text stringlengths 559 401k | source stringlengths 13 121 |
|---|---|
In thermodynamics, a critical point (or critical state) is the end point of a phase equilibrium curve. One example is the liquid–vapor critical point, the end point of the pressure–temperature curve that designates conditions under which a liquid and its vapor can coexist. At higher temperatures, the gas comes into a s... | Wikipedia/Critical_point_(thermodynamics) |
In signal processing and electronics, the frequency response of a system is the quantitative measure of the magnitude and phase of the output as a function of input frequency. The frequency response is widely used in the design and analysis of systems, such as audio and control systems, where they simplify mathematical... | Wikipedia/Response_function |
The Hubbard model is an approximate model used to describe the transition between conducting and insulating systems. It is particularly useful in solid-state physics. The model is named for John Hubbard.
The Hubbard model states that each electron experiences competing forces: one pushes it to tunnel to neighboring ato... | Wikipedia/Hubbard_model |
In physics and probability theory, Mean-field theory (MFT) or Self-consistent field theory studies the behavior of high-dimensional random (stochastic) models by studying a simpler model that approximates the original by averaging over degrees of freedom (the number of values in the final calculation of a statistic tha... | Wikipedia/Mean-field_theory |
In engineering, physics, and chemistry, the study of transport phenomena concerns the exchange of mass, energy, charge, momentum and angular momentum between observed and studied systems. While it draws from fields as diverse as continuum mechanics and thermodynamics, it places a heavy emphasis on the commonalities bet... | Wikipedia/Transport_theory_(statistical_physics) |
The Annual Review of Materials Research is a peer-reviewed journal that publishes review articles about materials science. It has been published by the nonprofit Annual Reviews since 1971, when it was first released under the title the Annual Review of Materials Science. Four people have served as editors, with the cur... | Wikipedia/Annual_Review_of_Materials_Science |
In solid-state physics, the nearly free electron model (or NFE model and quasi-free electron model) is a quantum mechanical model of physical properties of electrons that can move almost freely through the crystal lattice of a solid. The model is closely related to the more conceptual empty lattice approximation. The m... | Wikipedia/Nearly_free_electron_model |
In the field of optics, transparency (also called pellucidity or diaphaneity) is the physical property of allowing light to pass through the material without appreciable scattering of light. On a macroscopic scale (one in which the dimensions are much larger than the wavelengths of the photons in question), the photons... | Wikipedia/Transparent_materials |
In solid-state physics, the electronic band structure (or simply band structure) of a solid describes the range of energy levels that electrons may have within it, as well as the ranges of energy that they may not have (called band gaps or forbidden bands).
Band theory derives these bands and band gaps by examining the... | Wikipedia/Band_theory |
The Thomas–Fermi (TF) model, named after Llewellyn Thomas and Enrico Fermi, is a quantum mechanical theory for the electronic structure of many-body systems developed semiclassically shortly after the introduction of the Schrödinger equation. It stands separate from wave function theory as being formulated in terms of ... | Wikipedia/Thomas–Fermi_model |
Introduction to Solid State Physics, known colloquially as Kittel, is a classic condensed matter physics textbook written by American physicist Charles Kittel in 1953. The book has been highly influential and has seen widespread adoption; Marvin L. Cohen remarked in 2019 that Kittel's content choices in the original ed... | Wikipedia/Introduction_to_Solid_State_Physics |
The Drude model of electrical conduction was proposed in 1900 by Paul Drude to explain the transport properties of electrons in materials (especially metals). Basically, Ohm's law was well established and stated that the current J and voltage V driving the current are related to the resistance R of the material. The in... | Wikipedia/Drude_model |
Corneal topography, also known as photokeratoscopy or videokeratography, is a non-invasive medical imaging technique for mapping the anterior curvature of the cornea, the outer structure of the eye. Since the cornea is normally responsible for some 70% of the eye's refractive power, its topography is of critical import... | Wikipedia/Corneal_topography |
Areography, also known as the geography of Mars, is a subfield of planetary science that entails the delineation and characterization of regions on Mars. Areography is mainly focused on what is called physical geography on Earth; that is the distribution of physical features across Mars and their cartographic represent... | Wikipedia/Topography_of_Mars |
In computer graphics, a triangulated irregular network (TIN) is a representation of a continuous surface consisting entirely of triangular facets (a triangle mesh), used mainly as Discrete Global Grid in primary elevation modeling.
