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Osmium tetrasulfide is an inorganic compound , a salt of osmium metal and hydrogen sulfide acid with the chemical formula OsS 4 . [ 1 ]
Osmium tetrasulfide can be made by passing hydrogen sulfide through acidified solutions of osmium tetroxide : [ 2 ]
Osmium tetrasulfide forms dark brown crystals. It does not dissolve in cold water. It is soluble in dilute nitric acid . [ 3 ] It forms hydrates. [ 4 ]
Osmium tetrasulfide decomposes upon melting. [ 5 ]
This inorganic compound –related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/OsS4 |
Osama Kamal ( Arabic : أسامة كمال ; born 5 April 1959) was the Egyptian minister of petroleum and mineral resources. He was sworn into President Mohamad Morsi's cabinet, the Qandil Cabinet , on 2 August 2012, following the 2011–2012 Egyptian uprising that deposed President Hosni Mubarak . [ 1 ] [ 2 ] [ 3 ] He was in office until 6 May 2013.
Kamal was born on 5 April 1959 in the Nile Delta city Zagazig . [ 4 ] He attended Cairo University , faculty of engineering, and graduated from the chemical engineering department in 1982. [ 5 ] Kamal began his career at the state-owned Engineering for Petroleum and Process Industries (Enppi) as an engineer. A few years later, Kamal switched to another public-sector energy company, known as Petrojet. [ 5 ]
Osama Kamal helped establish Egyptian Petrochemicals Holding Company (Echem) in 2002, which belongs to Egypt's petroleum ministry, and is mandated with managing and developing Egypt's petrochemicals industry. Kamal was appointed chairman of Echem in 2009. [ 5 ] While Kamal was in charge of Echem, a crisis erupted pertaining the Agrium fertiliser plant in the Nile Delta city of Damietta . In 2008, Agrium, which is 30% owned by Echem, was faced with a popular campaign that alleged that the factory was bad for the environment, causing Egypt's government to abruptly cancel the project. Agrium was offered a 26% stake in the state-owned Egyptian fertilizer producer MOPCO instead. However, the project resumed in 2011, which dragged Agrium and Echem into more controversy. The project was completed and started operation in 2016. [ 6 ]
Currently Eng. Osama Kamal resides on the board of the Egyptian industrial company, Carbon Holdings. In addition he acts as a Chief Strategy Officer.
On 2 August 2012, Kamal was sworn into the Qandil Cabinet , as the minister of petroleum and mineral resources. [ 1 ] [ 2 ] His term ended on 7 May 2013 and he was replaced by Sherif Haddara in the post. [ 7 ] [ 8 ]
Kamal is married and has three daughters and three grandchildren. [ 4 ] | https://en.wikipedia.org/wiki/Osama_Kamal |
Osamu Shimomura ( 下村 脩 , Shimomura Osamu , August 27, 1928 – October 19, 2018 [ 1 ] ) was a Japanese organic chemist and marine biologist , and professor emeritus at Marine Biological Laboratory (MBL) in Woods Hole, Massachusetts and Boston University School of Medicine . He was awarded the Nobel Prize in Chemistry in 2008 for the discovery and development of green fluorescent protein (GFP) with two American scientists: Martin Chalfie of Columbia University and Roger Tsien of the University of California-San Diego. [ 2 ]
Born in Fukuchiyama, Kyoto in 1928, Shimomura was brought up in Manchukuo ( Manchuria , China) and Osaka, Japan while his father served as an officer in the Imperial Japanese Army . Later, his family moved to Isahaya, Nagasaki , [ 3 ] 25 km from the epicenter of the August 1945 atomic bombing of the city . He recalled hearing, as a 16-year-old boy, the bomber plane Bockscar before the atom bomb exploded. [ 4 ] The explosion flash blinded Shimomura for about thirty seconds, and he was later drenched by the "black rain" bomb fallout. [ 5 ] He overcame great odds in the following 11 years to earn an education and achieve academic success. [ 3 ]
Shimomura's education opportunities were starkly limited in devastated, post-war Japan. Although he later recalled having no interest in the subject, [ 4 ] he enrolled in the College of Pharmaceutical Sciences of Nagasaki Medical College (now Nagasaki University School of Pharmaceutical Sciences). [ 6 ] The Medical College campus had been entirely destroyed by the atomic bomb blast, forcing the pharmacy school to relocate to a temporary campus near Shimomura's home. This proximity was the fortuitous reason he embarked upon the studies and career which would ultimately lead to unanticipated rewards. [ 4 ] Shimomura was awarded a BS degree in pharmacy in 1951, and he stayed on as a lab assistant through 1955. [ 4 ]
Shimomura's mentor at Nagasaki helped him find employment as an assistant to Professor Yoshimasa Hirata at Nagoya University in 1956. [ 6 ] While working for Professor Hirata, he received a MS degree in organic chemistry in 1958 and, before leaving Japan for an appointment at Princeton University, a Ph.D. in organic chemistry in 1960 at Nagoya University. [ 7 ] [ 8 ] At Nagoya, Hirata assigned Shimomura the challenging task of determining what made the crushed remains of a type of crustacean (Jp. umi-hotaru , lit. "sea-firefly", Vargula hilgendorfii ) glow when moistened with water. This assignment led Shimomura to the successful identification of the protein causing the phenomenon, and he published the preliminary findings in the Bulletin of the Chemical Society of Japan in a paper titled "Crystalline Cypridina luciferin." The article caught the attention of Professor Frank Johnson at Princeton University , and Johnson successfully recruited Shimomura to work with him in 1960.
Shimomura worked in the department of biology at Princeton for Professor Johnson to study the bioluminescent jellyfish Aequorea victoria , which they collected during many summers at the Friday Harbor Laboratories of the University of Washington . [ 9 ] In 1962, their work culminated in the discovery of the proteins aequorin and green fluorescent protein (GFP) in A. victoria ; [ 10 ] for this work, he was awarded a third of the Nobel Prize in Chemistry in 2008.
His wife, Akemi, whom Shimomura met at Nagasaki University , is also an organic chemist and was a partner in his research activities. Their son, Tsutomu Shimomura , is a computer security expert who was involved in the arrest of Kevin Mitnick . Their daughter, Sachi Shimomura, is director of Undergraduate Studies for the English Department at Virginia Commonwealth University and the author of Odd Bodies and Visible Ends in Medieval Literature .
Shimomura died on October 19, 2018, of cancer in Nagasaki.
Books
Papers | https://en.wikipedia.org/wiki/Osamu_Shimomura |
Osazone are a class of carbohydrate derivatives found in organic chemistry formed when reducing sugars are reacted with excess of phenylhydrazine at boiling temperatures. [ 1 ] [ 2 ]
Osazone formation was developed by Emil Fischer , [ 3 ] who used the reaction as a test to identify monosaccharides .
The formation of a pair of hydrazone functionalities involves both oxidation and condensation reactions. [ 4 ] Since the reaction requires a free carbonyl group, only "reducing sugars" participate. Sucrose , which is nonreducing, does not form an osazone.
Osazones are highly coloured and crystalline compounds. Osazones are readily distinguished. [ 5 ] | https://en.wikipedia.org/wiki/Osazone |
Oscar Paul Kuipers ( Rotterdam , May 12, 1956) is a Dutch professor of molecular genetics at the University of Groningen . His areas of expertise include microbiology , biochemistry , molecular and cell biology , and biotechnology .
The research of Kuipers and his research group is focused on curiosity driven research with a keen eye on biotechnological applications. Kuipers has so far supervised 21 postdoctoral researchers and 35 PhD students in their doctorate research . At the moment (October 2015) he supervises 18 PhD students and 7 postdoctoral researchers. He has been invited more than 180 times to national and international conferences, seminars and congresses to give a lecture.
An important topic of research is the study of the genetics and physiology of bacteria . Among others, the cellular differentiation of bacteria is investigated. Bacteria growing in a culture can develop different characteristics, while their genome remains unchanged. Kuipers stated: ‘Our research has many applications, for example the improvement of protein production in industrial fermentation '. [ 5 ] Another important subject is the production and modification of peptides . These modified peptides (called lantibiotics ), are made by bacteria, and can serve as antibiotic . Modified peptides are chemically more stable and retain their function longer than unmodified peptides. This is beneficial for medical applications as a novel class of antibiotics.
Further areas of focus are the molecular biology of: competence, sporulation and bistability in Bacillus subtilis , the reconstruction of gene networks, antimicrobial peptides , antibiotics, mechanisms of pathogenesis , cell wall anchoring, controlled gene expression systems, the subcellular localization of protein , stress response, quorum sensing , regulation of the C- and N- metabolism , natural gene transfer methodologies, plant-biocontrol by Bacilli and Biotechnology applications.
From 2008 Kuipers is supervisor, coach and coordinator of the iGEM student team Groningen . [ 6 ] iGEM is a worldwide competition in the field of synthetic biology between teams of students from all over the world. The team from Groningen won in the annual in Boston ( United States ) organized international competition several gold medals (2008 to 2016). In 2012, when the team from Groningen became world champion, the research concerned a study of an alternative method to determine whether food is spoiled: the Food warden . This method makes use of genetic engineered bacteria responsive to the volatiles of the decomposing meat . A pigment makes the decomposition visible.
Prof. dr. O.P. Kuipers published 365 articles in scientific papers, 2 books and 18 chapters in books. 34 publications appeared in 2015. [ 7 ] Kuipers articles have been cited over 26 148 times and he has a H-index of 84 and an i10 index of 278. [ 8 ] This means that 80 of his articles have been cited at least 80 times and that 278 articles have been cited more than 10 times. | https://en.wikipedia.org/wiki/Oscar_Kuipers |
Oscarella carmela , commonly known as the slime sponge , is a species of sponge in the order Homosclerophorida that was first described in 2004 by G. Muricy and J.S. Pearse. It is believed to be native to intertidal waters in the north east temperate Pacific Ocean and was first found in seawater aquaria in that region. It is used as a model organism in evolutionary biology .
Oscarella carmela is either encrusting or massive and forms a slimy covering or a thicker layer of spongy matter with an uneven, lumpy, lobed surface. It grows in patches on hard substrates up to 20 to 30 cm (8 to 12 in) in diameter and overgrows other organisms. The colour is variable and ranges from orange-brown to tan or beige. This sponge does not contain spicules or spongin to reinforce its body wall and has a simple structure with only two types of cell with inclusions. [ 2 ] [ 3 ]
Oscarella carmela is believed to be a native of northern and central Californian marine waters. It was first observed in Monterey Bay Aquarium and several research seawater aquaria in western California. [ 3 ] It was later searched for, and eventually found, in the sea on the underside of boulders in rock pools in the high intertidal zone in Carmel Bay . Although it was not described until 2004, it is not believed to be an invasive species in the United States but is more likely to be indigenous and have been overlooked previously because it is uncommon and very similar to more common Halisarca species. It is in fact the only member of its genus Oscarella to be found in the eastern Pacific. It is hypothesized that in the wild it may be limited in its distribution by predation , whereas in the protected environment of an aquarium it grows profusely. [ 3 ]
Like other sponges, Oscarella carmela is a filter feeder . It creates a current of water through its interior from which it extracts bacteria and planktonic food particles. [ 4 ] Reproduction is viviparous and the planktonic larvae are the oval type known as amphiblastulae. This type of larval form is quite common in calcareous sponges but is unusual in other sponge groups. [ 5 ]
The genome of Oscarella carmela has been sequenced and it is used as a model in evolutionary developmental biology. Analysis of the genome suggests that the last common ancestor of sponges and eumetazoan animals (a clade that contains all the higher animals except the sponges and placozoans ) was more complex both genetically and morphologically than had previously been thought. The data suggest that homoscleromorph sponges have retained certain features that have been lost in other demosponges . [ 6 ] | https://en.wikipedia.org/wiki/Oscarella_carmela |
Oscheius is a genus of nematode .
O. tipulae is a satellite developmental genetic model organism used to study vulva formation. [ 7 ]
In phylogenetic studies, based on the analysis of sequences of three nuclear genes, Oscheius groups with Caenorhabditis species and the Diploscapter , Protorhabditis and Prodontorhabditis 'Protorhabditis' group, all included in the 'Eurhabditis' group of Rhabditidae genera. [ 8 ]
This Rhabditida roundworm-related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Oscheius |
The oscillating U-tube is a technique to determine the density of liquids and gases based on an electronic measurement of the frequency of oscillation, from which the density value is calculated. This measuring principle is based on the Mass-Spring Model.
The sample is filled into a container with oscillation capacity. The eigenfrequency of this container is influenced by the sample's mass . This container with oscillation capacity is a hollow, U-shaped glass tube (oscillating U-tube) which is electronically excited into undamped oscillation. The two branches of the U-shaped oscillator function as its spring elements.
The direction of oscillation is normal to the level of the two branches. The oscillator's eigenfrequency is only influenced by the part of the sample that is actually involved in the oscillation. The volume involved in the oscillation is limited by the stationary oscillation knots at the bearing points of the oscillator. If the oscillator is at least filled up to its bearing points, the same precisely defined volume always participates in the oscillation, thus the measured value of the sample's mass can be used to calculate its density.
Overfilling the oscillator beyond the bearing points is irrelevant to the measurement. For this reason the oscillator can also be employed to measure the density of sample media that flow through the tube (Continuous Measurement).
In modern digital density meters, Piezo elements are used to excite the U-tube whereby optical pickups determine the period of oscillation. This period τ can be measured with high resolution and stands in simple relation to the density ρ of the sample in the oscillator:
A and B are the respective instrument constants of each oscillator. Their values are determined by calibrating with two substances of the precisely known densities ρ1 and ρ2. Modern instruments calculate and store the constants A and B after the two calibration measurements, which are mostly performed with air and water . They employ suitable measures to compensate various parasitic influences on the measuring result, e.g. the influence of the sample's viscosity and the non-linearity caused by the measuring instrument's finite mass as well as aging effects of the glass (reference oscillator).
In 1967 the company Anton Paar GmbH presented the first digital density meter for liquids and gases employing the oscillating U-tube principle at ACHEMA . | https://en.wikipedia.org/wiki/Oscillating_U-tube |
In molecular biology , an oscillating gene is a gene that is expressed in a rhythmic pattern or in periodic cycles. [ 1 ] [ 2 ] Oscillating genes are usually circadian and can be identified by periodic changes in the state of an organism. Circadian rhythms , controlled by oscillating genes, have a period of approximately 24 hours. For example, plant leaves opening and closing at different times of the day or the sleep-wake schedule of animals can all include circadian rhythms. Other periods are also possible, such as 29.5 days resulting from circalunar rhythms or 12.4 hours resulting from circatidal rhythms. [ 3 ] Oscillating genes include both core clock component genes and output genes. A core clock component gene is a gene necessary for to the pacemaker. However, an output oscillating gene, such as the AVP gene, is rhythmic but not necessary to the pacemaker. [ 4 ]
The first recorded observations of oscillating genes come from the marches of Alexander the Great in the fourth century B.C. [ 5 ] At this time, one of Alexander's generals, Androsthenes , wrote that the tamarind tree would open its leaves during the day and close them at nightfall. [ 5 ] Until 1729, the rhythms associated with oscillating genes were assumed to be "passive responses to a cyclic environment". [ 3 ] In 1729, Jean-Jacques d'Ortous de Mairan demonstrated that the rhythms of a plant opening and closing its leaves continued even when placed somewhere where sunlight could not reach it. This was one of the first indications that there was an active element to the oscillations. In 1923, Ingeborg Beling published her paper "Über das Zeitgedächtnis der Bienen" ("On the Time Memory of Bees") which extended oscillations to animals, specifically bees [ 6 ] In 1971, Ronald Konopka and Seymour Benzer discovered that mutations of the PERIOD gene caused changes in the circadian rhythm of flies under constant conditions. They hypothesized that the mutation of the gene was affecting the basic oscillator mechanism. [ 7 ] Paul Hardin , Jeffrey Hall, and Michael Rosbash demonstrated that relationship by discovering that within the PERIOD gene, there was a feedback mechanism that controlled the oscillation. [ 8 ] The mid-1990s saw an outpouring of discoveries, with CLOCK , CRY , and others being added to the growing list of oscillating genes. [ 9 ] [ 10 ]
The primary molecular mechanism behind an oscillating gene is best described as a transcription/translation feedback loop. [ 11 ] This loop contains both positive regulators, which increase gene expression, and negative regulators, which decrease gene expression. [ 12 ] The fundamental elements of these loops are found across different phyla. In the mammalian circadian clock, for example, transcription factors CLOCK and BMAL1 are the positive regulators. [ 12 ] CLOCK and BMAL1 bind to the E-box of oscillating genes, such as Per1, Per2, and Per3 and Cry1 and Cry2, and upregulate their transcription. [ 12 ] When the PERs and CRYs form a heterocomplex in the cytoplasm and enter the nucleus again, they inhibit their own transcription. [ 13 ] This means that over time the mRNA and protein levels of PERs and CRYs, or any other oscillating gene under this mechanism, will oscillate.
There also exists a secondary feedback loop, or 'stabilizing loop', which regulates the cyclic expression of Bmal1. [ 12 ] This is caused by two nuclear receptors, REV-ERB and ROR, which suppresses and activates Bmal1 transcription, respectively. [ 12 ]
In addition to these feedback loops, post-translational modifications also play a role in changing the characteristics of the circadian clock, such as its period. [ 13 ] Without any type of feedback repression, the molecular clock would have a period of just a few hours. [ 12 ] Casein kinase members CK1ε and CK1δ were both found to be mammalian protein kinases involved in circadian regulation. [ 12 ] Mutations in these kinases are associated with familial advanced sleep phase syndrome ( FASPS ). [ 14 ] In general, phosphorylation is necessary for the degradation of PERs via ubiquitin ligases. [ 15 ] In contrast, phosphorylation of BMAL1 via CK2 is important for accumulation of BMAL1. [ 16 ]
The genes provided in this section are only a small number of the vast amount of oscillating genes found in the world. These genes were selected because they were determined to be the some of most important genes in regulating the circadian rhythm of their respective classification. | https://en.wikipedia.org/wiki/Oscillating_gene |
In mathematics , the oscillator representation is a projective unitary representation of the symplectic group , first investigated by Irving Segal , David Shale , and André Weil . A natural extension of the representation leads to a semigroup of contraction operators , introduced as the oscillator semigroup by Roger Howe in 1988. The semigroup had previously been studied by other mathematicians and physicists, most notably Felix Berezin in the 1960s. The simplest example in one dimension is given by SU(1,1) . It acts as Möbius transformations on the extended complex plane , leaving the unit circle invariant. In that case the oscillator representation is a unitary representation of a double cover of SU(1,1) and the oscillator semigroup corresponds to a representation by contraction operators of the semigroup in SL(2, C ) corresponding to Möbius transformations that take the unit disk into itself.
The contraction operators, determined only up to a sign, have kernels that are Gaussian functions . On an infinitesimal level the semigroup is described by a cone in the Lie algebra of SU(1,1) that can be identified with a light cone . The same framework generalizes to the symplectic group in higher dimensions, including its analogue in infinite dimensions. This article explains the theory for SU(1,1) in detail and summarizes how the theory can be extended.
The mathematical formulation of quantum mechanics by Werner Heisenberg and Erwin Schrödinger was originally in terms of unbounded self-adjoint operators on a Hilbert space . The fundamental operators corresponding to position and momentum satisfy the Heisenberg commutation relations . Quadratic polynomials in these operators, which include the harmonic oscillator , are also closed under taking commutators.
A large amount of operator theory was developed in the 1920s and 1930s to provide a rigorous foundation for quantum mechanics. Part of the theory was formulated in terms of unitary groups of operators, largely through the contributions of Hermann Weyl , Marshall Stone and John von Neumann . In turn these results in mathematical physics were subsumed within mathematical analysis, starting with the 1933 lecture notes of Norbert Wiener , who used the heat kernel for the harmonic oscillator to derive the properties of the Fourier transform .
The uniqueness of the Heisenberg commutation relations, as formulated in the Stone–von Neumann theorem , was later interpreted within group representation theory , in particular the theory of induced representations initiated by George Mackey . The quadratic operators were understood in terms of a projective unitary representation of the group SU(1,1) and its Lie algebra . Irving Segal and David Shale generalized this construction to the symplectic group in finite and infinite dimensions—in physics, this is often referred to as bosonic quantization : it is constructed as the symmetric algebra of an infinite-dimensional space. Segal and Shale have also treated the case of fermionic quantization , which is constructed as the exterior algebra of an infinite-dimensional Hilbert space. In the special case of conformal field theory in 1+1 dimensions, the two versions become equivalent via the so-called "boson-fermion correspondence." Not only does this apply in analysis where there are unitary operators between bosonic and fermionic Hilbert spaces, but also in the mathematical theory of vertex operator algebras . Vertex operators themselves originally arose in the late 1960s in theoretical physics , particularly in string theory .
André Weil later extended the construction to p-adic Lie groups , showing how the ideas could be applied in number theory , in particular to give a group theoretic explanation of theta functions and quadratic reciprocity . Several physicists and mathematicians observed the heat kernel operators corresponding to the harmonic oscillator were associated to a complexification of SU(1,1): this was not the whole of SL(2, C ), but instead a complex semigroup defined by a natural geometric condition. The representation theory of this semigroup, and its generalizations in finite and infinite dimensions, has applications both in mathematics and theoretical physics. [ 1 ]
The group:
is a subgroup of G c = SL(2, C ), the group of complex 2 × 2 matrices with determinant 1. If G 1 = SL(2, R ) then
This follows since the corresponding Möbius transformation is the Cayley transform which carries the upper half plane onto the unit disk and the real line onto the unit circle.
The group SL(2, R ) is generated as an abstract group by
and the subgroup of lower triangular matrices
Indeed, the orbit of the vector
under the subgroup generated by these matrices is easily seen to be the whole of R 2 and the stabilizer of v in G 1 lies in inside this subgroup.
The Lie algebra g {\displaystyle {\mathfrak {g}}} of SU(1,1) consists of matrices
The period 2 automorphism σ of G c
with
has fixed point subgroup G since
Similarly the same formula defines a period two automorphism σ of the Lie algebra g c {\displaystyle {\mathfrak {g}}_{c}} of G c , the complex matrices with trace zero. A standard basis of g c {\displaystyle {\mathfrak {g}}_{c}} over C is given by
Thus for −1 ≤ m , n ≤ 1
There is a direct sum decomposition
where g {\displaystyle {\mathfrak {g}}} is the +1 eigenspace of σ and i g {\displaystyle i{\mathfrak {g}}} the –1 eigenspace.
The matrices X in i g {\displaystyle i{\mathfrak {g}}} have the form
Note that
The cone C in i g {\displaystyle i{\mathfrak {g}}} is defined by two conditions. The first is det X < 0. {\displaystyle \det X<0.} By definition this condition is preserved under conjugation by G . Since G is connected it leaves the two components with x > 0 and x < 0 invariant. The second condition is x < 0. {\displaystyle x<0.}
The group G c acts by Möbius transformations on the extended complex plane. The subgroup G acts as automorphisms of the unit disk D . A semigroup H of G c , first considered by Olshanskii (1981) , can be defined by the geometric condition:
The semigroup can be described explicitly in terms of the cone C : [ 2 ]
In fact the matrix X can be conjugated by an element of G to the matrix
with
Since the Möbius transformation corresponding to exp Y sends z to e −2 y z , it follows that the right hand side lies in the semigroup. Conversely if g lies in H it carries the closed unit disk onto a smaller closed disk in its interior. Conjugating by an element of G , the smaller disk can be taken to have centre 0. But then for appropriate y , the element e − Y g {\displaystyle e^{-Y}g} carries D onto itself so lies in G .
A similar argument shows that the closure of H , also a semigroup, is given by
From the above statement on conjugacy, it follows that
where
If
then
since the latter is obtained by taking the transpose and conjugating by the diagonal matrix with entries ±1. Hence H also contains
which gives the inverse matrix if the original matrix lies in SU(1,1).
A further result on conjugacy follows by noting that every element of H must fix a point in D , which by conjugation with an element of G can be taken to be 0. Then the element of H has the form
The set of such lower triangular matrices forms a subsemigroup H 0 of H .
Since
every matrix in H 0 is conjugate to a diagonal matrix by a matrix M in H 0 .
Similarly every one-parameter semigroup S ( t ) in H fixes the same point in D so is conjugate by an element of G to a one-parameter semigroup in H 0 .
