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11,900 | Mechanochemistry – coupling of mechanical and chemical phenomena on a molecular scale and can be seen as a coupling of chemistry and mechanical engineering. Molecular biology – study of interactions between the various systems of a cell. It overlaps with biochemistry. | Outline of chemistry | 0.817509 |
11,901 | Chemical oceanography – study of ocean chemistry: the behavior of the chemical elements within the Earth's oceans Materials science – is an interdisciplinary field investigating the relationship between the structure of materials at atomic or molecular scales and their macroscopic properties. Mathematical chemistry – a... | Outline of chemistry | 0.817509 |
11,902 | Flow chemistry – study of chemical reactions in continuous flow, not as stationary batches, in industry and macro processing equipment. Immunohistochemistry – involves the process of detecting antigens (e.g., proteins) in cells of a tissue section by exploiting the principle of antibodies binding specifically to antige... | Outline of chemistry | 0.817509 |
11,903 | Chemical engineering – branch of engineering that applies the physical sciences (e.g., chemistry and physics) and/or life sciences (e.g., biology, microbiology and biochemistry) together with mathematics and economics to processes that convert raw materials or chemicals into more useful or valuable forms. Chemical biol... | Outline of chemistry | 0.817509 |
11,904 | Wet chemistry – is a form of analytical chemistry that uses classical methods such as observation to analyze materials usually in liquid phase. Agrochemistry – study and application of both chemistry and biochemistry for agricultural production, the processing of raw products into foods and beverages, and environmental... | Outline of chemistry | 0.817509 |
11,905 | In particular the application of quantum mechanics to chemistry is called quantum chemistry. Since the end of the Second World War, the development of computers has allowed a systematic development of computational chemistry, which is the art of developing and applying computer programs for solving chemical problems. T... | Outline of chemistry | 0.817509 |
11,906 | Green chemistry is a philosophy of chemical research and engineering that encourages the design of products and processes that minimize the use and generation of hazardous substances. Supramolecular chemistry – refers to the domain of chemistry beyond that of molecules and focuses on the chemical systems made up of a d... | Outline of chemistry | 0.817509 |
11,907 | Cosmochemistry – study of the chemical composition of matter in the universe and the processes that led to those compositions. Computational chemistry Environmental chemistry – study of chemical and biochemical phenomena that occur diverse aspects of the environment such the air, soil, and water. It also studies the ef... | Outline of chemistry | 0.817509 |
11,908 | Analytical chemistry incorporates standardized experimental methods in chemistry. These methods may be used in all subdisciplines of chemistry, excluding purely theoretical chemistry. Other Astrochemistry – study of the abundance and reactions of chemical elements and molecules in the universe, and their interaction wi... | Outline of chemistry | 0.817509 |
11,909 | Nuclear chemistry – study of how subatomic particles come together and make nuclei. Modern Transmutation is a large component of nuclear chemistry, and the table of nuclides is an important result and tool for this field. Analytical chemistry – analysis of material samples to gain an understanding of their chemical com... | Outline of chemistry | 0.817509 |
11,910 | Cluster chemistry Materials chemistry – preparation, characterization, and understanding of substances with a useful function. The field is a new breadth of study in graduate programs, and it integrates elements from all classical areas of chemistry with a focus on fundamental issues that are unique to materials. Prima... | Outline of chemistry | 0.817509 |
11,911 | Inorganic chemistry – study of the properties and reactions of inorganic compounds. The distinction between organic and inorganic disciplines is not absolute and there is much overlap, most importantly in the sub-discipline of organometallic chemistry. Bioinorganic chemistry – is a field that examines the role of metal... | Outline of chemistry | 0.817509 |
