# kb

id: sense-of-place-20260505-0
file: data/en.wikipedia.org/wiki/Sense_of_place-0.md
category: reference
title: Sense of place
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Sense_of_place
date_saved: 2026-05-05T07:21:24.143517+00:00
instance: kb-cron
parent: sense-of-place-20260505-0
summary:
The term sense of place refers to a multidimensional, complex construct used to characterize the relationship between people and spatial settings. It is a characteristic that some geographic places have and some do not, while to others it is a feeling or perception held by people (not by the place itself). It is often used in relation to those characteristics that make a place special or unique, as well as to those that foster a sense of authentic human attachment and belonging. Others, such as geographer Yi-Fu Tuan, have pointed to senses of place that are not "positive," such as fear. Some students and educators engage in "place-based education" in order to improve their "sense(s) of place," as well as to use various aspects of place as educational tools in general. The term is used in urban and rural studies in relation to place-making and place-attachment of communities to their environment or homeland. The term sense of place is used to describe how someone perceives and experiences a place or environment. Anthropologists Steven Feld and Keith Basso define sense of place as: 'the experiential and expressive ways places are known, imagined, yearned for, held, remembered, voiced, lived, contested and struggled


id: sense-of-place-20260505-1
file: data/en.wikipedia.org/wiki/Sense_of_place-1.md
category: reference
title: Sense of place
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Sense_of_place
date_saved: 2026-05-05T07:21:24.143517+00:00
instance: kb-cron
parent: sense-of-place-20260505-0
summary:
== Music and place == Ethnomusicologists, among other social scientists (like anthropologists, sociologists, and urban geographers), have begun to point toward music’s role in defining people’s “sense of place.” British ethnomusicologist Martin Stokes suggests that humans can construct an idea of “place” through music that signals their position in the world in terms of social boundaries and moral and political hierarchies. Stokes argues that music does not simply serve as a reflection of existing social structures, but yields the potential to actively transform a given space. Music denoting place can “preform” a knowledge of social boundaries and hierarchies that people use to negotiate and understand the identities of themselves and others and their relation to place. Examples of music’s role in defining a sense of place include ethnomusicologist George Lipsitz’s research on the performance of Mexican-American cultural identity in Los Angeles. In response to mechanical reproduction and increasingly commodified forms of culture, Walter Benjamin once argued that cultural objects have become increasingly removed from their original context and place of creation. In this context, ethnomusicologist George Lipsitz suggests that a consciousness of invisibility and alienation marks the cultural identity of minority groups excluded from political power and cultural recognition. Lipsitz


id: side-valley-20260505-0
file: data/en.wikipedia.org/wiki/Side_valley-0.md
category: reference
title: Side valley
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Side_valley
date_saved: 2026-05-05T07:21:25.338337+00:00
instance: kb-cron
parent: side-valley-20260505-0
summary:
Side valleys and tributary valleys are valleys whose brooks or rivers flow into greater ones. Upstream, the valleys can be classified in an increasing order which is equivalent to the usual orographic order: the tributaries are ordered from those nearest to the source of the river to those nearest to the mouth of the river. A confluence is where two or more tributaries or rivers flow together. == Orographic order (e.g. Humber) == In the orographic classification (order of rivers) the tributary river has order n+1, if n describes the primary (or main) river. A river which flows directly into the ocean (e.g. the English rivers Thames or Humber) has the orographic order n=1, the River Ouse n=2, the Wharfe n=3 and so on. == Geomorphology == The term "side valley" is used for higher order valleys near mountains (example above: the Pennines), as opposed to lower valleys that do not have a strong relief. This is because the "main stem river" (into which the secondary river flows) passes much more water than its tributaries and therefore has a lower current. has finer sediments that are deposited in a more flat manner. The higher the order of a valley, the


id: singular-population-entity-20260505-0
file: data/en.wikipedia.org/wiki/Singular_population_entity-0.md
category: reference
title: Singular population entity
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Singular_population_entity
date_saved: 2026-05-05T07:21:26.566087+00:00
instance: kb-cron
parent: singular-population-entity-20260505-0
summary:
From a statistical point of view, in Spain, a singular population entity (in Spanish: entidad singular de población) is defined as "any inhabited area within a municipality, inhabited or exceptionally uninhabited, clearly differentiated within that municipality and that is known by a specific name that identifies it without the possibility of confusion", according to the definition of the National Statistics Institute. == Population entities == The gazetteer published by the INE, usually on the dates when the population census was carried out, distinguishes, within each municipality, the following population entities, each of which has a delimited territorial scope. Collective population entity (in Spanish: Entidad colectiva de población): a grouping of two or more singular population entities. Singular population entity (in Spanish: Entidad singular de población): any inhabited area of a municipality, inhabited or exceptionally uninhabited, clearly differentiated within the municipality and known by a specific denomination that identifies it without the possibility of confusion. Population nucleus (in Spanish: Núcleo de población): is a group of at least ten buildings that form streets, squares and other urban roads. By exception, the number of buildings may be less than 10, as long as the population that inhabits them exceeds 50 inhabitants. The


id: small-state-20260505-0
file: data/en.wikipedia.org/wiki/Small_state-0.md
category: reference
title: Small state
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Small_state
date_saved: 2026-05-05T07:21:27.863702+00:00
instance: kb-cron
parent: small-state-20260505-0
summary:
Small state is a generic term for a small country, that has limited land, population, or resources. The term "small state" is similar to the term microstate or ministate, a sovereign state having a very small population or land area, usually both. However, the meanings of "state" and "very small" are not well-defined in international law. Two important definitions for a small state are World Bank, which defines small state as those with less than 1.5 million people, which is about 40-50 countries, and the U.N. Forum on Small States which has higher threshold and includes about 100 countries, most of which are under 5 million people. The term small state is a usually inclusive to microstates; while microstates can be considered small states, not all small states are microstates. There is different criteria for defining small states. A small state may be considered a small power if they are able to project political power regionally. == Definitions of a Small State == Metrics for defining a small state look at aspects of country including area, population, size of the government, territory, and political situation. One of the most common metrics to look at is population, which constraints other aspects,


id: spatial-scale-20260505-0
file: data/en.wikipedia.org/wiki/Spatial_scale-0.md
category: reference
title: Spatial scale
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Spatial_scale
date_saved: 2026-05-05T07:21:29.115608+00:00
instance: kb-cron
parent: spatial-scale-20260505-0
summary:
Spatial scale is a specific application of the term scale for describing or categorizing (e.g. into orders of magnitude) the size of a space (hence spatial), or the extent of it at which a phenomenon or process occurs. For instance, in physics an object or phenomenon can be called microscopic if too small to be visible. In climatology, a micro-climate is a climate which might occur in a mountain, valley or near a lake shore. In statistics, a megatrend is a political, social, economical, environmental or technological trend which involves the whole planet or is supposed to last a very large amount of time. The concept is also used in geography, astronomy, and meteorology. These divisions are somewhat arbitrary; where, on this table, mega- is assigned global scope, it may only apply continentally or even regionally in other contexts. The interpretations of meso- and macro- must then be adjusted accordingly. == See also == == References ==


id: spatiomap-20260505-0
file: data/en.wikipedia.org/wiki/Spatiomap-0.md
category: reference
title: Spatiomap
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Spatiomap
date_saved: 2026-05-05T07:21:30.307161+00:00
instance: kb-cron
parent: spatiomap-20260505-0
summary:
A spatiomap is a document similar to a map, but based on an orthophoto. Often, some annotations are added to the orthophoto. Similar to normal maps, can display a north arrow, a scale bar and cartographical information like the used projection. Spatiomaps are useful when other reliable source are missing for a certain area and/or when a map must be produced in very short time (e.g. for disaster management). Spatiomaps are frequently used during disaster relief. An image map or orthophotomap is a similar document, but is mostly regarded as an orthophotomosaic with some points, lines or polygon layers of a traditional map drawn over the orthophoto. An image map resembles a standard general purpose map but adds the use of an orthophotomosaic as a background. == References ==


id: synekism-20260505-0
file: data/en.wikipedia.org/wiki/Synekism-0.md
category: reference
title: Synekism
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Synekism
date_saved: 2026-05-05T07:21:31.623643+00:00
instance: kb-cron
parent: synekism-20260505-0
summary:
Synekism is a concept in urban studies coined by Edward Soja. It refers to the dynamic formation of the polis state — the union of several small urban settlements under the rule of a "capital" city (or so-called city-state or urban system). Soja's definition of synekism, mentioned in Writing the city spatially, is "the stimulus of urban agglomeration." == Social science == From the social sciences' view, it is also a "nucleated and hierarchically nested process of political governance, economic development, social order, and cultural identity" Soja. In densely settled urban places, a critical mass provides potential for innovation that is not typically available in rural environments, therefore synekism can be thought of as the geographical relationships that create and give importance to cities. == Notes == == Further reading == Bell, Thomas L.; Muller, Peter O. (March 2003). "Book Review". Annals of the Association of American Geographers. 93 (1): 248–250. doi:10.1111/1467-8306.93122. S2CID 144744015. (A review of Soja's Postmetropolis: Critical Studies of Cities and Regions, ISBN 1-57718-001-1.) Soja, Edward (2000). Postmetropolis: Critical Studies of Cities and Regions. Oxford: Basil Blackwell. Soja, Edward (November 2003). "Writing the city spatially". City. 7 (3): 269–280. doi:10.1080/1360481032000157478. ISSN 1360-4813. Wikidata Q29997800.


id: tableland-20260505-0
file: data/en.wikipedia.org/wiki/Tableland-0.md
category: reference
title: Tableland
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Tableland
date_saved: 2026-05-05T07:21:32.903996+00:00
instance: kb-cron
parent: tableland-20260505-0
summary:
A tableland is an area containing elevated landforms characterized by a distinct, flat, nearly level, or gently undulating surface. They often exhibit steep, cliff-like edges, known as escarpments, that separate them from surrounding lowlands. Depending on either their size, other physical characteristics, or geographic location, the landforms comprising a tableland are individually referred to by a number of names including butte, mesa, plateau, potrero, tepui, or tuya. A homologous landform under the sea is called either a tablemount or guyot. == Sedimentary tablelands == Sedimentary tablelands are tablelands that typically have developed from the erosion of coarse-grained, clastic, sedimentary rocks in the form of relatively flat-lying sandstones and conglomerates that have not been strongly deformed by tectonics. The primary control on the geomorphology of sedimentary tablelands is the dip of the layers of the sandstones, conglomerates, and associated sedimentary strata. Sedimentary tablelands only form if the dip of the sedimentary layers is negligible. If the sedimentary layers are tilted, although otherwise little deformed, asymmetric ridges known as cuestas develop. A really extensive sedimentary tablelands are often known as plateaus. As plateaus are dissected by the headward erosion and incision of river and stream courses and the retreat of their bounding


id: topocide-20260505-0
file: data/en.wikipedia.org/wiki/Topocide-0.md
category: reference
title: Topocide
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Topocide
date_saved: 2026-05-05T07:21:34.092905+00:00
instance: kb-cron
parent: topocide-20260505-0
summary:
Topocide is the deliberate alteration or destruction of a locale through industrial expansion and change, so that its original landscape and character are destroyed. Topocide can be the result of deliberate industrial expansion, as when industries form, then the people's center of life revolves around that industry. New jobs are formed and the environmental and cultural landscape is forever changed. A related term is domicide (from Latin domus, meaning home or abode, and caedo, meaning deliberate killing) the destruction of home; the two may be viewed as synonyms or they may be opposed, with topocide referring to destruction from the point of view of the perpetrator(s) and domicide from the perspective of those affected. == See also == Domicide == References == == Further reading == Bibliography of genocide studies


id: tourist-city-20260505-0
file: data/en.wikipedia.org/wiki/Tourist_city-0.md
category: reference
title: Tourist city
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Tourist_city
date_saved: 2026-05-05T07:21:35.302388+00:00
instance: kb-cron
parent: tourist-city-20260505-0
summary:
A tourist city is a place where the economy is mainly run by the tourism industry, and a large part of the city consists of areas dedicated to visitors. Some cities, such as New York City, Singapore, and London are not exclusively tourist cities, but do have tourist districts and are well known for their reputation as having a lot of international tourists. A tourist city differs from a regular city in the sense that it is trying to appeal to everyone who is not native to that area. This causes there to be a lot of advertising and attractions to try to draw people to the city. To regulate tourism, tourist cities also establish regions designed specifically for visitors on vacation. One example of this would be Cancún and its isolated Hotel Zone. == Examples of notable tourist cities == Entries are listed in the form of "name of city (country name)." Data from Euromonitor. == See also == List of cities by international visitors - cities ranked by number of visitors in a year Honeypot (tourism) == References ==


id: trailhead-20260505-0
file: data/en.wikipedia.org/wiki/Trailhead-0.md
category: reference
title: Trailhead
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Trailhead
date_saved: 2026-05-05T07:21:36.613774+00:00
instance: kb-cron
parent: trailhead-20260505-0
summary:
A trailhead is the point where a trail begins or is accessed, where the trail is often intended for hiking, biking, horseback riding, or off-road vehicles. Modern trailheads often contain restrooms, maps, signposts, and distribution centers for informational brochures about the trail and its features and parking areas for vehicles and trailers. The United States Access Board defines a trailhead "as an outdoor space that is designated by an entity responsible for administering or maintaining a trail to serve as an access point to the trail." The intersection of two trails is a trail junction and does not constitute a trailhead. Historically, the cities located at the terminus of major pathways for foot traffic, such as the Natchez Trace and the Chisholm Trail, were also known as trailheads. For mountain climbing and hiking, the elevation of the trailhead above sea level is posted to give an idea of how high the mountain is above the average terrain. A trailhead may also feature a trail grade, which determines the walking difficulty of the trail. == See also == Trail blazing == References == == External links == Trailheads, a crowd-sourced database of trailheads in the United States


id: tunnel-valley-20260505-0
file: data/en.wikipedia.org/wiki/Tunnel_valley-0.md
category: reference
title: Tunnel valley
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Tunnel_valley
date_saved: 2026-05-05T07:21:37.968444+00:00
instance: kb-cron
parent: tunnel-valley-20260505-0
summary:
A tunnel valley is a U-shaped valley originally cut under the glacial ice near the margin of continental ice sheets such as that now covering Antarctica and formerly covering portions of all continents during past glacial ages. They can be as long as 100 km (62 mi), 4 km (2.5 mi) wide, and 400 m (1,300 ft) deep. Tunnel valleys were formed by subglacial erosion by water and served as subglacial drainage pathways carrying large volumes of meltwater. Their cross-sections often exhibit steep-sided flanks similar to fjord walls. They presently appear as dry valleys, lakes, seabed depressions, and as areas filled with sediment. If they are filled with sediment, their lower layers are filled primarily with glacial, glaciofluvial or glaciolacustrine sediment, supplemented by upper layers of temperate infill. They can be found in areas formerly covered by glacial ice sheets including Africa, Asia, North America, Europe, Australia and offshore in the North Sea, the Atlantic and in waters near Antarctica. Tunnel valleys appear in the technical literature under several terms, including tunnel channels, subglacial valleys, iceways and linear incisions. == Significance == Tunnel valleys play a role in identifying oil-rich areas in Arabia and North Africa. The Upper Ordovician–Lower Silurian


id: tunnel-valley-20260505-1
file: data/en.wikipedia.org/wiki/Tunnel_valley-1.md
category: reference
title: Tunnel valley
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Tunnel_valley
date_saved: 2026-05-05T07:21:37.968444+00:00
instance: kb-cron
parent: tunnel-valley-20260505-0
summary:
=== Evidence of erosion mechanisms === Evidence suggests that erosion in a tunnel valley is primarily the result of water flow. They erode by meltwater, which it has been argued, episodically drains in repeated jökulhlaups from subglacial lakes and reservoirs; examples of such motion have been observed in Antarctica. Although there is evidence of ice erosion such as linear striations in the bedrock, these are observed only in the widest valleys, and are believed to have played a secondary role. The subglacial layout of valley tunnels is predominantly oriented parallel to glacial ice flow lines – essentially they stretch from areas of thicker sheet ice toward areas of thinner sheet ice. They can exhibit reverse gradients, which result when pressurized meltwater flows over obstacles such as ridges or hills along the glacier bed. Tunnel valleys can be formed under extremely thick glacial ice – examples have been observed on the bottom of Lake Superior and in the oceans offshore in Antarctica. The course of a tunnel valley typically runs from thickest glacial ice to the glacier margin; as a result the glacial ice pressurizes the water such that it runs uphill toward its end. == Formation of tunnel valleys ==


id: tunnel-valley-20260505-2
file: data/en.wikipedia.org/wiki/Tunnel_valley-2.md
category: reference
title: Tunnel valley
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Tunnel_valley
date_saved: 2026-05-05T07:21:37.968444+00:00
instance: kb-cron
parent: tunnel-valley-20260505-0
summary:
=== Jökulhlaup driven erosion === There is evidence that meltwater discharges are episodic. This can result because as water continues to collect, more ice is lifted, and the water moves outward in a growing under-ice lake. Areas where the ice is most easily lifted (i.e., areas with thinner overlying ice sheets) are lifted first. Hence the water may move up the terrain underlying the glacier if it moves toward areas of lower overlying ice. As water collects, additional ice is lifted until a release path is created. If no preexisting channel is present, the water is initially released in a broad-front jökulhlaup which can have a flow front that is tens of kilometers wide, spreading out in a thin front. As the flow continues, it tends to erode the underlying materials and the overlying ice, creating a channel even as the reduced pressure allows most of the glacial ice to settle back to the underlying surface, sealing off the broad front release and channelizing the flow. The direction of the channel is defined primarily by the overlying ice thickness and secondarily by the gradient of the underlying earth, and may be observed to “run uphill” as the pressure of the


id: tunnel-valley-20260505-3
file: data/en.wikipedia.org/wiki/Tunnel_valley-3.md
category: reference
title: Tunnel valley
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Tunnel_valley
date_saved: 2026-05-05T07:21:37.968444+00:00
instance: kb-cron
parent: tunnel-valley-20260505-0
summary:
=== Large-scale structure === Ice flow within glaciers results from an increase in the surface slope of the glacier, which result from geographic features combined with an imbalance between the amounts of ice accumulated through precipitation and lost through ablation. The increased gradient increases the shear stress on a glacier until it begins to flow. The flow velocity and deformation are also affected by the slope of the ice, the ice thickness and temperature. Punkari identified that continental ice sheets typically flow in fan-shaped lobes, which converge from separate sources and move at differing speeds. Lobes are separated by interlobate zones, which have thinner ice coverage. Water collects in this interlobate area. The hydraulic head (pressure) is lower in areas of thinner ice; hence subglacial water tends to converge on the interlobate joint. The separate lobes move at different speeds, generating friction at the ice boundary; the heat released melts ice to release additional water. The surface of the interlobate area is crevassed, allowing surface meltwater, which runs down the ice surface to the lower area, to penetrate into the ice. As a result, the ice-flow patterns and the debris accumulation are different in interlobate zones. Specifically, tunnel valleys and


id: tunnel-valley-20260505-4
file: data/en.wikipedia.org/wiki/Tunnel_valley-4.md
category: reference
title: Tunnel valley
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Tunnel_valley
date_saved: 2026-05-05T07:21:37.968444+00:00
instance: kb-cron
parent: tunnel-valley-20260505-0
summary:
Tunnel valleys have been observed in Minnesota, Wisconsin and Michigan at the margins of the Laurentide Ice Sheet. Examples of bedrock tunnel valleys in Minnesota include River Warren Falls and several valleys which lie deep beneath till deposited by the glaciers which created them, but can be traced in many places by the Chain of Lakes in Minneapolis and lakes and dry valleys in St. Paul. The Kawartha lakes of Ontario formed in the Late Wisconsinan glacial period. Ice melt from the Niagara Escarpment flowed through tunnel valleys beneath the ice expanded to form a west-to-east passage between the main Laurentide Ice Sheet and a mass of ice in the Lake Ontario basin. Cedar Creek Canyon is a tunnel valley located in Allen County, Indiana. It is a very straight, narrow gorge about 50 to 100 ft (15 to 30 m) deep that contains part of the lower segment of Cedar Creek, the largest tributary of the St. Joseph River. In the Laurentian Channel offshore eastern Canada, numerous tunnel valleys have been identified originating from the submerged valley of the St. Lawrence River, which is also of glacial origin. Seismic reflection profiles of the fill of tunnel valleys suggest that


id: upstate-20260505-0
file: data/en.wikipedia.org/wiki/Upstate-0.md
category: reference
title: Upstate
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Upstate
date_saved: 2026-05-05T07:21:39.233282+00:00
instance: kb-cron
parent: upstate-20260505-0
summary:
The term upstate may refer to the northerly portions of several U.S. states. On the east coast, upstate generally refers to places away from the Atlantic Ocean. It also can refer to parts of states that have a higher elevation, away from sea level. These regions tend to be rural; exceptions include Delaware and Illinois. Places Maine, except for "Down East" Upstate California, a 2001 marketing campaign to promote the northern half of Northern California Upstate New York, much of New York north of the New York City metropolitan area Upstate South Carolina, the northwestern "corner" of South Carolina Upstate Pennsylvania, a tourism region that includes much of Northeastern Pennsylvania Other SUNY Upstate Medical University, often referred to as "Upstate" Upstate University Hospital, Syracuse, New York A term used to refer to going to the penitentiary in New York, as all of New York's state prisons are upstate. == See also == Articles beginning with "Upstate" Downstate (disambiguation) Up North (disambiguation) Down south (disambiguation)


id: chronostratigraphy-20260505-0
file: data/en.wikipedia.org/wiki/Chronostratigraphy-0.md
category: reference
title: Chronostratigraphy
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Chronostratigraphy
date_saved: 2026-05-05T07:21:42.400088+00:00
instance: kb-cron
parent: chronostratigraphy-20260505-0
summary:
Chronostratigraphy is the branch of stratigraphy that studies the ages of rock strata in relation to time. The ultimate aim of chronostratigraphy is to arrange the sequence of deposition and the time of deposition of all rocks within a geological region, and eventually, the entire geologic record of the Earth. The standard stratigraphic nomenclature is a chronostratigraphic system based on palaeontological intervals of time defined by recognised fossil assemblages (biostratigraphy). The aim of chronostratigraphy is to give a meaningful age date to these fossil assemblage intervals and interfaces. == Methodology == Chronostratigraphy relies heavily upon isotope geology and geochronology to derive hard dating of known and well defined rock units which contain the specific fossil assemblages defined by the stratigraphic system. In practice, as it is very difficult to isotopically date most fossils and sedimentary rocks directly, inferences must be made in order to arrive at an age date which reflects the beginning of the interval. The methodology used is derived from the law of superposition and the principles of cross-cutting relationships. Because igneous rocks occur at specific intervals in time and are essentially instantaneous on a geologic time scale, and because they contain mineral assemblages which may be dated


id: chronozone-20260505-0
file: data/en.wikipedia.org/wiki/Chronozone-0.md
category: reference
title: Chronozone
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Chronozone
date_saved: 2026-05-05T07:21:43.591424+00:00
instance: kb-cron
parent: chronozone-20260505-0
summary:
A chronozone or chron is a unit in chronostratigraphy, defined by events such as geomagnetic reversals (magnetozones), or based on the presence of specific fossils (biozone or biochronozone). According to the International Commission on Stratigraphy, the term "chronozone" refers to the rocks formed during a particular time period, while "chron" refers to that time period. Although non-hierarchical, chronozones have been recognized as useful markers or benchmarks of time in the rock record. Chronozones are non-hierarchical in that chronozones do not need to correspond across geographic or geologic boundaries, nor be equal in length. Although a former, early constraint required that a chronozone be defined as smaller than a geological stage. Another early use was hierarchical in that Harland et al. (1989) used "chronozone" for the slice of time smaller than a faunal stage defined in biostratigraphy. The ICS superseded these earlier usages in 1994. The key factor in designating an internationally acceptable chronozone is whether the overall fossil column is clear, unambiguous, and widespread. Some accepted chronozones contain others, and certain larger chronozones have been designated which span whole defined geological time units, both large and small. For example, the chronozone Pliocene is a subset of the chronozone Neogene, and


id: depocenter-20260505-0
file: data/en.wikipedia.org/wiki/Depocenter-0.md
category: reference
title: Depocenter
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Depocenter
date_saved: 2026-05-05T07:21:46.122412+00:00
instance: kb-cron
parent: depocenter-20260505-0
summary:
A depocenter or depocentre in geology is the part of a sedimentary basin where a particular rock unit has its maximum thickness. Depending on the controls on subsidence and the sedimentary environment, the location of a basin's depocenter may vary with time, such as in active rift basins as extensional faults grow, link or become abandoned. == References ==


id: discrete-debris-accumulation-20260505-0
file: data/en.wikipedia.org/wiki/Discrete_debris_accumulation-0.md
category: reference
title: Discrete debris accumulation
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Discrete_debris_accumulation
date_saved: 2026-05-05T07:21:48.559798+00:00
instance: kb-cron
parent: discrete-debris-accumulation-20260505-0
summary:
Discrete debris accumulation (DDA) is a non-genetic term in mountain glacial geology to aid identification of non-lithified sediments on a valley or mountain slope or floor. It is intended that the debris accumulation is discrete such that it can be mapped, in the field and/or from aerial or satellite imagery. The origin or formative process may well not be known clearly or be changed by subsequent investigators it is advisable to have a non-genetic field reference so that discussion can then be used to ascertain, if possible, the origin. Mountain areas may currently have glaciers (glacierized) or have had glaciers (glaciated) or be subject to forms of periglacial activity. A moraine would be an easily identified DDA as would an esker. Although scree (talus) is generally easily identified and mapped, these deposits may be modified by ice, avalanches or downslope movement to create essentially new landforms. Many small slope failures and landslides can give the appearance of moraines or protalus ramparts on slopes. After mapping as a DDA, further investigation might draw light on the origin of the feature. The term was apparently first used by Sven Lukas for a very specific feature in Svalbard. Independently, it was suggested in


id: eonothem-20260505-0
file: data/en.wikipedia.org/wiki/Eonothem-0.md
category: reference
title: Eonothem
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Eonothem
date_saved: 2026-05-05T07:21:49.745512+00:00
instance: kb-cron
parent: eonothem-20260505-0
summary:
In stratigraphy and geology, an eonothem is the totality of rock strata laid down in the stratigraphic record deposited during a certain eon of the continuous geologic timescale. The eonothem is not to be confused with the eon itself, which is a corresponding division of geologic time spanning a specific number of (hundreds of millions of) years, during which rocks were formed that are classified within the eonothem. Eonothems have the same names as their corresponding eons, which means during the history of the Earth only four eonothems were formed. Oldest to newest these are the Hadean, Archean, Proterozoic, and Phanerozoic. A rock stratum, fossil or feature present in the "upper Phanerozoic" eonothem would therefore have originated within the "later Phanerozoic" eon. In practice, the rock column is discontinuous: Technically, a complete geologic record doesn't occur anywhere. For such a record to develop would require the area to have been receiving sedimentary deposits continually ever since the origin of the earth. Nowhere is such a situation known to exist. If it did exist, we could not effectively look at the strata because they would still be buried, and modern strata would continue to be deposited on top of them. The


id: erathem-20260505-0
file: data/en.wikipedia.org/wiki/Erathem-0.md
category: reference
title: Erathem
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Erathem
date_saved: 2026-05-05T07:21:50.980515+00:00
instance: kb-cron
parent: erathem-20260505-0
summary:
In stratigraphy, paleontology, geology, and geobiology, an erathem is the total stratigraphic unit deposited during a certain corresponding span of time during an era in the geologic timescale. It can therefore be used as a chronostratigraphic unit of time which delineates a large span of years – less than a geological eon, but greater than its successively smaller and more refined subdivisions (geologic periods, epochs, and geologic ages). By 3,500 million years ago (Mya) simple life had developed on earth (the oldest known microbial fossils in Australia are dated to this figure). The atmosphere was a mix of noxious and poisonous gases (methane, ammonia, sulfur compounds, etc. – a so-called reducing atmosphere lacking much free oxygen which was bound up in compounds). These simple organisms, cyanobacteria ruled the still cooling earth for approximately a billion years and gradually transformed the atmosphere to one containing free oxygen. These changes, along with tectonic activity left chemical trails (red bed formation, etc.) and other physical clues (magnetic orientation, layer formation factors) in the rock record, and it is these changes along with the later richer fossil record which specialists use to demarcate times early in planet earth's history in various disciplines. Erathems are


id: friability-20260505-0
file: data/en.wikipedia.org/wiki/Friability-0.md
category: reference
title: Friability
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Friability
date_saved: 2026-05-05T07:21:52.287947+00:00
instance: kb-cron
parent: friability-20260505-0
summary:
In materials science, friability ( FRY-ə-BIL-ə-tee), the condition of being friable, describes the tendency of a solid substance to break into smaller pieces under stress or contact, especially by rubbing. The opposite of friable is indurate. Substances that are designated hazardous, such as asbestos or crystalline silica, are often said to be friable if small particles are easily dislodged and become airborne, and hence respirable (able to enter human lungs), thereby posing a health hazard. Tougher substances, such as concrete, may also be mechanically ground down and reduced to finely divided mineral dust. However, such substances are not generally considered friable because of the degree of difficulty involved in breaking the substance's chemical bonds through mechanical means. Some substances, such as polyurethane foams, show an increase in friability with exposure to ultraviolet radiation, as in sunlight. Friable is sometimes used metaphorically to describe "brittle" personalities who can be "rubbed" by seemingly-minor stimuli to produce extreme emotional responses. == General == A friable substance is any substance that can be reduced to fibers or finer particles by the action of a small amount of pressure or friction, such as rubbing or inadvertently brushing up against the substance. The term could also


id: geological-event-20260505-0
file: data/en.wikipedia.org/wiki/Geological_event-0.md
category: reference
title: Geological event
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Geological_event
date_saved: 2026-05-05T07:21:53.471312+00:00
instance: kb-cron
parent: geological-event-20260505-0
summary:
A geological event is a temporary and spatially heterogeneous and dynamic (diachronous) happening in Earth history that contributes to the transformation of Earth system and the formation of geological strata. Event stratigraphy was first proposed as a system for the recognition, study and correlation of the effects of important physical or biological events on the broader stratigraphical record. Geological events range in time span by orders of magnitude, from seconds to millions of years, and in spatial scale from local to regional and, ultimately, global. In contrast to chronostratigraphic or geochronological units, that define the boundaries between periods, epochs and other units of the geologic time scale, complex dynamic diachronous changes are inherent to the event-stratigraphy paradigm. The lithostratigraphic or biostratigraphic boundaries that mark the onset and termination of geological events in the stratigraphic record may be diachronous, whereas those of formal chronostratigraphic or geochronologic units have basal boundaries that are isochronous. Examples of geological events include a single footprint, an earthquake, a series of volcanic eruptions, the formation of mountains (orogenies), the Great Oxidation Event (GOE) of 2.4-2.0 billion years ago and the Great Ordovician Biodiversification Event (GOBE) ~500 Ma. The event paradigm is firmly embedded in Quaternary science,


id: stratigraphic-unit-20260505-0
file: data/en.wikipedia.org/wiki/Stratigraphic_unit-0.md
category: reference
title: Stratigraphic unit
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Stratigraphic_unit
date_saved: 2026-05-05T07:21:54.692438+00:00
instance: kb-cron
parent: stratigraphic-unit-20260505-0
summary:
A stratigraphic unit is a volume of rock of identifiable origin and relative age range that is defined by the distinctive and dominant, easily mapped and recognizable petrographic, lithologic or paleontologic features (facies) that characterize it. Units must be mappable and distinct from one another, but the contact need not be particularly distinct. For instance, a unit may be defined by terms such as "when the sandstone component exceeds 75%". == Lithostratigraphic units == Sequences of sedimentary and volcanic rocks are subdivided on the basis of their shared or associated lithology. Formally identified lithostratigraphic units are structured in a hierarchy of lithostratigraphic rank, higher rank units generally comprising two or more units of lower rank. Going from smaller to larger in rank, the main lithostratigraphic ranks are bed, member, formation, group and supergroup. Formal names of lithostratigraphic units are assigned by geological surveys. Units of formation or higher rank are usually named for the unit's type location, and the formal name usually also states the unit's rank or lithology. A lithostratigraphic unit may have a change in rank over a some distance; a group may thin to a formation in another region and a formation may reduce in rank for


id: geon-geology-20260505-0
file: data/en.wikipedia.org/wiki/Geon_(geology)-0.md
category: reference
title: Geon (geology)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Geon_(geology)
date_saved: 2026-05-05T07:21:55.910880+00:00
instance: kb-cron
parent: geon-geology-20260505-0
summary:
The term geon (for geological eon) refers to large, geologic units of time. Geologists traditionally subdivide Earth history into a hierarchy of named intervals: eons, eras, periods, etc. (e.g., the Jurassic Period of the Mesozoic Era). Historians subdivide the history of human activity into intervals that are comparatively much shorter. In both geological and historical scales, the divisions of equal rank are characteristically of unequal duration, and the identification of a particular interval is primarily based on its fossil, artifact, or cultural content (e.g., Carboniferous, Neolithic, Dark Ages, Ming Dynasty). Both scales are calibrated against numerical ages obtained separately. An alternative way of referring to the past is to use a scale with intervals of equal duration. We speak of a given decade, century, or millennium. For the enormously long geologic time frame, it is advantageous to use corresponding large, equal time intervals encompassing the events and processes that have shaped our planet. The development of mountain ranges, ocean basins, and continents takes tens to hundreds of millions of years, and large time units thus are convenient for discussing long-term trends. Astronomers use light years and parsecs to deal with huge distances, rather than kilometres. Geologists have geons to refer


id: geosphere-20260505-0
file: data/en.wikipedia.org/wiki/Geosphere-0.md
category: reference
title: Geosphere
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Geosphere
date_saved: 2026-05-05T07:21:57.160011+00:00
instance: kb-cron
parent: geosphere-20260505-0
summary:
There are several conflicting usages of geosphere, variously defined. == History == === Aristotelian physics === In Aristotelian physics, the term was applied to four spherical natural places, concentrically nested around the center of the Earth, as described in the lectures Physica and Meteorologica. They were believed to explain the motions of the four terrestrial elements: Earth, Water, Air, and Fire. === Modern era === In modern texts and in Earth system science, geosphere refers to the solid parts of the Earth; it is used along with atmosphere, hydrosphere, and biosphere to describe the systems of the Earth (the interaction of these systems with the magnetosphere is sometimes listed). In that context, sometimes the term lithosphere is used instead of geosphere or solid Earth. The lithosphere, however, only refers to the uppermost layers of the solid Earth (oceanic and continental crustal rocks and uppermost mantle). == Other usages == "Geosphere" may also be taken as the collective name for the lithosphere, the hydrosphere, the cryosphere, and the atmosphere. The different collectives of the geosphere are able to exchange different mass and/or energy fluxes (the measurable amount of change). The exchange of these fluxes affects the balance of the different spheres


id: graviquake-20260505-0
file: data/en.wikipedia.org/wiki/Graviquake-0.md
category: reference
title: Graviquake
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Graviquake
date_saved: 2026-05-05T07:21:59.590808+00:00
instance: kb-cron
parent: graviquake-20260505-0
summary:
A graviquake is an earthquake occurring in an extensional tectonic setting, where the gravitational energy stored during the interseismic period is delivered by the collapse of a brittle upper crustal volume, slipping along a normal fault and generating the double couple recorded in the moment tensor solution. == References ==


id: joint-geology-20260505-0
file: data/en.wikipedia.org/wiki/Joint_(geology)-0.md
category: reference
title: Joint (geology)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Joint_(geology)
date_saved: 2026-05-05T07:22:00.828531+00:00
instance: kb-cron
parent: joint-geology-20260505-0
summary:
In geology, and more specifically in structural geology, a joint is a break (fracture) of natural origin in a layer or body of rock that lacks visible or measurable movement parallel to the surface (plane) of the fracture ("Mode 1" Fracture). Although joints can occur singly, they most frequently appear as joint sets and systems. A joint set is a family of parallel, evenly spaced joints that can be identified through mapping and analysis of their orientations, spacing, and physical properties. A joint system consists of two or more intersecting joint sets. The distinction between joints and faults hinges on the terms visible or measurable, a difference that depends on the scale of observation. Faults differ from joints in that they exhibit visible or measurable lateral movement between the opposite surfaces of the fracture ("Mode 2" and "Mode 3" Fractures). Thus a joint may be created by either strict movement of a rock layer or body perpendicular to the fracture or by varying degrees of lateral displacement parallel to the surface (plane) of the fracture that remains "invisible" at the scale of observation. Joints are among the most universal geologic structures, found in almost every exposure of rock. They vary


id: joint-geology-20260505-1
file: data/en.wikipedia.org/wiki/Joint_(geology)-1.md
category: reference
title: Joint (geology)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Joint_(geology)
date_saved: 2026-05-05T07:22:00.828531+00:00
instance: kb-cron
parent: joint-geology-20260505-0
summary:
==== Nonsystematic ==== Nonsystematic joints are joints that are so irregular in form, spacing, and orientation that they cannot be readily grouped into distinctive, through-going joint sets. === By formation === Joints can be classified according to their origin, under the labels of tectonics, hydraulics, exfoliation, unloading (release), and cooling. Different authors have proposed contradictory hypotheses for the same joint sets and types. And, joints in the same outcrop may form at different times under varied circumstances. ==== Tectonic ==== Tectonic joints are joints formed when the relative displacement of the joint walls is normal to its plane as the result of brittle deformation of bedrock in response to regional or local tectonic deformation of bedrock. Such joints form when directed tectonic stress causes the tensile strength of bedrock to be exceeded as the result of the stretching of rock layers under conditions of elevated pore fluid pressure and directed tectonic stress. Tectonic joints often reflect local tectonic stresses associated with local folding and faulting. Tectonic joints occur as both nonsystematic and systematic joints, including orthogonal and conjugate joint sets. ==== Hydraulic ==== Hydraulic joints are formed when pore fluid pressure becomes elevated as a result of vertical gravitational loading.


id: joint-geology-20260505-2
file: data/en.wikipedia.org/wiki/Joint_(geology)-2.md
category: reference
title: Joint (geology)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Joint_(geology)
date_saved: 2026-05-05T07:22:00.828531+00:00
instance: kb-cron
parent: joint-geology-20260505-0
summary:
== Importance == Joints are important not only in understanding the local and regional geology and geomorphology but also in developing natural resources, in the safe design of structures, and in environmental protection. Joints have a profound control on weathering and erosion of bedrock. As a result, they exert a strong control on how topography and morphology of landscapes develop. Understanding the local and regional distribution, physical character, and origin of joints is a significant part of understanding the geology and geomorphology of an area. Joints often impart a well-develop fracture-induced permeability to bedrock. As a result, joints strongly influence, even control, the natural circulation (hydrogeology) of fluids, e.g. groundwater and pollutants within aquifers, petroleum in reservoirs, and hydrothermal circulation at depth, within bedrock. Thus, joints are important to the economic and safe development of petroleum, hydrothermal, and groundwater resources and the subject of intensive research relative to these resources. Regional and local joint systems exert a strong control on how ore-forming hydrothermal fluids (consisting largely of H2O, CO2, and NaCl — which formed most of Earth's ore deposits) circulated within its crust. As a result, understanding their genesis, structure, chronology, and distribution is an important part of finding and


id: landmass-20260505-0
file: data/en.wikipedia.org/wiki/Landmass-0.md
category: reference
title: Landmass
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Landmass
date_saved: 2026-05-05T07:22:02.140535+00:00
instance: kb-cron
parent: landmass-20260505-0
summary:
A landmass, or land mass, is a large region or area of land that is in one piece and not noticeably broken up by oceans. The term is often used to refer to lands surrounded by an ocean or sea, such as a continent or a large island. In the field of geology, a landmass is a defined section of continental crust extending above sea level. Continents are often thought of as distinct landmasses and may include any islands that are part of the associated continental shelf. When multiple continents form a single contiguous land connection, the connected continents may be viewed as a single landmass. Earth's largest landmasses are (starting with largest): Afro-Eurasia (main landmass of the geoscheme region of the same name and its continental parts Africa and Eurasia, or Europe and Asia; the center of Earth's land hemisphere, comprising more than half of Earth's landmass) Americas (main landmass of the geo-region of the same name and its continental parts North and South America; comprising most of the landmass of the Western Hemisphere) Antarctica (main landmass of the geo-region and continent of the same name) Mainland Australia (main landmass of the geo-region Oceania, its subregion Australasia, the continent


id: leaverite-20260505-0
file: data/en.wikipedia.org/wiki/Leaverite-0.md
category: reference
title: Leaverite
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Leaverite
date_saved: 2026-05-05T07:22:03.396042+00:00
instance: kb-cron
parent: leaverite-20260505-0
summary:
Leaverite is a slang term used by geologists, mineralogists, archaeologists, and amateur rock collectors to identify a specimen in the field that may look interesting or valuable but not worth removing. Rocks identified as such are generally left where they were found, as there is not sufficient reason to attempt to remove them. The term leaverite derives from the phrase "leave 'er right there". There are several kinds of leaverites: Man-made features Unrocks; objects that may initially appear to be geologic in origin (also referred to as anthropogenicite). Large fossils and pseudofossils Additional forms of leaverites include: Specimens that are too large to move or transport Legality of the removal of the specimen is questionable or outright illegal Collection of the specimen may result in death or dismemberment (e.g. the specimen is hanging on the side of a 300 ft cliff, removal of the specimen would result in a boulder falling on the collector, a rattlesnake is sunning itself on the specimen) == References ==


id: paralithic-20260505-0
file: data/en.wikipedia.org/wiki/Paralithic-0.md
category: reference
title: Paralithic
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Paralithic
date_saved: 2026-05-05T07:22:06.822156+00:00
instance: kb-cron
parent: paralithic-20260505-0
summary:
A paralithic horizon is a weathered layer of bedrock. The term comes from the Greek words para, meaning "akin to", and lithic, meaning "stony". == References == Pietrasiak, Nicole (7 October 2008). "Mineralogy of a Paralithic Horizon (Well Weathered Bedrock) in the Mojave Desert". The 2008 Joint Annual Meeting. Soil Survey Staff (2010). "Paralithic contact". Keys to Soil Taxonomy, 2010 (11th ed.). Washington: Agriculture Department. p. 26. ISBN 9780160854279. Soil Survey Staff (2010). "Paralithic material". Keys to Soil Taxonomy, 2010 (11th ed.). Washington: Agriculture Department. p. 26. ISBN 9780160854279.


