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Lewis Norman Mander , AC , FAA , FRS (8 September 1939 – 8 February 2020) was a New Zealand -born Australian organic chemist . He has widely explored the synthesis and chemistry of the gibberellin class of diterpenes over a 20-year period at the Australian National University (ANU). [ 1 ] [ 2 ] In particular, he studied the effect of these hormones on stem growth and on the reasons why plant undergo bolting during plant development . The July 2004 edition of the Australian Journal of Chemistry was dedicated to Mander on the occasion of his 65th birthday. He retired in 2002 but remained active at the ANU until 2014. In 2018 Mander was made a Companion in the General Division in the Order of Australia which "...is awarded for eminent achievement and merit of the highest degree in service to Australia or humanity at large". [ 3 ] In an interview he gave after winning his award, Mander said that his goal was to improve the efficiency of extracting food from plants with the possibility of reducing food shortages in the future. [ 4 ]
Mander completed a BSc degree at the University of Auckland , New Zealand in 1960, followed by an MSc degree in 1961 from the same institution.
He then moved to Australia in 1962 to undertake a PhD degree at the University of Sydney before committing to an initial postdoctoral fellowship at the University of Michigan . Mander then moved to Caltech in 1965 (after his PhD had been conferred) for an additional two years.
Mander returned to Australia in 1966 to become a lecturer in organic chemistry at the University of Adelaide . He was promoted to Senior Lecturer in organic chemistry in 1970, where he remained until 1975. During this time Mander visited the University of Cambridge to research "...pathways to the pigments of life". [ 5 ] In 1977, he served as a Fulbright Senior Scholar at the California Institute of Technology . He was a distinguished Alumnus Professor at the University of Auckland in 1992 and an Eminent Scientist of RIKEN at Wako , in Saitama Prefecture , Japan from 1995 to 1996.
In Australia, he relocated to the Australian National University Research School of Chemistry as a Senior Fellow. He retired in 2002 but retained the title of Professor Emeritus at the Australian National University .
Notable students of Mander's include Jacqueline Whalley , professor at Auckland University of Technology . [ 6 ]
Mander died at home in Canberra, Australia on 8 February 2020, at age 80. [ 7 ]
In the early days, Mander was involved in extracting chemicals in plants that might help fight against cancer. [ 4 ] Eventually, he turned his research skills to “...the gibberellin family of plant bioregulators". [ 8 ] He further developed his interest in this chemical group to include an understanding of their role in plant development . Professor Sir Alan R. Battersby said that Mander's “...synthesis of gibberellic acid was a brilliant landmark achievement. This molecule is of daunting complexity and he developed two flexible routes to it, both depending on many ingenious and novel synthetic procedures". [ 9 ]
Amongst his many scholarly activities, Mander contributed a chapter on 'Stereoselective Synthesis' to the classic text 'Stereochemistry of Organic Compounds' by Professors Ernest L. Eliel and Samuel H. Wilen.
Other interests include: | https://en.wikipedia.org/wiki/Lew_Mander |
Lewis' law gives a relationship between the size and the shape of epithelial cells . It states that the average apical area A ¯ n {\displaystyle {\bar {A}}_{n}} of an epithelial cell is linearly related to its neighbor number n {\displaystyle n} . It is a phenomenological law that was first described in the cucumber epidermis by the morphologist Frederic Thomas Lewis in 1928. [ 1 ] The simplest version of Lewis' law can be expressed as A n ¯ A ¯ = n − 2 4 {\textstyle {\frac {\overline {A_{n}}}{\overline {A}}}={\frac {n-2}{4}}} , which reads: The average apical area of a cell with n {\displaystyle n} neighbors (divided by the average apical area of all cells) is proportional to its shape. While neighbor number distributions change throughout organogenesis , the average neighbor number of epithelial cells is n ¯ ≈ 6 {\textstyle {\bar {n}}\approx 6} , which can be traced back to Euler's formula for polygons. [ 2 ]
Frederic Thomas Lewis noticed that epidermal cells display a patterning similar to froths , which led him to quantify and analyze the sizes and shapes of epidermal cells. [ 1 ]
A variety of empirical studies in different epithelial tissues have confirmed Lewis' law. [ 3 ] [ 4 ] [ 5 ]
It has been suggested [ 6 ] that the emergence of Lewis' law on the apical surface of epithelia is a result of the concurrence of
According to this theory, the observed tissue-specific polygon distributions and Lewis' law arise as a compromise in order to maintain tissue integrity.
In order to understand morphogenetic events, i.e. the growth and shaping of tissues and organs, it is necessary to analyze the packing of cells into tissues. In that context, an analysis of patterning processes can help to identify the underlying mechanisms that drive morphogenesis . | https://en.wikipedia.org/wiki/Lewis'_law |
Lewis Paul (died 1759) was the original inventor of roller spinning, the basis of the water frame for spinning cotton in a cotton mill .
Lewis Paul was of Huguenot descent. His father was physician to Lord Shaftesbury . He may have begun work on designing a spinning machine for cotton as early as 1729, but probably did not make practical progress until after 1732 when he met John Wyatt , a carpenter then working in Birmingham for a gun barrel forger. Wyatt had designed a machine, probably for cutting files, in which Paul took an interest.
Roller spinning was certainly Paul's idea, and Wyatt built a machine (or model) for him. Paul obtained a patent for this on 24 June 1738. He then set about trying to license his machine, though some licences were granted in satisfaction of debts. In 1741, he set up a machine powered by two asses in the Upper Priory in Birmingham, near his house in Old Square.
Edward Cave , a publisher , obtained a licence and set up machines in a warehouse in London . In 1742, he acquired Marvel's Mill on the River Nene at Northampton . He rebuilt the mill to hold four or five water-powered spinning machines, each with 50 spindles. This was thus the first cotton mill . Cave died on 10 January 1754, so that the mill passed to his brother William and his nephew Paul. Samuel Touchet, a London merchant had the mill until 1755, but made no profit. It may then have been let to Lewis Paul, but he died in 1759. The Caves forfeited the lease for non-payment of rent in March 1761 and advertised the mill to let in November 1761. By 1768, the mill had reverted to being a corn mill .
Another mill that operated under Paul's patent was at Leominster . This was built in 1744 by John Bourn in partnership with Henry Morris of Lancashire . The mill burnt down in November 1754.
In 1748, Daniel Bourn and Lewis Paul separately obtained patents for carding machines, which were presumably used in the Leominster and Northampton mills respectively. This carding technology of Lewis Paul and Daniel Bourn seems to be the basis of later carding machines.
The principle of his rolling spinning process was perfected by John Kay and Thomas Highs and promoted by Richard Arkwright . [ 1 ] [ 2 ] Paul's machine seems only to have been modestly profitable, and it is not clear to what extent his work is reflected in Arkwright's much more successful machine, the water frame , patented in 1769. Like Paul and Bourn, Arkwright subsequently added a carding stage to his machinery, but his use of this as a means of continuing his patent rights beyond the expiry of his original patent failed, because the improvement was not his invention. | https://en.wikipedia.org/wiki/Lewis_Paul |
In organic chemistry , Lewis acid catalysis is the use of metal-based Lewis acids as catalysts for organic reactions . The acids act as an electron pair acceptor to increase the reactivity of a substrate . Common Lewis acid catalysts are based on main group metals such as aluminum , boron , silicon , and tin , as well as many early ( titanium , zirconium ) and late ( iron , copper , zinc ) d-block metals. The metal atom forms an adduct with a lone-pair bearing electronegative atom in the substrate, such as oxygen (both sp 2 or sp 3 ), nitrogen , sulfur , and halogens . The complexation has partial charge-transfer character and makes the lone-pair donor effectively more electronegative , activating the substrate toward nucleophilic attack , heterolytic bond cleavage , or cycloaddition with 1,3-dienes and 1,3-dipoles. [ 1 ]
Many classical reactions involving carbon–carbon or carbon–heteroatom bond formation can be catalyzed by Lewis acids. Examples include the Friedel-Crafts reaction , the aldol reaction , and various pericyclic processes that proceed slowly at room temperature, such as the Diels-Alder reaction and the ene reaction . In addition to accelerating the reactions, Lewis acid catalysts are able to impose regioselectivity and stereoselectivity in many cases.
Early developments in Lewis acid reagents focused on easily available compounds such as TiCl 4 , BF 3 , SnCl 4 , and AlCl 3 . Over the years, versatile catalysts bearing ligands designed for specific applications have facilitated improvement in both reactivity and selectivity of Lewis acid-catalyzed reactions. More recently, Lewis acid catalysts with chiral ligands have become an important class of tools for asymmetric catalysis . [ 2 ]
Challenges in the development of Lewis acid catalysis include inefficient catalyst turnover (caused by catalyst affinity for the product) and the frequent requirement of two-point binding for stereoselectivity, which often necessitates the use of auxiliary groups.
In reactions with polar mechanisms, Lewis acid catalysis often involves binding of the catalyst to Lewis basic heteroatoms and withdrawing electron density, which in turn facilitates heterolytic bond cleavage (in the case of Friedel-Crafts reaction ) or directly activates the substrate toward nucleophilic attack (in the case of carbonyl addition reactions). The dichotomy can have important consequences in some reactions, as in the case of Lewis acid-promoted acetal substitution reactions, where the S N 1 and S N 2 mechanisms shown below may give different stereochemical outcomes. Studying the product ratio in a bicyclic system, Denmark and colleagues showed that both mechanisms could be operative depending on the denticity of the Lewis acid and the identity of the R' group. [ 3 ]
In Diels-Alder and 1,3-dipolar cycloaddition reactions, Lewis acids lower the LUMO energy of the dienophile or dipolarphile, respectively, making it more reactive toward the diene or the dipole.
Among the types of reactions that can be catalyzed by Lewis acids , those with carbonyl -containing substrates have received the greatest amount of attention. The first major discovery in this area was in 1960, when Yates and Eaton reported the significant acceleration of the Diels-Alder reaction by AlCl 3 when maleic anhydride is the dienophile. [ 4 ]
Early theoretical studies that depended on frontier orbital analysis established that Lewis acid catalysis operates via lowering of the dienophile's LUMO energy,. [ 5 ] Recent studies, however, have shown that this rationale behind Lewis acid-catalyzed Diels-Alder reactions is incorrect. [ 6 ] [ 7 ] [ 8 ] [ 9 ] It is found that Lewis acids accelerate the Diels-Alder reaction by reducing the destabilizing steric Pauli repulsion between the interacting diene and dienophile and not by lowering the energy of the dienophile's LUMO and consequently, enhancing the normal electron demand orbital interaction. The Lewis acid bind via a donor-acceptor interaction to the dienophile and via that mechanism polarizes occupied orbital density away from the reactive C=C double bond of the dienophile towards the Lewis acid. This reduced occupied orbital density on C=C double bond of the dienophile will, in turn, engage in a less repulsive closed-shell-closed-shell orbital interaction with the incoming diene, reducing the destabilizing steric Pauli repulsion and hence lowers the Diels-Alder reaction barrier. In addition, the Lewis acid catalyst also increases the asynchronicity of the Diels-Alder reaction, making the occupied π-orbital located on the C=C double bond of the dienophile asymmetric. As a result, this enhanced asynchronicity leads to an extra reduction of the destabilizing steric Pauli repulsion as well as a diminishing pressure on the reactants to deform, in other words, it reduced the destabilizing activation strain (also known as distortion energy). [ 10 ] This working catalytic mechanism is known as Pauli-lowering catalysis , [ 11 ] which is operative in a variety of organic reactions. [ 12 ] [ 13 ] [ 14 ]
The original rationale behind Lewis acid-catalyzed Diels-Alder reactions is incorrect, [ 15 ] [ 16 ] [ 17 ] [ 18 ] because besides lowering the energy of the dienophile's LUMO, the Lewis acid also lowers the energy of the HOMO of the dienophile and hence increases the inverse electron demand LUMO-HOMO orbital energy gap. Thus, indeed Lewis acid catalysts strengthen the normal electron demand orbital interaction by lowering the LUMO of the dienophile, but, they simultaneously weaken the inverse electron demand orbital interaction by also lowering the energy of the dienophile's HOMO. These two counteracting phenomena effectively cancel each other, resulting in nearly unchanged orbital interactions when compared to the corresponding uncatalyzed Diels-Alder reactions and making this not the active mechanism behind Lewis acid-catalyzed Diels-Alder reactions.
In addition to rate acceleration, Lewis acid-catalyzed reactions sometimes exhibit enhanced stereoselectivity, which stimulated the development of stereoinduction models. The models have their roots in knowledge of the structures of Lewis acid-carbonyl complexes which, through decades of research in theoretical calculations , NMR spectroscopy, and X-ray crystallography , were fairly firmly established in the early 1990s: [ 19 ]
The Mukaiyama aldol reaction and the Sakurai reaction refer to the addition of silyl enol ethers and allylsilanes to carbonyl compounds, respectively. Only under Lewis acid catalysis are the reactions useful for synthesis. Acyclic transition states are believed to be operating in both reactions for either 1,2- or 1,4- addition, and steric factors control stereoselectivity. This is in contrast with the rigid Zimmerman-Traxler cyclic transition state that has been widely accepted for the aldol reaction with lithium, boron, and titanium enolates . As a consequence, the double bond geometry in the silyl enol ether or allylsilane does not translate well into product stereochemistry. A model for the Sakurai 1,2-addition, proposed by Kumada, is presented in the scheme below; [ 21 ] the syn diastereomer is predominant when the (E) silane is used, and also slightly favored when the (Z) silane is used. A similar analysis by Heathcock [ 22 ] explains the fact that, with simple substrates, there is essentially no diastereoselectivity for the intermolecular Mukaiyama aldol reaction.
The Lewis acid catalyst plays a role in stereoselectivity when the aldehyde can chelate onto the metal center and form a rigid cyclic intermediate. The stereochemical outcome is then consistent with approach of the nucleophile anti to the more bulky substituent on the ring. [ 23 ] [ 24 ]
Lewis acids such as ZnCl 2 , BF 3 , SnCl 4 , AlCl 3 , and MeAlCl 2 can catalyze both normal and inverse electron demand Diels-Alder reactions . The enhancement in rate is often dramatic, and regioselectivity towards ortho- or para-like products is often improved, as shown in the reaction between isoprene and methyl acrylate . [ 25 ]
The catalyzed Diels-Alder reaction is believed to be concerted . A computational study at the B3LYP/6-31G(d) level has shown, however, that the transition state of the BF 3 -catalyzed Diels-Alder reaction between propenal and 1,3-butadiene is more asynchronous than that of the thermal reaction – the bond further from the carbonyl group is formed ahead of the other bond. [ 26 ]
The carbonyl-ene reaction is almost always catalyzed by Lewis acids in synthetic applications. [ 27 ] A stepwise or a largely asynchronous mechanism has been proposed for the catalyzed reaction based on kinetic isotope effect studies. [ 28 ] Nonetheless, cyclic transition states are frequently invoked to interpret diastereoselectivity. In a seminal review in the early 1990s, Mikami and colleagues [ 29 ] proposed a late, chair-like transition state, which could rationalize many observed stereochemical results, including the role of steric bulk in diastereoselectivity: [ 30 ]
More recently, however, the same group carried out HF/6-31G* calculations on tin or aluminum Lewis acid-catalyzed ene reactions. Citing that methyl gloxylate chelates tin Lewis acids but not aluminum ones, they invoked an early, envelope-like transition state and rationalized the divergent stereochemical outcome of the ene reaction between (E) -2-butene and methyl glyoxylate. [ 31 ]
Lewis-acid catalyzed carbonyl addition reactions are routinely used to form carbon–carbon bonds in natural product synthesis. The first two reactions shown below are from the syntheses of (+)-lycoflexine [ 32 ] and zaragozic acid C , [ 33 ] respectively, which are direct applications of Sakurai and Mukaiyama reactions. The third reaction, en route to (+)-fawcettimine, is a Lewis-acid catalyzed cyclopropane opening that is analogous to a Mukaiyama- Michael reaction . [ 34 ]
The Diels-Alder reaction catalyzed or promoted by Lewis acids is a powerful and widely used method in natural product synthesis to attain scaffold complexity in a single step with stereochemical control. The two reactions shown below are an intramolecular Diels-Alder reaction towards (−)-fusarisetin A [ 35 ] and an intermolecular hetero-Diels-Alder reaction towards (−)-epibatidine, [ 36 ] respectively.
In Friedel–Crafts alkylation, a Lewis acid – usually a simple metal halide salt – promotes heterolytic cleavage of a carbon–halogen bond in an alkyl halide and generates a carbocation , which undergoes electrophilic aromatic substitution . Although vastly useful in synthesis, the reaction often suffers from side reactions that arise from carbocation rearrangement, alkyl migration, and over-alkylation. Similarly, in Friedel-Crafts acylation, a Lewis acid assists in the generation of an acylium ion from an acid chloride (or occasionally acid anhydride). Although the acylium ion is often assumed to be the active intermediate, [ 37 ] there is evidence that the protonated acylium dication is the active electrophile that undergoes subsequent electrophilic aromatic substitution. [ 38 ]
Important variants of the Friedel–Crafts reaction include chloromethylation (with formaldehyde and HCl), formylation (with HCl and CO or CN − ), and acylation with a nitrile as the acyl source. The nitrile-based acylation is particularly useful because it allows direct ortho-acylation of aniline without protecting the amine group. [ 39 ] A combination of a weak and a strong Lewis acid is necessary for the reaction to proceed, through the mechanism shown below. Guided by this mechanism, and equipped with knowledge that gallium trihalides are among the strongest Lewis acids, [ 40 ] process chemists at Merck were able to develop highly efficient conditions for this condition towards a drug candidate. [ 41 ]
Asymmetric catalysis by Lewis acids rely on catalysts with chiral ligands coordinated to the metal center. Over the years, a small number of chiral ligand scaffolds have stood out as having "privileged" catalytic properties suitable for a wide range of applications, often of unrelated mechanisms. Current research efforts in asymmetric Lewis acid catalysis mostly utilize or modify those ligands rather than create new scaffolds de novo . The "privileged" scaffolds share a few common features, including chemical stability and relative ease of elaboration. The majority of the scaffolds are multidentate . Most of them also have high scaffold rigidity within the ligand. Several of them have fairly mature stereoinduction models available. Some "privileged" scaffolds, as identified by Jacobsen [ 42 ] and Zhou, [ 43 ] are introduced below.
Most common chiral bisoxazoline (BOX) ligands consist of two identical chiral oxazoline moieties, substituted by a bulky group at the 4-positions, joined by a linker. The ligand is bidentate when the linker is a single carbon unit, but is tridentate (usually meridional) when the linker bears an additional coordinating atom, such as a pyridine nitrogen in the case of PyBOX ligands. The impact of ligand denticity and active intermediate geometry on the stereochemical outcome has been thoroughly reviewed. [ 44 ]
Many bidentate BOX-based Lewis acid-catalyzed reactions are based on copper(II) catalysts with substrates that are suitable for two-point binding. The stereochemical outcome is consistent with a twisted square planar intermediate that was proposed based on related crystal structures. [ 45 ] [ 46 ] The substituent at the oxazoline's 4-position blocks one enantiotopic face of the substrate, leading to enantioselectivity. This is demonstrated in the following aldol -type reaction, [ 47 ] but is applicable to a wide variety of reactions such as Mannich -type reactions, [ 48 ] ene reaction , [ 49 ] Michael addition , [ 50 ] Nazarov cyclization , [ 51 ] and hetero- Diels-Alder reaction . [ 52 ]
On the other hand, two-point binding on a Lewis acid bearing the meridionally tridentate PyBOX ligand would result in a square pyramidal complex. A study using benzyloxyacetaldehyde as the electrophile showed that the stereochemical outcome is consistent with the carbonyl oxygen binding equatorially and the ether oxygen binding axially. [ 53 ]
Developed by Noyori, BINAP (2,2'-diphenylphosphino-1,1'-binaphthyl) is a family of chiral diphosphine ligands featuring two triarylphosphine moieties installed on a binaphthalene backbone. [ 54 ] BINAP chelates onto a metal (usually a late transition metal) to form a C 2 -symmetric complex. As shown below in the structure of an (R) -BINAP ruthenium complex, [ 55 ] among the four remaining coordination sites on an octahedral metal center, the two equatorial sites (purple) are strongly influenced by the equatorial phenyl groups, while the two axial sites (green) are influenced by the axial phenyl groups.
Based on the structure, models for the observed enantioselectivity in many BINAP-based Lewis acid-catalyzed reactions have been proposed. For example, in the palladium-catalyzed enantioselective Diels-Alder reaction shown below, the dienophile is thought to coordinate the metal center at the equatorial sites. Thus the equatorial phenyl group on phosphorus obstructs the Si -face , resulting in excellent enantioselectivity. [ 56 ] A very similar model was used to rationalize the outcome of a nickel-catalyzed asymmetric enolate alkylation reaction, where the substrate also bears an auxiliary that allows it to chelate onto the metal. [ 57 ] On the other hand, a copper(I)-catalyzed hetero-ene reaction is thought to proceed through a tetrahedral intermediate, [ 58 ] offering an alternative mode of stereoinduction by changing the metal center.
BINOL (1,1'-binaphthyl-2,2'-diol) is usually used in conjunction with oxophilic Lewis acidic metals such as aluminum, titanium, zirconium, and various rare earth metals. In cases where BINOL itself does not provide ideal enantioselective control, it can be readily elaborated by substitution at the 3,3' positions (via lithiation ) and 6,6' positions (via the 6,6'-dibromide compound prepared by electrophilic aromatic substitution ) to modulate steric bulk and electronic properties. [ 59 ] For instance, aluminum catalysts based on bulky 3,3'-disilyl substituted BINOL have been developed as early examples of catalytic asymmetric hetero- Diels-Alder reaction [ 60 ] and Claisen rearrangement , [ 61 ] while introduction of electron-withdrawing groups at the 6,6'-positions was crucial for increasing the Lewis acidity, and hence catalytic activity, of zirconium(IV) catalysts toward a Mannich -type reaction. [ 62 ] To date, however, no model for the crucial factors governing BINOL-directed stereoinduction has been generally accepted.
TADDOL stands for tetraaryl-1,3-dioxolane-4,5-dimethanol. The broad application of titanium TADDOLate catalysts towards carbonyl additions and cycloadditions has been introduced by Seebach and coworkers, and has been thoroughly summarized in a seminal review, in which a working stereoinduction model that agreed with the observed selectivity in a wide variety of reactions was put forth, despite the lack of a clear picture of the mechanism. [ 63 ]
Lewis acid catalysis has been used in the asymmetry-setting step for the syntheses of many natural products . The first reaction shown below, from the synthesis of taxane skeleton, uses a copper-based catalyst supported by a chiral phosphoramidite ligand for a conjugate carbonyl addition reaction. [ 64 ] The second reaction, from the synthesis of ent - hyperforin , uses an iron-PyBOX catalyst for an asymmetric Diels-Alder reaction . [ 65 ] | https://en.wikipedia.org/wiki/Lewis_acid_catalysis |
In fluid dynamics and thermodynamics , the Lewis number (denoted Le ) is a dimensionless number defined as the ratio of thermal diffusivity to mass diffusivity . It is used to characterize fluid flows where there is simultaneous heat and mass transfer . The Lewis number puts the thickness of the thermal boundary layer in relation to the concentration boundary layer. [ 1 ] The Lewis number is defined as [ 2 ]
where:
In the field of fluid mechanics , many sources define the Lewis number to be the inverse of the above definition. [ 3 ] [ 4 ]
The Lewis number can also be expressed in terms of the Prandtl number ( Pr ) and the Schmidt number ( Sc ): [ 5 ]
It is named after Warren K. Lewis (1882–1975), [ 6 ] [ 7 ] who was the first head of the Chemical Engineering Department at MIT . Some workers in the field of combustion assume (incorrectly) that the Lewis number was named for Bernard Lewis (1899–1993), who for many years was a major figure in the field of combustion research. [ citation needed ]
The Lewis number is large for water ( L e ≈ 90 ≫ 1 ) {\displaystyle (\mathrm {Le} \approx 90\gg 1)} , and this is likely the reason why mammals do not have gills . [ 8 ] In gills, oxygen is extracted from seawater into the mammal. Since the Lewis number for water is high, this means that during this diffusion process, a relatively large amount of heat would also be extracted from the animal, as heat diffuses faster than oxygen. This would cause the animal to cool down too much while breathing. | https://en.wikipedia.org/wiki/Lewis_number |
Lewis structures – also called Lewis dot formulas , Lewis dot structures , electron dot structures , or Lewis electron dot structures ( LEDs ) – are diagrams that show the bonding between atoms of a molecule , as well as the lone pairs of electrons that may exist in the molecule. [ 1 ] [ 2 ] [ 3 ] Introduced by Gilbert N. Lewis in his 1916 article The Atom and the Molecule , a Lewis structure can be drawn for any covalently bonded molecule, as well as coordination compounds . [ 4 ] Lewis structures extend the concept of the electron dot diagram by adding lines between atoms to represent shared pairs in a chemical bond.
Lewis structures show each atom and its position in the structure of the molecule using its chemical symbol. Lines are drawn between atoms that are bonded to one another (pairs of dots can be used instead of lines). Excess electrons that form lone pairs are represented as pairs of dots, and are placed next to the atoms.
Although main group elements of the second period and beyond usually react by gaining, losing, or sharing electrons until they have achieved a valence shell electron configuration with a full octet of (8) electrons, hydrogen instead obeys the duplet rule , forming one bond for a complete valence shell of two electrons.
For a neutral molecule, the total number of electrons represented in a Lewis structure is equal to the sum of the numbers of valence electrons on each individual atom, not the maximum possible. Non-valence electrons are not represented in Lewis structures as they do not bond.
Once the total number of valence electrons has been determined, they are placed into the structure according to these steps:
Lewis structures for polyatomic ions may be drawn by the same method. However when counting electrons, negative ions should have extra electrons placed in their Lewis structures; positive ions should have fewer electrons than an uncharged molecule. When the Lewis structure of an ion is written, the entire structure is placed in brackets, and the charge is written as a superscript on the upper right, outside the brackets.
A simpler method has been proposed for constructing Lewis structures, eliminating the need for electron counting: the atoms are drawn showing the valence electrons; bonds are then formed by pairing up valence electrons of the atoms involved in the bond-making process, and anions and cations are formed by adding or removing electrons to/from the appropriate atoms. [ 5 ]
A trick is to count up valence electrons, then count up the number of electrons needed to complete the octet rule (or with hydrogen just 2 electrons), then take the difference of these two numbers. The answer is the number of electrons that make up the bonds. The rest of the electrons just go to fill all the other atoms' octets.
Another simple and general procedure to write Lewis structures and resonance forms has been proposed. [ 6 ] [ example needed ]
This system works in nearly all cases, however there are 3 instances where it will not work [ citation needed ] . These exceptions are outlined in the table below.
In terms of Lewis structures, formal charge is used in the description, comparison, and assessment of likely topological and resonance structures [ 7 ] by determining the apparent electronic charge of each atom within, based upon its electron dot structure, assuming exclusive covalency or non-polar bonding. It has uses in determining possible electron re-configuration when referring to reaction mechanisms , and often results in the same sign as the partial charge of the atom, with exceptions. In general, the formal charge of an atom can be calculated using the following formula, assuming non-standard definitions for the markup used:
where:
The formal charge of an atom is computed as the difference between the number of valence electrons that a neutral atom would have and the number of electrons that belong to it in the Lewis structure. Electrons in covalent bonds are split equally between the atoms involved in the bond. The total of the formal charges on an ion should be equal to the charge on the ion, and the total of the formal charges on a neutral molecule should be equal to zero.
For some molecules and ions, it is difficult to determine which lone pairs should be moved to form double or triple bonds, and two or more different resonance structures may be written for the same molecule or ion. In such cases it is usual to write all of them with two-way arrows in between (see § Example below) . This is sometimes the case when multiple atoms of the same type surround the central atom, and is especially common for polyatomic ions.
When this situation occurs, the molecule's Lewis structure is said to be a resonance structure , and the molecule exists as a resonance hybrid. Each of the different possibilities is superimposed on the others, and the molecule is considered to have a Lewis structure equivalent to some combination of these states.
The nitrate ion ( NO − 3 ), for instance, must form a double bond between nitrogen and one of the oxygens to satisfy the octet rule for nitrogen. However, because the molecule is symmetrical, it does not matter which of the oxygens forms the double bond. In this case, there are three possible resonance structures. Expressing resonance when drawing Lewis structures may be done either by drawing each of the possible resonance forms and placing double-headed arrows between them or by using dashed lines to represent the partial bonds (although the latter is a good representation of the resonance hybrid which is not, formally speaking, a Lewis structure).
When comparing resonance structures for the same molecule, usually those with the fewest formal charges contribute more to the overall resonance hybrid. When formal charges are necessary, resonance structures that have negative charges on the more electronegative elements and positive charges on the less electronegative elements are favored.
Single bonds can also be moved in the same way to create resonance structures for hypervalent molecules such as sulfur hexafluoride , which is the correct description according to quantum chemical calculations instead of the common expanded octet model.
The resonance structure should not be interpreted to indicate that the molecule switches between forms, but that the molecule acts as the average of multiple forms.
The formula of the nitrite ion is NO − 2 .
Chemical structures may be written in more compact forms, particularly when showing organic molecules . In condensed structural formulas, many or even all of the covalent bonds may be left out, with subscripts indicating the number of identical groups attached to a particular atom.
Another shorthand structural diagram is the skeletal formula (also known as a bond-line formula or carbon skeleton diagram). In a skeletal formula, carbon atoms are not signified by the symbol C but by the vertices of the lines. Hydrogen atoms bonded to carbon are not shown—they can be inferred by counting the number of bonds to a particular carbon atom—each carbon is assumed to have four bonds in total, so any bonds not shown are, by implication, to hydrogen atoms.
Other diagrams may be more complex than Lewis structures, showing bonds in 3D using various forms such as space-filling diagrams .
Despite their simplicity and development in the early twentieth century, when understanding of chemical bonding was still rudimentary, Lewis structures capture many of the key features of the electronic structure of a range of molecular systems, including those of relevance to chemical reactivity. Thus, they continue to enjoy widespread use by chemists and chemistry educators. This is especially true in the field of organic chemistry , where the traditional valence-bond model of bonding still dominates, and mechanisms are often understood in terms of curve-arrow notation superimposed upon skeletal formulae , which are shorthand versions of Lewis structures. Due to the greater variety of bonding schemes encountered in inorganic and organometallic chemistry , many of the molecules encountered require the use of fully delocalized molecular orbitals to adequately describe their bonding, making Lewis structures comparatively less important (although they are still common).
There are simple and archetypal molecular systems for which a Lewis description, at least in unmodified form, is misleading or inaccurate. Notably, the naive drawing of Lewis structures for molecules known experimentally to contain unpaired electrons (e.g., O 2 , NO, and ClO 2 ) leads to incorrect inferences of bond orders, bond lengths, and/or magnetic properties. A simple Lewis model also does not account for the phenomenon of aromaticity . For instance, Lewis structures do not offer an explanation for why cyclic C 6 H 6 (benzene) experiences special stabilization beyond normal delocalization effects, while C 4 H 4 (cyclobutadiene) actually experiences a special destabilization . [ citation needed ] Molecular orbital theory provides the most straightforward explanation for these phenomena. [ original research? ] | https://en.wikipedia.org/wiki/Lewis_structure |
Lex petrolea is a proposed sub branch of lex mercatoria that would be based on the body of international arbitral awards related to the petroleum industry. The first use of the term was in the landmark arbitration case Government of the State of Kuwait v. American Independent Oil Co. (AMINOIL) where the argument was made that the past disputes had "generated a customary rule valid for the oil industry - a lex petrolea that was in some sort a particular branch of a general universal lex mercatoria ". [ 1 ]
This law -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Lex_petrolea |
In spherical geometry , Lexell's theorem holds that every spherical triangle with the same surface area on a fixed base has its apex on a small circle , called Lexell's circle or Lexell's locus , [ 1 ] passing through each of the two points antipodal to the two base vertices .
A spherical triangle is a shape on a sphere consisting of three vertices (corner points ) connected by three sides, each of which is part of a great circle (the analog on the sphere of a straight line in the plane , for example the equator and meridians of a globe). Any of the sides of a spherical triangle can be considered the base , and the opposite vertex is the corresponding apex . Two points on a sphere are antipodal if they are diametrically opposite, as far apart as possible.
The theorem is named for Anders Johan Lexell , who presented a paper about it c. 1777 (published 1784) including both a trigonometric proof and a geometric one. [ 2 ] Lexell's colleague Leonhard Euler wrote another pair of proofs in 1778 (published 1797), and a variety of proofs have been written since by Adrien-Marie Legendre (1800), Jakob Steiner (1827), Carl Friedrich Gauss (1841), Paul Serret (1855), and Joseph-Émile Barbier (1864), among others. [ 3 ]
The theorem is the analog of propositions 37 and 39 in Book I of Euclid's Elements , which prove that every planar triangle with the same area on a fixed base has its apex on a straight line parallel to the base. [ 4 ] An analogous theorem can also be proven for hyperbolic triangles , for which the apex lies on a hypercycle .
Given a fixed base A B , {\displaystyle AB,} an arc of a great circle on a sphere, and two apex points C {\displaystyle C} and X {\displaystyle X} on the same side of great circle A B , {\displaystyle AB,} Lexell's theorem holds that the surface area of the spherical triangle △ A B X {\displaystyle \triangle ABX} is equal to that of △ A B C {\displaystyle \triangle ABC} if and only if X {\displaystyle X} lies on the small-circle arc B ∗ C A ∗ , {\displaystyle B^{*}CA^{*}\!,} where A ∗ {\displaystyle A^{*}} and B ∗ {\displaystyle B^{*}} are the points antipodal to A {\displaystyle A} and B , {\displaystyle B,} respectively.
As one analog of the planar formula area = 1 2 base ⋅ height {\displaystyle {\text{area}}={\tfrac {1}{2}}\,{\text{base}}\cdot {\text{height}}} for the area of a triangle , the spherical excess ε {\displaystyle \varepsilon } of spherical triangle △ A B C {\displaystyle \triangle ABC} can be computed in terms of the base c {\displaystyle c} (the angular length of arc A B {\displaystyle AB} ) and "height" h c {\displaystyle h_{c}} (the angular distance between the parallel small circles A ∗ B ∗ C {\displaystyle A^{*}B^{*}C} and A B C ∗ {\displaystyle ABC^{*}} ): [ 5 ]
This formula is based on consideration of a sphere of radius 1 {\displaystyle 1} , on which arc length is called angle measure and surface area is called spherical excess or solid angle measure . The angle measure of a complete great circle is 2 π {\displaystyle 2\pi } radians , and the spherical excess of a hemisphere (half-sphere) is 2 π {\displaystyle 2\pi } steradians , where π {\displaystyle \pi } is the circle constant .
In the limit for triangles much smaller than the radius of the sphere, this reduces to the planar formula.
The small circles A ∗ B ∗ C {\displaystyle A^{*}B^{*}C} and A B C ∗ {\displaystyle ABC^{*}} each intersect the great circle A B {\displaystyle AB} at an angle of 1 2 ε . {\displaystyle {\tfrac {1}{2}}\varepsilon .} [ 6 ]
There are several ways to prove Lexell's theorem, each illuminating a different aspect of the relationships involved.
The main idea in Lexell's c. 1777 geometric proof – also adopted by Eugène Catalan (1843), Robert Allardice (1883), Jacques Hadamard (1901), Antoine Gob (1922), and Hiroshi Maehara (1999) – is to split the triangle △ A ∗ B ∗ C {\displaystyle \triangle A^{*}B^{*}C} into three isosceles triangles with common apex at the circumcenter P {\displaystyle P} and then chase angles to find the spherical excess ε {\displaystyle \varepsilon } of triangle △ A B C . {\displaystyle \triangle ABC.} In the figure, points A {\displaystyle A} and B {\displaystyle B} are on the far side of the sphere so that we can clearly see their antipodal points and all of Lexell's circle l . {\displaystyle l.} [ 7 ]
Let the base angles of the isosceles triangles △ B ∗ C P {\displaystyle \triangle B^{*}CP} (shaded red in the figure), △ C A ∗ P {\displaystyle \triangle CA^{*}P} (blue), and △ A ∗ B ∗ P {\displaystyle \triangle A^{*}B^{*}P} (purple) be respectively α , {\displaystyle \alpha ,} β , {\displaystyle \beta ,} and δ . {\displaystyle \delta .} (In some cases P {\displaystyle P} is outside △ A ∗ B ∗ C {\displaystyle \triangle A^{*}B^{*}C} ; then one of the quantities α , β , δ {\displaystyle \alpha ,\beta ,\delta } will be negative.) We can compute the internal angles of △ A B C {\displaystyle \triangle ABC} (orange) in terms of these angles: ∠ A = π − β − δ {\displaystyle \angle A=\pi -\beta -\delta } (the supplement of ∠ A ∗ {\displaystyle \angle A^{*}} ) and likewise ∠ B = π − α − δ , {\displaystyle \angle B=\pi -\alpha -\delta ,} and finally ∠ C = α + β . {\displaystyle \angle C=\alpha +\beta .}
By Girard's theorem the spherical excess of △ A B C {\displaystyle \triangle ABC} is
If base A B {\displaystyle AB} is fixed, for any third vertex C {\displaystyle C} falling on the same arc of Lexell's circle, the point P {\displaystyle P} and therefore the quantity δ {\displaystyle \delta } will not change, so the excess ε {\displaystyle \varepsilon } of △ A B C , {\displaystyle \triangle ABC,} which depends only on δ , {\displaystyle \delta ,} will likewise be constant. And vice versa: if ε {\displaystyle \varepsilon } remains constant when the point C {\displaystyle C} is changed, then so must δ {\displaystyle \delta } be, and therefore P {\displaystyle P} must be fixed, so C {\displaystyle C} must remain on Lexell's circle.
Jakob Steiner (1827) wrote a proof in similar style to Lexell's, also using Girard's theorem, but demonstrating the angle invariants in the triangle △ A ∗ B ∗ C {\displaystyle \triangle A^{*}B^{*}C} by constructing a cyclic quadrilateral inside the Lexell circle, using the property that pairs of opposite angles in a spherical cyclic quadrilateral have the same sum. [ 8 ] [ 9 ]
Starting with a triangle △ A B C {\displaystyle \triangle ABC} , let l {\displaystyle l} be the Lexell circle circumscribing △ A ∗ B ∗ C , {\displaystyle \triangle A^{*}B^{*}C,} and let D {\displaystyle D} be another point on l {\displaystyle l} separated from C {\displaystyle C} by the great circle B ∗ A ∗ . {\displaystyle B^{*}A^{*}\!.} Let α 1 = ∠ C A ∗ B ∗ , {\displaystyle \alpha _{1}=\angle CA^{*}B^{*}\!,} β 1 = ∠ A ∗ B ∗ C , {\displaystyle \beta _{1}=\angle A^{*}B^{*}C,} α 2 = ∠ B ∗ A ∗ D , {\displaystyle \alpha _{2}=\angle B^{*}A^{*}D,} β 2 = ∠ D A ∗ B ∗ . {\displaystyle \beta _{2}=\angle DA^{*}B^{*}\!.}
Because the quadrilateral ◻ A ∗ D B ∗ C {\displaystyle \square A^{*}DB^{*}C} is cyclic, the sum of each pair of its opposite angles is equal, ∠ C + ∠ D = {\displaystyle \angle C+\angle D={}\!} α 1 + α 2 + β 1 + β 2 , {\displaystyle \alpha _{1}+\alpha _{2}+\beta _{1}+\beta _{2},} or rearranged α 1 + β 1 − ∠ C = {\displaystyle \alpha _{1}+\beta _{1}-\angle C={}\!} ∠ D − α 2 − β 2 . {\displaystyle \angle D-\alpha _{2}-\beta _{2}.}
By Girard's theorem the spherical excess ε {\displaystyle \varepsilon } of △ A B C {\displaystyle \triangle ABC} is
The quantity ∠ D − α 2 − β 2 {\displaystyle \angle D-\alpha _{2}-\beta _{2}} does not depend on the choice of C , {\displaystyle C,} so is invariant when C {\displaystyle C} is moved to another point on the same arc of l . {\displaystyle l.} Therefore ε {\displaystyle \varepsilon } is also invariant.
Conversely, if C {\displaystyle C} is changed but ε {\displaystyle \varepsilon } is invariant, then the opposite angles of the quadrilateral ◻ A ∗ D B ∗ C {\displaystyle \square A^{*}DB^{*}C} will have the same sum, which implies C {\displaystyle C} lies on the small circle A ∗ D B ∗ . {\displaystyle A^{*}DB^{*}\!.}
Euler in 1778 proved Lexell's theorem analogously to Euclid's proof of Elements I.35 and I.37 , as did Victor-Amédée Lebesgue independently in 1855, using spherical parallelograms – spherical quadrilaterals with congruent opposite sides, which have parallel small circles passing through opposite pairs of adjacent vertices and are in many ways analogous to Euclidean parallelograms . There is one complication compared to Euclid's proof, however: The four sides of a spherical parallelogram are the great-circle arcs through the vertices rather than the parallel small circles. Euclid's proof does not need to account for the small lens-shaped regions sandwiched between the great and small circles, which vanish in the planar case. [ 10 ]
A lemma analogous to Elements I.35: two spherical parallelograms on the same base and between the same parallels have equal area.
Proof : Let ◻ A B C 1 D 1 {\displaystyle \square ABC_{1}D_{1}} and ◻ A B C 2 D 2 {\displaystyle \square ABC_{2}D_{2}} be spherical parallelograms with the great circle m {\displaystyle m} (the "midpoint circle") passing through the midpoints of sides B C 1 {\displaystyle BC_{1}} and A D 1 {\displaystyle AD_{1}} coinciding with the corresponding midpoint circle in ◻ A B C 2 D 2 . {\displaystyle \square ABC_{2}D_{2}.} Let F {\displaystyle F} be the intersection point between sides A D 2 {\displaystyle AD_{2}} and B C 1 . {\displaystyle BC_{1}.} Because the midpoint circle m {\displaystyle m} is shared, the two top sides C 1 D 1 {\displaystyle C_{1}D_{1}} and C 2 D 2 {\displaystyle C_{2}D_{2}} lie on the same small circle l {\displaystyle l} parallel to m {\displaystyle m} and antipodal to a small circle l ∗ {\displaystyle l^{*}} passing through A {\displaystyle A} and B . {\displaystyle B.}
Two arcs of l {\displaystyle l} are congruent, D 1 D 2 ≅ C 1 C 2 , {\displaystyle D_{1}D_{2}\cong C_{1}C_{2},} thus the two curvilinear triangles △ B C 1 C 2 {\displaystyle \triangle BC_{1}C_{2}} and △ A D 1 D 2 , {\displaystyle \triangle AD_{1}D_{2},} each bounded by l {\displaystyle l} on the top side, are congruent. Each parallelogram is formed from one of these curvilinear triangles added to the triangle △ A B F {\displaystyle \triangle ABF} and to one of the congruent lens-shaped regions between each top side and l , {\displaystyle l,} with the curvilinear triangle △ D 2 C 1 F {\displaystyle \triangle D_{2}C_{1}F} cut away. Therefore the parallelograms have the same area. (As in Elements , the case where the parallelograms do not intersect on the sides is omitted, but can be proven by a similar argument.)
Proof of Lexell's theorem : Given two spherical triangles △ A B C 1 {\displaystyle \triangle ABC_{1}} and △ A B C 2 {\displaystyle \triangle ABC_{2}} each with its apex on the same small circle l {\displaystyle l} through points A ∗ {\displaystyle A^{*}} and B ∗ , {\displaystyle B^{*}\!,} construct new segments C 1 D 1 {\displaystyle C_{1}D_{1}} and C 2 D 2 {\displaystyle C_{2}D_{2}} congruent to A B {\displaystyle AB} with vertices D 1 {\displaystyle D_{1}} and D 2 {\displaystyle D_{2}} on l . {\displaystyle l.} The two quadrilaterals ◻ A B C 1 D 1 {\displaystyle \square ABC_{1}D_{1}} and ◻ A B C 2 D 2 {\displaystyle \square ABC_{2}D_{2}} are spherical parallelograms, each formed by pasting together the respective triangle and a congruent copy. By the lemma, the two parallelograms have the same area, so the original triangles must also have the same area.
Proof of the converse : If two spherical triangles have the same area and the apex of the second is assumed to not lie on the Lexell circle of the first, then the line through one side of the second triangle can be intersected with the Lexell circle to form a new triangle which has a different area from the second triangle but the same area as the first triangle, a contradiction. This argument is the same as that found in Elements I.39.
Another proof using the midpoint circle which is more visually apparent in a single picture is due to Carl Friedrich Gauss (1841), who constructs the Saccheri quadrilateral (a quadrilateral with two adjacent right angles and two other equal angles) formed between the side of the triangle and its perpendicular projection onto the midpoint circle m , {\displaystyle m,} [ 11 ] which has the same area as the triangle. [ 12 ]
Let m {\displaystyle m} be the great circle through the midpoints M 1 {\displaystyle M_{1}} of A C {\displaystyle AC} and M 2 {\displaystyle M_{2}} of B C , {\displaystyle BC,} and let A ′ , {\displaystyle A',} B ′ , {\displaystyle B',} and C ′ {\displaystyle C'} be the perpendicular projections of the triangle vertices onto m . {\displaystyle m.} The resulting pair of right triangles △ A A ′ M 1 {\displaystyle \triangle AA'M_{1}} and △ C C ′ M 1 {\displaystyle \triangle CC'M_{1}} (shaded red) have equal angles at M 1 {\displaystyle M_{1}} ( vertical angles ) and equal hypotenuses , so they are congruent ; so are the triangles △ B B ′ M 2 {\displaystyle \triangle BB'M_{2}} and △ C C ′ M 2 {\displaystyle \triangle CC'M_{2}} (blue). Therefore, the area of triangle △ A B C {\displaystyle \triangle ABC} is equal to the area of Saccheri quadrilateral ◻ A B B ′ A ′ , {\displaystyle \square ABB'A',} as each consists of one red triangle, one blue triangle, and the green quadrilateral ◻ A B M 2 M 1 {\displaystyle \square ABM_{2}M_{1}} pasted together. (If C ′ {\displaystyle C'} falls outside the arc A ′ B ′ , {\displaystyle A'B',} then either the red or blue triangles will have negative signed area.) Because the great circle m , {\displaystyle m,} and therefore the quadrilateral ◻ A B B ′ A ′ , {\displaystyle \square ABB'A',} is the same for any choice of C {\displaystyle C} lying on the Lexell circle l , {\displaystyle l,} the area of the corresponding triangle △ A B C {\displaystyle \triangle ABC} is constant.
The stereographic projection maps the sphere to the plane. A designated great circle is mapped onto the primitive circle in the plane, and its poles are mapped to the origin (center of the primitive circle) and the point at infinity , respectively. Every circle on the sphere is mapped to a circle or straight line in the plane, with straight lines representing circles through the second pole. The stereographic projection is conformal , meaning it preserves angles.
To prove relationships about a general spherical triangle △ A B C , {\displaystyle \triangle ABC,} without loss of generality vertex A {\displaystyle A} can be taken as the point which projects to the origin. The sides of the spherical triangle then project to two straight segments and a circular arc. If the tangent lines to the circular side at the other two vertices intersect at point E , {\displaystyle E,} a planar straight-sided quadrilateral ◻ A B E C {\displaystyle \square ABEC} can be formed whose external angle at E {\displaystyle E} is the spherical excess ε = ∠ A + ∠ B + ∠ C − π {\displaystyle \varepsilon =\angle A+\angle B+\angle C-\pi } of the spherical triangle. This is sometimes called the Cesàro method of spherical trigonometry, after crystallographer Giuseppe Cesàro [ de ; fr ] who popularized it in two 1905 papers. [ 13 ]
Paul Serret (in 1855, a half century before Cesàro), and independently Aleksander Simonič (2019), used Cesàro's method to prove Lexell's theorem. Let O {\displaystyle O} be the center in the plane of the circular arc to which side B C {\displaystyle BC} projects. Then ◻ O B E C {\displaystyle \square OBEC} is a right kite , so the central angle ∠ B O C {\displaystyle \angle BOC} is equal to the external angle at E , {\displaystyle E,} the triangle's spherical excess ε . {\displaystyle \varepsilon .} Planar angle ∠ B B ∗ C {\displaystyle \angle BB^{*}C} is an inscribed angle subtending the same arc, so by the inscribed angle theorem has measure 1 2 ε . {\displaystyle {\tfrac {1}{2}}\varepsilon .} This relationship is preserved for any choice of C {\displaystyle C} ; therefore, the spherical excess of the triangle is constant whenever C {\displaystyle C} remains on the Lexell circle l , {\displaystyle l,} which projects to a line through B ∗ {\displaystyle B^{*}} in the plane. (If the area of the triangle is greater than a half-hemisphere, a similar argument can be made, but the point E {\displaystyle E} is no longer internal to the angle ∠ B O C . {\displaystyle \angle BOC.} ) [ 14 ]
Every spherical triangle has a dual , its polar triangle ; if triangle △ A ′ B ′ C ′ {\displaystyle \triangle A'B'C'} (shaded purple) is the polar triangle of △ A B C {\displaystyle \triangle ABC} (shaded orange) then the vertices A ′ , B ′ , C ′ {\displaystyle A'\!,B'\!,C'} are the poles of the respective sides B C , C A , A B , {\displaystyle BC,CA,AB,} and vice versa, the vertices A , B , C {\displaystyle A,B,C} are the poles of the sides B ′ C ′ , C ′ A ′ , A ′ B ′ . {\displaystyle B'C'\!,C'A'\!,A'B'\!.} The polar duality exchanges the sides ( central angles ) and external angles (dihedral angles) between the two triangles.
Because each side of the dual triangle is the supplement of an internal angle of the original triangle, the spherical excess ε {\displaystyle \varepsilon } of △ A B C {\displaystyle \triangle ABC} is a function of the perimeter p ′ {\displaystyle p'} of the dual triangle △ A ′ B ′ C ′ {\displaystyle \triangle A'B'C'} :
where the notation | P Q | {\displaystyle |PQ|} means the angular length of the great-circle arc P Q . {\displaystyle PQ.}
In 1854 Joseph-Émile Barbier – and independently László Fejes Tóth (1953) – used the polar triangle in his proof of Lexell's theorem, which is essentially dual to the proof by isosceles triangles above , noting that under polar duality the Lexell circle l {\displaystyle l} circumscribing △ A ∗ B ∗ C {\displaystyle \triangle A^{*}B^{*}C} becomes an excircle l ′ {\displaystyle l'} of △ A ′ B ′ C ′ {\displaystyle \triangle A'B'C'} ( incircle of a colunar triangle ) externally tangent to side A ′ B ′ . {\displaystyle A'B'.} [ 15 ]
If vertex C {\displaystyle C} is moved along l , {\displaystyle l,} the side A ′ B ′ {\displaystyle A'B'} changes but always remains tangent to the same circle l ′ . {\displaystyle l'.} Because the arcs from each vertex to either adjacent touch point of an incircle or excircle are congruent, A ′ T B ≅ A ′ T C {\displaystyle A'T_{B}\cong A'T_{C}} (blue segments) and B ′ T A ≅ B ′ T C {\displaystyle B'T_{A}\cong B'T_{C}} (red segments), the perimeter p ′ {\displaystyle p'} is
which remains constant, depending only on the circle l ′ {\displaystyle l'} but not on the changing side A ′ B ′ . {\displaystyle A'B'.} Conversely, if the point C {\displaystyle C} moves off of l , {\displaystyle l,} the associated excircle l ′ {\displaystyle l'} will change in size, moving the points T A {\displaystyle T_{A}} and T B {\displaystyle T_{B}} both toward or both away from C ′ ∗ {\displaystyle C'^{*}} and changing the perimeter p ′ {\displaystyle p'} of △ A ′ B ′ C ′ {\displaystyle \triangle A'B'C'\!} and thus changing ε . {\displaystyle \varepsilon .}
The locus of points C {\displaystyle C} for which ε {\displaystyle \varepsilon } is constant is therefore l . {\displaystyle l.}
Both Lexell ( c. 1777 ) and Euler (1778) included trigonometric proofs in their papers, and several later mathematicians have presented trigonometric proofs, including Adrien-Marie Legendre (1800), Louis Puissant (1842), Ignace-Louis-Alfred Le Cointe (1858), and Joseph-Alfred Serret (1862). Such proofs start from known triangle relations such as the spherical law of cosines or a formula for spherical excess, and then proceed by algebraic manipulation of trigonometric identities . [ 16 ]
The sphere is separated into two hemispheres by the great circle A B , {\displaystyle AB,} and any Lexell circle through A ∗ {\displaystyle A^{*}} and B ∗ {\displaystyle B^{*}} is separated into two arcs, one in each hemisphere. If the point X {\displaystyle X} is on the opposite arc from C , {\displaystyle C,} then the areas of △ A B C {\displaystyle \triangle ABC} and △ A B X {\displaystyle \triangle ABX} will generally differ. However, if spherical surface area is interpreted to be signed, with sign determined by boundary orientation, then the areas of triangle △ A B C {\displaystyle \triangle ABC} and △ A B X {\displaystyle \triangle ABX} have opposite signs and differ by the area of a hemisphere.
Lexell suggested a more general framing. Given two distinct non-antipodal points A {\displaystyle A} and B , {\displaystyle B,} there are two great-circle arcs joining them: one shorter than a semicircle and the other longer. Given a triple A , B , C {\displaystyle A,B,C} of points, typically △ A B C {\displaystyle \triangle ABC} is interpreted to mean the area enclosed by the three shorter arcs joining each pair. However, if we allow choice of arc for each pair, then 8 distinct generalized spherical triangles can be made, some with self intersections, of which four might be considered to have the same base A B . {\displaystyle AB.}
These eight triangles do not all have the same surface area, but if area is interpreted to be signed, with sign determined by boundary orientation, then those which differ differ by the area of a hemisphere. [ 17 ]
In this context, given four distinct, non-antipodal points A , {\displaystyle A,} B , {\displaystyle B,} C , {\displaystyle C,} and X {\displaystyle X} on a sphere, Lexell's theorem holds that the signed surface area of any generalized triangle △ A B C {\displaystyle \triangle ABC} differs from that of any generalized triangle △ A B X {\displaystyle \triangle ABX} by a whole number of hemispheres if and only if A ∗ , {\displaystyle A^{*}\!,} B ∗ , {\displaystyle B^{*}\!,} C , {\displaystyle C,} and X {\displaystyle X} are concyclic .
As the apex C {\displaystyle C} approaches either of the points antipodal to the base vertices – say B ∗ {\displaystyle B^{*}} – along Lexell's circle l , {\displaystyle l,} in the limit the triangle degenerates to a lune tangent to l {\displaystyle l} at B ∗ {\displaystyle B^{*}} and tangent to the antipodal small circle l ∗ {\displaystyle l^{*}} at B , {\displaystyle B,} and having the same excess ε {\displaystyle \varepsilon } as any of the triangles with apex on the same arc of l . {\displaystyle l.} As a degenerate triangle, it has a straight angle at A {\displaystyle A} (i.e. ∠ A = π , {\displaystyle \angle A=\pi ,} a half turn) and equal angles B = B ∗ = 1 2 ε . {\displaystyle B=B^{*}={\tfrac {1}{2}}\varepsilon .} [ 18 ]
As C {\displaystyle C} approaches B ∗ {\displaystyle B^{*}} from the opposite direction (along the other arc of Lexell's circle), in the limit the triangle degenerates to the co-hemispherical lune tangent to the Lexell circle at B ∗ {\displaystyle B^{*}} with the opposite orientation and angles ∠ B = ∠ B ⋆ = π − 1 2 ε . {\displaystyle \angle B=\angle B^{\star }=\pi -{\tfrac {1}{2}}\varepsilon .}
The area of a spherical triangle is equal to half a hemisphere (excess ε = π {\displaystyle \varepsilon =\pi } ) if and only if the Lexell circle A ∗ B ∗ C {\displaystyle A^{*}B^{*}C} is orthogonal to the great circle A B , {\displaystyle AB,} that is if arc A ∗ B ∗ {\displaystyle A^{*}B^{*}} is a diameter of circle A ∗ B ∗ C {\displaystyle A^{*}B^{*}C} and arc A B {\displaystyle AB} is a diameter of A B C ∗ . {\displaystyle ABC^{*}\!.}
In this case, letting D {\displaystyle D} be the point diametrically opposed to C {\displaystyle C} on the Lexell circle A ∗ B ∗ C {\displaystyle A^{*}B^{*}C} then the four triangles △ A B C , {\displaystyle \triangle ABC,} △ B A D , {\displaystyle \triangle BAD,} △ C D A , {\displaystyle \triangle CDA,} and △ D C B {\displaystyle \triangle DCB} are congruent, and together form a spherical disphenoid A B C D {\displaystyle ABCD} (the central projection of a disphenoid onto a concentric sphere). The eight points A A ∗ B B ∗ C C ∗ D D ∗ {\displaystyle AA^{*}BB^{*}CC^{*}DD^{*}} are the vertices of a rectangular cuboid . [ 19 ]
A spherical parallelogram is a spherical quadrilateral ◻ A B C D {\displaystyle \square ABCD} whose opposite sides and opposite angles are congruent ( A B ≅ C D , {\displaystyle AB\cong CD,} B C ≅ D A , {\displaystyle BC\cong DA,} ∠ A = ∠ C , {\displaystyle \angle A=\angle C,} ∠ B = ∠ D {\displaystyle \angle B=\angle D} ). It is in many ways analogous to a planar parallelogram . The two diagonals A C {\displaystyle AC} and B D {\displaystyle BD} bisect each-other and the figure has 2-fold rotational symmetry about the intersection point (so the diagonals each split the parallelogram into two congruent spherical triangles, △ A B C ≅ △ C D A {\displaystyle \triangle ABC\cong \triangle CDA} and △ A B D ≅ △ C D B {\displaystyle \triangle ABD\cong \triangle CDB} ); if the midpoints of either pair of opposite sides are connected by a great circle m {\displaystyle m} , the four vertices fall on two parallel small circles equidistant from it. More specifically, any vertex (say D {\displaystyle D} ) of the spherical parallelogram lies at the intersection of the two Lexell circles ( l c d {\displaystyle l_{cd}} and l d a {\displaystyle l_{da}} ) passing through one of the adjacent vertices and the points antipodal to the other two vertices.
As with spherical triangles, spherical parallelograms with the same base and the apex vertices lying on the same Lexell circle have the same area; see § Spherical parallelograms above. Starting from any spherical triangle, a second congruent triangle can be formed via a (spherical) point reflection across the midpoint of any side. When combined, these two triangles form a spherical parallelogram with twice the area of the original triangle. [ 20 ]
The polar dual to Lexell's theorem, sometimes called Sorlin's theorem after A. N. J. Sorlin who first proved it trigonometrically in 1825, holds that for a spherical trilateral △ a b c {\displaystyle \triangle abc} with sides on fixed great circles a , b {\displaystyle a,b} (thus fixing the angle between them) and a fixed perimeter p = | a | + | b | + | c | {\displaystyle p=|a|+|b|+|c|} (where | a | {\displaystyle |a|} means the length of the triangle side a {\displaystyle a} ), the envelope of the third side c {\displaystyle c} is a small circle internally tangent to a , b {\displaystyle a,b} and externally tangent to c , {\displaystyle c,} the excircle to trilateral △ a b c . {\displaystyle \triangle abc.} Joseph-Émile Barbier later wrote a geometrical proof (1864) which he used to prove Lexell's theorem, by duality; see § Perimeter of the polar triangle above. [ 21 ]
This result also applies in Euclidean and hyperbolic geometry: Barbier's geometrical argument can be transplanted directly to the Euclidean or hyperbolic plane.
Lexell's loci for any base A B {\displaystyle AB} make a foliation of the sphere (decomposition into one-dimensional leaves ). These loci are arcs of small circles with endpoints at A ∗ {\displaystyle A^{*}} and B ∗ , {\displaystyle B^{*}\!,} on which any intermediate point C {\displaystyle C} is the apex of a triangle A B C {\displaystyle ABC} of a fixed signed area. That area is twice the signed angle between the Lexell circle and the great circle A B A ∗ B ∗ {\displaystyle ABA^{*}B^{*}} at either of the points A ∗ {\displaystyle A^{*}} or B ∗ {\displaystyle B^{*}} ; see § Lunar degeneracy above. In the figure, the Lexell circles are in green, except for those whose triangles' area is a multiple of a half hemisphere, which are black, with area labeled; see § Half-hemisphere area above. [ 22 ]
These Lexell circles through A ∗ {\displaystyle A^{*}} and B ∗ {\displaystyle B^{*}} are the spherical analog of the family of Apollonian circles through two points in the plane.
In 1784 Nicolas Fuss posed and solved the problem of finding the triangle △ A B C {\displaystyle \triangle ABC} of maximal area on a given base A B {\displaystyle AB} with its apex C {\displaystyle C} on a given great circle g . {\displaystyle g.} Fuss used an argument involving infinitesimal variation of C , {\displaystyle C,} but the solution is also a straightforward corollary of Lexell's theorem: the Lexell circle A ∗ B ∗ C {\displaystyle A^{*}B^{*}C} through the apex must be tangent to g {\displaystyle g} at C . {\displaystyle C.}
If g {\displaystyle g} crosses the great circle through A B {\displaystyle AB} at a point P {\displaystyle P} , then by the spherical analog of the tangent–secant theorem , the angular distance P C {\displaystyle PC} to the desired point of tangency satisfies
from which we can explicitly construct the point C {\displaystyle C} on g {\displaystyle g} such that △ A B C {\displaystyle \triangle ABC} has maximum area. [ 23 ]
In 1786 Theodor von Schubert posed and solved the problem of finding the spherical triangles of maximum and minimum area of a given base and altitude (the spherical length of a perpendicular dropped from the apex to the great circle containing the base); spherical triangles with constant altitude have their apex on a common small circle (the "altitude circle") parallel to the great circle containing the base. Schubert solved this problem by a calculus-based trigonometric approach to show that the triangle of minimal area has its apex at the nearest intersection of the altitude circle and the perpendicular bisector of the base, and the triangle of maximal area has its apex at the far intersection. However, this theorem is also a straightforward corollary of Lexell's theorem: the Lexell circles through the points antipodal to the base vertices representing the smallest and largest triangle areas are those tangent to the altitude circle. In 2019 Vincent Alberge and Elena Frenkel solved the analogous problem in the hyperbolic plane. [ 24 ]
In the Euclidean plane, a median of a triangle is the line segment connecting a vertex to the midpoint of the opposite side. The three medians of a triangle all intersect at its centroid . Each median bisects the triangle's area.