11,912 | Physical organic chemistry – study of the interrelationships between structure and reactivity in organic molecules. Polymer chemistry – multidisciplinary science that deals with the chemical synthesis and chemical properties of polymers or macromolecules. Click chemistry – "click" chemistry is a class of biocompatible ... | Outline of chemistry | 0.817509 |
11,913 | Biophysical chemistry – is a physical science that uses the concepts of physics and physical chemistry for the study of biological systems. Medicinal chemistry – discipline which applies chemistry for medical or drug related purposes. Organometallic chemistry – is the study of organometallic compounds, chemical compoun... | Outline of chemistry | 0.817509 |
11,914 | Biochemistry – study of the chemicals, chemical reactions and chemical interactions that take place in living organisms. Biochemistry and organic chemistry are closely related, as in medicinal chemistry or neurochemistry. Biochemistry is also associated with molecular biology and genetics. | Outline of chemistry | 0.817509 |
11,915 | Calorimetry – The study of heat changes in physical and chemical processes. Organic chemistry (outline) – study of the structure, properties, composition, mechanisms, and reactions of organic compounds. An organic compound is defined as any compound based on a carbon skeleton. | Outline of chemistry | 0.817509 |
11,916 | Spectroscopy – study of the interaction between matter and radiated energy. Stereochemistry – study of the relative spatial arrangement of atoms that form the structure of molecules Surface science – study of physical and chemical phenomena that occur at the interface of two phases, including solid–liquid interfaces, s... | Outline of chemistry | 0.817509 |
11,917 | Photochemistry – study of chemical reactions that proceed with the absorption of light by atoms or molecules. Quantum chemistry – branch of chemistry whose primary focus is the application of quantum mechanics in physical models and experiments of chemical systems. Solid-state chemistry – study of the synthesis, struct... | Outline of chemistry | 0.817509 |
11,918 | Electrochemistry – branch of chemistry that studies chemical reactions which take place in a solution at the interface of an electron conductor (the electrode: a metal or a semiconductor) and an ionic conductor (the electrolyte), and which involve electron transfer between the electrode and the electrolyte or species i... | Outline of chemistry | 0.817509 |
11,919 | Physical chemistry is a distinct discipline from chemical physics, but again, there is very strong overlap. Chemical kinetics – study of rates of chemical processes. Chemical physics – investigates physicochemical phenomena using techniques from atomic and molecular physics and condensed matter physics; it is the branc... | Outline of chemistry | 0.817509 |
11,920 | Physical chemistry has large overlap with molecular physics. Physical chemistry involves the use of infinitesimal calculus in deriving equations. It is usually associated with quantum chemistry and theoretical chemistry. | Outline of chemistry | 0.817509 |
11,921 | Physical chemistry – study of the physical and fundamental basis of chemical systems and processes. In particular, the energetics and dynamics of such systems and processes are of interest to physical chemists. Important areas of study include chemical thermodynamics, chemical kinetics, electrochemistry, statistical me... | Outline of chemistry | 0.817509 |
11,922 | Thermochemistry – Chemical kinetics – the study of the rates of chemical reactions and investigates how different experimental conditions can influence the speed of a chemical reaction and yield information about the reaction's mechanism and transition states, as well as the construction of mathematical models that can... | Outline of chemistry | 0.817509 |
11,923 | History of computational chemistry – history of the branch of chemistry that uses principles of computer science to assist in solving chemical problems. History of chemo-informatics – history of the use of computer and informational techniques, applied to a range of problems in the field of chemistry. History of molecu... | Outline of chemistry | 0.817509 |