id: primary-rock-20260505-0
file: data/en.wikipedia.org/wiki/Primary_rock-0.md
category: reference
title: Primary rock
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Primary_rock
date_saved: 2026-05-05T07:22:09.228469+00:00
instance: kb-cron
parent: primary-rock-20260505-0
summary:
Primary rock is an early term in geology that refers to crystalline rock formed first in geologic time, containing no organic remains, such as granite, gneiss and schist as well as igneous and magmatic formations from all ages. Webster's Revised Unabridged Dictionary published in 1913 provides the following term as used in geology: Primary rocks (geol.) a term early used for rocks supposed to have been first formed, being crystalline and containing no organic remains, as granite, gneiss, etc.; – called also primitive rocks. The terms Secondary, Tertiary, and Quaternary rocks have also been used in like manner, but of these the last two only are now in use. Ninety years later the McGraw-Hill Dictionary of Scientific & Technical Terms published in 2003 places the term in the geologic field of petrology: Primary rocks [′prī‚mer·ē ′räks] (petrology) Rocks whose constituents are newly formed particles that have never been constituents of previously formed rocks and that are not the products of alteration or replacement, such as limestones formed by precipitation from solution. The term dates from the late 18th century (see Giovanni Arduino and Abraham Gottlob Werner) when the first attempts to formulate a geologic time scale divided crustal rocks into


id: relict-20260505-0
file: data/en.wikipedia.org/wiki/Relict-0.md
category: reference
title: Relict
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Relict
date_saved: 2026-05-05T07:22:10.570192+00:00
instance: kb-cron
parent: relict-20260505-0
summary:
A relict is a surviving remnant of a natural phenomenon. == Biology == A relict is an organism that at an earlier time was abundant in a large area but now occurs at only one or a few small areas. == Geology and geomorphology == In geology, a relict is a structure or mineral from a parent rock that did not undergo metamorphosis when the surrounding rock did, or a rock that survived a destructive geologic process. In geomorphology, a relict landform is a landform formed by either erosive or constructive surficial processes that are no longer active as they were in the past. A glacial relict is a cold-adapted organism that is a remnant of a larger distribution that existed in the ice ages. == Human populations == As revealed by DNA testing, a relict population is an ancient people in an area, who have been largely supplanted by a later group of migrants and their descendants. In various places around the world, minority ethnic groups represent lineages of ancient human migrations in places now occupied by more populous ethnic groups, whose ancestors arrived later. For example, the first human groups to inhabit the Caribbean islands were hunter-gatherer tribes


id: sag-pond-20260505-0
file: data/en.wikipedia.org/wiki/Sag_pond-0.md
category: reference
title: Sag pond
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Sag_pond
date_saved: 2026-05-05T07:22:11.868316+00:00
instance: kb-cron
parent: sag-pond-20260505-0
summary:
A sag pond is a body of fresh water collected in the lowest parts of a depression formed between two sides of an active strike-slip, transtensional or normal fault zone. == Formation == A sag pond is formed along a strike-slip fault, which may create a depression in the earth. When water enters the depression from rivers, streams, rainfall or snowfall, it fills the low-lying area, and a pond is the result. == Scientific significance == Sag pond formation is common in California, and many of them can be found on the San Andreas Fault, which runs through the western part of the state. Aerial photography or high-resolution topography can easily determine their locations. == Uses == Sag ponds have been converted into stock reservoirs. Larger sag ponds are converted into reservoirs for public water sources, such as San Andreas Lake == Examples == Sag ponds of various sizes can be found along the San Andreas Fault, most notably within the Carrizo Plain and Sierra Pelona Mountains. Some have been converted into reservoirs like the San Andreas Reservoir near San Bruno, California. Various lagoons on the plateau of Pico Island in the Azores are the result of a graben. ==


id: sector-collapse-20260505-0
file: data/en.wikipedia.org/wiki/Sector_collapse-0.md
category: reference
title: Sector collapse
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Sector_collapse
date_saved: 2026-05-05T07:22:13.195652+00:00
instance: kb-cron
parent: sector-collapse-20260505-0
summary:
A sector collapse or lateral collapse is the structural failure and subsequent collapse of a minimum volume of 1 km3 (0.24 cu mi) of a volcano. Unlike smaller flank collapses, a sector collapse can involve the central volcanic pipe and historically this term had been restricted by some writers to such events in arc stratovolcanoes, but is now used for large events in any volcano. Sector collapses are one of the most hazardous volcanic events, often resulting in lateral blasts, landslides, and changes in volcanic eruptive behavior. Sector collapse can be caused by earthquakes, volcanic eruptions, gradual volcanic deformation, and other processes. Sector collapse events can occur on volcanoes at convergent and divergent plate boundaries. Sector collapses are generally very sudden; however, some attempts have been made to predict collapse events. == Causes == === Internal === Sector collapse can result from internal volcanic processes. Volcanic eruption can damage originally stable magma chambers, causing a portion of the volcano to collapse. While eruption is one cause, sector collapse can occur without any eruption. Magmatic intrusions can also lead to sector collapse. Dikes fracture and deform rock, leaving the volcano weaker and more susceptible to collapse. Hydrothermal activity is another internal


id: series-stratigraphy-20260505-0
file: data/en.wikipedia.org/wiki/Series_(stratigraphy)-0.md
category: reference
title: Series (stratigraphy)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Series_(stratigraphy)
date_saved: 2026-05-05T07:22:14.371039+00:00
instance: kb-cron
parent: series-stratigraphy-20260505-0
summary:
Series are subdivisions of rock layers based on the age of the rock and formally defined by international conventions of the geological timescale. A series is therefore a sequence of strata defining a chronostratigraphic unit. Series are subdivisions of systems and are themselves divided into stages. Series is a term defining a unit of rock layers formed during a certain interval of time (a chronostratigraphic unit); it is equivalent (but not synonymous) to the term geological epoch (see epoch criteria) which defines the interval of time itself, although the two words are sometimes confused in informal literature. == Series in the geological timescale == The geological timescale has all systems in the Phanerozoic eonothem subdivided into series. Some of these have their own names; in other cases a system is simply divided into a Lower, Middle and Upper series, with official series being capitalized and unofficial designations (such as "middle Cretaceous") being left uncapitalized. The Cretaceous system is, for example, divided into the Upper Cretaceous and Lower Cretaceous Series, while the Carboniferous System is divided into the Pennsylvanian and Mississippian Series. As of 2008, the International Commission on Stratigraphy had not yet named all four series of the Cambrian. Currently


id: stage-stratigraphy-20260505-0
file: data/en.wikipedia.org/wiki/Stage_(stratigraphy)-0.md
category: reference
title: Stage (stratigraphy)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Stage_(stratigraphy)
date_saved: 2026-05-05T07:22:15.598010+00:00
instance: kb-cron
parent: stage-stratigraphy-20260505-0
summary:
In chronostratigraphy, a stage is a succession of rock strata laid down in a single age on the geologic timescale, which usually represents millions of years of deposition. A given stage of rock and the corresponding age of time will by convention have the same name, and the same boundaries. Rock series are divided into stages, just as geological epochs are divided into ages. Stages are divided into smaller stratigraphic units called chronozones or substages, and added together into superstages. The term faunal stage is sometimes used, referring to the fact that the same fauna (animals) are found throughout the layer (by definition). == Definition == Stages are primarily defined by a consistent set of fossils (biostratigraphy) or a consistent magnetic polarity (see paleomagnetism) in the rock. Usually one or more index fossils that are common, found worldwide, easily recognized, and limited to a single, or at most a few, stages are used to define the stage's bottom. Thus, for example in the local North American subdivision, a paleontologist finding fragments of the trilobite Olenellus would identify the beds as being from the Waucoban Stage whereas fragments of a later trilobite such as Elrathia would identify the stage as Albertan.


id: subaerial-20260505-0
file: data/en.wikipedia.org/wiki/Subaerial-0.md
category: reference
title: Subaerial
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Subaerial
date_saved: 2026-05-05T07:22:17.882294+00:00
instance: kb-cron
parent: subaerial-20260505-0
summary:
In natural science, subaerial (literally "under the air") has been used since 1833, notably in geology and botany, to describe features and events occurring or formed on or near the Earth's land surface. They are thus exposed to Earth's atmosphere. This may be contrasted with subaqueous events or features located below a water surface, submarine events or features located below a sea surface, subterranean events or features located below ground, or subglacial events or features located below glacial ice such as ice sheets. == Geology == For example, a subaerial eruption of a volcano is one that ejects material in the open but "under the air" (under the atmosphere). Subaerial weathering is weathering by rain, frost, rivers etc. The term "subaerial" may exclude processes occurring in caves. The term is often used in sedimentology. == Botany == Leaves are subaerial organs of plants. Some plants may have subaerial roots, either totally (epiphytic plants such as some orchids) or more commonly only partly so. The oil palm tree can grow roots into accumulations of decaying leaves on the soil surface; these roots are said to be subaerial. Epiphyte plants growing above ground that do not feed from their tree support (for


id: subbasin-20260505-0
file: data/en.wikipedia.org/wiki/Subbasin-0.md
category: reference
title: Subbasin
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Subbasin
date_saved: 2026-05-05T07:22:19.116933+00:00
instance: kb-cron
parent: subbasin-20260505-0
summary:
A subbasin or sub-basin is a structural geologic feature where a larger basin is divided into a series of smaller basins with intervening intrabasinal highs. The term subbasin has common use in geologic literature, but has yet to be included in the API Glossary of Geology. == References ==


id: system-stratigraphy-20260505-0
file: data/en.wikipedia.org/wiki/System_(stratigraphy)-0.md
category: reference
title: System (stratigraphy)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/System_(stratigraphy)
date_saved: 2026-05-05T07:22:20.304965+00:00
instance: kb-cron
parent: system-stratigraphy-20260505-0
summary:
A system in stratigraphy is a sequence of strata (rock layers) that were laid down together within the same corresponding geological period. The associated period is a chronological time unit, a part of the geological time scale, while the system is a unit of chronostratigraphy. Systems are unrelated to lithostratigraphy, which subdivides rock layers on their lithology. Systems are subdivisions of erathems and are themselves divided into series and stages. == Systems in the geological timescale == The systems of the Phanerozoic were defined during the 19th century, beginning with the Cretaceous (by Belgian geologist Jean d'Omalius d'Halloy in the Paris Basin) and the Carboniferous (by British geologists William Conybeare and William Phillips in 1822). The Paleozoic and Mesozoic were divided into the currently used systems before the second half of the 19th century, except for a minor revision when the Ordovician system was added in 1879. The Cenozoic has seen more recent revisions by the International Commission on Stratigraphy. It has been divided into three systems with the Paleogene and Neogene replacing the former Tertiary System though the succeeding Quaternary remains. The one-time system names of Paleocene, Eocene, Oligocene, Miocene and Pliocene are now series within the Paleogene and


id: upper-mantle-body-20260505-0
file: data/en.wikipedia.org/wiki/Upper_mantle_body-0.md
category: reference
title: Upper mantle body
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Upper_mantle_body
date_saved: 2026-05-05T07:22:21.562858+00:00
instance: kb-cron
parent: upper-mantle-body-20260505-0
summary:
An upper mantle body is a geological region where upper mantle rocks (peridotite) outcrop on the surface of the Earth (including the ocean floor). Upper mantle outcrops include: upper mantle made at constructive plate boundaries, but preserved in ophiolites, for example Isabela ophiolite in the Philippines upper mantle above subduction zones, so called suprasubduction ophiolites (such as Troodos Ophiolite, Cyprus) upper mantle exposed by thinning of continental crust by extension to continental crust removal (Ligurian "Ophiolites" and conjugate margin of Iberia and Newfoundland) upper mantle exposures on earth's surface above sea-water level in Oceans (whose ocean floor is covered with oceanic crust). Examples are Macquarie Island in the Pacific and the St. Peter and St. Paul Islands in the Atlantic. upper mantle exposures on earth's surface on the ocean floor. Examples include Gakkel Ridge and Lena Trough. upper mantle exposures on earth's surface of disputed origin upper mantle exposure on earth's surface of not understood environment == References ==


id: vug-20260505-0
file: data/en.wikipedia.org/wiki/Vug-0.md
category: reference
title: Vug
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Vug
date_saved: 2026-05-05T07:22:22.821989+00:00
instance: kb-cron
parent: vug-20260505-0
summary:
A vug, vugh, or vugg () is a small- to medium-sized cavity inside rock. It may be formed through a variety of processes. Most commonly, cracks and fissures opened by tectonic activity (folding and faulting) are partially filled by quartz, calcite, and other secondary minerals. Open spaces within breccias formed by an ancient collapse are another important source of vugs. Vugs may also form when mineral crystals or fossils inside a rock matrix are later removed through erosion or dissolution processes, leaving behind irregular voids. The inner surfaces of such vugs are often coated with a crystal druse. Fine crystals are often found in vugs where the open space allows the free development of external crystal form. The term vug is not applied to veins and fissures that have become completely filled, but may be applied to any small cavities within such veins. Geodes are a vug-formed rock, although that term is usually reserved for more rounded crystal-lined cavities in sedimentary rocks and ancient lavas. The word vug was introduced to the English language by Cornish miners, from the days when Cornwall was a major supplier of tin. The Cornish word was vooga, which meant "cave". == Images == ==


id: aboutness-20260505-0
file: data/en.wikipedia.org/wiki/Aboutness-0.md
category: reference
title: Aboutness
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Aboutness
date_saved: 2026-05-05T07:22:24.487269+00:00
instance: kb-cron
parent: aboutness-20260505-0
summary:
Aboutness is a term used in library and information science (LIS), linguistics, philosophy of language, and philosophy of mind. In general, the term refers to the concept that a text, utterance, image, or action is on or of something. In LIS, it is often considered synonymous with a document's subject. In the philosophy of mind, it has been often considered synonymous with intentionality, perhaps since John Searle (1983). In the philosophy of logic and language, it is understood as the way a piece of text relates to a subject matter or topic. R. A. Fairthorne (1969) is credited with coining the exact term "aboutness", which became popular in LIS since the late 1970s, perhaps due to arguments put forward by William John Hutchins (1975, 1977, 1978). Hutchins argued that "aboutness" was to be preferred to "subject" because it removed some epistemological problems. Birger Hjørland (1992, 1997) argued, however, that the same epistemological problems also were present in Hutchins' proposal, why "aboutness" and "subject" should be considered synonymous. While information scientists may well be concerned with the literary aboutness (John Hutchins, 1975, 1977, 1978), philosophers of mind and psychologists with the psychological or intentional aboutness (John Searle, 1983) and language of


id: authority-control-20260505-0
file: data/en.wikipedia.org/wiki/Authority_control-0.md
category: reference
title: Authority control
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Authority_control
date_saved: 2026-05-05T07:22:25.700102+00:00
instance: kb-cron
parent: authority-control-20260505-0
summary:
In information science, authority control is a process that organizes information, for example in library catalogs, by using a single, distinct spelling of a name (heading) or an identifier (generally persistent and alphanumeric) for each topic or concept. The word authority in authority control derives from the idea that the names of people, places, things, and concepts are authorized, i.e., they are established in one particular form. These one-of-a-kind headings or identifiers are applied consistently throughout catalogs which make use of the respective authority file, and are applied for other methods of organizing data such as linkages and cross references. Each controlled entry is described in an authority record in terms of its scope and usage, and this organization helps the library staff maintain the catalog and make it user-friendly for researchers. Catalogers assign each subject—such as author, topic, series, or corporation—a particular unique identifier or heading term which is then used consistently, uniquely, and unambiguously for all references to that same subject, which removes variations from different spellings, transliterations, pen names, or aliases. The unique header can guide users to all relevant information including related or collocated subjects. Authority records can be combined into a database and called an


id: authority-control-20260505-1
file: data/en.wikipedia.org/wiki/Authority_control-1.md
category: reference
title: Authority control
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Authority_control
date_saved: 2026-05-05T07:22:25.700102+00:00
instance: kb-cron
parent: authority-control-20260505-0
summary:
Generally, there are different authority file headings and identifiers used by different libraries in different countries, possibly inviting confusion, but there are different approaches internationally to try to lessen the confusion. One international effort to prevent such confusion is the Virtual International Authority File which is a collaborative attempt to provide a single heading for a particular subject. It is a way to standardize information from different authority files around the world such as the Integrated Authority File (GND) maintained and used cooperatively by many libraries in German-speaking countries and the United States Library of Congress. The idea is to create a single worldwide virtual authority file. For example, the ID for Princess Diana in the GND is 118525123 (preferred name: Diana < Wales, Prinzessin>) while the United States Library of Congress uses the term Diana, Princess of Wales, 1961–1997; other authority files have other choices. The Virtual International Authority File choice for all of these variations is VIAF ID: 107032638 — that is, a common number representing all of these variations. The English Wikipedia prefers the term "Diana, Princess of Wales", but at the bottom of the article about her, there are links to various international cataloging efforts for


id: authority-control-20260505-2
file: data/en.wikipedia.org/wiki/Authority_control-2.md
category: reference
title: Authority control
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Authority_control
date_saved: 2026-05-05T07:22:25.700102+00:00
instance: kb-cron
parent: authority-control-20260505-0
summary:
For example, the Irish writer Brian O'Nolan, who lived from 1911 to 1966, wrote under many pen names such as Flann O'Brien and Myles na Gopaleen. Catalogers at the United States Library of Congress chose one form—"O'Brien, Flann, 1911–1966"—as the official heading. The example contains all three elements of a valid authority record: the first heading O'Brien, Flann, 1911–1966 is the form of the name that the Library of Congress chose as authoritative. In theory, every record in the catalog that represents a work by this author should have this form of the name as its author heading. What follows immediately below the heading beginning with Na Gopaleen, Myles, 1911–1966 are the see references. These forms of the author's name will appear in the catalog, but only as transcriptions and not as headings. If a user queries the catalog under one of these variant forms of the author's name, he or she would receive the response: "See O'Brien, Flann, 1911–1966." There is an additional spelling variant of the Gopaleen name: "Na gCopaleen, Myles, 1911–1966" has an extra C inserted because the author also employed the non-anglicized Irish spelling of his pen-name, in which the capitalized C shows the correct root


id: bibliomining-20260505-0
file: data/en.wikipedia.org/wiki/Bibliomining-0.md
category: reference
title: Bibliomining
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Bibliomining
date_saved: 2026-05-05T07:22:26.830125+00:00
instance: kb-cron
parent: bibliomining-20260505-0
summary:
Bibliomining is the use of a combination of data mining, data warehousing, and bibliometrics for the purpose of analyzing library services. The term was created in 2003 by Scott Nicholson, Assistant Professor, Syracuse University School of Information Studies, in order to distinguish data mining in a library setting from other types of data mining. == How bibliomining works == First a data warehouse must be created. This is done by compiling information on the resources, such as titles and authors, subject headings, and descriptions of the collections. Then the demographic surrogate information is organized. Finally the library information (such as the librarian, whether or not the information came from the reference desk or circulation desk, and the location of the library) is obtained. Once this is organized, the data can be processed and analyzed. This can be done via a few methods, such as online analytical processing (OLAP), using a data mining program, or through data visualization. == Uses of bibliomining == Bibliomining is used to discover patterns in what people are reading and researching and allows librarians to target their community better. Bibliomining can also help library directors focus their budgets on resources that will be utilized. Another use


id: born-digital-20260505-0
file: data/en.wikipedia.org/wiki/Born-digital-0.md
category: reference
title: Born-digital
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Born-digital
date_saved: 2026-05-05T07:22:28.033797+00:00
instance: kb-cron
parent: born-digital-20260505-0
summary:
The term born-digital refers to materials that originate in a digital form. This is in contrast to digital reformatting, through which analog materials become digital, as in the case of files created by scanning physical paper records. It is most often used in relation to digital libraries and the issues that go along with said organizations, such as digital preservation and intellectual property. However, as technologies have advanced and spread, the concept of being born-digital has also been discussed in relation to personal consumer-based sectors, with the rise of e-books and evolving digital music. Other terms that might be encountered as synonymous include "natively digital", "digital-first", and "digital-exclusive". == Discrepancies in definition == There exists some inconsistency in defining born-digital materials. Some believe such materials must exist in digital form exclusively; in other words, if they can be transferred into a physical, analog form, they are not truly born-digital. However, others maintain that while these materials will often not have a subsequent physical counterpart, having one does not bar them from being classified as 'born-digital'. For instance, Mahesh and Mittal identify two types of born-digital content, "exclusive digital" and "digital for print", allowing for a broader base of classification than


id: born-digital-20260505-1
file: data/en.wikipedia.org/wiki/Born-digital-1.md
category: reference
title: Born-digital
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Born-digital
date_saved: 2026-05-05T07:22:28.033797+00:00
instance: kb-cron
parent: born-digital-20260505-0
summary:
==== Sound recordings ==== Digital sound recordings have played a role since the 1970s with the acceptance of pulse-code modulation (PCM) in the recording process. Since then, numerous means of storing and delivering digital audio have been developed, including web streams, compact discs and mp3 audio files. Increasingly, digital audio are only available via download, lacking any kind of tangible counterpart. One example of this trend is the 2008 recording of Hector Berlioz's Symphonie fantastique by Los Angeles Philharmonic under Gustavo Dudamel. Available through download only, it has presented problems for libraries which may want to carry this work but cannot due to licensing limitations. Another example is Radiohead's 2007 release In Rainbows, released initially as a digital download, before later receiving a physical release. ==== Other media ==== WebExhibits are websites that act as virtual museums for any variety of content. These often use both primary and secondary historical sources, maps, timelines, infographics, and other data visualizations to showcase the historical past. One example is Clio Visualizing History's Click! The Ongoing Feminist Revolution, a web exhibit about the American women's movement from the 1940s to the present. Clio Visualizing History was founded by Lola Van Wagenen in 1996 to


id: born-digital-20260505-2
file: data/en.wikipedia.org/wiki/Born-digital-2.md
category: reference
title: Born-digital
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Born-digital
date_saved: 2026-05-05T07:22:28.033797+00:00
instance: kb-cron
parent: born-digital-20260505-0
summary:
=== Licensing === Laws created to protect the intellectual property were written for analog works; as such, provisions such as the first-sale doctrine of US copyright law, which enables libraries to lend materials to patrons, have not been applied to the digital realm. Therefore, certain copyrighted digital content that is licensed rather than owned, as is common with many digital materials, is often of limited use since it cannot be transmitted to patrons at various computers or lent through an interloan agreement. However, with regards to the preservation functions of libraries and archives and the subsequent need to make copies of born-digital materials, the laws of many countries have been changing, allowing for agreements to be made between these institutions and the rights holders of born-digital content. Consumers have also had to deal with intellectual property as it concerns their ownership of and ability to control the born-digital material that they buy. Piracy proves to be a bigger problem with digital objects, including those that are born-digital, because such materials can be copied and spread in perfect condition with speed and distance on a scale inconceivable for traditional print and physical materials. Again, the first-sale doctrine, which, from a consumer


id: discoverability-20260505-0
file: data/en.wikipedia.org/wiki/Discoverability-0.md
category: reference
title: Discoverability
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Discoverability
date_saved: 2026-05-05T07:22:29.265231+00:00
instance: kb-cron
parent: discoverability-20260505-0
summary:
Discoverability is the degree to which something, especially a piece of content or information, can be found in a search of a file, database, or other information system. Discoverability is a concern in library and information science, many aspects of digital media, software and web development, and in marketing, since products and services cannot be used if people cannot find it or do not understand what it can be used for. Metadata, or "information about information", such as a book's title, a product's description, or a website's keywords, affects how discoverable something is on a database or online. Adding metadata to a product that is available online can make it easier for end users to find the product. For example, if a song file is made available online, making the title, band name, genre, year of release, and other pertinent information available in connection with this song means the file can be retrieved more easily. The organization of information through the implementation of alphabetical structures or the integration of content into search engines exemplifies strategies employed to enhance the discoverability of information. The concept of discoverability, while related to but distinct from accessibility and usability, which are other qualities that


id: discoverability-20260505-1
file: data/en.wikipedia.org/wiki/Discoverability-1.md
category: reference
title: Discoverability
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Discoverability
date_saved: 2026-05-05T07:22:29.265231+00:00
instance: kb-cron
parent: discoverability-20260505-0
summary:
=== Specific to digital media === In relation to audiovisual content, according to the meaning given by the Canadian Radio-television and Telecommunications Commission (CRTC) for the purpose of its 2016 Discoverability Summit, discoverability can be summed up to the intrinsic ability of given content to "stand out of the lot", or to position itself so as to be easily found and discovered. A piece of audiovisual content can be a movie, a TV series, music, a book (eBook), an audio book or podcast. When audiovisual content such as a digital file for a TV show, movie, or song, is made available online, if the content is "tagged" with identifying information such as the names of the key artists (e.g., actors, directors and screenwriters for TV shows and movies; singers, musicians and record producers for songs) and the genres (for movies genres, music genres, etc.). When users interact with online content, algorithms typically determine what types of content the user is interested in, and then a computer program suggests "more like this", which is other content that the user may be interested in. Different websites and systems have different algorithms, but one approach, used by Amazon (company) for its online store,


id: discoverability-20260505-2
file: data/en.wikipedia.org/wiki/Discoverability-2.md
category: reference
title: Discoverability
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Discoverability
date_saved: 2026-05-05T07:22:29.265231+00:00
instance: kb-cron
parent: discoverability-20260505-0
summary:
=== Internet search === For items that are searched for online, the goal of discoverability is to be at or near the top of the search results. Organizations may make efforts to make it more likely, that "their" content or webpages are at the top, or close to the top, of search results; these approaches are often collectively called search engine optimization (SEO). Note that when an organization takes action to increase the SEO of its website, this does not normally involve changes to the search engine itself; rather, it involves adding metadata tags and original content, among other strategies, to increase the "visibility" of the website to search engine algorithms. Recent discussions of discoverability have also considered AI-generated answer systems, in which visibility may depend not only on ranking in conventional search results but also on whether sources are selected, cited, and synthesized in generated responses. ==== Services ==== In a service delivery context, the application of this principle requires collecting information about the service during the service analysis phase as during this phase; maximum information is available about the service's functional context and the capabilities of the service. At this stage, the domain knowledge of the business experts


id: discoverability-20260505-3
file: data/en.wikipedia.org/wiki/Discoverability-3.md
category: reference
title: Discoverability
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Discoverability
date_saved: 2026-05-05T07:22:29.265231+00:00
instance: kb-cron
parent: discoverability-20260505-0
summary:
=== Algorithms === In the digital economy, sophisticated algorithms are required for the analysis of the ways that end users search for, access and use different content or products online. Thus, not only is metadata created regarding the content or product, but also data about specific users' interaction with this content. If a social media website has a user profile for a given person, indicating demographic information (age, gender, location of residence, employment status, education, etc.), then this website can collect and analyse information about tendencies and preferences of a given user or a subcategory of users. This raises potential privacy concerns. Algorithms have been called “black boxes”, because the factors used by the leading websites in their algorithms are typically proprietary information which is not released to the public. While a number of search engine optimization (SEO) firms offer the services of attempting to increase the ranking of a client's web content or website, these SEO firms do not typically know the exact algorithms used by Google and Facebook. Web crawlers can only access 26% of new online content "...by recrawling a constant fraction of the entire web". One concern raised with the increasing role of algorithms in search


id: genreflecting-20260505-0
file: data/en.wikipedia.org/wiki/Genreflecting-0.md
category: reference
title: Genreflecting
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Genreflecting
date_saved: 2026-05-05T07:22:30.577856+00:00
instance: kb-cron
parent: genreflecting-20260505-0
summary:
Genreflecting is the process of examining and analyzing the patterns and characteristics of literary genres—both fiction and recreational nonfiction—and using that analysis to identify titles with similar appeals to readers (i.e., read-alikes), in order to make reading suggestions to individuals who are looking for something to read. The term "genreflecting" was first coined by Betty Rosenberg, a prominent library science educator, in 1982. Since that time, the term has been adopted by readers' advisory in libraries and extended to nonfiction genres as well as fiction. A similar practice in retail bookstores is called "hand-selling." == History == At the time the first edition of Rosenberg's Genreflecting was written, adding popular reading materials to library collections and recommending those titles to readers were controversial practices. Dr. Rosenberg's First Law of Reading—"Never apologize for your reading taste"—has since been adopted by growing numbers of librarians, and the field of readers' advisory has become increasingly central to the practice of librarianship. Nancy Pearl, the well known model of the librarian action figure and author of Book Lust and More Book Lust, has promoted and further popularized the practice of readers' advisory within the library community and beyond. In recent years, a number of


id: information-grazing-20260505-0
file: data/en.wikipedia.org/wiki/Information_grazing-0.md
category: reference
title: Information grazing
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Information_grazing
date_saved: 2026-05-05T07:22:31.783951+00:00
instance: kb-cron
parent: information-grazing-20260505-0
summary:
Information grazing refers to the ability to quickly obtain knowledge and facts just in time to solve new problems or answer questions. "Information grazing" can also be "information jumping", jumping from site to site and cherry-picking information seems to "rewire" the brain to deleterious effects or focus on something long enough to fully understand all its implications. Unlike traditional learning, where learning a subject in depth was necessary to draw enough pertinent knowledge to solve new problems or answer questions, information grazing assumes some subjects are so large, fast-changing, interdependent, or esoteric, that traditional methods of learning may be unable to solve new problems or answer questions as efficiently. Information grazing is also one of the most commonly used coping techniques for stress and experts have noticed a trend with people many under the age of 35. The change from a traditional in-depth learning and memorizing of facts to a mentality of quickly finding, using, and then forgetting knowledge, stems from the technological singularity concept that information is growing so fast (see information explosion or information overload) that an individual can no longer hope to be a "renaissance man" or effectively keep up with some fields of knowledge. Examples of


id: knowledge-organization-system-20260505-0
file: data/en.wikipedia.org/wiki/Knowledge_organization_system-0.md
category: reference
title: Knowledge organization system
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Knowledge_organization_system
date_saved: 2026-05-05T07:22:33.029933+00:00
instance: kb-cron
parent: knowledge-organization-system-20260505-0
summary:
Knowledge organization system (KOS), concept system, or concept scheme is the generic term used in knowledge organization (KO) for the selection of concepts with an indication of selected semantic relations. Despite their differences in type, coverage, and application, all KOS aim to support the organization of knowledge and information to facilitate their management and retrieval. KOS vary in complexity from simple sorted lists to complex relational networks. They represent both structural and functional features, and serve to eliminate ambiguity, control synonyms, establish relationships, and present properties. From their origins in library and information science (LIS), KOS have been applied to other domains and disciplines within science and industry, although scholarly research and debate remain primarily within the KO field. Challenges of KOS include ambiguity of terminology, repercussions of biased systems, and potential obsolescence. KOS can be expressed in RDF and RDFS as per the Simple Knowledge Organization System (SKOS) recommendation by W3C, which aims to enable the sharing and linking of KOS via the Web. One of the largest collections of KOS is the BARTOC registry. == Types == While different schema of KOS have been proposed, most are generally arranged in terms of the complexity of their construction and


id: preservation-of-meaning-20260505-0
file: data/en.wikipedia.org/wiki/Preservation_of_meaning-0.md
category: reference
title: Preservation of meaning
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Preservation_of_meaning
date_saved: 2026-05-05T07:22:34.255393+00:00
instance: kb-cron
parent: preservation-of-meaning-20260505-0
summary:
Preservation of meaning in library, archival or museum collections involves understanding spiritual, ritual, or cultural perceptions of value for specific objects, and ensuring these values are maintained and respected. Meaning is something assigned to objects of cultural or spiritual significance based on interpretations and perceived values by user populations, a process known as social construction of an object. When moved to memory institutions such as libraries or museums, these objects of social construction require unique approaches to preservation and maintenance in order to remain relevant as representations of cultural or spiritual societies. In many memory institutions of the Western World, including libraries and museums, focus is often placed on the informational content and physical attributes, or artifactual value, of collected materials. Preservation policies are primarily concerned with the maintenance of these two things, either through reformatting to preserve textual information, or repairs and environmental controls to ensure continued existence of their physical structure. However, it is necessary to look beyond the physical and informational aspects of objects in order to ensure we are also preserving the integrity of the spiritual or cultural values which may be fundamental in defining the object. Concerns arise when actions taken to preserve the physical


id: provenance-20260505-0
file: data/en.wikipedia.org/wiki/Provenance-0.md
category: reference
title: Provenance
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Provenance
date_saved: 2026-05-05T07:22:35.468635+00:00
instance: kb-cron
parent: provenance-20260505-0
summary:
Provenance (from French provenir 'to come from/forth') is the chronology of the ownership, custody or location of a historical object. The term was originally mostly used in relation to works of art, but is now used in similar senses in a wide range of fields, including archaeology, paleontology, archival science, economy, computing, and scientific enquiry in general. The primary purpose of tracing the provenance of an object or entity is normally to provide contextual and circumstantial evidence for its original production or discovery, by establishing, as far as practicable, its later history, especially the sequences of its formal ownership, custody and places of storage. The practice has a particular value in helping authenticate objects. Comparative techniques, expert opinions and the results of scientific tests may also be used to these ends, but establishing provenance is essentially a matter of documentation. The term dates to the 1780s in English. Provenance is conceptually comparable to the legal term chain of custody. For museums and the art trade, in addition to helping establish the authorship and authenticity of an object, provenance has become increasingly important in helping establish the moral and legal validity of a chain of custody, given the increasing amount of


id: provenance-20260505-1
file: data/en.wikipedia.org/wiki/Provenance-1.md
category: reference
title: Provenance
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Provenance
date_saved: 2026-05-05T07:22:35.468635+00:00
instance: kb-cron
parent: provenance-20260505-0
summary:
The objective of provenance research is to produce a complete list of owners (together, where possible, with the supporting documentary proof) from when the painting was commissioned or in the artist's studio through to the present time. In practice, there are likely to be gaps in the list and documents that are missing or lost. The documented provenance should also list when the painting has been part of an exhibition and a bibliography of when it has been discussed, or illustrated in print. Where the research is proceeding backwards, to discover the previous provenance of a painting whose current ownership and location are known, it is important to record the physical details of the painting – style, subject, signature, materials, dimensions, frame, etc. The titles of paintings and the attribution to a particular artist may change over time. The size of the work and its description can be used to identify earlier references to the painting. The back of a painting can contain significant provenance information. There may be exhibition marks, dealer stamps, gallery labels and other indications of previous ownership. There may be shipping labels. In the BBC TV programme Fake or Fortune? the provenance of the painting Bords


id: provenance-20260505-2
file: data/en.wikipedia.org/wiki/Provenance-2.md
category: reference
title: Provenance
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Provenance
date_saved: 2026-05-05T07:22:35.468635+00:00
instance: kb-cron
parent: provenance-20260505-0
summary:
== Books == In the case of books, the study of provenance refers to the study of the ownership of individual copies of books. It is usually extended to include the study of the circumstances in which individual copies of books have changed ownership, and of evidence left in books that shows how readers interacted with them. Provenance studies may shed light on the books themselves, providing evidence of the role particular titles have played in social, intellectual and literary history. Such studies may also add to our knowledge of particular owners of books. For instance, looking at the books owned by a writer may help to show which works influenced him or her. Many provenance studies are historically focused, and concentrated on books owned by writers, politicians and public figures. The recent ownership of books is studied, however, as is evidence of how ordinary or anonymous readers have interacted with books. Provenance can be studied both by examining the books themselves, for instance looking at inscriptions, marginalia, bookplates, book rhymes, and bindings, and by reference to external sources of information such as auction catalogues. == Pianos == Provenance for pianos is authenticated before a piano is inducted into a


id: provenance-20260505-3
file: data/en.wikipedia.org/wiki/Provenance-3.md
category: reference
title: Provenance
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Provenance
date_saved: 2026-05-05T07:22:35.468635+00:00
instance: kb-cron
parent: provenance-20260505-0
summary:
Archaeology and anthropology researchers use the word provenience (or alternatively find-spot) to refer to the exact location of discovery of an artifact, a bone or other remains, a soil sample, or a feature within an ancient site, whereas the word provenance covers an object's complete documented history. Ideally, in modern excavations, the provenience is recorded in three dimensions on a site grid with great precision, and may also be recorded on video to provide additional proof and context. In older work, often undertaken by amateurs, only the general site or approximate area may be known, especially when an artifact was found outside a professional excavation and its specific position not recorded. The term provenience appeared in the 1880s, about a century after provenance. Outside of academic contexts, it has been used as a synonymous variant spelling of provenance, especially in American English. Digital analytical methods have also been applied to the reconstruction of archaeological provenience. For example, a 2025 study used high-resolution three-dimensional surface comparison to identify ancient Egyptian artefacts produced from the same mould, contributing to the reassessment of their original context. Any given antiquity may have both a provenience, where it was found, and a provenance, where it


id: provenance-20260505-4
file: data/en.wikipedia.org/wiki/Provenance-4.md
category: reference
title: Provenance
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Provenance
date_saved: 2026-05-05T07:22:35.468635+00:00
instance: kb-cron
parent: provenance-20260505-0
summary:
=== Computer science === Within computer science, informatics uses the term "provenance" to mean the lineage of data, as per data provenance, with research in the last decade extending the conceptual model of causality and relation to include processes that act on data and agents that are responsible for those processes. See, for example, the proceedings of the International Provenance Annotation Workshop (IPAW) and Theory and Practice of Provenance (TaPP). Semantic web standards bodies, including the World Wide Web Consortium in 2014, have ratified a standard data model for provenance representation known as PROV which draws from many of the better-known provenance representation systems that preceded it, such as the Proof Markup Language and the Open Provenance Model. Interoperability is a design goal of most recent computer science provenance theories and models, for example the Open Provenance Model (OPM) 2008 generation workshop aimed at "establishing inter-operability of systems" through information exchange agreements. Data models and serialisation formats for delivering provenance information typically reuse existing metadata models where possible to enable this. Both the OPM Vocabulary and the PROV Ontology make extensive use of metadata models such as Dublin Core and Semantic Web technologies such as the Web Ontology Language (OWL).