It follows that there is a matrix M in H 0 such that
with S 0 ( t ) diagonal. Similarly there is a matrix N in H 0 such that
The semigroup H 0 generates the subgroup L of complex lower triangular matrices with determinant 1 (given by the above formula with a ≠ 0). Its Lie algebra consists of matrices of the form
In particular the one parameter semigroup exp tZ lies in H 0 for all t > 0 if and only if ℜ z < 0 {\displaystyle \Re z<0} and | ℜ z | > 1 2 | w | . {\displaystyle |\Re z|>{\tfrac {1}{2}}|w|.}
This follows from the criterion for H or directly from the formula
The exponential map is known not to be surjective in this case, even though it is surjective on the whole group L . This follows because the squaring operation is not surjective in H . Indeed, since the square of an element fixes 0 only if the original element fixes 0, it suffices to prove this in H 0 . Take α with |α| < 1 and
If a = α 2 and
with
then the matrix
has no square root in H 0 . For a square root would have the form
On the other hand,
The closed semigroup H ¯ {\displaystyle {\overline {H}}} is maximal in SL(2, C ): any larger semigroup must be the whole of SL(2, C ). [ 3 ] [ 4 ] [ 5 ] [ 6 ] [ 7 ]
Using computations motivated by theoretical physics, Ferrara et al. (1973) introduced the semigroup H {\displaystyle H} , defined through a set of inequalities. Without identification H {\displaystyle H} as a compression semigroup, they established the maximality of H ¯ {\displaystyle {\overline {H}}} . Using the definition as a compression semigroup, maximality reduces to checking what happens when adding a new fractional transformation g {\displaystyle g} to H ¯ {\displaystyle {\overline {H}}} . The idea of the proof depends on considering the positions of the two discs g ( D ) {\displaystyle g(D)} and D {\displaystyle D} . In the key cases, either one disc contains the other or they are disjoint. In the simplest cases, g {\displaystyle g} is the inverse of a scaling transformation or g ( z ) = − 1 / z {\displaystyle g(z)=-1/z} . In either case g {\displaystyle g} and H {\displaystyle H} generate an open neighbourhood of 1 and hence the whole of SL(2,C)
Later Lawson (1998) gave another more direct way to prove maximality by first showing that there is a g in S sending D onto the disk D c , | z | > 1. In fact if x ∈ S ∖ H ¯ , {\displaystyle x\in S\setminus {\overline {H}},} then there is a small disk D 1 in D such that xD 1 lies in D c . Then for some h in H , D 1 = hD . Similarly yxD 1 = D c for some y in H . So g = yxh lies in S and sends D onto D c . It follows that g 2 fixes the unit disc D so lies in SU(1,1). So g −1 lies in S . If t lies in H then tgD contains gD . Hence g − 1 t − 1 g ∈ H ¯ . {\displaystyle g^{-1}t^{-1}g\in {\overline {H}}.} So t −1 lies in S and therefore S contains an open neighbourhood of 1. Hence S = SL(2, C ).
Exactly the same argument works for Möbius transformations on R n and the open semigroup taking the closed unit sphere || x || ≤ 1 into the open unit sphere || x || < 1. The closure is a maximal proper semigroup in the group of all Möbius transformations. When n = 1, the closure corresponds to Möbius transformations of the real line taking the closed interval [–1,1] into itself. [ 8 ]
The semigroup H and its closure have a further piece of structure inherited from G , namely inversion on G extends to an antiautomorphism of H and its closure, which fixes the elements in exp C and its closure. For
the antiautomorphism is given by
and extends to an antiautomorphism of SL(2, C ).
Similarly the antiautomorphism
leaves G 1 invariant and fixes the elements in exp C 1 and its closure, so it has analogous properties for the semigroup in G 1 .
Let S {\displaystyle {\mathcal {S}}} be the space of Schwartz functions on R . It is dense in the Hilbert space L 2 ( R ) of square-integrable functions on R . Following the terminology of quantum mechanics , the "momentum" operator P and "position" operator Q are defined on S {\displaystyle {\mathcal {S}}} by
There operators satisfy the Heisenberg commutation relation
Both P and Q are self-adjoint for the inner product on S {\displaystyle {\mathcal {S}}} inherited from L 2 ( R ).
Two one parameter unitary groups U ( s ) and V ( t ) can be defined on S {\displaystyle {\mathcal {S}}} and L 2 ( R ) by
By definition
for f ∈ S {\displaystyle f\in {\mathcal {S}}} , so that formally
It is immediate from the definition that the one parameter groups U and V satisfy the Weyl commutation relation
The realization of U and V on L 2 ( R ) is called the Schrödinger representation .
The Fourier transform is defined on S {\displaystyle {\mathcal {S}}} by [ 9 ]
It defines a continuous map of S {\displaystyle {\mathcal {S}}} into itself for its natural topology.
Contour integration shows that the function
is its own Fourier transform.
On the other hand, integrating by parts or differentiating under the integral,
It follows that the operator on S {\displaystyle {\mathcal {S}}} defined by
commutes with both Q (and P ). On the other hand,
and since
lies in S {\displaystyle {\mathcal {S}}} , it follows that
and hence
This implies the Fourier inversion formula :
and shows that the Fourier transform is an isomorphism of S {\displaystyle {\mathcal {S}}} onto itself.
By Fubini's theorem
When combined with the inversion formula this implies that the Fourier transform preserves the inner product
so defines an isometry of S {\displaystyle {\mathcal {S}}} onto itself.
By density it extends to a unitary operator on L 2 ( R ), as asserted by Plancherel's theorem .
Suppose U ( s ) and V ( t ) are one parameter unitary groups on a Hilbert space H {\displaystyle {\mathcal {H}}} satisfying the Weyl commutation relations
For F ( s , t ) ∈ S ( R × R ) , {\displaystyle F(s,t)\in {\mathcal {S}}(\mathbf {R} \times \mathbf {R} ),} let [ 10 ] [ 11 ]
and define a bounded operator on H {\displaystyle {\mathcal {H}}} by
Then
where
The operators T ( F ) have an important non-degeneracy property : the linear span of all vectors T ( F )ξ is dense in H {\displaystyle {\mathcal {H}}} .
Indeed, if fds and gdt define probability measures with compact support, then the smeared operators
satisfy
and converge in the strong operator topology to the identity operator if the supports of the measures decrease to 0.
Since U ( f ) V ( g ) has the form T ( F ), non-degeneracy follows.
When H {\displaystyle {\mathcal {H}}} is the Schrödinger representation on L 2 ( R ), the operator T ( F ) is given by
It follows from this formula that U and V jointly act irreducibly on the Schrödinger representation since this is true for the operators given by kernels that are Schwartz functions. A concrete description is provided by Linear canonical transformations .
Conversely given a representation of the Weyl commutation relations on H {\displaystyle {\mathcal {H}}} , it gives rise to a non-degenerate representation of the *-algebra of kernel operators. But all such representations are on an orthogonal direct sum of copies of L 2 ( R ) with the action on each copy as above. This is a straightforward generalisation of the elementary fact that the representations of the N × N matrices are on direct sums of the standard representation on C N . The proof using matrix units works equally well in infinite dimensions.
The one parameter unitary groups U and V leave each component invariant, inducing the standard action on the Schrödinger representation.
In particular this implies the Stone–von Neumann theorem : the Schrödinger representation is the unique irreducible representation of the Weyl commutation relations on a Hilbert space.
Given U and V satisfying the Weyl commutation relations, define
Then
so that W defines a projective unitary representation of R 2 with cocycle given by
where z = x + i y = ( x , y ) {\displaystyle z=x+iy=(x,y)} and B is the symplectic form on R 2 given by
By the Stone–von Neumann theorem, there is a unique irreducible representation corresponding to this cocycle.
It follows that if g is an automorphism of R 2 preserving the form B , i.e. an element of SL(2, R ), then there is a unitary π( g ) on L 2 ( R ) satisfying the covariance relation
By Schur's lemma the unitary π( g ) is unique up to multiplication by a scalar ζ with |ζ| = 1, so that π defines a projective unitary representation of SL(2, R ).
This can be established directly using only the irreducibility of the Schrödinger representation. Irreducibility was a direct consequence of the fact the operators
with K a Schwartz function correspond exactly to operators given by kernels with Schwartz functions.
These are dense in the space of Hilbert–Schmidt operators , which, since it contains the finite rank operators, acts irreducibly.
The existence of π can be proved using only the irreducibility of the Schrödinger representation. The operators are unique up to a sign with
so that the 2-cocycle for the projective representation of SL(2, R ) takes values ±1.
In fact the group SL(2, R ) is generated by matrices of the form
and it can be verified directly that the following operators satisfy the covariance relations above:
The generators g i satisfy the following Bruhat relations , which uniquely specify the group SL(2, R ): [ 12 ]
It can be verified by direct calculation that these relations are satisfied up to a sign by the corresponding operators, which establishes that the cocycle takes values ±1.
There is a more conceptual explanation using an explicit construction of the metaplectic group as a double cover of SL(2, R ). [ 13 ] SL(2, R ) acts by Möbius transformations on the upper half plane H . Moreover, if
then
The function
satisfies the 1-cocycle relation
For each g , the function m ( g , z ) is non-vanishing on H and therefore has two possible holomorphic square roots. The metaplectic group is defined as the group
By definition it is a double cover of SL(2, R ) and is connected. Multiplication is given by
where
Thus for an element g of the metaplectic group there is a uniquely determined function m ( g , z ) 1/2 satisfying the 1-cocycle relation.
If ℑ z > 0 {\displaystyle \Im z>0} , then
lies in L 2 and is called a coherent state .
These functions lie in a single orbit of SL(2, R ) generated by
since for g in SL(2, R )
More specifically if g lies in Mp(2, R ) then
Indeed, if this holds for g and h , it also holds for their product. On the other hand, the formula is easily checked if g t has the form g i and these are generators.
This defines an ordinary unitary representation of the metaplectic group.
The element (1,–1) acts as multiplication by –1 on L 2 ( R ), from which it follows that the cocycle on SL(2, R ) takes only values ±1.
As explained in Lion & Vergne (1980) , the 2-cocycle on SL(2, R ) associated with the metaplectic representation, taking values ±1, is determined by the Maslov index .
Given three non-zero vectors u , v , w in the plane, their Maslov index τ ( u , v , w ) {\displaystyle \tau (u,v,w)} is defined as the signature of the quadratic form on R 3 defined by
Properties of the Maslov index :
Picking a non-zero vector u 0 , it follows that the function
defines a 2-cocycle on SL(2, R ) with values in the eighth roots of unity.
A modification of the 2-cocycle can be used to define a 2-cocycle with values in ±1 connected with the metaplectic cocycle. [ 14 ]
In fact given non-zero vectors u , v in the plane, define f ( u , v ) to be
If
then
The representatives π( g ) in the metaplectic representation can be chosen so that
where the 2-cocycle ω is given by
with
Holomorphic Fock space (also known as the Segal–Bargmann space ) is defined to be the vector space F {\displaystyle {\mathcal {F}}} of holomorphic functions f ( z ) on C with
finite. It has inner product
F {\displaystyle {\mathcal {F}}} is a Hilbert space with orthonormal basis
Moreover, the power series expansion of a holomorphic function in F {\displaystyle {\mathcal {F}}} gives its expansion with respect to this basis. [ 15 ] Thus for z in C
so that evaluation at z is gives a continuous linear functional on F . {\displaystyle {\mathcal {F}}.} In fact
where [ 16 ]
Thus in particular F {\displaystyle {\mathcal {F}}} is a reproducing kernel Hilbert space .
For f in F {\displaystyle {\mathcal {F}}} and z in C define
Then
so this gives a unitary representation of the Weyl commutation relations. [ 17 ] Now
It follows that the representation W F {\displaystyle W_{\mathcal {F}}} is irreducible.
Indeed, any function orthogonal to all the E a must vanish, so that their linear span is dense in F {\displaystyle {\mathcal {F}}} .
If P is an orthogonal projection commuting with W ( z ), let f = PE 0 . Then
The only holomorphic function satisfying this condition is the constant function. So
with λ = 0 or 1. Since E 0 is cyclic, it follows that P = 0 or I .
By the Stone–von Neumann theorem there is a unitary operator U {\displaystyle {\mathcal {U}}} from L 2 ( R ) onto F {\displaystyle {\mathcal {F}}} , unique up to multiplication by a scalar, intertwining the two representations of the Weyl commutation relations. By Schur's lemma and the Gelfand–Naimark construction , the matrix coefficient of any vector determines the vector up to a scalar multiple. Since the matrix coefficients of F = E 0 and f = H 0 are equal, it follows that the unitary U {\displaystyle {\mathcal {U}}} is uniquely determined by the properties
and
Hence for f in L 2 ( R )
so that
where
The operator U {\displaystyle {\mathcal {U}}} is called the Segal–Bargmann transform [ 18 ] and B is called the Bargmann kernel . [ 19 ]
The adjoint of U {\displaystyle {\mathcal {U}}} is given by the formula:
The action of SU(1,1) on holomorphic Fock space was described by Bargmann (1970) and Itzykson (1967) .
The metaplectic double cover of SU(1,1) can be constructed explicitly as pairs ( g , γ) with
and
If g = g 1 g 2 , then
using the power series expansion of (1 + z ) 1/2 for | z | < 1.
The metaplectic representation is a unitary representation π( g , γ) of this group satisfying the covariance relations
where
Since F {\displaystyle {\mathcal {F}}} is a reproducing kernel Hilbert space , any bounded operator T on it corresponds to a kernel given by a power series of its two arguments. In fact if
and F in F {\displaystyle {\mathcal {F}}} , then
The covariance relations and analyticity of the kernel imply that for S = π( g , γ),
for some constant C . Direct calculation shows that
leads to an ordinary representation of the double cover. [ 20 ]
Coherent states can again be defined as the orbit of E 0 under the metaplectic group.
For w complex, set
Then F w ∈ F {\displaystyle F_{w}\in {\mathcal {F}}} if and only if | w | < 1. In particular F 0 = 1 = E 0 . Moreover,
where
Similarly the functions zF w lie in F {\displaystyle {\mathcal {F}}} and form an orbit of the metaplectic group:
Since ( F w , E 0 ) = 1, the matrix coefficient of the function E 0 = 1 is given by [ 21 ]
The projective representation of SL(2, R ) on L 2 ( R ) or on F {\displaystyle {\mathcal {F}}} break up as a direct sum of two irreducible representations, corresponding to even and odd functions of x or z . The two representations can be realized on Hilbert spaces of holomorphic functions on the unit disk; or, using the Cayley transform, on the upper half plane. [ 22 ] [ 23 ]
The even functions correspond to holomorphic functions F + for which
is finite; and the odd functions to holomorphic functions F – for which
is finite. The polarized forms of these expressions define the inner products.
The action of the metaplectic group is given by
Irreducibility of these representations is established in a standard way. [ 24 ] Each representation breaks up as a direct sum of one dimensional eigenspaces of the rotation group each of which is generated by a C ∞ vector for the whole group. It follows that any closed invariant subspace is generated by the algebraic direct sum of eigenspaces it contains and that this sum is invariant under the infinitesimal action of the Lie algebra g {\displaystyle {\mathfrak {g}}} . On the other hand, that action is irreducible.
The isomorphism with even and odd functions in F {\displaystyle {\mathcal {F}}} can be proved using the Gelfand–Naimark construction since the matrix coefficients associated to 1 and z in the corresponding representations are proportional. Itzykson (1967) gave another method starting from the maps
from the even and odd parts to functions on the unit disk. These maps intertwine the actions of the metaplectic group given above and send z n to a multiple of w n . Stipulating that U ± should be unitary determines the inner products on functions on the disk, which can expressed in the form above. [ 25 ]
Although in these representations the operator L 0 has positive spectrum—the feature that distinguishes the holomorphic discrete series representations of SU(1,1)—the representations do not lie in the discrete series of the metaplectic group. Indeed, Kashiwara & Vergne (1978) noted that the matrix coefficients are not square integrable, although their third power is. [ 26 ]
Consider the following subspace of L 2 ( R ):
The operators
act on H . {\displaystyle {\mathcal {H}}.} X is called the annihilation operator and Y the creation operator . They satisfy
Define the functions
We claim they are the eigenfunctions of the harmonic oscillator, D . To prove this we use the commutation relations above:
Next we have:
This is known for n = 0 and the commutation relation above yields
The n th Hermite function is defined by
p n is called the n th Hermite polynomial .
Let
Thus
The operators P , Q or equivalently A , A * act irreducibly on H {\displaystyle {\mathcal {H}}} by a standard argument. [ 27 ] [ 28 ]
Indeed, under the unitary isomorphism with holomorphic Fock space H {\displaystyle {\mathcal {H}}} can be identified with C [ z ], the space of polynomials in z , with
If a subspace invariant under A and A* contains a non-zero polynomial p ( z ), then, applying a power of A *, it contains a non-zero constant; applying then a power of A , it contains all z n .
Under the isomorphism F n is sent to a multiple of z n and the operator D is given by
Let
so that
In the terminology of physics A , A * give a single boson and L 0 is the energy operator. It is diagonalizable with eigenvalues 1/2, 1, 3/2, ...., each of multiplicity one. Such a representation is called a positive energy representation .
Moreover,
so that the Lie bracket with L 0 defines a derivation of the Lie algebra spanned by A , A * and I . Adjoining L 0 gives the semidirect product . The infinitesimal version of the Stone–von Neumann theorem states that the above representation on C [ z ] is the unique irreducible positive energy representation of this Lie algebra with L 0 = A * A + 1/2. For A lowers energy and A * raises energy. So any lowest energy vector v is annihilated by A and the module is exhausted by the powers of A * applied to v . It is thus a non-zero quotient of C [ z ] and hence can be identified with it by irreducibility.
Let
so that
These operators satisfy:
and act by derivations on the Lie algebra spanned by A , A * and I .
They are the infinitesimal operators corresponding to the metaplectic representation of SU(1,1).
The functions F n are defined by
It follows that the Hermite functions are the orthonormal basis obtained by applying the Gram-Schmidt orthonormalization process to the basis x n exp - x 2 /2 of H {\displaystyle {\mathcal {H}}} .
The completeness of the Hermite functions follows from the fact that the Bargmann transform is unitary and carries the orthonormal basis e n ( z ) of holomorphic Fock space onto the H n ( x ).
The heat operator for the harmonic oscillator is the operator on L 2 ( R ) defined as the diagonal operator
It corresponds to the heat kernel given by Mehler's formula :
This follows from the formula
To prove this formula note that if s = σ 2 , then by Taylor's formula
Thus F σ, x lies in holomorphic Fock space and
an inner product that can be computed directly.
Wiener (1933 , pp. 51–67) establishes Mehler's formula directly and uses a classical argument to prove that
tends to f in L 2 ( R ) as t decreases to 0. This shows the completeness of the Hermite functions and also, since
can be used to derive the properties of the Fourier transform.
There are other elementary methods for proving the completeness of the Hermite functions, for example using Fourier series . [ 29 ]
The Sobolev spaces H s , sometimes called Hermite-Sobolev spaces , are defined to be the completions of S {\displaystyle {\mathcal {S}}} with respect to the norms
where
is the expansion of f in Hermite functions. [ 30 ]
Thus
The Sobolev spaces are Hilbert spaces. Moreover, H s and H – s are in duality under the pairing
For s ≥ 0,
for some positive constant C s .
Indeed, such an inequality can be checked for creation and annihilation operators acting on Hermite functions H n and this implies the general inequality. [ 31 ]
It follows for arbitrary s by duality.
Consequently, for a quadratic polynomial R in P and Q
The Sobolev inequality holds for f in H s with s > 1/2:
for any k ≥ 0.
Indeed, the result for general k follows from the case k = 0 applied to Q k f .
For k = 0 the Fourier inversion formula
implies
If s < t , the diagonal form of D , shows that the inclusion of H t in H s is compact (Rellich's lemma).
It follows from Sobolev's inequality that the intersection of the spaces H s is S {\displaystyle {\mathcal {S}}} . Functions in S {\displaystyle {\mathcal {S}}} are characterized by the rapid decay of their Hermite coefficients a n .
Standard arguments show that each Sobolev space is invariant under the operators W ( z ) and the metaplectic group. [ 32 ] Indeed, it is enough to check invariance when g is sufficiently close to the identity. In that case
with D + A an isomorphism from H t + 2 {\displaystyle H_{t+2}} to H t . {\displaystyle H_{t}.}
It follows that
If f ∈ H s , {\displaystyle f\in H_{s},} then
where the derivatives lie in H s − 1 / 2 . {\displaystyle H_{s-1/2}.}
Similarly the partial derivatives of total degree k of U ( s ) V ( t ) f lie in Sobolev spaces of order s – k /2.
Consequently, a monomial in P and Q of order 2k applied to f lies in H s – k and can be expressed as a linear combination of partial derivatives of U(s)V(t)f of degree ≤ 2k evaluated at 0.
The smooth vectors for the Weyl commutation relations are those u in L 2 ( R ) such that the map
is smooth. By the uniform boundedness theorem , this is equivalent to the requirement that each matrix coefficient (W(z)u,v) be smooth.
A vector is smooth if and only it lies in S {\displaystyle {\mathcal {S}}} . [ 33 ] Sufficiency is clear. For necessity, smoothness implies that the partial derivatives of W(z)u lie in L 2 ( R ) and hence also D k u for all positive k . Hence u lies in the intersection of the H k , so in S {\displaystyle {\mathcal {S}}} .
It follows that smooth vectors are also smooth for the metaplectic group.
Moreover, a vector is in S {\displaystyle {\mathcal {S}}} if and only if it is a smooth vector for the rotation subgroup of SU(1,1).
If Π( t ) is a one parameter unitary group and for f in S {\displaystyle {\mathcal {S}}}
then the vectors Π( f )ξ form a dense set of smooth vectors for Π.
In fact taking
the vectors v = Π( f ε )ξ converge to ξ as ε decreases to 0 and
is an analytic function of t that extends to an entire function on C .
The vector is called an entire vector for Π.
The wave operator associated to the harmonic oscillator is defined by
The operator is diagonal with the Hermite functions H n as eigenfunctions:
Since it commutes with D , it preserves the Sobolev spaces.
The analytic vectors constructed above can be rewritten in terms of the Hermite semigroup as
The fact that v is an entire vector for Π is equivalent to the summability condition
for all r > 0.
Any such vector is also an entire vector for U(s)V(t) , that is the map
defined on R 2 extends to an analytic map on C 2 .
This reduces to the power series estimate
So these form a dense set of entire vectors for U(s)V(t) ; this can also be checked directly using Mehler's formula.
The spaces of smooth and entire vectors for U(s)V(t) are each by definition invariant under the action of the metaplectic group as well as the Hermite semigroup.
Let
be the analytic continuation of the operators W ( x , y ) from R 2 to C 2 such that
Then W leaves the space of entire vectors invariant and satisfies
Moreover, for g in SL(2, R )
using the natural action of SL(2, R ) on C 2 .
Formally
There is a natural double cover of the Olshanski semigroup H , and its closure H ¯ {\displaystyle {\overline {H}}} that extends the double cover of SU(1,1) corresponding to the metaplectic group. It is given by pairs ( g , γ) where g is an element of H or its closure
and γ is a square root of a .
Such a choice determines a unique branch of
for | z | < 1.
The unitary operators π( g ) for g in SL(2, R ) satisfy
for u in C 2 .
An element g of the complexification SL(2, C ) is said to implementable if there is a bounded operator T such that it and its adjoint leave the space of entire vectors for W invariant, both have dense images and satisfy the covariance relations
for u in C 2 . The implementing operator T is uniquely determined up to multiplication by a non-zero scalar.
The implementable elements form a semigroup, containing SL(2, R ). Since the representation has positive energy, the bounded compact self-adjoint operators
for t > 0 implement the group elements in exp C 1 .
It follows that all elements of the Olshanski semigroup and its closure are implemented.
Maximality of the Olshanki semigroup implies that no other elements of SL(2, C ) are implemented. Indeed, otherwise every element of SL(2, C ) would be implemented by a bounded operator, which would contradict the non-invertibility of the operators S 0 ( t ) for t > 0.
In the Schrödinger representation the operators S 0 ( t ) for t > 0 are given by Mehler's formula. They are contraction operators , positive and in every Schatten class . Moreover, they leave invariant each of the Sobolev spaces. The same formula is true for ℜ t > 0 {\displaystyle \Re \,t>0} by analytic continuation.
It can be seen directly in the Fock model that the implementing operators can be chosen so that they define an ordinary representation of the double cover of H constructed above. The corresponding semigroup of contraction operators is called the oscillator semigroup . The extended oscillator semigroup is obtained by taking the semidirect product with the operators W ( u ). These operators lie in every Schatten class and leave invariant the Sobolev spaces and the space of entire vectors for W .
The decomposition
corresponds at the operator level to the polar decomposition of bounded operators .
Moreover, since any matrix in H is conjugate to a diagonal matrix by elements in H or H −1 , every operator in the oscillator semigroup is quasi-similar to an operator S 0 ( t ) with ℜ t > 0 {\displaystyle \Re t>0} . In particular it has the same spectrum consisting of simple eigenvalues.
In the Fock model, if the element g of the Olshanki semigroup H corresponds to the matrix
the corresponding operator is given by
where
and γ is a square root of a . Operators π( g ,γ) for g in the semigroup H are exactly those that are Hilbert–Schmidt operators and correspond to kernels of the form
for which the complex symmetric matrix
has operator norm strictly less than one.
Operators in the extended oscillator semigroup are given by similar expressions with additional linear terms in z and w appearing in the exponential.