The vertices of these triangles are created from field recorded spot elevations through ... | Wikipedia/Triangulated_irregular_network |
Environmental science is an interdisciplinary academic field that integrates physics, biology, meteorology, mathematics and geography (including ecology, chemistry, plant science, zoology, mineralogy, oceanography, limnology, soil science, geology and physical geography, and atmospheric science) to the study of the env... | Wikipedia/Environmental_science |
Terrain cartography or relief mapping is the depiction of the shape of the surface of the Earth on a map, using one or more of several techniques that have been developed. Terrain or relief is an essential aspect of physical geography, and as such its portrayal presents a central problem in cartographic design, and mor... | Wikipedia/Cartographic_relief_depiction |
In modern mapping, a topographic map or topographic sheet is a type of map characterized by large-scale detail and quantitative representation of relief features, usually using contour lines (connecting points of equal elevation), but historically using a variety of methods. Traditional definitions require a topographi... | Wikipedia/Topographic_map |
The U.S. Army Corps of Topographical Engineers was a branch of the United States Army authorized on 4 July 1838. It consisted only of officers who were handpicked from West Point and was used for mapping and the design and construction of federal civil works such as lighthouses and other coastal fortifications and navi... | Wikipedia/Corps_of_Topographical_Engineers |
Surveying or land surveying is the technique, profession, art, and science of determining the terrestrial two-dimensional or three-dimensional positions of points and the distances and angles between them. These points are usually on the surface of the Earth, and they are often used to establish maps and boundaries for... | Wikipedia/Topographical_surveys |
A satellite navigation or satnav system is a system that uses satellites to provide autonomous geopositioning. A satellite navigation system with global coverage is termed global navigation satellite system (GNSS). As of 2024, four global systems are operational: the United States's Global Positioning System (GPS), Rus... | Wikipedia/Global_navigation_satellite_systems |
A global relief model, sometimes also denoted as global topography model or composite model, combines digital elevation model (DEM) data over land with digital bathymetry model (DBM) data over water-covered areas (oceans, lakes) to describe Earth's relief. A relief model thus shows how Earth's surface would look like i... | Wikipedia/Global_Relief_Model |
Photography is the art, application, and practice of creating images by recording light, either electronically by means of an image sensor, or chemically by means of a light-sensitive material such as photographic film. It is employed in many fields of science, manufacturing (e.g., photolithography), and business, as w... | Wikipedia/Photographic |
Orography is the study of the topographic relief of mountains, and can more broadly include hills, and any part of a region's elevated terrain. Orography (also known as oreography, orology, or oreology) falls within the broader discipline of geomorphology. The term orography comes from the Greek: όρος, hill, γράφω, to ... | Wikipedia/Orography |
A digital elevation model (DEM) or digital surface model (DSM) is a 3D computer graphics representation of elevation data to represent terrain or overlaying objects, commonly of a planet, moon, or asteroid. A "global DEM" refers to a discrete global grid. DEMs are used often in geographic information systems (GIS), an... | Wikipedia/Digital_Elevation_Model |
The Ordnance Survey (OS) is the national mapping agency for Great Britain. The agency's name indicates its original military purpose (see ordnance and surveying), which was to map Scotland in the wake of the Jacobite rising of 1745. There was also a more general and nationwide need in light of the potential threat of i... | Wikipedia/Ordnance_Survey |
Ocean surface topography or sea surface topography, also called ocean dynamic topography, are highs and lows on the ocean surface, similar to the hills and valleys of Earth's land surface depicted on a topographic map.