id: provenance-20260505-5
file: data/en.wikipedia.org/wiki/Provenance-5.md
category: reference
title: Provenance
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Provenance
date_saved: 2026-05-05T07:22:35.468635+00:00
instance: kb-cron
parent: provenance-20260505-0
summary:
== Computers and law == The term provenance is used when ascertaining the source of goods such as computer hardware to assess if they are genuine or counterfeit. Chain of custody is an equivalent term used in law, especially for evidence in criminal or commercial cases. Software provenance encompasses the origin of software and its licensing terms. For example, when incorporating a free, open source or proprietary software component in an application, one may wish to understand its provenance to ensure that licensing requirements are fulfilled and that other software characteristics can be understood. Data provenance covers the provenance of computerized data. There are two main aspects of data provenance: ownership of the data and data usage. Ownership will tell the user who is responsible for the source of the data, ideally including information on the originator of the data. Data usage gives details regarding how the data has been used and modified and often includes information on how to cite the data source or sources. Data provenance is of particular concern with electronic data, as data sets are often modified and copied without proper citation or acknowledgement of the originating data set. Databases make it easy to select specific


id: realia-library-science-20260505-0
file: data/en.wikipedia.org/wiki/Realia_(library_science)-0.md
category: reference
title: Realia (library science)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Realia_(library_science)
date_saved: 2026-05-05T07:22:36.750177+00:00
instance: kb-cron
parent: realia-library-science-20260505-0
summary:
In library classification systems, realia are three-dimensional objects from real life such as coins, tools, and textiles, that do not fit into the traditional categories of library material. They can be either human-made (artifacts, tools, utensils, etc.) or naturally occurring (specimens, samples, etc.), usually borrowed, purchased, or received as donation by a teacher, library, or museum for use in classroom instruction or in exhibits. Archival and manuscript collections often receive items of memorabilia such as badges, emblems, insignias, jewelry, leather goods, needlework, etc., in connection with gifts of personal papers. Most government or institutional archives reject gifts of non-documentary objects, unless they have a documentary value. When accepting large bequests of mixed objects, they normally have the donors sign legal documents giving permission to the archive to destroy, exchange, sell, or dispose in any way those objects which, according to the best judgement of the archivist, are not manuscripts (which can include typescripts or printouts) or are not immediately useful for understanding the manuscripts. Recently, the usage of this term has been criticized by librarians based on the usage of the term realia to refer to artistic and historical artifacts and objects. The noun phrase "real world object" has been


id: realia-library-science-20260505-1
file: data/en.wikipedia.org/wiki/Realia_(library_science)-1.md
category: reference
title: Realia (library science)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Realia_(library_science)
date_saved: 2026-05-05T07:22:36.750177+00:00
instance: kb-cron
parent: realia-library-science-20260505-0
summary:
=== Hair, wool, and silk === Little research has been conducted regarding the preservation and conservation of human hair within the context of the archival world. However, much is known of the chemical structure and behavior of human hair, thanks in large part to the cosmetology industry; there is certainly a great deal of scholarship surrounding the care of other protein-based fibers, silk, and wool. Loose, braided, tied lovingly in silk ribbons, hair was memorialized especially by the Victorians. Often, hair would be encased within a glass or woven with metallic threads, paired with a metal setting to be worn as mourning jewelry, a memento of a friend or loved one. Hairwork, an art still practiced today by the Victorian Hairwork Society Archived 2016-08-23 at the Wayback Machine, was a popular needlework method, often resulting in two-dimensional mourning wreaths and graveyard scenes, worked in hair on a fabric ground. Wool, because of its ability to wick moisture and resist flame, was commonly used in Western textiles throughout history, and might present itself in the form of bed linens, clothing, and needlework. Hair (both human and animal) mainly consists of a protein, keratin, the fibers of which give the inner core


id: relevance-20260505-0
file: data/en.wikipedia.org/wiki/Relevance-0.md
category: reference
title: Relevance
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Relevance
date_saved: 2026-05-05T07:22:38.060447+00:00
instance: kb-cron
parent: relevance-20260505-0
summary:
Relevance is the connection between topics that makes one useful for dealing with the other. Relevance is studied in many different fields, including cognitive science, logic, and library and information science. Epistemology studies it in general, and different theories of knowledge have different implications for what is considered relevant. == Definition == "Something (A) is relevant to a task (T) if it increases the likelihood of accomplishing the goal (G), which is implied by T." A thing might be relevant, a document or a piece of information may be relevant. Relevance does not depend on whether we speak of "things" or "information". == Epistemology == If you believe that schizophrenia is caused by bad communication between mother and child, then family interaction studies become relevant. If, on the other hand, you subscribe to a genetic theory of relevance then the study of genes becomes relevant. If you subscribe to the epistemology of empiricism, then only intersubjectively controlled observations are relevant. If, on the other hand, you subscribe to feminist epistemology, then the sex of the observer becomes relevant. == Logic == In formal reasoning, relevance has proved an important but elusive concept. It is important because the solution of any


id: relevance-20260505-1
file: data/en.wikipedia.org/wiki/Relevance-1.md
category: reference
title: Relevance
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Relevance
date_saved: 2026-05-05T07:22:38.060447+00:00
instance: kb-cron
parent: relevance-20260505-0
summary:
In 1986, Dan Sperber and Deirdre Wilson drew attention to the central importance of relevance decisions in reasoning and communication. They proposed an account of the process of inferring relevant information from any given utterance. To do this work, they used what they called the "Principle of Relevance": namely, the position that any utterance addressed to someone automatically conveys the presumption of its own optimal relevance. The central idea of Sperber and Wilson's theory is that all utterances are encountered in some context, and the correct interpretation of a particular utterance is the one that allows most new implications to be made in that context on the basis of the least amount of information necessary to convey it. For Sperber and Wilson, relevance is conceived as relative or subjective, as it depends upon the state of knowledge of a hearer when they encounter an utterance. Sperber and Wilson stress that this theory is not intended to account for every intuitive application of the English word "relevance". Relevance, as a technical term, is restricted to relationships between utterances and interpretations, and so the theory cannot account for intuitions such as the one that relevance relationships obtain in problems involving physical objects.


id: search-retrieve-web-service-20260505-0
file: data/en.wikipedia.org/wiki/Search/Retrieve_Web_Service-0.md
category: reference
title: Search/Retrieve Web Service
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Search/Retrieve_Web_Service
date_saved: 2026-05-05T07:22:39.231473+00:00
instance: kb-cron
parent: search-retrieve-web-service-20260505-0
summary:
Search/Retrieve Web service (SRW) is a web service for search and retrieval. SRW provides a SOAP interface to queries, to augment the URL interface provided by its companion protocol Search/Retrieve via URL (SRU). Queries in SRU and SRW are expressed using the Contextual Query Language (CQL). Standards for SRW, SRU, and CQL are promulgated by the United States Library of Congress. The SRW service and SRU protocol were both created by as part of the ZING (Z39.50 International: Next Generation) initiative as successors to the Z39.50 protocol. == Example usage == == See also == Z39.50 === Implementations === refbase RefDB Te Ara: The Encyclopedia of New Zealand == External links == SRU: Search/Retrieve via URL SRW: Search/Retrieve Web Service CQL: Contextual Query Language


id: secondary-data-20260505-0
file: data/en.wikipedia.org/wiki/Secondary_data-0.md
category: reference
title: Secondary data
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Secondary_data
date_saved: 2026-05-05T07:22:40.477953+00:00
instance: kb-cron
parent: secondary-data-20260505-0
summary:
Secondary data refers to data that is collected by someone other than the primary user. Common sources of secondary data for social science include censuses, information collected by government departments, organizational records and data that was originally collected for other research purposes. Primary data, by contrast, are collected by the investigator conducting the research. Secondary data analysis can save time that would otherwise be spent collecting data and, particularly in the case of quantitative data, can provide larger and higher-quality databases that would be unfeasible for any individual researcher to collect on their own. In addition, analysts of social and economic change consider secondary data essential, since it is impossible to conduct a new survey that can adequately capture past change and/or developments. However, secondary data analysis can be less useful in marketing research, as data may be outdated or inaccurate. == Sources of secondary data == Secondary data can be obtained from many sources: censuses and government departments like housing, social security, electoral statistics, tax records internet searches and libraries GPS and remote sensing km progress reports journals, newspapers and magazines === Administrative data and census === Government departments and agencies routinely collect information when registering people or carrying


id: secondary-source-20260505-0
file: data/en.wikipedia.org/wiki/Secondary_source-0.md
category: reference
title: Secondary source
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Secondary_source
date_saved: 2026-05-05T07:22:41.737889+00:00
instance: kb-cron
parent: secondary-source-20260505-0
summary:
In scholarship, a secondary source is a document or recording that relates or discusses information originally presented elsewhere. A secondary source contrasts with a primary, or original, source of the information being discussed. A primary source can be a person with direct knowledge of a situation or it may be a document created by such a person. A secondary source is one that gives information about a primary source. In a secondary source, the original information is selected, modified and arranged in a suitable format. Secondary sources involve generalization, analysis, interpretation, or evaluation of the original information. The most accurate classification for any given source is not always obvious. "Primary" and "secondary" are relative terms, and some sources may be classified as primary or secondary, depending on how they are used. A third level, the tertiary source, such as an encyclopedia or dictionary, resembles a secondary source in that it contains analysis, but a tertiary source has a different purpose: it aims to elaborate a broad introductory overview of the topic at hand. == Classification of sources == Making distinctions between primary and secondary symbolic sources (objects meant to communicate information) is both subjective and contextual, such that precise definitions


id: secondary-source-20260505-1
file: data/en.wikipedia.org/wiki/Secondary_source-1.md
category: reference
title: Secondary source
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Secondary_source
date_saved: 2026-05-05T07:22:41.737889+00:00
instance: kb-cron
parent: secondary-source-20260505-0
summary:
=== Humanities and history === Secondary sources in history and humanities are usually books or scholarly journals, from the perspective of a later interpreter, especially by a later scholar. In the humanities, a peer reviewed article is always a secondary source. The delineation of sources as primary and secondary first arose in the field of historiography, as historians attempted to identify and classify the sources of historical writing. In scholarly writing, an important objective of classifying sources is to determine the independence and reliability of sources. In original scholarly writing, historians rely on primary sources, read in the context of the scholarly interpretations. Following the Rankean model established by German scholarship in the 19th century, historians use archives of primary sources. Most undergraduate research projects rely on secondary source material, with perhaps snippets of primary sources. ==== Law ==== In the legal field, source classification is important because the persuasiveness of a source usually depends upon its history. Primary sources may include cases, constitutions, statutes, administrative regulations, and other sources of binding legal authority, while secondary legal sources may include books, the headnotes of case reports, articles, and encyclopedias. Legal writers usually prefer to cite primary sources because only primary


id: selective-dissemination-of-information-20260505-0
file: data/en.wikipedia.org/wiki/Selective_dissemination_of_information-0.md
category: reference
title: Selective dissemination of information
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Selective_dissemination_of_information
date_saved: 2026-05-05T07:22:42.915152+00:00
instance: kb-cron
parent: selective-dissemination-of-information-20260505-0
summary:
Selective dissemination of information (SDI) was originally a phrase related to library and information science. SDI refers to tools and resources used to keep a user informed of new resources on specified topics, including the current-awareness services used to inform about new library acquisitions. SDI services pre-date the World Wide Web, and the term itself is somewhat dated. Contemporary analogous systems for SDI services include alerts, current awareness tools or trackers. These systems provide automated searches that inform the user of the availability of new resources meeting the user's specified keywords and search parameters. Alerts can be received a number of ways, including email, RSS feeds, voice mail, Instant messaging, and text messaging. Selective dissemination of information was a concept first described by Hans Peter Luhn of IBM in the 1950s. Software was developed in many companies and in government to provide this service in the 1950s and 60s, which allowed distribution of items recently published in abstract journals to be routed to individuals who are likely to be interested in the contents. For example, the system at Ft. Monmouth automatically sent out (by mail) a different set of abstracts to each of about 1,000 scientists and engineers in the


id: shelfmark-20260505-0
file: data/en.wikipedia.org/wiki/Shelfmark-0.md
category: reference
title: Shelfmark
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Shelfmark
date_saved: 2026-05-05T07:22:44.090781+00:00
instance: kb-cron
parent: shelfmark-20260505-0
summary:
A shelfmark is a mark in a book or manuscript that denotes the cupboard or bookcase where it is kept as well as the shelf and possibly even its location on the shelf. The closely related term pressmark (from press, meaning cupboard) denotes only the cupboard or case. It is distinct from a call number, which is the code under which a book or manuscript is registered and which is used to identify it when ordering it. Sometimes a shelfmark or pressmark may be used as a call number, but in other cases the call number contains no information about the book's physical location. In certain American institutions, shelfmark and call number are combined to create a long code containing information on location, classification, size, binding, author and date. Shelfmarks and pressmarks were usually written, inscribed or stamped on the pastedowns. When a book was moved, the old shelfmark was usually crossed out and a new one added. Old shelfmarks can sometimes provide valuable information about a manuscript's provenance. Shelfmarks originated in the early Middle Ages, usually as combinations of numbers and letters, probably indicating the cupboard and shelf. Letters later came to be assigned to specific batch acquisitions. In


id: source-literature-20260505-0
file: data/en.wikipedia.org/wiki/Source_literature-0.md
category: reference
title: Source literature
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Source_literature
date_saved: 2026-05-05T07:22:45.321851+00:00
instance: kb-cron
parent: source-literature-20260505-0
summary:
Source literature (understood as printed texts) is a kind of information source. It might, for example, be cited and used as sources in academic writings, and then called the literature on the subject. The meaning of "source literature" is relative. From the point of view of a bibliographic index the indexed papers are "source literature". For example, the Social Sciences Citation Index is a "source index" covering the journals being indexed. These journals are the source literature from the point of view of this index. But from the point of view of the indexed papers they are the bibliographical references contained in the single papers "source literature". In the humanities, the term "source literature" has a more precise meaning than "published sources": Many archives, for example, publish important sources to be used by historians and other scholars as reliable editions of formerly unpublished sources. The publishing of such sources requires knowledge of text philology and other fields. But this kind of expertise put into the publishing of source literature should be differentiated from the kind of expertise needed in order to use the sources in, for example, historical research. A historian may or may not use such "source literature" and


id: subject-documents-20260505-0
file: data/en.wikipedia.org/wiki/Subject_(documents)-0.md
category: reference
title: Subject (documents)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Subject_(documents)
date_saved: 2026-05-05T07:22:46.645452+00:00
instance: kb-cron
parent: subject-documents-20260505-0
summary:
In library and information science documents (such as books, articles and pictures) are classified and searched by subject – as well as by other attributes such as author, genre and document type. This makes "subject" a fundamental term in this field. Library and information specialists assign subject labels to documents to make them findable. There are many ways to do this and in general there is not always consensus about which subject should be assigned to a given document. To optimize subject indexing and searching, we need to have a deeper understanding of what a subject is. The question: "what is to be understood by the statement 'document A belongs to subject category X'?" has been debated in the field for more than 100 years (see below) == Theoretical view == === Charles Ammi Cutter (1837–1903) === For Cutter the stability of subjects depends on a social process in which their meaning is stabilized in a name or a designation. A subject "referred [...] to those intellections [...] that had received a name that itself represented a distinct consensus in usage" (Miksa, 1983a, p. 60) and: the "systematic structure of established subjects" is "resident in the public realm" (Miksa, 1983a,


id: subject-documents-20260505-1
file: data/en.wikipedia.org/wiki/Subject_(documents)-1.md
category: reference
title: Subject (documents)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Subject_(documents)
date_saved: 2026-05-05T07:22:46.645452+00:00
instance: kb-cron
parent: subject-documents-20260505-0
summary:
Ranganathan's definition of "subject" is strongly influenced by his Colon Classification system. The colon system is based on the combination of single elements from facets to subject designation. This is the reason why the combined nature of subjects are emphasized so strongly. It leads, however, to absurdities such as the claim that gold cannot be a subject (but is alternatively termed "an isolate"). This aspect of the theory has been criticized by Metcalfe (1973, p. 318). Metcalfe's skepticism regarding Ranganathan's theory is formulated in hard words (op. cit., p. 317): "This pseudo-science imposed itself on British disciples from about 1950 on...". It seems unacceptable that Ranganathan defines the word subject in a way that favors his own system. A scientific concept like "subject" should make it possible to compare different ways of establishing access to information. Whether or not subjects are combined or not should be examined once their definition has been given, it should not determined a priori, in the definition. Besides the emphasis on the combined, organizing and systematizing nature of subjects contains Ranganathan's definition of subject the pragmatic demand, that a subject should be determined in a way that suits a normal person's competency or specialization. Again


id: subject-documents-20260505-2
file: data/en.wikipedia.org/wiki/Subject_(documents)-2.md
category: reference
title: Subject (documents)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Subject_(documents)
date_saved: 2026-05-05T07:22:46.645452+00:00
instance: kb-cron
parent: subject-documents-20260505-0
summary:
=== Other views and definitions === In the ISO-standard for topic maps the concept of subject is defined this way: "Subject Anything whatsoever, regardless of whether it exists or has any other specific characteristics, about which anything whatsoever may be asserted by any means whatsoever." ISO 13250-1, here cited from draft: http://www1.y12.doe.gov/capabilities/sgml/sc34/document/0446.htm#overview) This definition may work well with the closed system of concepts provided by the topic maps standard. In broader contexts, however, is not fruitful because it does not contain any specification of what to identify in a document or in a discourse when ascribing subject identification terms or symbols to it. If different methods of subject analysis imply different results, which of these results can then be said to reflect the (true) subject? (Given that the expression "a true subject assignment" is meaningful at all, which is an important part of the problem). Different persons may have different opinions about what the subject of a specific document is. How can a theoretical understanding of the term "subject" be helpful deciding principles of subject analysis? == Related concepts == === Indexing words versus concepts versus subjects === A proposal for the differentiation between concept indexing and subject indexing was


id: transliteracy-20260505-0
file: data/en.wikipedia.org/wiki/Transliteracy-0.md
category: reference
title: Transliteracy
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Transliteracy
date_saved: 2026-05-05T07:22:47.829817+00:00
instance: kb-cron
parent: transliteracy-20260505-0
summary:
Transliteracy is "a fluidity of movement across a range of technologies, media and contexts". It is an ability to use diverse techniques to collaborate across different social groups. Transliteracy combines a range of capabilities required to move across a range of contexts, media, technologies and genres. Conceptually, transliteracy is situated across five capabilities: information capabilities (see information literacy), ICT (information and communication technologies), communication and collaboration, creativity and critical thinking. It is underpinned by literacy and numeracy. (See figure below) The concept of transliteracy is impacting the system of education and libraries. == History == While the term appears to come from the prefix trans- ('across') and the word literacy, the scholars who coined it say they developed it from the practice of transliteration, which means to use the letters of one language to write down a different language. The study of transliteracy was first developed in 2005 by the Transliteracies Research Project, directed by University of California at Santa Barbara Professor Alan Liu. The concept of 'transliteracies' was developed as part of research into online reading. It was shared and refined at the Transliteracies conference, held at UC Santa Barbara in 2005. The conference inspired the at the time


id: transliteracy-20260505-1
file: data/en.wikipedia.org/wiki/Transliteracy-1.md
category: reference
title: Transliteracy
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Transliteracy
date_saved: 2026-05-05T07:22:47.829817+00:00
instance: kb-cron
parent: transliteracy-20260505-0
summary:
== Transliteracy in libraries == In 2009, Dr. Susie Andretta, senior lecturer in Information Management at London Metropolitan University, conducted interviews with four different information professionals including an academic librarian, an outreach librarian, a content manager, and a scholar within the library science and information discipline. She was aiming to explore how transliteracy was colliding and combining with the print-world of libraries. Dr. Andretta defines transliteracy as "an umbrella term encompassing different literacies and multiple communication channels that require active participation with and across a range of platforms, and embracing both linear and non-linear messages (3)." The goals of these interviews ranged from the following: to test the information professional's awareness of transliteracy, to have them identify transliteracy and how it is integrated into their work, and to explain the impact transliteracy has had on they library they work at. Andretta found that out of all the information professionals interviewed, it was only the academic librarian who was vaguely familiar with the concept of transliteracy. Bernadette Daly Swanson, an Academic Librarian at UC Davis, expresses in her interview with Dr. Andretta how she would "like to think that the transliterate library is more of an environment where we do different


id: glossary-of-mathematical-jargon-20260505-0
file: data/en.wikipedia.org/wiki/Glossary_of_mathematical_jargon-0.md
category: reference
title: Glossary of mathematical jargon
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Glossary_of_mathematical_jargon
date_saved: 2026-05-05T07:22:49.499904+00:00
instance: kb-cron
parent: glossary-of-mathematical-jargon-20260505-0
summary:
The language of mathematics has a wide vocabulary of specialist and technical terms. It also has a certain amount of jargon: commonly used phrases which are part of the culture of mathematics, rather than of the subject. Jargon often appears in lectures, and sometimes in print, as informal shorthand for rigorous arguments or precise ideas. Much of this uses common English words, but with a specific non-obvious meaning when used in a mathematical sense. Some phrases, like "in general", appear below in more than one section. == Philosophy of mathematics == abstract nonsense A tongue-in-cheek reference to category theory, using which one can employ arguments that establish a (possibly concrete) result without reference to any specifics of the present problem. For that reason, it is also known as general abstract nonsense or generalized abstract nonsense. [The paper of Eilenberg and Mac Lane (1942)] introduced the very abstract idea of a 'category' — a subject then called 'general abstract nonsense'! [Grothendieck] raised algebraic geometry to a new level of abstraction...if certain mathematicians could console themselves for a time with the hope that all these complicated structures were 'abstract nonsense'...the later papers of Grothendieck and others showed that classical problems...which had resisted


id: glossary-of-mathematical-jargon-20260505-1
file: data/en.wikipedia.org/wiki/Glossary_of_mathematical_jargon-1.md
category: reference
title: Glossary of mathematical jargon
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Glossary_of_mathematical_jargon
date_saved: 2026-05-05T07:22:49.499904+00:00
instance: kb-cron
parent: glossary-of-mathematical-jargon-20260505-0
summary:
== Descriptive informalities == Although ultimately every mathematical argument must meet a high standard of precision, mathematicians use descriptive but informal statements to discuss recurring themes or concepts with unwieldy formal statements. Note that many of the terms are completely rigorous in context.


id: glossary-of-mathematical-jargon-20260505-2
file: data/en.wikipedia.org/wiki/Glossary_of_mathematical_jargon-2.md
category: reference
title: Glossary of mathematical jargon
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Glossary_of_mathematical_jargon
date_saved: 2026-05-05T07:22:49.499904+00:00
instance: kb-cron
parent: glossary-of-mathematical-jargon-20260505-0
summary:
almost all A shorthand term for "all except for a set of measure zero", when there is a measure to speak of, with the phrases almost surely and almost everywhere having related meanings. For example, "almost all real numbers are transcendental" because the algebraic real numbers form a countable subset of the real numbers with measure zero. One can also speak of "almost all" integers having a property to mean "all except finitely many", despite the integers not admitting a measure for which this agrees with the previous usage, such as "almost all prime numbers are odd". There is a more complicated meaning for integers as well, discussed in the main article. Finally, this term is sometimes used synonymously with generic, below. arbitrarily large Notions which arise mostly in the context of limits, referring to the recurrence of a phenomenon as the limit is approached. A statement such as that predicate P is satisfied by arbitrarily large values, can be expressed in more formal notation by ∀x : ∃y ≥ x : P(y). See also frequently. The statement that quantity f(x) depending on x "can be made" arbitrarily large, corresponds to ∀y : ∃x : f(x) ≥ y. arbitrary A


id: glossary-of-mathematical-jargon-20260505-3
file: data/en.wikipedia.org/wiki/Glossary_of_mathematical_jargon-3.md
category: reference
title: Glossary of mathematical jargon
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Glossary_of_mathematical_jargon
date_saved: 2026-05-05T07:22:49.499904+00:00
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parent: glossary-of-mathematical-jargon-20260505-0
summary:
These properties might include possessing a specified number of derivatives, with the function and its derivatives exhibiting some nice property (see nice above), such as Hölder continuity. Informally, this term is sometimes used synonymously with smooth, below. These imprecise uses of the word regular are not to be confused with the notion of a regular topological space, which is rigorously defined. resp. (Respectively) A convention to shorten parallel expositions. "A (resp. B) [has some relationship to] X (resp. Y)" means that A [has some relationship to] X and also that B [has (the same) relationship to] Y. For example, squares (resp. triangles) have 4 sides (resp. 3 sides); or compact (resp. Lindelöf) spaces are ones where every open cover has a finite (resp. countable) open subcover. sharp Often, a mathematical theorem will establish constraints on the behavior of some object; for example, a function will be shown to have an upper or lower bound. The constraint is sharp (sometimes optimal) if it cannot be made more restrictive without failing in some cases. For example, for arbitrary non-negative real numbers x, the exponential function ex, where e = 2.7182818..., gives an upper bound on the values of the quadratic function x2.


id: glossary-of-mathematical-jargon-20260505-4
file: data/en.wikipedia.org/wiki/Glossary_of_mathematical_jargon-4.md
category: reference
title: Glossary of mathematical jargon
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Glossary_of_mathematical_jargon
date_saved: 2026-05-05T07:22:49.499904+00:00
instance: kb-cron
parent: glossary-of-mathematical-jargon-20260505-0
summary:
== Proof terminology == The formal language of proof draws repeatedly from a small pool of ideas, many of which are invoked through various lexical shorthands in practice. aliter An obsolescent term which is used to announce to the reader an alternative method, or proof of a result. In a proof, it therefore flags a piece of reasoning that is superfluous from a logical point of view, but has some other interest. by way of contradiction (BWOC), or "for, if not, ..." The rhetorical prelude to a proof by contradiction, preceding the negation of the statement to be proved. if and only if (iff) An abbreviation for logical equivalence of statements. in general In the context of proofs, this phrase is often seen in induction arguments when passing from the base case to the induction step, and similarly, in the definition of sequences whose first few terms are exhibited as examples of the formula giving every term of the sequence. necessary and sufficient A minor variant on "if and only if"; "A is necessary (and sufficient) for B" means "A if (only if) B". For example, "For a field K to be algebraically closed it is necessary and sufficient that


id: glossary-of-mathematical-jargon-20260505-5
file: data/en.wikipedia.org/wiki/Glossary_of_mathematical_jargon-5.md
category: reference
title: Glossary of mathematical jargon
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Glossary_of_mathematical_jargon
date_saved: 2026-05-05T07:22:49.499904+00:00
instance: kb-cron
parent: glossary-of-mathematical-jargon-20260505-0
summary:
angle chasing Used to describe a geometrical proof that involves finding relationships between the various angles in a diagram. back-of-the-envelope calculation An informal computation omitting much rigor without sacrificing correctness. Often this computation is "proof of concept" and treats only an accessible special case. brute force Rather than finding underlying principles or patterns, this is a method where one would evaluate as many cases as needed to sufficiently prove or provide convincing evidence that the thing in question is true. Sometimes this involves evaluating every possible case (where it is also known as proof by exhaustion). by example A proof by example is an argument whereby a statement is not proved but instead illustrated by an example. If done well, the specific example would easily generalize to a general proof. by inspection A rhetorical shortcut made by authors who invite the reader to verify, at a glance, the correctness of a proposed expression or deduction. If an expression can be evaluated by straightforward application of simple techniques and without recourse to extended calculation or general theory, then it can be evaluated by inspection. It is also applied to solving equations; for example to find roots of a quadratic equation by


id: glossary-of-mathematical-jargon-20260505-6
file: data/en.wikipedia.org/wiki/Glossary_of_mathematical_jargon-6.md
category: reference
title: Glossary of mathematical jargon
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Glossary_of_mathematical_jargon
date_saved: 2026-05-05T07:22:49.499904+00:00
instance: kb-cron
parent: glossary-of-mathematical-jargon-20260505-0
summary:
multivalued A "multivalued function” from a set A to a set B is a function from A to the subsets of B. It has typically the property that, for almost all points x of B, there is a neighbourhood of x such that the restriction of the function to the neighbourhood can be considered as a set of functions from the neighbourhood to B. projection A projection is, roughly, a map from some space or object to another that omits some information on the object or space. For example, R 2 → R , ( x , y ) ↦ x {\displaystyle \mathbb {R} ^{2}\to \mathbb {R} ,(x,y)\mapsto x} is a projection and its restriction to a graph of a function, say, is also a projection. The terms “idempotent operator” and “forgetful map” are also synonyms for a projection. structure A mathematical structure on an object is an additional set of objects or data attached to the object (e.g., relation, operation, metric, topology). == See also == Glossary of areas of mathematics List of mathematical constants List of mathematical symbols Category:Mathematical terminology == Notes == == References == Eilenberg, Samuel; Mac Lane, Saunders (1942), "Natural Isomorphisms in Group Theory", Proc.


id: partial-differential-20260505-0
file: data/en.wikipedia.org/wiki/Partial_differential-0.md
category: reference
title: Partial differential
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Partial_differential
date_saved: 2026-05-05T07:22:50.785154+00:00
instance: kb-cron
parent: partial-differential-20260505-0
summary:
The character ∂ (Unicode: U+2202) is a stylized cursive d mainly used as a mathematical symbol, usually to denote a partial derivative such as ∂ z / ∂ x {\displaystyle {\partial z}/{\partial x}} (read as "the partial derivative of z with respect to x"). It is also used for boundary of a set, the boundary operator in a chain complex, and the conjugate of the Dolbeault operator on smooth differential forms over a complex manifold. It should be distinguished from other similar-looking symbols such as lowercase Greek letter delta (δ) or the lowercase Latin letter eth (ð). == History == The symbol was introduced originally in 1770 by Nicolas de Condorcet, who used it for a partial differential, and adopted for the partial derivative by Adrien-Marie Legendre in 1786. It represents a specialized cursive type of the letter d, just as the integral sign originates as a specialized type of a long s (first used in print by Leibniz in 1686). Use of the symbol was discontinued by Legendre, but it was taken up again by Carl Gustav Jacob Jacobi in 1841, whose usage became widely adopted. == Names and coding == The symbol is variously referred to as "partial",


id: abstract-nonsense-20260505-0
file: data/en.wikipedia.org/wiki/Abstract_nonsense-0.md
category: reference
title: Abstract nonsense
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Abstract_nonsense
date_saved: 2026-05-05T07:22:52.017389+00:00
instance: kb-cron
parent: abstract-nonsense-20260505-0
summary:
In mathematics, abstract nonsense, general abstract nonsense, generalized abstract nonsense, and general nonsense are nonderogatory terms used by mathematicians to describe long, theoretical parts of a proof they skip over when readers are expected to be familiar with them. These terms are mainly used for abstract methods related to category theory and homological algebra. More generally, "abstract nonsense" may refer to a proof that relies on category-theoretic methods, or even to the study of category theory itself. == Background == Roughly speaking, category theory is the study of the general form, that is, categories of mathematical theories, without regard to their content. As a result, mathematical proofs that rely on category-theoretic ideas often seem out-of-context, somewhat akin to a non sequitur. Authors sometimes dub these proofs "abstract nonsense" as a light-hearted way of alerting readers to their abstract nature. Labeling an argument "abstract nonsense" is usually not intended to be derogatory, and is instead used jokingly, in a self-deprecating way, affectionately, or even as a compliment to the generality of the argument. Alexander Grothendieck was critical of this notion, and stated that: The introduction of the cipher 0 or the group concept was general nonsense too, and mathematics was more


id: abstract-structure-20260505-0
file: data/en.wikipedia.org/wiki/Abstract_structure-0.md
category: reference
title: Abstract structure
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Abstract_structure
date_saved: 2026-05-05T07:22:53.223922+00:00
instance: kb-cron
parent: abstract-structure-20260505-0
summary:
In mathematics and related fields, an abstract structure is a way of describing a set of mathematical objects and the relationships between them, focusing on the essential rules and properties rather than any specific meaning or example. For example, in a game such as chess, the rules of how the pieces move and interact define the structure of the game, regardless of whether the pieces are made of wood or plastic. Similarly, an abstract structure defines a framework of objects, operations, and relationships. These structures are studied in their own right, revealing fundamental mathematical principles. While a real-world object or computer program might represent, instantiate, or implement an abstract structure, the structure itself exists as an abstract concept, independent of any particular representation. This abstraction allows to see common patterns across seemingly different areas of mathematics and to apply the same reasoning and tools to analyze them. Abstract structures are studied not only in logic and mathematics but in the fields that apply them, as computer science and computer graphics, and in the studies that reflect on them, such as philosophy (especially the philosophy of mathematics). An abstract structure has a richer structure than a concept or an idea. An


id: abstraction-mathematics-20260505-0
file: data/en.wikipedia.org/wiki/Abstraction_(mathematics)-0.md
category: reference
title: Abstraction (mathematics)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Abstraction_(mathematics)
date_saved: 2026-05-05T07:22:54.412897+00:00
instance: kb-cron
parent: abstraction-mathematics-20260505-0
summary:
Abstraction in mathematics is the process of extracting the underlying structures, patterns or properties of a mathematical concept, removing any dependence on real world objects with which it might originally have been connected, and generalizing it so that it has wider applications or matching among other abstract descriptions of equivalent phenomena. In other words, to be abstract is to remove context and application. Two of the most highly abstract areas of modern mathematics are category theory and model theory. == Description == Many areas of mathematics began with the study of real world problems, before the underlying rules and concepts were identified and defined as abstract structures. For example, geometry has its origins in the calculation of distances and areas in the real world, and algebra started with methods of solving problems in arithmetic. Abstraction is an ongoing process in mathematics and the historical development of many mathematical topics exhibits a progression from the concrete to the abstract. For example, the first steps in the abstraction of geometry were historically made by the ancient Greeks, with Euclid's Elements being the earliest extant documentation of the axioms of plane geometry—though Proclus tells of an earlier axiomatisation by Hippocrates of Chios. In


id: abuse-of-notation-20260505-0
file: data/en.wikipedia.org/wiki/Abuse_of_notation-0.md
category: reference
title: Abuse of notation
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Abuse_of_notation
date_saved: 2026-05-05T07:22:55.629926+00:00
instance: kb-cron
parent: abuse-of-notation-20260505-0
summary:
In mathematics, abuse of notation occurs when an author uses a mathematical notation in a way that is not entirely formally correct, but which might help to simplify the exposition or to suggest the correct intuition, while possibly minimizing errors and confusion at the same time. A closely related concept is abuse of language or abuse of terminology, where a term — rather than a notation — is used informally. For example, while the word representation properly designates a group homomorphism from a group G to GL(V), where V is a vector space, it is common to call V itself a "representation of G." Since both mathematical notation and terminology vary across time and context, what is flagged as abuse in one context could be formally correct in another. Furthermore, abuse of notation should not be conflated with misuse of notation, which does not have the presentational benefits of the former and should be avoided, such as the misuse of constants of integration. == Examples == === Suppression of parameters === Many mathematical objects consist of a set, often called the underlying set, equipped with some additional structure, such as a mathematical operation or a topology. It is a common


id: abuse-of-notation-20260505-1
file: data/en.wikipedia.org/wiki/Abuse_of_notation-1.md
category: reference
title: Abuse of notation
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Abuse_of_notation
date_saved: 2026-05-05T07:22:55.629926+00:00
instance: kb-cron
parent: abuse-of-notation-20260505-0
summary:
( E × F ) × G = E × ( F × G ) = E × F × G {\displaystyle (E\times F)\times G=E\times (F\times G)=E\times F\times G} . But this is strictly speaking not true: if x ∈ E {\displaystyle x\in E} , y ∈ F {\displaystyle y\in F} and z ∈ G {\displaystyle z\in G} , the identity ( ( x , y ) , z ) = ( x , ( y , z ) ) {\displaystyle ((x,y),z)=(x,(y,z))} would imply that ( x , y ) = x {\displaystyle (x,y)=x} and z = ( y , z ) {\displaystyle z=(y,z)} , and so ( ( x , y ) , z ) = ( x , y , z ) {\displaystyle ((x,y),z)=(x,y,z)} would mean nothing. However, these equalities can be legitimized and made rigorous in category theory—using the idea of a natural isomorphism. Another example of similar abuses occurs in statements such as "there are two non-Abelian groups of order 8", which more strictly stated means "there are two isomorphism classes of non-Abelian groups of order 8". === Equivalence classes === Referring to an equivalence class of an equivalence relation by x {\displaystyle x} instead of


id: active-and-passive-transformation-20260505-0
file: data/en.wikipedia.org/wiki/Active_and_passive_transformation-0.md
category: reference
title: Active and passive transformation
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Active_and_passive_transformation
date_saved: 2026-05-05T07:22:56.857340+00:00
instance: kb-cron
parent: active-and-passive-transformation-20260505-0
summary:
Geometric transformations can be distinguished into two types: active or alibi transformations which change the physical position of a set of points relative to a fixed frame of reference or coordinate system (alibi meaning "being somewhere else at the same time"); and passive or alias transformations which leave points fixed but change the frame of reference or coordinate system relative to which they are described (alias meaning "going under a different name"). For instance, active transformations are useful to describe successive positions of a rigid body. On the other hand, passive transformations may be useful in human motion analysis to observe the motion of the tibia relative to the femur, that is, its motion relative to a (local) coordinate system which moves together with the femur, rather than a (global) coordinate system which is fixed to the floor. In three-dimensional Euclidean space, any proper rigid transformation, whether active or passive, can be represented as a screw displacement, the composition of a translation along an axis and a rotation about that axis. The terms active transformation and passive transformation were first introduced in 1957 by Valentine Bargmann for describing Lorentz transformations in special relativity. == Example == As an example, let


id: active-and-passive-transformation-20260505-1
file: data/en.wikipedia.org/wiki/Active_and_passive_transformation-1.md
category: reference
title: Active and passive transformation
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Active_and_passive_transformation
date_saved: 2026-05-05T07:22:56.857340+00:00
instance: kb-cron
parent: active-and-passive-transformation-20260505-0
summary:
The distinction between active and passive transformations can be seen mathematically by considering abstract vector spaces. Fix a finite-dimensional vector space V {\displaystyle V} over a field K {\displaystyle K} (thought of as R {\displaystyle \mathbb {R} } or C {\displaystyle \mathbb {C} } ), and a basis B = { e i } 1 ≤ i ≤ n {\displaystyle {\mathcal {B}}=\{e_{i}\}_{1\leq i\leq n}} of V {\displaystyle V} . This basis provides an isomorphism C : K n → V {\displaystyle C:K^{n}\rightarrow V} via the component map ( v i ) 1 ≤ i ≤ n = ( v 1 , ⋯ , v n ) ↦ ∑ i v i e i {\textstyle (v_{i})_{1\leq i\leq n}=(v_{1},\cdots ,v_{n})\mapsto \sum _{i}v_{i}e_{i}} . An active transformation is then an endomorphism on V {\displaystyle V} , that is, a linear map from V {\displaystyle V} to itself. Taking such a transformation τ ∈ End ( V ) {\displaystyle \tau \in {\text{End}}(V)} , a vector v ∈ V {\displaystyle v\in V} transforms as v ↦ τ v {\displaystyle v\mapsto \tau v} . The components of τ {\displaystyle \tau } with respect to the basis B {\displaystyle {\mathcal {B}}} are defined via the equation τ


id: adequality-20260505-0
file: data/en.wikipedia.org/wiki/Adequality-0.md
category: reference
title: Adequality
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Adequality
date_saved: 2026-05-05T07:22:58.075678+00:00
instance: kb-cron
parent: adequality-20260505-0
summary:
Adequality is a technique developed by Pierre de Fermat in his treatise Methodus ad disquirendam maximam et minimam (a Latin treatise circulated in France c. 1636 ) to calculate maxima and minima of functions, tangents to curves, area, center of mass, least action, and other problems in calculus. According to André Weil, Fermat "introduces the technical term adaequalitas, adaequare, etc., which he says he has borrowed from Diophantus. As Diophantus V.11 shows, it means an approximate equality, and this is indeed how Fermat explains the word in one of his later writings." (Weil 1973). Diophantus coined the word παρισότης (parisotēs) to refer to an approximate equality. Claude Gaspard Bachet de Méziriac translated Diophantus's Greek word into Latin as adaequalitas. Paul Tannery's French translation of Fermat's Latin treatises on maxima and minima used the words adéquation and adégaler. == Fermat's method == Fermat used adequality first to find maxima of functions, and then adapted it to find tangent lines to curves. To find the maximum of a term p ( x ) {\displaystyle p(x)} , Fermat equated (or more precisely adequated) p ( x ) {\displaystyle p(x)} and p ( x + e ) {\displaystyle p(x+e)} and after doing algebra he


id: adequality-20260505-1
file: data/en.wikipedia.org/wiki/Adequality-1.md
category: reference
title: Adequality
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Adequality
date_saved: 2026-05-05T07:22:58.075678+00:00
instance: kb-cron
parent: adequality-20260505-0
summary:
This, I believe, was the real significance of his use of Diophantos' πἀρισον, stressing the smallness of the variation. The ordinary translation of 'adaequalitas' seems to be "approximate equality", but I much prefer "pseudo-equality" to present Fermat's thought at this point.He further notes that "there was never in M1 (Method 1) any question of the variation E being put equal to zero. The words Fermat used to express the process of suppressing terms containing E was 'elido', 'deleo', and 'expungo', and in French 'i'efface' and 'i'ôte'. We can hardly believe that a sane man wishing to express his meaning and searching for words, would constantly hit upon such tortuous ways of imparting the simple fact that the terms vanished because E was zero.(p. 51) Claus Jensen (1969) wrote:Moreover, in applying the notion of adégalité – which constitutes the basis of Fermat's general method of constructing tangents, and by which is meant a comparition of two magnitudes as if they were equal, although they are in fact not ("tamquam essent aequalia, licet revera aequalia non sint") – I will employ the nowadays more usual symbol ≈ {\displaystyle \scriptstyle \approx } . The Latin quotation comes from Tannery's 1891 edition of Fermat,


id: adjoint-20260505-0
file: data/en.wikipedia.org/wiki/Adjoint-0.md
category: reference
title: Adjoint
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Adjoint
date_saved: 2026-05-05T07:22:59.225743+00:00
instance: kb-cron
parent: adjoint-20260505-0
summary:
In mathematics, the term adjoint applies in several situations. Several of these share a similar formalism: if A is adjoint to B, then there is typically some formula of the type (Ax, y) = (x, By). Specifically, adjoint or adjunction may mean: Adjoint of a linear map, also called its transpose in case of matrices Hermitian adjoint (adjoint of a linear operator) in functional analysis Adjoint endomorphism of a Lie algebra Adjoint representation of a Lie group Adjoint functors in category theory Adjunction (field theory) Adjunction formula (algebraic geometry) Adjunction space in topology Conjugate transpose of a matrix in linear algebra Adjugate matrix, related to its inverse Adjoint equation The upper and lower adjoints of a Galois connection in order theory The adjoint of a differential operator with general polynomial coefficients Kleisli adjunction Monoidal adjunction Quillen adjunction Axiom of adjunction in set theory Adjunction (rule of inference)


id: almost-20260505-0
file: data/en.wikipedia.org/wiki/Almost-0.md
category: reference
title: Almost
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Almost
date_saved: 2026-05-05T07:23:11.268039+00:00
instance: kb-cron
parent: almost-20260505-0
summary:
In set theory, when dealing with sets of infinite size, the term almost or nearly is used to refer to all but a negligible amount of elements in the set. The notion of "negligible" depends on the context, and may mean "of measure zero" (in a measure space), "finite" (when infinite sets are involved), or "countable" (when uncountably infinite sets are involved). For example: The set S = { n ∈ N | n ≥ k } {\displaystyle S=\{n\in \mathbb {N} \,|\,n\geq k\}} is almost N {\displaystyle \mathbb {N} } for any k {\displaystyle k} in N {\displaystyle \mathbb {N} } , because only finitely many natural numbers are less than k {\displaystyle k} . The set of prime numbers is not almost N {\displaystyle \mathbb {N} } , because there are infinitely many natural numbers that are not prime numbers. The set of transcendental numbers are almost R {\displaystyle \mathbb {R} } , because the algebraic real numbers form a countable subset of the set of real numbers (which is uncountable). The Cantor set is uncountably infinite, but has Lebesgue measure zero. So almost all real numbers in (0, 1) are members of the complement of the Cantor set.


