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Lilienfeld radiation, named after Julius Edgar Lilienfeld , is electromagnetic radiation produced when electrons hit a metal surface. [ 1 ]
The Smith–Purcell effect is believed to be a variant of Lilienfeld radiation. [ 2 ]
Lilienfeld radiation is shown as Transition radiation by Vitaly Ginzburg and Ilya Frank in 1945 [ 3 ] [ 4 ]
This particle physics –related article is a stub . You can help Wikipedia by expanding it .
This electromagnetism -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Lilienfeld_radiation |
Lilingyan ( Chinese : 戾陵堰 ; pinyin : lìlíngyàn ) was an ancient irrigation system built in 250 AD during the Three Kingdoms period to irrigate the Beijing Plain around Jicheng (modern-day Beijing ). The irrigation system consisted of Lilingyan, a dam across the Shishui ( Yongding River ) at the foot of Liangshan ( Shijingshan ), and the Chexiangqu, a diversion channel that carried the water west into the Beijing Plain. The diversion channel fed water to the fields north, east and southeast of Jicheng and emptied into the Gaoliang River, which flowed back into the Yongding River. The irrigation system was an important development in Beijing's early history and helped increase food production and population in and around the city. [ 1 ] Lilingyan was named after Liling , the tomb of Liu Dan, Prince of Yan, who was buried in Liangshan after his death in 80 BC. [ 2 ]
Lilingyan was built in 250 AD by Liu Jing, a military commander of Youzhou , a prefecture in north China based in Jicheng . [ 1 ] Liu Jing was the son of Liu Fu, who had built irrigation systems along the Huai River . [ 3 ] As the commander of the local garrison, Liu Jing had to purchase grain from afar due to limited food production locally. [ 3 ] To improve local agricultural output, Liu Jing diverted the waters of the Yongding River, which flowed south of Jicheng to irrigate the plains to the north of the Yongding River. [ 3 ] He deployed soldiers to dam the Yongding in the hills west of Ji and channeled the water eastward along the Chexiangqu into the Beijing Plain. [ 3 ] The dam was 2.4 meters high and piled from woven baskets of stone. [ 3 ] To withstand the flash flood of theYongding as the river flows out of the Western Hills , the dam was 72 meters thick and gently sloped so that flood water can flow over the dam. The Chexiangqu made use of the old river bed of the Yongding, which used to flow north of Jicheng before it changed course and flowed south of the city. [ 1 ]
In 262, a sluice gate was added to control the flow of water into the diversion channel. [ 3 ] The irrigation system greatly improved agricultural output of the region and helped increase the population of Ji. [ 3 ] In 295, Liu Jing's son, Liu Hong, repaired and expanded the irrigation system by extending the aqueduct further east to modern-day Tongzhou . [ 3 ]
During the Northern Qi dynasty , the irrigation system was repaired in 519 and further expanded in 565 by connecting channels to the Sha River further north. [ 3 ] The system was again repaired in 650-655 during the Tang dynasty , and later became part of subsequent irrigation systems of Beijing. [ 3 ] Much of the land irrigated by the Lilingyan is now under urban Beijing, after the city shifted northward from Jicheng to Dadu in the 13th century. | https://en.wikipedia.org/wiki/Lilingyan |
In mathematics , Lill's method is a visual method of finding the real roots of a univariate polynomial of any degree . [ 1 ] It was developed by Austrian engineer Eduard Lill in 1867. [ 2 ] A later paper by Lill dealt with the problem of complex roots. [ 3 ]
Lill's method involves drawing a path of straight line segments making right angles , with lengths equal to the coefficients of the polynomial. The roots of the polynomial can then be found as the slopes of other right-angle paths, also connecting the start to the terminus, but with vertices on the lines of the first path.
To employ the method, a diagram is drawn starting at the origin. A line segment is drawn rightwards by the magnitude of the leading coefficient, so that with a negative coefficient, the segment will end left of the origin. From the end of the first segment, another segment is drawn upwards by the magnitude of the second coefficient, then left by the magnitude of the third, then down by the magnitude of the fourth, and so on. The sequence of directions (not turns) is always rightward, upward, leftward, downward, then repeating itself. Thus, each turn is counterclockwise. The process continues for every coefficient of the polynomial, including zeros, with negative coefficients "walking backwards." The final point reached, at the end of the segment corresponding to the equation's constant term, is the terminus.
A line is then launched from the origin at some angle θ , reflected off of each line segment at a right angle (not necessarily the "natural" angle of reflection), and refracted at a right angle through the line through each segment (including a line for the zero coefficients) when the angled path does not hit the line segment on that line. [ 4 ] The vertical and horizontal lines are reflected off or refracted through in the following sequence: the line containing the segment corresponding to the coefficient of x n −1 , then of x n −2 etc. Choosing θ so that the path lands on the terminus, −tan( θ ) is a root of this polynomial. For every real zero of the polynomial, there will be one unique initial angle and path that will land on the terminus. A quadratic with two real roots, for example, will have exactly two angles that satisfy the above conditions.
For complex roots, one must also find a series of similar triangles, but with the vertices of the root path displaced from the polynomial path by a distance equal to the imaginary part of the root. In this case, the root path will not be rectangular. [ 5 ] [ 3 ]
The construction in effect evaluates the polynomial according to Horner's method . For the polynomial a n x n + a n − 1 x n − 1 + a n − 2 x n − 2 + ⋯ {\displaystyle a_{n}x^{n}+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+\cdots } , the values of a n x {\displaystyle a_{n}x} , ( a n x + a n − 1 ) x {\displaystyle (a_{n}x+a_{n-1})x} , ( ( a n x + a n − 1 ) x + a n − 2 ) x {\displaystyle ((a_{n}x+a_{n-1})x+a_{n-2})x} , ... are successively generated as distances between the vertices of the polynomial and root paths. For a root of the polynomial, the final value is zero, so the last vertex coincides with the polynomial path terminus.
A solution line giving a root is similar to the Lill's construction for the polynomial with that root removed, because the visual construction is analogous to the synthetic division of the polynomial by a linear (root) monic ( Ruffini's rule ).
From the symmetry of the diagram, it can easily be seen that the roots of the reversed polynomial are the reciprocals of the original roots.
The construction can also be done using clockwise turns instead of counterclockwise turns. When a path is interpreted using the other convention, it corresponds to the mirrored polynomial (every odd coefficient's sign is changed), and the roots are negated.
When the right-angle path is traversed in the other direction but with the same direction convention, it corresponds to the reversed mirrored polynomial, and the roots are the negative reciprocals of the original roots. [ 4 ]
Lill's method can be used with Thales's theorem to find the real roots of a quadratic polynomial.
In this example with 3 x 2 + 5 x − 2 , the polynomial's line segments are first drawn in black, as above. A circle is drawn with the straight line segment joining the start and end points forming a diameter.
According to Thales's theorem, the triangle containing these points and any other point on the circle is a right triangle . Intersects of this circle with the middle segment of Lill's method, extended if needed, thus define the two angled paths in Lill's method, colored blue and red.
The negative of the gradients of their first segments, m , yield the real roots 1/3 and −2 .
In 1936, Margherita Piazzola Beloch showed how Lill's method could be adapted to solve cubic equations using paper folding . [ 6 ] If simultaneous folds are allowed, then any n th-degree equation with a real root can be solved using n − 2 simultaneous folds. [ 7 ]
In this example with 3 x 3 + 2 x 2 − 7 x + 2 , the polynomial's line segments are first drawn on a sheet of paper (black). Lines passing through reflections of the start and end points in the second and third segments, respectively (faint circle and square), and parallel to them (grey lines), are drawn.
For each root, the paper is folded until the start point (black circle) and end point (black square) are reflected onto these lines. The axis of reflection (dash-dot line) defines the angled path corresponding to the root (blue, purple, and red). The negative of the gradients of their first segments, m , yield the real roots 1/3 , 1 , and −2 . | https://en.wikipedia.org/wiki/Lill's_method |
Lillie's trichrome is a combination of dyes used in histology .
It is similar to Masson's trichrome stain , but it uses Biebrich scarlet for the plasma stain. It was initially published by Ralph D. Lillie in 1940. [ 1 ] It is applied by submerging the fixated sample into the following three solutions: [ 2 ] Weigert's iron hematoxylin working solution, Biebrich scarlet solution, and Fast Green FCF solution.
The resulting stains are black cell nuclei, brown cytoplasm, red muscle and myelinated fibers, blue collagen, and scarlet erythrocytes. [ 3 ]
Trichrome stains are normally used to differentiate between collagen and muscle tissues. Some studies that benefit from its application include end stage liver disease ( cirrhosis ), myocardial infarction , muscular dystrophy , and tumor analysis. [ 4 ]
This biochemistry article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Lillie's_trichrome |
The Lilliput effect is an observed decrease in animal body size in genera that have survived a major extinction . [ 1 ] There are several hypotheses as to why these patterns appear in the fossil record , some of which are: [ 2 ]
The term was coined in by Urbanek (1993) in a paper concerning the end-Silurian extinction of graptoloids [ 3 ] and is derived from an island in Gulliver’s Travels , Lilliput , inhabited by a race of miniature people. The size decrease may just be a temporary phenomenon restricted to the survival period of the extinction event. Atkinson et al. (2019) coined the term Brobdingnag effect [ 4 ] to describe a related phenomenon, operating in the opposite direction, whereby new species evolving after the Triassic-Jurassic mass extinction that began the period with small body sizes underwent substantial size increases. [ 4 ] The term is also from Gulliver's Travels , where Brobdingnag is a place inhabited by a race of giants.
Trends in body size changes are seen throughout the fossil record in many organisms, and major changes (shrinking and dwarfing) in body size can significantly affect the morphology of the animal itself as well as how it interacts with the environment. [ 2 ] Since Urbanek's publication several researchers have described a decrease in body size in fauna post-extinction event, although not all use the term "Lilliput effect" when discussing this trend in body size decrease. [ 5 ] [ 6 ] [ 7 ]
The Lilliput effect has been noted by several authors to have occurred after the Permian-Triassic mass extinction : Early Triassic fauna, both marine and terrestrial, is notably smaller than those preceding and following in the geologic record. [ 1 ]
The extinction event may have been more severe for the larger-bodied species, leaving only species of smaller-bodied animals behind. [ 1 ] As such, organisms in the smaller species which then make up the recovering ecosystem, will take time to evolve larger bodies to replace the extinct species and re-occupy the vacant ecological niche for a large-bodied animal. [ 1 ] Taxa whose animals are larger may be evolutionarily selected against for several reasons, including [ 1 ]
Stanley (1973) hypothesized that newly emerged animal taxa tend to develop at an originally small size, hence a sudden proliferation of new species would tend to produce many initially small organisms. [ 8 ]
It is possible that the extinction event selectively removed larger individuals within any single lineage, without extinguishing the entire species, but leaving as survivors only the individuals with a naturally smaller body size. The smaller survivors then form the new breeding population, and pass on that trait to their descendents. Because of the selection during the extinction, compared to the previously "normal"-sized members of the species who lived before the extinction event occurred, later members of that species living after the extinction, who are descended only from the smaller survivors, would be reduced in size, constituting a "new-normal". [ 1 ] | https://en.wikipedia.org/wiki/Lilliput_effect |
The Lily mottle virus (LMoV) , is a plant virus of the Potyviridae virus family that causes asymptomatic to mild diseases of individual plant parts in plants of the lily family ( Liliaceae ). However, a frequently occurring simultaneous infection with other plant viruses, which on their own only cause moderate or no disease, can cause an entire plant to perish. This coinfection leads to considerable crop damage in lily cultivation and is therefore of great economic importance. Lily mottle virus is spread by aphids and in horticulture during vegetative propagation by splitting the lily bulb . LMoV was regarded as a synonym for a subtype of the Tulip Breaking Virus (TBV) that occurs in lilies, although since 2005 it has been classified as a closely related but independent virus species of the genus Potyvirus .
The symptoms of the plant disease caused by LMoV were already known in the 19th century. Yet it was not until 1944 that P. Brierley and F. F. Smith succeeded in proving a coinfection with two viruses as the cause through infection experiments on several tulip and lily species. [ 1 ] They were able to detect the Lily symptomless virus (LSV, order Tymovirales : Betaflexiviridae : Carlavirus ) in several lily species grown in the USA ( Lilium auratum , L. speciosum , L. longiflorum ), which showed streaky brightening ( chlorosis ) or individual necrotic spots on the leaves, which was always present simultaneously with the Cucumber mosaic virus (CMV) or the Lily mottle virus. [ 2 ] They were also able to demonstrate that all three viruses are transmitted by aphids of the species Aphis gossypii .
Virus particles (virions) of the Lily mottle virus consist of a thread-like capsid with helical symmetry , in which a single-stranded RNA is packed as a genome ; a viral envelope is not present. [ 3 ] The capsid is 13 nm thick and about 740 nm long. The length of the capsid increases in the presence of divalent cations (especially calcium ions ) in the preparation and decreases after binding by the addition of EDTA . The individual capsomeres that make up the capsid require a pitch of 3.4 nm for one helix turn . Compared to viruses with rigid rods and a comparable structure (e.g. the Tobacco mosaic virus- TMV ), this duct height is relatively large and enables the LMoV capsid to be flexible and bendable. One turn requires 7.7 capsomeres, so that the entire capsid is composed of about 1700 capsomeres. [ 4 ] The individual capsomeres consist of only one molecule of the LMoV capsid protein (CP, coat/capsid protein) with a length of 274 amino acids (33 kDa). The CP is folded several times in such a way that the N- and C-terminus point outwards. These outer ends of the capsid protein are very variable. The protruding N-terminus mainly determines the specific attachment to the host cell and enables the serological differentiation of different virus isolates . The highly conserved sections in the middle of the CP (216 amino acids) within the different members of the Potyviridae point inwards in the capsid and interact with the viral RNA. [ 5 ]
The virions are stable against ethanol and lose their infectivity in the plant sap only after 10 minutes at 65-70 °C. [ 1 ] The LMoV has a density of 1.31 g/ml in density gradient centrifugation ( caesium chloride ) and a sedimentation coefficient of 137 to 160 S . [ 6 ]
The genome of LMoV is a linear, single-stranded RNA with positive polarity [(+)ssRNA] and a length of 9644 nucleotides . A viral protein (VPg) is covalently bound to the 5'-end of the RNA. As with cellular messenger RNAs , a poly(A) tail of 20 to 160 adenosines is located at the 3' end of the viral genome. Between the two non- coding ends (NCR: non-coding region) is an open reading frame (ORF), which codes for a polyprotein of 3095 amino acids. This polyprotein is cleaved into the individual viral proteins by proteases during translation.
An IRES structure was suspected in the 5'-NCR of potyviruses, as translation is initiated without a 5'-cap structure. [ 7 ] LMoV does not have a cap structure, nor could an IRES be confirmed from sequence data. The VPg protein bound to the 5'-NCR possibly serves as a primer for the RNA polymerase to amplify the RNA. However, the VPg of other potyviruses also interacts directly with the translation initiation factors eIF4E and eIFiso4E. [ 8 ] This could represent an as yet uncharacterized, Cap- and IRES-independent translation pathway. [ 9 ]
After infection, the virus enters the plant via the vascular bundle and is taken up by the cells through membrane vesicles ( endocytosis ). In the cytoplasm , the capsid disintegrates and the RNA is released. The viral RNA can also enter the cell very effectively via infected neighboring cells through cell contact sites ( plasmodesmata ). This direct transport of naked, infectious RNA is controlled by several viral proteins, including the so-called HC (helper component), which form a so-called movement complex. As with all (+)ssRNA viruses , the ingested RNA is first translated into protein at the ribosomes , as at least one copy of the viral RNA-dependent RNA polymerase is required for the replication of the RNA. After this has synthesized several copies of the viral RNA, the LMoV proteins are produced in large quantities. These accumulate at the synthesis sites of the viroplasm to form morphologically visible inclusion bodies . When infected with LMoV, these inclusion bodies have a characteristic, cylindrical to spiral-like shape in the cytoplasm; the virus protein that predominantly forms these cylinders is therefore also referred to as CI (cylindrical inclusion). Amorphous inclusion bodies are formed in the cell nucleus, which consist of two viral proteins NIa and NIb (nuclear inclusions). As the viral proteins are always formed in the same ratio during translation of the RNA and larger quantities of the capsid protein are required in comparison to other proteins, these proteins, which are not required in many copies, form inclusion bodies, are degraded or excreted from the cell.
The LMoV polyprotein is cleaved into eight individual proteins by viral proteases. At the N-terminus, viral protease 1 (P1) cleaves itself from the polyprotein. Next comes the HC protein, which is important for transmission by aphids; however, the mechanism is unclear. The HC has a papain -like protein domain at the C-terminus, with which the HC also splits off independently from the polyprotein. All other proteins are cleaved by the NIa protease. This is followed by a further protease (P3) with an as yet unknown function and the CI, from which a small peptide 6K1 is cleaved (possibly for activation). The CI is active as a helicase during RNA replication. Together with a protease component, the VPg forms the NIa. The NIb is the viral RNA polymerase from which the viral capsid protein CP is cleaved. Once sufficient viral (+)ssRNA and CP have been formed, packaging into the capsid can take place and mature viruses can be released into the plant sap by exocytosis . The much more effective infection of the naked RNA from cell to cell explains the appearance of patchy lesions on the leaves.
The genus Potyvirus is currently the largest group of all plant viruses with 168 virus species. [ 10 ] This large number of Potyviruses makes it difficult to distinguish and delimit individual species or subtypes, especially the Lily mottle virus and the Tulip breaking virus (TBV), which were long considered synonyms of a single species. LMoV was regarded as the subtype of TBV (TBV subtype Lily) that was widespread in lilies. This distinction was made even more complex by the fact that the true species TBV can also cause disease in lilies. With more and more comparative sequences of the genome of different virus isolates, incorrect assignments have so far been detected. [ 11 ] In a study of 187 complete genome sequences and 1220 partial sequences for the capsid protein of potyviruses, several subgroups within the genus were identified and the criteria for the species limits were also redefined for LMoV and TBV. [ 12 ] Accordingly, a match in the nucleotide sequence between two complete genomes of more than 76% is considered a species limit (corresponds to 82% match in the amino acid sequence). The part of the nucleotide sequence coding for the capsid protein CP showed a species limit of 76-77%. The sequence of the CI protein appeared to be the most suitable for differentiation. Several sequences of potyviruses (including TBV and LMoV), which were published in the international gene bank GenBank , had to be assigned to other species as a result.
The taxonomy defined by the " International Committee on Taxonomy of Viruses " and valid since 2005 includes subtypes of LMoV previously classified as TBV: [ 13 ]
About two weeks after being infected with LMoV, a light green mottle appears on young leaves . The discoloration can also appear in stripes along the leaf veins . Over the course of a few days, the leaf becomes thinner in the light spots and in severe cases the plant cells can die in these areas; the irregularly defined spots now appear dark brown and dried out. All new shoots and flowers that sprout after infection are reduced in size and often deformed.
However, the severity of the disease symptoms varies greatly between different lily species and hybrids . Even the disease of identical species in a single cultivation area varies in severity. This phenomenon can be explained by the influence of the growth phase at the time of infection, the point of entry and the infectious dose of the virus. In the Easter lily ( L. longiflorum ), no disease regularly develops, although the virus multiplies in the plant. In the tiger lily ( L. lancifolium ), only a very slight light green mottling occurs. In some LMoV infections, only reduced length growth and smaller flowers and bulbs can be observed. The economically important species L. formosanum always falls ill after an LMoV infection; this also applies to the wild varieties found in Taiwan . Only the specially selected variety Lilium formosanum "Little Snow White" has increased virus resistance . Very susceptible to LMoV and other plant viruses found in lilies is the hybrid "Enchantment" bred by Jan de Graaff in 1941 and all cultivars derived from it, such as the cultivar Lilium Asia . Hybrid cv. Enchantment. [ 11 ]
Infection with LMoV alone never leads to the death of the entire plant, but remains locally limited to some parts of the plant. However, a coinfection of LMoV with the Lily symptomless virus is particularly common, which alone does not cause any symptoms of the disease, but only reduced plant growth. If a plant is infected by both viruses, the disease progresses much more severely and quickly. After the initial typical symptoms of a pronounced LMoV infection, larger vascular bundles such as the entire phloem are affected, which ultimately causes the entire plant to die. A double-infected lily bulb can already be severely damaged during storage, lose its ability to sprout and die.
Reed aphids ingesting plant sap]]
The Lily Mottle virus is transmitted during the feeding act of aphids ( Aphididae ). The aphids ingest the virus, which is present in high concentrations in the plant sap, during the feeding act and can infect other plants with a delay of a few hours. The virus cannot multiply in the aphid itself. After absorption of the plant sap into the aphid's midgut, the virus is distributed in the bloodstream and enters the saliva of the sucking apparatus; a new plant can then be infected during the next feeding. The aphid species that predominantly transmit LMoV are Aphis gossypii , Myzus persicae , Macrosiphum euphorbiae and Doralis fabae . Stored bulbs can also be infected with the virus by Anuraphis (Yezabura) tulipae . [ 15 ] Winged specimens of the aphid population enable transmission over long distances.
During plant cultivation, the virus is transmitted when the plants are cut and injured with contaminated knives and scissors. Experimentally, this route of infection is used by targeted scoring of the plants. Splitting the lily bulbs during vegetative propagation spreads the virus to all daughter plants. The same applies to vegetative propagation by cuttings in tissue culture, which is very common in industrial horticulture. The virus is not spread by seeds; if a new plant germinates from the seed of an LMoV-infected plant, it is not infected.
The natural geographical spread of the virus is not known, as when it was discovered in the US in 1944, it was already being spread by man through the global trade in flowers and bulbs. The cultivation of lilies in large greenhouses and fields as a monoculture particularly favors transmission compared to the natural occurrence of wild plants. The virus is spread worldwide and is endemic in countries with significant lily cultivation. In addition to the United States, this includes the Netherlands , Poland , North and South Korea , Japan , Taiwan, China and Israel . The Lily mild mottle virus, a subtype of LMoV, was detected in 26.3% of all plants in a study of 185 lily samples from South Korean crops, and a co-infection of LMoV and the Tomato ringspot virus was observed in a further 23.2%. [ 16 ]
In the Netherlands, LMoV was detected several times in all plants of individual lily fields of the cultivar "Enchantment". [ 17 ] Often there was also an infection with the Lily symptomless virus. In plantations affected in this way, necrosis of the stem and leaves is increasingly observed, which is usually followed by the death of the plant. If all the lilies in a plantation are only infected with LMoV, this does not usually result in the loss of the entire flower crop; smaller flowers or plants with reduced growth are then offered at lower prices.
LMoV was detected in all of the approximately 340 lily cultivars grown on a large scale. [ 18 ] The undetected spread through worldwide transportation is particularly prevalent in those lily species that show no or only minor symptoms of infection, but can propagate the virus, such as the Easter lily and Tiger lily. The virus has a wider host range than was assumed in earlier studies. For example, LMoV was also detected in the winter endive ( C. endivia L. var. latifolium Lam.). [ 19 ]
The spread of LMoV in industrial production is primarily prevented by controlling the aphids as carriers. [ 18 ] The virus is mainly transmitted by spreading aphid populations in June and July, less so in May and August. Weekly control of the insects from May and biweekly in August and September is carried out on an industrial scale. The lilies are most often treated with kerosene oil or pyrethroids as aerosols .
To prevent infection, it is important to avoid further spread through seed bulbs and the global plant trade. Those lily species with no or only mild symptoms are a particular source of infection outbreaks, as the infection remains undetected. For this reason, the simultaneous cultivation of resistant and susceptible lily varieties is often avoided, as the virus can spread unnoticed in the resistant varieties without developing disease symptoms. These form a permanent reservoir for the infection of the susceptible varieties. In a monoculture of susceptible varieties, infected plants can be sorted out and thus the spread of the virus can be controlled to a certain extent. As the virus is not transmitted by seed like other members of the Potyvirus genus, a crop can be freed from infection with LMoV by more complex, renewed breeding from seed.
The transportation and trade of plant parts such as flowers, cuttings or bulbs from cultivation areas in which LMoV has been detected is subject to legal restrictions or an import ban in many countries. In particular, plant parts traded for propagation and breeding have had to be tested for LMoV in Germany since 1998 in accordance with the implementation of several EU directives . [ 20 ] To detect LMoV, immunological tests for LMoV virus proteins ( ELISA ) and, rarely, detection of the virus genome by PCR are used. Both the leaves ("leaf test") and the harvested bulbs ("bulb test") are used as test samples for diagnostics. Newer methods for the simultaneous detection of several plant viruses from one sample by DNA hybridization (macroarray) are currently being tested. [ 21 ] | https://en.wikipedia.org/wiki/Lily_mottle_virus |
A Liman (from Greek λιμήν ) in Israel is the name for an artificial earthen construction used to collect floodwater by damming a desert wadi . The runoff water is slowed by the dam, thus flooding a small area and allowing the water to infiltrate into the soil. This way, a small groves of trees can be sustained in the desert. The JNF-KKL has been funding the construction of limans in the Negev Desert . [ 1 ]
Limans were built in order to fight desertification without depleting groundwater resources, which are becoming increasingly rare in arid ecosystems. [ 2 ] Remaining soil humidity can be found in dry river beds ( wadis ) after rains occur, but these wadis are prone to flash floods . The result is massive soil erosion and the destruction of infrastructure. Also, the infiltration is insufficient because of the water's velocity, even though the runoff would be able to allow for the growth of trees in appropriate places. [ 3 ] The aim of building limans is to stop flash floods and to increase water infiltration, thus sustaining the growth of drought-hardy tree species and vegetation underneath them.
Limans are structures with small dams which catch runoff from a wadi to hold about 400-600mm of water, which suffices for the growth of drought-hardy tree species. [ 3 ] Limans can be built wherever tributary wadis [...] widen or come onto a large plain with potential arable land [...]. A check-dam [...] is built to retain runoff waters [...]. A spillway regulates the level of the water [...] to prevent the destruction of the check-dam . . [ 4 ] The embankment height should be 3–4 times the designed water depth, and the outlet should be to the side of the main flow to prevent direct through flow. Also, grazers should be excluded from the site to prevent soil compaction which would in turn decrease water infiltration . [ 3 ]
Because of its fast growth, Eucalyptus occidentalis was mostly used in the past. [ 3 ] However, some scientific studies have had better results with Eucalyptus sargentii . [ 5 ] Overall, any drought-hardy species are suitable, such as tamarind , acacia , prosopis , pistachio , eucalyptus , date palm and carob [ citation needed ] .
Before the invention of Limans, their creators as well as scientists expected several positive and/or negative impacts: | https://en.wikipedia.org/wiki/Liman_irrigation_system |
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Limbin (LBN) is a protein that is part of the EvC complex that consists of EvC and EvC2 genes, the activity of which is critical in bone formation and skeletal development. The complex positively regulates ciliary Hedgehog (Hh) signaling due to the ciliary localization. [ 1 ]
A mutation in these genes is associated with The Ellis-van Creveld (EvC) syndrome. EvC or otherwise known as Chondroestodermal dysplasia is a disorder inherited by the offspring of carriers of the mutated recessive gene and a non-mutated dominant gene leading to expression of chondrodysplasia and dwarfism . Bone growth occurs due to continuous proliferation and differentiation of chondrocytes along with endochondral ossification at both ends of a long bone. [ 2 ] The mutations in LBN cause premature termination of encoded proteins resulting in shortening of long bones.
Other characteristics accredited to a mutation in LBN include difficulty breathing due to shorted ribs, shortened tongue, dysplastic fingernails, and postaxial polydactyly . [ 1 ]
This biochemistry article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Limbin |
Limbitless Solutions is a 501(c)(3) non-profit organization founded in the United States that uses additive manufacturing ( 3D printing ) to create accessible, yet affordable personalized bionics and prosthetic partial arms for children with limb differences. [ 1 ] The organization says their bionic arms are manufactured for under $400, 1% of the standard production cost. [ 2 ] Headquartered on the University of Central Florida campus in Orlando, Florida , [ 3 ] the organization was founded by a team of engineering students, led by CEO and Executive Director Albert Manero.
The idea of Limbitless Solutions came to life in 2014 after a team of engineering students at the University of Central Florida led an initiative to provide bionic 3D printed limbs to children. In their free time, the students took advantage of a donated Stratasys Dimension 3D printer in the engineering manufacturing lab on campus to create an affordable prosthetic that displayed their ideas of art and engineering all into one. Their method was the first of its kind and minimized the cost and time of traditional prosthetic manufacturing processes like CNC milling . [ 4 ] [ 5 ] The first 3D printed arm the students created was run with off-the-shelf servomechanisms and batteries which are activated by the electromyography muscle energy on the child's limb. [ 6 ] Most prosthetic arms are mechanical , which presents a challenge for children without elbows because they have to open and close their mechanical prosthetic by bending their elbow. That led the Limbitless team to come up with the idea for an electronic arm with a muscle sensor that allows the child to open and close their prosthetic hand by flexing their biceps . [ 7 ]
Before creating the bionic arm, the child is measured carefully to ensure that the length, width, and size of their new 3D prosthetic is as similar to their residual arm as possible. The model of the arm is then appropriately scaled and adjusted using Fusion 360 before being printed, assembled, and fitted. Electromyography (EMG) sensors are then calibrated before the arm is ready for use. [ 8 ] The time to create one bionic arm varies depending on several factors, the most significant being the type of limb difference the child has.
Children who have been given bionics from Limbitless Solutions include a 7-year-old boy who received a 3D printed Iron Man themed arm, presented by actor Robert Downey Jr. [ 9 ] (facilitated by Microsoft 's The Collective Project), [ 10 ] a 12-year-old from Vero Beach, Florida , who was the recipient of a bionic arm presented by the Blue Man Group at Universal Studios in Orlando, Florida , an 11-year-old girl originally from California who was presented a floral themed arm at the Clearwater Marine Aquarium, an 8-year-old boy from Seattle, Washington, who received his arm as part of the 12 Arms of Christmas delivery, a 10-year-old girl from Texas who was the recipient of a UCF themed arm, presented by the UCF Cheerleading team and Knightro, the UCF mascot, and a 22-year old model from Hawaii who wore the arm she received on the runway. [ citation needed ]
Project Xavier is the name for the production of a wheelchair that is controlled by the same EMG sensors as the 3-D printed arms. These EMG sensors are placed on the temporalis muscles, allowing for those with limited or no hand dexterity the ability to control the wheelchair by clenching their jaw in different ways. This wheelchair reduces the need for the user to be pushed around by someone, making tasks easier and less time-consuming for them. This increased independence enhances their quality of life immediately. [ citation needed ]
In December 2018, Limbitless Solutions released a comic book entitled The Bionic Kid . The comic book was written by Zachary, one of the Limbitless Solutions bionic kids, his brother Christo, and their dad Niko. The visuals were created by student artists at the University of Central Florida with assistance from professors at UCF School of Visual Arts and Design, The Bionic Kid is being sold in order to support those with limb difference. This comic tells the story of Zachary, one of the bionic kids. They attend the 8-Bit-World Finals where Zachary ends up playing the accessible video game Bash Bro against a bully named Norman. After both are electrocuted in an accident, the each receive special powers. In the comic, Zachary is referred to as The Bionic Kid, Norman is called Aquarius, and Limbitless Solutions Executive Director Albert Manero is a character as well. [ citation needed ]
Limbitless Solutions also created custom video game controllers that have been created to utilize the same EMG input that is used to operate the prosthetic arms. Typically, traditional controllers have not fully considered disabled user-experience, but Limbitless is creating new accessibility tools for not only their Bionic Kids, but many others in the same situation. Inclusive gaming not only trains Bionic Kids, but empowers through creativity. [ citation needed ] | https://en.wikipedia.org/wiki/Limbitless_Solutions |
Lime is an inorganic material composed primarily of calcium oxides and hydroxides . It is also the name for calcium oxide which is used as an industrial mineral and is made by heating calcium carbonate in a kiln. Calcium oxide can occur as a product of coal-seam fires and in altered limestone xenoliths in volcanic ejecta. [ 1 ] The International Mineralogical Association recognizes lime as a mineral with the chemical formula of CaO. [ 2 ] The word lime originates with its earliest use as building mortar and has the sense of sticking or adhering . [ 3 ]
These materials are still used in large quantities in the manufacture of steel and as building and engineering materials (including limestone products, cement , concrete , and mortar ), as chemical feedstocks, for sugar refining , and other uses. Lime industries and the use of many of the resulting products date from prehistoric times in both the Old World and the New World. [ citation needed ] Lime is used extensively for wastewater treatment with ferrous sulfate .
The rocks and minerals from which these materials are derived, typically limestone or chalk , are composed primarily of calcium carbonate . They may be cut, crushed, or pulverized and chemically altered. Burning ( calcination ) of calcium carbonate in a lime kiln above 900 °C (1,650 °F) [ 4 ] converts it into the highly caustic and reactive material burnt lime , unslaked lime or quicklime ( calcium oxide ) and, through subsequent addition of water, into the less caustic (but still strongly alkaline ) slaked lime or hydrated lime ( calcium hydroxide , Ca(OH) 2 ), the process of which is called slaking of lime .
When the term lime is encountered in an agricultural context, it usually refers to agricultural lime , which today is usually crushed limestone, not a product of a lime kiln. Otherwise it most commonly means slaked lime , as the more reactive form is usually described more specifically as quicklime or burnt lime .
In the lime industry, limestone is a general term for rocks that contain 80% or more of calcium or magnesium carbonate , including marble , chalk , oolite , and marl . Further classification is done by composition as high calcium, argillaceous (clayey), silicious , conglomerate , magnesian , dolomite , and other limestones . [ 5 ] Uncommon sources of lime include coral, sea shells, calcite and ankerite .
Limestone is extracted from quarries or mines . Part of the extracted stone, selected according to its chemical composition and optical granulometry , is calcinated at about 900 °C (1,650 °F) in lime kilns to produce quicklime according to the reaction: [ 6 ]
Before use, quicklime is hydrated , that is combined with water, called slaking, so hydrated lime is also known as slaked lime, and is produced according to the reaction:
Dry slaking is slaking quicklime with just enough water to hydrate the quicklime, but to keep it as a powder; it is referred to as hydrated lime. In wet slaking , a slight excess of water is added to hydrate the quicklime to a form referred to as lime putty.
Lime is commonly used as a binding mortar in masonry due to its adhesive properties with bricks and stones. It is also used in whitewashing as a wall-coat to allow the whitewash to adhere the wall.
The process by which limestone (calcium carbonate) is converted to quicklime by heating, then to slaked lime by hydration, and naturally reverts to calcium carbonate by carbonation is called the lime cycle . [ 7 ] The conditions and compounds present during each step of the lime cycle have a strong influence of the end product, [ 8 ] thus the complex and varied physical nature of lime products.
An example is when slaked lime (calcium hydroxide) is mixed into a thick slurry with sand and water to form mortar for building purposes. When the masonry has been laid, the slaked lime in the mortar slowly begins to react with carbon dioxide to form calcium carbonate (limestone) according to the reaction:
The carbon dioxide that takes part in this reaction is principally available in the air or dissolved in rainwater [ 9 ] so pure lime mortar will not recarbonate under water or inside a thick masonry wall.
The lime cycle for dolomitic and magnesium lime is not well understood [ 8 ] but more complex because the magnesium compounds also slake to periclase which slake more slowly than calcium oxide and when hydrated produce several other compounds. Thus, these limes contain inclusions of portlandite , brucite , magnesite , and other magnesium hydroxycarbonate compounds. These magnesium compounds have very limited, contradictory research which questions whether they "...may be significantly reactive with acid rain, which could lead to the formation of magnesium sulfate salts." [ 10 ] Magnesium sulfate salts may damage the mortar when they dry and recrystallize due to expansion of the crystals as they form, which is known as sulfate attack .
Lime used in building materials is broadly classified as "pure", "hydraulic", and "poor" lime; [ 11 ] can be natural or artificial ; and may be further identified by its magnesium content such as dolomitic or magnesium lime. Uses include lime mortar , lime plaster , lime render , lime-ash floors , tabby concrete , whitewash , silicate mineral paint , and limestone blocks which may be of many types . The qualities of the many types of processed lime affect how they are used. The Romans used two types of lime mortar to make Roman concrete , which allowed them to revolutionize architecture, sometimes called the Concrete revolution .
Lime has many complex qualities as a building product including workability which includes cohesion, adhesion, air content, water content, crystal shape, board-life, spreadability, and flowability; bond strength; comprehensive strength; setting time; sand-carrying capacity; hydraulicity; free lime content; vapor permeability; flexibility; and resistance to sulfates. These qualities are affected by many factors during each step of manufacturing and installation, including the original ingredients of the source of lime; added ingredients before and during firing including inclusion of compounds from the fuel exhaust; firing temperature and duration; method of slaking including a hot mix (quicklime added to sand and water to make mortar), dry slaking and wet slaking; ratio of the mixture with aggregates and water; the sizes and types of aggregate; contaminants in the mixing water; workmanship; and rate of drying during curing. [ 12 ]
Pure lime is also known as rich, common, air, slaked, slack, pickling, hydrated, and high calcium lime. It consists primarily of calcium hydroxide which is derived by slaking quicklime (calcium oxide), and may contain up to 5% of other ingredients. Pure lime sets very slowly through contact with carbon dioxide in the air and moisture; it is not a hydraulic lime so it will not set under water. Pure lime is pure white and can be used for whitewash, plaster, and mortar. Pure lime is soluble in water containing carbonic acid , a natural, weak acid which is a solution of carbon dioxide in water and acid rain so it will slowly wash away, but this characteristic also produces autogenous or self-healing process where the dissolved lime can flow into cracks in the material and be redeposited, automatically repairing the crack.
Semi-hydraulic lime, also called partially hydraulic and grey lime, sets initially with water and then continues to set with air. This lime is similar to hydraulic lime but has less soluble silica (usually minimum 6%) and aluminates , and will set under water but will never harden. [ 13 ]
Hydraulic lime is also called water lime . Hydraulic lime contains lime with silica or alumina and sets with exposure to water and can set under water. [ 14 ] Natural hydraulic lime (NHL) is made from a limestone which naturally contains some clay . Artificial hydraulic lime is made by adding forms of silica or alumina such as clay to the limestone during firing, or by adding a pozzolana to pure lime. [ 13 ] Hydraulic limes are classified by their strength: feebly , moderately and eminently hydraulic lime. Feebly hydraulic lime contains 5-10% clay, slakes in minutes, and sets in about three weeks. It is used for less expensive work and in mild climates. Moderately hydraulic lime contains 11-20% clay, slakes in one to two hours, and sets in approximately one week. It is used for better quality work and exterior walls in freezing climates. Eminently hydraulic lime contains 21-30% clay, slakes very slowly, and sets in approximately a day. It is used in harsh environments such as damp locations and near saltwater. Hydraulic lime is off-white in color. "The degree of hydraulicity of mortars will affect many characteristics. By selecting an appropriate ratio of clay to limestone mortars that carbonate or set hydraulically to varying extents can be designed for particular application requirements such as setting time, strength, colour, durability, frost resistance, workability, speed of set in the presence of water, vapour permeability etc." [ 14 ]
Poor lime is also known as lean or meager lime. Poor lime sets and cures very slowly and has weak bonding. Poor lime is grey in color.
Magnesium lime contains more than 5% magnesium oxide (BS 6100) or 5-35% magnesium carbonate (ASTM C 59-91). [ 15 ] Dolomitic lime has a high magnesium content of 35-46% magnesium carbonate (ASTM C 59-91). [ 15 ] Dolomitic lime is named for the Dolomite Mountains in the Italian and Austrian Alps. [ 16 ]
In the United States the most commonly used masonry lime is Type S hydrated lime which is intended to be added to Portland cement to improve plasticity , water retention and other qualities. The S in type S stands for special which distinguishes it from Type N hydrated lime where the N stands for normal. The special attributes of Type S are its "...ability to develop high, early plasticity and higher water retentivity and by a limitation on its unhydrated oxide content." [ 17 ] The term Type S originated in 1946 in ASTM C 207 Hydrated Lime for Masonry Purposes. Type S lime is almost always dolomitic lime, hydrated under heat and pressure in an autoclave, and used in mortar, render , stucco , and plaster . Type S lime is not considered reliable as a pure binder in mortar due to high burning temperatures during production.
Kankar lime, a lime made from kankar which is a form of calcium carbonate.
Selenitic lime, also known as Scotts' cement after Henry Young Darracott Scott , is a cement of grey chalk or similar lime, such as in the Lias Group , with about 5% added gypsum plaster (calcined gypsum ). [ 13 ] Selenite is a type of gypsum, but selenitic cement may be made using any form of sulfate or sulfuric acid . [ 18 ] Sulfate arrests slaking, causes the cement to set quickly and stronger.
The Romans made concrete by mixing lime and volcanic ash to create a pozzolanic reaction . When this was mixed with volcanic tuff and placed under seawater, the seawater hydrated the lime in an exothermic reaction that solidified the mixture. [ 19 ] | https://en.wikipedia.org/wiki/Lime_(material) |
A lime kiln is a kiln used for the calcination of limestone ( calcium carbonate ) to produce the form of lime called quicklime ( calcium oxide ). The chemical equation for this reaction is: CaCO 3 + heat → CaO + CO 2
This reaction can take place at anywhere above 840 °C (1,540 °F), but is generally considered to occur at 900 °C (1,650 °F) (at which temperature the partial pressure of CO 2 is 1 atmosphere ), but a temperature around 1,000 °C (1,830 °F) (at which temperature the partial pressure of CO 2 is 3.8 atmospheres [ 1 ] ) is usually used to make the reaction proceed quickly. [ 2 ] Excessive temperature is avoided because it produces unreactive, "dead-burned" lime.
Slaked lime ( calcium hydroxide ) can be formed by mixing quicklime with water.
Because it is so readily made by heating limestone, lime must have been known from the earliest times, and all the early civilizations used it in building mortars and as a stabilizer in mud renders and floors. [ 3 ] According to finds at 'Ain Ghazal in Jordan, Yiftahel in Israel, [ 4 ] and Abu Hureyra in Syria dating to 7500–6000 BCE, the earliest use of lime was mostly as a binder on floors and in plaster for coating walls. [ 5 ] This use of plaster may in turn have led to the development of proto-pottery, made from lime and ash. [ 5 ] In mortar, the oldest binder was mud. [ 5 ] According to finds at Catal Hüyük in Turkey, mud was soon followed by clay, and then by lime in the 6th millennium BCE. [ 5 ]
Knowledge of its value in agriculture is also ancient, but agricultural use only became widely possible when the use of coal made it cheap [ 6 ] in the coalfields in the late 13th century, and an account of agricultural use was given in 1523. [ 7 ] The earliest descriptions of lime kilns differ little from those used for small-scale manufacture a century ago. [ citation needed ] Because land transportation of minerals like limestone and coal was difficult in the pre-industrial era, they were distributed by sea, and lime was most often manufactured at small coastal ports. [ citation needed ] Many preserved kilns are still to be seen on quaysides around the coasts of Britain.
Permanent lime kilns fall into two broad categories: "flare kilns" also known as "intermittent" or "periodic" kilns; and "draw kilns" also known as "perpetual" or "running" kilns. In a flare kiln, a bottom layer of coal was built up and the kiln above filled solely with chalk. The fire was alight for several days, and then the entire kiln was emptied of the lime.
In a draw kiln, usually a stone structure, the chalk or limestone was layered with wood, coal or coke and lit. As it burnt through, lime was extracted from the bottom of the kiln, through the draw hole. Further layers of stone and fuel were added to the top. [ 8 ] [ 9 ]
The common feature of early kilns was an egg-cup shaped burning chamber, with an air inlet at the base (the "eye"), constructed of brick. Limestone was crushed (often by hand) to fairly uniform 20–60 mm (1– 2 + 1 ⁄ 2 in) lumps – fine stone was rejected. Successive dome-shaped layers of limestone and wood or coal were built up in the kiln on grate bars across the eye. When loading was complete, the kiln was kindled at the bottom, and the fire gradually spread upwards through the charge. When burnt through, the lime was cooled and raked out through the base. Fine ash dropped out and was rejected with the "riddlings".
Only lump stone could be used, because the charge needed to "breathe" during firing. This also limited the size of kilns and explains why kilns were all much the same size. Above a certain diameter, the half-burned charge would be likely to collapse under its own weight, extinguishing the fire. So kilns always made 25–30 tonnes of lime in a batch. Typically the kiln took a day to load, three days to fire, two days to cool and a day to unload, so a one-week turnaround was normal. The degree of burning was controlled by trial and error from batch to batch by varying the amount of fuel used. Because there were large temperature differences between the center of the charge and the material close to the wall, a mixture of underburned (i.e. high loss on ignition ), well-burned and dead-burned lime was normally produced. Typical fuel efficiency was low, with 0.5 tonnes or more of coal being used per tonne of finished lime (15 MJ/kg).
Lime production was sometimes carried out on an industrial scale. One example at Annery in North Devon , England , near Great Torrington , was made up of three kilns grouped together in an 'L' shape and was situated beside the Torrington canal and the River Torridge to bring in the limestone and coal, and to transport away the calcined lime in the days before properly metalled roads existed. [ 10 ]
Sets of seven kilns were common. A loading gang and an unloading gang would work the kilns in rotation through the week.
A rarely used kiln was known as a "lazy kiln". [ 11 ]
In the late 19th and early 20th centuries the town of Waratah in Gippsland , Victoria , Australia produced a majority of the quicklime used in the city of Melbourne as well as around other parts of Gippsland. The town, now called Walkerville , was set on an isolated part of the Victorian coastline and exported the lime by ship. When this became unprofitable in 1926 the kilns were shut down. The present-day area, though having no town amenities as such, markets itself as a tourist destination. The ruins of the lime kilns can still be seen today.
A lime kiln also existed in Wool Bay , South Australia .
The large kiln at Crindledykes near Haydon Bridge , Northumbria, was one of more than 300 in the county. It was unique to the area in having four draw arches to a single pot. As production was cut back, the two side arches were blocked up, but were restored in 1989 by English Heritage .
The development of the national rail network made the local small-scale kilns increasingly unprofitable, and they gradually died out through the 19th century. They were replaced by larger industrial plants. At the same time, new uses for lime in the chemical , steel and sugar industries led to large-scale plants. These also saw the development of more efficient kilns.
A lime kiln erected at Dudley , West Midlands (formerly Worcestershire ) in 1842 survives as part of the Black Country Living Museum which opened in 1976, although the kilns were last used during the 1920s. It is now among the last in a region which was dominated by coalmining and limestone mining for generations until the 1960s. [ 12 ]
The theoretical heat (the standard enthalpy ) of reaction required to make high-calcium lime is around 3.15 MJ per kg of lime, so the batch kilns were only around 20% efficient. The key to development in efficiency was the invention of continuous kilns, avoiding the wasteful heat-up and cool-down cycles of the batch kilns. The first were simple shaft kilns, similar in construction to blast furnaces . These are counter-current shaft kilns. Modern variants include regenerative and annular kilns. Output is usually in the range 100–500 tonnes per day.
The fuel is injected part-way up the shaft, producing maximum temperature at this point. The fresh feed fed in at the top is first dried then heated to 800 °C, where de-carbonation begins, and proceeds progressively faster as the temperature rises. Below the burner, the hot lime transfers heat to, and is cooled by, the combustion air. A mechanical grate withdraws the lime at the bottom. A fan draws the gases through the kiln, and the level in the kiln is kept constant by adding feed through an airlock. As with batch kilns, only large, graded stone can be used, in order to ensure uniform gas-flows through the charge. The degree of burning can be adjusted by changing the rate of withdrawal of lime. Heat consumption as low as 4 MJ/kg is possible, but 4.5 to 5 MJ/kg is more typical. Due to temperature peak at the burners up to 1200 °C in a shaft kiln conditions are ideal to produce medium and hard burned lime.
These typically consist of a pair of shafts, operated alternately. First, when shaft A is the "primary" and B the "secondary" shaft, the combustion air is added from the top of shaft A, while fuel somewhat below via burner lances. The flame is top-bottom. The hot gases pass downward, cross to shaft B via the so-called "channel" and pass upward to exhaust of shaft B. At same time in both shafts cooling air is added from the bottom to cool the lime and to make exhaust of gases via the bottom of the kiln impossible via maintaining a positive pressure. The combustion air and cooling air leave the kiln jointly via exhaust on top of shaft B, preheating the stone. The direction of flow is reversed periodically (typically 5–10 times per hour) shaft A and B changing the role of "primary" and "secondary" shaft. The kiln has three zones: preheating zone on the top, burning zone in the middle, and cooling zone close to the bottom. The cycling produces a long burning zone of constant, relatively low temperature (around 950 °C) that is ideal for the production of high quality soft burned reactive lime. With exhaust gas temperatures as low as 120 °C and lime temperature at kiln outlet in 80 °C range the heat loss of the regenerative kiln is minimal, fuel consumption is as low as 3.6 MJ/kg. Due to these features the regenerative kilns are today mainstream technology under conditions of substantial fuel costs. Regenerative kilns are built with 150 to 800 t/day output, 300 to 450 being typical.
These contain a concentric internal cylinder. This gathers pre-heated air from the cooling zone, which is then used to pressurize the middle annular zone of the kiln. Air spreading outward from the pressurized zone causes counter-current flow upwards, and co-current flow downwards. This again produces a long, relatively cool calcining zone. Fuel consumption is in 4 to 4.5 MJ/kg range and the lime is typically medium burned.
Rotary kilns started to be used for lime manufacture at the start of the 20th century and now account for a large proportion of new installations if energy costs are less important. The early use of simple rotary kilns had the advantages that a much wider range of limestone size could be used, from fines upwards, and undesirable elements such as sulfur can be removed. On the other hand, fuel consumption was relatively high because of poor heat exchange compared with shaft kilns, leading to excessive heat loss in exhaust gases. Old fashioned "long" rotary kilns operate at 7 to 10 MJ/kg. Modern installations partially overcome this disadvantage by adding a preheater, which has the same good solids/gas contact as a shaft kiln, but fuel consumption is still somewhat higher, typically in range of 4.5 to 6 MJ/kg. In the design shown, a circle of shafts (typically 8–15) is arranged around the kiln riser duct. Hot limestone is discharged from the shafts in sequence, by the action of a hydraulic "pusher plate". Kilns of 1000 tonnes per day output are typical. The rotary kiln is the most flexible of any lime kilns able to produce soft, medium, or hard burned as well as dead-burned lime or dolime.
All the above kiln designs produce exhaust gas that carries an appreciable amount of dust. Lime dust is particularly corrosive. Equipment is installed to trap this dust, typically in the form of electrostatic precipitators or bag filters. The dust usually contains a high concentration of elements such as alkali metals , halogens and sulfur.
The lime industry is a significant carbon dioxide emitter. The manufacture of one tonne of calcium oxide involves decomposing calcium carbonate, with the formation of 785 kg of CO 2 in some applications, such as when used as mortar ; this CO 2 is later re-absorbed as the mortar goes off.
If the heat supplied to form the lime (3.75 MJ/kg in an efficient kiln) is obtained by burning fossil fuel it will release CO 2 : in the case of coal fuel 295 kg/t; in the case of natural gas fuel 206 kg/t. The electric power consumption of an efficient plant is around 20 kWh per tonne of lime. This additional input is the equivalent of around 20 kg CO 2 per ton if the electricity is coal-generated. Thus, total emission may be around 1 tonne of CO 2 for every tonne of lime even in efficient industrial plants, but is typically 1.3 t/t. [ 14 ] However, if the source of heat energy used in its manufacture is a fully renewable power source, such as solar, wind, hydro or even nuclear; there may be no net emission of CO 2 from the calcination process. Less energy is required in production per weight than portland cement , primarily because a lower temperature is required.
Wainmans Double Arched Lime Kiln – Made Grade II Listed Building – 1 February 2005
Details & Image: Wainmans Double Arched Lime Kiln - Made Grade II Listed Building | https://en.wikipedia.org/wiki/Lime_kiln |
Lime softening (also known as lime buttering , lime-soda treatment , or Clark 's process ) [ 1 ] is a type of water treatment used for water softening , which uses the addition of limewater ( calcium hydroxide ) to remove hardness (deposits of calcium and magnesium salts ) by precipitation . The process is also effective at removing a variety of microorganisms and dissolved organic matter by flocculation . [ 2 ]
Lime softening was first used in 1841 to treat Thames River water. The process expanded in use as the other benefits of the process were discovered. Lime softening greatly expanded in use during the early 1900s as industrial water use expanded. Lime softening provides soft water that can, in some cases, be used more effectively for heat transfer and various other industrial uses.
As lime in the form of limewater is added to raw water, the pH is raised and the equilibrium of carbonate species in the water is shifted. Dissolved carbon dioxide (CO 2 ) is changed into bicarbonate (HCO − 3 ) and then carbonate (CO 2- 3 ). This action causes calcium carbonate to precipitate due to exceeding the solubility product. Additionally, magnesium can be precipitated as magnesium hydroxide in a double displacement reaction. [ 3 ]
In the process both the calcium (and to an extent magnesium) in the raw water as well as the calcium added with the lime are precipitated. This is in contrast to ion exchange softening where sodium is exchanged for calcium and magnesium ions. In lime softening, there is a substantial reduction in total dissolved solids (TDS) whereas in ion exchange softening (sometimes referred to as zeolite softening), there is no significant change in the level of TDS.
Lime softening can also be used to remove iron , manganese , radium and arsenic from water.
Lime softening is now often combined with newer membrane processes to reduce waste streams. Lime softening can be applied to the concentrate (or reject stream) of membrane processes, thereby providing a stream of substantially reduced hardness (and thus TDS), that may be used in the finished stream. Also, in cases with very hard source water (often the case in Midwestern USA ethanol production plants), lime softening can be used to pre-treat the membrane feed water.
Lime softening produces large volumes of a mixture of calcium carbonate and magnesium hydroxide in a very finely divided white precipitate which may also contain some organic matter flocculated out of the raw water. Processing or disposal of this sludge material may be an additional cost to the process. Drying and re-calcining the waste allows the lime to be almost fully re-cycled, but drying and re-calcining is more expensive than producing new lime from limestone. | https://en.wikipedia.org/wiki/Lime_softening |
In horticulture , lime sulfur ( lime sulphur in British English , see American and British English spelling differences ) is mainly a mixture of calcium polysulfides and thiosulfate [ 1 ] (plus other reaction by-products as sulfite and sulfate ) formed by reacting calcium hydroxide with elemental sulfur , used in pest control . It can be prepared by boiling in water a suspension of poorly soluble calcium hydroxide (lime) and solid sulfur together with a small amount of surfactant to facilitate the dispersion of these solids in water. After elimination of residual solids (flocculation, decantation, and filtration), it is normally used as an aqueous solution, which is reddish-yellow in colour and has a distinctive offensive odor of hydrogen sulfide (H 2 S, rotten eggs).
The exact chemical reaction leading to the synthesis of lime sulfur is generally written as:
as reported in a document of the US Department of Agriculture (USDA). [ 2 ]
This vague reaction is poorly understood, because it involves the reduction of elemental sulfur and no reductant appears in the equation while sulfur oxidation products are also mentioned as products. The initial pH of the solution imposed by poorly soluble hydrated lime is alkaline (pH = 12.5) while the final pH is in range 11–12, typical for sulfides which are also strong bases .
When the hydrolysis of calcium sulfide is accounted for, the individual reactions for each of the by-products are:
However, elemental sulfur can undergo a disproportionation reaction, also called dismutation. The first reaction resembles a disproportionation reaction. The inverse comproportionation reaction is the reaction occurring in the Claus process used for desulfurization of oil and gas crude products in the refining industry:
By rewriting the last reaction in the inverse direction one obtains a reaction consistent with what is observed in the lime sulfur global reaction:
In alkaline conditions, it gives:
and after simplification, or more exactly recycling, of water molecules in the above reaction:
adding back 6 Ca 2+ cations from hydrated lime for the sake of electroneutrality, one obtains the global reaction.
This last reaction is consistent with the global lime sulfur reaction mentioned in the USDA document. [ 2 ] However, it does not account of all the details, a.o., the production of thiosulfate and sulfate amongst the end-products of the reaction. Meanwhile, it is a good first order approximation and it usefully highlights the overall lime sulfur reaction scheme because the chemistry of reduced or partially oxidized forms of sulfur is particularly complex and all the intermediate steps or involved mechanisms are hard to unravel. Moreover, once exposed to atmospheric oxygen and microbial activity, the lime sulfur system will undergo a fast oxidation and its different products will continue to evolve and finally enter the natural sulfur cycle .
The presence of thiosulfate in the lime sulfur reaction can be accounted by the reaction between sulfite and elemental sulfur (or with sulfide and polysulfides) and that of sulfate by the complete oxidation of sulfite or thiosulfate following a more complex reaction scheme. More information on calcium thiosulfate production has been described in a patent registered by Hajjatie et al. (2006). [ 3 ]
Hajjatie et al. (2006) wrote the lime sulfur reaction in various ways depending on the degree of polymerisation of calcium polysulfides, but the following reaction is probably the simplest of their series:
They also managed to successfully control this reaction to achieve the conversion of elemental sulfur in a quasi-pure solution of calcium thiosulfate.
The New York State Agricultural Experiment Station recipe for the concentrate suggests starting with 80 lb. of sulfur , 36 lb. of quicklime , and 50 gal. of water, equivalent to 19.172 kg of sulfur and 8.627 kg of calcium oxide per 100 litres of water. About 2.2:1 is the ratio (by weight) for compounding sulfur and quicklime; this makes the highest proportion of calcium pentasulfide. If calcium hydroxide (builders or hydrated lime ) is used, an increase by 1/3 or more (to 115 g/L or more) might be used with the 192 g/L of sulfur. If the quicklime is 85%, 90%, or 95% pure, use 101 g/L, 96 g/L, or 91 g/L; if impure hydrated lime is used, its quantity is increased to compensate, though in practice lime with a purity lower than 90% is rarely used. The mixture is then boiled for one hour while being stirred while small amounts of water are added for evaporation.
In agriculture and horticulture, lime sulfur is sold as a spray to control fungi , bacteria , and insects . On deciduous trees it can be sprayed during the winter on the surface of the bark in high concentrations, but as lime sulfur can burn foliage, it must be heavily diluted before spraying onto herbaceous crops, especially during warm weather. Lime sulfur is approved for use on organic crops in the European Union and the United Kingdom. [ 4 ]
Bonsai enthusiasts use undiluted lime sulfur to bleach, sterilize, and preserve deadwood on bonsai trees while giving an aged look. [ 5 ] Rather than spraying the entire tree, as with the pesticidal usage, lime sulfur is painted directly onto the exposed deadwood, and is often colored with a small amount of dark paint to make it look more natural. Without paint pigments, the lime sulfur solution bleaches wood to a bone-white color that takes time to weather and become natural-looking. [ 6 ] In the very specific case of the bonsai culture, if the lime sulfur is carefully and very patiently applied by hand with a small brush and does not enter into direct contact with the leaves or needles, this technique can be used on evergreen bonsai trees as well as other types of green trees. However, this does not apply for a normal use on common trees with green leaves.
Diluted solutions of lime sulfur (between 1:16 and 1:32) are also used as a dip for pets to help control ringworm (a fungus) , mange and other dermatoses and parasites . Undiluted lime sulfur is corrosive to skin and eyes and can cause serious injury like blindness.
Lime sulfur reacts with strong acids (including stomach acid ) to produce highly toxic hydrogen sulfide (rotten egg gas) and indeed usually has a distinct "rotten egg" odor to it. Lime sulfur is not flammable but can release highly irritating sulfur dioxide gas when in a fire.
Safety goggles and impervious gloves must be worn while handling lime sulfur. Lime sulfur solutions are strongly alkaline (typical commercial concentrates have a pH over 11.5 because of the presence of dissolved sulfides and hydroxide anions), and are harmful for living organisms and can cause blindness if splashed in the eyes.
The corrosive nature of lime sulfur is due to the reduced species of sulfur it contains, in particular the sulfides responsible for stress corrosion cracking and the thiosulfates causing pitting corrosion . Localized corrosion by the reduced species of sulfur can be dramatic, even the mere presence of elemental sulfur in contact with metals is enough to corrode them considerably, including so-called stainless steels .
Lime sulfur is believed to be the earliest synthetic chemical used as a pesticide, being used in the 1840s in France to control grape vine powdery mildew Uncinula necator , which had been introduced from the USA in 1845 and reduced wine production by 80%. [ citation needed ] In 1886 it was first used in California to control San Jose scale . Commencing around 1904, commercial suppliers began to manufacture lime sulfur; prior to that time, gardeners were expected to manufacture their own. By the 1920s essentially all commercial orchards in western countries were protected by regular spraying with lime sulfur. However by the 1940s, lime sulfur began to be replaced by synthetic organic fungicides which risked less damage to the crop's foliage. | https://en.wikipedia.org/wiki/Lime_sulfur |
In physiology , psychology , or psychophysics , a limen or a liminal point is a sensory threshold of a physiological or psychological response. Such points delineate boundaries of perception; that is, a limen defines a sensory threshold beyond which a particular stimulus becomes perceivable, and below which it remains unperceivable. [ 1 ]
Liminal , as an adjective, means situated at a sensory threshold, hence barely perceptible. Subliminal means below perception.
The absolute threshold is the lowest amount of sensation detectable by a sense organ.
This neuroscience article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Limen |
A limepit is either a place where limestone is quarried, or a man-made pit used to burn lime stones in the same way that modern-day kilns and furnaces constructed of brick are now used above ground for the calcination of limestone ( calcium carbonate , CaCO 3 ) and by which quicklime ( calcium oxide , CaO) is produced, an essential component in waterproofing and in wall plastering (plaster skim).
The production of lime in the Land of Israel has been dated as far back as the Canaanite period , and has continued in successive generations ever since. The man-made limepit was usually dug in ground near the place where limestone could be quarried. Remnants of old limepits have been unearthed in archaeological digs all throughout the Levant . In a country where hundreds of such limepits or limekilns for burning limestone were found, the Israel Antiquities Authority (IAA) describes dozens of them ( Hebrew : בור סיד / כבשן סיד ), one discovered in Kiryat Ye'arim , [ 1 ] another in Har Giora - East (2 km. north of Bar-Giora ), [ 2 ] as well as in Neve Yaakov , [ 3 ] among other places. Two lime kilns, stratigraphically dated to the late Hellenistic period were excavated at Ramat Rachel , the latter of which being circular in shape (3.6 metres in diameter) and built into the ruins of a large pool, using earlier walls. [ 4 ] A rounded kiln (2.5–2.8 metres in diameter) was found northeast of Jerusalem dating back to the Iron Age (seventh–sixth century BCE), and was built of stones and had a rectangular unit adjacent to it. [ 5 ] In the Lachish area, several lime kilns were excavated by a team on behalf of the IAA , and which kilns were partially hewn in the bedrock and partially built of fieldstones, and last used at some point between the mid-15th century and the mid-17th century CE . [ 6 ]
In Bedouin-Arab culture in Israel , the limepit was dug to a depth of about 2.5 metres (8.2 ft) and about 3 metres (9.8 ft) in diameter.
By all appearances, the pit was made after the same basic principle used in a " Dakota fire pit ," which is made with an air inlet at the base, allowing for air-ventilation, but on a larger scale. Air intake was achieved by digging an adjacent channel which ran from a short distance into the limepit, or else an underground shaft (shafts) at floor level of limepit leading from an open area, allowing for a steady, free-flowing draught of air to be drawn into the limepit as it burns. In this way, there was no need for the use of bellows to reach a high temperature, but only to stoke the fire with wood continuously for several days for it to reach a temperature of 900° Celsius (1650° F ). Its mode of operation was similar to that of a shaft kiln . After cooling, wood ashes that had accumulated were then separated from the burnt blocks of limestone. The limestone blocks were then crushed, afterwards slaked (the process of adding water and constantly turning the lime to create a chemical reaction, whereby the burnt lime, or what is known also as calcium oxide , [ 7 ] is changed into calcium hydroxide ), and mixed with an aggregate to form an adhesive paste (plaster) used in construction and for daubing buildings.
When properly burnt, limestone loses its carbonic acid ( H 2 CO 3 ) and becomes converted into caustic or quicklime ( CaO ). [ 8 ] One-hundred parts of raw limestone yields about 56 parts of quicklime. [ 8 ] In the West, quicklime was formerly a major component in common mortar , besides its predominant use in plastering. In some Middle-Eastern countries where rain-fall was scarce in the dry season, lime production for use in plastering home-made cisterns (in making them impermeable by adding thereto a pozzolanic agent) was especially important. This enabled them to collect the winter run-off of rain water and to have it stored for later use, whether for personal or agricultural needs. [ 9 ] Lime is also an important component in the production of Nabulsi soap , [ 10 ] in dyeing fabrics, and in use as a depilatory .
Many limepits were sunken in the ground at a depth of between 2.5 and 5 meters and 3 to 4.5 meters in diameter, in a circular fashion, and some were built with a retaining wall along the inside for support, usually constructed of uncut field-stones. Simpler limepits were made without supportive walls. In the following account, Abu-Rabiʻa describes the practice of Bedouins in the Negev , during the late 19th and early 20th-century:
Lime is derived from chalk [ sic ] by burning. The Bedouins used it in plastering their cisterns. Burning chalk stone was performed in simple kilns in close proximity to where the chalk was found. Lime kilns were made by digging a round hole, three metres wide, two and a half metres deep. After the hole was dug, the chalk and fuel for a fire would be brought to it. Stones of chalk (limestone) would be arranged in a circular dome in the pit. The burning process would last three to six days, without letup. After the burning was finished, the kiln would be left to cool for four to six days. The lime would then be taken out. The large lime blocks along the edge of the pit were considered of the highest quality, while the small pieces towards the center of the pit were considered grade B. One camel load, or cantur ( qentar / quntar = 100 ratels , or 250–300 kilograms), of lime would fetch 40 grush on the Jerusalem market in the early 1880s. [ 11 ]
In Israel , the principal fuel used to keep the lime-kiln burning was the dried brushwood of prickly burnet ( Sarcopoterium spinosum ) and savory ( Satureja thymbra ), where often camel loads of this dried wood would be hauled to the lime-kiln. [ 12 ] Monolithic stone structures were already in use for burning limestone ( nāri ) during the Ottoman period , throughout the Levant. [ 5 ] Modern kilns for burning lime first appeared in Palestine during the British Mandate . [ 11 ]
The lime stones selected were those that had the least amount of impurities within them. Limepits were almost always built near the supply of limestone, and a sufficient pile of wood kindling was heaped in great store before the actual burning process began, a supply that was to last between 3 and 7 days of continual burning, both, by night and day. In the southern Mediterranean regions, one of the favorite wood sources was thorny burnet ( Sarcopoterium spinosum ). [ 13 ] [ 14 ] The fire was attended by men with long staves and pitchforks who pushed the burning material into the pit. Initially, a cloud of smoke billowed from the pit. After several days of burning, when the uppermost stone in the fire pit began to glow a fiery red, it signaled that the burning process of the lime was finished, and that the process of carbon dioxide emissions from the limestone has been completed, and that the lime was now ready for marketing as lime or powder. [ 13 ] After being allowed to cool, the burnt limestone was extracted from the pit when it was light and brittle. During the burning process, the limestone loses about 50% of its original anatomical weight. [ 13 ] The lime becomes ready for use only after water has been added. | https://en.wikipedia.org/wiki/Limepit |
Liminal BioSciences Inc. , [ 1 ] formerly known as Prometic Life Sciences Inc., [ 2 ] is a Canadian biopharmaceutical company.
Liminal BioSciences development programs target seven transmembrane GPCRs (7TM GPCR’s), where the receptor protein passes through the cell membrane seven times. These receptors are easily accessible to hydrophilic drugs due to their presence on the cell surface, and their non-uniform expression enables selectivity in modulating physiological processes. Agonists and antagonists of 7TM GPCRs receptors are utilized for treating various diseases in all organ systems. An agonist is a drug that binds to a target and mimics the action of the natural ligand. An antagonist is a drug that binds to a receptor and prevents other molecules (such as the natural ligand) from binding.
Liminal BioSicences' believe that their drug discovery platform and deep understanding of GPCRs allows the Company to identify small molecule candidates that can accurately target GPCRs where other drug discovery approaches have been unsuccessful. Its drug discovery platform leverages a fully integrated chemistry and biology expertise supported by our broad in vivo capabilities, which allows it to investigate preclinical drug candidates’ efficacy in a wide variety of animal models and enables the Company to develop small molecule therapeutic candidates for the treatment of various metabolic, inflammatory and fibrotic diseases. Liminal BioSciences' aim to develop best or first-in-class therapies targeting indications with significant unmet needs, where a novel small molecule approach may be better suited using its drug discovery platform, specialized know-how and data-driven development plans.