11,924 | After the 1990s, supporters of the later started defending a so-called "mixed method" (also known as Balanced literacy) in which approaches from both methods are used. Influential researchers in psycho-pedagogy, cognitive sciences and neurosciences, such as Stanislas Dehaene and Michel Fayol have put their heavy scient... | Reading speed | 0.817509 |
11,925 | In 2019, using data from the UNESCO Institute for Statistics, they published a report entitled Ending Learning Poverty: What will it take?. Learning poverty is defined as being unable to read and understand a simple text by age 10. Although they say that all foundational skills are important, include reading, numeracy,... | Reading speed | 0.817509 |
11,926 | The Bat algorithm is a metaheuristic algorithm for global optimization. It was inspired by the echolocation behaviour of microbats, with varying pulse rates of emission and loudness. The Bat algorithm was developed by Xin-She Yang in 2010. | Bat algorithm | 0.817507 |
11,927 | In elliptic geometry, an elliptic rectangle is a figure in the elliptic plane whose four edges are elliptic arcs which meet at equal angles greater than 90°. Opposite arcs are equal in length. In hyperbolic geometry, a hyperbolic rectangle is a figure in the hyperbolic plane whose four edges are hyperbolic arcs which m... | Crossed rectangle | 0.817502 |
11,928 | In spherical geometry, a spherical rectangle is a figure whose four edges are great circle arcs which meet at equal angles greater than 90°. Opposite arcs are equal in length. The surface of a sphere in Euclidean solid geometry is a non-Euclidean surface in the sense of elliptic geometry. Spherical geometry is the simp... | Crossed rectangle | 0.817502 |
11,929 | The Extreme Ultraviolet Variability Experiment (EVE) measures the Sun's extreme ultraviolet irradiance with improved spectral resolution, "temporal cadence", accuracy, and precision over preceding measurements made by TIMED SEE, SOHO, and SORCE XPS. The instrument incorporates physics-based models in order to further s... | Extreme Ultraviolet Variability Experiment | 0.817502 |
11,930 | The Atmospheric Imaging Assembly (AIA), led from the Lockheed Martin Solar and Astrophysics Laboratory (LMSAL), provides continuous full-disk observations of the solar chromosphere and corona in seven extreme ultraviolet (EUV) channels, spanning a temperature range from approximately 20,000 Kelvin to in excess of 20 mi... | Extreme Ultraviolet Variability Experiment | 0.817502 |
11,931 | Raw data is a relative term (see data), because even once raw data have been "cleaned" and processed by one team of researchers, another team may consider these processed data to be "raw data" for another stage of research. Raw data can be inputted to a computer program or used in manual procedures such as analyzing st... | Raw Data | 0.817501 |
11,932 | A homogeneous binary relation is found in the power set of X × X for some set X, while a heterogeneous relation is found in the power set of X × Y, where X ≠ Y. Whether a given relation holds for two individuals is one bit of information, so relations are studied with Boolean arithmetic. Elements of the power set are p... | Calculus of relations | 0.817497 |
11,933 | To express a tautology, equate a formula with a truth value; The rules of proof are the substitution of equals for equals, and uniform replacement. Modus ponens remains valid, but is seldom employed.In the table below, the left column contains one or more logical or mathematical systems, and the algebraic structure whi... | Calculus of relations | 0.817497 |
11,934 | Algebraic logic treats algebraic structures, often bounded lattices, as models (interpretations) of certain logics, making logic a branch of order theory. In algebraic logic: Variables are tacitly universally quantified over some universe of discourse. There are no existentially quantified variables or open formulas; T... | Calculus of relations | 0.817497 |
11,935 | Lassen. It continues to support on-campus laboratory facilities in Campbell Hall. From 1998 to 2012, the RAL collaborated with the SETI Institute of Mountain View California to design, build and operate the Allen Telescope Array (ATA). RAL has been central to the creation of several radio observatories, including: Hat ... | Radio Astronomy Laboratory | 0.817497 |