These variations are expressed in terms of average sea surface height (SSH) relative to Earth's geoi... | Wikipedia/Sea-surface_topography |
Nanotopography refers to specific surface features which form or are generated at the nanoscopic scale. While the term can be used to describe a broad range of applications ranging from integrated circuits to microfluidics, in practice it typically applied to sub-micron textured surfaces as used in biomaterials resear... | Wikipedia/Nanotopography |
Earth science or geoscience includes all fields of natural science related to the planet Earth. This is a branch of science dealing with the physical, chemical, and biological complex constitutions and synergistic linkages of Earth's four spheres: the biosphere, hydrosphere/cryosphere, atmosphere, and geosphere (or lit... | Wikipedia/Geoscience |
A digital elevation model (DEM) or digital surface model (DSM) is a 3D computer graphics representation of elevation data to represent terrain or overlaying objects, commonly of a planet, moon, or asteroid. A "global DEM" refers to a discrete global grid. DEMs are used often in geographic information systems (GIS), an... | Wikipedia/Digital_elevation_model |
The seabed (also known as the seafloor, sea floor, ocean floor, and ocean bottom) is the bottom of the ocean. All floors of the ocean are known as seabeds.
The structure of the seabed of the global ocean is governed by plate tectonics. Most of the ocean is very deep, where the seabed is known as the abyssal plain. Seaf... | Wikipedia/Marine_topography |
Hypsometry (from Ancient Greek ὕψος (húpsos) 'height' and μέτρον (métron) 'measure') is the measurement of the elevation and depth of features of Earth's surface relative to mean sea level.
On Earth, the elevations can take on either positive or negative (below sea level) values. The distribution is theorised to be b... | Wikipedia/Hypsography |
Tomography is imaging by sections or sectioning that uses any kind of penetrating wave. The method is used in radiology, archaeology, biology, atmospheric science, geophysics, oceanography, plasma physics, materials science, cosmochemistry, astrophysics, quantum information, and other areas of science. The word tomogra... | Wikipedia/Tomography |
Atomic force microscopy (AFM) or scanning force microscopy (SFM) is a very-high-resolution type of scanning probe microscopy (SPM), with demonstrated resolution on the order of fractions of a nanometer, more than 1000 times better than the optical diffraction limit.
== Overview ==
Atomic force microscopy (AFM) gathe... | Wikipedia/Atomic_force_microscopy |
The term conceptual model refers to any model that is formed after a conceptualization or generalization process. Conceptual models are often abstractions of things in the real world, whether physical or social. Semantic studies are relevant to various stages of concept formation. Semantics is fundamentally a study of ... | Wikipedia/Model_(abstract) |
Topography may refer to:
== Cartography, geology and oceanography ==
Topography, the study of the current terrain features of a region and the graphic representation of the landform on a map
Inverted topography, landscape features that have reversed their elevation relative to other features
Karst topography, a lands... | Wikipedia/Topography_(disambiguation) |
A fall line refers to the line down a mountain or hill which is most directly downhill; that is, the direction a ball or other body would accelerate if it were free to move on the slope under gravity. Mathematically the fall line, the line of greatest slope, is the negative of the gradient (which points uphill) and per... | Wikipedia/Fall_line_(topography) |
In mathematics, an open set is a generalization of an open interval in the real line.
In a metric space (a set with a distance defined between every two points), an open set is a set that, with every point P in it, contains all points of the metric space that are sufficiently near to P (that is, all points whose distan... | Wikipedia/Open_(topology) |
In theoretical physics, geometrodynamics is an attempt to describe spacetime and associated phenomena completely in terms of geometry. Technically, its goal is to unify the fundamental forces and reformulate general relativity as a configuration space of three-metrics, modulo three-dimensional diffeomorphisms. The ori... | Wikipedia/Geometrodynamics |
In mathematics, a topological space is called separable if it contains a countable, dense subset; that is, there exists a sequence
(
x
n
)
n
=
1
... | Wikipedia/Separable_(topology) |
The topological entanglement entropy or topological entropy, usually denoted by
γ
{\displaystyle \gamma }
, is a number characterizing many-body states that possess topological order.
A non-zero topological entanglement entropy reflects the presence of long range quantum entan... | Wikipedia/Topological_entropy_in_physics |
In mathematics, especially in the fields of representation theory and module theory, a Frobenius algebra is a finite-dimensional unital associative algebra with a special kind of bilinear form which gives the algebras particularly nice duality theories. Frobenius algebras began to be studied in the 1930s by Richard Br... | Wikipedia/Frobenius_algebra |
In gauge theory, topological Yang–Mills theory, also known as the theta term or
θ
{\displaystyle \theta }
-term is a gauge-invariant term which can be added to the action for four-dimensional field theories, first introduced by Edward Witten. It does not change the classical ... | Wikipedia/Topological_Yang–Mills_theory |
A conformal field theory (CFT) is a quantum field theory that is invariant under conformal transformations. In two dimensions, there is an infinite-dimensional algebra of local conformal transformations, and conformal field theories can sometimes be exactly solved or classified.