Liminal BioSciences is a development stage biopharmaceutical company focused on discovering and developing novel and distinctive small molecule therapeutics that modulate G protein-coupled receptors, or GPCR, pathways. The Company is designing proprietary novel small molecule therapeutic candidates with the intent of developing best/first in class therapeutics for the treatment of metabolic, inflammatory and fibrotic diseases with significant unmet medical needs, using our integrated drug discovery platform, medicinal chemistry expertise and deep understanding of the GPCR biology.
Liminal BioSciences' pipeline is currently made up of three development programs. The candidate selected for clinical development, LMNL6511, a selective antagonist for the GPR84 receptor, is expected to commence a Phase 1 clinical trial in the second half of 2023. The Company is also developing potential OXER1 antagonists and GPR40 agonists, both of which are at the preclinical stage. In addition to these priority development programs.
In 2023, Liminal BioSciences' updates its goals in its 20-F Filing, which were to leverage its drug discovery platform and to develop distinctive novel small molecule therapeutics to treat the complex biology of metabolic, inflammatory and fibrotic diseases to address a wide range of significant unmet needs. The key activities to achieve this goal include:
• Investing in and leveraging its GPCR knowledge and drug discovery platform to develop differentiated GPCR targeted therapies for unmet medical needs;
• Advancing the candidate selected for its GPR84 antagonist development program, LMNL6511, to clinical stage targeting the treatment of fibrosis and metabolic diseases;
• Progressing the development of its OXER1 antagonist program and nominating a lead candidate for further development in eosinophil mediated diseases;
• Pursue the development of its GPR40 agonist program aiming to identify and develop a novel liver-safe GPR40 agonist for the treatment of type 2 diabetes, or T2D; and
• Identifying potential opportunities to monetize non-core assets and to streamline costs overall.
From early 2018 to early 2019, Prometic Life Sciences' Share price dropped 79%. [ 7 ] The company's sought permission from the Toronto Stock Exchange to proceed without a shareholder vote for reasons of "financial hardship". [ 8 ] In October 2019, the company allowed its shareholders to vote on a name change. [ 9 ]
Official website | https://en.wikipedia.org/wiki/Liminal_BioSciences |
Liming is the application of calcium - (Ca) and magnesium (Mg)-rich materials in various forms, including marl , chalk , limestone , burnt lime or hydrated lime to soil . [ 1 ] In acid soils , these materials react as a base and neutralize soil acidity . This often improves plant growth and increases the activity of soil bacteria , [ 1 ] but oversupply may result in harm to plant life. Modern liming was preceded by marling, a process of spreading raw chalk and lime debris across soil, in an attempt to modify pH or aggregate size. [ 2 ] Evidence of these practices dates to the 1200's and the earliest examples are taken from the modern British Isles. [ 2 ]
Liming can also improve aggregate stability on clay soils . For this purpose structure lime, products containing calcium oxide (CaO) or hydroxide (Ca(OH) 2 ) in mixes with calcium carbonate (CaCO 3 ), are often used. Structure liming can reduce losses of clay and nutrients from soil aggregates. [ 3 ] The degree to which a given amount of lime per unit of soil volume will increase soil pH depends on the buffer capacity of the soil (this is generally related to soil cation exchange capacity or CEC).
Most acid soils are saturated with aluminum rather than hydrogen ions . Soil acidity generally results from hydrolysis of aluminum. [ 4 ] This concept of "corrected lime potential" [ 5 ] to define the degree of base saturation in soils became the basis for procedures now used in soil testing laboratories to determine the "lime requirement" of soils. [ 6 ]
Soils with low CEC will usually show a more marked pH increase than soils with high CEC. But the low-CEC soils will witness more rapid leaching of the added bases, and so will see a quicker return to original acidity unless additional liming is done. Over-liming is most likely to occur on soil that has low CEC, such as sand which is deficient in buffering agents such as organic matter and clay . [ 7 ]
The net effect of soil liming on soil organic carbon is primarily the result of three processes. [ 8 ]
An agricultural study at the Faculty of Forestry in Freising , Germany, that compared tree stocks two and twenty years after liming found that liming promotes nitrate leaching and decreases the phosphorus content of some leaves. [ 11 ] | https://en.wikipedia.org/wiki/Liming_(soil) |
In mathematics , a limit is the value that a function (or sequence ) approaches as the argument (or index) approaches some value. [ 1 ] Limits of functions are essential to calculus and mathematical analysis , and are used to define continuity , derivatives , and integrals .
The concept of a limit of a sequence is further generalized to the concept of a limit of a topological net , and is closely related to limit and direct limit in category theory .
The limit inferior and limit superior provide generalizations of the concept of a limit which are particularly relevant when the limit at a point may not exist.
In formulas, a limit of a function is usually written as
and is read as "the limit of f of x as x approaches c equals L ". This means that the value of the function f can be made arbitrarily close to L , by choosing x sufficiently close to c . Alternatively, the fact that a function f approaches the limit L as x approaches c is sometimes denoted by a right arrow (→ or → {\displaystyle \rightarrow } ), as in
which reads " f {\displaystyle f} of x {\displaystyle x} tends to L {\displaystyle L} as x {\displaystyle x} tends to c {\displaystyle c} ".
According to Hankel (1871), the modern concept of limit originates from Proposition X.1 of Euclid's Elements , which forms the basis of the Method of exhaustion found in Euclid and Archimedes: "Two unequal magnitudes being set out, if from the greater there is subtracted a magnitude greater than its half, and from that which is left a magnitude greater than its half, and if this process is repeated continually, then there will be left some magnitude less than the lesser magnitude set out." [ 2 ] [ 3 ]
Grégoire de Saint-Vincent gave the first definition of limit (terminus) of a geometric series in his work Opus Geometricum (1647): "The terminus of a progression is the end of the series, which none progression can reach, even not if she is continued in infinity, but which she can approach nearer than a given segment." [ 4 ]
In the Scholium to Principia in 1687, Isaac Newton had a clear definition of a limit, stating that "Those ultimate ratios... are not actually ratios of ultimate quantities, but limits... which they can approach so closely that their difference is less than any given quantity". [ 5 ]
The modern definition of a limit goes back to Bernard Bolzano who, in 1817, developed the basics of the epsilon-delta technique to define continuous functions. However, his work remained unknown to other mathematicians until thirty years after his death. [ 6 ]
Augustin-Louis Cauchy in 1821, [ 7 ] followed by Karl Weierstrass , formalized the definition of the limit of a function which became known as the (ε, δ)-definition of limit .
The modern notation of placing the arrow below the limit symbol is due to G. H. Hardy , who introduced it in his book A Course of Pure Mathematics in 1908. [ 8 ]
The expression 0.999... should be interpreted as the limit of the sequence 0.9, 0.99, 0.999, ... and so on. This sequence can be rigorously shown to have the limit 1, and therefore this expression is meaningfully interpreted as having the value 1. [ 9 ]
Formally, suppose a 1 , a 2 , ... is a sequence of real numbers . When the limit of the sequence exists, the real number L is the limit of this sequence if and only if for every real number ε > 0 , there exists a natural number N such that for all n > N , we have | a n − L | < ε . [ 10 ] The common notation lim n → ∞ a n = L {\displaystyle \lim _{n\to \infty }a_{n}=L} is read as:
The formal definition intuitively means that eventually, all elements of the sequence get arbitrarily close to the limit, since the absolute value | a n − L | is the distance between a n and L .
Not every sequence has a limit. A sequence with a limit is called convergent ; otherwise it is called divergent . One can show that a convergent sequence has only one limit.
The limit of a sequence and the limit of a function are closely related. On one hand, the limit as n approaches infinity of a sequence { a n } is simply the limit at infinity of a function a ( n ) —defined on the natural numbers { n } . On the other hand, if X is the domain of a function f ( x ) and if the limit as n approaches infinity of f ( x n ) is L for every arbitrary sequence of points { x n } in X − x 0 which converges to x 0 , then the limit of the function f ( x ) as x approaches x 0 is equal to L . [ 11 ] One such sequence would be { x 0 + 1/ n } .
There is also a notion of having a limit "tend to infinity", rather than to a finite value L {\displaystyle L} . A sequence { a n } {\displaystyle \{a_{n}\}} is said to "tend to infinity" if, for each real number M > 0 {\displaystyle M>0} , known as the bound, there exists an integer N {\displaystyle N} such that for each n > N {\displaystyle n>N} , a n > M . {\displaystyle a_{n}>M.} That is, for every possible bound, the sequence eventually exceeds the bound. This is often written lim n → ∞ a n = ∞ {\displaystyle \lim _{n\rightarrow \infty }a_{n}=\infty } or simply a n → ∞ {\displaystyle a_{n}\rightarrow \infty } .
It is possible for a sequence to be divergent, but not tend to infinity. Such sequences are called oscillatory . An example of an oscillatory sequence is a n = ( − 1 ) n {\displaystyle a_{n}=(-1)^{n}} .
There is a corresponding notion of tending to negative infinity, lim n → ∞ a n = − ∞ {\displaystyle \lim _{n\rightarrow \infty }a_{n}=-\infty } , defined by changing the inequality in the above definition to a n < M , {\displaystyle a_{n}<M,} with M < 0. {\displaystyle M<0.}
A sequence { a n } {\displaystyle \{a_{n}\}} with lim n → ∞ | a n | = ∞ {\displaystyle \lim _{n\rightarrow \infty }|a_{n}|=\infty } is called unbounded , a definition equally valid for sequences in the complex numbers , or in any metric space . Sequences which do not tend to infinity are called bounded . Sequences which do not tend to positive infinity are called bounded above , while those which do not tend to negative infinity are bounded below .
The discussion of sequences above is for sequences of real numbers. The notion of limits can be defined for sequences valued in more abstract spaces, such as metric spaces . If M {\displaystyle M} is a metric space with distance function d {\displaystyle d} , and { a n } n ≥ 0 {\displaystyle \{a_{n}\}_{n\geq 0}} is a sequence in M {\displaystyle M} , then the limit (when it exists) of the sequence is an element a ∈ M {\displaystyle a\in M} such that, given ε > 0 {\displaystyle \varepsilon >0} , there exists an N {\displaystyle N} such that for each n > N {\displaystyle n>N} , we have d ( a , a n ) < ε . {\displaystyle d(a,a_{n})<\varepsilon .} An equivalent statement is that a n → a {\displaystyle a_{n}\rightarrow a} if the sequence of real numbers d ( a , a n ) → 0 {\displaystyle d(a,a_{n})\rightarrow 0} .
An important example is the space of n {\displaystyle n} -dimensional real vectors, with elements x = ( x 1 , ⋯ , x n ) {\displaystyle \mathbf {x} =(x_{1},\cdots ,x_{n})} where each of the x i {\displaystyle x_{i}} are real, an example of a suitable distance function is the Euclidean distance , defined by d ( x , y ) = ‖ x − y ‖ = ∑ i ( x i − y i ) 2 . {\displaystyle d(\mathbf {x} ,\mathbf {y} )=\|\mathbf {x} -\mathbf {y} \|={\sqrt {\sum _{i}(x_{i}-y_{i})^{2}}}.} The sequence of points { x n } n ≥ 0 {\displaystyle \{\mathbf {x} _{n}\}_{n\geq 0}} converges to x {\displaystyle \mathbf {x} } if the limit exists and ‖ x n − x ‖ → 0 {\displaystyle \|\mathbf {x} _{n}-\mathbf {x} \|\rightarrow 0} .
In some sense the most abstract space in which limits can be defined are topological spaces . If X {\displaystyle X} is a topological space with topology τ {\displaystyle \tau } , and { a n } n ≥ 0 {\displaystyle \{a_{n}\}_{n\geq 0}} is a sequence in X {\displaystyle X} , then the limit (when it exists) of the sequence is a point a ∈ X {\displaystyle a\in X} such that, given a (open) neighborhood U ∈ τ {\displaystyle U\in \tau } of a {\displaystyle a} , there exists an N {\displaystyle N} such that for every n > N {\displaystyle n>N} , a n ∈ U {\displaystyle a_{n}\in U} is satisfied. In this case, the limit (if it exists) may not be unique. However it must be unique if X {\displaystyle X} is a Hausdorff space .
This section deals with the idea of limits of sequences of functions, not to be confused with the idea of limits of functions, discussed below.
The field of functional analysis partly seeks to identify useful notions of convergence on function spaces. For example, consider the space of functions from a generic set E {\displaystyle E} to R {\displaystyle \mathbb {R} } . Given a sequence of functions { f n } n > 0 {\displaystyle \{f_{n}\}_{n>0}} such that each is a function f n : E → R {\displaystyle f_{n}:E\rightarrow \mathbb {R} } , suppose that there exists a function such that for each x ∈ E {\displaystyle x\in E} , f n ( x ) → f ( x ) or equivalently lim n → ∞ f n ( x ) = f ( x ) . {\displaystyle f_{n}(x)\rightarrow f(x){\text{ or equivalently }}\lim _{n\rightarrow \infty }f_{n}(x)=f(x).}
Then the sequence f n {\displaystyle f_{n}} is said to converge pointwise to f {\displaystyle f} . However, such sequences can exhibit unexpected behavior. For example, it is possible to construct a sequence of continuous functions which has a discontinuous pointwise limit.
Another notion of convergence is uniform convergence . The uniform distance between two functions f , g : E → R {\displaystyle f,g:E\rightarrow \mathbb {R} } is the maximum difference between the two functions as the argument x ∈ E {\displaystyle x\in E} is varied. That is, d ( f , g ) = max x ∈ E | f ( x ) − g ( x ) | . {\displaystyle d(f,g)=\max _{x\in E}|f(x)-g(x)|.} Then the sequence f n {\displaystyle f_{n}} is said to uniformly converge or have a uniform limit of f {\displaystyle f} if f n → f {\displaystyle f_{n}\rightarrow f} with respect to this distance. The uniform limit has "nicer" properties than the pointwise limit. For example, the uniform limit of a sequence of continuous functions is continuous.
Many different notions of convergence can be defined on function spaces. This is sometimes dependent on the regularity of the space. Prominent examples of function spaces with some notion of convergence are Lp spaces and Sobolev space .
Suppose f is a real-valued function and c is a real number . Intuitively speaking, the expression
means that f ( x ) can be made to be as close to L as desired, by making x sufficiently close to c . [ 12 ] In that case, the above equation can be read as "the limit of f of x , as x approaches c , is L ".
Formally, the definition of the "limit of f ( x ) {\displaystyle f(x)} as x {\displaystyle x} approaches c {\displaystyle c} " is given as follows. The limit is a real number L {\displaystyle L} so that, given an arbitrary real number ε > 0 {\displaystyle \varepsilon >0} (thought of as the "error"), there is a δ > 0 {\displaystyle \delta >0} such that, for any x {\displaystyle x} satisfying 0 < | x − c | < δ {\displaystyle 0<|x-c|<\delta } , it holds that | f ( x ) − L | < ε {\displaystyle |f(x)-L|<\varepsilon } . This is known as the (ε, δ)-definition of limit .
The inequality 0 < | x − c | {\displaystyle 0<|x-c|} is used to exclude c {\displaystyle c} from the set of points under consideration, but some authors do not include this in their definition of limits, replacing 0 < | x − c | < δ {\displaystyle 0<|x-c|<\delta } with simply | x − c | < δ {\displaystyle |x-c|<\delta } . This replacement is equivalent to additionally requiring that f {\displaystyle f} be continuous at c {\displaystyle c} .
It can be proven that there is an equivalent definition which makes manifest the connection between limits of sequences and limits of functions. [ 13 ] The equivalent definition is given as follows. First observe that for every sequence { x n } {\displaystyle \{x_{n}\}} in the domain of f {\displaystyle f} , there is an associated sequence { f ( x n ) } {\displaystyle \{f(x_{n})\}} , the image of the sequence under f {\displaystyle f} . The limit is a real number L {\displaystyle L} so that, for all sequences x n → c {\displaystyle x_{n}\rightarrow c} , the associated sequence f ( x n ) → L {\displaystyle f(x_{n})\rightarrow L} .
It is possible to define the notion of having a "left-handed" limit ("from below"), and a notion of a "right-handed" limit ("from above"). These need not agree. An example is given by the positive indicator function , f : R → R {\displaystyle f:\mathbb {R} \rightarrow \mathbb {R} } , defined such that f ( x ) = 0 {\displaystyle f(x)=0} if x ≤ 0 {\displaystyle x\leq 0} , and f ( x ) = 1 {\displaystyle f(x)=1} if x > 0 {\displaystyle x>0} . At x = 0 {\displaystyle x=0} , the function has a "left-handed limit" of 0, a "right-handed limit" of 1, and its limit does not exist. Symbolically, this can be stated as, for this example, lim x → c − f ( x ) = 0 {\displaystyle \lim _{x\to c^{-}}f(x)=0} , and lim x → c + f ( x ) = 1 {\displaystyle \lim _{x\to c^{+}}f(x)=1} , and from this it can be deduced lim x → c f ( x ) {\displaystyle \lim _{x\to c}f(x)} doesn't exist, because lim x → c − f ( x ) ≠ lim x → c + f ( x ) {\displaystyle \lim _{x\to c^{-}}f(x)\neq \lim _{x\to c^{+}}f(x)} .
It is possible to define the notion of "tending to infinity" in the domain of f {\displaystyle f} , lim x → + ∞ f ( x ) = L . {\displaystyle \lim _{x\rightarrow +\infty }f(x)=L.}
This could be considered equivalent to the limit as a reciprocal tends to 0: lim x ′ → 0 + f ( 1 / x ′ ) = L . {\displaystyle \lim _{x'\rightarrow 0^{+}}f(1/x')=L.}
or it can be defined directly: the "limit of f {\displaystyle f} as x {\displaystyle x} tends to positive infinity" is defined as a value L {\displaystyle L} such that, given any real ε > 0 {\displaystyle \varepsilon >0} , there exists an M > 0 {\displaystyle M>0} so that for all x > M {\displaystyle x>M} , | f ( x ) − L | < ε {\displaystyle |f(x)-L|<\varepsilon } . The definition for sequences is equivalent: As n → + ∞ {\displaystyle n\rightarrow +\infty } , we have f ( x n ) → L {\displaystyle f(x_{n})\rightarrow L} .
In these expressions, the infinity is normally considered to be signed ( + ∞ {\displaystyle +\infty } or − ∞ {\displaystyle -\infty } ) and corresponds to a one-sided limit of the reciprocal. A two-sided infinite limit can be defined, but an author would explicitly write ± ∞ {\displaystyle \pm \infty } to be clear.
It is also possible to define the notion of "tending to infinity" in the value of f {\displaystyle f} , lim x → c f ( x ) = ∞ . {\displaystyle \lim _{x\rightarrow c}f(x)=\infty .}
Again, this could be defined in terms of a reciprocal: lim x → c 1 f ( x ) = 0. {\displaystyle \lim _{x\rightarrow c}{\frac {1}{f(x)}}=0.}
Or a direct definition can be given as follows: given any real number M > 0 {\displaystyle M>0} , there is a δ > 0 {\displaystyle \delta >0} so that for 0 < | x − c | < δ {\displaystyle 0<|x-c|<\delta } , the absolute value of the function | f ( x ) | > M {\displaystyle |f(x)|>M} . A sequence can also have an infinite limit: as n → ∞ {\displaystyle n\rightarrow \infty } , the sequence f ( x n ) → ∞ {\displaystyle f(x_{n})\rightarrow \infty } .
This direct definition is easier to extend to one-sided infinite limits. While mathematicians do talk about functions approaching limits "from above" or "from below", there is not a standard mathematical notation for this as there is for one-sided limits.
In non-standard analysis (which involves a hyperreal enlargement of the number system), the limit of a sequence ( a n ) {\displaystyle (a_{n})} can be expressed as the standard part of the value a H {\displaystyle a_{H}} of the natural extension of the sequence at an infinite hypernatural index n=H . Thus,
Here, the standard part function "st" rounds off each finite hyperreal number to the nearest real number (the difference between them is infinitesimal ). This formalizes the natural intuition that for "very large" values of the index, the terms in the sequence are "very close" to the limit value of the sequence. Conversely, the standard part of a hyperreal a = [ a n ] {\displaystyle a=[a_{n}]} represented in the ultrapower construction by a Cauchy sequence ( a n ) {\displaystyle (a_{n})} , is simply the limit of that sequence:
In this sense, taking the limit and taking the standard part are equivalent procedures.
Let { a n } n > 0 {\displaystyle \{a_{n}\}_{n>0}} be a sequence in a topological space X {\displaystyle X} . For concreteness, X {\displaystyle X} can be thought of as R {\displaystyle \mathbb {R} } , but the definitions hold more generally. The limit set is the set of points such that if there is a convergent subsequence { a n k } k > 0 {\displaystyle \{a_{n_{k}}\}_{k>0}} with a n k → a {\displaystyle a_{n_{k}}\rightarrow a} , then a {\displaystyle a} belongs to the limit set. In this context, such an a {\displaystyle a} is sometimes called a limit point.
A use of this notion is to characterize the "long-term behavior" of oscillatory sequences. For example, consider the sequence a n = ( − 1 ) n {\displaystyle a_{n}=(-1)^{n}} . Starting from n=1, the first few terms of this sequence are − 1 , + 1 , − 1 , + 1 , ⋯ {\displaystyle -1,+1,-1,+1,\cdots } . It can be checked that it is oscillatory, so has no limit, but has limit points { − 1 , + 1 } {\displaystyle \{-1,+1\}} .
This notion is used in dynamical systems , to study limits of trajectories. Defining a trajectory to be a function γ : R → X {\displaystyle \gamma :\mathbb {R} \rightarrow X} , the point γ ( t ) {\displaystyle \gamma (t)} is thought of as the "position" of the trajectory at "time" t {\displaystyle t} . The limit set of a trajectory is defined as follows. To any sequence of increasing times { t n } {\displaystyle \{t_{n}\}} , there is an associated sequence of positions { x n } = { γ ( t n ) } {\displaystyle \{x_{n}\}=\{\gamma (t_{n})\}} . If x {\displaystyle x} is the limit set of the sequence { x n } {\displaystyle \{x_{n}\}} for any sequence of increasing times, then x {\displaystyle x} is a limit set of the trajectory.
Technically, this is the ω {\displaystyle \omega } -limit set. The corresponding limit set for sequences of decreasing time is called the α {\displaystyle \alpha } -limit set.
An illustrative example is the circle trajectory: γ ( t ) = ( cos ( t ) , sin ( t ) ) {\displaystyle \gamma (t)=(\cos(t),\sin(t))} . This has no unique limit, but for each θ ∈ R {\displaystyle \theta \in \mathbb {R} } , the point ( cos ( θ ) , sin ( θ ) ) {\displaystyle (\cos(\theta ),\sin(\theta ))} is a limit point, given by the sequence of times t n = θ + 2 π n {\displaystyle t_{n}=\theta +2\pi n} . But the limit points need not be attained on the trajectory. The trajectory γ ( t ) = t / ( 1 + t ) ( cos ( t ) , sin ( t ) ) {\displaystyle \gamma (t)=t/(1+t)(\cos(t),\sin(t))} also has the unit circle as its limit set.
Limits are used to define a number of important concepts in analysis.
A particular expression of interest which is formalized as the limit of a sequence is sums of infinite series. These are "infinite sums" of real numbers, generally written as ∑ n = 1 ∞ a n . {\displaystyle \sum _{n=1}^{\infty }a_{n}.} This is defined through limits as follows: [ 13 ] given a sequence of real numbers { a n } {\displaystyle \{a_{n}\}} , the sequence of partial sums is defined by s n = ∑ i = 1 n a i . {\displaystyle s_{n}=\sum _{i=1}^{n}a_{i}.} If the limit of the sequence { s n } {\displaystyle \{s_{n}\}} exists, the value of the expression ∑ n = 1 ∞ a n {\displaystyle \sum _{n=1}^{\infty }a_{n}} is defined to be the limit. Otherwise, the series is said to be divergent.
A classic example is the Basel problem , where a n = 1 / n 2 {\displaystyle a_{n}=1/n^{2}} . Then ∑ n = 1 ∞ 1 n 2 = π 2 6 . {\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n^{2}}}={\frac {\pi ^{2}}{6}}.}
However, while for sequences there is essentially a unique notion of convergence, for series there are different notions of convergence. This is due to the fact that the expression ∑ n = 1 ∞ a n {\displaystyle \sum _{n=1}^{\infty }a_{n}} does not discriminate between different orderings of the sequence { a n } {\displaystyle \{a_{n}\}} , while the convergence properties of the sequence of partial sums can depend on the ordering of the sequence.
A series which converges for all orderings is called unconditionally convergent . It can be proven to be equivalent to absolute convergence . This is defined as follows. A series is absolutely convergent if ∑ n = 1 ∞ | a n | {\displaystyle \sum _{n=1}^{\infty }|a_{n}|} is well defined. Furthermore, all possible orderings give the same value.
Otherwise, the series is conditionally convergent . A surprising result for conditionally convergent series is the Riemann series theorem : depending on the ordering, the partial sums can be made to converge to any real number, as well as ± ∞ {\displaystyle \pm \infty } .
A useful application of the theory of sums of series is for power series. These are sums of series of the form f ( z ) = ∑ n = 0 ∞ c n z n . {\displaystyle f(z)=\sum _{n=0}^{\infty }c_{n}z^{n}.} Often z {\displaystyle z} is thought of as a complex number, and a suitable notion of convergence of complex sequences is needed. The set of values of z ∈ C {\displaystyle z\in \mathbb {C} } for which the series sum converges is a circle, with its radius known as the radius of convergence .
The definition of continuity at a point is given through limits.
The above definition of a limit is true even if f ( c ) ≠ L {\displaystyle f(c)\neq L} . Indeed, the function f need not even be defined at c . However, if f ( c ) {\displaystyle f(c)} is defined and is equal to L {\displaystyle L} , then the function is said to be continuous at the point c {\displaystyle c} .
Equivalently, the function is continuous at c {\displaystyle c} if f ( x ) → f ( c ) {\displaystyle f(x)\rightarrow f(c)} as x → c {\displaystyle x\rightarrow c} , or in terms of sequences, whenever x n → c {\displaystyle x_{n}\rightarrow c} , then f ( x n ) → f ( c ) {\displaystyle f(x_{n})\rightarrow f(c)} .
An example of a limit where f {\displaystyle f} is not defined at c {\displaystyle c} is given below.
Consider the function
f ( x ) = x 2 − 1 x − 1 . {\displaystyle f(x)={\frac {x^{2}-1}{x-1}}.}
then f (1) is not defined (see Indeterminate form ), yet as x moves arbitrarily close to 1, f ( x ) correspondingly approaches 2: [ 14 ]
Thus, f ( x ) can be made arbitrarily close to the limit of 2—just by making x sufficiently close to 1 .
In other words, lim x → 1 x 2 − 1 x − 1 = 2. {\displaystyle \lim _{x\to 1}{\frac {x^{2}-1}{x-1}}=2.}
This can also be calculated algebraically, as x 2 − 1 x − 1 = ( x + 1 ) ( x − 1 ) x − 1 = x + 1 {\textstyle {\frac {x^{2}-1}{x-1}}={\frac {(x+1)(x-1)}{x-1}}=x+1} for all real numbers x ≠ 1 .
Now, since x + 1 is continuous in x at 1, we can now plug in 1 for x , leading to the equation lim x → 1 x 2 − 1 x − 1 = 1 + 1 = 2. {\displaystyle \lim _{x\to 1}{\frac {x^{2}-1}{x-1}}=1+1=2.}
In addition to limits at finite values, functions can also have limits at infinity. For example, consider the function f ( x ) = 2 x − 1 x {\displaystyle f(x)={\frac {2x-1}{x}}} where:
As x becomes extremely large, the value of f ( x ) approaches 2 , and the value of f ( x ) can be made as close to 2 as one could wish—by making x sufficiently large. So in this case, the limit of f ( x ) as x approaches infinity is 2 , or in mathematical notation, lim x → ∞ 2 x − 1 x = 2. {\displaystyle \lim _{x\to \infty }{\frac {2x-1}{x}}=2.}
An important class of functions when considering limits are continuous functions . These are precisely those functions which preserve limits , in the sense that if f {\displaystyle f} is a continuous function, then whenever a n → a {\displaystyle a_{n}\rightarrow a} in the domain of f {\displaystyle f} , then the limit f ( a n ) {\displaystyle f(a_{n})} exists and furthermore is f ( a ) {\displaystyle f(a)} .
In the most general setting of topological spaces, a short proof is given below:
Let f : X → Y {\displaystyle f:X\rightarrow Y} be a continuous function between topological spaces X {\displaystyle X} and Y {\displaystyle Y} . By definition, for each open set V {\displaystyle V} in Y {\displaystyle Y} , the preimage f − 1 ( V ) {\displaystyle f^{-1}(V)} is open in X {\displaystyle X} .
Now suppose a n → a {\displaystyle a_{n}\rightarrow a} is a sequence with limit a {\displaystyle a} in X {\displaystyle X} . Then f ( a n ) {\displaystyle f(a_{n})} is a sequence in Y {\displaystyle Y} , and f ( a ) {\displaystyle f(a)} is some point.
Choose a neighborhood V {\displaystyle V} of f ( a ) {\displaystyle f(a)} . Then f − 1 ( V ) {\displaystyle f^{-1}(V)} is an open set (by continuity of f {\displaystyle f} ) which in particular contains a {\displaystyle a} , and therefore f − 1 ( V ) {\displaystyle f^{-1}(V)} is a neighborhood of a {\displaystyle a} . By the convergence of a n {\displaystyle a_{n}} to a {\displaystyle a} , there exists an N {\displaystyle N} such that for n > N {\displaystyle n>N} , we have a n ∈ f − 1 ( V ) {\displaystyle a_{n}\in f^{-1}(V)} .
Then applying f {\displaystyle f} to both sides gives that, for the same N {\displaystyle N} , for each n > N {\displaystyle n>N} we have f ( a n ) ∈ V {\displaystyle f(a_{n})\in V} . Originally V {\displaystyle V} was an arbitrary neighborhood of f ( a ) {\displaystyle f(a)} , so f ( a n ) → f ( a ) {\displaystyle f(a_{n})\rightarrow f(a)} . This concludes the proof.
In real analysis, for the more concrete case of real-valued functions defined on a subset E ⊂ R {\displaystyle E\subset \mathbb {R} } , that is, f : E → R {\displaystyle f:E\rightarrow \mathbb {R} } , a continuous function may also be defined as a function which is continuous at every point of its domain.
In topology , limits are used to define limit points of a subset of a topological space, which in turn give a useful characterization of closed sets .
In a topological space X {\displaystyle X} , consider a subset S {\displaystyle S} . A point a {\displaystyle a} is called a limit point if there is a sequence { a n } {\displaystyle \{a_{n}\}} in S ∖ { a } {\displaystyle S\backslash \{a\}} such that a n → a {\displaystyle a_{n}\rightarrow a} .
The reason why { a n } {\displaystyle \{a_{n}\}} is defined to be in S ∖ { a } {\displaystyle S\backslash \{a\}} rather than just S {\displaystyle S} is illustrated by the following example. Take X = R {\displaystyle X=\mathbb {R} } and S = [ 0 , 1 ] ∪ { 2 } {\displaystyle S=[0,1]\cup \{2\}} . Then 2 ∈ S {\displaystyle 2\in S} , and therefore is the limit of the constant sequence 2 , 2 , ⋯ {\displaystyle 2,2,\cdots } . But 2 {\displaystyle 2} is not a limit point of S {\displaystyle S} .
A closed set, which is defined to be the complement of an open set, is equivalently any set C {\displaystyle C} which contains all its limit points.
The derivative is defined formally as a limit. In the scope of real analysis , the derivative is first defined for real functions f {\displaystyle f} defined on a subset E ⊂ R {\displaystyle E\subset \mathbb {R} } . The derivative at x ∈ E {\displaystyle x\in E} is defined as follows. If the limit of f ( x + h ) − f ( x ) h {\displaystyle {\frac {f(x+h)-f(x)}{h}}} as h → 0 {\displaystyle h\rightarrow 0} exists, then the derivative at x {\displaystyle x} is this limit.
Equivalently, it is the limit as y → x {\displaystyle y\rightarrow x} of f ( y ) − f ( x ) y − x . {\displaystyle {\frac {f(y)-f(x)}{y-x}}.}
If the derivative exists, it is commonly denoted by f ′ ( x ) {\displaystyle f'(x)} .
For sequences of real numbers, a number of properties can be proven. [ 13 ] Suppose { a n } {\displaystyle \{a_{n}\}} and { b n } {\displaystyle \{b_{n}\}} are two sequences converging to a {\displaystyle a} and b {\displaystyle b} respectively.
a n + b n → a + b . {\displaystyle a_{n}+b_{n}\rightarrow a+b.}
a n ⋅ b n → a ⋅ b . {\displaystyle a_{n}\cdot b_{n}\rightarrow a\cdot b.}
1 a n → 1 a . {\displaystyle {\frac {1}{a_{n}}}\rightarrow {\frac {1}{a}}.} Equivalently, the function f ( x ) = 1 / x {\displaystyle f(x)=1/x} is continuous about nonzero x {\displaystyle x} .
A property of convergent sequences of real numbers is that they are Cauchy sequences . [ 13 ] The definition of a Cauchy sequence { a n } {\displaystyle \{a_{n}\}} is that for every real number ε > 0 {\displaystyle \varepsilon >0} , there is an N {\displaystyle N} such that whenever m , n > N {\displaystyle m,n>N} , | a m − a n | < ε . {\displaystyle |a_{m}-a_{n}|<\varepsilon .}
Informally, for any arbitrarily small error ε {\displaystyle \varepsilon } , it is possible to find an interval of diameter ε {\displaystyle \varepsilon } such that eventually the sequence is contained within the interval.
Cauchy sequences are closely related to convergent sequences. In fact, for sequences of real numbers they are equivalent: any Cauchy sequence is convergent.
In general metric spaces, it continues to hold that convergent sequences are also Cauchy. But the converse is not true: not every Cauchy sequence is convergent in a general metric space. A classic counterexample is the rational numbers , Q {\displaystyle \mathbb {Q} } , with the usual distance. The sequence of decimal approximations to 2 {\displaystyle {\sqrt {2}}} , truncated at the n {\displaystyle n} th decimal place is a Cauchy sequence, but does not converge in Q {\displaystyle \mathbb {Q} } .
A metric space in which every Cauchy sequence is also convergent, that is, Cauchy sequences are equivalent to convergent sequences, is known as a complete metric space .
One reason Cauchy sequences can be "easier to work with" than convergent sequences is that they are a property of the sequence { a n } {\displaystyle \{a_{n}\}} alone, while convergent sequences require not just the sequence { a n } {\displaystyle \{a_{n}\}} but also the limit of the sequence a {\displaystyle a} .
Beyond whether or not a sequence { a n } {\displaystyle \{a_{n}\}} converges to a limit a {\displaystyle a} , it is possible to describe how fast a sequence converges to a limit. One way to quantify this is using the order of convergence of a sequence.
A formal definition of order of convergence can be stated as follows. Suppose { a n } n > 0 {\displaystyle \{a_{n}\}_{n>0}} is a sequence of real numbers which is convergent with limit a {\displaystyle a} . Furthermore, a n ≠ a {\displaystyle a_{n}\neq a} for all n {\displaystyle n} . If positive constants λ {\displaystyle \lambda } and α {\displaystyle \alpha } exist such that lim n → ∞ | a n + 1 − a | | a n − a | α = λ {\displaystyle \lim _{n\to \infty }{\frac {\left|a_{n+1}-a\right|}{\left|a_{n}-a\right|^{\alpha }}}=\lambda } then a n {\displaystyle a_{n}} is said to converge to a {\displaystyle a} with order of convergence α {\displaystyle \alpha } . The constant λ {\displaystyle \lambda } is known as the asymptotic error constant.
Order of convergence is used for example the field of numerical analysis , in error analysis.
Limits can be difficult to compute. There exist limit expressions whose modulus of convergence is undecidable . In recursion theory , the limit lemma proves that it is possible to encode undecidable problems using limits. [ 15 ]
There are several theorems or tests that indicate whether the limit exists. These are known as convergence tests . Examples include the ratio test and the squeeze theorem . However they may not tell how to compute the limit. | https://en.wikipedia.org/wiki/Limit_(mathematics) |
In mathematics , the limit comparison test (LCT) (in contrast with the related direct comparison test ) is a method of testing for the convergence of an infinite series .
Suppose that we have two series Σ n a n {\displaystyle \Sigma _{n}a_{n}} and Σ n b n {\displaystyle \Sigma _{n}b_{n}} with a n ≥ 0 , b n > 0 {\displaystyle a_{n}\geq 0,b_{n}>0} for all n {\displaystyle n} .
Then if lim n → ∞ a n b n = c {\displaystyle \lim _{n\to \infty }{\frac {a_{n}}{b_{n}}}=c} with 0 < c < ∞ {\displaystyle 0<c<\infty } , then either both series converge or both series diverge. [ 1 ]
Because lim n → ∞ a n b n = c {\displaystyle \lim _{n\to \infty }{\frac {a_{n}}{b_{n}}}=c} we know that for every ε > 0 {\displaystyle \varepsilon >0} there is a positive integer n 0 {\displaystyle n_{0}} such that for all n ≥ n 0 {\displaystyle n\geq n_{0}} we have that | a n b n − c | < ε {\displaystyle \left|{\frac {a_{n}}{b_{n}}}-c\right|<\varepsilon } , or equivalently
As c > 0 {\displaystyle c>0} we can choose ε {\displaystyle \varepsilon } to be sufficiently small such that c − ε {\displaystyle c-\varepsilon } is positive.
So b n < 1 c − ε a n {\displaystyle b_{n}<{\frac {1}{c-\varepsilon }}a_{n}} and by the direct comparison test , if ∑ n a n {\displaystyle \sum _{n}a_{n}} converges then so does ∑ n b n {\displaystyle \sum _{n}b_{n}} .
Similarly a n < ( c + ε ) b n {\displaystyle a_{n}<(c+\varepsilon )b_{n}} , so if ∑ n b n {\displaystyle \sum _{n}b_{n}} diverges, again by the direct comparison test, so does ∑ n a n {\displaystyle \sum _{n}a_{n}} .
That is, both series converge or both series diverge.
We want to determine if the series ∑ n = 1 ∞ 1 n 2 + 2 n {\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n^{2}+2n}}} converges. For this we compare it with the convergent series ∑ n = 1 ∞ 1 n 2 = π 2 6 {\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n^{2}}}={\frac {\pi ^{2}}{6}}}
As lim n → ∞ 1 n 2 + 2 n n 2 1 = 1 > 0 {\displaystyle \lim _{n\to \infty }{\frac {1}{n^{2}+2n}}{\frac {n^{2}}{1}}=1>0} we have that the original series also converges.
One can state a one-sided comparison test by using limit superior . Let a n , b n ≥ 0 {\displaystyle a_{n},b_{n}\geq 0} for all n {\displaystyle n} . Then if lim sup n → ∞ a n b n = c {\displaystyle \limsup _{n\to \infty }{\frac {a_{n}}{b_{n}}}=c} with 0 ≤ c < ∞ {\displaystyle 0\leq c<\infty } and Σ n b n {\displaystyle \Sigma _{n}b_{n}} converges, necessarily Σ n a n {\displaystyle \Sigma _{n}a_{n}} converges.
Let a n = 1 − ( − 1 ) n n 2 {\displaystyle a_{n}={\frac {1-(-1)^{n}}{n^{2}}}} and b n = 1 n 2 {\displaystyle b_{n}={\frac {1}{n^{2}}}} for all natural numbers n {\displaystyle n} . Now lim n → ∞ a n b n = lim n → ∞ ( 1 − ( − 1 ) n ) {\displaystyle \lim _{n\to \infty }{\frac {a_{n}}{b_{n}}}=\lim _{n\to \infty }(1-(-1)^{n})} does not exist, so we cannot apply the standard comparison test. However, lim sup n → ∞ a n b n = lim sup n → ∞ ( 1 − ( − 1 ) n ) = 2 ∈ [ 0 , ∞ ) {\displaystyle \limsup _{n\to \infty }{\frac {a_{n}}{b_{n}}}=\limsup _{n\to \infty }(1-(-1)^{n})=2\in [0,\infty )} and since ∑ n = 1 ∞ 1 n 2 {\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n^{2}}}} converges, the one-sided comparison test implies that ∑ n = 1 ∞ 1 − ( − 1 ) n n 2 {\displaystyle \sum _{n=1}^{\infty }{\frac {1-(-1)^{n}}{n^{2}}}} converges.
Let a n , b n ≥ 0 {\displaystyle a_{n},b_{n}\geq 0} for all n {\displaystyle n} . If Σ n a n {\displaystyle \Sigma _{n}a_{n}} diverges and Σ n b n {\displaystyle \Sigma _{n}b_{n}} converges, then necessarily lim sup n → ∞ a n b n = ∞ {\displaystyle \limsup _{n\to \infty }{\frac {a_{n}}{b_{n}}}=\infty } , that is, lim inf n → ∞ b n a n = 0 {\displaystyle \liminf _{n\to \infty }{\frac {b_{n}}{a_{n}}}=0} . The essential content here is that in some sense the numbers a n {\displaystyle a_{n}} are larger than the numbers b n {\displaystyle b_{n}} .
Let f ( z ) = ∑ n = 0 ∞ a n z n {\displaystyle f(z)=\sum _{n=0}^{\infty }a_{n}z^{n}} be analytic in the unit disc D = { z ∈ C : | z | < 1 } {\displaystyle D=\{z\in \mathbb {C} :|z|<1\}} and have image of finite area. By Parseval's formula the area of the image of f {\displaystyle f} is proportional to ∑ n = 1 ∞ n | a n | 2 {\displaystyle \sum _{n=1}^{\infty }n|a_{n}|^{2}} . Moreover, ∑ n = 1 ∞ 1 / n {\displaystyle \sum _{n=1}^{\infty }1/n} diverges. Therefore, by the converse of the comparison test, we have lim inf n → ∞ n | a n | 2 1 / n = lim inf n → ∞ ( n | a n | ) 2 = 0 {\displaystyle \liminf _{n\to \infty }{\frac {n|a_{n}|^{2}}{1/n}}=\liminf _{n\to \infty }(n|a_{n}|)^{2}=0} , that is, lim inf n → ∞ n | a n | = 0 {\displaystyle \liminf _{n\to \infty }n|a_{n}|=0} . | https://en.wikipedia.org/wiki/Limit_comparison_test |
In mathematics , in the study of dynamical systems with two-dimensional phase space , a limit cycle is a closed trajectory in phase space having the property that at least one other trajectory spirals into it either as time approaches infinity or as time approaches negative infinity. Such behavior is exhibited in some nonlinear systems . Limit cycles have been used to model the behavior of many real-world oscillatory systems. The study of limit cycles was initiated by Henri Poincaré (1854–1912).
We consider a two-dimensional dynamical system of the form x ′ ( t ) = V ( x ( t ) ) {\displaystyle x'(t)=V(x(t))} where V : R 2 → R 2 {\displaystyle V:\mathbb {R} ^{2}\to \mathbb {R} ^{2}} is a smooth function. A trajectory of this system is some smooth function x ( t ) {\displaystyle x(t)} with values in R 2 {\displaystyle \mathbb {R} ^{2}} which satisfies this differential equation. Such a trajectory is called closed (or periodic ) if it is not constant but returns to its starting point, i.e. if there exists some t 0 > 0 {\displaystyle t_{0}>0} such that x ( t + t 0 ) = x ( t ) {\displaystyle x(t+t_{0})=x(t)} for all t ∈ R {\displaystyle t\in \mathbb {R} } . An orbit is the image of a trajectory, a subset of R 2 {\displaystyle \mathbb {R} ^{2}} . A closed orbit , or cycle , is the image of a closed trajectory. A limit cycle is a cycle which is the limit set of some other trajectory.
By the Jordan curve theorem , every closed trajectory divides the plane into two regions, the interior and the exterior of the curve.
Given a limit cycle and a trajectory in its interior that approaches the limit cycle for time approaching + ∞ {\displaystyle +\infty } , then there is a neighborhood around the limit cycle such that all trajectories in the interior that start in the neighborhood approach the limit cycle for time approaching + ∞ {\displaystyle +\infty } . The corresponding statement holds for a trajectory in the interior that approaches the limit cycle for time approaching − ∞ {\displaystyle -\infty } , and also for trajectories in the exterior approaching the limit cycle.
In the case where all the neighboring trajectories approach the limit cycle as time approaches infinity, it is called a stable or attractive limit cycle (ω-limit cycle). If instead, all neighboring trajectories approach it as time approaches negative infinity, then it is an unstable limit cycle (α-limit cycle). If there is a neighboring trajectory which spirals into the limit cycle as time approaches infinity, and another one which spirals into it as time approaches negative infinity, then it is a semi-stable limit cycle. There are also limit cycles that are neither stable, unstable nor semi-stable: for instance, a neighboring trajectory may approach the limit cycle from the outside, but the inside of the limit cycle is approached by a family of other cycles (which would not be limit cycles).
Stable limit cycles are examples of attractors . They imply self-sustained oscillations : the closed trajectory describes the perfect periodic behavior of the system, and any small perturbation from this closed trajectory causes the system to return to it, making the system stick to the limit cycle.
Every closed trajectory contains within its interior a stationary point of the system, i.e. a point p {\displaystyle p} where V ′ ( p ) = 0 {\displaystyle V'(p)=0} . The Bendixson–Dulac theorem and the Poincaré–Bendixson theorem predict the absence or existence, respectively, of limit cycles of two-dimensional nonlinear dynamical systems.
Finding limit cycles, in general, is a very difficult problem. The number of limit cycles of a polynomial differential equation in the plane is the main object of the second part of Hilbert's sixteenth problem . It is unknown, for instance, whether there is any system x ′ = V ( x ) {\displaystyle x'=V(x)} in the plane where both components of V {\displaystyle V} are quadratic polynomials of the two variables, such that the system has more than 4 limit cycles.
Limit cycles are important in many scientific applications where systems with self-sustained oscillations are modelled. Some examples include: | https://en.wikipedia.org/wiki/Limit_cycle |
Limit load is the maximum load that a structure can safely carry. It's the load at which the structure is in a state of incipient plastic collapse. As the load on the structure increases, the displacements increases linearly in the elastic range until the load attains the yield value. Beyond this, the load-displacement response becomes non-linear and the plastic or irreversible part of the displacement increases steadily with the applied load. Plasticity spreads throughout the solid and at the limit load, the plastic zone becomes very large and the displacements become unbounded and the component is said to have collapsed.
Any load above the limit load will lead to the formation of plastic hinge in the structure. Engineers use limit states to define and check a structure's performance.
Bounding Theorems of Plastic-Limit Load Analysis : Plastic limit theorems provide a way to calculate limit loads without having to solve the boundary value problem in continuum mechanics . Finite element analysis provides an alternative way to estimate limit loads. They are:
The Upper Bound Plastic Collapse Theorem states that an upper bound to the collapse loads can be obtained by postulating a collapse mechanism and computing the ratio of its plastic dissipation to the work done by the applied loads.
This standards - or measurement -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Limit_load_(physics) |
Limit State Design ( LSD ), also known as Load And Resistance Factor Design ( LRFD ), refers to a design method used in structural engineering . A limit state is a condition of a structure beyond which it no longer fulfills the relevant design criteria. [ 1 ] The condition may refer to a degree of loading or other actions on the structure, while the criteria refer to structural integrity, fitness for use, durability or other design requirements. A structure designed by LSD is proportioned to sustain all actions likely to occur during its design life, and to remain fit for use, with an appropriate level of reliability for each limit state. Building codes based on LSD implicitly define the appropriate levels of reliability by their prescriptions.
The method of limit state design, developed in the USSR and based on research led by Professor N.S. Streletski, was introduced in USSR building regulations in 1955.
Limit state design requires the structure to satisfy two principal criteria: the ultimate limit state (ULS) and the serviceability limit state (SLS). [ 2 ]
Any design process involves a number of assumptions. First: the loads to which a structure will be subjected 2: foreseeable or cognizable possible exceptional scenarios and the stresses these may impress, and 3) the individual and collective strengths pertaining to any constituent part or sum of parts as a group and as a whole.
A clear distinction is made between the ultimate state (US) and the ultimate limit state (ULS). The Ultimate State is a physical situation that involves either excessive deformations sufficient to cause collapse of the component under consideration or the structure as a whole, or deformations exceeding values considered to be the acceptable tolerance.
A structure is deemed to satisfy the ultimate limit based upon arbitrary criteria, per the nominal, not physical, intentions or goals set forth by human actors, and that, as such, have nothing to do with engineering strictly speaking, but instead exist "on paper" to conceal, distort, or otherwise obfuscate the true fundamental behaviors applicable to a structure.
Complying with the design criteria of the ULS is not sufficient to perform the minimum requisite steps necessary for proper structural safety.
In addition to the ULS check mentioned above, a Service Limit State (SLS) computational check must be performed. To satisfy the serviceability limit state criterion, a structure must remain functional for the duration of its intended use subject to routine (everyday) loading.
The load and resistance factors are determined using statistics and a pre-selected probability of failure. Variability in the quality of construction, consistency of the construction material are accounted for in the factors. Generally, a factor of unity (one) or less is applied to the resistances of the material, and a factor of unity or greater to the loads. Not often used, but in some load cases a factor may be less than unity due to a reduced probability of the combined loads.
The aforementioned factors can differ for different materials or even between differing grades of the same material. For example, wood has larger factor of variability than steel. The factors applied to resistance also account for the degree of scientific confidence in the derivation of the values.
In determining the specific magnitude of the factors, more deterministic loads (e.g., dead load - the weight of the structure and permanent attachments like walls, floor treatments, ceiling finishes) are given lower factors (for example 1.4) than highly variable loads like earthquake, wind, or live (occupancy) loads (1.6). Impact loads are typically given higher factors still (say 2.0) in order to account for both their unpredictable magnitudes and the dynamic nature of the loading vs. the static nature of most models.
Limit states design has the potential to produce a more consistently designed structure as each element is intended to have the same probability of failure. In practical terms this normally results in a more efficient structure, and as such, it can be argued that LSD is superior from a practical engineering viewpoint.
The following is the treatment of LSD found in the National Building Code of Canada :
Limit state design has replaced the older concept of permissible stress design in most forms of civil engineering . A notable exception is transportation engineering . Even so, new codes are currently being developed for both geotechnical and transportation engineering which are LSD based. As a result, most modern buildings are designed in accordance with a code which is based on limit state theory. For example, in Europe, structures are designed to conform with the Eurocodes : Steel structures are designed in accordance with EN 1993 , and reinforced concrete structures to EN 1992 . Australia, Canada, China, France, Indonesia, and New Zealand (among many others) utilise limit state theory in the development of their design codes. In the purest sense, it is now considered inappropriate to discuss safety factors when working with LSD, as there are concerns that this may lead to confusion. Previously, it has been shown that the LRFD and ASD can produce significantly different designs of steel gable frames. [ 3 ]
There are few situations where ASD produces significantly lighter weight steel gable frame designs. Additionally, it has been shown that in high snow regions, the difference between the methods is more dramatic. [ 4 ]
The United States has been particularly slow to adopt limit state design (known as Load and Resistance Factor Design in the US). Design codes and standards are issued by diverse organizations, some of which have adopted limit state design, and others have not.
The ACI 318 Building Code Requirements for Structural Concrete uses Limit State design.
The ANSI/ AISC 360 Specification for Structural Steel Buildings , the ANSI/ AISI S-100 North American Specification for the Design of Cold Formed Steel Structural Members , and The Aluminum Association 's Aluminum Design Manual contain two methods of design side by side:
In contrast, the ANSI/ AWWA D100 Welded Carbon Steel Tanks for Water Storage and API 650 Welded Tanks for Oil Storage still use allowable stress design .
In Europe, the limit state design is enforced by the Eurocodes . | https://en.wikipedia.org/wiki/Limit_state_design |
In the philosophy of mathematics , specifically the philosophical foundations of set theory , limitation of size is a concept developed by Philip Jourdain and/or Georg Cantor to avoid Cantor's paradox . It identifies certain "inconsistent multiplicities", in Cantor's terminology, that cannot be sets because they are "too large". In modern terminology these are called proper classes .
The axiom of limitation of size is an axiom in some versions of von Neumann–Bernays–Gödel set theory or Morse–Kelley set theory . This axiom says that any class that is not "too large" is a set, and a set cannot be "too large". "Too large" is defined as being large enough that the class of all sets can be mapped one-to-one into it.
This logic -related article is a stub . You can help Wikipedia by expanding it .
This set theory -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Limitation_of_size |
In mathematics , a limiting case of a mathematical object is a special case that arises when one or more components of the object take on their most extreme possible values. [ 1 ] For example:
A limiting case is sometimes a degenerate case in which some qualitative properties differ from the corresponding properties of the generic case . For example:
This mathematics -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Limiting_case_(mathematics) |
A limiting factor is a variable of a system that causes a noticeable change in output or another measure of a type of system. The limiting factor is in a pyramid shape of organisms going up from the producers to consumers and so on. A factor not limiting over a certain domain of starting conditions may yet be limiting over another domain of starting conditions, including that of the factor.
The identification of a factor as limiting is possible only in distinction to one or more other factors that are non-limiting. Disciplines differ in their use of the term as to whether they allow the simultaneous existence of more than one limiting factor (which may then be called "co-limiting"), but they all require the existence of at least one non-limiting factor when the terms are used. There are several different possible scenarios of limitation when more than one factor is present. The first scenario, called single limitation occurs when only one factor, the one with maximum demand, limits the System. Serial co-limitation is when one factor has no direct limiting effects on the system, but must be present to increase the limitation of a second factor. A third scenario, independent limitation, occurs when two factors both have limiting effects on the system but work through different mechanisms. Another scenario, synergistic limitation, occurs when both factors contribute to the same limitation mechanism, but in different ways. [ 1 ]
In 1905 Frederick Blackman articulated the role of limiting factors as follows: "When a process is conditioned as to its rapidity by several separate factors the rate of the process is limited by the pace of the slowest factor." In terms of the magnitude of a function, he wrote, "When the magnitude of a function is limited by one of a set of possible factors, increase of that factor, and of that one alone, will be found to bring about an increase of the magnitude of the function." [ 2 ]
In population ecology , a regulating factor , also known as a limiting factor , [ 3 ] is something that keeps a population at equilibrium (neither increasing nor decreasing in size over time). [ citation needed ] Common limiting factor resources are environmental features that limit the growth, abundance, or distribution of an organism or a population of organisms in an ecosystem. [ 4 ] : G-11 [ 5 ] The concept of limiting factors is based on Liebig's Law of the Minimum , which states that growth is controlled not by the total amount of resources available, but by the scarcest resource. In other words, a factor is limiting if a change in the factor produces increased growth, abundance, or distribution of an organism when other factors necessary to the organism's life do not. Limiting factors may be physical or biological. [ 4 ] : 417, 8
Limiting factors are not limited to the condition of the species. Some factors may be increased or reduced based on circumstances. An example of a limiting factor is sunlight in the rain forest , where growth is limited to all plants on the forest floor unless more light becomes available. This decreases the number of potential factors that could influence a biological process, but only one is in effect at any one place and time. This recognition that there is always a single limiting factor is vital in ecology , and the concept has parallels in numerous other processes. The limiting factor also causes competition between individuals of a species population. For example, space is a limiting factor. Many predators and prey need a certain amount of space for survival: food, water, and other biological needs. If the population of a species is too high, they start competing for those needs. Thus the limiting factors hold down population in an area by causing some individuals to seek better prospects elsewhere and others to stay and starve. Some other limiting factors in biology include temperature and other weather related factors. Species can also be limited by the availability of macro- and micronutrients. There has even been evidence of co-limitation in prairie ecosystems. A study published in 2017 showed that sodium (a micronutrient) had no effect on its own, but when in combination with nitrogen and phosphorus (macronutrients), it did show positive effects, which is evidence of serial co-limitation. [ 1 ]
In oceanography, a prime example of a limiting factor is a limiting nutrient . Nutrient availability in freshwater and marine environments plays a critical role in determining what organisms survive and thrive. Nutrients are the building blocks of all living organisms, as they support biological activity. They are required to make proteins, DNA, membranes, organelles, and exoskeletons. The major elements that constitute >95% of organic matter mass are carbon, hydrogen, nitrogen, oxygen, sulfur, and phosphorus. Minor elements are iron, manganese, cobalt, zinc and copper. These minor elements are often only present in trace amounts but they are key as co-limiting factors as parts of enzymes, transporters, vitamins and amino acids. Within aquatic environments, nitrogen and phosphorus are leading contenders for most limiting nutrients.
Discovery of the Redfield ratio was a major insight that helped understand the relationship between nutrient availability in seawater and their relative abundance in organisms. Redfield was able to notice elemental consistencies between carbon, nitrogen and phosphorus when looking at larger organisms living in the ocean (C:N:P = 106:16:1). [ 6 ] He also observed consistencies in nutrients within the water column; nitrate to phosphate ratio was 16:1. The overarching idea was that the environment fundamentally influences the organisms that grow in it and the growing organisms fundamentally influence the environment. Redfield's opening statement in his 1934 paper explains "It is now well recognized that the growth of plankton in the surface layers of the sea is limited in part by the quantities of phosphate and nitrate available for their use and that the changes in the relative quantities of certain substances in seawater are determined in their relative proportions by biological activity". [ 7 ] Deviations from Redfield can be used to infer elemental limitations. Limiting nutrients can be discussed in terms of dissolved nutrients, suspended particles and sinking particles, among others. When discussing dissolved nutrient stoichiometry, large deviations from the original Redfield ratio can determine if an environment is phosphorus limited or nitrogen limited. When discussing suspended particle stoichiometry, higher N:P ratios are noted in oligotrophic waters (environments dominated by cyanobacteria ; low latitudes/equator) and lower N:P ratios are noted in nutrient rich ecosystems (environments dominated by diatoms ; high latitudes/poles). [ 8 ]
Many areas are severely nitrogen limited, but phosphorus limitation has also been observed. In many instances trace metals or co-limitation occur. Co-limitations refer to where two or more nutrients simultaneously limit a process. Pinpointing a single limiting factor can be challenging, as nutrient demand varies between organisms, life cycles, and environmental conditions (e.g. thermal stress can increase demand on nutrients for biological repairs).
AllBusiness.com defines a limiting (constraining) factor as an "item that restricts or limits production or sale of a given product". The examples provided include: "limited machine hours and labor-hours and shortage of materials and skilled labor. Other limiting factors may be cubic feet of display or warehouse space, or working capital." [ 9 ] The term is also frequently used in technology literature. [ 10 ] [ 11 ]
The analysis of limiting business factors is part of the program evaluation and review technique , critical path analysis , and theory of constraints as presented in The Goal .
In stoichiometry of a chemical reaction to produce a chemical product, it may be observed or predicted that with amounts supplied in specified proportions, one of the reactants will be consumed by the reaction before the others. The supply of this reagent thus limits the amount of product. This limiting reagent determines the theoretical yield of the reaction. The other reactants are said to be non-limiting or in excess. This distinction makes sense only when the chemical equilibrium so favors the products to cause the complete consumption of one of the reactants.
In studies of reaction kinetics , the rate of progress of the reaction may be limited by the concentration of one of the reactants or catalyst . In multi-step reactions, a step may be rate-limiting in terms of the production of the final product. In vivo , in an organism or an ecologic system , such factors as those may be rate-limiting, or in the overall analysis of a multi-step process including biologic , geologic , hydrologic , or atmospheric transport and chemical reactions , transport of a reactant may be limiting. | https://en.wikipedia.org/wiki/Limiting_factor |
The limiting oxygen concentration ( LOC ), [ 1 ] also known as the minimum oxygen concentration ( MOC ), [ 2 ] is defined as the limiting concentration of oxygen below which combustion is not possible, independent of the concentration of fuel. It is expressed in units of volume percent of oxygen. The LOC varies with pressure and temperature. It is also dependent on the type of inert (non-flammable) gas.
Limiting oxygen concentration for solid materials [ 4 ]
The effect of increasing the concentration of inert gas can be understood by viewing the inert as thermal ballast that quenches the flame temperature to a level below which the flame cannot exist. [ 5 ] Carbon dioxide is therefore more effective than nitrogen due to its higher molar heat capacity . [ 6 ]
The concept has important practical use in fire safety engineering . For instance, to safely fill a new container or a pressure vessel with flammable gases, the atmosphere of normal air (containing 20.9 volume percent of oxygen) in the vessel would first be flushed (purged) with nitrogen or another non-flammable inert gas, thereby reducing the oxygen concentration inside the container. When the oxygen concentration is below the LOC, flammable gas can then be safely admitted to the vessel, because the possibility of internal explosion has been eliminated.
The limiting oxygen concentration is a necessary parameter when designing hypoxic air fire prevention systems .
Monographs | https://en.wikipedia.org/wiki/Limiting_oxygen_concentration |
Limiting pressure velocity is a tribological term relating to the maximum temperature and compression that an assembly with rubbing surfaces can bear without failing. Pressure-limiting valves are a type of pressure control valve. They safeguard the system against excessive system pressure or limit the operation pressure.
Pre-load valves, also called sequence valves are a type of pressure control valve. They generate a largely constant pressure drop between the inlet and outlet on the valve. In the opposite direction, the flow can pass freely. In the normal position, the valve has minor leakage.
This engineering-related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Limiting_pressure_velocity |
The limiting reagent (or limiting reactant or limiting agent ) in a chemical reaction is a reactant that is totally consumed when the chemical reaction is completed. [ 1 ] [ 2 ] The amount of product formed is limited by this reagent, since the reaction cannot continue without it. If one or more other reagents are present in excess of the quantities required to react with the limiting reagent, they are described as excess reagents or excess reactants (sometimes abbreviated as "xs"), or to be in abundance . [ 3 ]
The limiting reagent must be identified in order to calculate the percentage yield of a reaction since the theoretical yield is defined as the amount of product obtained when the limiting reagent reacts completely. Given the balanced chemical equation , which describes the reaction, there are several equivalent ways to identify the limiting reagent and evaluate the excess quantities of other reagents.
This method is most useful when there are only two reactants. One reactant (A) is chosen, and the balanced chemical equation is used to determine the amount of the other reactant (B) necessary to react with A. If the amount of B actually present exceeds the amount required, then B is in excess and A is the limiting reagent. If the amount of B present is less than required, then B is the limiting reagent.
Consider the combustion of benzene , represented by the following chemical equation :
This means that 15 moles of molecular oxygen (O 2 ) is required to react with 2 moles of benzene (C 6 H 6 )
The amount of oxygen required for other quantities of benzene can be calculated using cross-multiplication (the rule of three). For example,
if 1.5 mol C 6 H 6 is present, 11.25 mol O 2 is required:
If in fact 18 mol O 2 are present, there will be an excess of (18 - 11.25) = 6.75 mol of unreacted oxygen when all the benzene is consumed. Benzene is then the limiting reagent.
This conclusion can be verified by comparing the mole ratio of O 2 and C 6 H 6 required by the balanced equation with the mole ratio actually present:
Since the actual ratio is larger than required, O 2 is the reagent in excess, which confirms that benzene is the limiting reagent.
In this method the chemical equation is used to calculate the amount of one product which can be formed from each reactant in the amount present. The limiting reactant is the one which can form the smallest amount of the product considered. This method can be extended to any number of reactants more easily than the first method.
20.0 g of iron (III) oxide (Fe 2 O 3 ) are reacted with 8.00 g aluminium (Al) in the following thermite reaction :
Since the reactant amounts are given in grams, they must be first converted into moles for comparison with the chemical equation, in order to determine how many moles of Fe can be produced from either reactant.
There is enough Al to produce 0.297 mol Fe, but only enough Fe 2 O 3 to produce 0.250 mol Fe. This means that the amount of Fe actually produced is limited by the Fe 2 O 3 present, which is therefore the limiting reagent.
It can be seen from the example above that the amount of product (Fe) formed from each reagent X (Fe 2 O 3 or Al) is proportional to the quantity
Moles of Reagent X Stoichiometric Coefficient of Reagent X {\displaystyle {\frac {\mbox{Moles of Reagent X }}{\mbox{Stoichiometric Coefficient of Reagent X}}}}
This suggests a shortcut which works for any number of reagents. Just calculate this formula for each reagent, and the reagent that has the lowest value of this formula is the limiting reagent. We can apply this shortcut in the above example. | https://en.wikipedia.org/wiki/Limiting_reagent |
Limiting similarity (informally "limsim") is a concept in theoretical ecology and community ecology that proposes the existence of a maximum level of niche overlap between two given species that will allow continued coexistence.
This concept is a corollary of the competitive exclusion principle , which states that, controlling for all else, two species competing for exactly the same resources cannot stably coexist. It assumes normally-distributed resource utilization curves ordered linearly along a resource axis, and as such, it is often considered to be an oversimplified model of species interactions. Moreover, it has theoretical weakness, and it is poor at generating real-world predictions or falsifiable hypotheses. Thus, the concept has fallen somewhat out of favor except in didactic settings (where it is commonly referenced), and has largely been replaced by more complex and inclusive theories.
In 1932, Georgii Gause created the competitive exclusion principle based on experiments with cultures of yeast and paramecium . [ 1 ] The principle maintains that two species with the same ecological niches cannot stably coexist. That is to say, when two species compete for identical resource access, one will be competitively superior and it will ultimately supplant the other. Over the next half century, limiting similarity slowly emerged as a natural outgrowth of this principle, aiming (but not necessarily succeeding) to be more quantitative and specific.
Noted ecologist and evolutionary biologist David Lack said retrospectively that he had already begun to mull around with the ideas of limiting similarity as early as the 1940s, but it wasn't until the end of the 1950s that the theory began to be built up and articulated. [ 2 ] G. Evelyn Hutchinson 's famous "Homage to Santa Rosalia" was the next foundational paper in the history of the theory. Its subtitle famously asks, "Why are there so many kinds of animals?", and the address attempts to answer this question by suggesting theoretical bounds to speciation and niche overlap. For the purposes of understanding limiting similarity, the key portion of Hutchinson's address is the end where he presents the observation that a seemingly ubiquitous ratio (1.3:1) defines the upper bound of morphological character similarity between closely related species. [ 3 ] While this so-called Hutchinson ratio and the idea of a universal limit have been overturned by later research, the address was still foundational to the theory of limiting similarity.
MacArthur and Levins were the first to introduce the term 'limiting similarity' in their 1967 paper. They attempted to lay out a rigorous quantitative basis for the theory using probability theory and the Lotka–Volterra competition equations . [ 4 ] In doing so, they provided the ultimate theoretical framework on which many subsequent studies were based.
As proposed by MacArthur and Levins in 1967, the theory of limiting similarity is rooted in the Lotka–Volterra competition model. This model describes two or more populations with logistic dynamics, adding in an additional term to account for their biological interactions. Thus for two populations, x 1 and x 2 :
where
MacArthur and Levins examine this system applied to three populations, also visualized as resource utilization curves, depicted below. In this model, at some upper limit of competition α , between two species x 1 and x 3 , the survival of a third species x 2 between the other two is not possible. This phenomenon is termed limiting similarity. Evolutionary, if two species are more similar than some limit L , a third species will converge towards the nearer of the two competitors. If the two species are less similar than some limit L , a third species will evolve an intermediate phenotype.
[embedded graph: U v R. x1, x2, x3 curves.]
For each resource R, U represents the probability of utilization per unit time by an individual. At some level of overlap between species x 1 and x 3 , the survival of a third species x 2 is no longer possible.
May [ 5 ] extended this theory when considering species with different carrying capacities, concluding that coexistence was unlikely if the distance between the modes of competing resource utilization curves d was less than the standard deviation of the curves w .
It is of note that the theory of limiting similarity does not easily generate falsifiable predictions about natural phenomenon. However, many studies have tried to test the theory by making the highly suspect assumption that character displacement can be used as a close proxy for niche incongruence. [ 6 ] One recent paleoecological study, for example, used fossil proxies of gastropod body size to determine levels of character displacement over 42,500 years during the Quaternary . They found little evidence of character displacement, and they concluded that "limiting similarity, as seen in both ecological character displacement and community-wide character displacement, is a transient ecological phenomenon rather than a long-term evolutionary process". [ 7 ] Other theoretical and empirical studies tend to find results that similarly play down the strength and role of limiting similarity in ecology and evolution. For example, Abrams (who is prolific on the subject of limiting similarity) and Rueffler find in 2009 that "there is no absolute limit to similarity; there is always some range of mortality rates of one species allowing coexistence, given a fixed mortality of the other species". [ 8 ]
What a lot of studies examining limiting similarity find are the weaknesses in the original theory that are addressed below.