11,936 | The Radio Astronomy Lab (RAL) is an Organized Research Unit (ORU) within the Astronomy Department at the University of California, Berkeley. It was founded by faculty member Harold Weaver in 1958. Until 2012, RAL maintained a radio astronomy observatory at Hat Creek, near Mt. | Radio Astronomy Laboratory | 0.817497 |
11,937 | Crude (the enumeration of instances) or gradual (new estimate of the proportion of truth in the hypothesis after each test). Gradual induction is qualitative or quantitative; if qualitative, then dependent on weightings of qualities or characters; if quantitative, then dependent on measurements, or on statistics, or on... | Experimental confirmation | 0.817491 |
11,938 | Explicit deduction of hypothesis's consequences as predictions, for induction to test, about evidence to be found. Corollarial or, if needed, theorematic. Induction. | Experimental confirmation | 0.817491 |
11,939 | Experiment: Rosalind Franklin used pure DNA to perform X-ray diffraction to produce photo 51. The results showed an X-shape. Analysis: When Watson saw the detailed diffraction pattern, he immediately recognized it as a helix. He and Crick then produced their model, using this information along with the previously known... | Experimental confirmation | 0.817491 |
11,940 | Reduce is a general-purpose computer algebra system geared towards applications in physics. The development of the Reduce computer algebra system was started in the 1960s by Anthony C. Hearn. Since then, many scientists from all over the world have contributed to its development under his direction. Reduce is written e... | Reduce computer algebra system | 0.817491 |
11,941 | This sort of coordinate transformation was commonplace in preparing tables of positions of the planets for both astronomy and astrology: it is equivalent to finding each of the positions L (or K) for the various eclipses which might be observable. Finally, the distance between Earth and Jupiter can be calculated using ... | Rømer's determination of the speed of light | 0.817488 |
11,942 | Laurent Nottale (born 29 July 1952) is an astrophysicist, a retired director of research at CNRS, and a researcher at the Paris Observatory. He is the author and inventor of the theory of scale relativity, which aims to unify quantum physics and relativity theory. | Scale relativity | 0.817482 |
11,943 | Scale relativity claims to extend the concept of relativity to physical scales (of time, length, energy, or momentum). Proponents have made wide-ranging claims on its behalf, including applications to the existence of dark matter and the formation of planetary systems, as well as to biology, geology, and the technologi... | Scale relativity | 0.817482 |
11,944 | Nottale began his professional work in the domain of general relativity. He defended his PhD Thesis in June 1980, entitled "Perturbation of the Hubble relation by clusters of galaxies", in which he showed that clusters of galaxies as a whole may act as gravitational lenses on distant sources. Some of these results were... | Scale relativity | 0.817482 |
11,945 | Available under a Creative Commons Attribution 4.0 International License. Flynn, Kevin J.; Mitra, Aditee; Anestis, Konstantinos; Anschütz, Anna A.; Calbet, Albert; Ferreira, Guilherme Duarte; Gypens, Nathalie; Hansen, Per J.; John, Uwe; Martin, Jon Lapeyra; Mansour, Joost S.; Maselli, Maira; Medić, Nikola; Norlin, Andr... | Marine protozoans | 0.817481 |
11,946 | "Marine Protists Are Not Just Big Bacteria". Current Biology. Elsevier BV. | Marine protozoans | 0.817481 |
11,947 | (A) represents transport processes include the transport into the cell from the surrounding seawater of primary calcification substrates Ca2+ and HCO3− (black arrows) and the removal of the end product H+ from the cell (gray arrow). The transport of Ca2+ through the cytoplasm to the coccolith vesicle (CV) is the domina... | Marine protozoans | 0.817481 |
11,948 | 1. Fréchet algebras are examples of associative topological algebras with jointly continuous multiplication. 2. Banach algebras are special cases of Fréchet algebras. 3. Stereotype algebras are examples of associative topological algebras with stereotype continuous multiplication. | Stereotype algebra | 0.81748 |
11,949 | In mathematics, a topological algebra A {\displaystyle A} is an algebra and at the same time a topological space, where the algebraic and the topological structures are coherent in a specified sense. | Stereotype algebra | 0.81748 |