Conformal field theory has important app... | Wikipedia/Conformal_Field_Theory |
In topology and related branches of mathematics, separated sets are pairs of subsets of a given topological space that are related to each other in a certain way: roughly speaking, neither overlapping nor touching. The notion of when two sets are separated or not is important both to the notion of connected spaces (an... | Wikipedia/Separated_by_neighbourhoods |
In topology and related branches of mathematics, separated sets are pairs of subsets of a given topological space that are related to each other in a certain way: roughly speaking, neither overlapping nor touching. The notion of when two sets are separated or not is important both to the notion of connected spaces (an... | Wikipedia/Separated_by_a_function |
In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space. It is closely related to the concepts of open set and interior. Intuitively speaking, a neighbourhood of a point is a set of points containing that point where one can move some amount in... | Wikipedia/Neighbourhood_(topology) |
In topology and related branches of mathematics, separated sets are pairs of subsets of a given topological space that are related to each other in a certain way: roughly speaking, neither overlapping nor touching. The notion of when two sets are separated or not is important both to the notion of connected spaces (an... | Wikipedia/Precisely_separated_by_a_function |
In non-technical terms, M-theory presents an idea about the basic substance of the universe. Although a complete mathematical formulation of M-theory is not known, the general approach is the leading contender for a universal "Theory of Everything" that unifies gravity with other forces such as electromagnetism. M-theo... | Wikipedia/Introduction_to_M-theory |
In physics, the fundamental interactions or fundamental forces are interactions in nature that appear not to be reducible to more basic interactions. There are four fundamental interactions known to exist:
gravity
electromagnetism
weak interaction
strong interaction
The gravitational and electromagnetic interactions ... | Wikipedia/Fundamental_physics |
In physics, the term swampland refers to effective low-energy physical theories which are not compatible with quantum gravity. This is in contrast with the so-called "string theory landscape" that are known to be compatible with string theory, which is hypothesized to be a consistent quantum theory of gravity. In other... | Wikipedia/Swampland_(physics) |
The Trouble with Physics: The Rise of String Theory, the Fall of a Science, and What Comes Next is a 2006 book by the theoretical physicist Lee Smolin about the problems with string theory. The book strongly criticizes string theory and its prominence in contemporary theoretical physics, on the grounds that string theo... | Wikipedia/The_Trouble_With_Physics |
In string theory, the string theory landscape (or landscape of vacua) is the collection of possible false vacua, together comprising a collective "landscape" of choices of parameters governing compactifications.
The term "landscape" comes from the notion of a fitness landscape in evolutionary biology. It was first app... | Wikipedia/String_theory_landscape |
In theoretical physics, compactification means changing a theory with respect to one of its space-time dimensions. Instead of having a theory with this dimension being infinite, one changes the theory so that this dimension has a finite length, and may also be periodic.
Compactification plays an important part in therm... | Wikipedia/Compactification_(physics) |
In algebraic geometry, a complex algebraic variety is an algebraic variety (in the scheme sense or otherwise) over the field of complex numbers.
== Chow's theorem ==
Chow's theorem states that a projective complex analytic variety, i.e., a closed analytic subvariety of the complex projective space
... | Wikipedia/Complex_algebraic_variety |
In mathematics, a vertex operator algebra (VOA) is an algebraic structure that plays an important role in two-dimensional conformal field theory and string theory. In addition to physical applications, vertex operator algebras have proven useful in purely mathematical contexts such as monstrous moonshine and the geome... | Wikipedia/Vertex_algebra |
In physics, matrix string theory is a set of equations that describe superstring theory in a non-perturbative framework. Type IIA string theory can be shown to be equivalent to a maximally supersymmetric two-dimensional gauge theory, the gauge group of which is U(N) for a large value of N. This matrix string theory was... | Wikipedia/Matrix_string_theory |
In theoretical physics, the anti-de Sitter/conformal field theory correspondence (frequently abbreviated as AdS/CFT) is a conjectured relationship between two kinds of physical theories. On one side are anti-de Sitter spaces (AdS) that are used in theories of quantum gravity, formulated in terms of string theory or M-t... | Wikipedia/Anti-de_Sitter/conformal_field_theory_correspondence |
In the general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of matter within it.