id: almost-all-20260505-0
file: data/en.wikipedia.org/wiki/Almost_all-0.md
category: reference
title: Almost all
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Almost_all
date_saved: 2026-05-05T07:23:12.516857+00:00
instance: kb-cron
parent: almost-all-20260505-0
summary:
In mathematics, the term "almost all" means "all but a negligible quantity". More precisely, if X {\displaystyle X} is a set, "almost all elements of X {\displaystyle X} " means "all elements of X {\displaystyle X} but those in a negligible subset of X {\displaystyle X} ". The meaning of "negligible" depends on the mathematical context; for instance, it can mean finite, countable, or null. In contrast, "almost no" means "a negligible quantity"; that is, "almost no elements of X {\displaystyle X} " means "a negligible quantity of elements of X {\displaystyle X} ". == Meanings in different areas of mathematics == === Prevalent meaning === Throughout mathematics, "almost all" is sometimes used to mean "all (elements of an infinite set) except for finitely many". This use occurs in philosophy as well. Similarly, "almost all" can mean "all (elements of an uncountable set) except for countably many". Examples: Almost all positive integers are greater than 1012. Almost all prime numbers are odd (2 is the only exception). Almost all polyhedra are irregular (as there are only nine exceptions: the five platonic solids and the four Kepler–Poinsot polyhedra). If P is a nonzero polynomial, then P(x) ≠ 0 for almost all


id: almost-everywhere-20260505-0
file: data/en.wikipedia.org/wiki/Almost_everywhere-0.md
category: reference
title: Almost everywhere
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Almost_everywhere
date_saved: 2026-05-05T07:23:13.771862+00:00
instance: kb-cron
parent: almost-everywhere-20260505-0
summary:
In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to the concept of measure zero, and is analogous to the notion of almost surely in probability theory. More specifically, a property holds almost everywhere if it holds for all elements in a set except a subset of measure zero, or equivalently, if the set of elements for which the property holds is conull. In cases where the measure is not complete, it is sufficient that the set be contained within a set of measure zero. When discussing sets of real numbers, the Lebesgue measure is usually assumed unless otherwise stated. The term almost everywhere is abbreviated a.e.; in older literature p.p. is used, to stand for the equivalent French phrase presque partout. A set with full measure is one whose complement is of measure zero. In probability theory, the terms almost surely, almost certain and almost always refer to events with probability 1 not necessarily including all of the outcomes. These are exactly the sets of full measure in a


id: almost-everywhere-20260505-1
file: data/en.wikipedia.org/wiki/Almost_everywhere-1.md
category: reference
title: Almost everywhere
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Almost_everywhere
date_saved: 2026-05-05T07:23:13.771862+00:00
instance: kb-cron
parent: almost-everywhere-20260505-0
summary:
== Definition using ultrafilters == Outside of the context of real analysis, the notion of a property true almost everywhere is sometimes defined in terms of an ultrafilter. An ultrafilter on a set X is a maximal collection F of subsets of X such that: If U ∈ F and U ⊆ V then V ∈ F The intersection of any two sets in F is in F The empty set is not in F A property P of points in X holds almost everywhere, relative to an ultrafilter F, if the set of points for which P holds is in F. For example, one construction of the hyperreal number system defines a hyperreal number as an equivalence class of sequences that are equal almost everywhere as defined by an ultrafilter. The definition of almost everywhere in terms of ultrafilters is closely related to the definition in terms of measures, because each ultrafilter defines a finitely-additive measure taking only the values 0 and 1, where a set has measure 1 if and only if it is included in the ultrafilter. == See also == Dirichlet's function, a function that is equal to 0 almost everywhere. Cantor function == References ==


id: almost-surely-20260505-0
file: data/en.wikipedia.org/wiki/Almost_surely-0.md
category: reference
title: Almost surely
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Almost_surely
date_saved: 2026-05-05T07:23:15.024547+00:00
instance: kb-cron
parent: almost-surely-20260505-0
summary:
In probability theory, an event is said to happen almost surely (sometimes abbreviated as a.s.) if it happens with probability 1 (with respect to the probability measure). In other words, the set of outcomes on which the event does not occur has probability 0, even though the set might not be empty. The concept is analogous to the concept of "almost everywhere" in measure theory. In probability experiments on a finite sample space with a non-zero probability for each outcome, there is no difference between almost surely and surely (since having a probability of 1 entails including all the sample points); however, this distinction becomes important when the sample space is an infinite set, because an infinite set can have non-empty subsets of probability 0. Some examples of the use of this concept include the strong and uniform versions of the law of large numbers, the continuity of the paths of Brownian motion, and the infinite monkey theorem. The terms almost certainly (a.c.) and almost always (a.a.) are also used. Almost never describes the opposite of almost surely: an event that happens with probability zero happens almost never. == Formal definition == Let ( Ω , F , P )


id: almost-surely-20260505-1
file: data/en.wikipedia.org/wiki/Almost_surely-1.md
category: reference
title: Almost surely
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Almost_surely
date_saved: 2026-05-05T07:23:15.024547+00:00
instance: kb-cron
parent: almost-surely-20260505-0
summary:
== Asymptotically almost surely == In asymptotic analysis, a property is said to hold asymptotically almost surely (a.a.s.) if over a sequence of sets, the probability converges to 1. This is equivalent to convergence in probability. For instance, in number theory, a large number is asymptotically almost surely composite, by the prime number theorem; and in random graph theory, the statement " G ( n , p n ) {\displaystyle G(n,p_{n})} is connected" (where G ( n , p ) {\displaystyle G(n,p)} denotes the graphs on n {\displaystyle n} vertices with edge probability p {\displaystyle p} ) is true a.a.s. when, for some ε > 0 {\displaystyle \varepsilon >0} p n > ( 1 + ε ) ln ⁡ n n . {\displaystyle p_{n}>{\frac {(1+\varepsilon )\ln n}{n}}.} In number theory, this is referred to as "almost all", as in "almost all numbers are composite". Similarly, in graph theory, this is sometimes referred to as "almost surely". == See also == Almost Almost everywhere, the corresponding concept in measure theory Convergence of random variables, for "almost sure convergence" With high probability Cromwell's rule, which says that probabilities should almost never be set as zero or one Degenerate distribution, for "almost surely


id: ansatz-20260505-0
file: data/en.wikipedia.org/wiki/Ansatz-0.md
category: reference
title: Ansatz
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Ansatz
date_saved: 2026-05-05T07:23:16.301104+00:00
instance: kb-cron
parent: ansatz-20260505-0
summary:
In physics and mathematics, an ansatz (; German: [ˈʔanzats] , meaning: "initial placement of a tool at a work piece", plural ansatzes or, from German, ansätze ; German: [ˈʔanzɛtsə] ) is an educated guess or an additional assumption made to help solve a problem, and which may later be verified to be part of the solution by its results. == Use == An ansatz is the establishment of the starting equation(s), the theorem(s), or the value(s) describing a mathematical or physical problem or solution. It typically provides an initial estimate or framework to the solution of a mathematical problem. In fact, an ansatz is sometimes thought of as a "trial answer" and an important technique in solving differential equations, possibly with initial or boundary conditions. After an ansatz, which constitutes nothing more than an assumption, has been established, the equations are solved more precisely for the general function of interest, which then constitutes a confirmation of the assumption. In essence, an ansatz makes assumptions about the form of the solution to a problem so as to make the solution easier to find. It has been demonstrated that machine learning techniques can be applied to provide initial estimates similar to those


id: apotome-mathematics-20260505-0
file: data/en.wikipedia.org/wiki/Apotome_(mathematics)-0.md
category: reference
title: Apotome (mathematics)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Apotome_(mathematics)
date_saved: 2026-05-05T07:23:18.760476+00:00
instance: kb-cron
parent: apotome-mathematics-20260505-0
summary:
In the historical study of mathematics, an apotome is a line segment formed from a longer line segment by breaking it into two parts, one of which is commensurable only in power to the whole; the other part is the apotome. In this definition, two line segments are said to be "commensurable only in power" when the ratio of their lengths is an irrational number but the ratio of their squared lengths is rational. Translated into modern algebraic language, an apotome can be interpreted as a quadratic irrational number formed by subtracting one square root of a rational number from another. This concept of the apotome appears in Euclid's Elements beginning in book X, where Euclid defines two special kinds of apotomes. In an apotome of the first kind, the whole is rational, while in an apotome of the second kind, the part subtracted from it is rational; both kinds of apotomes also satisfy an additional condition. Euclid Proposition XIII.6 states that, if a rational line segment is split into two pieces in the golden ratio, then both pieces may be represented as apotomes. == References ==


id: arbitrarily-large-20260505-0
file: data/en.wikipedia.org/wiki/Arbitrarily_large-0.md
category: reference
title: Arbitrarily large
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Arbitrarily_large
date_saved: 2026-05-05T07:23:20.271509+00:00
instance: kb-cron
parent: arbitrarily-large-20260505-0
summary:
In mathematics, the phrases arbitrarily large, arbitrarily small and arbitrarily long are used in statements to make clear the fact that an object is large, small, or long with little limitation or restraint, respectively. The use of "arbitrarily" often occurs in the context of real numbers (and its subsets thereof), though its meaning can differ from that of "sufficiently" and "infinitely". == Examples == The statement " f ( x ) {\displaystyle f(x)} is non-negative for arbitrarily large x {\displaystyle x} ." is a shorthand for: "For every real number n {\displaystyle n} , f ( x ) {\displaystyle f(x)} is non-negative for some value of x {\displaystyle x} greater than n {\displaystyle n} ." In the common parlance, the term "arbitrarily long" is often used in the context of sequence of numbers. For example, to say that there are "arbitrarily long arithmetic progressions of prime numbers" does not mean that there exists any infinitely long arithmetic progression of prime numbers (there is not), nor that there exists any particular arithmetic progression of prime numbers that is in some sense "arbitrarily long". Rather, the phrase is used to refer to the fact that no matter how large a number n


id: arithmetic-and-geometric-frobenius-20260505-0
file: data/en.wikipedia.org/wiki/Arithmetic_and_geometric_Frobenius-0.md
category: reference
title: Arithmetic and geometric Frobenius
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Arithmetic_and_geometric_Frobenius
date_saved: 2026-05-05T07:23:21.492710+00:00
instance: kb-cron
parent: arithmetic-and-geometric-frobenius-20260505-0
summary:
In mathematics, the Frobenius endomorphism is defined in any commutative ring R that has characteristic p, where p is a prime number. Namely, the mapping φ that takes r in R to rp is a ring endomorphism of R. The image of φ is then Rp, the subring of R consisting of p-th powers. In some important cases, for example finite fields, φ is surjective. Otherwise φ is an endomorphism but not a ring automorphism. The terminology of geometric Frobenius arises by applying the spectrum of a ring construction to φ. This gives a mapping φ*: Spec(Rp) → Spec(R) of affine schemes. Even in cases where Rp = R this is not the identity, unless R is the prime field. Mappings created by fibre product with φ*, i.e. base changes, tend in scheme theory to be called geometric Frobenius. The reason for a careful terminology is that the Frobenius automorphism in Galois groups, or defined by transport of structure, is often the inverse mapping of the geometric Frobenius. As in the case of a cyclic group in which a generator is also the inverse of a generator, there are in many situations two possible definitions of Frobenius, and without a


id: brown-measure-20260505-0
file: data/en.wikipedia.org/wiki/Brown_measure-0.md
category: reference
title: Brown measure
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Brown_measure
date_saved: 2026-05-05T07:23:22.881835+00:00
instance: kb-cron
parent: brown-measure-20260505-0
summary:
In mathematics, the Brown measure of an operator in a finite factor is a probability measure on the complex plane which may be viewed as an analog of the spectral counting measure (based on algebraic multiplicity) of matrices. It is named after Lawrence G. Brown. == Definition == Let M {\displaystyle {\mathcal {M}}} be a finite factor with the canonical normalized trace τ {\displaystyle \tau } and let I {\displaystyle I} be the identity operator. For every operator A ∈ M , {\displaystyle A\in {\mathcal {M}},} the function λ ↦ τ ( log ⁡ | A − λ I | ) , λ ∈ C , {\displaystyle \lambda \mapsto \tau (\log \left|A-\lambda I\right|),\;\lambda \in \mathbb {C} ,} is a subharmonic function and its Laplacian in the distributional sense is a probability measure on C {\displaystyle \mathbb {C} } μ A ( d ( a + b i ) ) := 1 2 π ∇ 2 τ ( log ⁡ | A − ( a + b i ) I | ) d a d b {\displaystyle \mu _{A}(\mathrm {d} (a+b\mathbb {i} )):={\frac {1}{2\pi }}\nabla ^{2}\tau (\log \left|A-(a+b\mathbb {i} )I\right|)\mathrm {d} a\mathrm {d} b} which is called the Brown measure of


id: canonical-form-20260505-0
file: data/en.wikipedia.org/wiki/Canonical_form-0.md
category: reference
title: Canonical form
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Canonical_form
date_saved: 2026-05-05T07:23:25.219824+00:00
instance: kb-cron
parent: canonical-form-20260505-0
summary:
In mathematics and computer science, a canonical, normal, or standard form of a mathematical object is a standard way of presenting that object as a mathematical expression. Often, it is one which provides the simplest representation of an object and allows it to be identified in a unique way. The distinction between "canonical" and "normal" forms varies from subfield to subfield. In most fields, a canonical form specifies a unique representation for every object, while a normal form simply specifies its form, without the requirement of uniqueness. The canonical form of a positive integer in decimal representation is a finite sequence of digits that does not begin with zero. More generally, for a class of objects on which an equivalence relation is defined, a canonical form consists in the choice of a specific object in each class. For example: Jordan normal form is a canonical form for matrix similarity. The row echelon form is a canonical form, when one considers as equivalent a matrix and its left product by an invertible matrix. In computer science, and more specifically in computer algebra, when representing mathematical objects in a computer, there are usually many different ways to represent the same object. In


id: canonical-form-20260505-1
file: data/en.wikipedia.org/wiki/Canonical_form-1.md
category: reference
title: Canonical form
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Canonical_form
date_saved: 2026-05-05T07:23:25.219824+00:00
instance: kb-cron
parent: canonical-form-20260505-0
summary:
The equation of a line: Ax + By = C, with A2 + B2 = 1 and C ≥ 0 The equation of a circle: ( x − h ) 2 + ( y − k ) 2 = r 2 {\displaystyle (x-h)^{2}+(y-k)^{2}=r^{2}} By contrast, there are alternative forms for writing equations. For example, the equation of a line may be written as a linear equation in point-slope and slope-intercept form. Convex polyhedra can be put into canonical form such that: All faces are flat, All edges are tangent to the unit sphere, and The centroid of the polyhedron is at the origin. === Integrable systems === Every differentiable manifold has a cotangent bundle. That bundle can always be endowed with a certain differential form, called the canonical one-form. This form gives the cotangent bundle the structure of a symplectic manifold, and allows vector fields on the manifold to be integrated by means of the Euler-Lagrange equations, or by means of Hamiltonian mechanics. Such systems of integrable differential equations are called integrable systems. === Dynamical systems === The study of dynamical systems overlaps with that of integrable systems; there one has the idea of a normal form (dynamical systems). ===


id: canonical-map-20260505-0
file: data/en.wikipedia.org/wiki/Canonical_map-0.md
category: reference
title: Canonical map
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Canonical_map
date_saved: 2026-05-05T07:23:26.446232+00:00
instance: kb-cron
parent: canonical-map-20260505-0
summary:
In mathematics, a canonical map, also called a natural map, is a map or morphism between objects that arises naturally from the definition or the construction of the objects. Often, it is a map which preserves the widest amount of structure. A choice of a canonical map sometimes depends on a convention (e.g., a sign convention). A closely related notion is a structure map or structure morphism; the map or morphism that comes with the given structure on the object. These are also sometimes called canonical maps. A canonical isomorphism is a canonical map that is also an isomorphism (i.e., invertible). In some contexts, it might be necessary to address an issue of choices of canonical maps or canonical isomorphisms; for a typical example, see prestack. == Examples == If N is a normal subgroup of a group G, then there is a canonical surjective group homomorphism from G to the quotient group G / N, that sends an element g to the coset determined by g. If I is an ideal of a ring R, then there is a canonical surjective ring homomorphism from R onto the quotient ring R / I, that sends an element r to its


id: cauchy-wavelet-20260505-0
file: data/en.wikipedia.org/wiki/Cauchy_wavelet-0.md
category: reference
title: Cauchy wavelet
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Cauchy_wavelet
date_saved: 2026-05-05T07:23:27.637906+00:00
instance: kb-cron
parent: cauchy-wavelet-20260505-0
summary:
In mathematics, Cauchy wavelets are a family of continuous wavelets, used in the continuous wavelet transform. == Definition == The Cauchy wavelet of order p {\displaystyle p} is defined as: ψ p ( t ) = Γ ( p + 1 ) 2 π ( j t + j ) p + 1 {\displaystyle \psi _{p}(t)={\frac {\Gamma (p+1)}{2\pi }}\left({\frac {j}{t+j}}\right)^{p+1}} where p > 0 {\displaystyle p>0} and j = − 1 {\displaystyle j={\sqrt {-1}}} therefore, its Fourier transform is defined as ψ p ^ ( ξ ) = ξ p e − ξ I [ ξ ≥ 0 ] {\displaystyle {\hat {\psi _{p}}}(\xi )=\xi ^{p}e^{-\xi }I_{[\xi \geq 0]}} . Sometimes it is defined as a function with its Fourier transform ψ p ^ ( ξ ) = ρ ( ξ ) ξ p e − ξ I [ ξ ≥ 0 ] {\displaystyle {\hat {\psi _{p}}}(\xi )=\rho (\xi )\xi ^{p}e^{-\xi }I_{[\xi \geq 0]}} where ρ ( ξ ) ∈ L ∞ ( R ) {\displaystyle \rho (\xi )\in L^{\infty }(\mathbb {R} )} and ρ ( ξ ) = ρ ( a ξ ) {\displaystyle \rho (\xi )=\rho (a\xi )} for ξ ∈ R {\displaystyle \xi \in \mathbb {R} } almost


id: cauchy-wavelet-20260505-1
file: data/en.wikipedia.org/wiki/Cauchy_wavelet-1.md
category: reference
title: Cauchy wavelet
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Cauchy_wavelet
date_saved: 2026-05-05T07:23:27.637906+00:00
instance: kb-cron
parent: cauchy-wavelet-20260505-0
summary:
f ( t ) = ∑ n = − ∞ ∞ L n ( f ) e j n t {\displaystyle f(t)=\sum _{n=-\infty }^{\infty }L_{n}(f)e^{jnt}} . and in fact we have Parseval's identity | | f | | 2 = ∑ n = − ∞ ∞ | L n ( f ) | 2 {\displaystyle ||f||^{2}=\sum _{n=-\infty }^{\infty }|L_{n}(f)|^{2}} . where | | f | | 2 = 1 2 π ∫ − π π | f ( t ) | 2 d t {\displaystyle ||f||^{2}={\frac {1}{2\pi }}\int _{-\pi }^{\pi }|f(t)|^{2}\,dt} i.e. the norm defined in L 2 ( [ − π , π ] ) {\displaystyle L^{2}([-\pi ,\pi ])} . Hence, in this example, the index set I {\displaystyle I} is the integer Z {\displaystyle \mathbb {Z} } , the vector space V {\displaystyle V} is L 2 ( [ − π , π ] ) {\displaystyle L^{2}([-\pi ,\pi ])} and the linear form L n {\displaystyle L_{n}} is the Fourier coefficient. Furthermore, the absolute value of Fourier coefficients { | L n ( f ) | } n ∈ Z {\displaystyle \{|L_{n}(f)|\}_{n\in \mathbb {Z} }} can only determine the norm of f {\displaystyle f} defined in L 2


id: characterization-mathematics-20260505-0
file: data/en.wikipedia.org/wiki/Characterization_(mathematics)-0.md
category: reference
title: Characterization (mathematics)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Characterization_(mathematics)
date_saved: 2026-05-05T07:23:28.881720+00:00
instance: kb-cron
parent: characterization-mathematics-20260505-0
summary:
In mathematics, a characterization of an object is a set of conditions that, while possibly different from the definition of the object, is logically equivalent to it. To say that "Property P characterizes object X" is to say that not only does X have property P, but that X is the only thing that has property P (i.e., P is a defining property of X). Similarly, a set of properties P is said to characterize X, when these properties distinguish X from all other objects. Even though a characterization identifies an object in a unique way, several characterizations can exist for a single object. Common mathematical expressions for a characterization of X in terms of P include "P is necessary and sufficient for X", and "X holds if and only if P". It is also common to find statements such as "Property Q characterizes Y up to isomorphism". The first type of statement says in different words that the extension of P is a singleton set, while the second says that the extension of Q is a single equivalence class (for isomorphism, in the given example — depending on how up to is being used, some other equivalence relation might


id: characterization-mathematics-20260505-1
file: data/en.wikipedia.org/wiki/Characterization_(mathematics)-1.md
category: reference
title: Characterization (mathematics)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Characterization_(mathematics)
date_saved: 2026-05-05T07:23:28.881720+00:00
instance: kb-cron
parent: characterization-mathematics-20260505-0
summary:
The least-upper-bound property The greatest-lower-bound property The nested interval property The Bolzano-Weierstrass theorem The convergence of Cauchy sequences A typical real analysis university course would begin with the first of these, the least-upper-bound property, as an axiomatic definition of the reals (sometimes called the "axiom of completeness" in texts), and gradually prove its way to the last, the convergence of Cauchy sequences. The proofs are quite nontrivial. Among these five characterizations, the Cauchy-sequence perspective turns out to be the easiest to generalize, and is chosen as the definition for the completeness of an abstract metric space. However, the least-upper-bound property is often the most useful to prove facts about real numbers themselves, such as the intermediate value theorem. Thus the most useful and most generalizable characterizations are at times different. Another example of this phenomenon is found in physics. Hamiltonian mechanics is a characterization of classical mechanics, being equivalent to Newton’s laws. However, it is much easier to generalize to quantum mechanics and statistical mechanics, which is its primary virtue. However, it is a different characterization, Lagrangian mechanics, that is often preferred for the study of classical mechanics itself. Since a characterization result is equivalent to the initial definition or


id: coefficient-20260505-0
file: data/en.wikipedia.org/wiki/Coefficient-0.md
category: reference
title: Coefficient
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Coefficient
date_saved: 2026-05-05T07:23:30.073796+00:00
instance: kb-cron
parent: coefficient-20260505-0
summary:
In mathematics, a coefficient is a multiplicative factor involved in some term of a polynomial, a series, or any other type of expression. It may be a number without units, in which case it is known as a numerical factor. It may also be a constant with units of measurement, in which it is known as a constant multiplier. In general, coefficients may be any expression (including variables such as a, b and c). When the combination of variables and constants is not necessarily involved in a product, it may be called a parameter. For example, the polynomial 2 x 2 − x + 3 {\displaystyle 2x^{2}-x+3} has coefficients 2, −1, and 3, and the powers of the variable x {\displaystyle x} in the polynomial a x 2 + b x + c {\displaystyle ax^{2}+bx+c} have coefficient parameters a {\displaystyle a} , b {\displaystyle b} , and c {\displaystyle c} . A constant coefficient, also known as constant term or simply constant, is a quantity either implicitly attached to the zeroth power of a variable or not attached to other variables in an expression; for example, the constant coefficients of the expressions above are the number 3 and the parameter


id: coefficient-of-determination-20260505-0
file: data/en.wikipedia.org/wiki/Coefficient_of_determination-0.md
category: reference
title: Coefficient of determination
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Coefficient_of_determination
date_saved: 2026-05-05T07:23:31.318214+00:00
instance: kb-cron
parent: coefficient-of-determination-20260505-0
summary:
In statistics, the coefficient of determination, denoted R2 or r2 and pronounced "R squared", is the proportion of the variation in the dependent variable that is predictable from the independent variable(s). It is a statistic used in the context of statistical models whose main purpose is either the prediction of future outcomes or the testing of hypotheses, on the basis of other related information. It provides a measure of how well observed outcomes are replicated by the model, based on the proportion of total variation of outcomes explained by the model. There are several definitions of R2 that are only sometimes equivalent. In simple linear regression (which includes an intercept), r2 is simply the square of the sample correlation coefficient (r), between the observed outcomes and the observed predictor values. If additional regressors are included, R2 is the square of the coefficient of multiple correlation. In both such cases, the coefficient of determination is always smaller than 1 and usually greater than 0. Cases where R2 is negative can arise when the predictions that are being compared to the corresponding outcomes have not been derived from a model-fitting procedure using those data. Even if a model-fitting procedure has been used,


id: coefficient-of-determination-20260505-1
file: data/en.wikipedia.org/wiki/Coefficient_of_determination-1.md
category: reference
title: Coefficient of determination
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Coefficient_of_determination
date_saved: 2026-05-05T07:23:31.318214+00:00
instance: kb-cron
parent: coefficient-of-determination-20260505-0
summary:
where the qi are arbitrary values that may or may not depend on i or on other free parameters (the common choice qi = xi is just one special case), and the coefficient estimates α ^ {\displaystyle {\widehat {\alpha }}} and β ^ {\displaystyle {\widehat {\beta }}} are obtained by minimizing the residual sum of squares. This set of conditions is an important one and it has a number of implications for the properties of the fitted residuals and the modelled values. In particular, under these conditions: f ¯ = y ¯ . {\displaystyle {\bar {f}}={\bar {y}}.\,} === As squared correlation coefficient === In linear least squares multiple regression (with fitted intercept and slope), R2 equals ρ 2 ( y , f ) {\displaystyle \rho ^{2}(y,f)} the square of the Pearson correlation coefficient between the observed y {\displaystyle y} and modeled (predicted) f {\displaystyle f} data values of the dependent variable. In a linear least squares regression with a single explanator (with fitted intercept and slope), this is also equal to ρ 2 ( y , x ) {\displaystyle \rho ^{2}(y,x)} the squared Pearson correlation coefficient between the dependent variable y {\displaystyle y} and explanatory variable x {\displaystyle x} .


id: coefficient-of-determination-20260505-2
file: data/en.wikipedia.org/wiki/Coefficient_of_determination-2.md
category: reference
title: Coefficient of determination
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Coefficient_of_determination
date_saved: 2026-05-05T07:23:31.318214+00:00
instance: kb-cron
parent: coefficient-of-determination-20260505-0
summary:
where, for the ith case, Y i {\displaystyle {Y_{i}}} is the response variable, X i , 1 , … , X i , p {\displaystyle X_{i,1},\dots ,X_{i,p}} are p regressors, and ε i {\displaystyle \varepsilon _{i}} is a mean zero error term. The quantities β 0 , … , β p {\displaystyle \beta _{0},\dots ,\beta _{p}} are unknown coefficients, whose values are estimated by least squares. The coefficient of determination R2 is a measure of the global fit of the model. Specifically, R2 is an element of [0, 1] and represents the proportion of variability in Yi that may be attributed to some linear combination of the regressors (explanatory variables) in X. R2 is often interpreted as the proportion of response variation "explained" by the regressors in the model. Thus, R2 = 1 indicates that the fitted model explains all variability in y {\displaystyle y} , while R2 = 0 indicates no 'linear' relationship (for straight line regression, this means that the straight line model is a constant line (slope = 0, intercept = y ¯ {\displaystyle {\bar {y}}} ) between the response variable and regressors). An interior value such as R2 = 0.7 may be interpreted as follows: "Seventy


id: coefficient-of-determination-20260505-3
file: data/en.wikipedia.org/wiki/Coefficient_of_determination-3.md
category: reference
title: Coefficient of determination
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Coefficient_of_determination
date_saved: 2026-05-05T07:23:31.318214+00:00
instance: kb-cron
parent: coefficient-of-determination-20260505-0
summary:
This equation corresponds to the ordinary least squares regression model with two regressors. The prediction is shown as the blue vector in the figure on the right. Geometrically, it is the projection of true value onto a larger model space in R 2 {\displaystyle \mathbb {R} ^{2}} (without intercept). Noticeably, the values of β 0 {\displaystyle \beta _{0}} and β 0 {\displaystyle \beta _{0}} are not the same as in the equation for smaller model space as long as X 1 {\displaystyle X_{1}} and X 2 {\displaystyle X_{2}} are not zero vectors. Therefore, the equations are expected to yield different predictions (i.e., the blue vector is expected to be different from the red vector). The least squares regression criterion ensures that the residual is minimized. In the figure, the blue line representing the residual is orthogonal to the model space in R 2 {\displaystyle \mathbb {R} ^{2}} , giving the minimal distance from the space. The smaller model space is a subspace of the larger one, and thereby the residual of the smaller model is guaranteed to be larger. Comparing the red and blue lines in the figure, the blue line is orthogonal to the space, and any other line


id: coefficient-of-determination-20260505-4
file: data/en.wikipedia.org/wiki/Coefficient_of_determination-4.md
category: reference
title: Coefficient of determination
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Coefficient_of_determination
date_saved: 2026-05-05T07:23:31.318214+00:00
instance: kb-cron
parent: coefficient-of-determination-20260505-0
summary:
The adjusted R2 can be interpreted as an instance of the bias-variance tradeoff. When we consider the performance of a model, a lower error represents a better performance. When the model becomes more complex, the variance will increase whereas the square of bias will decrease, and these two metrics add up to be the total error. Combining these two trends, the bias-variance tradeoff describes a relationship between the performance of the model and its complexity, which is shown as a u-shape curve on the right. For the adjusted R2 specifically, the model complexity (i.e. number of parameters) affects the R2 and the term / frac and thereby captures their attributes in the overall performance of the model. R2 can be interpreted as the variance of the model, which is influenced by the model complexity. A high R2 indicates a lower bias error because the model can better explain the change of Y with predictors. For this reason, we make fewer (erroneous) assumptions, and this results in a lower bias error. Meanwhile, to accommodate fewer assumptions, the model tends to be more complex. Based on bias-variance tradeoff, a higher complexity will lead to a decrease in bias and a better performance


id: coefficient-of-determination-20260505-5
file: data/en.wikipedia.org/wiki/Coefficient_of_determination-5.md
category: reference
title: Coefficient of determination
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Coefficient_of_determination
date_saved: 2026-05-05T07:23:31.318214+00:00
instance: kb-cron
parent: coefficient-of-determination-20260505-0
summary:
It is assumed that the matrix X is standardized with Z-scores and that the column vector y {\displaystyle y} is centered to have a mean of zero. Let the column vector β 0 {\displaystyle \beta _{0}} refer to the hypothesized regression parameters and let the column vector b {\displaystyle b} denote the estimated parameters. We can then define R 2 = 1 − ( y − X b ) ′ ( y − X b ) ( y − X β 0 ) ′ ( y − X β 0 ) . {\displaystyle R^{2}=1-{\frac {(y-Xb)'(y-Xb)}{(y-X\beta _{0})'(y-X\beta _{0})}}.} An R2 of 75% means that the in-sample accuracy improves by 75% if the data-optimized b solutions are used instead of the hypothesized β 0 {\displaystyle \beta _{0}} values. In the special case that β 0 {\displaystyle \beta _{0}} is a vector of zeros, we obtain the traditional R2 again. The individual effect on R2 of deviating from a hypothesis can be computed with R ⊗ {\displaystyle R^{\otimes }} ('R-outer'). This p {\displaystyle p} times p {\displaystyle p} matrix is given by R ⊗ = ( X ′ y ~ 0 ) ( X ′ y ~ 0 ) ′ ( X ′


id: coefficient-of-variation-20260505-0
file: data/en.wikipedia.org/wiki/Coefficient_of_variation-0.md
category: reference
title: Coefficient of variation
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Coefficient_of_variation
date_saved: 2026-05-05T07:23:32.496013+00:00
instance: kb-cron
parent: coefficient-of-variation-20260505-0
summary:
In probability theory and statistics, the coefficient of variation (CV), also known as normalized root-mean-square deviation (NRMSD), and relative standard deviation (RSD), is a standardized measure of dispersion of a probability distribution or frequency distribution. It is defined as the ratio of the standard deviation σ {\displaystyle \sigma } to the mean μ {\displaystyle \mu } (or its absolute value, | μ | {\displaystyle |\mu |} ), and often expressed as a percentage ("%RSD"). The CV or RSD is widely used in analytical chemistry to express the precision and repeatability of an assay. It is also commonly used in fields such as engineering or physics when doing quality assurance studies and ANOVA gauge R&R, by economists and investors in economic models, in epidemiology, and in psychology/neuroscience. == Definition == The coefficient of variation (CV) is defined as the ratio of the standard deviation σ {\displaystyle \sigma } to the mean μ {\displaystyle \mu } , C V = σ μ . {\displaystyle CV={\frac {\sigma }{\mu }}.} It shows the extent of variability in relation to the mean of the population. The coefficient of variation should be computed only for data measured on scales that have a meaningful zero (ratio scale)


id: coefficient-of-variation-20260505-1
file: data/en.wikipedia.org/wiki/Coefficient_of_variation-1.md
category: reference
title: Coefficient of variation
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Coefficient_of_variation
date_saved: 2026-05-05T07:23:32.496013+00:00
instance: kb-cron
parent: coefficient-of-variation-20260505-0
summary:
This term was intended to be analogous to the coefficient of variation, for describing multiplicative variation in log-normal data, but this definition of GCV has no theoretical basis as an estimate of c v {\displaystyle c_{\rm {v}}\,} itself. For many practical purposes (such as sample size determination and calculation of confidence intervals) it is s l n {\displaystyle s_{ln}\,} which is of most use in the context of log-normally distributed data. If necessary, this can be derived from an estimate of c v {\displaystyle c_{\rm {v}}\,} or GCV by inverting the corresponding formula. == Comparison to standard deviation == === Advantages === The coefficient of variation is useful because the standard deviation of data must always be understood in the context of the mean of the data. In contrast, the actual value of the CV is independent of the unit in which the measurement has been taken, so it is a dimensionless number. For comparison between data sets with different units or widely different means, one should use the coefficient of variation instead of the standard deviation. === Disadvantages === When the mean value is close to zero, the coefficient of variation will approach infinity and is therefore sensitive to


id: coefficient-of-variation-20260505-2
file: data/en.wikipedia.org/wiki/Coefficient_of_variation-2.md
category: reference
title: Coefficient of variation
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Coefficient_of_variation
date_saved: 2026-05-05T07:23:32.496013+00:00
instance: kb-cron
parent: coefficient-of-variation-20260505-0
summary:
== Examples of misuse == Comparing coefficients of variation between parameters using relative units can result in differences that may not be real. If we compare the same set of temperatures in Celsius and Fahrenheit (both relative units, where kelvin and Rankine scale are their associated absolute values): Celsius: [0, 10, 20, 30, 40] Fahrenheit: [32, 50, 68, 86, 104] The sample standard deviations are 15.81 and 28.46, respectively. The CV of the first set is 15.81/20 = 79%. For the second set (which are the same temperatures) it is 28.46/68 = 42%. If, for example, the data sets are temperature readings from two different sensors (a Celsius sensor and a Fahrenheit sensor) and you want to know which sensor is better by picking the one with the least variance, then you will be misled if you use CV. The problem here is that you have divided by a relative value rather than an absolute. Comparing the same data set, now in absolute units: Kelvin: [273.15, 283.15, 293.15, 303.15, 313.15] Rankine: [491.67, 509.67, 527.67, 545.67, 563.67] The sample standard deviations are still 15.81 and 28.46, respectively, because the standard deviation is not affected by a constant offset. The coefficients of


id: mathematical-coincidence-20260505-0
file: data/en.wikipedia.org/wiki/Mathematical_coincidence-0.md
category: reference
title: Mathematical coincidence
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Mathematical_coincidence
date_saved: 2026-05-05T07:23:33.671306+00:00
instance: kb-cron
parent: mathematical-coincidence-20260505-0
summary:
A mathematical coincidence is said to occur when two expressions with no direct relationship show a near-equality which has no apparent theoretical explanation. For example, there is a near-equality close to the round number 1000 between powers of 2 and powers of 10: 2 10 = 1024 ≈ 1000 = 10 3 . {\displaystyle 2^{10}=1024\approx 1000=10^{3}.} Some mathematical coincidences are used in engineering when one expression is taken as an approximation of another. == Introduction == A mathematical coincidence often involves an integer, and the surprising feature is the fact that a real number arising in some context is considered by some standard as a "close" approximation to a small integer or to a multiple or power of ten, or more generally, to a rational number with a small denominator. Other kinds of mathematical coincidences, such as integers simultaneously satisfying multiple seemingly unrelated criteria or coincidences regarding units of measurement, may also be considered. In the class of those coincidences that are of a purely mathematical sort, some simply result from sometimes very deep mathematical facts, while others appear to come 'out of the blue'. Given the countably infinite number of ways of forming mathematical expressions using a finite number


id: mathematical-coincidence-20260505-1
file: data/en.wikipedia.org/wiki/Mathematical_coincidence-1.md
category: reference
title: Mathematical coincidence
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Mathematical_coincidence
date_saved: 2026-05-05T07:23:33.671306+00:00
instance: kb-cron
parent: mathematical-coincidence-20260505-0
summary:
The coincidence 2 19 ≈ 3 12 {\displaystyle 2^{19}\approx 3^{12}} , from log 2 ⁡ ( 3 ) = 1.5849 … ≈ 19 12 {\displaystyle \log _{2}(3)=1.5849\ldots \approx {\frac {19}{12}}} , closely relates the interval of 7 semitones in equal temperament to a perfect fifth of just intonation: 2 7 / 12 ≈ 3 / 2 {\displaystyle 2^{7/12}\approx 3/2} , correct to about 0.1%. The just fifth is the basis of Pythagorean tuning; the difference between twelve just fifths and seven octaves is the Pythagorean comma. The coincidence ( 3 / 2 ) 4 = ( 81 / 16 ) ≈ 5 {\displaystyle {(3/2)}^{4}=(81/16)\approx 5} permitted the development of meantone temperament, in which just perfect fifths (ratio 3 / 2 {\displaystyle 3/2} ) and major thirds ( 5 / 4 {\displaystyle 5/4} ) are "tempered" so that four 3 / 2 {\displaystyle 3/2} 's is approximately equal to 5 / 1 {\displaystyle 5/1} , or a 5 / 4 {\displaystyle 5/4} major third up two octaves. The difference ( 81 / 80 {\displaystyle 81/80} ) between these stacks of intervals is the syntonic comma. The coincidence 2 12 5 7 = 1.33333319 … ≈ 4 3 {\displaystyle {\sqrt[{12}]{2}}{\sqrt[{7}]{5}}=1.33333319\ldots \approx {\frac