On the sphere, a median of a triangle can also be defined as the great-circle arc connecting a vertex to the midpoint of the opposite side. The three medians all intersect at a point, the central projection onto the sphere of the triangle's extrinsic centroid – that is, centroid of the flat triangle containing the three points if the sphere is embedded in 3-dimensional Euclidean space. However, on the sphere the great-circle arc through one vertex and a point on the opposite side which bisects the triangle's area is, in general, distinct from the corresponding median.
Jakob Steiner used Lexell's theorem to prove that these three area-bisecting arcs (which he called "equalizers") all intersect in a point, one possible alternative analog of the planar centroid in spherical geometry. (A different spherical analog of the centroid is the apex of three triangles of equal area whose bases are the sides of the original triangle, the point with ( 1 3 , 1 3 , 1 3 ) {\displaystyle {\bigl (}{\tfrac {1}{3}},{\tfrac {1}{3}},{\tfrac {1}{3}}{\bigr )}} as its spherical area coordinates .) [ 25 ]
The barycentric coordinate system for points relative to a given triangle in affine space does not have a perfect analogy in spherical geometry; there is no single spherical coordinate system sharing all of its properties. One partial analogy is spherical area coordinates for a point P {\displaystyle P} relative to a given spherical triangle △ A B C , {\displaystyle \triangle ABC,}
where each quantity ε Q R S {\displaystyle \varepsilon _{QRS}} is the signed spherical excess of the corresponding spherical triangle △ Q R S . {\displaystyle \triangle QRS.} These coordinates sum to 1 , {\displaystyle 1,} and using the same definition in the plane results in barycentric coordinates.
By Lexell's theorem, the locus of points with one coordinate constant is the corresponding Lexell circle. It is thus possible to find the point corresponding to a given triple of spherical area coordinates by intersecting two small circles.
Using their respective spherical area coordinates, any spherical triangle can be mapped to any other, or to any planar triangle, using corresponding barycentric coordinates in the plane. This can be used for polyhedral map projections ; for the definition of discrete global grids ; or for parametrizing triangulations of the sphere or texture mapping any triangular mesh topologically equivalent to a sphere. [ 26 ]
The analog of Lexell's theorem in the Euclidean plane comes from antiquity, and can be found in Book I of Euclid's Elements , propositions 37 and 39, built on proposition 35. In the plane, Lexell's circle degenerates to a straight line (which could be called Lexell's line ) parallel to the base. [ 4 ]
Elements I.35 holds that parallelograms with the same base whose top sides are colinear have equal area. Proof : Let the two parallelograms be ◻ A B C 1 D 1 {\displaystyle \square ABC_{1}D_{1}} and ◻ A B C 2 D 2 , {\displaystyle \square ABC_{2}D_{2},} with common base A B {\displaystyle AB} and C 1 , {\displaystyle C_{1},} D 1 , {\displaystyle D_{1},} C 2 , {\displaystyle C_{2},} and D 2 {\displaystyle D_{2}} on a common line parallel to the base, and let F {\displaystyle F} be the intersection between B C 1 {\displaystyle BC_{1}} and A D 2 . {\displaystyle AD_{2}.} Then the two top sides are congruent C 1 D 1 ≅ C 2 D 2 {\displaystyle C_{1}D_{1}\cong C_{2}D_{2}} so, adding the intermediate segment to each, C 1 C 2 ≅ D 1 D 2 . {\displaystyle C_{1}C_{2}\cong D_{1}D_{2}.} Therefore the two triangles △ B C 1 C 2 {\displaystyle \triangle BC_{1}C_{2}} and △ A D 1 D 2 {\displaystyle \triangle AD_{1}D_{2}} have matching sides so are congruent. Now each of the parallelograms is formed from one of these triangles, added to the triangle △ A B F {\displaystyle \triangle ABF} with the triangle △ D 2 C 1 F {\displaystyle \triangle D_{2}C_{1}F} cut away, so therefore the two parallelograms ◻ A B C 1 D 1 {\displaystyle \square ABC_{1}D_{1}} and ◻ A B C 2 D 2 {\displaystyle \square ABC_{2}D_{2}} have equal area.
Elements I.37 holds that triangles with the same base and an apex on the same line parallel to the base have equal area. Proof : Let triangles △ A B C 1 {\displaystyle \triangle ABC_{1}} and △ A B C 2 {\displaystyle \triangle ABC_{2}} each have its apex on the same line l {\displaystyle l} parallel to the base A B . {\displaystyle AB.} Construct new segments C 1 D 1 {\displaystyle C_{1}D_{1}} and C 2 D 2 {\displaystyle C_{2}D_{2}} congruent to A B {\displaystyle AB} with vertices D 1 {\displaystyle D_{1}} and D 2 {\displaystyle D_{2}} on l . {\displaystyle l.} The two quadrilaterals ◻ A B C 1 D 1 {\displaystyle \square ABC_{1}D_{1}} and ◻ A B C 2 D 2 {\displaystyle \square ABC_{2}D_{2}} are parallelograms, each formed by pasting together the respective triangle and a congruent copy. By I.35, the two parallelograms have the same area, so the original triangles must also have the same area.
Elements I.39 is the converse: two triangles of equal area on the same side of the same base have their apexes on a line parallel to the base. Proof : If two triangles have the same base and same area and the apex of the second is assumed to not lie on the line parallel to the base (the "Lexell line") through the first, then the line through one side of the second triangle can be intersected with the Lexell line to form a new triangle which has a different area from the second triangle but the same area as the first triangle, a contradiction.
In the Euclidean plane, the area ε {\displaystyle \varepsilon } of triangle △ A B C {\displaystyle \triangle ABC} can be computed using any side length (the base ) and the distance between the line through the base and the parallel line through the apex (the corresponding height ). Using point C {\displaystyle C} as the apex, and multiplying both sides of the traditional identity by 1 2 {\displaystyle {\tfrac {1}{2}}} to make the analogy to the spherical case more obvious, this is:
The Euclidean theorem can be taken as a corollary of Lexell's theorem on the sphere. It is the limiting case as the curvature of the sphere approaches zero, i.e. for spherical triangles as which are infinitesimal in proportion to the radius of the sphere.
In the hyperbolic plane , given a triangle △ A B C , {\displaystyle \triangle ABC,} the locus of a variable point X {\displaystyle X} such that the triangle △ A B X {\displaystyle \triangle ABX} has the same area as △ A B C {\displaystyle \triangle ABC} is a hypercycle passing through the points antipodal to A {\displaystyle A} and B , {\displaystyle B,} which could be called Lexell's hypercycle . Several proofs from the sphere have straightforward analogs in the hyperbolic plane, including a Gauss-style proof via a Saccheri quadrilateral by Barbarin (1902) and Frenkel & Su (2019), an Euler-style proof via hyperbolic parallelograms by Papadopoulos & Su (2017), and a Paul Serret-style proof via stereographic projection by Shvartsman (2007). [ 27 ]
In spherical geometry, the antipodal transformation takes each point to its antipodal (diametrically opposite) point. For a sphere embedded in Euclidean space, this is a point reflection through the center of the sphere; for a sphere stereographically projected to the plane, it is an inversion across the primitive circle composed with a point reflection across the origin (or equivalently, an inversion in a circle of imaginary radius of the same magnitude as the radius of the primitive circle).
In planar hyperbolic geometry , there is a similar antipodal transformation, but any two antipodal points lie in opposite branches of a double hyperbolic plane. For a hyperboloid of two sheets embedded in Minkowski space of signature ( − , + , + ) , {\displaystyle (-,+,+),} known as the hyperboloid model , the antipodal transformation is a point reflection through the center of the hyperboloid which takes each point onto the opposite sheet; in the conformal half-plane model it is a reflection across the boundary line of ideal points taking each point into the opposite half-plane; in the conformal disk model it is an inversion across the boundary circle, taking each point in the disk to a point in its complement. As on the sphere, any generalized circle passing through a pair of antipodal points in hyperbolic geometry is a geodesic . [ 28 ]
Analogous to the planar and spherical triangle area formulas, the hyperbolic area ε {\displaystyle \varepsilon } of the triangle can be computed in terms of the base c {\displaystyle c} (the hyperbolic length of arc A B {\displaystyle AB} ) and "height" h c {\displaystyle h_{c}} (the hyperbolic distance between the parallel hypercycles A ∗ B ∗ C {\displaystyle A^{*}B^{*}C} and A B C ∗ {\displaystyle ABC^{*}} ):
As in the spherical case, in the small-triangle limit this reduces to the planar formula.
Chasles, Michel (1837), Aperçu historique sur l'origine et le développment des méthodes en géométrie [ Historical overview of the origin and development of methods in geometry ] (in French), Brussels: Hayez, Ch. 5, §§ 42–45 , "Géométrie de la sphère" [Spherical geometry], pp. 235–240
Note that h c {\displaystyle h_{c}} is the shortest angular distance from C {\displaystyle C} to the small circle A B C ∗ , {\displaystyle ABC^{*}\!,} not the shortest distance from C {\displaystyle C} to the great circle A B . {\displaystyle AB.}
Maehara, Hiroshi (1999), "Lexell's theorem via an inscribed angle theorem", American Mathematical Monthly , 106 (4): 352– 353, doi : 10.1080/00029890.1999.12005052
Lexell, Anders Johan (1786), "De proprietatibus circulorum in superficie sphaerica descriptorum" [On the properties of circles described on a spherical surface], Acta Academiae Scientiarum Imperialis Petropolitanae (in Latin), 6 : 1782 (1): 58– 103, figures tab. 3
Steiner, Jakob (1841), "Sur le maximum et le minimum des figures dans le plan, sur la sphère et dans l'espace général" [On the maximum and the minimum of figures in the plane, on the sphere and in general space], Journal de mathématiques pures et appliquées (in French), 6 : 105– 170, EuDML 234575
Lebesgue, Victor-Amédée (1855), "Démonstration du théorème de Lexell" [Proof of Lexell's theorem], Nouvelles annales de mathématiques (in French), 14 : 24– 26, EuDML 96674
Persson, Ulf (2012), "Lexell's Theorem" (PDF) , Normat , 60 (3): 133– 134
Van Brummelen, Glen (2012), "8. Stereographic Projection", Heavenly Mathematics , Princeton University Press, pp. 129– 150
Maehara, Hiroshi; Martini, Horst (2022), "On Cesàro triangles and spherical polygons", Aequationes Mathematicae , 96 (2): 361– 379, doi : 10.1007/s00010-021-00820-y
The polar dual to Lexell's theorem had been previously proved trigonometrically by A. N. J. Sorlin (1825); see § Sorlin's theorem below.
Serret, Joseph-Alfred (1862), "Expressions du rayon du cercle circonscrit et des rayons des cercles inscrit et exinscrits." [Expressions of the radius of the circumscribed circle and the radii of the inscribed and exscribed circles.], Traité de trigonométrie [ Treatise on trigonometry ] (in French) (3rd ed.), Mallet-Bachelier, § 94 , pp. 141–142
Study, Eduard (1896), "Some Researches in Spherical Trigonometry" , Mathematical Papers Read at the International Mathematical Congress , International Mathematical Congress , Chicago, 1893, MacMillan, pp. 382– 394
Brooks, Jeff; Strantzen, John (2005), "Spherical Triangles of Area π and Isosceles Tetrahedra" (PDF) , Mathematics Magazine , 78 (4): 311– 314, doi : 10.1080/0025570X.2005.11953347 , JSTOR 30044179
Sorlin, A. N. J.; Gergonne, Joseph Diez (1825), "Trigonométrie. Recherches de trigonométrie sphérique" [Trigonometry. Research on spherical trigonometry], Annales de Mathématiques Pures et Appliquées , 15 : 273– 304, EuDML 80036
Fuss, Nicolas (1788) [written 1784], "Problematum quorundam sphaericorum solutio" , Nova Acta Academiae Scientiarum Imperialis Petropolitanae , 2 : 70– 83
Alberge, Vincent; Frenkel, Elena (2019), "3. On a problem of Schubert in hyperbolic geometry", in Alberge, Vincent; Papadopoulos, Athanase (eds.), Eighteen Essays in Non-Euclidean Geometry , European Mathematical Society, pp. 27– 46, doi : 10.4171/196-1/2
Lei, Kin; Qi, Dongxu; Tian, Xiaolin (2020), "A new coordinate system for constructing spherical grid systems" , Applied Sciences , 10 (2): 655, doi : 10.3390/app10020655
Shvartsman, Osip Vladimirovich (2007), Комментарий к статье П. В. Бибикова и И. В. Ткаченко «О трисекции и бисекции треугольника на плоскости Лобачевского» [Comment on the article by P. V. Bibikov and I. V. Tkachenko 'On trisection and bisection of a triangle in the Lobachevsky plane'] (PDF) , Matematicheskoe Prosveschenie , ser. 3 (in Russian), 11 : 127– 130
Johnson, Norman W. (1981), "Absolute Polarities and Central Inversion" , in Davis, Chandler; Grünbaum, Branko; Sherk, F.A. (eds.), The Geometric Vein: The Coxeter Festschrift , Springer, pp. 443– 464, doi : 10.1007/978-1-4612-5648-9_28 | https://en.wikipedia.org/wiki/Lexell's_theorem |
Lexicographic codes or lexicodes are greedily generated error-correcting codes with remarkably good properties. They were produced independently by Vladimir Levenshtein [ 1 ] and by John Horton Conway and Neil Sloane . [ 2 ] The binary lexicographic codes are linear codes , and include the Hamming codes and the binary Golay codes . [ 2 ]
A lexicode of length n and minimum distance d over a finite field is generated by starting with the all-zero vector and iteratively adding the next vector (in lexicographic order ) of minimum Hamming distance d from the vectors added so far. As an example, the length-3 lexicode of minimum distance 2 would consist of the vectors marked by an "X" in the following example:
Here is a table of all n-bit lexicode by d-bit minimal hamming distance, resulting of maximum 2 m codewords dictionnary.
For example, F 4 code (n=4,d=2,m=3), extended Hamming code (n=8,d=4,m=4) and especially Golay code (n=24,d=8,m=12) shows exceptional compactness compared to neighbors.
All odd d-bit lexicode distances are exact copies of the even d+1 bit distances minus the last dimension, so
an odd-dimensional space can never create something new or more interesting than the d+1 even-dimensional space above.
Since lexicodes are linear, they can also be constructed by means of their basis . [ 3 ]
Following C generate lexicographic code and parameters are set for the Golay code (N=24, D=8).
The theory of lexicographic codes is closely connected to combinatorial game theory . In particular, the codewords in a binary lexicographic code of distance d encode the winning positions in a variant of Grundy's game , played on a collection of heaps of stones, in which each move consists of replacing any one heap by at most d − 1 smaller heaps, and the goal is to take the last stone. [ 2 ] | https://en.wikipedia.org/wiki/Lexicographic_code |
In mathematics , the lexicographic or lexicographical order (also known as lexical order , or dictionary order ) is a generalization of the alphabetical order of the dictionaries to sequences of ordered symbols or, more generally, of elements of a totally ordered set .
There are several variants and generalizations of the lexicographical ordering. One variant applies to sequences of different lengths by comparing the lengths of the sequences before considering their elements.
Another variant, widely used in combinatorics , orders subsets of a given finite set by assigning a total order to the finite set, and converting subsets into increasing sequences , to which the lexicographical order is applied.
A generalization defines an order on an n -ary Cartesian product of partially ordered sets ; this order is a total order if and only if all factors of the Cartesian product are totally ordered.
The words in a lexicon (the set of words used in some language) have a conventional ordering, used in dictionaries and encyclopedias , that depends on the underlying ordering of the alphabet of symbols used to build the words. The lexicographical order is one way of formalizing word order given the order of the underlying symbols.
The formal notion starts with a finite set A , often called the alphabet , which is totally ordered . That is, for any two symbols a and b in A that are not the same symbol, either a < b or b < a .
The words of A are the finite sequences of symbols from A , including words of length 1 containing a single symbol, words of length 2 with 2 symbols, and so on, even including the empty sequence ε {\displaystyle \varepsilon } with no symbols at all. The lexicographical order on the set of all these finite words orders the words as follows:
However, in combinatorics , another convention is frequently used for the second case, whereby a shorter sequence is always smaller than a longer sequence. This variant of the lexicographical order is sometimes called shortlex order .
In lexicographical order, the word "Thomas" appears before "Thompson" because they first differ at the fifth letter ('a' and 'p'), and letter 'a' comes before the letter 'p' in the alphabet. Because it is the first difference, in this case the 5th letter is the "most significant difference" for alphabetical ordering.
An important property of the lexicographical order is that for each n , the set of words of length n is well-ordered by the lexicographical order (provided the alphabet is finite); that is, every decreasing sequence of words of length n is finite (or equivalently, every non-empty subset has a least element). [ 1 ] [ 2 ] It is not true that the set of all finite words is well-ordered; for example, the infinite set of words {b, ab, aab, aaab, ... } has no lexicographically earliest element.
The lexicographical order is used not only in dictionaries, but also commonly for numbers and dates.
One of the drawbacks of the Roman numeral system is that it is not always immediately obvious which of two numbers is the smaller. On the other hand, with the positional notation of the Hindu–Arabic numeral system , comparing numbers is easy, because the natural order on natural numbers is the same as the variant shortlex of the lexicographic order. In fact, with positional notation, a natural number is represented by a sequence of numerical digits , and a natural number is larger than another one if either it has more digits (ignoring leading zeroes) or the number of digits is the same and the first (most significant) digit which differs is larger.
For real numbers written in decimal notation , a slightly different variant of the lexicographical order is used: the parts on the left of the decimal point are compared as before; if they are equal, the parts at the right of the decimal point are compared with the lexicographical order. The padding 'blank' in this context is a trailing "0" digit.
When negative numbers are also considered, one has to reverse the order for comparing negative numbers. This is not usually a problem for humans, but it may be for computers (testing the sign takes some time). This is one of the reasons for adopting two's complement representation for representing signed integers in computers.
Another example of a non-dictionary use of lexicographical ordering appears in the ISO 8601 standard for dates, which expresses a date as YYYY-MM-DD. This formatting scheme has the advantage that the lexicographical order on sequences of characters that represent dates coincides with the chronological order : an earlier CE date is smaller in the lexicographical order than a later date up to year 9999. This date ordering makes computerized sorting of dates easier by avoiding the need for a separate sorting algorithm.
The monoid of words over an alphabet A is the free monoid over A . That is, the elements of the monoid are the finite sequences (words) of elements of A (including the empty sequence, of length 0), and the operation (multiplication) is the concatenation of words. A word u is a prefix (or 'truncation') of another word v if there exists a word w such that v = uw . By this definition, the empty word ( ε {\displaystyle \varepsilon } ) is a prefix of every word, and every word is a prefix of itself (with w = ε {\displaystyle =\varepsilon } ); care must be taken if these cases are to be excluded.
With this terminology, the above definition of the lexicographical order becomes more concise: Given a partially or totally ordered set A , and two words a and b over A such that b is non-empty, then one has a < b under lexicographical order, if at least one of the following conditions is satisfied:
Notice that, due to the prefix condition in this definition, ε < b for all b ≠ ε , {\displaystyle \varepsilon <b\,\,{\text{ for all }}b\neq \varepsilon ,} where ε {\displaystyle \varepsilon } is the empty word.
If < {\displaystyle \,<\,} is a total order on A , {\displaystyle A,} then so is the lexicographic order on the words of A . {\displaystyle A.} However, in general this is not a well-order , even if the alphabet A {\displaystyle A} is well-ordered. For instance, if A = { a , b } , the language { a n b | n ≥ 0, b > ε } has no least element in the lexicographical order: ... < aab < ab < b .
Since many applications require well orders, a variant of the lexicographical orders is often used. This well-order, sometimes called shortlex or quasi-lexicographical order , consists in considering first the lengths of the words (if length( a ) < length( b ) , then a < b {\displaystyle a<b} ), and, if the lengths are equal, using the lexicographical order. If the order on A is a well-order, the same is true for the shortlex order. [ 2 ] [ 3 ]
The lexicographical order defines an order on an n -ary Cartesian product of ordered sets, which is a total order when all these sets are themselves totally ordered. An element of a Cartesian product E 1 × ⋯ × E n {\displaystyle E_{1}\times \cdots \times E_{n}} is a sequence whose i {\displaystyle i} th element belongs to E i {\displaystyle E_{i}} for every i . {\displaystyle i.} As evaluating the lexicographical order of sequences compares only elements which have the same rank in the sequences, the lexicographical order extends to Cartesian products of ordered sets.
Specifically, given two partially ordered sets A {\displaystyle A} and B , {\displaystyle B,} the lexicographical order on the Cartesian product A × B {\displaystyle A\times B} is defined as ( a , b ) ≤ ( a ′ , b ′ ) if and only if a < a ′ or ( a = a ′ and b ≤ b ′ ) , {\displaystyle (a,b)\leq \left(a^{\prime },b^{\prime }\right){\text{ if and only if }}a<a^{\prime }{\text{ or }}\left(a=a^{\prime }{\text{ and }}b\leq b^{\prime }\right),}
The result is a partial order. If A {\displaystyle A} and B {\displaystyle B} are each totally ordered , then the result is a total order as well. The lexicographical order of two totally ordered sets is thus a linear extension of their product order .
One can define similarly the lexicographic order on the Cartesian product of an infinite family of ordered sets, if the family is indexed by the natural numbers , or more generally by a well-ordered set. This generalized lexicographical order is a total order if each factor set is totally ordered.
Unlike the finite case, an infinite product of well-orders is not necessarily well-ordered by the lexicographical order. For instance, the set of countably infinite binary sequences (by definition, the set of functions from natural numbers to { 0 , 1 } , {\displaystyle \{0,1\},} also known as the Cantor space { 0 , 1 } ω {\displaystyle \{0,1\}^{\omega }} ) is not well-ordered; the subset of sequences that have precisely one 1 {\displaystyle 1} (that is, { 100000..., 010000..., 001000..., ... } ) does not have a least element under the lexicographical order induced by 0 < 1 , {\displaystyle 0<1,} because 100000... > 010000... > 001000... > ... is an infinite descending chain . [ 1 ] Similarly, the infinite lexicographic product is not Noetherian either because 011111... < 101111... < 110111 ... < ... is an infinite ascending chain.
The functions from a well-ordered set X {\displaystyle X} to a totally ordered set Y {\displaystyle Y} may be identified with sequences indexed by X {\displaystyle X} of elements of Y . {\displaystyle Y.} They can thus be ordered by the lexicographical order, and for two such functions f {\displaystyle f} and g , {\displaystyle g,} the lexicographical order is thus determined by their values for the smallest x {\displaystyle x} such that f ( x ) ≠ g ( x ) . {\displaystyle f(x)\neq g(x).}
If Y {\displaystyle Y} is also well-ordered and X {\displaystyle X} is finite, then the resulting order is a well-order. As shown above, if X {\displaystyle X} is infinite this is not the case.
In combinatorics , one has often to enumerate, and therefore to order the finite subsets of a given set S . {\displaystyle S.} For this, one usually chooses an order on S . {\displaystyle S.} Then, sorting a subset of S {\displaystyle S} is equivalent to convert it into an increasing sequence. The lexicographic order on the resulting sequences induces thus an order on the subsets, which is also called the lexicographical order .
In this context, one generally prefer to sort first the subsets by cardinality , such as in the shortlex order . Therefore, in the following, we will consider only orders on subsets of fixed cardinal.
For example, using the natural order of the integers, the lexicographical ordering on the subsets of three elements of S = { 1 , 2 , 3 , 4 , 5 , 6 } {\displaystyle S=\{1,2,3,4,5,6\}} is
For ordering finite subsets of a given cardinality of the natural numbers , the colexicographical order (see below) is often more convenient, because all initial segments are finite, and thus the colexicographical order defines an order isomorphism between the natural numbers and the set of sets of n {\displaystyle n} natural numbers. This is not the case for the lexicographical order, as, with the lexicographical order, we have, for example, 12 n < 134 {\displaystyle 12n<134} for every n > 2. {\displaystyle n>2.}
Let Z n {\displaystyle \mathbb {Z} ^{n}} be the free Abelian group of rank n , {\displaystyle n,} whose elements are sequences of n {\displaystyle n} integers, and operation is the addition . A group order on Z n {\displaystyle \mathbb {Z} ^{n}} is a total order , which is compatible with addition, that is a < b if and only if a + c < b + c . {\displaystyle a<b\quad {\text{ if and only if }}\quad a+c<b+c.}
The lexicographical ordering is a group order on Z n . {\displaystyle \mathbb {Z} ^{n}.}
The lexicographical ordering may also be used to characterize all group orders on Z n . {\displaystyle \mathbb {Z} ^{n}.} [ 4 ] [ 5 ] In fact, n {\displaystyle n} linear forms with real coefficients, define a map from Z n {\displaystyle \mathbb {Z} ^{n}} into R n , {\displaystyle \mathbb {R} ^{n},} which is injective if the forms are linearly independent (it may be also injective if the forms are dependent, see below). The lexicographic order on the image of this map induces a group order on Z n . {\displaystyle \mathbb {Z} ^{n}.} Robbiano's theorem is that every group order may be obtained in this way.
More precisely, given a group order on Z n , {\displaystyle \mathbb {Z} ^{n},} there exist an integer s ≤ n {\displaystyle s\leq n} and s {\displaystyle s} linear forms with real coefficients, such that the induced map φ {\displaystyle \varphi } from Z n {\displaystyle \mathbb {Z} ^{n}} into R s {\displaystyle \mathbb {R} ^{s}} has the following properties;
The colexicographic or colex order is a variant of the lexicographical order that is obtained by reading finite sequences from the right to the left instead of reading them from the left to the right. More precisely, whereas the lexicographical order between two sequences is defined by
the colexicographical order is defined by
In general, the difference between the colexicographical order and the lexicographical order is not very significant. However, when considering increasing sequences, typically for coding subsets, the two orders differ significantly.
For example, for ordering the increasing sequences (or the sets) of two natural integers, the lexicographical order begins by
and the colexicographic order begins by
The main property of the colexicographical order for increasing sequences of a given length is that every initial segment is finite. In other words, the colexicographical order for increasing sequences of a given length induces an order isomorphism with the natural numbers, and allows enumerating these sequences. This is frequently used in combinatorics , for example in the proof of the Kruskal–Katona theorem .
When considering polynomials , the order of the terms does not matter in general, as the addition is commutative. However, some algorithms , such as polynomial long division , require the terms to be in a specific order. Many of the main algorithms for multivariate polynomials are related with Gröbner bases , concept that requires the choice of a monomial order , that is a total order , which is compatible with the monoid structure of the monomials . Here "compatible" means that a < b implies a c < b c , {\displaystyle a<b{\text{ implies }}ac<bc,} if the monoid operation is denoted multiplicatively. This compatibility implies that the product of a polynomial by a monomial does not change the order of the terms. For Gröbner bases, a further condition must be satisfied, namely that every non-constant monomial is greater than the monomial 1 . However this condition is not needed for other related algorithms, such as the algorithms for the computation of the tangent cone .
As Gröbner bases are defined for polynomials in a fixed number of variables, it is common to identify monomials (for example x 1 x 2 3 x 4 x 5 2 {\displaystyle x_{1}x_{2}^{3}x_{4}x_{5}^{2}} ) with their exponent vectors (here [1, 3, 0, 1, 2] ). If n is the number of variables, every monomial order is thus the restriction to N n {\displaystyle \mathbb {N} ^{n}} of a monomial order of Z n {\displaystyle \mathbb {Z} ^{n}} (see above § Group orders of Zn Z n , {\displaystyle \mathbb {Z} ^{n},} for a classification).
One of these admissible orders is the lexicographical order. It is, historically, the first to have been used for defining Gröbner bases, and is sometimes called pure lexicographical order for distinguishing it from other orders that are also related to a lexicographical order.
Another one consists in comparing first the total degrees , and then resolving the conflicts by using the lexicographical order. This order is not widely used, as either the lexicographical order or the degree reverse lexicographical order have generally better properties.
The degree reverse lexicographical order consists also in comparing first the total degrees, and, in case of equality of the total degrees, using the reverse of the colexicographical order. That is, given two exponent vectors, one has [ a 1 , … , a n ] < [ b 1 , … , b n ] {\displaystyle [a_{1},\ldots ,a_{n}]<[b_{1},\ldots ,b_{n}]} if either a 1 + ⋯ + a n < b 1 + ⋯ + b n , {\displaystyle a_{1}+\cdots +a_{n}<b_{1}+\cdots +b_{n},} or a 1 + ⋯ + a n = b 1 + ⋯ + b n and a i > b i for the largest i for which a i ≠ b i . {\displaystyle a_{1}+\cdots +a_{n}=b_{1}+\cdots +b_{n}\quad {\text{ and }}\quad a_{i}>b_{i}{\text{ for the largest }}i{\text{ for which }}a_{i}\neq b_{i}.}
For this ordering, the monomials of degree one have the same order as the corresponding indeterminates (this would not be the case if the reverse lexicographical order would be used). For comparing monomials in two variables of the same total degree, this order is the same as the lexicographic order. This is not the case with more variables. For example, for exponent vectors of monomials of degree two in three variables, one has for the degree reverse lexicographic order: [ 0 , 0 , 2 ] < [ 0 , 1 , 1 ] < [ 1 , 0 , 1 ] < [ 0 , 2 , 0 ] < [ 1 , 1 , 0 ] < [ 2 , 0 , 0 ] {\displaystyle [0,0,2]<[0,1,1]<[1,0,1]<[0,2,0]<[1,1,0]<[2,0,0]}
For the lexicographical order, the same exponent vectors are ordered as [ 0 , 0 , 2 ] < [ 0 , 1 , 1 ] < [ 0 , 2 , 0 ] < [ 1 , 0 , 1 ] < [ 1 , 1 , 0 ] < [ 2 , 0 , 0 ] . {\displaystyle [0,0,2]<[0,1,1]<[0,2,0]<[1,0,1]<[1,1,0]<[2,0,0].}
A useful property of the degree reverse lexicographical order is that a homogeneous polynomial is a multiple of the least indeterminate if and only if its leading monomial (its greater monomial) is a multiple of this least indeterminate. | https://en.wikipedia.org/wiki/Lexicographic_order |
Lexitropsins are members of a family of semi-synthetic DNA -binding ligands . [ 1 ] They are structural analogs of the natural antibiotics netropsin and distamycin . Antibiotics of this group can bind in the minor groove of DNA with different sequence-selectivity. [ 2 ] [ 3 ] Lexitropsins form a complexes with DNA with stoichiometry 1:1 and 2:1. Based on the 2:1 complexes were obtained ligands with high sequence-selectivity. [ 4 ] This property is due to their selectivity towards AT-rich regions. [ 5 ]
Recently, carbocyclic derivatives based on pentamidine were shown to exhibit in vivo antiproliferative effects on human breast cancer cells, possibly because of their ability to inhibit topoisomerase activity. [ 7 ] [ 5 ] [ 8 ] | https://en.wikipedia.org/wiki/Lexitropsin |
Lexus Link , launched in October 2000, is a subscription-based safety and security service from Lexus. It has been offered as a factory-installed option, available on certain Lexus models (LX, GX, LS, and GS), offering call-center-based telematics services to owners with equipped vehicles in the United States and Canada. The second-generation Lexus Link system utilizes a dedicated cellular phone (dual-mode CDMA/analog), Global Positioning Satellite (GPS) technology and 24-hour live-operator support. In 2009, an expanded system with added functionality, Lexus Enform with Safety Connect , succeeded Lexus Link.
The first generation Lexus Link system was a private-labeled brand of OnStar , operating on Verizon Wireless ’ cellular network, available as a factory-installed option on the following vehicles in Model Years 2001-04: LS 430 ('01-'04), GX 470 ('03-'04), LX 470 ('03-'04), SC 430 ('03-'04) and RX 330 ('04). The first generation system was analog-only and is no longer operational. [ 1 ]
The second generation Lexus Link system was launched October 2005 as a private-label brand of OEM Telematics Services , available as a factory-installed option on MY 2006 and later LX, GX (vehicles produced October 1, 2005 and later) and MY 2007 and later LX, GX, LS, GS vehicles and uses dual-mode (digital/analog) technology operating on Verizon Wireless’ cellular network. [ citation needed ]
The differences between the first and second generation systems are as follows:
Lexus Link is offered in the continental U.S. and Alaska. Different service packages are offered to customers. While safety and security are the main purpose of the Lexus Link system, further services include driving directions, information assistance, traffic, weather, stock quotes, or Personal Calling. Depending on service package, potential services include: [ 2 ]
Due to the growth and acceptance of digital cellular systems, many cellular carriers have abandoned analog coverage in favor of digital service. The Federal Communications Commission (FCC) ruled that cellular telephone companies operating in the United States are no longer required to provide analog service after February 2008. As a result, beginning January 1, 2008, Lexus Link service in the U.S. and Canada was only made available to vehicles equipped with dual-mode (analog/digital) equipment.
Since the first-generation Lexus Link system uses analog cellular technology and cannot be modified to digital operation, Lexus offered to disable the Lexus Link system and remove the button panel from the vehicle at no cost for owners of model year 2001–2004 vehicles. [ 3 ] | https://en.wikipedia.org/wiki/Lexus_Link |
Tetrapropylammonium perruthenate ( TPAP or TPAPR ) is the chemical compound described by the formula N(C 3 H 7 ) 4 RuO 4 . Sometimes known as the Ley –Griffith reagent, this ruthenium compound is used as a reagent in organic synthesis . This salt consists of the tetrapropylammonium cation and the perruthenate anion, RuO − 4 .
Ruthenium tetroxide is a highly aggressive oxidant, but TPAP, which is its one-electron reduced derivative, is a mild oxidizing agent for the conversion of primary alcohols to aldehydes (the Ley oxidation ). [ 2 ] Secondary alcohols are similarly oxidized to ketones . [ 3 ] It can also be used to oxidize primary alcohols all the way to the carboxylic acid with a higher catalyst loading, larger amount of the cooxidant, and addition of two equivalents of water. In this situation, the aldehyde reacts with water to form the geminal diol hydrate , which is then oxidized again. [ 4 ]
The oxidation generates water that can be removed by adding molecular sieves . TPAP is expensive, but it can be used in catalytic amounts. The catalytic cycle is maintained by adding a stoichiometric amount of a co-oxidant such as N -methylmorpholine N -oxide [ 5 ] or molecular oxygen. [ 6 ]
TPAP is also used to cleave vicinal diols to form aldehydes. [ 3 ] | https://en.wikipedia.org/wiki/Ley_Oxidation |
The Leyland L60 was a British 19-litre (1,200 cu in) vertical six-cylinder opposed-piston two-stroke multi-fuel diesel engine designed by Leyland Motors in the late 1950s/early 1960s for the Chieftain main battle tank (MBT). The engine was also used in the Vickers MBT and its Indian-built derivative, the Vijayanta .
The initial engine choice in 1954 for what was known at the time as "Medium Gun Tank No.2", later designated the "FV4201" and given the service name 'Chieftain', was a Rolls-Royce diesel V8, however during the Chieftain's design phase NATO introduced a policy in 1957 requiring all armoured fighting vehicles to have a multi-fuel capability. [ citation needed ] This left the Rolls-Royce engine an unsuitable option and so a new engine with this capability was required. [ i ] [ ii ]
Leyland Motors, under the direction of the Fighting Vehicles Research and Development Establishment (FVRDE) at Chertsey , was asked to develop an opposed-piston two-stroke diesel of similar design to those previously produced by Napier [ iii ] and Tilling-Stevens , the latter's Commer TS3 [ iv ] engine being particularly highly regarded.
This configuration, apart from being well-suited to multi-fuel use, also had the advantages of being of simple design with a low parts count, had low bearing loads , and possessed good cold-starting characteristics. Some technical assistance was provided to Leyland by Rolls-Royce, who by that time was a parent to the Napier aero-engine company, Napier itself remained a subsidiary of English Electric . [ v ] Both Tilling-Stevens and Leyland produced single-cylinder prototype engines for the tank engine project and by 1959 the resulting complete engine design had become the Leyland 60, or L60, with the first engine running that same year. [ vi ]
One of the reasons the L60's unusual configuration was chosen was so as to obtain as compact a power plant as possible so allowing the height of the vehicle to be kept as low as was practicable, a requirement for the Chieftain's design philosophy which was also seen in the recumbent driver's position.
The use of the two-stroke cycle allowed for a greater power for a given displacement , a 19-litre diesel engine being expected to be capable of around the same power as the 600 hp 27-litre petrol Meteor tank engine [ vii ] whilst taking up less room in the engine compartment. Scavenging , necessary in a large two-stroke diesel for evacuating the cylinders of exhaust gases, was performed by a Roots blower .
The Chieftain's L60 engine and cooling system were designed into an integrated engine-pack which could be changed "in the field" using the crane of an FV434 Armoured Repair Vehicle, which had been designed for this purpose and a complete engine change took around one-and-a-half, to 2 hours.
The requirement for an easily changeable engine pack was the result of a British Army analysis of previous tank battles that concluded that a likely future tank battle would last no longer than two hours and so the most demanding requirement expected for any tank engine during wartime would be for it to be run at full power for this total amount of time only and so it would then be advantageous for it to be removed from the vehicle after the battle and exchanged for a fresh engine within a minimum of time. [ viii ]
This would also allow the engines to be worked on in properly equipped REME workshops rather than 'in the field', the engines being exchanged between vehicles and workshops as-and-when required. [ ix ] This philosophy was also applied to the contemporary FV430 series of vehicles.
The initial production L60 units were, at 585 bhp at 2,100 rpm, down on the designed initial power of 600 bhp and were plagued with reliability problems. These problems were exacerbated during the Chieftain's introduction by initially an inadequate spares stock and an insufficient spare engine 'float' . [ x ] The L60 reliability problem would have been far worse had it not been for the removable engine pack, which usually allowed a vehicle to be operational again with a replacement engine within a couple of hours of breaking down. A persistent source of trouble was the failure of the cylinder liner sealing resulting in coolant leakage into the cylinder bore. Fan drive belts overstressed fan bearing housings in the crankcase leading to cracking. Reliability did improve over time with modifications and improvement programmes, such as the "Sundance" programme which also improved power output. Sundance was carried out in five main phases between 1976 and 1979. Sundance had been preceded by "Dark Morn", "High Noon", and the initial "Fleetfoot" engine development programme -the person responsible for the choosing of code names apparently being an admirer of Western film . The Sundance programme was the subject of parliamentary questions in the House of Commons in 1978. With the final rectification of most of the L60's previous reliability and power problems, vehicle availability levels rose to 80%. In the 1990 Gulf War Chieftain AARV and CHAVRE availability levels exceeded those of the Challenger 1 tank [ 2 ] which had by that point replaced Chieftain, using more-conventional four-stroke V12 diesels. [ xi ]
Final production engines produced 750 bhp (560 kW) following a series of modifications to engines in service under the various improvement programmes.
Initially, due to unfamiliarity with the two-stroke engine's different exhaust note and power band compared to a four-stroke engine, and with the resulting difficulty in choosing the correct gear required for the particular driving task, trainee drivers tended to under-rev the engines and use inappropriate gear selections, leading to great difficulty climbing gradients, and when the Chieftain Mk 1 was first introduced some drivers had difficulty climbing the vehicle onto the trailers of Thornycroft Antar tank transporters.
In 1975 all British Chieftains were brought up to Chieftain Mark 5 standard as part of the "Totem Pole" programme which included the fitting of all vehicles with the 750 bhp L60 Mark 8A. On undergoing "Totem Pole" upgrades Chieftain Mk 2 vehicles were re-designated the Mark 6. Mk 3 vehicles became the Mk 7, and Mk 3/3 vehicles became the Mk 8.
The engine was mated with a Merritt-Brown TN12 [ xiv ] triple-differential epicyclic gearbox providing "regenerative" steering, a derivative of the system first used on the Churchill tank . The gearbox was semi-automatic foot-operated and had six forward, and two reverse gears. Like the engine, it was designed to be quickly replaceable. The TN12 had originally been developed for the cancelled FV300 light tank series . A scaled down version of the TN12, the TN15, was used in the CVR(T) series of vehicles. [ xv ] | https://en.wikipedia.org/wiki/Leyland_L60 |
Lhasa ( Japanese pronunciation: [ɾasa] ) is a Japanese computer program used for unpacking or decompressing files in various compressed formats, including LHA (LZH) and ZIP . Lhasa is an open-source utility specifically designed for handling LHA compression, which was a common format for archiving files in early computer systems.
The development of Lhasa likely began in the early days of computing when the LHA compression format gained popularity. The LHA format, also known as LZH, was widely used in the 1980s and 1990s for archiving files on systems like the Amiga and early versions of Microsoft Windows .
This software article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Lhasa_(computing) |
Lithium borate , [ 1 ] also known as lithium tetraborate , [ 2 ] dilithium tetraborate [ 3 ] or boron lithium oxide [ 2 ] is an inorganic compound with the formula Li 2 B 4 O 7 . A colorless solid, lithium borate is used in making glasses and ceramics . It is not to be confused with B 8 Li 2 O 13 , also called lithium borate. [ 4 ]
Its structure consists of a polymeric borate backbone. The Li + centers are bound to four and five oxygen ligands. Boron centers are trigonal and tetrahedral. [ 5 ] [ 6 ]
Lithium borate can be used in the laboratory as LB buffer for gel electrophoresis of DNA and RNA . It is also used in the borax fusion method to vitrify mineral powder specimens for analysis by WDXRF spectroscopy . [ 7 ]
This inorganic compound –related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Li2B4O7 |
Dilithium acetylide is an organometallic compound with the formula Li 2 C 2 . It is typically derived by double deprotonation of acetylene. X-ray crystallography confirms the presence of C≡C subunits attached to lithium, resulting in a polymeric structure. [ 3 ] Li 2 C 2 is one of an extensive range of lithium-carbon compounds, which include the lithium-rich Li 4 C , Li 6 C 2 , Li 8 C 3 , Li 6 C 3 , Li 4 C 3 , Li 4 C 5 , and the graphite intercalation compounds LiC 6 , LiC 12 , and LiC 18 . It is an intermediate compound produced during radiocarbon dating procedures.
Li 2 C 2 is the most thermodynamically-stable lithium-rich carbide [ 3 ] and the only one that can be obtained directly from the elements. It was first produced by Moissan , in 1896 [ 4 ] who reacted coal with lithium carbonate .
The other lithium-rich compounds are produced by reacting lithium vapor with chlorinated hydrocarbons , e.g. CCl 4 . Lithium carbide is sometimes confused with the drug lithium carbonate , Li 2 CO 3 , because of the similarity of its name.
In the laboratory samples may be prepared by treating acetylene with butyl lithium : [ 5 ]
Instead of butyl lithium, a solution of lithium in ammonia can be used to prepare Li 2 C 2 . In this case, a transient adduct Li 2 C 2 · C 2 H 2 ·2 NH 3 if formed. It decomposes with release of ammonia at room temperature.
Samples prepared from acetylene generally are poorly crystalline . Crystalline samples may be prepared by a reaction between molten lithium and graphite at over 1000 °C. [ 3 ] Li 2 C 2 can also be prepared by reacting CO 2 with molten lithium. [ citation needed ]
Other method for production of Li 2 C 2 is heating of metallic lithium in atmosphere of ethylene . Lithium hydride is a coproduction:
Lithium carbide hydrolyzes readily to form acetylene as well as Lithium hydroxide :
Lithium hydride reacts with graphite at 400°C forming lithium carbide.
Lithium carbide reacts with acetylene in liquid ammonia rapidly to give a lithium hydrogen acetylide.
Preparation of the reagent in this way sometimes improves the yield in an ethynylation over that obtained with reagent prepared from lithium and acetylene. [ citation needed ]
Li 2 C 2 could be viewed as a Zintl phase . It is not a salt. It adopts a distorted anti -fluorite crystal structure , similar to that of rubidium peroxide ( Rb 2 O 2 ) and caesium peroxide ( Cs 2 O 2 ). Each lithium atom is surrounded by six carbon atoms from 4 different acetylide anions , with two acetylides co-ordinating side -on and the other two end-on. [ 3 ] [ 6 ] The relatively short C-C distance of 120 pm indicates the presence of a C≡C triple bond . At high temperatures Li 2 C 2 transforms reversibly to a cubic anti-fluorite structure. [ 7 ]
There are a number of procedures employed, some that burn the sample producing CO 2 that is then reacted with lithium, and others where the carbon containing sample is reacted directly with lithium metal. [ 8 ] The outcome is the same: Li 2 C 2 is produced, which can then be used to create species easy to use in mass spectroscopy , like acetylene and benzene . [ 9 ] Note that lithium nitride may be formed and this produces ammonia when hydrolyzed , which contaminates the acetylene gas. | https://en.wikipedia.org/wiki/Li2C2 |
Lithium carbonate is an inorganic compound , the lithium salt of carbonic acid with the formula Li 2 CO 3 . This white salt is widely used in processing metal oxides. It is on the World Health Organization's List of Essential Medicines [ 7 ] for its efficacy in the treatment of mood disorders such as bipolar disorder . [ 8 ] [ 7 ]
Lithium carbonate is an important industrial chemical . Its main use is as a precursor to compounds used in lithium-ion batteries.
Glasses derived from lithium carbonate are useful in ovenware. Lithium carbonate is a common ingredient in both low-fire and high-fire ceramic glaze . It forms low-melting fluxes with silica and other materials. Its alkaline properties are conducive to changing the state of metal oxide colorants in glaze , particularly red iron oxide ( Fe 2 O 3 ). Cement sets more rapidly when prepared with lithium carbonate, and is useful for tile adhesives . When added to aluminium trifluoride , it forms LiF which yields a superior electrolyte for the processing of aluminium . [ 9 ]
Lithium carbonate-derived compounds are crucial to lithium-ion batteries . Lithium carbonate may be converted into lithium hydroxide as an intermediate. In practice, two components of the battery are made with lithium compounds: the cathode and the electrolyte . The electrolyte is a solution of lithium hexafluorophosphate , while the cathode uses one of several lithiated structures, the most popular of which are lithium cobalt oxide and lithium iron phosphate .
In 1843, lithium carbonate was used to treat stones in the bladder . In 1859, some doctors recommended a therapy with lithium salts for a number of ailments , including gout , urinary calculi , rheumatism , mania , depression , and headache .
In 1948, John Cade discovered the anti-manic effects of lithium ions. [ 10 ] This finding led to lithium carbonate's use as a psychiatric medication to treat mania, the elevated phase of bipolar disorder . Prescription lithium carbonate from a pharmacy is suitable for use as medicine in humans but industrial lithium carbonate is not since it may contain unsafe levels of toxic heavy metals or other toxicants . After ingestion, lithium carbonate is dissociated into pharmacologically active lithium ions (Li + ) and (non-therapeutic) carbonate , with 300 mg of lithium carbonate containing approximately 8 mEq (8 mmol ) of lithium ion. [ 8 ] According to the Food and Drug Administration (FDA), 300–600 mg of lithium carbonate taken two to three times daily is typical for maintenance of bipolar I disorder in adults, [ 8 ] where the exact dose given varies depending on factors such as the patient's serum lithium concentrations, which must be closely monitored by a physician to avoid lithium toxicity and potential kidney damage (or even kidney failure ) from lithium-induced nephrogenic diabetes insipidus . [ 11 ] [ 8 ] Dehydration and certain drugs, including NSAIDs such as ibuprofen , can increase serum lithium concentrations to unsafe levels whereas other drugs, such as caffeine , may decrease concentrations. In contrast to the elemental ions sodium , potassium , and calcium , there is no known cellular mechanism specifically dedicated to regulating intracellular lithium.
Lithium can enter cells through epithelial sodium channels . [ 12 ] Lithium ions interfere with ion transport processes (see " Sodium pump ") that relay and amplify messages carried to the cells of the brain. [ 13 ] Mania is associated with irregular increases in protein kinase C (PKC) activity within the brain. Lithium carbonate and sodium valproate , another drug traditionally used to treat the disorder, act in the brain by inhibiting PKC's activity and help to produce other compounds that also inhibit the PKC. [ 14 ] Lithium carbonate's mood-controlling properties are not fully understood. [ 15 ]
Taking lithium salts has risks and side effects. Extended use of lithium to treat mental disorders has been known to lead to acquired nephrogenic diabetes insipidus . [ 16 ] Lithium intoxication can affect the central nervous system and renal system and can be lethal. [ 17 ] Over a prolonged period, lithium can accumulate in the principal cells of the collecting duct and interfere with antidiuretic hormone (ADH), which regulates the water permeability of principal cells in the collecting tubule. [ 12 ] The medullary interstitium of the collecting duct system naturally has a high sodium concentration and attempts to maintain it. There is no known mechanism for cells to distinguish lithium ions from sodium ions, so damage to the kidney 's nephrons may occur if lithium concentrations become too high as a result of dehydration , hyponatremia , an unusually low sodium diet , or certain drugs.
Lithium carbonate is used to impart a red color to fireworks . [ 18 ]
Unlike sodium carbonate , which forms at least three hydrates , lithium carbonate exists only in the anhydrous form. Its solubility in water is low relative to other lithium salts. The isolation of lithium from aqueous extracts of lithium ores capitalizes on this poor solubility. Its apparent solubility increases 10-fold under a mild pressure of carbon dioxide ; this effect is due to the formation of the metastable lithium bicarbonate , which is more soluble: [ 9 ] [ 19 ]
The extraction of lithium carbonate at high pressures of CO 2 and its precipitation upon depressurizing is the basis of the Quebec process.
Lithium carbonate can also be purified by exploiting its diminished solubility in hot water. Thus, heating a saturated aqueous solution causes crystallization of Li 2 CO 3 . [ 20 ]
Lithium carbonate, and other carbonates of group 1 , do not decarboxylate readily. Li 2 CO 3 decomposes at temperatures around 1300 °C.
Lithium is extracted from primarily two sources: spodumene in pegmatite deposits, and lithium salts in underground brine pools . About 82,000 tons were produced in 2020, showing significant and consistent growth. [ 21 ]
In the Salar de Atacama in the Atacama Desert of Northern Chile, lithium carbonate and hydroxide are produced from brine. [ 22 ] [ 23 ]
The process pumps lithium rich brine from below ground into shallow pans for evaporation. The brine contains many different dissolved ions, and as their concentration increases, salts precipitate out of solution and sink. The remaining supernatant liquid is used for the next step. The sequence of pans may vary depending on the concentration of ions in a particular source of brine.
In the first pan, halite (sodium chloride or common salt) crystallises. This has little economic value and is discarded. The supernatant, with ever increasing concentration of dissolved solids, is transferred successively to the sylvinite (sodium potassium chloride) pan, the carnalite (potassium magnesium chloride) pan and finally a pan designed to maximise the concentration of lithium chloride. The process takes about 15 months. The concentrate (30-35% lithium chloride solution) is trucked to Salar del Carmen. There, boron and magnesium are removed (typically residual boron is removed by solvent extraction and/or ion exchange and magnesium by raising the pH above 10 with sodium hydroxide ) [ 24 ] then in the final step, by addition of sodium carbonate , the desired lithium carbonate is precipitated out, separated, and processed.
Some of the by-products from the evaporation process may also have economic value.
There is considerable attention to the use of water in this water poor region. SQM commissioned a life-cycle analysis (LCA) which concluded that water consumption for SQM's lithium hydroxide and carbonate is significantly lower than the average consumption by production from the main ore-based process, using spodumene . A more general LCA suggests the opposite for extraction from reservoirs. [ 25 ]
The majority of brine based production is in the " lithium triangle " in South America.
A potential source of lithium is the leachates of geothermal wells , carried to the surface. [ 26 ] Recovery of lithium has been demonstrated in the field; the lithium is separated by simple precipitation and filtration. [ 27 ] The process and environmental costs are primarily those of the already-operating well; net environmental impacts may thus be positive. [ 28 ]
The brine of United Downs Deep Geothermal Power project near Redruth is claimed by Cornish Lithium to be valuable due to its high lithium concentration (220 mg/L) with low magnesium (<5 mg/L) and total dissolved solids content of <29g/L, [ 29 ] and a flow rate of 40-60l/s. [ 25 ]
α-spodumene is roasted at 1100 °C for 1h to make β-spodumene, then roasted at 250 °C for 10 minutes with sulphuric acid. [ 30 ] [ 22 ]
As of 2020, Australia was the world's largest producer of lithium intermediates, [ 31 ] all based on spodumene.
In recent years mining companies have begun exploration of lithium projects throughout North America , South America and Australia to identify economic deposits that can potentially bring new supplies of lithium carbonate online to meet the growing demand for the product. [ 32 ]
In 2020 Tesla Motors announced a revolutionary process to extract lithium from clay in Nevada using only salt and no acid. This was met with scepticism. [ 33 ]
A few small companies are recycling spent batteries , focusing on recovering copper and cobalt. Some recover lithium carbonate alongside the compound Li 2 Al 4 (CO 3 )(OH) 12 ⋅3H 2 O
also. [ 34 ] [ 35 ] [ 36 ] [ 37 ]
In April 2017 MGX Minerals reported it had received independent confirmation of its rapid lithium extraction process to recover lithium and other valuable minerals from oil and gas wastewater brine . [ 38 ]
Electrodialysis has been proposed to extract lithium from seawater, but it is not commercially viable. [ 39 ]
Natural lithium carbonate is known as zabuyelite . [ 40 ] This mineral is connected with deposits of some salt lakes and some pegmatites . [ 41 ] | https://en.wikipedia.org/wiki/Li2CO3 |
Lithium hexafluorogermanate is the inorganic compound with the formula Li 2 GeF 6 . It forms a solid off-white deliquescent powder. When exposed to moisture, it easily hydrolyses to release hydrogen fluoride and germanium tetrafluoride gases. [ 1 ]
Lithium hexafluorogermanate can be dissolved in a solution of hydrogen fluoride, which forms a precipitate of lithium fluoride . [ 2 ]
It can be used as a densification aid in the sintering of gadolinium oxysulfide , [ 3 ] [ 4 ] and as a lithium salt additive in a lithium-ion battery electrolyte. [ 5 ] | https://en.wikipedia.org/wiki/Li2GeF6 |
Lithium iridate , Li 2 IrO 3 , is a chemical compound of lithium , iridium and oxygen. It forms black crystals with three slightly different layered atomic structures, α, β, and sometimes γ. Lithium iridate exhibits metal-like, temperature-independent electrical conductivity , and changes its magnetic ordering from paramagnetic to antiferromagnetic upon cooling to 15 K.
Li 2 IrO 3 typically crystallizes in the α or β phase, and a rare γ phase has been reported. The crystal structure of α-Li 2 IrO 3 consists of an alternate stacking of hexagonal Li layers and honeycombs of edge-sharing IrO 6 octahedra with Li in the center. The offset in adjacent layers results in a relatively low (monoclinic) crystal symmetry. Li 2 IrO 3 crystals have abundant twinning defects where the ab crystal planes are rotated by 120° around the c axis. [ 1 ]
Li 2 IrO 3 crystals can be grown by direct sintering of Ir and Li metals, which both oxidize during heating in ambient atmosphere. The α phase is formed at 750–1050 °C, while heating to higher temperatures results in the β phase. The use of Li metal instead of more traditional lithium carbonate , which is easier to handle and store, results in larger crystals. The γ phase can be obtained by the calcination of lithium carbonate and iridium(IV) oxide , followed by annealing in molten lithium hydroxide at 700–800 °C. [ 1 ]
Lithium iridate is black in color and has a relatively high, temperature-independent electrical conductivity characteristic of metals. [ 2 ] Its both α and β phases exhibit the Kitaev exchange coupling between magnetic spins originating from Ir 4+ ions. These spins form an antiferromagnetic lattice at temperatures below 15 K ( Néel temperature , T N ), while the material is paramagnetic above T N . [ 1 ]
Lithium iridate is a potential electrode material for the lithium-ion battery . [ 2 ] This application is hindered by the high costs of Ir, as compared to the cheaper Li 2 MnO 3 alternative. [ 3 ] | https://en.wikipedia.org/wiki/Li2IrO3 |
Lithium molybdate ( Li 2 Mo O 4 ) is a chemical compound . It is mainly used as an inhibitor in some types of industrial air conditioning.
Lithium molybdate is used as corrosion inhibitor in LiBr ( Lithium bromide ) absorption chiller for industrial central air conditioning . It is manufactured and shipped as either a colorless, transparent fluid or a white crystal powder. In either state it not classified as a hazardous material.
Li 2 MoO 4 crystals have been found applicable for cryogenic phonon-scintillation detectors, which are used to investigate some rare nuclear processes. [ 2 ] The use of Li 2 MoO 4 ceramics for antennas has been studied due to their low loss dielectric properties and the possibility to fabricate them by a room-temperature densification method instead of conventional sintering . [ 3 ] Li 2 MoO 4 (LMO) have also been used with hollow glass microspheres (HGMS) to make low permittivity composite which has been used to make lenses for lens antennas. [ 4 ]
This inorganic compound –related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Li2MoO4 |
Lithium imide is an inorganic compound with the chemical formula Li 2 N H . This white solid can be formed by a reaction between lithium amide and lithium hydride . [ 1 ]
The product is light-sensitive and can undergo disproportionation to lithium amide and characteristically red lithium nitride .
Lithium imide is thought to have a simple face-centered cubic structure with a Fm 3 m space group ; with N-H bond distances of 0.82(6) Å and a H–N–H bond angle of 109.5°, giving it a similar structure to lithium amide. [ 2 ] [ 3 ]
Lithium imide is strongly basic and deprotonates even some extremely weak acids such as methane and ammonia , due to the very localized negative charge on the nitrogen , which carries two formal charges . It has uses in organic and organometallic chemistry . It has been investigated as a material for hydrogen storage . [ 1 ]
This inorganic compound –related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Li2NH |
Lithium oxide ( Li 2 O ) or lithia is an inorganic chemical compound . It is a white solid. Although not specifically important, many materials are assessed on the basis of their Li 2 O content. For example, the Li 2 O content of the principal lithium mineral spodumene (LiAlSi 2 O 6 ) is 8.03%. [ 2 ]
Lithium oxide forms along with small amounts of lithium peroxide when lithium metal is burned in the air and combines with oxygen at temperatures above 100 °C: [ 3 ]
Pure Li 2 O can be produced by the thermal decomposition of lithium peroxide , Li 2 O 2 , at 450 °C [ 3 ] [ 2 ]
Solid lithium oxide adopts an antifluorite structure with four-coordinated Li+ centers and eight-coordinated oxides. [ 4 ]
The ground state gas phase Li 2 O molecule is linear with a bond length consistent with strong ionic bonding. [ 5 ] [ 6 ] VSEPR theory would predict a bent shape similar to H 2 O .
Lithium oxide is used as a flux in ceramic glazes; and creates blues with copper and pinks with cobalt . Lithium oxide reacts with water and steam , forming lithium hydroxide and should be isolated from them.
Its usage is also being investigated for non-destructive emission spectroscopy evaluation and degradation monitoring within thermal barrier coating systems. It can be added as a co-dopant with yttria in the zirconia ceramic top coat, without a large decrease in expected service life of the coating. At high heat, lithium oxide emits a very detectable spectral pattern, which increases in intensity along with degradation of the coating. Implementation would allow in situ monitoring of such systems, enabling an efficient means to predict lifetime until failure or necessary maintenance.
Lithium metal might be obtained from lithium oxide by electrolysis, releasing oxygen as by-product.
Lithium oxide absorbs carbon dioxide forming lithium carbonate :
The oxide reacts slowly with water, forming lithium hydroxide : | https://en.wikipedia.org/wiki/Li2O |
Lithium peroxide is the inorganic compound with the formula Li 2 O 2 . Lithium peroxide is a white solid, and unlike most other alkali metal peroxides, it is nonhygroscopic . Because of its high oxygen:mass and oxygen:volume ratios, the solid has been used to remove CO 2 from and release O 2 to the atmosphere in spacecraft . [ 4 ]
It is prepared by the reaction of hydrogen peroxide and lithium hydroxide . This reaction initially produces lithium hydroperoxide : [ 4 ] [ 5 ]
This lithium hydroperoxide may exist as lithium peroxide monoperoxohydrate trihydrate (Li 2 O 2 ·H 2 O 2 ·3H 2 O).
Dehydration of this material gives the anhydrous peroxide salt:
Li 2 O 2 decomposes at about 450 °C to give lithium oxide :
The structure of solid Li 2 O 2 has been determined by X-ray crystallography and density functional theory . The solid features eclipsed "ethane-like" Li 6 O 2 subunits with an O-O distance of around 1.5 Å. [ 6 ]
It is used in air purifiers where weight is important, e.g., spacecraft or other sealed spaces and apparatuses to absorb carbon dioxide and release oxygen in the reaction: [ 4 ]
Li 2 O 2 + CO 2 → Li 2 CO 3 + 1 ⁄ 2 O 2
Similar to the reaction of lithium hydroxide with carbon dioxide to release 1 Li 2 CO 3 and 1 H 2 O, lithium peroxide has high absorption capacity and absorbs more CO 2 than does the same weight of lithium hydroxide and offers the bonus of releasing oxygen instead of water. [ 7 ]
Lithium peroxide can also act as a catalyst for polymerization of styrene to polystyrene. The polymerization of styrene to polystyrene typically involves the use of radical initiators via the free radical chain mechanism but lithium peroxide can also initiate radical polymerization reactions under certain conditions, although not as widely used.
