhi is also regarded as the first person to free algebra from geometrical operations and replace them with the type of arithmetic operations which are at the core of algebra today. His work on algebra and polynomials gave the rules for arithmetic operations to manipulate polynomials. The historian of mathematics ... | Syncopated algebra | 0.81672 |
12,740 | Arabic mathematicians treated irrational numbers as algebraic objects. The Egyptian mathematician Abū Kāmil Shujā ibn Aslam (c. 850–930) was the first to accept irrational numbers (often in the form of a square root, cube root or fourth root) as solutions to quadratic equations or as coefficients in an equation. | Syncopated algebra | 0.81672 |
12,741 | This trichotomy comes about because quadratic equations of the form x 2 + p x + q = 0 , {\displaystyle x^{2}+px+q=0,} with p {\displaystyle p} and q {\displaystyle q} positive, have no positive roots.In between the rhetorical and syncopated stages of symbolic algebra, a geometric constructive algebra was developed by c... | Syncopated algebra | 0.81672 |
12,742 | Later, René Descartes (17th century) introduced the modern notation (for example, the use of x—see below) and showed that the problems occurring in geometry can be expressed and solved in terms of algebra (Cartesian geometry).Equally important as the use or lack of symbolism in algebra was the degree of the equations t... | Syncopated algebra | 0.81672 |
12,743 | Symbolic algebra, in which full symbolism is used. Early steps toward this can be seen in the work of several Islamic mathematicians such as Ibn al-Banna (13th-14th centuries) and al-Qalasadi (15th century), although fully symbolic algebra was developed by François Viète (16th century). | Syncopated algebra | 0.81672 |
12,744 | Syncopated algebra, in which some symbolism is used, but which does not contain all of the characteristics of symbolic algebra. For instance, there may be a restriction that subtraction may be used only once within one side of an equation, which is not the case with symbolic algebra. Syncopated algebraic expression fir... | Syncopated algebra | 0.81672 |
12,745 | Algebra did not always make use of the symbolism that is now ubiquitous in mathematics; instead, it went through three distinct stages. The stages in the development of symbolic algebra are approximately as follows: Rhetorical algebra, in which equations are written in full sentences. For example, the rhetorical form o... | Syncopated algebra | 0.81672 |
12,746 | Shu-shu chiu-chang, or Mathematical Treatise in Nine Sections, was written by the wealthy governor and minister Ch'in Chiu-shao (c. 1202 – c. 1261). With the introduction of a method for solving simultaneous congruences, now called the Chinese remainder theorem, it marks the high point in Chinese indeterminate analysis... | Syncopated algebra | 0.81672 |
12,747 | In mathematical logic, algebraic logic is the reasoning obtained by manipulating equations with free variables. What is now usually called classical algebraic logic focuses on the identification and algebraic description of models appropriate for the study of various logics (in the form of classes of algebras that cons... | Logic of relations | 0.816717 |
12,748 | The relation algebra structure, based in set theory, was transcended by Tarski with axioms describing it. Then he asked if every algebra satisfying the axioms could be represented by a set relation. The negative answer opened the frontier of abstract algebraic logic. | Logic of relations | 0.816717 |
12,749 | Planetary geodesy (also known as planetary geodetics) deals with the measurement and representation of the planets of the Solar System, their gravitational fields and geodynamic phenomena (polar motion in three-dimensional, time-varying space). The science of geodesy has elements of both astrophysics and planetary scie... | Planetary Science | 0.816709 |
12,750 | The converse is not true in general: for example, in the quadratic integer ring Z {\displaystyle \mathbb {Z} \left} the element 3 is irreducible (if it factored nontrivially, the factors would each have to have norm 3, but there are no norm 3 elements since a 2 + 5 b 2 = 3 {\displaystyle a^{2}+5b^{2}=3} has no integer... | Associate elements | 0.816709 |
12,751 | Display Menu consists of Show Only Selected Sequences, Use Identical Symbol, Color Cells, Sort Sequences, Restore Input Order, Show Sequence Name, Show Group Name, and Change Font. Highlight Menu consists of Conserved Sites, Variable Sites, Parsimony-Informative Sites, Singleton Sites, 0-fold Degenerate Sites, 2-fold D... | MEGA, Molecular Evolutionary Genetics Analysis | 0.816706 |
12,752 | MEGA provides a graphical interface for displaying and manipulating aligned nucleotide and protein sequences. The Sequence Data Explorer has multiple menu functionalities to help with exporting data, searching alignments, changing display features, highlighting sites, and computing statistics: Data Menu consists of Wri... | MEGA, Molecular Evolutionary Genetics Analysis | 0.816706 |
12,753 | The tree topology can be exported to a file in MEGA tree format, or for timetrees, exported in a tabular format with relevant information used when constructing the timetree. Other export options include the current timetree calibrations, analysis summary, partition list, and pairwise distances. The tree explorer also ... | MEGA, Molecular Evolutionary Genetics Analysis | 0.816706 |