The key weakness of the theory of limiting similarity is that it is highly system specific and thus difficult to test in practice. In actual environments, one resource axis is inadequate and a specific analysis must be done for each given pair of species. In practice it is necessary to take into account:
While these complications don't invalidate the concept, they render limiting similarity exceedingly difficult to test in practice and useful for little more than didacticism.
Furthermore, Hubbell and Foster point out that extinction via competition can take an extremely long time and the importance of limiting similarity in extinction may even be superseded by speciation. [ 9 ] Also, from a theoretical standpoint, small changes in carrying capacities can allow for nearly completely overlapping resource utilization curves and in practice carrying capacity can be difficult to determine. Many studies that attempt to explore limiting similarity (including Huntley et al. 2007) resort to examining character displacement as a proxy for niche overlap, which is suspect at best. While a useful-if simple-model, limiting similarity is nearly untestable in reality. | https://en.wikipedia.org/wiki/Limiting_similarity |
In mechanical engineering , limits and fits are a set of rules regarding the dimensions and tolerances of mating machined parts. Limits and Fits are given to a part's dimensions to gain the desired type of fit . This is seen most commonly in regulating shaft sizes with hole sizes. [ 1 ]
Limits and Fits are standardized by the International Organization for Standardization (ISO) [ 2 ] and the American National Standards Institute (ANSI). [ 3 ] Tables are used to quickly calculate required tolerances for bolt holes, shafts, mating parts, and many similar scenarios.
Units for limits and fits are typically specified in thousandths of an inch or hundredths of a millimeter. [ 4 ]
There are three main types of fit:
These main three types of fit are umbrella categories for different sub-categories of fits. Sub-categories include sliding fit, running fit, push fit, wringing fit, force fit, tight fit, and shrink fit. Every different type of fit is used for a different type of interaction between mating parts. [ 5 ]
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This engineering-related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Limits_and_fits |
In calculus and mathematical analysis the limits of integration (or bounds of integration ) of the integral ∫ a b f ( x ) d x {\displaystyle \int _{a}^{b}f(x)\,dx}
of a Riemann integrable function f {\displaystyle f} defined on a closed and bounded interval are the real numbers a {\displaystyle a} and b {\displaystyle b} , in which a {\displaystyle a} is called the lower limit and b {\displaystyle b} the upper limit . The region that is bounded can be seen as the area inside a {\displaystyle a} and b {\displaystyle b} .
For example, the function f ( x ) = x 3 {\displaystyle f(x)=x^{3}} is defined on the interval [ 2 , 4 ] {\displaystyle [2,4]} ∫ 2 4 x 3 d x {\displaystyle \int _{2}^{4}x^{3}\,dx} with the limits of integration being 2 {\displaystyle 2} and 4 {\displaystyle 4} . [ 1 ]
In Integration by substitution , the limits of integration will change due to the new function being integrated. With the function that is being derived, a {\displaystyle a} and b {\displaystyle b} are solved for f ( u ) {\displaystyle f(u)} . In general, ∫ a b f ( g ( x ) ) g ′ ( x ) d x = ∫ g ( a ) g ( b ) f ( u ) d u {\displaystyle \int _{a}^{b}f(g(x))g'(x)\ dx=\int _{g(a)}^{g(b)}f(u)\ du} where u = g ( x ) {\displaystyle u=g(x)} and d u = g ′ ( x ) d x {\displaystyle du=g'(x)\ dx} . Thus, a {\displaystyle a} and b {\displaystyle b} will be solved in terms of u {\displaystyle u} ; the lower bound is g ( a ) {\displaystyle g(a)} and the upper bound is g ( b ) {\displaystyle g(b)} .
For example, ∫ 0 2 2 x cos ( x 2 ) d x = ∫ 0 4 cos ( u ) d u {\displaystyle \int _{0}^{2}2x\cos(x^{2})dx=\int _{0}^{4}\cos(u)\,du}
where u = x 2 {\displaystyle u=x^{2}} and d u = 2 x d x {\displaystyle du=2xdx} . Thus, f ( 0 ) = 0 2 = 0 {\displaystyle f(0)=0^{2}=0} and f ( 2 ) = 2 2 = 4 {\displaystyle f(2)=2^{2}=4} . Hence, the new limits of integration are 0 {\displaystyle 0} and 4 {\displaystyle 4} . [ 2 ]
The same applies for other substitutions.
Limits of integration can also be defined for improper integrals , with the limits of integration of both lim z → a + ∫ z b f ( x ) d x {\displaystyle \lim _{z\to a^{+}}\int _{z}^{b}f(x)\,dx} and lim z → b − ∫ a z f ( x ) d x {\displaystyle \lim _{z\to b^{-}}\int _{a}^{z}f(x)\,dx} again being a and b . For an improper integral ∫ a ∞ f ( x ) d x {\displaystyle \int _{a}^{\infty }f(x)\,dx} or ∫ − ∞ b f ( x ) d x {\displaystyle \int _{-\infty }^{b}f(x)\,dx} the limits of integration are a and ∞, or −∞ and b , respectively. [ 3 ]
If c ∈ ( a , b ) {\displaystyle c\in (a,b)} , then [ 4 ] ∫ a b f ( x ) d x = ∫ a c f ( x ) d x + ∫ c b f ( x ) d x . {\displaystyle \int _{a}^{b}f(x)\ dx=\int _{a}^{c}f(x)\ dx\ +\int _{c}^{b}f(x)\ dx.} | https://en.wikipedia.org/wiki/Limits_of_integration |
Limulus amebocyte lysate ( LAL ) is an aqueous extract of motile blood cells ( amebocytes ) from the Atlantic horseshoe crab Limulus polyphemus . LAL reacts with bacterial endotoxins such as lipopolysaccharides (LPS), which are components of the bacterial capsule , the outermost membrane of cell envelope of gram-negative bacteria . This reaction is the basis of the LAL test , which is widely used for the detection and quantification of bacterial endotoxins .
In Asia, a similar Tachypleus amebocyte lysate ( TAL ) test based on the local horseshoe crabs Tachypleus gigas or Tachypleus tridentatus is occasionally used instead. [ 1 ] The recombinant factor C (rFC) assay is a replacement of LAL and TAL based on a similar reaction. [ 2 ]
The American medical researcher Fred Bang reported in 1956 that gram-negative bacteria, even if killed, will cause the blood of the horseshoe crab to turn into a gel , a type of semi-solid mass. It was later recognized that the animal's blood cells, mobile cells called amebocytes , contain granules with a clotting factor known as coagulogen ; this is released outside the cell when bacterial endotoxins are encountered. After coagulation and subsequent gelling , the resulting gel is thought to contain bacterial infections in the animal's semi-closed circulatory system . [ 3 ] Modern analysis of the lysate has led to understanding of this system of cascade, with multiple enzymes working in sequence to produce the gel. The entry point of endotoxin-induced clotting is Limulus clotting factor C. [ 4 ]
In 1977 the U.S. Food and Drug Administration (FDA) approved LAL for testing drugs, products and devices that come in contact with blood. Prior to that date, a much slower and more expensive test on rabbits had been used for this purpose. [ 5 ]
Horseshoe crabs are collected and blood is removed from the horseshoe crab's pericardium ; some crabs are then returned to the water, while others are sold to be eaten or used as bait. Companies extracting LAL from horseshoe crabs stated before 2008 that mortality rates were below 3%. [ 6 ] A 2009 Massachusetts Division of Marine Fisheries study stated that earlier studies found 5 to 15% mortality for males and one estimate of 29% for females. The study itself found 22% for females returned immediately to the water, and 30% for females kept overnight to represent commercial practice. [ 7 ] The blood cells are separated from the serum using centrifugation and are then placed in distilled water, which causes them to swell and burst ("lyse"). This releases the chemicals from the inside of the cell (the "lysate"), which is then purified and freeze-dried . To test a sample for endotoxins, it is mixed with lysate and water; endotoxins are present if coagulation occurs. [ 8 ]
There are three basic methodologies: gel-clot , turbidimetric , and chromogenic . The primary application for LAL is the testing of parenteral pharmaceuticals and medical devices that contact blood or cerebrospinal fluid . In the United States, the FDA has published a guideline for validation of the LAL test as an endotoxin test for such products. [ 9 ]
The LAL cascade is also triggered by (1,3)-β-D-glucan , via a different Factor G. Both bacterial endotoxins and (1,3)-β-D-glucan are considered pathogen-associated molecular patterns , or PAMPs, substances which elicit inflammatory responses in mammals . [ 10 ]
One of the most time-consuming aspects of endotoxin testing using LAL is pretreating samples to overcome assay inhibition that may interfere with the LAL test such that the recovery of endotoxin is affected. If the product being tested causes the endotoxin recovery to be less than expected, the product is inhibitory to the LAL test. Products which cause higher than expected values are enhancing. Overcoming the inhibition and enhancement properties of a product is required by the FDA as part of the validation of the LAL test for use in the final release testing of injectables and medical devices. Proper endotoxin recovery must be proven before LAL can be used to release product. [ 11 ]
The LAL test is a major source of animal product dependence in the biomedical industry, and a challenge to the Three Rs of science in relation to the use of animals in testing. With reports of higher-than anticipated mortality rates [ 7 ] it has been considered more ethical to devise alternatives to the test. [ 12 ] Since 2003, a recombinant protein substitute for use in the LAL test has been commercially available. Named the recombinant factor C (rFC) assay, it is based on the same Limulus clotting factor C protein, but produced by genetically modified insect cells (the specific factor C sequence used does not necessarily come from the Atlantic horseshoe crab). [ 5 ]
Instead of emulating the whole clotting pathway, rFC tests let factor C cleave a synthetic fluorogenic substrate, so that the sample lights up when endotoxin activates the factor. Since it does not contain factor G, (1,3)-β-D-glucan will not cause false-positives . As of 2018, available evidence shows that the rFC test is no worse than the LAL test. [ 13 ]
The adoption of the rFC test was slow, which began to change in 2012 when the US FDA and the European health ministry acknowledged it as an accepted alternative. Its lack of mention in Pharmacopeias remained an issue, as there was no good standard for running the test in production. [ 13 ] In 2016, it was added to the European Pharmacopoeia . [ 5 ] A patent on rFC also limited adoption until its expiration in 2018. [ 13 ]
On 1 June 2020, the United States Pharmacopeia (USP) decided to cancel the proposal to include recombinant technology for endotoxin testing in chapter 85, Bacterial Endotoxins , and start the development of a separate chapter that expands on the use, validation, and comparability of endotoxin tests based on recombinantly derived reagents. A separate guidance-only chapter 1085.1 was proposed by the USP, though comments and feedback published on 11 December 2020 show that pharmaceutical companies and the FDA do not support this chapter, and request for compendial status. [ 14 ]
The monocyte activation test (MAT) is another proposed method to test for endotoxins based on monocytes in human blood . It measures the release of cytokines from these due to the presence of pyrogens , basically mirroring the process by which these toxins cause fever in humans (and rabbits, as in the original pyrogen test). [ 15 ] A protocol for the MAT test, using cultured cells, is described in the European Pharmacopoeia. [ 16 ]
A recent study employing genetically engineered monocytes was able to significantly enhance the sensitivity of monocyte-based detection assays by bringing down the assay-completion time from more than 20 hours to 2–3 hours. [ 17 ] | https://en.wikipedia.org/wiki/Limulus_amebocyte_lysate |
The LinBi project [ 1 ] ran between February 2019 and October 2020 as an EU -funded INEA - CEF project which focused on biodiversity and documentation of the variety of life on Earth. This diversity is preserved in a wide range of formats – books, illustrations, specimen scans, glass plate photographs, sound recordings, herbarium sheets, video and more. LinBi brought together botanists, researchers, the media and the public in a collaborative effort to enhance and support appreciation and use of European biodiversity material. The project has provided 1.3 million items of cultural heritage content to Europeana .
LinBi partners were:
LinBi created a data enrichment platform [ 7 ] to link existing items of data with new items, creating ‘enriched’ information objects. These were then processed by the OpenUp! Natural History content aggregator and provided to Europeana. The project has created three virtual exhibitions for Europeana. These exhibitions interlinked existing content with new and external data. The exhibitions focussed on Edible Plants from the Americas [ 8 ] (curated by RJB-CSIC), François Crépin and the Study of Wild Roses [ 9 ] (curated by Meise) and Magical, Mystical and Medicinal – Psychoactive plants and fungi, [ 10 ] (curated by RJB-CSIC). | https://en.wikipedia.org/wiki/LinBi |
The Lincoln County Process is a step used in producing almost all Tennessee whiskeys . The whiskey is filtered through—or steeped in [ 1 ] —maple charcoal chips before going into the charred oak casks for aging. The process is named for Lincoln County, Tennessee , which was the location of the cave spring where in 1825 Alfred Eaton originated the method prior to that portion of Lincoln County becoming part of Moore County formed in 1871 (the cave spring is where Jack Daniel's distillery moved to in 1881). [ 2 ] The Lincoln County Process is no longer used in that county where no distilleries operated after Prohibition until recently, (the new distilleries in Lincoln County are Benjamin Prichard's , which doesn't use the process for their Tennessee Whiskey and Southern Pride Distillery which doesn't offer a Tennessee Whiskey).
There is some dispute on the origins of the Lincoln County Process and it is possible that there is no single inventor. What is not disputed is that there were a great number of distillers in Lincoln County employing the Lincoln County Process (as early as 1825) prior to Jack Daniels as a company (initially the "Daniel & Call Company") being formed. [ 3 ] Certainly each distillery had their own approach of how to make whiskey, but as a group, the licensed distillers of old Lincoln County adopted the maple charcoal mellowing process en masse, as well as aging in charred oak barrels. It is interesting to note that in the early 1820s and into the 1890s, there was significantly more whiskey being produced in the country in the northeast than in Tennessee and Kentucky, yet, the adoption of using charred oak barrels for aging began being recognized simultaneously in Kentucky and Tennessee in the early 1820s, with Tennesseans in Lincoln County adding the maple charcoal filtering as an extra step. Was it coincidence or a common experience that compelled these isolated frontiersmen to embrace charred wood for their whiskey? It is worthy to consider that New Orleans was the main point of commerce at that time for both Tennessee and Kentucky and that the use of charred oak barrels for American whiskey is attributed to the Tarascon brothers adoption of the method for their shop on Bourbon Street in New Orleans. Not only were tradesmen traveling down river to New Orleans and back, but in 1814 men of Kentucky and Tennessee gathered (2500 in Lincoln County) to march into the Battle of New Orleans. For thousands of years, many different cultures have used charcoal as a filter. There is evidence that both the Hindus and the Phoenicians were using charcoal to filter water around 400 BC. [ 4 ]
For many years it was claimed that Alfred Eaton invented the process in 1825. [ 5 ] [ 6 ] [ 7 ] The author (R. Christopher) of the 1896 article for “The Nashville American” (now published as “The Tennessean”) titled "Lynchburg and Moore County" clearly states that Alfred Eaton was the originator of the Lincoln County Process. The author of the article made special note to thank the Hon. James J. Bean for his assistance and recognized his “untiring and zealous” work for Lynchburg. It is hard to imagine a more accurate accounting of the origin of Tennessee Whiskey and the Lincoln County Process than this published account which is based on living memory at the time of the zenith of Jack Daniels onto the world stage. The Hon. James J. Bean served Tennessee as the State Treasurer and was the son of C. H. Bean whom you will note had the largest distribution of Lincoln County Process whiskey in 1877 out of Lincoln and Moore Counties according to the published report of the 1877 meeting of the 4th District Distillers (where C. H. Bean was appointed chairman of the group). A better source more familiar with the distilling business history of the area and notable as trustworthy does not exist. However, The family of William "Billy" Pearson (1761-1844) claim to have documents that Billy sold his recipe for a "smooth sipping whiskey from a corn-mash, filtered through charcoal made from hard sugar-maple wood, and aged in Oak Barrels," to Eaton in 1825. [ 8 ] According to the Pearson family, the original recipe was created by Mary Stout Jacocks (b.c. 1715 - d.c. 1816) and that it traveled with the family from New Jersey, to Virginia, and then Pennsylvania where Mary's daughter Tabitha Jacocks Pearson (1734 - 1811) taught the recipe to Billy. Billy took the family recipe to South Carolina and on August 7, 1791 he was kicked out of the Padgett's Creek Baptist Church for making whiskey. [ 8 ] In 1812, Billy divorced his wife and moved with his four oldest children to a land he purchased not far from Lynchburg , TN in Bedford County . It is assumed that Billy farmed the land he purchased and continued to make whiskey until he sold his recipe along with the sugar maple charcoal filtration method to Eaton.
Recently it has been claimed that Nathan “Nearest” Green , the former slave, teacher of Jack Daniel and his eventual master distiller "was the one who decided to cut down sugar maple trees, create charcoal from it and filter his unaged whiskey [through it] before barreling." [ 9 ] Proponents of Green as the inventor of the Lincoln County Process propose that he learned the practice of using charcoal to filter water from his ancestors and adapted it to whiskey. They claim that slaves brought the practice to the United States from West Africa where there is a long history of people "using charcoal to filter water and purify their foods" [ 10 ] In this scenario, it is assumed that Green would have invented the Lincoln County Process sometime after his birth in 1820 and when he began working for Dan Call as a hired slave distiller around the 1850s. [ 11 ] [ 12 ]
Nearly every distillery creating Tennessee whiskey uses maple charcoal filtering, though the actual process for accomplishing this varies by company.
For Jack Daniel's, the charcoal used is created onsite from stacks (ricks) of two-by-two-inch (5 by 5 cm) sugar maple timbers. The timbers are primed with 140 proof (70% ABV ) Jack Daniel's and then ignited under large hoods to prevent sparks. Once they reach the char state, the ricks are sprayed with water to prevent complete combustion. The resulting charcoal is then fed through a grinder to produce bean-size pellets that are packed into 10-foot (3.0 m) vats used to filter impurities from the 140 proof whiskey. The whiskey is then reduced with water to 125 proof (62.5%) for aging. [ 13 ] The process was taught to Jack Daniel by Nearest Green , the namesake of Uncle Nearest Premium Whiskey . [ 14 ]
The George Dickel distillery uses deeper (13-foot [4.0 m]) vats and distills the whisky—the spelling used by Dickel—to 135 proof (67.5%). Dickel chills its whisky to 40 °F (4 °C) before it enters the vats and allows the liquid to fill the vats [ 1 ] instead of trickling it through.
Nelson's Green Brier Distillery uses the Lincoln County Process to make its wheated First 108 Tennessee whiskey and its white whiskey. [ 15 ]
Collier and McKeel , made in Nashville , uses a method that pumps the whiskey slowly through 10–13 feet (3–4 m) feet of sugar maple charcoal (instead of using gravity) made from trees cut by local sawmills. [ 16 ] [ 17 ]
Fugitives Tennessee Artisan Spirits uses wood harvested from old growth Lincoln County sugar maple trees grown on the Massey Farm in Lincoln County to create their proprietary charcoal. The Fugitives Lincoln County Process method takes Tennessee heritage grain Hickory Cane corn and Irish Barley distillate and steeps it slowly over 12 hours prior to aging in charred oak to make "Grandgousier Tennessee Whiskey" [ 18 ]
To be labeled as a straight whiskey , no flavoring or coloring compounds can be added to the spirit after the fermenting of the grain .
Some producers claim that according to a 1941 Internal Revenue Service ruling issued at the request of Jack Daniel Distillery, the Lincoln County Process is what distinguishes "Tennessee whiskey" from " bourbon ". [ 19 ] However, not all producers of products labeled as Tennessee whiskey use the process. (Specifically, it is not used in the production of Benjamin Prichard's Tennessee Whiskey . [ 20 ] )
The term "Tennessee whiskey" does not actually have a legal definition in the U.S. Federal regulations that define the Standards of Identity for Distilled Spirits . [ 21 ] The only legal definition of Tennessee whiskey in U.S. federally recognized legislation is the North American Free Trade Agreement (NAFTA) , which states only that Tennessee whiskey is "a straight Bourbon Whiskey authorized to be produced only in the State of Tennessee". [ 22 ] This definition is also recognized in the law of Canada , which states that Tennessee whiskey must be "a straight Bourbon Whiskey produced in the State of Tennessee". [ 23 ] None of these regulations requires the use of the Lincoln County filtering process (or any other filtering process).
On May 13, 2013, the governor of Tennessee signed House Bill 1084, requiring maple charcoal filtering to be used for products produced in the state labeling themselves as "Tennessee whiskey" (with a particular exception tailored to exempt Benjamin Prichard's) and including the existing requirements for bourbon. [ 24 ] [ 25 ] [ 26 ] As federal law requires statements of origin on labels to be accurate, the Tennessee law effectively gives a firm definition to Tennessee whiskey. [ 27 ] | https://en.wikipedia.org/wiki/Lincoln_County_Process |
The Lincoln index is a statistical measure used in several fields to estimate the population size of an animal species. Described by Frederick Charles Lincoln in 1930, it is also sometimes known as the Lincoln-Petersen method after C.G. Johannes Petersen who was the first to use the related mark and recapture method. [ 1 ]
Consider two observers who separately count the different species of plants or animals in a given area. If they each come back having found 100 species but only 5 particular species are found by both observers, then each observer clearly missed at least 95 species (that is, the 95 that only the other observer found). Thus, we know that both observers miss a lot. On the other hand, if 99 of the 100 species each observer found had been found by both, it is fair to expect that they have found a far higher percentage of the total species that are there to find.
The same reasoning applies to mark and recapture . If some animals in a given area are caught and marked, and later a second round of captures is done: the number of marked animals found in the second round can be used to generate an estimate of the total population. [ 2 ]
Another example arises in computational linguistics for estimating the total vocabulary of a language. Given two independent samples, the overlap between their vocabularies enables a useful estimate of how many more vocabulary items exist but did not happen to show up in either sample. A similar example involves estimating the number of typographical errors remaining in a text, from two proofreaders' counts.
The Lincoln Index formalizes this phenomenon. If E1 and E2 are the number of species (or words, or other phenomena) observed by two independent methods, and S is the number of observations in common, then the Lincoln Index is simply
L = E 1 E 2 S . {\displaystyle L={E_{1}E_{2} \over S}.}
For values of S < 10, this estimate is rough, and becomes extremely rough for values of S < 5. In the case where S = 0 (that is, there is no overlap at all) the Lincoln Index is formally undefined. This can arise if the observers only find a small percentage of the actual species (perhaps by not looking hard enough or long enough), if the observers are using methods that are not statistically independent (for example if one looks only for large creatures and the other only for small), or in other circumstances.
The Lincoln Index is merely an estimate.
For example, the species in a given area could tend to be either very common or very rare, or tend to be either very hard or very easy to see. [ 3 ] Then it would be likely that both observers would find a large share of the common species, and that both observers would miss a large share of the rare ones. Such distributions would throw off the consequent estimate. However, such distributions are unusual for natural phenomena, as suggested by Zipf's Law .
T. J. Gaskell and B. J. George propose an enhancement of the Lincoln Index that claims to reduce bias. [ 4 ] | https://en.wikipedia.org/wiki/Lincoln_index |
Linda-like systems are parallel and distributed programming models that use unstructured collections of tuples as a communication mechanism between different processes .
In addition to proper Linda implementations, these include other systems such as the following:
This computing article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Linda-like_systems |
Linda Faye Nazar OC FRSC FRS is a Senior Canada Research Chair in Solid State Materials and Distinguished Research Professor of Chemistry at the University of Waterloo . She develops materials for electrochemical energy storage and conversion. Nazar demonstrated that interwoven composites could be used to improve the energy density of lithium–sulphur batteries . She was awarded the 2019 Chemical Institute of Canada Medal .
Nazar studied chemistry at the University of British Columbia , where she earned a bachelor's degree in 1978. [ 1 ] She was inspired to study chemistry after being inspired by her first year professor. [ 2 ] Her father had trained as a scientist and ran his own jewellery making business. [ 2 ] Nazar joined the University of Toronto for her graduate studies, and completed a PhD under the supervision of Geoffrey Ozin in 1984. After obtaining her degree, she worked as a postdoctoral researcher working with Allan Jacobson at Exxon Research and Engineering Company , [ 3 ] before joining the University of Waterloo in the late 1980s, when she became interested in electrochemistry and Inorganic chemistry . [ 2 ]
Nazar works in materials chemistry at the University of Waterloo , where she designs energy storage devices and electrochemical systems. Her research group create new materials and nanostructures for lithium–sulfur batteries , including interwoven composites. She develops structural probes to understand how the morphology of materials that are capable of charge/ ionic redox processes impact their functions. These techniques include nuclear magnetic resonance ( NMR ), electrochemistry , AC Impedance Spectroscopy and X-ray diffraction measurements. [ 4 ] [ 5 ] Nazar was a founding member of the Waterloo Institute for Nanotechnology . [ 3 ] Nazar is recognised as being a "leading authority in advanced materials". [ 6 ] She was awarded a Canada Research Chair in 2004, which was renewed in 2008 and 2012. [ 7 ] [ 8 ] [ 9 ] In 2009 Nazar joined the California Institute of Technology as a More Distinguished Scholar . [ 3 ] [ 10 ] In 2013 she was awarded a $1.8 million fellowship from the National Research Council to investigate energy storage materials for automotive applications. [ 11 ]
Nazar is particularly interested in storage materials that go beyond lithium-ion batteries , sodium-ion batteries , zinc ion batteries and magnesium-ion batteries . [ 12 ] [ 13 ] [ 14 ] [ 15 ] Lithium-ion batteries are the battery of choice in hybrid electric vehicles, but concerns have arisen about the global supply of lithium . Her early work developed porous carbon architectures as frameworks for cathodes, enhancing their conductivity and discharge capacity. [ 16 ] She demonstrated that interwoven carbon composites could be used to improve the energy density of lithium–sulphur batteries . [ 4 ] She showed it was possible to create mesoporous carbon frameworks that constrain the grown of sulphur nanofillers, which improved energy storage and reversibility. [ 16 ]
Nazar calculated the low-cost lithium–sulphur batteries could take electric cars twice as far as current lithium-ion technologies. [ 2 ] Sulphur is an abundant material that can be used to replace cobalt oxide in lithium-ion batteries . [ 17 ] Unfortunately, sulphur can dissolve into the electrolyte solution, and be reduced by electrons to form polysulphides . [ 18 ] They are also susceptible to high internal resistance and capacity fading on cycling. [ 17 ] These challenges can be overcome by creating nanostructures in the electrodes. [ 17 ] Interwoven composites can also be made from manganese dioxide , which stabilise polysuplphides in lithium–sulphur batteries . [ 18 ] Manganese dioxide reduces sulphides via a surface-bound polythiosulphanates , and can withstand 2,000 discharge cycles without the loss of capacitance. [ 2 ] [ 18 ] [ 19 ] She has also developed lithium oxygen batteries , which are lightweight with high energy density. [ 20 ] [ 21 ] In lithium oxygen batteries , superoxide and peroxide can act to degrade the cells; limiting their lifetime. [ 21 ] If the electrolyte is replaced with a molten salt and the porous cathode with a bifunctional metal oxide, the peroxide does not form. [ 21 ] Nazar has worked on supercapacitors and polyanion materials. [ 22 ] [ 23 ]
She was made a Professor at the University of Waterloo in 2016 and holds a Tier 1 Canada Research Chair in Solid State Energy Materials. [ 24 ] Since 2014 Nazar has served on the board of directors of the International Meeting on Li-Batteries. [ 25 ] She serves on the editorial boards of the journals Angewandte Chemie , Energy & Environmental Science and the Journal of Materials Chemistry A . [ 26 ] [ 27 ]
Her awards and honours include; | https://en.wikipedia.org/wiki/Linda_Nazar |
Lindane , also known as gamma -hexachlorocyclohexane ( γ-HCH ), gammaxene , Gammallin and benzene hexachloride ( BHC ), [ 2 ] is an organochlorine chemical and an isomer of hexachlorocyclohexane that has been used both as an agricultural insecticide and as a pharmaceutical treatment for lice and scabies . [ 3 ] [ 4 ]
Lindane is a neurotoxin that interferes with GABA neurotransmitter function by interacting with the GABA A receptor -chloride channel complex at the picrotoxin binding site. In humans, lindane affects the nervous system , liver , and kidneys , and may well be a carcinogen . [ 5 ] [ 6 ] Whether lindane is an endocrine disruptor is unclear. [ 7 ] [ 8 ] [ 9 ]
The World Health Organization classifies lindane as "moderately hazardous", and its international trade is restricted and regulated under the Rotterdam Convention on Prior Informed Consent. [ 10 ] In 2009, the production and agricultural use of lindane was banned under the Stockholm Convention on persistent organic pollutants . [ 1 ] [ 11 ] A specific exemption to that ban allows it to continue to be used as a second-line pharmaceutical treatment for lice and scabies . [ 12 ]
The chemical was originally synthesized in 1825 by Faraday . It is named after the Dutch chemist Teunis van der Linden (1884–1965), the first to isolate and describe γ-hexachlorcyclohexane in 1912. The fact that mixtures of isomers of hexachlorocyclohexane have insecticidal activity is a case of multiple discovery . [ 13 ] Work in the 1930s at the Jealott's Hill laboratories of Imperial Chemical Industries Ltd (ICI) led in 1942 to the realization that the γ isomer was the key active component in the mixture which had hitherto been tested. Development work in the UK was accelerated because at that time in World War II imports of derris containing the insecticide rotenone were restricted owing to the Japanese occupation of Malaya and alternatives were urgently needed. In trials in 1943 it was found that a five-fold increase in the yield of oats and wheat was achieved using a dust formulation of the available material, owing to its efficacy against wireworm pests. By the end of 1945, γ-hexachlorcyclohexane of 98% purity became available and ICI commercialised a seed treatment launched in 1949 as Mergamma A , containing 1% mercury and 20% lindane. [ 14 ] Subsequently, lindane has been used to treat food crops and forestry products, as a seed or soil treatment, and to treat livestock and pets. It was used as a household pesticide as the active pesticide ingredient of an insecticidal floor wax product called "Freewax". [ 15 ] [ 16 ] It has also been used as pharmaceutical treatment for lice and scabies, formulated as a shampoo or lotion. [ 17 ] [ 18 ] [ 19 ] Between 1950 and 2000, an estimated 600,000 tonnes of lindane were produced globally, and the vast majority of which was used in agriculture. It has been manufactured by several countries, including the United States , China , Brazil , and several European countries .
By November 2006, the use of lindane had been banned in 52 countries and restricted in 33 others [ outdated statistic ] . Seventeen countries, including the US and Canada, allowed either limited agricultural or pharmaceutical use. [ 17 ] In 2009, an international ban on the use of lindane in agriculture was implemented under the Stockholm Convention on Persistent Organic Pollutants . A specific exemption allows for it to continue to be used in second-line treatments for the head lice and scabies for five more years. The production of the lindane isomers α- and β-hexachlorocyclohexane was also banned. [ 20 ] Although the US has not ratified the convention, it has similarly banned agricultural uses while still allowing its use as a second-line lice and scabies treatment. [ 12 ] [ 20 ] [ 21 ] [ 22 ]
In the US, lindane pesticide products were regulated by the U.S. Environmental Protection Agency (EPA), while lindane medications are regulated by the Food and Drug Administration (FDA). It was registered as an agricultural insecticide in the 1940s, and as pharmaceutical in 1951. [ 17 ] The EPA gradually began restricting its agricultural use in the 1970s due to concerns over its effects on human health and the environment. [ 23 ] By 2002, its use was limited to seed treatments for just six crops, [ 17 ] and in 2007, these last uses were cancelled. [ 24 ]
Lindane medications continue to be available in the US, though since 1995, they have been designated "second-line" treatments, meaning they should be prescribed when other "first-line" treatments have failed or cannot be used. [ 25 ] [ 26 ] [ 27 ] [ 28 ] In December 2007, the FDA sent a Warning Letter to Morton Grove Pharmaceuticals , the sole U.S. manufacturer of lindane products, [ 29 ] requesting that the company correct misleading information on two of its lindane websites. The letter said, in part, that the materials "are misleading in that they omit and/or minimize the most serious and important risk information associated with the use of Lindane Shampoo, particularly in pediatric patients; include a misleading dosing claim; and overstate the efficacy of Lindane Shampoo." [ 30 ]
California banned the pharmaceutical lindane, effective 2002, and the Michigan House of Representatives passed a bill in 2009 to restrict its use to doctors' offices. [ 31 ] A recent analysis of the California ban concluded that a majority of pediatricians had not experienced problems treating lice or scabies since that ban took effect. The study also documented a marked decrease in lindane wastewater contamination and a dramatic decline in lindane poisoning incidents reported to poison control centers. The authors concluded, "The California experience suggests elimination of pharmaceutical lindane produced environmental benefits, was associated with a reduction in reported unintentional exposures and did not adversely affect head lice and scabies treatment." [ 32 ] [ non-primary source needed ]
The Persistent Organic Pollutants Review Committee of the Stockholm Convention on Persistent Organic Pollutants considers the use of lindane in agriculture as largely redundant, with other, less toxic and less persistent pesticides. In the case of pharmaceutical use, the committee noted, "alternatives for pharmaceutical uses have often failed for scabies and lice treatment and the number of available alternative products for this use is scarce. For this particular case, a reasonable alternative would be to use lindane as a second-line treatment when other treatments fail, while potential new treatments are assessed." [ 33 ]
Lindane is a bird repellent . [ 34 ] Rudd & Genelly 1954 noticed that bird pests seemed uninterested in treated seeds, specifically pheasants and blackbirds around Davis, CA, US . [ 34 ] They tested its repellent effect on pheasants and found it effective, speculating that it may be usable as a general bird repellent. [ 34 ]
Lindane is not known to occur naturally. Hexachlorocyclohexane (HCH) was discovered in 1825. Its insecticidal properties were not known until the 1940s. Technical grade HCH, as a mixture of isomers is synthesized from benzene and chlorine in presence of ultraviolet light . The resulting product mixture comprises 65-70% α-HCH, 7-10% β-HCH, 14-15% lindane (γ-HCH), approximately 7% δ-HCH, 1-2% ε-HCH, and 1-2% other components. [ 35 ]
It can also be prepared by exposing a mixture of benzene and chlorine to alpha radiation . [ 36 ]
The EPA and WHO both classify lindane as "moderately" acutely toxic. It has an oral LD 50 of 88 mg/kg in rats and a dermal LD 50 of 1000 mg/kg. Most of the adverse human health effects reported for lindane have been related to agricultural uses and chronic, occupational exposure of seed-treatment workers. [ 33 ]
Exposure to large amounts of lindane can harm the nervous system , producing a range of symptoms from headache and dizziness to seizures, convulsions, and more rarely, death. [ 5 ] [ 37 ] Lindane has not been shown to affect the immune system in humans, and it is not considered to be genotoxic. [ 5 ] Prenatal exposure to β-HCH, an isomer of lindane and production byproduct, has been associated with altered thyroid hormone levels and could affect brain development. [ 38 ] [ non-primary source needed ]
The Occupational Safety and Health Administration and National Institute for Occupational Safety and Health have set occupational exposure limits ( permissible exposure and recommended exposure , respectively) for lindane at 0.5 mg/m 3 at a time-weighted average of eight hours for skin exposure. People can be exposed to lindane in the workplace by inhaling it, absorbing it through their skin, swallowing it, and eye contact. At levels of 50 mg/m 3 , lindane is immediately dangerous to life and health . [ 39 ]
It is classified as an extremely hazardous substance in the United States as defined in section 302 of the U.S. Emergency Planning and Community Right-to-Know Act (42 U.S.C. 11002), and is subject to strict reporting requirements by facilities which produce, store, or use it in significant quantities. [ 40 ]
Based primarily on evidence from animal studies, most evaluations of lindane have concluded that it may possibly cause cancer. In 2015, the International Agency for Research on Cancer classified lindane as a known human carcinogen, [ 41 ] [ 42 ] and in 2001 the EPA concluded there was "suggestive evidence of carcinogenicity, but not sufficient to assess human carcinogenic potential." [ 7 ] The U.S. Department of Health and Human Services determined that all isomers of hexachlorocyclohexane, including lindane, "may reasonably be anticipated to cause cancer in humans," [ 5 ] and in 1999, the EPA characterized the evidence carcinogenicity for lindane as "suggestive ... of carcinogenicity, but not sufficient to assess human carcinogenic potential." [ 19 ] Lindane and its isomers have also been on California's Proposition 65 list of known carcinogens since 1989. [ 43 ] In contrast, the World Health Organization concluded in 2004 that "lindane is not likely to pose a carcinogenic risk to humans." [ 44 ] India's BIS considers Lindane a "confirmed carcinogen". [ 45 ]
A variety of adverse reactions to lindane pharmaceuticals have been reported, ranging from skin irritation to seizures , and, in rare instances, death. The most common side effects are burning sensations, itching, dryness, and rash. [ 46 ] While serious effects are rare and have most often resulted from misuse, adverse reactions have occurred when used properly. [ 25 ] [ 47 ] [ 48 ] The FDA, therefore, requires a so-called black box warning on lindane products, which explains the risks of lindane products and their proper use. [ 46 ] [ 49 ]
The black box warning emphasizes that lindane should not be used on premature infants and individuals with known uncontrolled seizure disorders, and should be used with caution in infants, children, the elderly, and individuals with other skin conditions (e.g., dermatitis , psoriasis ) and people who weigh less than 110 lb (50 kg), as they may be at risk of serious neurotoxicity . [ 47 ]
Lindane is a persistent organic pollutant : it is relatively long-lived in the environment, it is transported long distances by natural processes like global distillation , and it can bioaccumulate in food chains , though it is rapidly eliminated when exposure is discontinued. [ 33 ]
The production and agricultural use of lindane are the primary causes of environmental contamination, [ 50 ] and levels of lindane in the environment have been decreasing in the U.S., consistent with decreasing agricultural usage patterns. [ 51 ] The production of lindane generates large amounts of waste hexachlorocyclohexane isomers, and "every ton of lindane manufactured produces about nine tons of toxic waste." [ 52 ] Modern manufacturing standards for lindane involve the treatment and conversion of waste isomers to less toxic molecules, a process known as "cracking". [ 51 ] [ 53 ]
When lindane is used in agriculture, an estimated 12–30% of it volatilizes into the atmosphere, where it is subject to long-range transport and can be deposited by rainfall. Lindane in soil can leach to surface and even ground water, and can bioaccumulate in the food chain. [ 23 ] However, biotransformation and elimination are relatively rapid when exposure is discontinued. [ 17 ] Most exposure of the general population to lindane has resulted from agricultural uses and the intake of foods, such as produce, meats, and milk, produced from treated agricultural commodities. Human exposure has decreased significantly since the cancellation of agricultural uses in 2006. Even so, the CDC published in 2005 its Third National Report on Human Exposures to Environmental Chemicals, which found no detectable amounts of lindane in human blood taken from a random sampling of about 5,000 people in the US as part of the NHANES study (National Health and Nutrition Examination Survey. [ 54 ] [ 55 ] The lack of detection of lindane in this large human "biomonitoring" study likely reflects the increasingly limited agricultural uses of lindane over the last two decades. The cancellation of agricultural uses in the United States will further reduce the amount of lindane introduced into the environment by more than 99%. [ 17 ] [ 23 ]
Over time, lindane is broken down in soil, sediment, and water into less harmful substances by algae, fungi, and bacteria; however, the process is relatively slow and dependent on ambient environmental conditions. [ 5 ] Lindane residues in honey and beeswax are reported to be the highest of any historical or current pesticide and to continue to pose a threat to honeybee health. [ 56 ] The ecological impact of lindane's environmental persistence continues to be debated.
The US EPA determined in 2002 that the agency does not believe that lindane contaminates drinking water in excess of levels considered safe. [ 5 ] U.S. Geological Survey teams concluded the same in 1999 and 2000. [ 57 ] With regard to lindane medications, the EPA conducted "down-the-drain" estimates of the amount of lindane reaching public water supplies and concluded that lindane levels from pharmaceutical sources were "extremely low" and not of concern. [ 19 ]
Note that the EPA has set the maximum contaminant level or "MCL" for lindane allowed in public water supplies and considered safe for drinking at 200 parts per trillion (ppt). [ 58 ] By comparison, the state of California imposes a lower MCL for lindane of 19 ppt. [ 32 ] However, the California standard is based on a dated 1988 national water criterion that was subsequently revised by the EPA in 2003 to 980 ppt. [ 7 ] [ 58 ] [ 59 ] [ 60 ] The EPA stated that the change resulted from "significant scientific advances made in the last two decades particularly in the areas of cancer and noncancer risk assessments." [ 58 ] While the EPA considered raising the MCL standard for lindane to 980 ppt at that time, the change was never implemented because states had little difficulty in maintaining lindane levels below the 200 ppt MCL limit already in place. [ 58 ] Today, the legally enforceable MCL standard for lindane is 200 ppt, while the national water criterion for lindane is 980 ppt. [ 58 ]
Lindane is the gamma isomer of hexachlorocyclohexane ("γ-HCH"). In addition to the issue of lindane pollution, some concerns are related to the other isomers of HCH, namely alpha-HCH and beta-HCH , which are notably more toxic than lindane, lack its insecticidal properties, and are byproducts of lindane production. [ 5 ] In the 1940s and 1950s, lindane producers stockpiled these isomers in open heaps, which led to ground and water contamination. The International HCH and Pesticide Forum has since been established to bring together experts to address the clean-up and containment of these sites. [ 53 ] [ 61 ] Modern manufacturing standards for lindane involve the treatment and conversion of waste isomers to less toxic industrial chemicals, a process known as " cracking ". [ 5 ] [ 53 ] Today, only a few production plants remain active worldwide to accommodate public-health uses of lindane and declining agricultural needs. [ 17 ] Lindane has not been manufactured in the U.S. since the mid-1970s, but continues to be imported. | https://en.wikipedia.org/wiki/Lindane |
A Lindblad resonance , named for the Swedish galactic astronomer Bertil Lindblad , [ 1 ] is an orbital resonance in which an object's epicyclic frequency (the rate at which one periapse follows another) is a simple multiple of some forcing frequency . Resonances of this kind tend to increase the object's orbital eccentricity [ 2 ] and to cause its longitude of periapse to line up in phase with the forcing. Lindblad resonances drive spiral density waves [ 3 ] both in galaxies (where stars are subject to forcing by the spiral arms themselves) and in Saturn's rings (where ring particles are subject to forcing by Saturn's moons ).
Lindblad resonances affect stars at such distances from a disc galaxy 's centre where the natural frequency of the radial component of a star's orbital velocity is close to the frequency of the gravitational potential maxima encountered during its course through the spiral arms. If a star's orbital speed around the galactic centre is greater than that of the part of the spiral arm through which it is passing, then an inner Lindblad resonance occurs—if smaller, then an outer Lindblad resonance. [ 4 ] At an inner resonance, a star's orbital speed is increased, moving the star outwards, and decreased for an outer resonance causing inward movement.
This astrophysics -related article is a stub . You can help Wikipedia by expanding it .
This scattering –related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Lindblad_resonance |
In quantum mechanics , the Gorini–Kossakowski–Sudarshan–Lindblad equation ( GKSL equation , named after Vittorio Gorini , Andrzej Kossakowski , George Sudarshan and Göran Lindblad ), master equation in Lindblad form , quantum Liouvillian , or Lindbladian is one of the general forms of Markovian master equations describing open quantum systems. It generalizes the Schrödinger equation to open quantum systems; that is, systems in contacts with their surroundings. The resulting dynamics are no longer unitary, but still satisfy the property of being trace-preserving and completely positive for any initial condition. [ 1 ]
The Schrödinger equation or, actually, the von Neumann equation, is a special case of the GKSL equation, which has led to some speculation that quantum mechanics may be productively extended and expanded through further application and analysis of the Lindblad equation. [ 2 ] The Schrödinger equation deals with state vectors , which can only describe pure quantum states and are thus less general than density matrices , which can describe mixed states as well.
Understanding the interaction of a quantum system with its environment is necessary for understanding many commonly observed phenomena like the spontaneous emission of light from excited atoms, or the performance of many quantum technological devices, like the laser .
In the canonical formulation of quantum mechanics, a system's time evolution is governed by unitary dynamics. This implies that there is no decay and phase coherence is maintained throughout the process, and is a consequence of the fact that all participating degrees of freedom are considered. However, any real physical system will interact with its environment, and is not absolutely isolated. The interaction with degrees of freedom that are external to the system results in dissipation of energy into the surroundings, causing decay and randomization of phase.
Certain mathematical techniques have been introduced to treat the interaction of a quantum system with its environment. One of these is the use of the density matrix , and its associated master equation. While in principle this approach to solving quantum dynamics is equivalent to the Schrödinger picture or Heisenberg picture , it allows more easily for the inclusion of incoherent processes, which represent environmental interactions. The density operator has the property that it can represent a classical mixture of quantum states, and is thus vital to accurately describe the dynamics of so-called open quantum systems.
The Lindblad master equation for system's density matrix ρ can be written as [ 1 ] (for a pedagogical introduction you may refer to [ 3 ] )
where { a , b } = a b + b a {\displaystyle \{a,b\}=ab+ba} is the anticommutator .
H {\displaystyle H} is the system Hamiltonian , describing the unitary aspects of the dynamics.
{ L i } i {\displaystyle \{L_{i}\}_{i}} are a set of jump operators , describing the dissipative part of the dynamics. The shape of the jump operators describes how the environment acts on the system, and must either be determined from microscopic models of the system-environment dynamics, or phenomenologically modelled .
γ i ≥ 0 {\displaystyle \gamma _{i}\geq 0} are a set of non-negative real coefficients called damping rates . If all γ i = 0 {\displaystyle \gamma _{i}=0} one recovers the von Neumann equation ρ ˙ = − ( i / ℏ ) [ H , ρ ] {\displaystyle {\dot {\rho }}=-(i/\hbar )[H,\rho ]} describing unitary dynamics, which is the quantum analog of the classical Liouville equation .
The entire equation can be written in superoperator form : ρ ˙ = L ( ρ ) {\displaystyle {\dot {\rho }}={\mathcal {L}}(\rho )} which resembles the classical Liouville equation ρ ˙ = { H , ρ } {\displaystyle {\dot {\rho }}=\{H,\rho \}} . For this reason, the superoperator L {\displaystyle {\mathcal {L}}} is called the Lindbladian superoperator or the Liouvillian superoperator . [ 3 ]
More generally, the GKSL equation has the form
where { A m } {\displaystyle \{A_{m}\}} are arbitrary operators and h is a positive semidefinite matrix. The latter is a strict requirement to ensure the dynamics is trace-preserving and completely positive. The number of A m {\displaystyle A_{m}} operators is arbitrary, and they do not have to satisfy any special properties. But if the system is N {\displaystyle N} -dimensional, it can be shown [ 1 ] that the master equation can be fully described by a set of N 2 − 1 {\displaystyle N^{2}-1} operators, provided they form a basis for the space of operators.
The general form is not in fact more general, and can be reduced to the special form. Since the matrix h is positive semidefinite, it can be diagonalized with a unitary transformation u :
where the eigenvalues γ i are non-negative. If we define another orthonormal operator basis
This reduces the master equation to the same form as before: ρ ˙ = − i ℏ [ H , ρ ] + ∑ i γ i ( L i ρ L i † − 1 2 { L i † L i , ρ } ) {\displaystyle {\dot {\rho }}=-{i \over \hbar }[H,\rho ]+\sum _{i}^{}\gamma _{i}\left(L_{i}\rho L_{i}^{\dagger }-{\frac {1}{2}}\left\{L_{i}^{\dagger }L_{i},\rho \right\}\right)}
The maps generated by a Lindbladian for various times are collectively referred to as a quantum dynamical semigroup —a family of quantum dynamical maps ϕ t {\displaystyle \phi _{t}} on the space of density matrices indexed by a single time parameter t ≥ 0 {\displaystyle t\geq 0} that obey the semigroup property
The Lindblad equation can be obtained by
which, by the linearity of ϕ t {\displaystyle \phi _{t}} , is a linear superoperator. The semigroup can be recovered as
The Lindblad equation is invariant under any unitary transformation v of Lindblad operators and constants,
and also under the inhomogeneous transformation
where a i are complex numbers and b is a real number.
However, the first transformation destroys the orthonormality of the operators L i (unless all the γ i are equal) and the second transformation destroys the tracelessness. Therefore, up to degeneracies among the γ i , the L i of the diagonal form of the Lindblad equation are uniquely determined by the dynamics so long as we require them to be orthonormal and traceless.
The Lindblad-type evolution of the density matrix in the Schrödinger picture can be equivalently described in the Heisenberg picture using the following (diagonalized) equation of motion [ 4 ] for each quantum observable X :
A similar equation describes the time evolution of the expectation values of observables, given by the Ehrenfest theorem .
Corresponding to the trace-preserving property of the Schrödinger picture Lindblad equation, the Heisenberg picture equation is unital , i.e. it preserves the identity operator.
The Lindblad master equation describes the evolution of various types of open quantum systems, e.g. a system weakly coupled to a Markovian reservoir. [ 1 ] Note that the H appearing in the equation is not necessarily equal to the bare system Hamiltonian, but may also incorporate effective unitary dynamics arising from the system-environment interaction.
A heuristic derivation, e.g. , in the notes by Preskill , [ 5 ] begins with a more general form of an open quantum system and converts it into Lindblad form by making the Markovian assumption and expanding in small time. A more physically motivated standard treatment [ 6 ] [ 7 ] covers three common types of derivations of the Lindbladian starting from a Hamiltonian acting on both the system and environment: the weak coupling limit (described in detail below), the low density approximation, and the singular coupling limit. Each of these relies on specific physical assumptions regarding, e.g., correlation functions of the environment. For example, in the weak coupling limit derivation, one typically assumes that (a) correlations of the system with the environment develop slowly, (b) excitations of the environment caused by system decay quickly, and (c) terms which are fast-oscillating when compared
to the system timescale of interest can be neglected. These three approximations are called Born,
Markov, and rotating wave, respectively. [ 8 ]
The weak-coupling limit derivation assumes a quantum system with a finite number of degrees of freedom coupled to a bath containing an infinite number of degrees of freedom. The system and bath each possess a Hamiltonian written in terms of operators acting only on the respective subspace of the total Hilbert space. These Hamiltonians govern the internal dynamics of the uncoupled system and bath. There is a third Hamiltonian that contains products of system and bath operators, thus coupling the system and bath. The most general form of this Hamiltonian is
The dynamics of the entire system can be described by the Liouville equation of motion, χ ˙ = − i [ H , χ ] {\displaystyle {\dot {\chi }}=-i[H,\chi ]} . This equation, containing an infinite number of degrees of freedom, is impossible to solve analytically except in very particular cases. What's more, under certain approximations, the bath degrees of freedom need not be considered, and an effective master equation can be derived in terms of the system density matrix, ρ = tr B χ {\displaystyle \rho =\operatorname {tr} _{B}\chi } . The problem can be analyzed more easily by moving into the interaction picture, defined by the unitary transformation M ~ = U 0 M U 0 † {\displaystyle {\tilde {M}}=U_{0}MU_{0}^{\dagger }} , where M {\displaystyle M} is an arbitrary operator, and U 0 = e i ( H S + H B ) t {\displaystyle U_{0}=e^{i(H_{S}+H_{B})t}} . Also note that U ( t , t 0 ) {\displaystyle U(t,t_{0})} is the total unitary operator of the entire system. It is straightforward to confirm that the Liouville equation becomes
where the Hamiltonian H ~ B S = e i ( H S + H B ) t H B S e − i ( H S + H B ) t {\displaystyle {\tilde {H}}_{BS}=e^{i(H_{S}+H_{B})t}H_{BS}e^{-i(H_{S}+H_{B})t}} is explicitly time dependent. Also, according to the interaction picture, χ ~ = U B S ( t , t 0 ) χ U B S † ( t , t 0 ) {\displaystyle {\tilde {\chi }}=U_{BS}(t,t_{0})\chi U_{BS}^{\dagger }(t,t_{0})} , where U B S = U 0 † U ( t , t 0 ) {\displaystyle U_{BS}=U_{0}^{\dagger }U(t,t_{0})} . This equation can be integrated directly to give
This implicit equation for χ ~ {\displaystyle {\tilde {\chi }}} can be substituted back into the Liouville equation to obtain an exact differo-integral equation
We proceed with the derivation by assuming the interaction is initiated at t = 0 {\displaystyle t=0} , and at that time there are no correlations between the system and the bath. This implies that the initial condition is factorable as χ ( 0 ) = ρ ( 0 ) R 0 {\displaystyle \chi (0)=\rho (0)R_{0}} , where R 0 {\displaystyle R_{0}} is the density operator of the bath initially.
Tracing over the bath degrees of freedom, tr R χ ~ = ρ ~ {\displaystyle \operatorname {tr} _{R}{\tilde {\chi }}={\tilde {\rho }}} , of the aforementioned differo-integral equation yields
This equation is exact for the time dynamics of the system density matrix but requires full knowledge of the dynamics of the bath degrees of freedom. A simplifying assumption called the Born approximation rests on the largeness of the bath and the relative weakness of the coupling, which is to say the coupling of the system to the bath should not significantly alter the bath eigenstates. In this case the full density matrix is factorable for all times as χ ~ ( t ) = ρ ~ ( t ) R 0 {\displaystyle {\tilde {\chi }}(t)={\tilde {\rho }}(t)R_{0}} . The master equation becomes
The equation is now explicit in the system degrees of freedom, but is very difficult to solve. A final assumption is the Born-Markov approximation that the time derivative of the density matrix depends only on its current state, and not on its past. This assumption is valid under fast bath dynamics, wherein correlations within the bath are lost extremely quickly, and amounts to replacing ρ ( t ′ ) → ρ ( t ) {\displaystyle \rho (t')\rightarrow \rho (t)} on the right hand side of the equation.
If the interaction Hamiltonian is assumed to have the form
for system operators α i {\displaystyle \alpha _{i}} and bath operators Γ i {\displaystyle \Gamma _{i}} then H ~ B S = ∑ i α ~ i Γ ~ i {\displaystyle {\tilde {H}}_{BS}=\sum _{i}{\tilde {\alpha }}_{i}{\tilde {\Gamma }}_{i}} . The master equation becomes
which can be expanded as
The expectation values ⟨ Γ i Γ j ⟩ = tr { Γ i Γ j R 0 } {\displaystyle \langle \Gamma _{i}\Gamma _{j}\rangle =\operatorname {tr} \{\Gamma _{i}\Gamma _{j}R_{0}\}} are with respect to the bath degrees of freedom.
By assuming rapid decay of these correlations (ideally ⟨ Γ i ( t ) Γ j ( t ′ ) ⟩ ∝ δ ( t − t ′ ) {\displaystyle \langle \Gamma _{i}(t)\Gamma _{j}(t')\rangle \propto \delta (t-t')} ), above form of the Lindblad superoperator L is achieved.
In the simplest case, there is just one jump operator F {\displaystyle F} and no unitary evolution. In this case, the Lindblad equation is
This case is often used in quantum optics to model either absorption or emission of photons from a reservoir.
To model both absorption and emission, one would need a jump operator for each. This leads to the most common Lindblad equation describing the damping of a quantum harmonic oscillator (representing e.g. a Fabry–Perot cavity ) coupled to a thermal bath , with jump operators:
Here n ¯ {\displaystyle {\overline {n}}} is the mean number of excitations in the reservoir damping the oscillator and γ is the decay rate.
To model the quantum harmonic oscillator Hamiltonian with frequency ω c {\displaystyle \omega _{c}} of the photons, we can add a further unitary evolution:
Additional Lindblad operators can be included to model various forms of dephasing and vibrational relaxation. These methods have been incorporated into grid-based density matrix propagation methods. | https://en.wikipedia.org/wiki/Lindbladian |
In probability theory , Lindeberg's condition is a sufficient condition (and under certain conditions also a necessary condition) for the central limit theorem (CLT) to hold for a sequence of independent random variables . [ 1 ] [ 2 ] [ 3 ] Unlike the classical CLT, which requires that the random variables in question have finite variance and be both independent and identically distributed , Lindeberg's CLT only requires that they have finite variance, satisfy Lindeberg's condition, and be independent . It is named after the Finnish mathematician Jarl Waldemar Lindeberg . [ 4 ]
Let ( Ω , F , P ) {\displaystyle (\Omega ,{\mathcal {F}},\mathbb {P} )} be a probability space , and X k : Ω → R , k ∈ N {\displaystyle X_{k}:\Omega \to \mathbb {R} ,\,\,k\in \mathbb {N} } , be independent random variables defined on that space. Assume the expected values E [ X k ] = μ k {\displaystyle \mathbb {E} \,[X_{k}]=\mu _{k}} and variances V a r [ X k ] = σ k 2 {\displaystyle \mathrm {Var} \,[X_{k}]=\sigma _{k}^{2}} exist and are finite. Also let s n 2 := ∑ k = 1 n σ k 2 . {\displaystyle s_{n}^{2}:=\sum _{k=1}^{n}\sigma _{k}^{2}.}
If this sequence of independent random variables X k {\displaystyle X_{k}} satisfies Lindeberg's condition :
for all ε > 0 {\displaystyle \varepsilon >0} , where 1 {…} is the indicator function , then the central limit theorem holds, i.e. the random variables
converge in distribution to a standard normal random variable as n → ∞ . {\displaystyle n\to \infty .}
Lindeberg's condition is sufficient, but not in general necessary (i.e. the inverse implication does not hold in general).
However, if the sequence of independent random variables in question satisfies
then Lindeberg's condition is both sufficient and necessary, i.e. it holds if and only if the result of central limit theorem holds.
Feller's theorem can be used as an alternative method to prove that Lindeberg's condition holds. [ 5 ] Letting S n := ∑ k = 1 n X k {\displaystyle S_{n}:=\sum _{k=1}^{n}X_{k}} and for simplicity E [ X k ] = 0 {\displaystyle \mathbb {E} \,[X_{k}]=0} , the theorem states
This theorem can be used to disprove the central limit theorem holds for X k {\displaystyle X_{k}} by using proof by contradiction . This procedure involves proving that Lindeberg's condition fails for X k {\displaystyle X_{k}} .
Because the Lindeberg condition implies max k = 1 , … , n σ k 2 s n 2 → 0 {\displaystyle \max _{k=1,\ldots ,n}{\frac {\sigma _{k}^{2}}{s_{n}^{2}}}\to 0} as n → ∞ {\displaystyle n\to \infty } , it guarantees that the contribution of any individual random variable X k {\displaystyle X_{k}} ( 1 ≤ k ≤ n {\displaystyle 1\leq k\leq n} ) to the variance s n 2 {\displaystyle s_{n}^{2}} is arbitrarily small, for sufficiently large values of n {\displaystyle n} .
Consider the following informative example which satisfies the Lindeberg condition. Let ξ i {\displaystyle \xi _{i}} be a sequence of zero mean, variance 1 iid random variables and a i {\displaystyle a_{i}} a non-random sequence satisfying:
max i n | a i | ‖ a ‖ 2 → 0 {\displaystyle \max _{i}^{n}{\frac {|a_{i}|}{\|a\|_{2}}}\rightarrow 0}
Now, define the normalized elements of the linear combination :
X n , i = a i ξ i ‖ a ‖ 2 {\displaystyle X_{n,i}={\frac {a_{i}\xi _{i}}{\|a\|_{2}}}}
which satisfies the Lindeberg condition:
∑ i n E [ | X i | 2 1 ( | X i | > ε ) ] ≤ ∑ i n E [ | X i | 2 1 ( | ξ i | > ε ‖ a ‖ 2 max i n | a i | ) ] = E [ | ξ i | 2 1 ( | ξ i | > ε ‖ a ‖ 2 max i n | a i | ) ] {\displaystyle \sum _{i}^{n}\mathbb {E} \left[\left|X_{i}\right|^{2}1(|X_{i}|>\varepsilon )\right]\leq \sum _{i}^{n}\mathbb {E} \left[\left|X_{i}\right|^{2}1\left(|\xi _{i}|>\varepsilon {\frac {\|a\|_{2}}{\max _{i}^{n}|a_{i}|}}\right)\right]=\mathbb {E} \left[\left|\xi _{i}\right|^{2}1\left(|\xi _{i}|>\varepsilon {\frac {\|a\|_{2}}{\max _{i}^{n}|a_{i}|}}\right)\right]}
but ξ i 2 {\displaystyle \xi _{i}^{2}} is finite so by DCT and the condition on the a i {\displaystyle a_{i}} we have that this goes to 0 for every ε {\displaystyle \varepsilon } . | https://en.wikipedia.org/wiki/Lindeberg's_condition |
In mathematics , Lindelöf's lemma is a simple but useful lemma in topology on the real line , named for the Finnish mathematician Ernst Leonard Lindelöf .
Let the real line have its standard topology. Then every open subset of the real line is a countable union of open intervals .
Lindelöf's lemma is also known as the statement that every open cover in a second-countable space has a countable subcover (Kelley 1955:49). This means that every second-countable space is also a Lindelöf space .
Let B {\displaystyle B} be a countable basis of X {\displaystyle X} . Consider an open cover, F = ⋃ α U α {\displaystyle {\mathcal {F}}=\bigcup _{\alpha }U_{\alpha }} . To get prepared for the following deduction, we define two sets for convenience, B α := { β ∈ B : β ⊂ U α } {\displaystyle B_{\alpha }:=\left\{\beta \in B:\beta \subset U_{\alpha }\right\}} , B ′ := ⋃ α B α {\displaystyle B':=\bigcup _{\alpha }B_{\alpha }} .
A straight-forward but essential observation is that, U α = ⋃ β ∈ B α β {\displaystyle U_{\alpha }=\bigcup _{\beta \in B_{\alpha }}\beta } which is from the definition of base. [ 1 ] Therefore, we can get that,
F = ⋃ α U α = ⋃ α ⋃ β ∈ B α β = ⋃ β ∈ B ′ β {\displaystyle {\mathcal {F}}=\bigcup _{\alpha }U_{\alpha }=\bigcup _{\alpha }\bigcup _{\beta \in B_{\alpha }}\beta =\bigcup _{\beta \in B'}\beta }
where B ′ ⊂ B {\displaystyle B'\subset B} , and is therefore at most countable. Next, by construction, for each β ∈ B ′ {\displaystyle \beta \in B'} there is some δ β {\displaystyle \delta _{\beta }} such that β ⊂ U δ β {\displaystyle \beta \subset U_{\delta _{\beta }}} . We can therefore write
F = ⋃ β ∈ B ′ U δ β {\displaystyle {\mathcal {F}}=\bigcup _{\beta \in B'}U_{\delta _{\beta }}}
completing the proof.
This topology-related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Lindelöf's_lemma |
The Lindemann index [ 1 ] is a simple measure of thermally driven disorder in atoms or molecules.
The local Lindemann index is defined as: [ 2 ]
q i = 1 N − 1 ∑ j ≠ i ⟨ r i j 2 ⟩ − ⟨ r i j ⟩ 2 ⟨ r i j ⟩ {\displaystyle q_{i}={\frac {1}{N-1}}\sum _{j\neq i}{\frac {\sqrt {\langle r_{ij}^{2}\rangle -\langle r_{ij}\rangle ^{2}}}{\langle r_{ij}\rangle }}}
where angle brackets indicate a time average. The global Lindemann index is a system average of this quantity.
Care must be taken if the molecule possesses globally defined dynamics, such as about a hinge or pivot, because these motions will obscure the local motions which the Lindemann index is designed to quantify. An appropriate tactic in this circumstance is to sum the r ij only over a small number of neighbouring atoms to arrive at each q i . A further variety of such modifications to the Lindemann index are available and have different merits, e.g. for the study of glassy vs crystalline materials. [ 4 ] | https://en.wikipedia.org/wiki/Lindemann_index |
In chemical kinetics , the Lindemann mechanism (also called the Lindemann–Christiansen mechanism [ 1 ] or the Lindemann–Hinshelwood mechanism [ 2 ] [ 3 ] ) is a schematic reaction mechanism for unimolecular reactions . Frederick Lindemann and J. A. Christiansen proposed the concept almost simultaneously in 1921, [ 4 ] [ 1 ] and Cyril Hinshelwood developed it to take into account the energy distributed among vibrational degrees of freedom for some reaction steps. [ 5 ] [ 6 ]
It breaks down an apparently unimolecular reaction into two elementary steps , with a rate constant for each elementary step. The rate law and rate equation for the entire reaction can be derived from the rate equations and rate constants for the two steps.
The Lindemann mechanism is used to model gas phase decomposition or isomerization reactions. Although the net formula for decomposition or isomerization appears to be unimolecular and suggests first-order kinetics in the reactant, the Lindemann mechanism shows that the unimolecular reaction step is preceded by a bimolecular activation step so that the kinetics may actually be second-order in certain cases. [ 7 ]
The overall equation for a unimolecular reaction may be written A → P, where A is the initial reactant molecule and P is one or more products (one for isomerization, more for decomposition).
A Lindemann mechanism typically includes an activated reaction intermediate , labeled A*. The activated intermediate is produced from the reactant only after a sufficient activation energy is acquired by collision with a second molecule M, which may or may not be similar to A. It then either deactivates from A* back to A by another collision, or reacts in a unimolecular step to produce the product(s) P.
The two-step mechanism is then
The rate equation for the rate of formation of product P may be obtained by using the steady-state approximation , in which the concentration of intermediate A* is assumed constant because its rates of production and consumption are (almost) equal. [ 8 ] This assumption simplifies the calculation of the rate equation.
For the schematic mechanism of two elementary steps above, rate constants are defined as k 1 {\displaystyle k_{1}} for the forward reaction rate of the first step, k − 1 {\displaystyle k_{-1}} for the reverse reaction rate of the first step, and k 2 {\displaystyle k_{2}} for the forward reaction rate of the second step. For each elementary step, the order of reaction is equal to the molecularity
The rate of production of the intermediate A* in the first elementary step is simply:
A* is consumed both in the reverse first step and in the forward second step. The respective rates of consumption of A* are:
According to the steady-state approximation, the rate of production of A* equals the rate of consumption. Therefore:
Solving for [ A ∗ ] {\displaystyle [{\ce {A}}^{*}]} , it is found that
The overall reaction rate is
Now, by substituting the calculated value for [A*], the overall reaction rate can be expressed in terms of the original reactants A and M: [ 9 ] [ 8 ]
The steady-state rate equation is of mixed order and predicts that a unimolecular reaction can be of either first or second order, depending on which of the two terms in the denominator is larger. At sufficiently low pressures, k − 1 [ M ] ≪ k 2 {\displaystyle k_{-1}[{\ce {M}}]\ll k_{2}} so that d [ P ] / d t = k 1 [ A ] [ M ] {\displaystyle \mathrm {d} [{\ce {P}}]/\mathrm {d} t=k_{1}[{\ce {A}}][{\ce {M}}]} , which is second order. That is, the rate-determining step is the first, bimolecular activation step. [ 8 ] [ 9 ]
At higher pressures, however, k − 1 [ M ] ≫ k 2 {\displaystyle k_{-1}[{\ce {M}}]\gg k_{2}} so that d [ P ] d t = k 1 k 2 k − 1 [ A ] {\displaystyle {\frac {\mathrm {d} [{\ce {P}}]}{\mathrm {d} t}}={\frac {k_{1}k_{2}}{k_{-1}}}[{\ce {A}}]} which is first order, and the rate-determining step is the second step, i.e. the unimolecular reaction of the activated molecule.
The theory can be tested by defining an effective rate constant (or coefficient) k u n i {\displaystyle k_{\rm {uni}}} which would be constant if the reaction were first order at all pressures: d [ P ] d t = k u n i [ A ] , k u n i = 1 [ A ] d [ P ] d t {\displaystyle {\frac {\mathrm {d} [{\ce {P}}]}{\mathrm {d} t}}=k_{\rm {uni}}[{\ce {A}}],\quad k_{\rm {uni}}={\frac {1}{[A]}}{\frac {\mathrm {d[P]} }{\mathrm {d} t}}} . The Lindemann mechanism predicts that k decreases with pressure, and that its reciprocal 1 k = k − 1 k 1 k 2 + 1 k 1 [ M ] {\displaystyle {\frac {1}{k}}={\frac {k_{-1}}{k_{1}k_{2}}}+{\frac {1}{k_{1}[{\ce {M}}]}}} is a linear function of 1 [ M ] {\displaystyle {\frac {1}{[{\ce {M}}]}}} or equivalently of 1 p {\displaystyle {\frac {1}{p}}} . Experimentally for many reactions, k {\displaystyle k} does decrease at low pressure, but the graph of 1 / k {\displaystyle 1/k} as a function of 1 / p {\displaystyle 1/p} is quite curved. To account accurately for the pressure-dependence of rate constants for unimolecular reactions, more elaborate theories are required such as the RRKM theory . [ 9 ] [ 8 ]
In the Lindemann mechanism for a true unimolecular reaction, the activation step is followed by a single step corresponding to the formation of products. Whether this is actually true for any given reaction must be established from the evidence.