The reversible lithium peroxide reaction is the basis for a prototype lithium–air battery . Using oxygen from the atmosphere allows the battery to eliminate storage of oxygen for its reaction, saving battery weight and size. [ 8 ] | https://en.wikipedia.org/wiki/Li2O2 |
Potassium sulfate Rubidium sulfate Caesium sulfate
Lithium sulfate is a white inorganic salt with the formula Li 2 S O 4 . It is the lithium salt of sulfuric acid .
Lithium sulfate is soluble in water, though it does not follow the usual trend of increasing solubility of most salts with temperature. To the contrary, its solubility in water decreases with increasing temperature, as its dissolution is an exothermic process. This relatively unusual property, also called retrograde solubility , is shared with few inorganic compounds , such as calcium hydroxide ( portlandite , an important mineral phase of hydrated cement paste), the calcium sulfates ( gypsum , bassanite , and anhydrite ) and lanthanoid sulfates whose dissolution reactions are also exothermic. The retrograde solubility is common for gases dissolution in water, but less frequently encountered for the dissolution of solids. Calcium carbonate also exhibits a retrograde solubility, but it also depends on the behavior of CO 2 dissolution in the calco-carbonate equilibria.
Lithium sulfate crystals, being piezoelectric , are also used in ultrasound-type non-destructive testing because they are very efficient sound receivers. However, they do suffer in this application because of their water solubility.
Since it has hygroscopic properties , the most common form of lithium sulfate is lithium sulfate monohydrate. Anhydrous lithium sulfate has a density of 2.22 g/cm 3 but, weighing lithium sulfate anhydrous can become cumbersome as it must be done in a water lacking atmosphere.
Lithium sulfate has pyroelectric properties . When aqueous lithium sulfate is heated, the electrical conductivity also increases. The molarity of lithium sulfate also plays a role in the electrical conductivity; optimal conductivity is achieved at 2 M and then decreases. [ 4 ]
When solid lithium sulfate is dissolved in water it has an endothermic disassociation . This is different from sodium sulfate which has an exothermic disassociation. However, the exact energy of disassociation is difficult to quantify as it seems also to depend on the quantity (number of mols) of the salt added to water. Small amounts of dissolved lithium sulfate induce a much greater temperature change per mol than large amounts. [ 5 ]
Lithium sulfate has two different crystal phases . In common phase II form, Lithium sulfate has a sphenoidal monoclinic crystal system that has edge lengths of a = 8.23Å b = 4.95Å c = 8.47Å β = 107.98°. When lithium sulfate is heated passed 130 °C it changes to a water free state but retains its crystal structure. It is not until 575 °C when there is a transformation from phase II to phase I. The crystal structure changes to a face centered cubic crystal system , with an edge length of 7.07Å. [ 6 ] During this phase change, the density of lithium sulfate changes from 2.22 to 2.07 g/cm 3 . [ 7 ]
Lithium sulfate is used to treat bipolar disorder (see lithium pharmacology ).
Lithium sulfate is researched as a potential component of ion conducting glasses. Transparent conducting film is a highly investigated topic as they are used in applications such as solar panels and the potential for a new class of battery. In these applications, it is important to have a high lithium content; the more commonly known binary lithium borate (Li₂O · B₂O₃) is difficult to obtain with high lithium concentrations and difficult to keep as it is hygroscopic. With the addition of lithium sulfate into the system, an easily produced, stable, high lithium concentration glass is able to be formed. Most of the current transparent ionic conducting films are made of organic plastics, and it would be ideal if an inexpensive stable inorganic glass could be developed. [ 8 ]
Lithium sulfate has been tested as an additive for Portland cement to accelerate curing with positive results. Lithium sulfate serves to speed up the hydration reaction (see Cement ) which decreases the curing time. A concern with decreased curing time is the strength of the final product, but when tested, lithium sulfate doped Portland cement had no observable decrease in strength. [ 9 ]
Lithium sulphate monohydrate ( Li 2 SO 4 · H 2 O ) containing around 10% lithium is a useful chemical for the production of lithium hydroxide for the lithium-ion battery materials supply chain. It is a less reactive material than LiOH, and hence can be more easily stored and transported. [ 10 ] [ 11 ]
Feedstock of hard-rock spodumene concentrate is processed by acid roasting, followed by water leaching, achieving a lithium recovery of 84-88%. Evaporation is then applied to the purified leach solution resulting in a primary lithium sulphate solid product made up mostly of lithium sulphate monohydrate ( Li 2 SO 4 · H 2 O ).
Lithium ion (Li + ) is used in psychiatry for the treatment of mania , endogenous depression, and psychosis, and also for treatment of schizophrenia . Usually lithium carbonate ( Li 2 CO 3 ) is applied, but sometimes lithium citrate ( Li 3 C 6 H 5 O 7 ), lithium sulfate or lithium oxy- butyrate are used as alternatives. [ 12 ] Li + is not metabolized. Because of Li + chemical similarity to sodium (Na + ) and potassium (K + ) cations, it may interact or interfere with the biochemical pathways of these substances and displace these cations from intra- or extracellular compartments of the body. Li + seems to be transported out of nerve and muscle cells by the active sodium pump , although less efficiently.
Lithium sulfate has a rapid gastrointestinal absorption rate (within a few minutes), and complete following oral administration of tablets or the liquid form. [ 12 ] It quickly diffuses into the liver and kidneys but requires 8–10 days to reach a body equilibrium. Li + produces many metabolic and neuroendocrine changes, but no conclusive evidence favors one particular mode of action. [ 12 ] For example, Li + interacts with neurohormones , particularly the biogenic amines , serotonin (5-hydroxy tryptamine ) and norepinephrine , which provides a probable mechanism for the beneficial effects in psychiatric disorders , e.g. manias . In the central nervous system (CNS), Li + affects nerve excitation, synaptic transmission , and neuronal metabolism . [ 13 ] Li + stabilizes serotoninergic neurotransmission .
Lithium sulfate is being used as a catalyst for the elimination reaction for transforming n-butyl bromide to 1-butene at close to 100% yields at a range of 320 °C to 370 °C. The yields of this reaction change dramatically if heated beyond this range as higher yields of 2-butene is formed. [ 14 ] | https://en.wikipedia.org/wiki/Li2O4S |
Lithium polonide is a chemical compound with the formula Li 2 Po . It is a polonide , a set of very chemically stable compounds of polonium. [ 2 ] [ 3 ]
Lithium polonide may be produced from a redox reaction between aqueous polonium hydride and lithium metal [ 2 ] [ 3 ] or from an acid-base reaction of H 2 Po with strong lithium-containing bases:
It may also be produced by heating lithium and polonium together at 300–400 °C. [ 1 ]
Like sodium polonide , lithium polonide has the antifluorite structure. [ 2 ] [ 3 ]
This inorganic compound –related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Li2Po |
Lithium platinate , Li 2 PtO 3 , is a chemical compound of lithium , platinum and oxygen. It is a semiconductor with a layered honeycomb crystal structure and a band gap of 2.3 eV, and can be prepared by direct calcination of Pt metal and lithium carbonate at ca. 600 °C. [ 3 ] Lithium platinate is a potential lithium-ion battery electrode material, [ 2 ] [ 4 ] though this application is hindered by the high costs of Pt, as compared to the cheaper Li 2 MnO 3 alternative. [ 5 ] | https://en.wikipedia.org/wiki/Li2PtO3 |
Lithium ruthenate , Li 2 RuO 3 , is a chemical compound of lithium , ruthenium and oxygen. It has a layered honeycomb crystal structure, and can be prepared by direct calcination of Ru metal and lithium carbonate at ca. 700 °C. [ 2 ] It is a potential lithium-ion battery electrode material, [ 2 ] though this application is hindered by the high costs of Ru, as compared to the cheaper Li 2 MnO 3 alternative. [ 3 ] | https://en.wikipedia.org/wiki/Li2RuO3 |
Lithium sulfide is the inorganic compound with the formula Li 2 S . It crystallizes in the antifluorite motif, described as the salt (Li + ) 2 S 2− . It forms a solid yellow-white deliquescent powder. In air, it easily hydrolyses to release foul smelling hydrogen sulfide gas. [ 2 ]
Lithium sulfide is prepared by treating lithium with sulfur. This reaction is conveniently conducted in anhydrous ammonia . [ 3 ]
The THF-soluble triethylborane adduct of lithium sulfide can be generated using superhydride . [ 4 ]
Lithium sulfide has been considered for use in lithium–sulfur batteries . [ 5 ] | https://en.wikipedia.org/wiki/Li2S |
Lithium sulfite , or lithium sulphite , is an ionic compound with the formula Li 2 SO 3 . [ 1 ]
This inorganic compound –related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Li2SO3 |
Potassium sulfate Rubidium sulfate Caesium sulfate
Lithium sulfate is a white inorganic salt with the formula Li 2 S O 4 . It is the lithium salt of sulfuric acid .
Lithium sulfate is soluble in water, though it does not follow the usual trend of increasing solubility of most salts with temperature. To the contrary, its solubility in water decreases with increasing temperature, as its dissolution is an exothermic process. This relatively unusual property, also called retrograde solubility , is shared with few inorganic compounds , such as calcium hydroxide ( portlandite , an important mineral phase of hydrated cement paste), the calcium sulfates ( gypsum , bassanite , and anhydrite ) and lanthanoid sulfates whose dissolution reactions are also exothermic. The retrograde solubility is common for gases dissolution in water, but less frequently encountered for the dissolution of solids. Calcium carbonate also exhibits a retrograde solubility, but it also depends on the behavior of CO 2 dissolution in the calco-carbonate equilibria.
Lithium sulfate crystals, being piezoelectric , are also used in ultrasound-type non-destructive testing because they are very efficient sound receivers. However, they do suffer in this application because of their water solubility.
Since it has hygroscopic properties , the most common form of lithium sulfate is lithium sulfate monohydrate. Anhydrous lithium sulfate has a density of 2.22 g/cm 3 but, weighing lithium sulfate anhydrous can become cumbersome as it must be done in a water lacking atmosphere.
Lithium sulfate has pyroelectric properties . When aqueous lithium sulfate is heated, the electrical conductivity also increases. The molarity of lithium sulfate also plays a role in the electrical conductivity; optimal conductivity is achieved at 2 M and then decreases. [ 4 ]
When solid lithium sulfate is dissolved in water it has an endothermic disassociation . This is different from sodium sulfate which has an exothermic disassociation. However, the exact energy of disassociation is difficult to quantify as it seems also to depend on the quantity (number of mols) of the salt added to water. Small amounts of dissolved lithium sulfate induce a much greater temperature change per mol than large amounts. [ 5 ]
Lithium sulfate has two different crystal phases . In common phase II form, Lithium sulfate has a sphenoidal monoclinic crystal system that has edge lengths of a = 8.23Å b = 4.95Å c = 8.47Å β = 107.98°. When lithium sulfate is heated passed 130 °C it changes to a water free state but retains its crystal structure. It is not until 575 °C when there is a transformation from phase II to phase I. The crystal structure changes to a face centered cubic crystal system , with an edge length of 7.07Å. [ 6 ] During this phase change, the density of lithium sulfate changes from 2.22 to 2.07 g/cm 3 . [ 7 ]
Lithium sulfate is used to treat bipolar disorder (see lithium pharmacology ).
Lithium sulfate is researched as a potential component of ion conducting glasses. Transparent conducting film is a highly investigated topic as they are used in applications such as solar panels and the potential for a new class of battery. In these applications, it is important to have a high lithium content; the more commonly known binary lithium borate (Li₂O · B₂O₃) is difficult to obtain with high lithium concentrations and difficult to keep as it is hygroscopic. With the addition of lithium sulfate into the system, an easily produced, stable, high lithium concentration glass is able to be formed. Most of the current transparent ionic conducting films are made of organic plastics, and it would be ideal if an inexpensive stable inorganic glass could be developed. [ 8 ]
Lithium sulfate has been tested as an additive for Portland cement to accelerate curing with positive results. Lithium sulfate serves to speed up the hydration reaction (see Cement ) which decreases the curing time. A concern with decreased curing time is the strength of the final product, but when tested, lithium sulfate doped Portland cement had no observable decrease in strength. [ 9 ]
Lithium sulphate monohydrate ( Li 2 SO 4 · H 2 O ) containing around 10% lithium is a useful chemical for the production of lithium hydroxide for the lithium-ion battery materials supply chain. It is a less reactive material than LiOH, and hence can be more easily stored and transported. [ 10 ] [ 11 ]
Feedstock of hard-rock spodumene concentrate is processed by acid roasting, followed by water leaching, achieving a lithium recovery of 84-88%. Evaporation is then applied to the purified leach solution resulting in a primary lithium sulphate solid product made up mostly of lithium sulphate monohydrate ( Li 2 SO 4 · H 2 O ).
Lithium ion (Li + ) is used in psychiatry for the treatment of mania , endogenous depression, and psychosis, and also for treatment of schizophrenia . Usually lithium carbonate ( Li 2 CO 3 ) is applied, but sometimes lithium citrate ( Li 3 C 6 H 5 O 7 ), lithium sulfate or lithium oxy- butyrate are used as alternatives. [ 12 ] Li + is not metabolized. Because of Li + chemical similarity to sodium (Na + ) and potassium (K + ) cations, it may interact or interfere with the biochemical pathways of these substances and displace these cations from intra- or extracellular compartments of the body. Li + seems to be transported out of nerve and muscle cells by the active sodium pump , although less efficiently.
Lithium sulfate has a rapid gastrointestinal absorption rate (within a few minutes), and complete following oral administration of tablets or the liquid form. [ 12 ] It quickly diffuses into the liver and kidneys but requires 8–10 days to reach a body equilibrium. Li + produces many metabolic and neuroendocrine changes, but no conclusive evidence favors one particular mode of action. [ 12 ] For example, Li + interacts with neurohormones , particularly the biogenic amines , serotonin (5-hydroxy tryptamine ) and norepinephrine , which provides a probable mechanism for the beneficial effects in psychiatric disorders , e.g. manias . In the central nervous system (CNS), Li + affects nerve excitation, synaptic transmission , and neuronal metabolism . [ 13 ] Li + stabilizes serotoninergic neurotransmission .
Lithium sulfate is being used as a catalyst for the elimination reaction for transforming n-butyl bromide to 1-butene at close to 100% yields at a range of 320 °C to 370 °C. The yields of this reaction change dramatically if heated beyond this range as higher yields of 2-butene is formed. [ 14 ] | https://en.wikipedia.org/wiki/Li2SO4 |
Lithium metasilicate is an ionic compound with the formula Li 2 SiO 3
Lithium metasilicate is prepared by the reaction of lithium carbonate and silicon dioxide at temperatures between 515 and 565 °C. [ 1 ]
The melting of lithium metasilicate is used for the calibration of thermocouples . [ 2 ] | https://en.wikipedia.org/wiki/Li2SiO3 |
Lithium telluride (Li 2 Te) is an inorganic compound of lithium and tellurium . Along with LiTe 3 , it is one of the two intermediate solid phases in the lithium-tellurium system. [ 3 ] It can be prepared by directly reacting lithium and tellurium in a beryllium oxide crucible at 950°C. [ 4 ]
This inorganic compound –related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Li2Te |
Lithium nitride is an inorganic compound with the chemical formula Li 3 N . It is the only stable alkali metal nitride . It is a reddish-pink solid with a high melting point. [ 1 ]
Lithium nitride is prepared by direct reaction of elemental lithium with nitrogen gas: [ 2 ]
Instead of burning lithium metal in an atmosphere of nitrogen, a solution of lithium in liquid sodium metal can be treated with N 2 .
Lithium nitride is an extremely strong base, so it must be protected from moisture as it reacts violently with water to produce ammonia :
Two other forms are known:
Lithium nitride shows ionic conductivity for Li + , with a value of c. 2×10 −4 Ω −1 cm −1 , and an (intracrystal) activation energy of c. 0.26 eV (c. 24 kJ/mol). Hydrogen doping increases conductivity , whilst doping with metal ions ( Al , Cu , Mg ) reduces it. [ 5 ] [ 6 ] The activation energy for lithium transfer across lithium nitride crystals (intercrystalline) has been determined to be higher, at c. 68.5 kJ/mol. [ 7 ] The alpha form is a semiconductor with band gap of c. 2.1 eV . [ 4 ]
Reacting lithium nitride with carbon dioxide results in amorphous carbon nitride ( C 3 N 4 ), a semiconductor , and lithium cyanamide ( Li 2 CN 2 ), a precursor to fertilizers , in an exothermic reaction . [ 8 ] [ 9 ]
Under hydrogen at around 200°C, Li 3 N will react to form lithium amide . [ 10 ]
At higher temperatures it will react further to form ammonia and lithium hydride .
Lithium imide can also be formed under certain conditions. Some research has explored this as a possible industrial process to produce ammonia since lithium hydride can be thermally decomposed back to lithium metal.
Lithium nitride has been investigated as a storage medium for hydrogen gas, as the reaction is reversible at 270 °C. Up to 11.5% by weight absorption of hydrogen has been achieved. [ 11 ]
This inorganic compound –related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Li3N |
Lithium orthosilicate is a compound with the chemical formula Li 4 SiO 4 . It is a white ceramic compound, which melts congruently at a temperature of 1,258 °C (2,296 °F). [ 1 ]
Lithium orthosilicate is of primary interest towards carbon dioxide capture , as this compound reacts with CO 2 at elevated temperatures to form lithium carbonate , and has been implemented in limited scale in such applications. [ 2 ] Further applications of Li 4 SiO 4 include solid electrolytes for lithium-ion batteries, specifically solid state batteries , [ 3 ] and tritium breeding materials, as a component of the breeding blanket for planned fusion energy systems such as ITER . [ 1 ] [ 4 ] | https://en.wikipedia.org/wiki/Li4SiO4 |
Lithium titanates are chemical compounds of lithium , titanium and oxygen . They are mixed oxides and belong to the titanates . The most important lithium titanates are:
Other lithium titanates, i.e. mixed oxides of the system Li 2 O–TiO 2 , are:
Lithium metatitanate is a compound with the chemical formula Li 2 TiO 3 . It is a white powder with a melting point of 1,533 °C (2,791 °F). [ 4 ] It is also used as an additive in porcelain enamels and ceramic insulating bodies based on titanates. It is frequently utilized as a flux due to its good stability. [ 6 ] In recent years, along with other lithium ceramics, metatitanate pebbles have been the subject of research efforts towards tritium breeding materials in nuclear fusion applications. [ 7 ]
The most stable lithium titanate phase is β-Li 2 TiO 3 that belongs to the monoclinic system . [ 8 ] A high-temperature cubic phase exhibiting solid-solution type behavior is referred to as γ-Li 2 TiO 3 and is known to form reversibly above temperatures in the range 1150-1250 °C. [ 9 ] A metastable cubic phase, isostructural with γ-Li 2 TiO 3 is referred to as α-Li 2 TiO 3 ; it is formed at low temperatures, and transforms to the more stable β-phase upon heating to 400 °C. [ 10 ]
The sintering process is taking a powder, putting it into a mold and heating it to below its melting point . Sintering is based on atomic diffusion, the atoms in the powder particle diffuse into surrounding particles eventually forming a solid or porous material.
It has been discovered that Li 2 TiO 3 powders have a high purity and good sintering ability. [ 11 ]
Lithium titanate is used as a cathode in layer one of a double layer cathode for molten carbonate fuel cells . These fuel cells have two material layers, layer 1 and layer 2, which allow for the production of high power molten carbonate fuel cells that work more efficiently. [ 12 ]
Li 2 TiO 3 is used in the cathode of some lithium-ion batteries , along with an aqueous binder and a conducting agent. Li 2 TiO 3 is used because it is capable of stabilizing the high capacity cathode conducting agents; LiMO 2 (M=Fe, Mn, Cr, Ni). Li 2 TiO 3 and the conduction agents (LiMO 2 ) are layered in order to create the cathode material. These layers allow for the occurrence of lithium diffusion.
The lithium-titanate battery is a rechargeable battery that is much faster to charge than other lithium-ion batteries. It differs from other lithium-ion batteries because it uses lithium-titanate on the anode surface rather than carbon. This is advantageous because it does not create a solid electrolyte interface layer, which acts as a barrier to the ingress and egress of Li-ion to and from the anode. This allows lithium-titanate batteries to be recharged more quickly and provide higher currents when necessary. A disadvantage of the lithium-titanate battery is a much lower capacity and voltage than the conventional lithium-ion battery. The lithium-titanate battery is currently being used in battery electric vehicles [ citation needed ] and other specialist applications.
Fusion reactions, such as those in the proposed ITER thermonuclear demonstrator reactor, are fueled by tritium and deuterium . Tritium resources are extremely limited in their availability, with total resources currently estimated at twenty kilograms. Lithium-containing ceramic pebbles can be used as solid breeder materials in a component known as a helium-cooled breeder blanket for the production of tritium. [ 13 ] The breeding blanket constitutes a key component of the ITER reactor design. In such reactor designs tritium is produced by neutrons leaving the plasma and interacting with lithium in the blanket. Li 2 TiO 3 along with Li 4 SiO 4 are attractive as tritium breeding materials because they exhibit high tritium release, low activation, and chemical stability. [ 7 ]
Li 2 TiO 3 powder is most commonly prepared by the mixing of lithium carbonate , Ti-nitrate solution, and citric acid followed by calcination , compaction , and sintering . The nanocrystalline material created is used as a breeder powder due to its high purity and activity. [ 14 ] [ 12 ] [ 15 ] | https://en.wikipedia.org/wiki/Li4Ti5O12 |
Lithium aluminium hydride , commonly abbreviated to LAH , is an inorganic compound with the chemical formula Li [ Al H 4 ] or LiAlH 4 . It is a white solid, discovered by Finholt, Bond and Schlesinger in 1947. [ 4 ] This compound is used as a reducing agent in organic synthesis , especially for the reduction of esters , carboxylic acids , and amides . The solid is dangerously reactive toward water, releasing gaseous hydrogen (H 2 ). Some related derivatives have been discussed for hydrogen storage .
LAH is a colourless solid but commercial samples are usually gray due to contamination. [ 5 ] This material can be purified by recrystallization from diethyl ether . Large-scale purifications employ a Soxhlet extractor . Commonly, the impure gray material is used in synthesis, since the impurities are innocuous and can be easily separated from the organic products. The pure powdered material is pyrophoric , but not its large crystals. [ 6 ] Some commercial materials contain mineral oil to inhibit reactions with atmospheric moisture, but more commonly it is packed in moisture-proof plastic sacks. [ 7 ]
LAH violently reacts with water, including atmospheric moisture, to liberate hydrogen gas. The reaction proceeds according to the following idealized equation: [ 5 ]
This reaction provides a useful method to generate hydrogen in the laboratory. Aged, air-exposed samples often appear white because they have absorbed enough moisture to generate a mixture of the white compounds lithium hydroxide and aluminium hydroxide . [ 8 ]
LAH crystallizes in the monoclinic space group P 2 1 / c . The unit cell has the dimensions: a = 4.82, b = 7.81, and c = 7.92 Å, α = γ = 90° and β = 112°. In the structure, Li + cations are surrounded by five [AlH 4 ] − anions , which have tetrahedral molecular geometry . The Li + cations are bonded to one hydrogen atom from each of the surrounding tetrahedral [AlH 4 ] − anion creating a bipyramid arrangement. At high pressures (>2.2 GPa) a phase transition may occur to give β-LAH. [ 9 ]
Li[AlH 4 ] was first prepared from the reaction between lithium hydride (LiH) and aluminium chloride : [ 4 ] [ 5 ]
In addition to this method, the industrial synthesis entails the initial preparation of sodium aluminium hydride from the elements under high pressure and temperature: [ 10 ]
Li[AlH 4 ] is then prepared by a salt metathesis reaction according to:
which proceeds in a high yield. LiCl is removed by filtration from an ethereal solution of LAH, with subsequent precipitation of LAH to yield a product containing around 1% w / w LiCl. [ 10 ]
An alternative preparation starts from LiH, and metallic Al instead of AlCl 3 . Catalyzed by a small quantity of TiCl 3 (0.2%), the reaction proceeds well using dimethylether as solvent. This method avoids the cogeneration of salt. [ 11 ]
LAH is soluble in many ethereal solutions. However, it may spontaneously decompose due to the presence of catalytic impurities, though, it appears to be more stable in tetrahydrofuran (THF). Thus, THF is preferred over, e.g., diethyl ether , despite the lower solubility. [ 12 ]
LAH is metastable at room temperature. During prolonged storage it slowly decomposes to Li 3 [AlH 6 ] (lithium hexahydridoaluminate) and LiH . [ 13 ] This process can be accelerated by the presence of catalytic elements, such as titanium , iron or vanadium .
When heated LAH decomposes in a three-step reaction mechanism : [ 13 ] [ 14 ] [ 15 ]
R1 is usually initiated by the melting of LAH in the temperature range 150–170 °C, [ 16 ] [ 17 ] [ 18 ] immediately followed by decomposition into solid Li 3 [AlH 6 ] , although R1 is known to proceed below the melting point of Li[AlH 4 ] as well. [ 19 ] At about 200 °C, Li 3 [AlH 6 ] decomposes into LiH ( R2 ) [ 13 ] [ 15 ] [ 18 ] and Al which subsequently convert into LiAl above 400 °C ( R3 ). [ 15 ] Reaction R1 is effectively irreversible. R3 is reversible with an equilibrium pressure of about 0.25 bar at 500 °C. R1 and R2 can occur at room temperature with suitable catalysts. [ 20 ]
The table summarizes thermodynamic data for LAH and reactions involving LAH, [ 21 ] [ 22 ] in the form of standard enthalpy , entropy , and Gibbs free energy change, respectively.
Lithium aluminium hydride (LAH) is widely used in organic chemistry as a reducing agent . [ 5 ] It is more powerful than the related reagent sodium borohydride owing to the weaker Al-H bond compared to the B-H bond. [ 23 ] Often as a solution in diethyl ether and followed by an acid workup, it will convert esters , carboxylic acids , acyl chlorides , aldehydes , and ketones into the corresponding alcohols (see: carbonyl reduction ). Similarly, it converts amide , [ 24 ] [ 25 ] nitro , nitrile , imine , oxime , [ 26 ] and organic azides into the amines (see: amide reduction ). It reduces quaternary ammonium cations into the corresponding tertiary amines. Reactivity can be tuned by replacing hydride groups by alkoxy groups . Due to its pyrophoric nature, instability, toxicity, low shelf life and handling problems associated with its reactivity, it has been replaced in the last decade, both at the small-industrial scale and for large-scale reductions by the more convenient related reagent sodium bis (2-methoxyethoxy)aluminium hydride , which exhibits similar reactivity but with higher safety, easier handling and better economics. [ 27 ]
LAH is most commonly used for the reduction of esters [ 28 ] [ 29 ] and carboxylic acids [ 30 ] to primary alcohols; prior to the advent of LAH this was a difficult conversion involving sodium metal in boiling ethanol (the Bouveault-Blanc reduction ). Aldehydes and ketones [ 31 ] can also be reduced to alcohols by LAH, but this is usually done using milder reagents such as Na[BH 4 ] ; α, β-unsaturated ketones are reduced to allylic alcohols. [ 32 ] When epoxides are reduced using LAH, the reagent attacks the less hindered end of the epoxide, usually producing a secondary or tertiary alcohol. Epoxycyclohexanes are reduced to give axial alcohols preferentially. [ 33 ]
Partial reduction of acid chlorides to give the corresponding aldehyde product cannot proceed via LAH, since the latter reduces all the way to the primary alcohol. Instead, the milder lithium tri- tert -butoxyaluminum hydride , which reacts significantly faster with the acid chloride than with the aldehyde, must be used. For example, when isovaleric acid is treated with thionyl chloride to give isovaleroyl chloride, it can then be reduced via lithium tri- tert -butoxyaluminum hydride to give isovaleraldehyde in 65% yield. [ 34 ] [ 35 ]
Lithium aluminium hydride also reduces alkyl halides to alkanes . [ 36 ] [ 37 ] Alkyl iodides react the fastest, followed by alkyl bromides and then alkyl chlorides. Primary halides are the most reactive followed by secondary halides. Tertiary halides react only in certain cases. [ 38 ]
Lithium aluminium hydride does not reduce simple alkenes or arenes . Alkynes are reduced only if an alcohol group is nearby, [ 39 ] and alkenes are reduced in the presence of catalytic TiCl 4 . [ 40 ] It was observed that the LiAlH 4 reduces the double bond in the N -allylamides. [ 41 ]
LAH is widely used to prepare main group and transition metal hydrides from the corresponding metal halides .
LAH also reacts with many inorganic ligands to form coordinated alumina complexes associated with lithium ions. [ 21 ]
LiAlH 4 contains 10.6 wt% hydrogen, thereby making LAH a potential hydrogen storage medium for future fuel cell -powered vehicles . The high hydrogen content, as well as the discovery of reversible hydrogen storage in Ti-doped NaAlH 4 , [ 42 ] have sparked renewed research into LiAlH 4 during the last decade. A substantial research effort has been devoted to accelerating the decomposition kinetics by catalytic doping and by ball milling . [ 43 ] In order to take advantage of the total hydrogen capacity, the intermediate compound LiH must be dehydrogenated as well. Due to its high thermodynamic stability this requires temperatures in excess of 400 °C, which is not considered feasible for transportation purposes. Accepting LiH + Al as the final product, the hydrogen storage capacity is reduced to 7.96 wt%. Another problem related to hydrogen storage is the recycling back to LiAlH 4 which, owing to its relatively low stability, requires an extremely high hydrogen pressure in excess of 10000 bar. [ 43 ] Cycling only reaction R2 — that is, using Li 3 AlH 6 as starting material — would store 5.6 wt% hydrogen in a single step (vs. two steps for NaAlH 4 which stores about the same amount of hydrogen). However, attempts at this process have not been successful so far. [ citation needed ]
A variety of salts analogous to LAH are known. NaH can be used to efficiently produce sodium aluminium hydride (NaAlH 4 ) by metathesis in THF:
Potassium aluminium hydride (KAlH 4 ) can be produced similarly in diglyme as a solvent: [ 44 ]
The reverse, i.e., production of LAH from either sodium aluminium hydride or potassium aluminium hydride can be achieved by reaction with LiCl or lithium hydride in diethyl ether or THF : [ 44 ]
"Magnesium alanate" (Mg(AlH 4 ) 2 ) arises similarly using MgBr 2 : [ 45 ]
Red-Al (or SMEAH, NaAlH 2 (OC 2 H 4 OCH 3 ) 2 ) is synthesized by reacting sodium aluminum tetrahydride (NaAlH 4 ) and 2-methoxyethanol : [ 46 ] | https://en.wikipedia.org/wiki/LiAlH4 |
Lithium aluminum oxide
Lithium aluminate ( LiAlO 2 ), also called lithium aluminium oxide , is an inorganic chemical compound , an aluminate of lithium . In microelectronics , lithium aluminate is considered as a lattice matching substrate for gallium nitride . [ 3 ] [ 4 ] In nuclear technology , lithium aluminate is of interest as a solid tritium breeder material, for preparing tritium fuel for nuclear fusion . [ 5 ] Lithium aluminate is a layered double hydroxide (LDH) with a crystal structure resembling that of hydrotalcite . [ dubious – discuss ] [ clarification needed ] Lithium aluminate solubility at high pH (12.5 – 13.5) is much lower than that of aluminium oxides . In the conditioning of low- and intermediate level radioactive waste (LILW), lithium nitrate is sometimes used as additive to cement to minimise aluminium corrosion at high pH and subsequent hydrogen production. [ 6 ] Indeed, upon addition of lithium nitrate to cement, a passive layer of LiH(AlO 2 ) 2 · 5 H 2 O is formed onto the surface of metallic aluminium waste immobilised in mortar . The lithium aluminate layer is insoluble in cement pore water and protects the underlying aluminium oxide covering the metallic aluminium from dissolution at high pH . It is also a pore filler. [ 7 ] This hinders the aluminium oxidation by the protons of water and reduces the hydrogen evolution rate by a factor of 10. [ 8 ]
Lithium aluminate also finds its use as an inert electrolyte support material in molten carbonate fuel cells , where the electrolyte may be a mixture of lithium carbonate , potassium carbonate , and sodium carbonate . [ 9 ]
In 1906 Weyberg described his newly synthesized compound, lithium hydrogen aluminate. This was the first known synthesis of this unique compound. He asserted that this new compound had the corresponding chemical formula: [ 10 ]
In 1915 Allen and Rogers asserted that an insoluble aluminate of lithium is formed when aluminum is dissolved in a solution of lithium hydroxide. This air-dried substance had an atomic ratio of 2Li:5Al and the chemical formula: [ 11 ]
In 1929 Prociv recreated Allen and Rogers experiment and through a series of conductometric measurements on the saturated solution of the substance concluded that lithium and aluminum were present in the ratio of 0.8Li:2Al, which, he says, is an atomic ratio of approximately 1Li:2Al. According to him lithium aluminate may also be precipitated by the addition of a solution of lithium hydroxide to a solution of aluminum salt or by adding a solution of lithium salt to a solution of an alkali aluminate. Thus there was disagreement between Allen/Rogers and Prociv as to the composition of lithium aluminate. This may have been attributed to variations between their precipitation conditions. [ 11 ]
In 1932 Dobbins and Sanders described the formation of lithium aluminate by the addition of dilute ammonia to a solution containing lithium and aluminum salt, in the presence of phelphtalein as an indicator. In their preparation of acid lithium aluminate they dissolved strips of amalgamated aluminum in normal and tenth normal solutions of lithium hydroxide. The lithium aluminate was precipitated by the addition of a solution of lithium hydroxide to a solution of aluminum salts, or by adding a solution of lithium salt to a solution of alkaline aluminate. In all cases the composition of the compound of lithium aluminate was expressed by the formula: [ 12 ]
They claimed that the formed compound contained lithium and aluminum in the atomic ratio of 2Li:5Al. Their chemical formula was simplified into the modern formulation for lithium aluminate:
The fundamental compound of lithium aluminate has found attention in two different fields: nuclear physics and solid-state chemistry. At least five different phases of lithium aluminate have been found. [ 13 ] The lithium aluminate crystal structure may be found in either α, β, or γ phases. [ 14 ]
Nuclear physicists are interested in the γ-LiAlO 2 modification of lithium aluminate, because of its good performance under high neutron and electron radiation. This modification also exhibits the essential chemical, thermo physical and mechanical stability at high temperature along with the required irradiation behavior. This phase appears to be a promising lithium ceramic, suitable as an in site tritium breeding material in future fusion reactors. [ 13 ]
Solid-state chemists investigating preparational routes to lithium aluminate discovered its interesting acid-base chemistry. The α-LiAlO 2 modification (but neither β-LiAlO 2 or γ-LiAlO 2 ) reacts with molten benzoic acid leading to nearly total Li + proton exchange thus forming LiHAl 2 O 4 There is a lot of interest in the chemical reactivity among the three modifications of LiAlO 2 . The reasons for the α-LiAlO 2 modification being highly reactive and the β-LiAlO 2 or γ-LiAlO 2 modifications being totally unreactive is currently a mystery. [ 13 ]
Lithium aluminate powder preparation was based on the solid-state reactions between Al 2 O 3 and lithium-containing compounds like Li 2 CO 3 , LiOH, Li 2 O , LiAc, and reactions occurred at temperatures between 400Deg C to 1000 Deg C. Due to the evaporation of lithium at high temperatures and contamination from grinding operations, pure lithium aluminate with controlled particle size has been difficult to synthesize. [ 15 ]
Synthesis of lithium aluminate has been essentially performed by several methods: in the solid state, by wet chemical, sol-gel, with the use of templates, various precursors, and combustion processes. The main product in a solid state reaction is the α-LiAlO 2 phase; in a wet chemical reaction, the main product is a solid solution of α-LiAlO 2 and γ-LiAlO 2 phases. [ 14 ] The α-LiAlO 2 modification (low temperature phase), with a hexagonal structure, undergoes transformation to the γ-modification (High temperature phase), with a tetragonal structure, at about 900 °C. The metastable β-modification, with a monoclinic structure, is assumed to transform to the γ-modification at about 900 °C. [ 15 ]
The compound is unknown in the natural environment. However, a related compound, LiAl 5 O 8 , is known as the very recently discovered (as of 2020) and very rare mineral chukochenite. [ 16 ] [ 17 ] | https://en.wikipedia.org/wiki/LiAlO2 |
Lithium borohydride (LiBH 4 ) is a borohydride and known in organic synthesis as a reducing agent for esters . Although less common than the related sodium borohydride , the lithium salt offers some advantages, being a stronger reducing agent and highly soluble in ethers, whilst remaining safer to handle than lithium aluminium hydride . [ 3 ]
Lithium borohydride may be prepared by the metathesis reaction , which occurs upon ball-milling the more commonly available sodium borohydride and lithium bromide : [ 4 ]
Alternatively, it may be synthesized by treating boron trifluoride with lithium hydride in diethyl ether : [ 5 ]
Lithium borohydride is useful as a source of hydride (H – ). It can react with a range of carbonyl substrates and other polarized carbon structures to form a hydrogen–carbon bond. It can also react with Brønsted–Lowry -acidic substances (sources of H + ) to form hydrogen gas.
As a hydride reducing agent, lithium borohydride is stronger than sodium borohydride [ 6 ] [ 7 ] but weaker than lithium aluminium hydride. [ 7 ] Unlike the sodium analog, it can reduce esters to alcohols, nitriles and primary amides to amines , and can open epoxides . The enhanced reactivity in many of these cases is attributed to the polarization of the carbonyl substrate by complexation to the lithium cation. [ 3 ] Unlike the aluminium analog, it does not react with nitro groups , carbamic acids , alkyl halides , or secondary and tertiary amides.
Lithium borohydride reacts with water to produce hydrogen. This reaction can be used for hydrogen generation. [ 8 ]
Although this reaction is usually spontaneous and violent, somewhat-stable aqueous solutions of lithium borohydride can be prepared at low temperature if degassed , distilled water is used and exposure to oxygen is carefully avoided. [ 9 ]
Lithium borohydride is renowned as one of the highest- energy-density chemical energy carriers . Although presently of no practical importance, the solid liberates 65 MJ / kg heat upon treatment with atmospheric oxygen. Since it has a density of 0.67 g/cm 3 , oxidation of liquid lithium borohydride gives 43 MJ/L . In comparison, gasoline gives 44 MJ/kg (or 35 MJ/L), while liquid hydrogen gives 120 MJ/kg (or 8.0 MJ/L). [ nb 1 ] The high specific energy density of lithium borohydride has made it an attractive candidate to propose for automobile and rocket fuel, but despite the research and advocacy, it has not been used widely. As with all chemical-hydride-based energy carriers, lithium borohydride is very complex to recycle (i.e. recharge) and therefore suffers from a low energy conversion efficiency . While batteries such as lithium-ion carry an energy density of up to 0.72 MJ/kg and 2.0 MJ/L, their DC -to-DC conversion efficiency can be as high as 90%. [ 10 ] In view of the complexity of recycling mechanisms for metal hydrides, [ 11 ] such high energy-conversion efficiencies are not practical with present technology. | https://en.wikipedia.org/wiki/LiBH4 |
Lithium bromide ( LiBr ) is a chemical compound of lithium and bromine . Its extreme hygroscopic character makes LiBr useful as a desiccant in certain air conditioning systems. [ 9 ]
LiBr is prepared by treating an aqueous suspension of lithium carbonate with hydrobromic acid or by reacting lithium hydroxide with bromine. [ 9 ] It forms several crystalline hydrates , unlike the other alkali metal bromides. [ 10 ]
Lithium hydroxide and hydrobromic acid (aqueous solution of hydrogen bromide) will precipitate lithium bromide in the presence of water.
A 50–60% aqueous solution of lithium bromide is used in air-conditioning systems as desiccant . It is also used in absorption chilling along with water (see absorption refrigerator ). Solid LiBr is a useful reagent in organic synthesis . It is included into oxidation and hydroformylation catalysts; it is also used for deprotonation and dehydration of organic compounds containing acidic protons, and for the purification of steroids and prostaglandins . [ 9 ]
Lithium bromide was used as a sedative beginning in the early 1900s, but it fell into disfavor in the 1940s as newer sedatives became available and when some heart patients died after using the salt substitute lithium chloride. [ 11 ] Like lithium carbonate and lithium chloride , it was used as treatment for bipolar disorder .
Lithium salts are psychoactive and somewhat corrosive. Heat is quickly generated when lithium bromide is dissolved into water because it has a negative enthalpy of solution . | https://en.wikipedia.org/wiki/LiBr |
Lithium acetate (CH 3 COOLi) is a salt of lithium and acetic acid . It is often abbreviated as LiOAc.
Lithium acetate is used in the laboratory as buffer for gel electrophoresis of DNA and RNA . It has a lower electrical conductivity and can be run at higher speeds than can gels made from TAE buffer (5-30V/cm as compared to 5-10V/cm). At a given voltage, the heat generation and thus the gel temperature is much lower than with TAE buffers, therefore the voltage can be increased to speed up electrophoresis so that a gel run takes only a fraction of the usual time. Downstream applications, such as isolation of DNA from a gel slice or Southern blot analysis, work as expected when using lithium acetate gels.
Lithium boric acid or sodium boric acid are usually preferable to lithium acetate or TAE when analyzing smaller fragments of DNA (less than 500 bp) due to the higher resolution of borate-based buffers in this size range as compared to acetate buffers.
Lithium acetate is also used to permeabilize the cell wall of yeast for use in DNA transformation . It is believed that the beneficial effect of LiOAc is caused by its chaotropic effect ; denaturing DNA, RNA and proteins. [ 2 ] | https://en.wikipedia.org/wiki/LiCH3COO |
Lithium chloride is a chemical compound with the formula Li Cl . The salt is a typical ionic compound (with certain covalent characteristics), although the small size of the Li + ion gives rise to properties not seen for other alkali metal chlorides, such as extraordinary solubility in polar solvents (83.05 g/100 mL of water at 20 °C) and its hygroscopic properties. [ 5 ]
The salt forms crystalline hydrates , unlike the other alkali metal chlorides. [ 6 ] Mono-, tri-, and pentahydrates are known. [ 7 ] The anhydrous salt can be regenerated by heating the hydrates. LiCl also absorbs up to four equivalents of ammonia /mol. As with any other ionic chloride, solutions of lithium chloride can serve as a source of chloride ion, e.g., forming a precipitate upon treatment with silver nitrate :
Lithium chloride is produced by treatment of lithium carbonate with hydrochloric acid . [ 5 ] Anhydrous LiCl is prepared from the hydrate by heating in a stream of hydrogen chloride .
Lithium chloride is mainly used for the production of lithium metal by electrolysis of a LiCl/ KCl melt at 450 °C (842 °F). LiCl is also used as a brazing flux for aluminium in automobile parts. It is used as a desiccant for drying air streams. [ 5 ] In more specialized applications, lithium chloride finds some use in organic synthesis , e.g., as an additive in the Stille reaction . Also, in biochemical applications, it can be used to precipitate RNA from cellular extracts. [ 8 ]
Lithium chloride is also used as a flame colorant to produce dark red flames.
Lithium chloride is used as a relative humidity standard in the calibration of hygrometers . At 25 °C (77 °F) a saturated solution (45.8%) of the salt will yield an equilibrium relative humidity of 11.30%. Additionally, lithium chloride can be used as a hygrometer. This deliquescent salt forms a self-solution when exposed to air. The equilibrium LiCl concentration in the resulting solution is directly related to the relative humidity of the air. The percent relative humidity at 25 °C (77 °F) can be estimated, with minimal error in the range 10–30 °C (50–86 °F), from the following first-order equation: RH=107.93-2.11C, where C is solution LiCl concentration, percent by mass.
Molten LiCl is used for the preparation of carbon nanotubes , [ 9 ] graphene [ 10 ] and lithium niobate . [ 11 ]
Lithium chloride has been shown to have strong acaricidal properties, being effective against Varroa destructor in populations of honey bees . [ 12 ]
Lithium chloride is used as an aversive agent in lab animals to study conditioned place preference and aversion .
Lithium salts affect the central nervous system in a variety of ways. While the citrate , carbonate , and orotate salts are currently used to treat bipolar disorder , other lithium salts including the chloride were used in the past. For a short time in the 1940s lithium chloride was manufactured as a salt substitute for people with hypertension, but this was prohibited after the toxic effects of the compound ( tremors , fatigue , nausea ) were recognized. [ 13 ] [ 14 ] [ 15 ] It was, however, noted by J. H. Talbott that many symptoms attributed to lithium chloride toxicity may have also been attributable to sodium chloride deficiency , to the diuretics often administered to patients who were given lithium chloride, or to the patients' underlying conditions. [ 13 ] | https://en.wikipedia.org/wiki/LiCl |
Lithium hypochlorite is a chemical compound with the chemical formula of Li O Cl . It is the lithium salt of hypochlorous acid . It consists of lithium cations ( Li + ) and hypochlorite anions ( − OCl ). It is a colorless, crystalline compound. It has been used as a disinfectant for pools, and is also used as a reagent for some chemical reactions.
Doses of 500 mg/kg cause detrimental clinical signs and significant mortality in rats . [ 1 ] The use of chlorine -based disinfectants in domestic water , although widespread, has led to some controversy because of the formation of small quantities of harmful byproducts such as chloroform . Studies showed no uptake of lithium if pools with lithium hypochlorite have been used. [ 2 ]
Lithium hypochlorite has been used as a fast-acting disfinectant for vinyl swimming pools . However, due to the increasing demand for lithium in lithium-ion batteries , manufacturers have stopped producing lithium hypochlorite, making it much harder to find these days. [ 3 ]
This inorganic compound –related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/LiClO |
Lithium perchlorate is the inorganic compound with the formula LiClO 4 . This white or colourless crystalline salt is noteworthy for its high solubility in many solvents. It exists both in anhydrous form and as a trihydrate .
Lithium perchlorate is used as a source of oxygen in some chemical oxygen generators . It decomposes at about 400 °C, yielding lithium chloride and oxygen : [ 5 ]
Over 60% of the mass of the lithium perchlorate is released as oxygen. [ 2 ] It has both the highest oxygen to weight and oxygen to volume ratio of all practical perchlorate salts, and higher oxygen to volume ratio than liquid oxygen . [ 6 ]
Lithium perchlorate is used as an oxidizer in some experimental solid rocket propellants , and to produce red colored flame in pyrotechnic compositions. [ 2 ] [ 7 ]
LiClO 4 is highly soluble in organic solvents, even diethyl ether. Such solutions are employed in Diels–Alder reactions , where it is proposed that the Lewis acidic Li + binds to Lewis basic sites on the dienophile, thereby accelerating the reaction. [ 8 ]
Lithium perchlorate is also used as a co-catalyst in the coupling of α,β-unsaturated carbonyls with aldehydes, also known as the Baylis–Hillman reaction . [ 9 ]
Solid lithium perchlorate is found to be a mild and efficient Lewis acid for promoting cyanosilylation of carbonyl compounds under neutral conditions. [ 10 ]
Lithium perchlorate is also used as an electrolyte salt in lithium-ion batteries . Lithium perchlorate is chosen over alternative salts such as lithium hexafluorophosphate or lithium tetrafluoroborate when its superior electrical impedance , conductivity , hygroscopicity , and anodic stability properties are of importance to the specific application. [ 11 ] However, these beneficial properties are often overshadowed by the electrolyte's strong oxidizing properties, making the electrolyte reactive toward its solvent at high temperatures and/or high current loads. Due to these hazards the battery is often considered unfit for industrial applications. [ 11 ]
Concentrated solutions of lithium perchlorate (4.5 mol/L) are used as a chaotropic agent to denature proteins .
Lithium perchlorate can be manufactured by reaction of sodium perchlorate with lithium chloride . It can be also prepared by electrolysis of lithium chlorate at 200 mA/cm 2 at temperatures above 20 °C. [ 12 ]
Perchlorates often give explosive mixtures with organic compounds, finely divided metals, sulfur, and other reducing agents. [ 12 ] [ 2 ] | https://en.wikipedia.org/wiki/LiClO4 |
Lithium cobalt oxide , sometimes called lithium cobaltate [ 2 ] or lithium cobaltite , [ 3 ] is a chemical compound with formula LiCoO 2 . The cobalt atoms are formally in the +3 oxidation state, hence the IUPAC name lithium cobalt(III) oxide .
Lithium cobalt oxide is a dark blue or bluish-gray crystalline solid, [ 4 ] and is commonly used in the positive electrodes of lithium-ion batteries .
The structure of LiCoO 2 has been studied with numerous techniques including x-ray diffraction , electron microscopy , neutron powder diffraction , and EXAFS . [ 5 ]
The solid consists of layers of monovalent lithium cations ( Li + ) that lie between extended anionic sheets of cobalt and oxygen atoms, arranged as edge-sharing octahedra , with two faces parallel to the sheet plane. [ 6 ] The cobalt atoms are formally in the trivalent oxidation state ( Co 3+ ) and are sandwiched between two layers of oxygen atoms ( O 2− ).
In each layer (cobalt, oxygen, or lithium), the atoms are arranged in a regular triangular lattice. The lattices are offset so that the lithium atoms are farthest from the cobalt atoms, and the structure repeats in the direction perpendicular to the planes every three cobalt (or lithium) layers. The point group symmetry is R 3 ¯ m {\displaystyle R{\bar {3}}m} in Hermann-Mauguin notation, signifying a unit cell with threefold improper rotational symmetry and a mirror plane. The threefold rotational axis (which is normal to the layers) is termed improper because the triangles of oxygen (being on opposite sides of each octahedron) are anti-aligned. [ 7 ]
Fully reduced lithium cobalt oxide can be prepared by heating a stoichiometric mixture of lithium carbonate Li 2 CO 3 and cobalt(II,III) oxide Co 3 O 4 or metallic cobalt at 600–800 °C, then annealing the product at 900 °C for many hours, all under an oxygen atmosphere. [ 6 ] [ 3 ] [ 7 ]
Nanometer-size particles more suitable for cathode use can also be obtained by calcination of hydrated cobalt oxalate β- CoC 2 O 4 ·2 H 2 O , in the form of rod-like crystals about 8 μm long and 0.4 μm wide, with lithium hydroxide LiOH , up to 750–900 °C. [ 9 ]
A third method uses lithium acetate , cobalt acetate , and citric acid in equal molar amounts, in water solution. Heating at 80 °C turns the mixture into a viscous transparent gel. The dried gel is then ground and heated gradually to 550 °C. [ 10 ]
The usefulness of lithium cobalt oxide as an intercalation electrode was discovered in 1980 by an Oxford University research group led by John B. Goodenough and Tokyo University 's Koichi Mizushima . [ 11 ]
The compound is now used as the cathode in some rechargeable lithium-ion batteries , with particle sizes ranging from nanometers to micrometers . [ 10 ] [ 9 ] During charging, the cobalt is partially oxidized to the +4 state, with some lithium ions moving to the electrolyte, resulting in a range of compounds Li x CoO 2 with 0 < x < 1. [ 3 ]
Batteries produced with LiCoO 2 cathodes have very stable capacities, but have lower capacities and power than those with cathodes based on (especially nickel-rich) nickel-cobalt-aluminum (NCA) or nickel-cobalt-manganese (NCM) oxides. [ 12 ] Issues with thermostability are better for LiCoO 2 cathodes than other nickel-rich chemistries although not significantly. This makes LiCoO 2 batteries susceptible to thermal runaway in cases of abuse such as high temperature operation (>130 °C) or overcharging . At elevated temperatures, LiCoO 2 decomposition generates oxygen , which then reacts with the organic electrolyte of the cell, this reaction is often seen in Lithium-Ion batteries where the battery becomes highly volatile and must be recycled in a safe manner. The decomposition of LiCoO 2 is a safety concern due to the magnitude of this highly exothermic reaction , which can spread to adjacent cells or ignite nearby combustible material. [ 13 ] In general, this is seen for many lithium-ion battery cathodes.
The delithiation process is usually by chemical means, [ 14 ] although a novel physical process has been developed based on ion sputtering and annealing cycles, [ 15 ] leaving the material properties intact. | https://en.wikipedia.org/wiki/LiCoO2 |
Lithium fluoride is an inorganic compound with the chemical formula LiF. It is a colorless solid that transitions to white with decreasing crystal size.
Its structure is analogous to that of sodium chloride , but it is much less soluble in water. It is mainly used as a component of molten salts . [ 4 ] Partly because Li and F are both light elements, and partly because F 2 is highly reactive, formation of LiF from the elements releases one of the highest energies per mass of reactants , second only to that of BeO .
LiF is prepared from lithium hydroxide or lithium carbonate with hydrogen fluoride . [ 5 ]
Lithium fluoride is reacted with hydrogen fluoride (HF) and phosphorus pentachloride to make lithium hexafluorophosphate Li[PF 6 ] , an ingredient in lithium ion battery electrolyte .
The lithium fluoride alone does not absorb hydrogen fluoride to form a bifluoride salt. [ 6 ]
Fluorine is produced by the electrolysis of molten potassium bifluoride . This electrolysis proceeds more efficiently when the electrolyte contains a few percent of LiF, possibly because it facilitates formation of an Li-C-F interface on the carbon electrodes . [ 4 ] A useful molten salt, FLiNaK , consists of a mixture of LiF, together with sodium fluoride and potassium fluoride . The primary coolant for the Molten-Salt Reactor Experiment was FLiBe ; 2LiF·BeF 2 (66 mol% of LiF, 33 mol% of BeF 2 ).
Because of the large band gap for LiF, its crystals are transparent to short wavelength ultraviolet radiation , more so than any other material . LiF is therefore used in specialized optics for the vacuum ultraviolet spectrum. [ 7 ] (See also magnesium fluoride .) Lithium fluoride is used also as a diffracting crystal in X-ray spectrometry.
It is also used as a means to record ionizing radiation exposure from gamma rays , beta particles , and neutrons (indirectly, using the 6 3 Li (n,alpha) nuclear reaction ) in thermoluminescent dosimeters . 6 LiF nanopowder enriched to 96% has been used as the neutron reactive backfill material for microstructured semiconductor neutron detectors (MSND). [ 8 ]
Lithium fluoride (highly enriched in the common isotope lithium-7) forms the basic constituent of the preferred fluoride salt mixture used in liquid-fluoride nuclear reactors . Typically lithium fluoride is mixed with beryllium fluoride to form a base solvent ( FLiBe ), into which fluorides of uranium and thorium are introduced. Lithium fluoride is exceptionally chemically stable and LiF/ BeF 2 mixtures ( FLiBe ) have low melting points (360 to 459 °C or 680 to 858 °F) and the best neutronic properties of fluoride salt combinations appropriate for reactor use. MSRE used two different mixtures in the two cooling circuits.
Lithium fluoride is widely used in PLED and OLED as a coupling layer to enhance electron injection . The thickness of the LiF layer is usually around 1 nm . The dielectric constant (or relative permittivity, ε) of LiF is 9.0. [ 9 ]
Naturally occurring lithium fluoride is known as the extremely rare mineral griceite . [ 10 ] | https://en.wikipedia.org/wiki/LiF |
Lithium iron phosphate or lithium ferro-phosphate ( LFP ) is an inorganic compound with the formula LiFePO 4 . It is a gray, red-grey, brown or black solid that is insoluble in water. The material has attracted attention as a component of lithium iron phosphate batteries , [ 1 ] a type of Li-ion battery . [ 2 ] This battery chemistry is targeted for use in power tools , electric vehicles , solar energy installations [ 3 ] [ 4 ] and more recently large grid-scale energy storage . [ 5 ] [ 2 ]
Most lithium batteries (Li-ion) used in consumer electronics products use cathodes made of lithium compounds such as lithium cobalt oxide ( LiCoO 2 ), lithium manganese oxide ( LiMn 2 O 4 ), and lithium nickel oxide ( LiNiO 2 ). The anodes are generally made of graphite .
Lithium iron phosphate exists naturally in the form of the mineral triphylite , but this material has insufficient purity for use in batteries.
With general chemical formula of LiMPO 4 , compounds in the LiFePO 4 family adopt the olivine structure. M includes not only Fe but also Co, Mn and Ti. [ 6 ] As the first commercial LiMPO 4 was C/ LiFePO 4 , the whole group of LiMPO 4 is informally called “lithium iron phosphate” or “ LiFePO 4 ”. However, more than one olivine-type phase may be used as a battery's cathode material. Olivine compounds such as A y MPO 4 , Li 1− x MFePO 4 , and LiFePO 4− z M have the same crystal structures as LiMPO 4 , and may replace it in a cathode. All may be referred to as “LFP”. [ citation needed ]
Manganese, phosphate, iron, and lithium also form an olivine structure . This structure is a useful contributor to the cathode of lithium rechargeable batteries. [ 7 ] This is due to the olivine structure created when lithium is combined with manganese, iron, and phosphate (as described above). The olivine structures of lithium rechargeable batteries are significant, for they are affordable, stable, and can be safely used to store energy. [ 8 ]
Arumugam Manthiram and John B. Goodenough first identified the polyanion class of cathode materials for lithium ion batteries . [ 9 ] [ 10 ] [ 11 ] LiFePO 4 was then identified as a cathode material belonging to the polyanion class for use in batteries in 1996 by Padhi et al. [ 12 ] [ 13 ] Reversible extraction of lithium from LiFePO 4 and insertion of lithium into FePO 4 was demonstrated. Neutron diffraction confirmed that LFP was able to ensure the security of large input/output current of lithium batteries. [ 14 ]
The material can be produced by heating a variety of iron and lithium salts with phosphates or phosphoric acid . Many related routes have been described including those that use hydrothermal synthesis . [ 15 ]
In LiFePO 4 , lithium has a +1 charge, iron +2 charge balancing the −3 charge for phosphate. Upon removal of Li, the material converts to the ferric form FePO 4 . [ 16 ]
The iron atom and 6 oxygen atoms form an octahedral coordination sphere , described as FeO 6 , with the Fe ion at the center. The phosphate groups, PO 4 , are tetrahedral. The three-dimensional framework is formed by the FeO 6 octahedra sharing O corners. Lithium ions reside within the octahedral channels in a zigzag manner. In crystallography , this structure is thought to belong to the P mnb space group of the orthorhombic crystal system. The lattice constants are: a = 6.008 Å, b = 10.334 Å, and c = 4.693 Å. The volume of the unit cell is 291.4 Å 3 .
In contrast to two traditional cathode materials, LiMnO 4 and LiCoO 2 , lithium ions of LiFePO 4 migrate in the lattice's one-dimensional free volume. During charge/discharge, the lithium ions are extracted concomitant with oxidation of Fe:
Extraction of lithium from LiFePO 4 produces FePO 4 with a similar structure. FePO 4 adopts a P mnb space group with a unit cell volume of 272.4 Å 3 , only slightly smaller than that of its lithiated precursor. Extraction of lithium ions reduces the lattice volume, as is the case with lithium oxides. LiFePO 4 's corner-shared FeO 6 octahedra are separated by the oxygen atoms of the PO 3− 4 tetrahedra and cannot form a continuous FeO 6 network, reducing conductivity.
A nearly close-packed hexagonal array of oxides centers provides relatively little free volume for Li + ions to migrate within. For this reason, the ionic conductivity of Li + is relatively low at ambient temperature. The details of the lithiation of FePO 4 and the delithiation of LiFePO 4 have been examined. Two phases of the lithiated material are implicated. [ 16 ] [ 17 ]
LFP cells have an operating voltage of 3.3 V, charge density of 170 mAh/g, high power density , long cycle life and stability at high temperatures. [ 18 ]
LFP's major commercial advantages are that it poses few safety concerns such as overheating and explosion, as well as long cycle lifetimes, high power density and has a wider operating temperature range. Power plants and automobiles use LFP. [ 19 ] [ 20 ]
BAE has announced that their HybriDrive Orion 7 hybrid bus uses about 180 kW LFP battery cells. AES has developed multi-trillion watt battery systems that are capable of subsidiary services of the power network, including spare capacity and frequency adjustment. In China, BAK and Tianjin Lishen are active in the area.
The safety is a crucial property for certain applications. For example, in 2016 an LFP-based energy storage system was installed in Paiyun Lodge on Mt.Jade (Yushan) (the highest alpine lodge in Taiwan ). As of 2024, the system is still operating safely. [ 3 ]
Although LFP has 25% less specific energy (Wh/g) than lithium batteries with oxide (e.g. nickel-cobalt-manganese, NCM) cathode materials, primarily due to its operational voltage (3.2 volts vs 3.7 for NCM-type cathode chemistries), it has 70% more than nickel-hydrogen batteries .
The major differences between LFP batteries and other lithium-ion battery types is that LFP batteries contain no cobalt (removing ethical and economic questions about cobalt's availability) and have a flat discharge curve.
LFP batteries have drawbacks, originating from a high electronic resistivity of LFP, as well as the lower maximum charge/discharge voltage. The energy density is significantly lower than LiCoO 2 (although higher than the nickel–metal hydride battery ).
Lithium cobalt oxide based battery chemistries are more prone to thermal runaway if overcharged and cobalt is both expensive and not widely geographically available. Other chemistries such as nickel-manganese-cobalt (NMC) have supplanted LiCo chemistry cells in most applications. The original ratio of Ni to Mn to Co was 3:3:3, whereas today, cells are being made with ratios of 8:1:1 or 6:2:2, whereby the Co content has been drastically reduced.
LiFePO 4 batteries are comparable to sealed lead acid batteries and are often being touted as a drop-in replacement for lead acid applications. The most notable difference between lithium iron phosphate and lead acid is the fact that the lithium battery capacity shows only a small dependence on the discharge rate. With very high discharge rates, for instance 0.8C, the capacity of the lead acid battery is only 60% of the rated capacity. Therefore, in cyclic applications where the discharge rate is often greater than 0.1C, a lower rated lithium battery will often have a higher actual capacity than the comparable lead acid battery. This means that at the same capacity rating, the lithium will cost more, but a lower capacity lithium battery can be used for the same application at a lower price. The cost of ownership when considering the lifecycle further increases the value of the lithium battery when compared to a lead acid battery. [ 21 ] [ independent source needed ] , but they have much poorer performance at lower temperatures, as covered in the section on effects of temperature .