12,754 | MEGA provides a graphical interface for displaying a phylogenetic tree based on a variety of options. In the view menu, the tree can be displayed in three different styles: traditional, radiation, or circle. Traditional trees have three different branch styles: rectangular, straight, or curved. The view menu also offer... | MEGA, Molecular Evolutionary Genetics Analysis | 0.816706 |
12,755 | In 2016, with the proposal of DeepMind's WaveNet, deep-learning-based models for speech synthesis began to gain popularity as a method of modeling waveforms and generating human-like speech. Tacotron2, a neural network architecture for speech synthesis developed by Google AI, was published in 2018 and required tens of ... | 15.ai | 0.816705 |
12,756 | Pronunciations of unfamiliar words are automatically deduced using phonological rules learned by the deep learning model.The application supports a simplified version of a set of English phonetic transcriptions known as ARPABET to correct mispronunciations or to account for heteronyms—words that are spelled the same bu... | 15.ai | 0.816705 |
12,757 | Available characters include GLaDOS and Wheatley from Portal, characters from Team Fortress 2, Twilight Sparkle and a number of main, secondary, and supporting characters from My Little Pony: Friendship Is Magic, SpongeBob from SpongeBob SquarePants, Daria Morgendorffer and Jane Lane from Daria, the Tenth Doctor from D... | 15.ai | 0.816705 |
12,758 | A landmark case between Google and the Authors Guild in 2013 ruled that Google Books—a service that searches the full text of printed copyrighted books—was transformative, thus meeting all requirements for fair use. This case set an important legal precedent for the field of deep learning and artificial intelligence: u... | 15.ai | 0.816705 |
12,759 | The Pony Preservation Project, which began in 2019, is a "collaborative effort by /mlp/ to build and curate pony datasets" with the aim of creating applications in artificial intelligence. The Friendship Is Magic voices on 15.ai were trained on a large dataset crowdsourced by the Pony Preservation Project: audio and di... | 15.ai | 0.816705 |
12,760 | The algorithm used by the project to facilitate the cloning of voices with minimal viable data has been dubbed DeepThroat (a double entendre in reference to speech synthesis using deep neural networks and the sexual act of deep-throating). The project and algorithm—initially conceived as part of MIT's Undergraduate Res... | 15.ai | 0.816705 |
12,761 | Furthermore, 15.ai has inspired the use of 4chan's Pony Preservation Project in other generative artificial intelligence projects.Several commercial alternatives have spawned with the rising popularity of 15.ai, leading to cases of misattribution and theft. In January 2022, it was discovered that Voiceverse NFT, a comp... | 15.ai | 0.816705 |
12,762 | 15.ai is a non-commercial freeware artificial intelligence web application that generates natural emotive high-fidelity text-to-speech voices from an assortment of fictional characters from a variety of media sources. Developed by a pseudonymous MIT researcher under the name 15, the project uses a combination of audio ... | 15.ai | 0.816705 |
12,763 | "Reception has also been largely acclaimed overseas, especially in Japan. Takayuki Furushima of Den Fami Nico Gamer has described 15.ai as "like magic," and Yuki Kurosawa of Automaton Media called it "revolutionary. "Computer scientist and technology entrepreneur Andrew Ng commented in his newsletter The Batch that the... | 15.ai | 0.816705 |
12,764 | 15.ai has been met with largely positive reception. Liana Ruppert of Game Informer described 15.ai as "simplistically brilliant." Lauren Morton of Rock, Paper, Shotgun and Natalia Clayton of PCGamer called it "fascinating," and José Villalobos of LaPS4 wrote that it "works as easy as it looks." Users praised the abilit... | 15.ai | 0.816705 |
12,765 | Due to this misattribution and absence of proper credit, 15.ai's terms of service has a rule that forbids having 15.ai-and-non-15.ai-generated speech in the same videos and projects.The My Little Pony: Friendship Is Magic fandom has seen a resurgence in video and musical content creation as a direct result, inspiring a... | 15.ai | 0.816705 |
12,766 | In mathematics, a fact is a statement (called a theorem) that can be proven by logical argument from certain axioms and definitions. | Scientific fact | 0.816701 |
12,767 | Strategy video games are categorized based on whether they offer the continuous gameplay of real-time strategy (RTS), or the discrete phases of turn-based strategy (TBS). Often the computer is expected to emulate a strategically thinking "side" similar to that of a human player (such as directing armies and constructin... | Games of strategy | 0.8167 |
12,768 | The Theis equation was created by Charles Vernon Theis (working for the US Geological Survey) in 1935, from heat transfer literature (with the mathematical help of C.I. Lubin), for two-dimensional radial flow to a point sink in an infinite, homogeneous aquifer. It is simply s = Q 4 π T W ( u ) u = r 2 S 4 T t {\display... | Aquifer test | 0.8167 |
12,769 | Homotopy groups are similar to homology groups in that they can represent "holes" in a topological space. There is a close connection between the first homotopy group π 1 ( X ) {\displaystyle \pi _{1}(X)} and the first homology group H 1 ( X ) {\displaystyle H_{1}(X)}: the latter is the abelianization of the former. He... | Homology group | 0.816698 |