Much early experimental investigation of the Lindemann mechanism involved study of the gas-phase decomposition of dinitrogen pentoxide [ 10 ] 2 N 2 O 5 → 2 N 2 O 4 + O 2 . This reaction was studied by Farrington Daniels and coworkers, and initially assumed to be a true unimolecular reaction. However it is now known to be a multistep reaction whose mechanism was established by Ogg [ 10 ] as:
An analysis using the steady-state approximation shows that this mechanism can also explain the observed first-order kinetics and the fall-off of the rate constant at very low pressures. [ 10 ]
The Lindemann-Hinshelwood mechanism explains unimolecular reactions that take place in the gas phase . Usually, this mechanism is used in gas phase decomposition and also in isomerization reactions . An example of isomerization by a Lindemann mechanism is the isomerization of cyclopropane . [ 11 ]
Although it seems like a simple reaction, it is actually a multistep reaction:
This isomerization can be explained by the Lindemann mechanism, because once the cyclopropane, the reactant, is excited by collision it becomes an energized cyclopropane . And then, this molecule can be deactivated back to reactants or produce propene , the product. | https://en.wikipedia.org/wiki/Lindemann_mechanism |
In transcendental number theory , the Lindemann–Weierstrass theorem is a result that is very useful in establishing the transcendence of numbers. It states the following:
Lindemann–Weierstrass theorem — if α 1 , ..., α n are algebraic numbers that are linearly independent over the rational numbers Q {\displaystyle \mathbb {Q} } , then e α 1 , ..., e α n are algebraically independent over Q {\displaystyle \mathbb {Q} } .
In other words, the extension field Q ( e α 1 , … , e α n ) {\displaystyle \mathbb {Q} (e^{\alpha _{1}},\dots ,e^{\alpha _{n}})} has transcendence degree n over Q {\displaystyle \mathbb {Q} } .
An equivalent formulation from Baker 1990 , Chapter 1, Theorem 1.4, is the following:
An equivalent formulation — If α 1 , ..., α n are distinct algebraic numbers, then the exponentials e α 1 , ..., e α n are linearly independent over the algebraic numbers.
This equivalence transforms a linear relation over the algebraic numbers into an algebraic relation over Q {\displaystyle \mathbb {Q} } by using the fact that a symmetric polynomial whose arguments are all conjugates of one another gives a rational number.
The theorem is named for Ferdinand von Lindemann and Karl Weierstrass . Lindemann proved in 1882 that e α is transcendental for every non-zero algebraic number α, thereby establishing that π is transcendental (see below). [ 1 ] Weierstrass proved the above more general statement in 1885. [ 2 ]
The theorem, along with the Gelfond–Schneider theorem , is extended by Baker's theorem , [ 3 ] and all of these would be further generalized by Schanuel's conjecture .
The theorem is also known variously as the Hermite–Lindemann theorem and the Hermite–Lindemann–Weierstrass theorem . Charles Hermite first proved the simpler theorem where the α i exponents are required to be rational integers and linear independence is only assured over the rational integers, [ 4 ] [ 5 ] a result sometimes referred to as Hermite's theorem. [ 6 ] Although that appears to be a special case of the above theorem, the general result can be reduced to this simpler case. Lindemann was the first to allow algebraic numbers into Hermite's work in 1882. [ 1 ] Shortly afterwards Weierstrass obtained the full result, [ 2 ] and further simplifications have been made by several mathematicians, most notably by David Hilbert [ 7 ] and Paul Gordan . [ 8 ]
The transcendence of e and π are direct corollaries of this theorem.
Suppose α is a non-zero algebraic number; then {α} is a linearly independent set over the rationals, and therefore by the first formulation of the theorem { e α } is an algebraically independent set; or in other words e α is transcendental. In particular, e 1 = e is transcendental. (A more elementary proof that e is transcendental is outlined in the article on transcendental numbers .)
Alternatively, by the second formulation of the theorem, if α is a non-zero algebraic number, then {0, α} is a set of distinct algebraic numbers, and so the set { e 0 , e α } = {1, e α } is linearly independent over the algebraic numbers and in particular e α cannot be algebraic and so it is transcendental.
To prove that π is transcendental, we prove that it is not algebraic. If π were algebraic, π i would be algebraic as well, and then by the Lindemann–Weierstrass theorem e π i = −1 (see Euler's identity ) would be transcendental, a contradiction. Therefore π is not algebraic, which means that it is transcendental.
A slight variant on the same proof will show that if α is a non-zero algebraic number then sin(α), cos(α), tan(α) and their hyperbolic counterparts are also transcendental.
p -adic Lindemann–Weierstrass Conjecture. — Suppose p is some prime number and α 1 , ..., α n are p -adic numbers which are algebraic and linearly independent over Q {\displaystyle \mathbb {Q} } , such that | α i | p < 1/ p for all i ; then the p -adic exponentials exp p (α 1 ), . . . , exp p (α n ) are p -adic numbers that are algebraically independent over Q {\displaystyle \mathbb {Q} } .
An analogue of the theorem involving the modular function j was conjectured by Daniel Bertrand in 1997, and remains an open problem. [ 9 ] Writing q = e 2 π i τ for the square of the nome and j (τ) = J ( q ), the conjecture is as follows.
Modular conjecture — Let q 1 , ..., q n be non-zero algebraic numbers in the complex unit disc such that the 3 n numbers
are algebraically dependent over Q {\displaystyle \mathbb {Q} } . Then there exist two indices 1 ≤ i < j ≤ n such that q i and q j are multiplicatively dependent.
Lindemann–Weierstrass Theorem (Baker's reformulation). — If a 1 , ..., a n are algebraic numbers, not all zero, and α 1 , ..., α n are distinct algebraic numbers, then [ 10 ]
The proof relies on two preliminary lemmas . Notice that Lemma B itself is already sufficient to deduce the original statement of Lindemann–Weierstrass theorem.
Lemma A. — Let c (1), ..., c ( r ) be integers and, for every k between 1 and r , let { γ ( k ) 1 , ..., γ ( k ) m ( k ) } be the roots of a non-zero polynomial with integer coefficients T k ( x ) {\displaystyle T_{k}(x)} . If γ ( k ) i ≠ γ ( u ) v whenever ( k , i ) ≠ ( u , v ) , then
has only the trivial solution c ( i ) = 0 {\displaystyle c(i)=0} for all i = 1 , … , r . {\displaystyle i=1,\dots ,r.}
Proof of Lemma A. To simplify the notation set:
Then the statement becomes
Let p be a prime number and define the following polynomials:
where ℓ is a non-zero integer such that ℓ α 1 , … , ℓ α n {\displaystyle \ell \alpha _{1},\ldots ,\ell \alpha _{n}} are all algebraic integers . Define [ 11 ]
Using integration by parts we arrive at
where n p − 1 {\displaystyle np-1} is the degree of f i {\displaystyle f_{i}} , and f i ( j ) {\displaystyle f_{i}^{(j)}} is the j -th derivative of f i {\displaystyle f_{i}} . This also holds for s complex (in this case the integral has to be intended as a contour integral, for example along the straight segment from 0 to s ) because
is a primitive of e s − x f i ( x ) {\displaystyle e^{s-x}f_{i}(x)} .
Consider the following sum:
In the last line we assumed that the conclusion of the Lemma is false. In order to complete the proof we need to reach a contradiction. We will do so by estimating | J 1 ⋯ J n | {\displaystyle |J_{1}\cdots J_{n}|} in two different ways.
First f i ( j ) ( α k ) {\displaystyle f_{i}^{(j)}(\alpha _{k})} is an algebraic integer which is divisible by p ! for j ≥ p {\displaystyle j\geq p} and vanishes for j < p {\displaystyle j<p} unless j = p − 1 {\displaystyle j=p-1} and k = i {\displaystyle k=i} , in which case it equals
This is not divisible by p when p is large enough because otherwise, putting
(which is a non-zero algebraic integer) and calling d i ∈ Z {\displaystyle d_{i}\in \mathbb {Z} } the product of its conjugates (which is still non-zero), we would get that p divides ℓ p ( p − 1 ) ! d i p {\displaystyle \ell ^{p}(p-1)!d_{i}^{p}} , which is false.
So J i {\displaystyle J_{i}} is a non-zero algebraic integer divisible by ( p − 1)!. Now
Since each f i ( x ) {\displaystyle f_{i}(x)} is obtained by dividing a fixed polynomial with integer coefficients by ( x − α i ) {\displaystyle (x-\alpha _{i})} , it is of the form
where g m {\displaystyle g_{m}} is a polynomial (with integer coefficients) independent of i . The same holds for the derivatives f i ( j ) ( x ) {\displaystyle f_{i}^{(j)}(x)} .
Hence, by the fundamental theorem of symmetric polynomials ,
is a fixed polynomial with rational coefficients evaluated in α i {\displaystyle \alpha _{i}} (this is seen by grouping the same powers of α n t − 1 + 1 , … , α n t {\displaystyle \alpha _{n_{t-1}+1},\dots ,\alpha _{n_{t}}} appearing in the expansion and using the fact that these algebraic numbers are a complete set of conjugates). So the same is true of J i {\displaystyle J_{i}} , i.e. it equals G ( α i ) {\displaystyle G(\alpha _{i})} , where G is a polynomial with rational coefficients independent of i .
Finally J 1 ⋯ J n = G ( α 1 ) ⋯ G ( α n ) {\displaystyle J_{1}\cdots J_{n}=G(\alpha _{1})\cdots G(\alpha _{n})} is rational (again by the fundamental theorem of symmetric polynomials) and is a non-zero algebraic integer divisible by ( p − 1 ) ! n {\displaystyle (p-1)!^{n}} (since the J i {\displaystyle J_{i}} 's are algebraic integers divisible by ( p − 1 ) ! {\displaystyle (p-1)!} ). Therefore
However one clearly has:
where F i is the polynomial whose coefficients are the absolute values of those of f i (this follows directly from the definition of I i ( s ) {\displaystyle I_{i}(s)} ). Thus
and so by the construction of the f i {\displaystyle f_{i}} 's we have | J 1 ⋯ J n | ≤ C p {\displaystyle |J_{1}\cdots J_{n}|\leq C^{p}} for a sufficiently large C independent of p , which contradicts the previous inequality. This proves Lemma A. ∎
Lemma B. — If b (1), ..., b ( n ) are integers and γ (1), ..., γ ( n ), are distinct algebraic numbers , then
has only the trivial solution b ( i ) = 0 {\displaystyle b(i)=0} for all i = 1 , … , n . {\displaystyle i=1,\dots ,n.}
Proof of Lemma B: Assuming
we will derive a contradiction, thus proving Lemma B.
Let us choose a polynomial with integer coefficients which vanishes on all the γ ( k ) {\displaystyle \gamma (k)} 's and let γ ( 1 ) , … , γ ( n ) , γ ( n + 1 ) , … , γ ( N ) {\displaystyle \gamma (1),\ldots ,\gamma (n),\gamma (n+1),\ldots ,\gamma (N)} be all its distinct roots. Let b ( n + 1) = ... = b ( N ) = 0.
The polynomial
vanishes at ( e γ ( 1 ) , … , e γ ( N ) ) {\displaystyle (e^{\gamma (1)},\dots ,e^{\gamma (N)})} by assumption. Since the product is symmetric, for any τ ∈ S N {\displaystyle \tau \in S_{N}} the monomials x τ ( 1 ) h 1 ⋯ x τ ( N ) h N {\displaystyle x_{\tau (1)}^{h_{1}}\cdots x_{\tau (N)}^{h_{N}}} and x 1 h 1 ⋯ x N h N {\displaystyle x_{1}^{h_{1}}\cdots x_{N}^{h_{N}}} have the same coefficient in the expansion of P .
Thus, expanding P ( e γ ( 1 ) , … , e γ ( N ) ) {\displaystyle P(e^{\gamma (1)},\dots ,e^{\gamma (N)})} accordingly and grouping the terms with the same exponent, we see that the resulting exponents h 1 γ ( 1 ) + ⋯ + h N γ ( N ) {\displaystyle h_{1}\gamma (1)+\dots +h_{N}\gamma (N)} form a complete set of conjugates and, if two terms have conjugate exponents, they are multiplied by the same coefficient.
So we are in the situation of Lemma A. To reach a contradiction it suffices to see that at least one of the coefficients is non-zero. This is seen by equipping C with the lexicographic order and by choosing for each factor in the product the term with non-zero coefficient which has maximum exponent according to this ordering: the product of these terms has non-zero coefficient in the expansion and does not get simplified by any other term. This proves Lemma B. ∎
We turn now to prove the theorem: Let a (1), ..., a ( n ) be non-zero algebraic numbers , and α (1), ..., α ( n ) distinct algebraic numbers. Then let us assume that:
We will show that this leads to contradiction and thus prove the theorem. The proof is very similar to that of Lemma B, except that this time the choices are made over the a ( i )'s:
For every i ∈ {1, ..., n }, a ( i ) is algebraic, so it is a root of an irreducible polynomial with integer coefficients of degree d ( i ). Let us denote the distinct roots of this polynomial a ( i ) 1 , ..., a ( i ) d ( i ) , with a ( i ) 1 = a ( i ).
Let S be the functions σ which choose one element from each of the sequences (1, ..., d (1)), (1, ..., d (2)), ..., (1, ..., d ( n )), so that for every 1 ≤ i ≤ n , σ( i ) is an integer between 1 and d ( i ). We form the polynomial in the variables x 11 , … , x 1 d ( 1 ) , … , x n 1 , … , x n d ( n ) , y 1 , … , y n {\displaystyle x_{11},\dots ,x_{1d(1)},\dots ,x_{n1},\dots ,x_{nd(n)},y_{1},\dots ,y_{n}}
Since the product is over all the possible choice functions σ, Q is symmetric in x i 1 , … , x i d ( i ) {\displaystyle x_{i1},\dots ,x_{id(i)}} for every i . Therefore Q is a polynomial with integer coefficients in elementary symmetric polynomials of the above variables, for every i , and in the variables y i . Each of the latter symmetric polynomials is a rational number when evaluated in a ( i ) 1 , … , a ( i ) d ( i ) {\displaystyle a(i)_{1},\dots ,a(i)_{d(i)}} .
The evaluated polynomial Q ( a ( 1 ) 1 , … , a ( n ) d ( n ) , e α ( 1 ) , … , e α ( n ) ) {\displaystyle Q(a(1)_{1},\dots ,a(n)_{d(n)},e^{\alpha (1)},\dots ,e^{\alpha (n)})} vanishes because one of the choices is just σ( i ) = 1 for all i , for which the corresponding factor vanishes according to our assumption above. Thus, the evaluated polynomial is a sum of the form
where we already grouped the terms with the same exponent. So in the left-hand side we have distinct values β(1), ..., β( N ), each of which is still algebraic (being a sum of algebraic numbers) and coefficients b ( 1 ) , … , b ( N ) ∈ Q {\displaystyle b(1),\dots ,b(N)\in \mathbb {Q} } .
The sum is nontrivial: if α ( i ) {\displaystyle \alpha (i)} is maximal in the lexicographic order, the coefficient of e | S | α ( i ) {\displaystyle e^{|S|\alpha (i)}} is just a product of a ( i ) j 's (with possible repetitions), which is non-zero.
By multiplying the equation with an appropriate integer factor, we get an identical equation except that now b (1), ..., b ( N ) are all integers. Therefore, according to Lemma B, the equality cannot hold, and we are led to a contradiction which completes the proof. ∎
Note that Lemma A is sufficient to prove that e is irrational , since otherwise we may write e = p / q , where both p and q are non-zero integers, but by Lemma A we would have qe − p ≠ 0, which is a contradiction. Lemma A also suffices to prove that π is irrational, since otherwise we may write π = k / n , where both k and n are integers) and then ± i π are the roots of n 2 x 2 + k 2 = 0; thus 2 − 1 − 1 = 2 e 0 + e i π + e − i π ≠ 0; but this is false.
Similarly, Lemma B is sufficient to prove that e is transcendental, since Lemma B says that if a 0 , ..., a n are integers not all of which are zero, then
Lemma B also suffices to prove that π is transcendental, since otherwise we would have 1 + e i π ≠ 0.
Baker's formulation of the theorem clearly implies the first formulation. Indeed, if α ( 1 ) , … , α ( n ) {\displaystyle \alpha (1),\ldots ,\alpha (n)} are algebraic numbers that are linearly independent over Q {\displaystyle \mathbb {Q} } , and
is a polynomial with rational coefficients, then we have
and since α ( 1 ) , … , α ( n ) {\displaystyle \alpha (1),\ldots ,\alpha (n)} are algebraic numbers which are linearly independent over the rationals, the numbers i 1 α ( 1 ) + ⋯ + i n α ( n ) {\displaystyle i_{1}\alpha (1)+\cdots +i_{n}\alpha (n)} are algebraic and they are distinct for distinct n -tuples ( i 1 , … , i n ) {\displaystyle (i_{1},\dots ,i_{n})} . So from Baker's formulation of the theorem we get b i 1 , … , i n = 0 {\displaystyle b_{i_{1},\ldots ,i_{n}}=0} for all n -tuples ( i 1 , … , i n ) {\displaystyle (i_{1},\dots ,i_{n})} .
Now assume that the first formulation of the theorem holds. For n = 1 {\displaystyle n=1} Baker's formulation is trivial, so let us assume that n > 1 {\displaystyle n>1} , and let a ( 1 ) , … , a ( n ) {\displaystyle a(1),\ldots ,a(n)} be non-zero algebraic numbers, and α ( 1 ) , … , α ( n ) {\displaystyle \alpha (1),\ldots ,\alpha (n)} distinct algebraic numbers such that:
As seen in the previous section, and with the same notation used there, the value of the polynomial
at
has an expression of the form
where we have grouped the exponentials having the same exponent. Here, as proved above, b ( 1 ) , … , b ( M ) {\displaystyle b(1),\ldots ,b(M)} are rational numbers, not all equal to zero, and each exponent β ( m ) {\displaystyle \beta (m)} is a linear combination of α ( i ) {\displaystyle \alpha (i)} with integer coefficients. Then, since n > 1 {\displaystyle n>1} and α ( 1 ) , … , α ( n ) {\displaystyle \alpha (1),\ldots ,\alpha (n)} are pairwise distinct, the Q {\displaystyle \mathbb {Q} } -vector subspace V {\displaystyle V} of C {\displaystyle \mathbb {C} } generated by α ( 1 ) , … , α ( n ) {\displaystyle \alpha (1),\ldots ,\alpha (n)} is not trivial and we can pick α ( i 1 ) , … , α ( i k ) {\displaystyle \alpha (i_{1}),\ldots ,\alpha (i_{k})} to form a basis for V . {\displaystyle V.} For each m = 1 , … , M {\displaystyle m=1,\dots ,M} , we have
For each j = 1 , … , k , {\displaystyle j=1,\ldots ,k,} let d j {\displaystyle d_{j}} be the least common multiple of all the d m , j {\displaystyle d_{m,j}} for m = 1 , … , M {\displaystyle m=1,\ldots ,M} , and put v j = 1 d j α ( i j ) {\displaystyle v_{j}={\tfrac {1}{d_{j}}}\alpha (i_{j})} . Then v 1 , … , v k {\displaystyle v_{1},\ldots ,v_{k}} are algebraic numbers, they form a basis of V {\displaystyle V} , and each β ( m ) {\displaystyle \beta (m)} is a linear combination of the v j {\displaystyle v_{j}} with integer coefficients. By multiplying the relation
by e N ( v 1 + ⋯ + v k ) {\displaystyle e^{N(v_{1}+\cdots +v_{k})}} , where N {\displaystyle N} is a large enough positive integer, we get a non-trivial algebraic relation with rational coefficients connecting e v 1 , ⋯ , e v k {\displaystyle e^{v_{1}},\cdots ,e^{v_{k}}} , against the first formulation of the theorem. | https://en.wikipedia.org/wiki/Lindemann–Weierstrass_theorem |
In mathematical logic , Lindenbaum's lemma , named after Adolf Lindenbaum , states that any consistent theory of predicate logic can be extended to a complete consistent theory. The lemma is a special case of the ultrafilter lemma for Boolean algebras , applied to the Lindenbaum algebra of a theory.
It is used in the proof of Gödel's completeness theorem , among other places. [ citation needed ]
The effective version of the lemma's statement, "every consistent computably enumerable theory can be extended to a complete consistent computably enumerable theory," fails (provided Peano arithmetic is consistent) by Gödel's incompleteness theorem .
The lemma was not published by Adolf Lindenbaum ; it is originally attributed to him by Alfred Tarski . [ 1 ]
This logic -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Lindenbaum's_lemma |
The Linde–Frank–Caro process is a method for hydrogen production by removing hydrogen and carbon dioxide from water gas by condensation . [ 1 ] [ 2 ] The process was invented in 1909 by Adolf Frank and developed with Carl von Linde and Heinrich Caro . [ 3 ]
Water gas is compressed to 20 bar and pumped into the Linde–Frank–Caro reactor. A water column removes most of the carbon dioxide and sulfur . Tubes with caustic soda then remove the remaining carbon dioxide, sulphur, and water from the gas stream. The gas enters a chamber and is cooled to −190 °C, resulting in the condensation of most of the gas to a liquid. The remaining gas is pumped to the next vessel where the nitrogen is liquefied by cooling to −205 °C, resulting in hydrogen gas as an end product. | https://en.wikipedia.org/wiki/Linde–Frank–Caro_process |
Lindgren oxidation is a selective method for oxidizing aldehydes to carboxylic acids . [ 1 ] The reaction is named after Bengt O. Lindgren.
The oxidation takes place in water containing solvent mixtures under slightly acidic conditions (pH 3–5) with sodium chlorite as oxidizer. To avoid complicated oxidation reactions the hypochlorite , which is formed in the reaction, has to be removed from the reaction mixture by scavengers. In the original publication, sulfamic acid and resorcinol were used. [ 1 ] George A. Kraus and co-workers were the first to use 2-methyl-2-butene as scavenger under buffered conditions for the oxidation of an aliphatic and an α,β-unsaturated aldehyde. [ 2 ] [ 3 ] Later hydrogen peroxide also proved to work to remove the hypochlorite. [ 4 ]
This chemical reaction article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Lindgren_oxidation |
In condensed matter physics , Lindhard theory [ 1 ] is a method of calculating the effects of electric field screening by electrons in a solid. It is based on quantum mechanics (first-order perturbation theory) and the random phase approximation . It is named after Danish physicist Jens Lindhard , who first developed the theory in 1954. [ 2 ] [ 3 ] [ 4 ]
Thomas–Fermi screening and the plasma oscillations can be derived as a special case of the more general Lindhard formula. In particular, Thomas–Fermi screening is the limit of the Lindhard formula when the wavevector (the reciprocal of the length-scale of interest) is much smaller than the Fermi wavevector, i.e. the long-distance limit. [ 1 ] The Lorentz–Drude expression for the plasma oscillations are recovered in the dynamic case (long wavelengths, finite frequency).
This article uses cgs-Gaussian units .
The Lindhard formula for the longitudinal dielectric function is given by
Here, δ {\displaystyle \delta } is a positive infinitesimal constant, V q {\displaystyle V_{\mathbf {q} }} is V eff ( q ) − V ind ( q ) {\displaystyle V_{\text{eff}}(\mathbf {q} )-V_{\text{ind}}(\mathbf {q} )} and f k {\displaystyle f_{\mathbf {k} }} is the carrier distribution function which is the Fermi–Dirac distribution function for electrons in thermodynamic equilibrium.
However this Lindhard formula is valid also for nonequilibrium distribution functions. It can be obtained by first-order perturbation theory and the random phase approximation (RPA).
To understand the Lindhard formula, consider some limiting cases in 2 and 3 dimensions. The 1-dimensional case is also considered in other ways.
In the long wavelength limit ( q → 0 {\displaystyle \mathbf {q} \to 0} ), Lindhard function reduces to
where ω p l 2 = 4 π e 2 N L 3 m {\displaystyle \omega _{\rm {pl}}^{2}={\frac {4\pi e^{2}N}{L^{3}m}}} is the three-dimensional plasma frequency (in SI units, replace the factor 4 π {\displaystyle 4\pi } by 1 / ϵ 0 {\displaystyle 1/\epsilon _{0}} .) For two-dimensional systems,
This result recovers the plasma oscillations from the classical dielectric function from Drude model and from quantum mechanical free electron model .
For the denominator of the Lindhard formula, we get
and for the numerator of the Lindhard formula, we get
Inserting these into the Lindhard formula and taking the δ → 0 {\displaystyle \delta \to 0} limit, we obtain
where we used E k = ℏ ω k {\displaystyle E_{\mathbf {k} }=\hbar \omega _{\mathbf {k} }} and V q = 4 π e 2 ϵ q 2 L 3 {\displaystyle V_{\mathbf {q} }={\frac {4\pi e^{2}}{\epsilon q^{2}L^{3}}}} .
First, consider the long wavelength limit ( q → 0 {\displaystyle q\to 0} ).
For the denominator of the Lindhard formula,
and for the numerator,
Inserting these into the Lindhard formula and taking the limit of δ → 0 {\displaystyle \delta \to 0} , we obtain
where we used E k = ℏ ϵ k {\displaystyle E_{\mathbf {k} }=\hbar \epsilon _{\mathbf {k} }} , V q = 2 π e 2 ϵ q L 2 {\displaystyle V_{\mathbf {q} }={\frac {2\pi e^{2}}{\epsilon qL^{2}}}} and ω p l 2 ( q ) = 2 π e 2 n q ϵ m {\displaystyle \omega _{\rm {pl}}^{2}(\mathbf {q} )={\frac {2\pi e^{2}nq}{\epsilon m}}} .
Consider the static limit ( ω + i δ → 0 {\displaystyle \omega +i\delta \to 0} ).
The Lindhard formula becomes
Inserting the above equalities for the denominator and numerator, we obtain
Assuming a thermal equilibrium Fermi–Dirac carrier distribution, we get
here, we used E k = ℏ 2 k 2 2 m {\displaystyle E_{\mathbf {k} }={\frac {\hbar ^{2}k^{2}}{2m}}} and ∂ E k ∂ k i = ℏ 2 k i m {\displaystyle {\frac {\partial E_{\mathbf {k} }}{\partial k_{i}}}={\frac {\hbar ^{2}k_{i}}{m}}} .
Therefore,
Here, κ {\displaystyle \kappa } is the 3D screening wave number (3D inverse screening length) defined as
κ = 4 π e 2 ϵ ∂ n ∂ μ {\displaystyle \kappa ={\sqrt {{\frac {4\pi e^{2}}{\epsilon }}{\frac {\partial n}{\partial \mu }}}}} .
Then, the 3D statically screened Coulomb potential is given by
And the inverse Fourier transformation of this result gives
known as the Yukawa potential . Note that in this Fourier transformation, which is basically a sum over all q {\displaystyle \mathbf {q} } , we used the expression for small | q | {\displaystyle |\mathbf {q} |} for every value of q {\displaystyle \mathbf {q} } which is not correct.
For a degenerated Fermi gas ( T =0), the Fermi energy is given by
So the density is
At T =0, E F ≡ μ {\displaystyle E_{\rm {F}}\equiv \mu } , so ∂ n ∂ μ = 3 2 n E F {\displaystyle {\frac {\partial n}{\partial \mu }}={\frac {3}{2}}{\frac {n}{E_{\rm {F}}}}} .
Inserting this into the above 3D screening wave number equation, we obtain
This result recovers the 3D wave number from Thomas–Fermi screening .
For reference, Debye–Hückel screening describes the non-degenerate limit case. The result is κ = 4 π e 2 n β ϵ {\displaystyle \kappa ={\sqrt {\frac {4\pi e^{2}n\beta }{\epsilon }}}} , known as the 3D Debye–Hückel screening wave number.
In two dimensions, the screening wave number is
Note that this result is independent of n .
Consider the static limit ( ω + i δ → 0 {\displaystyle \omega +i\delta \to 0} ).
The Lindhard formula becomes
Inserting the above equalities for the denominator and numerator, we obtain
Assuming a thermal equilibrium Fermi–Dirac carrier distribution, we get
Therefore,
κ = 2 π e 2 ϵ ∂ n ∂ μ {\displaystyle \kappa ={\frac {2\pi e^{2}}{\epsilon }}{\frac {\partial n}{\partial \mu }}} .
It is known that the chemical potential of the 2-dimensional Fermi gas is given by
and ∂ μ ∂ n = ℏ 2 π m 1 1 − e − ℏ 2 β π n / m {\displaystyle {\frac {\partial \mu }{\partial n}}={\frac {\hbar ^{2}\pi }{m}}{\frac {1}{1-e^{-\hbar ^{2}\beta \pi n/m}}}} .
This time, consider some generalized case for lowering the dimension.
The lower the dimension is, the weaker the screening effect.
In lower dimension, some of the field lines pass through the barrier material wherein the screening has no effect.
For the 1-dimensional case, we can guess that the screening affects only the field lines which are very close to the wire axis.
In real experiment, we should also take the 3D bulk screening effect into account even though we deal with 1D case like the single filament. The Thomas–Fermi screening has been applied to an electron gas confined to a filament and a coaxial cylinder. [ 5 ] For a K 2 Pt(CN) 4 Cl 0.32 ·2.6H 2 0 filament, it was found that the potential within the region between the filament and cylinder varies as e − k e f f r / r {\displaystyle e^{-k_{\rm {eff}}r}/r} and its effective screening length is about 10 times that of metallic platinum . [ 5 ] | https://en.wikipedia.org/wiki/Lindhard_theory |
The Lindsey–Fox algorithm , named after Pat Lindsey and Jim Fox, is a numerical algorithm for finding the roots or zeros of a high-degree polynomial with real coefficients over the complex field . It is particularly designed for random coefficients but also works well on polynomials with coefficients from samples of speech, seismic signals, and other measured phenomena. A Matlab implementation of this has factored polynomials of degree over a million on a desktop computer.
The Lindsey–Fox algorithm uses the FFT (fast Fourier transform) to very efficiently conduct a grid search in the complex plane to find accurate approximations to the N roots (zeros) of an N th-degree polynomial. The power of this grid search allows a new polynomial factoring strategy that has proven to be very effective for a certain class of polynomials. This algorithm was conceived of by Pat Lindsey and implemented by Jim Fox in a package of computer programs created to factor high-degree polynomials. It was originally designed and has been further developed to be particularly suited to polynomials with real, random coefficients. In that form, it has proven to be very successful by factoring thousands of polynomials of degrees from one thousand to hundreds of thousand as well as several of degree one million and one each of degree two million and four million. In addition to handling very high degree polynomials, it is accurate, fast, uses minimum memory, and is programmed in the widely available language, Matlab. There are practical applications, often cases where the coefficients are samples of some natural signal such as speech or seismic signals, where the algorithm is appropriate and useful. However, it is certainly possible to create special, ill-conditioned polynomials that it cannot factor, even low degree ones. The basic ideas of the algorithm were first published by Lindsey and Fox in 1992 [ 1 ] and reprinted in 1996. [ 2 ] After further development, other papers were published in 2003 [ 3 ] [ 4 ] and an on-line booklet. [ 5 ] The program was made available to the public in March 2004 on the Rice University web site. [ 6 ] [ failed verification ] A more robust version-2 was released in March 2006 and updated later in the year.
The strategy implemented in the Lindsey–Fox algorithm to factor polynomials is organized in three stages. The first evaluates the polynomial over a grid on the complex plane and conducts a direct search for potential zeros. The second stage takes these potential zeros and “polishes” them by applying Laguerre's method to bring them close to the actual zeros of the polynomial. The third stage multiplies these zeros together or “unfactors” them to create a polynomial that is verified against the original. If some of the zeros were not found, the original polynomial is “deflated” by dividing it by the polynomial created from the found zeros. This quotient polynomial will generally be of low order and can be factored by conventional methods with the additional, new zeros added to the set of those first found. If there are still missing zeros, the deflation is carried out until all are found or the whole program needs to be restarted with a finer grid. This system has proven to be fast, accurate, and robust on the class of polynomials with real, random coefficients and other similar, well-conditioned polynomials.
Stage one is the reason this algorithm is so efficient and is what sets it apart from most other factoring algorithms. Because the FFT (fast Fourier transform) is used to evaluate the polynomial, a fast evaluation over a dense grid in the complex plane is possible. In order to use the FFT, the grid is structured in polar coordinates. In the first phase of this stage, a grid is designed with concentric circles of a particular radius intersected by a set of radial lines. The positions and spacing of the radial lines and the circles are chosen to give a grid that will hopefully separate the expected roots. Because the zeros of a polynomial with random coefficients have a fairly uniform angular distribution and are clustered close to the unit circle, it is possible to design an evaluation grid that fits the expected root density very well. In the second phase of this stage, the polynomial is evaluated at the nodes of the grid using the fast Fourier transform (FFT). It is because of the extraordinary efficiency and accuracy of the FFT that a direct evaluation is possible. In the third phase of this first stage, a search is conducted over all of the 3 by 3 node cells formed in the grid. For each 3 by 3 cell (see Figure below), if the value of the polynomial at the center node of the cell (the "x") is less than the values at all 8 of the nodes on the edges of the cell (the "o's"), the center is designated a candidate zero. This rule is based on the “Minimum Modulus Theorem” which states that if a relative minimum of the absolute value of an analytic function over an open region exists, the minimum must be a zero of the function. Finally, this set of prospective zeros is passed to the second stage. The number is usually slightly larger than the degree because some were found twice or mistakes were made. The number could be less if some zeros were missed.
Stage two is more traditional than the other two. It “polishes” each of the prospective zeros found by the grid search. The first phase consists of applying an iterative algorithm to improve the accuracy of the location found by the grid search. In earlier versions of the program, Newton's method was used but analysis and experiment showed that Laguerre's method was both more robust and more accurate. Even though it required more calculation than Newton's method for each iteration, it converged in fewer iterations. The second phase of the second stage checks for duplications. A “fuzzy” uniqueness test is applied to each zero to eliminate any cases where on two or more prospective zeros, iterations converged to the same zero. If the number of unique, polished zeros is less than the degree of the polynomial, deflation later will be necessary. If the number is greater, some error has occurred. This stage consumes the largest part of the execution time of the total factorization, but it is crucial to the final accuracy of the roots. One of the two criteria for success in factoring a polynomial is that each root must have been successfully polished against the original polynomial.
Stage three has several phases and possible iterations or even restarting. The first phase of the third stage takes all of the unique, polished zeros that were found in the first two stages and multiplies them together into the coefficient form of a candidate polynomial (“unfactors” the zeros). If the degree of this reconstructed polynomial is the same as that of the original polynomial and if the difference in their coefficients is small, the factorization is considered successful. Often, however, several zeros were missed by the grid search and polish processes of stage one and two, or the uniqueness test discarded a legitimate zero (perhaps it is multiple), so the original polynomial is “deflated” (divided) by the reconstructed polynomial and the resulting (low degree) quotient is factored for the missing zeros. If that doesn't find them all, the deflation process is repeated until they are all found. This allows the finding of multiple roots (or very tightly clustered roots), even if some of them were discarded earlier. If, in the unusual case, these further iterations of deflation do not find all of the missing zeros, a new, finer grid is constructed and the whole process started again at stage one. More details on the third stage are in another module.
Multiple order and clustered roots are unusual in random coefficient polynomials. But, if they happen or if factoring an ill-conditioned polynomial is attempted, the roots will be found with the Lindsey–Fox program in most cases but with reduced accuracy. If there are multiple order zeros (Mth order with M not too high), the grid search will find them, but with multiplicity one. The polishing will converge to the multiple order root but not as fast as to a distinct root. In the case of a cluster with Q zeros that fall within a single cell, they are erroneously identified as a single zero and the polishing will converge to only one of them. In some cases, two zeros can be close to each other in adjacent cells and polish to the same point. In all of these cases, after the uniqueness test and deflation, the quotient polynomial will contain a M − 1 order zero and/or Q − 1 zeros clustered together. Each of these zeros will be found after M − 1 or Q − 1 deflations. There can be problems here because Laguerre's polishing algorithm is not as accurate and does not converge as fast for a multiple zero and it may even diverge when applied to tightly clustered zeros. Also, the condition of the quotient polynomial will be poorer when multiple and clustered zeros are involved. If multiple order zeros are extremely far from the unit circle, the special methods for handling multiple roots developed by Zhonggang Zeng are used. Zeng's method is powerful but designed for polynomials of moderate degrees, and hence only used in special cases [6]. References
Successful completion of the factoring of a polynomial requires matching zeros on the complex plane measured by the convergence of Laguerre's algorithm on each of the zeros. It also requires matching the polynomial reconstructed from the found zeros with the original polynomial by measuring the maximum difference in each coefficient.
Because the FFT is such an efficient means of evaluating the polynomial, a very fine grid can be used which will separate all or almost all of the zeros in a reasonable time. And because of the fineness of the grid, the starting point is close to the actual zero and the polishing almost always converges in a small number of steps (convergence is often a serious problem in traditional approaches). And because the searching and polishing is done on the original polynomial, they can be done on each root simultaneously on a parallel architecture computer. Because the searching is done on a 3 by 3 cell of the grid, no more than three concentric circles of the grid need be kept in memory at a time, i.e., it is not necessary to have the entire grid in memory. And, some parallelization of the FFT calculations can be done.
Deflation is often a major source of error or failure in a traditional iterative algorithm. Here, because of the good starting points and the powerful polisher, very few stages of deflation are generally needed and they produce a low order quotient polynomial that is generally well-conditioned. Moreover, to control error, the unfactoring (multiplying the found roots together) is done in the FFT domain (for degree larger than 500) and the deflation is done partly in the FFT domain and partly in the coefficient domain, depending on a combination of stability, error accumulation, and speed factors.
For random coefficient polynomials, the number of zeros missed by the grid search and polish stages ranges from 0 to 10 or occasionally more. In factoring one 2 million degree polynomial, the search and polish stages found all 2 million zeros in one grid search and required no deflation which shows the power of the grid search on this class of polynomial. When deflation is needed, one pass is almost always sufficient. However, if you have a multiple zero or two (or more) very, very closely spaced zeros, the uniqueness test will discard a legitimate zero but it will be found by later deflation. Stage three has enough tests and alternatives to handle almost all possible conditions. But, by the very definition of random coefficients, it is hard to absolutely guarantee success.
The timings of the Lindsey–Fox program and examples of root distributions of polynomials with random coefficients are here . | https://en.wikipedia.org/wiki/Lindsey–Fox_algorithm |
Lindstrand Balloons was a manufacturer of hot air balloons and other aerostats . The company was started by Swedish -born pilot and aeronautical designer Per Lindstrand in Oswestry , England, as Colt Balloons (later Thunder & Colt Balloons, then Lindstrand Balloons) in 1978. Lindstrand Balloons was known for its leading-edge engineering, which included sophisticated testing and production facilities [ citation needed ] .
Lindstrand Balloons designed and built the hot air balloons flown by Per Lindstrand and Richard Branson on their record breaking flights; first across the Atlantic Ocean in 1987 and then the Pacific Ocean in 1990. Subsequently Lindstrand designed and built three Rozière balloons which Per Lindstrand and Branson and others used in their unsuccessful attempts to circumnavigate the Earth by balloon. Per Lindstrand played an instrumental role in making these flights possible [ citation needed ] , and was pilot for all of them.
In the late-1990s, Cameron Holdings and its owner Don Cameron acquired two-thirds ownership of Lindstrand Balloons. Cameron bought the majority stake in Lindstrand Balloons from Rory McCarthy, a British industrialist associated with Richard Branson, who had invested in Lindstrand to support Branson's series of record-setting balloon flights. The remaining third of the company was owned by its founder Per Lindstrand, until 2003 when Per sold his remaining share to Cameron Holdings.
Despite Cameron's ownership, Lindstrand Balloons continued to operate as an independent company with separate management and its own distinct designs and products.
Lindstrand Technologies Per Lindstrand independently operates a separate company, which designs and builds gas balloons , innovative buildings, specialized aerospace equipment (including an advanced parachute for the Beagle 2 Mars -lander) and cutting edge inflatable structures including aircraft hangars, plugs for fire-containment for road tunnels and flood prevention systems.
On 15 April 2015 it was reported [ 1 ] that Lindstrand Balloons had closed, with 19 workers losing their jobs, citing the strength of the pound and terrorism threats in the Middle East as factors in the slow sales of balloons. [ 2 ] [ 3 ] | https://en.wikipedia.org/wiki/Lindstrand_Balloons |
In mathematical logic , Lindström's theorem (named after Swedish logician Per Lindström , who published it in 1969) states that first-order logic is the strongest logic [ 1 ] (satisfying certain conditions, e.g. closure under classical negation ) having both the (countable) compactness property and the (downward) Löwenheim–Skolem property . [ 2 ]
Lindström's theorem is perhaps the best known result of what later became known as abstract model theory , [ 3 ] the basic notion of which is an abstract logic ; [ 4 ] the more general notion of an institution was later introduced, which advances from a set-theoretical notion of model to a category -theoretical one. [ 5 ] Lindström had previously obtained a similar result in studying first-order logics extended with Lindström quantifiers . [ 6 ]
Lindström's theorem has been extended to various other systems of logic, in particular modal logics by Johan van Benthem and Sebastian Enqvist.
This mathematical logic -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Lindström's_theorem |
In mathematics, the Lindström–Gessel–Viennot lemma provides a way to count the tuples of non-intersecting lattice paths , or, more generally, paths on a directed graph. It was proved by Gessel–Viennot in 1985, based on previous work of Lindström published in 1973. The lemma is named after Bernt Lindström , Ira Gessel and Gérard Viennot .
Let G be a locally finite directed acyclic graph . This means that each vertex has finite degree , and that G contains no directed cycles. Consider base vertices A = { a 1 , … , a n } {\displaystyle A=\{a_{1},\ldots ,a_{n}\}} and destination vertices B = { b 1 , … , b n } {\displaystyle B=\{b_{1},\ldots ,b_{n}\}} , and also assign a weight ω e {\displaystyle \omega _{e}} to each directed edge e . These edge weights are assumed to belong to some commutative ring . For each directed path P between two vertices, let ω ( P ) {\displaystyle \omega (P)} be the product of the weights of the edges of the path. For any two vertices a and b , write e ( a , b ) for the sum e ( a , b ) = ∑ P : a → b ω ( P ) {\displaystyle e(a,b)=\sum _{P:a\to b}\omega (P)} over all paths from a to b . This is well-defined if between any two points there are only finitely many paths; but even in the general case, this can be well-defined under some circumstances (such as all edge weights being pairwise distinct formal indeterminates, and e ( a , b ) {\displaystyle e(a,b)} being regarded as a formal power series ). If one assigns the weight 1 to each edge, then e ( a , b ) is the number of paths from a to b .
With this setup, write
An n -tuple of non-intersecting paths from A to B means an n -tuple ( P 1 , ..., P n ) of paths in G with the following properties:
Given such an n -tuple ( P 1 , ..., P n ), we denote by σ ( P ) {\displaystyle \sigma (P)} the permutation of σ {\displaystyle \sigma } from the first condition.
The Lindström–Gessel–Viennot lemma then states that the determinant of M is the signed sum over all n -tuples P = ( P 1 , ..., P n ) of non-intersecting paths from A to B :
That is, the determinant of M counts the weights of all n -tuples of non-intersecting paths starting at A and ending at B , each affected with the sign of the corresponding permutation of ( 1 , 2 , … , n ) {\displaystyle (1,2,\ldots ,n)} , given by P i {\displaystyle P_{i}} taking a i {\displaystyle a_{i}} to b σ ( i ) {\displaystyle b_{\sigma (i)}} .
In particular, if the only permutation possible is the identity (i.e., every n -tuple of non-intersecting paths from A to B takes a i to b i for each i ) and we take the weights to be 1, then det( M ) is exactly the number of non-intersecting n -tuples of paths starting at A and ending at B .
To prove the Lindström–Gessel–Viennot lemma, we first introduce some notation.
An n -path from an n -tuple ( a 1 , a 2 , … , a n ) {\displaystyle (a_{1},a_{2},\ldots ,a_{n})} of vertices of G to an n -tuple ( b 1 , b 2 , … , b n ) {\displaystyle (b_{1},b_{2},\ldots ,b_{n})} of vertices of G will mean an n -tuple ( P 1 , P 2 , … , P n ) {\displaystyle (P_{1},P_{2},\ldots ,P_{n})} of paths in G , with each P i {\displaystyle P_{i}} leading from a i {\displaystyle a_{i}} to b i {\displaystyle b_{i}} . This n -path will be called non-intersecting just in case the paths P i and P j have no two vertices in common (including endpoints) whenever i ≠ j {\displaystyle i\neq j} . Otherwise, it will be called entangled .
Given an n -path P = ( P 1 , P 2 , … , P n ) {\displaystyle P=(P_{1},P_{2},\ldots ,P_{n})} , the weight ω ( P ) {\displaystyle \omega (P)} of this n -path is defined as the product ω ( P 1 ) ω ( P 2 ) ⋯ ω ( P n ) {\displaystyle \omega (P_{1})\omega (P_{2})\cdots \omega (P_{n})} .
A twisted n -path from an n -tuple ( a 1 , a 2 , … , a n ) {\displaystyle (a_{1},a_{2},\ldots ,a_{n})} of vertices of G to an n -tuple ( b 1 , b 2 , … , b n ) {\displaystyle (b_{1},b_{2},\ldots ,b_{n})} of vertices of G will mean an n -path from ( a 1 , a 2 , … , a n ) {\displaystyle (a_{1},a_{2},\ldots ,a_{n})} to ( b σ ( 1 ) , b σ ( 2 ) , … , b σ ( n ) ) {\displaystyle \left(b_{\sigma (1)},b_{\sigma (2)},\ldots ,b_{\sigma (n)}\right)} for some permutation σ {\displaystyle \sigma } in the symmetric group S n {\displaystyle S_{n}} . This permutation σ {\displaystyle \sigma } will be called the twist of this twisted n -path, and denoted by σ ( P ) {\displaystyle \sigma (P)} (where P is the n -path). This, of course, generalises the notation σ ( P ) {\displaystyle \sigma (P)} introduced before.
Recalling the definition of M , we can expand det M as a signed sum of permutations; thus we obtain
It remains to show that the sum of s i g n ( σ ( P ) ) ω ( P ) {\displaystyle \mathrm {sign} (\sigma (P))\omega (P)} over all entangled twisted n -paths vanishes. Let E {\displaystyle {\mathcal {E}}} denote the set of entangled twisted n -paths. To establish this, we shall construct an involution f : E ⟶ E {\displaystyle f:{\mathcal {E}}\longrightarrow {\mathcal {E}}} with the properties ω ( f ( P ) ) = ω ( P ) {\displaystyle \omega (f(P))=\omega (P)} and s i g n ( σ ( f ( P ) ) ) = − s i g n ( σ ( P ) ) {\displaystyle \mathrm {sign} (\sigma (f(P)))=-\mathrm {sign} (\sigma (P))} for all P ∈ E {\displaystyle P\in {\mathcal {E}}} . Given such an involution, the rest-term
in the above sum reduces to 0, since its addends cancel each other out (namely, the addend corresponding to each P ∈ E {\displaystyle P\in {\mathcal {E}}} cancels the addend corresponding to f ( P ) {\displaystyle f(P)} ).
Construction of the involution: The idea behind the definition of the involution f {\displaystyle f} is to take choose two intersecting paths within an entangled path, and switch their tails after their point of intersection. There are in general several pairs of intersecting paths, which can also intersect several times; hence, a careful choice needs to be made. Let P = ( P 1 , P 2 , . . . , P n ) {\displaystyle P=\left(P_{1},P_{2},...,P_{n}\right)} be any entangled twisted n -path. Then f ( P ) {\displaystyle f(P)} is defined as follows. We call a vertex crowded if it belongs to at least two of the paths P 1 , P 2 , . . . , P n {\displaystyle P_{1},P_{2},...,P_{n}} . The fact that the graph is acyclic implies that this is equivalent to "appearing at least twice in all the paths". [ 1 ] Since P is entangled, there is at least one crowded vertex. We pick the smallest i ∈ { 1 , 2 , … , n } {\displaystyle i\in \{1,2,\ldots ,n\}} such that P i {\displaystyle P_{i}} contains a crowded vertex. Then, we pick the first crowded vertex v on P i {\displaystyle P_{i}} ("first" in sense of "encountered first when travelling along P i {\displaystyle P_{i}} "), and we pick the largest j such that v belongs to P j {\displaystyle P_{j}} . The crowdedness of v implies j > i . Write the two paths P i {\displaystyle P_{i}} and P j {\displaystyle P_{j}} as
where σ = σ ( P ) {\displaystyle \sigma =\sigma (P)} , and where α {\displaystyle \alpha } and β {\displaystyle \beta } are chosen such that v is the α {\displaystyle \alpha } -th vertex along P i {\displaystyle P_{i}} and the β {\displaystyle \beta } -th vertex along P j {\displaystyle P_{j}} (that is, v = u α = v β {\displaystyle v=u_{\alpha }=v_{\beta }} ). We set α P = α {\displaystyle \alpha _{P}=\alpha } and β P = β {\displaystyle \beta _{P}=\beta } and i P = i {\displaystyle i_{P}=i} and j P = j {\displaystyle j_{P}=j} . Now define the twisted n -path f ( P ) {\displaystyle f(P)} to coincide with P {\displaystyle P} except for components i {\displaystyle i} and j {\displaystyle j} , which are replaced by
It is immediately clear that f ( P ) {\displaystyle f(P)} is an entangled twisted n -path. Going through the steps of the construction, it is easy to see that i f ( P ) = i P {\displaystyle i_{f(P)}=i_{P}} , j f ( P ) = j P {\displaystyle j_{f(P)}=j_{P}} and furthermore that α f ( P ) = α P {\displaystyle \alpha _{f(P)}=\alpha _{P}} and β f ( P ) = β P {\displaystyle \beta _{f(P)}=\beta _{P}} , so that applying f {\displaystyle f} again to f ( P ) {\displaystyle f(P)} involves swapping back the tails of f ( P ) i , f ( P ) j {\displaystyle f(P)_{i},f(P)_{j}} and leaving the other components intact. Hence f ( f ( P ) ) = P {\displaystyle f(f(P))=P} . Thus f {\displaystyle f} is an involution. It remains to demonstrate the desired antisymmetry properties:
From the construction one can see that σ ( f ( P ) ) {\displaystyle \sigma (f(P))} coincides with σ = σ ( P ) {\displaystyle \sigma =\sigma (P)} except that it swaps σ ( i ) {\displaystyle \sigma (i)} and σ ( j ) {\displaystyle \sigma (j)} , thus yielding s i g n ( σ ( f ( P ) ) ) = − s i g n ( σ ( P ) ) {\displaystyle \mathrm {sign} (\sigma (f(P)))=-\mathrm {sign} (\sigma (P))} . To show that ω ( f ( P ) ) = ω ( P ) {\displaystyle \omega (f(P))=\omega (P)} we first compute, appealing to the tail-swap
Hence ω ( f ( P ) ) = ∏ k = 1 n ω ( f ( P ) k ) = ∏ k = 1 , k ≠ i , j n ω ( P k ) ⋅ ω ( P i ′ ) ω ( P j ′ ) = ∏ k = 1 , k ≠ i , j n ω ( P k ) ⋅ ω ( P i ) ω ( P j ) = ∏ k = 1 n ω ( P k ) = ω ( P ) {\displaystyle \omega (f(P))=\prod _{k=1}^{n}\omega (f(P)_{k})=\prod _{k=1,~k\neq i,j}^{n}\omega (P_{k})\cdot \omega (P'_{i})\omega (P'_{j})=\prod _{k=1,~k\neq i,j}^{n}\omega (P_{k})\cdot \omega (P_{i})\omega (P_{j})=\prod _{k=1}^{n}\omega (P_{k})=\omega (P)} .
Thus we have found an involution with the desired properties and completed the proof of the Lindström–Gessel–Viennot lemma.
Remark. Arguments similar to the one above appear in several sources, with variations regarding the choice of which tails to switch. A version with j smallest (unequal to i ) rather than largest appears in the Gessel-Viennot 1989 reference (proof of Theorem 1).
The Lindström–Gessel–Viennot lemma can be used to prove the equivalence of the following two different definitions of Schur polynomials . Given a partition λ = λ 1 + ⋯ + λ r {\displaystyle \lambda =\lambda _{1}+\cdots +\lambda _{r}} of n , the Schur polynomial s λ ( x 1 , … , x n ) {\displaystyle s_{\lambda }(x_{1},\ldots ,x_{n})} can be defined as:
where the sum is over all semistandard Young tableaux T of shape λ , and the weight of a tableau T is defined as the monomial obtained by taking the product of the x i indexed by the entries i of T . For instance, the weight of the tableau is x 1 x 3 x 4 3 x 5 x 6 x 7 {\displaystyle x_{1}x_{3}x_{4}^{3}x_{5}x_{6}x_{7}} .
where h i are the complete homogeneous symmetric polynomials (with h i understood to be 0 if i is negative). For instance, for the partition (3,2,2,1), the corresponding determinant is
To prove the equivalence, given any partition λ as above, one considers the r starting points a i = ( r + 1 − i , 1 ) {\displaystyle a_{i}=(r+1-i,1)} and the r ending points b i = ( λ i + r + 1 − i , n ) {\displaystyle b_{i}=(\lambda _{i}+r+1-i,n)} , as points in the lattice Z 2 {\displaystyle \mathbb {Z} ^{2}} , which acquires the structure of a directed graph by asserting that the only allowed directions are going one to the right or one up; the weight associated to any horizontal edge at height i is x i , and the weight associated to a vertical edge is 1. With this definition, r -tuples of paths from A to B are exactly semistandard Young tableaux of shape λ , and the weight of such an r -tuple is the corresponding summand in the first definition of the Schur polynomials. For instance, with the tableau ,
one gets the corresponding 4 -tuple
On the other hand, the matrix M is exactly the matrix written above for D . This shows the required equivalence. (See also §4.5 in Sagan's book, or the First Proof of Theorem 7.16.1 in Stanley's EC2, or §3.3 in Fulmek's arXiv preprint, or §9.13 in Martin's lecture notes, for slight variations on this argument.)
One can also use the Lindström–Gessel–Viennot lemma to prove the Cauchy–Binet formula , and in particular the multiplicativity of the determinant.
The acyclicity of G is an essential assumption in the Lindström–Gessel–Viennot lemma; it guarantees (in reasonable situations) that the sums e ( a , b ) {\displaystyle e(a,b)} are well-defined, and it advects into the proof (if G is not acyclic, then f might transform a self-intersection of a path into an intersection of two distinct paths, which breaks the argument that f is an involution). Nevertheless, Kelli Talaska's 2012 paper establishes a formula generalizing the lemma to arbitrary digraphs. The sums e ( a , b ) {\displaystyle e(a,b)} are replaced by formal power series, and the sum over nonintersecting path tuples now becomes a sum over collections of nonintersecting and non-self-intersecting paths and cycles, divided by a sum over collections of nonintersecting cycles. The reader is referred to Talaska's paper for details. | https://en.wikipedia.org/wiki/Lindström–Gessel–Viennot_lemma |
Line-in recording is a term often used by manufacturers of sound equipment to refer to the capability of a device to record line level audio feeds . Microphone and instrument inputs, by contrast, are designed for devices which require further amplification to be at line-level.
The common 3.5 mm line-in connector has the left channel on the tip and right channel in the middle. The port is used to connect with other devices. [ 1 ] Line-in is most commonly used for instruments.
This sound technology article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Line-in_recording |
In geometry , a straight line , usually abbreviated line , is an infinitely long object with no width, depth, or curvature , an idealization of such physical objects as a straightedge , a taut string, or a ray of light . Lines are spaces of dimension one, which may be embedded in spaces of dimension two, three, or higher. The word line may also refer, in everyday life, to a line segment , which is a part of a line delimited by two points (its endpoints ).
Euclid's Elements defines a straight line as a "breadthless length" that "lies evenly with respect to the points on itself", and introduced several postulates as basic unprovable properties on which the rest of geometry was established. Euclidean line and Euclidean geometry are terms introduced to avoid confusion with generalizations introduced since the end of the 19th century, such as non-Euclidean , projective , and affine geometry .
In the Greek deductive geometry of Euclid's Elements , a general line (now called a curve ) is defined as a "breadthless length", and a straight line (now called a line segment ) was defined as a line "which lies evenly with the points on itself". [ 1 ] : 291 These definitions appeal to readers' physical experience, relying on terms that are not themselves defined, and the definitions are never explicitly referenced in the remainder of the text. In modern geometry, a line is usually either taken as a primitive notion with properties given by axioms , [ 1 ] : 95 or else defined as a set of points obeying a linear relationship, for instance when real numbers are taken to be primitive and geometry is established analytically in terms of numerical coordinates .
In an axiomatic formulation of Euclidean geometry, such as that of Hilbert (modern mathematicians added to Euclid's original axioms to fill perceived logical gaps), [ 1 ] : 108 a line is stated to have certain properties that relate it to other lines and points . For example, for any two distinct points, there is a unique line containing them, and any two distinct lines intersect at most at one point. [ 1 ] : 300 In two dimensions (i.e., the Euclidean plane ), two lines that do not intersect are called parallel . In higher dimensions, two lines that do not intersect are parallel if they are contained in a plane , or skew if they are not.
On a Euclidean plane , a line can be represented as a boundary between two regions. [ 2 ] : 104 Any collection of finitely many lines partitions the plane into convex polygons (possibly unbounded); this partition is known as an arrangement of lines .
In three-dimensional space , a first degree equation in the variables x , y , and z defines a plane, so two such equations, provided the planes they give rise to are not parallel, define a line which is the intersection of the planes. More generally, in n -dimensional space n −1 first-degree equations in the n coordinate variables define a line under suitable conditions.
In more general Euclidean space , R n (and analogously in every other affine space ), the line L passing through two different points a and b is the subset L = { ( 1 − t ) a + t b ∣ t ∈ R } . {\displaystyle L=\left\{(1-t)\,a+tb\mid t\in \mathbb {R} \right\}.} The direction of the line is from a reference point a ( t = 0) to another point b ( t = 1), or in other words, in the direction of the vector b − a . Different choices of a and b can yield the same line.
Three or more points are said to be collinear if they lie on the same line. If three points are not collinear, there is exactly one plane that contains them.
In affine coordinates , in n -dimensional space the points X = ( x 1 , x 2 , ..., x n ), Y = ( y 1 , y 2 , ..., y n ), and Z = ( z 1 , z 2 , ..., z n ) are collinear if the matrix [ 1 x 1 x 2 ⋯ x n 1 y 1 y 2 ⋯ y n 1 z 1 z 2 ⋯ z n ] {\displaystyle {\begin{bmatrix}1&x_{1}&x_{2}&\cdots &x_{n}\\1&y_{1}&y_{2}&\cdots &y_{n}\\1&z_{1}&z_{2}&\cdots &z_{n}\end{bmatrix}}} has a rank less than 3. In particular, for three points in the plane ( n = 2), the above matrix is square and the points are collinear if and only if its determinant is zero.
Equivalently for three points in a plane, the points are collinear if and only if the slope between one pair of points equals the slope between any other pair of points (in which case the slope between the remaining pair of points will equal the other slopes). By extension, k points in a plane are collinear if and only if any ( k –1) pairs of points have the same pairwise slopes.
In Euclidean geometry , the Euclidean distance d ( a , b ) between two points a and b may be used to express the collinearity between three points by: [ 3 ] [ 4 ]
However, there are other notions of distance (such as the Manhattan distance ) for which this property is not true.
In the geometries where the concept of a line is a primitive notion , as may be the case in some synthetic geometries , other methods of determining collinearity are needed.
In Euclidean geometry, all lines are congruent , meaning that every line can be obtained by moving a specific line. However, lines may play special roles with respect to other geometric objects and can be classified according to that relationship.
For instance, with respect to a conic (a circle , ellipse , parabola , or hyperbola ), lines can be:
In the context of determining parallelism in Euclidean geometry, a transversal is a line that intersects two other lines that may or not be parallel to each other.
For more general algebraic curves , lines could also be:
With respect to triangles we have:
For a convex quadrilateral with at most two parallel sides, the Newton line is the line that connects the midpoints of the two diagonals . [ 7 ]
For a hexagon with vertices lying on a conic we have the Pascal line and, in the special case where the conic is a pair of lines, we have the Pappus line .
Parallel lines are lines in the same plane that never cross. Intersecting lines share a single point in common. Coincidental lines coincide with each other—every point that is on either one of them is also on the other.
Perpendicular lines are lines that intersect at right angles . [ 8 ]
In three-dimensional space , skew lines are lines that are not in the same plane and thus do not intersect each other.
In synthetic geometry , the concept of a line is often considered as a primitive notion , [ 1 ] : 95 meaning it is not being defined by using other concepts, but it is defined by the properties, called axioms , that it must satisfy. [ 9 ]
However, the axiomatic definition of a line does not explain the relevance of the concept and is often too abstract for beginners. So, the definition is often replaced or completed by a mental image or intuitive description that allows understanding what is a line. Such descriptions are sometimes referred to as definitions, but are not true definitions since they cannot used in mathematical proofs . The "definition" of line in Euclid's Elements falls into this category; [ 1 ] : 95 and is never used in proofs of theorems.
Lines in a Cartesian plane or, more generally, in affine coordinates , are characterized by linear equations. More precisely, every line L {\displaystyle L} (including vertical lines) is the set of all points whose coordinates ( x , y ) satisfy a linear equation; that is, L = { ( x , y ) ∣ a x + b y = c } , {\displaystyle L=\{(x,y)\mid ax+by=c\},} where a , b and c are fixed real numbers (called coefficients ) such that a and b are not both zero. Using this form, vertical lines correspond to equations with b = 0.
One can further suppose either c = 1 or c = 0 , by dividing everything by c if it is not zero.
There are many variant ways to write the equation of a line which can all be converted from one to another by algebraic manipulation. The above form is sometimes called the standard form . If the constant term is put on the left, the equation becomes a x + b y − c = 0 , {\displaystyle ax+by-c=0,} and this is sometimes called the general form of the equation. However, this terminology is not universally accepted, and many authors do not distinguish these two forms.
These forms are generally named by the type of information (data) about the line that is needed to write down the form. Some of the important data of a line is its slope, x-intercept , known points on the line and y-intercept.
The equation of the line passing through two different points P 0 ( x 0 , y 0 ) {\displaystyle P_{0}(x_{0},y_{0})} and P 1 ( x 1 , y 1 ) {\displaystyle P_{1}(x_{1},y_{1})} may be written as ( y − y 0 ) ( x 1 − x 0 ) = ( y 1 − y 0 ) ( x − x 0 ) . {\displaystyle (y-y_{0})(x_{1}-x_{0})=(y_{1}-y_{0})(x-x_{0}).} If x 0 ≠ x 1 , this equation may be rewritten as y = ( x − x 0 ) y 1 − y 0 x 1 − x 0 + y 0 {\displaystyle y=(x-x_{0})\,{\frac {y_{1}-y_{0}}{x_{1}-x_{0}}}+y_{0}} or y = x y 1 − y 0 x 1 − x 0 + x 1 y 0 − x 0 y 1 x 1 − x 0 . {\displaystyle y=x\,{\frac {y_{1}-y_{0}}{x_{1}-x_{0}}}+{\frac {x_{1}y_{0}-x_{0}y_{1}}{x_{1}-x_{0}}}\,.} In two dimensions , the equation for non-vertical lines is often given in the slope–intercept form :
y = m x + b {\displaystyle y=mx+b} where:
The slope of the line through points A ( x a , y a ) {\displaystyle A(x_{a},y_{a})} and B ( x b , y b ) {\displaystyle B(x_{b},y_{b})} , when x a ≠ x b {\displaystyle x_{a}\neq x_{b}} , is given by m = ( y b − y a ) / ( x b − x a ) {\displaystyle m=(y_{b}-y_{a})/(x_{b}-x_{a})} and the equation of this line can be written y = m ( x − x a ) + y a {\displaystyle y=m(x-x_{a})+y_{a}} .
As a note, lines in three dimensions may also be described as the simultaneous solutions of two linear equations a 1 x + b 1 y + c 1 z − d 1 = 0 {\displaystyle a_{1}x+b_{1}y+c_{1}z-d_{1}=0} a 2 x + b 2 y + c 2 z − d 2 = 0 {\displaystyle a_{2}x+b_{2}y+c_{2}z-d_{2}=0} such that ( a 1 , b 1 , c 1 ) {\displaystyle (a_{1},b_{1},c_{1})} and ( a 2 , b 2 , c 2 ) {\displaystyle (a_{2},b_{2},c_{2})} are not proportional (the relations a 1 = t a 2 , b 1 = t b 2 , c 1 = t c 2 {\displaystyle a_{1}=ta_{2},b_{1}=tb_{2},c_{1}=tc_{2}} imply t = 0 {\displaystyle t=0} ). This follows since in three dimensions a single linear equation typically describes a plane and a line is what is common to two distinct intersecting planes.
Parametric equations are also used to specify lines, particularly in those in three dimensions or more because in more than two dimensions lines cannot be described by a single linear equation.
In three dimensions lines are frequently described by parametric equations: x = x 0 + a t y = y 0 + b t z = z 0 + c t {\displaystyle {\begin{aligned}x&=x_{0}+at\\y&=y_{0}+bt\\z&=z_{0}+ct\end{aligned}}} where:
Parametric equations for lines in higher dimensions are similar in that they are based on the specification of one point on the line and a direction vector.
The normal form (also called the Hesse normal form , [ 10 ] after the German mathematician Ludwig Otto Hesse ), is based on the normal segment for a given line, which is defined to be the line segment drawn from the origin perpendicular to the line. This segment joins the origin with the closest point on the line to the origin. The normal form of the equation of a straight line on the plane is given by: x cos φ + y sin φ − p = 0 , {\displaystyle x\cos \varphi +y\sin \varphi -p=0,} where φ {\displaystyle \varphi } is the angle of inclination of the normal segment (the oriented angle from the unit vector of the x -axis to this segment), and p is the (positive) length of the normal segment. The normal form can be derived from the standard form a x + b y = c {\displaystyle ax+by=c} by dividing all of the coefficients by a 2 + b 2 . {\displaystyle {\sqrt {a^{2}+b^{2}}}.} and also multiplying through by − 1 {\displaystyle -1} if c < 0. {\displaystyle c<0.}
Unlike the slope-intercept and intercept forms, this form can represent any line but also requires only two finite parameters, φ {\displaystyle \varphi } and p , to be specified. If p > 0 , then φ {\displaystyle \varphi } is uniquely defined modulo 2 π . On the other hand, if the line is through the origin ( c = p = 0 ), one drops the c /| c | term to compute sin φ {\displaystyle \sin \varphi } and cos φ {\displaystyle \cos \varphi } , and it follows that φ {\displaystyle \varphi } is only defined modulo π .
The vector equation of the line through points A and B is given by r = O A + λ A B {\displaystyle \mathbf {r} =\mathbf {OA} +\lambda \,\mathbf {AB} } (where λ is a scalar ).
If a is vector OA and b is vector OB , then the equation of the line can be written: r = a + λ ( b − a ) {\displaystyle \mathbf {r} =\mathbf {a} +\lambda (\mathbf {b} -\mathbf {a} )} .
A ray starting at point A is described by limiting λ. One ray is obtained if λ ≥ 0, and the opposite ray comes from λ ≤ 0.
In a Cartesian plane , polar coordinates ( r , θ ) are related to Cartesian coordinates by the parametric equations: [ 11 ] x = r cos θ , y = r sin θ . {\displaystyle x=r\cos \theta ,\quad y=r\sin \theta .}
In polar coordinates, the equation of a line not passing through the origin —the point with coordinates (0, 0) —can be written r = p cos ( θ − φ ) , {\displaystyle r={\frac {p}{\cos(\theta -\varphi )}},} with r > 0 and φ − π / 2 < θ < φ + π / 2. {\displaystyle \varphi -\pi /2<\theta <\varphi +\pi /2.} Here, p is the (positive) length of the line segment perpendicular to the line and delimited by the origin and the line, and φ {\displaystyle \varphi } is the (oriented) angle from the x -axis to this segment.