In the disk model for the two irreducible components of the metaplectic representation, the corresponding operators are given by
It is also possible to give an explicit formula for the contraction operators corresponding to g in H in the Schrödinger representation, It was by this formula that Howe (1988) introduced the oscillator semigroup as an explicit family of operators on L 2 ( R ). [ 34 ]
In fact consider the Siegel upper half plane consisting of symmetric complex 2x2 matrices with positive definite real part:
and define the kernel
with corresponding operator
for f in L 2 ( R ).
Then direct computation gives
where
Moreover,
where
By Mehler's formula for ℜ t > 0 {\displaystyle \Re \,t>0}
with
The oscillator semigroup is obtained by taking only matrices with B ≠ 0. From the above, this condition is closed under composition.
A normalized operator can be defined by
The choice of a square root determines a double cover.
In this case S Z corresponds to the element
of the Olshankii semigroup H .
Moreover, S Z is a strict contraction:
It follows also that
For a function a ( x , y ) on R 2 = C , let
So
where
Defining in general
the product of two such operators is given by the formula
where the twisted convolution or Moyal product is given by
The smoothing operators correspond to W ( F ) or ψ( a ) with F or a Schwartz functions on R 2 . The corresponding operators T have kernels that are Schwartz functions. They carry each Sobolev space into the Schwartz functions. Moreover, every bounded operator on L 2 ( R ) having this property has this form.
For the operators ψ( a ) the Moyal product translates into the Weyl symbolic calculus . Indeed, if the Fourier transforms of a and b have compact support then
where
This follows because in this case b must extend to an entire function on C 2 by the Paley-Wiener theorem .
This calculus can be extended to a broad class of symbols, but the simplest corresponds to convolution by a class of functions or distributions that all have the form T + S where T is a distribution of compact with singular support concentrated at 0 and where S is a Schwartz function. This class contains the operators P , Q as well as D 1/2 and D −1/2 where D is the harmonic oscillator.
The m th order symbols S m are given by smooth functions a satisfying
for all α and Ψ m consists of all operators ψ( a ) for such a .
If a is in S m and χ is a smooth function of compact support equal to 1 near 0, then
with T and S as above.
These operators preserve the Schwartz functions and satisfy;
The operators P and Q lie in Ψ 1 and D lies in Ψ 2 .
Properties:
The proof of boundedness of Howe (1980) is particularly simple: if
then
where the bracketed operator has norm less than ‖ a ‖ ⋅ ‖ b ‖ {\displaystyle \|a\|\cdot \|b\|} . So if F is supported in | z | ≤ R , then
The property of D −1 is proved by taking
with
Then R = I – DS lies in Ψ −1 , so that
lies in Ψ −2 and T = DA – I is smoothing. Hence
lies in Ψ −2 since D −1 T is smoothing.
The property for D 1/2 is established similarly by constructing B in Ψ 1/2 with real symbol such that D – B 4 is a smoothing operator. Using the holomorphic functional calculus it can be checked that D 1/2 – B 2 is a smoothing operator.
The boundedness result above was used by Howe (1980) to establish the more general inequality of Alberto Calderón and Remi Vaillancourt for pseudodifferential operators . An alternative proof that applies more generally to Fourier integral operators was given by Howe (1988) . He showed that such operators can be expressed as integrals over the oscillator semigroup and then estimated using the Cotlar-Stein lemma . [ 35 ]
Weil (1964) noted that the formalism of the Stone–von Neumann theorem and the oscillator representation of the symplectic group extends from the real numbers R to any locally compact abelian group . A particularly simple example is provided by finite abelian groups , where the proofs are either elementary or simplifications of the proofs for R . [ 36 ] [ 37 ]
Let A be a finite abelian group, written additively, and let Q be a non-degenerate quadratic form on A with values in T . Thus
is a symmetric bilinear form on A that is non-degenerate, so permits an identification between A and its dual group A * = Hom ( A , T ).
Let V = ℓ 2 ( A ) {\displaystyle V=\ell ^{2}(A)} be the space of complex-valued functions on A with inner product
Define operators on V by
for x , y in A . Then U ( x ) and V ( y ) are unitary representations of A on V satisfying the commutation relations
This action is irreducible and is the unique such irreducible representation of these relations.
Let G = A × A and for z = ( x , y ) in G set
Then
where
a non-degenerate alternating bilinear form on G . The uniqueness result above implies that if W' ( z ) is another family of unitaries giving a projective representation of G such that
then there is a unitary U , unique up to a phase, such that
for some λ( z ) in T .
In particular if g is an automorphism of G preserving B , then there is an essentially unique unitary π( g ) such that
The group of all such automorphisms is called the symplectic group for B and π gives a projective representation of G on V .
The group SL(2. Z ) naturally acts on G = A x A by symplectic automorphisms. It is generated by the matrices
If Z = – I , then Z is central and
These automorphisms of G are implemented on V by the following operators:
It follows that
where μ lies in T . Direct calculation shows that μ is given by the Gauss sum
The metaplectic group was defined as the group
The coherent state
defines a holomorphic map of H into L 2 ( R ) satisfying
This is in fact a holomorphic map into each Sobolev space H k and hence also H ≈ = S {\displaystyle H_{\approx }={\mathcal {S}}} .
On the other hand, in H − ≈ = S ′ {\displaystyle H_{-\approx }={\mathcal {S}}'} (in fact in H –1 ) there is a finite-dimensional space of distributions invariant under SL(2, Z ) and isomorphic to the N -dimensional oscillator representation on ℓ 2 ( A ) {\displaystyle \ell ^{2}(A)} where A = Z / N Z .
In fact let m > 0 and set N = 2 m . Let
The operators U ( x ), V ( y ) with x and y in M all commute and have a finite-dimensional subspace of fixed vectors formed by the distributions
with b in M 1 , where
The sum defining Ψ b converges in H − 1 ⊂ S ′ {\displaystyle H_{-1}\subset {\mathcal {S}}'} and depends only on the class of b in M 1 / M . On the other hand, the operators U ( x ) and V ( y ) with ' x , y in M 1 commute with all the corresponding operators for M . So M 1 leaves the subspace V 0 spanned by the Ψ b invariant. Hence the group A = M 1 acts on V 0 . This action can immediately be identified with the action on V for the N -dimensional oscillator representation associated with A , since
Since the operators π( R ) and π( S ) normalise the two sets of operators U and V corresponding to M and M 1 , it follows that they leave V 0 invariant and on V 0 must be constant multiples of the operators associated with the oscillator representation of A . In fact they coincide. From R this is immediate from the definitions, which show that
For S it follows from the Poisson summation formula and the commutation properties with the operators U ) x ) and V ( y ). The Poisson summation is proved classically as follows. [ 38 ]
For a > 0 and f in S {\displaystyle {\mathcal {S}}} let
F is a smooth function on R with period a :
The theory of Fourier series shows that
with the sum absolutely convergent and the Fourier coefficients given by
Hence
the usual Poisson summation formula.
This formula shows that S acts as follows
and so agrees exactly with formula for the oscillator representation on A .
Identifying A with Z /2 m Z , with
assigned to an integer n modulo 2 m , the theta functions can be defined directly as matrix coefficients: [ 39 ]
For τ in H and z in C set
so that | q | < 1. The theta functions agree with the standard classical formulas for the Jacobi-Riemann theta functions:
By definition they define holomorphic functions on H × C . The covariance properties of the function f τ and the distribution Ψ b lead immediately to the following transformation laws:
Because the operators π( S ), π ( R ) and π( J ) on L 2 ( R ) restrict to the corresponding operators on V 0 for any choice of m , signs of cocycles can be determined by taking m = 1. In this case the representation is 2-dimensional and the relation
on L 2 ( R ) can be checked directly on V 0 .
But in this case
The relation can also be checked directly by applying both sides to the ground state exp - x 2 /2.
Consequently, it follows that for m ≥ 1 the Gauss sum can be evaluated: [ 40 ]
For m odd, define
If m is odd, then, splitting the previous sum up into two parts, it follows that G (1, m ) equals m 1/2 if m is congruent to 1 mod 4 and equals i m 1/2 otherwise. If p is an odd prime and c is not divisible by p , this implies
where ( c p ) {\displaystyle \left({c \over p}\right)} is the Legendre symbol equal to 1 if c is a square mod p and –1 otherwise. Moreover, if p and q are distinct odd primes, then
From the formula for G (1, p ) and this relation, the law of quadratic reciprocity follows:
The theory of the oscillator representation can be extended from R to R n with the group SL(2, R ) replaced by the symplectic group Sp(2n, R ). The results can be proved either by straightforward generalisations from the one-dimensional case as in Folland (1989) or by using the fact that the n -dimensional case is a tensor product of n one-dimensional cases, reflecting the decomposition:
Let S {\displaystyle {\mathcal {S}}} be the space of Schwartz functions on R n , a dense subspace of L 2 ( R n ). For s , t in R n , define U ( s ) and V ( t ) on S {\displaystyle {\mathcal {S}}} and L 2 ( R ) by
From the definition U and V satisfy the Weyl commutation relation
As before this is called the Schrödinger representation.
The Fourier transform is defined on S {\displaystyle {\mathcal {S}}} by
The Fourier inversion formula
shows that the Fourier transform is an isomorphism of S {\displaystyle {\mathcal {S}}} onto itself extending to a unitary mapping of L 2 ( R n ) onto itself ( Plancherel's theorem ).
The Stone–von Neumann theorem asserts that the Schrödinger representation is irreducible and is the unique irreducible representation of the commutation relations: any other representation is a direct sum of copies of this representation.
If U and V satisfying the Weyl commutation relations, define
Then
so that W defines a projective unitary representation of R 2 n with cocycle given by
where z = x + i y = ( x , y ) {\displaystyle z=x+iy=(x,y)} and B is the symplectic form on R 2 n given by
The symplectic group Sp (2 n , R ) is defined to be group of automorphisms g of R 2 n preserving the form B . It follows from the Stone–von Neumann theorem that for each such g there is a unitary π( g ) on L 2 ( R ) satisfying the covariance relation
By Schur's lemma the unitary π( g ) is unique up to multiplication by a scalar ζ with |ζ| = 1, so that π defines a projective unitary representation of Sp( n ). Representatives can be chosen for π( g ), unique up to a sign, which show that the 2-cocycle for the projective representation of Sp(2 n , R ) takes values ±1. In fact elements of the group Sp( n , R ) are given by 2 n × 2 n real matrices g satisfying
where
Sp(2 n , R ) is generated by matrices of the form
and the operators
satisfy the covariance relations above. This gives an ordinary unitary representation of the metaplectic group , a double cover of Sp(2 n , R ). Indeed, Sp( n , R ) acts by Möbius transformations on the generalised Siegel upper half plane H n consisting of symmetric complex n × n matrices Z with strictly imaginary part by
if
The function
satisfies the 1-cocycle relation
The metaplectic group Mp(2 n , R ) is defined as the group
and is a connected double covering group of Sp(2 n , R ).
If ℑ Z > 0 {\displaystyle \Im Z>0} , then it defines a coherent state
in L 2 , lying in a single orbit of Sp(2 n ) generated by
If g lies in Mp(2n, R ) then
defines an ordinary unitary representation of the metaplectic group, from which it follows that the cocycle on Sp(2 n , R ) takes only values ±1.
Holomorphic Fock space is the Hilbert space F n {\displaystyle {\mathcal {F}}_{n}} of holomorphic functions f ( z ) on C n with finite norm
inner product
and orthonormal basis
for α a multinomial . For f in F n {\displaystyle {\mathcal {F}}_{n}} and z in C n , the operators
define an irreducible unitary representation of the Weyl commutation relations. By the Stone–von Neumann theorem there is a unitary operator U {\displaystyle {\mathcal {U}}} from L 2 ( R n ) onto F n {\displaystyle {\mathcal {F}}_{n}} intertwining the two representations. It is given by the Bargmann transform
where
Its adjoint U ∗ {\displaystyle {\mathcal {U}}^{*}} is given by the formula:
Sobolev spaces, smooth and analytic vectors can be defined as in the one-dimensional case using the sum of n copies of the harmonic oscillator
The Weyl calculus similarly extends to the n -dimensional case.
The complexification Sp(2 n , C ) of the symplectic group is defined by the same relation, but allowing the matrices A , B , C and D to be complex. The subsemigroup of group elements that take the Siegel upper half plane into itself has a natural double cover. The representations of Mp(2 n , R ) on L 2 ( R n ) and F n {\displaystyle {\mathcal {F}}_{n}} extend naturally to a representation of this semigroup by contraction operators defined by kernels, which generalise the one-dimensional case (taking determinants where necessary). The action of Mp(2 n , R ) on coherent states applies equally well to operators in this larger semigroup. [ 41 ]
As in the 1-dimensional case, where the group SL(2, R ) has a counterpart SU(1,1) through the Cayley transform with the upper half plane replaced by the unit disc, the symplectic group has a complex counterpart. Indeed, if C is the unitary matrix
then C Sp(2n) C −1 is the group of all matrices
such that
or equivalently
where
The Siegel generalized disk D n is defined as the set of complex symmetric n x n matrices W with operator norm less than 1.
It consist precisely of Cayley transforms of points Z in the Siegel generalized upper half plane:
Elements g act on D n
and, as in the one dimensional case this action is transitive. The stabilizer subgroup of 0 consists of matrices with A unitary and B = 0.
For W in D n the metaplectic coherent states in holomorphic Fock space are defined by
The inner product of two such states is given by
Moreover, the metaplectic representation π satisfies
The closed linear span of these states gives the even part of holomorphic Fock space F n + {\displaystyle {\mathcal {F}}_{n}^{+}} . The embedding of Sp(2 n ) in Sp(2( n +1)) and the compatible identification
lead to an action on the whole of F n {\displaystyle {\mathcal {F}}_{n}} . It can be verified directly that it is compatible with the action of the operators W ( z ). [ 42 ]
Since the complex semigroup has as Shilov boundary the symplectic group, the fact that this representation has a well-defined contractive extension to the semigroup follows from the maximum modulus principle and the fact that the semigroup operators are closed under adjoints. Indeed, it suffices to check, for two such operators S , T and vectors v i proportional to metaplectic coherent states, that
which follows because the sum depends holomorphically on S and T , which are unitary on the boundary.
Let S denote the unit sphere in C n and define the Hardy space H 2 ( S ) be the closure in L 2 ( S ) of the restriction of polynomials in the coordinates z 1 , ..., z n . Let P be the projection onto Hardy space. It is known that if m ( f ) denotes multiplication by a continuous function f on S , then the commutator [P, m ( f )] is compact. Consequently, defining the Toeplitz operator by
on Hardy space, it follows that T ( fg ) – T ( f ) T ( g ) is compact for continuous f and g . The same holds if f and g are matrix-valued functions (so that the corresponding Toeplitz operators are matrices of operators on H 2 ( S )). In particular if f is a function on S taking values in invertible matrices, then
are compact and hence T ( f ) is a Fredholm operator with an index defined as
The index has been computed using the methods of K-theory by Coburn (1973) and coincides up to a sign with the degree of f as a continuous mapping from S into the general linear group.
Helton & Howe (1975) gave an analytic way to establish this index theorem, simplied later by Howe. Their proof relies on the fact if f is smooth then the index is given by the formula of McKean and Singer : [ 43 ]
Howe (1980) noticed that there was a natural unitary isomorphism between H 2 ( S ) and L 2 ( R n ) carrying the Toeplitz operators
onto the operators
These are examples of zeroth order operators constructed within the Weyl calculus. The traces in the McKean-Singer formula can be computed directly using the Weyl calculus, leading to another proof of the index theorem. [ 44 ] This method of proving index theorems was generalised by Alain Connes within the framework of cyclic cohomology . [ 45 ]
The theory of the oscillator representation in infinite dimensions is due to Irving Segal and David Shale. [ 46 ] Graeme Segal used it to give a mathematically rigorous construction of projective representations of loop groups and the group of diffeomorphisms of the circle. At an infinitesimal level the construction of the representations of the Lie algebras, in this case the affine Kac–Moody algebra and the Virasoro algebra , was already known to physicists, through dual resonance theory and later string theory . Only the simplest case will be considered here, involving the loop group LU(1) of smooth maps of the circle into U(1) = T . The oscillator semigroup, developed independently by Neretin and Segal, allows contraction operators to be defined for the semigroup of univalent holomorphic maps of the unit disc into itself, extending the unitary operators corresponding to diffeomorphisms of the circle. When applied to the subgroup SU(1,1) of the diffeomorphism group, this gives a generalization of the oscillator representation on L 2 ( R ) and its extension to the Olshanskii semigroup.
The representation of commutation on Fock space is generalized to infinite dimensions by replacing C n (or its dual space) by an arbitrary complex Hilbert space H . The symmetric group S k acts on H ⊗ k . S k ( H ) is defined to be the fixed point subspace of S k and the symmetric algebra is the algebraic direct sum
It has a natural inner product inherited from H ⊗ k :
Taking the components S k ( H ) to be mutually orthogonal, the symmetric Fock space S ( H ) is defined to be the Hilbert space completion of this direct sum.
For ξ in H define the coherent state e ξ by
It follows that their linear span is dense in S ( H ), that the coherent states corresponding to n distinct vectors are linearly independent and that
When H is finite-dimensional, S ( H ) can naturally be identified with holomorphic Fock space for H *, since in the standard way S k ( H ) are just homogeneous polynomials of degree k on H * and the inner products match up. Moreover, S ( H ) has functorial properties. Most importantly
A similar result hold for finite orthogonal direct sums and extends to infinite orthogonal direct sums, using von Neumman's definition of the infinite tensor product with 1 the reference unit vector in S 0 ( H i ). Any contraction operator between Hilbert spaces induces a contraction operator between the corresponding symmetric Fock spaces in a functorial way.
A unitary operator on S ( H ) is uniquely determined by it values on coherent states. Moreover, for any assignment v ξ such that
there is a unique unitary operator U on S ( H ) such that
As in the finite-dimensional case, this allows the unitary operators W ( x ) to be defined for x in H :
It follows immediately from the finite-dimensional case that these operators are unitary and satisfy
In particular the Weyl commutation relations are satisfied:
Taking an orthonormal basis e n of H , S ( H ) can be written as an infinite tensor product of the S ( C e n ). The irreducibility of W on each of these spaces implies the irreducibility of W on the whole of S ( H ). W is called the complex wave representation .
To define the symplectic group in infinite dimensions let H R be the underlying real vector space of H with the symplectic form
and real inner product
The complex structure is then defined by the orthogonal operator
so that
A bounded invertible operator real linear operator T on H R lies in the symplectic group if it and its inverse preserve B . This is equivalent to the conditions:
The operator T is said to be implementable on S ( H ) provided there is a unitary π( T ) such that
The implementable operators form a subgroup of the symplectic group, the restricted symplectic group . By Schur's lemma, π( T ) is uniquely determined up to a scalar in T , so π gives a projective unitary representation of this subgroup.
The Segal-Shale quantization criterion states that T is implementable, i.e. lies in the restricted symplectic group, if and only if the commutator TJ – JT is a Hilbert–Schmidt operator .
Unlike the finite-dimensional case where a lifting π could be chosen so that it was multiplicative up to a sign, this is not possible in the infinite-dimensional case. (This can be seen directly using the example of the projective representation of the diffeomorphism group of the circle constructed below.)
The projective representation of the restricted symplectic group can be constructed directly on coherent states as in the finite-dimensional case. [ 47 ]
In fact, choosing a real Hilbert subspace of H of which H is a complexification, for any operator T on H a complex conjugate of T is also defined. Then the infinite-dimensional analogue of SU(1,1) consists of invertible bounded operators
satisfying gKg * = K (or equivalently the same relations as in the finite-dimensional case). These belong to the restricted symplectic group if and only if B is a Hilbert–Schmidt operator. This group acts transitively on the infinite-dimensional analogue D ≈ of the Seigel generalized unit disk consisting of Hilbert–Schmidt operators W that are symmetric with operator norm less than 1 via the formula
Again the stabilizer subgroup of 0 consists of g with A unitary and B = 0. The metaplectic coherent states f W can be defined as before and their inner product is given by the same formula, using the Fredholm determinant :
Define unit vectors by
and set
where μ(ζ) = ζ/|ζ|. As before this defines a projective representation and, if g 3 = g 1 g 2 , the cocycle is given by
This representation extends by analytic continuation to define contraction operators for the complex semigroup by the same analytic continuation argument as in the finite-dimensional case. It can also be shown that they are strict contractions.
Example Let H R be the real Hilbert space consisting of real-valued functions on the circle with mean 0
and for which
The inner product is given by
An orthogonal basis is given by the function sin( n θ) and cos( n θ) for n > 0. The Hilbert transform on the circle defined by
defines a complex structure on H R . J can also be written
where sign n = ±1 denotes the sign of n . The corresponding symplectic form is proportional to
In particular if φ is an orientation-preserving diffeomorphism of the circle and
then T φ is implementable. [ 48 ]
The operators W ( f ) with f smooth correspond to a subgroup of the loop group L T invariant under the diffeomorphism group of the circle. The infinitesimal operators corresponding to the vector fields
can be computed explicitly. They satisfy the Virasoro relations
In particular they cannor be adjusted by addition of scalar operators to remove the second term on the right hand side. This shows that the cocycle on the restricted symplectic group is not equivalent to one taking only the values ±1. | https://en.wikipedia.org/wiki/Oscillator_representation |
In spectroscopy, oscillator strength is a dimensionless quantity that expresses the probability of absorption or emission of electromagnetic radiation in transitions between energy levels of an atom or molecule. [ 1 ] [ 2 ] For example, if an emissive state has a small oscillator strength, nonradiative decay will outpace radiative decay . Conversely, "bright" transitions will have large oscillator strengths. [ 3 ] The oscillator strength can be thought of as the ratio between the quantum mechanical transition rate and the classical absorption/emission rate of a single electron oscillator with the same frequency as the transition. [ 4 ]
An atom or a molecule can absorb light and undergo a transition from
one quantum state to another.
The oscillator strength f 12 {\displaystyle f_{12}} of a transition from a lower state | 1 ⟩ {\displaystyle |1\rangle } to an upper state | 2 ⟩ {\displaystyle |2\rangle } may be defined by
where m e {\displaystyle m_{e}} is the mass of an electron and ℏ {\displaystyle \hbar } is
the reduced Planck constant . The quantum states | n ⟩ , n = {\displaystyle |n\rangle ,n=} 1,2, are assumed to have several
degenerate sub-states, which are labeled by m n {\displaystyle m_{n}} . "Degenerate" means
that they all have the same energy E n {\displaystyle E_{n}} .
The operator R x {\displaystyle R_{x}} is the sum of the x-coordinates r i , x {\displaystyle r_{i,x}} of all N {\displaystyle N} electrons in the system, i.e.
The oscillator strength is the same for each sub-state | n m n ⟩ {\displaystyle |nm_{n}\rangle } .