12,770 | The KAM theorem established that periodic orbits can follow complex trajectories; in particular, they may form braids that can be investigated using Floer homology.In one class of finite element methods, boundary-value problems for differential equations involving the Hodge-Laplace operator may need to be solved on top... | Homology group | 0.816698 |
12,771 | In topological data analysis, data sets are regarded as a point cloud sampling of a manifold or algebraic variety embedded in Euclidean space. By linking nearest neighbor points in the cloud into a triangulation, a simplicial approximation of the manifold is created and its simplicial homology may be calculated. Findin... | Homology group | 0.816698 |
12,772 | Homology groups are finitely generated abelian groups, and homology classes are elements of these groups. The Betti numbers of the manifold are the rank of the free part of the homology group, and the non-orientable cycles are described by the torsion part. The subsequent spread of homology groups brought a change of t... | Homology group | 0.816698 |
12,773 | A 2-cycle corresponds to a collection of embedded surfaces such as a sphere or a torus, and so on. Emmy Noether and, independently, Leopold Vietoris and Walther Mayer further developed the theory of algebraic homology groups in the period 1925–28. The new combinatorial topology formally treated topological classes as a... | Homology group | 0.816698 |
12,774 | p-adic analysis a branch of number theory that deals with the analysis of functions of p-adic numbers. p-adic dynamics an application of p-adic analysis looking at p-adic differential equations. p-adic Hodge theory Parabolic geometry Paraconsistent mathematics sometimes called inconsistent mathematics, it is an attempt... | Glossary of areas of mathematics | 0.816687 |
12,775 | Valuation theory Variational analysis Vector algebra a part of linear algebra concerned with the operations of vector addition and scalar multiplication, although it may also refer to vector operations of vector calculus, including the dot and cross product. In this case it can be contrasted with geometric algebra whic... | Glossary of areas of mathematics | 0.816687 |
12,776 | Inventory theory Inversive geometry the study of invariants preserved by a type of transformation known as inversion Inversive plane geometry inversive geometry that is limited to two dimensions Inversive ring geometry Itô calculus extends the methods of calculus to stochastic processes such as Brownian motion (see Wie... | Glossary of areas of mathematics | 0.816687 |
12,777 | Integral calculus Integral geometry the theory of measures on a geometrical space invariant under the symmetry group of that space. Intersection theory a branch of algebraic geometry and algebraic topology Intuitionistic type theory a type theory and an alternative foundation of mathematics. Invariant theory studies ho... | Glossary of areas of mathematics | 0.816687 |
12,778 | Ideal theory once the precursor name for what is now known as commutative algebra; it is the theory of ideals in commutative rings. Idempotent analysis the study of idempotent semirings, such as the tropical semiring. Incidence geometry the study of relations of incidence between various geometric objects, like curves ... | Glossary of areas of mathematics | 0.816687 |
12,779 | Group-character theory the part of character theory dedicated to the study of characters of group representations. Group representation theory Group theory the study of algebraic structures known as groups. Gyrotrigonometry a form of trigonometry used in gyrovector space for hyperbolic geometry. (An analogy of the vect... | Glossary of areas of mathematics | 0.816687 |
12,780 | Global analysis the study of differential equations on manifolds and the relationship between differential equations and topology. Global arithmetic dynamics Graph theory a branch of discrete mathematics devoted to the study of graphs. It has many applications in physical, biological and social systems. | Glossary of areas of mathematics | 0.816687 |
12,781 | Geometry also gave rise to the modern discipline of topology. Geometry of numbers initiated by Hermann Minkowski, it is a branch of number theory studying convex bodies and integer vectors. | Glossary of areas of mathematics | 0.816687 |
12,782 | It was then put into an axiomatic form by Euclid, giving rise to what is now known as classical Euclidean geometry. The use of coordinates by René Descartes gave rise to Cartesian geometry enabling a more analytical approach to geometric entities. Since then many other branches have appeared including projective geomet... | Glossary of areas of mathematics | 0.816687 |
12,783 | Geometric number theory Geometric topology a branch of topology studying manifolds and mappings between them; in particular the embedding of one manifold into another. Geometry a branch of mathematics concerned with shape and the properties of space. Classically it arose as what is now known as solid geometry; this was... | Glossary of areas of mathematics | 0.816687 |
12,784 | Geometric graph theory a large and amorphous subfield of graph theory, concerned with graphs defined by geometric means. Geometric group theory the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which the... | Glossary of areas of mathematics | 0.816687 |