It may be useful to express the equation in terms of the angle α = φ + π / 2 {\displaystyle \alpha =\varphi +\pi /2} between the x -axis and the line. In this case, the equation becomes r = p sin ( θ − α ) , {\displaystyle r={\frac {p}{\sin(\theta -\alpha )}},} with r > 0 and 0 < θ < α + π . {\displaystyle 0<\theta <\alpha +\pi .}
These equations can be derived from the normal form of the line equation by setting x = r cos θ , {\displaystyle x=r\cos \theta ,} and y = r sin θ , {\displaystyle y=r\sin \theta ,} and then applying the angle difference identity for sine or cosine.
These equations can also be proven geometrically by applying right triangle definitions of sine and cosine to the right triangle that has a point of the line and the origin as vertices, and the line and its perpendicular through the origin as sides.
The previous forms do not apply for a line passing through the origin, but a simpler formula can be written: the polar coordinates ( r , θ ) {\displaystyle (r,\theta )} of the points of a line passing through the origin and making an angle of α {\displaystyle \alpha } with the x -axis, are the pairs ( r , θ ) {\displaystyle (r,\theta )} such that r ≥ 0 , and θ = α or θ = α + π . {\displaystyle r\geq 0,\qquad {\text{and}}\quad \theta =\alpha \quad {\text{or}}\quad \theta =\alpha +\pi .}
In modern mathematics, given the multitude of geometries, the concept of a line is closely tied to the way the geometry is described. For instance, in analytic geometry , a line in the plane is often defined as the set of points whose coordinates satisfy a given linear equation , but in a more abstract setting, such as incidence geometry , a line may be an independent object, distinct from the set of points which lie on it.
When a geometry is described by a set of axioms , the notion of a line is usually left undefined (a so-called primitive object). The properties of lines are then determined by the axioms which refer to them. One advantage to this approach is the flexibility it gives to users of the geometry. Thus in differential geometry , a line may be interpreted as a geodesic (shortest path between points), while in some projective geometries , a line is a 2-dimensional vector space (all linear combinations of two independent vectors). This flexibility also extends beyond mathematics and, for example, permits physicists to think of the path of a light ray as being a line.
In many models of projective geometry , the representation of a line rarely conforms to the notion of the "straight curve" as it is visualised in Euclidean geometry. In elliptic geometry we see a typical example of this. [ 1 ] : 108 In the spherical representation of elliptic geometry, lines are represented by great circles of a sphere with diametrically opposite points identified. In a different model of elliptic geometry, lines are represented by Euclidean planes passing through the origin. Even though these representations are visually distinct, they satisfy all the properties (such as, two points determining a unique line) that make them suitable representations for lines in this geometry.
The "shortness" and "straightness" of a line, interpreted as the property that the distance along the line between any two of its points is minimized (see triangle inequality ), can be generalized and leads to the concept of geodesics in metric spaces .
Given a line and any point A on it, we may consider A as decomposing this line into two parts.
Each such part is called a ray and the point A is called its initial point . It is also known as half-line (sometimes, a half-axis if it plays a distinct role, e.g., as part of a coordinate axis ). It is a one-dimensional half-space . The point A is considered to be a member of the ray. [ a ] Intuitively, a ray consists of those points on a line passing through A and proceeding indefinitely, starting at A , in one direction only along the line. However, in order to use this concept of a ray in proofs a more precise definition is required.
Given distinct points A and B , they determine a unique ray with initial point A . As two points define a unique line, this ray consists of all the points between A and B (including A and B ) and all the points C on the line through A and B such that B is between A and C . [ 12 ] This is, at times, also expressed as the set of all points C on the line determined by A and B such that A is not between B and C . [ 13 ] A point D , on the line determined by A and B but not in the ray with initial point A determined by B , will determine another ray with initial point A . With respect to the AB ray, the AD ray is called the opposite ray .
Thus, we would say that two different points, A and B , define a line and a decomposition of this line into the disjoint union of an open segment ( A , B ) and two rays, BC and AD (the point D is not drawn in the diagram, but is to the left of A on the line AB ). These are not opposite rays since they have different initial points.
In Euclidean geometry two rays with a common endpoint form an angle . [ 14 ]
The definition of a ray depends upon the notion of betweenness for points on a line. It follows that rays exist only for geometries for which this notion exists, typically Euclidean geometry or affine geometry over an ordered field . On the other hand, rays do not exist in projective geometry nor in a geometry over a non-ordered field, like the complex numbers or any finite field .
A line segment is a part of a line that is bounded by two distinct end points and contains every point on the line between its end points. Depending on how the line segment is defined, either of the two end points may or may not be part of the line segment. Two or more line segments may have some of the same relationships as lines, such as being parallel, intersecting, or skew, but unlike lines they may be none of these, if they are coplanar and either do not intersect or are collinear .
A point on number line corresponds to a real number and vice versa. [ 15 ] Usually, integers are evenly spaced on the line, with positive numbers are on the right, negative numbers on the left. As an extension to the concept, an imaginary line representing imaginary numbers can be drawn perpendicular to the number line at zero. [ 16 ] The two lines forms the complex plane , a geometrical representation of the set of complex numbers . | https://en.wikipedia.org/wiki/Line_(geometry) |
Line 10 of the Beijing Subway ( Chinese : 北京地铁10号线 ; pinyin : běijīng dìtiě shíhào xiàn ) is the second loop line in Beijing 's rapid transit network as well as the second longest and most widely used line. The line is 57.024 km (35.433 mi) in length, [ 3 ] and runs entirely underground through Haidian , Chaoyang and Fengtai Districts, either directly underneath or just beyond the 3rd Ring Road . [ 4 ] The Line 10 loop is situated between 2 and 6 kilometres (1.243 and 3.728 mi) outside the Line 2 loop, and intersects with every subways line through the city centre, with 24 transfer stations and 45 stations in all. [ 4 ] The line's color is capri.
Upon completion in May 2013, it held the record as the world's longest rapid transit loop line, an acolade it held until March 2023. [ 5 ] [ 4 ] [ a ] It remains one of the longest entirely underground subway lines in the world, requiring 104 minutes to complete one full journey in either direction. [ 5 ] [ 4 ]
The Beijing Subway network was originally conceived to have only one loop line . [ 7 ] The booming economy and explosive population growth of Beijing put huge demand on Line 2, surpassing its designed capacity. In 2001 and 2002, the China Academy of Urban Planning and Design proposed two "L-shaped" lines named Line 10 and 11. Together they would form a second loop around Beijing and relieve pressure on line 2. [ 8 ]
On December 27, 2003, in preparation for the 2008 Summer Olympics in Beijing, Phase 1 of Line 10 started construction. [ 9 ] On July 19, 2008, Phase I of Line 10 entered operation ahead of the opening of the Olympic Games. [ 10 ] It was 24.68 km (15.34 mi) in length and had 22 stations. Phase I consisted of the northern and eastern sides of Line 10's rectangular loop from Bagou to Jingsong forming an inverted L-shaped line.
Construction on Phase II began on December 28, 2007. which meant that the original plan for Line 11 was not incorporated into the final network design and was instead absorbed into Line 10. Line 10 formed the second full loop around Beijing. [ 11 ] In 2010, the Ministry of Railways proposed that Fengtai Railway Station was to be renovated and expanded to become a bigger intercity rail terminal for Beijing, with access to the Beijing-Guangzhou high-speed railway . The rationale was to ease intercity traffic pressure on Beijing West railway station . [ 12 ] Due to the need to reorganize the stations on Line 10 to better serve the new rail terminal, work stopped on 2 stations, namely Mengjiacun (孟家村) and Niwa (泥洼). [ 13 ] The planning department proposed that the original Mengjiacun and Niwa subway stations be merged into the new Fengtai railway station, known as the "three stations into one" program. Local residents, after realizing their travel to a subway station would be greatly lengthened, quickly opposed the plan. [ 14 ] Planners reconsidered and moved Niwa station 300 m (984 ft) north to its current position and Mengjiacun station 100 m (328 ft) north to be renamed as Fengtai Railway Station. [ 15 ] The original station shells were demolished and new stations built in their respective new locations. Niwa station started reconstruction in February 2012, [ 16 ] while Fengtai railway station started on April 11, 2012. [ 17 ] This made the late 2012 opening date for that section of Line 10 highly unlikely and was postponed to the next year. [ 18 ] On December 30, 2012, the first section of Phase II, consisting of the southern and western sides of the loop opened. With the opening of Phase I and Phase II, Line 10 became a "C" shape. [ 19 ]
The near completion of Line 10 led to rapid growth of Line 10's ridership. At the same time, some traffic from Line 1 was diverted to the parallel and newly opened Line 6 , allowing Line 10 to overtake Line 1 as Beijing's busiest subway line. [ 20 ] The Beijing Subway started operating express trains that ran non-stop between Songjiazhuang to Jinsong to alleviate traffic in the southeastern section of Line 10. These express trains stopped operating after the completion of the loop. [ 21 ]
The loop was fully enclosed on May 5, 2013 with the opening of Fengtai and Niwa stations, as well as the infill Jiaomen East . [ 4 ] [ 10 ] [ 22 ] [ 23 ] Initially, Line 10 services consisted of a "full-loop" service that make the journey through all 45 stations in 104 minutes, and "partial-loop" trains that run from Chedaogou in the north-west to Songjiazhuang in the south-east before turning back. [ 4 ] With the delivery of more rolling stock, "partial-loop" trains were removed and all trains are now serving the full loop at a headway of 2 minutes and 15 seconds. By 2014, the completed loop carried on average 1.69 million passengers per day. [ 24 ] By 2019, large sections of Line 10 operated above 100% capacity, particularly the eastern and northern sections. Beijing Subway has responded by increasing the frequency of trains to every two minutes and removing some seats on trains to increase capacity. [ 25 ] [ 26 ]
From Bagou near Wanliu Park in Haidian District , Line 10 runs straight east, between the northern 3rd and 4th Ring Roads. At Xitucheng, the line meets the northern section of the Yuan dynasty earthen city wall , called tucheng . Jiandemen and Anzhenmen stations are named after former gates in the wall. At Beitucheng, Line 8 (Phase 1) extends off Line 10 and provides access to the Beijing Olympic Green . Farther east, Line 10 turns south after the Sanyuanqiao and follows the eastern 3rd Ring Road straight south to Jingsong in Chaoyang District . The Bagou-Jingsong section constituted Phase I of Line 10, which first opened in July 2008, and connects the university district in Haidian with the embassy district and Beijing CBD . A trip from Bagou to Jingsong takes about 40 minutes. The full loop takes about 104 minutes. [ 5 ]
Starting fare of RMB(¥) 3 that increases according to the distance fare scheme introduced in December 2014.
Regular subway users can use a Yikatong card, which offers even cheaper journeys, as well as mobile phone apps, which deploy payment via a QR code .
The first train on the inner (clockwise) loop departs from Xiju towards Shoujingmao at 5:20 am. The first train on the outer (counter-clockwise) loop departs from Shoujingmao towards Xiju at 6:12 am. The last inner loop train leaves Xiju for Bagou at 11:29 pm. The last outer loop train leaves Shoujingmao for Chedaogou at 11:06 pm. For the official timetable, see. [ 27 ]
There are subway public security bureaus (police stations) located in the Bagou , Beitucheng and Liangmaqiao stations. Emergencies can be reported by calling 110 or 64011327. [ 28 ]
Some trains terminate at stations marked '*'.
Line 10 utilizes a fleet of 6-car DKZ15 trains manufactured by CRRC Changchun Railway Vehicles . Initially when Phase I opened the line was operated with a fleet of only 40 trainsets (240 cars). [ 30 ] Some sets operated on the Olympic section of Line 8 before Line 8 was extended and acquired its own dedicated rolling stock. [ 31 ] When Line 10 Phase II opened the fleet was expanded to 84 trains. [ 32 ] However the two existing depots serving Line 10 had insufficient capacity for the entire fleet. Therefore, only 76 trainsets could operate on the line with 8 being temporary stored in other Beijing Subway depots. [ 33 ] With the opening of the new depot in Songjiazhuang and the need to reduce the headway on line to decrease crowding, an additional 32 trainsets were ordered. The fleet grew to 116 trainsets allowing Line 10 to operate at a headway of every 2 minutes throughout the line during rush hour. [ 34 ] Some trains had some seats removed to increase capacity.
Siemens Transportation Systems and China Railway Signaling & Communication Corp. have equipped the entire line with Siemens's Trainguard MT Communication Based Train Control (CBTC) system. [ 35 ] As a fallback, ETCS Level 1 is also available. [ 36 ] | https://en.wikipedia.org/wiki/Line_10_(Beijing_Subway) |
Line 10 is an east-west line on the Nanjing Metro system, running from Andemen to Yushanlu . Line 10 opened on July 1, 2014 with 14 stations spanning a total of 35.9 km (22.3 mi). [ 2 ]
The eastern section of Line 10 is the former Line 1 section between Andemen and Olympic Sport Center stations. [ 3 ] Together with its west extension, it broke away from Line 1 to form a standalone line on July 1, 2014. [ 2 ] [ 4 ] | https://en.wikipedia.org/wiki/Line_10_(Nanjing_Metro) |
Line 1 of the Nanjing Metro ( Chinese : 南京地铁1号线 ; pinyin : Nánjīng Dìtiě Yī Hào Xiàn ) is a north-south line and the first operating metro line in the Nanjing Metro system, inaugurated on September 3, 2005. After the opening of the 25.08 km (15.58 mi)-long south extension line on May 28, 2010, [ 2 ] the total length of Line 1 is now 37.9 km (23.5 mi), [ 3 ] running from CPU to Baguazhoudaqiaonan .
This line mainly runs in a north-south direction. It starts at Baguazhoudaqiaonan station in the north, and continues southwards towards CPU station which is located at the southeastern side of Nanjing.
Of the total 21.72 km (13.50 mi) of the main line track, 14.33 km (8.90 mi) of the Line 1 track runs underground, while 7.39 km (4.59 mi) of the track run on or above the ground. Of the total 16 stations, 11 of them are underground stations while the other 5 are either ground or elevated stations.
The southern extension line of Metro Line 1 has a total length of 25.08 km (15.58 mi) with 15 stations (excluding Andemen station). Of the total 15 stations, 8 of them are underground and the other 7 are all elevated stations. [ 4 ]
The northern extension covers a length of 7.2 km (4.5 mi) (6.7 km (4.2 mi) underground and 0.5 km (0.31 mi) elevated) with 5 underground stations (excluding Maigaoqiao station). [ 5 ] It was completed in 2022. [ 6 ] | https://en.wikipedia.org/wiki/Line_1_(Nanjing_Metro) |
Line 2 of the Nanjing Metro ( Chinese : 南京地铁2号线 ; pinyin : Nánjīng Dìtiě ÈrHào Xiàn ) is a subway line that runs mainly in an east-west direction on the Nanjing Metro network, running from Jingtianlu to Youfangqiao ; it entered operation on May 28, 2010. [ 1 ] [ 3 ] It covers a length of 37.95 km (23.58 mi) with 26 stations. [ 1 ] [ 4 ] Of the 26 stations, 17 stations are underground, 2 stations are on the surface, and the other 7 station are either above ground or elevated stations.
The section between Maqun and Jingtianlu was originally planned as an east extension of Line 2, but it entered operation, together with the main line, on the same day. [ 4 ] | https://en.wikipedia.org/wiki/Line_2_(Nanjing_Metro) |
Line 2 is an east–west line in the Shanghai Metro network. With a length of nearly 64 km (40 mi), it is the second longest line in the metro system after line 11 . Line 2 runs from National Exhibition and Convention Center in the west to Pudong Airport Terminal 1&2 in the east, passing Hongqiao Airport , the Huangpu river , and the Lujiazui Financial District in Pudong . With a daily ridership of over 1.9 million, it is the busiest line on the Shanghai Metro. [ 4 ] The eastern portion of the line, Guanglan Road - Pudong International Airport section, was operated as an independent service route until April 19, 2019, when through service began. The line is colored light green on system maps.
The first section of line 2 was opened on October 28, 1999, from Zhongshan Park to Longyang Road . This section, which included 12 stations, totaled 16.3 kilometres (10.1 mi). A year later coinciding with the tenth anniversary of the development and opening up of Pudong, marking the official opening of the line, Zhangjiang High Technology Park was added to the eastern part of the line, adding 2.8 kilometres (1.7 mi). [ 5 ] Four new stations, located west of the Zhongshan Park station, opened in December 2006, extending the line to Songhong Road . This section added 6.15 kilometres (3.82 mi) to the line. Four years later, in preparation for the 2010 Shanghai World Expo , the line was significantly expanded. In February, the Zhangjiang High Technology Park station was rebuilt. In addition, another eastern segment took line 2 to Guanglan Road . [ 6 ] [ 7 ] [ 8 ] A month later, the line was extended westward to East Xujing , adding 8.0 kilometres (5.0 mi) to the line including a stop at Hongqiao Airport Terminal 2 . [ 9 ] On April 8, an eastward extension added 8 stations to the line, totaling 26.6 kilometres (16.5 mi) and taking line 2 to Pudong International Airport . [ 10 ] [ 11 ] On July 1, Hongqiao Railway Station opens to the public with the opening of the railway station of the same name.
In October 2006, it was decided to rename three stations on line 2 by the end of the year, [ 12 ] [ 13 ] adopting a new naming scheme: metro stations, unlike bus stops, are no longer supposed to be named after neighbouring vertical streets, but famous streets and sights in the vicinity, making it easier for visitors to find these places. The renamed stations are Century Avenue (formerly Dongfang Road), East Nanjing Road (formerly Middle Henan Road) and West Nanjing Road (formerly Shimen No. 1 Road).
Up to April 19, 2019, when an eight-car train started serving the whole line 2 in a regular schedule, [ 16 ] the east section of line 2 was served by a four-car fleet. Line 2 had a piecewise service pattern during morning peak hours whereby the suburban segment between Guanglan Road station and Pudong International Airport station (Now Pudong Airport Terminal 1&2 station) is partially served by a four-car fleet in addition to the regular eight-car fleet serving the whole line. Already since 28 December 2018, during off-peak times, an eight-car fleet from National Exhibition and Convention Center station or Songhong Road station may terminate at Pudong Airport Terminal 1&2 station , but most trains still terminate at Guanglan Road station or Tangzhen (only during peak hours).
The line begins at East Xujing at the intersection of Xumin East Road and Zhuguang Road. [ 28 ] [ 29 ] The line heads northeastward under Xumin East Road for about 0.8 kilometres (0.50 mi) before veering off the road and heading east, passing under Huaxiang Road. The line then enters the interchange station serving the Hongqiao Railway Station . [ 30 ] This station is an interchange with Line 10 and Line 17 . Shortly thereafter, the line enters the Hongqiao Airport Terminal 2 . [ 31 ] Line 2 then turns northward until it meets Tianshan Road and turns east again, roughly running parallel under Tianshan Road. Along this road, the line enters the Songhong Road , [ 32 ] Beixinjing , [ 33 ] and Weining Road stations. [ 34 ] At the Loushanguan Road , line 2 veers away from Tianshan Road, heading northeastward. [ 35 ] The line then enters the Zhongshan Park along Changning Road, an interchange with lines 3 and 4 . [ 36 ] [ 37 ] The subway line then runs parallel under Changning Road for a short distance before heading east under Yuyuan Road. Along Yuyuan Road, there is a station at Jiangsu Road , an interchange to line 11 . [ 38 ] [ 39 ] East of this station, the line swerves away from Yuyuan Road and runs under Yongyuan Road, which merges into West Nanjing Road, where line 2 enters the Jing'an Temple , an interchange to line 7 . [ 40 ] Just before entering the West Nanjing Road , the metro line veers away from West Nanjing Road to Wujiang Road. [ 41 ] Line 2 comes back under the road shortly thereafter.
East of the West Nanjing Road , line 2 heads eastward along West Nanjing Road, passing under the South-North Elevated Road. It then turns northeast into the People's Square , [ 42 ] and interchange with lines 1 and 8 . East of the People's Square station, the subway line moves under East Nanjing Road to the East Nanjing Road . [ 43 ] Leaving the Huangpu District of Shanghai, the line heads under the Huangpu River and enters the Pudong New Area of Shanghai. The metro line passes the Oriental Pearl TV Tower and the World Finance Center near its station at Lujiazui . The line then runs under Century Avenue and heads southeastward to the Dongchang Road . Line 2 then heads to the Century Avenue , the largest interchange station that serves lines 4 , 6 and 9 as well. Line 2 continues southeastward along Century Avenue to the Shanghai Science and Technology Museum . From here, the line turns southward through Century Park to the Century Park . From here, the line turns southeast and then east as it enters the Longyang Road , an interchange with lines 7 , 16 and 18 as well as the Shanghai Maglev Train .
From the Longyang Road , the metro line heads eastward. Line 2 heads eastward, running under Zuchongzhi Road to the Zhangjiang High Technology Park station. The line heads northeastward under Zuchongzhi Road to the Jinke Road and Guanglan Road stations. Line 2 then veers away from Zuchongzhi Road and heads under the Waihuan Expressway to the Tangzhen and Middle Chuangxin Road stations, turning south. Heading southward, it heads through the East Huaxia Road and turns back east, running under Chuanhuan Road. Line 2 then passes through the Chuansha . Heading away from Chuanhuan Road, the metro line then enters the Lingkong Road and Yuandong Avenue stations along Huazhou Road before turning southeast. Line 2 then begins running parallel to the Shanghai Maglev Train as it runs under the Yingbin Expressway and enters the Haitiansan Road . From here, line 2 continues south to its terminus at the Pudong International Airport , which serves Shanghai Pudong International Airport .
As part of the phase IV extension of Shanghai Pudong International Airport terminal T3 will be built south of the current terminals T1, T2 and satellite terminal. The terminal T3 will be served by extending line 2 south by one station.
Construction started on a one station west extension from East Xujing to Panxiang Road [ 44 ] in June 2021. [ 45 ] The Panxiang Road station is south of the Panlong Road on line 17 , these are separate stations and will not have an interchange. The extension of Metro line 2 to the west will improve the connection of the Hongqiao business hub to the city center and Pudong. It will open in 2025.
Siemens Transportation Systems equipped this line with an overhead contact line (cantilever material: galvanized steel) and 7 DC traction power supply substations. [ 47 ]
As the first part of Line 2 was opened 20 years ago and the line has been experiencing congestion after rapid ridership growth. In 2014, Shanghai Metro investigated upgrading the existing signal system of line 2 (US&S United Signal AF900, fixed block TBTC) to increase the frequency of trains and reduce congestion. In October 2020, a new CASCO Tranavi (moving block CBTC, DTO) signaling system was overlaid on the existing signaling system on Line 2. Therefore, line 2 will be the first in the world to have two signal systems, the new primary one for day-to-day operations and the existing older one serving as a backup system during signal faults. [ 48 ] Intervals of trains on Line 2 could be reduced to 90 seconds thanks to the new signal system with the backup system capable of maintaining two minute headways. For the new system, a total of 100 trains on Line 2 will have their onboard signal systems upgraded. As of 2020 upgrading work was finished on 31 trains (the new 02A05 trains). [ 49 ] [ 50 ] The new signalling system was put into operation on January 27, 2024, which made it the longest signalling system renovation project in China. [ 51 ]
When line 2 was opened to traffic, the AC02 trains were not delivered. Therefore, some of the DC01 and AC01 trains were borrowed from line 1 .
All are 8-car Class A [ i ] rolling stock.
2018-2020 (expansion cars) | https://en.wikipedia.org/wiki/Line_2_(Shanghai_Metro) |
Line 8 of the Beijing Subway ( Chinese : 北京地铁8号线 ; pinyin : běijīng dìtiě bāhàoxiàn ) is a rapid transit line in Beijing . It sits on the central axis of Beijing. Line 8's color is green. It is 49.5 km (30.8 mi) in length with 35 stations (34 in operation). The most recent extension is the central section from National Art Museum to Zhushikou , opened on 31 December 2021.
In the north, Line 8 begins at Zhuxinzhuang on the Changping Line and heads east to Huilongguan Residential Area and then south through the Line 13 arc at Huoying , to the Olympic Park station on Line 15 . The line enters the Line 10 loop at Beitucheng and then the Line 2 loop at Gulou Dajie before reaching Nanluogu Xiang on Line 6 , and then to National Art Museum .
Apart from the Zhuxinzhuang station and an 1.7 km (1.056 mi) section of elevated track leading therefrom, the entire line runs underground.
Three stations ( Jinyu Hutong , Wangfujing and Qianmen ) on the central section opened on 31 December 2021.
The south section of Line 8, from Zhushikou to Yinghai, is 16.4 km (10.190 mi) long and has a 2.06 km (1.280 mi) elevated section. All stations are underground, except for Demao and Yinghai stations, which are elevated.
Line 8 has been planned and built in several phases.
Line 8 was planned as the subway line that follows Beijing's central north–south axis . [ 4 ] [ 5 ] The first section of Line 8 to be built was the four-station segment from Beitucheng to Forest Park South Gate , 4.35 km (2.703 mi) in length, that serves the Olympic Green . This section was included in Beijing's bid for the 2008 Summer Olympics , which was awarded to the city in 2001. Originally, Beijing's subway planners also considered building a subway extension line off of Line 13 or Line 5 to serve the Olympic Green but ultimately chose to build Line 8 as a branch off of Line 10. [ 6 ] Construction began in 2004. [ 7 ] With other Olympic venues also under construction, Phase I of Line 8 was built using the cut-and-cover method to reduce the difficulty of construction. [ 8 ] The Olympic Branch Line, as Line 8 Phase I was known, entered into operation together with Line 10 on July 19, 2008. It serves the Olympic Green , located due north of the city centre, during the 2008 Summer Olympics . The Phase I only included 4 stations. Access was originally restricted to riders with an Olympic Register Card or a ticket to an event at the Olympic Games or Paralympic Games on the day of the event. In early October 2008, the line was fully opened to the public.
On December 8, 2007, while Phase I of Line 8 was still under construction, work began on Phase II to extend Line 8 in both directions along the city's north–south central axis. [ 9 ] In Phase II was estimated to cost ¥ 10.1 billion and was scheduled to be completed by 2012. [ 10 ] [ 11 ]
The northern extension to Huilongguan Dongdajie , 10.7 km (6.649 mi) in length with 6 stations, extended Line 8 from the South Gate of Forest Park to Huilongguan Dongdajie in Changping District beyond the Line 13 arc. Land clearing for Phase II began in December 2007. [ 10 ] [ 12 ] Tunnel boring machines began work on October 16, 2009. [ 13 ] In the fall of 2011, the entire Line 8 shut down and the entire line including the Phase II northern extension reopened on December 31, 2011. [ 13 ]
The Lincuiqiao station , just west of the Olympic Forest Park, was originally planned as an emergency stop, but was added at the behest of nearby residents and their municipal people's congress representative, Tian Yuan, who argued that the 5.1 km (3.169 mi) gap between South Gate of Forest Park and Yongtaizhuang made subway access inconvenient for residents along Lincui Road. [ 14 ] Lincuiqiao was officially added as a station to Phase II plans in December 2008.
The southern extension to Guloudajie, 3.28 km (2.04 mi) in length, opened on December 30, 2012. Travel time from Huilongguan to the Second Ring Road was reduced by a half-hour. [ 15 ] Daily ridership reached 203,000 in March 2013. [ 16 ]
On December 28, 2013, Line 8 reached 26.614 km (16.537 mi) in length with the opening of the Changping-Line 8 Connector and the southern extension to Nanluoguxiang. [ 17 ] [ 18 ]
The Changping-Line 8 Connector, also known as the Changba Connector Line (昌八联络线) or Changba Connector, is a 6.3-kilometre-long (3.9 mi) extension of Line 8 from Huilongguan Dongdajie to Zhuxinzhuang on the Changping Line. [ 17 ] The Changba Connector contains three stations: Pingxifu, Yuzhilu and Zhuxinzhuang, and forms the northernmost section of Line 8. [ 17 ] The Changba Connector was designed to alleviate passenger traffic on Line 13 by allowing Changping Line riders heading to destinations in eastern Beijing to switch to Line 8 at Zhuxingzhuang instead of transferring to Line 13 at Xi'erqi. The connector was built from April 2011 to September 2013 and entered operation at the end of 2013. [ 17 ] [ 18 ]
South of Guloudajie, Line 8 was extended a further 2.16 km (1.34 mi) through Shichahai station to Nanluogu Xiang station on December 28, 2013. [ 19 ]
The one-station extension from Nanluoguxiang to National Art Museum was opened on 30 December 2018. [ 20 ] [ 21 ]
In Phase III & IV, Line 8 will be extended further south from the National Art Museum through Qianmen and Yongdingmen to beyond the southern 5th Ring Road . The line will veer to the east of the central axis to avoid passing under the Forbidden City and Tian'anmen Square, before returning to the central axis alignment at Qianmen .
Planning began in 2009. [ 22 ] Plans of Line 8 in Phase III & IV showed 16 stations for 21.0 km (13.0 mi). [ 23 ] Phase III includes 14 stations and Phase IV includes 2 stations, Demao and Yinghai. Phase III was scheduled to be built by 2015 [ 24 ] but the commencement of construction was not set to begin until October 2013. [ 25 ] The section from National Art Museum to Zhushikou started construction in November 2016. [ 26 ]
On 30 December 2018, the southernmost section of Phase III, and Phase IV, from Zhushikou to Yinghai was opened (12 stations were opened, Dahongmen was not opened). [ 20 ] [ 21 ]
The section between National Art Museum and Zhushikou , which includes 3 stations ( Jinyu Hutong , Wangfujing and Qianmen ) opened on 31 December 2021. [ 26 ]
Through service between Line 8 and Changping line via Zhuxinzhuang station is under planning. [ 27 ] [ 28 ]
A further southern extension from Yinghai to the China-Japan Innovation Cooperation Demonstration Zone is under planning. [ 29 ] The extension is entirely in Daxing District of Beijing.
During the Olympics Line 8 borrowed DKZ15 trains from Line 10; after the opening of the first sections of Phase II new CSR Sifang Locomotive SFM12 trains dedicated to Line 8 were rolled out. [ 32 ] | https://en.wikipedia.org/wiki/Line_8_(Beijing_Subway) |
In geometry and topology , the line at infinity is a projective line that is added to the affine plane in order to give closure to, and remove the exceptional cases from, the incidence properties of the resulting projective plane . The line at infinity is also called the ideal line . [ 1 ]
In projective geometry, any pair of lines always intersects at some point, but parallel lines do not intersect in the real plane. The line at infinity is added to the real plane. This completes the plane, because now parallel lines intersect at a point which lies on the line at infinity. Also, if any pair of lines do not intersect at a point on the line, then the pair of lines are parallel.
Every line intersects the line at infinity at some point. The point at which the parallel lines intersect depends only on the slope of the lines, not at all on their y-intercept .
In the affine plane, a line extends in two opposite directions. In the projective plane, the two opposite directions of a line meet each other at a point on the line at infinity. Therefore, lines in the projective plane are closed curves , i.e., they are cyclical rather than linear. This is true of the line at infinity itself; it meets itself at its two endpoints (which are therefore not actually endpoints at all) and so it is actually cyclical.
If we consider the real affine plane, then the line at infinity can be visualized as a circle which surrounds it. However, diametrically opposite points of the circle are equivalent—they are the same point. The combination of the real affine plane and the line at infinity makes the real projective plane , R P 2 {\displaystyle \mathbb {R} P^{2}} .
A hyperbola can be seen as a closed curve which intersects the line at infinity in two different points. These two points are specified by the slopes of the two asymptotes of the hyperbola. Likewise, a parabola can be seen as a closed curve which intersects the line at infinity in a single point. This point is specified by the slope of the axis of the parabola. If the parabola is cut by its vertex into a symmetrical pair of "horns", then these two horns become more parallel to each other further away from the vertex, and are actually parallel to the axis and to each other at infinity, so that they intersect at the line at infinity.
The analogue for the complex projective plane is a 'line' at infinity that is (naturally) a complex projective line . Topologically this is quite different, in that it is a Riemann sphere , which is therefore a 2- sphere , being added to a complex affine space of two dimensions over C {\displaystyle \mathbb {C} } (so four real dimensions), resulting in a four-dimensional compact manifold . The result is orientable , while the real projective plane is not.
The complex line at infinity was much used in nineteenth century geometry. In fact one of the most applied tricks was to regard a circle as a conic constrained to pass through two points at infinity, the solutions of
This equation is the form taken by that of any circle when we drop terms of lower order in X and Y . More formally, we should use homogeneous coordinates
and note that the line at infinity is specified by setting
Making equations homogeneous by introducing powers of Z , and then setting Z = 0, does precisely eliminate terms of lower order.
Solving the equation, therefore, we find that all circles 'pass through' the circular points at infinity
These of course are complex points, for any representing set of homogeneous coordinates. Since the projective plane has a large enough symmetry group , they are in no way special, though. The conclusion is that the three-parameter family of circles can be treated as a special case of the linear system of conics passing through two given distinct points P and Q . | https://en.wikipedia.org/wiki/Line_at_infinity |
In telecommunications , a line code is a pattern of voltage, current, or photons used to represent digital data transmitted down a communication channel or written to a storage medium . This repertoire of signals is usually called a constrained code in data storage systems. [ 1 ] Some signals are more prone to error than others as the physics of the communication channel or storage medium constrains the repertoire of signals that can be used reliably. [ 2 ]
Common line encodings are unipolar , polar , bipolar , and Manchester code .
After line coding, the signal is put through a physical communication channel, either a transmission medium or data storage medium . [ 3 ] [ 4 ] The most common physical channels are:
Some of the more common binary line codes include:
Each line code has advantages and disadvantages. Line codes are chosen to meet one or more of the following criteria:
Most long-distance communication channels cannot reliably transport a DC component . The DC component is also called the disparity , the bias , or the DC coefficient . The disparity of a bit pattern is the difference in the number of one bits vs the number of zero bits. The running disparity is the running total of the disparity of all previously transmitted bits. [ 5 ] The simplest possible line code, unipolar , gives too many errors on such systems, because it has an unbounded DC component.
Most line codes eliminate the DC component – such codes are called DC-balanced , zero-DC, or DC-free. There are three ways of eliminating the DC component:
Bipolar line codes have two polarities, are generally implemented as RZ, and have a radix of three since there are three distinct output levels (negative, positive and zero). One of the principle advantages of this type of code is that it can eliminate any DC component. This is important if the signal must pass through a transformer or a long transmission line.
Unfortunately, several long-distance communication channels have polarity ambiguity. Polarity-insensitive line codes compensate in these channels. [ 6 ] [ 7 ] [ 8 ] [ 9 ] There are three ways of providing unambiguous reception of 0 and 1 bits over such channels:
For reliable clock recovery at the receiver, a run-length limitation may be imposed on the generated channel sequence, i.e., the maximum number of consecutive ones or zeros is bounded to a reasonable number. A clock period is recovered by observing transitions in the received sequence, so that a maximum run length guarantees sufficient transitions to assure clock recovery quality.
RLL codes are defined by four main parameters: m , n , d , k . The first two, m / n , refer to the rate of the code, while the remaining two specify the minimal d and maximal k number of zeroes between consecutive ones. This is used in both telecommunications and storage systems that move a medium past a fixed recording head . [ 10 ]
Specifically, RLL bounds the length of stretches (runs) of repeated bits during which the signal does not change. If the runs are too long, clock recovery is difficult; if they are too short, the high frequencies might be attenuated by the communications channel. By modulating the data , RLL reduces the timing uncertainty in decoding the stored data, which would lead to the possible erroneous insertion or removal of bits when reading the data back. This mechanism ensures that the boundaries between bits can always be accurately found (preventing bit slip ), while efficiently using the media to reliably store the maximal amount of data in a given space.
Early disk drives used very simple encoding schemes, such as RLL (0,1) FM code, followed by RLL (1,3) MFM code which were widely used in hard disk drives until the mid-1980s and are still used in digital optical discs such as CD , DVD , MD , Hi-MD and Blu-ray using EFM and EFMPLus codes. [ 11 ] Higher density RLL (2,7) and RLL (1,7) codes became the de facto standards for hard disks by the early 1990s. [ citation needed ]
Line coding should make it possible for the receiver to synchronize itself to the phase of the received signal. If the clock recovery is not ideal, then the signal to be decoded will not be sampled at the optimal times. This will increase the probability of error in the received data.
Biphase line codes require at least one transition per bit time. This makes it easier to synchronize the transceivers and detect errors, however, the baud rate is greater than that of NRZ codes.
A line code will typically reflect technical requirements of the transmission medium, such as optical fiber or shielded twisted pair . These requirements are unique for each medium, because each one has different behavior related to interference, distortion, capacitance and attenuation. [ 12 ] | https://en.wikipedia.org/wiki/Line_code |
In geometry , line coordinates are used to specify the position of a line just as point coordinates (or simply coordinates ) are used to specify the position of a point.
There are several possible ways to specify the position of a line in the plane. A simple way is by the pair ( m , b ) where the equation of the line is y = mx + b . Here m is the slope and b is the y -intercept . This system specifies coordinates for all lines that are not vertical. However, it is more common and simpler algebraically to use coordinates ( l , m ) where the equation of the line is lx + my + 1 = 0. This system specifies coordinates for all lines except those that pass through the origin. The geometrical interpretations of l and m are the negative reciprocals of the x and y -intercept respectively.
The exclusion of lines passing through the origin can be resolved by using a system of three coordinates ( l , m , n ) to specify the line with the equation lx + my + n = 0. Here l and m may not both be 0. In this equation, only the ratios between l , m and n are significant, in other words if the coordinates are multiplied by a non-zero scalar then line represented remains the same. So ( l , m , n ) is a system of homogeneous coordinates for the line.
If points in the real projective plane are represented by homogeneous coordinates ( x , y , z ) , the equation of the line is lx + my + nz = 0, provided ( l , m , n ) ≠ (0,0,0) . In particular, line coordinate (0, 0, 1) represents the line z = 0, which is the line at infinity in the projective plane . Line coordinates (0, 1, 0) and (1, 0, 0) represent the x and y -axes respectively.
Just as f ( x , y ) = 0 can represent a curve as a subset of the points in the plane, the equation φ( l , m ) = 0 represents a subset of the lines on the plane. The set of lines on the plane may, in an abstract sense, be thought of as the set of points in a projective plane, the dual of the original plane. The equation φ( l , m ) = 0 then represents a curve in the dual plane.
For a curve f ( x , y ) = 0 in the plane, the tangents to the curve form a curve in the dual space called the dual curve . If φ( l , m ) = 0 is the equation of the dual curve, then it is called the tangential equation , for the original curve. A given equation φ( l , m ) = 0 represents a curve in the original plane determined as the envelope of the lines that satisfy this equation. Similarly, if φ( l , m , n ) is a homogeneous function then φ( l , m , n ) = 0 represents a curve in the dual space given in homogeneous coordinates, and may be called the homogeneous tangential equation of the enveloped curve.
Tangential equations are useful in the study of curves defined as envelopes, just as Cartesian equations are useful in the study of curves defined as loci.
A linear equation in line coordinates has the form al + bm + c = 0, where a , b and c are constants. Suppose ( l , m ) is a line that satisfies this equation. If c is not 0 then lx + my + 1 = 0, where x = a / c and y = b / c , so every line satisfying the original equation passes through the point ( x , y ). Conversely, any line through ( x , y ) satisfies the original equation, so al + bm + c = 0 is the equation of set of lines through ( x , y ). For a given point ( x , y ), the equation of the set of lines though it is lx + my + 1 = 0, so this may be defined as the tangential equation of the point. Similarly, for a point ( x , y , z ) given in homogeneous coordinates, the equation of the point in homogeneous tangential coordinates is lx + my + nz = 0.
The intersection of the lines ( l 1 , m 1 ) and ( l 2 , m 2 ) is the solution to the linear equations
By Cramer's rule , the solution is
The lines ( l 1 , m 1 ), ( l 2 , m 2 ), and ( l 3 , m 3 ) are concurrent when the determinant
For homogeneous coordinates, the intersection of the lines ( l 1 , m 1 , n 1 ) and ( l 2 , m 2 , n 2 ) is the cross product :
The lines ( l 1 , m 1 , n 1 ), ( l 2 , m 2 , n 2 ) and ( l 3 , m 3 , n 3 ) are concurrent when the determinant
Dually, the coordinates of the line containing ( x 1 , y 1 , z 1 ) and ( x 2 , y 2 , z 2 ) can be obtained via the cross product:
For two given points in the real projective plane , ( x 1 , y 1 , z 1 ) and ( x 2 , y 2 , z 2 ), the three determinants
determine the projective line containing them.
Similarly, for two points in RP 3 , ( x 1 , y 1 , z 1 , w 1 ) and ( x 2 , y 2 , z 2 , w 2 ), the line containing them is determined by the six determinants
This is the basis for a system of homogeneous line coordinates in three-dimensional space called Plücker coordinates . Six numbers in a set of coordinates only represent a line when they satisfy an additional equation. This system maps the space of lines in three-dimensional space to projective space RP 5 , but with the additional requirement the space of lines corresponds to the Klein quadric , which is a manifold of dimension four.
More generally, the lines in n -dimensional projective space are determined by a system of n ( n − 1)/2 homogeneous coordinates that satisfy a set of ( n − 2)( n − 3)/2 conditions, resulting in a manifold of dimension 2 n − 2.
Isaak Yaglom has shown [ 1 ] how dual numbers provide coordinates for oriented lines in the Euclidean plane, and split-complex numbers form line coordinates for the hyperbolic plane . The coordinates depend on the presence of an origin and reference line on it. Then, given an arbitrary line its coordinates are found from the intersection with the reference line. The distance s from the origin to the intersection and the angle θ of inclination between the two lines are used:
Since there are lines ultraparallel to the reference line in the Lobachevski plane, they need coordinates too: There is a unique common perpendicular , say s is the distance from the origin to this perpendicular, and d is the length of the segment between reference and the given line.
The motions of the line geometry are described with linear fractional transformations on the appropriate complex planes. [ 1 ] : 87, 123 | https://en.wikipedia.org/wiki/Line_coordinates |
In mathematics , a line field on a manifold is a formation of a line being tangent to a manifold at each point, i.e. a section of the line bundle over the manifold. Line fields are of particular interest in the study of complex dynamical systems , where it is conventional to modify the definition slightly.
In general, let M be a manifold. A line field on M is a function μ that assigns to each point p of M a line μ ( p ) through the origin in the tangent space T p ( M ). Equivalently, one may say that μ ( p ) is an element of the projective tangent space PT p ( M ), or that μ is a section of the projective tangent bundle PT( M ).
In the study of complex dynamical systems, the manifold M is taken to be a Hersee surface . A line field on a subset A of M (where A is required to have positive two- dimensional Lebesgue measure ) is a line field on A in the general sense above that is defined almost everywhere in A and is also a measurable function . [ 1 ]
This mathematical analysis –related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Line_field |
In scientific visualization , line integral convolution ( LIC ) is a method to visualize a vector field (such as fluid motion ) at high spatial resolutions. [ 1 ] The LIC technique was first proposed by Brian Cabral and Leith Casey Leedom in 1993. [ 2 ]
In LIC, discrete numerical line integration is performed along the field lines (curves) of the vector field on a uniform grid . The integral operation is a convolution of a filter kernel and an input texture, often white noise . [ 1 ] In signal processing , this process is known as a discrete convolution . [ 3 ]
Traditional visualizations of vector fields use small arrows or lines to represent vector direction and magnitude. This method has a low spatial resolution, which limits the density of presentable data and risks obscuring characteristic features in the data. [ 1 ] [ 3 ] More sophisticated methods, such as streamlines and particle tracing techniques, can be more revealing but are highly dependent on proper seed points. [ 1 ] Texture-based methods, like LIC, avoid these problems since they depict the entire vector field at point-like (pixel) resolution. [ 1 ]
Compared to other integration-based techniques that compute field lines of the input vector field, LIC has the advantage that all structural features of the vector field are displayed, without the need to adapt the start and end points of field lines to the specific vector field. In other words, it shows the topology of the vector field. [ citation needed ]
In user testing, LIC was found to be particularly good for identifying critical points. [ 4 ]
LIC causes output values to be strongly correlated along the field lines, but uncorrelated in orthogonal directions. [ 1 ] As a result, the field lines contrast each other and stand out visually from the background.
Intuitively, the process can be understood with the following example: the flow of a vector field can be visualized by overlaying a fixed, random pattern of dark and light paint. As the flow passes by the paint, the fluid picks up some of the paint's color, averaging it with the color it has already acquired. The result is a randomly striped, smeared texture where points along the same streamline tend to have a similar color. Other physical examples include:
Although the input vector field and the result image are discretized, it pays to look at it from a continuous viewpoint. [ 1 ] Let v {\displaystyle \mathbf {v} } be the vector field given in some domain Ω {\displaystyle \Omega } . Although the input vector field is typically discretized, we regard the field v {\displaystyle \mathbf {v} } as defined in every point of Ω {\displaystyle \Omega } , i.e. we assume an interpolation . Streamlines, or more generally field lines, are tangent to the vector field in each point. They end either at the boundary of Ω {\displaystyle \Omega } or at critical points where v = 0 {\displaystyle \mathbf {v} =\mathbf {0} } . For the sake of simplicity, critical points and boundaries are ignored in the following.
A field line σ {\displaystyle {\boldsymbol {\sigma }}} , parametrized by arc length s {\displaystyle s} , is defined as
d σ ( s ) d s = v ( σ ( s ) ) | v ( σ ( s ) ) | . {\displaystyle {\frac {d{\boldsymbol {\sigma }}(s)}{ds}}={\frac {\mathbf {v} ({\boldsymbol {\sigma }}(s))}{|\mathbf {v} ({\boldsymbol {\sigma }}(s))|}}.}
Let σ r ( s ) {\displaystyle {\boldsymbol {\sigma }}_{\mathbf {r} }(s)} be the field line that passes through the point r {\displaystyle \mathbf {r} } for s = 0 {\displaystyle s=0} . Then the image gray value at r {\displaystyle \mathbf {r} } is set to
D ( r ) = ∫ − L / 2 L / 2 k ( s ) N ( σ r ( s ) ) d s {\displaystyle D(\mathbf {r} )=\int _{-L/2}^{L/2}k(s)N({\boldsymbol {\sigma }}_{\mathbf {r} }(s))ds}
where k ( s ) {\displaystyle k(s)} is the convolution kernel , N ( r ) {\displaystyle N(\mathbf {r} )} is the noise image, and L {\displaystyle L} is the length of field line segment that is followed.
D ( r ) {\displaystyle D(\mathbf {r} )} has to be computed for each pixel in the LIC image. If carried out naively, this is quite expensive. First, the field lines have to be computed using a numerical method for solving ordinary differential equations , like a Runge–Kutta method , and then for each pixel the convolution along a field line segment has to be calculated.
The final image will normally be colored in some way. Typically, some scalar field in Ω {\displaystyle \Omega } (like the vector length) is used to determine the hue, while the grayscale LIC output determines the brightness .
Different choices of convolution kernels and random noise produce different textures; for example, pink noise produces a cloudy pattern where areas of higher flow stand out as smearing, suitable for weather visualization. Further refinements in the convolution can improve the quality of the image. [ 6 ]
Algorithmically, LIC takes a vector field and noise texture as input, and outputs a texture. The process starts by generating in the domain of the vector field a random gray level image at the desired output resolution. Then, for every pixel in this image, the forward and backward streamline of a fixed arc length is calculated. The value assigned to the current pixel is computed by a convolution of a suitable convolution kernel with the gray levels of all the noise pixels lying on a segment of this streamline. This creates a gray level LIC image.
Basic LIC images are grayscale images, without color and animation. While such LIC images convey the direction of the field vectors, they do not indicate orientation; for stationary fields, this can be remedied by animation. Basic LIC images do not show the length of the vectors (or the strength of the field).
The length of the vectors (or the strength of the field) is usually coded in color; alternatively, animation can be used. [ 2 ] [ 1 ]
LIC images can be animated by using a kernel that changes over time. Samples at a constant time from the streamline would still be used, but instead of averaging all pixels in a streamline with a static kernel, a ripple-like kernel constructed from a periodic function multiplied by a Hann function acting as a window (in order to prevent artifacts) is used. The periodic function is then shifted along the period to create an animation.
The computation can be significantly accelerated by re-using parts of already computed field lines, specializing to a box function as convolution kernel k ( s ) {\displaystyle k(s)} and avoiding redundant computations during convolution. [ 1 ] The resulting fast LIC method can be generalized to convolution kernels that are arbitrary polynomials. [ 7 ]
Because LIC does not encode flow orientation, it cannot distinguish between streamlines of equal direction but opposite orientation. [ 8 ] Oriented Line Integral Convolution (OLIC) solves this issue by using a ramp-like asymmetric kernel and a low-density noise texture. [ 8 ] The kernel asymmetrically modulates the intensity along the streamline, producing a trace that encodes orientation; the low-density of the noise texture prevents smeared traces from overlapping, aiding readability.
Fast Rendering of Oriented Line Integral Convolution (FROLIC) is a variation that approximates OLIC by rendering each trace in discrete steps instead of as a continuous smear. [ 8 ] [ 9 ]
For time-dependent vector fields (unsteady flow), a variant called Unsteady Flow LIC has been designed that maintains the coherence of the flow animation. [ 10 ] An interactive GPU-based implementation of UFLIC has been presented. [ 11 ]
Since the computation of an LIC image is expensive but inherently parallel, the process has been parallelized [ 12 ] and, with availability of GPU-based implementations, interactive on PCs.
Note that the domain Ω {\displaystyle \Omega } does not have to be a 2D domain: the method is applicable to higher dimensional domains using multidimensional noise fields. However, the visualization of the higher-dimensional LIC texture is problematic; one way is to use interactive exploration with 2D slices that are manually positioned and rotated. The domain Ω {\displaystyle \Omega } does not have to be flat either; the LIC texture can be computed also for arbitrarily shaped 2D surfaces in 3D space. [ 13 ]
This technique has been applied to a wide range of problems since it first was published in 1993, both scientific and creative, including:
Representing vector fields:
Artistic effects for image generation and stylization:
Terrain generalization: | https://en.wikipedia.org/wiki/Line_integral_convolution |
Line level is the specified strength of an audio signal used to transmit analog sound between audio components such as CD and DVD players, television sets , audio amplifiers , and mixing consoles .
Generally, line level signals sit in the middle of the hierarchy of signal levels in audio engineering. There are weaker signals such as those from microphones (Mic Level/Microphone Level) and instrument pickups (Instrument Level), and stronger signals, such as those used to drive headphones and loudspeakers (Speaker Level). The "strength" of these various signals does not necessarily refer to the output voltage of the source device; it also depends on its output impedance and output power capability.
Consumer electronic devices concerned with audio (for example sound cards ) often have a connector labeled line in and/or line out . Line out provides an audio signal output and line in receives a signal input. The line in/out connections on consumer-oriented audio equipment are typically unbalanced , with a 3.5 mm (0.14 inch, but commonly called "eighth inch") 3-conductor TRS minijack connector providing ground, left channel, and right channel, or stereo RCA jacks . Professional equipment commonly uses balanced connections on 6.35 mm (1/4 inch) TRS phone jacks or XLR connectors . Professional equipment may also use unbalanced connections with (1/4 inch) TS phone jacks.
A line level describes a line's nominal signal level as a ratio, expressed in decibels , against a standard reference voltage. The nominal level and the reference voltage against which it is expressed depend on the line level being used. While the nominal levels themselves vary, only two reference voltages are common: decibel volts (dBV) for consumer applications, and decibels unloaded (dBu) for professional applications.
The decibel volt reference voltage is 1 V RMS = 0 dBV . [ 1 ] The decibel unloaded reference voltage, 0 dBu , is the AC voltage required to produce 1 mW of power across a 600 Ω impedance (approximately 0.7746 V RMS ). [ 2 ] This awkward unit is a holdover from the early telephone standards, which used 600 Ω sources and loads, and measured dissipated power in decibel-milliwatts ( dBm ). Modern audio equipment does not use 600 Ω matched loads, hence dBm unloaded ( dBu ).
The most common nominal level for professional equipment is +4 dBu (by convention, decibel values are written with an explicit sign symbol). For consumer equipment it is −10 dBV , which is used to reduce manufacturing costs. [ 3 ]
Expressed in absolute terms, a signal at −10 dBV is equivalent to a sine wave signal with a peak amplitude (V PK ) of approximately 0.447 volts , or any general signal at 0.316 volts root mean square (V RMS ). A signal at +4 dBu is equivalent to a sine wave signal with a peak amplitude of approximately 1.736 volts , or any general signal at approximately 1.228 V RMS .
Peak-to-peak (sometimes abbreviated as p-p ) amplitude (V PP ) refers to the total voltage swing of a signal, which is double the peak amplitude of the signal. For instance, a signal with a peak amplitude of ±0.5 V has a p-p amplitude of 1.0 V .
The line level signal is an alternating current signal without a DC offset, meaning that its voltage varies with respect to signal ground from the peak amplitude (for example +1.5 V ) to the equivalent negative voltage ( −1.5 V ). [ 4 ]
As cables between line output and line input are generally extremely short compared to the audio signal wavelength in the cable, transmission line effects can be disregarded and impedance matching need not be used.
Instead, line level circuits use the impedance bridging principle,
in which a low impedance output drives a high impedance input.
A typical line out connection has an output impedance from 100 to 600 Ω, with lower values being more common in newer equipment.
Line inputs present a much higher impedance, typically 10 kΩ or more. [ 5 ]
The two impedances form a voltage divider with a shunt element that is large relative to the size of the series element, which ensures that little of the signal is shunted to ground and that current requirements are minimized.
Most of the voltage asserted by the output appears across the input impedance and almost none of the voltage is dropped across the output. [ 5 ] The line input acts similarly to a high impedance voltmeter or oscilloscope input, measuring the voltage asserted by the output while drawing minimal current (and hence minimal power) from the source.
The high impedance of the line in circuit does not load down the output of the source device.
These are voltage signals (as opposed to current signals) and it is the signal information (voltage) that is desired, not power to drive a transducer , such as a speaker or antenna. The actual information that is exchanged between the devices is the variance in voltage; it is this alternating voltage signal that conveys the information, making the current irrelevant.
Line-out symbol. PC Guide color lime green.
Line outputs usually present a source impedance of from 100 to 600 ohms . The voltage can reach 2 volts peak-to-peak with levels referenced to −10 dBV (300 mV) at 10 kΩ . The frequency response of most modern equipment is advertised as at least 20 Hz to 20 kHz, which corresponds to the range of human hearing . Line outputs are intended to drive a load impedance of
10,000 ohms; with only a few volts, this requires only minimal current.
Connecting a low-impedance load such as a loudspeaker (usually 4 to 8 Ω ) to a line out will essentially short circuit the output circuit. Such loads are around 1/1000 the impedance a line out is designed to drive, so the line out is usually not designed to source the current that would be drawn by a 4 to 8 ohm load at normal line out signal voltages. The result will be very weak sound from the speaker and possibly a damaged line out circuit.
Headphone outputs and line outputs are sometimes confused.
Different make and model headphones have widely varying impedances, from as little as 20 Ω to a few hundred ohms; the lowest of these will have results similar to a speaker, while the highest may work acceptably if the line out impedance is low enough and the headphones are sensitive enough.
Conversely, a headphone output generally has a source impedance of only a few ohms (to provide a bridging connection with 32 ohm headphones) and will easily drive a line input.
For similar reasons, "wye"-cables (or "Y-splitters") should not be used to combine two line out signals into a single line in.
Each line output would be driving the other line output as well as the intended input,
again resulting in a much heavier load than designed for. This will result in signal loss and possibly even damage.
An active mixer, using for example op-amps ,
should be used instead. [ 6 ] A large resistor in series with each output can be used to safely mix them together, but must be appropriately designed for the load impedance and cable length.
Line-in symbol. PC Guide color light blue.
It is intended by designers that the line out of one device be connected to the line input of another. Line inputs are designed to accept voltage levels in the range provided by line outputs.
Impedances, on the other hand, are deliberately not matched from output to input.
The impedance of a line input is typically around 10 kΩ .
When driven by a line output's usual low impedance of 100 to 600 ohms, this forms a "bridging" connection in which most of the voltage generated by the source (the output) is dropped across the load (the input), and minimal current flows due to the load's relatively high impedance.
Although line inputs have a high impedance compared to that of line outputs,
they should not be confused with so-called "Hi-Z" inputs (Z being the symbol for impedance ) which have an impedance of 47 kΩ to over 1 MΩ . These "Hi-Z" or "instrument" inputs generally have higher gain than a line input.
They are designed to be used with, for example, electric guitar pickups and " direct injection " boxes.
Some of these sources can provide only minimal voltage and current and the high impedance input is designed to not load them excessively.
Acoustic sounds (such as voices or musical instruments ) are often recorded with transducers ( microphones and pickups ) that produce weak electrical signals. These signals must be amplified to line level, where they are more easily manipulated by other devices such as mixing consoles and tape recorders. Such amplification is performed by a device known as a preamplifier or "preamp",
which boosts the signal to line level.
After manipulation at line level, signals are then typically sent to a power amplifier , where they are amplified to levels that can drive headphones or loudspeakers . These convert the signals back into sounds that can be heard through the air.
Most phonograph cartridges also have a low output level and require a preamp; typically, a home stereo integrated amplifier or receiver will have a special phono input . This input passes the signal through a phono preamp, which applies RIAA equalization to the signal as well as boosting it to line level. | https://en.wikipedia.org/wiki/Line_level |
Line moiré is one type of moiré pattern ; a pattern that appears when superposing two transparent layers containing correlated opaque patterns. Line moiré is the case when the superposed patterns comprise straight or curved lines. When moving the layer patterns, the moiré patterns transform or move at a faster speed. This effect is called optical moiré speedup.
Simple moiré patterns can be observed when superposing two transparent layers comprising periodically repeating opaque parallel lines as shown in Figure 1. The lines of one layer are parallel to the lines of the second layer.
The superposition image does not change if transparent layers with their opaque patterns are inverted. When considering printed samples, one of the layers is denoted as the base layer and the other one as the revealing layer. It is assumed that the revealing layer is printed on a transparency and is superimposed on top of the base layer, which can be printed either on a transparency or on an opaque paper. The periods of the two layer patterns are close. We denote the period of the base layer as p b and the period of the revealing layer as p r .
The superposition image of Figure 1 outlines periodically repeating dark parallel bands, called moiré lines. Spacing between the moiré lines is much larger than the periods of lines in the two layers.
Light bands of the superposition image correspond to the zones where the lines of both layers overlap. The dark bands of the superposition image forming the moiré lines correspond to the zones where the lines of the two layers interleave, hiding the white background. The labels of Figure 2 show the passages from light zones with overlapping layer lines to dark zones with interleaving layer lines. The light and dark zones are periodically interchanging.
Figure 3 shows a detailed diagram of the superposition image between two adjacent zones with overlapping lines of the revealing and base layers (i.e., between two light bands). [ 1 ]
The period p m of moiré lines is the distance from one point where the lines of both layers overlap (at the bottom of the figure) to the next such point (at the top). Let us count the layer lines, starting from the bottom point. At the count 0 the lines of both layers overlap. Since in our case p r < p b , for the same number of counted lines, the base layer lines with a long period advance faster than the revealing layer lines with a short period. At the halfway of the distance p m , the base layer lines are ahead the revealing layer lines by a half a period ( p r /2) of the revealing layer lines, due to which the lines are interleaving, forming a dark moiré band. At the full distance p m , the base layer lines are ahead of the revealing layer lines by a full period p r , so the lines of the layers again overlap. The base layer lines gain the distance p m with as many lines ( p m / p b ) as the number of the revealing layer lines ( p m / p r ) for the same distance minus one: p m / p r = p m / p b + 1. From here we obtain the well known formula for the period p m of the superposition image: [ 2 ]
For the case when the revealing layer period is longer than the base layer period, the distance between moiré bands is the absolute value computed by the formula. The superposition of two layers comprising parallel lines forms an optical image comprising parallel moiré lines with a magnified period. According to the formula for computing p m , the closer the periods of the two layers, the stronger the magnification factor is.
The thicknesses of layer lines affect the overall darkness of the superposition image and the thickness of the moiré bands, but the period p m does not depend on the layer lines’ thickness.
The moiré bands of Figure 1 will move if we displace the revealing layer. When the revealing layer moves perpendicularly to layer lines, the moiré bands move along the same axis, but several times faster than the movement of the revealing layer.
The GIF animation shown in Figure 4 corresponds to a slow movement of the revealing layer. The GIF file repeatedly animates an upward movement of the revealing layer (perpendicular to layer lines) across a distance equal to p r . The animation demonstrates that the moiré lines of the superposition image move up at a speed, much faster than the movement speed of the revealing layer.
When the revealing layer is shifted up perpendicularly to the layer lines by one full period ( p r ) of its pattern, the superposition optical image must be the same as the initial one. It means that the moiré lines traverse a distance equal to the period of the superposition image p m while the revealing layer traverses the distance equal to its period p r . Assuming that the base layer is immobile ( v b =0), the following equation represents the ratio of the optical speed to the revealing layer’s speed:
By replacing p m with its formula, we have [ 3 ]
In case the period of the revealing layer is longer than the period of the base layer, the optical image moves in the opposite direction. The negative value of the ratio computed according to this formula signifies a movement in the reverse direction.
Here we present patterns with inclined lines. When we are interested in optical speedup we can represent the case of inclined patterns such that the formulas for computing moiré periods and optical speedups remain valid in their current simplest form. For this purpose, the values of periods p r , p b , and p m correspond to the distances between the lines along the axis of movements (the vertical axis in the animated example of Figure 4). When the layer lines are perpendicular to the movement axis, the periods ( p ) are equal to the distances (denoted as T ) between the lines (as in Figure 4). If the lines are inclined, the periods ( p ) along the axis of the movement are not equal to the distances ( T ) between the lines.
The superposition of two layers with identically inclined lines forms moiré lines inclined at the same angle. Figure 5 is obtained from Figure 1 with a vertical shearing. In Figure 5 the layer lines and the moiré lines are inclined by 10 degrees. Since the inclination is not a rotation, during the inclination the distance ( p ) between the layer lines along the vertical axis is conserved, but the true distance ( T ) between the lines (along an axis perpendicular to these lines) is changed. The difference between the vertical periods p b , p r , and the distances T b , T r is shown in the diagram of Figure 8.
The inclination degree of layer lines may change along the horizontal axis forming curves. The superposition of two layers with identical inclination pattern forms moiré curves with the same inclination pattern. In Figure 6 the inclination degree of layer lines gradually changes according to the following sequence of degrees (+30, –30, +30, –30, +30). Layer periods p b and p r represent the distances between the curves along the vertical axis. The presented formulas for computing the period p m (the vertical distance between the moiré curves) and the optical speedup (along the vertical axis) are valid for Figure 6.
More interesting is the case when the inclination degrees of layer lines are not the same for the base and revealing layers. Figure 7 shows an animation of a superposition images where the inclination degree of base layer lines is constant (10 degrees), but the inclination of the revealing layer lines oscillates between 5 and 15 degrees. The periods of layers along the vertical axis p b and p r are the same all the time. Correspondingly, the period p m (along the vertical axis) computed with the basic formula also remains the same.
Figure 8 helps to compute the inclination degree of moiré optical lines as a function of the inclination of the revealing and the base layer lines. We draw the layer lines schematically without showing their true thicknesses. The bold lines of the diagram inclined by α b degrees are the base layer lines. The bold lines inclined by α r degrees are the revealing layer lines. The base layer lines are vertically spaced by a distance equal to p b , and the revealing layer lines are vertically spaced by a distance equal to p r . The distances T b and T r represent the true space between the base layer and revealing layer lines, correspondingly. The intersections of the lines of the base and the revealing layers (marked in the figure by two arrows) lie on a central axis of a light moiré band. The dashed line of Figure 8 corresponds to the axis of the light moiré band. The inclination degree of moiré lines is therefore the inclination α m of the dashed line.
From Figure 8 we deduce the following two equations:
From these equations we deduce the equation for computing the inclination of moiré lines as a function of the inclinations of the base layer and the revealing layer lines:
The true pattern periods T b , T r , and T m (along the axes perpendicular to pattern lines) are computed as follows (see Figure 8):
From here, using the formula for computing tan( α m ) with periods p , we deduce a well known formula for computing the moiré angle α m with periods T : [ 4 ] [ 5 ] [ 6 ]
From formula for computing p m we deduce another well known formula for computing the period T m of moiré pattern (along the axis perpendicular to moiré bands):
In the particular case when T b = T r = T , the formula for the period T m is reduced into well known formula:
And the formula for computing α m is reduced to:
Here is the equation for computing the revealing layer line inclination α r for a given base layer line inclination α b , and a desired moiré line inclination α m :
For any given base layer line inclination, this equation permits us to obtain a desired moiré line inclination by properly choosing the revealing layer inclination. In Figure 6 we showed an example where the curves of layers follow an identical inclination pattern forming a superposition image with the same inclination pattern. The inclination degrees of the layers’ and moiré lines change along the horizontal axis according to the following sequence of alternating degree values (+30, –30, +30, –30, +30). In Figure 9 we obtain the same superposition pattern as in Figure 6, but with a base layer comprising straight lines inclined by –10 degrees. The revealing pattern of Figure 9 is computed by interpolating the curves into connected straight lines, where for each position along the horizontal axis, the revealing line’s inclination angle α r is computed as a function of α b and α m according to the equation above.
Figure 9 demonstrates that the difference between the inclination angles of revealing and base layer lines has to be several times smaller than the difference between inclination angles of moiré and base layer lines.
Another example forming the same superposition patterns as in Figure 6 and Figure 9 is shown in Figure 10. In Figure 10 the desired inclination pattern (+30, –30, +30, –30, +30) is obtained using a base layer with an inverted inclination pattern (–30, +30, –30, +30, –30).
Figure 11 shows an animation where we obtain a superposition image with a constant inclination pattern of moiré lines (+30, –30, +30, –30, +30) for continuously modifying pairs of base and revealing layers. The base layer inclination pattern gradually changes and the revealing layer inclination pattern correspondingly adapts such that the superposition image’s inclination pattern remains the same. | https://en.wikipedia.org/wiki/Line_moiré |
Line notation is a typographical notation system using ASCII characters, most often used for chemical nomenclature . | https://en.wikipedia.org/wiki/Line_notation |
The Line of Contact ( Armenian : շփման գիծ , shp’man gits , Azerbaijani : təmas xətti ) was the front line which separated Armenian forces (the Nagorno-Karabakh Defense Army and the Armenian Armed Forces ) and the Azerbaijan Armed Forces from the end of the First Nagorno-Karabakh War in 1994 until the 2020 Nagorno-Karabakh ceasefire agreement .
It was formed in the aftermath of the May 1994 ceasefire that ended the First Nagorno-Karabakh War (1988–94). [ 1 ] During its existence, the mountain range of Murovdag (Mrav) was the northern part of the line of contact and essentially served as a natural border between the two forces. [ 2 ] [ 3 ] The length of the line of contact was between 180 kilometres (110 mi) [ 4 ] and 200 kilometres (120 mi) until 2020. [ 5 ]
For the first time since the 1994 ceasefire, the front line slightly shifted during the 2016 Nagorno-Karabakh conflict , [ 6 ] when Azerbaijan recaptured some hectares of territory. [ 7 ] The Line of Contact effectively dissolved during the 2020 Nagorno-Karabakh war , after Azerbaijan made significant military gains against the internationally unrecognised Republic of Artsakh including capturing parts of the original Nagorno-Karabakh Autonomous Oblast . Armenian forces later withdrew from almost the entirety of Karabkh territories that it remained in control of as part of the 2020 ceasefire agreement, officially ending the existence of the Line of Contact. [ 8 ] A new line of contact then existed between the remaining Artsakhi zone in the former Autonomous Oblast controlled by Russian Armed Forces peacekeepers and the recaptured Azerbaijani territories until 2024, when the Republic of Artsakh was dissolved following an Azerbaijani offensive in September 2023 and a ceasefire agreement .