The definition can be recast by inserting the Rydberg energy Ry {\displaystyle {\text{Ry}}} and Bohr radius a 0 {\displaystyle a_{0}}
In case the matrix elements of R x , R y , R z {\displaystyle R_{x},R_{y},R_{z}} are the same, we can get rid of the sum and of the 1/3 factor
To make equations of the previous section applicable to the states belonging to the continuum spectrum, they should be rewritten in terms of matrix elements of the momentum p {\displaystyle {\boldsymbol {p}}} . In absence of magnetic field, the Hamiltonian can be written as H = 1 2 m p 2 + V ( r ) {\displaystyle H={\frac {1}{2m}}{\boldsymbol {p}}^{2}+V({\boldsymbol {r}})} , and calculating a commutator [ H , x ] {\displaystyle [H,x]} in the basis of eigenfunctions of H {\displaystyle H} results in the relation between matrix elements
Next, calculating matrix elements of a commutator [ p x , x ] {\displaystyle [p_{x},x]} in the same basis and eliminating matrix elements of x {\displaystyle x} , we arrive at
Because [ p x , x ] = − i ℏ {\displaystyle [p_{x},x]=-i\hbar } , the above expression results in a sum rule
where f n k {\displaystyle f_{nk}} are oscillator strengths for quantum transitions between the states n {\displaystyle n} and k {\displaystyle k} . This is the Thomas-Reiche-Kuhn sum rule, and the term with k = n {\displaystyle k=n} has been omitted because in confined systems such as atoms or molecules the diagonal matrix element ⟨ n | p x | n ⟩ = 0 {\displaystyle \langle n|p_{x}|n\rangle =0} due to the time inversion symmetry of the Hamiltonian H {\displaystyle H} . Excluding this term eliminates divergency because of the vanishing denominator. [ 5 ]
In crystals, the electronic energy spectrum has a band structure E n ( p ) {\displaystyle E_{n}({\boldsymbol {p}})} . Near the minimum of an isotropic energy band, electron energy can be expanded in powers of p {\displaystyle {\boldsymbol {p}}} as E n ( p ) = p 2 / 2 m ∗ {\displaystyle E_{n}({\boldsymbol {p}})={\boldsymbol {p}}^{2}/2m^{*}} where m ∗ {\displaystyle m^{*}} is the electron effective mass . It can be shown [ 6 ] that it satisfies the equation
Here the sum runs over all bands with k ≠ n {\displaystyle k\neq n} . Therefore, the ratio m / m ∗ {\displaystyle m/m^{*}} of the free electron mass m {\displaystyle m} to its effective mass m ∗ {\displaystyle m^{*}} in a crystal can be considered as the oscillator strength for the transition of an electron from the quantum state at the bottom of the n {\displaystyle n} band into the same state. [ 7 ] | https://en.wikipedia.org/wiki/Oscillator_strength |
A Continuous Oscillatory Baffled Reactor (COBR) is a specially designed chemical reactor to achieve plug flow under laminar flow conditions. Achieving plug flow has previously been limited to either a large number of continuous stirred-tank reactors (CSTR) in series or conditions with high turbulent flow. The technology incorporates annular baffles to a tubular reactor framework to create eddies when liquid is pushed up through the tube. Likewise, when liquid is on a downstroke through the tube, eddies are created on the other side of the baffles. Eddy generation on both sides of the baffles creates very effective mixing while still maintaining plug flow. By using COBR, potentially higher yields of product can be made with greater control and reduced waste. [ 1 ]
A standard COBR consists of a 10-150mm ID tube with equally spaced baffles throughout. There are typically two pumps in a COBR; one pump is reciprocating to generate continuous oscillatory flow and a second pump creates net flow through the tube. This design offers a control over mixing intensity that conventional tubular reactors cannot achieve. [ 2 ] Each baffled cell acts as a CSTR and because a secondary pump is creating a net laminar flow, much longer residence times can be achieved relative to turbulent flow systems. [ 3 ]
With conventional tubular reactors, mixing is accomplished through stirring mechanisms or turbulent flow conditions, which is difficult to control. By changing variable values such as baffle spacing or thickness, COBRs can operate with much better mixing control. For instance, it has been found that a spacing of 1.5 times tube diameter size is the most effective mixing condition; furthermore, vortex deformation increases with increase in baffle thickness greater than 3mm. [ 2 ]
The low shear rate and enhanced mass transfer provided by the COBR makes it an ideal reactor for various biological processes. For shear rate, it has been found that COBRs have an evenly distributed, five-fold reduction in shear rate relative to conventional tubular reactors; this is especially important for biological process given that high shear rates can damage microorganisms.
For the case of mass transfer, COBR fluid mechanics allows for an increase in oxygen gas residence time. Furthermore, the vortexes created in the COBRs causes a gas bubble break-up and thus an increase in surface area for gas transfer. For aerobic biological processes, therefore, COBRs again present an advantage. An especially promising aspect of the COBR technology is its ability to scale-up processes while still retaining the advantages in shear rate and mass transfer.
Though the prospect for COBR applications in fields like bioprocessing are very promising, there are a number of necessary improvements to be made before more global use. Clearly, there is additional complexity in the COBR design relative to other bioreactors, which can introduce complications in operation. Furthermore, for bioprocessing it is possible that fouling of baffles and internal surfaces becomes an issue. Perhaps the most significant needed advancement moving forward is further comprehensive studies that COBR technology can indeed be useful in industry. There are currently no COBRs in use at industrial bioprocessing plants and the evidence of its effectiveness, though very promising and theoretically an improvement relative to current reactors in industry, is limited to smaller laboratory-scale experiments. [ 3 ] | https://en.wikipedia.org/wiki/Oscillatory_baffled_reactor |
In mathematical analysis an oscillatory integral is a type of distribution . Oscillatory integrals make rigorous many arguments that, on a naive level, appear to use divergent integrals. It is possible to represent approximate solution operators for many differential equations as oscillatory integrals.
An oscillatory integral f ( x ) {\displaystyle f(x)} is written formally as
where ϕ ( x , ξ ) {\displaystyle \phi (x,\xi )} and a ( x , ξ ) {\displaystyle a(x,\xi )} are functions defined on R x n × R ξ N {\displaystyle \mathbb {R} _{x}^{n}\times \mathrm {R} _{\xi }^{N}} with the following properties:
When m < − N {\displaystyle m<-N} , the formal integral defining f ( x ) {\displaystyle f(x)} converges for all x {\displaystyle x} , and there is no need for any further discussion of the definition of f ( x ) {\displaystyle f(x)} . However, when m ≥ − N {\displaystyle m\geq -N} , the oscillatory integral is still defined as a distribution on R n {\displaystyle \mathbb {R} ^{n}} , even though the integral may not converge. In this case the distribution f ( x ) {\displaystyle f(x)} is defined by using the fact that a ( x , ξ ) ∈ S 1 , 0 m ( R x n × R ξ N ) {\displaystyle a(x,\xi )\in S_{1,0}^{m}(\mathbb {R} _{x}^{n}\times \mathrm {R} _{\xi }^{N})} may be approximated by functions that have exponential decay in ξ {\displaystyle \xi } . One possible way to do this is by setting
where the limit is taken in the sense of tempered distributions . Using integration by parts, it is possible to show that this limit is well defined, and that there exists a differential operator L {\displaystyle L} such that the resulting distribution f ( x ) {\displaystyle f(x)} acting on any ψ {\displaystyle \psi } in the Schwartz space is given by
where this integral converges absolutely. The operator L {\displaystyle L} is not uniquely defined, but can be chosen in such a way that depends only on the phase function ϕ {\displaystyle \phi } , the order m {\displaystyle m} of the symbol a {\displaystyle a} , and N {\displaystyle N} . In fact, given any integer M {\displaystyle M} , it is possible to find an operator L {\displaystyle L} so that the integrand above is bounded by C ( 1 + | ξ | ) − M {\displaystyle C(1+|\xi |)^{-M}} for | ξ | {\displaystyle |\xi |} sufficiently large. This is the main purpose of the definition of the symbol classes.
Many familiar distributions can be written as oscillatory integrals.
The Fourier inversion theorem implies that the delta function , δ ( x ) {\displaystyle \delta (x)} is equal to
If we apply the first method of defining this oscillatory integral from above, as well as the Fourier transform of the Gaussian , we obtain a well known sequence of functions which approximate the delta function:
An operator L {\displaystyle L} in this case is given for example by
where Δ x {\displaystyle \Delta _{x}} is the Laplacian with respect to the x {\displaystyle x} variables, and k {\displaystyle k} is any integer greater than ( n − 1 ) / 2 {\displaystyle (n-1)/2} . Indeed, with this L {\displaystyle L} we have
and this integral converges absolutely.
The Schwartz kernel of any differential operator can be written as an oscillatory integral. Indeed if
where D α = ∂ x α / i | α | {\displaystyle D^{\alpha }=\partial _{x}^{\alpha }/i^{|\alpha |}} , then the kernel of L {\displaystyle L} is given by
Any Lagrangian distribution [ clarification needed ] can be represented locally by oscillatory integrals, see Hörmander (1983) . Conversely, any oscillatory integral is a Lagrangian distribution. This gives a precise description of the types of distributions which may be represented as oscillatory integrals. | https://en.wikipedia.org/wiki/Oscillatory_integral |
In mathematics , in the field of harmonic analysis , an oscillatory integral operator is an integral operator of the form
where the function S ( x , y ) is called the phase of the operator and the function a ( x , y ) is called the symbol of the operator. λ is a parameter. One often considers S ( x , y ) to be real-valued and smooth, and a ( x , y ) smooth and compactly supported . Usually one is interested in the behavior of T λ for large values of λ .
Oscillatory integral operators often appear in many fields of mathematics ( analysis , partial differential equations , integral geometry , number theory ) and in physics. Properties of oscillatory integral operators have been studied by Elias Stein and his school. [ 1 ]
The following bound on the L 2 → L 2 action of oscillatory integral operators (or L 2 → L 2 operator norm ) was obtained by Lars Hörmander in his paper on Fourier integral operators : [ 2 ]
Assume that x,y ∈ R n , n ≥ 1. Let S ( x , y ) be real-valued and smooth, and let a ( x , y ) be smooth and compactly supported . If det j , k ∂ 2 S ∂ x j ∂ y k ( x , y ) ≠ 0 {\textstyle \det _{j,k}{\frac {\partial ^{2}S}{\partial x_{j}\,\partial y_{k}}}(x,y)\neq 0} everywhere on the support of a ( x , y ), then there is a constant C such that T λ , which is initially defined on smooth functions , extends to a continuous operator from L 2 ( R n ) to L 2 ( R n ), with the norm bounded by C λ − n / 2 {\displaystyle C\lambda ^{-n/2}} , for every λ ≥ 1: | https://en.wikipedia.org/wiki/Oscillatory_integral_operator |
In fluid dynamics , the Oseen equations (or Oseen flow ) describe the flow of a viscous and incompressible fluid at small Reynolds numbers , as formulated by Carl Wilhelm Oseen in 1910. Oseen flow is an improved description of these flows, as compared to Stokes flow , with the (partial) inclusion of convective acceleration . [ 1 ]
Oseen's work is based on the experiments of G.G. Stokes , who had studied the falling of a sphere through a viscous fluid. He developed a correction term, which included inertial factors, for the flow velocity used in Stokes' calculations, to solve the problem known as Stokes' paradox . His approximation leads to an improvement to Stokes' calculations.
The Oseen equations are, in case of an object moving with a steady flow velocity U through the fluid—which is at rest far from the object—and in a frame of reference attached to the object: [ 1 ] − ρ U ⋅ ∇ u = − ∇ p + μ ∇ 2 u , ∇ ⋅ u = 0 , {\displaystyle {\begin{aligned}-\rho \mathbf {U} \cdot \nabla \mathbf {u} &=-\nabla p\,+\,\mu \nabla ^{2}\mathbf {u} ,\\\nabla \cdot \mathbf {u} &=0,\end{aligned}}} where
The boundary conditions for the Oseen flow around a rigid object are: u = U at the object surface , u → 0 and p → p ∞ for r → ∞ , {\displaystyle {\begin{aligned}\mathbf {u} &=\mathbf {U} &&{\text{at the object surface}},\\\mathbf {u} &\to 0&&{\text{and}}\quad p\to p_{\infty }\quad {\text{for}}\quad r\to \infty ,\end{aligned}}} with r the distance from the object's center, and p ∞ the undisturbed pressure far from the object.
Source: [ 2 ]
A fundamental property of Oseen's equation is that the general solution can be split into longitudinal and transversal waves.
A solution ( u L , p ′ ) {\displaystyle \left(\mathbf {u} _{\text{L}},p'\right)} is a longitudinal wave if the velocity is irrotational and hence the viscous term drops out. The equations become u L t + U u L x + 1 ρ ∇ p = 0 , ∇ ⋅ u L = 0 , ∇ × u L = 0 {\displaystyle {\mathbf {u} _{\text{L}}}_{t}+U{\mathbf {u} _{\text{L}}}_{x}+{\frac {1}{\rho }}\nabla p=0,\quad \nabla \cdot \mathbf {u} _{\text{L}}=0,\quad \nabla \times \mathbf {u} _{\text{L}}=0}
In consequence u L = ∇ ϕ , ∇ 2 ϕ = 0 , p ′ = p − p ∞ = − ρ U u L {\displaystyle \mathbf {u} _{\text{L}}=\nabla \phi ,\quad \nabla ^{2}\phi =0,\quad p'=p-p_{\infty }=-\rho U\mathbf {u} _{\text{L}}}
Velocity is derived from potential theory and pressure is from linearized Bernoulli's equations.
A solution ( u T , 0 ) {\displaystyle (\mathbf {u} _{\text{T}},0)} is a transversal wave if the pressure p ′ {\displaystyle p'} is identically zero and the velocity field is solenoidal. The equations are u T t + U u T x = ν ∇ 2 u T , ∇ ⋅ u T = 0. {\displaystyle {\mathbf {u} _{\text{T}}}_{t}+U{\mathbf {u} _{\text{T}}}_{x}=\nu \nabla ^{2}\mathbf {u} _{\text{T}},\quad \nabla \cdot \mathbf {u_{T}} =0.}
Then the complete Oseen solution is given by u = u L + u T {\displaystyle \mathbf {u} =\mathbf {u} _{\text{L}}+\mathbf {u} _{\text{T}}}
a splitting theorem due to Horace Lamb . [ 3 ] The splitting is unique if conditions at infinity (say u = 0 , p = p ∞ {\displaystyle \mathbf {u} =0,\ p=p_{\infty }} ) are specified.
For certain Oseen flows, further splitting of transversal wave into irrotational and rotational component is possible u T = u 1 + u 2 . {\displaystyle \mathbf {u} _{\text{T}}=\mathbf {u} _{1}+\mathbf {u} _{2}.} Let χ {\displaystyle \chi } be the scalar function which satisfies U χ x = ν ∇ 2 χ {\displaystyle U\chi _{x}=\nu \nabla ^{2}\chi } and vanishes at infinity and conversely let u T = ( u T , v T ) {\displaystyle \mathbf {u} _{\text{T}}=(u_{\text{T}},v_{\text{T}})} be given such that ∫ − ∞ ∞ v T d y = 0 {\textstyle \int _{-\infty }^{\infty }v_{\text{T}}\,dy=0} , then the transversal wave is u T = − ν U ∇ χ + χ i , u 1 = − ν U ∇ χ , u 2 = χ i . {\displaystyle \mathbf {u} _{\text{T}}=-{\frac {\nu }{U}}\nabla \chi +\chi \mathbf {i} ,\quad \mathbf {u} _{\text{1}}=-{\frac {\nu }{U}}\nabla \chi ,\quad \mathbf {u} _{\text{2}}=\chi \mathbf {i} .} where χ {\displaystyle \chi } is determined from χ = U ν ∫ y ∞ v T d y {\textstyle \chi ={\frac {U}{\nu }}\int _{y}^{\infty }v_{T}\,dy} and i {\displaystyle \mathbf {i} } is the unit vector. Neither u 1 {\displaystyle \mathbf {u} _{1}} or u 2 {\displaystyle \mathbf {u} _{2}} are transversal by itself, but u 1 + u 2 {\displaystyle \mathbf {u} _{1}+\mathbf {u} _{2}} is transversal. Therefore, u = u L + u T = u L + u 1 + u 2 {\displaystyle \mathbf {u} =\mathbf {u} _{\text{L}}+\mathbf {u} _{\text{T}}=\mathbf {u} _{\text{L}}+\mathbf {u} _{1}+\mathbf {u} _{2}}
The only rotational component is being u 2 {\displaystyle \mathbf {u} _{2}} .
Source: [ 2 ]
The fundamental solution due to a singular point force embedded in an Oseen flow is the Oseenlet . The closed-form fundamental solutions for the generalized unsteady Stokes and Oseen flows associated with arbitrary time-dependent translational and rotational motions have been derived for the Newtonian [ 4 ] and micropolar [ 5 ] fluids.
Using the Oseen equation, Horace Lamb was able to derive improved expressions for the viscous flow around a sphere in 1911, improving on Stokes law towards somewhat higher Reynolds numbers. [ 1 ] Also, Lamb derived—for the first time—a solution for the viscous flow around a circular cylinder. [ 1 ]
The solution to the response of a singular force f {\displaystyle \mathbf {f} } when no external boundaries are present be written as U u x + 1 ρ ∇ p − ν ∇ 2 u = f , ∇ ⋅ u = 0 {\displaystyle U\mathbf {u} _{x}+{\frac {1}{\rho }}\nabla p-\nu \nabla ^{2}\mathbf {u} =\mathbf {f} ,\quad \nabla \cdot \mathbf {u} =0}
If f = δ ( q , q o ) a {\displaystyle \mathbf {f} =\delta (q,q_{o})\mathbf {a} } , where δ ( q , q o ) {\displaystyle \delta (q,q_{o})} is the singular force concentrated at the point q o {\displaystyle q_{o}} and q {\displaystyle q} is an arbitrary point and a {\displaystyle \mathbf {a} } is the given vector, which gives the direction of the singular force, then in the absence of boundaries, the velocity and pressure is derived from the fundamental tensor Γ ( q , q o ) {\displaystyle \Gamma (q,q_{o})} and the fundamental vector Π ( q , q o ) {\displaystyle \Pi (q,q_{o})} u ( q ) = Γ ( q , q o ) a , p ′ = p − p ∞ = Π ( q , q o ) ⋅ a {\displaystyle \mathbf {u} (q)=\Gamma (q,q_{o})\mathbf {a} ,\quad p'=p-p_{\infty }=\Pi (q,q_{o})\cdot \mathbf {a} }
Now if f {\displaystyle \mathbf {f} } is arbitrary function of space, the solution for an unbounded domain is u ( q ) = ∫ Γ ( q , q o ) f ( q o ) d q o , p ′ ( q ) = ∫ Π ( q , q o ) ⋅ f ( q o ) d q o {\displaystyle \mathbf {u} (q)=\int \Gamma (q,q_{o})\mathbf {f} (q_{o})dq_{o},\quad p'(q)=\int \Pi (q,q_{o})\cdot \mathbf {f} (q_{o})dq_{o}} where d q o {\displaystyle dq_{o}} is the infinitesimal volume/area element around the point q o {\displaystyle q_{o}} .
Without loss of generality q o = ( 0 , 0 ) {\displaystyle q_{o}=(0,0)} taken at the origin and q = ( x , y ) {\displaystyle q=(x,y)} . Then the fundamental tensor and vector are Γ = ( ∂ A ∂ x ∂ A ∂ y ∂ A ∂ y − ∂ A ∂ x ) + 1 2 π ν e λ x K o ( λ r ) ( 1 0 0 1 ) , Π = ρ 2 π ∇ ( ln r ) {\displaystyle \Gamma ={\begin{pmatrix}{\frac {\partial A}{\partial x}}&{\frac {\partial A}{\partial y}}\\{\frac {\partial A}{\partial y}}&-{\frac {\partial A}{\partial x}}\end{pmatrix}}+{\frac {1}{2\pi \nu }}e^{\lambda x}K_{o}(\lambda r){\begin{pmatrix}1&0\\0&1\end{pmatrix}},\quad \Pi ={\frac {\rho }{2\pi }}\nabla (\ln r)} where λ = U 2 ν , r 2 = x 2 + y 2 , A = − 1 2 π U [ ln r + e λ x K o ( λ r ) ] {\displaystyle \lambda ={\frac {U}{2\nu }},\quad r^{2}=x^{2}+y^{2},\quad A=-{\frac {1}{2\pi U}}\left[\ln r+e^{\lambda x}K_{o}(\lambda r)\right]} where K o ( λ r ) {\displaystyle K_{o}(\lambda r)} is the modified Bessel function of the second kind of order zero.
Without loss of generality q o = ( 0 , 0 , 0 ) {\displaystyle q_{o}=(0,0,0)} taken at the origin and q = ( x , y , z ) {\displaystyle q=(x,y,z)} . Then the fundamental tensor and vector are Γ = ( ∂ A ∂ x ∂ B ∂ x ∂ C ∂ x ∂ A ∂ y ∂ B ∂ y ∂ C ∂ y ∂ A ∂ z ∂ B ∂ z ∂ C ∂ z ) + 1 4 π ν e − λ ( r − x ) r ( 1 0 0 0 1 0 0 0 1 ) , Π = − ρ 4 π ∇ ( 1 r ) {\displaystyle \Gamma ={\begin{pmatrix}{\frac {\partial A}{\partial x}}&{\frac {\partial B}{\partial x}}&{\frac {\partial C}{\partial x}}\\{\frac {\partial A}{\partial y}}&{\frac {\partial B}{\partial y}}&{\frac {\partial C}{\partial y}}\\{\frac {\partial A}{\partial z}}&{\frac {\partial B}{\partial z}}&{\frac {\partial C}{\partial z}}\end{pmatrix}}+{\frac {1}{4\pi \nu }}{\frac {e^{-\lambda (r-x)}}{r}}{\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}},\quad \Pi =-{\frac {\rho }{4\pi }}\nabla \left({\frac {1}{r}}\right)} where
Oseen considered the sphere to be stationary and the fluid to be flowing with a flow velocity ( U {\displaystyle U} ) at an infinite distance from the sphere. Inertial terms were neglected in Stokes' calculations. [ 6 ] It is a limiting solution when the Reynolds number tends to zero. When the Reynolds number is small and finite, such as 0.1, correction for the inertial term is needed. Oseen substituted the following flow velocity values into the Navier-Stokes equations . u 1 = u + u 1 ′ , u 2 = u 2 ′ , u 3 = u 3 ′ . {\displaystyle u_{1}=u+u_{1}',\qquad u_{2}=u_{2}',\qquad u_{3}=u_{3}'.}
Inserting these into the Navier-Stokes equations and neglecting the quadratic terms in the primed quantities leads to the derivation of Oseen's approximation: u ∂ u 1 ′ ∂ x 1 = − 1 ρ ∂ p ∂ x 1 + ν ∇ 2 u i ′ ( i = 1 , 2 , 3 ) . {\displaystyle u{\partial u_{1}' \over \partial x_{1}}=-{1 \over \rho }{\partial p \over \partial x_{1}}+\nu \nabla ^{2}u_{i}'\qquad \left({i=1,2,3}\right).}
Since the motion is symmetric with respect to x {\displaystyle x} axis and the divergence of the vorticity vector is always zero we get: ( ∇ 2 − U 2 v ∂ ∂ x ) χ = G ( x ) = 0 {\displaystyle \left(\nabla ^{2}-{U \over 2v}{\partial \over \partial x}\right)\chi =G(x)=0} the function G ( x ) {\displaystyle G(x)} can be eliminated by adding to a suitable function in x {\displaystyle x} , is the vorticity function, and the previous function can be written as: U v ∂ u ′ ∂ x = ∇ 2 u ′ {\displaystyle {U \over v}{\partial u' \over \partial x}=\nabla ^{2}u'} and by some integration the solution for χ {\displaystyle \chi } is: e − U x 2 v χ = C e − U Re 2 v Re {\displaystyle e^{-Ux \over 2v}\chi ={{Ce^{-U\operatorname {Re} \over 2v}} \over \operatorname {Re} }} thus by letting x {\displaystyle x} be the "privileged direction" it produces: φ = A 0 Re + A 1 ∂ ∂ x 1 Re + A 2 ∂ 2 ∂ x 2 1 Re + … {\displaystyle \varphi ={A_{0} \over \operatorname {Re} }+A_{1}{\partial \over \partial x}{1 \over \operatorname {Re} }+A_{2}{\partial ^{2} \over \partial x^{2}}{1 \over \operatorname {Re} }+\ldots }
then by applying the three boundary conditions we obtain C = − 3 2 U a , A 0 = − 3 2 v a , A 1 = 1 4 U a 3 , etc. {\displaystyle C=-{3 \over 2}Ua,\ A_{0}=-{3 \over 2}va,\ A_{1}={1 \over 4}Ua^{3}\ {\text{, etc.}}} the new improved drag coefficient now become: C d = 12 Re ( 1 + 3 8 Re ) {\displaystyle C_{\text{d}}={12 \over \operatorname {Re} }\left(1+{3 \over 8}\operatorname {Re} \right)} and finally, when Stokes' solution was solved on the basis of Oseen's approximation, it showed that the resultant drag force is given by F = 6 π μ a u ( 1 + 3 8 Re ) , {\displaystyle F=6\pi \,\mu \,au\left(1+{3 \over 8}\operatorname {Re} \right),}
where:
The force from Oseen's equation differs from that of Stokes by a factor of 1 + 3 8 R e . {\displaystyle 1+{3 \over 8}\mathrm {Re} .}
The equations for the perturbation read: [ 7 ] ∇ u ′ = 0 u ⋅ ∇ u ′ = − ∇ p + ν ∇ 2 u ′ , {\displaystyle {\begin{aligned}\nabla u'&~=0\\u\cdot \nabla u'&~=-\nabla p+\nu \nabla ^{2}u',\end{aligned}}} but when the velocity field is: u y = u cos θ ( 1 + a 3 2 r 3 − 3 a 2 r ) u z = − u sin θ ( 1 − a 3 4 r 3 − 3 a 4 r ) . {\displaystyle {\begin{aligned}u_{y}&=u\cos \theta \left({1+{a^{3} \over 2r^{3}}-{3a \over 2r}}\right)\\u_{z}&=-u\sin \theta \left({1-{a^{3} \over 4r^{3}}-{3a \over 4r}}\right).\end{aligned}}}
In the far field r a {\displaystyle {r \over a}} ≫ 1, the viscous stress is dominated by the last term. That is: ∇ 2 u ′ = O ( a 3 r 3 ) . {\displaystyle \nabla ^{2}u'=O\left({a^{3} \over r^{3}}\right).}
The inertia term is dominated by the term: u ∂ u ′ ∂ z 1 ∼ O ( a 2 r 2 ) . {\displaystyle u{\partial u' \over \partial z_{1}}\sim O\left({a^{2} \over r^{2}}\right).}
The error is then given by the ratio: u ∂ u ′ ∂ z 1 ν ∇ 2 u ′ = O ( r a ) . {\displaystyle u{{\partial u' \over \partial z_{1}} \over {\nu \nabla ^{2}u'}}=O\left({r \over a}\right).}
This becomes unbounded for r a {\displaystyle {r \over a}} ≫ 1, therefore the inertia cannot be ignored in the far field. By taking the curl, Stokes equation gives ∇ 2 ζ = 0. {\displaystyle \nabla ^{2}\zeta \,=0.} Since the body is a source of vorticity , ζ {\displaystyle \zeta \,} would become unbounded logarithmically for large r a . {\displaystyle {r \over a}.} This is certainly unphysical and is known as Stokes' paradox .