The equations were published by Albert Einstein in 1915 in the form of a tensor equation which related the local spacetime curvature (expr... | Wikipedia/Einstein's_equation |
The SYZ conjecture is an attempt to understand the mirror symmetry conjecture, an issue in theoretical physics and mathematics. The original conjecture was proposed in a paper by Strominger, Yau, and Zaslow, entitled "Mirror Symmetry is T-duality".
Along with the homological mirror symmetry conjecture, it is one of th... | Wikipedia/SYZ_conjecture |
In theoretical physics, p-form electrodynamics is a generalization of Maxwell's theory of electromagnetism.
== Ordinary (via. one-form) Abelian electrodynamics ==
We have a 1-form
A
{\displaystyle \mathbf {A} }
, a gauge symmetry
... | Wikipedia/P-form_electrodynamics |
The Trouble with Physics: The Rise of String Theory, the Fall of a Science, and What Comes Next is a 2006 book by the theoretical physicist Lee Smolin about the problems with string theory. The book strongly criticizes string theory and its prominence in contemporary theoretical physics, on the grounds that string theo... | Wikipedia/The_Trouble_with_Physics:_The_Rise_of_String_Theory,_the_Fall_of_a_Science,_and_What_Comes_Next |
Twistor string theory is an equivalence between N = 4 supersymmetric Yang–Mills theory and the perturbative topological B model string theory in twistor space.
It was initially proposed by Edward Witten in 2003.
Twistor theory was introduced by Roger Penrose from the 1960s as a new approach to the unification of quantu... | Wikipedia/Twistor_string_theory |
In mathematics, a modular form is a holomorphic function on the complex upper half-plane,
H
{\displaystyle {\mathcal {H}}}
, that roughly satisfies a functional equation with respect to the group action of the modular group and a gro... | Wikipedia/Modular_function |
The term "bootstrap model" is used for a class of theories that use very general consistency criteria to determine the form of a quantum theory from some assumptions on the spectrum of particles. It is a form of S-matrix theory.
== Overview ==
In the 1960s and '70s, the ever-growing list of strongly interacting parti... | Wikipedia/Bootstrap_model |
In string theory, a heterotic string is a closed string (or loop) which is a hybrid ('heterotic') of a superstring and a bosonic string. There are two kinds of heterotic superstring theories, the heterotic SO(32) and the heterotic E8 × E8, abbreviated to HO and HE. Apart from that there exist seven more heterotic strin... | Wikipedia/Heterotic_string_theory |
In theoretical physics, F-theory is a branch of string theory developed by Iranian-American physicist Cumrun Vafa. The new vacua described by F-theory were discovered by Vafa and allowed string theorists to construct new realistic vacua — in the form of F-theory compactified on elliptically fibered Calabi–Yau four-fold... | Wikipedia/F-theory |
A chiral phenomenon is one that is not identical to its mirror image (see the article on mathematical chirality). The spin of a particle may be used to define a handedness, or helicity, for that particle, which, in the case of a massless particle, is the same as chirality. A symmetry transformation between the two is ... | Wikipedia/Chirality_(physics) |
String theory is a branch of theoretical physics.