11,950 | After the 'rediscovery' of Aristotle's Metaphysics in the mid-twelfth century, many scholastics wrote commentaries on this work (in particular Aquinas and Scotus). The problem of universals was one of the main problems engaged during that period. Other subjects included: Hylomorphism – development of the Aristotelian d... | Medieval logic | 0.81748 |
11,951 | : 35 Despite this view of philosophy as the servant of theology, this did not prevent the medievals from developing original and innovative philosophies against the backdrop of their theological projects. For instance, such thinkers as Augustine of Hippo and Thomas of Aquinas made monumental breakthroughs in the philos... | Medieval logic | 0.81748 |
11,952 | One of the most notable thinkers of the era, Thomas of Aquinas, never considered himself a philosopher, and criticized philosophers for always "falling short of the true and proper wisdom". The problems discussed throughout this period are the relation of faith to reason, the existence and simplicity of God, the purpos... | Medieval logic | 0.81748 |
11,953 | Medieval philosophy is the philosophy that existed through the Middle Ages, the period roughly extending from the fall of the Western Roman Empire in the 5th century until after the Renaissance in the 13th and 14th centuries. Medieval philosophy, understood as a project of independent philosophical inquiry, began in Ba... | Medieval logic | 0.81748 |
11,954 | Schools of translation grew up in Italy and Sicily, and eventually in the rest of Europe. Scholars such as Adelard of Bath travelled to Sicily and the Arab world, translating works on astronomy and mathematics, including the first complete translation of Euclid's Elements. Powerful Norman kings gathered men of knowledg... | Medieval logic | 0.817479 |
11,955 | Starting around 1927, Paul Dirac combined quantum mechanics with the relativistic theory of electromagnetism. Further work in the 1940s, by Richard Feynman, Freeman Dyson, Julian Schwinger, and Sin-Itiro Tomonaga, completed this theory, which is now called quantum electrodynamics, the revised theory of electromagnetism... | Fundamental forces | 0.817478 |
11,956 | Electrical and magnetic phenomena have been observed since ancient times, but it was only in the 19th century James Clerk Maxwell discovered that electricity and magnetism are two aspects of the same fundamental interaction. By 1864, Maxwell's equations had rigorously quantified this unified interaction. Maxwell's theo... | Fundamental forces | 0.817478 |
11,957 | Electromagnetism and weak interaction appear to be very different at everyday low energies. They can be modeled using two different theories. However, above unification energy, on the order of 100 GeV, they would merge into a single electroweak force. The electroweak theory is very important for modern cosmology, parti... | Fundamental forces | 0.817478 |
11,958 | The weak and electromagnetic forces have already been unified with the electroweak theory of Sheldon Glashow, Abdus Salam, and Steven Weinberg, for which they received the 1979 Nobel Prize in physics. Some physicists seek to unite the electroweak and strong fields within what is called a Grand Unified Theory (GUT). An ... | Fundamental forces | 0.817478 |
11,959 | The gravitational force is attributed to the curvature of spacetime, described by Einstein's general theory of relativity. The other three are discrete quantum fields, and their interactions are mediated by elementary particles described by the Standard Model of particle physics.Within the Standard Model, the strong in... | Fundamental forces | 0.817478 |
11,960 | In physics, the fundamental interactions or fundamental forces are the interactions that do not appear to be reducible to more basic interactions. There are four fundamental interactions known to exist: gravity electromagnetism weak interaction strong interactionThe gravitational and electromagnetic interactions produc... | Fundamental forces | 0.817478 |
11,961 | Office Delve allows Office 365 users to search and manage their emails, meetings, contacts, social networks and documents stored on OneDrive or Sites in Office 365. Delve uses machine learning and artificial intelligence In April 2015 Microsoft launched a mobile version of Office Delve in the App Store and Google Play ... | Equation Editor | 0.817478 |