Consider the case of a solid sphere moving in a stationary liquid with a constant velocity.
The liquid is modeled as an incompressible fluid (i.e. with constant density ), and being stationary means that its velocity tends towards zero as the distance from the sphere approaches infinity.
For a real body there will be a transient effect due to its acceleration as it begins its motion; however after enough time it will tend towards zero, so that the fluid velocity everywhere will approach the one obtained in the hypothetical case in which the body is already moving for infinite time.
Thus we assume a sphere of radius a moving at a constant velocity U → {\displaystyle {\vec {U}}} , in an incompressible fluid that is at rest at infinity. We will work in coordinates x → m {\displaystyle {\vec {x}}_{m}} that move along with the sphere with the coordinate center located at the sphere's center. We have: u → ( ‖ x → m ‖ = a ) = U → u → ( ‖ x → m ‖ → ∞ ) → 0 {\displaystyle {\begin{aligned}{\vec {u}}\left(\left\|{{\vec {x}}_{m}}\right\|=a\right)&={\vec {U}}\\{\vec {u}}\left(\left\|{{\vec {x}}_{m}}\right\|\rightarrow \infty \right)&\rightarrow 0\end{aligned}}}
Since these boundary conditions, as well as the equation of motions, are time invariant (i.e. they are unchanged by shifting the time t → t + Δ t {\displaystyle t\rightarrow t+\Delta t} ) when expressed in the x → m {\displaystyle {\vec {x}}_{m}} coordinates, the solution depends upon the time only through these coordinates.
The equations of motion are the Navier-Stokes equations defined in the resting frame coordinates x → = x → m − U → ⋅ t {\displaystyle {\vec {x}}={\vec {x}}_{m}-{\vec {U}}\cdot t} . While spatial derivatives are equal in both coordinate systems, the time derivative that appears in the equations satisfies: ∂ u → ( x → , t ) ∂ t = ∑ i d x m i d t ∂ u → ( x → m ) ∂ x m i = − ( U → ⋅ ∇ → m ) u → {\displaystyle {\frac {\partial {\vec {u}}\left({\vec {x}},t\right)}{\partial t}}=\sum _{i}{{\frac {d{x_{m}}_{i}}{dt}}{\frac {\partial {\vec {u}}\left({\vec {x}}_{m}\right)}{\partial {x_{m}}_{i}}}}=-\left({\vec {U}}\cdot {\vec {\nabla }}_{m}\right){\vec {u}}} where the derivative ∇ → m {\displaystyle {\vec {\nabla }}_{m}} is with respect to the moving coordinates x → m {\displaystyle {\vec {x}}_{m}} . We henceforth omit the m subscript.
Oseen's approximation sums up to neglecting the term non-linear in u → {\displaystyle {\vec {u}}} . Thus the incompressible Navier-Stokes equations become: ( U → ⋅ ∇ → ) u → + ν ∇ 2 u → = 1 ρ ∇ → p {\displaystyle \left({\vec {U}}\cdot {\vec {\nabla }}\right){\vec {u}}+\nu \nabla ^{2}{\vec {u}}={\frac {1}{\rho }}{\vec {\nabla }}p} for a fluid having density ρ and kinematic viscosity ν = μ/ρ (μ being the dynamic viscosity ). p is the pressure .
Due to the continuity equation for incompressible fluid ∇ → ⋅ u → = 0 {\displaystyle {\vec {\nabla }}\cdot {\vec {u}}=0} , the solution can be expressed using a vector potential ψ → {\displaystyle {\vec {\psi }}} . This turns out to be directed at the φ → {\displaystyle {\vec {\varphi }}} direction and its magnitude is equivalent to the stream function used in two-dimensional problems. It turns out to be: ψ = U a 2 ( − a 4 r 2 sin θ + 3 1 − cos θ r sin θ 1 − e − R r 4 a ( 1 + cos θ ) R ) u → = ∇ → × ( ψ φ ^ ) = 1 r sin θ ∂ ∂ θ ( ψ sin θ ) r ^ − 1 r ∂ ∂ r ( r ψ ) θ ^ {\displaystyle {\begin{aligned}\psi &=Ua^{2}\left(-{\frac {a}{4r^{2}}}\sin \theta +3{\frac {1-\cos \theta }{r\sin \theta }}{\frac {1-e^{-{\frac {Rr}{4a}}(1+\cos \theta )}}{R}}\right)\\{\vec {u}}&={\vec {\nabla }}\times (\psi {\hat {\varphi }})={\frac {1}{r\sin \theta }}{\frac {\partial }{\partial \theta }}\left(\psi \sin \theta \right){\hat {r}}-{\frac {1}{r}}{\frac {\partial }{\partial r}}\left(r\psi \right){\hat {\theta }}\end{aligned}}} where R = 2 a U / ν {\displaystyle R=2aU/\nu } is Reynolds number for the flow close to the sphere.
Note that in some notations ψ {\displaystyle \psi } is replaced by Ψ = ψ ⋅ r sin θ {\displaystyle \Psi =\psi \cdot r\sin \theta } so that the derivation of u → {\displaystyle {\vec {u}}} from Ψ {\displaystyle \Psi } is more similar to its derivation from the stream function in the two-dimensional case (in polar coordinates).
ψ {\displaystyle \psi } can be expressed as follows: ψ = ψ 1 + ψ 2 − ψ 2 e − k r ( 1 + cos θ ) {\displaystyle \psi =\psi _{1}+\psi _{2}-\psi _{2}e^{-kr(1+\cos \theta )}}
where: ψ 1 ≡ − U a 3 4 r 2 sin θ ψ 2 ≡ 3 U a 2 R r 1 − cos θ sin θ {\displaystyle {\begin{aligned}\psi _{1}&\equiv -{\frac {Ua^{3}}{4r^{2}}}\sin \theta \\\psi _{2}&\equiv {\frac {3Ua^{2}}{Rr}}{\frac {1-\cos \theta }{\sin \theta }}\end{aligned}}} k ≡ R 4 a {\displaystyle k\equiv {\frac {R}{4a}}} , so that U 2 k = 2 U a R = ν {\displaystyle {\frac {U}{2k}}={\frac {2Ua}{R}}=\nu } .
The vector Laplacian of a vector of the type V ( r , θ ) φ ^ {\displaystyle V(r,\theta ){\hat {\varphi }}} reads : ∇ 2 ( V ( r , θ ) φ ^ ) = φ ^ ⋅ ( ∇ 2 − 1 r 2 sin 2 θ ) V ( r , θ ) = φ ^ ⋅ [ 1 r 2 ∂ ∂ r ( r 2 ∂ ∂ r V ( r , θ ) ) + 1 r 2 sin θ ∂ ∂ θ ( sin θ ∂ ∂ θ V ( r , θ ) ) − V ( r , θ ) r 2 sin 2 θ ] {\displaystyle {\begin{aligned}&\nabla ^{2}\left(V(r,\theta ){\hat {\varphi }}\right)={\hat {\varphi }}\cdot \left(\nabla ^{2}-{\frac {1}{r^{2}\sin ^{2}\theta }}\right)V(r,\theta )=\\&{\hat {\varphi }}\cdot \left[{\frac {1}{r^{2}}}{\frac {\partial }{\partial r}}\left(r^{2}{\frac {\partial }{\partial r}}V(r,\theta )\right)+{\frac {1}{r^{2}\sin \theta }}{\frac {\partial }{\partial \theta }}\left(\sin \theta {\frac {\partial }{\partial \theta }}V(r,\theta )\right)-{\frac {V(r,\theta )}{r^{2}\sin ^{2}\theta }}\right]\end{aligned}}} .
It can thus be calculated that: ∇ 2 ( ψ 1 φ ^ ) = 0 ∇ 2 ( ψ 2 φ ^ ) = 0 {\displaystyle {\begin{aligned}\nabla ^{2}\left(\psi _{1}{\hat {\varphi }}\right)&=0\\\nabla ^{2}\left(\psi _{2}{\hat {\varphi }}\right)&=0\end{aligned}}}
Therefore: ∇ 2 ψ → = − ∇ 2 ( ψ 2 e − k r ( 1 + cos θ ) φ ^ ) = − ( ψ 2 ∇ 2 e − k r ( 1 + cos θ ) + 2 ∂ ψ 2 ∂ r ∂ ∂ r e − k r ( 1 + cos θ ) + 2 r 2 ∂ ψ 2 ∂ θ ∂ ∂ θ e − k r ( 1 + cos θ ) ) φ ^ = − 6 U a 2 R sin θ ( k 2 r + k r 2 ) e − k r ( 1 + cos θ ) φ ^ {\displaystyle {\begin{aligned}\nabla ^{2}{\vec {\psi }}&=-\nabla ^{2}\left(\psi _{2}e^{-kr(1+\cos \theta )}{\hat {\varphi }}\right)\\&=-\left(\psi _{2}\nabla ^{2}e^{-kr(1+\cos \theta )}+2{\frac {\partial \psi _{2}}{\partial r}}{\frac {\partial }{\partial r}}e^{-kr(1+\cos \theta )}+{\frac {2}{r^{2}}}{\frac {\partial \psi _{2}}{\partial \theta }}{\frac {\partial }{\partial \theta }}e^{-kr(1+\cos \theta )}\right){\hat {\varphi }}\\&=-{\frac {6Ua^{2}}{R}}\sin \theta \left({\frac {k^{2}}{r}}+{\frac {k}{r^{2}}}\right)e^{-kr(1+\cos \theta )}{\hat {\varphi }}\end{aligned}}}
Thus the vorticity is: ω → ≡ ∇ → × u → = − ∇ 2 ψ → = 6 U a 2 R sin θ ( k 2 r + k r 2 ) e − k r ( 1 + cos θ ) φ ^ {\displaystyle {\vec {\omega }}\equiv {\vec {\nabla }}\times {\vec {u}}=-\nabla ^{2}{\vec {\psi }}={\frac {6Ua^{2}}{R}}\sin \theta \left({\frac {k^{2}}{r}}+{\frac {k}{r^{2}}}\right)e^{-kr(1+\cos \theta )}{\hat {\varphi }}}
where we have used the vanishing of the divergence of ψ → {\displaystyle {\vec {\psi }}} to relate the vector laplacian and a double curl .
The equation of motion's left hand side is the curl of the following: ( U → ⋅ ∇ → ) ψ → + ν ∇ 2 ψ → = ( U → ⋅ ∇ → ) ψ → − ν ω → {\displaystyle \left({\vec {U}}\cdot {\vec {\nabla }}\right){\vec {\psi }}+\nu \nabla ^{2}{\vec {\psi }}=\left({\vec {U}}\cdot {\vec {\nabla }}\right){\vec {\psi }}-\nu {\vec {\omega }}}
We calculate the derivative separately for each term in ψ {\displaystyle \psi } .
Note that: U → = U ( cos θ r ^ − sin θ θ ^ ) {\displaystyle {\vec {U}}=U\left(\cos \theta {\hat {r}}-\sin \theta {\hat {\theta }}\right)}
And also: ∂ ψ 2 ∂ r = − 1 r ψ 2 sin θ ∂ ψ 2 ∂ θ = ψ 2 {\displaystyle {\begin{aligned}{\frac {\partial \psi _{2}}{\partial r}}&=-{\frac {1}{r}}\psi _{2}\\\sin \theta {\frac {\partial \psi _{2}}{\partial \theta }}&=\psi _{2}\end{aligned}}}
We thus have: ( U → ⋅ ∇ → ) ( ψ 1 φ ^ ) = U ( cos θ ∂ ψ 1 ∂ r − 1 r sin θ ∂ ψ 1 ∂ θ ) φ ^ = 3 U 2 a 3 4 r 3 sin θ cos θ φ ^ ( U → ⋅ ∇ → ) ( ψ 2 φ ^ ) = U ( cos θ ∂ ψ 2 ∂ r − 1 r sin θ ∂ ψ 2 ∂ θ ) φ ^ = − U 1 r ( 1 + cos θ ) ψ 2 φ ^ = − 3 U 2 a 2 R r 2 sin θ φ ^ ( U → ⋅ ∇ → ) ( − ψ 2 e − k r ( 1 + cos θ ) φ ^ ) = − e − k r ( 1 + cos θ ) ( ( U → ⋅ ∇ → ) ( ψ 2 φ ^ ) + ψ 2 ( U → ⋅ ∇ → ) ( − k r ( 1 + cos θ ) φ ^ ) ) = U ψ 2 e − k r ( 1 + cos θ ) ( 1 r ( 1 + cos θ ) + cos θ ∂ ( k r ( 1 + cos θ ) ) ∂ r − 1 r sin θ ∂ ( k r ( 1 + cos θ ) ) ∂ θ ) φ ^ = U ψ 2 ( 1 + cos θ ) ( 1 r + k ) e − k r ( 1 + cos θ ) φ ^ = 3 U 2 a 2 R sin θ ( 1 r 2 + k r ) e − k r ( 1 + cos θ ) φ ^ = U 2 k ω → = ν ω → {\displaystyle {\begin{aligned}\left({\vec {U}}\cdot {\vec {\nabla }}\right)\left(\psi _{1}{\hat {\varphi }}\right)&=U\left(\cos \theta {\frac {\partial \psi _{1}}{\partial r}}-{\frac {1}{r}}\sin \theta {\frac {\partial \psi _{1}}{\partial \theta }}\right){\hat {\varphi }}={\frac {3U^{2}a^{3}}{4r^{3}}}\sin \theta \cos \theta {\hat {\varphi }}\\\left({\vec {U}}\cdot {\vec {\nabla }}\right)\left(\psi _{2}{\hat {\varphi }}\right)&=U\left(\cos \theta {\frac {\partial \psi _{2}}{\partial r}}-{\frac {1}{r}}\sin \theta {\frac {\partial \psi _{2}}{\partial \theta }}\right){\hat {\varphi }}=-U{\frac {1}{r}}(1+\cos \theta )\psi _{2}{\hat {\varphi }}=-{\frac {3U^{2}a^{2}}{Rr^{2}}}\sin \theta {\hat {\varphi }}\\\left({\vec {U}}\cdot {\vec {\nabla }}\right)\left(-\psi _{2}e^{-kr(1+\cos \theta )}{\hat {\varphi }}\right)&=-e^{-kr(1+\cos \theta )}\left(\left({\vec {U}}\cdot {\vec {\nabla }}\right)\left(\psi _{2}{\hat {\varphi }})+\psi _{2}\left({\vec {U}}\cdot {\vec {\nabla }}\right)\left(-kr(1+\cos \theta \right){\hat {\varphi }}\right)\right)\\&=U\psi _{2}e^{-kr(1+\cos \theta )}\left({\frac {1}{r}}(1+\cos \theta )+\cos \theta {\frac {\partial (kr(1+\cos \theta ))}{\partial r}}-{\frac {1}{r}}\sin \theta {\frac {\partial (kr(1+\cos \theta ))}{\partial \theta }}\right){\hat {\varphi }}\\&=U\psi _{2}(1+\cos \theta )\left({\frac {1}{r}}+k\right)e^{-kr(1+\cos \theta )}{\hat {\varphi }}={\frac {3U^{2}a^{2}}{R}}\sin \theta \left({\frac {1}{r^{2}}}+{\frac {k}{r}}\right)e^{-kr(1+\cos \theta )}{\hat {\varphi }}\\&={\frac {U}{2k}}{\vec {\omega }}=\nu {\vec {\omega }}\end{aligned}}}
Combining all the terms we have: ( U → ⋅ ∇ → ) ψ → + ν ∇ 2 ψ → = ( 3 U 2 a 3 4 r 3 sin θ cos θ − 3 U 2 a 2 R r 2 sin θ ) φ ^ {\displaystyle \left({\vec {U}}\cdot {\vec {\nabla }}\right){\vec {\psi }}+\nu \nabla ^{2}{\vec {\psi }}=\left({\frac {3U^{2}a^{3}}{4r^{3}}}\sin \theta \cos \theta -{\frac {3U^{2}a^{2}}{Rr^{2}}}\sin \theta \right){\hat {\varphi }}}
Taking the curl, we find an expression that is equal to 1 / ρ {\displaystyle 1/\rho } times the gradient of the following function, which is the pressure: p = p 0 − 3 μ U a 2 r 2 cos θ + ρ U 2 a 3 4 r 3 ( 3 cos 2 θ − 1 ) {\displaystyle p=p_{0}-{\frac {3\mu Ua}{2r^{2}}}\cos \theta +{\frac {\rho U^{2}a^{3}}{4r^{3}}}\left(3\cos ^{2}\theta -1\right)}
where p 0 {\displaystyle p_{0}} is the pressure at infinity, θ {\displaystyle \theta } .is the polar angle originated from the opposite side of the front stagnation point ( θ = π {\displaystyle \theta =\pi } where is the front stagnation point).
Also, the velocity is derived by taking the curl of ψ → {\displaystyle {\vec {\psi }}} : u → = U [ − a 3 2 r 3 cos θ + 3 a 2 R r 2 − 3 a 2 R ( 1 r 2 + k r [ 1 − cos θ ] ) e − k r ( 1 + cos θ ) ] r ^ − U [ a 3 4 r 3 sin θ + 3 a 2 R r k sin θ e − k r ( 1 + cos θ ) ] θ ^ {\displaystyle {\vec {u}}=U\left[-{\frac {a^{3}}{2r^{3}}}\cos \theta +{\frac {3a^{2}}{Rr^{2}}}-{\frac {3a^{2}}{R}}\left({\frac {1}{r^{2}}}+{\frac {k}{r}}[1-\cos \theta ]\right)e^{-kr(1+\cos \theta )}\right]{\hat {r}}-U\left[{\frac {a^{3}}{4r^{3}}}\sin \theta +{\frac {3a^{2}}{Rr}}k\sin \theta e^{-kr(1+\cos \theta )}\right]{\hat {\theta }}}
These p and u satisfy the equation of motion and thus constitute the solution to Oseen's approximation.
One may question, however, whether the correction term was chosen by chance, because in a frame of reference moving with the sphere, the fluid near the sphere is almost at rest, and in that region inertial force is negligible and Stokes' equation is well justified. [ 6 ] Far away from the sphere, the flow velocity approaches u and Oseen's approximation is more accurate. [ 6 ] But Oseen's equation was obtained applying the equation for the entire flow field. This question was answered by Proudman and Pearson in 1957, [ 8 ] who solved the Navier-Stokes equations and gave an improved Stokes' solution in the neighborhood of the sphere and an improved Oseen's solution at infinity, and matched the two solutions in a supposed common region of their validity. They obtained:
F = 6 π μ a U ( 1 + 3 8 Re + 9 40 Re 2 ln Re + O ( Re 2 ) ) . {\displaystyle F=6\pi \,\mu \,aU\left(1+{3 \over 8}\operatorname {Re} +{9 \over 40}\operatorname {Re} ^{2}\ln \operatorname {Re} +{\mathcal {O}}\left(\operatorname {Re} ^{2}\right)\right).}
The method and formulation for analysis of flow at a very low Reynolds number is important. The slow motion of small particles in a fluid is common in bio-engineering . Oseen's drag formulation can be used in connection with flow of fluids under various special conditions, such as: containing particles, sedimentation of particles, centrifugation or ultracentrifugation of suspensions, colloids, and blood through isolation of tumors and antigens. [ 6 ] The fluid does not even have to be a liquid, and the particles do not need to be solid. It can be used in a number of applications, such as smog formation and atomization of liquids.
Blood flow in small vessels, such as capillaries , is characterized by small Reynolds and Womersley numbers . A vessel of diameter of 10 µm with a flow of 1 millimetre/second , viscosity of 0.02 poise for blood, density of 1 g/cm 3 and a heart rate of 2 Hz , will have a Reynolds number of 0.005 and a Womersley number of 0.0126. At these small Reynolds and Womersley numbers, the viscous effects of the fluid become predominant. Understanding the movement of these particles is essential for drug delivery and studying metastasis movements of cancers. | https://en.wikipedia.org/wiki/Oseen_equations |
Sea sorrel or in French oseille de mer can refer to different seaweeds: | https://en.wikipedia.org/wiki/Oseille_de_mer |
In mathematics , the multiplicative ergodic theorem , or Oseledets theorem provides the theoretical background for computation of Lyapunov exponents of a nonlinear dynamical system . It was proved by Valery Oseledets (also spelled "Oseledec") in 1965 and reported at the International Mathematical Congress in Moscow in 1966. A conceptually different proof of the multiplicative ergodic theorem was found by M. S. Raghunathan . [ citation needed ] [ 1 ] The theorem has been extended to semisimple Lie groups by V. A. Kaimanovich and further generalized in the works of David Ruelle , Grigory Margulis , Anders Karlsson , and François Ledrappier . [ citation needed ]
The multiplicative ergodic theorem is stated in terms of matrix cocycles of a dynamical system. The theorem states conditions for the existence of the defining limits and describes the Lyapunov exponents. It does not address the rate of convergence.
A cocycle of an autonomous dynamical system X is a map C : X×T → R n×n satisfying
where X and T (with T = Z⁺ or T = R⁺ ) are the phase space
and the time range, respectively, of the dynamical system,
and I n is the n -dimensional unit matrix.
The dimension n of the matrices C is not related to the phase space X .
Let μ be an ergodic invariant measure on X and C a cocycle
of the dynamical system such that for each t ∈ T , the maps x → log ‖ C ( x , t ) ‖ {\displaystyle x\rightarrow \log \|C(x,t)\|} and x → log ‖ C ( x , t ) − 1 ‖ {\displaystyle x\rightarrow \log \|C(x,t)^{-1}\|} are L 1 -integrable with respect to μ . Then for μ -almost all x and each non-zero vector u ∈ R n the limit
exists and assumes, depending on u but not on x , up to n different values.
These are the Lyapunov exponents.
Further, if λ 1 > ... > λ m are the different limits then there are subspaces R n = R 1 ⊃ ... ⊃ R m ⊃ R m +1 = {0}, depending on x , such that the limit is λ i for u ∈ R i \ R i +1 and i = 1, ..., m .
The values of the Lyapunov exponents are invariant with respect to a wide range of coordinate transformations. Suppose that g : X → X is a one-to-one map such that ∂ g / ∂ x {\displaystyle \partial g/\partial x} and its inverse exist; then the values of the Lyapunov exponents do not change.
Verbally, ergodicity means that time and space averages are equal, formally:
where the integrals and the limit exist.
Space average (right hand side, μ is an ergodic measure on X )
is the accumulation of f ( x ) values weighted by μ( dx ).
Since addition is commutative, the accumulation of the f ( x )μ( dx ) values may be done in arbitrary order.
In contrast, the time average (left hand side) suggests a specific ordering
of the f ( x ( s )) values along the trajectory.