The term "Line of Contact" was widely used in official documents and statements, including by the OSCE Minsk Group . [ 9 ]
Some Armenian analysts, including Ara Papian encouraged the Armenian side to avoid the term "line of contact", instead calling it a "state border" between Artsakh and Azerbaijan. [ 10 ] [ 11 ] Independent journalist and author Tatul Hakobyan writes of it as a state border of Azerbaijan and Artsakh and notes that it is called the "line of contact" in international lexicon. [ 12 ]
In Azerbaijan, the "line of contact" is often referred to as the "line of occupation". [ 13 ]
The line of contact was, immediately after the ceasefire, a "relatively quiet zone with barbed wire and lightly armed soldiers sitting in trenches", according to Thomas de Waal . [ 14 ] There was also a relatively large no-man's land after the ceasefire which was several kilometers wide in some places. It was reduced to a few hundred meters in most areas of the line of contact due to Azerbaijani redeployments into the former neutral zone. [ 15 ] In 2016, there were around 20,000 men on each side of the heavily militarized line of contact. [ 16 ] Since the ceasefire the line of contact had become a heavily militarized, fortified and mined no-man's-land and a buffer zone of trenches . [ 1 ] [ 17 ] [ 18 ] According to de Waal, it is the "most militarised zone in the wider Europe," [ 14 ] and one of the three most militarized zones in the world (along with Kashmir and Korea ). [ 5 ] The trenches along the line of contact were extensively compared to those of World War I . [ 5 ] [ 19 ] [ 20 ]
The line of contact was regularly monitored by a group of six OSCE observers, headed by Andrzej Kasprzyk of Poland. [ 21 ] There were exchanges of fire virtually on a daily basis. [ 22 ] There had been significant violations of the ceasefire on various occasions, [ 23 ] usually characterized by low-intensity fighting. [ 24 ] Significant fighting occurred in April 2016 , [ 25 ] when for the first time since the ceasefire the line of contact was shifted, though not significantly. [ 26 ] According to Laurence Broers of Chatham House , "Although slivers of territory changed hands for the first time since 1994, little of strategic significance appears to have altered on the ground." [ 27 ] The 2016 clashes also marked the first time since the 1994 ceasefire that heavy artillery was used, [ 28 ] while the 2020 Nagorno-Karabakh conflict saw use of heavy artillery, armoured warfare , and drone warfare . On October 9, 2020, when President of Azerbaijan Ilham Aliyev addressed the nation, he stated, "There is no status quo. There is no line of contact. We smashed it." [ 29 ]
According to Kolosov and Zotova (2020), "the deployment of military units along the separation line, the special regime of the border zone on both sides, constant skirmishes, and the destruction during the war and immediately after it of a number of cities and other settlements turned the border territories into an economic desert." [ 30 ]
According to the International Crisis Group , all of 150,000 Karabakh Armenians are "within reach of Azerbaijani missiles and artillery shells", while around twice the number of Azerbaijanis (300,000) "live in the 15km-wide zone along the Azerbaijani side of the line of contact." [ 31 ] | https://en.wikipedia.org/wiki/Line_of_Contact_(Nagorno-Karabakh) |
In physics , the line of action (also called line of application ) of a force ( F → ) is a geometric representation of how the force is applied. It is the straight line through the point at which the force is applied, and is in the same direction as the vector F → . [ 1 ] [ 2 ] The lever arm is the perpendicular distance from the axis of rotation to the line of action. [ 3 ]
The concept is essential, for instance, for understanding the net effect of multiple forces applied to a body . For example, if two forces of equal magnitude act upon a rigid body along the same line of action but in opposite directions, they cancel and have no net effect. But if, instead, their lines of action are not identical, but merely parallel , then their effect is to create a moment on the body, which tends to rotate it. [ citation needed ]
For the simple geometry associated with the figure, there are three equivalent equations for the magnitude of the torque associated with a force F → {\displaystyle {\vec {F}}} directed at displacement r → {\displaystyle {\vec {r}}} from the axis whenever the force is perpendicular to the axis:
where r → × F → {\displaystyle {\vec {r}}\times {\vec {F}}} is the cross-product , F ⊥ {\displaystyle F_{\perp }} is the component of F → {\displaystyle {\vec {F}}} perpendicular to r ^ {\displaystyle {\hat {r}}} , r ⊥ {\displaystyle r_{\perp }} is the moment arm , and θ {\displaystyle \theta } is the angle between r → {\displaystyle {\vec {r}}} and F → {\displaystyle {\vec {F}}} . [ citation needed ]
This classical mechanics –related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Line_of_action |
An orbital node is either of the two points where an orbit intersects a plane of reference to which it is inclined. [ 1 ] A non-inclined orbit , which is contained in the reference plane, has no nodes.
Common planes of reference include the following:
If a reference direction from one side of the plane of reference to the other is defined, the two nodes can be distinguished. For geocentric and heliocentric orbits, the ascending node (or north node ) is where the orbiting object moves north through the plane of reference, and the descending node (or south node ) is where it moves south through the plane. [ 4 ] In the case of objects outside the Solar System, the ascending node is the node where the orbiting secondary passes away from the observer, and the descending node is the node where it moves towards the observer. [ 5 ] , p. 137.
The position of the node may be used as one of a set of parameters, called orbital elements , which describe the orbit. This is done by specifying the longitude of the ascending node (or, sometimes, the longitude of the node .)
The line of nodes is the straight line resulting from the intersection of the object's orbital plane with the plane of reference; it passes through the two nodes. [ 2 ]
The symbol of the ascending node is ( Unicode : U+260A, ☊), and the symbol of the descending node is ( Unicode : U+260B, ☋).
In medieval and early modern times, the ascending and descending nodes of the Moon in the ecliptic plane were called the "dragon's head" ( Latin : caput draconis , Arabic : رأس الجوزهر ) and "dragon's tail" ( Latin : cauda draconis ), respectively. [ 6 ] : p.141, [ 7 ] : p.245 These terms originally referred to the times when the Moon crossed the apparent path of the sun in the sky (as in a solar eclipse ). Also, corruptions of the Arabic term such as ganzaar , genzahar , geuzaar and zeuzahar were used in the medieval West to denote either of the nodes. [ 8 ] : pp.196–197, [ 9 ] : p.65, [ 10 ] : pp.95–96
The Koine Greek terms αναβιβάζων and καταβιβάζων were also used for the ascending and descending nodes, giving rise to the English terms anabibazon and catabibazon . [ 11 ] [ 12 ] : ¶27
For the orbit of the Moon around Earth , the plane is taken to be the ecliptic , not the equatorial plane . The gravitational pull of the Sun upon the Moon causes its nodes to gradually precess westward, completing a cycle in approximately 18.6 years. [ 1 ] [ 13 ]
The image of the ascending and descending orbital nodes as the head and tail of a dragon, 180 degrees apart in the sky, goes back to the Chaldeans; it was used by the Zoroastrians, and then by Arabic astronomers and astrologers. In Middle Persian, its head and tail were respectively called gōzihr sar and gōzihr dumb ; in Arabic, al-ra's al-jawzihr and al-dhanab al-jawzihr — or in the case of the Moon, ___ al-tennin . [ 14 ] Among the arguments against astrologers made by Ibn Qayyim al-Jawziyya (1292–1350), in his Miftah Dar al-SaCadah: "Why is it that you have given an influence to al-Ra's [the head] and al-Dhanab [the tail], which are two imaginary points [ascending and descending nodes]?" [ 15 ] | https://en.wikipedia.org/wiki/Line_of_nodes |
The line of sight , also known as visual axis or sightline (also sight line ), is an imaginary line between a viewer/ observer / spectator 's eye (s) and a subject of interest, or their relative direction . [ 1 ] The subject may be any definable object taken note of or to be taken note of by the observer, at any distance more than least distance of distinct vision. In optics , refraction of a ray due to use of lenses can cause distortion . [ 2 ] Shadows, patterns and movement can also influence line of sight interpretation [ 3 ] [ 4 ] (as in optical illusions ).
The term "line" typically presumes that the light by which the observed object is seen travels as a straight ray , which is sometimes not the case as light can take a curved/angulated path when reflected from a mirror , [ 5 ] refracted by a lens or density changes in the traversed media , or deflected by a gravitational field . Fields of study feature specific targets, such as vessels in navigation, marker flags or natural features in surveying, celestial objects in astronomy, and so on. To have optimal observational outcome, it is preferable to have a completely unobstructed sightline. | https://en.wikipedia.org/wiki/Line_of_sight |
The line of thrust is the locus of the points, through which forces pass in a retaining wall or an arch . It is the line, along which internal forces flow, [1] , [2] .
In a stone structure , the line of thrust is a theoretical line that through the structure represents the path of the resultants of the compressive forces , [3] . For a structure to be stable, the line of thrust must lie entirely inside the structure, [4] , [5] .
The line of thrust is important in almost any architecture bearing weight. This includes aircraft , bridges , plus arches ; see catenary arch .
An arch won't collapse, when the line of thrust is entirely internal to the arch, [6] .
This classical mechanics –related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Line_of_thrust |
Line representations in robotics are used for the following:
When using such line it is needed to have conventions for the representations so they are clearly defined. This article discusses several of these methods.
A line L ( p , d ) {\displaystyle L(p,d)} is completely defined by the ordered set of two vectors:
Each point x {\displaystyle x} on the line is given a parameter value t {\displaystyle t} that satisfies: x = p + t d {\displaystyle x=p+td} . The parameter t is unique once p {\displaystyle p} and d {\displaystyle d} are chosen. The representation L ( p , d ) {\displaystyle L(p,d)} is not minimal, because it uses six parameters for only four degrees of freedom. The following two constraints apply:
Arthur Cayley and Julius Plücker introduced an alternative representation using two free vectors. This representation was finally named after Plücker. The Plücker representation is denoted by L p l ( d , m ) {\displaystyle L_{pl}(d,m)} . Both d {\displaystyle d} and m {\displaystyle m} are free vectors: d {\displaystyle d} represents the direction of the line and m {\displaystyle m} is the moment of d {\displaystyle d} about the chosen reference origin. m = p × d {\displaystyle m=p\times d} ( m {\displaystyle m} is independent of which point p {\displaystyle p} on the line is chosen!) The advantage of the Plücker coordinates is that they are homogeneous. A line in Plücker coordinates has still four out of six independent parameters, so it is not a minimal representation. The two constraints on the six Plücker coordinates are
A line representation is minimal if it uses four parameters, which is the minimum needed to represent all possible lines in the Euclidean Space (E³).
Jaques Denavit and Richard S. Hartenberg presented the first minimal representation for a line which is now widely used. The common normal between two lines was the main geometric concept that allowed Denavit and Hartenberg to find a minimal representation. Engineers use the Denavit–Hartenberg convention(D–H) to help them describe the positions of links and joints unambiguously. Every link gets its own coordinate system . There are a few rules to consider in choosing the coordinate system:
Once the coordinate frames are determined, inter-link transformations are uniquely described by the following four parameters:
The Hayati–Roberts line representation, denoted L h r ( e x , e y , l x , l y ) {\displaystyle L_{hr}(e_{x},e_{y},l_{x},l_{y})} , is another minimal line representation, with parameters:
This representation is unique for a directed line. The coordinate singularities are different from the DH singularities: it has singularities if the line becomes parallel to either the X {\displaystyle X} or Y {\displaystyle Y} axis of the world frame.
The product of exponentials formula represents the kinematics of an open-chain mechanism as the product of exponentials of twists , and may be used to describe a series of revolute, prismatic, and helical joints. [ 1 ] | https://en.wikipedia.org/wiki/Line_representations_in_robotics |
In geometry , a line segment is a part of a straight line that is bounded by two distinct endpoints (its extreme points ), and contains every point on the line that is between its endpoints. It is a special case of an arc , with zero curvature . The length of a line segment is given by the Euclidean distance between its endpoints. A closed line segment includes both endpoints, while an open line segment excludes both endpoints; a half-open line segment includes exactly one of the endpoints. In geometry , a line segment is often denoted using an overline ( vinculum ) above the symbols for the two endpoints, such as in AB . [ 1 ]
Examples of line segments include the sides of a triangle or square. More generally, when both of the segment's end points are vertices of a polygon or polyhedron , the line segment is either an edge (of that polygon or polyhedron) if they are adjacent vertices, or a diagonal . When the end points both lie on a curve (such as a circle ), a line segment is called a chord (of that curve).
If V is a vector space over R {\displaystyle \mathbb {R} } or C , {\displaystyle \mathbb {C} ,} and L is a subset of V , then L is a line segment if L can be parameterized as
for some vectors u , v ∈ V {\displaystyle \mathbf {u} ,\mathbf {v} \in V} where v is nonzero. The endpoints of L are then the vectors u and u + v .
Sometimes, one needs to distinguish between "open" and "closed" line segments. In this case, one would define a closed line segment as above, and an open line segment as a subset L that can be parametrized as
for some vectors u , v ∈ V . {\displaystyle \mathbf {u} ,\mathbf {v} \in V.}
Equivalently, a line segment is the convex hull of two points. Thus, the line segment can be expressed as a convex combination of the segment's two end points.
In geometry , one might define point B to be between two other points A and C , if the distance | AB | added to the distance | BC | is equal to the distance | AC | . Thus in R 2 , {\displaystyle \mathbb {R} ^{2},} the line segment with endpoints A = ( a x , a y ) {\displaystyle A=(a_{x},a_{y})} and C = ( c x , c y ) {\displaystyle C=(c_{x},c_{y})} is the following collection of points:
In an axiomatic treatment of geometry, the notion of betweenness is either assumed to satisfy a certain number of axioms, or defined in terms of an isometry of a line (used as a coordinate system).
Segments play an important role in other theories. For example, in a convex set , the segment that joins any two points of the set is contained in the set. This is important because it transforms some of the analysis of convex sets, to the analysis of a line segment. The segment addition postulate can be used to add congruent segment or segments with equal lengths, and consequently substitute other segments into another statement to make segments congruent.
A line segment can be viewed as a degenerate case of an ellipse , in which the semiminor axis goes to zero, the foci go to the endpoints, and the eccentricity goes to one. A standard definition of an ellipse is the set of points for which the sum of a point's distances to two foci is a constant; if this constant equals the distance between the foci, the line segment is the result. A complete orbit of this ellipse traverses the line segment twice. As a degenerate orbit, this is a radial elliptic trajectory .
In addition to appearing as the edges and diagonals of polygons and polyhedra , line segments also appear in numerous other locations relative to other geometric shapes .
Some very frequently considered segments in a triangle to include the three altitudes (each perpendicularly connecting a side or its extension to the opposite vertex ), the three medians (each connecting a side's midpoint to the opposite vertex), the perpendicular bisectors of the sides (perpendicularly connecting the midpoint of a side to one of the other sides), and the internal angle bisectors (each connecting a vertex to the opposite side). In each case, there are various equalities relating these segment lengths to others (discussed in the articles on the various types of segment), as well as various inequalities .
Other segments of interest in a triangle include those connecting various triangle centers to each other, most notably the incenter , the circumcenter , the nine-point center , the centroid and the orthocenter .
In addition to the sides and diagonals of a quadrilateral , some important segments are the two bimedians (connecting the midpoints of opposite sides) and the four maltitudes (each perpendicularly connecting one side to the midpoint of the opposite side).
Any straight line segment connecting two points on a circle or ellipse is called a chord . Any chord in a circle which has no longer chord is called a diameter , and any segment connecting the circle's center (the midpoint of a diameter) to a point on the circle is called a radius .
In an ellipse, the longest chord, which is also the longest diameter , is called the major axis , and a segment from the midpoint of the major axis (the ellipse's center) to either endpoint of the major axis is called a semi-major axis . Similarly, the shortest diameter of an ellipse is called the minor axis , and the segment from its midpoint (the ellipse's center) to either of its endpoints is called a semi-minor axis . The chords of an ellipse which are perpendicular to the major axis and pass through one of its foci are called the latera recta of the ellipse. The interfocal segment connects the two foci.
When a line segment is given an orientation ( direction ) it is called a directed line segment or oriented line segment . It suggests a translation or displacement (perhaps caused by a force ). The magnitude and direction are indicative of a potential change. Extending a directed line segment semi-infinitely produces a directed half-line and infinitely in both directions produces a directed line . This suggestion has been absorbed into mathematical physics through the concept of a Euclidean vector . [ 2 ] [ 3 ] The collection of all directed line segments is usually reduced by making equipollent any pair having the same length and orientation. [ 4 ] This application of an equivalence relation was introduced by Giusto Bellavitis in 1835.
Analogous to straight line segments above, one can also define arcs as segments of a curve .
In one-dimensional space, a ball is a line segment.
An oriented plane segment or bivector generalizes the directed line segment.
Beyond Euclidean geometry, geodesic segments play the role of line segments.
A line segment is a one-dimensional simplex ; a two-dimensional simplex is a triangle.
This article incorporates material from Line segment on PlanetMath , which is licensed under the Creative Commons Attribution/Share-Alike License . | https://en.wikipedia.org/wiki/Line_segment |
A line source , as opposed to a point source , area source , or volume source , is a source of air, noise, water contamination or electromagnetic radiation that emanates from a linear (one-dimensional) geometry. The most prominent linear sources are roadway air pollution , aircraft air emissions , roadway noise , certain types of water pollution sources that emanate over a range of river extent rather than from a discrete point, elongated light tubes, certain dose models in medical physics and electromagnetic antennas . While point sources of pollution were studied since the late nineteenth century, linear sources did not receive much attention from scientists until the late 1960s, when environmental regulations for highways and airports began to emerge. At the same time, computers with the processing power to accommodate the data processing needs of the computer models required to tackle these one-dimensional sources became more available.
In addition, this era of the 1960s saw the first emergence of environmental scientists who spanned the disciplines required to accomplish these studies. For example, meteorologists, chemists, and computer scientists in the air pollution field were required to build complex models to address roadway air dispersion modeling . Prior to the 1960s, these specialities tended to work within their own disciplines, but with the advent of NEPA , the Clean Air Act , the Noise Control Act in the United States, and other seminal legislation, the era of multidisciplinary environmental science had begun.
For electromagnetic linear sources, the principal early advances in computer modeling arose in the Soviet Union and USA when the end of World War II and the Cold War were fought partially by progress in electronic warfare , including the technologies of active antenna arrays.
Air pollution levels near major highways and urban arterials are in violation of U.S. National Ambient Air Quality Standards where millions of Americans live or work. Even the interior of a building does not really protect inhabitants from adverse exterior air quality, since the exterior air is the intake supply, and it is well known that indoor air quality is typically worse than exterior air.
A roadway travelled by motor vehicles can be idealized by a line source emitting air pollutants. This mathematical problem was first solved in 1970 by a collaboration of physics , mathematics and computer science . [ 1 ] [ 2 ] The original theory assumed steady-state traffic conditions and meteorology on a perfectly straight roadway. Currently the models have evolved to treat variable meteorology, time-variant traffic operations and complex roadbed geometries. Current technology allows highway designers and city planners to analyze alternative roadway development plans and assess air quality impacts. The same basic model theory can be applied to airport operations, since the linear source is merely an inclined line. In the early 1970s these ESL models were refined into area source models to account for the finite width of the roadway.
Roadway noise is the most important example of a linear noise source, since it comprises about 80 percent of the environmental noise exposure for humans worldwide. In the 1960s, when computer modeling of this phenomenon was perfected, the first applications of linear source noise modeling became systematic. After passage of the National Environmental Policy Act and Noise Control Act, [ 3 ] the demand for detailed analysis soared, and decision makers began to look to acoustical scientists for answers regarding the planning of new roadways and the design of noise mitigation . The intensity of roadway noise is governed by the following variables: traffic operations (speed, truck mix, age of vehicle fleet), roadway surface type, tire types, roadway geometrics, terrain, micrometeorology and the geometry of area structures.
Due to the complexity of the variables, a line source acoustic model must be a computer model that can analyze sound levels in the vicinity of roadways. The first meaningful models arose in the late 1960s and early 1970s. Two of the leading research teams were BBN in Boston and ESL Inc. of Sunnyvale, California. Both of these groups developed complex mathematical models to allow the study of alternate roadway designs, traffic operations and noise mitigation strategies in an arbitrary setting. [ 4 ] Later model alterations have come into widespread use among state Departments of Transportation and city planners, but the accuracy of early models has had little change in 40 years.
Generally line source acoustic models trace sound ray bundles and calculate spreading loss along with ray bundle divergence (or convergence} from refractive phenomena. Diffraction is usually addressed by establishing secondary emitters at any points of topographic or anthropomorphic “sharpness” (such as noise barriers or building surfaces. Meteorology can be addressed in a statistical manner allowing for actual wind rose and wind speed statistics (along with thermocline data).
Less common are line source applications in the field of water pollutant dispersal. This phenomenon generally arises when surface runoff scours soil contaminants from upper soil layers and transports these pollutants to a linear receiving water, such as a river. The underlying land management practices which lead to such sources of water pollution are logging , pesticide application, construction grading, slash-and-burn activity and urban stormwater runoff.
Again computer models are needed to address the complexity of such an extended linear discharge into a dynamic medium such as flowing water. The resulting surface runoff water carrying pollutants may be considered a line source discharging into a river or stream. The chemical composition of this surface runoff may be characterized by a surface runoff model such as the USGS runoff precipitation algorithm, [ 5 ] while the instream transport may be analyzed by a dynamic river pollutant model such as DSSAM .
In the study of illumination, a variety of sources are linear in nature, most commonly the fluorescent tube , During the process of interior lighting design it is important to calculate the light intensity at work stations or other user areas, not only to ensure sufficient light is present, but more importantly to avoid over-illumination and its attendant energy wastage as well as adverse health effects. Thus the scientists involved in light transmission calculations employ computer models that recognize linear sources when fluorescent fixtures are used. In a typical setting there may be hundreds of finite length light sources that comprise the light output in an office environment. A related concept are the ultraviolet tubes used in phototherapy , where output radiation from the tube can be accurately modeled by treating the tube as a line source. [ 6 ] On a larger scale, an illuminated roadway may act as a line source of light pollution . | https://en.wikipedia.org/wiki/Line_source |
Line spectral pairs ( LSP ) or line spectral frequencies ( LSF ) are used to represent linear prediction coefficients (LPC) for transmission over a channel. [ 1 ] LSPs have several properties (e.g. smaller sensitivity to quantization noise) that make them superior to direct quantization of LPCs. For this reason, LSPs are very useful in speech coding .
LSP representation was developed by Fumitada Itakura , [ 2 ] at Nippon Telegraph and Telephone (NTT) in 1975. [ 3 ] From 1975 to 1981, he studied problems in speech analysis and synthesis based on the LSP method. [ 4 ] In 1980, his team developed an LSP-based speech synthesizer chip. LSP is an important technology for speech synthesis and coding, and in the 1990s was adopted by almost all international speech coding standards as an essential component, contributing to the enhancement of digital speech communication over mobile channels and the internet worldwide. [ 3 ] LSPs are used in the code-excited linear prediction (CELP) algorithm, developed by Bishnu S. Atal and Manfred R. Schroeder in 1985.
The LP polynomial A ( z ) = 1 − ∑ k = 1 p a k z − k {\displaystyle A(z)=1-\sum _{k=1}^{p}a_{k}z^{-k}} can be expressed as A ( z ) = 0.5 [ P ( z ) + Q ( z ) ] {\displaystyle A(z)=0.5[P(z)+Q(z)]} , where:
By construction, P is a palindromic polynomial and Q an antipalindromic polynomial ; physically P ( z ) corresponds to the vocal tract with the glottis closed and Q ( z ) with the glottis open. [ 5 ] It can be shown that:
The Line Spectral Pair representation of the LP polynomial consists simply of the location of the roots of P and Q (i.e. ω {\displaystyle \omega } such that z = e i ω , P ( z ) = 0 {\displaystyle z=e^{i\omega },P(z)=0} ). As they occur in pairs, only half of the actual roots (conventionally between 0 and π {\displaystyle \pi } ) need be transmitted. The total number of coefficients for both P and Q is therefore equal to p , the number of original LP coefficients (not counting a 0 = 1 {\displaystyle a_{0}=1} ).
A common algorithm for finding these [ 6 ] is to evaluate the polynomial at a sequence of closely spaced points around the unit circle, observing when the result changes sign; when it does a root must lie between the points tested. Because the roots of P are interspersed with those of Q a single pass is sufficient to find the roots of both polynomials.
To convert back to LPCs, we need to evaluate A ( z ) = 0.5 [ P ( z ) + Q ( z ) ] {\displaystyle A(z)=0.5[P(z)+Q(z)]} by "clocking" an impulse through it N times (order of the filter), yielding the original filter, A ( z ).
Line spectral pairs have several interesting and useful properties. When the roots of P ( z ) and Q ( z ) are interleaved, stability of the filter is ensured if and only if the roots are monotonically increasing. Moreover, the closer two roots are, the more resonant the filter is at the corresponding frequency. Because LSPs are not overly sensitive to quantization noise and stability is easily ensured, LSP are widely used for quantizing LPC filters. Line spectral frequencies can be interpolated.
Includes an overview in relation to LPC. | https://en.wikipedia.org/wiki/Line_spectral_pairs |
An evolutionary lineage is a temporal series of populations, organisms, cells, or genes connected by a continuous line of descent from ancestor to descendant. [ 1 ] [ 2 ] Lineages are subsets of the evolutionary tree of life . Lineages are often determined by the techniques of molecular systematics .
Lineages are typically visualized as subsets of a phylogenetic tree . A lineage is a single line of descent or linear chain within the tree, while a clade is a (usually branched) monophyletic group, containing a single ancestor and all its descendants. [ 3 ] Phylogenetic trees are typically created from DNA , RNA or protein sequence data. Apart from this, morphological differences and similarities have been, and still are used to create phylogenetic trees. Sequences from different individuals are collected and their similarity is quantified. Mathematical procedures are used to cluster individuals by similarity. [ 4 ]
Members of a species are considered to evolve as a single unit (or lineage) when they repeatedly share the same genes. The nodes would represent a split in lineage due to a breaking of genetic connections: when a single lineage is divided into two subsets, with the individuals not exchanging genes, they will accumulate differences in genes. If they do not fuse back again, it will create a new distinct descendant clade. [ 4 ]
Just as a map is a scaled approximation of true geography , a phylogenetic tree is an approximation of the true complete evolutionary relationships. For example, in a full tree of life, the entire clade of animals can be collapsed to a single branch of the tree. However, this is merely a limitation of rendering space. In theory, a true and complete tree for all living organisms or for any DNA sequence could be generated. [ 4 ]
Nevertheless, phylogenies can sometimes appear in a non-treelike form. Branches on the tree of life may grow together, a phenomenon called reticulation , which occurs due to different biological processes. Another process, introgression , occurs when hybrids between distinct lineages transfer novel genetic material through subsequent crossing. In other cases, hybrid speciation takes place when lineages hybridize to form a new, distinct lineage. Horizontal gene transfer, involving the introgression of very few genes, usually appears as a treelike population history with some genes having a discordant history. Thus, the tree-like representation would be proper as long as introgression and hybrid speciation are rare or limited to closely related tips (of lineages). In some cases, evolutionary relantionships should be depicted better in the form of a network. [ 4 ]
Most species of multicellular plants , animals and fungi reproduce sexually as do many protists . Therefore the evolution of the lineages of such species has likely been substantially influenced by sexual interactions. In the fossil record, lineages with the capability for sexual reproduction first appeared about 2.0 billion years ago in the Proterozoic Eon, [ 5 ] [ 6 ] although a later date, 1.2 billion years ago has also been proposed. [ 7 ] [ 8 ] Lineages of sexually reproducing eukaryotic organisms may have evolved from a single-celled common ancestor. [ 9 ] [ 10 ] [ 11 ] | https://en.wikipedia.org/wiki/Lineage_(evolution) |
The lineage markers are characteristic molecules for cell lineages , e.g. cell surface markers , mRNAs , or internal proteins . Certain antibodies can be used to detect or purify cells with these markers by binding to their surface antigens. A standard cocktail of antibodies can be designed to remove or purify mature hematopoietic cells or to detect Cluster of differentiation from a sample. Those antibodies are e.g. targeted to CD2 , CD3 , CD4 , CD5 , CD8 , NK1.1 , B220 , TER-119, and Gr-1 in mice and CD3 (T lymphocytes), CD14 (Monocytes), CD16 (NK cells, granulocytes), CD19 (B lymphocytes), CD20 (B lymphocytes), and CD56 (NK cells) in humans. [ 1 ]
Lineage markers include mitochondrial DNA ( mtDNA ) and Y-chromosome short tandem repeat ( Y-STR ) haplotypes that are transferred directly from generation to generation either from mother to child in the case of mtDNA, or from father to son in the case of the Y-chromosome . X-chromosome markers are another tool that can be used for genetic identity testing. Lineage markers can be helpful in missing persons investigations, disaster victim identification, forensic casework where other evidence is limited, and some complex kinship situations. X-chromosome analysis is especially helpful in assessing some kinship scenarios. [ 2 ]
This genetics article is a stub . You can help Wikipedia by expanding it .
This article about biological engineering is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Lineage_markers |
Linear Algebra and its Applications is a biweekly peer-reviewed mathematics journal published by Elsevier and covering matrix theory and finite-dimensional linear algebra .
The journal was established in January 1968 with A.J. Hoffman , A.S. Householder , A.M. Ostrowski , H. Schneider , and O. Taussky Todd as founding editors-in-chief . [ 1 ] The current editors-in-chief are Richard A. Brualdi ( University of Wisconsin at Madison ), Volker Mehrmann ( Technische Universität Berlin ), and Peter Semrl ( University of Ljubljana ).
The journal is abstracted and indexed in:
According to the Journal Citation Reports , the journal has a 2020 impact factor of 1.401. [ 2 ] | https://en.wikipedia.org/wiki/Linear_Algebra_and_Its_Applications |
The Linear Executable (LE) format is a file format for executables , object code , and DLLs designed for 32-bit protected mode operating systems. Originally used by the OS/2 operating system and adopted by various DOS extenders , it also served as the file format for Virtual Device Drivers (VxD) in early versions of Windows, including Windows 3.x and the Windows 9x series. [ 1 ] The malleability of LE files attracted interest in using them for steganography . [ 2 ]
The LE format was first introduced in the early 1990s during a period of transition from 16-bit to 32-bit computing . It was developed as an extension of the older New Executable (NE) format, which was used for 16-bit applications. Limitations in memory management and addressing led to the development of LE as a 32-bit replacement. LE expanded on NE's functionality by allowing the system to operate in protected mode.
An extended version of the format, called LX, was developed specifically for OS/2 Warp and supported further extensions over the LE format.
Files in the LE format begin with an MZ header (the standard DOS executable header) for backward compatibility with DOS systems. Within the MZ header, at offset 0x3C , there is a 32-bit value referred to as the e_lfanew field, which contains a pointer to the extended header (the LE header). The LE header starts with the ASCII characters LE (or LX in OS/2 Warp ). In Linear Executables, file offsets and structures are typically defined relative to the start of the LE header or as absolute offsets within the file. [ 3 ]
This software article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Linear_Executable |
Linear acetylenic carbon ( LAC ), also known as carbyne or a Linear Carbon Chain ( LCC ), is an allotrope of carbon that has the chemical structure (−C≡C−) n as a repeat unit , with alternating single and triple bonds . [ 2 ] [ 3 ] It would thus be the ultimate member of the polyyne family.
This polymeric carbyne is of considerable interest to nanotechnology as its Young's modulus is 32.7 TPa – forty times that of diamond ; [ 4 ] this extraordinary number is, however, based on a novel definition of cross-sectional area that does not correspond to the space occupied by the structure. Carbyne has also been identified in interstellar space; however, its existence in condensed phases has been contested recently, as such chains would crosslink exothermically (and perhaps explosively) if they approached each other. [ 5 ]
The first claims of detection of this allotrope were made in 1960 [ 5 ] [ 6 ] and repeated in 1978. [ 7 ] A 1982 re-examination of samples from several previous reports determined that the signals originally attributed to carbyne were in fact due to silicate impurities in the samples. [ 8 ] Absence of carbyne crystalline rendered the direct observation of a pure carbyne-assembled solid still a major challenge, [ clarification needed ] because carbyne crystals with well-defined structures and sufficient sizes are not available to date. This is indeed the major obstacle to general acceptance of carbyne as a true carbon allotrope. The mysterious carbyne still attracted scientists with its possible extraordinary properties. [ 9 ]
During the past thirty five years an increasing body of experimental and theoretical work has been published in the scientific literature dealing with the preparation of carbyne and the study of its structure, properties and potential applications. [ 10 ] [ 11 ] In 1968 a silver-white new mineral was discovered in graphitic gneisses of the Ries Crater (Nordlingen, Bavaria, Germany). [ 12 ] This material was found to consist entirely of carbon and its hexagonal cell dimensions matched those reported earlier for carbyne by Russian scientists. [ 13 ] It was concluded that this novel form of natural carbon, chaoite , was generated from graphite by the combined action of high temperature and high pressure, presumably caused by the impact of meteorite. Soon afterwards this “white” carbon was synthesized by sublimation of pyrolytic graphite in vacuum. [ 14 ]
In 1984, a group at Exxon reported the detection of clusters with even numbers of carbons, between 30 and 180, in carbon evaporation experiments, and attributed them to polyyne carbon. [ 15 ] However, these clusters later were identified as fullerenes . [ 5 ]
In 1991, carbyne was allegedly detected among various other allotropes of carbon in samples of amorphous carbon black vaporized and quenched by shock waves produced by shaped explosive charges . [ 16 ]
In 1995, the preparation of carbyne chains with over 300 carbons was reported. They were claimed to be reasonably stable, even against moisture and oxygen , as long as the terminal alkynes on the chain are capped with inert groups (such as tert -butyl or trifluoromethyl ) rather than hydrogen atoms. The study claimed that the data specifically indicated a carbyne-like structures rather than fullerene-like ones. [ 17 ] However, according to H. Kroto , the properties and synthetic methods used in those studies are consistent with generation of fullerenes . [ 5 ]
Another 1995 report claimed detection of carbyne chains of indeterminate length in a layer of carbonized material, about 180 nm thick, resulting from the reaction of solid polytetrafluoroethylene (PTFE, Teflon) immersed in alkali metal amalgam at ambient temperature (with no hydrogen-bearing species present). [ 18 ] The assumed reaction was
where M is either lithium , sodium , or potassium . The authors conjectured that nanocrystals of the metal fluoride between the chains prevented their polymerization.
In 1999, it was reported that copper(I) acetylide ( Cu + 2 C 2− 2 ), after partial oxidation by exposure to air or copper(II) ions followed by decomposition with hydrochloric acid , leaves a "carbonaceous" residue with the spectral signature of (−C≡C−) n chains with n =2–6. The proposed mechanism involves oxidative polymerization of the acetylide anions C 2− 2 into carbyne-type anions C(≡C−C≡) n C 2− or cumulene-type anions C(=C=C=) m C 4− . [ 19 ] Also, thermal decomposition of copper acetylide in vacuum yielded a fluffy deposit of fine carbon powder on the walls of the flask, which, on the basis of spectral data, was claimed to be carbyne rather than graphite. [ 19 ] Finally, the oxidation of copper acetylide in ammoniacal solution ( Glaser's reaction ) produces a carbonaceous residue that was claimed to consist of "polyacetylide" anions capped with residual copper(I) ions,
On the basis of the residual amount of copper, the mean number of units n was estimated to be around 230. [ 20 ]
In 2004, an analysis of a synthesized linear carbon allotrope found it to have a cumulene electronic structure—sequential double bonds along an sp -hybridized carbon chain—rather than the alternating triple–single pattern of linear carbyne. [ 21 ]
In 2016, the synthesis of linear chains of up to 6,000 sp -hybridized carbon atoms was reported. The chains were grown inside double-walled carbon nanotubes , and are highly stable protected by their hosts. [ 22 ] [ 23 ]
While the existence of "carbyne" chains in pure neutral carbon material is still disputed, short (−C≡C−) n chains are well established as substructures of larger molecules ( polyynes ). [ 24 ] As of 2010, the longest such chain in a stable molecule had 22 acetylenic units (44 atoms), stabilized by rather bulky end groups. [ 25 ]
The carbon atoms in this form are each linear in geometry with sp orbital hybridisation . The estimated length of the bonds is 120.7 pm (triple) and 137.9 pm (single). [ 18 ]
Other possible configurations for a chain of carbon atoms include polycumulene (polyethylene-diylidene) chains with double bonds only ( 128.2 pm ). This chain is expected to have slightly higher energy, with a Peierls gap of 2–5 eV . For short C n molecules, however, the polycumulene structure seems favored. When n is even, two ground configurations, very close in energy, may coexist: one linear, and one cyclic (rhombic). [ 18 ]
The limits of flexibility of the carbyne chain are illustrated by a synthetic polyyne with a backbone of 8 acetylenic units, whose chain was found to be bent by 25° or more (about 3° at each carbon) in the solid state, to accommodate the bulky end groups of adjacent molecules. [ 26 ]
The highly symmetric carbyne chain is expected to have only one Raman -active mode with Σ g symmetry, due to stretching of bonds in each single-double pair [ clarification needed ] , with frequency typically between 1800 and 2300 cm −1 , [ 18 ] and affected by their environments. [ 27 ]
Carbyne chains have been claimed to be the strongest material known per density. Calculations indicate that carbyne's specific tensile strength (strength divided by density) of (6.0–7.5) × 10 7 ( N⋅m )/ kg beats graphene ( (4.7–5.5) × 10 7 (N⋅m)/kg ), carbon nanotubes ( (4.3–5.0) × 10 7 (N⋅m)/kg ), and diamond ( (2.5–6.5) × 10 7 (N⋅m)/kg ). [ 28 ] [ 29 ] [ 30 ] Its specific modulus ( Young's Modulus divided by density) of around 10 9 (N⋅m)/kg is also double that of graphene, which is around 4.5 × 10 8 (N⋅m)/kg . [ 28 ] [ 30 ]
Stretching carbyne 10% alters its electronic band gap from 3.2–4.4 eV . [ 31 ] Outfitted with molecular handles at chain's ends, it can also be twisted to alter its band gap. With a 90° end-to-end twist, carbyne turns into a magnetic semiconductor. [ 29 ]
In 2017, the band gaps of confined linear carbon chains (LCC) inside double-walled carbon nanotubes with lengths ranging from 36 up to 6000 carbon atoms were determined for the first time ranging from 2.253–1.848 eV , following a linear relation with Raman frequency. This lower bound is the smallest band gap of linear carbon chains observed so far. In 2020, the strength (Young's modulus) of linear carbon chains (LCC) was experimentally calculated to be about 20 TPa which is much higher than that of other carbon materials like graphene and carbon nanotubes. [ 32 ] The comparison with experimental data obtained for short chains in gas phase or in solution demonstrates the effect of the DWCNT encapsulation, leading to an essential downshift of the band gap. [ 33 ]
The LCCs inside double-walled carbon nanotubes lead to an increase of the photoluminescence (PL) signal of the inner tubes up to a factor of 6 for tubes with (8,3) chirality. This behavior can be attributed to a local charge transfer from the inner tubes to the carbon chains, counterbalancing quenching mechanisms induced by the outer tubes. [ 34 ]
Carbyne chains can take on side molecules that may make the chains suitable for energy [ 29 ] and hydrogen [ 35 ] storage.
With a differential Raman scattering cross section of 10 −22 cm 2 sr −1 per atom, carbyne chains confined inside carbon nanotubes are the strongest Raman scatterer ever reported, [ 36 ] exceeding any other know material by two orders of magnitude. | https://en.wikipedia.org/wiki/Linear_acetylenic_carbon |
Linear algebra is the branch of mathematics concerning linear equations such as
linear maps such as
and their representations in vector spaces and through matrices . [ 1 ] [ 2 ] [ 3 ]
Linear algebra is central to almost all areas of mathematics. For instance, linear algebra is fundamental in modern presentations of geometry , including for defining basic objects such as lines , planes and rotations . Also, functional analysis , a branch of mathematical analysis , may be viewed as the application of linear algebra to function spaces .
Linear algebra is also used in most sciences and fields of engineering because it allows modeling many natural phenomena, and computing efficiently with such models. For nonlinear systems , which cannot be modeled with linear algebra, it is often used for dealing with first-order approximations , using the fact that the differential of a multivariate function at a point is the linear map that best approximates the function near that point.
The procedure (using counting rods) for solving simultaneous linear equations now called Gaussian elimination appears in the ancient Chinese mathematical text Chapter Eight: Rectangular Arrays of The Nine Chapters on the Mathematical Art . Its use is illustrated in eighteen problems, with two to five equations. [ 4 ]
Systems of linear equations arose in Europe with the introduction in 1637 by René Descartes of coordinates in geometry . In fact, in this new geometry, now called Cartesian geometry , lines and planes are represented by linear equations, and computing their intersections amounts to solving systems of linear equations.
The first systematic methods for solving linear systems used determinants and were first considered by Leibniz in 1693. In 1750, Gabriel Cramer used them for giving explicit solutions of linear systems, now called Cramer's rule . Later, Gauss further described the method of elimination, which was initially listed as an advancement in geodesy . [ 5 ]
In 1844 Hermann Grassmann published his "Theory of Extension" which included foundational new topics of what is today called linear algebra. In 1848, James Joseph Sylvester introduced the term matrix , which is Latin for womb .
Linear algebra grew with ideas noted in the complex plane . For instance, two numbers w and z in C {\displaystyle \mathbb {C} } have a difference w – z , and the line segments wz and 0( w − z ) are of the same length and direction. The segments are equipollent . The four-dimensional system H {\displaystyle \mathbb {H} } of quaternions was discovered by W.R. Hamilton in 1843. [ 6 ] The term vector was introduced as v = x i + y j + z k representing a point in space. The quaternion difference p – q also produces a segment equipollent to pq . Other hypercomplex number systems also used the idea of a linear space with a basis .
Arthur Cayley introduced matrix multiplication and the inverse matrix in 1856, making possible the general linear group . The mechanism of group representation became available for describing complex and hypercomplex numbers. Crucially, Cayley used a single letter to denote a matrix, thus treating a matrix as an aggregate object. He also realized the connection between matrices and determinants and wrote "There would be many things to say about this theory of matrices which should, it seems to me, precede the theory of determinants". [ 5 ]
Benjamin Peirce published his Linear Associative Algebra (1872), and his son Charles Sanders Peirce extended the work later. [ 7 ]
The telegraph required an explanatory system, and the 1873 publication by James Clerk Maxwell of A Treatise on Electricity and Magnetism instituted a field theory of forces and required differential geometry for expression. Linear algebra is flat differential geometry and serves in tangent spaces to manifolds . Electromagnetic symmetries of spacetime are expressed by the Lorentz transformations , and much of the history of linear algebra is the history of Lorentz transformations .
The first modern and more precise definition of a vector space was introduced by Peano in 1888; [ 5 ] by 1900, a theory of linear transformations of finite-dimensional vector spaces had emerged. Linear algebra took its modern form in the first half of the twentieth century when many ideas and methods of previous centuries were generalized as abstract algebra . The development of computers led to increased research in efficient algorithms for Gaussian elimination and matrix decompositions, and linear algebra became an essential tool for modeling and simulations. [ 5 ]
Until the 19th century, linear algebra was introduced through systems of linear equations and matrices . In modern mathematics, the presentation through vector spaces is generally preferred, since it is more synthetic , more general (not limited to the finite-dimensional case), and conceptually simpler, although more abstract.
A vector space over a field F (often the field of the real numbers or of the complex numbers ) is a set V equipped with two binary operations . Elements of V are called vectors , and elements of F are called scalars . The first operation, vector addition , takes any two vectors v and w and outputs a third vector v + w . The second operation, scalar multiplication , takes any scalar a and any vector v and outputs a new vector a v . The axioms that addition and scalar multiplication must satisfy are the following. (In the list below, u , v and w are arbitrary elements of V , and a and b are arbitrary scalars in the field F .) [ 8 ]
The first four axioms mean that V is an abelian group under addition.
The elements of a specific vector space may have various natures; for example, they could be tuples , sequences , functions , polynomials , or a matrices . Linear algebra is concerned with the properties of such objects that are common to all vector spaces.
Linear maps are mappings between vector spaces that preserve the vector-space structure. Given two vector spaces V and W over a field F , a linear map (also called, in some contexts, linear transformation or linear mapping) is a map
that is compatible with addition and scalar multiplication, that is
for any vectors u , v in V and scalar a in F .
An equivalent condition is that for any vectors u , v in V and scalars a , b in F , one has
When V = W are the same vector space, a linear map T : V → V is also known as a linear operator on V .
A bijective linear map between two vector spaces (that is, every vector from the second space is associated with exactly one in the first) is an isomorphism . Because an isomorphism preserves linear structure, two isomorphic vector spaces are "essentially the same" from the linear algebra point of view, in the sense that they cannot be distinguished by using vector space properties. An essential question in linear algebra is testing whether a linear map is an isomorphism or not, and, if it is not an isomorphism, finding its range (or image) and the set of elements that are mapped to the zero vector, called the kernel of the map. All these questions can be solved by using Gaussian elimination or some variant of this algorithm .
The study of those subsets of vector spaces that are in themselves vector spaces under the induced operations is fundamental, similarly as for many mathematical structures. These subsets are called linear subspaces . More precisely, a linear subspace of a vector space V over a field F is a subset W of V such that u + v and a u are in W , for every u , v in W , and every a in F . (These conditions suffice for implying that W is a vector space.)
For example, given a linear map T : V → W , the image T ( V ) of V , and the inverse image T −1 ( 0 ) of 0 (called kernel or null space), are linear subspaces of W and V , respectively.
Another important way of forming a subspace is to consider linear combinations of a set S of vectors: the set of all sums
where v 1 , v 2 , ..., v k are in S , and a 1 , a 2 , ..., a k are in F form a linear subspace called the span of S . The span of S is also the intersection of all linear subspaces containing S . In other words, it is the smallest (for the inclusion relation) linear subspace containing S .
A set of vectors is linearly independent if none is in the span of the others. Equivalently, a set S of vectors is linearly independent if the only way to express the zero vector as a linear combination of elements of S is to take zero for every coefficient a i .
A set of vectors that spans a vector space is called a spanning set or generating set . If a spanning set S is linearly dependent (that is not linearly independent), then some element w of S is in the span of the other elements of S , and the span would remain the same if one were to remove w from S . One may continue to remove elements of S until getting a linearly independent spanning set . Such a linearly independent set that spans a vector space V is called a basis of V . The importance of bases lies in the fact that they are simultaneously minimal-generating sets and maximal independent sets. More precisely, if S is a linearly independent set, and T is a spanning set such that S ⊆ T , then there is a basis B such that S ⊆ B ⊆ T .
Any two bases of a vector space V have the same cardinality , which is called the dimension of V ; this is the dimension theorem for vector spaces . Moreover, two vector spaces over the same field F are isomorphic if and only if they have the same dimension. [ 9 ]
If any basis of V (and therefore every basis) has a finite number of elements, V is a finite-dimensional vector space . If U is a subspace of V , then dim U ≤ dim V . In the case where V is finite-dimensional, the equality of the dimensions implies U = V .
If U 1 and U 2 are subspaces of V , then
where U 1 + U 2 denotes the span of U 1 ∪ U 2 . [ 10 ]
Matrices allow explicit manipulation of finite-dimensional vector spaces and linear maps . Their theory is thus an essential part of linear algebra.
Let V be a finite-dimensional vector space over a field F , and ( v 1 , v 2 , ..., v m ) be a basis of V (thus m is the dimension of V ). By definition of a basis, the map
is a bijection from F m , the set of the sequences of m elements of F , onto V . This is an isomorphism of vector spaces, if F m is equipped with its standard structure of vector space, where vector addition and scalar multiplication are done component by component.
This isomorphism allows representing a vector by its inverse image under this isomorphism, that is by the coordinate vector ( a 1 , ..., a m ) or by the column matrix
If W is another finite dimensional vector space (possibly the same), with a basis ( w 1 , ..., w n ) , a linear map f from W to V is well defined by its values on the basis elements, that is ( f ( w 1 ), ..., f ( w n )) . Thus, f is well represented by the list of the corresponding column matrices. That is, if
for j = 1, ..., n , then f is represented by the matrix
with m rows and n columns.
Matrix multiplication is defined in such a way that the product of two matrices is the matrix of the composition of the corresponding linear maps, and the product of a matrix and a column matrix is the column matrix representing the result of applying the represented linear map to the represented vector. It follows that the theory of finite-dimensional vector spaces and the theory of matrices are two different languages for expressing the same concepts.
Two matrices that encode the same linear transformation in different bases are called similar . It can be proved that two matrices are similar if and only if one can transform one into the other by elementary row and column operations . For a matrix representing a linear map from W to V , the row operations correspond to change of bases in V and the column operations correspond to change of bases in W . Every matrix is similar to an identity matrix possibly bordered by zero rows and zero columns. In terms of vector spaces, this means that, for any linear map from W to V , there are bases such that a part of the basis of W is mapped bijectively on a part of the basis of V , and that the remaining basis elements of W , if any, are mapped to zero. Gaussian elimination is the basic algorithm for finding these elementary operations, and proving these results.
A finite set of linear equations in a finite set of variables, for example, x 1 , x 2 , ..., x n , or x , y , ..., z is called a system of linear equations or a linear system . [ 11 ] [ 12 ] [ 13 ] [ 14 ] [ 15 ]
Systems of linear equations form a fundamental part of linear algebra. Historically, linear algebra and matrix theory have been developed for solving such systems. In the modern presentation of linear algebra through vector spaces and matrices, many problems may be interpreted in terms of linear systems.
For example, let
be a linear system.
To such a system, one may associate its matrix
and its right member vector
Let T be the linear transformation associated with the matrix M . A solution of the system ( S ) is a vector
such that
that is an element of the preimage of v by T .
Let ( S′ ) be the associated homogeneous system , where the right-hand sides of the equations are put to zero:
The solutions of ( S′ ) are exactly the elements of the kernel of T or, equivalently, M .
The Gaussian-elimination consists of performing elementary row operations on the augmented matrix
for putting it in reduced row echelon form . These row operations do not change the set of solutions of the system of equations. In the example, the reduced echelon form is
showing that the system ( S ) has the unique solution
It follows from this matrix interpretation of linear systems that the same methods can be applied for solving linear systems and for many operations on matrices and linear transformations, which include the computation of the ranks , kernels , matrix inverses .
A linear endomorphism is a linear map that maps a vector space V to itself.
If V has a basis of n elements, such an endomorphism is represented by a square matrix of size n .
Concerning general linear maps, linear endomorphisms, and square matrices have some specific properties that make their study an important part of linear algebra, which is used in many parts of mathematics, including geometric transformations , coordinate changes , quadratic forms , and many other parts of mathematics.
The determinant of a square matrix A is defined to be [ 16 ]
where S n is the group of all permutations of n elements, σ is a permutation, and (−1) σ the parity of the permutation. A matrix is invertible if and only if the determinant is invertible (i.e., nonzero if the scalars belong to a field).
Cramer's rule is a closed-form expression , in terms of determinants, of the solution of a system of n linear equations in n unknowns . Cramer's rule is useful for reasoning about the solution, but, except for n = 2 or 3 , it is rarely used for computing a solution, since Gaussian elimination is a faster algorithm.
The determinant of an endomorphism is the determinant of the matrix representing the endomorphism in terms of some ordered basis. This definition makes sense since this determinant is independent of the choice of the basis.
If f is a linear endomorphism of a vector space V over a field F , an eigenvector of f is a nonzero vector v of V such that f ( v ) = av for some scalar a in F . This scalar a is an eigenvalue of f .
If the dimension of V is finite, and a basis has been chosen, f and v may be represented, respectively, by a square matrix M and a column matrix z ; the equation defining eigenvectors and eigenvalues becomes
Using the identity matrix I , whose entries are all zero, except those of the main diagonal, which are equal to one, this may be rewritten
As z is supposed to be nonzero, this means that M – aI is a singular matrix , and thus that its determinant det ( M − aI ) equals zero. The eigenvalues are thus the roots of the polynomial
If V is of dimension n , this is a monic polynomial of degree n , called the characteristic polynomial of the matrix (or of the endomorphism), and there are, at most, n eigenvalues.
If a basis exists that consists only of eigenvectors, the matrix of f on this basis has a very simple structure: it is a diagonal matrix such that the entries on the main diagonal are eigenvalues, and the other entries are zero. In this case, the endomorphism and the matrix are said to be diagonalizable . More generally, an endomorphism and a matrix are also said diagonalizable, if they become diagonalizable after extending the field of scalars. In this extended sense, if the characteristic polynomial is square-free , then the matrix is diagonalizable.
A symmetric matrix is always diagonalizable. There are non-diagonalizable matrices, the simplest being
(it cannot be diagonalizable since its square is the zero matrix , and the square of a nonzero diagonal matrix is never zero).
When an endomorphism is not diagonalizable, there are bases on which it has a simple form, although not as simple as the diagonal form. The Frobenius normal form does not need to extend the field of scalars and makes the characteristic polynomial immediately readable on the matrix. The Jordan normal form requires to extension of the field of scalar for containing all eigenvalues and differs from the diagonal form only by some entries that are just above the main diagonal and are equal to 1.
A linear form is a linear map from a vector space V over a field F to the field of scalars F , viewed as a vector space over itself. Equipped by pointwise addition and multiplication by a scalar, the linear forms form a vector space, called the dual space of V , and usually denoted V* [ 17 ] or V ′ . [ 18 ] [ 19 ]
If v 1 , ..., v n is a basis of V (this implies that V is finite-dimensional), then one can define, for i = 1, ..., n , a linear map v i * such that v i *( v i ) = 1 and v i *( v j ) = 0 if j ≠ i . These linear maps form a basis of V * , called the dual basis of v 1 , ..., v n . (If V is not finite-dimensional, the v i * may be defined similarly; they are linearly independent, but do not form a basis.)
For v in V , the map
is a linear form on V* . This defines the canonical linear map from V into ( V *)* , the dual of V* , called the double dual or bidual of V . This canonical map is an isomorphism if V is finite-dimensional, and this allows identifying V with its bidual. (In the infinite-dimensional case, the canonical map is injective, but not surjective.)
There is thus a complete symmetry between a finite-dimensional vector space and its dual. This motivates the frequent use, in this context, of the bra–ket notation
for denoting f ( x ) .
Let
be a linear map. For every linear form h on W , the composite function h ∘ f is a linear form on V . This defines a linear map
between the dual spaces, which is called the dual or the transpose of f .
If V and W are finite-dimensional, and M is the matrix of f in terms of some ordered bases, then the matrix of f* over the dual bases is the transpose M T of M , obtained by exchanging rows and columns.
If elements of vector spaces and their duals are represented by column vectors, this duality may be expressed in bra–ket notation by
To highlight this symmetry, the two members of this equality are sometimes written
Besides these basic concepts, linear algebra also studies vector spaces with additional structure, such as an inner product . The inner product is an example of a bilinear form , and it gives the vector space a geometric structure by allowing for the definition of length and angles. Formally, an inner product is a map.
that satisfies the following three axioms for all vectors u , v , w in V and all scalars a in F : [ 20 ] [ 21 ]
We can define the length of a vector v in V by
and we can prove the Cauchy–Schwarz inequality :
In particular, the quantity
and so we can call this quantity the cosine of the angle between the two vectors.
Two vectors are orthogonal if ⟨ u , v ⟩ = 0 . An orthonormal basis is a basis where all basis vectors have length 1 and are orthogonal to each other. Given any finite-dimensional vector space, an orthonormal basis could be found by the Gram–Schmidt procedure. Orthonormal bases are particularly easy to deal with, since if v = a 1 v 1 + ⋯ + a n v n , then
The inner product facilitates the construction of many useful concepts. For instance, given a transform T , we can define its Hermitian conjugate T* as the linear transform satisfying
If T satisfies TT* = T*T , we call T normal . It turns out that normal matrices are precisely the matrices that have an orthonormal system of eigenvectors that span V .
There is a strong relationship between linear algebra and geometry , which started with the introduction by René Descartes , in 1637, of Cartesian coordinates . In this new (at that time) geometry, now called Cartesian geometry , points are represented by Cartesian coordinates , which are sequences of three real numbers (in the case of the usual three-dimensional space ). The basic objects of geometry, which are lines and planes are represented by linear equations. Thus, computing intersections of lines and planes amounts to solving systems of linear equations. This was one of the main motivations for developing linear algebra.
Most geometric transformation , such as translations , rotations , reflections , rigid motions , isometries , and projections transform lines into lines. It follows that they can be defined, specified, and studied in terms of linear maps. This is also the case of homographies and Möbius transformations when considered as transformations of a projective space .
Until the end of the 19th century, geometric spaces were defined by axioms relating points, lines, and planes ( synthetic geometry ). Around this date, it appeared that one may also define geometric spaces by constructions involving vector spaces (see, for example, Projective space and Affine space ). It has been shown that the two approaches are essentially equivalent. [ 22 ] In classical geometry, the involved vector spaces are vector spaces over the reals, but the constructions may be extended to vector spaces over any field, allowing considering geometry over arbitrary fields, including finite fields .
Presently, most textbooks introduce geometric spaces from linear algebra, and geometry is often presented, at the elementary level, as a subfield of linear algebra.
Linear algebra is used in almost all areas of mathematics, thus making it relevant in almost all scientific domains that use mathematics. These applications may be divided into several wide categories.
Functional analysis studies function spaces . These are vector spaces with additional structure, such as Hilbert spaces . Linear algebra is thus a fundamental part of functional analysis and its applications, which include, in particular, quantum mechanics ( wave functions ) and Fourier analysis ( orthogonal basis ).
Nearly all scientific computations involve linear algebra. Consequently, linear algebra algorithms have been highly optimized. BLAS and LAPACK are the best known implementations. For improving efficiency, some of them configure the algorithms automatically, at run time, to adapt them to the specificities of the computer ( cache size, number of available cores , ...).
Since the 1960s there have been processors with specialized instructions [ 23 ] for optimizing the operations of linear algebra, optional array processors [ 24 ] under the control of a conventional processor, supercomputers [ 25 ] [ 26 ] [ 27 ] designed for array processing and conventional processors augmented [ 28 ] with vector registers.
Some contemporary processors , typically graphics processing units (GPU), are designed with a matrix structure, for optimizing the operations of linear algebra. [ 29 ]
The modeling of ambient space is based on geometry . Sciences concerned with this space use geometry widely. This is the case with mechanics and robotics , for describing rigid body dynamics ; geodesy for describing Earth shape ; perspectivity , computer vision , and computer graphics , for describing the relationship between a scene and its plane representation; and many other scientific domains.
In all these applications, synthetic geometry is often used for general descriptions and a qualitative approach, but for the study of explicit situations, one must compute with coordinates . This requires the heavy use of linear algebra.
Most physical phenomena are modeled by partial differential equations . To solve them, one usually decomposes the space in which the solutions are searched into small, mutually interacting cells . For linear systems this interaction involves linear functions . For nonlinear systems , this interaction is often approximated by linear functions. [ b ] This is called a linear model or first-order approximation. Linear models are frequently used for complex nonlinear real-world systems because they make parametrization more manageable. [ 30 ] In both cases, very large matrices are generally involved. Weather forecasting (or more specifically, parametrization for atmospheric modeling ) is a typical example of a real-world application, where the whole Earth atmosphere is divided into cells of, say, 100 km of width and 100 km of height.
[ 31 ] [ 32 ] [ 33 ]
Linear algebra, a branch of mathematics dealing with vector spaces and linear mappings between these spaces, plays a critical role in various engineering disciplines, including fluid mechanics , fluid dynamics , and thermal energy systems. Its application in these fields is multifaceted and indispensable for solving complex problems.
In fluid mechanics , linear algebra is integral to understanding and solving problems related to the behavior of fluids. It assists in the modeling and simulation of fluid flow, providing essential tools for the analysis of fluid dynamics problems. For instance, linear algebraic techniques are used to solve systems of differential equations that describe fluid motion. These equations, often complex and non-linear , can be linearized using linear algebra methods, allowing for simpler solutions and analyses.
In the field of fluid dynamics, linear algebra finds its application in computational fluid dynamics (CFD), a branch that uses numerical analysis and data structures to solve and analyze problems involving fluid flows. CFD relies heavily on linear algebra for the computation of fluid flow and heat transfer in various applications. For example, the Navier–Stokes equations , fundamental in fluid dynamics , are often solved using techniques derived from linear algebra. This includes the use of matrices and vectors to represent and manipulate fluid flow fields.
Furthermore, linear algebra plays a crucial role in thermal energy systems, particularly in power systems analysis. It is used to model and optimize the generation, transmission , and distribution of electric power. Linear algebraic concepts such as matrix operations and eigenvalue problems are employed to enhance the efficiency, reliability, and economic performance of power systems . The application of linear algebra in this context is vital for the design and operation of modern power systems , including renewable energy sources and smart grids .
Overall, the application of linear algebra in fluid mechanics , fluid dynamics , and thermal energy systems is an example of the profound interconnection between mathematics and engineering . It provides engineers with the necessary tools to model, analyze, and solve complex problems in these domains, leading to advancements in technology and industry.
This section presents several related topics that do not appear generally in elementary textbooks on linear algebra but are commonly considered, in advanced mathematics, as parts of linear algebra.
The existence of multiplicative inverses in fields is not involved in the axioms defining a vector space. One may thus replace the field of scalars by a ring R , and this gives the structure called a module over R , or R -module.
The concepts of linear independence, span, basis, and linear maps (also called module homomorphisms ) are defined for modules exactly as for vector spaces, with the essential difference that, if R is not a field, there are modules that do not have any basis. The modules that have a basis are the free modules , and those that are spanned by a finite set are the finitely generated modules . Module homomorphisms between finitely generated free modules may be represented by matrices. The theory of matrices over a ring is similar to that of matrices over a field, except that determinants exist only if the ring is commutative , and that a square matrix over a commutative ring is invertible only if its determinant has a multiplicative inverse in the ring.
Vector spaces are completely characterized by their dimension (up to an isomorphism). In general, there is not such a complete classification for modules, even if one restricts oneself to finitely generated modules. However, every module is a cokernel of a homomorphism of free modules.
Modules over the integers can be identified with abelian groups , since the multiplication by an integer may be identified as a repeated addition. Most of the theory of abelian groups may be extended to modules over a principal ideal domain . In particular, over a principal ideal domain, every submodule of a free module is free, and the fundamental theorem of finitely generated abelian groups may be extended straightforwardly to finitely generated modules over a principal ring.
There are many rings for which there are algorithms for solving linear equations and systems of linear equations. However, these algorithms have generally a computational complexity that is much higher than similar algorithms over a field. For more details, see Linear equation over a ring .
In multilinear algebra , one considers multivariable linear transformations, that is, mappings that are linear in each of several different variables. This line of inquiry naturally leads to the idea of the dual space , the vector space V* consisting of linear maps f : V → F where F is the field of scalars. Multilinear maps T : V n → F can be described via tensor products of elements of V* .
If, in addition to vector addition and scalar multiplication, there is a bilinear vector product V × V → V , the vector space is called an algebra ; for instance, associative algebras are algebras with an associate vector product (like the algebra of square matrices, or the algebra of polynomials).
Vector spaces that are not finite-dimensional often require additional structure to be tractable. A normed vector space is a vector space along with a function called a norm , which measures the "size" of elements. The norm induces a metric , which measures the distance between elements, and induces a topology , which allows for a definition of continuous maps. The metric also allows for a definition of limits and completeness – a normed vector space that is complete is known as a Banach space . A complete metric space along with the additional structure of an inner product (a conjugate symmetric sesquilinear form ) is known as a Hilbert space , which is in some sense a particularly well-behaved Banach space. Functional analysis applies the methods of linear algebra alongside those of mathematical analysis to study various function spaces; the central objects of study in functional analysis are L p spaces , which are Banach spaces, and especially the L 2 space of square-integrable functions, which is the only Hilbert space among them. Functional analysis is of particular importance to quantum mechanics, the theory of partial differential equations, digital signal processing, and electrical engineering. It also provides the foundation and theoretical framework that underlies the Fourier transform and related methods. | https://en.wikipedia.org/wiki/Linear_algebra |
A linear biochemical pathway is a chain of enzyme-catalyzed reaction steps where the product of one reaction becomes the substrate for the next reaction . The molecules progress through the pathway sequentially from the starting substrate to the final product. Each step in the pathway is usually facilitated by a different specific enzyme that catalyzes the chemical transformation . An example includes DNA replication , which connects the starting substrate and the end product in a straightforward sequence.
Biological cells consume nutrients to sustain life. These nutrients are broken down to smaller molecules . Some of the molecules are used in the cells for various biological functions, and others are reassembled into more complex structures required for life. The breakdown and reassembly of nutrients is called metabolism . An individual cell contains thousands of different kinds of small molecules, such as sugars , lipids , and amino acids . The interconversion of these molecules is carried out by catalysts called enzymes . For example, the most widely studied bacterium, E. coli strain K-12, is able to produce about 2,338 metabolic enzymes. [ 1 ] These enzymes collectively form a complex web of reactions comprising pathways by which substrates (including nutients and intermediates) are converted to products (other intermediates and end-products).
The figure below shows a four step pathway, with intermediates, S 1 , S 2 , {\displaystyle S_{1},S_{2},} and S 3 {\displaystyle S_{3}} . To sustain a steady-state, the boundary species X o {\displaystyle X_{o}} and X 1 {\displaystyle X_{1}} are fixed. Each step is catalyzed by an enzyme, e i {\displaystyle e_{i}} .
Linear pathways follow a step-by-step sequence, where each enzymatic reaction results in the transformation of a substrate into an intermediate product. This intermediate is processed by subsequent enzymes until the final product is synthesized.
A linear pathway can be studied in various ways. Multiple computer simulations can be run to try to understand the pathway's behavior. Another way to understand the properties of a linear pathway is to take a more analytical approach. Analytical solutions can be derived for the steady-state if simple mass-action kinetics are assumed. [ 2 ] [ 3 ] [ 4 ] Analytical solutions for the steady-state when assuming Michaelis-Menten kinetics can be obtained [ 5 ] [ 6 ] but are quite often avoided. Instead, such models are linearized. The three approaches that are usually used are therefore:
It is possible to build a computer simulation of a linear biochemical pathway. This can be done by building a simple model that describes each intermediate through a differential equation . The differential equations can be written by invoking mass conservation . For example, for the linear pathway:
X o ⟶ v 1 S 1 ⟶ v 2 S 2 ⟶ v 3 S 3 ⟶ v 4 X 1 {\displaystyle X_{o}{\stackrel {v_{1}}{\longrightarrow }}S_{1}{\stackrel {v_{2}}{\longrightarrow }}S_{2}{\stackrel {v_{3}}{\longrightarrow }}S_{3}{\stackrel {v_{4}}{\longrightarrow }}X_{1}}
where X o {\displaystyle X_{o}} and X 1 {\displaystyle X_{1}} are fixed boundary species, the non-fixed intermediate S 1 {\displaystyle S_{1}} can be described using the differential equation:
d S 1 d t = v 1 − v 2 {\displaystyle {\frac {dS_{1}}{dt}}=v_{1}-v_{2}}
The rate of change of the non-fixed intermediates S 2 {\displaystyle S_{2}} and S 3 {\displaystyle S_{3}} can be written in the same way:
d S 2 d t = v 2 − v 3 {\displaystyle {\frac {dS_{2}}{dt}}=v_{2}-v_{3}}
d S 3 d t = v 3 − v 4 {\displaystyle {\frac {dS_{3}}{dt}}=v_{3}-v_{4}}
To run a simulation the rates, v i {\displaystyle v_{i}} need to be defined. If mass-action kinetics are assumed for the reaction rates, then the differential equation can be written as:
d S 1 d t = k 1 X o − k 2 S 1 d S 2 d t = k 2 S 1 − k 3 S 2 d S 3 d t = k 3 S 2 − k 4 S 3 {\displaystyle {\begin{array}{lcl}{\dfrac {dS_{1}}{dt}}&=&k_{1}X_{o}-k_{2}S_{1}\\[4pt]{\dfrac {dS_{2}}{dt}}&=&k_{2}S_{1}-k_{3}S_{2}\\[4pt]{\dfrac {dS_{3}}{dt}}&=&k_{3}S_{2}-k_{4}S_{3}\end{array}}}
If values are assigned to the rate constants, k i {\displaystyle k_{i}} , and the fixed species X o {\displaystyle X_{o}} and X 1 {\displaystyle X_{1}} the differential equations can be solved.
Computer simulations can only yield so much insight, as one would be required to run simulations on a wide range of parameter values, which can be unwieldy. A generally more powerful way to understand the properties of a model is to solve the differential equations analytically.