11,962 | In mathematics, and particularly in graph theory, the dimension of a graph is the least integer n such that there exists a "classical representation" of the graph in the Euclidean space of dimension n with all the edges having unit length. In a classical representation, the vertices must be distinct points, but the edg... | Dimension (graph theory) | 0.817476 |
11,963 | One case for non-commutative rings where localization has a clear interest is for rings of differential operators. It has the interpretation, for example, of adjoining a formal inverse D−1 for a differentiation operator D. This is done in many contexts in methods for differential equations. There is now a large mathema... | Noncommutative algebra | 0.817472 |
11,964 | If R is Noetherian, and U is finitely generated, then U is a Noetherian module over R, and the conclusion is satisfied. Somewhat remarkable is that the weaker assumption, namely that U is finitely generated as an R-module (and no finiteness assumption on R), is sufficient to guarantee the conclusion. This is essentiall... | Noncommutative algebra | 0.817472 |
11,965 | This is sufficient to guarantee that a right-Noetherian ring is right Goldie. The converse does not hold: every right Ore domain is a right Goldie domain, and hence so is every commutative integral domain. A consequence of Goldie's theorem, again due to Goldie, is that every semiprime principal right ideal ring is isom... | Noncommutative algebra | 0.817472 |
11,966 | In mathematics, Goldie's theorem is a basic structural result in ring theory, proved by Alfred Goldie during the 1950s. What is now termed a right Goldie ring is a ring R that has finite uniform dimension (also called "finite rank") as a right module over itself, and satisfies the ascending chain condition on right ann... | Noncommutative algebra | 0.817472 |
11,967 | Working within GA, Euclidean space E 3 {\displaystyle {\mathcal {E}}^{3}} (along with a conformal point at infinity) is embedded projectively in the CGA G ( 4 , 1 ) {\displaystyle {\mathcal {G}}(4,1)} via the identification of Euclidean points with 1D subspaces in the 4D null cone of the 5D CGA vector subspace. This al... | Geometric algebra | 0.817466 |
11,968 | Using an orthogonal basis, a graded vector space structure can be established. Elements of the geometric algebra that are scalar multiples of 1 {\displaystyle 1} are grade- 0 {\displaystyle 0} blades and are called scalars. Multivectors that are in the span of { e 1 , … , e n } {\displaystyle \{e_{1},\ldots ,e_{n}\}} a... | Geometric algebra | 0.817466 |
11,969 | The resemblance to the imaginary unit is not incidental: the subspace { α 0 + α 1 I ∣ α i ∈ R } {\displaystyle \{\alpha _{0}+\alpha _{1}I\mid \alpha _{i}\in \mathbb {R} \}} is R {\displaystyle \mathbb {R} } -algebra isomorphic to the complex numbers. In this way, a copy of the complex numbers is embedded in the geometr... | Geometric algebra | 0.817466 |
11,970 | Using the vector space isomorphism between the geometric algebra and exterior algebra, this gives the equivalence class of α I {\displaystyle \alpha I} for all α ≠ 0 {\displaystyle \alpha \neq 0} . Orthonormality gets rid of this ambiguity except for the signs above. Suppose the geometric algebra G ( n , 0 ) {\displays... | Geometric algebra | 0.817466 |
11,971 | Unit pseudoscalars are blades that play important roles in GA. A unit pseudoscalar for a non-degenerate subspace W {\displaystyle W} of V {\displaystyle V} is a blade that is the product of the members of an orthonormal basis for W {\displaystyle W} . It can be shown that if I {\displaystyle I} and I ′ {\displaystyle I... | Geometric algebra | 0.817466 |
11,972 | Two potential candidates are currently under investigation as the foundation for affine and projective geometry in three dimensions R ( 3 , 3 ) {\displaystyle {\mathcal {R}}(3,3)} and R ( 4 , 4 ) {\displaystyle {\mathcal {R}}(4,4)} which includes representations for shears and non-uniform scaling, as well as quadric su... | Geometric algebra | 0.817466 |
11,973 | A nonzero element of the algebra does not necessarily have a multiplicative inverse. For example, if u {\displaystyle u} is a vector in V {\displaystyle V} such that u 2 = 1 {\displaystyle u^{2}=1} , the element 1 2 ( 1 + u ) {\displaystyle \textstyle {\frac {1}{2}}(1+u)} is both a nontrivial idempotent element and a n... | Geometric algebra | 0.817466 |