Since matrix multiplication is, in general, not commutative,
accumulation of multiplied cocycle values (and limits thereof) according to C ( x ( t 0 ), t k ) = C ( x ( t k −1 ), t k − t k −1 ) ... C ( x ( t 0 ), t 1 − t 0 )
— for t k large and
the steps t i − t i −1 small — makes sense only for a prescribed ordering. Thus, the time average may exist (and the theorem states that it actually exists), but there is no space average counterpart. In other words, the Oseledets theorem differs from additive ergodic theorems (such as G. D. Birkhoff 's and J. von Neumann 's) in that it guarantees the existence of the time average, but makes no claim about the space average. | https://en.wikipedia.org/wiki/Oseledets_theorem |
Oseltamivir total synthesis concerns the total synthesis of the anti-influenza drug oseltamivir [ 1 ] marketed by Hoffmann-La Roche under the trade name Tamiflu . Its commercial production starts from the biomolecule shikimic acid harvested from Chinese star anise and from recombinant E. coli . [ 2 ] [ 3 ] Control of stereochemistry is important: the molecule has three stereocenters and the sought-after isomer is only 1 of 8 stereoisomers. [ citation needed ]
The current production method is based on the first scalable synthesis developed by Gilead Sciences [ 4 ] starting from naturally occurring quinic acid or shikimic acid . Due to lower yields and the extra steps required (because of the additional dehydration), the quinic acid route was dropped in favour of the one based on shikimic acid, which received further improvements by Hoffmann-La Roche . [ 5 ] [ 6 ] The current industrial synthesis is summarised below:
The current production method includes two reaction steps with potentially hazardous azides . A reported azide-free Roche synthesis of tamiflu is summarised graphically below: [ 7 ]
The synthesis commences from naturally available (−)- shikimic acid . The 3,4-pentylidene acetal mesylate is prepared in three steps: esterification with ethanol and thionyl chloride ; ketalization with p -toluenesulfonic acid and 3-pentanone ; and mesylation with triethylamine and methanesulfonyl chloride . Reductive opening of the ketal under modified Hunter conditions [ 8 ] in dichloromethane yields an inseparable mixture of isomeric mesylates. The corresponding epoxide is formed under basic conditions with potassium bicarbonate . Using the inexpensive Lewis acid magnesium bromide diethyl etherate (commonly prepared fresh by the addition of magnesium turnings to 1,2-dibromoethane in benzene : diethyl ether ), the epoxide is opened with allyl amine to yield the corresponding 1,2-amino alcohol. The water-immiscible solvents methyl tert-butyl ether and acetonitrile are used to simplify the workup procedure, which involved stirring with 1 M aqueous ammonium sulfate . Reduction on palladium , promoted by ethanolamine , followed by acidic workup yielded the deprotected 1,2-aminoalcohol. The aminoalcohol was converted directly to the corresponding allyl-diamine in an interesting cascade sequence that commences with the unselective imination of benzaldehyde with azeotropic water removal in methyl tert-butyl ether. Mesylation, followed by removal of the solid byproduct triethylamine hydrochloride , results in an intermediate that was poised to undergo aziridination upon transimination with another equivalent of allylamine. With the librated methanesulfonic acid , the aziridine opens cleanly to yield a diamine that immediately undergoes a second transimination. Acidic hydrolysis then removed the imine . Selective acylation with acetic anhydride (under buffered conditions, the 5-amino group is protonated owing to a considerable difference in p K a , 4.2 vs 7.9, preventing acetylation ) yields the desired N -acetylated product in crystalline form upon extractive workup. Finally, deallylation as above, yielded the freebase of oseltamivir, which was converted to the desired oseltamivir phosphate by treatment with phosphoric acid . The final product is obtained in high purity (99.7%) and an overall yield of 17-22% from (−)-shikimic acid. It is noted that the synthesis avoids the use of potentially explosive azide reagents and intermediates; however, the synthesis actually used by Roche uses azides. Roche has other routes to
oseltamivir that do not involve the use of (−)-shikimic acid as a chiral pool starting material, such as a Diels-Alder route involving furan and ethyl acrylate or an isophthalic acid route, which involves catalytic hydrogenation and enzymatic desymmetrization. [ citation needed ]
In 2006 the group of E.J. Corey published a novel route bypassing shikimic acid starting from butadiene and acrylic acid . [ 9 ] The inventors chose not to patent this procedure which is described below.
Butadiene 1 reacts in an asymmetric Diels-Alder reaction with the esterification product of acrylic acid and 2,2,2-trifluoroethanol 2 catalysed by the CBS catalyst . The ester 3 is converted into an amide in 4 by reaction with ammonia and the next step to lactam 5 is an iodolactamization with iodine initiated by trimethylsilyl triflate . The amide group is fitted with a BOC protecting group by reaction with Boc anhydride in 6 and the iodine substituent is removed in an elimination reaction with DBU to the alkene 7 . Bromine is introduced in 8 by an allylic bromination with NBS and the amide group is cleaved with ethanol and caesium carbonate accompanied by elimination of bromide to the diene ethyl ester 9 . The newly formed double bond is functionalized with N -bromoacetamide 10 catalyzed with tin(IV) bromide with complete control of stereochemistry . In the next step the bromine atom in 11 is displaced by the nitrogen atom in the amide group with the strong base KHMDS to the aziridine 12 which in turn is opened by reaction with 3-pentanol 13 to the ether 14 . In the final step the BOC group is removed with phosphoric acid and the oseltamivir phosphate 15 is formed.
Also in 2006 the group of Masakatsu Shibasaki of the University of Tokyo published a synthesis again bypassing shikimic acid. [ 10 ] [ 11 ]
An improved method published in 2007 starts with the enantioselective desymmetrization of aziridine 1 with trimethylsilyl azide (TMSN 3 ) and a chiral catalyst to the azide 2 . The amide group is protected as a BOC group with Boc anhydride and DMAP in 3 and iodolactamization with iodine and potassium carbonate first gives the unstable intermediate 4 and then stable cyclic carbamate 5 after elimination of hydrogen iodide with DBU .
The amide group is reprotected as BOC 6 and the azide group converted to the amide 7 by reductive acylation with thioacetic acid and 2,6-lutidine . Caesium carbonate accomplishes the hydrolysis of the carbamate group to the alcohol 8 which is subsequently oxidized to ketone 9 with Dess-Martin periodinane . Cyanophosphorylation with diethyl phosphorocyanidate (DEPC) modifies the ketone group to the cyanophosphate 10 paving the way for an intramolecular allylic rearrangement to unstable β-allyl phosphate 11 (toluene, sealed tube) which is hydrolyzed to alcohol 12 with ammonium chloride . This hydroxyl group has the wrong stereochemistry and is therefore inverted in a Mitsunobu reaction with p-nitrobenzoic acid followed by hydrolysis of the p-nitrobenzoate to 13 .
A second Mitsunobu reaction then forms the aziridine 14 available for ring-opening reaction with 3-pentanol catalyzed by boron trifluoride to ether 15 . In the final step the BOC group is removed (HCl) and phosphoric acid added to objective 16 .
An approach published in 2007 [ 12 ] like Corey's starts by an asymmetric Diels-Alder reaction this time with starting materials pyridine and acrolein .
Pyridine ( 1 ) is reduced with sodium borohydride in presence of benzyl chloroformate to the Cbz protected dihydropyridine 2 . The asymmetric Diels-Alder reaction with acrolein 3 is carried out with the McMillan catalyst to the aldehyde 4 as the endo isomer which is oxidized to the carboxylic acid 5 with sodium chlorite , monopotassium phosphate and 2-methyl-2-butene . Addition of bromine gives halolactonization product 6 and after replacement of the Cbz protective group by a BOC protective group in 7 ( hydrogenolysis in the presence of di- tert -butyl dicarbonate ) a carbonyl group is introduced in intermediate 8 by catalytic ruthenium(IV) oxide and sacrificial catalyst sodium periodate . Addition of ammonia cleaves the ester group to form amide 9 the alcohol group of which is mesylated to compound 10 . In the next step iodobenzene diacetate is added, converting the amide in a Hofmann rearrangement to the allyl carbamate 12 after capturing the intermediate isocyanate with allyl alcohol 11 . On addition of sodium ethoxide in ethanol three reactions take place simultaneously: cleavage of the amide to form new an ethyl ester group, displacement of the mesyl group by newly formed BOC protected amine to an aziridine group and an elimination reaction forming the alkene group in 13 with liberation of HBr. In the final two steps the aziridine ring is opened by 3-pentanol 14 and boron trifluoride to aminoether 15 with the BOC group replaced by an acyl group and on removal of the other amine protecting group ( Pd/C , Ph 3 P , and 1,3-dimethylbarbituric acid in ethanol) and addition of phosphoric acid oseltamivir 16 is obtained.
In 2008 the group of Barry M. Trost of Stanford University published the shortest synthetic route to date. [ 13 ]
In 2009, Hayashi et al. successfully produced an efficient, low cost synthetic route to prepare (-)-oseltamivir ( 1 ). Their goal was to design a procedure that would be suitable for large-scale production. Keeping cost, yield, and number of synthetic steps in mind, an enantioselective total synthesis of ( 1 ) was accomplished through three one-pot operations. [ 14 ] [ 5 ] Hayashi et al.'s use of one-pot operations allowed them to perform several reactions steps in a single pot, which ultimately minimized the number of purification steps needed, waste, and saved time.
In the first one-pot operation , Hayashi et al. begins by using diphenylprolinol silyl ether ( 4 ) [ 6 ] as an organocatalyst , along with alkoxyaldehyde ( 2 ) and nitroalkene ( 3 ) to perform an asymmetric Michael reaction , affording an enantioselective Michael adduct . Upon addition of a diethyl vinylphosphate derivative ( 5 ) to the Michael adduct , a domino Michael reaction and Horner-Wadsworth-Emmons reaction occurs due to the phosphonate group produced from ( 5 ) to give an ethyl cyclohexenecarboxylate derivative along with two unwanted by-products. To transform the undesired by-products into the desired ethyl cyclohexencarboxylate derivative, the mixture of the product and by-products was treated with Cs 2 CO 3 in ethanol. This induced a retro-Michael reaction on one by-product and a retro- aldol reaction accompanied with a Horner-Wadsworth-Emmons reaction for the other. Both by-products were successfully converted to the desired derivative. Finally, the addition of p -toluenethiol with Cs 2 CO 3 gives ( 6 ) in a 70% yield after being purified by column chromatography , with the desired isomer dominating. [ 14 ]
In the second one-pot operation , trifluoroacetic acid is employed first to deprotect the tert -butyl ester of ( 6 ); any excess reagent was removed via evaporation. The carboxylic acid produced as a result of the deprotection was then converted to an acyl chloride by oxalyl chloride and a catalytic amount of DMF . Finally, addition of sodium azide, in the last reaction of the second one-pot operation, produce the acyl azide ( 7 ) without any purification needed. [ 14 ]
The final one-pot operation begins with a Curtius Rearrangement of acyl azide ( 7 ) to produce an isocyanate functional group at room temperature. The isocyanate derivative then reacts with acetic acid to yield the desired acetylamino moiety found in ( 1 ). This domino Curtius rearrangement and amide formation occurs in the absence of heat, which is extremely beneficial for reducing any possible hazard. The nitro moiety of ( 7 ) is reduced to the desired amine observed in ( 1 ) with Zn/HCl. Due to the harsh conditions of the nitro reduction, ammonia was used to neutralize the reaction. Potassium carbonate was then added to give ( 1 ), via a retro-Michael reaction of the thiol . ( 1 ) was then purified by an acid/base extraction. The overall yield for the total synthesis of (-)-oseltamivir is 57%. [ 14 ] Hayashi et al. use of inexpensive, non-hazardous reagents has allowed for an efficient, high yielding synthetic route that can allow for vast amount of novel derivatives to be produced in hopes of combatting against viruses resistant to (-)-oseltamivir. | https://en.wikipedia.org/wiki/Oseltamivir_total_synthesis |
Osipkov–Merritt models (named for Leonid Osipkov and David Merritt ) are mathematical representations of spherical stellar systems ( galaxies , star clusters , globular clusters etc.). The Osipkov–Merritt formula generates a one-parameter family of phase-space distribution functions that reproduce a specified density profile (representing stars) in a specified gravitational potential (in which the stars move). The density and potential need not be self-consistently related.
A free parameter adjusts the degree of velocity anisotropy, from isotropic to completely radial motions. The method is a generalization of Eddington's formula [ 1 ] for constructing isotropic spherical models.
The method was derived independently by its two eponymous discoverers. [ 2 ] [ 3 ] The latter derivation includes two additional families of models (Type IIa, b) with tangentially anisotropic motions.
According to Jeans's theorem , the phase-space density of stars f must be expressible in terms of the isolating integrals of motion , which in a spherical stellar system are the energy E and the angular momentum J . The Osipkov-Merritt ansatz is
where r a , the "anisotropy radius", is a free parameter. This ansatz implies that f is constant on spheroids in velocity space since
where v r , v t are velocity components parallel and perpendicular to the radius vector r and Φ( r ) is the gravitational potential .
The density ρ is the integral over velocities of f :
which can be written
or
This equation has the form of an Abel integral equation and can be inverted to give f in terms of ρ :
Following a derivation similar to the one above, the velocity dispersions in an Osipkov–Merritt model satisfy
The motions are nearly radial ( σ r ≫ σ t {\displaystyle \sigma _{r}\gg \sigma _{t}} ) for r ≫ r a {\displaystyle r\gg r_{a}} and nearly isotropic ( σ r ≈ σ t {\displaystyle \sigma _{r}\approx \sigma _{t}} ) for r ≪ r a {\displaystyle r\ll r_{a}} . This is a desirable feature, since stellar systems that form via gravitational collapse have isotropic cores and radially-anisotropic envelopes. [ 4 ]
If r a is assigned too small a value, f may be negative for some Q . This is a consequence of the fact that spherical mass models can not always be reproduced by purely radial orbits. Since the number of stars on an orbit can not be negative, values of r a that generate negative f' s are unphysical. This result can be used to constrain the maximum degree of anisotropy of spherical galaxy models. [ 3 ]
In his 1985 paper, Merritt defined two additional families of models ("Type II") that have isotropic cores and tangentially anisotropic envelopes. Both families assume
In Type IIa models, the orbits become completely circular at r=r a and remain so at all larger radii.
In Type IIb models, stars beyond r a move on orbits of various eccentricities, although the motion is always biased toward circular. In both families, the tangential velocity dispersion undergoes a jump as r increases past r a .
C. M. Carollo et al. (1995) [ 5 ] derive many observable properties of Type I Osipkov–Merritt models.
Typical applications of Osipkov–Merritt models include: | https://en.wikipedia.org/wiki/Osipkov–Merritt_model |
Osman Kibar is a Turkish American billionaire. He is the founder of the biotechnology firm Biosplice . [ 1 ] [ 2 ] [ 3 ]
Kibar was born in İzmir , Turkey , as the grandson of Osman Kibar [ tr ] , who was the mayor of İzmir from 1964 to 1973. [ 4 ] His father, Seli Kibar, was an economist. [ 5 ]
He attended İzmir Gazipaşa Primary School [ 6 ] and then Robert College . For college, he moved to the United States to attend a 3-2 program , studying economics at Pomona College [ 7 ] and electrical engineering at Caltech , and receiving a bachelor's degree from both. He did his master's degree and doctorate on biophotonics at UC San Diego . [ citation needed ]
In the late 1990s, when he was finishing his doctorate, he invented a cancer diagnosis system, which he turned into a company called Genoptix. The company went public a few years later, and was purchased by Novartis in 2011 for $476 million. [ 8 ]
After completing his education, Kibar moved to New York City , where he worked for the hedge fund sponsor Pequot Capital . [ 9 ]
He subsequently moved back to San Diego and in August 2011 founded a biotechnology company named Wintherix with an investment of $3.5 million from his friend Cevdet Samikoğlu. [ 6 ] He later renamed the company Samumed, and subsequently Biosplice . [ 4 ] Its investors include Ali Sabancı and Ergun Özen. The company researches cures for articular cartilage damage , hair loss , degenerative disc disease , lung tissue regeneration , cancer , psoriasis , damaged tendons , and Alzheimer's disease . [ 4 ]
As of November 2020 [update] , Kibar has a net worth of $2.9 billion. [ 10 ] He ranked eighth on Forbes Turkey ' s 2017 list of the 100 richest Turks. [ 11 ]
Kibar is married and has four children. [ 1 ] [ 6 ]
Despite his wealth, he spends money extremely frugally, saying that all the activities he enjoys, such as reading, playing go , and moviegoing, are free. [ 6 ] Kibar is also a successful poker player, and won the first tournament he ever entered in 2006. [ 12 ] | https://en.wikipedia.org/wiki/Osman_Kibar |
Osmium heptafluoride is a possible inorganic chemical compound of osmium metal and fluorine with the chemical formula OsF 7 . [ 2 ] [ 3 ] It was first reported in 1966 by the reaction of fluorine and osmium at 600 °C and 400 atm, [ 4 ] but no purported synthesis could be reproduced in 2006, giving only osmium hexafluoride instead. [ 5 ]
If it exists, osmium(VII) fluoride is supposedly a bluish-yellow hygroscopic substance, extremely unstable. [ 6 ] The compound starts decomposing at –100 °C. It should be stored in a nickel vessel at the temperature of liquid nitrogen .
Osmium heptafluoride decomposes to osmium hexafluoride when slightly heated:
This inorganic compound –related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Osmium_heptafluoride |
Osmium octafluoride is an inorganic chemical compound of osmium metal and fluorine with the chemical formula Os F 8 . [ 3 ] [ 4 ] Some sources consider it to be a still hypothetical compound . [ 5 ] An early report of the synthesis of OsF 8 was much later shown to be a mistaken identification of OsF 6 . [ 6 ] Theoretical analysis indicates OsF 8 would have an approximately square antiprismatic molecular geometry . [ 7 ]
Rapid cooling of fluorine and osmium reaction products: [ 8 ]
This inorganic compound –related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Osmium_octafluoride |
Osmium tetroxide (also osmium(VIII) oxide ) is the chemical compound with the formula OsO 4 . The compound is noteworthy for its many uses, despite its toxicity and the rarity of osmium . It also has a number of unusual properties, one being that the solid is volatile . The compound is colourless, but most samples appear yellow. [ 6 ] This is most likely due to the presence of the impurity OsO 2 , which is yellow-brown in colour. [ 7 ] In biology, its property of binding to lipids has made it a widely used stain in electron microscopy.
Osmium(VIII) oxide forms monoclinic crystals. [ 4 ] [ 8 ] It has a characteristic acrid chlorine -like odor. The element name osmium is derived from osme , Greek for odor . OsO 4 is volatile: it sublimes at room temperature . It is soluble in a wide range of organic solvents. It is moderately soluble in water, with which it reacts reversibly to form osmic acid (see below). [ 9 ] Pure osmium(VIII) oxide is probably colourless; [ 10 ] it has been suggested that its yellow hue is attributable due to osmium dioxide (OsO 2 ) impurities. [ 11 ] The osmium tetroxide molecule is tetrahedral and therefore nonpolar. This nonpolarity helps OsO 4 penetrate charged cell membranes.
The osmium of OsO 4 has an oxidation number of VIII; however, the metal does not possess a corresponding 8+ charge as the bonding in the compound is largely covalent in character (the ionization energy required to produce a formal 8+ charge also far exceeds the energies available in normal chemical reactions). The osmium atom exhibits double bonds to the four oxide ligands , resulting in a 16 electron complex . OsO 4 is isoelectronic with permanganate and chromate ions.
OsO 4 is formed slowly when osmium powder reacts with O 2 at ambient temperature. Reaction of bulk solid requires heating to 400 °C. [ 12 ]
Alkenes add to OsO 4 to give diolate species that hydrolyze to cis -diols. The net process is called dihydroxylation. This proceeds via a [3 + 2] cycloaddition reaction between the OsO 4 and alkene to form an intermediate osmate ester that rapidly hydrolyses to yield the vicinal diol . As the oxygen atoms are added in a concerted step, the resulting stereochemistry is cis .
OsO 4 is expensive and highly toxic, making it an unappealing reagent to use in stoichiometric amounts. However, its reactions are made catalytic by adding reoxidants to reoxidise the Os(VI) by-product back to Os(VIII). Typical reagents include H 2 O 2 ( Milas hydroxylation ), N-methylmorpholine N-oxide ( Upjohn dihydroxylation ) and K 3 Fe(CN) 6 /water. These reoxidants do not react with the alkenes on their own. Other osmium compounds can be used as catalysts, including osmate(VI) salts ([OsO 2 (OH) 4 )] 2− , and osmium trichloride hydrate (OsCl 3 · x H 2 O). These species oxidise to osmium(VIII) in the presence of such oxidants. [ 13 ]
Lewis bases such as tertiary amines and pyridines increase the rate of dihydroxylation. This "ligand-acceleration" arises via the formation of adduct OsO 4 L, which adds more rapidly to the alkene. If the amine is chiral, then the dihydroxylation can proceed with enantioselectivity (see Sharpless asymmetric dihydroxylation ). [ 14 ] OsO 4 does not react with most carbohydrates. [ 15 ]
The process can be extended to give two aldehydes in the Lemieux–Johnson oxidation , which uses periodate to achieve diol cleavage and to regenerate the catalytic loading of OsO 4 . This process is equivalent to that of ozonolysis .
OsO 4 is a Lewis acid and a mild oxidant. It reacts with alkaline aqueous solution to give the perosmate anion OsO 4 (OH) 2− 2 . [ 17 ] This species is easily reduced to osmate anion, OsO 2 (OH) 2− 4 .
When the Lewis base is an amine , adducts are also formed. Thus OsO 4 can be stored in the form of osmeth , in which OsO 4 is complexed with hexamine . Osmeth can be dissolved in tetrahydrofuran (THF) and diluted in an aqueous buffer solution to make a dilute (0.25%) working solution of OsO 4 . [ 18 ]
With tert-BuNH 2 , the imido derivative is produced:
Similarly, with NH 3 one obtains the nitrido complex :
The [Os(N)O 3 ] − anion is isoelectronic and isostructural with OsO 4 .
OsO 4 is very soluble in tert-butyl alcohol . In solution, it is readily reduced by hydrogen to osmium metal. The suspended osmium metal can be used to catalytically hydrogenate a wide variety of organic chemicals containing double or triple bonds.
OsO 4 undergoes "reductive carbonylation" with carbon monoxide in methanol at 400 K and 200 sbar to produce the triangular cluster Os 3 (CO) 12 :
Osmium forms several oxofluorides, all of which are very sensitive to moisture.
Purple cis -OsO 2 F 4 forms at 77 K in an anhydrous HF solution: [ 19 ]
OsO 4 also reacts with F 2 to form yellow OsO 3 F 2 : [ 20 ]
OsO 4 reacts with one equivalent of [Me 4 N]F at 298 K and 2 equivalents at 253 K: [ 12 ]
In organic synthesis OsO 4 is widely used to oxidize alkenes to the vicinal diols, adding two hydroxyl groups at the same side ( syn addition ). See reaction and mechanism above. This reaction has been made both catalytic ( Upjohn dihydroxylation ) and asymmetric ( Sharpless asymmetric dihydroxylation ).
Osmium(VIII) oxide is also used in catalytic amounts in the Sharpless oxyamination to give vicinal amino-alcohols.
In combination with sodium periodate , OsO 4 is used for the oxidative cleavage of alkenes ( Lemieux-Johnson oxidation ) when the periodate serves both to cleave the diol formed by dihydroxylation, and to reoxidize the OsO 3 back to OsO 4 . The net transformation is identical to that produced by ozonolysis . Below an example from the total synthesis of Isosteviol. [ 21 ]
OsO 4 is a widely used staining agent used in transmission electron microscopy (TEM) to provide contrast to the image. [ 22 ] This staining method may also be known in the literature as the OTO [ 23 ] [ 24 ] (osmium-thiocarbohydrazide-osmium) method, or osmium impregnation [ 25 ] technique or simply as osmium staining. As a lipid stain, it is also useful in scanning electron microscopy (SEM) as an alternative to sputter coating . It embeds a heavy metal directly into cell membranes, creating a high electron scattering rate without the need for coating the membrane with a layer of metal, which can obscure details of the cell membrane. In the staining of the plasma membrane , osmium(VIII) oxide binds phospholipid head regions, thus creating contrast with the neighbouring protoplasm (cytoplasm). Additionally, osmium(VIII) oxide is also used for fixing biological samples in conjunction with HgCl 2 . Its rapid killing abilities are used to quickly kill live specimens such as protozoa. OsO 4 stabilizes many proteins by transforming them into gels without destroying structural features. Tissue proteins that are stabilized by OsO 4 are not coagulated by alcohols during dehydration. [ 15 ] Osmium(VIII) oxide is also used as a stain for lipids in optical microscopy. [ 26 ] OsO 4 also stains the human cornea (see safety considerations ).
It is also used to stain copolymers preferentially, the best known example being block copolymers where one phase can be stained so as to show the microstructure of the material. For example, styrene-butadiene block copolymers have a central polybutadiene chain with polystyrene end caps. When treated with OsO 4 , the butadiene matrix reacts preferentially and so absorbs the oxide. The presence of a heavy metal is sufficient to block the electron beam, so the polystyrene domains are seen clearly in thin films in TEM .
OsO 4 is an intermediate in the extraction of osmium from its ores. Osmium-containing residues are treated with sodium peroxide (Na 2 O 2 ) forming Na 2 [OsO 4 (OH) 2 ], which is soluble. When exposed to chlorine , this salt gives OsO 4 . In the final stages of refining, crude OsO 4 is dissolved in alcoholic NaOH forming Na 2 [OsO 2 (OH) 4 ], which, when treated with NH 4 Cl , to give (NH 4 ) 4 [OsO 2 Cl 2 ]. This salt is reduced under hydrogen to give osmium. [ 9 ]
OsO 4 allowed for the confirmation of the soccer ball model of buckminsterfullerene , a 60-atom carbon allotrope . The adduct , formed from a derivative of OsO 4 , was C 60 (OsO 4 )(4- tert - butyl pyridine ) 2 . The adduct broke the fullerene's symmetry, allowing for crystallization and confirmation of the structure of C 60 by X-ray crystallography . [ 27 ]
The only known clinical use of osmium tetroxide is for the treatment of arthritis. [ 28 ] The lack of reports of long-term side effects from the local administration of osmium tetroxide (OsO 4 ) suggest that osmium itself can be biocompatible , though this depends on the osmium compound administered.