Analytical solutions are possible if simple mass-action kinetics on each reaction step are assumed:
where k i {\displaystyle k_{i}} and k − 1 {\displaystyle k_{-1}} are the forward and reverse rate-constants, respectively. s i − 1 {\displaystyle s_{i-1}} is the substrate and s i {\displaystyle s_{i}} the product. If the equilibrium constant for this reaction is:
The mass-action kinetic equation can be modified to be:
Given the reaction rates, the differential equations describing the rates of change of the species can be described. For example, the rate of change of s 1 {\displaystyle s_{1}} will equal:
By setting the differential equations to zero, the steady-state concentration for the species can be derived. From here, the pathway flux equation can be determined. For the three-step pathway, the steady-state concentrations of s 1 {\displaystyle s_{1}} and s 2 {\displaystyle s_{2}} are given by:
s 1 = q 1 q 3 k 2 k 3 x 1 + k 1 k 2 q 3 x o + k 1 k 3 q 2 q 3 x o k 1 k 2 + k 1 k 3 q 2 + k 2 k 3 q 1 q 2 s 2 = q 2 q 3 k 1 k 3 x 1 + k 2 k 3 q 1 x 1 + k 1 k 2 q 1 q 3 x o k 1 k 2 + k 1 k 3 q 2 + k 2 k 3 q 1 q 2 {\displaystyle {\begin{aligned}&s_{1}={\frac {q_{1}}{q_{3}}}{\frac {k_{2}k_{3}x_{1}+k_{1}k_{2}q_{3}x_{o}+k_{1}k_{3}q_{2}q_{3}x_{o}}{k_{1}k_{2}+k_{1}k_{3}q_{2}+k_{2}k_{3}q_{1}q_{2}}}\\[6pt]&s_{2}={\frac {q_{2}}{q_{3}}}{\frac {k_{1}k_{3}x_{1}+k_{2}k_{3}q_{1}x_{1}+k_{1}k_{2}q_{1}q_{3}x_{o}}{k_{1}k_{2}+k_{1}k_{3}q_{2}+k_{2}k_{3}q_{1}q_{2}}}\end{aligned}}}
Inserting either s 1 {\displaystyle s_{1}} or s 2 {\displaystyle s_{2}} into one of the rate laws will give the steady-state pathway flux, J {\displaystyle J} :
A pattern can be seen in this equation such that, in general, for a linear pathway of n {\displaystyle n} steps, the steady-state pathway flux is given by:
J = x o ∏ i = 1 n q i − x 1 ∑ i = 1 n 1 k i ( ∏ j = i n q j ) {\displaystyle J={\frac {x_{o}\prod _{i=1}^{n}q_{i}-x_{1}}{\sum _{i=1}^{n}{\frac {1}{k_{i}}}\left(\prod _{j=i}^{n}q_{j}\right)}}}
Note that the pathway flux is a function of all the kinetic and thermodynamic parameters. This means there is no single parameter that determines the flux completely. If k i {\displaystyle k_{i}} is equated to enzyme activity, then every enzyme in the pathway has some influence over the flux.
Given the flux expression, it is possible to derive the flux control coefficients by differentiation and scaling of the flux expression. This can be done for the general case of n {\displaystyle n} steps:
C i J = 1 k i ∏ j = i n q j ∑ j = 1 n 1 k j ∏ k = j n q k {\displaystyle C_{i}^{J}={\frac {{\frac {1}{k_{i}}}\prod _{j=i}^{n}q_{j}}{\sum _{j=1}^{n}{\frac {1}{k_{j}}}\prod _{k=j}^{n}q_{k}}}}
This result yields two corollaries:
For the three-step linear chain, the flux control coefficients are given by:
C 1 J = 1 k 1 q 1 q 2 q 3 d ; C 2 J = 1 k 2 q 2 q 3 d ; C 3 J = 1 k 3 q 3 d {\displaystyle C_{1}^{J}={\frac {1}{k_{1}}}{\frac {q_{1}q_{2}q_{3}}{d}};\quad C_{2}^{J}={\frac {1}{k_{2}}}{\frac {q_{2}q_{3}}{d}};\quad C_{3}^{J}={\frac {1}{k_{3}}}{\frac {q_{3}}{d}}}
where d {\displaystyle d} is given by:
d = 1 k 1 q 1 q 2 q 3 + 1 k 2 q 2 q 3 + 1 k 3 q 3 {\displaystyle d={\frac {1}{k_{1}}}q_{1}q_{2}q_{3}+{\frac {1}{k_{2}}}q_{2}q_{3}+{\frac {1}{k_{3}}}q_{3}}
Given these results, there are some patterns:
With more moderate equilibrium constants, perturbations can travel upstream as well as downstream. For example, a perturbation at the last step, k 3 {\displaystyle k_{3}} , is better able to influence the reaction rates upstream, which results in an alteration in the steady-state flux.
An important result can be obtained if all k i {\displaystyle k_{i}} are set as equal to each other. Under these conditions, the flux control coefficient is proportional to the numerator. That is:
C 1 J ∝ q 1 q 2 q 3 C 2 J ∝ q 2 q 3 C 3 J ∝ q 3 {\displaystyle {\begin{aligned}C_{1}^{J}&\propto q_{1}q_{2}q_{3}\\C_{2}^{J}&\propto q_{2}q_{3}\\C_{3}^{J}&\propto q_{3}\\\end{aligned}}}
If it is assumed that the equilibrium constants are all greater than 1.0, as earlier steps have more q i {\displaystyle q_{i}} terms, it must mean that earlier steps will, in general, have high larger flux control coefficients. In a linear chain of reaction steps, flux control will tend to be biased towards the front of the pathway. From a metabolic engineering or drug-targeting perspective, preference should be given to targeting the earlier steps in a pathway since they have the greatest effect on pathway flux. Note that this rule only applies to pathways without negative feedback loops. [ 7 ] | https://en.wikipedia.org/wiki/Linear_biochemical_pathway |
In Hamiltonian mechanics , the linear canonical transformation ( LCT ) is a family of integral transforms that generalizes many classical transforms. It has 4 parameters and 1 constraint, so it is a 3-dimensional family, and can be visualized as the action of the special linear group SL 2 ( C ) on the time–frequency plane (domain). As this defines the original function up to a sign, this translates into an action of its double cover on the original function space.
The LCT generalizes the Fourier , fractional Fourier , Laplace , Gauss–Weierstrass , Bargmann and the Fresnel transforms as particular cases. The name "linear canonical transformation" is from canonical transformation , a map that preserves the symplectic structure, as SL 2 ( R ) can also be interpreted as the symplectic group Sp 2 , and thus LCTs are the linear maps of the time–frequency domain which preserve the symplectic form , and their action on the Hilbert space is given by the Metaplectic group .
The basic properties of the transformations mentioned above, such as scaling, shift, coordinate multiplication are considered. Any linear canonical transformation is related to affine transformations in phase space, defined by time-frequency or position-momentum coordinates.
The LCT can be represented in several ways; most easily, [ 1 ] it can be parameterized by a 2×2 matrix with determinant 1, i.e., an element of the special linear group SL 2 ( C ). Then for any such matrix ( a b c d ) , {\displaystyle {\bigl (}{\begin{smallmatrix}a&b\\c&d\end{smallmatrix}}{\bigr )},} with ad − bc = 1, the corresponding integral transform from a function x ( t ) {\displaystyle x(t)} to X ( u ) {\displaystyle X(u)} is defined as
X ( a , b , c , d ) ( u ) = { 1 i b ⋅ e i π d b u 2 ∫ − ∞ ∞ e − i 2 π 1 b u t e i π a b t 2 x ( t ) d t , when b ≠ 0 , d ⋅ e i π c d u 2 x ( d ⋅ u ) , when b = 0. {\displaystyle X_{(a,b,c,d)}(u)={\begin{cases}{\sqrt {\frac {1}{ib}}}\cdot e^{i\pi {\frac {d}{b}}u^{2}}\int _{-\infty }^{\infty }e^{-i2\pi {\frac {1}{b}}ut}e^{i\pi {\frac {a}{b}}t^{2}}x(t)\,dt,&{\text{when }}b\neq 0,\\{\sqrt {d}}\cdot e^{i\pi cdu^{2}}x(d\cdot u),&{\text{when }}b=0.\end{cases}}}
Many classical transforms are special cases of the linear canonical transform:
Scaling , x ( u ) ↦ σ x ( σ u ) {\displaystyle x(u)\mapsto {\sqrt {\sigma }}x(\sigma u)} , corresponds to scaling the time and frequency dimensions inversely (as time goes faster, frequencies are higher and the time dimension shrinks): [ 1 / σ 0 0 σ ] {\displaystyle {\begin{bmatrix}1/\sigma &0\\0&\sigma \end{bmatrix}}}
The Fourier transform corresponds to a clockwise rotation by 90° in the time–frequency plane, represented by the matrix [ a b c d ] = [ 0 1 − 1 0 ] . {\displaystyle {\begin{bmatrix}a&b\\c&d\end{bmatrix}}={\begin{bmatrix}0&1\\-1&0\end{bmatrix}}.}
The fractional Fourier transform corresponds to rotation by an arbitrary angle; they are the elliptic elements of SL 2 ( R ), represented by the matrices [ a b c d ] = [ cos θ sin θ − sin θ cos θ ] . {\displaystyle {\begin{bmatrix}a&b\\c&d\end{bmatrix}}={\begin{bmatrix}\cos \theta &\sin \theta \\-\sin \theta &\cos \theta \end{bmatrix}}.} The Fourier transform is the fractional Fourier transform when θ = 90 ∘ . {\displaystyle \theta =90^{\circ }.} The inverse Fourier transform corresponds to θ = − 90 ∘ . {\displaystyle \theta =-90^{\circ }.}
The Fresnel transform corresponds to shearing, and are a family of parabolic elements , represented by the matrices [ a b c d ] = [ 1 λ z 0 1 ] , {\displaystyle {\begin{bmatrix}a&b\\c&d\end{bmatrix}}={\begin{bmatrix}1&\lambda z\\0&1\end{bmatrix}},} where z is distance, and λ is wavelength.
The Laplace transform corresponds to rotation by 90° into the complex domain and can be represented by the matrix [ a b c d ] = [ 0 i i 0 ] . {\displaystyle {\begin{bmatrix}a&b\\c&d\end{bmatrix}}={\begin{bmatrix}0&i\\i&0\end{bmatrix}}.}
The fractional Laplace transform corresponds to rotation by an arbitrary angle into the complex domain and can be represented by the matrix [ 2 ] [ a b c d ] = [ i cos θ i sin θ i sin θ − i cos θ ] . {\displaystyle {\begin{bmatrix}a&b\\c&d\end{bmatrix}}={\begin{bmatrix}i\cos \theta &i\sin \theta \\i\sin \theta &-i\cos \theta \end{bmatrix}}.} The Laplace transform is the fractional Laplace transform when θ = 90 ∘ . {\displaystyle \theta =90^{\circ }.} The inverse Laplace transform corresponds to θ = − 90 ∘ . {\displaystyle \theta =-90^{\circ }.}
Chirp multiplication, x ( u ) ↦ e i π τ u 2 x ( u ) {\displaystyle x(u)\mapsto e^{i\pi \tau u^{2}}x(u)} , corresponds to b = 0 , c = τ {\displaystyle b=0,c=\tau } : [ citation needed ] [ a b c d ] = [ 1 0 τ 1 ] . {\displaystyle {\begin{bmatrix}a&b\\c&d\end{bmatrix}}={\begin{bmatrix}1&0\\\tau &1\end{bmatrix}}.}
Composition of LCTs corresponds to multiplication of the corresponding matrices; this is also known as the additivity property of the Wigner distribution function (WDF). Occasionally the product of transforms can pick up a sign factor due to picking a different branch of the square root in the definition of the LCT. In the literature, this is called the metaplectic phase .
If the LCT is denoted by O F ( a , b , c , d ) {\displaystyle O_{F}^{(a,b,c,d)}} , i.e.
X ( a , b , c , d ) ( u ) = O F ( a , b , c , d ) [ x ( t ) ] , {\displaystyle X_{(a,b,c,d)}(u)=O_{F}^{(a,b,c,d)}[x(t)],}
then
O F ( a 2 , b 2 , c 2 , d 2 ) { O F ( a 1 , b 1 , c 1 , d 1 ) [ x ( t ) ] } = O F ( a 3 , b 3 , c 3 , d 3 ) [ x ( t ) ] , {\displaystyle O_{F}^{(a_{2},b_{2},c_{2},d_{2})}\left\{O_{F}^{(a_{1},b_{1},c_{1},d_{1})}[x(t)]\right\}=O_{F}^{(a_{3},b_{3},c_{3},d_{3})}[x(t)],}
where
[ a 3 b 3 c 3 d 3 ] = [ a 2 b 2 c 2 d 2 ] [ a 1 b 1 c 1 d 1 ] . {\displaystyle {\begin{bmatrix}a_{3}&b_{3}\\c_{3}&d_{3}\end{bmatrix}}={\begin{bmatrix}a_{2}&b_{2}\\c_{2}&d_{2}\end{bmatrix}}{\begin{bmatrix}a_{1}&b_{1}\\c_{1}&d_{1}\end{bmatrix}}.}
If W X ( a , b , c , d ) ( u , v ) {\displaystyle W_{X(a,b,c,d)}(u,v)} is the X ( a , b , c , d ) ( u ) {\displaystyle X_{(a,b,c,d)}(u)} , where X ( a , b , c , d ) ( u ) {\displaystyle X_{(a,b,c,d)}(u)} is the LCT of x ( t ) {\displaystyle x(t)} , then
W X ( a , b , c , d ) ( u , v ) = W x ( d u − b v , − c u + a v ) , {\displaystyle W_{X(a,b,c,d)}(u,v)=W_{x}(du-bv,-cu+av),} W X ( a , b , c , d ) ( a u + b v , c u + d v ) = W x ( u , v ) . {\displaystyle W_{X(a,b,c,d)}(au+bv,cu+dv)=W_{x}(u,v).}
LCT is equal to the twisting operation for the WDF and the Cohen's class distribution also has the twisting operation.
We can freely use the LCT to transform the parallelogram whose center is at (0, 0) to another parallelogram which has the same area and the same center:
From this picture we know that the point (−1, 2) transform to the point (0, 1), and the point (1, 2) transform to the point (4, 3). As the result, we can write down the equations
{ − a + 2 b = 0 , − c + 2 d = 1 , { a + 2 b = 4 , c + 2 d = 3. {\displaystyle {\begin{cases}-a+2b=0,\\-c+2d=1,\end{cases}}\qquad {\begin{cases}a+2b=4,\\c+2d=3.\end{cases}}}
Solve these equations gives ( a , b , c , d ) = (2, 1, 1, 1).
Paraxial optical systems implemented entirely with thin lenses and propagation through free space and/or graded-index (GRIN) media, are quadratic-phase systems (QPS); these were known before Moshinsky and Quesne (1974) called attention to their significance in connection with canonical transformations in quantum mechanics. The effect of any arbitrary QPS on an input wavefield can be described using the linear canonical transform, a particular case of which was developed by Segal (1963) and Bargmann (1961) in order to formalize Fock's (1928) boson calculus. [ 3 ]
In quantum mechanics , linear canonical transformations can be identified with the linear transformations which mix the momentum operator with the position operator and leave invariant the canonical commutation relations .
Canonical transforms are used to analyze differential equations. These include diffusion , the Schrödinger free particle , the linear potential (free-fall), and the attractive and repulsive oscillator equations. It also includes a few others such as the Fokker–Planck equation . Although this class is far from universal, the ease with which solutions and properties are found makes canonical transforms an attractive tool for problems such as these. [ 4 ]
Wave propagation through air, a lens, and between satellite dishes are discussed here. All of the computations can be reduced to 2×2 matrix algebra. This is the spirit of LCT.
Assuming the system looks like as depicted in the figure, the wave travels from the ( x i , y i ) plane to the ( x , y ) plane. The Fresnel transform is used to describe electromagnetic wave propagation in free space:
U 0 ( x , y ) = − j λ e j k z z ∫ − ∞ ∞ ∫ − ∞ ∞ e j k 2 z [ ( x − x i ) 2 + ( y − y i ) 2 ] U i ( x i , y i ) d x i d y i , {\displaystyle U_{0}(x,y)=-{\frac {j}{\lambda }}{\frac {e^{jkz}}{z}}\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }e^{j{\frac {k}{2z}}\left[(x-x_{i})^{2}+(y-y_{i})^{2}\right]}U_{i}(x_{i},y_{i})\,dx_{i}\,dy_{i},}
where
This is equivalent to LCT (shearing), when
[ a b c d ] = [ 1 λ z 0 1 ] . {\displaystyle {\begin{bmatrix}a&b\\c&d\end{bmatrix}}={\begin{bmatrix}1&\lambda z\\0&1\end{bmatrix}}.}
When the travel distance ( z ) is larger, the shearing effect is larger.
With the lens as depicted in the figure, and the refractive index denoted as n , the result is [ 5 ]
U 0 ( x , y ) = e j k n Δ e − j k 2 f [ x 2 + y 2 ] U i ( x , y ) , {\displaystyle U_{0}(x,y)=e^{jkn\Delta }e^{-j{\frac {k}{2f}}[x^{2}+y^{2}]}U_{i}(x,y),}
where f is the focal length, and Δ is the thickness of the lens.
The distortion passing through the lens is similar to LCT, when
[ a b c d ] = [ 1 0 − 1 λ f 1 ] . {\displaystyle {\begin{bmatrix}a&b\\c&d\end{bmatrix}}={\begin{bmatrix}1&0\\{\frac {-1}{\lambda f}}&1\end{bmatrix}}.}
This is also a shearing effect: when the focal length is smaller, the shearing effect is larger.
The spherical mirror—e.g., a satellite dish—can be described as a LCT, with
[ a b c d ] = [ 1 0 − 1 λ R 1 ] . {\displaystyle {\begin{bmatrix}a&b\\c&d\end{bmatrix}}={\begin{bmatrix}1&0\\{\frac {-1}{\lambda R}}&1\end{bmatrix}}.}
This is very similar to lens, except focal length is replaced by the radius R of the dish. A spherical mirror with radius curvature of R is equivalent to a thin lens with the focal length f = − R /2 (by convention, R < 0 for concave mirror, R > 0 for convex mirror). Therefore, if the radius is smaller, the shearing effect is larger.
The relation between the input and output we can use LCT to represent
[ a b c d ] = [ 1 λ z 2 0 1 ] [ 1 0 − 1 / λ f 1 ] [ 1 λ z 1 0 1 ] = [ 1 − z 2 / f λ ( z 1 + z 2 ) − λ z 1 z 2 / f − 1 / λ f 1 − z 1 / f ] . {\displaystyle {\begin{bmatrix}a&b\\c&d\end{bmatrix}}={\begin{bmatrix}1&\lambda z_{2}\\0&1\end{bmatrix}}{\begin{bmatrix}1&0\\-1/\lambda f&1\end{bmatrix}}{\begin{bmatrix}1&\lambda z_{1}\\0&1\end{bmatrix}}={\begin{bmatrix}1-z_{2}/f&\lambda (z_{1}+z_{2})-\lambda z_{1}z_{2}/f\\-1/\lambda f&1-z_{1}/f\end{bmatrix}}\,.}
In this part, we show the basic properties of LCT
Given a two-dimensional column vector r = [ x y ] , {\displaystyle r={\begin{bmatrix}x\\y\end{bmatrix}},} we show some basic properties (result) for the specific input below:
The system considered is depicted in the figure to the right: two dishes – one being the emitter and the other one the receiver – and a signal travelling between them over a distance D .
First, for dish A (emitter), the LCT matrix looks like this:
[ 1 0 − 1 λ R A 1 ] . {\displaystyle {\begin{bmatrix}1&0\\{\frac {-1}{\lambda R_{A}}}&1\end{bmatrix}}.}
Then, for dish B (receiver), the LCT matrix similarly becomes:
[ 1 0 − 1 λ R B 1 ] . {\displaystyle {\begin{bmatrix}1&0\\{\frac {-1}{\lambda R_{B}}}&1\end{bmatrix}}.}
Last, for the propagation of the signal in air, the LCT matrix is:
[ 1 λ D 0 1 ] . {\displaystyle {\begin{bmatrix}1&\lambda D\\0&1\end{bmatrix}}.}
Putting all three components together, the LCT of the system is:
[ a b c d ] = [ 1 0 − 1 λ R B 1 ] [ 1 λ D 0 1 ] [ 1 0 − 1 λ R A 1 ] = [ 1 − D R A − λ D 1 λ ( R A − 1 + R B − 1 − R A − 1 R B − 1 D ) 1 − D R B ] . {\displaystyle {\begin{bmatrix}a&b\\c&d\end{bmatrix}}={\begin{bmatrix}1&0\\{\frac {-1}{\lambda R_{B}}}&1\end{bmatrix}}{\begin{bmatrix}1&\lambda D\\0&1\end{bmatrix}}{\begin{bmatrix}1&0\\{\frac {-1}{\lambda R_{A}}}&1\end{bmatrix}}={\begin{bmatrix}1-{\frac {D}{R_{A}}}&-\lambda D\\{\frac {1}{\lambda }}(R_{A}^{-1}+R_{B}^{-1}-R_{A}^{-1}R_{B}^{-1}D)&1-{\frac {D}{R_{B}}}\end{bmatrix}}\,.} | https://en.wikipedia.org/wiki/Linear_canonical_transformation |
In chemistry and materials science , linear chain compounds are materials composed of one-dimensional arrays of metal-metal bonded molecules or ions . Such materials exhibit anisotropic electrical conductivity . [ 1 ] [ 2 ]
Many linear chain compounds feature square planar complexes . One example is Rh(acac)(CO) 2 , which stack with Rh···Rh distances of about 326 pm . [ 3 ] Classic examples include Krogmann's salt and Magnus's green salt . Another example is the partially oxidized derivatives of [Pt(oxalate) 2 ] 2− . The otherwise ordinary complex IrBr(CO) 3 gives an electrically conductive derivative upon oxidation, e.g., with bromine to give IrBr 1+ x (CO) 3− x , where x ~0.05. [ 2 ] [ 4 ] Related chlorides have the formulae IrCl 1+ x (CO) 3 and K 0.6 Ir(CO) 2 Cl 2 ·½H 2 O . [ 5 ]
In contrast to linear chain compounds, extended metal atom chains (EMACs) are molecules or ions that consist of a finite, often short, linear strings of metal atoms, surrounded by organic ligands . [ 6 ]
One group of platinum chains is based on alternating cations and anions of [Pt(CNR) 4 ] 2+ (R = i Pr, c -C 12 H 23 , p -(C 2 H 5 )C 6 H 4 ) and [Pt(CN) 4 ] 2− . [ 1 ] These may be able to be used as vapochromic sensor materials, or materials which change color when exposed to different vapors. [ 8 ] [ 9 ] [ 10 ]
Linear chains of Pd-Pd bonds protected by a "π-electron sheath" are known. [ 1 ] [ 11 ]
Not only do these olefin-stabilized metal chains constitute a significant contribution to the field of organometallic chemistry , both the complex's metal atom structures and the olefin ligands themselves can conduct a current. [ 1 ] [ 12 ]
Some linear chain compounds are produced or fabricated by electrocrystallization . The technique is used to obtain single crystals of low-dimensional electrical conductors. [ 13 ] | https://en.wikipedia.org/wiki/Linear_chain_compound |
In mathematics , a linear combination or superposition is an expression constructed from a set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of x and y would be any expression of the form ax + by , where a and b are constants). [ 1 ] [ 2 ] [ 3 ] [ 4 ] The concept of linear combinations is central to linear algebra and related fields of mathematics. Most of this article deals with linear combinations in the context of a vector space over a field , with some generalizations given at the end of the article.
Let V be a vector space over the field K . As usual, we call elements of V vectors and call elements of K scalars .
If v 1 ,..., v n are vectors and a 1 ,..., a n are scalars, then the linear combination of those vectors with those scalars as coefficients is
There is some ambiguity in the use of the term "linear combination" as to whether it refers to the expression or to its value. In most cases the value is emphasized, as in the assertion "the set of all linear combinations of v 1 ,..., v n always forms a subspace". However, one could also say "two different linear combinations can have the same value" in which case the reference is to the expression. The subtle difference between these uses is the essence of the notion of linear dependence : a family F of vectors is linearly independent precisely if any linear combination of the vectors in F (as value) is uniquely so (as expression). In any case, even when viewed as expressions, all that matters about a linear combination is the coefficient of each v i ; trivial modifications such as permuting the terms or adding terms with zero coefficient do not produce distinct linear combinations.
In a given situation, K and V may be specified explicitly, or they may be obvious from context. In that case, we often speak of a linear combination of the vectors v 1 ,..., v n , with the coefficients unspecified (except that they must belong to K ). Or, if S is a subset of V , we may speak of a linear combination of vectors in S , where both the coefficients and the vectors are unspecified, except that the vectors must belong to the set S (and the coefficients must belong to K ). Finally, we may speak simply of a linear combination , where nothing is specified (except that the vectors must belong to V and the coefficients must belong to K ); in this case one is probably referring to the expression, since every vector in V is certainly the value of some linear combination.
Note that by definition, a linear combination involves only finitely many vectors (except as described in the § Generalizations section.
However, the set S that the vectors are taken from (if one is mentioned) can still be infinite ; each individual linear combination will only involve finitely many vectors.
Also, there is no reason that n cannot be zero ; in that case, we declare by convention that the result of the linear combination is the zero vector in V .
Let the field K be the set R of real numbers , and let the vector space V be the Euclidean space R 3 .
Consider the vectors e 1 = (1,0,0) , e 2 = (0,1,0) and e 3 = (0,0,1) .
Then any vector in R 3 is a linear combination of e 1 , e 2 , and e 3 .
To see that this is so, take an arbitrary vector ( a 1 , a 2 , a 3 ) in R 3 , and write:
Let K be the set C of all complex numbers , and let V be the set C C ( R ) of all continuous functions from the real line R to the complex plane C .
Consider the vectors (functions) f and g defined by f ( t ) := e it and g ( t ) := e − it .
(Here, e is the base of the natural logarithm , about 2.71828..., and i is the imaginary unit , a square root of −1.)
Some linear combinations of f and g are:
On the other hand, the constant function 3 is not a linear combination of f and g . To see this, suppose that 3 could be written as a linear combination of e it and e − it . This means that there would exist complex scalars a and b such that ae it + be − it = 3 for all real numbers t . Setting t = 0 and t = π gives the equations a + b = 3 and a + b = −3 , and clearly this cannot happen. See Euler's identity .
Let K be R , C , or any field, and let V be the set P of all polynomials with coefficients taken from the field K .
Consider the vectors (polynomials) p 1 := 1, p 2 := x + 1 , and p 3 := x 2 + x + 1 .
Is the polynomial x 2 − 1 a linear combination of p 1 , p 2 , and p 3 ?
To find out, consider an arbitrary linear combination of these vectors and try to see when it equals the desired vector x 2 − 1.
Picking arbitrary coefficients a 1 , a 2 , and a 3 , we want
Multiplying the polynomials out, this means
and collecting like powers of x , we get
Two polynomials are equal if and only if their corresponding coefficients are equal, so we can conclude
This system of linear equations can easily be solved.
First, the first equation simply says that a 3 is 1.
Knowing that, we can solve the second equation for a 2 , which comes out to −1.
Finally, the last equation tells us that a 1 is also −1.
Therefore, the only possible way to get a linear combination is with these coefficients.
Indeed,
so x 2 − 1 is a linear combination of p 1 , p 2 , and p 3 .
On the other hand, what about the polynomial x 3 − 1? If we try to make this vector a linear combination of p 1 , p 2 , and p 3 , then following the same process as before, we get the equation
However, when we set corresponding coefficients equal in this case, the equation for x 3 is
which is always false.
Therefore, there is no way for this to work, and x 3 − 1 is not a linear combination of p 1 , p 2 , and p 3 .
Take an arbitrary field K , an arbitrary vector space V , and let v 1 ,..., v n be vectors (in V ).
It is interesting to consider the set of all linear combinations of these vectors.
This set is called the linear span (or just span ) of the vectors, say S = { v 1 , ..., v n }. We write the span of S as span( S ) [ 5 ] [ 6 ] or sp( S ):
Suppose that, for some sets of vectors v 1 ,..., v n ,
a single vector can be written in two different ways as a linear combination of them:
This is equivalent, by subtracting these ( c i := a i − b i {\displaystyle c_{i}:=a_{i}-b_{i}} ), to saying a non-trivial combination is zero: [ 7 ] [ 8 ]
If that is possible, then v 1 ,..., v n are called linearly dependent ; otherwise, they are linearly independent .
Similarly, we can speak of linear dependence or independence of an arbitrary set S of vectors.
If S is linearly independent and the span of S equals V , then S is a basis for V .
By restricting the coefficients used in linear combinations, one can define the related concepts of affine combination , conical combination , and convex combination , and the associated notions of sets closed under these operations.
Because these are more restricted operations, more subsets will be closed under them, so affine subsets, convex cones, and convex sets are generalizations of vector subspaces: a vector subspace is also an affine subspace, a convex cone, and a convex set, but a convex set need not be a vector subspace, affine, or a convex cone.
These concepts often arise when one can take certain linear combinations of objects, but not any: for example, probability distributions are closed under convex combination (they form a convex set), but not conical or affine combinations (or linear), and positive measures are closed under conical combination but not affine or linear – hence one defines signed measures as the linear closure.
Linear and affine combinations can be defined over any field (or ring), but conical and convex combination require a notion of "positive", and hence can only be defined over an ordered field (or ordered ring ), generally the real numbers.
If one allows only scalar multiplication , not addition, one obtains a (not necessarily convex) cone ; one often restricts the definition to only allowing multiplication by positive scalars.
All of these concepts are usually defined as subsets of an ambient vector space (except for affine spaces, which are also considered as "vector spaces forgetting the origin"), rather than being axiomatized independently.
More abstractly, in the language of operad theory , one can consider vector spaces to be algebras over the operad R ∞ {\displaystyle \mathbf {R} ^{\infty }} (the infinite direct sum , so only finitely many terms are non-zero; this corresponds to only taking finite sums), which parametrizes linear combinations: the vector ( 2 , 3 , − 5 , 0 , … ) {\displaystyle (2,3,-5,0,\dots )} for instance corresponds to the linear combination 2 v 1 + 3 v 2 − 5 v 3 + 0 v 4 + ⋯ {\displaystyle 2\mathbf {v} _{1}+3\mathbf {v} _{2}-5\mathbf {v} _{3}+0\mathbf {v} _{4}+\cdots } . Similarly, one can consider affine combinations, conical combinations, and convex combinations to correspond to the sub-operads where the terms sum to 1, the terms are all non-negative, or both, respectively. Graphically, these are the infinite affine hyperplane, the infinite hyper-octant, and the infinite simplex. This formalizes what is meant by R n {\displaystyle \mathbf {R} ^{n}} being or the standard simplex being model spaces, and such observations as that every bounded convex polytope is the image of a simplex. Here suboperads correspond to more restricted operations and thus more general theories.
From this point of view, we can think of linear combinations as the most general sort of operation on a vector space – saying that a vector space is an algebra over the operad of linear combinations is precisely the statement that all possible algebraic operations in a vector space are linear combinations.
The basic operations of addition and scalar multiplication, together with the existence of an additive identity and additive inverses, cannot be combined in any more complicated way than the generic linear combination: the basic operations are a generating set for the operad of all linear combinations.
Ultimately, this fact lies at the heart of the usefulness of linear combinations in the study of vector spaces.
If V is a topological vector space , then there may be a way to make sense of certain infinite linear combinations, using the topology of V .
For example, we might be able to speak of a 1 v 1 + a 2 v 2 + a 3 v 3 + ⋯, going on forever.
Such infinite linear combinations do not always make sense; we call them convergent when they do.
Allowing more linear combinations in this case can also lead to a different concept of span, linear independence, and basis.
The articles on the various flavors of topological vector spaces go into more detail about these.
If K is a commutative ring instead of a field, then everything that has been said above about linear combinations generalizes to this case without change.
The only difference is that we call spaces like this V modules instead of vector spaces.
If K is a noncommutative ring , then the concept still generalizes, with one caveat:
since modules over noncommutative rings come in left and right versions, our linear combinations may also come in either of these versions, whatever is appropriate for the given module.
This is simply a matter of doing scalar multiplication on the correct side.
A more complicated twist comes when V is a bimodule over two rings, K L and K R .
In that case, the most general linear combination looks like
where a 1 ,..., a n belong to K L , b 1 ,..., b n belong to K R , and v 1 ,…, v n belong to V . | https://en.wikipedia.org/wiki/Linear_combination |
A linear combination of atomic orbitals or LCAO is a quantum superposition of atomic orbitals and a technique for calculating molecular orbitals in quantum chemistry . [ 1 ] In quantum mechanics, electron configurations of atoms are described as wavefunctions . In a mathematical sense, these wave functions are the basis set of functions, the basis functions, which describe the electrons of a given atom. In chemical reactions , orbital wavefunctions are modified, i.e. the electron cloud shape is changed, according to the type of atoms participating in the chemical bond .
It was introduced in 1929 by Sir John Lennard-Jones with the description of bonding in the diatomic molecules of the first main row of the periodic table , but had been used earlier by Linus Pauling for H 2 + . [ 2 ] [ 3 ]
An initial assumption is that the number of molecular orbitals is equal to the number of atomic orbitals included in the linear expansion. In a sense, n atomic orbitals combine to form n molecular orbitals, which can be numbered i = 1 to n and which may not all be the same. The expression (linear expansion) for the i th molecular orbital would be:
or
where ϕ i {\displaystyle \ \phi _{i}} is a molecular orbital represented as the sum of n atomic orbitals χ r {\displaystyle \ \chi _{r}} , each multiplied by a corresponding coefficient c r i {\displaystyle \ c_{ri}} , and r (numbered 1 to n ) represents which atomic orbital is combined in the term. The coefficients are the weights of the contributions of the n atomic orbitals to the molecular orbital. The Hartree–Fock method is used to obtain the coefficients of the expansion.
The orbitals are thus expressed as linear combinations of basis functions , and the basis functions are single- electron functions which may or may not be centered on the nuclei of the component atoms of the molecule . In either case the basis functions are usually also referred to as atomic orbitals (even though only in the former case this name seems to be adequate). The atomic orbitals used are typically those of hydrogen-like atoms since these are known analytically i.e. Slater-type orbitals but other choices are possible such as the Gaussian functions from standard basis sets or the pseudo-atomic orbitals from plane-wave pseudopotentials.
By minimizing the total energy of the system, an appropriate set of coefficients of the linear combinations is determined. This quantitative approach is now known as the Hartree–Fock method. However, since the development of computational chemistry , the LCAO method often refers not to an actual optimization of the wave function but to a qualitative discussion which is very useful for predicting and rationalizing results obtained via more modern methods. In this case, the shape of the molecular orbitals and their respective energies are deduced approximately from comparing the energies of the atomic orbitals of the individual atoms (or molecular fragments) and applying some recipes known as level repulsion and the like. The graphs that are plotted to make this discussion clearer are called correlation diagrams. The required atomic orbital energies can come from calculations or directly from experiment via Koopmans' theorem .
This is done by using the symmetry of the molecules and orbitals involved in bonding, and thus is sometimes called symmetry adapted linear combination (SALC). The first step in this process is assigning a point group to the molecule. Each operation in the point group is performed upon the molecule. The number of bonds that are unmoved is the character of that operation. This reducible representation is decomposed into the sum of irreducible representations. These irreducible representations correspond to the symmetry of the orbitals involved.
Molecular orbital diagrams provide simple qualitative LCAO treatment. The Hückel method , the extended Hückel method and the Pariser–Parr–Pople method , provide some quantitative theories. | https://en.wikipedia.org/wiki/Linear_combination_of_atomic_orbitals |
In mathematical optimization theory , the linear complementarity problem (LCP) arises frequently in computational mechanics and encompasses the well-known quadratic programming as a special case. It was proposed by Cottle and Dantzig in 1968. [ 1 ] [ 2 ] [ 3 ]
Given a real matrix M and vector q , the linear complementarity problem LCP( q , M ) seeks vectors z and w which satisfy the following constraints:
A sufficient condition for existence and uniqueness of a solution to this problem is that M be symmetric positive-definite . If M is such that LCP( q , M ) has a solution for every q , then M is a Q-matrix . If M is such that LCP( q , M ) have a unique solution for every q , then M is a P-matrix . Both of these characterizations are sufficient and necessary. [ 4 ]
The vector w is a slack variable , [ 5 ] and so is generally discarded after z is found. As such, the problem can also be formulated as:
Finding a solution to the linear complementarity problem is associated with minimizing the quadratic function
subject to the constraints
These constraints ensure that f is always non-negative. The minimum of f is 0 at z if and only if z solves the linear complementarity problem.
If M is positive definite , any algorithm for solving (strictly) convex QPs can solve the LCP. Specially designed basis-exchange pivoting algorithms, such as Lemke's algorithm and a variant of the simplex algorithm of Dantzig have been used for decades. Besides having polynomial time complexity, interior-point methods are also effective in practice.
Also, a quadratic-programming problem stated as minimize f ( x ) = c T x + 1 2 x T Q x {\displaystyle f(x)=c^{T}x+{\tfrac {1}{2}}x^{T}Qx} subject to A x ⩾ b {\displaystyle Ax\geqslant b} as well as x ⩾ 0 {\displaystyle x\geqslant 0} with Q symmetric
is the same as solving the LCP with
This is because the Karush–Kuhn–Tucker conditions of the QP problem can be written as:
with v the Lagrange multipliers on the non-negativity constraints, λ the multipliers on the inequality constraints, and s the slack variables for the inequality constraints. The fourth condition derives from the complementarity of each group of variables ( x , s ) with its set of KKT vectors (optimal Lagrange multipliers) being ( v , λ ) . In that case,
If the non-negativity constraint on the x is relaxed, the dimensionality of the LCP problem can be reduced to the number of the inequalities, as long as Q is non-singular (which is guaranteed if it is positive definite ). The multipliers v are no longer present, and the first KKT conditions can be rewritten as:
or:
pre-multiplying the two sides by A and subtracting b we obtain:
The left side, due to the second KKT condition, is s . Substituting and reordering:
Calling now
we have an LCP, due to the relation of complementarity between the slack variables s and their Lagrange multipliers λ . Once we solve it, we may obtain the value of x from λ through the first KKT condition.
Finally, it is also possible to handle additional equality constraints:
This introduces a vector of Lagrange multipliers μ , with the same dimension as b e q {\displaystyle b_{eq}} .
It is easy to verify that the M and Q for the LCP system s = M λ + Q {\displaystyle s=M{\lambda }+Q} are now expressed as:
From λ we can now recover the values of both x and the Lagrange multiplier of equalities μ :
In fact, most QP solvers work on the LCP formulation, including the interior point method , principal / complementarity pivoting, and active set methods. [ 1 ] [ 2 ] LCP problems can be solved also by the criss-cross algorithm , [ 6 ] [ 7 ] [ 8 ] [ 9 ] conversely, for linear complementarity problems, the criss-cross algorithm terminates finitely only if the matrix is a sufficient matrix. [ 8 ] [ 9 ] A sufficient matrix is a generalization both of a positive-definite matrix and of a P-matrix , whose principal minors are each positive. [ 8 ] [ 9 ] [ 10 ] Such LCPs can be solved when they are formulated abstractly using oriented-matroid theory. [ 11 ] [ 12 ] [ 13 ] | https://en.wikipedia.org/wiki/Linear_complementarity_problem |
In the mathematical field of order theory , a continuum or linear continuum is a generalization of the real line .
Formally, a linear continuum is a linearly ordered set S of more than one element that is densely ordered , i.e., between any two distinct elements there is another (and hence infinitely many others), and complete , i.e., which "lacks gaps" in the sense that every nonempty subset with an upper bound has a least upper bound in the set. More symbolically:
A set has the least upper bound property, if every nonempty subset of the set that is bounded above has a least upper bound in the set. Linear continua are particularly important in the field of topology where they can be used to verify whether an ordered set given the order topology is connected or not. [ 1 ]
Unlike the standard real line, a linear continuum may be bounded on either side: for example, any (real) closed interval is a linear continuum.
Examples in addition to the real numbers:
Even though linear continua are important in the study of ordered sets , they do have applications in the mathematical field of topology . In fact, we will prove that an ordered set in the order topology is connected if and only if it is a linear continuum. We will prove one implication, and leave the other one as an exercise. (Munkres explains the second part of the proof in [ 3 ] )
Theorem
Let X be an ordered set in the order topology. If X is connected, then X is a linear continuum.
Proof:
Suppose that x and y are elements of X with x < y . If there exists no z in X such that x < z < y , consider the sets:
These sets are disjoint (If a is in A , a < y so that if a is in B , a > x and a < y which is impossible by hypothesis), nonempty ( x is in A and y is in B ) and open (in the order topology), and their union is X . This contradicts the connectedness of X .
Now we prove the least upper bound property. If C is a subset of X that is bounded above and has no least upper bound, let D be the union of all open rays of the form ( b , +∞) where b is an upper bound for C . Then D is open (since it is the union of open sets), and closed (if a is not in D , then a < b for all upper bounds b of C so that we may choose q > a such that q is in C (if no such q exists, a is the least upper bound of C ), then an open interval containing a may be chosen that doesn't intersect D ). Since D is nonempty (there is more than one upper bound of D for if there was exactly one upper bound s , s would be the least upper bound. Then if b 1 and b 2 are two upper bounds of D with b 1 < b 2 , b 2 will belong to D ), D and its complement together form a separation on X . This contradicts the connectedness of X . | https://en.wikipedia.org/wiki/Linear_continuum |
Linear dichroism ( LD ) or diattenuation is the difference between absorption of light polarized parallel and polarized perpendicular to an orientation axis. [ 1 ] It is the property of a material whose transmittance depends on the orientation of linearly polarized light incident upon it. As a technique, it is primarily used to study the functionality and structure of molecules . LD measurements are based on the interaction between matter and light and thus are a form of electromagnetic spectroscopy .
This effect has been applied across the EM spectrum , where different wavelengths of light can probe a host of chemical systems. The predominant use of LD currently is in the study of bio- macromolecules (e.g. DNA ) as well as synthetic polymers .
LD uses linearly polarized light, which is light that has been polarized in one direction only. This produces a wave, the electric field vector , which oscillates in only one plane, giving rise to a classic sinusoidal wave shape as the light travels through space. By using light parallel and perpendicular to the orientation direction it is possible to measure how much more energy is absorbed in one dimension of the molecule relative to the other, providing information to the experimentalist.
As light interacts with the molecule being investigated, should the molecule start absorbing the light then electron density inside the molecule will be shifted as the electron becomes photoexcited . This movement of charge is known as an electronic transition , the direction of which is called the electric transition polarisation. It is this property for which LD is a measurement.
The LD of an oriented molecule can be calculated using the following equation:-
Where A ║ is the absorbance parallel to the orientation axis and A ┴ is the absorbance perpendicular to the orientation axis.
Note that light of any wavelength can be used to generate an LD signal.
The LD signal generated therefore has two limits upon the signal that can be generated. For a chemical system whose electric transition is parallel to the orientation axis, the following equation can be written:
For most chemical systems this represents an electric transition polarised across the length of the molecule (i.e. parallel to the orientation axis).
Alternatively, the electric transition polarisation can be found to be perfectly perpendicular to the orientation of the molecule, giving rise to the following equation:
This equation represents the LD signal recorded if the electric transition is polarised across the width of the molecule (i.e. perpendicular to the orientation axis), which in the case of LD is the smaller of the two investigable axes.
LD can therefore be used in two ways. If the orientation of the molecules in flow [ clarification needed ] is known, then the experimentalist can look at the direction of polarisation in the molecule (which gives an insight into the chemical structure of a molecule), or if the polarisation direction is unknown it can be used as a means of working out how oriented in flow a molecule is.
Ultraviolet (UV) LD is typically employed in the analysis of biological molecules, especially large, flexible, long molecules that prove difficult to structurally determine by such methods as NMR and X-ray diffraction .
DNA is almost ideally suited for UV LD detection. The molecule is very long and very thin, making it very easy to orient in flow. This gives rise to a strong LD signal. DNA systems that have been studied using UV LD include DNA- enzyme complexes and DNA- ligand complexes, [ 2 ] the formation of the latter being easily observable through kinetic experiments.
Fibrous proteins , such as proteins involved in Alzheimer's disease and prion proteins fulfil the requirements for UV LD in that they are a class of long, thin molecules. In addition, cytoskeletal proteins [ 3 ] can also be measured using LD.
The insertion of membrane proteins into a lipid membrane has been monitored using LD, supplying the experimentalist with information about the orientation of the protein relative to the lipid membrane at different time points.
In addition, other types of molecule have been analysed by UV LD, including carbon nanotubes [ 4 ] and their associated ligand complexes.
The Couette flow orientation system is the most widely used method of sample orientation for UV LD. It has a number of characteristics which make it highly suitable as a method of sample alignment. Couette flow is currently the only established means of orientating molecules in the solution phase. This method also requires only very small amounts of analysis sample ( 20 - 40 μL) in order to generate an LD spectrum. The constant recirculation of sample is another useful property of the system, allowing many repeat measurements to be taken of each sample, decreasing the effect of noise on the final recorded spectrum.
Its mode of operation is very simple, with the sample sandwiched between a spinning tube and a stationary rod. As the sample is spun inside the cell, the light beam is shone through the sample, the parallel absorbance calculated from horizontally polarised light, the perpendicular absorbance from the vertically polarised light.
Couette flow UV LD is currently the only commercially available means of LD orientation.
Stretched film linear dichroism is a method of orientation based on incorporating the sample molecules into a polyethylene film. [ 5 ] The polyethylene film is then stretched, causing the randomly oriented molecules on the film to ‘follow’ the movement of the film. The stretching of the film results in the sample molecules being oriented in the direction of the stretch.
LD is very similar to Circular Dichroism (CD), but with two important differences. (i) CD spectroscopy uses circularly polarized light whereas LD uses linearly polarized light. (ii) In CD experiments molecules are usually free in solution so they are randomly oriented. The observed spectrum is then a function only of the chiral or asymmetric nature of the molecules in the solution. With biomacromolecules CD is particularly useful for determining the secondary structure. By way of contrast, in LD experiments the molecules need to have a preferential orientation otherwise the LD=0. With biomacromolecules flow orientation is often used, other methods include stretched films, magnetic fields, and squeezed gels. Thus LD gives information such as alignment on a surface or the binding of a small molecule to a flow-oriented macromolecule, endowing it with different functionality from other spectroscopic techniques. The differences between LD and CD are complementary and can be a potent means for elucidating the structure of biological molecules when used in conjunction with one another, the combination of techniques revealing far more information than a single technique in isolation. For example, CD tells us when a membrane peptide or protein folds whereas LD tells when it inserts into a membrane. [ 6 ]
Fluorescence -detected linear dichroism (FDLD) is a very useful technique to the experimentalist as it combines the advantages of UV LD whilst also offering the confocal detection of the fluorescence emission. [ 7 ] FDLD has applications in microscopy, where can be used as a means of two-dimensional surface mapping through differential polarisation spectroscopy (DPS) where the anisotropy of the scanned object allows an image to be recorded. FDLD can also be used in conjunction with intercalating fluorescent dyes (which can also be monitored using UV LD). The intensity difference recorded between the two types of polarised light for the fluorescence reading is proportional to the UV LD signal, allowing the use of DPS to image surfaces | https://en.wikipedia.org/wiki/Linear_dichroism |
In mathematics , a linear differential equation is a differential equation that is linear in the unknown function and its derivatives, so it can be written in the form a 0 ( x ) y + a 1 ( x ) y ′ + a 2 ( x ) y ″ ⋯ + a n ( x ) y ( n ) = b ( x ) {\displaystyle a_{0}(x)y+a_{1}(x)y'+a_{2}(x)y''\cdots +a_{n}(x)y^{(n)}=b(x)} where a 0 ( x ) , ..., a n ( x ) and b ( x ) are arbitrary differentiable functions that do not need to be linear, and y ′, ..., y ( n ) are the successive derivatives of an unknown function y of the variable x .
Such an equation is an ordinary differential equation (ODE). A linear differential equation may also be a linear partial differential equation (PDE), if the unknown function depends on several variables, and the derivatives that appear in the equation are partial derivatives .
A linear differential equation or a system of linear equations such that the associated homogeneous equations have constant coefficients may be solved by quadrature , which means that the solutions may be expressed in terms of integrals . This is also true for a linear equation of order one, with non-constant coefficients. An equation of order two or higher with non-constant coefficients cannot, in general, be solved by quadrature. For order two, Kovacic's algorithm allows deciding whether there are solutions in terms of integrals, and computing them if any.
The solutions of homogeneous linear differential equations with polynomial coefficients are called holonomic functions . This class of functions is stable under sums, products, differentiation , integration , and contains many usual functions and special functions such as exponential function , logarithm , sine , cosine , inverse trigonometric functions , error function , Bessel functions and hypergeometric functions . Their representation by the defining differential equation and initial conditions allows making algorithmic (on these functions) most operations of calculus , such as computation of antiderivatives , limits , asymptotic expansion , and numerical evaluation to any precision, with a certified error bound.
The highest order of derivation that appears in a (linear) differential equation is the order of the equation. The term b ( x ) , which does not depend on the unknown function and its derivatives, is sometimes called the constant term of the equation (by analogy with algebraic equations ), even when this term is a non-constant function. If the constant term is the zero function , then the differential equation is said to be homogeneous , as it is a homogeneous polynomial in the unknown function and its derivatives. The equation obtained by replacing, in a linear differential equation, the constant term by the zero function is the associated homogeneous equation . A differential equation has constant coefficients if only constant functions appear as coefficients in the associated homogeneous equation.
A solution of a differential equation is a function that satisfies the equation.
The solutions of a homogeneous linear differential equation form a vector space . In the ordinary case, this vector space has a finite dimension, equal to the order of the equation. All solutions of a linear differential equation are found by adding to a particular solution any solution of the associated homogeneous equation.
A basic differential operator of order i is a mapping that maps any differentiable function to its i th derivative , or, in the case of several variables, to one of its partial derivatives of order i . It is commonly denoted d i d x i {\displaystyle {\frac {d^{i}}{dx^{i}}}} in the case of univariate functions, and ∂ i 1 + ⋯ + i n ∂ x 1 i 1 ⋯ ∂ x n i n {\displaystyle {\frac {\partial ^{i_{1}+\cdots +i_{n}}}{\partial x_{1}^{i_{1}}\cdots \partial x_{n}^{i_{n}}}}} in the case of functions of n variables. The basic differential operators include the derivative of order 0, which is the identity mapping.
A linear differential operator (abbreviated, in this article, as linear operator or, simply, operator ) is a linear combination of basic differential operators, with differentiable functions as coefficients. In the univariate case, a linear operator has thus the form [ 1 ] a 0 ( x ) + a 1 ( x ) d d x + ⋯ + a n ( x ) d n d x n , {\displaystyle a_{0}(x)+a_{1}(x){\frac {d}{dx}}+\cdots +a_{n}(x){\frac {d^{n}}{dx^{n}}},} where a 0 ( x ), ..., a n ( x ) are differentiable functions, and the nonnegative integer n is the order of the operator (if a n ( x ) is not the zero function ).
Let L be a linear differential operator. The application of L to a function f is usually denoted Lf or Lf ( X ) , if one needs to specify the variable (this must not be confused with a multiplication). A linear differential operator is a linear operator , since it maps sums to sums and the product by a scalar to the product by the same scalar.
As the sum of two linear operators is a linear operator, as well as the product (on the left) of a linear operator by a differentiable function, the linear differential operators form a vector space over the real numbers or the complex numbers (depending on the nature of the functions that are considered). They form also a free module over the ring of differentiable functions.
The language of operators allows a compact writing for differentiable equations: if L = a 0 ( x ) + a 1 ( x ) d d x + ⋯ + a n ( x ) d n d x n , {\displaystyle L=a_{0}(x)+a_{1}(x){\frac {d}{dx}}+\cdots +a_{n}(x){\frac {d^{n}}{dx^{n}}},} is a linear differential operator, then the equation a 0 ( x ) y + a 1 ( x ) y ′ + a 2 ( x ) y ″ + ⋯ + a n ( x ) y ( n ) = b ( x ) {\displaystyle a_{0}(x)y+a_{1}(x)y'+a_{2}(x)y''+\cdots +a_{n}(x)y^{(n)}=b(x)} may be rewritten L y = b ( x ) . {\displaystyle Ly=b(x).}
There may be several variants to this notation; in particular the variable of differentiation may appear explicitly or not in y and the right-hand and of the equation, such as Ly ( x ) = b ( x ) or Ly = b .
The kernel of a linear differential operator is its kernel as a linear mapping, that is the vector space of the solutions of the (homogeneous) differential equation Ly = 0 .
In the case of an ordinary differential operator of order n , Carathéodory's existence theorem implies that, under very mild conditions, the kernel of L is a vector space of dimension n , and that the solutions of the equation Ly ( x ) = b ( x ) have the form S 0 ( x ) + c 1 S 1 ( x ) + ⋯ + c n S n ( x ) , {\displaystyle S_{0}(x)+c_{1}S_{1}(x)+\cdots +c_{n}S_{n}(x),} where c 1 , ..., c n are arbitrary numbers. Typically, the hypotheses of Carathéodory's theorem are satisfied in an interval I , if the functions b , a 0 , ..., a n are continuous in I , and there is a positive real number k such that | a n ( x ) | > k for every x in I .
A homogeneous linear differential equation has constant coefficients if it has the form a 0 y + a 1 y ′ + a 2 y ″ + ⋯ + a n y ( n ) = 0 {\displaystyle a_{0}y+a_{1}y'+a_{2}y''+\cdots +a_{n}y^{(n)}=0} where a 1 , ..., a n are (real or complex) numbers. In other words, it has constant coefficients if it is defined by a linear operator with constant coefficients.
The study of these differential equations with constant coefficients dates back to Leonhard Euler , who introduced the exponential function e x , which is the unique solution of the equation f ′ = f such that f (0) = 1 . It follows that the n th derivative of e cx is c n e cx , and this allows solving homogeneous linear differential equations rather easily.
Let a 0 y + a 1 y ′ + a 2 y ″ + ⋯ + a n y ( n ) = 0 {\displaystyle a_{0}y+a_{1}y'+a_{2}y''+\cdots +a_{n}y^{(n)}=0} be a homogeneous linear differential equation with constant coefficients (that is a 0 , ..., a n are real or complex numbers).
Searching solutions of this equation that have the form e αx is equivalent to searching the constants α such that a 0 e α x + a 1 α e α x + a 2 α 2 e α x + ⋯ + a n α n e α x = 0. {\displaystyle a_{0}e^{\alpha x}+a_{1}\alpha e^{\alpha x}+a_{2}\alpha ^{2}e^{\alpha x}+\cdots +a_{n}\alpha ^{n}e^{\alpha x}=0.} Factoring out e αx (which is never zero), shows that α must be a root of the characteristic polynomial a 0 + a 1 t + a 2 t 2 + ⋯ + a n t n {\displaystyle a_{0}+a_{1}t+a_{2}t^{2}+\cdots +a_{n}t^{n}} of the differential equation, which is the left-hand side of the characteristic equation a 0 + a 1 t + a 2 t 2 + ⋯ + a n t n = 0. {\displaystyle a_{0}+a_{1}t+a_{2}t^{2}+\cdots +a_{n}t^{n}=0.}
When these roots are all distinct , one has n distinct solutions that are not necessarily real, even if the coefficients of the equation are real. These solutions can be shown to be linearly independent , by considering the Vandermonde determinant of the values of these solutions at x = 0, ..., n – 1 . Together they form a basis of the vector space of solutions of the differential equation (that is, the kernel of the differential operator).
y ⁗ − 2 y ‴ + 2 y ″ − 2 y ′ + y = 0 {\displaystyle y''''-2y'''+2y''-2y'+y=0} has the characteristic equation z 4 − 2 z 3 + 2 z 2 − 2 z + 1 = 0. {\displaystyle z^{4}-2z^{3}+2z^{2}-2z+1=0.} This has zeros, i , − i , and 1 (multiplicity 2). The solution basis is thus e i x , e − i x , e x , x e x . {\displaystyle e^{ix},\;e^{-ix},\;e^{x},\;xe^{x}.} A real basis of solution is thus cos x , sin x , e x , x e x . {\displaystyle \cos x,\;\sin x,\;e^{x},\;xe^{x}.}
In the case where the characteristic polynomial has only simple roots , the preceding provides a complete basis of the solutions vector space. In the case of multiple roots , more linearly independent solutions are needed for having a basis. These have the form x k e α x , {\displaystyle x^{k}e^{\alpha x},} where k is a nonnegative integer, α is a root of the characteristic polynomial of multiplicity m , and k < m . For proving that these functions are solutions, one may remark that if α is a root of the characteristic polynomial of multiplicity m , the characteristic polynomial may be factored as P ( t )( t − α ) m . Thus, applying the differential operator of the equation is equivalent with applying first m times the operator d d x − α {\textstyle {\frac {d}{dx}}-\alpha } , and then the operator that has P as characteristic polynomial. By the exponential shift theorem , ( d d x − α ) ( x k e α x ) = k x k − 1 e α x , {\displaystyle \left({\frac {d}{dx}}-\alpha \right)\left(x^{k}e^{\alpha x}\right)=kx^{k-1}e^{\alpha x},}
and thus one gets zero after k + 1 application of d d x − α {\textstyle {\frac {d}{dx}}-\alpha } .
As, by the fundamental theorem of algebra , the sum of the multiplicities of the roots of a polynomial equals the degree of the polynomial, the number of above solutions equals the order of the differential equation, and these solutions form a basis of the vector space of the solutions.
In the common case where the coefficients of the equation are real, it is generally more convenient to have a basis of the solutions consisting of real-valued functions . Such a basis may be obtained from the preceding basis by remarking that, if a + ib is a root of the characteristic polynomial, then a – ib is also a root, of the same multiplicity. Thus a real basis is obtained by using Euler's formula , and replacing x k e ( a + i b ) x {\displaystyle x^{k}e^{(a+ib)x}} and x k e ( a − i b ) x {\displaystyle x^{k}e^{(a-ib)x}} by x k e a x cos ( b x ) {\displaystyle x^{k}e^{ax}\cos(bx)} and x k e a x sin ( b x ) {\displaystyle x^{k}e^{ax}\sin(bx)} .
A homogeneous linear differential equation of the second order may be written y ″ + a y ′ + b y = 0 , {\displaystyle y''+ay'+by=0,} and its characteristic polynomial is r 2 + a r + b . {\displaystyle r^{2}+ar+b.}
If a and b are real , there are three cases for the solutions, depending on the discriminant D = a 2 − 4 b . In all three cases, the general solution depends on two arbitrary constants c 1 and c 2 .
Finding the solution y ( x ) satisfying y (0) = d 1 and y ′(0) = d 2 , one equates the values of the above general solution at 0 and its derivative there to d 1 and d 2 , respectively. This results in a linear system of two linear equations in the two unknowns c 1 and c 2 . Solving this system gives the solution for a so-called Cauchy problem , in which the values at 0 for the solution of the DEQ and its derivative are specified.
A non-homogeneous equation of order n with constant coefficients may be written y ( n ) ( x ) + a 1 y ( n − 1 ) ( x ) + ⋯ + a n − 1 y ′ ( x ) + a n y ( x ) = f ( x ) , {\displaystyle y^{(n)}(x)+a_{1}y^{(n-1)}(x)+\cdots +a_{n-1}y'(x)+a_{n}y(x)=f(x),} where a 1 , ..., a n are real or complex numbers, f is a given function of x , and y is the unknown function (for sake of simplicity, " ( x ) " will be omitted in the following).
There are several methods for solving such an equation. The best method depends on the nature of the function f that makes the equation non-homogeneous. If f is a linear combination of exponential and sinusoidal functions, then the exponential response formula may be used. If, more generally, f is a linear combination of functions of the form x n e ax , x n cos( ax ) , and x n sin( ax ) , where n is a nonnegative integer, and a a constant (which need not be the same in each term), then the method of undetermined coefficients may be used. Still more general, the annihilator method applies when f satisfies a homogeneous linear differential equation, typically, a holonomic function .
The most general method is the variation of constants , which is presented here.
The general solution of the associated homogeneous equation y ( n ) + a 1 y ( n − 1 ) + ⋯ + a n − 1 y ′ + a n y = 0 {\displaystyle y^{(n)}+a_{1}y^{(n-1)}+\cdots +a_{n-1}y'+a_{n}y=0} is y = u 1 y 1 + ⋯ + u n y n , {\displaystyle y=u_{1}y_{1}+\cdots +u_{n}y_{n},} where ( y 1 , ..., y n ) is a basis of the vector space of the solutions and u 1 , ..., u n are arbitrary constants. The method of variation of constants takes its name from the following idea. Instead of considering u 1 , ..., u n as constants, they can be considered as unknown functions that have to be determined for making y a solution of the non-homogeneous equation. For this purpose, one adds the constraints 0 = u 1 ′ y 1 + u 2 ′ y 2 + ⋯ + u n ′ y n 0 = u 1 ′ y 1 ′ + u 2 ′ y 2 ′ + ⋯ + u n ′ y n ′ ⋮ 0 = u 1 ′ y 1 ( n − 2 ) + u 2 ′ y 2 ( n − 2 ) + ⋯ + u n ′ y n ( n − 2 ) , {\displaystyle {\begin{aligned}0&=u'_{1}y_{1}+u'_{2}y_{2}+\cdots +u'_{n}y_{n}\\0&=u'_{1}y'_{1}+u'_{2}y'_{2}+\cdots +u'_{n}y'_{n}\\&\;\;\vdots \\0&=u'_{1}y_{1}^{(n-2)}+u'_{2}y_{2}^{(n-2)}+\cdots +u'_{n}y_{n}^{(n-2)},\end{aligned}}} which imply (by product rule and induction ) y ( i ) = u 1 y 1 ( i ) + ⋯ + u n y n ( i ) {\displaystyle y^{(i)}=u_{1}y_{1}^{(i)}+\cdots +u_{n}y_{n}^{(i)}} for i = 1, ..., n – 1 , and y ( n ) = u 1 y 1 ( n ) + ⋯ + u n y n ( n ) + u 1 ′ y 1 ( n − 1 ) + u 2 ′ y 2 ( n − 1 ) + ⋯ + u n ′ y n ( n − 1 ) . {\displaystyle y^{(n)}=u_{1}y_{1}^{(n)}+\cdots +u_{n}y_{n}^{(n)}+u'_{1}y_{1}^{(n-1)}+u'_{2}y_{2}^{(n-1)}+\cdots +u'_{n}y_{n}^{(n-1)}.}
Replacing in the original equation y and its derivatives by these expressions, and using the fact that y 1 , ..., y n are solutions of the original homogeneous equation, one gets f = u 1 ′ y 1 ( n − 1 ) + ⋯ + u n ′ y n ( n − 1 ) . {\displaystyle f=u'_{1}y_{1}^{(n-1)}+\cdots +u'_{n}y_{n}^{(n-1)}.}
This equation and the above ones with 0 as left-hand side form a system of n linear equations in u ′ 1 , ..., u ′ n whose coefficients are known functions ( f , the y i , and their derivatives). This system can be solved by any method of linear algebra . The computation of antiderivatives gives u 1 , ..., u n , and then y = u 1 y 1 + ⋯ + u n y n .
As antiderivatives are defined up to the addition of a constant, one finds again that the general solution of the non-homogeneous equation is the sum of an arbitrary solution and the general solution of the associated homogeneous equation.
The general form of a linear ordinary differential equation of order 1, after dividing out the coefficient of y ′( x ) , is: y ′ ( x ) = f ( x ) y ( x ) + g ( x ) . {\displaystyle y'(x)=f(x)y(x)+g(x).}
If the equation is homogeneous, i.e. g ( x ) = 0 , one may rewrite and integrate: y ′ y = f , log y = k + F , {\displaystyle {\frac {y'}{y}}=f,\qquad \log y=k+F,} where k is an arbitrary constant of integration and F = ∫ f d x {\displaystyle F=\textstyle \int f\,dx} is any antiderivative of f . Thus, the general solution of the homogeneous equation is y = c e F , {\displaystyle y=ce^{F},} where c = e k is an arbitrary constant.
For the general non-homogeneous equation, it is useful to multiply both sides of the equation by the reciprocal e − F of a solution of the homogeneous equation. [ 2 ] This gives y ′ e − F − y f e − F = g e − F . {\displaystyle y'e^{-F}-yfe^{-F}=ge^{-F}.} As − f e − F = d d x ( e − F ) , {\displaystyle -fe^{-F}={\tfrac {d}{dx}}\left(e^{-F}\right),} the product rule allows rewriting the equation as d d x ( y e − F ) = g e − F . {\displaystyle {\frac {d}{dx}}\left(ye^{-F}\right)=ge^{-F}.} Thus, the general solution is y = c e F + e F ∫ g e − F d x , {\displaystyle y=ce^{F}+e^{F}\int ge^{-F}dx,} where c is a constant of integration, and F is any antiderivative of f (changing of antiderivative amounts to change the constant of integration).
Solving the equation y ′ ( x ) + y ( x ) x = 3 x . {\displaystyle y'(x)+{\frac {y(x)}{x}}=3x.} The associated homogeneous equation y ′ ( x ) + y ( x ) x = 0 {\displaystyle y'(x)+{\frac {y(x)}{x}}=0} gives y ′ y = − 1 x , {\displaystyle {\frac {y'}{y}}=-{\frac {1}{x}},} that is y = c x . {\displaystyle y={\frac {c}{x}}.}
Dividing the original equation by one of these solutions gives x y ′ + y = 3 x 2 . {\displaystyle xy'+y=3x^{2}.} That is ( x y ) ′ = 3 x 2 , {\displaystyle (xy)'=3x^{2},} x y = x 3 + c , {\displaystyle xy=x^{3}+c,} and y ( x ) = x 2 + c / x . {\displaystyle y(x)=x^{2}+c/x.} For the initial condition y ( 1 ) = α , {\displaystyle y(1)=\alpha ,} one gets the particular solution y ( x ) = x 2 + α − 1 x . {\displaystyle y(x)=x^{2}+{\frac {\alpha -1}{x}}.}
A system of linear differential equations consists of several linear differential equations that involve several unknown functions. In general one restricts the study to systems such that the number of unknown functions equals the number of equations.
An arbitrary linear ordinary differential equation and a system of such equations can be converted into a first order system of linear differential equations by adding variables for all but the highest order derivatives. That is, if y ′ , y ″ , … , y ( k ) {\displaystyle y',y'',\ldots ,y^{(k)}} appear in an equation, one may replace them by new unknown functions y 1 , … , y k {\displaystyle y_{1},\ldots ,y_{k}} that must satisfy the equations y ′ = y 1 {\displaystyle y'=y_{1}} and y i ′ = y i + 1 , {\displaystyle y_{i}'=y_{i+1},} for i = 1, ..., k – 1 .
A linear system of the first order, which has n unknown functions and n differential equations may normally be solved for the derivatives of the unknown functions. If it is not the case this is a differential-algebraic system , and this is a different theory. Therefore, the systems that are considered here have the form y 1 ′ ( x ) = b 1 ( x ) + a 1 , 1 ( x ) y 1 + ⋯ + a 1 , n ( x ) y n ⋮ y n ′ ( x ) = b n ( x ) + a n , 1 ( x ) y 1 + ⋯ + a n , n ( x ) y n , {\displaystyle {\begin{aligned}y_{1}'(x)&=b_{1}(x)+a_{1,1}(x)y_{1}+\cdots +a_{1,n}(x)y_{n}\\[1ex]&\;\;\vdots \\[1ex]y_{n}'(x)&=b_{n}(x)+a_{n,1}(x)y_{1}+\cdots +a_{n,n}(x)y_{n},\end{aligned}}} where b n {\displaystyle b_{n}} and the a i , j {\displaystyle a_{i,j}} are functions of x . In matrix notation, this system may be written (omitting " ( x ) ") y ′ = A y + b . {\displaystyle \mathbf {y} '=A\mathbf {y} +\mathbf {b} .}
The solving method is similar to that of a single first order linear differential equations, but with complications stemming from noncommutativity of matrix multiplication.
Let u ′ = A u . {\displaystyle \mathbf {u} '=A\mathbf {u} .} be the homogeneous equation associated to the above matrix equation.
Its solutions form a vector space of dimension n , and are therefore the columns of a square matrix of functions U ( x ) {\displaystyle U(x)} , whose determinant is not the zero function. If n = 1 , or A is a matrix of constants, or, more generally, if A commutes with its antiderivative B = ∫ A d x {\displaystyle \textstyle B=\int Adx} , then one may choose U equal the exponential of B . In fact, in these cases, one has d d x exp ( B ) = A exp ( B ) . {\displaystyle {\frac {d}{dx}}\exp(B)=A\exp(B).} In the general case there is no closed-form solution for the homogeneous equation, and one has to use either a numerical method , or an approximation method such as Magnus expansion .
Knowing the matrix U , the general solution of the non-homogeneous equation is y ( x ) = U ( x ) y 0 + U ( x ) ∫ U − 1 ( x ) b ( x ) d x , {\displaystyle \mathbf {y} (x)=U(x)\mathbf {y_{0}} +U(x)\int U^{-1}(x)\mathbf {b} (x)\,dx,} where the column matrix y 0 {\displaystyle \mathbf {y_{0}} } is an arbitrary constant of integration .