String theory may also refer to:
Concatenation theory, a topic in symbolic logic dealing with strings of characters
Music
String Theory (band), an American electronic music band
String Theory (Hanson album), 2018
String Theory (The Selecter album), 2013
Other media
"... | Wikipedia/String_theory_(disambiguation) |
In general relativity, a vacuum solution is a Lorentzian manifold whose Einstein tensor vanishes identically. According to the Einstein field equation, this means that the stress–energy tensor also vanishes identically, so that no matter or non-gravitational fields are present. These are distinct from the electrovacuu... | Wikipedia/Vacuum_solution |
The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory is a book by Brian Greene published in 1999, which introduces string and superstring theory, and provides a comprehensive though non-technical assessment of the theory and some of its shortcomings. In 2000, it won the Royal Soc... | Wikipedia/The_Elegant_Universe:_Superstrings,_Hidden_Dimensions,_and_the_Quest_for_the_Ultimate_Theory |
The non-critical string theory describes the relativistic string without enforcing the critical dimension. Although this allows the construction of a string theory in 4 spacetime dimensions, such a theory usually does not describe a Lorentz invariant background. However, there are recent developments which make possibl... | Wikipedia/Non-critical_string_theory |
In theoretical physics, the superconformal algebra is a graded Lie algebra or superalgebra that combines the conformal algebra and supersymmetry. In two dimensions, the superconformal algebra is infinite-dimensional. In higher dimensions, superconformal algebras are finite-dimensional and generate the superconformal gr... | Wikipedia/Superconformal_algebra |
In theoretical physics, Nordström's theory of gravitation was a predecessor of general relativity. Strictly speaking, there were actually two distinct theories proposed by the Finnish theoretical physicist Gunnar Nordström, in 1912 and 1913, respectively. The first was quickly dismissed, but the second became the first... | Wikipedia/Nordström's_theory_of_gravitation |
Holography is a technique that enables a wavefront to be recorded and later reconstructed. It is best known as a method of generating three-dimensional images, and has a wide range of other uses, including data storage, microscopy, and interferometry. In principle, it is possible to make a hologram for any type of wave... | Wikipedia/Holography |
String field theory (SFT) is a formalism in string theory in which the dynamics of relativistic strings is reformulated in the language of quantum field theory. This is accomplished at the level of perturbation theory by finding a collection of vertices for joining and splitting strings, as well as string propagators,... | Wikipedia/String_field_theory |
In geometry, the Poincaré disk model, also called the conformal disk model, is a model of 2-dimensional hyperbolic geometry in which all points are inside the unit disk, and straight lines are either circular arcs contained within the disk that are orthogonal to the unit circle or diameters of the unit circle.
The grou... | Wikipedia/Poincaré_disk_model |
In theoretical physics, type I string theory is one of five consistent supersymmetric string theories in ten dimensions. It is the only one whose strings are unoriented (both orientations of a string are equivalent) and the only one which perturbatively contains not only closed strings, but also open strings. The termi... | Wikipedia/Type_I_string_theory |
The mathematical formulations of quantum mechanics are those mathematical formalisms that permit a rigorous description of quantum mechanics. This mathematical formalism uses mainly a part of functional analysis, especially Hilbert spaces, which are a kind of linear space. Such are distinguished from mathematical forma... | Wikipedia/Postulates_of_quantum_mechanics |
In theoretical physics, the matrix theory is a quantum mechanical model proposed in 1997 by Tom Banks, Willy Fischler, Stephen Shenker, and Leonard Susskind; it is also known as BFSS matrix model, after the authors' initials.
== Overview ==
This theory describes the behavior of a set of nine large matrices. In their ... | Wikipedia/Matrix_theory_(physics) |
The circuit topology of an electronic circuit is the form taken by the network of interconnections of the circuit components. Different specific values or ratings of the components are regarded as being the same topology. Topology is not concerned with the physical layout of components in a circuit, nor with their po... | Wikipedia/Circuit_topology_(electrical) |
In chemistry, topology provides a way of describing and predicting the molecular structure within the constraints of three-dimensional (3-D) space. Given the determinants of chemical bonding and the chemical properties of the atoms, topology provides a model for explaining how the atoms ethereal wave functions must fi... | Wikipedia/Topology_(chemistry) |
In category theory, a branch of mathematics, a presheaf on a category
C
{\displaystyle C}
is a functor
F
:
C
o
p
→
... | Wikipedia/Presheaf_(category_theory) |
In mathematics, a Lawvere–Tierney topology is an analog of a Grothendieck topology for an arbitrary topos, used to construct a topos of sheaves. A Lawvere–Tierney topology is also sometimes also called a local operator or coverage or topology or geometric modality. They were introduced by William Lawvere (1971) and My... | Wikipedia/Lawvere–Tierney_topology |
In mathematics, the idea of descent extends the intuitive idea of 'gluing' in topology. Since the topologists' glue is the use of equivalence relations on topological spaces, the theory starts with some ideas on identification.