OsO 4 will irreversibly stain the human cornea , which can lead to blindness. The permissible exposure limit for osmium(VIII) oxide (8 hour time-weighted average) is 2 μg/m 3 . [ 8 ] Osmium(VIII) oxide can penetrate plastics and food packaging, and therefore must be stored in glass under refrigeration. [ 15 ] | https://en.wikipedia.org/wiki/Osmium_tetroxide |
Osmoconformers are marine organisms that maintain an internal environment which is isotonic to their external environment. [ 1 ] This means that the osmotic pressure of the organism's cells is equal to the osmotic pressure of their surrounding environment. By minimizing the osmotic gradient, this subsequently minimizes the net influx and efflux of water into and out of cells. Even though osmoconformers have an internal environment that is isosmotic to their external environment, the types of ions in the two environments differ greatly in order to allow critical biological functions to occur. [ 2 ]
An advantage of osmoconformation is that such organisms don’t need to expend as much energy as osmoregulators in order to regulate ion gradients . However, to ensure that the correct types of ions are in the desired location, a small amount of energy is expended on ion transport . A disadvantage to osmoconformation is that the organisms are subject to changes in the osmolarity of their environment. [ 3 ]
Most osmoconformers are marine invertebrates such as echinoderms (such as starfish), mussels , marine crabs, lobsters , jellyfish , ascidians ( sea squirts - primitive chordates), and scallops . Some insects are also osmoconformers. [ 3 ] Some osmoconformers, such as echinoderms, are stenohaline , which means they can only survive in a limited range of external osmolarities. The survival of such organisms is thus contingent on their external osmotic environment remaining relatively constant. [ 3 ] On the other hand, some osmoconformers are classified as euryhaline , which means they can survive in a broad range of external osmolarities. Mussels are a prime example of a euryhaline osmoconformer. Mussels have adapted to survive in a broad range of external salinities due to their ability to close their shells which allows them to seclude themselves from unfavorable external environments. [ 3 ]
There are a couple of examples of osmoconformers that are craniates such as hagfish , skates and sharks . Their body fluid is isosmotic with seawater, but their high osmolarity is maintained by making the concentration of organic solutes unnaturally high. Sharks concentrate urea in their bodies, and since urea denatures proteins at high concentrations, they also accumulate trimethylamine N-oxide (TMAO) to counter the effect. Sharks adjust their internal osmolarity according to the osmolarity of the sea water surrounding them. Rather than ingesting sea water in order to change their internal salinity, sharks are able to absorb sea water directly. This is due to the high concentration of urea kept inside their bodies. This high concentration of urea creates a diffusion gradient which permits the shark to absorb water in order to equalize the concentration difference. [ 4 ] The crab-eating frog , or Rana cancrivora, is an example of a vertebrate osmoconformer. The crab-eating frog also regulates its rates of urea retention and excretion, which allows them to survive and maintain their status as osmoconformers in a wide range of external salinities. [ 3 ] Hagfish maintain an internal ion composition plasma that differs from that of seawater. The internal ionic environment of hagfish contains a lower concentration of divalent ions (Ca2+, Mg2+, SO4 2-) and a slightly higher concentration of monovalent ions . [ 5 ] Hagfish therefore have to expend some energy for osmoregulation.
Ion gradients are crucial to many major biological functions on a cellular level. Consequently, the ionic composition of an organism's internal environment is highly regulated with respect to its external environment. Osmoconformers have adapted so that they utilize the ionic composition of their external environment, which is typically seawater, in order to support important biological functions. For instance, seawater has a high concentration of sodium ions , which helps support muscle contraction and neuronal signaling when paired with high internal concentrations of potassium ions . [ 3 ] | https://en.wikipedia.org/wiki/Osmoconformer |
An osmometer is a device for measuring the osmotic strength of a solution , colloid , or compound .
There are several different techniques employed in osmometry:
Osmometers are useful for determining the total concentration of dissolved salts and sugars in blood or urine samples. Osmometry is also useful in determining the molecular weight of unknown compounds and polymers.
Osmometry is the measurement of the osmotic strength of a substance. [ 2 ] This is often used by chemists for the determination of average molecular weight .
Osmometry is also useful for estimating the drought tolerance of plant leaves. [ 3 ] | https://en.wikipedia.org/wiki/Osmometer |
Osmond iron (also spelt osmund and also called osborn) was wrought iron made by a particular process. This is associated with the first European production of cast iron in furnaces such as Lapphyttan in Sweden . [ 1 ]
Osmonds appear in some of the earliest English Customs accounts, for example in 1325. [ 2 ] The kappe was a Swedish weight used for osmond which occurs in a commercial treaty in Novgorod in 1203, and this implies the production of osmond iron. [ 3 ]
Osmond iron was made by melting pig iron in a hearth that is narrower and deeper than a typical finery in an English finery forge . The hearth had a charcoal fire blown with bellows through a tuyere. As the iron melted, the drops fell through the blast and congealed. They were then lifted with an iron bar into the blast. As they melted they were caught on the end of a large staff, held in the fire and turned rapidly so that the drops spread out, forming a ball. [ 4 ]
Osmonds reached England during the later Middle Ages through the port of Gdańsk . However, there were hammer mills in its hinterland and that of Lübeck , which made the osmonds into bar iron . In the 1620s, Gustav II Adolf of Sweden prohibited his subjects from exporting unfinished iron, and all trade in osmonds ceased. [ 5 ]
The osmond process was also used in the county of Mark in Westphalia , in southern Germany and Switzerland . [ 6 ]
The process was introduced to Wales in connection with the establishment by William Humfrey and others of wireworks at Tintern in 1566, an enterprise that was shortly afterwards taken over by the Company of Mineral and Battery Works . [ 7 ] Humfrey arranged to bring an expert maker of Osmond iron, Corslett Tinkhaus, from southwest Westphalia, where the production had reached a high level of technical proficiency. Tinkhaus arrived in Wales in 1567 and began working at Rhydygwern in the Glamorgan part of the lordship of Machen . This was where the first Machen Forge was, and he was evidently making osmond iron there. The iron was apparently forged with a tilt hammer , rather than the helve hammer, usual in finery forges . This was the raw material for the wireworks at Tintern. Osmond iron was made at Pontypool in the 18th century to supply wireworks there, and one of the forges there was still called the 'Osborn Forge' in the 19th century. [ 8 ] | https://en.wikipedia.org/wiki/Osmond_process |
Osmoprotectants or compatible solutes are small organic molecules with neutral charge and low toxicity at high concentrations that act as osmolytes and help organisms to survive in extreme osmotic stress . [ 1 ] Osmoprotectants can be placed in three chemical classes: betaines and associated molecules, sugars and polyols , and amino acids. These molecules accumulate in cells and balance the osmotic difference between the cell's surroundings and the cytosol . [ 2 ] In plants, their accumulation can increase survival during stresses such as drought. In extreme cases, such as in bdelloid rotifers , tardigrades , brine shrimp , and nematodes , these molecules can allow cells to survive being completely dried out and let them enter a state of suspended animation called cryptobiosis . [ 3 ]
Intracellular osmoprotectant concentrations are regulated in response to environmental conditions such as osmolarity and temperature via regulation of specific transcription factors and transporters . They have been shown to play a protective role by maintaining enzyme activity through freeze-thaw cycles and at higher temperatures. It is currently believed that they function by stabilizing protein structures by promoting preferential exclusion from the water layers on the surface of hydrated proteins. This favors the native conformation and displaces inorganic salts that would otherwise cause misfolding. [ 4 ]
Compatible solutes have a functional role in agriculture. In high stress conditions, such as drought or high salinity, plants that naturally create or take up osmoprotectants show increased survival rates. By inducing expression or uptake of these molecules in crops in which they are naturally not present, there is an increase in the areas in which they are able to be grown. One documented reason for increased growth is regulation of toxic reactive oxygen species (ROS). In high salinity ROS production is stimulated by the photosystems of the plant. Osmoprotectants can prevent the photosystem-salt interactions, reducing ROS production. For these reasons, introduction of biosynthetic pathways which result in the creation of osmoprotectants in crops is a current area of research, but inducing expression at significant amounts is currently posing a barrier in this area of research. [ 5 ]
Osmoprotectants are also important for the maintenance of top soil bacteria populations. Desiccation of top soils results in increased salinity. In these situations, the soil microbes increase the concentration of these molecule in their cytoplasm into the molar range allowing them to persist until conditions approve. [ 2 ] In extreme cases, osmoprotectants allow cells to enter cryptobiosis. In this state the cytosol and osmoprotectants become a glass-like solid that helps stabilize proteins and cell membranes from the damaging effects of desiccation. [ 6 ]
Osmoprotectants regulate gene expression in response to environmental osmolarity as compatible solutes even in small concentrations affect gene expression, ranging from inducing production of more compatible solutes to regulating components involved in infection, such as phospholipase C in Pseudomonas aeruginosa . [ 7 ]
This cell biology article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Osmoprotectant |
Osmoregulation is the active regulation of the osmotic pressure of an organism 's body fluids , detected by osmoreceptors , to maintain the homeostasis of the organism's water content; that is, it maintains the fluid balance and the concentration of electrolytes ( salts in solution which in this case is represented by body fluid) to keep the body fluids from becoming too diluted or concentrated. Osmotic pressure is a measure of the tendency of water to move into one solution from another by osmosis . [ 1 ] The higher the osmotic pressure of a solution, the more water tends to move into it. Pressure must be exerted on the hypertonic side of a selectively permeable membrane to prevent diffusion of water by osmosis from the side containing pure water.
Although there may be hourly and daily variations in osmotic balance, an animal is generally in an osmotic steady state over the long term. Organisms in aquatic and terrestrial environments must maintain the right concentration of solutes and amount of water in their body fluids; this involves excretion (getting rid of metabolic nitrogen wastes and other substances such as hormones that would be toxic if allowed to accumulate in the blood ) through organs such as the skin and the kidneys .
Two major types of osmoregulation are osmoconformers and osmoregulators . Osmoconformers match their body osmolarity to their environment actively or passively. Most marine invertebrates are osmoconformers, although their ionic composition may be different from that of seawater. In a strictly osmoregulating animal, the amounts of internal salt and water are held relatively constant in the face of environmental changes. It requires that intake and outflow of water and salts be equal over an extended period of time.
Organisms that maintain an internal osmolarity different from the medium in which they are immersed have been termed osmoregulators. They tightly regulate their body osmolarity , maintaining constant internal conditions. They are more common in the animal kingdom. Osmoregulators actively control salt concentrations despite the salt concentrations in the environment. An example is freshwater fish. The gills actively uptake salt from the environment by the use of mitochondria-rich cells. Water will diffuse into the fish, so it excretes a very hypotonic (dilute) urine to expel all the excess water. A marine fish has an internal osmotic concentration lower than that of the surrounding seawater, so it tends to lose water and gain salt. It actively excretes salt out from the gills . Most fish are stenohaline , which means they are restricted to either salt or fresh water and cannot survive in water with a different salt concentration than they are adapted to. However, some fish show an ability to effectively osmoregulate across a broad range of salinities; fish with this ability are known as euryhaline species, e.g., flounder . Flounder have been observed to inhabit two disparate environments—marine and fresh water—and it is inherent to adapt to both by bringing in behavioral and physiological modifications.
Some marine fish, like sharks, have adopted a different, efficient mechanism to conserve water, i.e., osmoregulation. They retain urea in their blood in relatively higher concentration. Urea damages living tissues so, to cope with this problem, some fish retain trimethylamine oxide , which helps to counteract urea's destabilizing effects on cells. Sharks, having slightly higher solute concentration (i.e., above 1000 mOsm which is sea solute concentration), do not drink water like fresh water fish.
While there are no specific osmoregulatory organs in higher plants , the stomata are important in regulating water loss through evapotranspiration , and on the cellular level the vacuole is crucial in regulating the concentration of solutes in the cytoplasm . Strong winds , low humidity and high temperatures all increase evapotranspiration from leaves. Abscisic acid is an important hormone in helping plants to conserve water—it causes stomata to close and stimulates root growth so that more water can be absorbed.
Plants share with animals the problems of obtaining water but, unlike in animals, the loss of water in plants is crucial to create a driving force to move nutrients from the soil to tissues. Certain plants have evolved methods of water conservation.
Xerophytes are plants that can survive in dry habitats, such as deserts, and are able to withstand prolonged periods of water shortage. Succulent plants such as the cacti store water in the vacuoles of large parenchyma tissues. Other plants have leaf modifications to reduce water loss, such as needle-shaped leaves, sunken stomata , and thick, waxy cuticles as in the pine . The sand-dune marram grass has rolled leaves with stomata on the inner surface.
Hydrophytes are plants that grow in aquatic habitats; they may be floating, submerged, or emergent, and may grow in seasonal (rather than permanent) wetlands. In these plants the water absorption may occur through the whole surface of the plant, e.g., the water lily , or solely through the roots, as in sedges. These plants do not face major osmoregulatory challenges from water scarcity , but aside from species adapted for seasonal wetlands, have few defenses against desiccation.
Halophytes are plants living in soils with high salt concentrations, such as salt marshes or alkaline soils in desert basins. They have to absorb water from such a soil which has higher salt concentration and therefore lower water potential(higher osmotic pressure). Halophytes cope with this situation by activating salts in their roots. As a consequence, the cells of the roots develop lower water potential which brings in water by osmosis. The excess salt can be stored in cells or excreted out from salt glands on leaves. The salt thus secreted by some species help them to trap water vapours from the air, which is absorbed in liquid by leaf cells. Therefore, this is another way of obtaining additional water from air, e.g., glasswort and cord-grass .
Mesophytes are plants living in lands of temperate zone, which grow in well-watered soil. They can easily compensate the water lost by transpiration through absorbing water from the soil. To prevent excessive transpiration they have developed a waterproof external covering called cuticle.
Kidneys play a very large role in human osmoregulation by regulating the amount of water reabsorbed from glomerular filtrate in kidney tubules, which is controlled by hormones such as antidiuretic hormone (ADH), aldosterone , and angiotensin II . For example, a decrease in water potential is detected by osmoreceptors in the hypothalamus , which stimulates ADH release from the pituitary gland to increase the permeability of the walls of the collecting ducts in the kidneys. Therefore, a large proportion of water is reabsorbed from fluid in the kidneys to prevent too much water from being excreted . [ 2 ]
Drinking is not common behavior in pinnipeds and cetaceans . Water balance is maintained in marine mammals by metabolic and dietary water, while accidental ingestion and dietary salt may help maintain homeostasis of electrolytes. The kidneys of pinnipeds and cetaceans are lobed in structure, unlike those of non- bears among terrestrial mammals, but this specific adaptation does not confer any greater concentrating ability. Unlike most other aquatic mammals, manatees frequently drink fresh water and sea otters frequently drink saltwater. [ 3 ]
In teleost (advanced ray-finned) fishes, the gills, kidney and digestive tract are involved in maintenance of body fluid balance, as the main osmoregulatory organs. Gills in particular are considered the primary organ by which ionic concentration is controlled in marine teleosts.
Unusually, the catfishes in the eeltail family Plotosidae have an extra-branchial salt-secreting dendritic organ. The dendritic organ is likely a product of convergent evolution with other vertebrate salt-secreting organs. The role of this organ was discovered by its high NKA and NKCC activity in response to increasing salinity. However, the Plotosidae dendritic organ may be of limited use under extreme salinity conditions, compared to more typical gill-based ionoregulation. [ 4 ]
Amoeba makes use of contractile vacuoles to collect excretory wastes, such as ammonia , from the intracellular fluid by diffusion and active transport . As osmotic action pushes water from the environment into the cytoplasm, the vacuole moves to the surface and pumps the contents into the environment.
Bacteria respond to osmotic stress by rapidly accumulating electrolytes or small organic solutes via transporters whose activities are stimulated by increases in osmolarity. The bacteria may also turn on genes encoding transporters of osmolytes and enzymes that synthesize osmoprotectants. [ 5 ] The EnvZ/OmpR two-component system , which regulates the expression of porins , is well characterized in the model organism E. coli . [ 6 ]
Ammonia is a toxic by-product of protein metabolism and is generally converted to less toxic substances after it is produced then excreted; mammals convert ammonia to urea, whereas birds and reptiles form uric acid to be excreted with other wastes via their cloacas .
Four processes occur: | https://en.wikipedia.org/wiki/Osmoregulation |
The osmostat is the regulatory center in the hypothalamus that controls the osmolality of the extracellular fluid. The area in the anterior region of the hypothalamus contains the osmoreceptors , cells that control osmolality via the secretion of antidiuretic hormone (ADH).
In neurological conditions such as epilepsy or paraplegia , the osmostat can be pathologically reset, secreting ADH at a lower osmolality, which may cause hyponatremia . A reset osmostat is also a feature of SIADH . [ 1 ]
This neuroanatomy article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Osmostat |
The osmotic-controlled release oral delivery system (OROS) is an advanced controlled release oral drug delivery system in the form of a rigid tablet with a semi-permeable outer membrane and one or more small laser drilled holes in it. As the tablet passes through the body , water is absorbed through the semipermeable membrane via osmosis , and the resulting osmotic pressure is used to push the active drug through the laser drilled opening(s) in the tablet and into the gastrointestinal tract . OROS is a trademarked name owned by ALZA Corporation , which pioneered the use of osmotic pumps for oral drug delivery. [ 1 ] [ 2 ] [ 3 ]
Osmotic release systems have a number of major advantages over other controlled-release mechanisms. They are significantly less affected by factors such as pH , food intake, GI motility , and differing intestinal environments. Using an osmotic pump to deliver drugs has additional inherent advantages regarding control over drug delivery rates. This allows for much more precise drug delivery over an extended period of time, which results in much more predictable pharmacokinetics . However, osmotic release systems are relatively complicated, somewhat difficult to manufacture, and may cause irritation or even blockage of the GI tract due to prolonged release of irritating drugs from the non-deformable tablet. [ 1 ] [ 4 ] [ 5 ] [ 6 ] [ 7 ] [ 8 ] [ 9 ]
The Elementary Osmotic Pump (EOP) was developed by ALZA in 1974, and was the first practical example of an osmotic pump based drug release system for oral use. [ 1 ] [ 2 ] [ 10 ] [ 11 ] [ 12 ] It was introduced to the market in the early 1980s in Osmosin ( indomethacin ) and Acutrim ( phenylpropanolamine ), but unexpectedly severe issues with GI irritation and cases of GI perforation led to the withdrawal of Osmosin. [ 1 ]
Merck & Co. later developed the Controlled-Porosity Osmotic Pump (CPOP) with the intention of addressing some of the issues that led to Osmosin's withdrawal via a new approach to the final stage of the release mechanism. [ 1 ] Unlike the EOP, the CPOP had no pre-formed hole in the outer shell for the drug to be expelled out of. Instead, the CPOP's semipermeable membrane was designed to form numerous small pores upon contact with water through which the drug would be expelled via osmotic pressure. The pores were formed via the use of a pH insensitive leachable or dissolvable additive such as sorbitol . [ 13 ]
Both the EOP and CPOP were relatively simple designs, and were limited by their inability to deliver poorly soluble drugs. [ 1 ] This led to the development of an additional internal "push layer" composed of material (a swellable polymer ) that would expand as it absorbed water, which then pushed the drug layer (which incorporates a viscous polymer for suspension of poorly soluble drugs) out of the exit hole at a controlled rate. [ 1 ] [ 4 ] Osmotic agents such as sodium chloride , potassium chloride , or xylitol are added to both the drug and push layers to increase the osmotic pressure . [ 1 ] [ 4 ] [ 5 ] The initial design developed in 1982 by ALZA researchers was designated the Push-Pull Osmotic Pump (PPOP), and Procardia XL ( nifedipine ) was one of the first drugs to utilize this PPOP design. [ 1 ] [ 2 ]
In the early 1990s, an ALZA-funded research program began to develop a new dosage form of methylphenidate for the treatment of children with attention deficit hyperactivity disorder (ADHD). [ 14 ] Methylphenidate's short half-life required multiple doses to be administered each day to attain long-lasting coverage, which made it an ideal candidate for the OROS technology. Multiple candidate pharmacokinetic profiles were evaluated and tested in an attempt to determine the optimal way to deliver the drug, which was especially important given the puzzling failure of an existing extended-release formulation of methylphenidate (Ritalin SR) to act as expected. The zero-order (flat) release profile that the PPOP was optimal at delivering failed to maintain its efficacy over time, which suggested that acute tolerance to methylphenidate formed over the course of the day. This explained why Ritalin SR was inferior to twice-daily Ritalin IR, and led to the hypothesis that an ascending pattern of drug delivery was necessary to maintain clinical effect. Trials designed to test this hypothesis were successful, and ALZA subsequently developed a modified PPOP design that utilized an overcoat of methylphenidate designed to release immediately and rapidly raise serum levels, followed by 10 hours of first-order (ascending) drug delivery from the modified PPOP design. This design was called the Push-Stick Osmotic Pump (PSOP), and utilized two separate drug layers with different concentrations of methylphenidate in addition to the (now quite robust) push layer. [ 1 ] [ 14 ]
OROS medications include: [ 1 ] [ 3 ] [ 4 ] [ 7 ] | https://en.wikipedia.org/wiki/Osmotic-controlled_release_oral_delivery_system |
Osmotic blistering is a chemical phenomenon where two substances attempt to reach equilibrium through a semi-permeable membrane . [ 1 ] The phenomenon appears as a bubble or "blister" in the coating.
Water will flow from one solution to another, trying to create equilibrium between both solutions. Usually, the two solutions are concrete and the coating application on top of the concrete. Concrete is very porous, so water beneath the concrete, will force itself through the concrete, typically through vapor transmission. The water will then try to break through the semi-permeable membrane (either the surface of the concrete or the primer). Most epoxies or urethanes or polymer applications are not permeable so the water will stop at the polymer coating. However, the pressure from the water does not stop, forcing the water to collect directly in between the concrete and the layer of epoxy/urethane. This collection creates the notorious “osmotic blister” that is commonly feared by coating specialists.
For steel substrates:
The presence of soluble salts (particularly sulfates and chlorides) at the metal/paint interface is known to have a detrimental effect on the integrity of most paint systems including fluorocoatings. The salts come from atmospheric pollution and contamination during blasting or other substrate preparation processes. These salts promote osmotic blistering of the coating and underfilm metallic corrosion. As a result, loss of adhesion, cathodic disbondment, and scribe creep can be observed.
A coating behaves as a semi-impermeable membrane; allowing moisture to pass through but not salts4,5. When a paint coating is applied on a metallic surface contaminated with soluble salts, an osmotic blistering process takes place (Figure 8.10). Osmosis is the spontaneous net movement of solvent molecules (water) through a semipermeable membrane (coating film) into a region of higher solute concentration (the salt contaminated substrate). The process drives to equalize the solute concentrations on the two sides, but because salt cannot pass through the membrane (coating) it can never equalize. Water continues to permeate into the region. As the soluble substance dissolves under the paint layer, the pressure caused by the increase in volume can exert a greater force than the paint adhesion and cohesion forces, giving rise to the formation of a blister; the process called osmotic blistering. The blisters are first filled with water and later with corrosion products from the corrosion of the metallic substrate. | https://en.wikipedia.org/wiki/Osmotic_blistering |
An osmotic coefficient ϕ {\displaystyle \phi } is a quantity which characterises the deviation of a solvent from ideal behaviour , referenced to Raoult's law . It can be also applied to solutes. Its definition depends on the ways of expressing chemical composition of mixtures.
The osmotic coefficient based on molality m is defined by: ϕ = μ A ∗ − μ A R T M A ∑ i m i {\displaystyle \phi ={\frac {\mu _{A}^{*}-\mu _{A}}{RTM_{A}\sum _{i}m_{i}}}}
and on a mole fraction basis by:
ϕ = − μ A ∗ − μ A R T ln x A {\displaystyle \phi =-{\frac {\mu _{A}^{*}-\mu _{A}}{RT\ln x_{A}}}}
where μ A ∗ {\displaystyle \mu _{A}^{*}} is the chemical potential of the pure solvent and μ A {\displaystyle \mu _{A}} is the chemical potential of the solvent in a solution, M A is its molar mass , x A its mole fraction , R the gas constant and T the temperature in Kelvin . [ 1 ] The latter osmotic
coefficient is sometimes called the rational osmotic coefficient . The values for the two definitions are different, but since
ln x A = − ln ( 1 + M A ∑ i m i ) ≈ − M A ∑ i m i , {\displaystyle \ln x_{A}=-\ln \left(1+M_{A}\sum _{i}m_{i}\right)\approx -M_{A}\sum _{i}m_{i},}
the two definitions are similar, and in fact both approach 1 as the concentration goes to zero.
For liquid solutions, the osmotic coefficient is often used to calculate the salt activity coefficient from the solvent activity, or vice versa. For example, freezing point depression measurements, or measurements of deviations from ideality for other colligative properties , allows calculation of the salt activity coefficient through the osmotic coefficient.