There are 4 groups of patents on LFP battery materials:
These patents underlie mature mass production technologies. The largest production capacity is up to 250 tons per month.
In patent lawsuits in the US in 2005 and 2006, UT and Hydro-Québec claimed that LiFePO 4 as the cathode infringed their patents, US 5910382 and US 6514640 . The patent claims involved a unique crystal structure and a chemical formula of the battery cathode material.
On April 7, 2006, A123 filed an action seeking a declaration of non-infringement and invalidity UT's patents. A123 separately filed two ex parte Reexamination Proceedings before the United States Patent and Trademark Office (USPTO), in which they sought to invalidate the patents based upon prior art.
In a parallel court proceeding, UT sued Valence Technology , a company that commercializes LFP products that alleged infringement.
The USPTO issued a Reexamination Certificate for the '382 patent on April 15, 2008, and for the '640 patent on May 12, 2009, by which the claims of these patents were amended. This allowed the current patent infringement suits filed by Hydro-Quebec against Valence and A123 to proceed. After a Markman hearing, on April 27, 2011, the Western District Court of Texas held that the claims of the reexamined patents had a narrower scope than as originally granted. The key question was whether the earlier Goodenough 's patents from the UT (licensed to Hydro-Quebec) were infringed by A123, that had its own improved versions of LiFePO4 patents, that contained cobalt dopant. The end results was licensing of Goodenough's patents by A123 under undisclosed terms. [ 23 ]
On December 9, 2008, the European Patent Office revoked Dr. Goodenough’s patent numbered 0904607. This decision basically reduced the patent risk of using LFP in European automobile applications. The decision is believed to be based on the lack of novelty. [ 24 ]
The first major settlement was the lawsuit between NTT and the UT. In October 2008, [ 25 ] NTT announced that they would settle the case in the Japan Supreme Civil Court for $30 million. As part of the agreement, UT agreed that NTT did not steal the information and that NTT would share its LFP patents with UT. NTT’s patent is also for an olivine LFP, with the general chemical formula of A y MPO 4 (A is for alkali metal and M for the combination of Co and Fe), now used by BYD Company . Although chemically the materials are nearly the same, from the viewpoint of patents, A y MPO 4 of NTT is different from the materials covered by UT. A y MPO 4 has higher capacity than LiMPO 4 . At the heart of the case was that NTT engineer Okada Shigeto, who had worked in the UT labs developing the material, was accused of stealing UT’s intellectual property .
As of 2020, an organization named LifePO+C claims to own the key IP and offers licenses. It is a consortium between Johnson Matthey, the CNRS, University of Montreal, and Hydro Quebec.
LFP has two shortcomings: low conductivity (high overpotential) and low lithium diffusion constant, both of which limit the charge/discharge rate. Adding conducting particles in delithiated FePO 4 raises its electron conductivity. For example, adding conducting particles with good diffusion capability like graphite and carbon [ 26 ] to LiMPO 4 powders significantly improves conductivity between particles, increases the efficiency of LiMPO 4 and raises its reversible capacity up to 95% of the theoretical values. However, addition of conductive additives also increases the "dead mass" present in the cell that does not contribute to energy storage. LiMPO 4 shows good cycling performance even under charge/discharge current as large as 5C. [ 27 ]
Coating LFP with inorganic oxides can make LFP’s structure more stable and increase conductivity. Traditional LiCoO 2 with oxide coating shows improved cycling performance. This coating also inhibits dissolution of Co and slows the decay of LiCoO 2 capacity. Similarly, LiMPO 4 with an inorganic coating such as ZnO [ 28 ] and ZrO 2 , [ 29 ] has a better cycling lifetime, larger capacity and better characteristics under rapid discharge. The addition of a conductive carbon increases efficiency. Mitsui Zosen and Aleees reported that addition of conducting metal particles such as copper and silver increased efficiency. [ 30 ] LiMPO 4 with 1 wt% of metal additives has a reversible capacity up to 140 mAh/g and better efficiency under high discharge current.
Substituting other materials for the iron or lithium in LiMPO 4 can also raise efficiency. Substituting zinc for iron increases crystallinity of LiMPO 4 because zinc and iron have similar ionic radii. [ 31 ] Cyclic voltammetry confirms that LiFe 1− x M x PO 4 , after metal substitution, has higher reversibility of lithium ion insertion and extraction. During lithium extraction, Fe (II) is oxidized to Fe (III) and the lattice volume shrinks. The shrinking volume changes lithium’s returning paths.
Mass production with stability and high quality still faces many challenges.
Similar to lithium oxides, LiMPO 4 may be synthesized by a variety of methods, including: solid-phase synthesis , emulsion drying, sol-gel process , solution coprecipitation, vapor-phase deposition , electrochemical synthesis, electron beam irradiation, microwave process [ vague ] , hydrothermal synthesis, ultrasonic pyrolysis and spray pyrolysis .
In the emulsion drying process, the emulsifier is first mixed with kerosene. Next, the solutions of lithium salts and iron salts are added to this mixture. This process produces nanocarbon particles. [ 32 ] Hydrothermal synthesis produces LiMPO 4 with good crystallinity. Conductive carbon is obtained by adding polyethylene glycol to the solution followed by thermal processing. [ 33 ] Vapor phase deposition produces a thin film LiMPO 4 . [ 34 ] In flame spray pyrolysis FePO 4 is mixed with lithium carbonate and glucose and charged with electrolytes . The mixture is then injected inside a flame and filtered to collect the synthesized LiFePO 4 . [ 35 ]
The effects of temperature on lithium iron phosphate batteries can be divided into the effects of high temperature and low temperature.
Generally, LFP chemistry batteries are less susceptible to thermal runaway reactions like those that occur in lithium cobalt batteries; LFP batteries exhibit better performance at an elevated temperature. Research has shown that at room temperature (23 °C), the initial capacity loss approximates 40-50 mAh/g. However, at 40 °C and 60 °C, the capacity losses approximate 25 and 15 mAh/g respectively, but these capacity losses were spread over 20 cycles instead of a bulk loss like that in the case of room temperature capacity loss. [ 36 ]
However, this is only true for a short cycling timeframe. Later yearlong study has shown that despite LFP batteries having double the equivalent full cycle, the capacity fade rate increased with increasing temperature for LFP cells but the increasing temperature does not impact NCA cells or have a negligible impact on the aging of NMC cells. [ 37 ] This capacity fade is primarily due to the solid electrolyte interface (SEI) formation reaction being accelerated by increasing temperature.
LFP batteries are especially affected by decreasing temperature which possibly hamper their application in high-latitude areas. The initial discharge capacities for LFP/C samples at temperatures of 23, 0, -10, and -20 °C are 141.8, 92.7, 57.9 and 46.7 mAh/g with coulombic efficiency 91.2%, 74.5%, 63.6% and 61.3%. These losses are accounted for by the slow diffusion of lithium ions within electrodes and the formation of SEI that come with lower temperatures which subsequently increase the charge-transfer resistance on the electrolyte-electrode interfaces. [ 38 ] Another possible cause of the lowered capacity formation is lithium plating. As mentioned above, low temperature lowers the diffusion rate of lithium ions within the electrodes, allowing for the lithium plating rate to compete with that of intercalation rate. The colder condition leads to higher growth rates and shifts the initial point to lower state of charge which means that the plating process starts earlier. [ 39 ] Lithium plating uses up lithium which then compete with the intercalation of lithium into graphite, decreasing the capacity of the batteries. The aggregated lithium ions are deposited on the surface of electrodes in the form of “plates” or even dendrites which may penetrate the separators, short-circuiting the battery completely. [ 40 ] | https://en.wikipedia.org/wiki/LiFePO4 |
Lithium hydride is an inorganic compound with the formula Li H . This alkali metal hydride is a colorless solid, although commercial samples are grey. Characteristic of a salt-like (ionic) hydride , it has a high melting point, and it is not soluble but reactive with all protic organic solvents . It is soluble and nonreactive with certain molten salts such as lithium fluoride , lithium borohydride , and sodium hydride . With a molar mass of 7.95 g/mol, it is the lightest ionic compound .
LiH is a diamagnetic and an ionic conductor with a conductivity gradually increasing from 2 × 10 −5 Ω −1 cm −1 at 443 °C to 0.18 Ω −1 cm −1 at 754 °C; there is no discontinuity in this increase through the melting point. [ 3 ] : 36 The dielectric constant of LiH decreases from 13.0 (static, low frequencies) to 3.6 (visible-light frequencies). [ 3 ] : 35 LiH is a soft material with a Mohs hardness of 3.5. [ 3 ] : 42 Its compressive creep (per 100 hours) rapidly increases from < 1% at 350 °C to > 100% at 475 °C, meaning that LiH cannot provide mechanical support when heated. [ 3 ] : 39
The thermal conductivity of LiH decreases with temperature and depends on morphology: the corresponding values are 0.125 W/(cm·K) for crystals and 0.0695 W/(cm·K) for compacts at 50 °C, and 0.036 W/(cm·K) for crystals and 0.0432 W/(cm·K) for compacts at 500 °C. [ 3 ] : 60 The linear thermal expansion coefficient is 4.2 × 10 −5 /°C at room temperature. [ 3 ] : 49
LiH is produced by treating lithium metal with hydrogen gas:
This reaction is especially rapid at temperatures above 600 °C. Addition of 0.001–0.003% carbon, and/or increasing temperature/pressure, increases the yield up to 98% at 2-hour residence time. [ 3 ] : 147 However, the reaction proceeds at temperatures as low as 29 °C. The yield is 60% at 99 °C and 85% at 125 °C, and the rate depends significantly on the surface condition of LiH. [ 3 ] : 5
Less common ways of LiH synthesis include thermal decomposition of lithium aluminium hydride (200 °C), lithium borohydride (300 °C), n -butyllithium (150 °C), or ethyllithium (120 °C), as well as several reactions involving lithium compounds of low stability and available hydrogen content. [ 3 ] : 144–145
Chemical reactions yield LiH in the form of lumped powder , which can be compressed into pellets without a binder . More complex shapes can be produced by casting from the melt . [ 3 ] : 160 ff. Large single crystals (about 80 mm long and 16 mm in diameter) can be then grown from molten LiH powder in hydrogen atmosphere by the Bridgman–Stockbarger technique . They often have bluish color owing to the presence of colloidal Li. This color can be removed by post-growth annealing at lower temperatures (~550 °C) and lower thermal gradients. [ 3 ] : 154 Major impurities in these crystals are Na (20–200 ppm ), O (10–100 ppm), Mg (0.5–6 ppm), Fe (0.5-2 ppm) and Cu (0.5-2 ppm). [ 3 ] : 155
Bulk cold-pressed LiH parts can be easily machined using standard techniques and tools to micrometer precision. However, cast LiH is brittle and easily cracks during processing. [ 3 ] : 171
A more energy efficient route to form lithium hydride powder is by ball milling lithium metal under high hydrogen pressure. To prevent cold welding of lithium metal (due to its high ductility ), small amounts of lithium hydride powder are added during this process. [ 7 ]
LiH powder reacts rapidly with air of low humidity , forming LiOH , Li 2 O and Li 2 CO 3 . In moist air the powder ignites spontaneously, forming a mixture of products including some nitrogenous compounds. The lump material reacts with humid air, forming a superficial coating, which is a viscous fluid. This inhibits further reaction, although the appearance of a film of "tarnish" is quite evident. Little or no nitride is formed on exposure to humid air. The lump material, contained in a metal dish, may be heated in air to slightly below 200 °C without igniting, although it ignites readily when touched by an open flame. The surface condition of LiH, presence of oxides on the metal dish, etc., have a considerable effect on the ignition temperature. Dry oxygen does not react with crystalline LiH unless heated strongly, when an almost explosive combustion occurs. [ 3 ] : 6
LiH is highly reactive towards water and other protic reagents: [ 3 ] : 7
LiH is less reactive with water than Li and thus is a much less powerful reducing agent for water, alcohols , and other media containing reducible solutes . This is true for all the binary saline hydrides . [ 3 ] : 22
LiH pellets slowly expand in moist air, forming LiOH ; however, the expansion rate is below 10% within 24 hours in a pressure of 2 Torr of water vapor. [ 3 ] : 7 If moist air contains carbon dioxide , then the product is lithium carbonate . [ 3 ] : 8 LiH reacts with ammonia , slowly at room temperature, but the reaction accelerates significantly above 300 °C. [ 3 ] : 10 LiH reacts slowly with higher alcohols and phenols , but vigorously with lower alcohols. [ 3 ] : 14
LiH reacts with sulfur dioxide to give the dithionite :
though above 50 °C the product is lithium sulfide instead. [ 3 ] : 9
LiH reacts with acetylene to form lithium carbide and hydrogen . With anhydrous organic acids , phenols and acid anhydrides , LiH reacts slowly, producing hydrogen gas and the lithium salt of the acid. With water-containing acids, LiH reacts faster than with water. [ 3 ] : 8 Many reactions of LiH with oxygen-containing species yield LiOH, which in turn irreversibly reacts with LiH at temperatures above 300 °C: [ 3 ] : 10
Lithium hydride is rather unreactive at moderate temperatures with O 2 or Cl 2 . It is, therefore, used in the synthesis of other useful hydrides, [ 8 ] e.g.,
With a hydrogen content in proportion to its mass three times that of NaH, LiH has the highest hydrogen content of any hydride. LiH is periodically of interest for hydrogen storage, but applications have been thwarted by its stability to decomposition. Thus removal of H 2 requires temperatures above the 700 °C used for its synthesis, such temperatures are expensive to create and maintain. The compound was once tested as a fuel component in a model rocket. [ 9 ] [ 10 ]
LiH is not usually a hydride-reducing agent, except in the synthesis of hydrides of certain metalloids. For example, silane is produced in the reaction of lithium hydride and silicon tetrachloride by the Sundermeyer process:
Lithium hydride is used in the production of a variety of reagents for organic synthesis , such as lithium aluminium hydride ( Li[AlH 4 ] ) and lithium borohydride ( Li[BH 4 ] ). Triethylborane reacts to give superhydride ( Li[BH(CH 2 CH 3 ) 3 ] ). [ 11 ]
Lithium hydride (LiH) is sometimes a desirable material for the shielding of nuclear reactors , with the isotope lithium-6 (Li-6), and it can be fabricated by casting. [ 12 ] [ 13 ]
Lithium deuteride, in the form of lithium-7 deuteride ( 7 Li 2 H or 7 LiD), is a good moderator for nuclear reactors , because deuterium ( 2 H or D) has a lower neutron absorption cross-section than ordinary hydrogen or protium ( 1 H) does, and the cross-section for 7 Li is also low, decreasing the absorption of neutrons in a reactor. 7 Li is preferred for a moderator because it has a lower neutron capture cross-section, and it also forms less tritium ( 3 H or T) under bombardment with neutrons. [ 14 ]
The corresponding lithium-6 deuteride ( 6 Li 2 H or 6 LiD) is the primary fusion fuel in thermonuclear weapons . [ citation needed ] In hydrogen warheads of the Teller–Ulam design , a nuclear fission trigger explodes to heat and compress the lithium-6 deuteride, and to bombard the 6 LiD with neutrons to produce tritium in an exothermic reaction:
The deuterium and tritium then fuse to produce helium , one neutron, and 17.59 MeV of free energy in the form of gamma rays , kinetic energy , etc. Tritium has a favorable reaction cross section . The helium is an inert byproduct. [ citation needed ]
Before the Castle Bravo nuclear weapons test in 1954, it was thought that only the less common isotope 6 Li would breed tritium when struck with fast neutrons. The Castle Bravo test showed (accidentally) that the more plentiful 7 Li also does so under extreme conditions, albeit by an endothermic reaction.
LiH reacts violently with water to give hydrogen gas and LiOH, which is caustic. Consequently, LiH dust can explode in humid air, or even in dry air due to static electricity. At concentrations of 5–55 mg/m 3 in air the dust is extremely irritating to the mucous membranes and skin and may cause an allergic reaction. Because of the irritation, LiH is normally rejected rather than accumulated by the body. [ 3 ] : 157, 182
Some lithium salts, which can be produced in LiH reactions, are toxic. LiH fire should not be extinguished using carbon dioxide, carbon tetrachloride, or aqueous fire extinguishers; it should be smothered by covering with a metal object or graphite or dolomite powder. Sand is less suitable, as it can explode when mixed with burning LiH, especially if not dry. LiH is normally transported in oil, using containers made of ceramic, certain plastics or steel, and is handled in an atmosphere of dry argon or helium. [ 3 ] : 156 Whilst nitrogen can be used, it will react with lithium at elevated temperatures. [ 3 ] : 157 LiH normally contains some metallic lithium, which corrodes steel or silica containers at elevated temperatures. [ 3 ] : 173–174, 179 | https://en.wikipedia.org/wiki/LiH |
Lithium iodide , or LiI, is a compound of lithium and iodine . When exposed to air , it becomes yellow in color, due to the oxidation of iodide to iodine. [ 2 ] It crystallizes in the NaCl motif . [ 3 ] It can participate in various hydrates . [ 4 ]
Lithium iodide is used as a solid-state electrolyte for high-temperature batteries. It is also the standard electrolyte in artificial pacemakers [ 6 ] due to the long cycle life it enables. [ 7 ] The solid is used as a phosphor for neutron detection. [ 8 ] It is also used, in a complex with Iodine , in the electrolyte of dye-sensitized solar cells .
In organic synthesis , LiI is useful for cleaving C-O bonds. For example, it can be used to convert methyl esters to carboxylic acids : [ 9 ]
Similar reactions apply to epoxides and aziridines .
Lithium iodide was used as a radiocontrast agent for CT scans . Its use was discontinued due to renal toxicity. Inorganic iodine solutions suffered from hyperosmolarity and high viscosities. Current iodinated contrast agents are organoiodine compounds . [ 10 ]
It is also useful in MALDI imaging mass spectrometry of lipids by adding lithium salts to the matrix solution. [ 11 ] | https://en.wikipedia.org/wiki/LiI |
LiMETER stands for l ight- i nducible me mbrane- t ethered peripheral e ndoplasmic r eticulum (ER). LiMETER is an optogenetics tool designed to reversibly label cortical ER [ 1 ] or the apposition between plasma membrane (PM) and endoplasmic reticulum (ER) membranes (termed as ER-PM junctions ).
The ER luminal domain of LiMETER contains a signal peptide and the transmembrane domain derived from STIM1 , [ 2 ] with GFP placed in between as a reporter. STIM1 is an ER-resident calcium sensor protein responsible for sensing calcium changes in internal calcium stores and communicate with ORAI calcium channels in the plasma membrane. [ 3 ] [ 4 ] [ 5 ] [ 6 ] [ 7 ] [ 8 ] The cytoplasmic region of LiMETER contains a flexible linker and a genetically encoded lightswitch LOV2 domain (light oxygen voltage-sensing domain, residues 404–546) derived from Avena sativa phototropin 1, followed by a C-terminal PM-targeting polybasic tail that associates with negative charged phosphoinositides in the inner half of the leaflet of plasma membrane. [ 1 ]
In the dark, the Jα helix docks to the LOV2 domain and cages the polybasic tail to prevent its interaction with negatively charged PM-resident phosphoinositides. Following blue light illumination, photoexcitation generates a covalent adduct between a cysteine residue and the flavin cofactor in LOV2, and subsequently promotes the undocking and unwinding of the Jα helix, thereby exposing the polybasic C-tail to enable translocation of the protein towards PM to form puncta-like structures.
As a result, LiMETER undergoes photo-inducible translocation toward ER–PM junctions to specifically label cER. This process can be reversibly repeated with multiple light–dark cycles without significant loss in the magnitude of response. [ 1 ]
This optical tool enables cell biologists to quantitatively examine the effect of regulators that modulate the dynamics of cER accumulation at defined spatiotemporal resolution in living cells. | https://en.wikipedia.org/wiki/LiMETER |
Lithiophilite is a mineral containing the element lithium . It is lithium manganese (II) phosphate with chemical formula LiMnPO 4 . It occurs in pegmatites often associated with triphylite, the iron end member in a solid solution series. The mineral with intermediate composition is known as sicklerite and has the chemical formula Li(Mn,Fe)PO 4 ). The name lithiophilite is derived from the Greek philos ( φιλός ) "friend", as lithiophilite is usually found with lithium. [ 3 ]
Lithiophylite is a resinous reddish to yellowish brown mineral crystallizing in the orthorhombic system often as slender prisms. It is usually associated with lepidolite , beryl , quartz , albite , amblygonite , and spodumene of pegmatitic origin. It rather readily weathers to a variety of secondary manganese phosphates and oxides. It is a late-stage mineral in some complex granite pegmatites . [ 4 ] Members of the triphylite-lithiophilite series readily alter to secondary minerals.
The type locality is the Branchville Quarry, Branchville , Fairfield County, Connecticut where it was first reported in 1878. [ 3 ] The largest documented single crystal of lithiophilite was found in New Hampshire , US, measured 2.44×1.83×1.22 m 3 and weighed about 20 tonnes. [ 5 ]
The synthetic form of triphylite, lithium iron phosphate , is a promising material for the production of lithium-ion batteries . | https://en.wikipedia.org/wiki/LiMnPO4 |
Lithium azide is the lithium salt of hydrazoic acid . It is an unstable and toxic compound that decomposes into lithium and nitrogen when heated.
It can be prepared by metathesis reaction between sodium azide and lithium nitrate or lithium sulfate solutions:
It can also be prepared by reacting lithium sulfate with barium azide .
This inorganic compound –related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/LiN3 |
Lithium nitrite is the lithium salt of nitrous acid, with formula LiNO 2 . This compound is hygroscopic and very soluble in water. It is used as a corrosion inhibitor in mortar . [ 4 ] It is also used in the production of explosives , due to its ability to nitrosate ketones under certain conditions. [ 5 ]
Lithium nitrate (LiNO 3 ) will undergo thermal decomposition above 500 °C to yield the evolution of lithium nitrite and oxygen as in the following reaction: [ 6 ]
Lithium nitrite can also be prepared by the reaction of nitric oxide (NO) with lithium hydroxide (LiOH) as shown below: [ 6 ]
Lithium nitrite crystals can be obtained most efficiently by reacting lithium sulfate and barium nitrite in an aqueous solution. However, these crystals can also be prepared by mixing equal amounts of lithium sulfate and potassium nitrite in highly concentrated aqueous solution. This is followed by considerable evaporation and filtration, which removes the resulting precipitate of potassium sulfate and lithium potassium sulfate after further evaporation and extraction with absolute alcohol. [ 7 ]
Lithium nitrite is exceptionally soluble in absolute alcohol. However, potassium nitrite is not very soluble. This makes absolute alcohol a choice solvent for the crystallization of lithium nitrite because the crystals can be extracted in a substantially pure state. The alcoholic solution will leave a white residue of small crystals upon evaporation. The addition of a small amount of water to this residue will yield the larger needle-shaped crystals of lithium nitrite monohydrate (LiNO 2 ·H 2 O). [ 7 ]
The above methods will result in flat, needle-shaped crystals. These crystals are white and typically 1–2 cm. in length. Below 100 °C, these crystals will melt in their own water of crystallization and will tend to lose water slowly. Rapid dehydration will occur at temperatures above 160 °C as well as a minuscule loss of nitrogen oxide. This rapid dehydration leaves behind a residue which consists almost entirely of the anhydrous salt. [ 7 ] This anhydrous salt is extraordinarily soluble in water and will readily form a supersaturated solution. Monohydrate crystals will deposit from this supersaturated solution upon cooling or with the addition of ready formed salt crystals. [ 7 ]
Reinforcement bars, ready mixed concrete materials, and repair materials are often subject to corrosion. These resources will rapidly degrade due to chloride attack and carbonatation . This not only affects the service lives of such materials, but it also requires a considerable cost for the repair of such defects. Lithium nitrite and calcium nitrite are generally used in the construction industry as a means to protect reinforced concrete structures from corrosion. Unlike calcium nitrite inhibitors, lithium nitrite is particularly valued for corrosion inhibition and resistance of carbonation when an accelerated hardening process is not used and when a high concentration of 10% or more cement is added by weight. [ 4 ]
Generally speaking, studying the effectiveness of such inhibitors has been done using destructive methods. These studies require placing specimens to accelerated corrosion and measuring the degree of corrosion. "However, it is extremely difficult to measure the effect of corrosion inhibitors in actual structures using a destructive method."
Recently, sensors that can measure changes in electrical resistance due to the corrosion in iron and thus indicate the degree of corrosion of a material have been developed. These sensors provide a non-destructive way to evaluate the degree of corrosion in concrete materials. Therefore, the effect of lithium nitrite as a corrosion inhibitor has also been studied by non-destructive means. [ 4 ]
A study was conducted in Korea to experimentally determine the most effective dose and performance of lithium nitrite corrosion inhibitors. This experiment employed the molar ratio of nitrite ions to chloride ions (NO 2 − /Cl − ) as a test parameter. This study concluded that a lithium nitrite dosage of 0.6 in the nitrite-chloride ion molar ratio is a successful dosage for mortar containing chlorides. [ 4 ] | https://en.wikipedia.org/wiki/LiNO2 |
Lithium nitrate is an inorganic compound with the formula LiNO 3 . It is the lithium salt of nitric acid (an alkali metal nitrate ). The salt is deliquescent , absorbing water to form the hydrated form, lithium nitrate trihydrate. Its eutectics are of interest for heat transfer fluids. [ 2 ]
It is made by treating lithium carbonate or lithium hydroxide with nitric acid .
This deliquescent colourless salt is an oxidizing agent used in the manufacture of red-colored fireworks and flares .
The hydrated form, lithium nitrate trihydrate, has an extremely high specific heat of fusion , 287 ± 7 J/g , [ 3 ] and hence can be used for thermal energy storage at its melt temperature of 303.3 K. [ 4 ]
Lithium nitrate has been proposed as a medium to store heat collected from the sun for cooking. A Fresnel lens would be used to melt solid lithium nitrate, which would then function as a "solar battery", allowing heat to be redistributed later by convection. [ 5 ]
Lithium nitrate can be synthesized by reacting nitric acid and lithium carbonate.
Generally when forming LiNO 3 , a pH indicator is used to determine when all of the acid has been neutralized. However, this neutralization can also be recognized with the loss of carbon dioxide production. [ 6 ] In order to rid the final product of excess water, the sample is heated.
Lithium nitrate can be toxic to the body when ingested by targeting the central nervous system, thyroids, kidneys, and cardio-vascular system. [ 7 ] When exposed to the skin, eyes, and mucous membranes, lithium nitrate can cause irritation to these areas. [ 8 ] | https://en.wikipedia.org/wiki/LiNO3 |
Lithium niobate ( Li Nb O 3 ) is a synthetic salt consisting of niobium , lithium , and oxygen . Its single crystals are an important material for optical waveguides, mobile phones, piezoelectric sensors, optical modulators and various other linear and non-linear optical applications. [ 6 ] Lithium niobate is sometimes referred to by the brand name linobate . [ 7 ]
Lithium niobate is a colorless solid, and it is insoluble in water. It has a trigonal crystal system , which lacks inversion symmetry and displays ferroelectricity , the Pockels effect , the piezoelectric effect, photoelasticity and nonlinear optical polarizability. Lithium niobate has negative uniaxial birefringence which depends slightly on the stoichiometry of the crystal and on temperature. It is transparent for wavelengths between 350 and 5200 nanometers .
Lithium niobate can be doped with magnesium oxide , which increases its resistance to optical damage (also known as photorefractive damage). Other available dopants are iron , zinc , hafnium , copper , gadolinium , erbium , yttrium , manganese and boron .
Single crystals of lithium niobate can be grown using the Czochralski process . [ 8 ]
After a crystal is grown, it is sliced into wafers of different orientation. Common orientations are Z-cut, X-cut, Y-cut, and cuts with rotated angles of the previous axes. [ 9 ]
Thin-film lithium niobate (e.g. for optical wave guides ) can be transferred to or grown on sapphire and other substrates, using the smart cut (ion slicing) process [ 10 ] [ 11 ] or MOCVD process. [ 12 ] The technology is known as lithium niobate on insulator (LNOI). [ 13 ]
Nanoparticles of lithium niobate and niobium pentoxide can be produced at low temperature. [ 14 ] The complete protocol implies a LiH induced reduction of NbCl 5 followed by in situ spontaneous oxidation into low-valence niobium nano-oxides. These niobium oxides are exposed to air atmosphere resulting in pure Nb 2 O 5 . Finally, the stable Nb 2 O 5 is converted into lithium niobate LiNbO 3 nanoparticles during the controlled hydrolysis of the LiH excess. [ 15 ] Spherical nanoparticles of lithium niobate with a diameter of approximately 10 nm can be prepared by impregnating a mesoporous silica matrix with a mixture of an aqueous solution of LiNO 3 and NH 4 NbO(C 2 O 4 ) 2 followed by 10 min heating in an infrared furnace. [ 16 ]
Lithium niobate is used extensively in the telecommunications market, e.g. in mobile telephones and optical modulators . [ 17 ] Due to its large electro-mechanical coupling, it is the material of choice for surface acoustic wave (SAW) devices. [ 18 ] For some uses it can be replaced by lithium tantalate ( LiTaO 3 ) . Other uses are in laser frequency doubling , nonlinear optics , Pockels cells , optical parametric oscillators , Q-switching devices for lasers, other acousto-optic devices, optical switches for gigahertz frequencies, etc. It is an excellent material for manufacture of optical waveguides . It's also used in the making of optical spatial low-pass ( anti-aliasing ) filters. Additionally, it is used in pyroelectric infrared (IR) detectors, where it detects temperature changes by generating electric charges. [ 19 ]
In the past few years lithium niobate is finding applications as a kind of electrostatic tweezers, an approach known as optoelectronic tweezers as the effect requires light excitation to take place. [ 20 ] [ 21 ] This effect allows for fine manipulation of micrometer-scale particles with high flexibility since the tweezing action is constrained to the illuminated area. The effect is based on the very high electric fields generated during light exposure (1–100 kV/cm) within the illuminated spot. These intense fields are also finding applications in biophysics and biotechnology, as they can influence living organisms in a variety of ways. [ 22 ] For example, iron-doped lithium niobate excited with visible light has been shown to produce cell death in tumoral cell cultures. [ 23 ]
Periodically poled lithium niobate ( PPLN ) is a domain-engineered lithium niobate crystal, used mainly for achieving quasi-phase-matching in nonlinear optics . The ferroelectric domains point alternatively to the +c and the −c direction, with a period of typically between 5 and 35 μm . The shorter periods of this range are used for second-harmonic generation , while the longer ones for optical parametric oscillation . Periodic poling can be achieved by electrical poling with periodically structured electrode. Controlled heating of the crystal can be used to fine-tune phase matching in the medium due to a slight variation of the dispersion with temperature.
Periodic poling uses the largest value of lithium niobate's nonlinear tensor, d 33 = 27 pm/V. Quasi-phase-matching gives maximum efficiencies that are 2/π (64%) of the full d 33 , about 17 pm/V. [ 24 ]
Other materials used for periodic poling are wide- band-gap inorganic crystals like KTP (resulting in periodically poled KTP , PPKTP ), lithium tantalate , and some organic materials.
The periodic-poling technique can also be used to form surface nanostructures . [ 25 ] [ 26 ]
However, due to its low photorefractive damage threshold, PPLN only finds limited applications, namely, at very low power levels. MgO-doped lithium niobate is fabricated by periodically poled method. Periodically poled MgO-doped lithium niobate (PPMgOLN) therefore expands the application to medium power level.
The Sellmeier equations for the extraordinary index are used to find the poling period and approximate temperature for quasi-phase-matching. Jundt [ 27 ] gives
valid from 20 to 250 °C for wavelengths from 0.4 to 5 micrometers , whereas for longer wavelengths, [ 28 ]
which is valid for T = 25 to 180 °C, for wavelengths λ between 2.8 and 4.8 micrometers.
In these equations f = ( T − 24.5)( T + 570.82), λ is in micrometers, and T is in °C.
More generally for ordinary and extraordinary index for MgO-doped LiNbO 3 :
with:
for congruent LiNbO 3 (CLN) and stochiometric LiNbO 3 (SLN). [ 29 ] | https://en.wikipedia.org/wiki/LiNbO3 |
Lithium superoxide is an unstable inorganic salt with formula Li O 2 . A radical compound, it can be produced at low temperature in matrix isolation experiments, or in certain nonpolar , non-protic solvents . Lithium superoxide is also a transient species during the reduction of oxygen in a lithium–air galvanic cell , and serves as a main constraint on possible solvents for such a battery. For this reason, it has been investigated thoroughly using a variety of methods, both theoretical and spectroscopic.
The LiO 2 molecule is a misnomer: the bonds between lithium and oxygen are highly ionic , with almost complete electron-transfer. [ 1 ] The force constant between the two oxygen atoms matches the constants measured for the superoxide anion ( O − 2 ) in other contexts. The bond length for the O-O bond was determined to be 1.34 Å . Using a simple crystal structure optimization, the Li-O bond was calculated to be approximately 2.10 Å. [ 2 ]
There have been quite a few studies regarding the clusters formed by LiO 2 molecules. The most common dimer has been found to be the cage isomer. Second to it is the singlet bypyramidal structure. Studies have also been done on the chair complex and the planar ring, but these two are less favorable, though not necessarily impossible. [ 3 ]
Lithium superoxide is extremely reactive because of the odd number of electrons present in the π* molecular orbital of the superoxide anion. [ 4 ] Matrix isolation techniques can produce pure samples of the compound, but they are only stable at 15-40 K . [ 3 ]
At higher (but still cryogenic) temperatures, lithium superoxide can be produced by ozonating lithium peroxide ( Li 2 O 2 ) in freon 12 :
The resulting product is only stable up to −35 °C. [ 5 ]
Alternatively, lithium electride dissolved in anhydrous ammonia will reduce oxygen gas to yield the same product:
Lithium superoxide is, however, only metastable in ammonia, gradually oxidizing the solvent to water and nitrogen gas:
Unlike other known decompositions of LiO 2 , this reaction bypasses lithium peroxide. [ 6 ]
Like other superoxides, lithium superoxide is the product of a one-electron reduction of an oxygen molecule . It thus appears whenever oxygen is mixed with single-electron redox catalysts , such as p -benzoquinone . [ 7 ]
Lithium superoxide also appears at the cathode of a lithium-air galvanic cell during discharge, as in the following reaction: [ 8 ]
This product typically then reacts and proceed to form lithium peroxide , Li 2 O 2
The mechanism for this last reaction has not been confirmed and developing a complete theory of the oxygen reduction process remains a theoretical challenge as of 2022 [update] . [ 9 ] Indeed, recent work suggests that LiO 2 can be stabilized via a suitable cathode made of graphene with iridium nanoparticles. [ 10 ]
A significant challenge when investigating these batteries is finding an ideal solvent in which to perform these reactions; current candidates are ether - and amide -based, but these compounds readily react with the superoxide and decompose. [ 9 ] Nevertheless, lithium-air cells remain the focus of intense research, because of their large energy density —comparable to the internal combustion engine. [ 8 ]
Lithium superoxide can also form for extended periods of time in low-density, high-energy environments, such as the upper atmosphere. The mesosphere contains a persistent layer of alkali metal cations ablated from meteors . For sodium and potassium , many of the ions bond to form particles of the corresponding superoxide. It is currently unclear whether lithium should react analogously. [ 11 ] | https://en.wikipedia.org/wiki/LiO2 |
Lithium hydroxide is an inorganic compound with the formula LiOH. It can exist as anhydrous or hydrated, and both forms are white hygroscopic solids. They are soluble in water and slightly soluble in ethanol . Both are available commercially. While classified as a strong base , lithium hydroxide is the weakest known alkali metal hydroxide.
The preferred feedstock is hard-rock spodumene , where the lithium content is expressed as % lithium oxide .
Lithium hydroxide is often produced industrially from lithium carbonate in a metathesis reaction with calcium hydroxide : [ 7 ]
The initially produced hydrate is dehydrated by heating under vacuum up to 180 °C.
An alternative route involves the intermediacy of lithium sulfate : [ 8 ] [ 9 ]
The main by-products are gypsum and sodium sulphate , which have some market value.
According to Bloomberg, Ganfeng Lithium Co. Ltd. [ 10 ] (GFL or Ganfeng) [ 11 ] and Albemarle were the largest producers in 2020 with around 25kt/y, followed by Livent Corporation (FMC) and SQM . [ 10 ] Significant new capacity is planned, to keep pace with demand driven by vehicle electrification. Ganfeng are to expand lithium chemical capacity to 85,000 tons, adding the capacity leased from Jiangte, Ganfeng will become the largest lithium hydroxide producer globally in 2021. [ 10 ]
Albemarle's Kemerton, Western Australia plant, originally planned to deliver 100kt/y has been scaled back to 50kt/y. [ 12 ]
In 2020 Tianqi Lithium's , plant in Kwinana, Western Australia was the largest producer, with a capacity of 48kt/y. [ 13 ]
Lithium hydroxide is mainly consumed in the production of cathode materials for lithium-ion batteries such as lithium cobalt oxide ( LiCoO 2 ) and lithium iron phosphate . It is preferred over lithium carbonate as a precursor for lithium nickel manganese cobalt oxides . [ 14 ]
A popular lithium grease thickener is lithium 12-hydroxystearate , which produces a general-purpose lubricating grease due to its high resistance to water and usefulness at a range of temperatures.
Lithium hydroxide is used in breathing gas purification systems for spacecraft , submarines , and rebreathers to remove carbon dioxide from exhaled gas by producing lithium carbonate and water: [ 15 ]
or
The latter, anhydrous hydroxide, is preferred for its lower mass and lesser water production for respirator systems in spacecraft. One gram of anhydrous lithium hydroxide can remove 450 cm 3 of carbon dioxide gas. The monohydrate loses its water at 100–110 °C.
Lithium hydroxide, together with lithium carbonate , is a key intermediates used for the production of other lithium compounds, illustrated by its use in the production of lithium fluoride : [ 7 ]
It is also used in ceramics and some Portland cement formulations, where it is also used to suppress ASR ( concrete cancer ). [ 16 ]
Lithium hydroxide ( isotopically enriched in lithium-7 ) is used to alkalize the reactor coolant in pressurized water reactors for corrosion control. [ 17 ] It is good radiation protection against free neutrons.
In 2012, the price of lithium hydroxide was about US$5–6/kg. [ 18 ]
In December 2020, it had risen to $9/kg [ 19 ]
On 18 March 2021, the price had risen to $11.50/kg [ 20 ] | https://en.wikipedia.org/wiki/LiOH |
Lithium hexafluorophosphate is an inorganic compound with the formula Li PF 6 . It is a white crystalline powder.
LiPF 6 is manufactured by reacting phosphorus pentachloride with hydrogen fluoride and lithium fluoride [ 1 ] [ 2 ]
Suppliers include Targray and Morita Chemical Industries Co., Ltd.
The salt is relatively stable thermally, but loses 50% weight at 200 °C (392 °F). It hydrolyzes near 70 °C (158 °F) [ 3 ] according to the following equation forming highly toxic HF gas:
Owing to the Lewis acidity of the Li + ions, LiPF 6 also catalyses the tetrahydropyranylation of tertiary alcohols . [ 4 ]
In lithium-ion batteries , LiPF 6 reacts with Li 2 CO 3 , which may be catalysed by small amounts of HF: [ 5 ]
The main use of LiPF 6 is in commercial secondary batteries, an application that exploits its high solubility in polar aprotic solvents . Specifically, solutions of lithium hexafluorophosphate in carbonate blends of ethylene carbonate , dimethyl carbonate , diethyl carbonate and/or ethyl methyl carbonate, with a small amount of one or many additives such as fluoroethylene carbonate and vinylene carbonate , serve as state-of-the-art electrolytes in lithium-ion batteries . [ 6 ] [ 7 ] [ 8 ] This application takes advantage of the inertness of the hexafluorophosphate anion toward strong reducing agents, such as lithium metal, as well as of the ability of [PF6-] to passivate the positive aluminium current collector. [ 9 ] | https://en.wikipedia.org/wiki/LiPF6 |
LiSiCA ( Li gand Si milarity using C lique A lgorithm) is a ligand-based virtual screening software that searches for 2D and 3D similarities between a reference compound and a database of target compounds which should be represented in a Mol2 format. The similarities are expressed using the Tanimoto coefficients and the target compounds are ranked accordingly. LiSiCA is also available as LiSiCA PyMOL plugin both on Linux and Windows operating systems.
As an input LiSiCA requires at least one reference compound and database of target compounds. For 3D screening this database has to be a pregenerated database of conformations of target and for 2D screening a topology, that is, a list of atoms and bonds, for each target compound. On each step the algorithm compares reference compound to one of the compounds from target compounds based on their 2D or 3D representation. Both compounds(molecules) are converted to molecular graphs. In 2D and 3D screening the molecular graph vertices represent atoms. In 2D screening edges of molecular graph represent covalent bonds while in 3D screening edges are drawn between every pair of vertices and have no chemical meaning. A product graph generated from molecular graphs is then searched using fast maximum clique algorithm [ 1 ] [ 2 ] to find the largest substructure common to both compounds. The similarity between compounds is calculated using Tanimoto coefficients and target compounds are ranked according to their Tanimoto coefficients.
LiSiCA can search 2D and 3D similarities between a reference compound and a database of target compounds. It takes as an input at least one reference compound and a database of target compounds. By default it returns only the compound most similar to the reference compound out of all compounds in database of target compounds.
Other optional parameters LiSiCA uses are:
In addition LiSiCA PyMOL plugin also offers to load saved results.
The Slovene word lisica means 'fox', which is why the logo of LiSiCA software is a fox holding two molecules. | https://en.wikipedia.org/wiki/LiSiCA |
Lithium tantalate is the inorganic compound with the formula Li Ta O 3 . It is a white, diamagnetic , water-insoluble solid. The compound has the perovskite structure . It has optical , piezoelectric , and pyroelectric properties. Considerable information is available from commercial sources about this material. [ 3 ]
Lithium tantalate is produced by treating tantalum(V) oxide with lithium oxide. The use of excess alkali gives water-soluble polyoxotantalates. Single crystals of Lithium tantalate are pulled from the melt using the Czochralski method . [ 3 ]
Lithium tantalate is used for nonlinear optics , passive infrared sensors such as motion detectors , terahertz generation and detection, surface acoustic wave applications, cell phones.
Lithium tantalate is a standard detector element in infrared spectrophotometers . [ 4 ]
The phenomenon of pyroelectric fusion has been demonstrated using a lithium tantalate crystal producing a large enough charge to generate and accelerate a beam of deuterium nuclei into a deuterated target resulting in the production of a small flux of helium-3 and neutrons through nuclear fusion without extreme heat or pressure. [ 5 ]
A difference between positively and negatively charged parts of pyroelectric LiTaO 3 crystals was observed when water freezes to them. [ 6 ]
This inorganic compound –related article is a stub . You can help Wikipedia by expanding it .
This crystallography -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/LiTaO3 |
In mathematics , in combinatorics , the Li Shanlan identity (also called Li Shanlan's summation formula ) is a certain combinatorial identity attributed to the nineteenth century Chinese mathematician Li Shanlan . [ 1 ] Since Li Shanlan is also known as Li Renshu (his courtesy name ), this identity is also referred to as the Li Renshu identity . [ 2 ] This identity appears in the third chapter of Duoji bilei (垛积比类 / 垛積比類, meaning summing finite series ), a mathematical text authored by Li Shanlan and published in 1867 as part of his collected works. A Czech mathematician Josef Kaucky published an elementary proof of the identity along with a history of the identity in 1964. [ 3 ] Kaucky attributed the identity to a certain Li Jen-Shu. From the account of the history of the identity, it has been ascertained that Li Jen-Shu is in fact Li Shanlan. [ 1 ] Western scholars had been studying Chinese mathematics for its historical value; but the attribution of this identity to a nineteenth century Chinese mathematician sparked a rethink on the mathematical value of the writings of Chinese mathematicians. [ 2 ]
The Li Shanlan identity states that
Li Shanlan did not present the identity in this way. He presented it in the traditional Chinese algorithmic and rhetorical way. [ 4 ]
Li Shanlan had not given a proof of the identity in Duoji bilei . The first proof using differential equations and Legendre polynomials, concepts foreign to Li, was published by Pál Turán in 1936, and the proof appeared in Chinese in Yung Chang 's paper published in 1939. [ 2 ] Since then at least fifteen different proofs have been found. [ 2 ] The following is one of the simplest proofs. [ 5 ]
The proof begins by expressing ( n q ) {\displaystyle n \choose q} as Vandermonde's convolution :
Pre-multiplying both sides by ( n p ) {\displaystyle n \choose p} ,
Using the following relation
the above relation can be transformed to
Next the relation
is used to get
Another application of Vandermonde's convolution yields
and hence
Since ( p j ) {\displaystyle p \choose j} is independent of k , this can be put in the form
Next, the result
gives
Setting p = q and replacing j by k ,
Li's identity follows from this by replacing n by n + p and doing some rearrangement of terms in the resulting expression:
The term duoji denotes a certain traditional Chinese method of computing sums of piles. Most of the mathematics that was developed in China since the sixteenth century is related to the duoji method. Li Shanlan was one of the greatest exponents of this method and Duoji bilei is an exposition of his work related to this method. Duoji bilei consists of four chapters: Chapter 1 deals with triangular piles, Chapter 2 with finite power series, Chapter 3 with triangular self-multiplying piles and Chapter 4 with modified triangular piles. [ 6 ] | https://en.wikipedia.org/wiki/Li_Shanlan_identity |
A liana is a long- stemmed woody vine that is rooted in the soil at ground level and uses trees, as well as other means of vertical support, to climb up to the canopy in search of direct sunlight. [ 1 ] The word liana does not refer to a taxonomic grouping, but rather a habit of plant growth – much like tree or shrub . It comes from standard French liane , itself from an Antilles French dialect word meaning to sheave . [ citation needed ]
Lianas are characteristic of tropical moist broadleaf forests (especially seasonal forests ), but may be found in temperate rainforests and temperate deciduous forests. There are also temperate lianas, for example the members of the Clematis or Vitis (wild grape) genera. Lianas can form bridges in the forest canopy, providing arboreal animals — including ants and many other invertebrates, lizards, rodents, sloths, monkeys, and lemurs — with paths through the forest. For example, in the Eastern tropical forests of Madagascar many lemurs achieve higher mobility from the web of lianas draped among the vertical tree species. Many lemurs prefer trees with lianas because of their roots. [ 2 ]
Lianas do not derive nutrients directly from trees, but live on and indirectly derive nutrients at the expense of trees. [ 3 ] [ 4 ] Specifically, they greatly reduce tree growth [ 5 ] and tree reproduction, [ 6 ] greatly increase tree mortality, [ 7 ] prevent tree seedlings from establishing, [ 5 ] alter the course of regeneration in forests, [ 8 ] and ultimately decrease tree population growth rates. [ 9 ] For example, forests without lianas grow 150% more fruit, and trees with lianas have twice the probability of dying. [ 10 ]
Lianas are uniquely adapted to living in forests as they use host trees, for stability, to reach to top of the canopy. Lianas directly damage their hosts by mechanical abrasion and strangulation, render hosts more susceptible to ice and wind damage, [ citation needed ] and increase the probability that the host tree falls. [ citation needed ] Lianas also provide support for weaker trees when strong winds blow by laterally anchoring them to stronger trees. [ 11 ] However, this anchoring can also be destructive because when one tree falls, the connections made by the lianas can cause many other trees to fall. [ 11 ] Because of these negative effects, trees that remain free of lianas are at an advantage; some species have evolved characteristics which help them avoid or shed lianas. [ 12 ]
Some lianas attain great length, such as Bauhinia sp. in Surinam which has grown as long as 600 meters (2000'). [ 13 ] [ 14 ] Hawkins has accepted a length of 1.5 km (1 mile) for an Entada phaseoloides . [ 15 ] The longest monocot liana is Calamus manan (or Calamus ornatus ) at 240 meters (787'). [ 16 ] Dr. Francis E. Putz states that lianas (species not indicated) have weighed "hundreds of tons" and been a half mile (0.8 km) in length. [ 17 ] One way of distinguishing lianas from trees and shrubs is their stiffness , specifically, the Young's modulus of various parts of the stem. Trees and shrubs have young twigs and smaller branches that are quite flexible and older growth such as trunks and large branches that are stiffer. A liana often has stiff young growths and older, more flexible growth at the base of the stem. [ 18 ]
Some families and genera containing liana species include:
List of Longest Vines | https://en.wikipedia.org/wiki/Liana |
The liberal paradox , also Sen paradox or Sen's paradox , is a logical paradox proposed by Amartya Sen which shows that no means of aggregating individual preferences into a single, social choice, can simultaneously fulfill the following, seemingly mild conditions:
Sen's result shows that this is impossible. The three, rather minimalistic, assumptions cannot all hold together. The paradox—more properly called a proof of contradiction, and a paradox only in the sense of informal logic—is contentious because it appears to contradict the classical liberal idea that markets are both Pareto-efficient and respect individual freedoms . [ 1 ] [ 2 ] [ 3 ]
Sen's proof, set in the context of social choice theory, is similar in many respects to Arrow's impossibility theorem and the Gibbard–Satterthwaite theorem . As a mathematical construct, it also has much wider applicability: it is essentially about cyclical majorities between partially ordered sets, of which at least three must participate in order to give rise to the phenomenon. Since the idea is about pure mathematics and logic, similar arguments abound much further afield. They, for example, lead to the necessity of the fifth normal form in relational database design. The history of the argument also goes deeper, Condorcet's paradox perhaps being the first example of the finite sort.
A particular distribution of goods or outcome of any social process is regarded as Pareto-efficient if there is no way to improve one or more people's situations without harming another. Put another way, an outcome is not Pareto-efficient if there is a way to improve at least one person's situation without harming anyone else.
For example, suppose a mother has ten dollars which she intends to give to her two children Carlos and Shannon. Suppose the children each want only money, and they do not get jealous of one another. The following distributions are Pareto-efficient:
However, a distribution where the mother gives each of them $2 and wastes the remaining $6 is not Pareto-efficient, because she could have given the wasted money to either child and made that child better off without harming the other.
In this example, it was presumed that a child was made better or worse off by gaining or losing money, respectively, and that neither child gained or lost by evaluating her share in comparison to the other. To be more precise, we must evaluate all possible preferences that the child might have and consider a situation as Pareto-efficient if there is no other social state that at least one person favors (or prefers) and no one disfavors.
Pareto efficiency is often used in economics as a minimal sense of economic efficiency . If a mechanism does not result in Pareto-efficient outcomes, it is regarded as inefficient, since there was another outcome that could have made some people better off without harming anyone else.
The view that markets produce Pareto-efficient outcomes is regarded as an important and central justification for capitalism . This result was established (with certain assumptions) in an area of study known as general equilibrium theory and is known as the first fundamental theorem of welfare economics . As a result, these results often feature prominently in conservative libertarian justifications of unregulated markets.
Sen's original example [ 4 ] used a simple society with only two people and only one social issue to consider. The two members of society are named "Lewd" and "Prude". In this society there is a copy of a Lady Chatterley's Lover and it must be given either to Lewd to read, to Prude to read, or disposed of - unread. Suppose that Lewd enjoys this sort of reading and would prefer to read it rather than have it disposed of. However, they would get even more enjoyment out of Prude being forced to read it.
Prude thinks that the book is indecent and that it should be disposed of, unread. However, if someone must read it, Prude would prefer to read it rather than Lewd since Prude thinks it would be even worse for someone to read and enjoy the book rather than read it in disgust.
Given these preferences of the two individuals in the society, a social planner must decide what to do. Should the planner force Lewd to read the book, force Prude to read the book or let it go unread? More particularly, the social planner must rank all three possible outcomes in terms of their social desirability. The social planner decides that they should be committed to individual rights, each individual should get to choose whether they, themself will read the book. Lewd should get to decide whether the outcome "Lewd reads" will be ranked higher than "No one reads", and similarly Prude should get to decide whether the outcome "Prude reads" will be ranked higher than "No one reads".
Following this strategy, the social planner declares that the outcome "Lewd reads" will be ranked higher than "No one reads" (because of Lewd's preferences) and that "No one reads" will be ranked higher than "Prude reads" (because of Prude's preferences). Consistency then requires that "Lewd reads" be ranked higher than "Prude reads", and so the social planner gives the book to Lewd to read.
Notice that this outcome is regarded as worse than "Prude reads" by both Prude and Lewd, and the chosen outcome is therefore Pareto inferior to another available outcome—the one where Prude is forced to read the book.
Another example was provided by philosopher Allan Gibbard . [ 5 ] Suppose there are two individuals Alice and Bob who live next door to each other. Alice loves the color blue and hates red. Bob loves the color green and hates yellow. If each were free to choose the color of their house independently of the other, they would choose their favorite colors. But Alice hates Bob with a passion, and she would gladly endure a red house if it meant that Bob would have to endure his house being yellow. Bob similarly hates Alice, and would gladly endure a yellow house if that meant that Alice would live in a red house.
If each individual is free to choose their own house color, independently of the other, Alice would choose a blue house and Bob would choose a green one. But, this outcome is not Pareto efficient, because both Alice and Bob would prefer the outcome where Alice's house is red and Bob's is yellow. As a result, giving each individual the freedom to choose their own house color has led to an inefficient outcome—one that is inferior to another outcome where neither is free to choose their own color.
Mathematically, we can represent Alice's preferences with this symbol: ≻ A {\displaystyle \succ _{A}} and Bob's preferences with this one: ≻ B {\displaystyle \succ _{B}} . We can represent each outcome as a pair: ( Color of Alice's house , Color of Bob's house ). As stated Alice's preferences are:
And Bob's are:
If we allow free and independent choices of both parties we end up with the outcome (Blue, Green) which is dispreferred by both parties to the outcome (Red, Yellow) and is therefore not Pareto efficient.
Suppose there is a society N consisting of two or more individuals and a set X of two or more social outcomes. (For example, in the Alice and Bob case, N consisted of Alice and Bob, and X consisted of the four color options ⟨Blue, Yellow⟩, ⟨Blue, Green⟩, ⟨Red, Yellow⟩, and ⟨Red, Green⟩.)
Suppose each individual in the society has a total and transitive preference relation on the set of social outcomes X . For notation, the preference relation of an individual i ∊ N is denoted by ≼ i . Each preference relation belongs to the set Rel(X) of all total and transitive relations on X .
A social choice function is a map which can take any configuration of preference relations of N as input and produce a subset of ("chosen") social outcomes as output. Formally, a social choice function is a map
F : R e l ( X ) N → P ( X ) {\displaystyle F:{Rel}(X)^{N}\rightarrow {\mathcal {P}}(X)}
from the set of functions between N → Rel(X) , to the power set of X . (Intuitively, the social choice function represents a societal principle for choosing one or more social outcomes based on individuals' preferences. By representing the social choice process as a function on Rel(X) N , we are tacitly assuming that the social choice function is defined for any possible configuration of preference relations; this is sometimes called the Universal Domain assumption.)
The liberal paradox states that every social choice function satisfies at most one of the following properties, never both:
In other words, the liberal paradox states that for every social choice function F , there is a configuration of preference relations p ∊ Rel(X) N for which F violates either Pareto optimality or Minimal liberalism (or both). In the examples of Sen and Gibbard noted above, the social choice function satisfies minimal liberalism at the expense of Pareto optimality.
Because the paradox relies on very few conditions, there are a limited number of ways to escape the paradox. Essentially one must either reject the universal domain assumption, the Pareto principle , or the minimal liberalism principle . Sen himself suggested two ways out, one a rejection of universal domain another a rejection of the Pareto principle.
Julian Blau proves that Sen's paradox can only arise when individuals have "nosy" preferences—that is when their preference depends not only on their own action but also on others' actions. [ 6 ] In the example of Alice and Bob above, Alice has a preference over how Bob paints his house, and Bob has a preference over Alice's house color as well.
Most arguments which demonstrate market efficiency assume that individuals care about only their own consumption and not others' consumption and therefore do not consider the situations that give rise to Sen's paradox. In fact, this shows a strong relationship between Sen's paradox and the well known result that markets fail to produce Pareto outcomes in the presence of externalities . [ 7 ] Externalities arise when the choices of one party affect another. Classic examples of externalities include pollution or overfishing . Because of their nosy preferences, Alice's choice imposes a negative externality on Bob and vice versa.
To prevent the paradox, Sen suggests that "The ultimate guarantee for individual liberty may rest not on rules for social choice but on developing individual values that respect each other's personal choices." [ 4 ] Doing so would amount to limiting certain types of nosy preferences, or alternatively restricting the application of the Pareto principle only to those situations where individuals fail to have nosy preferences.
Note that if we consider the case of cardinal preferences—for instance, if Alice and Bob both had to state, within certain bounds, how much happiness they would get for each color of each house separately, and the situation which produced the most happiness were chosen—a minimally-liberal solution does not require that they have no nosiness at all, but just that the sum of all "nosy" preferences about one house's color are below some threshold, while the "non-nosy" preferences are all above that threshold. Since there are generally some questions for which this will be true—Sen's classic example is an individual's choice of whether to sleep on their back or their side—the goal of combining minimal liberalism with Pareto efficiency, while impossible to guarantee in all theoretical cases, may not in practice be impossible to obtain.
Alternatively, one could remain committed to the universality of the rules for social choice and to individual rights and instead reject the universal application of the Pareto principle. Sen also hints that this should be how one escapes the paradox:
What is the moral? It is that in a very basic sense liberal values conflict with the Pareto principle. If someone takes the Pareto principle seriously, as economists seem to do, then he has to face problems of consistency in cherishing liberal values, even very mild ones. Or, to look at it in another way, if someone does have certain liberal values, then he may have to eschew his adherence to Pareto optimality. While the Pareto criterion has been thought to be an expression of individual liberty, it appears that in choices involving more than two alternatives it can have consequences that are, in fact, deeply illiberal. [ 4 ]
Most commentators on Sen's paradox have argued that Sen's minimal liberalism condition does not adequately capture the notion of individual rights. [ 5 ] [ 8 ] [ 9 ] [ 10 ] Essentially what is excluded from Sen's characterization of individual rights is the ability to voluntarily form contracts that lay down one's claim to a right.
For example, in the example of Lewd and Prude, although each has a right to refuse to read the book, Prude would voluntarily sign a contract with Lewd promising to read the book on condition that Lewd refrain from doing so. In such a circumstance there was no violation of Prude's or Lewd's rights because each entered the contract willingly. Similarly, Alice and Bob might sign a contract to each paint their houses their dispreferred color on condition that the other does the same.
In this vein, Gibbard provides a weaker version of the minimal liberalism claim which he argues is consistent with the possibility of contracts and which is also consistent with the Pareto principle given any possible preferences of the individuals. [ 5 ]
Alternatively, instead of both Lewd and Prude deciding what to do at the same time, they should do it one after the other. If Prude decides not to read, then Lewd will decide to read. This yields the same outcome. However, if Prude decides to read, Lewd won't. "Prude reads" is preferred by Prude (and also Lewd) to "Lewd reads", so he will decide to read (with no obligation, voluntarily) to get this Pareto efficient outcome. Marc Masat hints that this should be another way out of the paradox:
If there's, at least, one player without dominant strategy, the game will be played sequentially where players with dominant strategy and need to change it (if they are in the Pareto optimal they don't have to) will be the firsts to choose, allowing to reach the Pareto Efficiency without dictatorship nor restricted domain and also avoiding contract's costs such as time, money or other people. If all players present a dominant strategy, contracts may be used. [ 11 ] | https://en.wikipedia.org/wiki/Liberal_paradox |
Liberman's lemma is a theorem used in studying intrinsic geometry of convex surface .
It is named after Joseph Liberman .
If γ {\displaystyle \gamma } is a unit-speed minimizing geodesic on the surface of a convex body K in Euclidean space then for any point p ∈ K , the function
is concave.
This differential geometry -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Liberman's_lemma |
libevent is a software library that provides asynchronous event notification. The libevent API provides a mechanism to execute a callback function when a specific event occurs on a file descriptor or after a timeout has been reached. libevent also supports callbacks triggered by signals and regular timeouts.
libevent is meant to replace the event loop found in event-driven network servers . An application can just call event_dispatch() and then add or remove events dynamically without having to change the event loop.
Currently, libevent supports /dev/poll , kqueue(2) , POSIX select(2) , Windows IOCP , poll(2) , epoll(7) and Solaris event ports . It also has experimental support for real-time signals. The exposed event API is uniform over all of the supported platforms. As a result, libevent allows for portable application development and provides "the most scalable event notification mechanism available on an operating system". [ 1 ]
Using callbacks on signals, libevent makes it possible to write "secure" signal handlers as none of the user supplied signal handling code runs in the signal's context .
libevent was created by Niels Provos , and is maintained primarily by Azat Khuzhin. It is released under a BSD license . [ 3 ]
Some of the notable applications that take advantage of libevent are: | https://en.wikipedia.org/wiki/Libevent |
liblzg is a compression library for performing lossless data compression . It implements an algorithm that is a variation of the LZ77 algorithm, called the LZG algorithm, with the primary focus of providing a very simple and fast decoding method. One of the key features of the algorithm is that it requires no memory during decompression. The software library is free software , distributed under the zlib license .
If a duplicate series of bytes (a repeated string) is spotted in the uncompressed data stream, then a back- reference is inserted, linking to the previous location of that identical string instead. An encoded match to an earlier string consists of a length (3–128 bytes) and a distance (1–526,341 bytes). The level of compression can be controlled by specifying the maximum distance for which duplicated strings will be searched (this is the size of the sliding window ).
The data format consists of a header, followed by the compressed data. The header contains an identifier and house keeping information, such as compressed and decompressed data sizes and a 32-bit checksum (a variant of the Fletcher checksum ).