12,785 | It includes a number of subareas such as polyhedral combinatorics (the study of faces of convex polyhedra), convex geometry (the study of convex sets, in particular combinatorics of their intersections), and discrete geometry, which in turn has many applications to computational geometry. Geometric function theory the ... | Glossary of areas of mathematics | 0.816687 |
12,786 | Geometric analysis a discipline that uses methods from differential geometry to study partial differential equations as well as the applications to geometry. Geometric calculus extends the geometric algebra to include differentiation and integration. Geometric combinatorics a branch of combinatorics. | Glossary of areas of mathematics | 0.816687 |
12,787 | This includes spherical trigonometry, hyperbolic trigonometry, gyrotrigonometry, and universal hyperbolic trigonometry. Geometric algebra an alternative approach to classical, computational and relativistic geometry. It shows a natural correspondence between geometric entities and elements of algebra. | Glossary of areas of mathematics | 0.816687 |
12,788 | Galois cohomology an application of homological algebra, it is the study of group cohomology of Galois modules. Galois theory named after Évariste Galois, it is a branch of abstract algebra providing a connection between field theory and group theory. Galois geometry a branch of finite geometry concerned with algebraic... | Glossary of areas of mathematics | 0.816687 |
12,789 | Function theory an ambiguous term that generally refers to mathematical analysis. Functional analysis a branch of mathematical analysis, the core of which is formed by the study of function spaces, which are some sort of topological vector spaces. Functional calculus historically the term was used synonymously with cal... | Glossary of areas of mathematics | 0.816687 |
12,790 | Experimental mathematics an approach to mathematics in which computation is used to investigate mathematical objects and identify properties and patterns. Exterior algebra Exterior calculus Extraordinary cohomology theory Extremal combinatorics a branch of combinatorics, it is the study of the possible sizes of a colle... | Glossary of areas of mathematics | 0.816687 |
12,791 | Euclidean differential geometry also known as classical differential geometry. See differential geometry. Euler calculus a methodology from applied algebraic topology and integral geometry that integrates constructible functions and more recently definable functions by integrating with respect to the Euler characterist... | Glossary of areas of mathematics | 0.816687 |
12,792 | Epidemiology Equivariant noncommutative algebraic geometry Ergodic Ramsey theory a branch where problems are motivated by additive combinatorics and solved using ergodic theory. Ergodic theory the study of dynamical systems with an invariant measure, and related problems. Euclidean geometry An area of geometry based on... | Glossary of areas of mathematics | 0.816687 |
12,793 | Enumerative combinatorics an area of combinatorics that deals with the number of ways that certain patterns can be formed. Enumerative geometry a branch of algebraic geometry concerned with counting the number of solutions to geometric questions. This is usually done by means of intersection theory. | Glossary of areas of mathematics | 0.816687 |
12,794 | It is a part of commutative algebra and algebraic geometry. Elliptic geometry a type of non-Euclidean geometry (it violates Euclid's parallel postulate) and is based on spherical geometry. It is constructed in elliptic space. | Glossary of areas of mathematics | 0.816687 |
12,795 | Elementary mathematics parts of mathematics frequently taught at the primary and secondary school levels. This includes elementary arithmetic, geometry, probability and statistics, elementary algebra and trigonometry. (calculus is not usually considered a part) Elementary group theory the study of the basics of group t... | Glossary of areas of mathematics | 0.816687 |
12,796 | Econometrics the application of mathematical and statistical methods to economic data. Effective descriptive set theory a branch of descriptive set theory dealing with set of real numbers that have lightface definitions. It uses aspects of computability theory. Elementary algebra a fundamental form of algebra extending... | Glossary of areas of mathematics | 0.816687 |
12,797 | Discrete Morse theory a combinatorial adaption of Morse theory. Distance geometry Domain theory a branch that studies special kinds of partially ordered sets (posets) commonly called domains. Donaldson theory the study of smooth 4-manifolds using gauge theory. Dyadic algebra Dynamical systems theory an area used to des... | Glossary of areas of mathematics | 0.816687 |
12,798 | Diffiety theory Diophantine geometry in general the study of algebraic varieties over fields that are finitely generated over their prime fields. Discrepancy theory Discrete differential geometry Discrete exterior calculus Discrete geometry a branch of geometry that studies combinatorial properties and constructive met... | Glossary of areas of mathematics | 0.816687 |
12,799 | Differential geometry a form of geometry that uses techniques from integral and differential calculus as well as linear and multilinear algebra to study problems in geometry. Classically, these were problems of Euclidean geometry, although now it has been expanded. It is generally concerned with geometric structures on... | Glossary of areas of mathematics | 0.816687 |
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