id: mathematical-coincidence-20260505-2
file: data/en.wikipedia.org/wiki/Mathematical_coincidence-2.md
category: reference
title: Mathematical coincidence
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Mathematical_coincidence
date_saved: 2026-05-05T07:23:33.671306+00:00
instance: kb-cron
parent: mathematical-coincidence-20260505-0
summary:
π 9 / e 8 ≈ 10 {\displaystyle \pi ^{9}/e^{8}\approx 10} , to about 5 decimal places. That is, ln ⁡ ( π ) ≈ ln ⁡ ( 10 ) + 8 9 {\displaystyle \ln(\pi )\approx {\ln(10)+8 \over 9}} , within 0.0002%. 2 π + e ≈ 9 {\displaystyle 2\pi +e\approx 9} , within 0.02%. e − π 9 + e − 4 π 9 + e − 9 π 9 + e − 16 π 9 + e − 25 π 9 + e − 36 π 9 + e − 49 π 9 + e − 64 π 9 = 1.00000000000105 … ≈ 1 {\textstyle e^{-{\frac {\pi }{9}}}+e^{-4{\frac {\pi }{9}}}+e^{-9{\frac {\pi }{9}}}+e^{-16{\frac {\pi }{9}}}+e^{-25{\frac {\pi }{9}}}+e^{-36{\frac {\pi }{9}}}+e^{-49{\frac {\pi }{9}}}+e^{-64{\frac {\pi }{9}}}=1.00000000000105\ldots \approx 1} . In fact, this generalizes to the approximate identity ∑ k = 1 n − 1 e − k 2 π n ≈ − 1 + n 2 , {\displaystyle \textstyle \sum _{k=1}^{n-1}{e^{-{\frac {k^{2}\pi }{n}}}}\approx {\frac {-1+{\sqrt {n}}}{2}},} which can be explained by the Jacobian theta functional identity. Ramanujan's constant: e π 163 ≈ 262537412640768744 = 12 3 ( 231 2 − 1 ) 3 + 744 {\displaystyle e^{\pi {\sqrt {163}}}\approx 262537412640768744=12^{3}(231^{2}-1)^{3}+744} , within


id: mathematical-coincidence-20260505-3
file: data/en.wikipedia.org/wiki/Mathematical_coincidence-3.md
category: reference
title: Mathematical coincidence
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Mathematical_coincidence
date_saved: 2026-05-05T07:23:33.671306+00:00
instance: kb-cron
parent: mathematical-coincidence-20260505-0
summary:
1 3 + 5 3 + 3 3 = 153 {\displaystyle \,1^{3}+5^{3}+3^{3}=153} , 3 3 + 7 3 + 0 3 = 370 {\displaystyle \,3^{3}+7^{3}+0^{3}=370} , 3 3 + 7 3 + 1 3 = 371 {\displaystyle \,3^{3}+7^{3}+1^{3}=371} , and 4 3 + 0 3 + 7 3 = 407 {\displaystyle \,4^{3}+0^{3}+7^{3}=407} are all narcissistic numbers. 588 2 + 2353 2 = 5882353 {\displaystyle \,588^{2}+2353^{2}=5882353} , a prime number. The fraction 1/17 also produces 0.05882353 when rounded to 8 digits. 2 1 + 6 2 + 4 3 + 6 4 + 7 5 + 9 6 + 8 7 = 2646798 {\displaystyle \,2^{1}+6^{2}+4^{3}+6^{4}+7^{5}+9^{6}+8^{7}=2646798} . The largest number with this pattern is 12157692622039623539 = 1 1 + 2 2 + 1 3 + … + 9 20 {\displaystyle \,12157692622039623539=1^{1}+2^{2}+1^{3}+\ldots +9^{20}} . 13532385396179 = 13 × 53 2 × 3853 × 96179 {\displaystyle 13532385396179=13\times 53^{2}\times 3853\times 96179} . This number, found in 2017, answers a question by John Conway whether the digits of a composite number could be the same as its prime factorization. A similar example (in fact the smallest) in binary is 255987 = 3 3 × 19 × 499 {\displaystyle 255987=3^{3}\times 19\times 499} , whose prime factorization in


id: mathematical-coincidence-20260505-4
file: data/en.wikipedia.org/wiki/Mathematical_coincidence-4.md
category: reference
title: Mathematical coincidence
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Mathematical_coincidence
date_saved: 2026-05-05T07:23:33.671306+00:00
instance: kb-cron
parent: mathematical-coincidence-20260505-0
summary:
The ratio of a mile to a kilometre is also approximately the golden ratio φ = 1 + 5 2 ≈ 1.618 . {\displaystyle \varphi ={\frac {1+{\sqrt {5}}}{2}}\approx 1.618\,.} As a consequence, a Fibonacci number of miles is approximately the next Fibonacci number of kilometres. The ratio of a mile to a kilometre is also very close to ln ⁡ ( 5 ) ≈ 1.6094379 , {\displaystyle \ln(5)\approx 1.6094379,} within 0.006%. This means that 5 m ≈ e k , {\displaystyle 5^{m}\approx e^{k},} where m {\displaystyle m} is the number of miles, k {\displaystyle k} is the number of kilometres and e {\displaystyle e} is Euler's number. A density of one ounce per cubic foot is very close to one kilogram per cubic metre: 1 oz / ft3 = 0.028349523125 kg / (0.3048 m)3 ≈ 1.0012 kg/m3. The conversion factor between troy ounces and grams (1 troy ounce = 31.1034768 g) is approximately equal to 10 π − π 10 = 99 10 π ≈ 31.1018 . {\displaystyle 10\pi -{\frac {\pi }{10}}={\frac {99}{10}}\pi \approx 31.1018\,.} ==== Fine-structure constant ==== The fine-structure constant α {\displaystyle \alpha } is close to, and was once conjectured to be precisely equal to ⁠1/137⁠. Its


id: commutative-diagram-20260505-0
file: data/en.wikipedia.org/wiki/Commutative_diagram-0.md
category: reference
title: Commutative diagram
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Commutative_diagram
date_saved: 2026-05-05T07:23:35.026376+00:00
instance: kb-cron
parent: commutative-diagram-20260505-0
summary:
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. == Description == A commutative diagram often consists of three parts: objects (also known as vertices) morphisms (also known as arrows or edges) paths or composites === Arrow symbols === In algebra texts, the type of morphism can be denoted with different arrow usages: A monomorphism may be labeled with a ↪ {\displaystyle \hookrightarrow } or a ↣ {\displaystyle \rightarrowtail } . An epimorphism may be labeled with a ↠ {\displaystyle \twoheadrightarrow } . An isomorphism may be labeled with a → ∼ {\displaystyle {\overset {\sim }{\rightarrow }}} . The dashed arrow typically represents the claim that the indicated morphism exists (whenever the rest of the diagram holds); the arrow may be optionally labeled as ∃ {\displaystyle \exists } . If the morphism is in addition unique, then the dashed arrow may be labeled ! {\displaystyle !} or ∃ ! {\displaystyle \exists !} . If the morphism acts between two arrows


id: complete-set-of-invariants-20260505-0
file: data/en.wikipedia.org/wiki/Complete_set_of_invariants-0.md
category: reference
title: Complete set of invariants
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Complete_set_of_invariants
date_saved: 2026-05-05T07:23:36.291616+00:00
instance: kb-cron
parent: complete-set-of-invariants-20260505-0
summary:
In mathematics, a complete set of invariants for a classification problem is a collection of maps f i : X → Y i {\displaystyle f_{i}:X\to Y_{i}} (where X {\displaystyle X} is the collection of objects being classified, up to some equivalence relation ∼ {\displaystyle \sim } , and the Y i {\displaystyle Y_{i}} are some sets), such that x ∼ x ′ {\displaystyle x\sim x'} if and only if f i ( x ) = f i ( x ′ ) {\displaystyle f_{i}(x)=f_{i}(x')} for all i {\displaystyle i} . In words, such that two objects are equivalent if and only if all invariants are equal. Symbolically, a complete set of invariants is a collection of maps such that ( ∏ f i ) : ( X / ∼ ) → ( ∏ Y i ) {\displaystyle \left(\prod f_{i}\right):(X/\sim )\to \left(\prod Y_{i}\right)} is injective. As invariants are, by definition, equal on equivalent objects, equality of invariants is a necessary condition for equivalence; a complete set of invariants is a set such that equality of these is also sufficient for equivalence. In the context of a group action, this may be stated as: invariants are functions of coinvariants (equivalence classes, orbits), and


id: conjecture-20260505-0
file: data/en.wikipedia.org/wiki/Conjecture-0.md
category: reference
title: Conjecture
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Conjecture
date_saved: 2026-05-05T07:23:37.531234+00:00
instance: kb-cron
parent: conjecture-20260505-0
summary:
In mathematics, a conjecture is a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis or Fermat's conjecture (now a theorem, proven in 1995 by Andrew Wiles), have shaped much of mathematical history as new areas of mathematics are developed in order to prove them. == Resolution of conjectures == === Proof === Formal mathematics is based on provable truth. In mathematics, any number of cases supporting a universally quantified conjecture, no matter how large, is insufficient for establishing the conjecture's veracity, since a single counterexample could immediately bring down the conjecture. Mathematical journals sometimes publish the minor results of research teams having extended the search for a counterexample farther than previously done. For instance, the Collatz conjecture, which concerns whether or not certain sequences of integers terminate, has been tested for all integers up to 1.2 × 1012 (1.2 trillion). However, the failure to find a counterexample after extensive search does not constitute a proof that the conjecture is true—because the conjecture might be false but with a very large minimal counterexample. Nevertheless, mathematicians often regard a conjecture as strongly supported by evidence even though not yet proved. That evidence may


id: conjecture-20260505-1
file: data/en.wikipedia.org/wiki/Conjecture-1.md
category: reference
title: Conjecture
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Conjecture
date_saved: 2026-05-05T07:23:37.531234+00:00
instance: kb-cron
parent: conjecture-20260505-0
summary:
In mathematics, the four color theorem, or the four color map theorem, states that given any separation of a plane into contiguous regions, producing a figure called a map, no more than four colors are required to color the regions of the map—so that no two adjacent regions have the same color. Two regions are called adjacent if they share a common boundary that is not a corner, where corners are the points shared by three or more regions. For example, in the map of the United States of America, Utah and Arizona are adjacent, but Utah and New Mexico, which only share a point that also belongs to Arizona and Colorado, are not. Möbius mentioned the problem in his lectures as early as 1840. The conjecture was first proposed on October 23, 1852 when Francis Guthrie, while trying to color the map of counties of England, noticed that only four different colors were needed. The five color theorem, which has a short elementary proof, states that five colors suffice to color a map and was proven in the late 19th century; however, proving that four colors suffice turned out to be significantly harder. A number of false proofs and


id: conjecture-20260505-2
file: data/en.wikipedia.org/wiki/Conjecture-2.md
category: reference
title: Conjecture
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Conjecture
date_saved: 2026-05-05T07:23:37.531234+00:00
instance: kb-cron
parent: conjecture-20260505-0
summary:
In mathematics, the Riemann hypothesis, proposed by Bernhard Riemann (1859), is a conjecture that the non-trivial zeros of the Riemann zeta function all have real part 1/2. The name is also used for some closely related analogues, such as the Riemann hypothesis for curves over finite fields. The Riemann hypothesis implies results about the distribution of prime numbers. Along with suitable generalizations, some mathematicians consider it the most important unresolved problem in pure mathematics. The Riemann hypothesis, along with the Goldbach conjecture, is part of Hilbert's eighth problem in David Hilbert's list of 23 unsolved problems; it is also one of the Clay Mathematics Institute Millennium Prize Problems. === P versus NP problem === The P versus NP problem is a major unsolved problem in computer science. Informally, it asks whether every problem whose solution can be quickly verified by a computer can also be quickly solved by a computer; it is widely conjectured that the answer is no. It was essentially first mentioned in a 1956 letter written by Kurt Gödel to John von Neumann. Gödel asked whether a certain NP-complete problem could be solved in quadratic or linear time. The precise statement of the P=NP problem was introduced


id: connectedness-20260505-0
file: data/en.wikipedia.org/wiki/Connectedness-0.md
category: reference
title: Connectedness
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Connectedness
date_saved: 2026-05-05T07:23:38.743176+00:00
instance: kb-cron
parent: connectedness-20260505-0
summary:
In mathematics, connectedness is used to refer to various properties meaning, in some sense, "all one piece". When a mathematical object has such a property, we say it is connected; otherwise it is disconnected. When a disconnected object can be split naturally into connected pieces, each piece is usually called a component (or connected component). == Connectedness in topology == A topological space is said to be connected if it is not the union of two disjoint nonempty open sets. A set is open if it contains no point lying on its boundary; thus, in an informal, intuitive sense, the fact that a space can be partitioned into disjoint open sets suggests that the boundary between the two sets is not part of the space, and thus splits it into two separate pieces. == Other notions of connectedness == Fields of mathematics are typically concerned with special kinds of objects. Often such an object is said to be connected if, when it is considered as a topological space, it is a connected space. Thus, manifolds, Lie groups, and graphs are all called connected if they are connected as topological spaces, and their components are the topological components. Sometimes it is


id: continuous-or-discrete-variable-20260505-0
file: data/en.wikipedia.org/wiki/Continuous_or_discrete_variable-0.md
category: reference
title: Continuous or discrete variable
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Continuous_or_discrete_variable
date_saved: 2026-05-05T07:23:40.034121+00:00
instance: kb-cron
parent: continuous-or-discrete-variable-20260505-0
summary:
In mathematics and statistics, a quantitative variable may be continuous or discrete. If it can take on two real values and all the values between them, the variable is continuous in that interval. If it can take on a value such that there is a non-infinitesimal gap on each side of it containing no values that the variable can take on, then it is discrete around that value. In some contexts, a variable can be discrete in some ranges of the number line and continuous in others. In statistics, continuous and discrete variables are distinct statistical data types which are described with different probability distributions. == Continuous variable == A continuous variable is a variable such that there are possible values between any two values. For example, a variable over a non-empty range of the real numbers is continuous if it can take on any value in that range. Methods of calculus are often used in problems in which the variables are continuous, for example in continuous optimization problems. In statistical theory, the probability distributions of continuous variables can be expressed in terms of probability density functions. In continuous-time dynamics, the variable time is treated as continuous, and the equation


id: corollary-20260505-0
file: data/en.wikipedia.org/wiki/Corollary-0.md
category: reference
title: Corollary
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Corollary
date_saved: 2026-05-05T07:23:41.286404+00:00
instance: kb-cron
parent: corollary-20260505-0
summary:
In mathematics and logic, a corollary (US: , KORR-ə-lair-ee; UK: , kər-OL-ər-ee) is a proposition which can be readily deduced from a previous, already proven proposition. A corollary could be a proposition that is incidentally proved while proving another proposition; it might also be used more casually to refer to something which naturally or incidentally accompanies something else. == Overview == In mathematics, a corollary is a theorem connected by a short proof to an existing theorem. The use of the term corollary, rather than proposition or theorem, is intrinsically subjective. More formally, proposition B is a corollary of proposition A, if B can be readily deduced from A or is self-evident from its proof. In many cases, a corollary corresponds to a special case of a larger theorem, which makes the theorem easier to use and apply, even though its importance is generally considered to be secondary to that of the theorem. In particular, B is unlikely to be termed a corollary if its mathematical consequences are as significant as those of A. A corollary might have a proof that explains its derivation, even though such a derivation might be considered rather self-evident in some occasions (e.g., the Pythagorean


id: correlation-coefficient-20260505-0
file: data/en.wikipedia.org/wiki/Correlation_coefficient-0.md
category: reference
title: Correlation coefficient
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Correlation_coefficient
date_saved: 2026-05-05T07:23:42.530685+00:00
instance: kb-cron
parent: correlation-coefficient-20260505-0
summary:
A correlation coefficient is a numerical measure of some type of linear correlation, meaning a linear function between two variables. The variables may be two columns of a given data set of observations, often called a sample, or two components of a multivariate random variable with a known distribution. Several types of correlation coefficient exist, each with their own definition and range of usability and characteristics. They all assume values in the range from −1 to +1, where ±1 indicates the strongest possible correlation and 0 indicates no correlation. As tools of analysis, correlation coefficients present certain problems, including the propensity of some types to be distorted by outliers and the possibility of incorrectly being used to infer a causal relationship between the variables (for more, see Correlation does not imply causation). == Types == There are several different measures for the degree of correlation in data, depending on the kind of data: principally, whether the data is a measurement, ordinal, or categorical. === Pearson === The Pearson product-moment correlation coefficient, also known as r, R, or Pearson's r, is a measure of the strength and direction of the linear relationship between two variables that is defined as the covariance


id: counterexample-20260505-0
file: data/en.wikipedia.org/wiki/Counterexample-0.md
category: reference
title: Counterexample
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Counterexample
date_saved: 2026-05-05T07:23:43.842156+00:00
instance: kb-cron
parent: counterexample-20260505-0
summary:
A counterexample is a specific example that contradicts a claim, hypothesis, or generalization. In logic a counterexample disproves a universally stated claim, and does so rigorously in the fields of mathematics and philosophy. For example, the statement that "student John Smith is not lazy" is a counterexample to the generalization "students are lazy", and both a counterexample to, and disproof of, the universally quantified "all students are lazy." == In mathematics == In mathematics, counterexamples are often used to prove the boundaries of possible theorems. By using counterexamples to show that certain conjectures are false, mathematical researchers can then avoid going down blind alleys and learn to modify conjectures to produce provable theorems. It is sometimes said that mathematical development consists primarily in finding (and proving) theorems and counterexamples. === Rectangle example === Suppose that a mathematician is studying geometry and shapes, and she wishes to prove certain theorems about them. She conjectures that "All rectangles are squares", and she is interested in knowing whether this statement is true or false. In this case, she can either attempt to prove the truth of the statement using deductive reasoning, or she can attempt to find a counterexample of the statement if


id: cryptomorphism-20260505-0
file: data/en.wikipedia.org/wiki/Cryptomorphism-0.md
category: reference
title: Cryptomorphism
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Cryptomorphism
date_saved: 2026-05-05T07:23:45.040735+00:00
instance: kb-cron
parent: cryptomorphism-20260505-0
summary:
In mathematics, two objects, especially systems of axioms or semantics for them, are called cryptomorphic if they are equivalent but not obviously equivalent. In particular, two definitions or axiomatizations of the same object are "cryptomorphic" if it is not obvious that they define the same object. Examples of cryptomorphic definitions abound in matroid theory and others can be found elsewhere, e.g., in group theory the definition of a group by a single operation of division, which is not obviously equivalent to the usual three "operations" of identity element, inverse, and multiplication. This word is a play on the many morphisms in mathematics, but "cryptomorphism" is only very distantly related to "isomorphism", "homomorphism", or "morphisms". The equivalence may in a cryptomorphism, if it is not actual identity, be informal, or may be formalized in terms of a bijection or equivalence of categories between the mathematical objects defined by the two cryptomorphic axiom systems. == Etymology == The word was coined by Garrett Birkhoff before 1967, for use in the third edition of his book Lattice Theory. Birkhoff did not give it a formal definition, though others working in the field have made some attempts since. == Use in matroid theory ==


id: definition-20260505-0
file: data/en.wikipedia.org/wiki/Definition-0.md
category: reference
title: Definition
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Definition
date_saved: 2026-05-05T07:23:46.235974+00:00
instance: kb-cron
parent: definition-20260505-0
summary:
A definition is a semantic statement of the meaning of a term (a word, phrase, or other set of symbols). Definitions can be classified into two large categories: intensional definitions (which try to give the sense of a term), and extensional definitions (which try to list the objects that a term describes). Another important category of definitions is the class of ostensive definitions, which convey the meaning of a term by pointing out examples. A term may have many different senses and multiple meanings, and thus require multiple definitions. In mathematics, a definition is used to give a precise meaning to a new term, by describing a condition which unambiguously qualifies what the mathematical term is and is not. Definitions and axioms form the basis on which all of modern mathematics is to be constructed. == Basic terminology == In modern usage, a definition is something, typically expressed in words, that attaches a meaning to a word or group of words. The word or group of words that is to be defined is called the definiendum, and the word, group of words, or action that defines it is called the definiens. For example, in the definition "An elephant is a


id: definition-20260505-1
file: data/en.wikipedia.org/wiki/Definition-1.md
category: reference
title: Definition
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Definition
date_saved: 2026-05-05T07:23:46.235974+00:00
instance: kb-cron
parent: definition-20260505-0
summary:
In classical thought, a definition was taken to be a statement of the essence of a thing. Aristotle had it that an object's essential attributes form its "essential nature", and that a definition of the object must include these essential attributes. The idea that a definition should state the essence of a thing led to the distinction between nominal and real essence—a distinction originating with Aristotle. In the Posterior Analytics, he says that the meaning of a made-up name can be known (he gives the example "goat stag") without knowing what he calls the "essential nature" of the thing that the name would denote (if there were such a thing). This led medieval logicians to distinguish between what they called the quid nominis, or the "whatness of the name", and the underlying nature common to all the things it names, which they called the quid rei, or the "whatness of the thing". The name "hobbit", for example, is perfectly meaningful. It has a quid nominis, but one could not know the real nature of hobbits, and so the quid rei of hobbits cannot be known. By contrast, the name "man" denotes real things (men) that have a certain quid rei.


id: definition-20260505-2
file: data/en.wikipedia.org/wiki/Definition-2.md
category: reference
title: Definition
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Definition
date_saved: 2026-05-05T07:23:46.235974+00:00
instance: kb-cron
parent: definition-20260505-0
summary:
At least one thing is stated to be a member of the set being defined; this is sometimes called a "base set". All things bearing a certain relation to other members of the set are also to count as members of the set. It is this step that makes the definition recursive. All other things are excluded from the set For instance, we could define a natural number as follows (after Peano): "0" is a natural number. Each natural number has a unique successor, such that: the successor of a natural number is also a natural number; distinct natural numbers have distinct successors; no natural number is succeeded by "0". Nothing else is a natural number. So "0" will have exactly one successor, which for convenience can be called "1". In turn, "1" will have exactly one successor, which could be called "2", and so on. The second condition in the definition itself refers to natural numbers, and hence involves self-reference. Although this sort of definition involves a form of circularity, it is not vicious, and the definition has been quite successful. In the same way, we can define ancestor as follows: A parent is an ancestor. A parent of


id: definition-20260505-3
file: data/en.wikipedia.org/wiki/Definition-3.md
category: reference
title: Definition
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Definition
date_saved: 2026-05-05T07:23:46.235974+00:00
instance: kb-cron
parent: definition-20260505-0
summary:
=== Limitations of definition === Given that a natural language such as English contains, at any given time, a finite number of words, any comprehensive list of definitions must either be circular or rely upon primitive notions. If every term of every definiens must itself be defined, "where at last should we stop?" A dictionary, for instance, insofar as it is a comprehensive list of lexical definitions, must resort to circularity. Many philosophers have chosen instead to leave some terms undefined. The scholastic philosophers claimed that the highest genera (called the ten generalissima) cannot be defined, since a higher genus cannot be assigned under which they may fall. Thus being, unity and similar concepts cannot be defined. Locke supposes in An Essay Concerning Human Understanding that the names of simple concepts do not admit of any definition. More recently Bertrand Russell sought to develop a formal language based on logical atoms. Other philosophers, notably Wittgenstein, rejected the need for any undefined simples. Wittgenstein pointed out in his Philosophical Investigations that what counts as a "simple" in one circumstance might not do so in another. He rejected the very idea that every explanation of the meaning of a term needed itself


id: dependent-and-independent-variables-20260505-0
file: data/en.wikipedia.org/wiki/Dependent_and_independent_variables-0.md
category: reference
title: Dependent and independent variables
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Dependent_and_independent_variables
date_saved: 2026-05-05T07:23:47.442630+00:00
instance: kb-cron
parent: dependent-and-independent-variables-20260505-0
summary:
A variable is considered dependent if it depends on (or is hypothesized to depend on) an independent variable. Dependent variables are the outcome of the test they depend on, by some law or rule (e.g., by a mathematical function). Independent variables, on the other hand, are not seen as depending on any other variable in the scope of the experiment in question. Rather, they are controlled by the experimenter. == In pure mathematics == In mathematics, a function is a rule for taking an input (in the simplest case, a number or set of numbers) and providing an output (which may also be a number or set of numbers). A symbol that stands for an arbitrary input is called an independent variable, while a symbol that stands for an arbitrary output is called a dependent variable. The most common symbol for the input is x, and the most common symbol for the output is y; the function itself is commonly written y = f(x). It is possible to have multiple independent variables or multiple dependent variables. For instance, in multivariable calculus, one often encounters functions of the form z = f(x,y), where z is a dependent variable and x and


id: dependent-and-independent-variables-20260505-1
file: data/en.wikipedia.org/wiki/Dependent_and_independent_variables-1.md
category: reference
title: Dependent and independent variables
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Dependent_and_independent_variables
date_saved: 2026-05-05T07:23:47.442630+00:00
instance: kb-cron
parent: dependent-and-independent-variables-20260505-0
summary:
== Other variables == A variable may be thought to alter the dependent or independent variables, but may not actually be the focus of the experiment. So that the variable will be kept constant or monitored to try to minimize its effect on the experiment. Such variables may be designated as either a "controlled variable", "control variable", or "fixed variable". Extraneous variables are candidate independent variables which may be included in a regression analysis to aid a researcher with accurate response parameter estimation, prediction, and goodness of fit, but are not of substantive interest to the hypothesis under examination. For example, in a study examining the effect of post-secondary education on lifetime earnings, some extraneous variables might be gender, ethnicity, social class, genetics, intelligence, age, and so forth. A variable is extraneous only when it can be assumed (or shown) to influence the dependent variable. If included in a regression, it can improve the fit of the model. If it is excluded from the regression and if it has a non-zero covariance with one or more of the independent variables of interest, its omission will bias the regression's result for the effect of that independent variable of interest. This effect


id: differential-mathematics-20260505-0
file: data/en.wikipedia.org/wiki/Differential_(mathematics)-0.md
category: reference
title: Differential (mathematics)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Differential_(mathematics)
date_saved: 2026-05-05T07:23:48.842935+00:00
instance: kb-cron
parent: differential-mathematics-20260505-0
summary:
In mathematics, differential refers to several related notions derived from the early days of calculus, put on a rigorous footing, such as infinitesimal differences and the derivatives of functions. The term is used in various branches of mathematics such as calculus, differential geometry, algebraic geometry and algebraic topology. == Introduction == The term differential is used nonrigorously in calculus to refer to an infinitesimal ("infinitely small") change in some varying quantity. For example, if x is a variable, then a change in the value of x is often denoted Δx (pronounced delta x). The differential dx represents an infinitely small change in the variable x. The idea of an infinitely small or infinitely slow change is, intuitively, extremely useful, and there are a number of ways to make the notion mathematically precise. Using calculus, it is possible to relate the infinitely small changes of various variables to each other mathematically using derivatives. If y is a function of x, then the differential dy of y is related to dx by the formula d y = d y d x d x , {\displaystyle dy={\frac {dy}{dx}}\,dx,} where d y d x {\displaystyle {\frac {dy}{dx}}\,} denotes not 'dy divided by dx' as


id: differential-mathematics-20260505-1
file: data/en.wikipedia.org/wiki/Differential_(mathematics)-1.md
category: reference
title: Differential (mathematics)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Differential_(mathematics)
date_saved: 2026-05-05T07:23:48.842935+00:00
instance: kb-cron
parent: differential-mathematics-20260505-0
summary:
Differentials as linear maps. This approach underlies the definition of the derivative and the exterior derivative in differential geometry. Differentials as nilpotent elements of commutative rings. This approach is popular in algebraic geometry. Differentials in smooth models of set theory. This approach is known as synthetic differential geometry or smooth infinitesimal analysis and is closely related to the algebraic geometric approach, except that ideas from topos theory are used to hide the mechanisms by which nilpotent infinitesimals are introduced. Differentials as infinitesimals in hyperreal number systems, which are extensions of the real numbers that contain invertible infinitesimals and infinitely large numbers. This is the approach of nonstandard analysis pioneered by Abraham Robinson. These approaches are very different from each other, but they have in common the idea of being quantitative, i.e., saying not just that a differential is infinitely small, but how small it is. === Differentials as linear maps === There is a simple way to make precise sense of differentials, first used on the Real line by regarding them as linear maps. It can be used on R {\displaystyle \mathbb {R} } , R n {\displaystyle \mathbb {R} ^{n}} , a Hilbert space, a Banach space, or more


id: differential-mathematics-20260505-2
file: data/en.wikipedia.org/wiki/Differential_(mathematics)-2.md
category: reference
title: Differential (mathematics)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Differential_(mathematics)
date_saved: 2026-05-05T07:23:48.842935+00:00
instance: kb-cron
parent: differential-mathematics-20260505-0
summary:
d f p = ∑ j = 1 n D j f ( p ) ( d x j ) p . {\displaystyle df_{p}=\sum _{j=1}^{n}D_{j}f(p)\,(dx_{j})_{p}.} The coefficients D j f ( p ) {\displaystyle D_{j}f(p)} are (by definition) the partial derivatives of f {\displaystyle f} at p {\displaystyle p} with respect to x 1 , x 2 , … , x n {\displaystyle x_{1},x_{2},\ldots ,x_{n}} . Hence, if f {\displaystyle f} is differentiable on all of R n {\displaystyle \mathbb {R} ^{n}} , we can write, more concisely: d f = ∂ f ∂ x 1 d x 1 + ∂ f ∂ x 2 d x 2 + ⋯ + ∂ f ∂ x n d x n . {\displaystyle df={\frac {\partial f}{\partial x_{1}}}\,dx_{1}+{\frac {\partial f}{\partial x_{2}}}\,dx_{2}+\cdots +{\frac {\partial f}{\partial x_{n}}}\,dx_{n}.} In the one-dimensional case this becomes d f = d f d x d x {\displaystyle df={\frac {df}{dx}}dx} as before. This idea generalizes straightforwardly to functions from R n {\displaystyle \mathbb {R} ^{n}} to R m {\displaystyle \mathbb {R} ^{m}} . Furthermore, it has the decisive advantage over other definitions of the derivative that it is invariant under changes of coordinates. This means that the same idea can


id: differential-mathematics-20260505-3
file: data/en.wikipedia.org/wiki/Differential_(mathematics)-3.md
category: reference
title: Differential (mathematics)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Differential_(mathematics)
date_saved: 2026-05-05T07:23:48.842935+00:00
instance: kb-cron
parent: differential-mathematics-20260505-0
summary:
=== Algebraic geometry === In algebraic geometry, differentials and other infinitesimal notions are handled in a very explicit way by accepting that the coordinate ring or structure sheaf of a space may contain nilpotent elements. The simplest example is the ring of dual numbers R[ε], where ε2 = 0. This can be motivated by the algebro-geometric point of view on the derivative of a function f from R to R at a point p. For this, note first that f − f(p) belongs to the ideal Ip of functions on R which vanish at p. If the derivative f vanishes at p, then f − f(p) belongs to the square Ip2 of this ideal. Hence the derivative of f at p may be captured by the equivalence class [f − f(p)] in the quotient space Ip/Ip2, and the 1-jet of f (which encodes its value and its first derivative) is the equivalence class of f in the space of all functions modulo Ip2. Algebraic geometers regard this equivalence class as the restriction of f to a thickened version of the point p whose coordinate ring is not R (which is the quotient space of functions on R modulo Ip) but


id: essentially-unique-20260505-0
file: data/en.wikipedia.org/wiki/Essentially_unique-0.md
category: reference
title: Essentially unique
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Essentially_unique
date_saved: 2026-05-05T07:23:50.018300+00:00
instance: kb-cron
parent: essentially-unique-20260505-0
summary:
In mathematics, the term essentially unique is used to describe a weaker form of uniqueness, where an object satisfying a property is "unique" only in the sense that all objects satisfying the property are equivalent to each other. The notion of essential uniqueness presupposes some form of "sameness", which is often formalized using an equivalence relation. A related notion is a universal property, where an object is not only essentially unique, but unique up to a unique isomorphism (meaning that it has trivial automorphism group). In general there can be more than one isomorphism between examples of an essentially unique object. == Examples == === Set theory === At the most basic level, there is an essentially unique set of any given cardinality, whether one labels the elements { 1 , 2 , 3 } {\displaystyle \{1,2,3\}} or { a , b , c } {\displaystyle \{a,b,c\}} . In this case, the non-uniqueness of the isomorphism (e.g., match 1 to a {\displaystyle a} or 1 to c {\displaystyle c} ) is reflected in the symmetric group. On the other hand, there is an essentially unique totally ordered set of any given finite cardinality that is unique up to unique isomorphism:


id: eventually-mathematics-20260505-0
file: data/en.wikipedia.org/wiki/Eventually_(mathematics)-0.md
category: reference
title: Eventually (mathematics)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Eventually_(mathematics)
date_saved: 2026-05-05T07:23:51.308179+00:00
instance: kb-cron
parent: eventually-mathematics-20260505-0
summary:
In the mathematical areas of number theory and analysis, an infinite sequence or a function is said to eventually have a certain property, if it does not have the said property across all its ordered instances, but will after some instances have passed. The use of the term "eventually" can be often rephrased as "for sufficiently large numbers", and can be also extended to the class of properties that apply to elements of any ordered set (such as sequences and subsets of R {\displaystyle \mathbb {R} } ). == Notation == The general form where the phrase eventually (or sufficiently large) is found appears as follows: P {\displaystyle P} is eventually true for x {\displaystyle x} ( P {\displaystyle P} is true for sufficiently large x {\displaystyle x} ), where ∀ {\displaystyle \forall } and ∃ {\displaystyle \exists } are the universal and existential quantifiers, which is actually a shorthand for: ∃ a ∈ R {\displaystyle \exists a\in \mathbb {R} } such that P {\displaystyle P} is true ∀ x ≥ a {\displaystyle \forall x\geq a} or somewhat more formally: ∃ a ∈ R : ∀ x ∈ R : x ≥ a ⇒ P ( x ) {\displaystyle \exists


id: exceptional-object-20260505-0
file: data/en.wikipedia.org/wiki/Exceptional_object-0.md
category: reference
title: Exceptional object
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Exceptional_object
date_saved: 2026-05-05T07:23:52.521982+00:00
instance: kb-cron
parent: exceptional-object-20260505-0
summary:
In mathematics, an exceptional object is one of finitely many exceptions to some classification of objects. Many branches of mathematics study objects of a given type and prove a classification theorem. A common theme is that the classification results in a number of series of objects and a finite number of exceptions — often with desirable properties — that do not fit into any series; these are known as exceptional objects. In many cases, these exceptional objects play a further and important role in the subject. Furthermore, the exceptional objects in one branch of mathematics often relate to the exceptional objects in others. A related phenomenon is exceptional isomorphism, when two series are in general different, but agree for some small values. For example, spin groups in low dimensions are isomorphic to other classical Lie groups. == Regular polytopes == The prototypical examples of exceptional objects arise in the classification of regular polytopes: in two dimensions, there is a series of regular n-gons for n ≥ 3. In every dimension above 2, one can find analogues of the cube, tetrahedron and octahedron. In three dimensions, one finds two more regular polyhedra — the dodecahedron (12-hedron) and the icosahedron (20-hedron) —


id: exceptional-object-20260505-1
file: data/en.wikipedia.org/wiki/Exceptional_object-1.md
category: reference
title: Exceptional object
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Exceptional_object
date_saved: 2026-05-05T07:23:52.521982+00:00
instance: kb-cron
parent: exceptional-object-20260505-0
summary:
== Block designs == An exceptional block design is the Steiner system S(5,8,24) whose automorphism group is the sporadic simple Mathieu group M 24 {\displaystyle M_{24}} . The codewords of the extended binary Golay code have a length of 24 bits and have weights 0, 8, 12, 16, or 24. This code can correct up to three errors. So every 24-bit word with weight 5 can be corrected to a codeword with weight 8. The bits of a 24-bit word can be thought of as specifying the possible subsets of a 24 element set. So the extended binary Golay code gives a unique 8 element subset for each 5 element subset. In fact, it defines S(5,8,24). == Outer automorphisms == Certain families of groups often have a certain outer automorphism group, but in particular cases, they have other exceptional outer automorphisms. Among families of finite simple groups, the only example is in the automorphisms of the symmetric and alternating groups: for n ≥ 3 , n ≠ 6 {\displaystyle n\geq 3,n\neq 6} the alternating group A n {\displaystyle A_{n}} has one outer automorphism (corresponding to conjugation by an odd element of S n {\displaystyle S_{n}} ) and the symmetric group


id: exceptional-object-20260505-2
file: data/en.wikipedia.org/wiki/Exceptional_object-2.md
category: reference
title: Exceptional object
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Exceptional_object
date_saved: 2026-05-05T07:23:52.521982+00:00
instance: kb-cron
parent: exceptional-object-20260505-0
summary:
The Leech lattice is 24-dimensional. Most sporadic simple groups can be related to the Leech lattice, or more broadly the Monster. The exceptional Jordan algebra has a representation in terms of 24×24 real matrices together with the Jordan product rule. These objects are connected to various other phenomena in math which may be considered surprising but not themselves "exceptional". For example, in algebraic topology, 8-fold real Bott periodicity can be seen as coming from the octonions. In the theory of modular forms, the 24-dimensional nature of the Leech lattice underlies the presence of 24 in the formulas for the Dedekind eta function and the modular discriminant, which connection is deepened by Monstrous moonshine, a development that related modular functions to the Monster group. === Physics === In string theory and superstring theory we often find that particular dimensions are singled out as a result of exceptional algebraic phenomena. For example, bosonic string theory requires a spacetime of dimension 26 which is directly related to the presence of 24 in the Dedekind eta function. Similarly, the possible dimensions of supergravity are related to the dimensions of the division algebras. === Monstrous moonshine === Many of the exceptional objects in mathematics and


id: formulario-mathematico-20260505-0
file: data/en.wikipedia.org/wiki/Formulario_mathematico-0.md
category: reference
title: Formulario mathematico
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Formulario_mathematico
date_saved: 2026-05-05T07:23:53.723865+00:00
instance: kb-cron
parent: formulario-mathematico-20260505-0
summary:
Formulario Mathematico (Latino sine flexione: Formulary for Mathematics) is a book by Giuseppe Peano which expresses fundamental theorems of mathematics in a symbolic language developed by Peano. The author was assisted by Giovanni Vailati, Mario Pieri, Alessandro Padoa, Giovanni Vacca, Vincenzo Vivanti, Gino Fano and Cesare Burali-Forti. The Formulario was first published in 1894. The fifth and last edition was published in 1908. Nicolas Bourbaki described Peano's notation in the Formulario as "following current mathematical usage, and introducing many well-chosen abbreviating symbols, his language succeeded moreover in being fairly readable, ..." Hubert Kennedy wrote "the development and use of mathematical logic is the guiding motif of the project". He also explains the variety of Peano's publication under the title: the five editions of the Formulario [are not] editions in the usual sense of the word. Each is essentially a new elaboration, although much material is repeated. Moreover, the title and language varied: the first three, titled Formulaire de Mathématiques, and the fourth, titled, Formulaire Mathématique, were written in French, while Latino sine flexione, Peano's own invention, was used for the fifth edition, titled Formulario Mathematico. ... Ugo Cassina lists no less than twenty separately published items as being parts of


id: generalized-inverse-20260505-0
file: data/en.wikipedia.org/wiki/Generalized_inverse-0.md
category: reference
title: Generalized inverse
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Generalized_inverse
date_saved: 2026-05-05T07:23:55.090630+00:00
instance: kb-cron
parent: generalized-inverse-20260505-0
summary:
In mathematics, and in particular, algebra, a generalized inverse (or, g-inverse) of an element x is an element y that has some properties of an inverse element but not necessarily all of them. The purpose of constructing a generalized inverse of a matrix is to obtain a matrix that can serve as an inverse in some sense for a wider class of matrices than invertible matrices. Generalized inverses can be defined in any mathematical structure that involves associative multiplication, that is, in a semigroup. This article describes generalized inverses of a matrix A {\displaystyle A} . A matrix A g ∈ R n × m {\displaystyle A^{\mathrm {g} }\in \mathbb {R} ^{n\times m}} is a generalized inverse of a matrix A ∈ R m × n {\displaystyle A\in \mathbb {R} ^{m\times n}} if A A g A = A . {\displaystyle AA^{\mathrm {g} }A=A.} A generalized inverse exists for an arbitrary matrix, and when a matrix has a regular inverse, this inverse is its unique generalized inverse. == Motivation == Consider the linear system A x = y {\displaystyle Ax=y} where A {\displaystyle A} is an m × n {\displaystyle m\times n} matrix and y ∈ C ( A )