The compressed data starts with four bytes, identifying four unique 8-bit marker symbols ( m1 , m2 , m3 and m4 ). These are used to separate literal data bytes from various forms of length-distance pair encodings.
Any symbol that is not a marker byte is considered a literal byte, and will be copied as is to the decompressed data buffer. However, if the decoder encounters any of the four marker bytes, it will decode a length-distance pair that is used as a back reference into the previously decompressed data.
The marker bytes are interpreted as follows (% denotes a binary number):
m1 represents the most general form of a copy operation, and it occupies four bytes in the compressed data stream:
...where length= DECODELENGTH(%lllll+2) , and offset= %ooommmmmmmmnnnnnnnn + 2056 .
m2 is a shorter form of a copy operation, occupying three bytes in the compressed data stream:
...where length= DECODELENGTH(%lllll+2) , and offset= %ooommmmmmmm + 8 .
m3 requires only two bytes, and is used for short lengths, close to the marker:
...where length= %ll+3 , and offset= %oooooo + 8 .
m4 requires only two bytes, and is used for nearby copies (including RLE , when the offset is 1):
...where length= DECODELENGTH(%lllll+2) , and offset= %ooo + 1 .
As a special case, if any of the marker symbols are followed by a zero byte (0), the marker symbol itself is written to the decompressed buffer.
The DECODELENGTH function implements a non-linear mapping of a number in the range 3-33 to a number in the range 3-128, according to the following table:
As the marker symbols are chosen as the four least common symbols in the uncompressed data stream (with a probability of at most 1 256 {\displaystyle {\tfrac {1}{256}}} each), and a single occurrence of a marker symbol requires two bytes to encode, the compressed data may grow by at most 4 256 {\displaystyle {\tfrac {4}{256}}} < 1.6% compared to the decompressed data (worst case).
The liblzg library compensates for this by using a plain 1:1 copy mode if the encoder identifies that the compressed data will be larger than the original uncompressed data. Hence, in practice, the maximum data growth is 0% (plus the size of the data header, which is 16 bytes).
Both the compression and the decompression algorithms are implemented in an open source library, written in the C programming language . There are also several alternate implementations of the decompression algorithm available (for instance in JavaScript and 8-bit assembly language ). | https://en.wikipedia.org/wiki/Liblzg |
In molecular biology , a library is a collection of genetic material fragments that are stored and propagated in a population of microbes through the process of molecular cloning . There are different types of DNA libraries, including cDNA libraries (formed from reverse-transcribed RNA ), genomic libraries (formed from genomic DNA) and randomized mutant libraries (formed by de novo gene synthesis where alternative nucleotides or codons are incorporated). DNA library technology is a mainstay of current molecular biology , genetic engineering , and protein engineering , and the applications of these libraries depend on the source of the original DNA fragments. There are differences in the cloning vectors and techniques used in library preparation, but in general each DNA fragment is uniquely inserted into a cloning vector and the pool of recombinant DNA molecules is then transferred into a population of bacteria (a Bacterial Artificial Chromosome or BAC library) or yeast such that each organism contains on average one construct (vector + insert). As the population of organisms is grown in culture, the DNA molecules contained within them are copied and propagated (thus, "cloned").
The term "library" can refer to a population of organisms, each of which carries a DNA molecule inserted into a cloning vector, or alternatively to the collection of all of the cloned vector molecules.
A cDNA library represents a sample of the mRNA purified from a particular source (either a collection of cells, a particular tissue, or an entire organism), which has been converted back to a DNA template by the use of the enzyme reverse transcriptase . It thus represents the genes that were being actively transcribed in that particular source under the physiological, developmental, or environmental conditions that existed when the mRNA was purified. cDNA libraries can be generated using techniques that promote "full-length" clones or under conditions that generate shorter fragments used for the identification of " expressed sequence tags ".
cDNA libraries are useful in reverse genetics, but they only represent a very small (less than 1%) portion of the overall genome in a given organism.
Applications of cDNA libraries include:
A genomic library is a set of clones that together represents the entire genome of a given organism. The number of clones that constitute a genomic library depends on (1) the size of the genome in question and (2) the insert size tolerated by the particular cloning vector system. For most practical purposes, the tissue source of the genomic DNA is unimportant because each cell of the body contains virtually identical DNA (with some exceptions).
Applications of genomic libraries include:
In contrast to the library types described above, a variety of artificial methods exist for making libraries of variant genes. [ 1 ] Variation throughout the gene can be introduced randomly by either error-prone PCR , [ 2 ] DNA shuffling to recombine parts of similar genes together, [ 3 ] or transposon-based methods to introduce indels . [ 4 ] Alternatively, mutations can be targeted to specific codons during de novo synthesis or saturation mutagenesis to construct one or more point mutants of a gene in a controlled way. [ 5 ] This results in a mixture of double stranded DNA molecules which represent variants of the original gene.
The expressed proteins from these libraries can then be screened for variants which exhibit favorable properties (e.g. stability, binding affinity or enzyme activity). This can be repeated in cycles of creating gene variants and screening the expression products in a directed evolution process. [ 1 ]
If creating an mRNA library (i.e. with cDNA clones), there are several possible protocols for isolating full length mRNA. To extract DNA for genomic DNA (also known as gDNA) libraries, a DNA mini-prep may be useful.
cDNA libraries require care to ensure that full length clones of mRNA are captured as cDNA (which will later be inserted into vectors). Several protocols have been designed to optimise the synthesis of the 1st cDNA strand and the 2nd cDNA strand for this reason, and also to make directional cloning into the vector more likely.
gDNA fragments are generated from the extracted gDNA by using non-specific frequent cutter restriction enzymes.
The nucleotide sequences of interest are preserved as inserts to a plasmid or the genome of a bacteriophage that has been used to infect bacterial cells.
Vectors are propagated most commonly in bacterial cells, but if using a YAC (Yeast Artificial Chromosome) then yeast cells may be used. Vectors could also be propagated in viruses, but this can be time-consuming and tedious. However, the high transfection efficiency achieved by using viruses (often phages) makes them useful for packaging the vector (with the ligated insert) and then introducing them into the bacterial (or yeast) cell.
Additionally, for cDNA libraries, a system using the Lambda Zap II phage, ExAssist, and 2 E. coli species has been developed. A Cre-Lox system using loxP sites and the in vivo expression of the recombinase enzyme can also be used instead. These are examples of in vivo excision systems. In vitro excision involves subcloning often using traditional restriction enzymes and cloning strategies. In vitro excision can be more time-consuming and may require more "hands-on" work than in vivo excision systems. In either case, the systems allow the movement of the vector from the phage into a live cell, where the vector can replicate and propagate until the library is to be used.
This involves "screening" for the sequences of interest. There are multiple possible methods to achieve this. | https://en.wikipedia.org/wiki/Library_(biology) |
A library and information scientist , also known as a library scholar , is a researcher or academic who specializes in the field of library and information science and often participates in scholarly writing about and related to library and information science. A library and information scientist is neither limited to any one subfield of library and information science nor any one particular type of library. These scientists come from all information-related sectors including library and book history.
The University of Chicago Graduate Library School was established in 1928 to grant a graduate degree in librarianship with an emphasis on research. [ 1 ] The program expanded the concept of librarianship, focused on scientific inquiry and established it as a domain for scientific study. [ 2 ]
In The Spirit of Inquiry: The Graduate Library School at Chicago, 1921-51 Richardson reviewed the history of the School and its impact on the discipline. [ 3 ]
Bibliometric methods have been used to create maps of library and information science, thus identifying the most important researchers as well as their relative connections (or distances) and identifying emerging trends related to LIS publications within the field.
White and McCain (1998) [ 4 ] made a map of information science and Åström (2002), [ 5 ] Chen, Ibekwe-SanJuan, and Hou (2010), [ 6 ] Janssens, Leta, Glanzel, and De Moor (2006), [ 7 ] and Zhao and Strotmann (2008) [ 8 ] constructed some later maps of library and information science.
Jabeen, Yun, Rafiq, and Jabeen (2015) [ 9 ] mapped the growth and trends of LIS publications.
See also Beta Phi Mu Award , Award of Merit - Association for Information Science and Technology , Justin Winsor Prize (library) | https://en.wikipedia.org/wiki/Library_and_information_scientist |
The Library of Latin Texts ( LLT ) is a subscription-based database of Latin texts, from antiquity up to the present day. Started in 1991 as the Cetedoc Library of Christian Latin Texts (CLCLT), it continues to be developed by the Centre ‘Traditio Litterarum Occidentalium’ and is hosted by Brepols Publishers .
In 1991, development of the Cetedoc Library of Christian Latin Texts (CLCLT) started, with the aim of encompassing the entirety of Christian Latin literature. This digital database, initially released as a CD-ROM , was produced by the Cetedoc, led by prof. Paul Tombeur at the Université catholique de Louvain . Since 2001, the activities of Cetedoc have been continued in Turnhout , Belgium by the Centre ‘Traditio Litterarum Occidentalium’ (CTLO), still led by Paul Tombeur.
In 2002, it was decided to expand the database's chronological scope beyond medieval and patristic times, its name was changed to Library of Latin Texts . In 2009, a B series (LLT-B) was added to the original LLT (hence known as LLT-A). The LLT-B's scope was to accelerate the growth of the database by directly adopting the text of existing editions, without the intensive research work that is applied to the texts of the LLT-A. This work includes verifying facts related to the text and correcting errors in the printed edition. [ 1 ]
Unlike similar initiatives, like Corpus Corporum and The Latin Library , the LLT is not an open-access database. This allows for the adoption of copyrighted editions. In fact, while open-access initiatives have to rely on out-of-copyright, possibly outdated editions, the LLT's policy is to select texts that have "been edited according to best contemporary scholarly practice". [ 1 ] The texts edited in Brepols' Corpus Christianorum series form the core of the LLT, even though numerically, they are outnumbered by texts edited in other publishers' series and, if no modern edition is available, by out-of-copyright editions. Nevertheless, for scholarly purposes, the LLT should be used in conjunction with the printed editions, as the critical apparatus is not included in the database. [ 2 ] | https://en.wikipedia.org/wiki/Library_of_Latin_Texts |
Libration (from the Latin verb librare "to balance, to sway"; cf. libra "scales") is a type of reciprocating motion in which an object with a nearly fixed orientation repeatedly rotates slightly back and forth. In physics and chemistry , a molecule (or other group of atoms ) can undergo libration if it is subject to external forces or constraints that restrict its orientation.
For example, in liquid water , any given water molecule is attracted to neighboring molecules, so that it has a preferred orientation and cannot freely rotate. (Of course, over time, the neighboring molecules move around and the preferred orientation changes.) However, it can undergo librational motions, which are measureable in an infrared absorption spectrum [ 1 ] and contribute to motional narrowing of other peaks, for instance the OH stretch.
Another example is a molecular crystal : Each molecular unit has a preferred orientation due to interactions with the nearby molecules, but they have librational modes corresponding to small rotations about this preferred orientation. [ 2 ]
This physical chemistry -related article is a stub . You can help Wikipedia by expanding it .
This spectroscopy -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Libration_(molecule) |
Other:
The Librem 5 is a smartphone manufactured by Purism that is part of their Librem line of products. The phone is designed with the goal of using free software whenever possible and includes PureOS , a Linux operating system, by default. [ 3 ] Like other Librem products, the Librem 5 focuses on privacy and freedom and includes features like hardware kill switches and easily-replaceable components. Its name, with a numerical "5", refers to its screen size, not a release version. After an announcement on 24 August 2017, the distribution of developer kits and limited pre-release models occurred throughout 2019 and most of 2020. The first mass-production version of the Librem 5 was shipped on 18 November 2020.
On August 24, 2017, Purism started a crowdfunding campaign for the Librem 5, [ 4 ] [ 5 ] a smartphone aimed not only to run purely on free software provided in PureOS but to "[focus] on security by design and privacy protection by default". Purism claimed that the phone would become "the world's first ever IP-native mobile handset, using end-to-end encrypted decentralized communication". [ 6 ] Purism has cooperated with GNOME in its development of the Librem 5 software. It is planned that KDE and Ubuntu Touch will also be offered as optional interfaces. [ 7 ]
The release of the Librem 5 was delayed several times. It was originally planned to launch in January 2019. Purism announced on September 4, 2018 that the launch date would be postponed until April 2019, [ 8 ] due to two power management bugs in the silicon and the Europe/North America holiday season. Development kits for software developers, which were shipped out in December 2018 [ 9 ] were unaffected by the bugs, since developers normally connect the device to a power outlet rather than rely on the phone battery. In February, the launch date was postponed again to the third quarter of 2019, because of the necessity of further CPU tests. [ 10 ]
Specifications and pre-orders, for $649, to increase to $699, were announced in July 2019. [ 11 ] On September 5, 2019, Purism announced that shipping was scheduled to occur later that month, but that it would be done as an "iterative" process. [ 12 ] The iterative release plan included the announcement of six different "batches" of Librem 5 releases, of which the first four would be limited pre-production models. Each consecutive batch, which consisted of different arboreal-themed code names and release dates, would feature hardware, mechanical, and software improvements. Purism contacted each customer that had pre-ordered to allow them to choose which batch they'd prefer to receive. Pre-mass production batches, in order of release, included code names "Aspen", "Birch", "Chestnut", and "Dogwood". The fifth batch, "Evergreen", would be the official mass-production model, while the sixth batch, "Fir", would be the second mass-production model.
On September 24, 2019, Purism announced that the first batch of limited-production Librem 5 phones (Aspen) had started shipping. [ 13 ] [ 14 ] A video of an early phone was produced [ 15 ] and a shipping and status update was released soon after. [ 16 ] [ 17 ] However, it was later reported that the Aspen batch had been shipped only to employees and developers. On November 22, 2019, it was reported that the second batch (Birch) would consist of around 100 phones and would be in the hands of backers by the first week of December. [ 18 ] In December 2019, Jim Salter of Ars Technica reported "prototype" devices were being received; however, they were not really a "phone" yet. There was no audio when attempting to place a phone call (which was fixed with a software update a few weeks later [ 19 ] ), and cameras didn't work yet. [ 20 ] Reports of the third batch of limited pre-mass-production models (Chestnut) being received by customers and reviewers occurred in January 2020. [ 21 ] By May 2020, TechRadar reported that the call quality was fine, though the speaker mode was "a bit quiet", and volume adjustment did not work. According to TechRadar, the 3 to 5-hour battery time and the inability of the phone to charge while turned on was "A stark reminder of the Librem 5's beta status". [ 22 ]
On November 18, 2020, Purism announced via press release that they had begun shipping the finished version of the Librem 5, known as "Evergreen". [ 23 ] [ 24 ] Following its release, in December 2019, Purism announced that it will offer a "Librem 5 USA" version of the phone for the price of $1999, which is assembled in the United States for extra supply chain security . [ 25 ] According to Purism CEO Todd Weaver, "having a secure auditable US based supply chain including parts procurement, fabrication, testing, assembly, and fulfillment all from within the same facility is the best possible security story." [ 26 ]
The Librem 5 features an i.MX 8M Quad Core processor with an integrated GPU which supports OpenGL 3.0, OpenGL ES 3.1, Vulkan 1.0 and OpenCL 1.2 with default drivers; [ 27 ] however, since the driver used is the open source Etnaviv driver, it currently only supports OpenGL 2.1 and OpenGL ES 2.0. [ 28 ] [ 2 ] It has 3 GB of RAM, 32 GB of eMMC storage, a 13 MP rear camera, and an 8 MP front camera. The left side of the phone features three hardware kill switches , which cut power to the camera and microphone, Wi-Fi and Bluetooth modem, and the baseband modem. [ 29 ] ) The device uses a USB-C connector for charging. The 144 mm (5.7-inch) IPS display has a resolution of 1440×720 pixels. It also has a 3.5 mm TRRS headphone/mic jack, a single SIM slot, and a microSD card slot. [ 14 ]
The Librem 5 is powered by a lithium-ion battery . The capacity of the battery was 2000 mAh in earliest development batches, [ 30 ] which was increased to 4500 mAh in the mass-production batch. The battery is designed to be user-replaceable. The battery is unique to Librem 5 and cannot be replaced by any other battery type. In addition, Purism ships replacement batteries only within the US unless combined with another device. [ 31 ]
The hardware features three hardware kill switches that physically cut off power from both cameras and the microphone, Wi-Fi and Bluetooth, and baseband processor , respectively. Further precautionary measures can be used with Lockdown Mode , which, in addition to powering off the cameras, microphone, WiFi, Bluetooth and cellular baseband, also cuts power to the GNSS , IMU , ambient light and proximity sensor . This is possible due to the fact that these components are not integrated into the system on a chip (SoC) like they are in conventional smartphones. Instead, the cellular baseband and Wi-Fi/Bluetooth components are located on two replaceable M.2 cards, which means that they can be changed to support different wireless standards. [ 14 ] [ 32 ] The kill switch to cut the circuit to the microphone will prevent the 3.5 mm audio jack being used for acoustic cryptanalysis . [ 33 ]
In place of an integrated mobile SoC found in most smartphones, the Librem 5 uses six separate chips: i.MX 8M Quad, Silicon Labs RS9116, Broadmobi BM818 / Gemalto PLS8, STMicroelectronics Teseo-LIV3F, Wolfson Microelectronics WM8962, and Texas Instruments bq25895. [ 2 ]
The downside to having dedicated chips instead of an integrated system-on-chip is that it takes more energy to operate separate chips, and the phone's circuit boards are much larger. On the other hand, using separate components means longer support from the manufacturers than with mobile SoCs, which have short support timelines. [ 34 ] According to Purism, the Librem 5 is designed to avoid planned obsolescence and will receive lifetime software updates. [ 35 ]
The Librem 5 is the first phone to contain a smartcard reader, in which an OpenPGP card can be inserted for secure cryptographic operations. [ 14 ] Purism plans to use OpenPGP cards to implement storage of GPG keys, disk unlocking, secure authentication, a local password vault, protection of sensitive files, user persons, and travel persons. [ 36 ]
To promote better security, all the source code in the root file system is free/open source software and can be reviewed by the user. Purism publishes the schematics of the Librem 5's printed circuit boards (PCBs) under the GPL 3.0+ license, [ 37 ] and publishes x-rays of the phone, [ 38 ] so that the user can verify that there haven't been any changes to the hardware, such as inserted spy chips. [ 39 ]
The Librem 5 ships with Purism's PureOS , a Debian GNU/Linux derivative. The operating system uses a new mobile user interface developed by Purism called Phosh, a portmanteau from " pho ne sh ell". It is based on Wayland , wlroots, GTK 3, and GNOME . [ 41 ] Unlike other mobile Linux interfaces, such as Ubuntu Touch and KDE Plasma Mobile , Phosh is based on tight integration with the desktop Linux software stack, which Purism developers believe will make it easier to maintain in the long-term and incorporate into existing desktop Linux distributions. Phosh has been packaged in a number of desktop distros (Debian, Arch, Manjaro, Fedora and openSUSE) and is used by eight of the sixteen Linux ports for the PinePhone . [ 42 ]
The phone is a convergence device: [ 43 ] [ 44 ] if connected to a keyboard, monitor, and mouse, it can run Linux applications as a desktop computer would. Many desktop Linux applications can run on the phone as well, albeit possibly without a touch-friendly UI. [ 14 ]
Purism is taking a unique approach to convergence by downsizing existing desktop software to reuse it in a mobile environment. Purism has developed the libhandy library (now replaced with Libadwaita ) to make GTK software adaptive so its interface elements adjust to smaller mobile screens. [ 45 ] In contrast, other companies such as Microsoft and Samsung with Ubuntu (and Canonical before Unity8) tried to achieve convergence by having separate sets of software for the mobile and desktop PC environments. Most iOS apps, Android apps and Plasma Mobile's Kirigami implement convergence by upsizing existing mobile apps to use them in a desktop interface. [ 42 ]
Purism claims that the "Librem 5 will be the first ever Matrix -powered smartphone, natively using end-to-end encrypted decentralised communication in its dialer and messaging app". [ 46 ] [ 47 ]
Purism was unable to find a free/open-source cellular modem , so the phone uses a modem with proprietary hardware, but isolates it from the rest of the components rather than having it integrated with the system on a chip (SoC). This prevents code on the modem from being able to read or modify data going to and from the SoC. [ 14 ] [ 48 ] | https://en.wikipedia.org/wiki/Librem_5 |
Librestream Technologies Inc. is a privately owned, venture capital–backed company based in Winnipeg, Canada. Librestream provides technologies that enable mobile and remote enterprise collaboration.
Librestream is known for its unique hand-held mobile devices and accompanying software which help extend traditional video conferencing and collaborative services to locations previously unreachable.
Mobile collaboration is a technology-based process of communicating utilizing electronic assets and accompanying software designed for use in remote locations. Newest generation hand-held electronic devices include video, audio, and telestration (on-screen drawing) capabilities broadcast over secure networks, enabling multi-party conferencing in real-time.
Differing from traditional video conferencing , mobile collaboration utilizes wireless , cellular , and broadband technologies enabling effective collaboration independent of location. Where traditional video conferencing has been limited to boardrooms, offices, and lecture theatres, recent technological advancements have extended the capabilities of video conferencing for use with discreet, hand-held mobile devices, permitting true mobile collaborative possibilities. [ 1 ]
The origins of Librestream date back to the late 1980s in Winnipeg, Canada, when Kerry Thacher co-founded Ubitrex Corporation with Robert Nickel, a small high-tech start-up that designed and developed a clinical information system for use in hospitals to capture patient data. [ 2 ] The system included a hand-held device designed for use by clinicians. Subsequent to successful clinical use of this system, Ubitrex was sold in 1994 to U.S.-based Continental Healthcare Systems.
Two years later, Thacher bought back the device side of the business from Continental and formed AirWire Mobile Technologies Inc. which continued to support the substantial base of hospitals using the technology. Only a few weeks into AirWire’s operation, Symbol Technologies Inc. , a large U.S.-based manufacturer of mobile devices, took notice of AirWire’s unique expertise and contracted with Airwire to create a product to help Symbol pursue the healthcare market.
In 1999, Symbol bought AirWire. Over the next few years, the team, continuing to operate out of Winnipeg, successfully designed and developed a number of mobile devices, one of which delivered Voice over IP (VoIP), a significant advancement in mobile device technology at that time. This activity led to the development of high-volume hand-held mobile expertise in Winnipeg for the first time, a capability that would later form the foundation for the creation of Librestream.
In August 2003, Thacher left Symbol, and weeks later Symbol closed its Winnipeg office. Although many of the engineers received offers from Symbol to re-locate to New York, all declined. Later in 2003, a group of eight professionals from the former Winnipeg arm of Symbol gathered to plan their next venture. In addition to Thacher, this group included Bill Gillanders, Rob McConnell, Don Freiling, Tim Braun, Kent Wotherspoon, Conway Wieler, and Chris Kavanagh.
Two things were clear: the emergence and continued growth of the Internet was certain, and there existed a tangible opportunity to develop enhanced video-handling capabilities for a next generation of hand-held devices. [ 3 ] The group was convinced that together they could design and develop the necessary technology to allow individuals, regardless of location, the ability to collaborate in new ways. The technology would extend the boundaries of traditional videoconferencing beyond the boardroom to workplaces previously unreachable such as a manufacturing plant floor a continent away. [ 4 ]
Librestream (a combination of ‘free’ and stream’) was formed in 2003 by Kerry Thacher. [ 5 ] Backed by venture capital and individual investors, Librestream developed its first alpha product, the MCD-1000, coupled with desktop collaboration software, MCA, in 2006. Eventually, the mobile collaborative device evolved into the Onsight 1000, one of multiple rugged hand-held devices in the Onsight product line. The desktop collaboration software evolved into the Onsight Expert application.
In 2006, Librestream formed a marketing partnership with Tandberg , one of the leading video conferencing industry players at that time. The arrangement offered Librestream industry exposure and allowed the breadth and depth of the product to grow in response to evolving requirements.
By 2010, Librestream’s customer base grew to include global enterprises, many of them Fortune 500 firms, in industries such as manufacturing, [ 6 ] energy, [ 7 ] healthcare, [ 8 ] insurance, [ 9 ] government and public safety. [ 10 ]
The impact of mobile collaboration technology is significant in its potential to change the way people work. Live, visual interaction removes traditional restrictions of distance and time. [ 11 ] Business processes are optimized through accelerated problem resolution, reductions in downtimes and travel, improvements in customer service and increased productivity. [ 12 ]
Librestream works with three strategic partners. With Cisco Systems , Librestream is a Registered Cisco Developer Network member, and Onsight is a core component of the Cisco Manufacturing Mobile Video Collaboration (MMVC) solution. [ 13 ] With Inmarsat , Librestream has successfully tested the Onsight system over the Inmarsat BGAN satellite network to provide mobile collaboration to land and maritime satellite customers. With Verizon Wireless , Librestream has tested and optimized the Onsight mobile devices for 4G LTE networks. [ 14 ] | https://en.wikipedia.org/wiki/Librestream |
libRoadRunner is a C / C++ software library that supports simulation of SBML based models.. [ 1 ] It uses LLVM to generate extremely high-performance code and is the fastest SBML-based simulator currently available. [ 2 ] Its main purpose is for use as a reusable library that can be hosted by other applications, particularly on large compute clusters for doing parameter optimization where performance is critical. It also has a set of Python bindings that allow it to be easily used from Python as well as a set of bindings for Julia . [ 3 ]
libroadrunner is often paired with Tellurium , [ 4 ] which adds additional functionality such as Antimony [ 5 ] scripting.
libroadrunner has been widely used in the systems biology community for doing research in systems biology modeling, as well as being a host for other simulation platforms.
libroadrunner has been used in a large variety of research projects. The following lists a small number of those studies:
A number of reviews and commentaries have been written that discuss libroadrunner:
Development of libroadrunner is primarily funded through research grants from the National Institutes of Health [ 30 ] | https://en.wikipedia.org/wiki/Libroadrunner |
libuv is a multi-platform C library that provides support for asynchronous I/O based on event loops . It supports epoll(4) , kqueue(2) , Windows IOCP , Solaris event ports and Linux io_uring . It is primarily designed for use in Node.js but it is also used by other software projects. [ 3 ] It was originally an abstraction around libev or Microsoft IOCP , as libev does not support IOCP on Windows. In node-v0.9.0's version of libuv, the dependency on libev was removed. [ 4 ]
From: [ 2 ]
According to libuv developer Ben Noordhuis , the name libuv originally had no specific meaning, but as people kept asking about it, so they made something up. They came up with Unicorn Velociraptor , which became the logo of the library. [ 5 ] | https://en.wikipedia.org/wiki/Libuv |
License borrowing is a feature that allows a user to run software on a computer that is not continuously connected to the license server on the network .
When making a borrow request, the user is either connected to the server over the network, or with some systems the license can be borrowed via secure file exchange between the disconnected user's system and the server . After the license has been borrowed, the user can then disconnect the computer from the network and continue to use the software for the length of the borrow period, which is typically determined by the software vendor. During this time, the borrowed license is removed from the pool of available licenses. After the borrow period expires the license is then checked back into the pool. | https://en.wikipedia.org/wiki/License_borrowing |
A lichen ( / ˈ l aɪ k ən / LIE -kən , UK also / ˈ l ɪ tʃ ən / LI -chən ) is a hybrid colony of algae or cyanobacteria living symbiotically among filaments of multiple fungus species, along with yeasts and bacteria [ 1 ] [ 2 ] embedded in the cortex or "skin", in a mutualistic relationship. [ 3 ] [ 4 ] [ 5 ] [ 6 ] [ 7 ] Lichens are the lifeform that first brought the term symbiosis (as Symbiotismus [ 8 ] ) into biological context.
Lichens have since been recognized as important actors in nutrient cycling and producers which many higher trophic feeders feed on, such as reindeer, gastropods, nematodes, mites, and springtails. [ 9 ] [ 10 ] [ 11 ] [ 12 ] Lichens have properties different from those of their component organisms. They come in many colors, sizes, and forms and are sometimes plant-like, but are not plants . They may have tiny, leafless branches ( fruticose ); flat leaf-like structures ( foliose ); grow crust-like, adhering tightly to a surface ( substrate ) like a thick coat of paint ( crustose ); [ 13 ] have a powder-like appearance ( leprose ); or other growth forms . [ 14 ]
A macrolichen is a lichen that is either bush-like or leafy; all other lichens are termed microlichens . [ 4 ] Here, "macro" and "micro" do not refer to size, but to the growth form. [ 4 ] Common names for lichens may contain the word moss (e.g., " reindeer moss ", " Iceland moss "), and lichens may superficially look like and grow with mosses , but they are not closely related to mosses or any plant. [ 6 ] : 3 Lichens do not have roots that absorb water and nutrients as plants do, [ 15 ] : 2 but like plants, they produce their own energy by photosynthesis . [ 16 ] When they grow on plants, they do not live as parasites , but instead use the plant's surface as a substrate.
Lichens occur from sea level to high alpine elevations, in many environmental conditions, and can grow on almost any surface. [ 16 ] [ 17 ] They are abundant growing on bark, leaves , mosses, or other lichens [ 15 ] and hanging from branches "living on thin air" ( epiphytes ) in rainforests and in temperate woodland . They grow on rock, walls, gravestones , roofs , exposed soil surfaces, rubber, bones, and in the soil as part of biological soil crusts . Various lichens have adapted to survive in some of the most extreme environments on Earth: arctic tundra , hot dry deserts , rocky coasts , and toxic slag heaps. They can even live inside solid rock, growing between the grains ( endolithic ).
There are about 20,000 known species. [ 18 ] Some lichens have lost the ability to reproduce sexually, yet continue to speciate . [ 15 ] [ 19 ] They can be seen as being relatively self-contained miniature ecosystems , where the fungi, algae, or cyanobacteria have the potential to engage with other microorganisms in a functioning system that may evolve as an even more complex composite organism . [ 20 ] [ 21 ] [ 22 ] [ 23 ] Lichens may be long-lived , with some considered to be among the oldest living things. [ 6 ] [ 24 ] They are among the first living things to grow on fresh rock exposed after an event such as a landslide. The long life-span and slow and regular growth rate of some species can be used to date events ( lichenometry ). Lichens are a keystone species in many ecosystems and benefit trees and birds . [ 25 ]
The English word lichen derives from the Greek λειχήν leichēn ("tree moss, lichen, lichen-like eruption on skin") via Latin lichen . [ 26 ] [ 27 ] [ 28 ] The Greek noun, which literally means "licker", derives from the verb λείχειν leichein , "to lick". [ 29 ] [ 30 ] In American English, "lichen" is pronounced the same as the verb "liken" ( / ˈ l aɪ k ən / ). In British English, both this pronunciation and one rhyming with "kitchen" ( / ˈ l ɪ tʃ ən / ) are used. [ 31 ] [ 32 ] [ 33 ]
Lichens grow in a wide range of shapes and forms; this external appearance is known as their morphology . The shape of a lichen is usually determined by the organization of the fungal filaments. [ 34 ] The nonreproductive tissues, or vegetative body parts, are called the thallus . Lichens are grouped by thallus type, since the thallus is usually the most visually prominent part of the lichen. Thallus growth forms typically correspond to a few basic internal structure types. Common names for lichens often come from a growth form or color that is typical of a lichen genus .
Common groupings of lichen thallus growth forms are:
There are variations in growth types in a single lichen species, grey areas between the growth type descriptions, and overlapping between growth types, so some authors might describe lichens using different growth type descriptions.
When a crustose lichen gets old, the center may start to crack up like old-dried paint, old-broken asphalt paving, or like the polygonal "islands" of cracked-up mud in a dried lakebed. This is called being rimose or areolate , and the "island" pieces separated by the cracks are called areolas. [ 35 ] The areolas appear separated, but are (or were) [ citation needed ] connected by an underlying prothallus or hypothallus . [ 38 ] When a crustose lichen grows from a center and appears to radiate out, it is called crustose placodioid. When the edges of the areolas lift up from the substrate, it is called squamulose . [ 39 ] : 159 [ 37 ]
These growth form groups are not precisely defined. Foliose lichens may sometimes branch and appear to be fruticose. Fruticose lichens may have flattened branching parts and appear leafy. Squamulose lichens may appear where the edges lift up. Gelatinous lichens may appear leafy when dry. [ 39 ] : 159
The thallus is not always the part of the lichen that is most visually noticeable. Some lichens can grow inside solid rock between the grains ( endolithic lichens ), with only the sexual fruiting part visible growing outside the rock. [ 35 ] These may be dramatic in color or appearance. [ 35 ] Forms of these sexual parts are not in the above growth form categories. [ 35 ] The most visually noticeable reproductive parts are often circular, raised, plate-like or disc-like outgrowths, with crinkly edges, and are described in sections below.
Lichens come in many colors. [ 15 ] : 4 Coloration is usually determined by the photosynthetic component. [ 34 ] Special pigments, such as yellow usnic acid , give lichens a variety of colors, including reds, oranges, yellows, and browns, especially in exposed, dry habitats. [ 40 ] In the absence of special pigments, lichens are usually bright green to olive gray when wet, gray or grayish-green to brown when dry. [ 40 ] This is because moisture causes the surface skin ( cortex ) to become more transparent, exposing the green photobiont layer. [ 40 ] Different colored lichens covering large areas of exposed rock surfaces, or lichens covering or hanging from bark can be a spectacular display when the patches of diverse colors "come to life" or "glow" in brilliant displays following rain.
Different colored lichens may inhabit different adjacent sections of a rock face, depending on the angle of exposure to light. [ 40 ] Colonies of lichens may be spectacular in appearance, dominating much of the surface of the visual landscape in forests and natural places, such as the vertical "paint" covering the vast rock faces of Yosemite National Park . [ 41 ]
Color is used in identification. [ 42 ] : 4 The color of a lichen changes depending on whether the lichen is wet or dry. [ 42 ] Color descriptions used for identification are based on the color that shows when the lichen is dry. [ 42 ] Dry lichens with a cyanobacterium as the photosynthetic partner tend to be dark grey, brown, or black. [ 42 ]
The underside of the leaf-like lobes of foliose lichens is a different color from the top side ( dorsiventral ), often brown or black, sometimes white. A fruticose lichen may have flattened "branches", appearing similar to a foliose lichen, but the underside of a leaf-like structure on a fruticose lichen is the same color as the top side. The leaf-like lobes of a foliose lichen may branch, giving the appearance of a fruticose lichen, but the underside will be a different color from the top side. [ 38 ]
The sheen on some jelly-like gelatinous lichens is created by mucilaginous secretions. [ 34 ]
A lichen consists of a simple photosynthesizing organism, usually a green alga or cyanobacterium , surrounded by filaments of a fungus. Generally, most of a lichen's bulk is made of interwoven fungal filaments, [ 43 ] but this is reversed in filamentous and gelatinous lichens. [ 34 ] The fungus is called a mycobiont . The photosynthesizing organism is called a photobiont . Algal photobionts are called phycobionts . [ 44 ] Cyanobacteria photobionts are called cyanobionts . [ 44 ]
The part of a lichen that is not involved in reproduction, the "body" or "vegetative tissue" of a lichen, is called the thallus . The thallus form is very different from any form where the fungus or alga are growing separately. The thallus is made up of filaments of the fungus called hyphae . The filaments grow by branching then rejoining to create a mesh, which is called being " anastomosed ". The mesh of fungal filaments may be dense or loose.
Generally, the fungal mesh surrounds the algal or cyanobacterial cells, often enclosing them within complex fungal tissues that are unique to lichen associations. The thallus may or may not have a protective "skin" of densely packed fungal filaments, often containing a second fungal species, [ 3 ] which is called a cortex. Fruticose lichens have one cortex layer wrapping around the "branches". Foliose lichens have an upper cortex on the top side of the "leaf", and a separate lower cortex on the bottom side. Crustose and squamulose lichens have only an upper cortex, with the "inside" of the lichen in direct contact with the surface they grow on (the substrate ). Even if the edges peel up from the substrate and appear flat and leaf-like, they lack a lower cortex, unlike foliose lichens. Filamentous, byssoid, leprose, [ 38 ] gelatinous, and other lichens do not have a cortex; in other words, they are ecorticate . [ 45 ]
Fruticose, foliose, crustose, and squamulose lichens generally have up to three different types of tissue, differentiated by having different densities of fungal filaments. [ 43 ] The top layer, where the lichen contacts the environment, is called a cortex . [ 43 ] The cortex is made of densely tightly woven, packed, and glued together ( agglutinated ) fungal filaments. [ 43 ] The dense packing makes the cortex act like a protective "skin", keeping other organisms out, and reducing the intensity of sunlight on the layers below. [ 43 ] The cortex layer can be up to several hundred micrometers (μm) in thickness (less than a millimeter). [ 46 ] The cortex may be further topped by an epicortex of secretions, not cells, 0.6–1 μm thick in some lichens . [ 46 ] This secretion layer may or may not have pores. [ 46 ]
Below the cortex layer is a layer called the photobiontic layer or symbiont layer . [ 36 ] [ 43 ] The symbiont layer has less densely packed fungal filaments, with the photosynthetic partner embedded in them. [ 43 ] The less dense packing allows air circulation during photosynthesis, similar to the anatomy of a leaf. [ 43 ] Each cell or group of cells of the photobiont is usually individually wrapped by hyphae, and in some cases penetrated by a haustorium . [ 34 ] In crustose and foliose lichens, algae in the photobiontic layer are diffuse among the fungal filaments, decreasing in gradation into the layer below. In fruticose lichens, the photobiontic layer is sharply distinct from the layer below. [ 34 ]
The layer beneath the symbiont layer is called the medulla . The medulla is less densely packed with fungal filaments than the layers above. In foliose lichens, as in Peltigera , [ 39 ] : 159 there is usually another densely packed layer of fungal filaments called the lower cortex. [ 38 ] [ 43 ] Root-like fungal structures called rhizines ( usually ) [ 39 ] : 159 grow from the lower cortex to attach or anchor the lichen to the substrate. [ 4 ] [ 38 ] Fruticose lichens have a single cortex wrapping all the way around the "stems" and "branches". [ 39 ] The medulla is the lowest layer, and may form a cottony white inner core for the branchlike thallus, or it may be hollow. [ 39 ] : 159 Crustose and squamulose lichens lack a lower cortex, and the medulla is in direct contact with the substrate that the lichen grows on.
In crustose areolate lichens, the edges of the areolas peel up from the substrate and appear leafy. In squamulose lichens the part of the lichen thallus that is not attached to the substrate may also appear leafy. But these leafy parts lack a lower cortex, which distinguishes crustose and squamulose lichens from foliose lichens. [ 43 ] Conversely, foliose lichens may appear flattened against the substrate like a crustose lichen, but most of the leaf-like lobes can be lifted up from the substrate because it is separated from it by a tightly packed lower cortex. [ 38 ]
Gelatinous, [ 39 ] : 159 byssoid, and leprose lichens lack a cortex (are ecorticate ), and generally have only undifferentiated tissue, similar to only having a symbiont layer. [ citation needed ]
In lichens that include both green algal and cyanobacterial symbionts, the cyanobacteria may be held on the upper or lower surface in small pustules called cephalodia .
Pruinia is a whitish coating on top of an upper surface. [ 47 ] An epinecral layer is "a layer of horny dead fungal hyphae with indistinct lumina in or near the cortex above the algal layer". [ 47 ]
In August 2016, it was reported that some macrolichens have more than one species of fungus in their tissues. [ 3 ]
Lichens are fungi that have discovered agriculture
A lichen is a composite organism that emerges from algae or cyanobacteria living among the filaments ( hyphae ) of the fungi in a mutually beneficial symbiotic relationship. [ 49 ] The fungi benefit from the carbohydrates produced by the algae or cyanobacteria via photosynthesis . The algae or cyanobacteria benefit by being protected from the environment by the filaments of the fungi, which also gather moisture and nutrients from the environment, and (usually) provide an anchor to it. Although some photosynthetic partners in a lichen can survive outside the lichen, the lichen symbiotic association extends the ecological range of both partners, whereby most descriptions of lichen associations describe them as symbiotic. Both partners gain water and mineral nutrients mainly from the atmosphere, through rain and dust. The fungal partner protects the alga by retaining water, serving as a larger capture area for mineral nutrients and, in some cases, provides minerals obtained from the substrate . If a cyanobacterium is present, as a primary partner or another symbiont in addition to a green alga as in certain tripartite lichens, they can fix atmospheric nitrogen , complementing the activities of the green alga.
In three different lineages the fungal partner has independently lost the mitochondrial gene atp9, which has key functions in mitochondrial energy production. The loss makes the fungi completely dependent on their symbionts. [ 50 ]
The algal or cyanobacterial cells are photosynthetic and, as in plants, they reduce atmospheric carbon dioxide into organic carbon sugars to feed both symbionts. Phycobionts (algae) produce sugar alcohols ( ribitol , sorbitol , and erythritol ), which are absorbed by the mycobiont (fungus). [ 44 ] Cyanobionts produce glucose . [ 44 ] Lichenized fungal cells can make the photobiont "leak" out the products of photosynthesis, where they can then be absorbed by the fungus. [ 15 ] : 5
It appears many, probably the majority, of lichen also live in a symbiotic relationship with an order of basidiomycete yeasts called Cyphobasidiales . The absence of this third partner could explain why growing lichen in the laboratory is difficult. The yeast cells are responsible for the formation of the characteristic cortex of the lichen thallus, and could also be important for its shape. An example of this lichen-yeast symbiosis is the North American beard-like lichens. [ 51 ]
The lichen combination of alga or cyanobacterium with a fungus has a very different form (morphology), physiology, and biochemistry than the component fungus, alga, or cyanobacterium growing by itself, naturally or in culture. The body ( thallus ) of most lichens is different from those of either the fungus or alga growing separately. When grown in the laboratory in the absence of its photobiont, a lichen fungus develops as a structureless, undifferentiated mass of fungal filaments ( hyphae ). If combined with its photobiont under appropriate conditions, its characteristic form associated with the photobiont emerges, in the process called morphogenesis . [ 6 ] In a few remarkable cases, a single lichen fungus can develop into two very different lichen forms when associating with either a green algal or a cyanobacterial symbiont. Quite naturally, these alternative forms were at first considered to be different species, until they were found growing in a conjoined manner. [ citation needed ]
Evidence that lichens are examples of successful symbiosis is the fact that lichens can be found in almost every habitat and geographic area on the planet. [ 20 ] Two species in two genera of green algae are found in over 35% of all lichens, but can only rarely be found living on their own outside of a lichen. [ 52 ]
In a case where one fungal partner simultaneously had two green algae partners that outperform each other in different climates, this might indicate having more than one photosynthetic partner at the same time might enable the lichen to exist in a wider range of habitats and geographic locations. [ 20 ]
Phycobionts can have a net output of sugars with only water vapor. [ 44 ] The thallus must be saturated with liquid water for cyanobionts to photosynthesize. [ 44 ]
Algae produce sugars that are absorbed by the fungus by diffusion into special fungal hyphae called appressoria or haustoria in contact with the wall of the algal cells. [ 53 ] The appressoria or haustoria may produce a substance that increases permeability of the algal cell walls, and may penetrate the walls. [ 53 ] The algae may contribute up to 80% of their sugar production to the fungus. [ 53 ]
Lichen associations may be examples of mutualism or commensalism , but the lichen relationship can be considered parasitic [ 54 ] under circumstances where the photosynthetic partner can exist in nature independently of the fungal partner, but not vice versa. Photobiont cells are routinely destroyed in the course of nutrient exchange. The association continues because reproduction of the photobiont cells matches the rate at which they are destroyed. [ 54 ] The fungus surrounds the algal cells, [ 16 ] often enclosing them within complex fungal tissues unique to lichen associations. In many species the fungus penetrates the algal cell wall, [ 16 ] forming penetration pegs ( haustoria ) similar to those produced by pathogenic fungi that feed on a host. [ 37 ] [ 55 ] Cyanobacteria in laboratory settings can grow faster when they are alone rather than when they are part of a lichen.
Symbiosis in lichens is so well-balanced that lichens have been considered to be relatively self-contained miniature ecosystems in and of themselves. [ 20 ] [ 21 ] It is thought that lichens may be even more complex symbiotic systems that include non-photosynthetic bacterial communities performing other functions as partners in a holobiont . [ 22 ] [ 23 ]
Many lichens are very sensitive to environmental disturbances and can be used to cheaply [ 16 ] assess air pollution , [ 56 ] [ 57 ] [ 58 ] ozone depletion, and metal contamination. Lichens have been used in making dyes , perfumes ( oakmoss ), [ 59 ] and in traditional medicines . A few lichen species are eaten by insects [ 16 ] or larger animals, such as reindeer. [ 60 ] Lichens are widely used as environmental indicators or bio-indicators. When air is very badly polluted with sulphur dioxide, there may be no lichens present; only some green algae can tolerate those conditions. If the air is clean, then shrubby, hairy and leafy lichens become abundant. A few lichen species can tolerate fairly high levels of pollution, and are commonly found in urban areas, on pavements, walls and tree bark. The most sensitive lichens are shrubby and leafy, while the most tolerant lichens are all crusty in appearance. Since industrialisation, many of the shrubby and leafy lichens such as Ramalina , Usnea and Lobaria species have very limited ranges, often being confined to the areas which have the cleanest air.
Some fungi can only be found living on lichens as obligate parasites . These are referred to as lichenicolous fungi , and are a different species from the fungus living inside the lichen; thus they are not considered to be part of the lichen. [ 61 ]
Moisture makes the cortex become more transparent. [ 15 ] : 4 This way, the algae can conduct photosynthesis when moisture is available, and is protected at other times. When the cortex is more transparent, the algae show more clearly and the lichen looks greener.
Lichens can show intense antioxidant activity. [ 62 ] [ 63 ] Secondary metabolites are often deposited as crystals in the apoplast . [ 64 ] Secondary metabolites are thought to play a role in preference for some substrates over others. [ 64 ]
Lichens often have a regular but very slow growth rate of less than a millimeter per year.
In crustose lichens, the area along the margin is where the most active growth is taking place. [ 39 ] : 159 Most crustose lichens grow only 1–2 mm in diameter per year.
Lichens may be long-lived , with some considered to be among the oldest living organisms. [ 6 ] [ 24 ] Lifespan is difficult to measure because what defines the "same" individual lichen is not precise. [ 65 ] Lichens grow by vegetatively breaking off a piece, which may or may not be defined as the "same" lichen, and two lichens can merge, then becoming the "same" lichen. [ 65 ] One specimen of Rhizocarpon geographicum on East Baffin Island has an estimated age of 9500 years. [ 66 ] [ 67 ] Thalli of Rhizocarpon geographicum and Rhizocarpon eupetraeoides / inarense in the central Brooks Range of northern Alaska have been given a maximum possible age of 10,000–11,500 years. [ 68 ] [ 69 ]
Unlike simple dehydration in plants and animals, lichens may experience a complete loss of body water in dry periods. [ 16 ] Lichens are capable of surviving extremely low levels of water content ( poikilohydric ). [ 70 ] : 5–6 They quickly absorb water when it becomes available again, becoming soft and fleshy. [ 16 ]
In tests, lichen survived and showed remarkable results on the adaptation capacity of photosynthetic activity within the simulation time of 34 days under Martian conditions in the Mars Simulation Laboratory (MSL) maintained by the German Aerospace Center (DLR). [ 71 ] [ 72 ]
The European Space Agency has discovered that lichens can survive unprotected in space. In an experiment led by Leopoldo Sancho from the Complutense University of Madrid, two species of lichen— Rhizocarpon geographicum and Rusavskia elegans —were sealed in a capsule and launched on a Russian Soyuz rocket 31 May 2005. Once in orbit, the capsules were opened and the lichens were directly exposed to the vacuum of space with its widely fluctuating temperatures and cosmic radiation. After 15 days, the lichens were brought back to earth and were found to be unchanged in their ability to photosynthesize. [ 73 ] [ 74 ]
Many lichens reproduce asexually, either by a piece breaking off and growing on its own ( vegetative reproduction ) or through the dispersal of diaspores containing a few algal cells surrounded by fungal cells. [ 4 ] Because of the relative lack of differentiation in the thallus, the line between diaspore formation and vegetative reproduction is often blurred. Fruticose lichens can fragment, and new lichens can grow from the fragment ( vegetative reproduction ). Many lichens break up into fragments when they dry, dispersing themselves by wind action, to resume growth when moisture returns. [ 75 ] [ 76 ] Soredia (singular: "soredium") are small groups of algal cells surrounded by fungal filaments that form in structures called soralia, from which the soredia can be dispersed by wind. [ 4 ] Isidia (singular: "isidium") are branched, spiny, elongated, outgrowths from the thallus that break off for mechanical dispersal. [ 4 ] Lichen propagules ( diaspores ) typically contain cells from both partners, although the fungal components of so-called "fringe species" rely instead on algal cells dispersed by the "core species". [ 77 ]
Structures involved in reproduction often appear as discs, bumps, or squiggly lines on the surface of the thallus. [ 15 ] : 4 Though it has been argued that sexual reproduction in photobionts is selected against, there is strong evidence that suggests meiotic activities (sexual reproduction) in Trebouxia . [ 78 ] [ 79 ] Many lichen fungi reproduce sexually like other fungi, producing spores formed by meiosis and fusion of gametes. Following dispersal, such fungal spores must meet with a compatible algal partner before a functional lichen can form.
Some lichen fungi belong to the phylum Basidiomycota ( basidiolichens ) and produce mushroom -like reproductive structures resembling those of their nonlichenized relatives.
Most lichen fungi belong to Ascomycetes ( ascolichens ). Among the ascolichens, spores are produced in spore-producing structures called ascomata . [ 15 ] The most common types of ascomata are the apothecium (plural: apothecia) and perithecium (plural: perithecia). [ 15 ] : 14 Apothecia are usually cups or plate-like discs located on the top surface of the lichen thallus. When apothecia are shaped like squiggly line segments instead of like discs, they are called lirellae . [ 15 ] : 14 Perithecia are shaped like flasks that are immersed in the lichen thallus tissue, which has a small hole for the spores to escape the flask, and appear like black dots on the lichen surface. [ 15 ] : 14
The three most common spore body types are raised discs called apothecia (singular: apothecium), bottle-like cups with a small hole at the top called perithecia (singular: perithecium), and pycnidia (singular: pycnidium), shaped like perithecia but without asci (an ascus is the structure that contains and releases the sexual spores in fungi of the Ascomycota ). [ 80 ]
The apothecium has a layer of exposed spore-producing cells called asci (singular: ascus), and is usually a different color from the thallus tissue. [ 15 ] : 14 When the apothecium has an outer margin, the margin is called the exciple . [ 15 ] : 14 When the exciple has a color similar to colored thallus tissue the apothecium or lichen is called lecanorine , meaning similar to members of the genus Lecanora . [ 15 ] : 14 When the exciple is blackened like carbon it is called lecideine meaning similar to members of the genus Lecidea . [ 15 ] : 14 When the margin is pale or colorless it is called biatorine . [ 15 ] : 14
A " podetium " (plural: podetia ) is a lichenized stalk-like structure of the fruiting body rising from the thallus, associated with some fungi that produce a fungal apothecium . [ 36 ] Since it is part of the reproductive tissue, podetia are not considered part of the main body (thallus), but may be visually prominent. [ 36 ] The podetium may be branched, and sometimes cup-like. They usually bear the fungal pycnidia or apothecia or both. [ 36 ] Many lichens have apothecia that are visible to the naked eye. [ 4 ]
Most lichens produce abundant sexual structures. [ 81 ] Many species appear to disperse only by sexual spores. [ 81 ] For example, the crustose lichens Graphis scripta and Ochrolechia parella produce no symbiotic vegetative propagules. Instead, the lichen-forming fungi of these species reproduce sexually by self-fertilization (i.e. they are homothallic ). This breeding system may enable successful reproduction in harsh environments. [ 81 ]
Mazaedia (singular: mazaedium) are apothecia shaped like a dressmaker's pin in pin lichens , where the fruiting body is a brown or black mass of loose ascospores enclosed by a cup-shaped exciple, which sits on top of a tiny stalk. [ 15 ] : 15
Lichens are classified by the fungal component. Lichen species are given the same scientific name ( binomial name ) as the fungus species in the lichen. Lichens are being integrated into the classification schemes for fungi. The alga bears its own scientific name, which bears no relationship to that of the lichen or fungus. [ 82 ] There are about 20,000 identified lichen species, [ 83 ] [ 84 ] and taxonomists have estimated that the total number of lichen species (including those yet undiscovered) might be as high as 28,000. [ 85 ] Nearly 20% of known fungal species are associated with lichens. [ 53 ]
" Lichenized fungus " may refer to the entire lichen, or to just the fungus. This may cause confusion without context. A particular fungus species may form lichens with different algae species, giving rise to what appear to be different lichen species, but which are still classified (as of 2014) as the same lichen species. [ 86 ]
Formerly, some lichen taxonomists placed lichens in their own division, the Mycophycophyta , but this practice is no longer accepted because the components belong to separate lineages . Neither the ascolichens nor the basidiolichens form monophyletic lineages in their respective fungal phyla, but they do form several major solely or primarily lichen-forming groups within each phylum. [ 87 ] Even more unusual than basidiolichens is the fungus Geosiphon pyriforme , a member of the Glomeromycota that is unique in that it encloses a cyanobacterial symbiont inside its cells. Geosiphon is not usually considered to be a lichen, and its peculiar symbiosis was not recognized for many years. The genus is more closely allied to endomycorrhizal genera. Fungi from Verrucariales also form marine lichens with the brown algae Petroderma maculiforme , [ 88 ] and have a symbiotic relationship with seaweed (such as rockweed ) and Blidingia minima , where the algae are the dominant components. The fungi is thought to help the rockweeds to resist desiccation when exposed to air. [ 89 ] [ 90 ] In addition, lichens can also use yellow-green algae ( Heterococcus ) as their symbiotic partner. [ 91 ]
Lichens independently emerged from fungi associating with algae and cyanobacteria multiple times throughout history. [ 92 ]
The fungal component of a lichen is called the mycobiont . The mycobiont may be an Ascomycete or Basidiomycete . [ 18 ] The associated lichens are called either ascolichens or basidiolichens , respectively. Living as a symbiont in a lichen appears to be a successful way for a fungus to derive essential nutrients, since about 20% of all fungal species have acquired this mode of life. [ 93 ]
Thalli produced by a given fungal symbiont with its differing partners may be similar, [ citation needed ] and the secondary metabolites identical, [ citation needed ] indicating [ citation needed ] that the fungus has the dominant role in determining the morphology of the lichen. But the same mycobiont with different photobionts may also produce very different growth forms. [ 86 ] Lichens are known in which there is one fungus associated with two or even three algal species.
Although each lichen thallus generally appears homogeneous, some evidence seems to suggest that the fungal component may consist of more than one genetic individual of that species. [ citation needed ]
Two or more fungal species can interact to form the same lichen. [ 94 ]
The following table lists the orders and families of fungi that include lichen-forming species.
The photosynthetic partner in a lichen is called a photobiont . The photobionts in lichens come from a variety of simple prokaryotic and eukaryotic organisms. In the majority of lichens the photobiont is a green alga ( Chlorophyta ) or a cyanobacterium . In some lichens both types are present; in such cases, the alga is typically the primary partner, with the cyanobacteria being located in cryptic pockets. [ 95 ] Algal photobionts are called phycobionts , while cyanobacterial photobionts are called cyanobionts . [ 44 ] About 90% of all known lichens have phycobionts, and about 10% have cyanobionts. [ 44 ] Approximately 100 species of photosynthetic partners from 40 [ 44 ] genera and five distinct classes (prokaryotic: Cyanophyceae ; eukaryotic: Trebouxiophyceae , Phaeophyceae , Chlorophyceae ) have been found to associate with the lichen-forming fungi. [ 96 ]
Common algal photobionts are from the genera Trebouxia , Trentepohlia , Pseudotrebouxia , or Myrmecia . Trebouxia is the most common genus of green algae in lichens, occurring in about 40% of all lichens. "Trebouxioid" means either a photobiont that is in the genus Trebouxia , or resembles a member of that genus, and is therefore presumably a member of the class Trebouxiophyceae . [ 36 ] The second most commonly represented green alga genus is Trentepohlia . [ 37 ] Overall, about 100 species of eukaryotes are known to occur as photobionts in lichens. All the algae are probably able to exist independently in nature as well as in the lichen. [ 94 ]
A " cyanolichen " is a lichen with a cyanobacterium as its main photosynthetic component (photobiont). [ 97 ] Most cyanolichen are also ascolichens, but a few basidiolichen like Dictyonema and Acantholichen have cyanobacteria as their partner. [ 98 ]
The most commonly occurring cyanobacterium genus is Nostoc . [ 94 ] Other [ 37 ] common cyanobacterium photobionts are from Scytonema . [ 18 ] Many cyanolichens are small and black, and have limestone as the substrate. [ citation needed ] Another cyanolichen group, the jelly lichens of the genera Collema or Leptogium are gelatinous and live on moist soils. Another group of large and foliose species including Peltigera , Lobaria , and Degelia are grey-blue, especially when dampened or wet. Many of these characterize the Lobarion communities of higher rainfall areas in western Britain, e.g., in the Celtic rain forest . Strains of cyanobacteria found in various cyanolichens are often closely related to one another. [ 99 ] They differ from the most closely related free-living strains. [ 99 ]
The lichen association is a close symbiosis. It extends the ecological range of both partners but is not always obligatory for their growth and reproduction in natural environments, since many of the algal symbionts can live independently. A prominent example is the alga Trentepohlia , which forms orange-coloured populations on tree trunks and suitable rock faces. Lichen propagules ( diaspores ) typically contain cells from both partners, although the fungal components of so-called "fringe species" rely instead on algal cells dispersed by the "core species". [ 77 ]
The same cyanobiont species can occur in association with different fungal species as lichen partners. [ 100 ] The same phycobiont species can occur in association with different fungal species as lichen partners. [ 44 ] More than one phycobiont may be present in a single thallus. [ 44 ]
A single lichen may contain several algal genotypes . [ 101 ] [ 102 ] These multiple genotypes may better enable response to adaptation to environmental changes, and enable the lichen to inhabit a wider range of environments. [ 103 ]
There are about 20,000 known lichen species . [ 18 ] But what is meant by "species" is different from what is meant by biological species in plants, animals, or fungi, where being the same species implies that there is a common ancestral lineage . [ 18 ] Because lichens are combinations of members of two or even three different biological kingdoms , these components must have a different ancestral lineage from each other. By convention, lichens are still called "species" anyway, and are classified according to the species of their fungus, not the species of the algae or cyanobacteria. Lichens are given the same scientific name ( binomial name ) as the fungus in them, which may cause some confusion. The alga bears its own scientific name, which has no relationship to the name of the lichen or fungus. [ 82 ]
Depending on context, "lichenized fungus" may refer to the entire lichen, or to the fungus when it is in the lichen, which can be grown in culture in isolation from the algae or cyanobacteria. Some algae and cyanobacteria are found naturally living outside of the lichen. The fungal, algal, or cyanobacterial component of a lichen can be grown by itself in culture. When growing by themselves, the fungus, algae, or cyanobacteria have very different properties than those of the lichen. Lichen properties such as growth form, physiology, and biochemistry, are very different from the combination of the properties of the fungus and the algae or cyanobacteria.
The same fungus growing in combination with different algae or cyanobacteria, can produce lichens that are very different in most properties, meeting non-DNA criteria for being different "species". Historically, these different combinations were classified as different species. When the fungus is identified as being the same using modern DNA methods, these apparently different species get reclassified as the same species under the current (2014) convention for classification by fungal component. This has led to debate about this classification convention. These apparently different "species" have their own independent evolutionary history. [ 4 ] [ 86 ]
There is also debate as to the appropriateness of giving the same binomial name to the fungus, and to the lichen that combines that fungus with an alga or cyanobacterium ( synecdoche ). This is especially the case when combining the same fungus with different algae or cyanobacteria produces dramatically different lichen organisms, which would be considered different species by any measure other than the DNA of the fungal component. If the whole lichen produced by the same fungus growing in association with different algae or cyanobacteria, were to be classified as different "species", the number of "lichen species" would be greater.
The largest number of lichenized fungi occur in the Ascomycota , with about 40% of species forming such an association. [ 82 ] Some of these lichenized fungi occur in orders with nonlichenized fungi that live as saprotrophs or plant parasites (for example, the Leotiales , Dothideales , and Pezizales ). Other lichen fungi occur in only five orders in which all members are engaged in this habit (Orders Graphidales , Gyalectales , Peltigerales , Pertusariales , and Teloschistales ). Overall, about 98% of lichens have an ascomycetous mycobiont. [ 104 ] Next to the Ascomycota, the largest number of lichenized fungi occur in the unassigned fungi imperfecti , a catch-all category for fungi whose sexual form of reproduction has never been observed. [ citation needed ] Comparatively few basidiomycetes are lichenized, but these include agarics , such as species of Lichenomphalia , clavarioid fungi , such as species of Multiclavula , and corticioid fungi , such as species of Dictyonema .
Lichen identification uses growth form, microscopy and reactions to chemical tests.