11,974 | The above definition of the geometric algebra is abstract, so we summarize the properties of the geometric product by the following set of axioms. The geometric product has the following properties, for A , B , C ∈ G ( p , q ) {\displaystyle A,B,C\in {\mathcal {G}}(p,q)}: A B ∈ G ( p , q ) {\displaystyle AB\in {\mathca... | Geometric algebra | 0.817466 |
11,975 | The notation G ( p , q ) {\displaystyle {\mathcal {G}}(p,q)} (respectively G ( p , q , r ) {\displaystyle {\mathcal {G}}(p,q,r)} ) will be used to denote a geometric algebra for which the bilinear form g {\displaystyle g} has the signature ( p , q ) {\displaystyle (p,q)} (respectively ( p , q , r ) {\displaystyle (p,q,... | Geometric algebra | 0.817466 |
11,976 | There are a number of different ways to define a geometric algebra. Hestenes's original approach was axiomatic, "full of geometric significance" and equivalent to the universal Clifford algebra. Given a finite-dimensional vector space V {\displaystyle V} over a field F {\displaystyle F} with a symmetric bilinear form (... | Geometric algebra | 0.817466 |
11,977 | Geometric algebra represents subspaces of V {\displaystyle V} as blades, and so they coexist in the same algebra with vectors from V {\displaystyle V} . A k {\displaystyle k} -dimensional subspace W {\displaystyle W} of V {\displaystyle V} is represented by taking an orthogonal basis { b 1 , b 2 , … , b k } {\displayst... | Geometric algebra | 0.817466 |
11,978 | The Pfaffian provides a metric for the exterior algebra and, as pointed out by Claude Chevalley, Clifford algebra reduces to the exterior algebra with a zero quadratic form. The role the Pfaffian plays can be understood from a geometric viewpoint by developing Clifford algebra from simplices. This derivation provides a... | Geometric algebra | 0.817466 |
11,979 | Since every element of the algebra can be expressed as the sum of products of this form, this defines the exterior product for every pair of elements of the algebra. It follows from the definition that the exterior product forms an alternating algebra. The equivalent structure equation for Clifford algebra is a 1 a 2 a... | Geometric algebra | 0.817466 |
11,980 | The exterior product is naturally extended as an associative bilinear binary operator between any two elements of the algebra, satisfying the identities 1 ∧ a i = a i ∧ 1 = a i a 1 ∧ a 2 ∧ ⋯ ∧ a r = 1 r ! ∑ σ ∈ S r sgn ( σ ) a σ ( 1 ) a σ ( 2 ) ⋯ a σ ( r ) , {\displaystyle {\begin{aligned}1\wedge a_{i}&=a_{i}\wedge 1... | Geometric algebra | 0.817466 |
11,981 | If the quadratic form is fully degenerate, the inner product of any two vectors is always zero, and the geometric algebra is then simply an exterior algebra. Unless otherwise stated, this article will treat only nondegenerate geometric algebras. | Geometric algebra | 0.817466 |
11,982 | The exterior product of two vectors can be identified with the signed area enclosed by a parallelogram the sides of which are the vectors. The cross product of two vectors in 3 {\displaystyle 3} dimensions with positive-definite quadratic form is closely related to their exterior product. Most instances of geometric al... | Geometric algebra | 0.817466 |
11,983 | Geometric Multiplication of Vectors: An Introduction to Geometric Algebra in Physics. Springer International Publishing;Birkhäuser. p. | Geometric algebra | 0.817466 |
11,984 | A Geometric Algebra Invitation to Space-Time Physics, Robotics and Molecular Geometry. Springer. pp. | Geometric algebra | 0.817466 |
11,985 | Computer Vision, Graphics and Neurocomputing. Geometric Algebra Applications. Vol. | Geometric algebra | 0.817466 |
11,986 | (2000), "A Unified Mathematical Language for Physics and Engineering in the 21st Century" (PDF), Philosophical Transactions of the Royal Society A, 358 (1765): 21–39, Bibcode:2000RSPTA.358...21L, doi:10.1098/rsta.2000.0517, S2CID 91884543, archived (PDF) from the original on 2015-03-19 Baylis, W. E. (2002), Electrodyna... | Geometric algebra | 0.817466 |
11,987 | ; Rodríguez, M.A. (1997), "Clifford Algebras and Geometric Algebra", Advances in Applied Clifford Algebras, 7 (2): 91–102, doi:10.1007/BF03041220, S2CID 120860757 Hestenes, David (1999), New Foundations for Classical Mechanics (2nd ed. ), Springer Verlag, ISBN 978-0-7923-5302-7 Lasenby, Joan; Lasenby, Anthony N.; Doran... | Geometric algebra | 0.817466 |