id: generalized-inverse-20260505-1
file: data/en.wikipedia.org/wiki/Generalized_inverse-1.md
category: reference
title: Generalized inverse
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Generalized_inverse
date_saved: 2026-05-05T07:23:55.090630+00:00
instance: kb-cron
parent: generalized-inverse-20260505-0
summary:
( A A g ) ∗ = A A g {\displaystyle (AA^{\mathrm {g} })^{*}=AA^{\mathrm {g} }} ( A g A ) ∗ = A g A , {\displaystyle (A^{\mathrm {g} }A)^{*}=A^{\mathrm {g} }A,} where ∗ {\displaystyle {}^{*}} denotes conjugate transpose. If A g {\displaystyle A^{\mathrm {g} }} satisfies the first condition, then it is a generalized inverse of A {\displaystyle A} . If it satisfies the first two conditions, then it is a reflexive generalized inverse of A {\displaystyle A} . If it satisfies all four conditions, then it is the pseudoinverse of A {\displaystyle A} , which is denoted by A + {\displaystyle A^{+}} and also known as the Moore–Penrose inverse, after the pioneering works by E. H. Moore and Roger Penrose. It is convenient to define an I {\displaystyle I} -inverse of A {\displaystyle A} as an inverse that satisfies the subset I ⊂ { 1 , 2 , 3 , 4 } {\displaystyle I\subset \{1,2,3,4\}} of the Penrose conditions listed above. Relations, such as A ( 1 , 4 ) A A ( 1 , 3 ) = A + {\displaystyle A^{(1,4)}AA^{(1,3)}=A^{+}} , can be established between these different classes of I {\displaystyle I} -inverses. When A


id: generalized-inverse-20260505-2
file: data/en.wikipedia.org/wiki/Generalized_inverse-2.md
category: reference
title: Generalized inverse
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Generalized_inverse
date_saved: 2026-05-05T07:23:55.090630+00:00
instance: kb-cron
parent: generalized-inverse-20260505-0
summary:
A right inverse of a non-square matrix A {\displaystyle A} is given by A R − 1 = A ⊺ ( A A ⊺ ) − 1 {\displaystyle A_{\mathrm {R} }^{-1}=A^{\intercal }\left(AA^{\intercal }\right)^{-1}} , provided A {\displaystyle A} has full row rank. A left inverse of a non-square matrix A {\displaystyle A} is given by A L − 1 = ( A ⊺ A ) − 1 A ⊺ {\displaystyle A_{\mathrm {L} }^{-1}=\left(A^{\intercal }A\right)^{-1}A^{\intercal }} , provided A {\displaystyle A} has full column rank. If A = B C {\displaystyle A=BC} is a rank factorization, then G = C R − 1 B L − 1 {\displaystyle G=C_{\mathrm {R} }^{-1}B_{\mathrm {L} }^{-1}} is a g-inverse of A {\displaystyle A} , where C R − 1 {\displaystyle C_{\mathrm {R} }^{-1}} is a right inverse of C {\displaystyle C} and B L − 1 {\displaystyle B_{\mathrm {L} }^{-1}} is left inverse of B {\displaystyle B} . If A = P [ I r 0 0 0 ] Q {\displaystyle A=P{\begin{bmatrix}I_{r}&0\\0&0\end{bmatrix}}Q} for any non-singular matrices P {\displaystyle P} and Q {\displaystyle Q} , then G = Q − 1 [ I r U W V ] P − 1 {\displaystyle G=Q^{-1}{\begin{bmatrix}I_{r}&U\\W&V\end{bmatrix}}P^{-1}} is a


id: generalized-space-20260505-0
file: data/en.wikipedia.org/wiki/Generalized_space-0.md
category: reference
title: Generalized space
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Generalized_space
date_saved: 2026-05-05T07:23:56.343111+00:00
instance: kb-cron
parent: generalized-space-20260505-0
summary:
In mathematics, a generalized space is a generalization of a topological space. Impetuses for such a generalization comes at least in two forms: A desire to apply concepts like cohomology for objects that are not traditionally viewed as spaces. For example, a topos was originally introduced for this reason. A practical need to remedy the deficiencies that some naturally occurring categories of spaces (e.g., ones in functional analysis) tend not to be abelian, a standard requirement to do homological algebra. Alexander Grothendieck's dictum says a topos is a generalized space; precisely, he and his followers write in exposé 4 of SGA I: On peut done dire que la notion de topos, dérivé naturel du point de vue faisceautique en Topologie, constitue à son tour un élargissement substantiel de la notion d'espace topologique, un grand nombre de situations qui autrefois n'étaient pas considérées comme relevant de intuition topologique However, William Lawvere argues in his 1975 paper that this dictum should be turned backward; namely, "a topos is the 'algebra of continuous (set-valued) functions' on a generalized space, not the generalized space itself." A generalized space should not be confused with a geometric object that can substitute the role of spaces. For


id: hand-waving-20260505-0
file: data/en.wikipedia.org/wiki/Hand-waving-0.md
category: reference
title: Hand-waving
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Hand-waving
date_saved: 2026-05-05T07:23:57.540774+00:00
instance: kb-cron
parent: hand-waving-20260505-0
summary:
Hand-waving is a pejorative label for attempting to be seen as effective – in word, reasoning, or deed – while actually doing nothing effective or substantial. It is often applied to debating techniques that involve fallacies, misdirection and the glossing over of details. It is also used academically to indicate unproven claims and skipped steps in proofs (sometimes intentionally, as in lectures and instructional materials), with some specific meanings in particular fields, including literary criticism, speculative fiction, mathematics, logic, science and engineering. The term can additionally be used in work situations, when attempts are made to display productivity or assure accountability without actually resulting in them. The term can also be used as a self-admission of, and suggestion to defer discussion about, an allegedly unimportant weakness in one's own argument's evidence, to forestall an opponent dwelling on it. In debate competition, certain cases of this form of hand-waving may be explicitly permitted. Hand-waving is an idiomatic metaphor, derived in part from the use of excessive gesticulation, perceived as unproductive, distracting or nervous, in communication or other effort. The term also evokes the sleight-of-hand distraction techniques of stage magic, and suggests that the speaker or writer seems to believe that if


id: hand-waving-20260505-1
file: data/en.wikipedia.org/wiki/Hand-waving-1.md
category: reference
title: Hand-waving
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Hand-waving
date_saved: 2026-05-05T07:23:57.540774+00:00
instance: kb-cron
parent: hand-waving-20260505-0
summary:
If someone starts a sentence with "Clearly..." or "Obviously..." or "It is self-evident that...", it is a good bet he is about to handwave (alternatively, use of these constructions in a sarcastic tone before a paraphrase of someone else's argument suggests that it is a handwave). The theory behind this term is that if you wave your hands at the right moment, the listener may be sufficiently distracted to not notice that what you have said is bogus [i.e., incorrect]. Failing that, if a listener does object, you might try to dismiss the objection with a wave of your hand. The implication that hand-waving is done with the specific intent to mislead has long been attached to the term, due to the use of literal waving of a hand – either natural-looking or showy, but never desperate – by illusionists to distract audiences and misdirect their attention from the mechanisms of the sleight-of-hand, gimmicked props or other trick being used in the performance. This meaning has become reinforced in recent decades by the influence of Star Wars (1977) and its sequels, in which the fictional Jedi mind trick involves a subtle hand wave with mystical powers – that only work


id: hand-waving-20260505-2
file: data/en.wikipedia.org/wiki/Hand-waving-2.md
category: reference
title: Hand-waving
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Hand-waving
date_saved: 2026-05-05T07:23:57.540774+00:00
instance: kb-cron
parent: hand-waving-20260505-0
summary:
Hand-waving arguments in engineering and other applied sciences often include order-of-magnitude estimates and dimensional analysis, especially in the use of Fermi problems in physics and engineering education. However, competent, well-intentioned researchers and professors also rely on explicitly declared hand-waving when, given a limited time, a large result must be shown and minor technical details cannot be given much attention—e.g., "it can be shown that z is an even number", as an intermediary step in reaching a conclusion. Another example of hand-waving can be found in the oversimplifications of the geologic representations commonly used in groundwater models created in support of land-development applications, especially those involving metal mining and aggregate extraction. Back-of-the-envelope calculations are approximate ways to get an answer by over-simplification, and are comparable to hand-waving in this sense. == In business == Hand-waving has been used to describe work-related situations where productivity is seemingly displayed, but deliverables are not produced, especially in the context of intentional engagement in busy work or pretend-work, vague claims of overwork or complications, impenetrably buzzword-laden rationalizations for delays or otherwise poor performance, and plausible-sounding but weak excuse-making and attention-deflecting tactics. In employment situations, as in political discourse, a hand-waving effort may seek to shift


id: harmonic-mathematics-20260505-0
file: data/en.wikipedia.org/wiki/Harmonic_(mathematics)-0.md
category: reference
title: Harmonic (mathematics)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Harmonic_(mathematics)
date_saved: 2026-05-05T07:23:58.780140+00:00
instance: kb-cron
parent: harmonic-mathematics-20260505-0
summary:
In mathematics, a number of concepts employ the word harmonic. The similarity of this terminology to that of music is not accidental: the equations of motion of vibrating strings, drums and columns of air are given by formulas involving Laplacians; the solutions to which are given by eigenvalues corresponding to their modes of vibration. Thus, the term "harmonic" is applied when one is considering functions with sinusoidal variations, or solutions of Laplace's equation and related concepts. Mathematical terms whose names include "harmonic" include: Projective harmonic conjugate Cross-ratio Harmonic analysis Harmonic conjugate Harmonic form Harmonic function Harmonic mean Harmonic mode Harmonic number Harmonic series Alternating harmonic series Harmonic tremor Spherical harmonics


id: if-and-only-if-20260505-0
file: data/en.wikipedia.org/wiki/If_and_only_if-0.md
category: reference
title: If and only if
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/If_and_only_if
date_saved: 2026-05-05T07:24:00.078315+00:00
instance: kb-cron
parent: if-and-only-if-20260505-0
summary:
In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either both statements are true or both are false. The connective is biconditional (a statement of material equivalence), and can be likened to the standard material conditional ("only if", equal to "if ... then") combined with its converse ("if"); hence the name. The result is that the truth of either one of the connected statements requires the truth of the other (i.e. either both statements are true, or both are false), though it is controversial whether the connective thus defined is properly rendered by the English "if and only if"—with its pre-existing meaning. For example, P if and only if Q means that P is true whenever Q is true, and the only case in which P is true is if Q is also true, whereas in the case of P if Q, there could be other scenarios where P is true and Q is false. In writing, phrases commonly used as alternatives to P "if and only if" Q include: Q is necessary


id: if-and-only-if-20260505-1
file: data/en.wikipedia.org/wiki/If_and_only_if-1.md
category: reference
title: If and only if
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/If_and_only_if
date_saved: 2026-05-05T07:24:00.078315+00:00
instance: kb-cron
parent: if-and-only-if-20260505-0
summary:
== More general usage == Iff is used outside the field of logic as well. Wherever logic is applied, especially in mathematical discussions, it has the same meaning as above: it is an abbreviation for if and only if, indicating that one statement is both necessary and sufficient for the other. This is an example of mathematical jargon (although, as noted above, if is more often used than iff in statements of definition). The elements of X are all and only the elements of Y means: "For any z in the domain of discourse, z is in X if and only if z is in Y." == When "if" means "if and only if" == In their Artificial Intelligence: A Modern Approach, Russell and Norvig note (page 282), in effect, that it is often more natural to express if and only if as if together with a "database (or logic programming) semantics". They give the example of the English sentence "Richard has two brothers, Geoffrey and John". In a database or logic program, this could be represented simply by two sentences: Brother(Richard, Geoffrey). Brother(Richard, John). The database semantics interprets the database (or program) as containing all and only the knowledge


id: inequality-mathematics-20260505-0
file: data/en.wikipedia.org/wiki/Inequality_(mathematics)-0.md
category: reference
title: Inequality (mathematics)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Inequality_(mathematics)
date_saved: 2026-05-05T07:24:01.502525+00:00
instance: kb-cron
parent: inequality-mathematics-20260505-0
summary:
In mathematics, an inequality is a relation which makes a non-equal comparison between two numbers or other mathematical expressions. It is used most often to compare two numbers on the number line by their size. The main types of inequality are less than and greater than (denoted by < and >, respectively the less-than and greater-than signs). == Notation == There are several different notations used to represent different kinds of inequalities: The notation a < b means that a is less than b. The notation a > b means that a is greater than b. In either case, a is not equal to b. These relations are known as strict inequalities, meaning that a is strictly less than or strictly greater than b. Equality is excluded. In contrast to strict inequalities, there are two types of inequality relations that are not strict: The notation a ≤ b or a ⩽ b or a ≦ b means that a is less than or equal to b (or, equivalently, at most b). The notation a ≥ b or a ⩾ b or a ≧ b means that a is greater than or equal to b (or, equivalently, at least b). In


id: inequality-mathematics-20260505-1
file: data/en.wikipedia.org/wiki/Inequality_(mathematics)-1.md
category: reference
title: Inequality (mathematics)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Inequality_(mathematics)
date_saved: 2026-05-05T07:24:01.502525+00:00
instance: kb-cron
parent: inequality-mathematics-20260505-0
summary:
== Formal definitions and generalizations == A (non-strict) partial order is a binary relation ≤ over a set P which is reflexive, antisymmetric, and transitive. That is, for all a, b, and c in P, it must satisfy the three following clauses: a ≤ a (reflexivity) if a ≤ b and b ≤ a, then a = b (antisymmetry) if a ≤ b and b ≤ c, then a ≤ c (transitivity) A set with a partial order is called a partially ordered set. Those are the very basic axioms that every kind of order has to satisfy. A strict partial order is a relation < that satisfies a ≮ a (irreflexivity), if a < b, then b ≮ a (asymmetry), if a < b and b < c, then a < c (transitivity), where ≮ means that < does not hold. Some types of partial orders are specified by adding further axioms, such as: Total order: For every a and b in P, a ≤ b or b ≤ a . Dense order: For all a and b in P for which a < b, there is a c in P such that a < c < b. Least-upper-bound


id: inequality-mathematics-20260505-2
file: data/en.wikipedia.org/wiki/Inequality_(mathematics)-2.md
category: reference
title: Inequality (mathematics)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Inequality_(mathematics)
date_saved: 2026-05-05T07:24:01.502525+00:00
instance: kb-cron
parent: inequality-mathematics-20260505-0
summary:
| ⟨ u , v ⟩ | 2 ≤ ⟨ u , u ⟩ ⋅ ⟨ v , v ⟩ , {\displaystyle |\langle \mathbf {u} ,\mathbf {v} \rangle |^{2}\leq \langle \mathbf {u} ,\mathbf {u} \rangle \cdot \langle \mathbf {v} ,\mathbf {v} \rangle ,} where ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle } is the inner product. Examples of inner products include the real and complex dot product; In Euclidean space Rn with the standard inner product, the Cauchy–Schwarz inequality is ( ∑ i = 1 n u i v i ) 2 ≤ ( ∑ i = 1 n u i 2 ) ( ∑ i = 1 n v i 2 ) . {\displaystyle {\biggl (}\sum _{i=1}^{n}u_{i}v_{i}{\biggr )}^{2}\leq {\biggl (}\sum _{i=1}^{n}u_{i}^{2}{\biggr )}{\biggl (}\sum _{i=1}^{n}v_{i}^{2}{\biggr )}.} == Power inequalities == A power inequality is an inequality containing terms of the form ab, where a and b are real positive numbers or variable expressions. They often appear in mathematical olympiads exercises. Examples: For any real x, e x ≥ 1 + x . {\displaystyle e^{x}\geq 1+x.} If x > 0 and p > 0, then 1 p ( x p − 1 ) ≥ ln ⁡ ( x


id: inequation-20260505-0
file: data/en.wikipedia.org/wiki/Inequation-0.md
category: reference
title: Inequation
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Inequation
date_saved: 2026-05-05T07:24:02.687697+00:00
instance: kb-cron
parent: inequation-20260505-0
summary:
In mathematics, an inequation is a statement that either an inequality (relations "greater than" and "less than", < and >) or a relation "not equal to" (≠) holds between two values. It is usually written in the form of a pair of expressions denoting the values in question, with a relational sign between the two sides, indicating the specific inequality relation. Some examples of inequations are: a < b {\displaystyle a<b} x + y + z ≤ 1 {\displaystyle x+y+z\leq 1} n > 1 {\displaystyle n>1} x ≠ 0 {\displaystyle x\neq 0} In some cases, the term "inequation" has a more restricted definition, reserved only for statements whose inequality relation is "not equal to" (or "distinct"). == Chains of inequations == A shorthand notation is used for the conjunction of several inequations involving common expressions, by chaining them together. For example, the chain 0 ≤ a < b ≤ 1 {\displaystyle 0\leq a<b\leq 1} is shorthand for 0 ≤ a a n d a < b a n d b ≤ 1 {\displaystyle 0\leq a~~\mathrm {and} ~~a<b~~\mathrm {and} ~~b\leq 1} which also implies that 0 < b {\displaystyle 0<b} and a < 1 {\displaystyle a<1} . In rare cases, chains


id: proof-by-infinite-descent-20260505-0
file: data/en.wikipedia.org/wiki/Proof_by_infinite_descent-0.md
category: reference
title: Proof by infinite descent
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Proof_by_infinite_descent
date_saved: 2026-05-05T07:24:03.923311+00:00
instance: kb-cron
parent: proof-by-infinite-descent-20260505-0
summary:
In mathematics, a proof by infinite descent, also known as Fermat's method of descent, is a particular kind of proof by contradiction used to show that a statement cannot possibly hold for any number, by showing that if the statement were to hold for a number, then the same would be true for a smaller number, leading to an infinite descent and ultimately a contradiction. It is a method which relies on the well-ordering principle, and is often used to show that a given equation, such as a Diophantine equation, has no solutions. Typically, one shows that if a solution to a problem existed, which in some sense was related to one or more natural numbers, it would necessarily imply that a second solution existed, which was related to one or more 'smaller' natural numbers. This in turn would imply a third solution related to smaller natural numbers, implying a fourth solution, therefore a fifth solution, and so on. However, there cannot be an infinity of ever-smaller natural numbers, and therefore by mathematical induction, the original premise—that any solution exists—is incorrect: its correctness produces a contradiction. An alternative way to express this is to assume one or more solutions or


id: proof-by-infinite-descent-20260505-1
file: data/en.wikipedia.org/wiki/Proof_by_infinite_descent-1.md
category: reference
title: Proof by infinite descent
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Proof_by_infinite_descent
date_saved: 2026-05-05T07:24:03.923311+00:00
instance: kb-cron
parent: proof-by-infinite-descent-20260505-0
summary:
p q = 2 r 2 s = r s {\displaystyle {\frac {p}{q}}={\frac {2r}{2s}}={\frac {r}{s}}} . Therefore, if √2 could be written as a rational number, then it could always be written as a rational number with smaller parts, which itself could be written with yet-smaller parts, ad infinitum. But this is impossible in the set of natural numbers. Since √2 is a real number, which can be either rational or irrational, the only option left is for √2 to be irrational. (Alternatively, this proves that if √2 were rational, no "smallest" representation as a fraction could exist, as any attempt to find a "smallest" representation p/q would imply that a smaller one existed, which is a similar contradiction.) === Irrationality of √k if it is not an integer === For positive integer k, suppose that √k is not an integer, but is rational and can be expressed as ⁠m/n⁠ for natural numbers m and n, and let q be the largest integer less than √k (that is, q is the floor of √k). Then k = m n = m ( k − q ) n ( k − q ) = m k − m q n k


id: invariant-mathematics-20260505-0
file: data/en.wikipedia.org/wiki/Invariant_(mathematics)-0.md
category: reference
title: Invariant (mathematics)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Invariant_(mathematics)
date_saved: 2026-05-05T07:24:05.176600+00:00
instance: kb-cron
parent: invariant-mathematics-20260505-0
summary:
In mathematics, an invariant is a property of a mathematical object (or a class of mathematical objects) which remains unchanged after operations or transformations of a certain type are applied to the objects. The particular class of objects and type of transformations are usually indicated by the context in which the term is used. For example, the area of a triangle is an invariant with respect to isometries of the Euclidean plane. The phrases "invariant under" and "invariant to" a transformation are both used. More generally, an invariant with respect to an equivalence relation is a property that is constant on each equivalence class. Invariants are used in diverse areas of mathematics such as geometry, topology, algebra and discrete mathematics. Some important classes of transformations are defined by an invariant they leave unchanged. For example, conformal maps are defined as transformations of the plane that preserve angles. The discovery of invariants is an important step in the process of classifying mathematical objects. == Examples == A simple example of invariance is expressed in our ability to count. For a finite set of objects of any kind, there is a number to which we always arrive, regardless of the order in


id: invariant-mathematics-20260505-1
file: data/en.wikipedia.org/wiki/Invariant_(mathematics)-1.md
category: reference
title: Invariant (mathematics)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Invariant_(mathematics)
date_saved: 2026-05-05T07:24:05.176600+00:00
instance: kb-cron
parent: invariant-mathematics-20260505-0
summary:
The table above shows clearly that the invariant holds for each of the possible transformation rules, which means that whichever rule one picks, at whatever state, if the number of I's was not a multiple of three before applying the rule, then it will not be afterwards either. Given that there is a single I in the starting string MI, and one is not a multiple of three, one can then conclude that it is impossible to go from MI to MU (as the number of I's will never be a multiple of three). == Invariant set == A subset S of the domain U of a mapping T: U → U is an invariant set under the mapping when x ∈ S ⟺ T ( x ) ∈ S . {\displaystyle x\in S\iff T(x)\in S.} The elements of S are not necessarily fixed, even though the set S is fixed in the power set of U. (Some authors use the terminology setwise invariant, vs. pointwise invariant, to distinguish between these cases.) For example, a circle is an invariant subset of the plane under a rotation about the circle's center. Further, a conical surface is invariant as a set under


id: invariant-mathematics-20260505-2
file: data/en.wikipedia.org/wiki/Invariant_(mathematics)-2.md
category: reference
title: Invariant (mathematics)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Invariant_(mathematics)
date_saved: 2026-05-05T07:24:05.176600+00:00
instance: kb-cron
parent: invariant-mathematics-20260505-0
summary:
=== Unchanged under perturbation === Thirdly, if one is studying an object that varies in a family, as is common in algebraic geometry and differential geometry, one may ask if the property is unchanged under perturbation (for example, if an object is constant on families or invariant under change of metric). == Invariants in computer science == In computer science, an invariant is a logical assertion that is always held to be true during a certain phase of execution of a computer program. For example, a loop invariant is a condition that is true at the beginning and the end of every iteration of a loop. Invariants are especially useful when reasoning about the correctness of a computer program. The theory of optimizing compilers, the methodology of design by contract, and formal methods for determining program correctness, all rely heavily on invariants. Programmers often use assertions in their code to make invariants explicit. Some object oriented programming languages have a special syntax for specifying class invariants. === Automatic invariant detection in imperative programs === Abstract interpretation tools can compute simple invariants of given imperative computer programs. The kinds of properties that can be found depend on the abstract domains used.


id: irreducibility-mathematics-20260505-0
file: data/en.wikipedia.org/wiki/Irreducibility_(mathematics)-0.md
category: reference
title: Irreducibility (mathematics)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Irreducibility_(mathematics)
date_saved: 2026-05-05T07:24:06.491146+00:00
instance: kb-cron
parent: irreducibility-mathematics-20260505-0
summary:
In mathematics, the concept of irreducibility is used in several ways. A polynomial over a field may be an irreducible polynomial if it cannot be factored over that field. In abstract algebra, irreducible can be an abbreviation for irreducible element of an integral domain; for example an irreducible polynomial. In representation theory, an irreducible representation is a nontrivial representation with no nontrivial proper subrepresentations. Similarly, an irreducible module is another name for a simple module. Absolutely irreducible is a term applied to mean irreducible, even after any finite extension of the field of coefficients. It applies in various situations, for example to irreducibility of a linear representation, or of an algebraic variety; where it means just the same as irreducible over an algebraic closure. In commutative algebra, a commutative ring R is irreducible if its prime spectrum, that is, the topological space Spec R, is an irreducible topological space. A matrix is irreducible if it is not similar via a permutation to a block upper triangular matrix (that has more than one block of positive size). (Replacing non-zero entries in the matrix by one, and viewing the matrix as the adjacency matrix of a directed graph, the matrix is irreducible


id: jacobian-20260505-0
file: data/en.wikipedia.org/wiki/Jacobian-0.md
category: reference
title: Jacobian
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Jacobian
date_saved: 2026-05-05T07:24:07.753500+00:00
instance: kb-cron
parent: jacobian-20260505-0
summary:
In mathematics, a Jacobian, named for Carl Gustav Jacob Jacobi, may refer to: Jacobian matrix and determinant Jacobian elliptic functions Jacobian variety Jacobian ideal Intermediate Jacobian Generalized Jacobian Robot Jacobian


id: law-mathematics-20260505-0
file: data/en.wikipedia.org/wiki/Law_(mathematics)-0.md
category: reference
title: Law (mathematics)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Law_(mathematics)
date_saved: 2026-05-05T07:24:08.972603+00:00
instance: kb-cron
parent: law-mathematics-20260505-0
summary:
In mathematics, a law is a formula that is always true within a given context. Laws describe a relationship, between two or more expressions or terms (which may contain variables), usually using equality or inequality, or between formulas themselves, for instance, in mathematical logic. For example, the formula a 2 ≥ 0 {\displaystyle a^{2}\geq 0} is true for all real numbers a, and is therefore a law. Laws over an equality are called identities. For example, ( a + b ) 2 = a 2 + 2 a b + b 2 {\displaystyle (a+b)^{2}=a^{2}+2ab+b^{2}} and cos 2 ⁡ θ + sin 2 ⁡ θ = 1 {\displaystyle \cos ^{2}\theta +\sin ^{2}\theta =1} are identities. Mathematical laws are distinguished from scientific laws which are based on observations, and try to describe or predict a range of natural phenomena. The more significant laws are often called theorems. == Notable examples == === Geometric laws === Triangle inequality: If a, b, and c are the lengths of the sides of a triangle then the triangle inequality states that c ≤ a + b , {\displaystyle c\leq a+b,} with equality only in the degenerate case of a triangle with zero area. In Euclidean geometry


id: law-mathematics-20260505-1
file: data/en.wikipedia.org/wiki/Law_(mathematics)-1.md
category: reference
title: Law (mathematics)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Law_(mathematics)
date_saved: 2026-05-05T07:24:08.972603+00:00
instance: kb-cron
parent: law-mathematics-20260505-0
summary:
=== Logical laws === De Morgan's laws: In propositional logic and Boolean algebra, De Morgan's laws, also known as De Morgan's theorem, are a pair of transformation rules that are both valid rules of inference. They are named after Augustus De Morgan, a 19th-century British mathematician. The rules allow the expression of conjunctions and disjunctions purely in terms of each other via negation. The rules can be expressed in English as: not (A or B) = (not A) and (not B) not (A and B) = (not A) or (not B) where "A or B" is an "inclusive or" meaning at least one of A or B rather than an "exclusive or" that means exactly one of A or B. In formal language, the rules are written as ¬ ( P ∨ Q ) ⟺ ( ¬ P ) ∧ ( ¬ Q ) , and ¬ ( P ∧ Q ) ⟺ ( ¬ P ) ∨ ( ¬ Q ) , {\displaystyle {\begin{aligned}\neg (P\lor Q)&\iff (\neg P)\land (\neg Q),\quad {\text{ and }}\\\neg (P\land Q)&\iff (\neg P)\lor (\neg Q),\end{aligned}}} where P and Q are propositions, ¬ {\displaystyle \neg } is the negation logic operator (NOT), ∧ {\displaystyle \land }


id: left-and-right-algebra-20260505-0
file: data/en.wikipedia.org/wiki/Left_and_right_(algebra)-0.md
category: reference
title: Left and right (algebra)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Left_and_right_(algebra)
date_saved: 2026-05-05T07:24:10.169286+00:00
instance: kb-cron
parent: left-and-right-algebra-20260505-0
summary:
In algebra, the terms left and right denote the order of a binary operation (usually, but not always, called "multiplication") in non-commutative algebraic structures. A binary operation ∗ is usually written in the infix form: s ∗ t The argument s is placed on the left side, and the argument t is on the right side. Even if the symbol of the operation is omitted, the order of s and t does matter (unless ∗ is commutative). A two-sided property is fulfilled on both sides. A one-sided property is related to one (unspecified) of two sides. Although the terms are similar, left–right distinction in algebraic parlance is not related either to left and right limits in calculus, or to left and right in geometry. == Binary operation as an operator == A binary operation ∗ may be considered as a family of unary operators through currying: Rt(s) = s ∗ t, depending on t as a parameter – this is the family of right operations. Similarly, Ls(t) = s ∗ t defines the family of left operations parametrized with s. If for some e, the left operation Le is the identity operation, then e is called a left identity. Similarly,


id: lemma-mathematics-20260505-0
file: data/en.wikipedia.org/wiki/Lemma_(mathematics)-0.md
category: reference
title: Lemma (mathematics)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Lemma_(mathematics)
date_saved: 2026-05-05T07:24:11.535656+00:00
instance: kb-cron
parent: lemma-mathematics-20260505-0
summary:
In mathematics and other fields, a lemma (pl.: lemmas or lemmata) is a generally minor, proven proposition which is used to prove a larger statement. For that reason, it is also known as a "helping theorem" or an "auxiliary theorem". In many cases, a lemma derives its importance from the theorem it aims to prove; however, a lemma can also turn out to be more important than originally thought. == Etymology == From the Ancient Greek λῆμμα, (perfect passive εἴλημμαι) something received or taken. Thus, something taken for granted in an argument. == Comparison with theorem == There is no formal distinction between a lemma and a theorem, only one of intention (see Theorem terminology). However, a lemma can be considered a minor result whose sole purpose is to help prove a more substantial theorem – a step in the direction of proof. == Well-known lemmas == Some powerful results in mathematics are known as lemmas, first named for their originally minor purpose. These include, among others: While these results originally seemed too simple or too technical to warrant independent interest, they have eventually turned out to be central to the theories in which they occur. == See also ==


id: lemniscate-20260505-0
file: data/en.wikipedia.org/wiki/Lemniscate-0.md
category: reference
title: Lemniscate
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Lemniscate
date_saved: 2026-05-05T07:24:12.760643+00:00
instance: kb-cron
parent: lemniscate-20260505-0
summary:
In algebraic geometry, a lemniscate ( or ) is any of several figure-eight or ∞-shaped curves. The word comes from the Latin lēmniscātus, meaning "decorated with ribbons", from the Greek λημνίσκος (lēmnískos), meaning "ribbon", or which alternatively may refer to the wool from which the ribbons were made. Curves that have been called a lemniscate include three quartic plane curves: the hippopede or lemniscate of Booth, the lemniscate of Bernoulli, and the lemniscate of Gerono. The hippopede was studied by Proclus (5th century), but the term "lemniscate" was not used until the work of Jacob Bernoulli in the late 17th century. == History and examples == === Lemniscate of Booth === The consideration of curves with a figure-eight shape can be traced back to Proclus, a Greek Neoplatonist philosopher and mathematician who lived in the 5th century AD. Proclus considered the cross-sections of a torus by a plane parallel to the axis of the torus. As he observed, for most such sections the cross section consists of either one or two ovals; however, when the plane is tangent to the inner surface of the torus, the cross-section takes on a figure-eight shape, which Proclus called a horse fetter (a device


id: level-logarithmic-quantity-20260505-0
file: data/en.wikipedia.org/wiki/Level_(logarithmic_quantity)-0.md
category: reference
title: Level (logarithmic quantity)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Level_(logarithmic_quantity)
date_saved: 2026-05-05T07:24:14.200490+00:00
instance: kb-cron
parent: level-logarithmic-quantity-20260505-0
summary:
In science and engineering, a power level and a field level (also called a root-power level) are logarithmic magnitudes of certain quantities referenced to a standard reference value of the same type. A power level is a logarithmic quantity used to measure power, power density or sometimes energy, with commonly used unit decibel (dB). A field level (or root-power level) is a logarithmic quantity used to measure quantities of which the square is typically proportional to power (for instance, the square of voltage is proportional to power multiplied by the conductor's resistance), with commonly used units neper (Np) or decibel (dB). The type of level and choice of units indicate the scaling of the logarithm of the ratio between the quantity and its reference value, though a logarithm may be considered to be a dimensionless quantity. The reference values for each type of quantity are often specified by international standards. Power and field levels are used in electronic engineering, telecommunications, acoustics and related disciplines. Power levels are used for signal power, noise power, sound power, sound exposure, etc. Field levels are used for voltage, current, sound pressure. == Power level == Level of a power quantity, denoted LP, is defined


id: like-terms-20260505-0
file: data/en.wikipedia.org/wiki/Like_terms-0.md
category: reference
title: Like terms
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Like_terms
date_saved: 2026-05-05T07:24:15.379757+00:00
instance: kb-cron
parent: like-terms-20260505-0
summary:
In mathematics, like terms are summands in a sum that differ only by a numerical factor. Like terms can be regrouped by adding their coefficients. Typically, in a polynomial expression, like terms are those that contain the same variables to the same powers, possibly with different coefficients. More generally, when some variable are considered as parameters, like terms are defined similarly, but "numerical factors" must be replaced by "factors depending only on the parameters". For example, when considering a quadratic equation, one considers often the expression ( x − r ) ( x − s ) , {\displaystyle (x-r)(x-s),} where r {\displaystyle r} and s {\displaystyle s} are the roots of the equation and may be considered as parameters. Then, expanding the above product and regrouping the like terms gives x 2 − ( r + s ) x + r s . {\displaystyle x^{2}-(r+s)x+rs.} == Generalization == In this discussion, a "term" will refer to a string of numbers being multiplied or divided (that division is simply multiplication by a reciprocal) together. Terms are within the same expression and are combined by either addition or subtraction. For example, take the expression: a x + b x {\displaystyle ax+bx} There


id: linearity-20260505-0
file: data/en.wikipedia.org/wiki/Linearity-0.md
category: reference
title: Linearity
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Linearity
date_saved: 2026-05-05T07:24:16.618385+00:00
instance: kb-cron
parent: linearity-20260505-0
summary:
In mathematics, the term linear is used in two distinct senses for two different properties: linearity of a function (or mapping); linearity of a polynomial. An example of a linear function is the function defined by f ( x ) = ( a x , b x ) {\displaystyle f(x)=(ax,bx)} that maps the real line to a line in the Euclidean plane R2 that passes through the origin. An example of a linear polynomial in the variables X , {\displaystyle X,} Y {\displaystyle Y} and Z {\displaystyle Z} is a X + b Y + c Z + d . {\displaystyle aX+bY+cZ+d.} Linearity of a mapping is closely related to proportionality. Examples in physics include the linear relationship of voltage and current in an electrical conductor (Ohm's law), and the relationship of mass and weight. By contrast, more complicated relationships, such as between velocity and kinetic energy, are nonlinear. Generalized for functions in more than one dimension, linearity means the property of a function of being compatible with addition and scaling, also known as the superposition principle. Linearity of a polynomial means that its degree is less than two. The use of the term for polynomials stems from the fact


id: linearity-20260505-1
file: data/en.wikipedia.org/wiki/Linearity-1.md
category: reference
title: Linearity
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Linearity
date_saved: 2026-05-05T07:24:16.618385+00:00
instance: kb-cron
parent: linearity-20260505-0
summary:
In every row in which the truth value of the function is T, there are an odd number of Ts assigned to the arguments, and in every row in which the function is F there is an even number of Ts assigned to arguments. Specifically, f(F, F, ..., F) = F, and these functions correspond to linear maps over the Boolean vector space. In every row in which the value of the function is T, there is an even number of Ts assigned to the arguments of the function; and in every row in which the truth value of the function is F, there are an odd number of Ts assigned to arguments. In this case, f(F, F, ..., F) = T. Another way to express this is that each variable always makes a difference in the truth value of the operation or it never makes a difference. Negation, Logical biconditional, exclusive or, tautology, and contradiction are linear functions. == Physics == In physics, linearity is a property of the differential equations governing many systems; for instance, the Maxwell equations or the diffusion equation. Linearity of a homogeneous differential equation means that if two functions f and g are solutions


id: mathematical-beauty-20260505-0
file: data/en.wikipedia.org/wiki/Mathematical_beauty-0.md
category: reference
title: Mathematical beauty
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Mathematical_beauty
date_saved: 2026-05-05T07:24:17.882469+00:00
instance: kb-cron
parent: mathematical-beauty-20260505-0
summary:
Mathematical beauty is a type of aesthetic value that is experienced in doing or contemplating mathematics. The testimonies of mathematicians indicate that various aspects of mathematics—including results, formulae, proofs and theories—can trigger subjective responses similar to the beauty of art, music, or nature. The pleasure in this experience can serve as a motivation for doing mathematics, and some mathematicians, such as G.H. Hardy, have characterized mathematics as an art form that seeks beauty. Beauty in mathematics has been subject to examination by mathematicians themselves and by philosophers, psychologists, and neuroscientists. Understanding beauty in general can be difficult because it is a subjective response to sense-experience but is perceived as a property of an external object, and because it may be shaped by cultural influence or personal experience. Mathematical beauty presents additional problems, since the aesthetic response is evoked by abstract ideas which can be communicated symbolically, and which may only be available to a minority of people with mathematical ability and training. The appreciation of mathematics may also be less passive than (for example) listening to music. Furthermore, beauty in mathematics may be connected to other aesthetic or non-aesthetic values. Some authors identify mathematical elegance with mathematical beauty; others distinguish


id: mathematical-beauty-20260505-1
file: data/en.wikipedia.org/wiki/Mathematical_beauty-1.md
category: reference
title: Mathematical beauty
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Mathematical_beauty
date_saved: 2026-05-05T07:24:17.882469+00:00
instance: kb-cron
parent: mathematical-beauty-20260505-0
summary:
== Properties of beautiful mathematics == Many mathematicians and philosophers who have written about mathematical beauty have tried to identify properties or criteria that are conducive to the perception of beauty in a piece of mathematics. It is debated whether beauty can be clarified or explained by such properties: Paul Erdős thought that it was no more possible to convince someone of the beauty of a piece of mathematics than to convince them of the beauty of Beethoven's Ninth Symphony, if they couldn't see it for themselves. === Results === In his 1940 essay A Mathematician's Apology, G. H. Hardy said that a beautiful result, including its proof, possesses three "purely aesthetic qualities", namely "inevitability", "unexpectedness", and "economy". He particularly excluded enumeration of cases as "one of the duller forms of mathematical argument". In 1997, Gian-Carlo Rota disagreed with unexpectedness as a sufficient condition for beauty and proposed a counterexample: A great many theorems of mathematics, when first published, appear to be surprising; thus for example some twenty years ago [from 1977] the proof of the existence of non-equivalent differentiable structures on spheres of high dimension was thought to be surprising, but it did not occur to anyone to call


id: mathematical-beauty-20260505-2
file: data/en.wikipedia.org/wiki/Mathematical_beauty-2.md
category: reference
title: Mathematical beauty
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Mathematical_beauty
date_saved: 2026-05-05T07:24:17.882469+00:00
instance: kb-cron
parent: mathematical-beauty-20260505-0
summary:
=== Information-theory model === In the 1970s, Abraham Moles and Frieder Nake analyzed links between beauty, information processing, and information theory. In the 1990s, Jürgen Schmidhuber formulated a mathematical theory of observer-dependent subjective beauty based on algorithmic information theory: the most beautiful objects among subjectively comparable objects have short algorithmic descriptions (i.e., Kolmogorov complexity) relative to what the observer already knows. Schmidhuber explicitly distinguishes between beautiful and interesting. The latter corresponds to the first derivative of subjectively perceived beauty: the observer continually tries to improve the predictability and compressibility of the observations by discovering regularities such as repetitions and symmetries and fractal self-similarity. Whenever the observer's learning process (possibly a predictive artificial neural network) leads to improved data compression such that the observation sequence can be described by fewer bits than before, the temporary interesting-ness of the data corresponds to the compression progress, and is proportional to the observer's internal curiosity reward. === Neural correlates === Brain imaging experiments conducted by Semir Zeki, Michael Atiyah and collaborators show that the experience of mathematical beauty has, as a neural correlate, activity in field A1 of the medial orbito-frontal cortex (mOFC) of the brain and that this activity is parametrically related to


id: mathematical-model-20260505-0
file: data/en.wikipedia.org/wiki/Mathematical_model-0.md
category: reference
title: Mathematical model
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Mathematical_model
date_saved: 2026-05-05T07:24:19.245253+00:00
instance: kb-cron
parent: mathematical-model-20260505-0
summary:
A mathematical model is an abstract description of a concrete system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used in many fields, including applied mathematics, natural sciences, social sciences and engineering. In particular, the field of operations research studies the use of mathematical modelling and related tools to solve problems in business or military operations. A model may help to characterize a system by studying the effects of different components, which may be used to make predictions about behavior or solve specific problems. == Elements of a mathematical model == Mathematical models can take many forms, including dynamical systems, statistical models, differential equations, or game theoretic models. These and other types of models can overlap, with a given model involving a variety of abstract structures. In many cases, the quality of a scientific field depends on how well the mathematical models developed on the theoretical side agree with results of repeatable experiments. Lack of agreement between theoretical mathematical models and experimental measurements often leads to important advances as better theories are developed. In the physical sciences, a traditional mathematical model contains most of the following elements: Governing equations


id: mathematical-model-20260505-1
file: data/en.wikipedia.org/wiki/Mathematical_model-1.md
category: reference
title: Mathematical model
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Mathematical_model
date_saved: 2026-05-05T07:24:19.245253+00:00
instance: kb-cron
parent: mathematical-model-20260505-0
summary:
== Construction == In business and engineering, mathematical models may be used to maximize a certain output. The system under consideration will require certain inputs. The system relating inputs to outputs depends on other variables too: decision variables, state variables, exogenous variables, and random variables. Decision variables are sometimes known as independent variables. Exogenous variables are sometimes known as parameters or constants. The variables are not independent of each other as the state variables are dependent on the decision, input, random, and exogenous variables. Furthermore, the output variables are dependent on the state of the system (represented by the state variables). Objectives and constraints of the system and its users can be represented as functions of the output variables or state variables. The objective functions will depend on the perspective of the model's user. Depending on the context, an objective function is also known as an index of performance, as it is some measure of interest to the user. Although there is no limit to the number of objective functions and constraints a model can have, using or optimizing the model becomes more involved (computationally) as the number increases. For example, economists often apply linear algebra when using input–output models.