The outcome of the "Pd test" is called "Pd", which is also used as an abbreviation for the chemical used in the test, para-phenylenediamine . [ 36 ] If putting a drop on a lichen turns an area bright yellow to orange, this helps identify it as belonging to either the genus Cladonia or Lecanora . [ 36 ]
The fossil record for lichens is poor. [ 105 ] The extreme habitats that lichens dominate, such as tundra, mountains, and deserts, are not ordinarily conducive to producing fossils. [ 105 ] [ 106 ] There are fossilized lichens embedded in amber. The fossilized Anzia is found in pieces of amber in northern Europe and dates back approximately 40 million years. [ 107 ] Lichen fragments are also found in fossil leaf beds, such as Lobaria from Trinity County in northern California, US, dating back to the early to middle Miocene . [ 108 ]
The oldest fossil lichen in which both symbiotic partners have been recovered is Winfrenatia , an early zygomycetous ( Glomeromycotan ) lichen symbiosis that may have involved controlled parasitism, [ citation needed ] is permineralized in the Rhynie Chert of Scotland, dating from early Early Devonian , about 400 million years ago. [ 109 ] The slightly older fossil Spongiophyton has also been interpreted as a lichen on morphological [ 110 ] and isotopic [ 111 ] grounds, although the isotopic basis is decidedly shaky. [ 112 ] It has been demonstrated that Silurian - Devonian fossils Nematothallus [ 113 ] and Prototaxites [ 114 ] were lichenized. Thus lichenized Ascomycota and Basidiomycota were a component of Early Silurian - Devonian terrestrial ecosystems. [ 115 ] [ 116 ] Newer research suggests that lichen evolved after the evolution of land plants. [ 117 ]
The ancestral ecological state of both Ascomycota and Basidiomycota was probably saprobism , and independent lichenization events may have occurred multiple times. [ 118 ] [ 119 ] In 1995, Gargas and colleagues proposed that there were at least five independent origins of lichenization; three in the basidiomycetes and at least two in the Ascomycetes. [ 120 ] Lutzoni et al. (2001) suggest lichenization probably evolved earlier and was followed by multiple independent losses. Some non-lichen-forming fungi may have secondarily lost the ability to form a lichen association. As a result, lichenization has been viewed as a highly successful nutritional strategy. [ 121 ] [ 122 ]
Lichenized Glomeromycota may extend well back into the Precambrian. Lichen-like fossils consisting of coccoid cells ( cyanobacteria ?) and thin filaments (mucoromycotinan Glomeromycota ?) are permineralized in marine phosphorite of the Doushantuo Formation in southern China. These fossils are thought to be 551 to 635 million years old or Ediacaran . [ 123 ] Ediacaran acritarchs also have many similarities with Glomeromycotan vesicles and spores. [ 124 ] It has also been claimed that Ediacaran fossils including Dickinsonia , [ 125 ] were lichens, [ 126 ] although this claim is controversial. [ 127 ] Endosymbiotic Glomeromycota comparable with living Geosiphon may extend back into the Proterozoic in the form of 1500 million year old Horodyskia [ 128 ] and 2200 million year old Diskagma . [ 129 ] Discovery of these fossils suggest that fungi developed symbiotic partnerships with photoautotrophs long before the evolution of vascular plants, though the Ediacaran lichen hypothesis is largely rejected due to an inappropriate definition of lichens based on taphonomy and substrate ecology. [ 130 ] However, a 2019 study by the same scientist who rejected the Ediacaran lichen hypothesis, Nelsen, used new time-calibrated phylogenies to conclude that there is no evidence of lichen before the existence of vascular plants. [ 131 ]
Lecanoromycetes, one of the most common classes of lichen-forming fungi, diverged from its ancestor, which may have also been lichen forming, around 258 million years ago, during the late Paleozoic period. However, the closely related clade Euritiomycetes appears to have become lichen-forming only 52 million years ago, during the early Cenozoic period. [ 132 ]
Lichens grow on and in a wide range of substrates and habitats, including some of the most extreme conditions on earth. [ 133 ] They are abundant growing on bark, leaves, and hanging from epiphyte branches in rain forests and in temperate woodland . They grow on bare rock, walls, gravestones, roofs, and exposed soil surfaces. They can survive in some of the most extreme environments on Earth: arctic tundra , hot dry deserts , rocky coasts, and toxic slag heaps . They can live inside solid rock, growing between the grains, and in the soil as part of a biological soil crust in arid habitats such as deserts. Some lichens do not grow on anything, living out their lives blowing about the environment. [ 4 ]
When growing on mineral surfaces, some lichens slowly decompose their substrate by chemically degrading and physically disrupting the minerals, contributing to the process of weathering by which rocks are gradually turned into soil. While this contribution to weathering is usually benign, it can cause problems for artificial stone structures. For example, there is an ongoing lichen growth problem on Mount Rushmore National Memorial that requires the employment of mountain-climbing conservators to clean the monument. [ 134 ]
Lichens are not parasites on the plants they grow on, but only use them as a substrate. The fungi of some lichen species may "take over" the algae of other lichen species. [ 16 ] [ 135 ] Lichens make their own food from their photosynthetic parts and by absorbing minerals from the environment. [ 16 ] Lichens growing on leaves may have the appearance of being parasites on the leaves, but they are not. Some lichens in Diploschistes parasitise other lichens. Diploschistes muscorum starts its development in the tissue of a host Cladonia species. [ 55 ] : 30 [ 37 ] : 171
In the arctic tundra, lichens, together with mosses and liverworts , make up the majority of the ground cover , which helps insulate the ground and may provide forage for grazing animals. An example is " reindeer moss ", which is a lichen, not a moss. [ 16 ]
There are only two species of known permanently submerged lichens; Hydrothyria venosa is found in fresh water environments, and Verrucaria serpuloides is found in marine environments. [ 136 ]
A crustose lichen that grows on rock is called a saxicolous lichen . [ 36 ] [ 39 ] : 159 Crustose lichens that grow on the rock are epilithic , and those that grow immersed inside rock, growing between the crystals with only their fruiting bodies exposed to the air, are called endolithic lichens . [ 35 ] [ 39 ] : 159 [ 97 ] A crustose lichen that grows on bark is called a corticolous lichen . [ 39 ] : 159 A lichen that grows on wood from which the bark has been stripped is called a lignicolous lichen . [ 45 ] Lichens that grow immersed inside plant tissues are called endophloidic lichens or endophloidal lichens . [ 35 ] [ 39 ] : 159 Lichens that use leaves as substrates, whether the leaf is still on the tree or on the ground, are called epiphyllous or foliicolous . [ 44 ] A terricolous lichen grows on the soil as a substrate. Many squamulose lichens are terricolous. [ 39 ] : 159 Umbilicate lichens are foliose lichens that are attached to the substrate at only one point. [ 35 ] A vagrant lichen is not attached to a substrate at all, and lives its life being blown around by the wind.
In addition to distinct physical mechanisms by which lichens break down raw stone, studies indicate lichens attack stone chemically, entering newly chelated minerals into the ecology. The substances exuded by lichens, known for their strong ability to bind and sequester metals, along with the common formation of new minerals, especially metal oxalates , and the traits of the substrates they alter, all highlight the important role lichens play in the process of chemical weathering . [ 137 ] Over time, this activity creates new fertile soil from stone.
Lichens may be important in contributing nitrogen to soils in some deserts through being eaten, along with their rock substrate, by snails, which then defecate, putting the nitrogen into the soils. [ 138 ] Lichens help bind and stabilize soil sand in dunes. [ 4 ] In deserts and semi-arid areas, lichens are part of extensive, living biological soil crusts , essential for maintaining the soil structure. [ 4 ]
Lichens are pioneer species , among the first living things to grow on bare rock or areas denuded of life by a disaster. [ 4 ] Lichens may have to compete with plants for access to sunlight, but because of their small size and slow growth, they thrive in places where higher plants have difficulty growing. Lichens are often the first to settle in places lacking soil, constituting the sole vegetation in some extreme environments such as those found at high mountain elevations and at high latitudes. [ 139 ] Some survive in the tough conditions of deserts, and others on frozen soil of the Arctic regions. [ 140 ]
A major ecophysiological advantage of lichens is that they are poikilohydric ( poikilo - variable, hydric - relating to water), meaning that though they have little control over the status of their hydration, they can tolerate irregular and extended periods of severe desiccation . Like some mosses , liverworts , ferns and a few resurrection plants , upon desiccation, lichens enter a metabolic suspension or stasis (known as cryptobiosis ) in which the cells of the lichen symbionts are dehydrated to a degree that halts most biochemical activity. In this cryptobiotic state, lichens can survive wider extremes of temperature, radiation and drought in the harsh environments they often inhabit.
Lichens do not have roots and do not need to tap continuous reservoirs of water like most higher plants, thus they can grow in locations impossible for most plants, such as bare rock, sterile soil or sand, and various artificial structures such as walls, roofs, and monuments. Many lichens also grow as epiphytes ( epi - on the surface, phyte - plant) on plants, particularly on the trunks and branches of trees. When growing on plants, lichens are not parasites ; they do not consume any part of the plant nor poison it. Lichens produce allelopathic chemicals that inhibit the growth of mosses. Some ground-dwelling lichens, such as members of the subgenus Cladina (reindeer lichens), produce allelopathic chemicals that leach into the soil and inhibit the germination of seeds, spruce and other plants. [ 141 ] Stability (that is, longevity) of their substrate is a major factor of lichen habitats. Most lichens grow on stable rock surfaces or the bark of old trees, but many others grow on soil and sand. In these latter cases, lichens are often an important part of soil stabilization; indeed, in some desert ecosystems, vascular (higher) plant seeds cannot become established except in places where lichen crusts stabilize the sand and help retain water.
Lichens may be eaten by some animals, such as reindeer , living in arctic regions. The larvae of a number of Lepidoptera species feed exclusively on lichens. These include common footman and marbled beauty . They are very low in protein and high in carbohydrates, making them unsuitable for some animals. The Northern flying squirrel uses it for nesting, food and winter water.
If lichens are exposed to air pollutants at all times, without any deciduous parts, they are unable to avoid the accumulation of pollutants. Also lacking stomata and a cuticle , lichens may absorb aerosols and gases over the entire thallus surface from which they may readily diffuse to the photobiont layer. [ 142 ] Because lichens do not possess roots, their primary source of most elements is the air, and therefore elemental levels in lichens often reflect the accumulated composition of ambient air. The processes by which atmospheric deposition occurs include fog and dew , gaseous absorption, and dry deposition. [ 143 ] Consequently, environmental studies with lichens emphasize their feasibility as effective biomonitors of atmospheric quality. [ 142 ]
Not all lichens are equally sensitive to air pollutants , so different lichen species show different levels of sensitivity to specific atmospheric pollutants. [ 144 ] The sensitivity of a lichen to air pollution is directly related to the energy needs of the mycobiont, so that the stronger the dependency of the mycobiont on the photobiont, the more sensitive the lichen is to air pollution. [ 145 ] Upon exposure to air pollution, the photobiont may use metabolic energy for repair of its cellular structures that would otherwise be used for maintenance of its photosynthetic activity, therefore leaving less metabolic energy available for the mycobiont. The alteration of the balance between the photobiont and mycobiont can lead to the breakdown of the symbiotic association. Therefore, lichen decline may result not only from the accumulation of toxic substances, but also from altered nutrient supplies that favor one symbiont over the other. [ 142 ]
This interaction between lichens and air pollution has been used as a means of monitoring air quality since 1859, with more systematic methods developed by William Nylander in 1866. [ 4 ]
Lichens are eaten by many different cultures across the world. Although some lichens are only eaten in times of famine , others are a staple food or even a delicacy . Two obstacles are often encountered when eating lichens: lichen polysaccharides are generally indigestible to humans, and lichens usually contain mildly toxic secondary compounds that should be removed before eating. Very few lichens are poisonous, but those high in vulpinic acid or usnic acid are toxic. [ 146 ] Most poisonous lichens are yellow. [ citation needed ]
In the past, Iceland moss ( Cetraria islandica ) was an important source of food for humans in northern Europe, and was cooked as a bread, porridge, pudding, soup, or salad. It is also fed to cattle, pigs and ponies. Bryoria fremontii (edible horsehair lichen) was an important food in parts of North America, where it was usually pitcooked . Northern peoples in North America and Siberia traditionally eat the partially digested reindeer lichen ( Cladina spp.) after they remove it from the rumen of caribou or reindeer that have been killed. Rock tripe ( Umbilicaria spp. and Lasalia spp.) is a lichen that has frequently been used as an emergency food in North America, and one species, Umbilicaria esculenta , ( iwatake in Japanese) is used in a variety of traditional Korean and Japanese foods. [ 147 ]
Lichenometry is a technique used to determine the age of exposed rock surfaces based on the size of lichen thalli. Introduced by Beschel in the 1950s, [ 148 ] the technique has found many applications. it is used in archaeology , palaeontology , and geomorphology . It uses the presumed regular but slow rate of lichen growth to determine the age of exposed rock . [ 41 ] : 9 [ 149 ] Measuring the diameter (or other size measurement) of the largest lichen of a species on a rock surface indicates the length of time since the rock surface was first exposed. Lichen can be preserved on old rock faces for up to [ citation needed ] 10,000 years, providing the maximum age limit of the technique, though it is most accurate (within 10% error) when applied to surfaces that have been exposed for less than 1,000 years. [ 150 ] Lichenometry is especially useful for dating surfaces less than 500 years old, as radiocarbon dating techniques are less accurate over this period. [ 151 ] The lichens most commonly used for lichenometry are those of the genera Rhizocarpon (e.g. the species Rhizocarpon geographicum , map lichen) and Xanthoria .
Lichens have been shown to degrade polyester resins , as can be seen in archaeological sites in the Roman city of Baelo Claudia in Spain. [ 152 ] Lichens can accumulate several environmental pollutants such as lead, copper, and radionuclides . [ 153 ] Some species of lichen, such as Parmelia sulcata (called a hammered shield lichen, among other names) and Lobaria pulmonaria (lung lichen), and many in the Cladonia genus , have been shown to produce serine proteases capable of the degradation of pathogenic forms of prion protein (PrP), which may be useful in treating contaminated environmental reservoirs. [ 154 ] [ 155 ] [ 156 ]
Many lichens produce secondary compounds, including pigments that reduce harmful amounts of sunlight and powerful toxins that deter herbivores or kill bacteria. These compounds are very useful for lichen identification, and have had economic importance as dyes such as cudbear or primitive antibiotics .
A pH indicator (which can indicate acidic or basic substances) called litmus is a dye extracted from the lichen Roccella tinctoria ("dyer's weed") [ 157 ] by boiling. It gives its name to the well-known litmus test .
Traditional dyes of the Scottish Highlands for Harris tweed and other traditional cloths were made from lichens, including the orange Xanthoria parietina ("common orange lichen") and the grey foliaceous Parmelia saxatilis common on rocks and known colloquially as "crottle".
There are reports dating almost 2,000 years old of lichens being used to make purple and red dyes. [ 158 ] Of great historical and commercial significance are lichens belonging to the family Roccellaceae , commonly called orchella weed or orchil. Orcein and other lichen dyes have largely been replaced by synthetic versions .
Historically, in traditional medicine of Europe, Lobaria pulmonaria was collected in large quantities as "lungwort", due to its lung-like appearance (the " doctrine of signatures " suggesting that herbs can treat body parts that they physically resemble).Similarly, Peltigera leucophlebia ("ruffled freckled pelt") was used as a supposed cure for thrush , due to the resemblance of its cephalodia to the appearance of the disease. [ 37 ]
Lichens produce metabolites being researched for their potential therapeutic or diagnostic value. [ 159 ] Some metabolites produced by lichens are structurally and functionally similar to broad-spectrum antibiotics while few are associated respectively to antiseptic similarities. [ 160 ] Usnic acid is the most commonly studied metabolite produced by lichens. [ 160 ] It is also under research as a bactericidal agent against Escherichia coli and Staphylococcus aureus . [ 161 ]
Colonies of lichens may be spectacular in appearance, dominating the surface of the visual landscape as part of the aesthetic appeal to visitors of Yosemite National Park , Sequoia National Park , and the Bay of Fires . [ 41 ] : 2 Orange and yellow lichens add to the ambience of desert trees, tundras, and rocky seashores. Intricate webs of lichens hanging from tree branches add a mysterious aspect to forests. Fruticose lichens are used in model railroading [ 162 ] and other modeling hobbies as a material for making miniature trees and shrubs.
In early Midrashic literature, the Hebrew word " vayilafeth " in Ruth 3:8 is explained as referring to Ruth entwining herself around Boaz like lichen. [ 163 ] The 10th century Arab physician Al-Tamimi mentions lichens dissolved in vinegar and rose water being used in his day for the treatment of skin diseases and rashes. [ 164 ]
The plot of John Wyndham 's science fiction novel Trouble with Lichen revolves around an anti-aging chemical extracted from a lichen.
Although lichens had been recognized as organisms for quite some time, it was not until 1867, when Swiss botanist Simon Schwendener proposed his dual theory of lichens, that lichens are a combination of fungi with algae or cyanobacteria, whereby the true nature of the lichen association began to emerge. [ 165 ] Schwendener's hypothesis, which at the time lacked experimental evidence, arose from his extensive analysis of the anatomy and development in lichens, algae, and fungi using a light microscope . Many of the leading lichenologists at the time, such as James Crombie and Nylander , rejected Schwendener's hypothesis because the consensus was that all living organisms were autonomous. [ 165 ]
Other prominent biologists, such as Heinrich Anton de Bary , Albert Bernhard Frank , Beatrix Potter , Melchior Treub and Hermann Hellriegel , were not so quick to reject Schwendener's ideas and the concept soon spread into other areas of study, such as microbial, plant, animal and human pathogens. [ 165 ] [ 166 ] [ 167 ] When the complex relationships between pathogenic microorganisms and their hosts were finally identified, Schwendener's hypothesis began to gain popularity. Further experimental proof of the dual nature of lichens was obtained when Eugen Thomas published his results in 1939 on the first successful re-synthesis experiment. [ 165 ]
In the 2010s, a new facet of the fungi–algae partnership was discovered. Toby Spribille and colleagues found that many types of lichen that were long thought to be ascomycete –algae pairs were actually ascomycete– basidiomycete –algae trios. The third symbiotic partner in many lichens is a basidiomycete yeast. [ 3 ] [ 168 ] | https://en.wikipedia.org/wiki/Lichen |
Lichen anatomy and physiology is very different from the anatomy and physiology of the fungus and/or algae and/or cyanobacteria that make up the lichen when growing apart from the lichen, either naturally, or in culture. The fungal partner is called the mycobiont . The photosynthetic partner, algae or cyanobacteria, is called the photobiont . The body of a lichens that does not contain reproductive parts of the fungus is called the thallus . The thallus is different from those of either the fungus or alga growing separately. The fungus surrounds the algal cells, often enclosing them within complex fungal tissues unique to lichen associations. In many species the fungus penetrates the algal cell wall, forming penetration pegs or haustoria similar to those produced by pathogenic fungi. [ 1 ] [ 2 ] Lichens are capable of surviving extremely low levels of water content ( poikilohydric ). [ 3 ] However, the re-configuration of membranes following a period of dehydration requires several minutes at least.
The algal or cyanobacterial cells are photosynthetic , and as in plants they reduce atmospheric carbon dioxide into organic carbon sugars to feed both symbionts. Both partners gain water and mineral nutrients mainly from the atmosphere, through rain and dust. The fungal partner protects the alga by retaining water, serving as a larger capture area for mineral nutrients and, in some cases, provides minerals obtained from the substrate . If a cyanobacterium is present, as a primary partner or another symbiont in addition to green alga as in certain tripartite lichens, they can fix atmospheric nitrogen , complementing the activities of the green alga.
Although strains of cyanobacteria found in various cyanolichens are often closely related to one another, they differ from the most closely related free-living strains. [ 4 ] The lichen association is a close symbiosis. It extends the ecological range of both partners but is not always obligatory for their growth and reproduction in natural environments, since many of the algal symbionts can live independently. A prominent example is the alga Trentepohlia which forms orange-coloured populations on tree trunks and suitable rock faces. Lichen propagules ( diaspores ) typically contain cells from both partners, although the fungal components of so-called "fringe species" rely instead on algal cells dispersed by the "core species". [ 5 ]
Lichen associations may be examples of mutualism , commensalism or even parasitism , [ citation needed ] depending on the species. Cyanobacteria in laboratory settings can grow faster when they are alone rather than when they are part of a lichen.
In tests, lichen survived and showed remarkable results on the adaptation capacity of photosynthetic activity within the simulation time of 34 days under Martian conditions in the Mars Simulation Laboratory (MSL) maintained by the German Aerospace Center (DLR). [ 6 ] [ 7 ]
Living as a symbiont in a lichen appears to be a successful way for a fungus to derive essential nutrients, as about 20% of all fungal species have acquired this mode of life. The fungal partner may be an Ascomycete or Basidiomycete . [ 9 ] Common algal partners are Trebouxia , Pseudotrebouxia , or Myrmecia . Common cyanobacterium partners include are Nostoc [ 1 ] or Scytonema . [ 9 ]
The largest number of lichenized fungi occur in the Ascomycota , with about 40% of species forming such an association. [ 10 ] Some of these lichenized fungi occur in orders with nonlichenized fungi that live as saprotrophs or plant parasites (for example, the Leotiales , Dothideales , and Pezizales ). Other lichen fungi occur in only five orders in which all members are engaged in this habit (Orders Graphidales , Gyalectales , Peltigerales , Pertusariales , and Teloschistales ). Lichenized and nonlichenized fungi can even be found in the same genus or species. [ citation needed ] Overall, about 98% of lichens have an ascomycetous mycobiont. Next to the Ascomycota, the largest number of lichenized fungi occur in the unassigned fungi imperfecti . Comparatively few basidiomycetes are lichenized, but these include agarics , such as species of Lichenomphalia , clavarioid fungi , such as species of Multiclavula , and corticioid fungi , such as species of Dictyonema .
The autotrophic symbionts occurring in lichens are a wide variety of simple, photosynthetic organisms commonly and traditionally known as algae. These symbionts include both prokaryotic and eukaryotic organisms. Approximately 100 species of photosynthetic partners from 40 genera and five distinct classes (prokaryotic: Cyanophyceae ; eukaryotic: Trebouxiophyceae , Phaeophyceae , Chlorophyceae ) have been found to associate with the lichen-forming fungi. [ 11 ] The prokaryotes belong to the Cyanobacteria , whose representatives are often called bluegreen algae. The bluegreen algae occur as symbionts in about 8% of the known lichens. The most commonly occurring genus is Nostoc . [ 12 ] The majority of the lichens contain eukaryotic autotrophs belonging to the Chlorophyta (green algae) or to the Xanthophyta ( yellow-green algae ). About 90% of all known lichens have a green alga as a symbiont, and among these, Trebouxia is the most common genus, occurring in about 40% of all lichens. The second most commonly represented green alga genus is Trentepohlia . Overall, about 100 species are known to occur as autotrophs in lichens. All the algae are probably able to exist independently in nature as well as in the lichen. [ 12 ]
A particular fungus species and algal species are not necessarily always associated together in a lichen. One fungus, for example, can form lichens with a variety of different algae. The thalli produced by a given fungal symbiont with its differing partners will be similar, and the secondary metabolites identical, indicating that the fungus has the dominant role in determining the morphology of the lichen. Further, the same algal species can occur in association with different fungal partners. Lichens are known in which there is one fungus associated with two or even three algal species. Rarely, the reverse can occur, and two or more fungal species can interact to form the same lichen. [ 12 ]
Both the lichen and the fungus partner bear the same scientific name, and the lichens are being integrated into the classification schemes for fungi. The alga bears its own scientific name, which bears no relationship to that of the lichen or fungi. [ 10 ]
Depending on context, the entire lichen, or just the fungus that is part of the lichen. Both the lichen and the fungus that is a part of the lichen are currently (2014) given the same species name, which creates an ambiguity. An example of when "lichenized fungus" refers to just the fungus is when the fungus is grown in culture without a phycobiont. An example where "lichenized fungus" refers to the entire lichen is in a list of classified lichens.
Some fungi can only be found living on lichens ( obligate parasites ), but are not considered part of the lichen. These are referred to as lichenolous fungi .
The photosynthetic component of a lichen is called the photobiont or phycobiont . [ 13 ] Sometimes the photobiont is a green alga ( Chlorophyta ), sometimes a 'blue-green alga' ( cyanobacterium , a photosynthetic bacterium rather than an alga in the strict sense), and sometimes both. The layer of tissue containing the cells of the photobiont is called the "photobiontic layer". [ 13 ]
" Clorococcoid " means a green alga (Chlorophyta) that has single cells that are globose , which is common in lichens. [ 14 ] These were once classified in the order Chlorococcales , which you may find stated in older literature, but new DNA data shows many independent lines of evolution exist among this formerly large taxonomic group. Chlorococcales is now a relatively small order and may no longer include any lichen photobionts. Trebouxia , once included here, is now considered to be in a separate class, Trebouxiophyceae . " Trebouxioid " refers to members of this class or algae resembling them.
" Trebouxioid " means a clorococcoid green algal photobiont belongs to the genus Trebouxia , or resembles a member of that genus, and is therefore presumably a member of the class Trebouxiophyceae . [ 13 ]
A cyanolichen is a lichen with a cyanobacteria as its main photosynthetic component ( photobiont ). [ 14 ] Many cyanolichens are small and black, and have limestone as the substrate. Another cyanolichen group, the jelly lichens ( e.g., from the genera Collema or Leptogium ) are large and foliose (e.g., species of Peltigera , Lobaria , and Degelia ). These lichen species are grey-blue, especially when dampened or wet. Many of these characterize the Lobarion communities of higher rainfall areas in western Britain, e.g., in the Celtic Rainforest . | https://en.wikipedia.org/wiki/Lichen_anatomy_and_physiology |
Lichen morphology describes the external appearance and structures of a lichen . These can vary considerably from species to species. Lichen growth forms are used to group lichens by "vegetative" thallus types, and forms of "non-vegetative" reproductive parts. Some lichen thalli have the aspect of leaves (foliose lichens); others cover the substrate like a crust (crustose lichens) ( illustration, right ), others such as the genus Ramalina adopt shrubby forms (fruticose lichens), and there are gelatinous lichens such as the genus Collema . [ 1 ]
Although the form of a lichen is determined by the genetic material of the fungal partner, association with a photobiont is required for the development of that form. When grown in the laboratory in the absence of its photobiont, a lichen fungus develops as an undifferentiated mass of hyphae . If combined with its photobiont under appropriate conditions, its characteristic form emerges, in the process called morphogenesis . [ 2 ] In a few remarkable cases, a single lichen fungus can develop into two very different lichen forms when associating with either a green algal or a cyanobacterial symbiont. Quite naturally, these alternative forms were at first considered to be different species, until they were found growing in a conjoined manner.
Under magnification, a section through a typical foliose lichen thallus reveals four layers of interlaced fungal filaments. The uppermost layer is formed by densely agglutinated fungal hyphae building a protective outer layer called the cortex , which can reach several hundred μm in thickness. [ 3 ] This cortex may be further topped by an epicortex 0.6-1μm thick in some Parmeliaceae, which may be with or without pores, and is secreted by cells—it is not itself cellular. [ 3 ] In lichens that include both green algal and cyanobacterial symbionts, the cyanobacteria may be held on the upper or lower surface in small pustules called cephalodia . Beneath the upper cortex is an algal layer composed of algal cells embedded in rather densely interwoven fungal hyphae. Each cell or group of cells of the photobiont is usually individually wrapped by hyphae, and in some cases penetrated by an haustorium . Beneath this algal layer is a third layer of loosely interwoven fungal hyphae without algal cells. This layer is called the medulla . Beneath the medulla, the bottom surface resembles the upper surface and is called the lower cortex , again consisting of densely packed fungal hyphae. The lower cortex of foliose lichens often bears rootlike fungal structures known as rhizines , which serve to attach the thallus to the substrate on which it grows. Lichens also sometimes contain structures made from fungal metabolites , for example crustose lichens sometimes have a polysaccharide layer in the cortex. Although each lichen thallus generally appears homogeneous, some evidence seems to suggest that the fungal component may consist of more than one genetic individual of that species. This seems to also be true of the photobiont species involved.
A podetium (plural podetia ) is a lichenized stem-like structure of an apothecium rising from the primary body of the thallus. [ 4 ] Since it is part of the reproductive tissue, it is not considered part of the thallus. [ 4 ] The podetium may be branched, and sometimes cup-like. It usually bears the pycnidia or apothecia or both. | https://en.wikipedia.org/wiki/Lichen_morphology |
Lichen products , also known as lichen substances , are organic compounds produced by a lichen . Specifically, they are secondary metabolites . Lichen products are represented in several different chemical classes, including terpenoids , orcinol derivatives, chromones , xanthones , depsides , and depsidones . Over 800 lichen products of known chemical structure have been reported in the scientific literature, and most of these compounds are exclusively found in lichens. [ 1 ] Examples of lichen products include usnic acid (a dibenzofuran ), atranorin (a depside), lichexanthone (a xanthone), salazinic acid (a depsidone), and isolichenan , an α-glucan . Many lichen products have biological activity , and research into these effects is ongoing. [ 2 ]
Most lichen products are biochemically synthesized via the acetyl-polymalonyl pathway (also known as polyketide pathway), while only a few originate from the mevalonate and shikimate biosynthetic pathways. [ 3 ]
Lichen products accumulate on the outer walls of the fungal hyphae , and are quite stable. Crystal deposits can be visualised using scanning electron microscopy . [ 4 ] For this reason, even very old herbarium specimens can be analysed. [ 5 ] The amount of lichen products in lichen (as a percentage of dry weight ) is typically between 0.1%–10%, although in some instances it may be as high as 30%. [ 6 ] They are usually found in the medulla , or less commonly, the cortex . [ 7 ]
In 1907, Wilhelm Zopf identified and classified about 150 lichen products. Seventy years later, this number had risen to 300, and by 1995, 850 lichen products were known; [ 8 ] as of 2021, more than 1000 have been identified. [ 9 ] Analytical methods were developed in the 1970s using thin-layer chromatography for the routine identification of lichen products. [ 10 ] [ 11 ] More recently, published techniques demonstrate ways to more efficiently harvest secondary metabolites from lichen samples. [ 12 ]
Lichen products play a crucial role in differentiating lichenised fungi, particularly in groups where morphological characteristics are less distinct. This approach is notably applied in the genus Lepraria , which lacks sexual reproduction and ascomata (fruiting bodies), typically key features for species identification. [ 20 ] Similarly, in genera with more complex structures like the crustose genus Ochrolechia , [ 21 ] and the fruticose Cladonia , [ 22 ] [ 23 ] the presence, absence, or substitution of specific lichen products is frequently used to distinguish species, especially when these variations align with differences in geographical distribution. [ 24 ] | https://en.wikipedia.org/wiki/Lichen_product |
Lichen stromatolites are laminar calcretes that are proposed as being formed by a sequence of repetitions of induration followed by lichen colonization. Endolithic lichens inhabit areas between grains of rock, chemically and physically weathering that rock, leaving a rind, which is then indurated (hardened), then recolonized. [ 1 ] [ 2 ]
This article about lichens or lichenology is a stub . You can help Wikipedia by expanding it .
This geology article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Lichen_stromatolite |
A lichenicolous fungus is a member of a specialised group of fungi that live exclusively on lichens as their host organisms. These fungi, comprising over 2,000 known species across 280 genera , exhibit a wide range of ecological strategies, including parasitism , commensalism , and mutualism . They can be found in diverse environments worldwide, from tropical to polar regions , and play important roles in lichen ecology and biodiversity . Lichenicolous fungi are classified into several taxonomic groups, with the majority belonging to the Ascomycota and a smaller portion to the Basidiomycota . Their interactions with host lichens range from mild parasitism to severe pathogenicity , sometimes causing significant damage to lichen communities.
While the study of lichenicolous fungi dates back to the mid-18th century, recent decades have seen significant advancements through modern research methods, including molecular techniques, metagenomics , and sophisticated imaging . These fungi show varying levels of host specificity, with some species restricted to a single lichen genus or species, while others can colonise multiple hosts. A unique subset, known as lichenicolous lichens, initiates its lifecycle as parasites but eventually becomes lichenised through a process called kleptosymbiosis . Various ecological and environmental factors, including altitude, microhabitat availability, and host specificity, influence the diversity and distribution of lichenicolous fungi.
The study of lichenicolous fungi presents unique challenges due to their microscopic size and intimate association with their hosts. Researchers employ various methods, from traditional culture techniques to advanced molecular approaches. Isolating and culturing of these fungi can be difficult, often requiring specialised media and growth conditions. Molecular methods have revolutionised the field, enabling more accurate identification and phylogenetic analysis. Nevertheless, distinguishing foreign hyphae within lichen thalli from the mycobiont proper (the fungal component of the lichen) remains a significant challenge.
Recent research has broadened our understanding of lichenicolous fungi, particularly within groups such as black fungi and the genus Cladophialophora . These studies have not only revealed new species but also highlighted the potential for lichens to serve as refugia for specialised fungal organisms. Advancements in isolation techniques, culturing methods, and molecular analyses have significantly advanced the field. The actual number of lichenicolous fungal species may be much higher than currently described, potentially reaching 3,000–5,000 species.
Lichenicolous fungi are a specialised group of fungi that live exclusively on lichens as their host organisms. The term "lichenicolous" comes from Latin, with "lichen" referring to the host and "cola" meaning "inhabitant". [ 1 ] These fungi are distinct from the fungal component of lichens themselves, which are known as lichenised fungi. [ 2 ] The study of lichenicolous fungi dates back to the mid-18th century, predating the recognition of lichens as symbiotic organisms. [ 3 ]
Lichenicolous fungi represent a highly diverse group, with over 2,000 known species across 280 genera, reflecting a wide range of ecological strategies and relationships with their lichen hosts. As of 1981, it was estimated that there might be as many as 300 genera and 1,000 species of lichenicolous fungi. [ 3 ] These relationships vary widely, including parasitism , where the fungus harms the lichen; commensalism , where the fungus benefits without affecting the lichen; and mutualism , where both organisms benefit. [ 1 ] Lichen thalli provide a complex and varied habitat for lichenicolous fungi, characterised by biological gradients that range from actively growing to decaying parts. This diversity within a single thallus creates a spectrum of microenvironments, which may contribute to the diversification of fungal life strategies. For example, certain lichen-associated fungi specialise in colonising epinecral layers or epicortices , structures lacking living host cells. Others prefer decaying parts of lichens, linking necrotic to saprobic life styles. The presence of these biological gradients within lichen thalli may act as a catalyst for the evolution and adaptation of lichenicolous fungi, potentially explaining the wide range of ecological relationships observed, from commensalism to parasitism. This complexity within the lichen habitat has led some researchers to suggest that lichens may serve as a 'cradle' for fungal evolution, fostering the development of diverse fungal life strategies. [ 4 ] Some lichenicolous fungi are pathogenic , while others are saprotrophic , decomposing dead lichen tissues. A few lichenicolous fungi eventually become lichenised themselves, integrating into the lichen structure. [ 1 ] Lichenicolous fungi can be non-lichenised or lichenised, obligate or facultative, and may or may not induce symptoms on the host. [ 5 ] They are part of the complex microbiome associated with lichens, which includes bacteria , algae , and other fungi. [ 6 ]
Lichenicolous basidiomycetes , which make up about 5% of all known lichenicolous fungi as of 2018, include both homobasidiomycetes and heterobasidiomycetes . [ 7 ] Homobasidiomycetes typically have basidia (spore-producing structures) that lack septa (internal partitions) and non-gelatinous basidiomata (fruiting bodies), while heterobasidiomycetes have septate basidia and gelatinous basidiomata. [ 7 ]
Morphologically , lichenicolous fungi can take diverse forms. They may produce visible fruiting bodies on the surface of the host lichen, or exist as hyphae within the host thallus . Some species can cause gall -like structures or other deformations in their hosts. [ 3 ]
By 2018, there were 2,319 accepted species of lichenicolous fungi, including around 1,250 species of lichenicolous ascomycetes in 280 genera, and 62 species of lichenicolous basidiomycetes in ten genera. [ 8 ] These can be categorised into three main groups: [ 8 ]
Advancements in molecular techniques and increased exploration have significantly expanded the known diversity of lichenicolous basidiomycetes, nearly doubling the species count since 2018. [ 7 ]
The known diversity of lichenicolous fungi has increased significantly in recent years. This growth in known diversity reflects a resurgence of interest in lichenicolous fungi in the late 20th century. [ 3 ] For example, Clauzade and Roux compiled 457 species in 1976, [ 9 ] which increased to 686 species by 1989. [ 10 ] In 1997, there were 894 accepted species of obligately lichenicolous fungi. By 2018, this number had grown to over 1,800, [ 8 ] with a rapid rate of discovery, averaging 45 new infrageneric taxa added per year over the past two decades. In total, about 1,170 infrageneric taxa have been added since 1990. [ 5 ]
Lichenicolous basidiomycetes , a diverse group within the Basidiomycota, are primarily found in the subdivisions Agaricomycotina and Pucciniomycotina . These subdivisions encompass a wide range of fungal forms, including those that produce complex fruiting bodies and those with simpler, often parasitic forms. [ 7 ]
Within the Agaricomycotina, lichenicolous species are present in two classes . In the Agaricomycetes , lichenicolous species are found in five of its eighteen orders : Agaricales , Atheliales , Boletales , Cantharellales , and Corticiales . In the Tremellomycetes, lichenicolous species are found in two of its five orders: Filobasidiales and Tremellales . Within the Pucciniomycotina, lichenicolous species are found in three classes: Agaricostilbomycetes , Cystobasidiomycetes , and Microbotryomycetes . [ 7 ]
As of 2018, there were 93 known species of lichenicolous basidiomycetes (plus 7 facultatively lichenicolous taxa). [ 8 ] Studies published since then have reported the discovery of many new species and even new genera. By 2022, the number of known species had nearly doubled to almost 200. [ 7 ] To put this diversity in context, the estimated 300 genera and 1000 species of lichenicolous fungi (as of 1981) can be compared with the 150 genera and 700 species of Gasteromycetes , or 90 genera and 600 species of Pezizales . [ 3 ]
The description of the new genus Crittendenia in the Pucciniomycotina represents a significant taxonomic development, as it redefines the classification of several host-specific parasitic species previously placed under Chionosphaera . [ 13 ] Another development is the recognition of several species complexes , such as Syzygospora physciacearum and Tremella parmeliarum , which have been split into multiple distinct species, often with high host specificity. [ 7 ]
Lichenicolous species are also found among black fungi , a group characterised by dark-coloured mycelia due to melanin in their cell walls. These fungi, known for their ability to colonise extreme habitats including lichen thalli, are represented by the genus Cladophialophora . In 2023, nine new species of Cladophialophora were described from lichens collected in China, exemplifying the ongoing discovery of novel lichenicolous fungi. These species were isolated from the medullary tissues of various lichen hosts, suggesting potential biotrophic or commensal relationships. Cladophialophora represents a significant lineage of lichenicolous fungi, with many species likely awaiting discovery. This research underscores the diversity of lichenicolous fungi and highlights the role of lichens as refugia for these specialised organisms. [ 14 ]
A biological analysis of lichenicolous fungi genera reveals distinct trends. A significant proportion of genera with lichenicolous representatives are exclusively lichenicolous, while others include saprophytic or lichen-forming species as well. For instance, among the lichenicolous Hyphomycetes, 43% of genera are confined to lichens, while for lichenicolous Coelomycetes , this figure rises to 75%. [ 3 ]
While the number of known lichenicolous basidiomycete species has significantly increased in recent years, it is believed that many more species remain undiscovered. As of 2022, estimates of global diversity suggest that there are over 1000 species of lichenicolous heterobasidiomycetes, more than 60 species of lichenicolous homobasidiomycetes, and more than 30 species of endolichenic homobasidiomycetes. These estimates are based on analyses of host specificity, current knowledge gaps, and the rate of new species discoveries. The actual number of species may be even higher, especially in under-explored regions and taxonomic groups. [ 7 ] Some researchers estimate that the total number of lichenicolous fungal species could potentially reach 3,000. [ 5 ]
The taxonomy and nomenclature of lichenicolous fungi have changed significantly with the adoption of the "one fungus–one name" principle in the International Code of Nomenclature for algae, fungi, and plants (Melbourne Code). This principle has important implications for classifying and naming lichenicolous fungi. [ 5 ] Under the new rules, all legitimate fungal names are treated equally for establishing priority , regardless of the life history stage. This change is significant for lichenicolous fungi, as many species were previously described separately in their sexual ( teleomorph ) and asexual (anamorph) states. [ 5 ]
The implementation of this principle has led to several taxonomic changes. When genetically identical teleomorphs and anamorphs have different names, the older name takes priority, unless the younger name is conserved. For species known only in their anamorphic state, if the teleomorph is discovered, the species is described or combined in the appropriate teleomorph genus, even if teleomorph characters are not mentioned. Anamorphic species recognised as undescribed are now placed in teleomorph genera when their phylogenetic placement is known, rather than in separate anamorphic genera. [ 5 ]
These changes have reorganised several groups of lichenicolous fungi. For example, the genus Vouauxiomyces , which included anamorphs of Abrothallus species, has been reduced to synonymy with Abrothallus . [ 5 ] Several Phoma -like lichenicolous fungi have been found to belong to various lineages within Dothideomycetes and have been reclassified accordingly. For instance, some species previously placed in Phoma are now recognised as members of Didymocyrtis in the Phaeosphaeriaceae . [ 5 ]
These taxonomic and nomenclatural changes present challenges but also opportunities for a more accurate and phylogenetically informed classification of lichenicolous fungi. However, caution is needed when describing new genera, as names may already exist in traditional generic synonymy. [ 5 ]
Lichenicolous fungi exhibit diverse ecological relationships with their lichen hosts, ranging from harmful parasitism to neutral commensalism and potentially beneficial (mutualistic) interactions. Host specificity, environmental conditions, and fungal adaptability collectively shape these relationships. While many lichenicolous fungi are often considered parasitic or commensalistic, recent research suggests that some species may play important ecological roles within the lichen thallus. Certain lichen-associated fungi might contribute to nutrient cycling by degrading and recycling older parts of the lichen. This process could potentially benefit the host lichen by facilitating the redistribution of nutrients within the thallus. Additionally, these fungi may be involved in the breakdown of dead organic matter, contributing to the overall nutrient dynamics of the lichen community. The extent and significance of these potential ecological functions are still being investigated, but they highlight the complex and potentially mutualistic relationships that can exist between lichenicolous fungi and their host lichens. These findings suggest that the ecological impact of lichenicolous fungi may be more nuanced and potentially beneficial than previously understood, contributing to the health and longevity of lichen communities in various environments. [ 4 ]
Lichenicolous fungi can be broadly categorised into two main types based on their impact on the host: [ 15 ]
While many lichenicolous fungi are parasitic, others establish commensalistic or potentially mutualistic relationships. Some species, the lichenicolous lichens, can even develop their own lichenised thalli using the host lichen's photobiont. [ 8 ] Certain lichenicolous fungi may play important roles in nutrient cycling within the lichen thallus, potentially benefiting the host by degrading and recycling older parts of the lichen. [ 6 ]
Host specificity is a crucial factor in the distribution and ecology of lichenicolous fungi. An estimated 95% of species associate with only a single lichen species or genus, suggesting potential coevolutionary relationships. [ 15 ] This high degree of specificity has significant evolutionary implications, indicating long-term adaptations between fungi and their lichen hosts. However, the range of host specificity can vary greatly among lichenicolous fungi. While many are highly specific, others can colonise multiple unrelated lichen species. [ 3 ]
The varying levels of host specificity have important evolutionary implications. Highly specific parasites risk co-extinction with their hosts, while generalists may have more stable long-term prospects. This may explain why it is unusual for a single genus of lichenicolous fungi to include more than one species that attacks a particular host lichen. When this does occur, the symptoms are often distinct, possibly reflecting different ecological strategies. [ 3 ]
Within a single genus, one species might be parasitic on a host while another is parasymbiotic on the same host. This variation in relationships even among closely related species highlights the complex nature of lichenicolous fungi ecology. [ 3 ]
Molecular studies suggest that some lichenicolous fungi may have broader host ranges than previously thought based on morphological observations alone. Some species have been found to be present in asymptomatic lichens or even in lichen species not previously known to harbour these fungi. [ 8 ]
Lichenicolous fungi can also be involved in complex cases of hyperparasitism , where multiple levels of parasitic relationships occur. In some instances, non-lichenised fungi can parasitise lichenicolous lichens. For example, species such as Stigmidium arthrorhaphidis , Cercidospora trypetheliza , and C. soror have been observed infecting Arthrorhaphis citrinella , which itself grows parasitically on Baeomyces , Cladonia squamules, or decaying lichens. Even more intricate relationships exist where lichens develop hyperparasitically on lichenicolous lichens. An example is Rhizocarpon diploschistidina , which parasitizes Diploschistes muscorum , a lichenicolous lichen that initially parasitizes Cladonia species. These multi-tiered parasitic relationships highlight the complex ecological web that can develop within lichen communities. [ 4 ]
The distribution of lichenicolous fungi is influenced by various ecological and environmental factors. For instance, the Koralpe mountain range in Austria, with its stable conditions and variety of microhabitats , supports a high diversity of both lichens and lichenicolous fungi. The presence of exposed boulders and cliffs provides numerous microniches for these fungi to exploit, ranging from the outer cortex to the inner medulla of lichen thalli. [ 16 ]
In alpine and polar regions , the stability and longevity of lichen thalli provide consistent microhabitats, allowing lichenicolous fungi to establish long-term populations. These environmental conditions contribute to the observed patterns of beta diversity – the variation in species composition between different habitats. Research in the Koralpe Mountain area revealed high beta diversity due to numerous microenvironmental conditions supporting various fungal species. Alpha diversity (diversity within subplots) was found to be higher than beta diversity (diversity within plots), indicating substantial habitat differentiation even within small areas. [ 16 ]
Environmental factors such as pollution can also influence the distribution of lichenicolous fungi, affecting both the lichens and their associated fungi. This can lead to changes in lichen and lichenicolous fungi communities in response to environmental changes. [ 3 ]
The impact of lichenicolous fungi on their hosts varies widely, ranging from minor, localised effects to extensive damage or even death of the host lichen. These effects can manifest in various forms, including discolourations, thallus damage, and gall-like malformations.
Lichenicolous fungi can cause different types of discolourations on their host lichens. These can appear as brownish or whitish necrotic patches, with the extent and pattern of discolouration often depending on the specific fungus-host interaction. For example, Nectriella tincta on Anaptychia fusca and Nesolechia oxyspora on Parmelia saxatilis can cause extensive bleaching of the thallus. Brownish discolourations are characteristic of other species, such as those caused by Lichenoconium echinosporum on Heterodea muelleri . [ 3 ]
Some fungi can cause extensive discolourations when luxuriantly developed. A striking example is Xanthoriicola physciae on Xanthoria parietina , which can give the host a soot -spattered appearance and potentially reduce its photosynthetic area to the point of local death. [ 3 ]
In contrast, other lichenicolous fungi have very localised effects. For instance, some species only affect the apothecia (fruiting bodies) of their hosts, such as Vouauxiella lichenicola and V. verrucosa on red-fruited Lecanora species, which can result in a piebald appearance of the apothecial discks. [ 3 ]
Some lichenicolous fungi can cause significant damage to their host lichens. For example, the broad-spectrum pathogen Athelia arachnoidea is known to cause extensive damage in European lichen communities, particularly those affected by air pollution. [ 8 ] This species, along with others like Erythricium aurantiacum , Marchandiomyces corallinus , and Parmeliicida pandemica , can severely damage or kill entire lichen populations. [ 7 ] In one documented instance, Lecanora conizaeoides recovered from parasitic infection by Licheniconium lecanorae . The lichen accomplished this through vigorous growth at the apothecium's margin, which effectively buried both the damaged hymenium and the parasite's pycnidia within its thallus. [ 17 ]
A significant group of lichenicolous fungi are the cecidogenous (gall-inducing) taxa. Approximately 40 species of lichenicolous ascomycetes and basidiomycetes are known to induce gall formations on their host thalli, often in a species-specific manner. Some of these fungi stimulate the growth of both the mycobiont and photobiont, although the exact mechanisms are not fully understood, while others parasitise the photobiont. [ 3 ] [ 15 ]
Gall-like malformations can take various forms, from slight swellings to complex structures. For instance, Lichenomyces lichenum forms stipitate apothecium-like galls on Lobaria and Pseudocyphellaria species, sometimes even with thalline margins . [ 3 ] Gall formation by lichenicolous fungi exhibits considerable diversity in structure and developmental patterns. These fungal-induced structures can create complex microhabitats, often initiating an intricate ecological succession of various organisms within the gall. For instance, detailed histochemical studies of Biatoropsis usnearum infections on Usnea thalli have revealed that the infection process initiates in the cortical layer of the host. As the gall develops, the parasite forms tremelloid haustoria , primarily in the central part of mature galls. These fully developed galls can then serve as microhabitats for other lichen colonisers, such as species from the genus Cyphobasidium . [ 4 ]
Gall-like malformations on lichens are not always caused by fungi. Other organisms, such as mites ( Acari , Eriophyoidea ) and nematodes , can also induce gall formation in lichens, adding to the complexity of lichen-microorganism interactions. [ 3 ]
The effects of lichenicolous fungi on their hosts can be influenced by various factors, including the species involved, environmental conditions, and the health of the host lichen. Some homobasidiomycetes, such as Athelia arachnoidea , show seasonal peaks in their development and can survive as small sclerotia or bulbils on bark or mosses after killing their lichen hosts, appearing to have a facultatively lichenicolous lifestyle. [ 7 ]
Lichenicolous fungi demonstrate remarkable adaptability in their ecological strategies. While many are exclusively lichenicolous, some species can transition between different lifestyles. For instance, Chaenothecopsis consociata ( Mycocaliciales ) typically invades thalli of Chaenotheca chrysocephala ( Caliciaceae , Lecanorales ), but can also associate with Dictyochloropsis symbiontica to form its own crustose thallus. Similarly, Athelia arachnoidea is necrotrophic on various lichen taxa, free-living algae, and bryophytes , but has also been identified as the sexual state of Rhizoctonia carotae , a postharvest disease of carrots. [ 15 ]
Lichenicolous lichens, a subset of lichenicolous fungi, start as parasites on other lichens and eventually become lichenised. This process, called kleptosymbiosis , involves the fungus acquiring photobionts from its host lichen. Diploschistes muscorum exemplifies this phenomenon. [ 18 ] Lichenicolous lichens are relatively common; a study in Italy found that 189 of 3005 lichenised species (about 6%) were lichenicolous. [ 19 ]
These lichens show distinct biological and ecological characteristics. They are predominantly crustose, mostly have green, non- trentepohlioid algae as photobionts , and primarily reproduce sexually. Ecologically, they tend to occupy specific niches, being mostly saxicolous (growing on rocks) and preferring dry, well-lit habitats across various altitudes. This preference may explain their strategy of "stealing" photobionts, possibly an adaptation to harsh environments where forming new symbioses is challenging. [ 19 ] Lichenicolous lichens can have their algal component positioned in two ways: either internally within the host lichen, or externally as distinct thalli on the host's surface. [ 20 ] An example of the former is Tetramelas pulverulentus , which grows on Physconia distorta . [ 21 ] In contrast, Erichansenia epithallina has been documented growing on the surface or more than a dozen different host lichen species. [ 22 ]
Although often called "parasites", many lichenicolous lichens do not strictly fit this definition, as they eventually develop their own thallus. [ 19 ] True parasitism frequently occurs in lichenicolous lichens from the genera Acarospora , Diploschistes , Rhizocarpon , and Verrucaria . [ 20 ] Researchers suggest that known lichenicolous lichens are only a small part of the total number, and that this strategy might be more widespread than currently recognised. Further DNA sequencing studies could reveal more species of lichenicolous lichens, ranging from obligate to occasional forms. [ 19 ]
Lichenicolous basidiomycetes are globally distributed, inhabiting diverse environments across all continents. Their presence ranges from the tropics to polar regions, demonstrating their remarkable adaptability. Recent discoveries even extend to the harsh climates of Antarctica, demonstrating the extensive ecological niches these fungi occupy. The current knowledge of their distribution is heavily influenced by the varying levels of exploration and research across different regions. Europe, North America, and to a lesser extent, South America appear to be the best-explored regions. However, this may reflect research effort rather than actual diversity. For instance, while Europe has fewer known species than North America, it has a lower proportion of newly described species, suggesting it may be more thoroughly studied. South America, despite having a significant number of known species, is considered under-explored. Systematic studies of lichenicolous fungi in this region only began around 2000, and experts anticipate a substantial increase in known diversity in the coming years. Africa and much of Asia remain poorly explored for lichenicolous basidiomycetes, and the current low numbers of known species likely underrepresent the true diversity in these regions. Similarly, while Oceania has a moderate number of known species, most discoveries have been incidental, and thorough explorations in Australia and New Zealand are expected to yield many more species. [ 7 ]
Studies have shown that natural, unpolluted habitats, such as alpine regions, support a high diversity of lichenicolous fungi. For instance, the Koralpe Mountain area in Austria hosts numerous species due to its stable environmental conditions and variety of microhabitats. A study conducted in this region identified 63 lichen and 41 lichenicolous fungal species within a relatively small area, illustrating the rich biodiversity of these communities. [ 16 ]
The diversity of lichenicolous fungi is not uniformly distributed across all environments. Habitats that have remained unpolluted and stable over long periods are particularly rich in these fungi. In regions such as Hungary, India, and parts of the Holarctic, including North America, Russia, and Sweden, national checklists have documented numerous species of lichenicolous fungi. These checklists, based on detailed morphological recognition of environmental samples, showcase the species diversity in different environments and on different lichen hosts. [ 16 ]
Furthermore, alpine and polar regions, which served as refugia during past climatic events, also show high diversity of lichenicolous fungi. The stability and longevity of lichen thalli in these regions create suitable niches for the diverse fungal communities. For example, the alpine lichen communities of the Koralpe Massive in Austria, which is a nunatak area, support an exceptionally rich lichen species diversity due to the varied microhabitats provided by the gneiss and marble outcrops scattered across the landscape. [ 16 ]
The cross-taxon analysis approach has shown that lichen abundance and diversity significantly influence the diversity patterns of lichenicolous fungi. This method has demonstrated that lichen communities can serve as reliable surrogates for predicting lichenicolous fungal diversity, aiding in the conservation and study of these specialised fungi. [ 16 ]
The evolutionary relationships between lichens and lichenicolous fungi offer insights into the development of fungal symbioses and parasitism. While not all aspects of their co-evolution are fully understood, several trends and patterns have been observed that shed light on the evolutionary processes at play.
The high degree of host specificity observed in many lichenicolous fungi suggests a long history of co-evolution with their lichen hosts. This co-evolutionary relationship has likely led to the development of specialised adaptations in both the lichenicolous fungi and their hosts. However, the evolutionary trajectory is not always straightforward, as evidenced by the existence of generalist species that can colonise multiple unrelated lichen hosts. [ 3 ]
The evolution of lichenicolous fungi appears to favour strategies that ensure long-term survival without eliminating the host population. This is reflected in the relatively low number of highly pathogenic lichenicolous fungi compared to those that establish more stable, less destructive relationships with their hosts. Such an evolutionary strategy makes sense, as fungi that rapidly kill their hosts risk their own extinction if they are highly host-specific. [ 3 ]
Within genera of lichenicolous fungi, interesting trends have been observed in terms of host range and the symptoms they produce. It is relatively uncommon for a single genus to include more than one species that attacks a particular host lichen. When this does occur, the symptoms produced by each species are almost invariably distinct. This differentiation in symptoms may represent evolutionary adaptations to exploit different niches within the same host, reducing direct competition between closely related species. [ 3 ]
For example, within the genus Lichenoconium , L. lecanorae causes the apothecial discs of Lecanora conizaeoides to become blackened while the thallus retains its normal colour. In contrast, L. erodens causes the apothecial disks of the same host to be slightly decolourised and extensive whitish lesions to form in the thallus. This differentiation in symptoms allows both species to coexist on the same host species by exploiting different parts of the lichen thallus. [ 3 ]
Another evolutionary trend observed is the tendency for one fungus in a genus to be parasymbiotic and another parasitic when on the same host. For instance, Corticifraga peltigerae is parasitic on Peltigera thalli, forming bleached circular patches, whereas Corticifraga fuckelii is apparently parasymbiotic on the same host. This suggests that even within a single genus, different species may evolve varying degrees of pathogenicity or mutualism with their hosts. [ 3 ]
The study of lichenicolous fungi dates back to the mid-18th century, predating the recognition of lichens as symbiotic organisms. However, the field has seen significant growth and development in recent decades. [ 8 ]
One of the earliest documented observations of a lichenicolous fungus was Biatoropsis usnearum , a heterobasidiomycete forming gall -like structures on Usnea thalli. Johann Jakob Dillenius illustrated this phenomenon in his 1742 work Historia Muscorum . Erik Acharius further discussed and illustrated Usnea specimens infected by B. usnearum in 1795. [ 24 ] In 1810, Acharius published detailed colour illustrations distinguishing between normal disc -shaped apothecia and the nodule-like, bulging structures he termed cephalodia in Usnea . [ 25 ] These "cephalodia" are now recognised as the basidiomata of Biatoropsis usnearum . [ 26 ]
The 19th century saw an increase in the description of lichenicolous fungi species. William Lauder Lindsay presented the first overview of the group in 1869. Friedrich Wilhelm Zopf provided a list of lichen hosts and their associated fungi in 1896. [ 27 ] Several researchers made significant contributions during this period, preparing detailed illustrated critical accounts of both the taxonomy and biology of selected species. Notable works include those by Charles Tulasne (1852), William Lauder Lindsay (1869), [ 28 ] and Friedrich Wilhelm Zopf (1897). Henri Olivier provided a detailed account of lichenicolous fungi from France (1905–1907), while Léon Vouaux published a worldwide flora with keys and descriptions of all known species (1912–1914). Karl von Keissler revised the Central European species in 1930. [ 3 ] [ 4 ]
Henri Olivier gave a detailed account of lichenicolous fungi from France (1905–1907), while Léon Vouaux published a worldwide flora with keys and descriptions of all known species (1912–1914). [ 29 ] [ 30 ] [ 31 ] Karl von Keissler revised the Central European species in 1930. [ 24 ]
In the early 20th century, Werner and his co-workers carried out developmental and biological investigations on a few species, contributing to the growing body of knowledge about lichenicolous fungi. [ 3 ] Despite the early interest, lichenicolous fungi were often overlooked due to their inconspicuous nature and the specialised knowledge required to study them. This led to a period of relative neglect in the mid-20th century. [ 3 ] The study of these organisms was challenging, as researchers needed expertise in both mycology and lichenology to accurately identify and characterise them.
A resurgence of interest in lichenicolous fungi occurred in the late 20th century. Georges Clauzade and Claude Roux compiled 457 species in 1976, [ 9 ] which increased to 686 species by 1989. [ 10 ] David Hawksworth's 1983 publication [ 32 ] of keys to 218 lichenicolous species from the British Isles stimulated further research. Comprehensive revisions of major groups followed, including lichenicolous hyphomycetes (Hawksworth 1979 [ 33 ] ), Coelomycetes (Hawksworth 1981), [ 34 ] and heterobasidiomycetes (Diederich 1996). [ 3 ] [ 24 ]
The inclusion of lichenicolous fungi in national lichen checklists, starting with Rolf Santesson 's 1993 work on Sweden and Norway, [ 35 ] further encouraged their study. From 1989 to 2003, the number of known species in this group approximately doubled. [ 24 ] In their 2024 book The Lives of Lichens , the lichenologists Robert Lücking and Toby Spribille highlighted the growing community around lichenicolous fungi research:
"Hunting for, collecting, and identifying lichenicolous fungi is a lichenology subculture of its own, with its own literature, websites, and social media groups". [ 36 ]
Early research on lichens and lichenicolous fungi faced significant challenges in distinguishing between the conidiomata of the lichen mycobiont and those of lichenicolous fungi. David L. Hawksworth summarised the complexity of these structures and the potential for misinterpretation. [ 37 ]
Conidiomata in lichens can take various forms, including roughly spherical (" globose ") or flask-shaped (" pycnidia "), cupuliform (" acervular "), cushion-like (" sporodochia "), hooded or peltate (" campylidia "), or erect (" synnemata ", " hyphophores "). The most common type in lichen-forming fungi is the pycnidial conidiomata, which opens by a single pore. These structures were often termed " spermogonia " in early literature when presumed to have a sexual role. [ 37 ]
Hawksworth noted that the study of conidiophores and conidiogenous cells was challenging due to their small size. Advancements in microscopy techniques, such as the use of biologically active washing powders prior to fixing and critical point drying for scanning electron microscopy , have allowed for more detailed examinations of these structures. [ 37 ]
A significant issue in early studies was the potential confusion between conidiomata of the lichen mycobiont and those of lichenicolous fungi. This was particularly problematic when the invading fungus caused little or no damage to the host lichen. In some cases, new names for conidial fungi were erroneously based on the normal conidiomata of the host lichen. For example, the genus Pyrenotrichum was based on what are now known to be the campylidia of various foliicolous (leaf-dwelling) lichens. These historical challenges emphasise how our understanding of these organisms has evolved over time, with many early misconceptions being corrected through subsequent research. [ 37 ]
The study of lichenicolous fungi employs a range of methodologies, from traditional culture techniques to advanced molecular approaches. Each method contributes unique insights into the biology, ecology, and diversity of these organisms.