11,988 | 2, Springer, ISBN 3-540-57063-2 Doran, Chris J. L. (1994), Geometric Algebra and its Application to Mathematical Physics (PhD thesis), University of Cambridge, doi:10.17863/CAM.16148, hdl:1810/251691, OCLC 53604228 Baylis, W. E., ed. (2011) , Clifford (Geometric) Algebra with Applications to Physics, Mathematics, and E... | Geometric algebra | 0.817466 |
11,989 | 29, pp. 2338–2345 Wilmot, G.P. (1988b), "Clifford algebra and the Pfaffian expansion", Journal of Mathematical Physics, 29: 2346–2350 Chevalley, Claude (1991), The Algebraic Theory of Spinors and Clifford Algebras, Collected Works, vol. | Geometric algebra | 0.817466 |
11,990 | ), "A Unified Language for Mathematics and Physics", Clifford Algebras and Their Applications in Mathematical Physics, NATO ASI Series (Series C), Springer, 183: 1–23, doi:10.1007/978-94-009-4728-3_1, ISBN 978-94-009-4728-3 Wilmot, G.P. (1988a), The Structure of Clifford algebra. Journal of Mathematical Physics, vol. | Geometric algebra | 0.817466 |
11,991 | Algèbre, Hermann, ISBN 9782225655166 Hestenes, David; Sobczyk, Garret (1984), Clifford Algebra to Geometric Calculus, a Unified Language for Mathematics and Physics, Springer Netherlands, ISBN 9789027716736 Hestenes, David (1986), J.S.R. Chisholm; A.K. Commons (eds. | Geometric algebra | 0.817466 |
11,992 | doi:10.2307/2369379. JSTOR 2369379. Artin, Emil (1988) , Geometric algebra, Wiley Classics Library, Wiley, doi:10.1002/9781118164518, ISBN 978-0-471-60839-4, MR 1009557 Hestenes, David (1966), Space–time Algebra, Gordon and Breach, ISBN 978-0-677-01390-9, OCLC 996371 Wheeler, J. A.; Misner, C.; Thorne, K. S. | Geometric algebra | 0.817466 |
11,993 | Arranged chronologicallyGrassmann, Hermann (1844), Die lineale Ausdehnungslehre ein neuer Zweig der Mathematik: dargestellt und durch Anwendungen auf die übrigen Zweige der Mathematik, wie auch auf die Statik, Mechanik, die Lehre vom Magnetismus und die Krystallonomie erläutert, Leipzig: O. Wigand, OCLC 20521674 Cliffo... | Geometric algebra | 0.817466 |
11,994 | In more detail, there have been three approaches to geometric algebra: quaternionic analysis, initiated by Hamilton in 1843 and geometrized as rotors by Clifford in 1878; geometric algebra, initiated by Grassmann in 1844; and vector analysis, developed out of quaternionic analysis in the late 19th century by Gibbs and ... | Geometric algebra | 0.817466 |
11,995 | Vector analysis was motivated by James Clerk Maxwell's studies of electromagnetism, and specifically the need to express and manipulate conveniently certain differential equations. Vector analysis had a certain intuitive appeal compared to the rigors of the new algebras. Physicists and mathematicians alike readily adop... | Geometric algebra | 0.817466 |
11,996 | Subsequently, Rudolf Lipschitz in 1886 generalized Clifford's interpretation of the quaternions and applied them to the geometry of rotations in n {\displaystyle n} dimensions. Later these developments would lead other 20th-century mathematicians to formalize and explore the properties of the Clifford algebra. Neverthe... | Geometric algebra | 0.817466 |
11,997 | Following Grassmann, in 1878 William Kingdon Clifford examined Grassmann's algebraic system alongside the quaternions of William Rowan Hamilton in (Clifford 1878). From his point of view, the quaternions described certain transformations (which he called rotors), whereas Grassmann's algebra described certain properties... | Geometric algebra | 0.817466 |
11,998 | Although the connection of geometry with algebra dates as far back at least to Euclid's Elements in the third century B.C. (see Greek geometric algebra), GA in the sense used in this article was not developed until 1844, when it was used in a systematic way to describe the geometrical properties and transformations of ... | Geometric algebra | 0.817466 |
11,999 | A general linear transformation from vectors to vectors is of interest. With the natural restriction to preserving the induced exterior algebra, the outermorphism of the linear transformation is the unique extension of the versor. If f {\displaystyle f} is a linear function that maps vectors to vectors, then its outerm... | Geometric algebra | 0.817466 |
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