id: mathematical-model-20260505-2
file: data/en.wikipedia.org/wiki/Mathematical_model-2.md
category: reference
title: Mathematical model
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Mathematical_model
date_saved: 2026-05-05T07:24:19.245253+00:00
instance: kb-cron
parent: mathematical-model-20260505-0
summary:
=== Complexity === In general, model complexity involves a trade-off between simplicity and accuracy of the model. Occam's razor is a principle particularly relevant to modeling, its essential idea being that among models with roughly equal predictive power, the simplest one is the most desirable. While added complexity usually improves the realism of a model, it can make the model difficult to understand and analyze, and can also pose computational problems, including numerical instability. Thomas Kuhn argues that as science progresses, explanations tend to become more complex before a paradigm shift offers radical simplification. For example, when modeling the flight of an aircraft, we could embed each mechanical part of the aircraft into our model and would thus acquire an almost white-box model of the system. However, the computational cost of adding such a huge amount of detail would effectively inhibit the usage of such a model. Additionally, the uncertainty would increase due to an overly complex system, because each separate part induces some amount of variance into the model. It is therefore usually appropriate to make some approximations to reduce the model to a sensible size. Engineers often can accept some approximations in order to get a more robust


id: mathematical-model-20260505-3
file: data/en.wikipedia.org/wiki/Mathematical_model-3.md
category: reference
title: Mathematical model
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Mathematical_model
date_saved: 2026-05-05T07:24:19.245253+00:00
instance: kb-cron
parent: mathematical-model-20260505-0
summary:
== Significance in the natural sciences == Mathematical models are of great importance in the natural sciences, particularly in physics. Physical theories are almost invariably expressed using mathematical models. Throughout history, more and more accurate mathematical models have been developed. Newton's laws accurately describe many everyday phenomena, but at certain limits theory of relativity and quantum mechanics must be used. It is common to use idealized models in physics to simplify things. Massless ropes, point particles, ideal gases and the particle in a box are among the many simplified models used in physics. The laws of physics are represented with simple equations such as Newton's laws, Maxwell's equations and the Schrödinger equation. These laws are a basis for making mathematical models of real situations. Many real situations are very complex and thus modeled approximately on a computer, a model that is computationally feasible to compute is made from the basic laws or from approximate models made from the basic laws. For example, molecules can be modeled by molecular orbital models that are approximate solutions to the Schrödinger equation. In engineering, physics models are often made by mathematical methods such as finite element analysis. Different mathematical models use different geometries that


id: mathematical-model-20260505-4
file: data/en.wikipedia.org/wiki/Mathematical_model-4.md
category: reference
title: Mathematical model
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Mathematical_model
date_saved: 2026-05-05T07:24:19.245253+00:00
instance: kb-cron
parent: mathematical-model-20260505-0
summary:
The state S 1 {\displaystyle S_{1}} represents that there has been an even number of 0s in the input so far, while S 2 {\displaystyle S_{2}} signifies an odd number. A 1 in the input does not change the state of the automaton. When the input ends, the state will show whether the input contained an even number of 0s or not. If the input did contain an even number of 0s, M {\displaystyle M} will finish in state S 1 , {\displaystyle S_{1},} an accepting state, so the input string will be accepted. The language recognized by M {\displaystyle M} is the regular language given by the regular expression 1*( 0 (1*) 0 (1*) )*, where "*" is the Kleene star, e.g., 1* denotes any non-negative number (possibly zero) of symbols "1". Many everyday activities carried out without a thought are uses of mathematical models. A geographical map projection of a region of the earth onto a small, plane surface is a model which can be used for many purposes such as planning travel. Another simple activity is predicting the position of a vehicle from its initial position, direction and speed of travel, using the equation that distance traveled


id: metatheorem-20260505-0
file: data/en.wikipedia.org/wiki/Metatheorem-0.md
category: reference
title: Metatheorem
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Metatheorem
date_saved: 2026-05-05T07:24:20.538671+00:00
instance: kb-cron
parent: metatheorem-20260505-0
summary:
In logic, a metatheorem is a statement about a formal system proven in a metalanguage. Unlike theorems proved within a given formal system, a metatheorem is proved within a metatheory, and may reference concepts that are present in the metatheory but not the object theory. A formal system is determined by a formal language and a deductive system (axioms and rules of inference). The formal system can be used to prove particular sentences of the formal language with that system. Metatheorems, however, are proved externally to the system in question, in its metatheory. Common metatheories used in logic are set theory (especially in model theory) and primitive recursive arithmetic (especially in proof theory). Rather than demonstrating particular sentences to be provable, metatheorems may show that each of a broad class of sentences can be proved, or show that certain sentences cannot be proved. == Examples == Examples of metatheorems include: The deduction theorem for first-order logic says that a sentence of the form φ→ψ is provable from a set of axioms A if and only if the sentence ψ is provable from the system whose axioms consist of φ and all the axioms of A. The class existence theorem of


id: minimal-counterexample-20260505-0
file: data/en.wikipedia.org/wiki/Minimal_counterexample-0.md
category: reference
title: Minimal counterexample
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Minimal_counterexample
date_saved: 2026-05-05T07:24:21.770105+00:00
instance: kb-cron
parent: minimal-counterexample-20260505-0
summary:
In mathematics, a minimal counterexample is the smallest example which falsifies a claim. It is also sometimes called a minimal criminal, smallest criminal, or least criminal, especially (but not exclusively) in the context of the four-color theorem. A proof by minimal counterexample (or by minimal/smallest/least criminal) is a method of proof which combines the use of a minimal counterexample with the methods of proof by induction and proof by contradiction. More specifically, in trying to prove a proposition P, one first assumes by contradiction that it is false, and that therefore there must be at least one counterexample. With respect to some idea of size (which may need to be chosen carefully), one then concludes that there is such a counterexample C that is minimal. In regard to the argument, C is generally something quite hypothetical (since the truth of P excludes the possibility of C), but it may be possible to argue that if C existed, then it would have some definite properties which, after applying some reasoning similar to that in an inductive proof, would lead to a contradiction, thereby showing that the proposition P is indeed true. If the form of the contradiction is that we can


id: modulo-mathematics-20260505-0
file: data/en.wikipedia.org/wiki/Modulo_(mathematics)-0.md
category: reference
title: Modulo (mathematics)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Modulo_(mathematics)
date_saved: 2026-05-05T07:24:33.217058+00:00
instance: kb-cron
parent: modulo-mathematics-20260505-0
summary:
In mathematics, the term modulo ("with respect to a modulus of", the Latin ablative of modulus which itself means "a small measure") is often used to assert that two distinct mathematical objects can be regarded as equivalent—if their difference is accounted for by an additional factor. It was initially introduced into mathematics in the context of modular arithmetic by Carl Friedrich Gauss in 1801. Since then, the term has gained many meanings—some exact and some imprecise (such as equating "modulo" with "except for"). For the most part, the term often occurs in statements of the form: A is the same as B modulo C which is often equivalent to "A is the same as B up to C", and means A and B are the same—except for differences accounted for or explained by C. == History == Modulo is a mathematical jargon that was introduced into mathematics in the book Disquisitiones Arithmeticae by Carl Friedrich Gauss in 1801. Given the integers a, b and n, the expression "a ≡ b (mod n)", pronounced "a is congruent to b modulo n", means that a − b is an integer multiple of n, or equivalently, a and b both share the same


id: necessity-and-sufficiency-20260505-0
file: data/en.wikipedia.org/wiki/Necessity_and_sufficiency-0.md
category: reference
title: Necessity and sufficiency
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Necessity_and_sufficiency
date_saved: 2026-05-05T07:24:34.458153+00:00
instance: kb-cron
parent: necessity-and-sufficiency-20260505-0
summary:
In logic and mathematics, necessity and sufficiency are terms used to describe a conditional or implicational relationship between two statements. For example, in the conditional statement: "If P then Q", Q is necessary for P, because the truth of Q is "necessarily" guaranteed by the truth of P. (Equivalently, it is impossible to have P without Q, or the falsity of Q ensures the falsity of P.) Similarly, P is sufficient for Q, because P being true always or "sufficiently" implies that Q is true, but P not being true does not always imply that Q is not true. In general, a necessary condition is one (possibly one of several conditions) that must be present in order for another condition to occur, while a sufficient condition is one that produces the said condition. The assertion that a statement is a "necessary and sufficient" condition of another means that the former statement is true if and only if the latter is true. That is, the two statements must be either simultaneously true, or simultaneously false. In ordinary English (also natural language) "necessary" and "sufficient" often indicate relations between conditions or states of affairs, not statements. For example, being round is a


id: necessity-and-sufficiency-20260505-1
file: data/en.wikipedia.org/wiki/Necessity_and_sufficiency-1.md
category: reference
title: Necessity and sufficiency
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Necessity_and_sufficiency
date_saved: 2026-05-05T07:24:34.458153+00:00
instance: kb-cron
parent: necessity-and-sufficiency-20260505-0
summary:
Example 1 For it to be true that "John is a bachelor", it is necessary that it be also true that he is unmarried, male, adult, since to state "John is a bachelor" implies John has each of those three additional predicates. Example 2 For the whole numbers greater than two, being odd is necessary to being prime, since two is the only whole number that is both even and prime. Example 3 Consider thunder, the sound caused by lightning. One says that thunder is necessary for lightning, since lightning never occurs without thunder. Whenever there is lightning, there is thunder. The thunder does not cause the lightning (since lightning causes thunder), but because lightning always comes with thunder, we say that thunder is necessary for lightning. (That is, in its formal sense, necessity doesn't imply causality.) Example 4 Being at least 30 years old is necessary for serving in the U.S. Senate. If you are under 30 years old, then it is impossible for you to be a senator. That is, if you are a senator, it follows that you must be at least 30 years old. Example 5 In algebra, for some set S together with an operation


id: necessity-and-sufficiency-20260505-2
file: data/en.wikipedia.org/wiki/Necessity_and_sufficiency-2.md
category: reference
title: Necessity and sufficiency
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Necessity_and_sufficiency
date_saved: 2026-05-05T07:24:34.458153+00:00
instance: kb-cron
parent: necessity-and-sufficiency-20260505-0
summary:
that P is necessary for Q, P ⇐ Q {\displaystyle P\Leftarrow Q} , and that P is sufficient for Q, P ⇒ Q {\displaystyle P\Rightarrow Q} . equivalently, it may be understood to say that P and Q is necessary for the other, P ⇒ Q ∧ Q ⇒ P {\displaystyle P\Rightarrow Q\land Q\Rightarrow P} , which can also be stated as each is sufficient for or implies the other. One may summarize any, and thus all, of these cases by the statement "P if and only if Q", which is denoted by P ⇔ Q {\displaystyle P\Leftrightarrow Q} , whereas cases tell us that P ⇔ Q {\displaystyle P\Leftrightarrow Q} is identical to P ⇒ Q ∧ Q ⇒ P {\displaystyle P\Rightarrow Q\land Q\Rightarrow P} . For example, in graph theory a graph G is called bipartite if it is possible to assign to each of its vertices the color black or white in such a way that every edge of G has one endpoint of each color. And for any graph to be bipartite, it is a necessary and sufficient condition that it contain no odd-length cycles. Thus, discovering whether a graph has any odd cycles tells one


id: null-mathematics-20260505-0
file: data/en.wikipedia.org/wiki/Null_(mathematics)-0.md
category: reference
title: Null (mathematics)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Null_(mathematics)
date_saved: 2026-05-05T07:24:35.707916+00:00
instance: kb-cron
parent: null-mathematics-20260505-0
summary:
In mathematics, the word null (from German: null meaning "zero", which is from Latin: nullus meaning "none") is often associated with the concept of zero, or with the concept of nothing. It is used in varying contexts from "having zero members in a set" (e.g., null set) to "having a value of zero" (e.g., null vector). In a vector space, the null vector is the neutral element of vector addition; depending on the context, a null vector may also be a vector mapped to some null by a function under consideration (such as a quadratic form coming with the vector space, see null vector, a linear mapping given as matrix product or dot product, a seminorm in a Minkowski space, etc.). In set theory, the empty set, that is, the set with zero elements, denoted "{}" or "∅", may also be called null set. In measure theory, a null set is a (possibly nonempty) set with zero measure. A null space of a mapping is the part of the domain that is mapped into the null element of the image (the inverse image of the null element). For example, in linear algebra, the null space of a linear mapping, also


id: of-the-form-20260505-0
file: data/en.wikipedia.org/wiki/Of_the_form-0.md
category: reference
title: Of the form
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Of_the_form
date_saved: 2026-05-05T07:24:37.077999+00:00
instance: kb-cron
parent: of-the-form-20260505-0
summary:
In mathematics, the phrase "of the form" indicates that a mathematical object, or (more frequently) a collection of objects, follows a certain pattern of expression. It is frequently used to reduce the formality of mathematical proofs. == Example of use == Here is a proof which should be appreciable with limited mathematical background: Statement: The product of any two even natural numbers is also even. Proof: Any even natural number is of the form 2n, where n is a natural number. Therefore, let us assume that we have two even numbers which we will denote by 2k and 2l. Their product is (2k)(2l) = 4(kl) = 2(2kl). Since 2kl is also a natural number, the product is even. Note: In this case, both exhaustivity and exclusivity were needed. That is, it was not only necessary that every even number is of the form 2n (exhaustivity), but also that every expression of the form 2n is an even number (exclusivity). This will not be the case in every proof, but normally, at least exhaustivity is implied by the phrase of the form. == References == == External links == Weisstein, Eric W. "Of the Form". MathWorld.


id: order-mathematics-20260505-0
file: data/en.wikipedia.org/wiki/Order_(mathematics)-0.md
category: reference
title: Order (mathematics)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Order_(mathematics)
date_saved: 2026-05-05T07:24:38.280206+00:00
instance: kb-cron
parent: order-mathematics-20260505-0
summary:
Order in mathematics may refer to: == Set theory == Total order and partial order, a binary relation generalizing the usual ordering of numbers and of words in a dictionary Ordered set Order in Ramsey theory, uniform structures in consequence to critical set cardinality == Algebra == Order (group theory), the cardinality of a group or period of an element Order of a polynomial (disambiguation) Order of a square matrix, its dimension Order (ring theory), an algebraic structure Ordered group Ordered field == Analysis == Order (differential equation) or order of highest derivative, of a differential equation Leading-order terms NURBS order, a number one greater than the degree of the polynomial representation of a non-uniform rational B-spline Order of convergence, a measurement of convergence Order of derivation Order of an entire function Order of a power series, the lowest degree of its terms Ordered list, a sequence or tuple Orders of approximation in Big O notation Z-order (curve), a space-filling curve == Arithmetic == Multiplicative order in modular arithmetic Order of operations Orders of magnitude, a class of scale or magnitude of any amount == Combinatorics == Order in the Josephus permutation Ordered selections and partitions of the twelvefold way


id: orthomorphism-20260505-0
file: data/en.wikipedia.org/wiki/Orthomorphism-0.md
category: reference
title: Orthomorphism
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Orthomorphism
date_saved: 2026-05-05T07:24:39.493560+00:00
instance: kb-cron
parent: orthomorphism-20260505-0
summary:
In abstract algebra, an orthomorphism is a certain kind of mapping from a group into itself. Let G be a group, and let θ be a permutation of G. Then θ is an orthomorphism of G if the mapping f defined by f(x) = x−1 θ(x) is also a permutation of G. A permutation φ of G is a complete mapping if the mapping g defined by g(x) = xφ(x) is also a permutation of G. Orthomorphisms and complete mappings are closely related. == References ==


id: parameter-20260505-0
file: data/en.wikipedia.org/wiki/Parameter-0.md
category: reference
title: Parameter
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Parameter
date_saved: 2026-05-05T07:24:40.755713+00:00
instance: kb-cron
parent: parameter-20260505-0
summary:
A parameter (from Ancient Greek παρά (pará) 'beside, subsidiary' and μέτρον (métron) 'measure'), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when identifying the system, or when evaluating its performance, status, condition, etc. Parameter has more specific meanings within various disciplines, including mathematics, computer programming, engineering, statistics, logic, linguistics, and electronic musical composition. In addition to its technical uses, there are also extended uses, especially in non-scientific contexts, where it is used to mean defining characteristics or boundaries, as in the phrases 'test parameters' or 'game play parameters'. == Modelization == When a system is modeled by equations, the values that describe the system are called parameters. For example, in mechanics, the masses, the dimensions and shapes (for solid bodies), the densities and the viscosities (for fluids), appear as parameters in the equations modeling movements. There are often several choices for the parameters, and choosing a convenient set of parameters is called parametrization. For example, if one were considering the movement of an object on the surface of a sphere much larger


id: parameter-20260505-1
file: data/en.wikipedia.org/wiki/Parameter-1.md
category: reference
title: Parameter
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Parameter
date_saved: 2026-05-05T07:24:40.755713+00:00
instance: kb-cron
parent: parameter-20260505-0
summary:
[Kilpatrick quoting Woods] "Now ... the engineers ... change the lever arms of the linkage ... the speed of the car ... will still depend on the pedal position ... but in a ... different manner. You have changed a parameter" A parametric equaliser is an audio filter that allows the frequency of maximum cut or boost to be set by one control, and the size of the cut or boost by another. These settings, the frequency level of the peak or trough, are two of the parameters of a frequency response curve, and in a two-control equaliser they completely describe the curve. More elaborate parametric equalisers may allow other parameters to be varied, such as skew. These parameters each describe some aspect of the response curve seen as a whole, over all frequencies. A graphic equaliser provides individual level controls for various frequency bands, each of which acts only on that particular frequency band. If asked to imagine the graph of the relationship y = ax2, one typically visualizes a range of values of x, but only one value of a. Of course a different value of a can be used, generating a different relation between x and y.


id: parameter-20260505-2
file: data/en.wikipedia.org/wiki/Parameter-2.md
category: reference
title: Parameter
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Parameter
date_saved: 2026-05-05T07:24:40.755713+00:00
instance: kb-cron
parent: parameter-20260505-0
summary:
In statistics and econometrics, the probability framework above still holds, but attention shifts to estimating the parameters of a distribution based on observed data, or testing hypotheses about them. In frequentist estimation parameters are considered "fixed but unknown", whereas in Bayesian estimation they are treated as random variables, and their uncertainty is described as a distribution. In estimation theory of statistics, "statistic" or estimator refers to samples, whereas "parameter" or estimand refers to populations, where the samples are taken from. A statistic is a numerical characteristic of a sample that can be used as an estimate of the corresponding parameter, the numerical characteristic of the population from which the sample was drawn. For example, the sample mean (estimator), denoted X ¯ {\displaystyle {\overline {X}}} , can be used as an estimate of the mean parameter (estimand), denoted μ, of the population from which the sample was drawn. Similarly, the sample variance (estimator), denoted S2, can be used to estimate the variance parameter (estimand), denoted σ2, of the population from which the sample was drawn. (Note that the sample standard deviation (S) is not an unbiased estimate of the population standard deviation (σ): see Unbiased estimation of standard deviation.) It is


id: parameter-20260505-3
file: data/en.wikipedia.org/wiki/Parameter-3.md
category: reference
title: Parameter
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Parameter
date_saved: 2026-05-05T07:24:40.755713+00:00
instance: kb-cron
parent: parameter-20260505-0
summary:
== Engineering == In engineering (especially involving data acquisition) the term parameter sometimes loosely refers to an individual measured item. This usage is not consistent, as sometimes the term channel refers to an individual measured item, with parameter referring to the setup information about that channel. "Speaking generally, properties are those physical quantities which directly describe the physical attributes of the system; parameters are those combinations of the properties which suffice to determine the response of the system. Properties can have all sorts of dimensions, depending upon the system being considered; parameters are dimensionless, or have the dimension of time or its reciprocal." The term can also be used in engineering contexts, however, as it is typically used in the physical sciences. == Environmental science == In environmental science and particularly in chemistry and microbiology, a parameter is used to describe a discrete chemical or microbiological entity that can be assigned a value: commonly a concentration, but may also be a logical entity (present or absent), a statistical result such as a 95 percentile value or in some cases a subjective value. == Linguistics == Within linguistics, the word "parameter" is almost exclusively used to denote a binary switch in


id: parameter-space-20260505-0
file: data/en.wikipedia.org/wiki/Parameter_space-0.md
category: reference
title: Parameter space
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Parameter_space
date_saved: 2026-05-05T07:24:41.985731+00:00
instance: kb-cron
parent: parameter-space-20260505-0
summary:
The parameter space is the space of all possible parameter values that define a particular mathematical model. It is also sometimes called weight space, and is often a subset of finite-dimensional Euclidean space. In statistics, parameter spaces are particularly useful for describing parametric families of probability distributions. They also form the background for parameter estimation. In the case of extremum estimators for parametric models, a certain objective function is maximized or minimized over the parameter space. Theorems of existence and consistency of such estimators require some assumptions about the topology of the parameter space. For instance, compactness of the parameter space, together with continuity of the objective function, suffices for the existence of an extremum estimator. Sometimes, parameters are analyzed to view how they affect their statistical model. In that context, they can be viewed as inputs of a function, in which case the technical term for the parameter space is domain of a function. The ranges of values of the parameters may form the axes of a plot, and particular outcomes of the model may be plotted against these axes to illustrate how different regions of the parameter space produce different types of behavior in the model. == Examples


id: parametric-family-20260505-0
file: data/en.wikipedia.org/wiki/Parametric_family-0.md
category: reference
title: Parametric family
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Parametric_family
date_saved: 2026-05-05T07:24:43.335955+00:00
instance: kb-cron
parent: parametric-family-20260505-0
summary:
In mathematics and its applications, a parametric family or a parameterized family is a family of objects (a set of related objects) whose differences depend only on the chosen values for a set of parameters. Common examples are parametrized (families of) functions, probability distributions, curves, shapes, etc. == In probability and its applications == For example, the probability density function fX of a random variable X may depend on a parameter θ. In that case, the function may be denoted f X ( ⋅ ; θ ) {\displaystyle f_{X}(\cdot \,;\theta )} to indicate the dependence on the parameter θ. θ is not a formal argument of the function as it is considered to be fixed. However, each different value of the parameter gives a different probability density function. Then the parametric family of densities is the set of functions { f X ( ⋅ ; θ ) ∣ θ ∈ Θ } {\displaystyle \{f_{X}(\cdot \,;\theta )\mid \theta \in \Theta \}} , where Θ denotes the parameter space, the set of all possible values that the parameter θ can take. As an example, the normal distribution is a family of similarly shaped distributions parametrized by their mean and their variance. In


id: parts-per-notation-20260505-0
file: data/en.wikipedia.org/wiki/Parts-per_notation-0.md
category: reference
title: Parts-per notation
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Parts-per_notation
date_saved: 2026-05-05T07:24:44.742123+00:00
instance: kb-cron
parent: parts-per-notation-20260505-0
summary:
In science and engineering, parts-per notation is a set of pseudo-units to describe the small values of miscellaneous dimensionless quantities, e.g. mole fraction or mass fraction. Since these fractions are quantity-per-quantity measures, they are pure numbers with no associated units of measurement. Commonly used are parts-per-million – ppm, 10−6 parts-per-billion – ppb, 10−9 parts-per-trillion – ppt, 10−12 parts-per-quadrillion – ppq, 10−15 This notation is not part of the International System of Units (the SI system) and its meaning is ambiguous. In chemistry, ambiguity arises because parts-per notation is able to be used to refer to a mole fraction and a mass fraction, which are unequal depending on the molar mass of the substance. == Applications == Parts-per notation is often used describing dilute solutions in chemistry, for instance, the relative abundance of dissolved minerals or pollutants in water. The quantity "1 ppm" can be used for a mass fraction if a water-borne pollutant is present at one-millionth of a gram per gram of sample solution. When working with aqueous solutions, it is common to assume that the density of water is 1.00 g/mL. Therefore, it is common to equate 1 kilogram of water with 1 L of water. Consequently, 1


id: parts-per-notation-20260505-1
file: data/en.wikipedia.org/wiki/Parts-per_notation-1.md
category: reference
title: Parts-per notation
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Parts-per_notation
date_saved: 2026-05-05T07:24:44.742123+00:00
instance: kb-cron
parent: parts-per-notation-20260505-0
summary:
One part per billion (ppb) denotes one part per 1,000,000,000 (109) parts, and a value of 10−9. This is equivalent to about three seconds out of a century. One part per trillion (ppt) denotes one part per 1,000,000,000,000 (1012) parts, and a value of 10−12. This is equivalent to about thirty seconds out of every million years. One part per quadrillion (ppq) denotes one part per 1,000,000,000,000,000 (1015) parts, and a value of 10−15. This is equivalent to about two and a half minutes out of the age of the Earth (4.5 billion years). Although relatively uncommon in analytical chemistry, measurements at the ppq level are sometimes performed. == Criticism == Although the International Bureau of Weights and Measures (an international standards organization known also by its French-language initials BIPM) recognizes the use of parts-per notation, it is not formally part of the International System of Units (SI). Note that although "percent" (%) is not formally part of the SI, both the BIPM and the International Organization for Standardization (ISO) take the position that "in mathematical expressions, the internationally recognized symbol % (percent) may be used with the SI to represent the number 0.01" for dimensionless quantities. According to IUPAP,


id: pathological-mathematics-20260505-0
file: data/en.wikipedia.org/wiki/Pathological_(mathematics)-0.md
category: reference
title: Pathological (mathematics)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Pathological_(mathematics)
date_saved: 2026-05-05T07:24:45.941558+00:00
instance: kb-cron
parent: pathological-mathematics-20260505-0
summary:
In mathematics, when a mathematical phenomenon runs counter to some intuition, then the phenomenon is sometimes called pathological. On the other hand, if a phenomenon does not run counter to intuition, it is sometimes called well-behaved or nice. These terms are sometimes useful in mathematical research and teaching, but there is no strict mathematical definition of pathological or well-behaved. == In analysis == A classic example of a pathology is the Weierstrass function, a function that is continuous everywhere but differentiable nowhere. The sum of a differentiable function and the Weierstrass function is again continuous but nowhere differentiable; so there are at least as many such functions as differentiable functions. In fact, using the Baire category theorem, one can show that continuous functions are generically nowhere differentiable. Such examples were deemed pathological when they were first discovered. To quote Henri Poincaré: Logic sometimes breeds monsters. For half a century there has been springing up a host of weird functions, which seem to strive to have as little resemblance as possible to honest functions that are of some use. No more continuity, or else continuity but no derivatives, etc. More than this, from the point of view of logic, it is


id: pathological-mathematics-20260505-1
file: data/en.wikipedia.org/wiki/Pathological_(mathematics)-1.md
category: reference
title: Pathological (mathematics)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Pathological_(mathematics)
date_saved: 2026-05-05T07:24:45.941558+00:00
instance: kb-cron
parent: pathological-mathematics-20260505-0
summary:
== Well-behaved == Mathematicians (and those in related sciences) very frequently speak of whether a mathematical object—a function, a set, a space of one sort or another—is "well-behaved". While the term has no fixed formal definition, it generally refers to the quality of satisfying a list of prevailing conditions, which might be dependent on context, mathematical interests, fashion, and taste. To ensure that an object is "well-behaved", mathematicians introduce further axioms to narrow down the domain of study. This has the benefit of making analysis easier, but produces a loss of generality of any conclusions reached. In both pure and applied mathematics (e.g., optimization, numerical integration, mathematical physics), well-behaved also means not violating any assumptions needed to successfully apply whatever analysis is being discussed. The opposite case is usually labeled "pathological". It is not unusual to have situations in which most cases (in terms of cardinality or measure) are pathological, but the pathological cases will not arise in practice—unless constructed deliberately. The term "well-behaved" is generally applied in an absolute sense—either something is well-behaved or it is not. For example: In algorithmic inference, a well-behaved statistic is monotonic, well-defined, and sufficient. In Bézout's theorem, two polynomials are well-behaved, and thus


id: pathological-mathematics-20260505-2
file: data/en.wikipedia.org/wiki/Pathological_(mathematics)-2.md
category: reference
title: Pathological (mathematics)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Pathological_(mathematics)
date_saved: 2026-05-05T07:24:45.941558+00:00
instance: kb-cron
parent: pathological-mathematics-20260505-0
summary:
The Cantor set is a subset of the interval [ 0 , 1 ] {\displaystyle [0,1]} that has measure zero but is uncountable. The fat Cantor set is nowhere dense but has positive measure. The Fabius function is everywhere smooth but nowhere analytic. Volterra's function is differentiable with bounded derivative everywhere, but the derivative is not Riemann-integrable. The Peano space-filling curve is a continuous surjective function that maps the unit interval [ 0 , 1 ] {\displaystyle [0,1]} onto [ 0 , 1 ] × [ 0 , 1 ] {\displaystyle [0,1]\times [0,1]} . The Dirichlet function, which is the indicator function for rationals, is a bounded function that is not Riemann integrable. The Cantor function is a monotonic continuous surjective function that maps [ 0 , 1 ] {\displaystyle [0,1]} onto [ 0 , 1 ] {\displaystyle [0,1]} , but has zero derivative almost everywhere. The Minkowski question-mark function is continuous and strictly increasing but has zero derivative almost everywhere. Satisfaction classes containing "intuitively false" arithmetical statements can be constructed for countable, recursively saturated models of Peano arithmetic. The Osgood curve is a Jordan curve (unlike most space-filling curves) of positive area. An exotic sphere is homeomorphic but not


id: percentage-point-20260505-0
file: data/en.wikipedia.org/wiki/Percentage_point-0.md
category: reference
title: Percentage point
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Percentage_point
date_saved: 2026-05-05T07:24:47.195404+00:00
instance: kb-cron
parent: percentage-point-20260505-0
summary:
A percentage point or percent point is the description for the arithmetic difference between two percentages. For example, moving up from 40 percent to 44 percent is an increase of 4 percentage points (although it is a 10-percent increase in the quantity being measured, if the total amount remains the same). In written text, the unit (the percentage point) is usually either written out, or abbreviated as pp, p.p., or %pt. to avoid confusion with percentage increase or decrease in the actual quantity. After the first occurrence, some writers abbreviate by using just "point" or "points". == Differences between percentages and percentage points == Consider the following hypothetical example: In 1980, 50 percent of the population smoked, and in 1990 only 40 percent of the population smoked. One can thus say that from 1980 to 1990, the prevalence of smoking decreased by 10 percentage points (or by 10 percent of the population) or by 20 percent when talking about smokers only – percentages indicate proportionate part of a total. Percentage-point differences are one way to express a risk or probability. Consider a drug that cures a given disease in 70 percent of all cases, while without the drug, the disease


id: pointwise-20260505-0
file: data/en.wikipedia.org/wiki/Pointwise-0.md
category: reference
title: Pointwise
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Pointwise
date_saved: 2026-05-05T07:24:48.491777+00:00
instance: kb-cron
parent: pointwise-20260505-0
summary:
In mathematics, the qualifier pointwise is used to indicate that a certain property is defined by considering each value f ( x ) {\displaystyle f(x)} of some function f . {\displaystyle f.} An important class of pointwise concepts are the pointwise operations, that is, operations defined on functions by applying the operations to function values separately for each point in the domain of definition. Important relations can also be defined pointwise. == Pointwise operations == === Formal definition === A binary operation o: Y × Y → Y on a set Y can be lifted pointwise to an operation O: (X→Y) × (X→Y) → (X→Y) on the set X → Y of all functions from X to Y as follows: Given two functions f1: X → Y and f2: X → Y, define the function O(f1, f2): X → Y by Commonly, o and O are denoted by the same symbol. A similar definition is used for unary operations o, and for operations of other arity. === Examples === The pointwise addition f + g {\displaystyle f+g} of two functions f {\displaystyle f} and g {\displaystyle g} with the same domain and codomain is defined by: The pointwise product or


id: porism-20260505-0
file: data/en.wikipedia.org/wiki/Porism-0.md
category: reference
title: Porism
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Porism
date_saved: 2026-05-05T07:24:49.702070+00:00
instance: kb-cron
parent: porism-20260505-0
summary:
A porism is a mathematical proposition or corollary. It has been used to refer to a direct consequence of a proof, analogous to how a corollary refers to a direct consequence of a theorem. In modern usage, it is a relationship that holds for an infinite range of values but only if a certain condition is assumed, such as Steiner's porism. The term originates from three books of Euclid that have been lost. A proposition may not have been proven, so a porism may not be a theorem or true. == Origins == The book that talks about porisms first is Euclid's Porisms. What is known of it is in Pappus of Alexandria's Collection, who mentions it along with other geometrical treatises, and gives several lemmas necessary for understanding it. Pappus states: The porisms of all classes are neither theorems nor problems, but occupy a position intermediate between the two, so that their enunciations can be stated either as theorems or problems, and consequently some geometers think that they are theorems, and others that they are problems, being guided solely by the form of the enunciation. But it is clear from the definitions that the old geometers understood better the


id: porism-20260505-1
file: data/en.wikipedia.org/wiki/Porism-1.md
category: reference
title: Porism
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Porism
date_saved: 2026-05-05T07:24:49.702070+00:00
instance: kb-cron
parent: porism-20260505-0
summary:
under certain conditions a problem becomes impossible; under certain other conditions, indeterminate or capable of an infinite number of solutions. These cases could be defined separately, were in a manner intermediate between theorems and problems, and were called "porisms." Playfair defined a porism as "[a] proposition affirming the possibility of finding such conditions as will render a certain problem indeterminate or capable of innumerable solutions." Although Playfair's definition of a porism appears to be most favoured in England, Simson's view has been most generally accepted abroad, and had the support of Michel Chasles. However, in Liouville's Journal de mathematiques pures et appliquées (vol. xx., July, 1855), P. Breton published Recherches nouvelles sur les porismes d'Euclide, in which he gave a new translation of the text of Pappus, and sought to base a view of the nature of a porism that conforms more closely to Pappus's definition. This was followed in the same journal and in La Science by a controversy between Breton and A. J. H. Vincent, who disputed the interpretation given by the former of Pappus's text, and declared himself in favour of Frans van Schooten's idea, put forward in his Mathematicae exercitationes (1657). According to Schooten, if the


id: power-root-power-and-field-quantities-20260505-0
file: data/en.wikipedia.org/wiki/Power,_root-power,_and_field_quantities-0.md
category: reference
title: Power, root-power, and field quantities
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Power,_root-power,_and_field_quantities
date_saved: 2026-05-05T07:24:50.916584+00:00
instance: kb-cron
parent: power-root-power-and-field-quantities-20260505-0
summary:
A power quantity is a power or a quantity directly proportional to power, e.g., energy density, acoustic intensity, and luminous intensity. Energy quantities may also be labelled as power quantities in this context. A root-power quantity is a quantity such as voltage, current, sound pressure, electric field strength, speed, or charge density, the square of which, in linear systems, is proportional to power. The term root-power quantity refers to the square root that relates these quantities to power. The term was introduced in ISO 80000-1 § Annex C; it replaces and deprecates the term field quantity. == Implications == It is essential to know which category a measurement belongs to when using decibels (dB) for comparing the levels of such quantities. A change of one bel in the level corresponds to a 10× change in power, so when comparing power quantities x and y, the difference is defined to be 10×log10(y/x) decibel. With root-power quantities, however the difference is defined as 20×log10(y/x) dB. In the analysis of signals and systems using sinusoids, field quantities and root-power quantities may be complex-valued, as in the propagation constant. == "Root-power quantity" vs. "field quantity" == In justifying the deprecation of the term "field


id: projection-mathematics-20260505-0
file: data/en.wikipedia.org/wiki/Projection_(mathematics)-0.md
category: reference
title: Projection (mathematics)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Projection_(mathematics)
date_saved: 2026-05-05T07:24:52.122463+00:00
instance: kb-cron
parent: projection-mathematics-20260505-0
summary:
In mathematics, a projection is a mapping from a set to itself—or an endomorphism of a mathematical structure—that is idempotent, that is, equals its composition with itself. The image of a point or a subset ⁠ S {\displaystyle S} ⁠ under a projection is called the projection of ⁠ S {\displaystyle S} ⁠. An everyday example of a projection is the casting of shadows onto a plane (sheet of paper): the projection of a point is its shadow on the sheet of paper, and the projection (shadow) of a point on the sheet of paper is that point itself (idempotency). The shadow of a three-dimensional sphere is a disk. Originally, the notion of projection was introduced in Euclidean geometry to denote the projection of the three-dimensional Euclidean space onto a plane in it, like the shadow example. The two main projections of this kind are: The projection from a point onto a plane or central projection: If C is a point, called the centre of projection, then the projection of a point P different from C onto a plane that does not contain C is the intersection of the line CP with the plane. The points P such that the


id: projection-mathematics-20260505-1
file: data/en.wikipedia.org/wiki/Projection_(mathematics)-1.md
category: reference
title: Projection (mathematics)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Projection_(mathematics)
date_saved: 2026-05-05T07:24:52.122463+00:00
instance: kb-cron
parent: projection-mathematics-20260505-0
summary:
In set theory: An operation typified by the j-th projection map, written projj, that takes an element x = (x1, ..., xj, ..., xn) of the Cartesian product X1 × ⋯ × Xj × ⋯ × Xn to the value projj(x) = xj. This map is always surjective and, when each space Xk has a topology, this map is also continuous and open. A mapping that takes an element to its equivalence class under a given equivalence relation is known as the canonical projection. The evaluation map sends a function f to the value f(x) for a fixed x. The space of functions YX can be identified with the Cartesian product ∏ i ∈ X Y {\textstyle \prod _{i\in X}Y} , and the evaluation map is a projection map from the Cartesian product. For relational databases and query languages, the projection is a unary operation written as Π a 1 , … , a n ( R ) {\displaystyle \Pi _{a_{1},\ldots ,a_{n}}(R)} where a 1 , … , a n {\displaystyle a_{1},\ldots ,a_{n}} is a set of attribute names. The result of such projection is defined as the set that is obtained when all tuples in R are restricted to


id: mathematical-proof-20260505-0
file: data/en.wikipedia.org/wiki/Mathematical_proof-0.md
category: reference
title: Mathematical proof
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Mathematical_proof
date_saved: 2026-05-05T07:24:53.330614+00:00
instance: kb-cron
parent: mathematical-proof-20260505-0
summary:
A mathematical proof is a deductive argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms, along with the accepted rules of inference. Proofs are examples of exhaustive deductive reasoning that establish logical certainty, to be distinguished from empirical arguments or non-exhaustive inductive reasoning that establish "reasonable expectation". Presenting many cases in which the statement holds is not enough for a proof, which must demonstrate that the statement is true in all possible cases. A proposition that has not been proved but is believed to be true is known as a conjecture, or a hypothesis if frequently used as an assumption for further mathematical work. Proofs employ logic expressed in mathematical symbols, along with natural language that usually admits some ambiguity. In most mathematical literature, proofs are written in terms of rigorous informal logic. Purely formal proofs, written fully in symbolic language without the involvement of natural language, are considered in proof theory. The distinction between formal and informal proofs has led to much examination of current


id: mathematical-proof-20260505-1
file: data/en.wikipedia.org/wiki/Mathematical_proof-1.md
category: reference
title: Mathematical proof
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Mathematical_proof
date_saved: 2026-05-05T07:24:53.330614+00:00
instance: kb-cron
parent: mathematical-proof-20260505-0
summary:
In direct proof, the conclusion is established by logically combining the axioms, definitions, and earlier theorems. For example, direct proof can be used to prove that the sum of two even integers is always even: Consider two even integers x and y. Since they are even, they can be written as x = 2a and y = 2b, respectively, for some integers a and b. Then the sum is x + y = 2a + 2b = 2(a+b). Therefore x+y has 2 as a factor and, by definition, is even. Hence, the sum of any two even integers is even. This proof uses the definition of even integers, the integer properties of closure under addition and multiplication, and the distributive property. === Proof by mathematical induction === Despite its name, mathematical induction is a method of deduction, not a form of inductive reasoning. In proof by mathematical induction, a single "base case" is proved, and an "induction rule" is proved that establishes that any arbitrary case implies the next case. Since in principle the induction rule can be applied repeatedly (starting from the proven base case), it follows that all (usually infinitely many) cases are provable. This avoids having to