== Descent of vector bundles ==
The case of the construction of vector bundles from data o... | Wikipedia/Descent_(category_theory) |
In ring theory, a branch of mathematics, a ring is called a reduced ring if it has no non-zero nilpotent elements. Equivalently, a ring is reduced if it has no non-zero elements with square zero, that is, x2 = 0 implies x = 0. A commutative algebra over a commutative ring is called a reduced algebra if its underlying ... | Wikipedia/Reduced_(ring_theory) |
In mathematics, and more particularly in set theory, a cover (or covering) of a set
X
{\displaystyle X}
is a family of subsets of
X
{\displaystyle X}
whose union is all of
X
{\displaystyle X}
... | Wikipedia/Cover_(topology) |
In category theory, a branch of mathematics, a sieve is a way of choosing arrows with a common codomain. It is a categorical analogue of a collection of open subsets of a fixed open set in topology. In a Grothendieck topology, certain sieves become categorical analogues of open covers in topology. Sieves were introdu... | Wikipedia/Sieve_(category_theory) |
In mathematics, the flat topology is a Grothendieck topology used in algebraic geometry. It is used to define the theory of flat cohomology; it also plays a fundamental role in the theory of descent (faithfully flat descent). The term flat here comes from flat modules.
There are several slightly different flat topologi... | Wikipedia/Flat_topology |
In algebraic geometry, the h topology is a Grothendieck topology introduced by Vladimir Voevodsky to study the homology of schemes. It combines several good properties possessed by its related "sub"topologies, such as the qfh and cdh topologies. It has subsequently been used by Beilinson to study p-adic Hodge theory, ... | Wikipedia/Cdh_topology |
The Science Citation Index Expanded (SCIE) is a citation index owned by Clarivate and previously by Thomson Reuters.
It was created by the Eugene Garfield at the Institute for Scientific Information, launched in 1964 as Science Citation Index (SCI). It was later distributed via CD/DVD and became available online in 19... | Wikipedia/Science_Citation_Index |
Geometry (from Ancient Greek γεωμετρία (geōmetría) 'land measurement'; from γῆ (gê) 'earth, land' and μέτρον (métron) 'a measure') is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest b... | Wikipedia/geometry |
In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point. The tan... | Wikipedia/Derivative_(calculus) |
Quantum information is the information of the state of a quantum system. It is the basic entity of study in quantum information theory, and can be manipulated using quantum information processing techniques. Quantum information refers to both the technical definition in terms of Von Neumann entropy and the general comp... | Wikipedia/Quantum_information_theory |
Bass–Serre theory is a part of the mathematical subject of group theory that deals with analyzing the algebraic structure of groups acting by automorphisms on simplicial trees. The theory relates group actions on trees with decomposing groups as iterated applications of the operations of free product with amalgamation ... | Wikipedia/Bass–Serre_theory |
Forced perspective is a technique that employs optical illusion to make an object appear farther away, closer, larger or smaller than it actually is. It manipulates human visual perception through the use of scaled objects and the correlation between them and the vantage point of the spectator or camera. It has uses in... | Wikipedia/Forced_perspective |
In the mathematical subject of group theory, small cancellation theory studies groups given by group presentations satisfying small cancellation conditions, that is where defining relations have "small overlaps" with each other. Small cancellation conditions imply algebraic, geometric and algorithmic properties of the ... | Wikipedia/Small_cancellation_theory |
In the mathematical subject of geometric group theory, a Dehn function, named after Max Dehn, is an optimal function associated to a finite group presentation which bounds the area of a relation in that group (that is a freely reduced word in the generators representing the identity element of the group) in terms of th... | Wikipedia/Dehn_function |
In topology, an area of mathematics, the virtually Haken conjecture states that every compact, orientable, irreducible three-dimensional manifold with infinite fundamental group is virtually Haken. That is, it has a finite cover (a covering space with a finite-to-one covering map) that is a Haken manifold.
After the pr... | Wikipedia/Virtually_Haken_conjecture |
In algebra, a finitely generated group is a group G that has some finite generating set S so that every element of G can be written as the combination (under the group operation) of finitely many elements of S and of inverses of such elements.
By definition, every finite group is finitely generated, since S can be ta... | Wikipedia/Finitely_generated_group |
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