Traditional mycological methods form the foundation for studying lichenicolous fungi. These techniques are crucial for understanding their biology and interactions with lichen hosts. Researchers typically begin by collecting infected lichen tissues from the field. They identify fungal parasites by their fruiting structures or the discolouration they cause on lichen thalli. In the laboratory, these structures are isolated and cultured on solid agar media such as Sabouraud's medium , potato dextrose agar , or cornmeal agar. To reduce contamination , surface sterilisation with ethanol or sodium hypochlorite is often necessary. [ 38 ] For accurate identification and genetic studies, researchers prefer single- spore or single- conidium cultures. This process involves:
Some lichenicolous fungi require specific growth conditions, such as low-nitrogen media or the presence of lichen tissues. These cultured fungi are valuable for various experiments, including studies on degradative abilities, interactions with lichen secondary metabolites , and genetic analyses . To ensure future availability and identity verification, researchers often deposit cultures in recognised fungal culture collections . [ 38 ]
The study of lichenicolous fungi employs a range of methodologies, from traditional culture techniques to advanced molecular approaches. Researchers typically begin by collecting infected lichen tissues from the field. They identify fungal parasites by their fruiting structures or the discolouration they cause on lichen thalli. In the laboratory, these structures are isolated and cultured on solid agar media such as Sabouraud's medium , potato dextrose agar , or cornmeal agar. To reduce contamination , surface sterilisation with ethanol or sodium hypochlorite is often necessary. [ 38 ]
For accurate identification and genetic studies, researchers prefer single- spore or single- conidium cultures. This process involves:
Some lichenicolous fungi require specific growth conditions, such as low-nitrogen media or the presence of lichen tissues. These cultured fungi are valuable for various experiments, including studies on degradative abilities, interactions with lichen secondary metabolites , and genetic analyses . To ensure future availability and identity verification, researchers often deposit cultures in recognised fungal culture collections . [ 38 ]
In recent years, culture-independent methods, particularly those employing molecular techniques, have revolutionised the study of lichenicolous fungi. These include:
These modern techniques have revealed a high diversity of lichen-associated fungi, many of which remain undetected by traditional methods. For instance, DNA fingerprinting techniques have shown a high diversity of lichen-associated fungi that does not necessarily correlate with the presence of externally visible lichenicolous fungi. [ 39 ] Next-generation sequencing studies of lichen mycobiomes in various habitats, from Arctic to alpine environments, have further expanded our understanding of the diversity and distribution of these fungi. [ 40 ] [ 41 ]
The application of these advanced methods suggests that the actual number of lichenicolous fungal species may far exceed current estimates, potentially reaching 3,000–5,000 species. This indicates substantial potential for discovering new species and genera through the continued application of molecular methods. [ 5 ] [ 8 ]
Despite these advances, traditional microscopy remains essential in the study of lichenicolous fungi, particularly for understanding their physical interactions with host lichens. However, distinguishing foreign hyphae within lichen thalli from the mycobiont proper remains a significant challenge, highlighting the need for integrating both traditional and modern research approaches. [ 15 ]
Cross-taxon analysis is a research method used to study the correlation between different taxonomic groups by analysing their diversity patterns. [ 42 ] In the context of lichenicolous fungi, this method helps understand how the diversity of these fungi relates to their lichen hosts. By collecting data on both lichens and lichenicolous fungi from the same habitats, researchers can identify patterns and relationships between the two groups. For example, studies in the Koralpe Mountains of Austria have shown that the diversity of lichenicolous fungi closely follows the diversity of their lichen hosts, suggesting that lichens can be used as indicators to predict the presence and diversity of these fungi. [ 16 ]
This approach uses statistical techniques like co-correspondence analysis to create predictive models , which have shown that certain lichen species, especially those that are abundant and widely distributed, can reliably indicate the diversity of lichenicolous fungi. This method not only enhances our understanding of the ecological relationships between lichens and lichenicolous fungi but also provides a practical tool for biodiversity conservation. By using lichens as surrogate indicators, researchers can more efficiently identify and protect areas with high biodiversity, particularly in underexplored regions. [ 16 ]
Molecular studies have revealed that lichenicolous fungi may possess broader host ranges than previously inferred from morphological observations alone. Researchers have detected some species in asymptomatic lichens and even in lichen species not previously known to host these fungi. [ 8 ]
Recent estimates indicate that the actual number of lichenicolous fungal species may far exceed current descriptions, potentially reaching 3,000–5,000 species. [ 8 ] This suggests substantial potential for discovering new species and genera through the application of molecular methods. [ 5 ] | https://en.wikipedia.org/wiki/Lichenicolous_fungus |
Lichenin , also known as lichenan or moss starch , is a complex glucan occurring in certain species of lichens . It can be extracted from Cetraria islandica ( Iceland moss ). [ 1 ] It has been studied since about 1957. [ 2 ]
Chemically, lichenin is a mixed-linkage glucan , consisting of repeating glucose units linked by β-1,3 and β-1,4 glycosidic bonds. [ 1 ]
It is an important carbohydrate for reindeers and northern flying squirrels , which eat the lichen Bryoria fremontii .
It can be extracted by digesting Iceland moss in a cold, weak solution of carbonate of soda for some time, and then boiling. By this process the lichenin is dissolved and on cooling separates as a colorless jelly. Iodine imparts no color to it. [ 3 ]
In his 1960 novel Trouble with Lichen , John Wyndham gives the name Lichenin to a biochemical extract of lichen used to extend life expectancy beyond 300 years.
This biochemistry article is a stub . You can help Wikipedia by expanding it .
This article about lichens or lichenology is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Lichenin |
Lichenology is the branch of mycology that studies the lichens , symbiotic organisms made up of an intimate symbiotic association of a microscopic alga (or a cyanobacterium ) with a filamentous fungus . Lichens are chiefly characterized by this symbiosis.
Study of lichens draws knowledge from several disciplines: mycology , phycology , microbiology and botany . Scholars of lichenology are known as lichenologists . Study of lichens is conducted by both professional and amateur lichenologists.
Methods for species identification include reference to single-access keys on lichens. An example reference work is Lichens of North America (2001) by Irwin M. Brodo , Sylvia Sharnoff and Stephen Sharnoff and that book's 2016 expansion, Keys to Lichens of North America: Revised and Expanded by the same three authors joined by Susan Laurie-Bourque . [ 1 ]
A chemical spot test can be used to detect the presence of certain lichen products which can be characteristic of a given lichen species . Some components of certain lichens may also fluoresce under ultraviolet light , providing another form of lichen identification test.
Lichenologists may also study the growth and growth rate of lichens, lichenometry , the role of lichens in nutrient cycling , the ecological role of lichens in biological soil crusts , the morphology of lichens , their anatomy and physiology , and ethnolichenology topics including the study of edible lichens . As with any other field of study, lichenology has its own set of rules for taxonomic nomenclature and its own set of other terminology .
Lichens as a group have received less attention in classical treatises on botany than other groups although the relationship between humans and some species has been documented from early times. Several species have appeared in the works of Dioscorides , Pliny the Elder and Theophrastus although the studies are not very deep. During the first centuries of the modern age they were usually put forward as examples of spontaneous generation and their reproductive mechanisms were totally ignored. [ 2 ] For centuries naturalists had included lichens in diverse groups until in the early 18th century a French researcher Joseph Pitton de Tournefort in his Institutiones Rei Herbariae grouped them into their own genus. He adopted the Latin term lichen, which had already been used by Pliny who had imported it from Theophrastus but up until then this term had not been widely employed. [ 3 ] The original meaning of the Greek word λειχήν (leichen) was moss that in its turn derives from the Greek verb λείχω (liekho) to suck because of the great ability of these organisms to absorb water. In its original use the term signified mosses , liverworts as well as lichens . Some forty years later Dillenius in his Historia Muscorum made the first division of the group created by Tournefort separating the sub-families Usnea , Coralloides [ a ] and Lichens in response to the morphological characteristics of the lichen thallus . [ 5 ]
After the revolution in taxonomy brought in by Linnaeus and his new system of classification lichens are retained in the Plant Kingdom forming a single group Lichen with eight divisions within the group according to the morphology of the thallus . [ 6 ] The taxonomy of lichens was first intensively investigated by the Swedish botanist Erik Acharius (1757–1819), who is therefore sometimes named the "father of lichenology". Acharius was a student of Carl Linnaeus . Some of his more important works on the subject, which marked the beginning of lichenology as a discipline, are:
Later lichenologists include the American scientists Vernon Ahmadjian and Edward Tuckerman and the Russian evolutionary biologist Konstantin Merezhkovsky , as well as amateurs such as Louisa Collings .
Over the years research shed new light into the nature of these organisms still classified as plants. A controversial issue surrounding lichens since the early 19th century is their reproduction. In these years a group of researchers faithful to the tenets of Linnaeus considered that lichens reproduced sexually and had sexual reproductive organs, as in other plants, independent of whether asexual reproduction also occurred. Other researchers only considered asexual reproduction by means of Propagules . [ 7 ]
Against this background appeared the Swedish botanist Erik Acharius disciple of Linnaeus, who is today considered the father of lichenology, starting the taxonomy of lichens with his pioneering study of Swedish lichens in Lichenographiae Suecicae Prodromus of 1798 or in his Synopsis Methodica Lichenum, Sistens omnes hujus Ordinis Naturalis of 1814. [ 8 ] These studies and classifications are the cornerstone of subsequent investigations. In these early years of structuring the new discipline various works of outstanding scientific importance appeared such as Lichenographia Europaea Reformata published in 1831 by Elias Fries or Enumeratio Critico Lichenum Europaeorum 1850 by Ludwig Schaerer [ 9 ] in Germany. [ 10 ]
But these works suffer from being superficial and mere lists of species without further physiological studies. [ 11 ] It took until the middle of the 19th century for research to catch up using biochemical and physiological methods. In Germany Hermann Itzigsohn [ de ] [ 12 ] and Johann Bayrhoffer , [ 13 ] in France Edmond Tulasne and Camille Montagne , in Russia Fedor Buhse , [ 14 ] in England William Allport Leighton and in the United States Edward Tuckerman began to publish works of great scientific importance.
Scientific publications settled many unknown facts about lichens. In the French publication Annales des Sciences Naturelles in an article of 1852 "Memorie pour servir a l'Histoire des Lichens Organographique et Physiologique" by Edmond Tulasne , the reproductive organs or apothecia of lichens was identified. [ 15 ] [ 16 ]
These new discoveries were becoming increasingly contradictory for scientists. The apothecium reproductive organ being unique to fungi but absent in other photosynthetic organisms. With improvements in microscopy , algae were identified in the lichen structure, which heightened the contradictions. At first the presence of algae was taken as being due to contamination due to collection of samples in damp conditions and they were not considered as being in a symbiotic relation with the fungal part of the thallus. That the algae continued to multiply showed that they were not mere contaminants.
It was Anton de Bary a German mycologist who specialised in phytopathology who first suggested in 1865 that lichens were merely the result of parasitism of various fungi of the ascomycetes group by nostoc type algae and others. Successive studies such as those carried out by Andrei Famintsyn and Baranetzky [ 17 ] in 1867 showed no dependence of the algal component upon the lichen thallus and that the algal component could live independently of the thallus. [ 18 ] It was in 1869 that Simon Schwendener demonstrated that all lichens were the result of fungal attack on the cells of algal cells and that all these algae also exist free in nature. This researcher was the first to recognise the dual nature of lichens as a result of the capture of the algal component by the fungal component. [ 19 ] In 1873 Jean-Baptiste Edouard Bornet concluded form studying many different lichen species that the relationship between fungi and algae was purely symbiotic . It was also established that algae could associate with many different fungi to form different lichen phenotypes .
In 1909 the Russian lichenologist Konstantin Mereschkowski presented a research paper "The Theory of two Plasms as the basis of Symbiogenesis , A new study on the Origin of Organisms", which aims to explain a new theory of Symbiogenesis by lichens and other organisms as evidenced by his earlier work "Nature and Origin of Chromatophores in the Plant Kingdom". These new ideas can be studied today under the title of the Theory of Endosymbiosis . [ 20 ]
Despite the above studies the dual nature of lichens remained no more than a theory until in 1939 the Swiss researcher Eugen A Thomas [ 21 ] was able to reproduce in the laboratory the phenotype of the lichen Cladonia pyxidata [ 22 ] by combining its two identified components.
During the 20th century botany and mycology were still attempting to solve the two main problems surrounding lichens. On the one hand the definition of lichens and the relationship between the two symbionts and the taxonomic position of these organisms within the plant and fungal kingdoms. There appeared numerous renowned researchers within the field of lichenology such as Henry Nicollon des Abbayes , William Alfred Weber , Antonina Georgievna Borissova , Irwin M. Brodo , and George Albert Llano .
Lichenology has found applications beyond biology itself in the field of geology in a technique known as lichenometry where the age of an exposed surface can be found by studying the age of lichens growing on them. Age dating in this way can be absolute or relative because the growth of these organisms can be arrested under various conditions. The technique provides an average age of the older individual lichens providing a minimum age of the medium being studied. [ 23 ] Lichenometry relies upon the fact that the maximum diameter of the largest thallus of an epilithic lichen growing on a substrate is directly proportional to the time from first exposure of the area to the environment as seen in studies by Roland Beschel [ 24 ] in 1950 and is especially useful in areas exposed for less than 1000 years. Growth is greatest in the first 20 to 100 years with 15–50 mm growth per year and less in the following years with average growth of 2–4 mm per year. [ 25 ]
The difficulty of giving a definition applicable to every known lichen has been debated since lichenologists first recognised the dual nature of lichens. In 1982 the International Association for Lichenology convened a meeting to adopt a single definition of lichen drawing on the proposals of a committee. The chairman of this committee was the renowned researcher Vernon Ahmadjian . The definition finally adopted was that lichen could be considered as the association between a fungus and a photosynthetic symbiont resulting in a thallus of specific structure. [ 26 ]
Such a simple a priori definition soon brought criticism from various lichenologists and there soon emerged reviews and suggestions for amendments. For example, David L. Hawksworth considered the definition imperfect because it is impossible to determine which one thallus is of a specific structure since thalli changed depending upon the substrate and conditions in which they developed. This researcher represents one of the main trends among lichenologists who consider it impossible to give a single definition to lichens since they are a unique type of organism. [ 26 ]
Today studies in lichenology are not restricted to the description and taxonomy of lichens but have application in various scientific fields. Especially important are studies on environmental quality that are made through the interaction of lichens with their environment. Lichen is extremely sensitive to various air pollutants, especially to sulphur dioxide , which causes acid rain and prevents water absorption.
Although several species of lichen have been used in traditional medicine it was not until the early 20th century that modern science became interested in them. The discovery of various substances with antibacterial action in lichen thalli was essential for scientists to become aware of the possible importance of these organisms to medicine . [ 27 ] From the 1940s there appeared various works by the noted microbiologist Rufus Paul Burkholder who demonstrated antibacterial action of lichens of the genus Usnea against Bacillus subtilis and Sarcina lutea . [ 28 ] Studies showed that the substance that inhibited growth of bacteria was usnic acid . Something similar occurred with the substance Ramelina synthesised by the lichen Ramalina reticulata , [ 29 ] nevertheless, these substances proved ineffective against Gram negative bacteria such as Escherichia coli and Pseudomonas . With these investigations the number of antibacterial substances and possible drug targets known to be produced by lichens increased ergosterol , usnic acid etc. [ 30 ]
Interest in the potential of substances synthesised by lichens increased with the end of World War II along with the growing interest in all antibiotic substances. In 1947 antibacterial action was identified in extracts of Cetraria islandica and the compounds identified as responsible for bacterial inhibition were shown to be d-protolichosteric acid and d-1-usnic acid . [ 31 ] Further investigations have identified novel antibacterial substances, Alectosarmentin [ 32 ] or Atranorin . [ 33 ]
Antibacterial action of substances produced by lichens is related to their ability to disrupt bacterial proteins with a subsequent loss of bacterial metabolic capacity. This is possible due to the action of lichen phenolics such as usnic acid derivatives. [ 34 ]
From the 1950s the lichen product usnic acid was the object of most antitumour research. These studies revealed some in vitro antitumour activity by substances identified in two common lichens Peltigera leucophlebia and Collema flaccidum . [ 35 ]
Recent work in the field of applied biochemistry has shown some antiviral activity with some lichen substances. In 1989 K Hirabayashi [ 36 ] presented his investigations on inhibitory lichen polysaccharides in HIV infection. [ 37 ] | https://en.wikipedia.org/wiki/Lichenologist |
Lichenology is the branch of mycology that studies the lichens , symbiotic organisms made up of an intimate symbiotic association of a microscopic alga (or a cyanobacterium ) with a filamentous fungus . Lichens are chiefly characterized by this symbiosis.
Study of lichens draws knowledge from several disciplines: mycology , phycology , microbiology and botany . Scholars of lichenology are known as lichenologists . Study of lichens is conducted by both professional and amateur lichenologists.
Methods for species identification include reference to single-access keys on lichens. An example reference work is Lichens of North America (2001) by Irwin M. Brodo , Sylvia Sharnoff and Stephen Sharnoff and that book's 2016 expansion, Keys to Lichens of North America: Revised and Expanded by the same three authors joined by Susan Laurie-Bourque . [ 1 ]
A chemical spot test can be used to detect the presence of certain lichen products which can be characteristic of a given lichen species . Some components of certain lichens may also fluoresce under ultraviolet light , providing another form of lichen identification test.
Lichenologists may also study the growth and growth rate of lichens, lichenometry , the role of lichens in nutrient cycling , the ecological role of lichens in biological soil crusts , the morphology of lichens , their anatomy and physiology , and ethnolichenology topics including the study of edible lichens . As with any other field of study, lichenology has its own set of rules for taxonomic nomenclature and its own set of other terminology .
Lichens as a group have received less attention in classical treatises on botany than other groups although the relationship between humans and some species has been documented from early times. Several species have appeared in the works of Dioscorides , Pliny the Elder and Theophrastus although the studies are not very deep. During the first centuries of the modern age they were usually put forward as examples of spontaneous generation and their reproductive mechanisms were totally ignored. [ 2 ] For centuries naturalists had included lichens in diverse groups until in the early 18th century a French researcher Joseph Pitton de Tournefort in his Institutiones Rei Herbariae grouped them into their own genus. He adopted the Latin term lichen, which had already been used by Pliny who had imported it from Theophrastus but up until then this term had not been widely employed. [ 3 ] The original meaning of the Greek word λειχήν (leichen) was moss that in its turn derives from the Greek verb λείχω (liekho) to suck because of the great ability of these organisms to absorb water. In its original use the term signified mosses , liverworts as well as lichens . Some forty years later Dillenius in his Historia Muscorum made the first division of the group created by Tournefort separating the sub-families Usnea , Coralloides [ a ] and Lichens in response to the morphological characteristics of the lichen thallus . [ 5 ]
After the revolution in taxonomy brought in by Linnaeus and his new system of classification lichens are retained in the Plant Kingdom forming a single group Lichen with eight divisions within the group according to the morphology of the thallus . [ 6 ] The taxonomy of lichens was first intensively investigated by the Swedish botanist Erik Acharius (1757–1819), who is therefore sometimes named the "father of lichenology". Acharius was a student of Carl Linnaeus . Some of his more important works on the subject, which marked the beginning of lichenology as a discipline, are:
Later lichenologists include the American scientists Vernon Ahmadjian and Edward Tuckerman and the Russian evolutionary biologist Konstantin Merezhkovsky , as well as amateurs such as Louisa Collings .
Over the years research shed new light into the nature of these organisms still classified as plants. A controversial issue surrounding lichens since the early 19th century is their reproduction. In these years a group of researchers faithful to the tenets of Linnaeus considered that lichens reproduced sexually and had sexual reproductive organs, as in other plants, independent of whether asexual reproduction also occurred. Other researchers only considered asexual reproduction by means of Propagules . [ 7 ]
Against this background appeared the Swedish botanist Erik Acharius disciple of Linnaeus, who is today considered the father of lichenology, starting the taxonomy of lichens with his pioneering study of Swedish lichens in Lichenographiae Suecicae Prodromus of 1798 or in his Synopsis Methodica Lichenum, Sistens omnes hujus Ordinis Naturalis of 1814. [ 8 ] These studies and classifications are the cornerstone of subsequent investigations. In these early years of structuring the new discipline various works of outstanding scientific importance appeared such as Lichenographia Europaea Reformata published in 1831 by Elias Fries or Enumeratio Critico Lichenum Europaeorum 1850 by Ludwig Schaerer [ 9 ] in Germany. [ 10 ]
But these works suffer from being superficial and mere lists of species without further physiological studies. [ 11 ] It took until the middle of the 19th century for research to catch up using biochemical and physiological methods. In Germany Hermann Itzigsohn [ de ] [ 12 ] and Johann Bayrhoffer , [ 13 ] in France Edmond Tulasne and Camille Montagne , in Russia Fedor Buhse , [ 14 ] in England William Allport Leighton and in the United States Edward Tuckerman began to publish works of great scientific importance.
Scientific publications settled many unknown facts about lichens. In the French publication Annales des Sciences Naturelles in an article of 1852 "Memorie pour servir a l'Histoire des Lichens Organographique et Physiologique" by Edmond Tulasne , the reproductive organs or apothecia of lichens was identified. [ 15 ] [ 16 ]
These new discoveries were becoming increasingly contradictory for scientists. The apothecium reproductive organ being unique to fungi but absent in other photosynthetic organisms. With improvements in microscopy , algae were identified in the lichen structure, which heightened the contradictions. At first the presence of algae was taken as being due to contamination due to collection of samples in damp conditions and they were not considered as being in a symbiotic relation with the fungal part of the thallus. That the algae continued to multiply showed that they were not mere contaminants.
It was Anton de Bary a German mycologist who specialised in phytopathology who first suggested in 1865 that lichens were merely the result of parasitism of various fungi of the ascomycetes group by nostoc type algae and others. Successive studies such as those carried out by Andrei Famintsyn and Baranetzky [ 17 ] in 1867 showed no dependence of the algal component upon the lichen thallus and that the algal component could live independently of the thallus. [ 18 ] It was in 1869 that Simon Schwendener demonstrated that all lichens were the result of fungal attack on the cells of algal cells and that all these algae also exist free in nature. This researcher was the first to recognise the dual nature of lichens as a result of the capture of the algal component by the fungal component. [ 19 ] In 1873 Jean-Baptiste Edouard Bornet concluded form studying many different lichen species that the relationship between fungi and algae was purely symbiotic . It was also established that algae could associate with many different fungi to form different lichen phenotypes .
In 1909 the Russian lichenologist Konstantin Mereschkowski presented a research paper "The Theory of two Plasms as the basis of Symbiogenesis , A new study on the Origin of Organisms", which aims to explain a new theory of Symbiogenesis by lichens and other organisms as evidenced by his earlier work "Nature and Origin of Chromatophores in the Plant Kingdom". These new ideas can be studied today under the title of the Theory of Endosymbiosis . [ 20 ]
Despite the above studies the dual nature of lichens remained no more than a theory until in 1939 the Swiss researcher Eugen A Thomas [ 21 ] was able to reproduce in the laboratory the phenotype of the lichen Cladonia pyxidata [ 22 ] by combining its two identified components.
During the 20th century botany and mycology were still attempting to solve the two main problems surrounding lichens. On the one hand the definition of lichens and the relationship between the two symbionts and the taxonomic position of these organisms within the plant and fungal kingdoms. There appeared numerous renowned researchers within the field of lichenology such as Henry Nicollon des Abbayes , William Alfred Weber , Antonina Georgievna Borissova , Irwin M. Brodo , and George Albert Llano .
Lichenology has found applications beyond biology itself in the field of geology in a technique known as lichenometry where the age of an exposed surface can be found by studying the age of lichens growing on them. Age dating in this way can be absolute or relative because the growth of these organisms can be arrested under various conditions. The technique provides an average age of the older individual lichens providing a minimum age of the medium being studied. [ 23 ] Lichenometry relies upon the fact that the maximum diameter of the largest thallus of an epilithic lichen growing on a substrate is directly proportional to the time from first exposure of the area to the environment as seen in studies by Roland Beschel [ 24 ] in 1950 and is especially useful in areas exposed for less than 1000 years. Growth is greatest in the first 20 to 100 years with 15–50 mm growth per year and less in the following years with average growth of 2–4 mm per year. [ 25 ]
The difficulty of giving a definition applicable to every known lichen has been debated since lichenologists first recognised the dual nature of lichens. In 1982 the International Association for Lichenology convened a meeting to adopt a single definition of lichen drawing on the proposals of a committee. The chairman of this committee was the renowned researcher Vernon Ahmadjian . The definition finally adopted was that lichen could be considered as the association between a fungus and a photosynthetic symbiont resulting in a thallus of specific structure. [ 26 ]
Such a simple a priori definition soon brought criticism from various lichenologists and there soon emerged reviews and suggestions for amendments. For example, David L. Hawksworth considered the definition imperfect because it is impossible to determine which one thallus is of a specific structure since thalli changed depending upon the substrate and conditions in which they developed. This researcher represents one of the main trends among lichenologists who consider it impossible to give a single definition to lichens since they are a unique type of organism. [ 26 ]
Today studies in lichenology are not restricted to the description and taxonomy of lichens but have application in various scientific fields. Especially important are studies on environmental quality that are made through the interaction of lichens with their environment. Lichen is extremely sensitive to various air pollutants, especially to sulphur dioxide , which causes acid rain and prevents water absorption.
Although several species of lichen have been used in traditional medicine it was not until the early 20th century that modern science became interested in them. The discovery of various substances with antibacterial action in lichen thalli was essential for scientists to become aware of the possible importance of these organisms to medicine . [ 27 ] From the 1940s there appeared various works by the noted microbiologist Rufus Paul Burkholder who demonstrated antibacterial action of lichens of the genus Usnea against Bacillus subtilis and Sarcina lutea . [ 28 ] Studies showed that the substance that inhibited growth of bacteria was usnic acid . Something similar occurred with the substance Ramelina synthesised by the lichen Ramalina reticulata , [ 29 ] nevertheless, these substances proved ineffective against Gram negative bacteria such as Escherichia coli and Pseudomonas . With these investigations the number of antibacterial substances and possible drug targets known to be produced by lichens increased ergosterol , usnic acid etc. [ 30 ]
Interest in the potential of substances synthesised by lichens increased with the end of World War II along with the growing interest in all antibiotic substances. In 1947 antibacterial action was identified in extracts of Cetraria islandica and the compounds identified as responsible for bacterial inhibition were shown to be d-protolichosteric acid and d-1-usnic acid . [ 31 ] Further investigations have identified novel antibacterial substances, Alectosarmentin [ 32 ] or Atranorin . [ 33 ]
Antibacterial action of substances produced by lichens is related to their ability to disrupt bacterial proteins with a subsequent loss of bacterial metabolic capacity. This is possible due to the action of lichen phenolics such as usnic acid derivatives. [ 34 ]
From the 1950s the lichen product usnic acid was the object of most antitumour research. These studies revealed some in vitro antitumour activity by substances identified in two common lichens Peltigera leucophlebia and Collema flaccidum . [ 35 ]
Recent work in the field of applied biochemistry has shown some antiviral activity with some lichen substances. In 1989 K Hirabayashi [ 36 ] presented his investigations on inhibitory lichen polysaccharides in HIV infection. [ 37 ] | https://en.wikipedia.org/wiki/Lichenology |
In archaeology , palaeontology , and geomorphology , lichenometry is a geomorphic method of geochronologic dating that uses lichen growth to determine the age of exposed rock , based on a presumed specific rate of increase in radial size over time. [ 1 ] [ 2 ] : 9 Measuring the diameter of the largest lichen of a species on a rock surface can therefore be used to determine the length of time the rock has been exposed. Lichen can be preserved on old rock faces for up to 10,000 years, [ 3 ] providing the maximum age limit of the technique, but it is most accurate (within 10% error) when applied to surfaces that have been exposed for less than 1,000 years. [ 4 ] (The practical limit of the technique might be 4,000 to 5,000 years. [ 3 ] )
Lichenometry is especially useful for dating surfaces less than 500 years old, as radiocarbon dating techniques are less accurate over this period. [ 5 ] The lichens most commonly used for lichenometry are those of the genera Rhizocarpon (such as the species Rhizocarpon geographicum ) and Xanthoria . The measured growth rates of R. geographicum tends to fall within the range of 0.9–0.3 millimeter per year, depending on several factors, including the size of the lichen patch. [ 6 ]
The technique was first employed by Knut Fægri in 1933, though the first exclusively lichenometric paper was not published until 1950, by Austrian Roland Beschel , [ 7 ] in a paper concerning the European Alps . [ 8 ]
Lichenometry can provide dates for glacial deposits in tundra environments, lake level changes, glacial moraines , trim lines , palaeofloods, [ 9 ] rockfalls, seismic events associated with the rockfalls, [ 2 ] talus ( scree ) stabilization and former extent of permafrost or very persistent snow cover. [ 10 ] It has also been explored as a tool in assessing the speed of glacier retreat due to climate change . [ 11 ]
Among the potential problems of the technique are the difficulty of correctly identifying the species, the delay between exposure and colonization, the varying growth rates from region to region, growth rates not always being constant over time and depend on substrate texture and composition, the climate, and determining the lichen that is the largest. [ 5 ]
Several methods exist for dating surfaces with help of lichenometry; the most simple relies on a single largest lichen though other methods use more. There are also differences in the way the lichen is measured; some scientists suggest that the largest diameter should be measured, but others prefer the diameter of the largest inscribed circle. A problem in dating lichens is the fact that several thalli can fuse together, making several minor lichens appears as a larger one of older age. [ 12 ] The lichenometrist Tom Bradwell has listed the following five method families as the principal ones into which most other methods can be classified: | https://en.wikipedia.org/wiki/Lichenometry |
Some types of lichen are able to fix nitrogen from the atmosphere. This process relies on the presence of cyanobacteria as a partner species within the lichen. The ability to fix nitrogen enables lichen to live in nutrient-poor environments. Lichen can also extract nitrogen from the rocks on which they grow.
Nitrogen fixation, and hence the abundance of lichen and their host plants, may be decreased by application of nitrogen-based agricultural fertilizer and by atmospheric pollution.
The nitrogen cycle is one of the Earth's biogeochemical cycles . It involves the conversion of nitrogen into different chemical forms. The main processes of the nitrogen cycle are the fixation, ammonification, nitrification, and denitrification. As one of the macronutrients, nitrogen plays an important role in plant growth. The nitrogen cycle is affected by environmental factors. For example, in the subarctic heath, increase in temperature can cause nitrogen fixation to increase or decrease based on season, while overall climate warming indirectly caused the vegetation change which in turn affected the nitrogen fixation process. [ 1 ]
Lichens are symbiotic organisms that play an important role in the biogeochemical cycle on Earth. The characteristics of lichens, such as strong resistance to factors such as desiccation, ability to grow and break down rocks allow lichen to grow in different types of environment including highly nitrogen limited area such as subarctic heath . [ 1 ] [ 2 ] While it does not occur often, formation of akinetes (type of cell formed by cyanobacteria which are resistant to cold and desiccation) was observed in nitrogen fixing lichen. [ 2 ] Depending on its partner, lichens derive the carbon and nitrogen from algal and cyanobacteria photobionts (which fixes nitrogen from the air). [ 3 ] Lichen fungi can fix nitrogen during the day and night, as long the dark period is not too long. [ 2 ]
Both nitrogen-fixing lichens and non-nitrogen-fixing lichens take up nitrogen from the environment as a nutrient. [ 4 ] Both type of lichens secrete many different organic compounds to absorb minerals from the substrates.
Main difference between nitrogen fixing lichen and non-nitrogen fixing lichen is their photosynthetic partner: nitrogen fixing lichen partner with cyanobacteria which can fix nitrogen from the air, while green alga, partner of non nitrogen fixing lichen, does not perform the same process. [ 5 ] The nitrogen fixation is energetically costly due to chemical transformation and only about 10% of lichen are partnered with cyanobacteria. [ 5 ] [ 6 ] In agricultural regions, non nitrogen fixing lichen reflect uptake of ammonia emission indicating that it have lower nitrogen value. [ 7 ]
Some lichens such as Placopsis gelada contain both nitrogen fixing phototrophs and non nitrogen-fixing phototrophs in which Nostoc (cyanobacteria, the phototrophic nitrogen fixer) was dwelling within cephalodia (small gall like structure within lichen; contains cyanobacteria symbionts). [ 4 ] In such cases, heterocyst differentiation was greater in cephalodia when compared to having Nostoc as the primary symbionts in lichens, showing that, in the presence of non nitrogen-fixing phototroph, Nostoc specialize for nitrogen fixation. [ 4 ]
A lichen's response to nutrient enrichment depends on not only on species and environmental factors but also partially on thallus concentrations of nutrients such as nitrogen and phosphorus. [ 8 ]
Ammonium, nitrate and organic nitrogen can be assimilated by lichen along with phosphorus as an important stimulant for cyanolichens . The photobiont will become less dependent on fungal nutrient supply when nitrogen deposition increases as it will be able to access its own nitrogen and it will stimulate the photobiont, causing it to build up, resulting in increased photosynthesis which increases carbon input. [ 8 ] However, for lichens that cannot increase their photobiont growth, nitrogen deposition can be damaging due to higher nitrogen concentration than their biological requirements. [ 8 ]
Generally, when a lichenized algal cell is nitrogen limited, the addition of nitrogen caused the growth of algal cells. [ 8 ] Under nitrogen limiting condition, chlorophyll concentration was positively correlated with the growth of algal cells indicating that should the concentration of chlorophyll increase, the photobiont population will also increase. [ 8 ] As lichens absorb nitrogen through fixation, it will have a very strong negative reaction if the nitrogen availability changes, indicating its sensitivity to environmental changes. According to the experiment by Sparrius et al., when nitrogen fertilizer was added into the soil, lichen cover was reduced by ~50%, while the addition of phosphorus showed opposite result. [ 9 ] In the region such as boreal forest , where nitrogen and phosphorus are limiting nutrients and for symbiotic interaction to occur properly, their ratio must be balanced. [ 8 ] General pollution of climate that is indicated by the concentration of nitrogen oxides can also affect the growth of lichen. [ 10 ] When compared to bryophyte (non-vascular land plant), which is also sensitive to nitrogen fertilizer, lichen showed a much stronger response. [ 9 ]
There are many different species of lichens and each has its own way of allocating nitrogen. The non nitrogen fixing lichen invests a large amount of nitrogen into photosynthetic tissue, whereas nitrogen fixing lichen will invest into the fungal tissue. [ 3 ] Nitrogen-fixing lichen species can only attain a certain amount of nitrogen, as the addition of ammonium decreases its rate of nitrogen-fixation, which decreases the amount of nitrogen that is exported into the adjacent hyphae . [ 3 ] Nitrogen fixation is energy dependent and very costly for lichens. [ 11 ] In a region where nitrogen deposition is high, lichens have a lower uptake of nitrogen in comparison to the Antarctic green algal lichen, which takes up 90% of nitrogen deposition in both nitrate and ammonium form. [ 3 ] Some lichen species are able to refrain from assimilating excessive amount of nitrogen in order to maintain a balanced tissue concentration. [ 3 ] Majority of lichen species absorbs more NH4+ than NO3- and the impact of temperature on the rate of fixation is "consonant to the normal enzymatic kinetics of them". [ 3 ] [ 11 ]
Nitrogen fixing lichens actively fix atmospheric nitrogen using the nostoc , located in the cephalodia . Lichens are sensitive to nitrogen availability. [ 11 ] Upon nitrogen fixation, there will be an increase of algal cell growth, chlorophyll concentration, and photobiont population. While costly, in regions where nitrogen availability is low, fixation process is the main way for the lichen to absorb nitrogen which is macronutrient (essential nutrient).
Nitrogen, as a macronutrient and a biogeochemical cycle, also affects the ecology. Through the nitrogen cycle, it breaks down into the chemical form that allows plants to absorb as nutrients. There are certain regions in the world that most plants cannot live due to harsh environments as well as lack of nutrients such as nitrogen. That means that in some regions, the biogeochemical cycle (including nitrogen cycle and carbon cycle ) is unlikely to run smoothly. Lichen is able to absorb nitrogen in multiple forms from soil, rock, and air, taking a part in carbon cycle at the same time. Even though only a small fraction of lichens have the ability to fix nitrogen, it helps the lichen to spread throughout the world and survive even in the harsh environment. [ 5 ] [ 6 ]
The industrial nitrogen fertilizer greatly affected the vegetation and agriculture throughout the world, resulting significantly increased the amount of food with better quality, but it has a negative impact on ecology in the long run. [ 12 ] Deposition of nitrogen causes soil acidification, and the nitrogen in the fertilizer are often leached through soil and water, running off the different area. [ 13 ] [ 14 ] Soil acidification increases toxicity of the soil which reduces plant biodiversity and based on the toxic level of soil acidification, heavy metal such as aluminum and iron can be related to soil water. [ 14 ]
Earth's mantle contains non-atmospheric nitrogen in the form of rocks and in the soil. [ 15 ] Weathering of the rocks and stone are normally caused by physical, chemical and biological processes. Plants cannot absorb nitrogen from rocks, but fungi can. Fungi within lichens can extract nutrients from mineral surfaces by secreting organic acids. The organic acids (e.g. phenolic acids) are important in solubilizing nutrients from inorganic substrates. [ 4 ] A study was conducted to test rock phosphate solubilization by lichen-forming fungi. Bacteria that were attached to biotic or abiotic surfaces stimulate exopolysaccharide synthesis. [ 4 ] While lichens have the ability to absorb nitrogen from rock, this only accounts for a small portion of the nitrogen cycle compared to the conversion of atmospheric nitrogen as it is more easily available.
Photobionts will become less dependent on fungal nutrient supply when nitrogen deposition increases, as it will be able to access its own nitrogen, and primary producers' nutrient limit will also be reduced. [ 8 ]
Nitrogen is one of the more limiting nutrients and the addition of nitrogen stimulates the photobiont, building up its cell, which subsequently increases its photosynthesis and its carbon input. Multiple nitrogen compounds can be assimilated by lichens, such as NH 4 + , NO 3 − and organic nitrogen compounds. [ 8 ] Nitrogen deposition reduces the nutrient limitation of primary production. Increase in nitrogen deposition will allow the photobiont to access its own nitrogen which makes it less fungal dependent but only up to certain point. [ 8 ]
Depending on the environmental nitrogen availability, the addition of nitrogen can either increase and decrease the growth of the lichen. If the lichen cannot increase its photobiont growth, high nitrogen uptake may result in a higher concentration than it physiologically requires which will negatively affect the lichen and its host plant as the other nutrients are too limiting.
Lichen's response to nutrient enrichment is both species-specific and dependent on environmental factors such as nutrient concentration, light availability and water supply. [ 8 ]
Lichen is nitrogen sensitive and change in nitrogen availability can affect its health greatly.
Two main nitrogen stress factors for lichens are nitrogen deficiency and high nitrogen deposition. [ 3 ] Both types of nitrogen stress result in the reduction of the rate of thallus expansion in lichen. Nitrogen stressed lichen did not show a significant change in chitin:chlorophyll ratios, but ergosterol concentration showed significant increase indicating a higher demand on the respiratory system.
According to an experiment, the ammonium toxicity due to nitrogen deposition reduced the vitality of lichen greatly at different regions such as inland dunes, boreal conditions, and subarctic heaths. [ 3 ] [ 9 ] | https://en.wikipedia.org/wiki/Lichens_and_nitrogen_cycling |
Lichens are a composite organism consisting of fungi and algae living in symbiotic relationship . They are well adapted to survive in harsh conditions. One of the many places they can be found is the Namib , the desert that gave Namibia its name. Fog in the coastal parts of the desert provides the necessary moisture for the organisms' survival. In the Namib they grow on shrubs, rocks and pebbles of the gravel plains. These small organisms can densely cover large areas, forming lichen fields.
The desert hosts 120 lichen species . Most of them are rare and a significant number of them occur only there. "Many are endemic to this region and others show affinities between the Namib lichen biota and other fog deserts of the world, such as the Atacama in South America and Baja California in Mexico and California". [ 1 ]
Besides the Namib Desert, lichens also occur in suitable places elsewhere in Namibia, with at least 232 species recorded from the country as a whole. [ 2 ]
Lichen vegetation is very vulnerable to pollution and mechanical damage. Lichen fields in the Namib are at risk from off-road driving and mining. [ 1 ]
However, the Wlotzkasbaken lichen field north of Swakopmund was considered for protection after an Environmental Impact Assessment was done before the development of a desalination plant serving Trekopje uranium mine. [ 3 ] Fourteen kilometers of fencing was erected around the northeastern side of the field to protect them from damage caused by vehicles taking shortcuts through the desert. Signs were set up by the Ministry of Environment and Tourism to announce the site location and vulnerability, including several colorful information boards on lichens. The mine also put up an information stand. [ 4 ] | https://en.wikipedia.org/wiki/Lichens_of_Namibia |
Lichtheimia is a genus of fungi belonging to the family Lichtheimiaceae . [ 1 ]
The genus has cosmopolitan distribution . [ 1 ]
Species: [ 1 ] | https://en.wikipedia.org/wiki/Lichtheimia |
Licking is the action of passing the tongue over a surface, typically either to deposit saliva onto the surface, or to collect liquid, food or minerals onto the tongue for ingestion , or to communicate with other animals . Many animals both groom themselves , eat or drink by licking.
Grooming: Animals commonly clean themselves through licking. In mammals , licking helps keep the fur clean and untangled. The tongues of many mammals have a rough upper surface that acts like a brush when the animal licks its fur. [ 2 ] Certain reptiles, such as geckos , clean their eyes by licking them, due to them not having eye lids. [ 3 ]
Mammals typically lick their offspring clean immediately after birth ; in many species this is necessary to free the newborn from the amniotic sac . The licking not only cleans and dries the offspring's fur, but also stimulates its breathing and digestive processes. [ 4 ] Canids also stimulate their pups to urinate by licking their preputial gland secretions. [ 1 ]
Food and water acquisition: Hummingbirds are often said to "sip" nectar, but in fact they lap up nectar on their long tongues. [ 5 ] [ 6 ] Their tongues have fringed edges, which help both in nectar-eating and in catching tiny insects. Mother hummingbirds also lick their chicks after a rainstorm to dry them by licking water droplets from the coats of the chicks to avoid them chilling. Many animals drink by licking. While young mammals drink milk from their mothers' teats by sucking , the typical method of drinking for adult mammals involves dipping the tongue repeatedly into water and using it to scoop water into the mouth. [ 7 ] This method of drinking relies in part on the water adhering to the surface of the tongue and in part on muscular control of the tongue to form it into a spoonlike shape. [ citation needed ] Cattle, horses and other animals lick rocks, salt licks or other objects to obtain mineral nutrients. [ 8 ] [ 9 ]
Gustation: Animals also use their tongues to enhance their sense of smell . [ 10 ] By licking a surface or extending the tongue beyond the mouth, molecules are transferred via the tongue to the olfactory receptors in the nose and in some animals, to the vomeronasal organ . In some mammals, the tongue is used to "lick" the air during the flehmen response to assist transfer of pheremones . Similarly, snakes use smell to track their prey. They smell by using their forked tongues to collect airborne particles, then passing them to the vomeronasal organ. They keep their tongues constantly in motion, sampling particles from the air, ground, and water, analyzing the chemicals found, and determining the presence of prey or predators in the local environment. [ 11 ]
Communication: Dogs and cats use licking both to clean and to show affection among themselves or to humans, typically licking their faces. [ 12 ] Many animals use licking as a submissive or appeasement signal in dominance hierarchies . [ 13 ] [ 14 ]
Thermoregulation: Some animals use licking to cool themselves. Cats do not sweat the way humans do and the saliva deposited by licking provides a similar means of evaporative cooling . [ 15 ] Some animals spread saliva over areas of the body with little or no fur to maximise heat loss. For example, kangaroos lick their wrists and rats lick their testicles. [ 16 ] [ 17 ]
Mating behavior: Male mammals often lick the genitals of females before copulation . [ 18 ] Post-copulatory genital grooming often occurs in male rats and prosimian primates. [ 19 ] This behavior may prevent disease transmission. [ 20 ] [ 21 ]
Ring-tailed lemurs lick each other's babies as a means of collective grooming and of reinforcing social cohesion within the community. [ 22 ] Macaques and other primates lick leaves for water in addition to dipping their arms into tree crevices and licking the water off. [ 23 ] Chimpanzees use licking in a variety of ways: licking objects, such as dead trees, that others in their community have licked, [ 24 ] licking each other's body parts for grooming and sex [ 24 ] and licking rocks for salt. [ 25 ] Gorillas use licking in addition to other senses to determine the nature of an object. [ 26 ]
Compared to most other mammals, licking has a minor role for humans . The human tongue is relatively short and inflexible, and is not well adapted for either grooming or drinking. Instead, humans prefer to wash themselves using their hands and drink by sucking or pouring fluid into their mouth. Humans have much less hair over their skin than most other mammals, and much of that hair is in places which they cannot reach with their own mouth. The presence of sweat glands all over the human body makes licking as a cooling method unnecessary.
Nonetheless, licking does play a role for humans. Even though humans cannot effectively drink water by licking, the human tongue is quite sufficient for licking more viscous fluids. Some foods are sold in a form intended to be consumed mainly by licking, e.g. ice cream cones and lollipops . Though useful, in some cultures it is considered improper table manners to clean one's fingers by licking during a meal.
Some people in the Afar tribe of Ethiopia have been reported to have used their tongues to lick other humans, as a way of cleaning them from the dust that accumulates on them in a very water-scarce region. [ 27 ] [ failed verification ]
Humans use licking for a number of other purposes. For example, licking can moisten the adhesive surfaces of stamps or envelopes . Many people lick a fingertip (usually that of the index finger ) for some extra grip when turning a page, taking a sheet of paper from the top of a pile or opening a plastic bag. In sewing , thread ends are commonly wet by licking to make the fibres stick together and thus make threading them through the eye of a needle easier. Another practice considered uncivilized is licking one's hand and using it to groom one's hair.
Humans also use their tongues for sexual purposes, such as during cunnilingus , anilingus , fellatio , breast licking, [ 28 ] foot licking , and whilst French kissing , where two people lick each other's tongues.
Self-licking can sometimes become abnormally frequent [ 31 ] occasionally resulting in a lick granuloma . The most common cause of lick granuloma appears to be psychological, related to stress, anxiety, separation anxiety , boredom, or compulsiveness. [ 32 ] Lick granulomae are especially seen in active dogs left alone for long periods of time. One theory is that excessive licking causes endorphin release, which reduces pain and makes the dog feel temporarily euphoric . This provides the animal with positive feedback from the licking, and subsequent addiction to the behaviour.
Animals in captivity sometimes develop a licking stereotypy during which surfaces (walls, bars, gates, etc.) are repeatedly licked for no apparent reason. This has been observed in captive giraffes and camels. [ 33 ] [ 34 ] | https://en.wikipedia.org/wiki/Licking |
In mathematics , the Lickorish–Wallace theorem in the theory of 3-manifolds states that any closed , orientable , connected 3-manifold may be obtained by performing Dehn surgery on a framed link in the 3-sphere with ±1 surgery coefficients. Furthermore, each component of the link can be assumed to be unknotted.
The theorem was proved in the early 1960s by W. B. R. Lickorish and Andrew H. Wallace , independently and by different methods. Lickorish's proof rested on the Lickorish twist theorem , which states that any orientable automorphism of a closed orientable surface is generated by Dehn twists along 3 g − 1 specific simple closed curves in the surface, where g denotes the genus of the surface. Wallace's proof was more general and involved adding handles to the boundary of a higher-dimensional ball.
A corollary of the theorem is that every closed, orientable 3-manifold bounds a simply-connected compact 4-manifold .
By using his work on automorphisms of non-orientable surfaces, Lickorish also showed that every closed, non-orientable, connected 3-manifold is obtained by Dehn surgery on a link in the non-orientable 2-sphere bundle over the circle. Similar to the orientable case, the surgery can be done in a special way which allows the conclusion that every closed, non-orientable 3-manifold bounds a compact 4-manifold. | https://en.wikipedia.org/wiki/Lickorish–Wallace_theorem |
Lidia Angeleri Hügel (born 1960) is an Italian mathematician whose research in abstract algebra and representation theory focuses on tilting theory and its offshoot, silting theory . She is a professor of algebra at the University of Verona . [ 1 ]
Angeleri Hügel was born in Milan , [ 2 ] in 1960. [ 3 ] She studied mathematics at the Ludwig Maximilian University of Munich , completing a Ph.D. there in 1991 under the supervision of Wolfgang Zimmermann. [ 4 ]
She continued at Ludwig Maximilian University of Munich as a postdoctoral researcher from 1992 to 2002, earning a habilitation there in 2000. In 2002, she was Ramon y Cajal Fellow at the Autonomous University of Barcelona , and briefly held an associate professorship at the University of Insubria , before moving to the University of Verona as an associate professor. She became full professor at the University of Verona in 2016. [ 5 ]
At the University of Verona, she served as Vice-Rector for International Relations from 2013 to 2019. [ 5 ]
Angeleri Hügel is the co-editor of the Handbook of Tilting Theory (Cambridge University Press, London Mathematical Society Lecture Note Series 332, 2007, with Dieter Happel and Henning Krause). [ 6 ] | https://en.wikipedia.org/wiki/Lidia_Angeleri_Hügel |
Lidia Vallarino (1930–2017) was an inorganic chemist who was chemistry lecturer at the University of Milan. In the 1950s and 19960s, she was a rare example of a well-published female active in coordination chemistry and organometallic chemistry . [ 1 ] [ 2 ]
Vallarino received her PhD in 1954 from the University of Milan under the supervision of L. Malatesta for work on isocyanide complexes . [ 3 ] She later took a position at ICI Laboratories under Joseph Chatt , where she worked on diene complexes of the platinum group metals . [ 4 ] As an independent scientist, she worked on both organorhodium chemistry [ 5 ] and macrocyclic complexes of the lanthanides. [ 6 ] She retired as professor of chemistry at Virginia Commonwealth University . | https://en.wikipedia.org/wiki/Lidia_Vallarino |
In mathematics , a Lidstone series , named after George James Lidstone , is a kind of polynomial expansion that can express certain types of entire functions .
Let ƒ ( z ) be an entire function of exponential type less than ( N + 1) π , as defined below. Then ƒ ( z ) can be expanded in terms of polynomials A n as follows:
Here A n ( z ) is a polynomial in z of degree n , C k a constant, and ƒ ( n ) ( a ) the n th derivative of ƒ at a .
A function is said to be of exponential type of less than t if the function
is bounded above by t . Thus, the constant N used in the summation above is given by
with
This mathematical analysis –related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Lidstone_series |
In mathematics, a Lie-* algebra is a D-module with a Lie* bracket. They were introduced by Alexander Beilinson and Vladimir Drinfeld ( Beilinson & Drinfeld (2004 , section 2.5.3)), and are similar to the conformal algebras discussed by Kac (1998) and to vertex Lie algebras .
This algebra -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Lie-*_algebra |
In mathematics , a Lie group (pronounced / l iː / LEE ) is a group that is also a differentiable manifold , such that group multiplication and taking inverses are both differentiable.
A manifold is a space that locally resembles Euclidean space , whereas groups define the abstract concept of a binary operation along with the additional properties it must have to be thought of as a "transformation" in the abstract sense, for instance multiplication and the taking of inverses (to allow division), or equivalently, the concept of addition and subtraction. Combining these two ideas, one obtains a continuous group where multiplying points and their inverses is continuous. If the multiplication and taking of inverses are smooth (differentiable) as well, one obtains a Lie group.
Lie groups provide a natural model for the concept of continuous symmetry , a celebrated example of which is the circle group . Rotating a circle is an example of a continuous symmetry. For any rotation of the circle, there exists the same symmetry, [ 1 ] and concatenation of such rotations makes them into the circle group, an archetypal example of a Lie group. Lie groups are widely used in many parts of modern mathematics and physics .
Lie groups were first found by studying matrix subgroups G {\displaystyle G} contained in GL n ( R ) {\displaystyle {\text{GL}}_{n}(\mathbb {R} )} or GL n ( C ) {\displaystyle {\text{GL}}_{n}(\mathbb {C} )} , the groups of n × n {\displaystyle n\times n} invertible matrices over R {\displaystyle \mathbb {R} } or C {\displaystyle \mathbb {C} } . These are now called the classical groups , as the concept has been extended far beyond these origins. Lie groups are named after Norwegian mathematician Sophus Lie (1842–1899), who laid the foundations of the theory of continuous transformation groups . Lie's original motivation for introducing Lie groups was to model the continuous symmetries of differential equations , in much the same way that finite groups are used in Galois theory to model the discrete symmetries of algebraic equations .
Sophus Lie considered the winter of 1873–1874 as the birth date of his theory of continuous groups. [ 2 ] Thomas Hawkins, however, suggests that it was "Lie's prodigious research activity during the four-year period from the fall of 1869 to the fall of 1873" that led to the theory's creation. [ 2 ] Some of Lie's early ideas were developed in close collaboration with Felix Klein . Lie met with Klein every day from October 1869 through 1872: in Berlin from the end of October 1869 to the end of February 1870, and in Paris, Göttingen and Erlangen in the subsequent two years. [ 3 ] Lie stated that all of the principal results were obtained by 1884. But during the 1870s all his papers (except the very first note) were published in Norwegian journals, which impeded recognition of the work throughout the rest of Europe. [ 4 ] In 1884 a young German mathematician, Friedrich Engel , came to work with Lie on a systematic treatise to expose his theory of continuous groups. From this effort resulted the three-volume Theorie der Transformationsgruppen , published in 1888, 1890, and 1893. The term groupes de Lie first appeared in French in 1893 in the thesis of Lie's student Arthur Tresse. [ 5 ]
Lie's ideas did not stand in isolation from the rest of mathematics. In fact, his interest in the geometry of differential equations was first motivated by the work of Carl Gustav Jacobi , on the theory of partial differential equations of first order and on the equations of classical mechanics . Much of Jacobi's work was published posthumously in the 1860s, generating enormous interest in France and Germany. [ 6 ] Lie's idée fixe was to develop a theory of symmetries of differential equations that would accomplish for them what Évariste Galois had done for algebraic equations: namely, to classify them in terms of group theory. Lie and other mathematicians showed that the most important equations for special functions and orthogonal polynomials tend to arise from group theoretical symmetries. In Lie's early work, the idea was to construct a theory of continuous groups , to complement the theory of discrete groups that had developed in the theory of modular forms , in the hands of Felix Klein and Henri Poincaré . The initial application that Lie had in mind was to the theory of differential equations . On the model of Galois theory and polynomial equations , the driving conception was of a theory capable of unifying, by the study of symmetry , the whole area of ordinary differential equations . However, the hope that Lie theory would unify the entire field of ordinary differential equations was not fulfilled. Symmetry methods for ODEs continue to be studied, but do not dominate the subject. There is a differential Galois theory , but it was developed by others, such as Picard and Vessiot, and it provides a theory of quadratures , the indefinite integrals required to express solutions.
Additional impetus to consider continuous groups came from ideas of Bernhard Riemann , on the foundations of geometry, and their further development in the hands of Klein. Thus three major themes in 19th century mathematics were combined by Lie in creating his new theory:
Although today Sophus Lie is rightfully recognized as the creator of the theory of continuous groups, a major stride in the development of their structure theory, which was to have a profound influence on subsequent development of mathematics, was made by Wilhelm Killing , who in 1888 published the first paper in a series entitled Die Zusammensetzung der stetigen endlichen Transformationsgruppen ( The composition of continuous finite transformation groups ). [ 7 ] The work of Killing, later refined and generalized by Élie Cartan , led to classification of semisimple Lie algebras , Cartan's theory of symmetric spaces , and Hermann Weyl 's description of representations of compact and semisimple Lie groups using highest weights .
In 1900 David Hilbert challenged Lie theorists with his Fifth Problem presented at the International Congress of Mathematicians in Paris.
Weyl brought the early period of the development of the theory of Lie groups to fruition, for not only did he classify irreducible representations of semisimple Lie groups and connect the theory of groups with quantum mechanics, but he also put Lie's theory itself on firmer footing by clearly enunciating the distinction between Lie's infinitesimal groups (i.e., Lie algebras) and the Lie groups proper, and began investigations of topology of Lie groups. [ 8 ] The theory of Lie groups was systematically reworked in modern mathematical language in a monograph by Claude Chevalley .
Lie groups are smooth differentiable manifolds and as such can be studied using differential calculus , in contrast with the case of more general topological groups . One of the key ideas in the theory of Lie groups is to replace the global object, the group, with its local or linearized version, which Lie himself called its "infinitesimal group" and which has since become known as its Lie algebra .
Lie groups play an enormous role in modern geometry , on several different levels. Felix Klein argued in his Erlangen program that one can consider various "geometries" by specifying an appropriate transformation group that leaves certain geometric properties invariant . Thus Euclidean geometry corresponds to the choice of the group E(3) of distance-preserving transformations of the Euclidean space R 3 {\displaystyle \mathbb {R} ^{3}} , conformal geometry corresponds to enlarging the group to the conformal group , whereas in projective geometry one is interested in the properties invariant under the projective group . This idea later led to the notion of a G -structure , where G is a Lie group of "local" symmetries of a manifold.
Lie groups (and their associated Lie algebras) play a major role in modern physics, with the Lie group typically playing the role of a symmetry of a physical system. Here, the representations of the Lie group (or of its Lie algebra ) are especially important. Representation theory is used extensively in particle physics . Groups whose representations are of particular importance include the rotation group SO(3) (or its double cover SU(2) ), the special unitary group SU(3) and the Poincaré group .
On a "global" level, whenever a Lie group acts on a geometric object, such as a Riemannian or a symplectic manifold , this action provides a measure of rigidity and yields a rich algebraic structure. The presence of continuous symmetries expressed via a Lie group action on a manifold places strong constraints on its geometry and facilitates analysis on the manifold. Linear actions of Lie groups are especially important, and are studied in representation theory .
In the 1940s–1950s, Ellis Kolchin , Armand Borel , and Claude Chevalley realised that many foundational results concerning Lie groups can be developed completely algebraically, giving rise to the theory of algebraic groups defined over an arbitrary field . This insight opened new possibilities in pure algebra, by providing a uniform construction for most finite simple groups , as well as in algebraic geometry . The theory of automorphic forms , an important branch of modern number theory , deals extensively with analogues of Lie groups over adele rings ; p -adic Lie groups play an important role, via their connections with Galois representations in number theory.
A real Lie group is a group that is also a finite-dimensional real smooth manifold , in which the group operations of multiplication and inversion are smooth maps . Smoothness of the group multiplication
means that μ is a smooth mapping of the product manifold G × G into G . The two requirements can be combined to the single requirement that the mapping
be a smooth mapping of the product manifold into G .
We now present an example of a group with an uncountable number of elements that is not a Lie group under a certain topology. The group given by
with a ∈ R ∖ Q {\displaystyle a\in \mathbb {R} \setminus \mathbb {Q} } a fixed irrational number , is a subgroup of the torus T 2 {\displaystyle \mathbb {T} ^{2}} that is not a Lie group when given the subspace topology . [ 9 ] If we take any small neighborhood U {\displaystyle U} of a point h {\displaystyle h} in H {\displaystyle H} , for example, the portion of H {\displaystyle H} in U {\displaystyle U} is disconnected. The group H {\displaystyle H} winds repeatedly around the torus without ever reaching a previous point of the spiral and thus forms a dense subgroup of T 2 {\displaystyle \mathbb {T} ^{2}} .
The group H {\displaystyle H} can, however, be given a different topology, in which the distance between two points h 1 , h 2 ∈ H {\displaystyle h_{1},h_{2}\in H} is defined as the length of the shortest path in the group H {\displaystyle H} joining h 1 {\displaystyle h_{1}} to h 2 {\displaystyle h_{2}} . In this topology, H {\displaystyle H} is identified homeomorphically with the real line by identifying each element with the number θ {\displaystyle \theta } in the definition of H {\displaystyle H} . With this topology, H {\displaystyle H} is just the group of real numbers under addition and is therefore a Lie group.
The group H {\displaystyle H} is an example of a " Lie subgroup " of a Lie group that is not closed. See the discussion below of Lie subgroups in the section on basic concepts.
Let GL ( n , C ) {\displaystyle \operatorname {GL} (n,\mathbb {C} )} denote the group of n × n {\displaystyle n\times n} invertible matrices with entries in C {\displaystyle \mathbb {C} } . Any closed subgroup of GL ( n , C ) {\displaystyle \operatorname {GL} (n,\mathbb {C} )} is a Lie group; [ 10 ] Lie groups of this sort are called matrix Lie groups. Since most of the interesting examples of Lie groups can be realized as matrix Lie groups, some textbooks restrict attention to this class, including those of Hall, [ 11 ] Rossmann, [ 12 ] and Stillwell. [ 13 ] Restricting attention to matrix Lie groups simplifies the definition of the Lie algebra and the exponential map. The following are standard examples of matrix Lie groups.
All of the preceding examples fall under the heading of the classical groups .
A complex Lie group is defined in the same way using complex manifolds rather than real ones (example: SL ( 2 , C ) {\displaystyle \operatorname {SL} (2,\mathbb {C} )} ), and holomorphic maps. Similarly, using an alternate metric completion of Q {\displaystyle \mathbb {Q} } , one can define a p -adic Lie group over the p -adic numbers , a topological group which is also an analytic p -adic manifold, such that the group operations are analytic. In particular, each point has a p -adic neighborhood.
Hilbert's fifth problem asked whether replacing differentiable manifolds with topological or analytic ones can yield new examples. The answer to this question turned out to be negative: in 1952, Gleason , Montgomery and Zippin showed that if G is a topological manifold with continuous group operations, then there exists exactly one analytic structure on G which turns it into a Lie group (see also Hilbert–Smith conjecture ). If the underlying manifold is allowed to be infinite-dimensional (for example, a Hilbert manifold ), then one arrives at the notion of an infinite-dimensional Lie group. It is possible to define analogues of many Lie groups over finite fields , and these give most of the examples of finite simple groups .
The language of category theory provides a concise definition for Lie groups: a Lie group is a group object in the category of smooth manifolds. This is important, because it allows generalization of the notion of a Lie group to Lie supergroups . This categorical point of view leads also to a different generalization of Lie groups, namely Lie groupoids , which are groupoid objects in the category of smooth manifolds with a further requirement.
A Lie group can be defined as a ( Hausdorff ) topological group that, near the identity element, looks like a transformation group, with no reference to differentiable manifolds. [ 14 ] First, we define an immersely linear Lie group to be a subgroup G of the general linear group GL ( n , C ) {\displaystyle \operatorname {GL} (n,\mathbb {C} )} such that
(For example, a closed subgroup of GL ( n , C ) {\displaystyle \operatorname {GL} (n,\mathbb {C} )} ; that is, a matrix Lie group satisfies the above conditions.)
Then a Lie group is defined as a topological group that (1) is locally isomorphic near the identities to an immersely linear Lie group and (2) has at most countably many connected components. Showing the topological definition is equivalent to the usual one is technical (and the beginning readers should skip the following) but is done roughly as follows:
The topological definition implies the statement that if two Lie groups are isomorphic as topological groups, then they are isomorphic as Lie groups. In fact, it states the general principle that, to a large extent, the topology of a Lie group together with the group law determines the geometry of the group.