id: mathematical-proof-20260505-2
file: data/en.wikipedia.org/wiki/Mathematical_proof-2.md
category: reference
title: Mathematical proof
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Mathematical_proof
date_saved: 2026-05-05T07:24:53.330614+00:00
instance: kb-cron
parent: mathematical-proof-20260505-0
summary:
In proof by exhaustion, the conclusion is established by dividing it into a finite number of cases and proving each one separately. The number of cases sometimes can become very large. For example, the first proof of the four color theorem was a proof by exhaustion with 1,936 cases. This proof was controversial because the majority of the cases were checked by a computer program, not by hand. === Closed chain inference === A closed chain inference shows that a collection of statements are pairwise equivalent. In order to prove that the statements φ 1 , … , φ n {\displaystyle \varphi _{1},\ldots ,\varphi _{n}} are each pairwise equivalent, proofs are given for the implications φ 1 ⇒ φ 2 {\displaystyle \varphi _{1}\Rightarrow \varphi _{2}} , φ 2 ⇒ φ 3 {\displaystyle \varphi _{2}\Rightarrow \varphi _{3}} , … {\displaystyle \dots } , φ n − 1 ⇒ φ n {\displaystyle \varphi _{n-1}\Rightarrow \varphi _{n}} and φ n ⇒ φ 1 {\displaystyle \varphi _{n}\Rightarrow \varphi _{1}} . The pairwise equivalence of the statements then results from the transitivity of the material conditional. === Probabilistic proof === A probabilistic proof is one in which an example is shown to exist, with certainty,


id: mathematical-proof-20260505-3
file: data/en.wikipedia.org/wiki/Mathematical_proof-3.md
category: reference
title: Mathematical proof
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Mathematical_proof
date_saved: 2026-05-05T07:24:53.330614+00:00
instance: kb-cron
parent: mathematical-proof-20260505-0
summary:
== Undecidable statements == A statement that is neither provable nor disprovable from a set of axioms is called undecidable (from those axioms). One example is the parallel postulate, which is neither provable nor refutable from the remaining axioms of Euclidean geometry. Mathematicians have shown there are many statements that are neither provable nor disprovable in Zermelo–Fraenkel set theory with the axiom of choice (ZFC), the standard system of set theory in mathematics (assuming that ZFC is consistent); see List of statements undecidable in ZFC. Gödel's (first) incompleteness theorem shows that many axiom systems of mathematical interest will have undecidable statements. == Heuristic mathematics and experimental mathematics == While early mathematicians such as Eudoxus of Cnidus did not use proofs, from Euclid to the foundational mathematics developments of the late 19th and 20th centuries, proofs were an essential part of mathematics. With the increase in computing power in the 1960s, significant work began to be done investigating mathematical objects beyond the proof-theorem framework, in experimental mathematics. Early pioneers of these methods intended the work ultimately to be resolved into a classical proof-theorem framework, e.g. the early development of fractal geometry, which was ultimately so resolved. == Related concepts == ===


id: universal-property-20260505-0
file: data/en.wikipedia.org/wiki/Universal_property-0.md
category: reference
title: Universal property
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Universal_property
date_saved: 2026-05-05T07:24:54.601173+00:00
instance: kb-cron
parent: universal-property-20260505-0
summary:
In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently from the method chosen for constructing them. For example, the definitions of the integers from the natural numbers, of the rational numbers from the integers, of the real numbers from the rational numbers, and of polynomial rings from the field of their coefficients can all be done in terms of universal properties. In particular, the concept of universal property allows a simple proof that all constructions of real numbers are equivalent: it suffices to prove that they satisfy the same universal property. Technically, a universal property is defined in terms of categories and functors by means of a universal morphism (see § Formal definition, below). Universal morphisms can also be thought more abstractly as initial or terminal objects of a comma category (see § Connection with comma categories, below). Universal properties occur almost everywhere in mathematics, and the use of the concept allows the use of general properties of universal properties for easily proving some properties that would need boring verifications otherwise. For example, given


id: universal-property-20260505-1
file: data/en.wikipedia.org/wiki/Universal_property-1.md
category: reference
title: Universal property
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Universal_property
date_saved: 2026-05-05T07:24:54.601173+00:00
instance: kb-cron
parent: universal-property-20260505-0
summary:
== Connection with comma categories == Universal morphisms can be described more concisely as initial and terminal objects in a comma category (i.e. one where morphisms are seen as objects in their own right). Let F : C → D {\displaystyle F:{\mathcal {C}}\to {\mathcal {D}}} be a functor and X {\displaystyle X} an object of D {\displaystyle {\mathcal {D}}} . Then recall that the comma category ( X ↓ F ) {\displaystyle (X\downarrow F)} is the category where Objects are pairs of the form ( B , f : X → F ( B ) ) {\displaystyle (B,f:X\to F(B))} , where B {\displaystyle B} is an object in C {\displaystyle {\mathcal {C}}} A morphism from ( B , f : X → F ( B ) ) {\displaystyle (B,f:X\to F(B))} to ( B ′ , f ′ : X → F ( B ′ ) ) {\displaystyle (B',f':X\to F(B'))} is given by a morphism h : B → B ′ {\displaystyle h:B\to B'} in C {\displaystyle {\mathcal {C}}} such that the diagram commutes: Now suppose that the object ( A , u : X → F ( A ) ) {\displaystyle (A,u:X\to F(A))} in ( X ↓ F ) {\displaystyle


id: universal-property-20260505-2
file: data/en.wikipedia.org/wiki/Universal_property-2.md
category: reference
title: Universal property
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Universal_property
date_saved: 2026-05-05T07:24:54.601173+00:00
instance: kb-cron
parent: universal-property-20260505-0
summary:
such that for any other object Z {\displaystyle Z} of C {\displaystyle {\mathcal {C}}} and morphisms f : Z → X {\displaystyle f:Z\to X} and g : Z → Y {\displaystyle g:Z\to Y} there exists a unique morphism h : Z → X × Y {\displaystyle h:Z\to X\times Y} such that f = π 1 ∘ h {\displaystyle f=\pi _{1}\circ h} and g = π 2 ∘ h {\displaystyle g=\pi _{2}\circ h} . To understand this characterization as a universal property, take the category D {\displaystyle {\mathcal {D}}} to be the product category C × C {\displaystyle {\mathcal {C}}\times {\mathcal {C}}} and define the diagonal functor Δ : C → C × C {\displaystyle \Delta :{\mathcal {C}}\to {\mathcal {C}}\times {\mathcal {C}}} by Δ ( X ) = ( X , X ) {\displaystyle \Delta (X)=(X,X)} and Δ ( f : X → Y ) = ( f , f ) {\displaystyle \Delta (f:X\to Y)=(f,f)} . Then ( X × Y , ( π 1 , π 2 ) ) {\displaystyle (X\times Y,(\pi _{1},\pi _{2}))} is a universal morphism from Δ {\displaystyle \Delta } to the object ( X , Y ) {\displaystyle (X,Y)} of C × C {\displaystyle {\mathcal {C}}\times


id: universal-property-20260505-3
file: data/en.wikipedia.org/wiki/Universal_property-3.md
category: reference
title: Universal property
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Universal_property
date_saved: 2026-05-05T07:24:54.601173+00:00
instance: kb-cron
parent: universal-property-20260505-0
summary:
k : A → A ′ {\displaystyle k:A\to A'} such that u ′ = F ( k ) ∘ u {\displaystyle u'=F(k)\circ u} . This is easily seen by substituting ( A , u ′ ) {\displaystyle (A,u')} in the definition of a universal morphism. It is the pair ( A , u ) {\displaystyle (A,u)} which is essentially unique in this fashion. The object A {\displaystyle A} itself is only unique up to isomorphism. Indeed, if ( A , u ) {\displaystyle (A,u)} is a universal morphism and k : A → A ′ {\displaystyle k:A\to A'} is any isomorphism then the pair ( A ′ , u ′ ) {\displaystyle (A',u')} , where u ′ = F ( k ) ∘ u {\displaystyle u'=F(k)\circ u} is also a universal morphism. === Equivalent formulations === The definition of a universal morphism can be rephrased in a variety of ways. Let F : C → D {\displaystyle F:{\mathcal {C}}\to {\mathcal {D}}} be a functor and let X {\displaystyle X} be an object of D {\displaystyle {\mathcal {D}}} . Then the following statements are equivalent: ( A , u ) {\displaystyle (A,u)} is a universal morphism from X {\displaystyle X} to


id: universal-property-20260505-4
file: data/en.wikipedia.org/wiki/Universal_property-4.md
category: reference
title: Universal property
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Universal_property
date_saved: 2026-05-05T07:24:54.601173+00:00
instance: kb-cron
parent: universal-property-20260505-0
summary:
For each object X {\displaystyle X} in C {\displaystyle {\mathcal {C}}} , ( F ( X ) , η X ) {\displaystyle (F(X),\eta _{X})} is a universal morphism from X {\displaystyle X} to G {\displaystyle G} . That is, for all f : X → G ( Y ) {\displaystyle f:X\to G(Y)} there exists a unique g : F ( X ) → Y {\displaystyle g:F(X)\to Y} for which the following diagrams commute. For each object Y {\displaystyle Y} in D {\displaystyle {\mathcal {D}}} , ( G ( Y ) , ϵ Y ) {\displaystyle (G(Y),\epsilon _{Y})} is a universal morphism from F {\displaystyle F} to Y {\displaystyle Y} . That is, for all g : F ( X ) → Y {\displaystyle g:F(X)\to Y} there exists a unique f : X → G ( Y ) {\displaystyle f:X\to G(Y)} for which the following diagrams commute. Universal constructions are more general than adjoint functor pairs: a universal construction is like an optimization problem; it gives rise to an adjoint pair if and only if this problem has a solution for every object of C {\displaystyle {\mathcal {C}}} (equivalently, every object of D {\displaystyle {\mathcal {D}}} ). == History == Universal


id: proportion-mathematics-20260505-0
file: data/en.wikipedia.org/wiki/Proportion_(mathematics)-0.md
category: reference
title: Proportion (mathematics)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Proportion_(mathematics)
date_saved: 2026-05-05T07:24:55.781739+00:00
instance: kb-cron
parent: proportion-mathematics-20260505-0
summary:
A proportion is a mathematical statement expressing equality of two ratios. a : b = c : d {\displaystyle a:b=c:d} a and d are called extremes, b and c are called means. Proportion can be written as a b = c d {\displaystyle {\frac {a}{b}}={\frac {c}{d}}} , where ratios are expressed as fractions. Such a proportion is known as geometrical proportion, not to be confused with arithmetical proportion and harmonic proportion. == Properties of proportions == Fundamental rule of proportion. This rule is sometimes called Means‐Extremes Property. If the ratios are expressed as fractions, then the same rule can be phrased in terms of the equality of "cross-products" and is called Cross‐Products Property. If a b = c d {\displaystyle \ {\frac {a}{b}}={\frac {c}{d}}} , then a d = b c {\displaystyle \ ad=bc} If a b = c d {\displaystyle \ {\frac {a}{b}}={\frac {c}{d}}} , then b a = d c {\displaystyle \ {\frac {b}{a}}={\frac {d}{c}}} If a b = c d {\displaystyle \ {\frac {a}{b}}={\frac {c}{d}}} , then a c = b d {\displaystyle \ {\frac {a}{c}}={\frac {b}{d}}} , d b = c a {\displaystyle \ {\frac {d}{b}}={\frac {c}{a}}} . If a b = c d {\displaystyle \


id: proportionality-mathematics-20260505-0
file: data/en.wikipedia.org/wiki/Proportionality_(mathematics)-0.md
category: reference
title: Proportionality (mathematics)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Proportionality_(mathematics)
date_saved: 2026-05-05T07:24:56.943613+00:00
instance: kb-cron
parent: proportionality-mathematics-20260505-0
summary:
In mathematics, two sequences of numbers, often experimental data, are proportional or directly proportional if their corresponding elements have a constant ratio. The ratio is called coefficient of proportionality (or proportionality constant) and its reciprocal is known as constant of normalization (or normalizing constant). Two sequences are inversely proportional if corresponding elements have a constant product. Two functions f ( x ) {\displaystyle f(x)} and g ( x ) {\displaystyle g(x)} are proportional if their ratio f ( x ) g ( x ) {\textstyle {\frac {f(x)}{g(x)}}} is a constant function. If several pairs of variables share the same direct proportionality constant, the equation expressing the equality of these ratios is called a proportion, e.g., ⁠a/b⁠ = ⁠x/y⁠ = ⋯ = k (for details see Ratio). Proportionality is closely related to linearity. == Direct proportionality == Given an independent variable x and a dependent variable y, y is directly proportional to x if there is a positive constant k such that: y = k x . {\displaystyle y=kx.} The relation is often denoted using the symbols ∝ (not to be confused with the Greek letter alpha) or ~, with exception of Japanese texts, where ~ is reserved for intervals: y


id: q-e-d-20260505-0
file: data/en.wikipedia.org/wiki/Q.E.D.-0.md
category: reference
title: Q.E.D.
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Q.E.D.
date_saved: 2026-05-05T07:24:58.133655+00:00
instance: kb-cron
parent: q-e-d-20260505-0
summary:
Q.E.D. or QED is an initialism of the Latin phrase quod erat demonstrandum, meaning "that which was to be demonstrated". Literally, it states "what was to be shown". Traditionally, the abbreviation is placed at the end of mathematical proofs and philosophical arguments in print publications, to indicate that the proof or the argument is complete. == Etymology and early use == The phrase quod erat demonstrandum is a translation into Latin from the Greek ὅπερ ἔδει δεῖξαι (hoper edei deixai; abbreviated as ΟΕΔ). The meaning of the Latin phrase is "that [thing] which was to be demonstrated" (with demonstrandum in the gerundive). The Greek phrase was used by many early Greek mathematicians, including Euclid and Archimedes. The Latin phrase is attested in a 1501 Euclid translation of Giorgio Valla. Its abbreviation q.e.d. is used once in 1598 by Johannes Praetorius, more in 1643 by Anton Deusing, extensively in 1655 by Isaac Barrow in the form Q.E.D., and subsequently by many post-Renaissance mathematicians and philosophers. == Modern philosophy == During the European Renaissance, scholars often wrote in Latin, and phrases such as Q.E.D. were often used to conclude proofs. Perhaps the most famous use of Q.E.D. in a philosophical argument is


id: quadrature-mathematics-20260505-0
file: data/en.wikipedia.org/wiki/Quadrature_(mathematics)-0.md
category: reference
title: Quadrature (mathematics)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Quadrature_(mathematics)
date_saved: 2026-05-05T07:24:59.521729+00:00
instance: kb-cron
parent: quadrature-mathematics-20260505-0
summary:
In mathematics, quadrature is a historic term for the computation of areas and is thus used for computation of integrals. The word is derived from the Latin quadratus meaning "square". The reason is that, for Ancient Greek mathematicians, the computation of an area consisted of constructing a square of the same area. In this sense, the modern term is squaring. For example, the quadrature of the circle, (or squaring the circle) is a famous old problem that has been shown, in the 19th century, to be impossible with the methods available to the Ancient Greeks. Integral calculus, introduced in the 17th century, is a general method for computation of areas. Quadrature came to refer to the computation of any integral; such a computation is presently called more often "integral" or "integration". However, the computation of solutions of differential equations and differential systems is also called integration, and quadrature remains useful for distinguishing integrals from solutions of differential equations, in contexts where both problems are considered. This is the case in numerical analysis; see numerical quadrature. Also, reduction to quadratures and solving by quadratures means expressing solutions of differential equations in terms of integrals. The remainder of this article is devoted


id: qualitative-property-20260505-0
file: data/en.wikipedia.org/wiki/Qualitative_property-0.md
category: reference
title: Qualitative property
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Qualitative_property
date_saved: 2026-05-05T07:25:00.803798+00:00
instance: kb-cron
parent: qualitative-property-20260505-0
summary:
Qualitative properties are properties that are observed and can generally not be measured with a numerical result, unlike quantitative properties, which have numerical characteristics. == Description == Qualitative properties are properties that are observed and can generally not be measured with a numerical result. They are contrasted to quantitative properties which have numerical characteristics. === Evaluation === Although measuring something in qualitative terms is difficult, most people can (and will) make a judgement about a behaviour on the basis of how they feel treated. This indicates that qualitative properties are closely related to emotional impressions. A test method can result in qualitative data about something. This can be a categorical result or a binary classification (e.g., pass/fail, go/no go, conform/non-conform). It can sometimes be an engineering judgement. === Categorization === The data that all share a qualitative property form a nominal category. A variable which codes for the presence or absence of such a property is called a binary categorical variable, or equivalently a dummy variable. == Types == Some engineering and scientific properties are qualitative. Some important qualitative properties that concern businesses are: Human factors, human work capital is important issue that deals with qualitative properties. Some common aspects


id: reduction-mathematics-20260505-0
file: data/en.wikipedia.org/wiki/Reduction_(mathematics)-0.md
category: reference
title: Reduction (mathematics)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Reduction_(mathematics)
date_saved: 2026-05-05T07:25:01.964455+00:00
instance: kb-cron
parent: reduction-mathematics-20260505-0
summary:
In mathematics, reduction refers to the rewriting of an expression into a simpler form. For example, the process of rewriting a fraction into one with the smallest whole-number denominator possible (while keeping the numerator a whole number) is called "reducing a fraction". Rewriting a radical (or "root") expression with the smallest possible whole number under the radical symbol is called "reducing a radical". Minimizing the number of radicals that appear underneath other radicals in an expression is called denesting radicals. == Algebra == In linear algebra, reduction refers to applying simple rules to a series of equations or matrices to change them into a simpler form. In the case of matrices, the process involves manipulating either the rows or the columns of the matrix and so is usually referred to as row-reduction or column-reduction, respectively. Often the aim of reduction is to transform a matrix into its "row-reduced echelon form" or "row-echelon form"; this is the goal of Gaussian elimination. == Calculus == In calculus, reduction refers to using the technique of integration by parts to evaluate integrals by reducing them to simpler forms. == Static (Guyan) reduction == In dynamic analysis, static reduction refers to reducing the number of


id: resilience-mathematics-20260505-0
file: data/en.wikipedia.org/wiki/Resilience_(mathematics)-0.md
category: reference
title: Resilience (mathematics)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Resilience_(mathematics)
date_saved: 2026-05-05T07:25:03.234631+00:00
instance: kb-cron
parent: resilience-mathematics-20260505-0
summary:
In mathematical modeling, resilience refers to the ability of a dynamical system to recover from perturbations and return to its original stable steady state. It is a measure of the stability and robustness of a system in the face of changes or disturbances. If a system is not resilient enough, it is more susceptible to perturbations and can more easily undergo a critical transition. A common analogy used to explain the concept of resilience of an equilibrium is one of a ball in a valley. A resilient steady state corresponds to a ball in a deep valley, so any push or perturbation will very quickly lead the ball to return to the resting point where it started. On the other hand, a less resilient steady state corresponds to a ball in a shallow valley, so the ball will take a much longer time to return to the equilibrium after a perturbation. The concept of resilience is particularly useful in systems that exhibit tipping points, whose study has a long history that can be traced back to catastrophe theory. While this theory was initially overhyped and fell out of favor, its mathematical foundation remains strong and is now recognized as relevant


id: rigidity-mathematics-20260505-0
file: data/en.wikipedia.org/wiki/Rigidity_(mathematics)-0.md
category: reference
title: Rigidity (mathematics)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Rigidity_(mathematics)
date_saved: 2026-05-05T07:25:04.445143+00:00
instance: kb-cron
parent: rigidity-mathematics-20260505-0
summary:
In mathematics, a rigid collection C of mathematical objects c (for instance sets or functions) is one in which every c ∈ C is uniquely determined by less information about c than one would expect. The above statement does not define a mathematical property; instead, it describes in what sense the adjective "rigid" is typically used in mathematics, by mathematicians. == Examples == Some examples include: Harmonic functions on the unit disk are rigid in the sense that they are uniquely determined by their boundary values. Holomorphic functions are determined by the set of all derivatives at a single point. A smooth function from the real line to the complex plane is not, in general, determined by all its derivatives at a single point, but it is if we require additionally that it be possible to extend the function to one on a neighbourhood of the real line in the complex plane. The Schwarz lemma is an example of such a rigidity theorem. By the fundamental theorem of algebra, polynomials in C are rigid in the sense that any polynomial is completely determined by its values on any infinite set, say N, or the unit disk. By the previous example,


id: scope-logic-20260505-0
file: data/en.wikipedia.org/wiki/Scope_(logic)-0.md
category: reference
title: Scope (logic)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Scope_(logic)
date_saved: 2026-05-05T07:25:06.412643+00:00
instance: kb-cron
parent: scope-logic-20260505-0
summary:
In logic, the scope of a quantifier or connective is the shortest formula in which it occurs, determining the range in the formula to which the quantifier or connective is applied. The notions of a free variable and bound variable are defined in terms of whether that formula is within the scope of a quantifier, and the notions of a dominant connective and subordinate connective are defined in terms of whether a connective includes another within its scope. == Connectives == The scope of a logical connective occurring within a formula is the smallest well-formed formula that contains the connective in question. The connective with the largest scope in a formula is called its dominant connective, main connective, main operator, major connective, or principal connective; a connective within the scope of another connective is said to be subordinate to it. For instance, in the formula ( ( ( P → Q ) ∨ ¬ Q ) ↔ ( ¬ ¬ P ∧ Q ) ) {\displaystyle (\left(\left(P\rightarrow Q\right)\lor \lnot Q\right)\leftrightarrow \left(\lnot \lnot P\land Q\right))} , the dominant connective is ↔, and all other connectives are subordinate to it; the → is subordinate to the ∨, but not to the ∧;


id: sides-of-an-equation-20260505-0
file: data/en.wikipedia.org/wiki/Sides_of_an_equation-0.md
category: reference
title: Sides of an equation
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Sides_of_an_equation
date_saved: 2026-05-05T07:25:07.638119+00:00
instance: kb-cron
parent: sides-of-an-equation-20260505-0
summary:
In mathematics, LHS is informal shorthand for the left-hand side of an equation. Similarly, RHS is the right-hand side. The two sides have the same value, expressed differently, since equality is symmetric. More generally, these terms may apply to an inequation or inequality; the right-hand side is everything on the right side of a test operator in an expression, with LHS defined similarly. == Example == The expression on the right side of the "=" sign is the right side of the equation and the expression on the left of the "=" is the left side of the equation. For example, in x + 5 = y + 8 {\displaystyle x+5=y+8} x + 5 is the left-hand side (LHS) and y + 8 is the right-hand side (RHS). == Homogeneous and inhomogeneous equations == In solving mathematical equations, particularly linear simultaneous equations, differential equations and integral equations, the terminology homogeneous is often used for equations with some linear operator L on the LHS and 0 on the RHS. In contrast, an equation with a non-zero RHS is called inhomogeneous or non-homogeneous, as exemplified by Lf = g, with g a fixed function, which equation is to be solved for f.


id: significand-20260505-0
file: data/en.wikipedia.org/wiki/Significand-0.md
category: reference
title: Significand
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Significand
date_saved: 2026-05-05T07:25:08.958728+00:00
instance: kb-cron
parent: significand-20260505-0
summary:
The significand (also coefficient, sometimes argument, or more ambiguously mantissa, fraction, or characteristic) is the first (left) part of a number in scientific notation or related concepts in floating-point representation, consisting of its significant digits. For negative numbers, it does not include the initial minus sign. Depending on the interpretation of the exponent, the significand may represent an integer or a fractional number, which may cause the term "mantissa" to be misleading, since the mantissa of a logarithm is always its fractional part. Although the other names mentioned are common, significand is the word used by IEEE 754, an important technical standard for floating-point arithmetic. In mathematics, the term "argument" may also be ambiguous, since "the argument of a number" sometimes refers to the length of a circular arc from 1 to a number on the unit circle in the complex plane. == Example == The number 123.45 can be represented as a decimal floating-point number with the integer 12345 as the significand and a 10−2 power term, also called characteristics, where −2 is the exponent (and 10 is the base). Its value is given by the following arithmetic: 123.45 = 12345 × 10−2. This same value can also be


id: stochastic-20260505-0
file: data/en.wikipedia.org/wiki/Stochastic-0.md
category: reference
title: Stochastic
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Stochastic
date_saved: 2026-05-05T07:25:10.291744+00:00
instance: kb-cron
parent: stochastic-20260505-0
summary:
Stochastic (; from Ancient Greek στόχος (stókhos) 'aim, guess') is the property of being well-described by a random probability distribution. Stochasticity and randomness are technically distinct concepts. Stochasticity refers to a modeling approach, while randomness describes phenomena. These terms are often used interchangeably. In probability theory, the formal concept of a stochastic process is also referred to as a random process. Stochasticity is used in many different fields, including actuarial science, image processing, signal processing, computer science, information theory, telecommunications, chemistry, ecology, neuroscience, physics, and cryptography. It is also used in finance, medicine, linguistics, music, media, colour theory, botany, manufacturing and geomorphology. == Etymology == The word stochastic in English was originally used as an adjective with the definition "pertaining to conjecturing", and stemming from a Greek word meaning "to aim at a mark, guess", and the Oxford English Dictionary gives the year 1662 as its earliest occurrence. In his work on probability Ars Conjectandi, originally published in Latin in 1713, Jakob Bernoulli used the phrase "Ars Conjectandi sive Stochastice", which has been translated to "the art of conjecturing or stochastics". This phrase was used, with reference to Bernoulli, by Ladislaus Bortkiewicz, who in 1917 wrote in German the word


id: stochastic-20260505-1
file: data/en.wikipedia.org/wiki/Stochastic-1.md
category: reference
title: Stochastic
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Stochastic
date_saved: 2026-05-05T07:25:10.291744+00:00
instance: kb-cron
parent: stochastic-20260505-0
summary:
== Finance == The financial markets use stochastic models to represent the seemingly random behaviour of various financial assets, including the random behavior of the price of one currency compared to that of another (such as the price of US Dollar compared to that of the Euro), and also to represent random behaviour of interest rates. These models are then used by financial analysts to value options on stock prices, bond prices, and on interest rates, see Markov models. Moreover, it is at the heart of the insurance industry. == Geomorphology == The formation of river meanders has been analyzed as a stochastic process. == Language and linguistics == Non-deterministic approaches in language studies are largely inspired by the work of Ferdinand de Saussure, for example, in functionalist linguistic theory, which argues that competence is based on performance. This distinction in functional theories of grammar should be carefully distinguished from the langue and parole distinction. To the extent that linguistic knowledge is constituted by experience with language, grammar is argued to be probabilistic and variable rather than fixed and absolute. This conception of grammar as probabilistic and variable follows from the idea that one's competence changes in accordance with one's


id: stratification-mathematics-20260505-0
file: data/en.wikipedia.org/wiki/Stratification_(mathematics)-0.md
category: reference
title: Stratification (mathematics)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Stratification_(mathematics)
date_saved: 2026-05-05T07:25:11.575927+00:00
instance: kb-cron
parent: stratification-mathematics-20260505-0
summary:
Stratification has several usages in mathematics. == In mathematical logic == In mathematical logic, stratification is any consistent assignment of numbers to predicate symbols guaranteeing that a unique formal interpretation of a logical theory exists. Specifically, we say that a set of clauses of the form Q 1 ∧ ⋯ ∧ Q n ∧ ¬ Q n + 1 ∧ ⋯ ∧ ¬ Q n + m → P {\displaystyle Q_{1}\wedge \dots \wedge Q_{n}\wedge \neg Q_{n+1}\wedge \dots \wedge \neg Q_{n+m}\rightarrow P} is stratified if and only if there is a stratification assignment S that fulfills the following conditions: If a predicate P is positively derived from a predicate Q (i.e., P is the head of a rule, and Q occurs positively in the body of the same rule), then the stratification number of P must be greater than or equal to the stratification number of Q, in short S ( P ) ≥ S ( Q ) {\displaystyle S(P)\geq S(Q)} . If a predicate P is derived from a negated predicate Q (i.e., P is the head of a rule, and Q occurs negatively in the body of the same rule), then the stratification number of P must be greater


id: strict-20260505-0
file: data/en.wikipedia.org/wiki/Strict-0.md
category: reference
title: Strict
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Strict
date_saved: 2026-05-05T07:25:12.769533+00:00
instance: kb-cron
parent: strict-20260505-0
summary:
In mathematical writing, the term strict refers to the property of excluding equality and equivalence and often occurs in the context of inequality and monotonic functions. It is often attached to a technical term to indicate that the exclusive meaning of the term is to be understood. The opposite is non-strict, which is often understood to be the case but can be put explicitly for clarity. In some contexts, the word "proper" can also be used as a mathematical synonym for "strict". == Use == This term is commonly used in the context of inequalities — the phrase "strictly less than" means "less than and not equal to" (likewise "strictly greater than" means "greater than and not equal to"). More generally, a strict partial order, strict total order, and strict weak order exclude equality and equivalence. When comparing numbers to zero, the phrases "strictly positive" and "strictly negative" mean "positive and not equal to zero" and "negative and not equal to zero", respectively. In the context of functions, the adverb "strictly" is used to modify the terms "monotonic", "increasing", and "decreasing". On the other hand, sometimes one wants to specify the inclusive meanings of terms. In the context of comparisons,


id: t-c-mits-20260505-0
file: data/en.wikipedia.org/wiki/T.C._Mits-0.md
category: reference
title: T.C. Mits
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/T.C._Mits
date_saved: 2026-05-05T07:25:14.020975+00:00
instance: kb-cron
parent: t-c-mits-20260505-0
summary:
T.C. Mits (acronym for "the celebrated man in the street"), is a term coined by Lillian Rosanoff Lieber to refer to an everyman. In Lieber's works, T.C. Mits was a character who made scientific topics more approachable to the public audience. The phrase has enjoyed sparse use by authors in fields such as molecular biology, secondary education, and general semantics. == The Education of T.C. MITS == Dr. Lillian Rosanoff Lieber wrote this treatise on mathematical thinking in twenty chapters. The writing took a form that resembled free-verse poetry, though Lieber included an introduction stating that the form was meant only to facilitate rapid reading, rather than emulate free-verse. Lieber's husband, a fellow professor at Long Island University, Hugh Gray Lieber, provided illustrations for the book. The title of the book was meant to emphasize that mathematics can be understood by anyone, which was further shown when a special "Overseas edition for the Armed Forces" was published in 1942, and approved by the Council on Books in Wartime to be sent to American troops fighting in World War II. == See also == John Doe The man on the Clapham omnibus == References ==


id: the-politics-of-modelling-numbers-between-science-and-policy-20260505-0
file: data/en.wikipedia.org/wiki/The_Politics_of_Modelling,_Numbers_Between_Science_and_Policy-0.md
category: reference
title: The Politics of Modelling, Numbers Between Science and Policy
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/The_Politics_of_Modelling,_Numbers_Between_Science_and_Policy
date_saved: 2026-05-05T07:25:15.216481+00:00
instance: kb-cron
parent: the-politics-of-modelling-numbers-between-science-and-policy-20260505-0
summary:
The Politics of Modelling, Numbers Between Science and Policy is a multi-authors book edited by Andrea Saltelli and Monica Di Fiore and published in August 2023 by Oxford University Press. == Synopsis == The Politics of Modelling elaborates and expands on themes of responsible modelling from a manifesto published in the journal Nature in 2020. The text is structured into three main sections: Meeting Models, The Rules, and The Rules in Practice. The combination of theory with policy relevant examples makes the book accessible to modellers, researchers of modelling, and policy makers. The volume comes with a foreword of Wendy Nelson Espeland and a preface of Daniel Sarewitz, with chapters from Andy Stirling, Wolfgang Drechsler, Philip B. Stark, Ting Xu, Paolo Vineis, Andrea Saltelli, and other scholars. == Reception == The book argues that models live in a "state of exception" provided by their access to a wealth of methodology of analysis, and by their epistemic authority borrowed from mathematics. This state allows models to better defend an appearance of neutrality that is appreciated by policymakers in search of a justification. A review published in the Science notes that the volume incorporated insights from science and technology studies to explore


id: theorem-20260505-0
file: data/en.wikipedia.org/wiki/Theorem-0.md
category: reference
title: Theorem
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Theorem
date_saved: 2026-05-05T07:25:16.454658+00:00
instance: kb-cron
parent: theorem-20260505-0
summary:
In mathematics and formal logic, a theorem is a statement that has been proven, or can be proven. The proof of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems. In mainstream mathematics, the axioms and the inference rules are commonly left implicit, and, in this case, they are almost always those of Zermelo–Fraenkel set theory with the axiom of choice (ZFC), or of a less powerful theory, such as Peano arithmetic. Generally, an assertion that is explicitly called a theorem is a proved result that is not an immediate consequence of other known theorems. Moreover, many authors qualify as theorems only the most important results, and use the terms lemma, proposition and corollary for less important theorems. In mathematical logic, the concepts of theorems and proofs have been formalized in order to allow mathematical reasoning about them. In this context, statements become well-formed formulas of some formal language. A theory consists of some basis statements called axioms, and some deducing rules (sometimes included in the axioms). The theorems of the theory are the statements that can be


id: theorem-20260505-1
file: data/en.wikipedia.org/wiki/Theorem-1.md
category: reference
title: Theorem
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Theorem
date_saved: 2026-05-05T07:25:16.454658+00:00
instance: kb-cron
parent: theorem-20260505-0
summary:
== Epistemological considerations == Many mathematical theorems are conditional statements, whose proofs deduce conclusions from conditions known as hypotheses or premises. In light of the interpretation of proof as justification of truth, the conclusion is often viewed as a necessary consequence of the hypotheses. Namely, that the conclusion is true in case the hypotheses are true—without any further assumptions. However, the conditional could also be interpreted differently in certain deductive systems, depending on the meanings assigned to the derivation rules and the conditional symbol (e.g., non-classical logic). Although theorems can be written in a completely symbolic form (e.g., as propositions in propositional calculus), they are often expressed informally in a natural language such as English for better readability. The same is true of proofs, which are often expressed as logically organized and clearly worded informal arguments, intended to convince readers of the truth of the statement of the theorem beyond any doubt, and from which a formal symbolic proof can in principle be constructed. In addition to the better readability, informal arguments are typically easier to check than purely symbolic ones—indeed, many mathematicians would express a preference for a proof that not only demonstrates the validity of a theorem, but


id: theorem-20260505-2
file: data/en.wikipedia.org/wiki/Theorem-2.md
category: reference
title: Theorem
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Theorem
date_saved: 2026-05-05T07:25:16.454658+00:00
instance: kb-cron
parent: theorem-20260505-0
summary:
Nonetheless, there is some degree of empiricism and data collection involved in the discovery of mathematical theorems. By establishing a pattern, sometimes with the use of a powerful computer, mathematicians may have an idea of what to prove, and in some cases even a plan for how to set about doing the proof. It is also possible to find a single counter-example and so establish the impossibility of a proof for the proposition as-stated, and possibly suggest restricted forms of the original proposition that might have feasible proofs. For example, both the Collatz conjecture and the Riemann hypothesis are well-known unsolved problems; they have been extensively studied through empirical checks, but remain unproven. The Collatz conjecture has been verified for start values up to about 2.88 × 1018. The Riemann hypothesis has been verified to hold for the first 10 trillion non-trivial zeroes of the zeta function. Although most mathematicians can tolerate supposing that the conjecture and the hypothesis are true, neither of these propositions is considered proved. Such evidence does not constitute proof. For example, the Mertens conjecture is a statement about natural numbers that is now known to be false, but no explicit counterexample (i.e., a natural number


id: theorem-20260505-3
file: data/en.wikipedia.org/wiki/Theorem-3.md
category: reference
title: Theorem
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Theorem
date_saved: 2026-05-05T07:25:16.454658+00:00
instance: kb-cron
parent: theorem-20260505-0
summary:
Theorem (name of the person who proved it, along with year of discovery or publication of the proof) Statement of theorem (sometimes called the proposition) Proof Description of proof End The end of the proof may be signaled by the letters Q.E.D. (quod erat demonstrandum) or by one of the tombstone marks, such as "□" or "∎", meaning "end of proof", introduced by Paul Halmos following their use in magazines to mark the end of an article. The exact style depends on the author or publication. Many publications provide instructions or macros for typesetting in the house style. It is common for a theorem to be preceded by definitions describing the exact meaning of the terms used in the theorem. It is also common for a theorem to be preceded by a number of propositions or lemmas which are then used in the proof. However, lemmas are sometimes embedded in the proof of a theorem, either with nested proofs, or with their proofs presented after the proof of the theorem. Corollaries to a theorem are either presented between the theorem and the proof, or directly after the proof. Sometimes, corollaries have proofs of their own that explain why they follow


id: theorem-20260505-4
file: data/en.wikipedia.org/wiki/Theorem-4.md
category: reference
title: Theorem
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Theorem
date_saved: 2026-05-05T07:25:16.454658+00:00
instance: kb-cron
parent: theorem-20260505-0
summary:
=== Works cited === Boolos, George; Burgess, John; Jeffrey, Richard (2007). Computability and Logic (5th ed.). Cambridge University Press. Enderton, Herbert (2001). A Mathematical Introduction to Logic (2nd ed.). Harcourt Academic Press. Heath, Sir Thomas Little (1897). The works of Archimedes. Dover. Retrieved 2009-11-15. Hedman, Shawn (2004). A First Course in Logic. Oxford University Press. Hinman, Peter (2005). Fundamentals of Mathematical Logic. Wellesley, MA: A K Peters. Hoffman, Paul (1998). The Man Who Loved Only Numbers: The Story of Paul Erdős and the Search for Mathematical Truth. Hyperion, New York. ISBN 1-85702-829-5. Hodges, Wilfrid (1993). Model Theory. Cambridge University Press. Johnstone, P. T. (1987). Notes on Logic and Set Theory. Cambridge University Press. Monk, J. Donald (1976). Mathematical Logic. Springer-Verlag. Petkovsek, Marko; Wilf, Herbert; Zeilberger, Doron (1996). A = B (PDF). A.K. Peters, Wellesley, Massachusetts. ISBN 1-56881-063-6. Rautenberg, Wolfgang (2010). A Concise Introduction to Mathematical Logic (3rd ed.). Springer. van Dalen, Dirk (1994). Logic and Structure (3rd ed.). Springer-Verlag. Wentworth, G.; Smith, D.E. (1913). Plane Geometry. Ginn & Co. == Further reading == Chiswell, Ian; Hodges, Wilfred (2007). Mathematical Logic. Oxford University Press. Hunter, Geoffrey (1996) [1971]. Metalogic: An Introduction to the Metatheory of Standard First-Order Logic. University of California


id: toy-model-20260505-0
file: data/en.wikipedia.org/wiki/Toy_model-0.md
category: reference
title: Toy model
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Toy_model
date_saved: 2026-05-05T07:25:17.746720+00:00
instance: kb-cron
parent: toy-model-20260505-0
summary:
In scientific modeling, a toy model is a deliberately simplistic model with many details removed so that it can be used to explain a mechanism concisely. It is also useful in a description of the fuller model. In toy mathematical models used in mathematical physics, this is usually done by reducing or extending the number of dimensions or reducing the number of fields/variables or restricting them to a particular symmetric form. In toy economic models, some may be only loosely based on theory, others more explicitly so. They allow for a quick first pass at some question, and present the essence of the answer from a more complicated model or from a class of models. For the researcher, they may come before writing a more elaborate model, or after, once the elaborate model has been worked out. Blanchard's list of examples includes the IS–LM model, the Mundell–Fleming model, the RBC model, and the New Keynesian model. The phrase "tinker-toy model" is also used, in reference to the Tinkertoys product used for children's constructivist learning. == Examples == Examples of toy models in physics include: the Ising model as a toy model for ferromagnetism, or lattice models more generally. It is


id: toy-theorem-20260505-0
file: data/en.wikipedia.org/wiki/Toy_theorem-0.md
category: reference
title: Toy theorem
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Toy_theorem
date_saved: 2026-05-05T07:25:18.952329+00:00
instance: kb-cron
parent: toy-theorem-20260505-0
summary:
In mathematics, a toy theorem is a simplified instance (special case) of a more general theorem, which can be useful in providing a handy representation of the general theorem, or a framework for proving the general theorem. One way of obtaining a toy theorem is by introducing some simplifying assumptions in a theorem. In many cases, a toy theorem is used to illustrate the claim of a theorem, while in other cases, studying the proofs of a toy theorem (derived from a non-trivial theorem) can provide insight that would be hard to obtain otherwise. Toy theorems can also have educational value as well. For example, after presenting a theorem (with, say, a highly non-trivial proof), one can sometimes give some assurance that the theorem really holds, by proving a toy version of the theorem. == Examples == A toy theorem of the Brouwer fixed-point theorem is obtained by restricting the dimension to one. In this case, the Brouwer fixed-point theorem follows almost immediately from the intermediate value theorem. Another example of toy theorem is Rolle's theorem, which is obtained from the mean value theorem by equating the function values at the endpoints. == See also == Corollary Fundamental theorem Lemma