If initial conditions are given as y ( x 0 ) = y 0 , {\displaystyle \mathbf {y} (x_{0})=\mathbf {y} _{0},} the solution that satisfies these initial conditions is y ( x ) = U ( x ) U − 1 ( x 0 ) y 0 + U ( x ) ∫ x 0 x U − 1 ( t ) b ( t ) d t . {\displaystyle \mathbf {y} (x)=U(x)U^{-1}(x_{0})\mathbf {y_{0}} +U(x)\int _{x_{0}}^{x}U^{-1}(t)\mathbf {b} (t)\,dt.}
A linear ordinary equation of order one with variable coefficients may be solved by quadrature , which means that the solutions may be expressed in terms of integrals . This is not the case for order at least two. This is the main result of Picard–Vessiot theory which was initiated by Émile Picard and Ernest Vessiot , and whose recent developments are called differential Galois theory .
The impossibility of solving by quadrature can be compared with the Abel–Ruffini theorem , which states that an algebraic equation of degree at least five cannot, in general, be solved by radicals. This analogy extends to the proof methods and motivates the denomination of differential Galois theory .
Similarly to the algebraic case, the theory allows deciding which equations may be solved by quadrature, and if possible solving them. However, for both theories, the necessary computations are extremely difficult, even with the most powerful computers.
Nevertheless, the case of order two with rational coefficients has been completely solved by Kovacic's algorithm .
Cauchy–Euler equations are examples of equations of any order, with variable coefficients, that can be solved explicitly. These are the equations of the form x n y ( n ) ( x ) + a n − 1 x n − 1 y ( n − 1 ) ( x ) + ⋯ + a 0 y ( x ) = 0 , {\displaystyle x^{n}y^{(n)}(x)+a_{n-1}x^{n-1}y^{(n-1)}(x)+\cdots +a_{0}y(x)=0,} where a 0 , … , a n − 1 {\displaystyle a_{0},\ldots ,a_{n-1}} are constant coefficients.
A holonomic function , also called a D-finite function , is a function that is a solution of a homogeneous linear differential equation with polynomial coefficients.
Most functions that are commonly considered in mathematics are holonomic or quotients of holonomic functions. In fact, holonomic functions include polynomials , algebraic functions , logarithm , exponential function , sine , cosine , hyperbolic sine , hyperbolic cosine , inverse trigonometric and inverse hyperbolic functions , and many special functions such as Bessel functions and hypergeometric functions .
Holonomic functions have several closure properties ; in particular, sums, products, derivative and integrals of holonomic functions are holonomic. Moreover, these closure properties are effective, in the sense that there are algorithms for computing the differential equation of the result of any of these operations, knowing the differential equations of the input. [ 3 ]
Usefulness of the concept of holonomic functions results of Zeilberger's theorem, which follows. [ 3 ]
A holonomic sequence is a sequence of numbers that may be generated by a recurrence relation with polynomial coefficients. The coefficients of the Taylor series at a point of a holonomic function form a holonomic sequence. Conversely, if the sequence of the coefficients of a power series is holonomic, then the series defines a holonomic function (even if the radius of convergence is zero). There are efficient algorithms for both conversions, that is for computing the recurrence relation from the differential equation, and vice versa . [ 3 ]
It follows that, if one represents (in a computer) holonomic functions by their defining differential equations and initial conditions, most calculus operations can be done automatically on these functions, such as derivative , indefinite and definite integral , fast computation of Taylor series (thanks of the recurrence relation on its coefficients), evaluation to a high precision with certified bound of the approximation error, limits , localization of singularities , asymptotic behavior at infinity and near singularities, proof of identities, etc. [ 4 ] | https://en.wikipedia.org/wiki/Linear_differential_equation |
Linear dynamical systems are dynamical systems whose evolution functions are linear . While dynamical systems, in general, do not have closed-form solutions , linear dynamical systems can be solved exactly, and they have a rich set of mathematical properties. Linear systems can also be used to understand the qualitative behavior of general dynamical systems, by calculating the equilibrium points of the system and approximating it as a linear system around each such point.
In a linear dynamical system, the variation of a state vector
(an N {\displaystyle N} -dimensional vector denoted x {\displaystyle \mathbf {x} } ) equals a constant matrix
(denoted A {\displaystyle \mathbf {A} } ) multiplied by x {\displaystyle \mathbf {x} } . This variation can take two forms: either
as a flow , in which x {\displaystyle \mathbf {x} } varies
continuously with time
or as a mapping, in which x {\displaystyle \mathbf {x} } varies in discrete steps
These equations are linear in the following sense: if x ( t ) {\displaystyle \mathbf {x} (t)} and y ( t ) {\displaystyle \mathbf {y} (t)} are two valid solutions, then so is any linear combination of the two solutions, e.g., z ( t ) = d e f α x ( t ) + β y ( t ) {\displaystyle \mathbf {z} (t)\ {\stackrel {\mathrm {def} }{=}}\ \alpha \mathbf {x} (t)+\beta \mathbf {y} (t)} where α {\displaystyle \alpha } and β {\displaystyle \beta } are any two scalars . The matrix A {\displaystyle \mathbf {A} } need not be symmetric .
Linear dynamical systems can be solved exactly, in contrast to most nonlinear ones. Occasionally, a nonlinear system can be solved exactly by a change of variables to a linear system. Moreover, the solutions of (almost) any nonlinear system can be well-approximated by an equivalent linear system near its fixed points . Hence, understanding linear systems and their solutions is a crucial first step to understanding the more complex nonlinear systems.
If the initial vector x 0 = d e f x ( t = 0 ) {\displaystyle \mathbf {x} _{0}\ {\stackrel {\mathrm {def} }{=}}\ \mathbf {x} (t=0)} is aligned with a right eigenvector r k {\displaystyle \mathbf {r} _{k}} of
the matrix A {\displaystyle \mathbf {A} } , the dynamics are simple
where λ k {\displaystyle \lambda _{k}} is the corresponding eigenvalue ;
the solution of this equation is
as may be confirmed by substitution.
If A {\displaystyle \mathbf {A} } is diagonalizable , then any vector in an N {\displaystyle N} -dimensional space can be represented by a linear combination of the right and left eigenvectors (denoted l k {\displaystyle \mathbf {l} _{k}} ) of the matrix A {\displaystyle \mathbf {A} } .
Therefore, the general solution for x ( t ) {\displaystyle \mathbf {x} (t)} is
a linear combination of the individual solutions for the right
eigenvectors
Similar considerations apply to the discrete mappings.
The roots of the characteristic polynomial det( A - λ I ) are the eigenvalues of A . The sign and relation of these roots, λ n {\displaystyle \lambda _{n}} , to each other may be used to determine the stability of the dynamical system
For a 2-dimensional system, the characteristic polynomial is of the form λ 2 − τ λ + Δ = 0 {\displaystyle \lambda ^{2}-\tau \lambda +\Delta =0} where τ {\displaystyle \tau } is the trace and Δ {\displaystyle \Delta } is the determinant of A . Thus the two roots are in the form:
and Δ = λ 1 λ 2 {\displaystyle \Delta =\lambda _{1}\lambda _{2}} and τ = λ 1 + λ 2 {\displaystyle \tau =\lambda _{1}+\lambda _{2}} . Thus if Δ < 0 {\displaystyle \Delta <0} then the eigenvalues are of opposite sign, and the fixed point is a saddle. If Δ > 0 {\displaystyle \Delta >0} then the eigenvalues are of the same sign. Therefore, if τ > 0 {\displaystyle \tau >0} both are positive and the point is unstable, and if τ < 0 {\displaystyle \tau <0} then both are negative and the point is stable. The discriminant will tell you if the point is nodal or spiral (i.e. if the eigenvalues are real or complex). | https://en.wikipedia.org/wiki/Linear_dynamical_system |
In dosimetry , linear energy transfer (LET) is the amount of energy that an ionizing particle transfers to the material traversed per unit distance. It describes the action of radiation into matter.
It is identical to the retarding force acting on a charged ionizing particle travelling through the matter. [ 1 ] By definition, LET is a positive quantity. LET depends on the nature of the radiation as well as on the material traversed.
A high LET will slow down the radiation more quickly, generally making shielding more effective and preventing deep penetration. On the other hand, the higher concentration of deposited energy can cause more severe damage to any microscopic structures near the particle track. If a microscopic defect can cause larger-scale failure, as is the case in biological cells and microelectronics , the LET helps explain why radiation damage is sometimes disproportionate to the absorbed dose . Dosimetry attempts to factor in this effect with radiation weighting factors .
Linear energy transfer is closely related to stopping power , since both equal the retarding force. The unrestricted linear energy transfer is identical to linear electronic stopping power, as discussed below. But the stopping power and LET concepts are different in the respect that total stopping power has the nuclear stopping power component, [ 2 ] and this component does not cause electronic excitations. Hence nuclear stopping power is not contained in LET.
The appropriate SI unit for LET is the newton , but it is most typically expressed in units of kiloelectronvolts per micrometre (keV/μm) or megaelectronvolts per centimetre (MeV/cm). While medical physicists and radiobiologists usually speak of linear energy transfer , most non-medical physicists talk about stopping power .
The secondary electrons produced during the process of ionization by the primary charged particle are conventionally called delta rays , if their energy is large enough so that they themselves can ionize. [ 3 ] Many studies focus upon the energy transferred in the vicinity of the primary particle track and therefore exclude interactions that produce delta rays with energies larger than a certain value Δ. [ 1 ] This energy limit is meant to exclude secondary electrons that carry energy far from the primary particle track, since a larger energy implies a larger range . This approximation neglects the directional distribution of secondary radiation and the non-linear path of delta rays, but simplifies analytic evaluation. [ 4 ]
In mathematical terms, Restricted linear energy transfer is defined by
where d E Δ {\displaystyle {\text{d}}E_{\Delta }} is the energy loss of the charged particle due to electronic collisions while traversing a distance d x {\displaystyle {{\text{d}}x}} , excluding all secondary electrons with kinetic energies larger than Δ. If Δ tends toward infinity, then there are no electrons with larger energy, and the linear energy transfer becomes the unrestricted linear energy transfer which is identical to the linear electronic stopping power . [ 1 ] Here, the use of the term "infinity" is not to be taken literally; it simply means that no energy transfers, however large, are excluded.
During his investigations of radioactivity, Ernest Rutherford coined the terms alpha rays , beta rays and gamma rays for the three types of emissions that occur during radioactive decay .
Linear energy transfer is best defined for monoenergetic ions, i.e. protons , alpha particles , and the heavier nuclei called HZE ions found in cosmic rays or produced by particle accelerators . These particles cause frequent direct ionizations within a narrow diameter around a relatively straight track, thus approximating continuous deceleration. As they slow down, the changing particle cross section modifies their LET, generally increasing it to a Bragg peak just before achieving thermal equilibrium with the absorber, i.e., before the end of range . At equilibrium, the incident particle essentially comes to rest or is absorbed, at which point LET is undefined.
Since the LET varies over the particle track, an average value is often used to represent the spread. Averages weighted by track length or weighted by absorbed dose are present in the literature, with the latter being more common in dosimetry. These averages are not widely separated for heavy particles with high LET, but the difference becomes more important in the other type of radiations discussed below. [ 4 ]
Often overlooked for alpha particles is the recoil-nucleus of the alpha emitter, which has significant ionization energy of roughly 5% of the alpha particle, but because of its high electric charge and large mass, has an ultra-short range of only a few Angstroms . This can skew results significantly if one is examining the Relative Biological Effectiveness of the alpha particle in the cytoplasm, while ignoring the recoil nucleus contribution, which alpha-parent being one of numerous heavy metals , is typically adhered to chromatic material such as chromosomes .
Electrons produced in nuclear decay are called beta particles . Because of their low mass relative to atoms, they are strongly scattered by nuclei (Coulomb or Rutherford scattering ), much more so than heavier particles. Beta particle tracks are therefore crooked. In addition to producing secondary electrons (delta rays) while ionizing atoms, they also produce bremsstrahlung photons. A maximum range of beta radiation can be defined experimentally [ 5 ] which is smaller than the range that would be measured along the particle path.
Gamma rays are photons, whose absorption cannot be described by LET. When a gamma quantum passes through matter, it may be absorbed in a single process ( photoelectric effect , Compton effect or pair production ), or it continues unchanged on its path. (Only in the case of the Compton effect, another gamma quantum of lower energy proceeds). Gamma ray absorption therefore obeys an exponential law (see Gamma rays ); the absorption is described by the absorption coefficient or by the half-value thickness .
LET has therefore no meaning when applied to photons. However, many authors speak of "gamma LET" anyway, [ 6 ] where they are actually referring to the LET of the secondary electrons , i.e., mainly Compton electrons, produced by the gamma radiation. [ 7 ] The secondary electrons will ionize far more atoms than the primary photon. This gamma LET has little relation to the attenuation rate of the beam, but it may have some correlation to the microscopic defects produced in the absorber. Even a monoenergetic gamma beam will produce a spectrum of electrons, and each secondary electron will have a variable LET as it slows down, as discussed above. The "gamma LET" is therefore an average.
The transfer of energy from an uncharged primary particle to charged secondary particles can also be described by using the mass energy-transfer coefficient . [ 1 ]
Many studies have attempted to relate linear energy transfer to the relative biological effectiveness (RBE) of radiation, with inconsistent results. The relationship varies widely depending on the nature of the biological material, and the choice of endpoint to define effectiveness. Even when these are held constant, different radiation spectra that shared the same LET have significantly different RBE. [ 4 ]
Despite these variations, some overall trends are commonly seen. The RBE is generally independent of LET for any LET less than 10 keV/μm, so a low LET is normally chosen as the reference condition where RBE is set to unity. Above 10 keV/μm, some systems show a decline in RBE with increasing LET, while others show an initial increase to a peak before declining. Mammalian cells usually experience a peak RBE for LET's around 100 keV/μm. [ 4 ] These are very rough numbers; for example, one set of experiments found a peak at 30 keV/μm.
The International Commission on Radiation Protection ( ICRP ) proposed a simplified model of RBE-LET relationships for use in dosimetry . They defined a quality factor of radiation as a function of dose-averaged unrestricted LET in water, and intended it as a highly uncertain, but generally conservative, approximation of RBE. Different iterations of their model are shown in the graph to the right. The 1966 model was integrated into their 1977 recommendations for radiation protection in ICRP 26. This model was largely replaced in the 1991 recommendations of ICRP 60 by radiation weighting factors that were tied to the particle type and independent of LET. ICRP 60 revised the quality factor function and reserved it for use with unusual radiation types that did not have radiation weighting factors assigned to them. [ 8 ]
When used to describe the dosimetry of ionizing radiation in the biological or biomedical setting, the LET (like linear stopping power ) is usually expressed in units of k eV / μm .
In space applications , electronic devices can be disturbed by the passage of energetic electrons, protons or heavier ions that may alter the state of a circuit , producing " single event effects ". [ 9 ] The effect of the radiation is described by the LET (which is here taken as synonymous with stopping power), typically expressed in units of MeV·cm 2 /mg of material, the units used for mass stopping power (the material in question is usually Si for MOS devices). The units of measurement arise from a combination of the energy lost by the particle to the material per unit path length (MeV/cm) divided by the density of the material (mg/cm 3 ). [ 10 ]
"Soft errors" of electronic devices due to cosmic rays on earth are, however, mostly due to neutrons which do not directly interact with the material and whose passage can therefore not be described by LET. Rather, one measures their effect in terms of neutrons per cm 2 per hour, see Soft error . | https://en.wikipedia.org/wiki/Linear_energy_transfer |
In mathematics , a linear equation is an equation that may be put in the form a 1 x 1 + … + a n x n + b = 0 , {\displaystyle a_{1}x_{1}+\ldots +a_{n}x_{n}+b=0,} where x 1 , … , x n {\displaystyle x_{1},\ldots ,x_{n}} are the variables (or unknowns ), and b , a 1 , … , a n {\displaystyle b,a_{1},\ldots ,a_{n}} are the coefficients , which are often real numbers . The coefficients may be considered as parameters of the equation and may be arbitrary expressions , provided they do not contain any of the variables. To yield a meaningful equation, the coefficients a 1 , … , a n {\displaystyle a_{1},\ldots ,a_{n}} are required to not all be zero.
Alternatively, a linear equation can be obtained by equating to zero a linear polynomial over some field , from which the coefficients are taken.
The solutions of such an equation are the values that, when substituted for the unknowns, make the equality true.
In the case of just one variable, there is exactly one solution (provided that a 1 ≠ 0 {\displaystyle a_{1}\neq 0} ). Often, the term linear equation refers implicitly to this particular case, in which the variable is sensibly called the unknown .
In the case of two variables, each solution may be interpreted as the Cartesian coordinates of a point of the Euclidean plane . The solutions of a linear equation form a line in the Euclidean plane, and, conversely, every line can be viewed as the set of all solutions of a linear equation in two variables. This is the origin of the term linear for describing this type of equation. More generally, the solutions of a linear equation in n variables form a hyperplane (a subspace of dimension n − 1 ) in the Euclidean space of dimension n .
Linear equations occur frequently in all mathematics and their applications in physics and engineering , partly because non-linear systems are often well approximated by linear equations.
This article considers the case of a single equation with coefficients from the field of real numbers , for which one studies the real solutions. All of its content applies to complex solutions and, more generally, to linear equations with coefficients and solutions in any field . For the case of several simultaneous linear equations, see system of linear equations .
A linear equation in one variable x can be written as a x + b = 0 , {\displaystyle ax+b=0,} with a ≠ 0 {\displaystyle a\neq 0} .
The solution is x = − b a {\displaystyle x=-{\frac {b}{a}}} .
A linear equation in two variables x and y can be written as a x + b y + c = 0 , {\displaystyle ax+by+c=0,} where a and b are not both 0 . [ 1 ]
If a and b are real numbers, it has infinitely many solutions.
If b ≠ 0 , the equation
is a linear equation in the single variable y for every value of x . It therefore has a unique solution for y , which is given by
This defines a function . The graph of this function is a line with slope − a b {\displaystyle -{\frac {a}{b}}} and y -intercept − c b . {\displaystyle -{\frac {c}{b}}.} The functions whose graph is a line are generally called linear functions in the context of calculus . However, in linear algebra , a linear function is a function that maps a sum to the sum of the images of the summands. So, for this definition, the above function is linear only when c = 0 , that is when the line passes through the origin. To avoid confusion, the functions whose graph is an arbitrary line are often called affine functions , and the linear functions such that c = 0 are often called linear maps .
Each solution ( x , y ) of a linear equation
may be viewed as the Cartesian coordinates of a point in the Euclidean plane . With this interpretation, all solutions of the equation form a line , provided that a and b are not both zero. Conversely, every line is the set of all solutions of a linear equation.
The phrase "linear equation" takes its origin in this correspondence between lines and equations: a linear equation in two variables is an equation whose solutions form a line.
If b ≠ 0 , the line is the graph of the function of x that has been defined in the preceding section. If b = 0 , the line is a vertical line (that is a line parallel to the y -axis) of equation x = − c a , {\displaystyle x=-{\frac {c}{a}},} which is not the graph of a function of x .
Similarly, if a ≠ 0 , the line is the graph of a function of y , and, if a = 0 , one has a horizontal line of equation y = − c b . {\displaystyle y=-{\frac {c}{b}}.}
There are various ways of defining a line. In the following subsections, a linear equation of the line is given in each case.
A non-vertical line can be defined by its slope m , and its y -intercept y 0 (the y coordinate of its intersection with the y -axis). In this case, its linear equation can be written
If, moreover, the line is not horizontal, it can be defined by its slope and its x -intercept x 0 . In this case, its equation can be written
or, equivalently,
These forms rely on the habit of considering a nonvertical line as the graph of a function . [ 2 ] For a line given by an equation
these forms can be easily deduced from the relations
A non-vertical line can be defined by its slope m , and the coordinates x 1 , y 1 {\displaystyle x_{1},y_{1}} of any point of the line. In this case, a linear equation of the line is
or
This equation can also be written
to emphasize that the slope of a line can be computed from the coordinates of any two points.
A line that is not parallel to an axis and does not pass through the origin cuts the axes into two different points. The intercept values x 0 and y 0 of these two points are nonzero, and an equation of the line is [ 3 ]
(It is easy to verify that the line defined by this equation has x 0 and y 0 as intercept values).
Given two different points ( x 1 , y 1 ) and ( x 2 , y 2 ) , there is exactly one line that passes through them. There are several ways to write a linear equation of this line.
If x 1 ≠ x 2 , the slope of the line is y 2 − y 1 x 2 − x 1 . {\displaystyle {\frac {y_{2}-y_{1}}{x_{2}-x_{1}}}.} Thus, a point-slope form is [ 3 ]
By clearing denominators , one gets the equation
which is valid also when x 1 = x 2 (to verify this, it suffices to verify that the two given points satisfy the equation).
This form is not symmetric in the two given points, but a symmetric form can be obtained by regrouping the constant terms:
(exchanging the two points changes the sign of the left-hand side of the equation).
The two-point form of the equation of a line can be expressed simply in terms of a determinant . There are two common ways for that.
The equation ( x 2 − x 1 ) ( y − y 1 ) − ( y 2 − y 1 ) ( x − x 1 ) = 0 {\displaystyle (x_{2}-x_{1})(y-y_{1})-(y_{2}-y_{1})(x-x_{1})=0} is the result of expanding the determinant in the equation
The equation ( y 1 − y 2 ) x + ( x 2 − x 1 ) y + ( x 1 y 2 − x 2 y 1 ) = 0 {\displaystyle (y_{1}-y_{2})x+(x_{2}-x_{1})y+(x_{1}y_{2}-x_{2}y_{1})=0} can be obtained by expanding with respect to its first row the determinant in the equation
Besides being very simple and mnemonic, this form has the advantage of being a special case of the more general equation of a hyperplane passing through n points in a space of dimension n − 1 . These equations rely on the condition of linear dependence of points in a projective space .
A linear equation with more than two variables may always be assumed to have the form
The coefficient b , often denoted a 0 is called the constant term (sometimes the absolute term in old books [ 4 ] [ 5 ] ). Depending on the context, the term coefficient can be reserved for the a i with i > 0 .
When dealing with n = 3 {\displaystyle n=3} variables, it is common to use x , y {\displaystyle x,\;y} and z {\displaystyle z} instead of indexed variables.
A solution of such an equation is a n -tuple such that substituting each element of the tuple for the corresponding variable transforms the equation into a true equality.
For an equation to be meaningful, the coefficient of at least one variable must be non-zero. If every variable has a zero coefficient, then, as mentioned for one variable, the equation is either inconsistent (for b ≠ 0 ) as having no solution, or all n -tuples are solutions.
The n -tuples that are solutions of a linear equation in n variables are the Cartesian coordinates of the points of an ( n − 1) -dimensional hyperplane in an n -dimensional Euclidean space (or affine space if the coefficients are complex numbers or belong to any field). In the case of three variables, this hyperplane is a plane .
If a linear equation is given with a j ≠ 0 , then the equation can be solved for x j , yielding
If the coefficients are real numbers , this defines a real-valued function of n real variables . | https://en.wikipedia.org/wiki/Linear_equation |
In algebra , linear equations and systems of linear equations over a field are widely studied. "Over a field" means that the coefficients of the equations and the solutions that one is looking for belong to a given field, commonly the real or the complex numbers . This article is devoted to the same problems where "field" is replaced by " commutative ring ", or "typically Noetherian integral domain ".
In the case of a single equation, the problem splits in two parts. First, the ideal membership problem , which consists, given a non-homogeneous equation
with a 1 , … , a k {\displaystyle a_{1},\ldots ,a_{k}} and b in a given ring R , to decide if it has a solution with x 1 , … , x k {\displaystyle x_{1},\ldots ,x_{k}} in R , and, if any, to provide one. This amounts to decide if b belongs to the ideal generated by the a i . The simplest instance of this problem is, for k = 1 and b = 1 , to decide if a is a unit in R .
The syzygy problem consists, given k elements a 1 , … , a k {\displaystyle a_{1},\ldots ,a_{k}} in R , to provide a system of generators of the module of the syzygies of ( a 1 , … , a k ) , {\displaystyle (a_{1},\ldots ,a_{k}),} that is a system of generators of the submodule of those elements ( x 1 , … , x k ) {\displaystyle (x_{1},\ldots ,x_{k})} in R k that are solutions of the homogeneous equation
The simplest case, when k = 1 amounts to find a system of generators of the annihilator of a 1 .
Given a solution of the ideal membership problem, one obtains all the solutions by adding to it the elements of the module of syzygies. In other words, all the solutions are provided by the solution of these two partial problems.
In the case of several equations, the same decomposition into subproblems occurs. The first problem becomes the submodule membership problem . The second one is also called the syzygy problem .
A ring such that there are algorithms for the arithmetic operations (addition, subtraction, multiplication) and for the above problems may be called a computable ring , or effective ring . One may also say that linear algebra on the ring is effective .
The article considers the main rings for which linear algebra is effective.
To be able to solve the syzygy problem, it is necessary that the module of syzygies is finitely generated , because it is impossible to output an infinite list. Therefore, the problems considered here make sense only for a Noetherian ring , or at least a coherent ring . In fact, this article is restricted to Noetherian integral domains because of the following result. [ 1 ]
This theorem is useful to prove the existence of algorithms. However, in practice, the algorithms for the systems are designed directly.
A field is an effective ring as soon one has algorithms for addition, subtraction, multiplication, and computation of multiplicative inverses . In fact, solving the submodule membership problem is what is commonly called solving the system , and solving the syzygy problem is the computation of the null space of the matrix of a system of linear equations . The basic algorithm for both problems is Gaussian elimination .
Let R be an effective commutative ring.
There are algorithms to solve all the problems addressed in this article over the integers . In other words, linear algebra is effective over the integers ; see Linear Diophantine system for details.
More generally, linear algebra is effective on a principal ideal domain if there are algorithms for addition, subtraction and multiplication, and
It is useful to extend to the general case the notion of a unimodular matrix by calling unimodular a square matrix whose determinant is a unit . This means that the determinant is invertible and implies that the unimodular matrices are exactly the invertible matrices such all entries of the inverse matrix belong to the domain.
The above two algorithms imply that given a and b in the principal ideal domain, there is an algorithm computing a unimodular matrix
such that
(This algorithm is obtained by taking for s and t the coefficients of Bézout's identity, and for u and v the quotient of − b and a by as + bt ; this choice implies that the determinant of the square matrix is 1 .)
Having such an algorithm, the Smith normal form of a matrix may be computed exactly as in the integer case, and this suffices to apply the described in Linear Diophantine system for getting an algorithm for solving every linear system.
The main case where this is commonly used is the case of linear systems over the ring of univariate polynomials over a field. In this case, the extended Euclidean algorithm may be used for computing the above unimodular matrix; see Polynomial greatest common divisor § Bézout's identity and extended GCD algorithm for details.
Linear algebra is effective on a polynomial ring k [ x 1 , … , x n ] {\displaystyle k[x_{1},\ldots ,x_{n}]} over a field k . This has been first proved in 1926 by Grete Hermann . [ 2 ] The algorithms resulting from Hermann's results are only of historical interest, as their computational complexity is too high for allowing effective computer computation.
Proofs that linear algebra is effective on polynomial rings and computer implementations are presently all based on Gröbner basis theory . | https://en.wikipedia.org/wiki/Linear_equation_over_a_ring |
In order theory , a branch of mathematics , a linear extension of a partial order is a total order (or linear order) that is compatible with the partial order. As a classic example, the lexicographic order of totally ordered sets is a linear extension of their product order .
A partial order is a reflexive , transitive and antisymmetric relation. Given any partial orders ≤ {\displaystyle \,\leq \,} and ≤ ∗ {\displaystyle \,\leq ^{*}\,} on a set X , {\displaystyle X,} ≤ ∗ {\displaystyle \,\leq ^{*}\,} is a linear extension of ≤ {\displaystyle \,\leq \,} exactly when
It is that second property that leads mathematicians to describe ≤ ∗ {\displaystyle \,\leq ^{*}\,} as extending ≤ . {\displaystyle \,\leq .}
Alternatively, a linear extension may be viewed as an order-preserving bijection from a partially ordered set P {\displaystyle P} to a chain C {\displaystyle C} on the same ground set.
A preorder is a reflexive and transitive relation. The difference between a preorder and a partial-order is that a preorder allows two different items to be considered "equivalent", that is, both x ≾ y {\displaystyle x\precsim y} and y ≾ x {\displaystyle y\precsim x} hold, while a partial-order allows this only when x = y {\displaystyle x=y} . A relation ≾ ∗ {\displaystyle \precsim ^{*}} is called a linear extension of a preorder ≾ {\displaystyle \precsim } if:
The difference between these definitions is only in condition 3. When the extension is a partial order, condition 3 need not be stated explicitly, since it follows from condition 2. Proof : suppose that x ≾ y {\displaystyle x\precsim y} and not y ≾ x {\displaystyle y\precsim x} . By condition 2, x ≾ ∗ y {\displaystyle x\precsim ^{*}y} . By reflexivity, "not y ≾ x {\displaystyle y\precsim x} " implies that y ≠ x {\displaystyle y\neq x} . Since ≾ ∗ {\displaystyle \precsim ^{*}} is a partial order, x ≾ ∗ y {\displaystyle x\precsim ^{*}y} and y ≠ x {\displaystyle y\neq x} imply "not y ≾ ∗ x {\displaystyle y\precsim ^{*}x} ". Therefore, x ≺ ∗ y {\displaystyle x\prec ^{*}y} .
However, for general preorders, condition 3 is needed to rule out trivial extensions. Without this condition, the preorder by which all elements are equivalent ( y ≾ x {\displaystyle y\precsim x} and x ≾ y {\displaystyle x\precsim y} hold for all pairs x , y ) would be an extension of every preorder.
The statement that every partial order can be extended to a total order is known as the order-extension principle . A proof using the axiom of choice was first published by Edward Marczewski (Szpilrajin) in 1930. Marczewski writes that the theorem had previously been proven by Stefan Banach , Kazimierz Kuratowski , and Alfred Tarski , again using the axiom of choice, but that the proofs had not been published. [ 1 ]
There is an analogous statement for preorders: every preorder can be extended to a total preorder. This statement was proved by Hansson. [ 2 ] : Lemma 3
In modern axiomatic set theory the order-extension principle is itself taken as an axiom, of comparable ontological status to the axiom of choice. The order-extension principle is implied by the Boolean prime ideal theorem or the equivalent compactness theorem , [ 3 ] but the reverse implication doesn't hold. [ 4 ]
Applying the order-extension principle to a partial order in which every two elements are incomparable shows that (under this principle) every set can be linearly ordered. This assertion that every set can be linearly ordered is known as the ordering principle , OP, and is a weakening of the well-ordering theorem . However, there are models of set theory in which the ordering principle holds while the order-extension principle does not. [ 5 ]
The order extension principle is constructively provable for finite sets using topological sorting algorithms, where the partial order is represented by a directed acyclic graph with the set's elements as its vertices . Several algorithms can find an extension in linear time . [ 6 ] Despite the ease of finding a single linear extension, the problem of counting all linear extensions of a finite partial order is #P-complete ; however, it may be estimated by a fully polynomial-time randomized approximation scheme . [ 7 ] [ 8 ] Among all partial orders with a fixed number of elements and a fixed number of comparable pairs, the partial orders that have the largest number of linear extensions are semiorders . [ 9 ]
The order dimension of a partial order is the minimum cardinality of a set of linear extensions whose intersection is the given partial order; equivalently, it is the minimum number of linear extensions needed to ensure that each critical pair of the partial order is reversed in at least one of the extensions.
Antimatroids may be viewed as generalizing partial orders; in this view, the structures corresponding to the linear extensions of a partial order are the basic words of the antimatroid. [ 10 ]
This area also includes one of order theory's most famous open problems, the 1/3–2/3 conjecture , which states that in any finite partially ordered set P {\displaystyle P} that is not totally ordered there exists a pair ( x , y ) {\displaystyle (x,y)} of elements of P {\displaystyle P} for which the linear extensions of P {\displaystyle P} in which x < y {\displaystyle x<y} number between 1/3 and 2/3 of the total number of linear extensions of P . {\displaystyle P.} [ 11 ] [ 12 ] An equivalent way of stating the conjecture is that, if one chooses a linear extension of P {\displaystyle P} uniformly at random, there is a pair ( x , y ) {\displaystyle (x,y)} which has probability between 1/3 and 2/3 of being ordered as x < y . {\displaystyle x<y.} However, for certain infinite partially ordered sets, with a canonical probability defined on its linear extensions as a limit of the probabilities for finite partial orders that cover the infinite partial order, the 1/3–2/3 conjecture does not hold. [ 13 ]
Counting the number of linear extensions of a finite poset is a common problem in algebraic combinatorics . This number is given by the leading coefficient of the order polynomial multiplied by | P | ! . {\displaystyle |P|!.}
Young tableau can be considered as linear extensions of a finite order-ideal in the infinite poset N × N , {\displaystyle \mathbb {N} \times \mathbb {N} ,} and they are counted by the hook length formula . | https://en.wikipedia.org/wiki/Linear_extension |
In mathematics , especially in the area of mathematical analysis known as dynamical systems theory , a linear flow on the torus is a flow on the n -dimensional torus T n = S 1 × S 1 × ⋯ × S 1 ⏟ n {\displaystyle \mathbb {T} ^{n}=\underbrace {S^{1}\times S^{1}\times \cdots \times S^{1}} _{n}} ,
which is represented by the following differential equations with respect to the standard angular coordinates ( θ 1 , θ 2 , … , θ n ) : {\displaystyle \left(\theta _{1},\theta _{2},\ldots ,\theta _{n}\right):} d θ 1 d t = ω 1 , d θ 2 d t = ω 2 , … , d θ n d t = ω n {\displaystyle {\frac {d\theta _{1}}{dt}}=\omega _{1},\quad {\frac {d\theta _{2}}{dt}}=\omega _{2},\quad \ldots ,\quad {\frac {d\theta _{n}}{dt}}=\omega _{n}} .
The solution of these equations can explicitly be expressed as Φ ω t ( θ 1 , θ 2 , … , θ n ) = ( θ 1 + ω 1 t , θ 2 + ω 2 t , … , θ n + ω n t ) mod 2 π {\displaystyle \Phi _{\omega }^{t}(\theta _{1},\theta _{2},\dots ,\theta _{n})=(\theta _{1}+\omega _{1}t,\theta _{2}+\omega _{2}t,\dots ,\theta _{n}+\omega _{n}t){\bmod {2}}\pi } .
If we represent the torus as T n = R n / Z n {\displaystyle \mathbb {T} ^{n}=\mathbb {R} ^{n}/\mathbb {Z} ^{n}} we see that a starting point is moved by the flow in the direction ω = ( ω 1 , ω 2 , … , ω n ) {\displaystyle \omega =\left(\omega _{1},\omega _{2},\ldots ,\omega _{n}\right)} at constant speed and when it reaches the border of the unitary n {\displaystyle n} -cube it jumps to the opposite face of the cube.
For a linear flow on the torus, all orbits are either periodic or dense on a subset of the n {\displaystyle n} -torus, which is a k {\displaystyle k} -torus. When the components of ω {\displaystyle \omega } are rationally independent all the orbits are dense on the whole space. This can be easily seen in the two-dimensional case: if the two components of ω {\displaystyle \omega } are rationally independent, the Poincaré section of the flow on an edge of the unit square is an irrational rotation on a circle and therefore its orbits are dense on the circle, as a consequence the orbits of the flow must be dense on the torus.
In topology , an irrational winding of a torus is a continuous injection of a line into a two-dimensional torus that is used to set up several counterexamples. [ 1 ] A related notion is the Kronecker foliation of a torus, a foliation formed by the set of all translates of a given irrational winding.
One way of constructing a torus is as the quotient space T 2 = R 2 / Z 2 {\displaystyle \mathbb {T^{2}} =\mathbb {R} ^{2}/\mathbb {Z} ^{2}} of a two-dimensional real vector space by the additive subgroup of integer vectors, with the corresponding projection π : R 2 → T 2 . {\displaystyle \pi :\mathbb {R} ^{2}\to \mathbb {T^{2}} .} Each point in the torus has as its preimage one of the translates of the square lattice Z 2 {\displaystyle \mathbb {Z} ^{2}} in R 2 , {\displaystyle \mathbb {R} ^{2},} and π {\displaystyle \pi } factors through a map that takes any point in the plane to a point in the unit square [ 0 , 1 ) 2 {\displaystyle [0,1)^{2}} given by the fractional parts of the original point's Cartesian coordinates.
Now consider a line in R 2 {\displaystyle \mathbb {R} ^{2}} given by the equation y = k x . {\displaystyle y=kx.} If the slope k {\displaystyle k} of the line is rational , it can be represented by a fraction and a corresponding lattice point of Z 2 . {\displaystyle \mathbb {Z} ^{2}.} It can be shown that then the projection of this line is a simple closed curve on a torus.
If, however, k {\displaystyle k} is irrational , it will not cross any lattice points except 0, which means that its projection on the torus will not be a closed curve, and the restriction of π {\displaystyle \pi } on this line is injective . Moreover, it can be shown that the image of this restricted projection as a subspace, called the irrational winding of a torus, is dense in the torus.
Irrational windings of a torus may be used to set up counter-examples related to monomorphisms . An irrational winding is an immersed submanifold but not a regular submanifold of the torus, which shows that the image of a manifold under a continuous injection to another manifold is not necessarily a (regular) submanifold. [ 2 ] Irrational windings are also examples of the fact that the topology of the submanifold does not have to coincide with the subspace topology of the submanifold. [ 2 ]
Secondly, the torus can be considered as a Lie group U ( 1 ) × U ( 1 ) {\displaystyle U(1)\times U(1)} , and the line can be considered as R {\displaystyle \mathbb {R} } . It is then easy to show that the image of the continuous and analytic group homomorphism x ↦ ( e i x , e i k x ) {\displaystyle x\mapsto \left(e^{ix},e^{ikx}\right)} is not a regular submanifold for irrational k , {\displaystyle k,} [ 2 ] [ 3 ] although it is an immersed submanifold, and therefore a Lie subgroup. It may also be used to show that if a subgroup H {\displaystyle H} of the Lie group G {\displaystyle G} is not closed, the quotient G / H {\displaystyle G/H} does not need to be a manifold [ 4 ] and might even fail to be a Hausdorff space .
^ a: As a topological subspace of the torus, the irrational winding is not a manifold at all because it is not locally homeomorphic to R {\displaystyle \mathbb {R} } . | https://en.wikipedia.org/wiki/Linear_flow_on_the_torus |
In mathematics , a linear form (also known as a linear functional , [ 1 ] a one-form , or a covector ) is a linear map [ nb 1 ] from a vector space to its field of scalars (often, the real numbers or the complex numbers ).
If V is a vector space over a field k , the set of all linear functionals from V to k is itself a vector space over k with addition and scalar multiplication defined pointwise . This space is called the dual space of V , or sometimes the algebraic dual space , when a topological dual space is also considered. It is often denoted Hom( V , k ) , [ 2 ] or, when the field k is understood, V ∗ {\displaystyle V^{*}} ; [ 3 ] other notations are also used, such as V ′ {\displaystyle V'} , [ 4 ] [ 5 ] V # {\displaystyle V^{\#}} or V ∨ . {\displaystyle V^{\vee }.} [ 2 ] When vectors are represented by column vectors (as is common when a basis is fixed), then linear functionals are represented as row vectors , and their values on specific vectors are given by matrix products (with the row vector on the left).
The constant zero function , mapping every vector to zero, is trivially a linear functional. Every other linear functional (such as the ones below) is surjective (that is, its range is all of k ).
Suppose that vectors in the real coordinate space R n {\displaystyle \mathbb {R} ^{n}} are represented as column vectors x = [ x 1 ⋮ x n ] . {\displaystyle \mathbf {x} ={\begin{bmatrix}x_{1}\\\vdots \\x_{n}\end{bmatrix}}.}
For each row vector a = [ a 1 ⋯ a n ] {\displaystyle \mathbf {a} ={\begin{bmatrix}a_{1}&\cdots &a_{n}\end{bmatrix}}} there is a linear functional f a {\displaystyle f_{\mathbf {a} }} defined by f a ( x ) = a 1 x 1 + ⋯ + a n x n , {\displaystyle f_{\mathbf {a} }(\mathbf {x} )=a_{1}x_{1}+\cdots +a_{n}x_{n},} and each linear functional can be expressed in this form.
This can be interpreted as either the matrix product or the dot product of the row vector a {\displaystyle \mathbf {a} } and the column vector x {\displaystyle \mathbf {x} } : f a ( x ) = a ⋅ x = [ a 1 ⋯ a n ] [ x 1 ⋮ x n ] . {\displaystyle f_{\mathbf {a} }(\mathbf {x} )=\mathbf {a} \cdot \mathbf {x} ={\begin{bmatrix}a_{1}&\cdots &a_{n}\end{bmatrix}}{\begin{bmatrix}x_{1}\\\vdots \\x_{n}\end{bmatrix}}.}
The trace tr ( A ) {\displaystyle \operatorname {tr} (A)} of a square matrix A {\displaystyle A} is the sum of all elements on its main diagonal . Matrices can be multiplied by scalars and two matrices of the same dimension can be added together; these operations make a vector space from the set of all n × n {\displaystyle n\times n} matrices. The trace is a linear functional on this space because tr ( s A ) = s tr ( A ) {\displaystyle \operatorname {tr} (sA)=s\operatorname {tr} (A)} and tr ( A + B ) = tr ( A ) + tr ( B ) {\displaystyle \operatorname {tr} (A+B)=\operatorname {tr} (A)+\operatorname {tr} (B)} for all scalars s {\displaystyle s} and all n × n {\displaystyle n\times n} matrices A and B . {\displaystyle A{\text{ and }}B.}
Linear functionals first appeared in functional analysis , the study of vector spaces of functions . A typical example of a linear functional is integration : the linear transformation defined by the Riemann integral I ( f ) = ∫ a b f ( x ) d x {\displaystyle I(f)=\int _{a}^{b}f(x)\,dx} is a linear functional from the vector space C [ a , b ] {\displaystyle C[a,b]} of continuous functions on the interval [ a , b ] {\displaystyle [a,b]} to the real numbers. The linearity of I {\displaystyle I} follows from the standard facts about the integral: I ( f + g ) = ∫ a b [ f ( x ) + g ( x ) ] d x = ∫ a b f ( x ) d x + ∫ a b g ( x ) d x = I ( f ) + I ( g ) I ( α f ) = ∫ a b α f ( x ) d x = α ∫ a b f ( x ) d x = α I ( f ) . {\displaystyle {\begin{aligned}I(f+g)&=\int _{a}^{b}[f(x)+g(x)]\,dx=\int _{a}^{b}f(x)\,dx+\int _{a}^{b}g(x)\,dx=I(f)+I(g)\\I(\alpha f)&=\int _{a}^{b}\alpha f(x)\,dx=\alpha \int _{a}^{b}f(x)\,dx=\alpha I(f).\end{aligned}}}
Let P n {\displaystyle P_{n}} denote the vector space of real-valued polynomial functions of degree ≤ n {\displaystyle \leq n} defined on an interval [ a , b ] . {\displaystyle [a,b].} If c ∈ [ a , b ] , {\displaystyle c\in [a,b],} then let ev c : P n → R {\displaystyle \operatorname {ev} _{c}:P_{n}\to \mathbb {R} } be the evaluation functional ev c f = f ( c ) . {\displaystyle \operatorname {ev} _{c}f=f(c).} The mapping f ↦ f ( c ) {\displaystyle f\mapsto f(c)} is linear since ( f + g ) ( c ) = f ( c ) + g ( c ) ( α f ) ( c ) = α f ( c ) . {\displaystyle {\begin{aligned}(f+g)(c)&=f(c)+g(c)\\(\alpha f)(c)&=\alpha f(c).\end{aligned}}}
If x 0 , … , x n {\displaystyle x_{0},\ldots ,x_{n}} are n + 1 {\displaystyle n+1} distinct points in [ a , b ] , {\displaystyle [a,b],} then the evaluation functionals ev x i , {\displaystyle \operatorname {ev} _{x_{i}},} i = 0 , … , n {\displaystyle i=0,\ldots ,n} form a basis of the dual space of P n {\displaystyle P_{n}} ( Lax (1996) proves this last fact using Lagrange interpolation ).
A function f {\displaystyle f} having the equation of a line f ( x ) = a + r x {\displaystyle f(x)=a+rx} with a ≠ 0 {\displaystyle a\neq 0} (for example, f ( x ) = 1 + 2 x {\displaystyle f(x)=1+2x} ) is not a linear functional on R {\displaystyle \mathbb {R} } , since it is not linear . [ nb 2 ] It is, however, affine-linear .
In finite dimensions, a linear functional can be visualized in terms of its level sets , the sets of vectors which map to a given value. In three dimensions, the level sets of a linear functional are a family of mutually parallel planes; in higher dimensions, they are parallel hyperplanes . This method of visualizing linear functionals is sometimes introduced in general relativity texts, such as Gravitation by Misner, Thorne & Wheeler (1973) .
If x 0 , … , x n {\displaystyle x_{0},\ldots ,x_{n}} are n + 1 {\displaystyle n+1} distinct points in [ a , b ] , then the linear functionals ev x i : f ↦ f ( x i ) {\displaystyle \operatorname {ev} _{x_{i}}:f\mapsto f\left(x_{i}\right)} defined above form a basis of the dual space of P n , the space of polynomials of degree ≤ n . {\displaystyle \leq n.} The integration functional I is also a linear functional on P n , and so can be expressed as a linear combination of these basis elements. In symbols, there are coefficients a 0 , … , a n {\displaystyle a_{0},\ldots ,a_{n}} for which I ( f ) = a 0 f ( x 0 ) + a 1 f ( x 1 ) + ⋯ + a n f ( x n ) {\displaystyle I(f)=a_{0}f(x_{0})+a_{1}f(x_{1})+\dots +a_{n}f(x_{n})} for all f ∈ P n . {\displaystyle f\in P_{n}.} This forms the foundation of the theory of numerical quadrature . [ 6 ]
Linear functionals are particularly important in quantum mechanics . Quantum mechanical systems are represented by Hilbert spaces , which are anti – isomorphic to their own dual spaces. A state of a quantum mechanical system can be identified with a linear functional. For more information see bra–ket notation .
In the theory of generalized functions , certain kinds of generalized functions called distributions can be realized as linear functionals on spaces of test functions .
Every non-degenerate bilinear form on a finite-dimensional vector space V induces an isomorphism V → V ∗ : v ↦ v ∗ such that v ∗ ( w ) := ⟨ v , w ⟩ ∀ w ∈ V , {\displaystyle v^{*}(w):=\langle v,w\rangle \quad \forall w\in V,}
where the bilinear form on V is denoted ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \,\cdot \,,\,\cdot \,\rangle } (for instance, in Euclidean space , ⟨ v , w ⟩ = v ⋅ w {\displaystyle \langle v,w\rangle =v\cdot w} is the dot product of v and w ).
The inverse isomorphism is V ∗ → V : v ∗ ↦ v , where v is the unique element of V such that ⟨ v , w ⟩ = v ∗ ( w ) {\displaystyle \langle v,w\rangle =v^{*}(w)} for all w ∈ V . {\displaystyle w\in V.}
The above defined vector v ∗ ∈ V ∗ is said to be the dual vector of v ∈ V . {\displaystyle v\in V.}
In an infinite dimensional Hilbert space , analogous results hold by the Riesz representation theorem . There is a mapping V ↦ V ∗ from V into its continuous dual space V ∗ .
Let the vector space V have a basis e 1 , e 2 , … , e n {\displaystyle \mathbf {e} _{1},\mathbf {e} _{2},\dots ,\mathbf {e} _{n}} , not necessarily orthogonal . Then the dual space V ∗ {\displaystyle V^{*}} has a basis ω ~ 1 , ω ~ 2 , … , ω ~ n {\displaystyle {\tilde {\omega }}^{1},{\tilde {\omega }}^{2},\dots ,{\tilde {\omega }}^{n}} called the dual basis defined by the special property that ω ~ i ( e j ) = { 1 if i = j 0 if i ≠ j . {\displaystyle {\tilde {\omega }}^{i}(\mathbf {e} _{j})={\begin{cases}1&{\text{if}}\ i=j\\0&{\text{if}}\ i\neq j.\end{cases}}}
Or, more succinctly, ω ~ i ( e j ) = δ i j {\displaystyle {\tilde {\omega }}^{i}(\mathbf {e} _{j})=\delta _{ij}}
where δ i j {\displaystyle \delta _{ij}} is the Kronecker delta . Here the superscripts of the basis functionals are not exponents but are instead contravariant indices.
A linear functional u ~ {\displaystyle {\tilde {u}}} belonging to the dual space V ~ {\displaystyle {\tilde {V}}} can be expressed as a linear combination of basis functionals, with coefficients ("components") u i , u ~ = ∑ i = 1 n u i ω ~ i . {\displaystyle {\tilde {u}}=\sum _{i=1}^{n}u_{i}\,{\tilde {\omega }}^{i}.}
Then, applying the functional u ~ {\displaystyle {\tilde {u}}} to a basis vector e j {\displaystyle \mathbf {e} _{j}} yields u ~ ( e j ) = ∑ i = 1 n ( u i ω ~ i ) e j = ∑ i u i [ ω ~ i ( e j ) ] {\displaystyle {\tilde {u}}(\mathbf {e} _{j})=\sum _{i=1}^{n}\left(u_{i}\,{\tilde {\omega }}^{i}\right)\mathbf {e} _{j}=\sum _{i}u_{i}\left[{\tilde {\omega }}^{i}\left(\mathbf {e} _{j}\right)\right]}
due to linearity of scalar multiples of functionals and pointwise linearity of sums of functionals. Then u ~ ( e j ) = ∑ i u i [ ω ~ i ( e j ) ] = ∑ i u i δ i j = u j . {\displaystyle {\begin{aligned}{\tilde {u}}({\mathbf {e} }_{j})&=\sum _{i}u_{i}\left[{\tilde {\omega }}^{i}\left({\mathbf {e} }_{j}\right)\right]\\&=\sum _{i}u_{i}{\delta }_{ij}\\&=u_{j}.\end{aligned}}}
So each component of a linear functional can be extracted by applying the functional to the corresponding basis vector.
When the space V carries an inner product , then it is possible to write explicitly a formula for the dual basis of a given basis. Let V have (not necessarily orthogonal) basis e 1 , … , e n . {\displaystyle \mathbf {e} _{1},\dots ,\mathbf {e} _{n}.} In three dimensions ( n = 3 ), the dual basis can be written explicitly ω ~ i ( v ) = 1 2 ⟨ ∑ j = 1 3 ∑ k = 1 3 ε i j k ( e j × e k ) e 1 ⋅ e 2 × e 3 , v ⟩ , {\displaystyle {\tilde {\omega }}^{i}(\mathbf {v} )={\frac {1}{2}}\left\langle {\frac {\sum _{j=1}^{3}\sum _{k=1}^{3}\varepsilon ^{ijk}\,(\mathbf {e} _{j}\times \mathbf {e} _{k})}{\mathbf {e} _{1}\cdot \mathbf {e} _{2}\times \mathbf {e} _{3}}},\mathbf {v} \right\rangle ,} for i = 1 , 2 , 3 , {\displaystyle i=1,2,3,} where ε is the Levi-Civita symbol and ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle } the inner product (or dot product ) on V .
In higher dimensions, this generalizes as follows ω ~ i ( v ) = ⟨ ∑ 1 ≤ i 2 < i 3 < ⋯ < i n ≤ n ε i i 2 … i n ( ⋆ e i 2 ∧ ⋯ ∧ e i n ) ⋆ ( e 1 ∧ ⋯ ∧ e n ) , v ⟩ , {\displaystyle {\tilde {\omega }}^{i}(\mathbf {v} )=\left\langle {\frac {\sum _{1\leq i_{2}<i_{3}<\dots <i_{n}\leq n}\varepsilon ^{ii_{2}\dots i_{n}}(\star \mathbf {e} _{i_{2}}\wedge \cdots \wedge \mathbf {e} _{i_{n}})}{\star (\mathbf {e} _{1}\wedge \cdots \wedge \mathbf {e} _{n})}},\mathbf {v} \right\rangle ,} where ⋆ {\displaystyle \star } is the Hodge star operator .
Modules over a ring are generalizations of vector spaces, which removes the restriction that coefficients belong to a field . Given a module M over a ring R , a linear form on M is a linear map from M to R , where the latter is considered as a module over itself. The space of linear forms is always denoted Hom k ( V , k ) , whether k is a field or not. It is a right module if V is a left module.
The existence of "enough" linear forms on a module is equivalent to projectivity . [ 8 ]
Dual Basis Lemma — An R - module M is projective if and only if there exists a subset A ⊂ M {\displaystyle A\subset M} and linear forms { f a ∣ a ∈ A } {\displaystyle \{f_{a}\mid a\in A\}} such that, for every x ∈ M , {\displaystyle x\in M,} only finitely many f a ( x ) {\displaystyle f_{a}(x)} are nonzero, and x = ∑ a ∈ A f a ( x ) a {\displaystyle x=\sum _{a\in A}{f_{a}(x)a}}
Suppose that X {\displaystyle X} is a vector space over C . {\displaystyle \mathbb {C} .} Restricting scalar multiplication to R {\displaystyle \mathbb {R} } gives rise to a real vector space [ 9 ] X R {\displaystyle X_{\mathbb {R} }} called the realification of X . {\displaystyle X.} Any vector space X {\displaystyle X} over C {\displaystyle \mathbb {C} } is also a vector space over R , {\displaystyle \mathbb {R} ,} endowed with a complex structure ; that is, there exists a real vector subspace X R {\displaystyle X_{\mathbb {R} }} such that we can (formally) write X = X R ⊕ X R i {\displaystyle X=X_{\mathbb {R} }\oplus X_{\mathbb {R} }i} as R {\displaystyle \mathbb {R} } -vector spaces.
Every linear functional on X {\displaystyle X} is complex-valued while every linear functional on X R {\displaystyle X_{\mathbb {R} }} is real-valued. If dim X ≠ 0 {\displaystyle \dim X\neq 0} then a linear functional on either one of X {\displaystyle X} or X R {\displaystyle X_{\mathbb {R} }} is non-trivial (meaning not identically 0 {\displaystyle 0} ) if and only if it is surjective (because if φ ( x ) ≠ 0 {\displaystyle \varphi (x)\neq 0} then for any scalar s , {\displaystyle s,} φ ( ( s / φ ( x ) ) x ) = s {\displaystyle \varphi \left((s/\varphi (x))x\right)=s} ), where the image of a linear functional on X {\displaystyle X} is C {\displaystyle \mathbb {C} } while the image of a linear functional on X R {\displaystyle X_{\mathbb {R} }} is R . {\displaystyle \mathbb {R} .} Consequently, the only function on X {\displaystyle X} that is both a linear functional on X {\displaystyle X} and a linear function on X R {\displaystyle X_{\mathbb {R} }} is the trivial functional; in other words, X # ∩ X R # = { 0 } , {\displaystyle X^{\#}\cap X_{\mathbb {R} }^{\#}=\{0\},} where ⋅ # {\displaystyle \,{\cdot }^{\#}} denotes the space's algebraic dual space .
However, every C {\displaystyle \mathbb {C} } -linear functional on X {\displaystyle X} is an R {\displaystyle \mathbb {R} } -linear operator (meaning that it is additive and homogeneous over R {\displaystyle \mathbb {R} } ), but unless it is identically 0 , {\displaystyle 0,} it is not an R {\displaystyle \mathbb {R} } -linear functional on X {\displaystyle X} because its range (which is C {\displaystyle \mathbb {C} } ) is 2-dimensional over R . {\displaystyle \mathbb {R} .} Conversely, a non-zero R {\displaystyle \mathbb {R} } -linear functional has range too small to be a C {\displaystyle \mathbb {C} } -linear functional as well.
If φ ∈ X # {\displaystyle \varphi \in X^{\#}} then denote its real part by φ R := Re φ {\displaystyle \varphi _{\mathbb {R} }:=\operatorname {Re} \varphi } and its imaginary part by φ i := Im φ . {\displaystyle \varphi _{i}:=\operatorname {Im} \varphi .} Then φ R : X → R {\displaystyle \varphi _{\mathbb {R} }:X\to \mathbb {R} } and φ i : X → R {\displaystyle \varphi _{i}:X\to \mathbb {R} } are linear functionals on X R {\displaystyle X_{\mathbb {R} }} and φ = φ R + i φ i . {\displaystyle \varphi =\varphi _{\mathbb {R} }+i\varphi _{i}.} The fact that z = Re z − i Re ( i z ) = Im ( i z ) + i Im z {\displaystyle z=\operatorname {Re} z-i\operatorname {Re} (iz)=\operatorname {Im} (iz)+i\operatorname {Im} z} for all z ∈ C {\displaystyle z\in \mathbb {C} } implies that for all x ∈ X , {\displaystyle x\in X,} [ 9 ] φ ( x ) = φ R ( x ) − i φ R ( i x ) = φ i ( i x ) + i φ i ( x ) {\displaystyle {\begin{alignedat}{4}\varphi (x)&=\varphi _{\mathbb {R} }(x)-i\varphi _{\mathbb {R} }(ix)\\&=\varphi _{i}(ix)+i\varphi _{i}(x)\\\end{alignedat}}} and consequently, that φ i ( x ) = − φ R ( i x ) {\displaystyle \varphi _{i}(x)=-\varphi _{\mathbb {R} }(ix)} and φ R ( x ) = φ i ( i x ) . {\displaystyle \varphi _{\mathbb {R} }(x)=\varphi _{i}(ix).} [ 10 ]
The assignment φ ↦ φ R {\displaystyle \varphi \mapsto \varphi _{\mathbb {R} }} defines a bijective [ 10 ] R {\displaystyle \mathbb {R} } -linear operator X # → X R # {\displaystyle X^{\#}\to X_{\mathbb {R} }^{\#}} whose inverse is the map L ∙ : X R # → X # {\displaystyle L_{\bullet }:X_{\mathbb {R} }^{\#}\to X^{\#}} defined by the assignment g ↦ L g {\displaystyle g\mapsto L_{g}} that sends g : X R → R {\displaystyle g:X_{\mathbb {R} }\to \mathbb {R} } to the linear functional L g : X → C {\displaystyle L_{g}:X\to \mathbb {C} } defined by L g ( x ) := g ( x ) − i g ( i x ) for all x ∈ X . {\displaystyle L_{g}(x):=g(x)-ig(ix)\quad {\text{ for all }}x\in X.} The real part of L g {\displaystyle L_{g}} is g {\displaystyle g} and the bijection L ∙ : X R # → X # {\displaystyle L_{\bullet }:X_{\mathbb {R} }^{\#}\to X^{\#}} is an R {\displaystyle \mathbb {R} } -linear operator, meaning that L g + h = L g + L h {\displaystyle L_{g+h}=L_{g}+L_{h}} and L r g = r L g {\displaystyle L_{rg}=rL_{g}} for all r ∈ R {\displaystyle r\in \mathbb {R} } and g , h ∈ X R # . {\displaystyle g,h\in X_{\mathbb {R} }^{\#}.} [ 10 ] Similarly for the imaginary part, the assignment φ ↦ φ i {\displaystyle \varphi \mapsto \varphi _{i}} induces an R {\displaystyle \mathbb {R} } -linear bijection X # → X R # {\displaystyle X^{\#}\to X_{\mathbb {R} }^{\#}} whose inverse is the map X R # → X # {\displaystyle X_{\mathbb {R} }^{\#}\to X^{\#}} defined by sending I ∈ X R # {\displaystyle I\in X_{\mathbb {R} }^{\#}} to the linear functional on X {\displaystyle X} defined by x ↦ I ( i x ) + i I ( x ) . {\displaystyle x\mapsto I(ix)+iI(x).}
This relationship was discovered by Henry Löwig in 1934 (although it is usually credited to F. Murray), [ 11 ] and can be generalized to arbitrary finite extensions of a field in the natural way. It has many important consequences, some of which will now be described.
Suppose φ : X → C {\displaystyle \varphi :X\to \mathbb {C} } is a linear functional on X {\displaystyle X} with real part φ R := Re φ {\displaystyle \varphi _{\mathbb {R} }:=\operatorname {Re} \varphi } and imaginary part φ i := Im φ . {\displaystyle \varphi _{i}:=\operatorname {Im} \varphi .}
Then φ = 0 {\displaystyle \varphi =0} if and only if φ R = 0 {\displaystyle \varphi _{\mathbb {R} }=0} if and only if φ i = 0. {\displaystyle \varphi _{i}=0.}
Assume that X {\displaystyle X} is a topological vector space . Then φ {\displaystyle \varphi } is continuous if and only if its real part φ R {\displaystyle \varphi _{\mathbb {R} }} is continuous, if and only if φ {\displaystyle \varphi } 's imaginary part φ i {\displaystyle \varphi _{i}} is continuous. That is, either all three of φ , φ R , {\displaystyle \varphi ,\varphi _{\mathbb {R} },} and φ i {\displaystyle \varphi _{i}} are continuous or none are continuous. This remains true if the word "continuous" is replaced with the word " bounded ". In particular, φ ∈ X ′ {\displaystyle \varphi \in X^{\prime }} if and only if φ R ∈ X R ′ {\displaystyle \varphi _{\mathbb {R} }\in X_{\mathbb {R} }^{\prime }} where the prime denotes the space's continuous dual space . [ 9 ]
Let B ⊆ X . {\displaystyle B\subseteq X.} If u B ⊆ B {\displaystyle uB\subseteq B} for all scalars u ∈ C {\displaystyle u\in \mathbb {C} } of unit length (meaning | u | = 1 {\displaystyle |u|=1} ) then [ proof 1 ] [ 12 ] sup b ∈ B | φ ( b ) | = sup b ∈ B | φ R ( b ) | . {\displaystyle \sup _{b\in B}|\varphi (b)|=\sup _{b\in B}\left|\varphi _{\mathbb {R} }(b)\right|.} Similarly, if φ i := Im φ : X → R {\displaystyle \varphi _{i}:=\operatorname {Im} \varphi :X\to \mathbb {R} } denotes the complex part of φ {\displaystyle \varphi } then i B ⊆ B {\displaystyle iB\subseteq B} implies sup b ∈ B | φ R ( b ) | = sup b ∈ B | φ i ( b ) | . {\displaystyle \sup _{b\in B}\left|\varphi _{\mathbb {R} }(b)\right|=\sup _{b\in B}\left|\varphi _{i}(b)\right|.} If X {\displaystyle X} is a normed space with norm ‖ ⋅ ‖ {\displaystyle \|\cdot \|} and if B = { x ∈ X : ‖ x ‖ ≤ 1 } {\displaystyle B=\{x\in X:\|x\|\leq 1\}} is the closed unit ball then the supremums above are the operator norms (defined in the usual way) of φ , φ R , {\displaystyle \varphi ,\varphi _{\mathbb {R} },} and φ i {\displaystyle \varphi _{i}} so that [ 12 ] ‖ φ ‖ = ‖ φ R ‖ = ‖ φ i ‖ . {\displaystyle \|\varphi \|=\left\|\varphi _{\mathbb {R} }\right\|=\left\|\varphi _{i}\right\|.} This conclusion extends to the analogous statement for polars of balanced sets in general topological vector spaces .
Below, all vector spaces are over either the real numbers R {\displaystyle \mathbb {R} } or the complex numbers C . {\displaystyle \mathbb {C} .}
If V {\displaystyle V} is a topological vector space , the space of continuous linear functionals — the continuous dual — is often simply called the dual space. If V {\displaystyle V} is a Banach space , then so is its (continuous) dual. To distinguish the ordinary dual space from the continuous dual space, the former is sometimes called the algebraic dual space . In finite dimensions, every linear functional is continuous, so the continuous dual is the same as the algebraic dual, but in infinite dimensions the continuous dual is a proper subspace of the algebraic dual.
A linear functional f on a (not necessarily locally convex ) topological vector space X is continuous if and only if there exists a continuous seminorm p on X such that | f | ≤ p . {\displaystyle |f|\leq p.} [ 13 ]
Continuous linear functionals have nice properties for analysis : a linear functional is continuous if and only if its kernel is closed, [ 14 ] and a non-trivial continuous linear functional is an open map , even if the (topological) vector space is not complete. [ 15 ]
A vector subspace M {\displaystyle M} of X {\displaystyle X} is called maximal if M ⊊ X {\displaystyle M\subsetneq X} (meaning M ⊆ X {\displaystyle M\subseteq X} and M ≠ X {\displaystyle M\neq X} ) and does not exist a vector subspace N {\displaystyle N} of X {\displaystyle X} such that M ⊊ N ⊊ X . {\displaystyle M\subsetneq N\subsetneq X.} A vector subspace M {\displaystyle M} of X {\displaystyle X} is maximal if and only if it is the kernel of some non-trivial linear functional on X {\displaystyle X} (that is, M = ker f {\displaystyle M=\ker f} for some linear functional f {\displaystyle f} on X {\displaystyle X} that is not identically 0 ). An affine hyperplane in X {\displaystyle X} is a translate of a maximal vector subspace. By linearity, a subset H {\displaystyle H} of X {\displaystyle X} is a affine hyperplane if and only if there exists some non-trivial linear functional f {\displaystyle f} on X {\displaystyle X} such that H = f − 1 ( 1 ) = { x ∈ X : f ( x ) = 1 } . {\displaystyle H=f^{-1}(1)=\{x\in X:f(x)=1\}.} [ 11 ] If f {\displaystyle f} is a linear functional and s ≠ 0 {\displaystyle s\neq 0} is a scalar then f − 1 ( s ) = s ( f − 1 ( 1 ) ) = ( 1 s f ) − 1 ( 1 ) . {\displaystyle f^{-1}(s)=s\left(f^{-1}(1)\right)=\left({\frac {1}{s}}f\right)^{-1}(1).} This equality can be used to relate different level sets of f . {\displaystyle f.} Moreover, if f ≠ 0 {\displaystyle f\neq 0} then the kernel of f {\displaystyle f} can be reconstructed from the affine hyperplane H := f − 1 ( 1 ) {\displaystyle H:=f^{-1}(1)} by ker f = H − H . {\displaystyle \ker f=H-H.}
Any two linear functionals with the same kernel are proportional (i.e. scalar multiples of each other).
This fact can be generalized to the following theorem.
Theorem [ 16 ] [ 17 ] — If f , g 1 , … , g n {\displaystyle f,g_{1},\ldots ,g_{n}} are linear functionals on X , then the following are equivalent:
If f is a non-trivial linear functional on X with kernel N , x ∈ X {\displaystyle x\in X} satisfies f ( x ) = 1 , {\displaystyle f(x)=1,} and U is a balanced subset of X , then N ∩ ( x + U ) = ∅ {\displaystyle N\cap (x+U)=\varnothing } if and only if | f ( u ) | < 1 {\displaystyle |f(u)|<1} for all u ∈ U . {\displaystyle u\in U.} [ 15 ]
Any (algebraic) linear functional on a vector subspace can be extended to the whole space; for example, the evaluation functionals described above can be extended to the vector space of polynomials on all of R . {\displaystyle \mathbb {R} .} However, this extension cannot always be done while keeping the linear functional continuous. The Hahn–Banach family of theorems gives conditions under which this extension can be done. For example,
Hahn–Banach dominated extension theorem [ 18 ] ( Rudin 1991 , Th. 3.2) — If p : X → R {\displaystyle p:X\to \mathbb {R} } is a sublinear function , and f : M → R {\displaystyle f:M\to \mathbb {R} } is a linear functional on a linear subspace M ⊆ X {\displaystyle M\subseteq X} which is dominated by p on M , then there exists a linear extension F : X → R {\displaystyle F:X\to \mathbb {R} } of f to the whole space X that is dominated by p , i.e., there exists a linear functional F such that F ( m ) = f ( m ) {\displaystyle F(m)=f(m)} for all m ∈ M , {\displaystyle m\in M,} and | F ( x ) | ≤ p ( x ) {\displaystyle |F(x)|\leq p(x)} for all x ∈ X . {\displaystyle x\in X.}
Let X be a topological vector space (TVS) with continuous dual space X ′ . {\displaystyle X'.}
For any subset H of X ′ , {\displaystyle X',} the following are equivalent: [ 19 ]
If H is an equicontinuous subset of X ′ {\displaystyle X'} then the following sets are also equicontinuous:
the weak-* closure, the balanced hull , the convex hull , and the convex balanced hull . [ 19 ] Moreover, Alaoglu's theorem implies that the weak-* closure of an equicontinuous subset of X ′ {\displaystyle X'} is weak-* compact (and thus that every equicontinuous subset weak-* relatively compact). [ 20 ] [ 19 ] | https://en.wikipedia.org/wiki/Linear_form |
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