Lie groups occur in abundance throughout mathematics and physics. Matrix groups or algebraic groups are (roughly) groups of matrices (for example, orthogonal and symplectic groups ), and these give most of the more common examples of Lie groups.
The only connected Lie groups with dimension one are the real line R {\displaystyle \mathbb {R} } (with the group operation being addition) and the circle group S 1 {\displaystyle S^{1}} of complex numbers with absolute value one (with the group operation being multiplication). The S 1 {\displaystyle S^{1}} group is often denoted as U ( 1 ) {\displaystyle \operatorname {U} (1)} , the group of 1 × 1 {\displaystyle 1\times 1} unitary matrices.
In two dimensions, if we restrict attention to simply connected groups, then they are classified by their Lie algebras. There are (up to isomorphism) only two Lie algebras of dimension two. The associated simply connected Lie groups are R 2 {\displaystyle \mathbb {R} ^{2}} (with the group operation being vector addition) and the affine group in dimension one, described in the previous subsection under "first examples".
There are several standard ways to form new Lie groups from old ones:
Some examples of groups that are not Lie groups (except in the trivial sense that any group having at most countably many elements can be viewed as a 0-dimensional Lie group, with the discrete topology ), are:
To every Lie group we can associate a Lie algebra whose underlying vector space is the tangent space of the Lie group at the identity element and which completely captures the local structure of the group. Informally we can think of elements of the Lie algebra as elements of the group that are " infinitesimally close" to the identity, and the Lie bracket of the Lie algebra is related to the commutator of two such infinitesimal elements. Before giving the abstract definition we give a few examples:
The concrete definition given above for matrix groups is easy to work with, but has some minor problems: to use it we first need to represent a Lie group as a group of matrices, but not all Lie groups can be represented in this way, and it is not even obvious that the Lie algebra is independent of the representation we use. [ 18 ] To get around these problems we give
the general definition of the Lie algebra of a Lie group (in 4 steps):
This Lie algebra g {\displaystyle {\mathfrak {g}}} is finite-dimensional and it has the same dimension as the manifold G . The Lie algebra of G determines G up to "local isomorphism", where two Lie groups are called locally isomorphic if they look the same near the identity element.
Problems about Lie groups are often solved by first solving the corresponding problem for the Lie algebras, and the result for groups then usually follows easily.
For example, simple Lie groups are usually classified by first classifying the corresponding Lie algebras.
We could also define a Lie algebra structure on T e using right invariant vector fields instead of left invariant vector fields. This leads to the same Lie algebra, because the inverse map on G can be used to identify left invariant vector fields with right invariant vector fields, and acts as −1 on the tangent space T e .
The Lie algebra structure on T e can also be described as follows:
the commutator operation
on G × G sends ( e , e ) to e , so its derivative yields a bilinear operation on T e G . This bilinear operation is actually the zero map, but the second derivative, under the proper identification of tangent spaces, yields an operation that satisfies the axioms of a Lie bracket , and it is equal to twice the one defined through left-invariant vector fields.
If G and H are Lie groups, then a Lie group homomorphism f : G → H is a smooth group homomorphism . In the case of complex Lie groups, such a homomorphism is required to be a holomorphic map . However, these requirements are a bit stringent; every continuous homomorphism between real Lie groups turns out to be (real) analytic . [ 19 ] [ b ]
The composition of two Lie homomorphisms is again a homomorphism, and the class of all Lie groups, together with these morphisms, forms a category . Moreover, every Lie group homomorphism induces a homomorphism between the corresponding Lie algebras. Let ϕ : G → H {\displaystyle \phi :G\to H} be a Lie group homomorphism and let ϕ ∗ {\displaystyle \phi _{*}} be its derivative at the identity. If we identify the Lie algebras of G and H with their tangent spaces at the identity elements, then ϕ ∗ {\displaystyle \phi _{*}} is a map between the corresponding Lie algebras:
which turns out to be a Lie algebra homomorphism (meaning that it is a linear map which preserves the Lie bracket ). In the language of category theory , we then have a covariant functor from the category of Lie groups to the category of Lie algebras which sends a Lie group to its Lie algebra and a Lie group homomorphism to its derivative at the identity.
Two Lie groups are called isomorphic if there exists a bijective homomorphism between them whose inverse is also a Lie group homomorphism. Equivalently, it is a diffeomorphism which is also a group homomorphism. Observe that, by the above, a continuous homomorphism from a Lie group G {\displaystyle G} to a Lie group H {\displaystyle H} is an isomorphism of Lie groups if and only if it is bijective.
Isomorphic Lie groups necessarily have isomorphic Lie algebras; it is then reasonable to ask how isomorphism classes of Lie groups relate to isomorphism classes of Lie algebras.
The first result in this direction is Lie's third theorem , which states that every finite-dimensional, real Lie algebra is the Lie algebra of some (linear) Lie group. One way to prove Lie's third theorem is to use Ado's theorem , which says every finite-dimensional real Lie algebra is isomorphic to a matrix Lie algebra. Meanwhile, for every finite-dimensional matrix Lie algebra, there is a linear group (matrix Lie group) with this algebra as its Lie algebra. [ 20 ]
On the other hand, Lie groups with isomorphic Lie algebras need not be isomorphic. Furthermore, this result remains true even if we assume the groups are connected. To put it differently, the global structure of a Lie group is not determined by its Lie algebra; for example, if Z is any discrete subgroup of the center of G then G and G / Z have the same Lie algebra (see the table of Lie groups for examples). An example of importance in physics are the groups SU(2) and SO(3) . These two groups have isomorphic Lie algebras, [ 21 ] but the groups themselves are not isomorphic, because SU(2) is simply connected but SO(3) is not. [ 22 ]
On the other hand, if we require that the Lie group be simply connected , then the global structure is determined by its Lie algebra: two simply connected Lie groups with isomorphic Lie algebras are isomorphic. [ 23 ] (See the next subsection for more information about simply connected Lie groups.) In light of Lie's third theorem, we may therefore say that there is a one-to-one correspondence between isomorphism classes of finite-dimensional real Lie algebras and isomorphism classes of simply connected Lie groups.
A Lie group G {\displaystyle G} is said to be simply connected if every loop in G {\displaystyle G} can be shrunk continuously to a point in G {\displaystyle G} . This notion is important because of the following result that has simple connectedness as a hypothesis:
Lie's third theorem says that every finite-dimensional real Lie algebra is the Lie algebra of a Lie group. It follows from Lie's third theorem and the preceding result that every finite-dimensional real Lie algebra is the Lie algebra of a unique simply connected Lie group.
An example of a simply connected group is the special unitary group SU(2) , which as a manifold is the 3-sphere. The rotation group SO(3) , on the other hand, is not simply connected. (See Topology of SO(3) .) The failure of SO(3) to be simply connected is intimately connected to the distinction between integer spin and half-integer spin in quantum mechanics. Other examples of simply connected Lie groups include the special unitary group SU(n) , the spin group (double cover of rotation group) Spin( n ) for n ≥ 3 {\displaystyle n\geq 3} , and the compact symplectic group Sp(n) . [ 25 ]
Methods for determining whether a Lie group is simply connected or not are discussed in the article on fundamental groups of Lie groups .
The exponential map from the Lie algebra M ( n ; C ) {\displaystyle \mathrm {M} (n;\mathbb {C} )} of the general linear group G L ( n ; C ) {\displaystyle \mathrm {GL} (n;\mathbb {C} )} to G L ( n ; C ) {\displaystyle \mathrm {GL} (n;\mathbb {C} )} is defined by the matrix exponential , given by the usual power series:
for matrices X {\displaystyle X} . If G {\displaystyle G} is a closed subgroup of G L ( n ; C ) {\displaystyle \mathrm {GL} (n;\mathbb {C} )} , then the exponential map takes the Lie algebra of G {\displaystyle G} into G {\displaystyle G} ; thus, we have an exponential map for all matrix groups. Every element of G {\displaystyle G} that is sufficiently close to the identity is the exponential of a matrix in the Lie algebra. [ 26 ]
The definition above is easy to use, but it is not defined for Lie groups that are not matrix groups, and it is not clear that the exponential map of a Lie group does not depend on its representation as a matrix group. We can solve both problems using a more abstract definition of the exponential map that works for all Lie groups, as follows.
For each vector X {\displaystyle X} in the Lie algebra g {\displaystyle {\mathfrak {g}}} of G {\displaystyle G} (i.e., the tangent space to G {\displaystyle G} at the identity), one proves that there is a unique one-parameter subgroup c : R → G {\displaystyle c:\mathbb {R} \rightarrow G} such that c ′ ( 0 ) = X {\displaystyle c'(0)=X} . Saying that c {\displaystyle c} is a one-parameter subgroup means simply that c {\displaystyle c} is a smooth map into G {\displaystyle G} and that
for all s {\displaystyle s} and t {\displaystyle t} . The operation on the right hand side is the group multiplication in G {\displaystyle G} . The formal similarity of this formula with the one valid for the exponential function justifies the definition
This is called the exponential map , and it maps the Lie algebra g {\displaystyle {\mathfrak {g}}} into the Lie group G {\displaystyle G} . It provides a diffeomorphism between a neighborhood of 0 in g {\displaystyle {\mathfrak {g}}} and a neighborhood of e {\displaystyle e} in G {\displaystyle G} . This exponential map is a generalization of the exponential function for real numbers (because R {\displaystyle \mathbb {R} } is the Lie algebra of the Lie group of positive real numbers with multiplication), for complex numbers (because C {\displaystyle \mathbb {C} } is the Lie algebra of the Lie group of non-zero complex numbers with multiplication) and for matrices (because M ( n , R ) {\displaystyle M(n,\mathbb {R} )} with the regular commutator is the Lie algebra of the Lie group G L ( n , R ) {\displaystyle \mathrm {GL} (n,\mathbb {R} )} of all invertible matrices).
Because the exponential map is surjective on some neighbourhood N {\displaystyle N} of e {\displaystyle e} , it is common to call elements of the Lie algebra infinitesimal generators of the group G {\displaystyle G} . The subgroup of G {\displaystyle G} generated by N {\displaystyle N} is the identity component of G {\displaystyle G} .
The exponential map and the Lie algebra determine the local group structure of every connected Lie group, because of the Baker–Campbell–Hausdorff formula : there exists a neighborhood U {\displaystyle U} of the zero element of g {\displaystyle {\mathfrak {g}}} , such that for X , Y ∈ U {\displaystyle X,Y\in U} we have
where the omitted terms are known and involve Lie brackets of four or more elements. In case X {\displaystyle X} and Y {\displaystyle Y} commute, this formula reduces to the familiar exponential law exp ( X ) exp ( Y ) = exp ( X + Y ) {\displaystyle \exp(X)\exp(Y)=\exp(X+Y)} .
The exponential map relates Lie group homomorphisms. That is, if ϕ : G → H {\displaystyle \phi :G\to H} is a Lie group homomorphism and ϕ ∗ : g → h {\displaystyle \phi _{*}:{\mathfrak {g}}\to {\mathfrak {h}}} the induced map on the corresponding Lie algebras, then for all x ∈ g {\displaystyle x\in {\mathfrak {g}}} we have
In other words, the following diagram commutes , [ 27 ]
(In short, exp is a natural transformation from the functor Lie to the identity functor on the category of Lie groups.)
The exponential map from the Lie algebra to the Lie group is not always onto , even if the group is connected (though it does map onto the Lie group for connected groups that are either compact or nilpotent). For example, the exponential map of SL(2, R ) is not surjective. Also, the exponential map is neither surjective nor injective for infinite-dimensional (see below) Lie groups modelled on C ∞ Fréchet space , even from arbitrary small neighborhood of 0 to corresponding neighborhood of 1.
A Lie subgroup H {\displaystyle H} of a Lie group G {\displaystyle G} is a Lie group that is a subset of G {\displaystyle G} and such that the inclusion map from H {\displaystyle H} to G {\displaystyle G} is an injective immersion and group homomorphism . According to Cartan's theorem , a closed subgroup of G {\displaystyle G} admits a unique smooth structure which makes it an embedded Lie subgroup of G {\displaystyle G} —i.e. a Lie subgroup such that the inclusion map is a smooth embedding.
Examples of non-closed subgroups are plentiful; for example take G {\displaystyle G} to be a torus of dimension 2 or greater, and let H {\displaystyle H} be a one-parameter subgroup of irrational slope , i.e. one that winds around in G . Then there is a Lie group homomorphism φ : R → G {\displaystyle \varphi :\mathbb {R} \to G} with i m ( φ ) = H {\displaystyle \mathrm {im} (\varphi )=H} . The closure of H {\displaystyle H} will be a sub-torus in G {\displaystyle G} .
The exponential map gives a one-to-one correspondence between the connected Lie subgroups of a connected Lie group G {\displaystyle G} and the subalgebras of the Lie algebra of G {\displaystyle G} . [ 28 ] Typically, the subgroup corresponding to a subalgebra is not a closed subgroup. There is no criterion solely based on the structure of G {\displaystyle G} which determines which subalgebras correspond to closed subgroups.
One important aspect of the study of Lie groups is their representations, that is, the way they can act (linearly) on vector spaces. In physics, Lie groups often encode the symmetries of a physical system. The way one makes use of this symmetry to help analyze the system is often through representation theory. Consider, for example, the time-independent Schrödinger equation in quantum mechanics, H ^ ψ = E ψ {\displaystyle {\hat {H}}\psi =E\psi } . Assume the system in question has the rotation group SO(3) as a symmetry, meaning that the Hamiltonian operator H ^ {\displaystyle {\hat {H}}} commutes with the action of SO(3) on the wave function ψ {\displaystyle \psi } . (One important example of such a system is the hydrogen atom , which has a spherically symmetric potential.) This assumption does not necessarily mean that the solutions ψ {\displaystyle \psi } are rotationally invariant functions. Rather, it means that the space of solutions to H ^ ψ = E ψ {\displaystyle {\hat {H}}\psi =E\psi } is invariant under rotations (for each fixed value of E {\displaystyle E} ). This space, therefore, constitutes a representation of SO(3). These representations have been classified and the classification leads to a substantial simplification of the problem , essentially converting a three-dimensional partial differential equation to a one-dimensional ordinary differential equation.
The case of a connected compact Lie group K (including the just-mentioned case of SO(3)) is particularly tractable. [ 29 ] In that case, every finite-dimensional representation of K decomposes as a direct sum of irreducible representations. The irreducible representations, in turn, were classified by Hermann Weyl . The classification is in terms of the "highest weight" of the representation. The classification is closely related to the classification of representations of a semisimple Lie algebra .
One can also study (in general infinite-dimensional) unitary representations of an arbitrary Lie group (not necessarily compact). For example, it is possible to give a relatively simple explicit description of the representations of the group SL(2, R ) and the representations of the Poincaré group .
Lie groups may be thought of as smoothly varying families of symmetries. Examples of symmetries include rotation about an axis. What must be understood is the nature of 'small' transformations, for example, rotations through tiny angles, that link nearby transformations. The mathematical object capturing this structure is called a Lie algebra ( Lie himself called them "infinitesimal groups"). It can be defined because Lie groups are smooth manifolds, so have tangent spaces at each point.
The Lie algebra of any compact Lie group (very roughly: one for which the symmetries form a bounded set) can be decomposed as a direct sum of an abelian Lie algebra and some number of simple ones. The structure of an abelian Lie algebra is mathematically uninteresting (since the Lie bracket is identically zero); the interest is in the simple summands. Hence the question arises: what are the simple Lie algebras of compact groups? It turns out that they mostly fall into four infinite families, the "classical Lie algebras" A n , B n , C n and D n , which have simple descriptions in terms of symmetries of Euclidean space. But there are also just five "exceptional Lie algebras" that do not fall into any of these families. E 8 is the largest of these.
Lie groups are classified according to their algebraic properties ( simple , semisimple , solvable , nilpotent , abelian ), their connectedness ( connected or simply connected ) and their compactness .
A first key result is the Levi decomposition , which says that every simply connected Lie group is the semidirect product of a solvable normal subgroup and a semisimple subgroup.
The identity component of any Lie group is an open normal subgroup , and the quotient group is a discrete group . The universal cover of any connected Lie group is a simply connected Lie group, and conversely any connected Lie group is a quotient of a simply connected Lie group by a discrete normal subgroup of the center. Any Lie group G can be decomposed into discrete, simple, and abelian groups in a canonical way as follows. Write
so that we have a sequence of normal subgroups
Then
This can be used to reduce some problems about Lie groups (such as finding their unitary representations) to the same problems for connected simple groups and nilpotent and solvable subgroups of smaller dimension.
Lie groups are often defined to be finite-dimensional, but there are many groups that resemble Lie groups, except for being infinite-dimensional. The simplest way to define infinite-dimensional Lie groups is to model them locally on Banach spaces (as opposed to Euclidean space in the finite-dimensional case), and in this case much of the basic theory is similar to that of finite-dimensional Lie groups. However this is inadequate for many applications, because many natural examples of infinite-dimensional Lie groups are not Banach manifolds . Instead one needs to define Lie groups modeled on more general locally convex topological vector spaces. In this case the relation between the Lie algebra and the Lie group becomes rather subtle, and several results about finite-dimensional Lie groups no longer hold.
The literature is not entirely uniform in its terminology as to exactly which properties of infinite-dimensional groups qualify the group for the prefix Lie in Lie group . On the Lie algebra side of affairs, things are simpler since the qualifying criteria for the prefix Lie in Lie algebra are purely algebraic. For example, an infinite-dimensional Lie algebra may or may not have a corresponding Lie group. That is, there may be a group corresponding to the Lie algebra, but it might not be nice enough to be called a Lie group, or the connection between the group and the Lie algebra might not be nice enough (for example, failure of the exponential map to be onto a neighborhood of the identity). It is the "nice enough" that is not universally defined.
Some of the examples that have been studied include: | https://en.wikipedia.org/wiki/Lie_group |
In mathematics , a Lie groupoid is a groupoid where the set Ob {\displaystyle \operatorname {Ob} } of objects and the set Mor {\displaystyle \operatorname {Mor} } of morphisms are both manifolds , all the category operations (source and target, composition, identity-assigning map and inversion) are smooth, and the source and target operations
are submersions .
A Lie groupoid can thus be thought of as a "many-object generalization" of a Lie group , just as a groupoid is a many-object generalization of a group . Accordingly, while Lie groups provide a natural model for (classical) continuous symmetries , Lie groupoids are often used as model for (and arise from) generalised, point-dependent symmetries. [ 1 ] Extending the correspondence between Lie groups and Lie algebras, Lie groupoids are the global counterparts of Lie algebroids .
Lie groupoids were introduced by Charles Ehresmann [ 2 ] [ 3 ] under the name differentiable groupoids .
A Lie groupoid consists of
such that
Using the language of category theory , a Lie groupoid can be more compactly defined as a groupoid (i.e. a small category where all the morphisms are invertible) such that the sets M {\displaystyle M} of objects and G {\displaystyle G} of morphisms are manifolds, the maps s {\displaystyle s} , t {\displaystyle t} , m {\displaystyle m} , i {\displaystyle i} and u {\displaystyle u} are smooth and s {\displaystyle s} and t {\displaystyle t} are submersions. A Lie groupoid is therefore not simply a groupoid object in the category of smooth manifolds : one has to ask the additional property that s {\displaystyle s} and t {\displaystyle t} are submersions.
Lie groupoids are often denoted by G ⇉ M {\displaystyle G\rightrightarrows M} , where the two arrows represent the source and the target. The notation G 1 ⇉ G 0 {\displaystyle G_{1}\rightrightarrows G_{0}} is also frequently used, especially when stressing the simplicial structure of the associated nerve .
In order to include more natural examples, the manifold G {\displaystyle G} is not required in general to be Hausdorff or second countable (while M {\displaystyle M} and all other spaces are).
The original definition by Ehresmann required G {\displaystyle G} and M {\displaystyle M} to possess a smooth structure such that only m {\displaystyle m} is smooth and the maps g ↦ 1 s ( g ) {\displaystyle g\mapsto 1_{s(g)}} and g ↦ 1 t ( g ) {\displaystyle g\mapsto 1_{t(g)}} are subimmersions (i.e. have locally constant rank ). Such definition proved to be too weak and was replaced by Pradines with the one currently used. [ 4 ]
While some authors [ 5 ] introduced weaker definitions which did not require s {\displaystyle s} and t {\displaystyle t} to be submersions, these properties are fundamental to develop the entire Lie theory of groupoids and algebroids.
The fact that the source and the target map of a Lie groupoid G ⇉ M {\displaystyle G\rightrightarrows M} are smooth submersions has some immediate consequences:
A Lie subgroupoid of a Lie groupoid G ⇉ M {\displaystyle G\rightrightarrows M} is a subgroupoid H ⇉ N {\displaystyle H\rightrightarrows N} (i.e. a subcategory of the category G {\displaystyle G} ) with the extra requirement that H ⊆ G {\displaystyle H\subseteq G} is an immersed submanifold. As for a subcategory, a (Lie) subgroupoid is called wide if N = M {\displaystyle N=M} . Any Lie groupoid G ⇉ M {\displaystyle G\rightrightarrows M} has two canonical wide subgroupoids:
A normal Lie subgroupoid is a wide Lie subgroupoid H ⊆ G {\displaystyle H\subseteq G} inside I G {\displaystyle IG} such that, for every h ∈ H , g ∈ G {\displaystyle h\in H,g\in G} with s ( h ) = t ( h ) = s ( g ) {\displaystyle s(h)=t(h)=s(g)} , one has g h g − 1 ∈ H {\displaystyle ghg^{-1}\in H} . The isotropy groups of H {\displaystyle H} are therefore normal subgroups of the isotropy groups of G {\displaystyle G} .
A Lie groupoid morphism between two Lie groupoids G ⇉ M {\displaystyle G\rightrightarrows M} and H ⇉ N {\displaystyle H\rightrightarrows N} is a groupoid morphism F : G → H , f : M → N {\displaystyle F:G\to H,f:M\to N} (i.e. a functor between the categories G {\displaystyle G} and H {\displaystyle H} ), where both F {\displaystyle F} and f {\displaystyle f} are smooth. The kernel ker ( F ) := { g ∈ G ∣ F ( g ) = 1 s ( g ) } {\displaystyle \ker(F):=\{g\in G\mid F(g)=1_{s(g)}\}} of a morphism between Lie groupoids over the same base manifold is automatically a normal Lie subgroupoid.
The quotient G / ker ( F ) ⇉ M {\displaystyle G/\ker(F)\rightrightarrows M} has a natural groupoid structure such that the projection G → G / ker ( F ) {\displaystyle G\to G/\ker(F)} is a groupoid morphism; however, unlike quotients of Lie groups , G / ker ( F ) {\displaystyle G/\ker(F)} may fail to be a Lie groupoid in general. Accordingly, the isomorphism theorems for groupoids cannot be specialised to the entire category of Lie groupoids, but only to special classes. [ 6 ]
A Lie groupoid is called abelian if its isotropy Lie groups are abelian . For similar reasons as above, while the definition of abelianisation of a group extends to set-theoretical groupoids, in the Lie case the analogue of the quotient G a b = G / ( I G , I G ) {\displaystyle G^{ab}=G/(IG,IG)} may not exist or be smooth. [ 7 ]
A bisection of a Lie groupoid G ⇉ M {\displaystyle G\rightrightarrows M} is a smooth map b : M → G {\displaystyle b:M\to G} such that s ∘ b = i d M {\displaystyle s\circ b=id_{M}} and t ∘ b {\displaystyle t\circ b} is a diffeomorphism of M {\displaystyle M} . In order to overcome the lack of symmetry between the source and the target, a bisection can be equivalently defined as a submanifold B ⊆ G {\displaystyle B\subseteq G} such that s ∣ B : B → M {\displaystyle s_{\mid B}:B\to M} and t ∣ B : B → M {\displaystyle t_{\mid B}:B\to M} are diffeomorphisms; the relation between the two definitions is given by B = b ( M ) {\displaystyle B=b(M)} . [ 8 ]
The set of bisections forms a group , with the multiplication b 1 ⋅ b 2 {\displaystyle b_{1}\cdot b_{2}} defined as ( b 1 ⋅ b 2 ) ( x ) := b 1 ( b 2 ( x ) ) b 2 ( x ) . {\displaystyle (b_{1}\cdot b_{2})(x):=b_{1}(b_{2}(x))b_{2}(x).} and inversion defined as b 1 − 1 ( x ) := i ∘ b 1 ( ( t ∘ b 2 ) − 1 ( x ) ) {\displaystyle b_{1}^{-1}(x):=i\circ b_{1}\left((t\circ b_{2})^{-1}(x)\right)} Note that the definition is given in such a way that, if t ∘ b 1 = ϕ 1 {\displaystyle t\circ b_{1}=\phi _{1}} and t ∘ b 2 = ϕ 2 {\displaystyle t\circ b_{2}=\phi _{2}} , then t ∘ ( b 1 ⋅ b 2 ) = ϕ 1 ∘ ϕ 2 {\displaystyle t\circ (b_{1}\cdot b_{2})=\phi _{1}\circ \phi _{2}} and t ∘ b 1 − 1 = ϕ 1 − 1 {\displaystyle t\circ b_{1}^{-1}=\phi _{1}^{-1}} .
The group of bisections can be given the compact-open topology , as well as an (infinite-dimensional) structure of Fréchet manifold compatible with the group structure, making it into a Fréchet-Lie group.
A local bisection b : U ⊆ M → G {\displaystyle b:U\subseteq M\to G} is defined analogously, but the multiplication between local bisections is of course only partially defined.
Note that some of the following classes make sense already in the category of set-theoretical or topological groupoids .
A Lie groupoid is transitive (in older literature also called connected) if it satisfies one of the following equivalent conditions:
Gauge groupoids constitute the prototypical examples of transitive Lie groupoids: indeed, any transitive Lie groupoid is isomorphic to the gauge groupoid of some principal bundle, namely the G x {\displaystyle G_{x}} -bundle t : s − 1 ( x ) → M {\displaystyle t:s^{-1}(x)\to M} , for any point x ∈ M {\displaystyle x\in M} . For instance:
As a less trivial instance of the correspondence between transitive Lie groupoids and principal bundles, consider the fundamental groupoid Π 1 ( M ) {\displaystyle \Pi _{1}(M)} of a (connected) smooth manifold M {\displaystyle M} . This is naturally a topological groupoid, which is moreover transitive; one can see that Π 1 ( M ) {\displaystyle \Pi _{1}(M)} is isomorphic to the gauge groupoid of the universal cover of M {\displaystyle M} . Accordingly, Π 1 ( M ) {\displaystyle \Pi _{1}(M)} inherits a smooth structure which makes it into a Lie groupoid.
Submersions groupoids M × μ M ⇉ M {\displaystyle M\times _{\mu }M\rightrightarrows M} are an example of non-transitive Lie groupoids, whose orbits are precisely the fibres of μ {\displaystyle \mu } .
A stronger notion of transitivity requires the anchor ( s , t ) : G → M × M {\displaystyle (s,t):G\to M\times M} to be a surjective submersion. Such condition is also called local triviality , because G {\displaystyle G} becomes locally isomorphic (as Lie groupoid) to a trivial groupoid over any open U ⊆ M {\displaystyle U\subseteq M} (as a consequence of the local triviality of principal bundles). [ 6 ]
When the space G {\displaystyle G} is second countable, transitivity implies local triviality. Accordingly, these two conditions are equivalent for many examples but not for all of them: for instance, if Γ {\displaystyle \Gamma } is a transitive pseudogroup, its germ groupoid G e r m ( Γ ) {\displaystyle \mathrm {Germ} (\Gamma )} is transitive but not locally trivial.
A Lie groupoid is called proper if ( s , t ) : G → M × M {\displaystyle (s,t):G\to M\times M} is a proper map. As a consequence
For instance:
As seen above, properness for Lie groupoids is the "right" analogue of compactness for Lie groups. One could also consider more "natural" conditions, e.g. asking that the source map s : G → M {\displaystyle s:G\to M} is proper (then G ⇉ M {\displaystyle G\rightrightarrows M} is called s-proper ), or that the entire space G {\displaystyle G} is compact (then G ⇉ M {\displaystyle G\rightrightarrows M} is called compact ), but these requirements turns out to be too strict for many examples and applications. [ 10 ]
A Lie groupoid is called étale if it satisfies one of the following equivalent conditions:
As a consequence, also the t {\displaystyle t} -fibres, the isotropy groups and the orbits become discrete.
For instance:
An étale groupoid is called effective if, for any two local bisections b 1 , b 2 {\displaystyle b_{1},b_{2}} , the condition t ∘ b 1 = t ∘ b 2 {\displaystyle t\circ b_{1}=t\circ b_{2}} implies b 1 = b 2 {\displaystyle b_{1}=b_{2}} . For instance:
In general, any effective étale groupoid arise as the germ groupoid of some pseudogroup. [ 11 ] However, a (more involved) definition of effectiveness, which does not assume the étale property, can also be given.
A Lie groupoid is called s {\displaystyle s} -connected if all its s {\displaystyle s} -fibres are connected . Similarly, one talks about s {\displaystyle s} -simply connected groupoids (when the s {\displaystyle s} -fibres are simply connected ) or source-k-connected groupoids (when the s {\displaystyle s} -fibres are k-connected , i.e. the first k {\displaystyle k} homotopy groups are trivial).
Note that the entire space of arrows G {\displaystyle G} is not asked to satisfy any connectedness hypothesis. However, if G {\displaystyle G} is a source- k {\displaystyle k} -connected Lie groupoid over a k {\displaystyle k} -connected manifold, then G {\displaystyle G} itself is automatically k {\displaystyle k} -connected.
For instanceː
Recall that an action of a groupoid G ⇉ M {\displaystyle G\rightrightarrows M} on a set P {\displaystyle P} along a function μ : P ⇉ M {\displaystyle \mu :P\rightrightarrows M} is defined via a collection of maps μ − 1 ( x ) → μ − 1 ( y ) , p ↦ g ⋅ p {\displaystyle \mu ^{-1}(x)\to \mu ^{-1}(y),\quad p\mapsto g\cdot p} for each morphism g ∈ G {\displaystyle g\in G} between x , y ∈ M {\displaystyle x,y\in M} . Accordingly, an action of a Lie groupoid G ⇉ M {\displaystyle G\rightrightarrows M} on a manifold P {\displaystyle P} along a smooth map μ : P ⇉ M {\displaystyle \mu :P\rightrightarrows M} consists of a groupoid action where the maps μ − 1 ( x ) → μ − 1 ( y ) {\displaystyle \mu ^{-1}(x)\to \mu ^{-1}(y)} are smooth. Of course, for every x ∈ M {\displaystyle x\in M} there is an induced smooth action of the isotropy group G x {\displaystyle G_{x}} on the fibre μ − 1 ( x ) {\displaystyle \mu ^{-1}(x)} .
Given a Lie groupoid G ⇉ M {\displaystyle G\rightrightarrows M} , a principal G {\displaystyle G} -bundle consists of a G {\displaystyle G} -space P {\displaystyle P} and a G {\displaystyle G} -invariant surjective submersion π : P → N {\displaystyle \pi :P\to N} such that P × N G → P × π P , ( p , g ) ↦ ( p , p ⋅ g ) {\displaystyle P\times _{N}G\to P\times _{\pi }P,\quad (p,g)\mapsto (p,p\cdot g)} is a diffeomorphism. Equivalent (but more involved) definitions can be given using G {\displaystyle G} -valued cocycles or local trivialisations.
When G {\displaystyle G} is a Lie groupoid over a point, one recovers, respectively, standard Lie group actions and principal bundles .
A representation of a Lie groupoid G ⇉ M {\displaystyle G\rightrightarrows M} consists of a Lie groupoid action on a vector bundle π : E → M {\displaystyle \pi :E\to M} , such that the action is fibrewise linear, i.e. each bijection π − 1 ( x ) → π − 1 ( y ) {\displaystyle \pi ^{-1}(x)\to \pi ^{-1}(y)} is a linear isomorphism. Equivalently, a representation of G {\displaystyle G} on E {\displaystyle E} can be described as a Lie groupoid morphism from G {\displaystyle G} to the general linear groupoid G L ( E ) {\displaystyle GL(E)} .
Of course, any fibre E x {\displaystyle E_{x}} becomes a representation of the isotropy group G x {\displaystyle G_{x}} . More generally, representations of transitive Lie groupoids are uniquely determined by representations of their isotropy groups, via the construction of the associated vector bundle .
Examples of Lie groupoids representations include the following:
The set R e p ( G ) {\displaystyle \mathrm {Rep} (G)} of isomorphism classes of representations of a Lie groupoid G ⇉ M {\displaystyle G\rightrightarrows M} has a natural structure of semiring , with direct sums and tensor products of vector bundles.
The notion of differentiable cohomology for Lie groups generalises naturally also to Lie groupoids: the definition relies on the simplicial structure of the nerve N ( G ) n = G ( n ) {\displaystyle N(G)_{n}=G^{(n)}} of G ⇉ M {\displaystyle G\rightrightarrows M} , viewed as a category.
More precisely, recall that the space G ( n ) {\displaystyle G^{(n)}} consists of strings of n {\displaystyle n} composable morphisms, i.e.
G ( n ) := { ( g 1 , … , g n ) ∈ G × … × G ∣ s ( g i ) = t ( g i + 1 ) ∀ i = 1 , … , n − 1 } , {\displaystyle G^{(n)}:=\{(g_{1},\ldots ,g_{n})\in G\times \ldots \times G\mid s(g_{i})=t(g_{i+1})\quad \forall i=1,\ldots ,n-1\},}
and consider the map t ( n ) = t ∘ p r 1 : G ( n ) → M , ( g 1 , … , g n ) ↦ t ( g 1 ) {\displaystyle t^{(n)}=t\circ \mathrm {pr} _{1}:G^{(n)}\to M,(g_{1},\ldots ,g_{n})\mapsto t(g_{1})} .
A differentiable n {\displaystyle n} -cochain of G ⇉ M {\displaystyle G\rightrightarrows M} with coefficients in some representation E → M {\displaystyle E\to M} is a smooth section of the pullback vector bundle ( t ( n ) ) ∗ E → G ( n ) {\displaystyle (t^{(n)})^{*}E\to G^{(n)}} . One denotes by C n ( G , E ) {\displaystyle C^{n}(G,E)} the space of such n {\displaystyle n} -cochains, and considers the differential d n : C n ( G , E ) → C n + 1 ( G , E ) {\displaystyle d_{n}:C^{n}(G,E)\to C^{n+1}(G,E)} , defined as
d n ( c ) ( g 1 , … , g n + 1 ) := g 1 ⋅ c ( g 2 , … , g n + 1 ) + ∑ i = 1 n ( − 1 ) i c ( g 1 , … , g i g i + 1 , … , g n + 1 ) + ( − 1 ) n + 1 c ( g 1 , … , g n ) . {\displaystyle d_{n}(c)(g_{1},\ldots ,g_{n+1}):=g_{1}\cdot c(g_{2},\ldots ,g_{n+1})+\sum _{i=1}^{n}(-1)^{i}c(g_{1},\ldots ,g_{i}g_{i+1},\ldots ,g_{n+1})+(-1)^{n+1}c(g_{1},\ldots ,g_{n}).}
Then ( C n ( G , E ) , d n ) {\displaystyle (C^{n}(G,E),d^{n})} becomes a cochain complex and its cohomology, denoted by H d n ( G , E ) {\displaystyle H_{d}^{n}(G,E)} , is called the differentiable cohomology of G ⇉ M {\displaystyle G\rightrightarrows M} with coefficients in E → M {\displaystyle E\to M} . Note that, since the differential at degree zero is d 0 ( c ) ( g ) = g ⋅ c ( s ( g ) ) − c ( t ( g ) ) {\displaystyle d_{0}(c)(g)=g\cdot c(s(g))-c(t(g))} , one has always H d 0 ( G , E ) = ker ( d 0 ) = Γ ( E ) G {\displaystyle H_{d}^{0}(G,E)=\ker(d_{0})=\Gamma (E)^{G}} .
Of course, the differentiable cohomology of G ⇉ ∗ {\displaystyle G\rightrightarrows {*}} as a Lie groupoid coincides with the standard differentiable cohomology of G {\displaystyle G} as a Lie group (in particular, for discrete groups one recovers the usual group cohomology ). On the other hand, for any proper Lie groupoid G ⇉ M {\displaystyle G\rightrightarrows M} , one can prove that H d n ( G , E ) = 0 {\displaystyle H_{d}^{n}(G,E)=0} for every n > 0 {\displaystyle n>0} . [ 12 ]
Any Lie groupoid G ⇉ M {\displaystyle G\rightrightarrows M} has an associated Lie algebroid A → M {\displaystyle A\to M} , obtained with a construction similar to the one which associates a Lie algebra to any Lie groupː
The Lie group–Lie algebra correspondence generalises to some extends also to Lie groupoids: the first two Lie's theorem (also known as the subgroups–subalgebras theorem and the homomorphisms theorem) can indeed be easily adapted to this setting.
In particular, as in standard Lie theory, for any s-connected Lie groupoid G {\displaystyle G} there is a unique (up to isomorphism) s-simply connected Lie groupoid G ~ {\displaystyle {\tilde {G}}} with the same Lie algebroid of G {\displaystyle G} , and a local diffeomorphism G ~ → G {\displaystyle {\tilde {G}}\to G} which is a groupoid morphism. For instance,
However, there is no analogue of Lie's third theorem ː while several classes of Lie algebroids are integrable, there are examples of Lie algebroids, for instance related to foliation theory , which do not admit an integrating Lie groupoid. [ 13 ] The general obstructions to the existence of such integration depend on the topology of G {\displaystyle G} . [ 14 ]
As discussed above, the standard notion of (iso)morphism of groupoids (viewed as functors between categories ) restricts naturally to Lie groupoids. However, there is a more coarse notion of equivalence, called Morita equivalence, which is more flexible and useful in applications.
First, a Morita map (also known as a weak equivalence or essential equivalence) between two Lie groupoids G 1 ⇉ G 0 {\displaystyle G_{1}\rightrightarrows G_{0}} and H 1 ⇉ H 0 {\displaystyle H_{1}\rightrightarrows H_{0}} consists of a Lie groupoid morphism from G to H which is moreover fully faithful and essentially surjective (adapting these categorical notions to the smooth context). We say that two Lie groupoids G 1 ⇉ G 0 {\displaystyle G_{1}\rightrightarrows G_{0}} and H 1 ⇉ H 0 {\displaystyle H_{1}\rightrightarrows H_{0}} are Morita equivalent if and only if there exists a third Lie groupoid K 1 ⇉ K 0 {\displaystyle K_{1}\rightrightarrows K_{0}} together with two Morita maps from G to K and from H to K .
A more explicit description of Morita equivalence (e.g. useful to check that it is an equivalence relation ) requires the existence of two surjective submersions P → G 0 {\displaystyle P\to G_{0}} and P → H 0 {\displaystyle P\to H_{0}} together with a left G {\displaystyle G} -action and a right H {\displaystyle H} -action, commuting with each other and making P {\displaystyle P} into a principal bi-bundle. [ 15 ]
Many properties of Lie groupoids, e.g. being proper, being Hausdorff or being transitive, are Morita invariant. On the other hand, being étale is not Morita invariant.
In addition, a Morita equivalence between G 1 ⇉ G 0 {\displaystyle G_{1}\rightrightarrows G_{0}} and H 1 ⇉ H 0 {\displaystyle H_{1}\rightrightarrows H_{0}} preserves their transverse geometry , i.e. it induces:
Last, the differentiable cohomologies of two Morita equivalent Lie groupoids are isomorphic. [ 12 ]
A concrete instance of the last example goes as follows. Let M be a smooth manifold and { U α } {\displaystyle \{U_{\alpha }\}} an open cover of M {\displaystyle M} . Its Čech groupoid G 1 ⇉ G 0 {\displaystyle G_{1}\rightrightarrows G_{0}} is defined by the disjoint unions G 0 := ⨆ α U α {\displaystyle G_{0}:=\bigsqcup _{\alpha }U_{\alpha }} and G 1 := ⨆ α , β U α β {\displaystyle G_{1}:=\bigsqcup _{\alpha ,\beta }U_{\alpha \beta }} , where U α β = U α ∩ U β ⊂ M {\displaystyle U_{\alpha \beta }=U_{\alpha }\cap U_{\beta }\subset M} . The source and target map are defined as the embeddings s : U α β → U α {\displaystyle s:U_{\alpha \beta }\to U_{\alpha }} and t : U α β → U β {\displaystyle t:U_{\alpha \beta }\to U_{\beta }} , and the multiplication is the obvious one if we read the U α β {\displaystyle U_{\alpha \beta }} as subsets of M (compatible points in U α β {\displaystyle U_{\alpha \beta }} and U β γ {\displaystyle U_{\beta \gamma }} actually are the same in M {\displaystyle M} and also lie in U α γ {\displaystyle U_{\alpha \gamma }} ). The Čech groupoid is in fact the pullback groupoid, under the obvious submersion p : G 0 → M {\displaystyle p:G_{0}\to M} , of the unit groupoid M ⇉ M {\displaystyle M\rightrightarrows M} . As such, Čech groupoids associated to different open covers of M {\displaystyle M} are Morita equivalent.
Investigating the structure of the orbit space of a Lie groupoid leads to the notion of a smooth stack. For instance, the orbit space is a smooth manifold if the isotropy groups are trivial (as in the example of the Čech groupoid), but it is not smooth in general. The solution is to revert the problem and to define a smooth stack as a Morita-equivalence class of Lie groupoids. The natural geometric objects living on the stack are the geometric objects on Lie groupoids invariant under Morita-equivalence: an example is the Lie groupoid cohomology.
Since the notion of smooth stack is quite general, obviously all smooth manifolds are smooth stacks. Other classes of examples include orbifolds , which are (equivalence classes of) proper étale Lie groupoids, and orbit spaces of foliations. | https://en.wikipedia.org/wiki/Lie_groupoid |
In mathematics, the Lie operad is an operad whose algebras are Lie algebras . The notion (at least one version) was introduced by Ginzburg & Kapranov (1994) in their formulation of Koszul duality .
Fix a base field k and let L i e ( x 1 , … , x n ) {\displaystyle {\mathcal {Lie}}(x_{1},\dots ,x_{n})} denote the free Lie algebra over k with generators x 1 , … , x n {\displaystyle x_{1},\dots ,x_{n}} and L i e ( n ) ⊂ L i e ( x 1 , … , x n ) {\displaystyle {\mathcal {Lie}}(n)\subset {\mathcal {Lie}}(x_{1},\dots ,x_{n})} the subspace spanned by all the bracket monomials containing each x i {\displaystyle x_{i}} exactly once. The symmetric group S n {\displaystyle S_{n}} acts on L i e ( x 1 , … , x n ) {\displaystyle {\mathcal {Lie}}(x_{1},\dots ,x_{n})} by permutations of the generators and, under that action, L i e ( n ) {\displaystyle {\mathcal {Lie}}(n)} is invariant. The operadic composition is given by substituting expressions (with renumbered variables) for variables. Then, L i e = { L i e ( n ) } {\displaystyle {\mathcal {Lie}}=\{{\mathcal {Lie}}(n)\}} is an operad. [ 1 ]
The Koszul-dual of L i e {\displaystyle {\mathcal {Lie}}} is the commutative-ring operad , an operad whose algebras are the commutative rings over k.
This algebra -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Lie_operad |
Lie point symmetry is a concept in advanced mathematics . Towards the end of the nineteenth century, Sophus Lie introduced the notion of Lie group in order to study the solutions of ordinary differential equations [ 1 ] [ 2 ] [ 3 ] (ODEs). He showed the following main property: the order of an ordinary differential equation can be reduced by one if it is invariant under one-parameter Lie group of point transformations . [ 4 ] This observation unified and extended the available integration techniques. Lie devoted the remainder of his mathematical career to developing these continuous groups that have now an impact on many areas of mathematically based sciences. The applications of Lie groups to differential systems were mainly established by Lie and Emmy Noether , and then advocated by Élie Cartan .
Roughly speaking, a Lie point symmetry of a system is a local group of transformations that maps every solution of the system to another solution of the same system. In other words, it maps the solution set of the system to itself. Elementary examples of Lie groups are translations , rotations and scalings .
The Lie symmetry theory is a well-known subject. In it are discussed continuous symmetries opposed to, for example, discrete symmetries . The literature for this theory can be found, among other places, in these notes. [ 5 ] [ 6 ] [ 7 ] [ 8 ] [ 9 ]
Lie groups and hence their infinitesimal generators can be naturally "extended" to act on the space of independent variables, state variables (dependent variables) and derivatives of the state variables up to any finite order. There are many other kinds of symmetries. For example, contact transformations let coefficients of the transformations infinitesimal generator depend also on first derivatives of the coordinates. Lie-Bäcklund transformations let them involve derivatives up to an arbitrary order. The possibility of the existence of such symmetries was recognized by Noether. [ 10 ] For Lie point symmetries, the coefficients of the infinitesimal generators depend only on coordinates, denoted by Z {\displaystyle Z} .
Lie symmetries were introduced by Lie in order to solve ordinary differential equations. Another application of symmetry methods is to reduce systems of differential equations, finding equivalent systems of differential equations of simpler form. This is called reduction. In the literature, one can find the classical reduction process, [ 4 ] and the moving frame -based reduction process. [ 11 ] [ 12 ] [ 13 ] Also symmetry groups can be used for classifying different symmetry classes of solutions.
Lie's fundamental theorems underline that Lie groups can be characterized by elements known as infinitesimal generators . These mathematical objects form a Lie algebra of infinitesimal generators. Deduced "infinitesimal symmetry conditions" (defining equations of the symmetry group) can be explicitly solved in order to find the closed form of symmetry groups, and thus the associated infinitesimal generators.
Let Z = ( z 1 , … , z n ) {\displaystyle Z=(z_{1},\dots ,z_{n})} be the set of coordinates on which a system is defined where n {\displaystyle n} is the cardinality of Z {\displaystyle Z} . An infinitesimal generator δ {\displaystyle \delta } in the field R ( Z ) {\displaystyle \mathbb {R} (Z)} is a linear operator δ : R ( Z ) → R ( Z ) {\displaystyle \delta :\mathbb {R} (Z)\rightarrow \mathbb {R} (Z)} that has R {\displaystyle \mathbb {R} } in its kernel and that satisfies the Leibniz rule :
In the canonical basis of elementary derivations { ∂ ∂ z 1 , … , ∂ ∂ z n } {\displaystyle \left\{{\frac {\partial }{\partial z_{1}}},\dots ,{\frac {\partial }{\partial z_{n}}}\right\}} , it is written as:
where ξ z i {\displaystyle \xi _{z_{i}}} is in R ( Z ) {\displaystyle \mathbb {R} (Z)} for all i {\displaystyle i} in { 1 , … , n } {\displaystyle \left\{1,\dots ,n\right\}} .
Lie algebras can be generated by a generating set of infinitesimal generators as defined above. To every Lie group, one can associate a Lie algebra. Roughly, a Lie algebra g {\displaystyle {\mathfrak {g}}} is an algebra constituted by a vector space equipped with Lie bracket as additional operation. The base field of a Lie algebra depends on the concept of invariant . Here only finite-dimensional Lie algebras are considered.
A dynamical system (or flow ) is a one-parameter group action . Let us denote by D {\displaystyle {\mathcal {D}}} such a dynamical system, more precisely, a (left-)action of a group ( G , + ) {\displaystyle (G,+)} on a manifold M {\displaystyle M} :
such that for all point Z {\displaystyle Z} in M {\displaystyle M} :
A continuous dynamical system is defined on a group G {\displaystyle G} that can be identified to R {\displaystyle \mathbb {R} } i.e. the group elements are continuous.
An invariant , roughly speaking, is an element that does not change under a transformation.
In this paragraph, we consider precisely expanded Lie point symmetries i.e. we work in an expanded space meaning that the distinction between independent variable, state variables and parameters are avoided as much as possible.
A symmetry group of a system is a continuous dynamical system defined on a local Lie group G {\displaystyle G} acting on a manifold M {\displaystyle M} . For the sake of clarity, we restrict ourselves to n-dimensional real manifolds M = R n {\displaystyle M=\mathbb {R} ^{n}} where n {\displaystyle n} is the number of system coordinates.
Let us define algebraic systems used in the forthcoming symmetry definition.
Let F = ( f 1 , … , f k ) = ( p 1 / q 1 , … , p k / q k ) {\displaystyle F=(f_{1},\dots ,f_{k})=(p_{1}/q_{1},\dots ,p_{k}/q_{k})} be a finite set of rational functions over the field R {\displaystyle \mathbb {R} } where p i {\displaystyle p_{i}} and q i {\displaystyle q_{i}} are polynomials in R [ Z ] {\displaystyle \mathbb {R} [Z]} i.e. in variables Z = ( z 1 , … , z n ) {\displaystyle Z=(z_{1},\dots ,z_{n})} with coefficients in R {\displaystyle \mathbb {R} } . An algebraic system associated to F {\displaystyle F} is defined by the following equalities and inequalities:
An algebraic system defined by F = ( f 1 , … , f k ) {\displaystyle F=(f_{1},\dots ,f_{k})} is regular (a.k.a. smooth ) if the system F {\displaystyle F} is of maximal rank k {\displaystyle k} , meaning that the Jacobian matrix ( ∂ f i / ∂ z j ) {\displaystyle (\partial f_{i}/\partial z_{j})} is of rank k {\displaystyle k} at every solution Z {\displaystyle Z} of the associated semi-algebraic variety .
The following theorem (see th. 2.8 in ch.2 of [ 5 ] ) gives necessary and sufficient conditions so that a local Lie group G {\displaystyle G} is a symmetry group of an algebraic system.
Theorem . Let G {\displaystyle G} be a connected local Lie group of a continuous dynamical system acting in the n-dimensional space R n {\displaystyle \mathbb {R} ^{n}} . Let F : R n → R k {\displaystyle F:\mathbb {R} ^{n}\rightarrow \mathbb {R} ^{k}} with k ≤ n {\displaystyle k\leq n} define a regular system of algebraic equations:
Then G {\displaystyle G} is a symmetry group of this algebraic system if, and only if,
for every infinitesimal generator δ {\displaystyle \delta } in the Lie algebra g {\displaystyle {\mathfrak {g}}} of G {\displaystyle G} .
Consider the algebraic system defined on a space of 6 variables, namely Z = ( P , Q , a , b , c , l ) {\displaystyle Z=(P,Q,a,b,c,l)} with:
The infinitesimal generator
is associated to one of the one-parameter symmetry groups. It acts on 4 variables, namely a , b , c {\displaystyle a,b,c} and P {\displaystyle P} . One can easily verify that δ f 1 = f 1 − f 2 {\displaystyle \delta f_{1}=f_{1}-f_{2}} and δ f 2 = 0 {\displaystyle \delta f_{2}=0} . Thus the relations δ f 1 = δ f 2 = 0 {\displaystyle \delta f_{1}=\delta f_{2}=0} are satisfied for any Z {\displaystyle Z} in R 6 {\displaystyle \mathbb {R} ^{6}} that vanishes the algebraic system.
Let us define systems of first-order ODEs used in the forthcoming symmetry definition.
Let d ⋅ / d t {\displaystyle d\cdot /dt} be a derivation w.r.t. the continuous independent variable t {\displaystyle t} . We consider two sets X = ( x 1 , … , x k ) {\displaystyle X=(x_{1},\dots ,x_{k})} and Θ = ( θ 1 , … , θ l ) {\displaystyle \Theta =(\theta _{1},\dots ,\theta _{l})} . The associated coordinate set is defined by Z = ( z 1 , … , z n ) = ( t , x 1 , … , x k , θ 1 , … , θ l ) {\displaystyle Z=(z_{1},\dots ,z_{n})=(t,x_{1},\dots ,x_{k},\theta _{1},\dots ,\theta _{l})} and its cardinal is n = 1 + k + l {\displaystyle n=1+k+l} . With these notations, a system of first-order ODEs is a system where:
and the set F = ( f 1 , … , f k ) {\displaystyle F=(f_{1},\dots ,f_{k})} specifies the evolution of state variables of ODEs w.r.t. the independent variable. The elements of the set X {\displaystyle X} are called state variables , these of Θ {\displaystyle \Theta } parameters .
One can associate also a continuous dynamical system to a system of ODEs by resolving its equations.
An infinitesimal generator is a derivation that is closely related to systems of ODEs (more precisely to continuous dynamical systems). For the link between a system of ODEs, the associated vector field and the infinitesimal generator, see section 1.3 of. [ 4 ] The infinitesimal generator δ {\displaystyle \delta } associated to a system of ODEs, described as above, is defined with the same notations as follows:
Here is a geometrical definition of such symmetries. Let D {\displaystyle {\mathcal {D}}} be a continuous dynamical system and δ D {\displaystyle \delta _{\mathcal {D}}} its infinitesimal generator. A continuous dynamical system S {\displaystyle {\mathcal {S}}} is a Lie point symmetry of D {\displaystyle {\mathcal {D}}} if, and only if, S {\displaystyle {\mathcal {S}}} sends every orbit of D {\displaystyle {\mathcal {D}}} to an orbit. Hence, the infinitesimal generator δ S {\displaystyle \delta _{\mathcal {S}}} satisfies the following relation [ 8 ] based on Lie bracket :
where λ {\displaystyle \lambda } is any constant of δ D {\displaystyle \delta _{\mathcal {D}}} and δ S {\displaystyle \delta _{\mathcal {S}}} i.e. δ D λ = δ S λ = 0 {\displaystyle \delta _{\mathcal {D}}\lambda =\delta _{\mathcal {S}}\lambda =0} . These generators are linearly independent.
One does not need the explicit formulas of D {\displaystyle {\mathcal {D}}} in order to compute the infinitesimal generators of its symmetries.
Consider Pierre François Verhulst 's logistic growth model with linear predation, [ 14 ] where the state variable x {\displaystyle x} represents a population. The parameter a {\displaystyle a} is the difference between the growth and predation rate and the parameter b {\displaystyle b} corresponds to the receptive capacity of the environment:
The continuous dynamical system associated to this system of ODEs is:
The independent variable t ^ {\displaystyle {\hat {t}}} varies continuously; thus the associated group can be identified with R {\displaystyle \mathbb {R} } .
The infinitesimal generator associated to this system of ODEs is:
The following infinitesimal generators belong to the 2-dimensional symmetry group of D {\displaystyle {\mathcal {D}}} :
There exist many software packages in this area. [ 15 ] [ 16 ] [ 17 ] For example, the package liesymm of Maple provides some Lie symmetry methods for PDEs . [ 18 ] It manipulates integration of determining systems and also differential forms . Despite its success on small systems, its integration capabilities for solving determining systems automatically are limited by complexity issues. The DETools package uses the prolongation of vector fields for searching Lie symmetries of ODEs. Finding Lie symmetries for ODEs, in the general case, may be as complicated as solving the original system. | https://en.wikipedia.org/wiki/Lie_point_symmetry |
In mathematics , the mathematician Sophus Lie ( / l iː / LEE ) initiated lines of study involving integration of differential equations , transformation groups , and contact of spheres that have come to be called Lie theory . [ 1 ] For instance, the latter subject is Lie sphere geometry . This article addresses his approach to transformation groups, which is one of the areas of mathematics , and was worked out by Wilhelm Killing and Élie Cartan .
The foundation of Lie theory is the exponential map relating Lie algebras to Lie groups which is called the Lie group–Lie algebra correspondence . The subject is part of differential geometry since Lie groups are differentiable manifolds . Lie groups evolve out of the identity (1) and the tangent vectors to one-parameter subgroups generate the Lie algebra. The structure of a Lie group is implicit in its algebra, and the structure of the Lie algebra is expressed by root systems and root data .
Lie theory has been particularly useful in mathematical physics since it describes the standard transformation groups: the Galilean group , the Lorentz group , the Poincaré group and the conformal group of spacetime .
The one-parameter groups are the first instance of Lie theory. The compact case arises through Euler's formula in the complex plane . Other one-parameter groups occur in the split-complex number plane as the unit hyperbola
and in the dual number plane as the line { exp ( ε t ) = 1 + ε t : t ∈ R } ε 2 = 0. {\displaystyle \lbrace \exp(\varepsilon t)=1+\varepsilon t:t\in R\rbrace \quad \varepsilon ^{2}=0.} In these cases the Lie algebra parameters have names: angle , hyperbolic angle , and slope . [ 2 ] These species of angle are useful for providing polar decompositions which describe the planar subalgebras of 2 x 2 real matrices.
There is a classical 3-parameter Lie group and algebra pair: the quaternions of unit length which can be identified with the 3-sphere . Its Lie algebra is the subspace of quaternion vectors. Since the commutator ij − ji = 2k, the Lie bracket in this algebra is twice the cross product of ordinary vector analysis .
Another elementary 3-parameter example is given by the Heisenberg group and its Lie algebra.
Standard treatments of Lie theory often begin with the classical groups .
Early expressions of Lie theory are found in books composed by Sophus Lie with Friedrich Engel and Georg Scheffers from 1888 to 1896.
In Lie's early work, the idea was to construct a theory of continuous groups , to complement the theory of discrete groups that had developed in the theory of modular forms , in the hands of Felix Klein and Henri Poincaré . The initial application that Lie had in mind was to the theory of differential equations . On the model of Galois theory and polynomial equations , the driving conception was of a theory capable of unifying, by the study of symmetry , the whole area of ordinary differential equations .
According to Thomas W. Hawkins Jr. , it was Élie Cartan that made Lie theory what it is:
In his work on transformation groups, Sophus Lie proved three theorems relating the groups and algebras that bear his name. The first theorem exhibited the basis of an algebra through infinitesimal transformations . [ 4 ] : 96 The second theorem exhibited structure constants of the algebra as the result of commutator products in the algebra. [ 4 ] : 100 The third theorem showed these constants are anti-symmetric and satisfy the Jacobi identity . [ 4 ] : 106 As Robert Gilmore wrote:
Lie theory is frequently built upon a study of the classical linear algebraic groups . Special branches include Weyl groups , Coxeter groups , and buildings . The classical subject has been extended to Groups of Lie type .
In 1900 David Hilbert challenged Lie theorists with his Fifth Problem presented at the International Congress of Mathematicians in Paris. | https://en.wikipedia.org/wiki/Lie_theory |
Lieb's square ice constant is a mathematical constant used in the field of combinatorics to approximately count Eulerian orientations of grid graphs . It was introduced by Elliott H. Lieb in 1967. [ 1 ] It is called the square ice constant because the orientations that it counts arise in statistical mechanics of crystalline structures as the states of an ice-type model on a square grid.
The value of Lieb's square ice constant is 8 3 9 ≈ 1.5396. {\displaystyle {\frac {8{\sqrt {3}}}{9}}\approx 1.5396.} Based on this, the number of Eulerian orientations of an n × n {\displaystyle n\times n} grid is ( 8 3 9 − o ( 1 ) ) n 2 , {\displaystyle \left({\frac {8{\sqrt {3}}}{9}}-o(1)\right)^{n^{2}},} where the o ( 1 ) {\displaystyle o(1)} term, an instance of little o notation , hides parts of the formula that tend to zero in the limit as n {\displaystyle n} grows.
An n × n {\displaystyle n\times n} grid graph has n 2 {\displaystyle n^{2}} vertices. When constructed with periodic boundary conditions (with edges that wrap around from left to right and from top to bottom) it has 2 n 2 {\displaystyle 2n^{2}} edges and is 4-regular , meaning that each vertex has exactly four neighbors. An orientation of this graph is an assignment of a direction to each edge. It is an Eulerian orientation if it gives each vertex exactly two incoming edges and exactly two outgoing edges. An Eulerian orientation can be constructed by orienting each row of the grid (including the wraparound edge) as a cycle, and each column as another cycle, but there are many more orientations that are not of this special form.
Denote the number of Eulerian orientations of this graph by f ( n ) {\displaystyle f(n)} . Then this number is approximately exponential in n 2 {\displaystyle n^{2}} , with Lieb's square ice constant as the base of the exponential. More precisely, lim n → ∞ f ( n ) n 2 = ( 4 3 ) 3 2 = 8 3 9 = 1.5396007 … {\displaystyle \lim _{n\to \infty }{\sqrt[{n^{2}}]{f(n)}}=\left({\frac {4}{3}}\right)^{\frac {3}{2}}={\frac {8{\sqrt {3}}}{9}}=1.5396007\dots } is Lieb's square ice constant. [ 2 ] Lieb used a transfer-matrix method to compute this exactly. [ 1 ]
The exact numbers of Eulerian orientations of an n × n {\displaystyle n\times n} grid graph, with periodic boundary conditions, for n = 2 , 3 , … {\displaystyle n=2,3,\dots } , are [ 3 ]
Lieb's original motivation for studying this counting problem comes from statistical mechanics . In this area, the ice-type models are used to model hydrogen bonds in crystalline structures such as water ice where each element of the structure (such as a water molecule) has bonds connecting it to four neighbors, with two bonds of each polarity . A state of this system describes the polarity of the hydrogen bond for each of the four neighbors. If the elements and their adjacencies are described by the vertices and edges of an undirected graph, the polarities of their bonds can be described by orienting this graph, with two edges of each direction at each vertex. With an additional assumption that all consistent choices of orientation have equal energy, the number of possible states equals the partition function , important for calculating the properties of a system at thermodynamic equilibrium . Different crystalline structures have different partition functions; the value calculated by Lieb is for an unrealistic model in which the water molecules or other elements are arranged in a square grid. [ 1 ] As well as for hydrogen bonds, analogous arrangements of elements with four neighbors, obeying the same two-in two-out rules, can occur in spin ice , and the same calculation of the partition function applies in that case.
The function f ( n ) {\displaystyle f(n)} also counts the number of 3-colorings of grid graphs, the number of nowhere-zero 3-flows in grid graphs, and the number of local flat foldings of the Miura fold . [ 4 ] | https://en.wikipedia.org/wiki/Lieb's_square_ice_constant |
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