In a single solute solution, the (molality based) osmotic coefficient and the solute activity coefficient γ {\displaystyle \gamma } are related to the excess Gibbs free energy G E {\displaystyle G^{E}} by the relations:
and there is thus a differential relationship between them (temperature and pressure held constant):
For a single salt solute with molal activity ( γ ± m {\displaystyle \gamma _{\pm }m} ), the osmotic coefficient can be written as ϕ = − ln ( a A ) ν m M A {\displaystyle \phi ={\frac {-\ln(a_{A})}{\nu mM_{A}}}} where ν {\displaystyle \nu } is the stochiometric number of salt and a A {\displaystyle a_{A}} the activity of the solvent. ϕ {\displaystyle \phi } can be calculated from the salt activity coefficient via: [ 2 ]
Moreover, the activity coefficient of the salt γ ± {\displaystyle \gamma _{\pm }} can be calculated from: [ 3 ]
According to Debye–Hückel theory , which is accurate only at low concentrations, ( ϕ − 1 ) ∑ i m i {\textstyle (\phi -1)\sum _{i}m_{i}} is asymptotic to − 2 3 A I 3 / 2 {\textstyle -{\frac {2}{3}}AI^{3/2}} , where I is ionic strength and A is the Debye–Hückel constant (equal to about 1.17 for water at 25 °C).
This means that, at least at low concentrations, the vapor pressure of the solvent will be greater than that predicted by Raoult's law. For instance, for solutions of magnesium chloride , the vapor pressure is slightly greater than that predicted by Raoult's law up to a concentration of 0.7 mol/kg, after which the vapor pressure is lower than Raoult's law predicts. For aqueous solutions, the osmotic coefficients can be calculated theoretically by Pitzer equations [ 4 ] or TCPC model. [ 5 ] [ 6 ] [ 7 ] [ 8 ] | https://en.wikipedia.org/wiki/Osmotic_coefficient |
Osmotic concentration , formerly known as osmolarity , [ 1 ] is the measure of solute concentration , defined as the number of osmoles (Osm) of solute per litre (L) of solution (osmol/L or Osm/L). The osmolarity of a solution is usually expressed as Osm/L (pronounced "osmolar"), in the same way that the molarity of a solution is expressed as "M" (pronounced "molar").
Whereas molarity measures the number of moles of solute per unit volume of solution, osmolarity measures the number of particles on dissociation of osmotically active material ( osmoles of solute particles) per unit volume of solution. [ 2 ] This value allows the measurement of the osmotic pressure of a solution and the determination of how the solvent will diffuse across a semipermeable membrane ( osmosis ) separating two solutions of different osmotic concentration.
The unit of osmotic concentration is the osmole . This is a non- SI unit of measurement that defines the number of moles of solute that contribute to the osmotic pressure of a solution. A milliosmole ( mOsm ) is one thousandth of an osmole. A microosmole ( μOsm ) (also spelled micro-osmole ) is one millionth of an osmole.
Osmolarity is distinct from molarity because it measures osmoles of solute particles rather than moles of solute. The distinction arises because some compounds can dissociate in solution, whereas others cannot. [ 2 ]
Ionic compounds , such as salts , can dissociate in solution into their constituent ions , so there is not a one-to-one relationship between the molarity and the osmolarity of a solution. For example, sodium chloride (NaCl) dissociates into Na + and Cl − ions. Thus, for every 1 mole of NaCl in solution, there are 2 osmoles of solute particles (i.e., a 1 mol/L NaCl solution is a 2 osmol/L NaCl solution). Both sodium and chloride ions affect the osmotic pressure of the solution. [ 2 ] [Note: NaCl does not dissociate completely in water at standard temperature and pressure, so the solution will be composed of Na+ ions, Cl- ions, and some NaCl molecules, with actual osmolality = Na+ concentration x 1.75]
Another example is magnesium chloride (MgCl 2 ), which dissociates into Mg 2+ and 2Cl − ions. For every 1 mole of MgCl 2 in the solution, there are 3 osmoles of solute particles.
Nonionic compounds do not dissociate, and form only 1 osmole of solute per 1 mole of solute. For example, a 1 mol/L solution of glucose is 1 osmol/L. [ 2 ]
Multiple compounds may contribute to the osmolarity of a solution. For example, a 3 Osm solution might consist of 3 moles glucose, or 1.5 moles NaCl, or 1 mole glucose + 1 mole NaCl, or 2 moles glucose + 0.5 mole NaCl, or any other such combination. [ 2 ]
The osmolarity of a solution, given in osmoles per liter (osmol/L) is calculated from the following expression: o s m o l a r i t y = ∑ i φ i n i C i {\displaystyle \mathrm {osmolarity} =\sum _{i}\varphi _{i}\,n_{i}C_{i}} where
Osmolarity can be measured using an osmometer which measures colligative properties , such as Freezing-point depression , Vapor pressure , or Boiling-point elevation .
Osmolarity and tonicity are related but distinct concepts. Thus, the terms ending in -osmotic (isosmotic, hyperosmotic, hypoosmotic) are not synonymous with the terms ending in -tonic (isotonic, hypertonic, hypotonic). The terms are related in that they both compare the solute concentrations of two solutions separated by a membrane. The terms are different because osmolarity takes into account the total concentration of penetrating solutes and non-penetrating solutes, whereas tonicity takes into account the total concentration of non-freely penetrating solutes only . [ 3 ] [ 2 ]
Penetrating solutes can diffuse through the cell membrane , causing momentary changes in cell volume as the solutes "pull" water molecules with them. Non-penetrating solutes cannot cross the cell membrane; therefore, the movement of water across the cell membrane (i.e., osmosis ) must occur for the solutions to reach equilibrium .
A solution can be both hyperosmotic and isotonic. [ 2 ] For example, the intracellular fluid and extracellular can be hyperosmotic, but isotonic – if the total concentration of solutes in one compartment is different from that of the other, but one of the ions can cross the membrane (in other words, a penetrating solute), drawing water with it, thus causing no net change in solution volume.
Plasma osmolarity, the osmolarity of blood plasma , can be calculated from plasma osmolality by the following equation: [ 4 ]
where:
According to IUPAC, osmolality is the quotient of the negative natural logarithm of the rational activity of water and the molar mass of water, whereas osmolarity is the product of the osmolality and the mass density of water (also known as osmotic concentration). [ 1 ]
In simpler terms, osmolality is an expression of solute osmotic concentration per mass of solvent, whereas osmolarity is per volume of solution (thus the conversion by multiplying with the mass density of solvent in solution (kg solvent/litre solution). osmolality = ∑ i φ i n i m i {\displaystyle {\text{osmolality}}=\sum _{i}\varphi _{i}\,n_{i}m_{i}}
where m i is the molality of component i .
Plasma osmolarity/osmolality is important for keeping proper electrolytic balance in the blood stream. Improper balance can lead to dehydration , alkalosis , acidosis or other life-threatening changes. Antidiuretic hormone (vasopressin) is partly responsible for this process by controlling the amount of water the body retains from the kidney when filtering the blood stream. [ 6 ]
A concentration of an osmotically active substance is said to be hyperosmolar if a high concentration causes a change in osmatic pressure in a tissue, organ, or system. Similarly, it is said to be hypoossmolar if the osmolarity, or osmatic concentration, is too low. For example, if the osmolarity of parenteral nutrition is too high, it can cause severe tissue damage. [ 7 ] One example of a condition caused by hypoosmolarity is water intoxication . [ 8 ] | https://en.wikipedia.org/wiki/Osmotic_concentration |
Osmotic dehydration is an operation used for the partial removal of water from plant tissues by immersion in a hypertonic ( osmotic ) solution.
Sugar or salt solutions are used to reduce the moisture content of foods before actual drying process. This technique is used to give the product quality improvement over conventional drying process.
Mild heat treatment after osmotic dehydration favours colour and flavour retention resulting in the product having superior organoleptic characteristics. It also increases resistance to heat treatment, prevents enzymatic browning and inhibits activities of polyphenol oxidase . The process is economical.
Osmotic dehydration depends on:
Water removal
is based on the natural and non-destructive phenomenon of osmosis across cell membranes . The driving force for the diffusion of water from the tissue into the solution is provided by the higher osmotic pressure of the hyper-tonic solution. The diffusion of water is accompanied by the simultaneous counter diffusion of solutes from the osmotic solution into the tissue. Since the cell membrane responsible for osmotic transport is not perfectly selective, solutes present in the cells ( organic acids , reducing sugars , minerals, flavors and pigment compounds) can also be leaked into the osmotic solution, which affects the organoleptic and nutritional characteristics of the product.
The rate of diffusion of water from any material made up of such tissues depends upon factors such as temperature and concentration of the osmotic solution, the size and geometry of the material, the solution-to-material mass ratio and, to a certain level, agitation of the solution. [ 1 ]
This biology article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Osmotic_dehydration |
Osmotic power , salinity gradient power or blue energy is the energy available from the difference in the salt concentration between seawater and river water . Two practical methods for this are reverse electrodialysis (RED) and pressure retarded osmosis (PRO). Both processes rely on osmosis with membranes . The key waste product is brackish water . This byproduct is the result of natural forces that are being harnessed: the flow of fresh water into seas that are made up of salt water.
In 1954, Pattle [ 1 ] suggested that there was an untapped source of power when a river mixes with the sea, in terms of the lost osmotic pressure, however it was not until the mid ‘70s where a practical method of harnessing it using selectively permeable membranes by Loeb [ 2 ] was outlined.
The method of generating power by pressure retarded osmosis was invented by Prof. Sidney Loeb in 1973 at the Ben-Gurion University of the Negev, Beersheba, Israel. [ 3 ] The idea came to Prof. Loeb, in part, as he observed the Jordan River flowing into the Dead Sea. He wanted to harvest the energy of mixing of the two aqueous solutions (the Jordan River being one and the Dead Sea being the other) that was going to waste in this natural mixing process. [ 4 ] In 1977 Prof. Loeb invented a method of producing power by a reverse electrodialysis heat engine. [ 5 ]
The technologies have been confirmed in laboratory conditions. They are being developed into commercial use in the Netherlands (RED) and Norway (PRO). The cost of the membrane has been an obstacle. A new, lower cost membrane, based on an electrically modified polyethylene plastic, made it fit for potential commercial use. [ 6 ] Other methods have been proposed and are currently under development. Among them, a method based on electric double-layer capacitor technology [ 7 ] and a method based on vapor pressure difference. [ 8 ]
Salinity gradient power is a specific renewable energy alternative that creates renewable and sustainable power by using naturally occurring processes. This practice does not contaminate or release carbon dioxide (CO 2 ) emissions (vapor pressure methods will release dissolved air containing CO 2 at low pressures—these non-condensable gases can be re-dissolved of course, but with an energy penalty). Also as stated by Jones and Finley within their article “Recent Development in Salinity Gradient Power”, there is basically no fuel cost.
Salinity gradient energy is based on using the resources of “osmotic pressure difference between fresh water and sea water.” [ 9 ] All energy that is proposed to use salinity gradient technology relies on the evaporation to separate water from salt. Osmotic pressure is the "chemical potential of concentrated and dilute solutions of salt". [ 10 ] When looking at relations between high osmotic pressure and low, solutions with higher concentrations of salt have higher pressure.
Differing salinity gradient power generations exist but one of the most commonly discussed is pressure-retarded osmosis (PRO). Within PRO seawater is pumped into a pressure chamber where the pressure is lower than the difference between fresh and salt water pressure. Fresh water moves in a semipermeable membrane and increases its volume in the chamber. As the pressure in the chamber is compensated a turbine spins to generate electricity. In Braun's article he states that this process is easy to understand in a more broken down manner. Two solutions, A being salt water and B being fresh water are separated by a membrane. He states "only water molecules can pass the semipermeable membrane. As a result of the osmotic pressure difference between both solutions, the water from solution B thus will diffuse through the membrane in order to dilute solution A". [ 10 ] The pressure drives the turbines and power the generator that produces the electrical energy. Osmosis might be used directly to "pump" fresh water out of The Netherlands into the sea. This is currently done using electric pumps.
A 2012 study on efficiency from Yale University concluded that the highest extractable work in constant-pressure PRO with a seawater draw solution and river water feed solution is 0.75 kWh/m 3 (2.7 kJ/L) while the free energy of mixing is 0.81 kWh/m 3 (2.9 kJ/L) — a thermodynamic extraction efficiency of 91.0%. [ 11 ]
While the mechanics and concepts of salinity gradient power are still being studied, the power source has been implemented in several different locations. Most of these are experimental, but thus far they have been predominantly successful. The various companies that have utilized this power have also done so in many different ways as there are several concepts and processes that harness the power from salinity gradient.
One method to utilize salinity gradient energy is called pressure-retarded osmosis . [ 12 ] In this method, seawater is pumped into a pressure chamber that is at a pressure lower than the difference between the pressures of saline water and fresh water. Freshwater is also pumped into the pressure chamber through a membrane, which increase both the volume and pressure of the chamber. As the pressure differences are compensated, a turbine is spun, providing kinetic energy. This method is being specifically studied by the Norwegian utility Statkraft , which has calculated that up to 2.85 GW would be available from this process in Norway. [ 13 ] Statkraft has built the world's first prototype PRO power plant on the Oslo fjord which was opened by Princess Mette-Marit of Norway [ 14 ] on November 24, 2009. It aimed to produce enough electricity to light and heat a small town within five years by osmosis. At first, it did produce a minuscule 4 kilowatts – enough to heat a large electric kettle, but by 2015 the target was 25 megawatts – the same as a small wind farm. [ 15 ] In January 2014 however Statkraft announced not to continue this pilot. [ 16 ] Statkraft found that with existing technology, the salt gradient was not high enough to be economic, which other studies have agreed on. [ 17 ] Higher salt gradients can be found in geothermal brines and desalination plant brines, [ 18 ] and SaltPower, a Danish company, is now building its first commercial plant with high salinity brine. [ 19 ] There is perhaps more potential in integrating Pressure Retarded Osmosis as an operating mode of reverse osmosis, rather than a stand-alone technology. [ 20 ]
A second method being developed and studied is reversed electrodialysis or reverse dialysis, which is essentially the creation of a salt battery. This method was described by Weinstein and Leitz as “an array of alternating anion and cation exchange membranes can be used to generate electric power from the free energy of river and sea water.”
The technology related to this type of power is still in its infant stages, even though the principle was discovered in the 1950s. Standards and a complete understanding of all the ways salinity gradients can be utilized are important goals to strive for in order to make this clean energy source more viable in the future.
A third method is Doriano Brogioli 's [ 7 ] capacitive method, which is relatively new and has so far only been tested on lab scale. With this method energy can be extracted out of the mixing of saline water and freshwater by cyclically charging up electrodes in contact with saline water, followed by a discharge in freshwater. Since the amount of electrical energy which is needed during the charging step is less than one gets out during the discharge step, each completed cycle effectively produces energy. An intuitive explanation of this effect is that the great number of ions in the saline water efficiently neutralizes the charge on each electrode by forming a thin layer of opposite charge very close to the electrode surface, known as an electric double layer . Therefore, the voltage over the electrodes remains low during the charge step and charging is relatively easy. In between the charge and discharge step, the electrodes are brought in contact with freshwater. After this, there are less ions available to neutralize the charge on each electrode such that the voltage over the electrodes increases. The discharge step which follows is therefore able to deliver a relatively high amount of energy. A physical explanation is that on an electrically charged capacitor, there is a mutually attractive electric force between the electric charge on the electrode, and the ionic charge in the liquid. In order to pull ions away from the charged electrode, osmotic pressure must do work . This work done increases the electrical potential energy in the capacitor. An electronic explanation is that capacitance is a function of ion density. By introducing a salinity gradient and allowing some of the ions to diffuse out of the capacitor, this reduces the capacitance, and so the voltage must increase, since the voltage equals the ratio of charge to capacitance.
Both of these methods do not rely on membranes, so filtration requirements are not as important as they are in the PRO & RED schemes.
Similar to the open cycle in ocean thermal energy conversion (OTEC). The disadvantage of this cycle is the cumbersome problem of a large diameter turbine (75 meters +) operating at below atmospheric pressure to extract the power between the water with less salinity & the water with greater salinity.
For the purpose of dehumidifying air, in a water-spray absorption refrigeration system, water vapor is dissolved into a deliquescent salt water mixture using osmotic power as an intermediary. The primary power source originates from a thermal difference, as part of a thermodynamic heat engine cycle.
At the Eddy Potash Mine in New Mexico, a technology called "salinity gradient solar pond " (SGSP) is being utilized to provide the energy needed by the mine. This method does not harness osmotic power , only solar power (see: solar pond ). Sunlight reaching the bottom of the saltwater pond is absorbed as heat. The effect of natural convection , wherein "heat rises", is blocked using density differences between the three layers that make up the pond, in order to trap heat. The upper convection zone is the uppermost zone, followed by the stable gradient zone, then the bottom thermal zone. The stable gradient zone is the most important. The saltwater in this layer can not rise to the higher zone because the saltwater above has lower salinity and is therefore less-dense and more buoyant; and it can not sink to the lower level because that saltwater is denser. This middle zone, the stable gradient zone, effectively becomes an "insulator" for the bottom layer (although the main purpose is to block natural convection, since water is a poor insulator). This water from the lower layer, the storage zone, is pumped out and the heat is used to produce energy, usually by turbine in an organic Rankine cycle . [ 21 ]
In theory a solar pond could be used to generate osmotic power if evaporation from solar heat is used to create a salinity gradient, and the potential energy in this salinity gradient is harnessed directly using one of the first three methods above, such as the capacitive method.
A research team built an experimental system using boron nitride that produced much greater power than the Statkraft prototype. It used an impermeable and electrically insulating membrane that was pierced by a single boron nitride nanotube with an external diameter of a few dozen nanometers. With this membrane separating a salt water reservoir and a fresh water reservoir, the team measured the electric current passing through the membrane using two electrodes immersed in the fluid either side of the nanotube.
The results showed the device was able to generate an electric current on the order of a nanoampere. The researchers claim this is 1,000 times the yield of other known techniques for harvesting osmotic energy and makes boron nitride nanotubes an extremely efficient solution for harvesting the energy of salinity gradients for usable electrical power.
The team claimed that a 1 square metre (11 sq ft) membrane could generate around 4 kW and be capable of generating up to 30 MWh per year. [ 22 ]
At the 2019 fall meeting of the Materials Research Society a team from Rutgers University reported creating a membrane that contained around 10 million BNNTs per cubic centimeter. [ 23 ] [ 24 ]
At Pennsylvania State University, Dr. Logan tries to use waste heat with low calority using the fact that ammonium bicarbonate decomposes into NH 3 and CO 2 in warm water to form ammonium bicarbonate again in cold water. So in a RED energy producing closed system the two different gradients of salinity are kept. [ 25 ]
Marine and river environments have obvious differences in water quality, namely salinity. Each species of aquatic plant and animal is adapted to survive in either marine, brackish, or freshwater environments. There are species that can tolerate both, but these species usually thrive best in a specific water environment. The main waste product of salinity gradient technology is brackish water. The discharge of brackish water into the surrounding waters, if done in large quantities and with any regularity, will cause salinity fluctuations. While some variation in salinity is usual, particularly where fresh water (rivers) empties into an ocean or sea anyway, these variations become less important for both bodies of water with the addition of brackish waste waters. Extreme salinity changes in an aquatic environment may result in findings of low densities of both animals and plants due to intolerance of sudden severe salinity drops or spikes. [ 26 ] According to the prevailing environmentalist opinions, the possibility of these negative effects should be considered by the operators of future large blue energy establishments.
The impact of brackish water on ecosystems can be minimized by pumping it out to sea and releasing it into the mid-layer, away from the surface and bottom ecosystems.
Impingement and entrainment at intake structures are a concern due to large volumes of both river and sea water utilized in both PRO and RED schemes. Intake construction permits must meet strict environmental regulations and desalination plants and power plants that utilize surface water are sometimes involved with various local, state and federal agencies to obtain permission that can take upwards to 18 months.
The Tethys database provides access to scientific literature and general information on the potential environmental effects of salinity gradient power. [ 27 ] | https://en.wikipedia.org/wiki/Osmotic_power |
Osmotic pressure is the minimum pressure which needs to be applied to a solution to prevent the inward flow of its pure solvent across a semipermeable membrane . [ 1 ] It is also defined as the measure of the tendency of a solution to take in its pure solvent by osmosis . [ not verified in body ] Potential osmotic pressure is the maximum osmotic pressure that could develop in a solution if it were separated from its pure solvent by a semipermeable membrane.
Osmosis occurs when two solutions containing different concentrations of solute are separated by a selectively permeable membrane. Solvent molecules pass preferentially through the membrane from the low-concentration solution to the solution with higher solute concentration. The transfer of solvent molecules will continue until osmotic equilibrium is attained. [ 1 ] [ 2 ]
Jacobus van 't Hoff found a quantitative relationship between osmotic pressure and solute concentration, expressed in the following equation:
where Π {\displaystyle \Pi } is osmotic pressure, i is the dimensionless van 't Hoff index , c is the molar concentration of solute, R is the ideal gas constant , and T is the absolute temperature (usually in kelvins ). This formula applies when the solute concentration is sufficiently low that the solution can be treated as an ideal solution . The proportionality to concentration means that osmotic pressure is a colligative property . Note the similarity of this formula to the ideal gas law in the form P = n V R T = c gas R T {\textstyle P={\frac {n}{V}}RT=c_{\text{gas}}RT} where n is the total number of moles of gas molecules in the volume V , and n / V is the molar concentration of gas molecules. Harmon Northrop Morse and Frazer showed that the equation applied to more concentrated solutions if the unit of concentration was molal rather than molar ; [ 3 ] so when the molality is used this equation has been called the Morse equation .
For more concentrated solutions the van 't Hoff equation can be extended as a power series in solute concentration, c . To a first approximation,
where Π 0 {\displaystyle \Pi _{0}} is the ideal pressure and A is an empirical parameter. The value of the parameter A (and of parameters from higher-order approximations) can be used to calculate Pitzer parameters . Empirical parameters are used to quantify the behavior of solutions of ionic and non-ionic solutes which are not ideal solutions in the thermodynamic sense.
The Pfeffer cell was developed for the measurement of osmotic pressure.
Osmotic pressure measurement may be used for the determination of molecular weights .
Osmotic pressure is an important factor affecting biological cells. [ 4 ] Osmoregulation is the homeostasis mechanism of an organism to reach balance in osmotic pressure.
When a biological cell is in a hypotonic environment, the cell interior accumulates water, water flows across the cell membrane into the cell, causing it to expand. In plant cells , the cell wall restricts the expansion, resulting in pressure on the cell wall from within called turgor pressure . Turgor pressure allows herbaceous plants to stand upright. It is also the determining factor for how plants regulate the aperture of their stomata . In animal cells excessive osmotic pressure can result in cytolysis due to the absence of a cell wall.
Osmotic pressure is the basis of filtering (" reverse osmosis "), a process commonly used in water purification . The water to be purified is placed in a chamber and put under an amount of pressure greater than the osmotic pressure exerted by the water and the solutes dissolved in it. Part of the chamber opens to a differentially permeable membrane that lets water molecules through, but not the solute particles. The osmotic pressure of ocean water is approximately 27 atm . Reverse osmosis desalinates fresh water from ocean salt water and is applied globally on a very large scale.
Consider the system at the point when it has reached equilibrium. The condition for this is that the chemical potential of the solvent (since only it is free to flow toward equilibrium) on both sides of the membrane is equal. The compartment containing the pure solvent has a chemical potential of μ 0 ( p ) {\displaystyle \mu ^{0}(p)} , where p {\displaystyle p} is the pressure. On the other side, in the compartment containing the solute, the chemical potential of the solvent depends on the mole fraction of the solvent, 0 < x v < 1 {\displaystyle 0<x_{v}<1} . Besides, this compartment can assume a different pressure, p ′ {\displaystyle p'} . We can therefore write the chemical potential of the solvent as μ v ( x v , p ′ ) {\displaystyle \mu _{v}(x_{v},p')} . If we write p ′ = p + Π {\displaystyle p'=p+\Pi } , the balance of the chemical potential is therefore:
Here, the difference in pressure of the two compartments Π ≡ p ′ − p {\displaystyle \Pi \equiv p'-p} is defined as the osmotic pressure exerted by the solutes. Holding the pressure, the addition of solute decreases the chemical potential (an entropic effect ). Thus, the pressure of the solution has to be increased in an effort to compensate the loss of the chemical potential.
In order to find Π {\displaystyle \Pi } , the osmotic pressure, we consider equilibrium between a solution containing solute and pure water.
We can write the left hand side as:
where γ v {\displaystyle \gamma _{v}} is the activity coefficient of the solvent. The product γ v x v {\displaystyle \gamma _{v}x_{v}} is also known as the activity of the solvent, which for water is the water activity a w {\displaystyle a_{w}} . The addition to the pressure is expressed through the expression for the energy of expansion:
where V m {\displaystyle V_{m}} is the molar volume (m³/mol). Inserting the expression presented above into the chemical potential equation for the entire system and rearranging will arrive at:
If the liquid is incompressible the molar volume is constant, V m ( p ′ ) ≡ V m {\displaystyle V_{m}(p')\equiv V_{m}} , and the integral becomes Π V m {\displaystyle \Pi V_{m}} . Thus, we get
The activity coefficient is a function of concentration and temperature, but in the case of dilute mixtures, it is often very close to 1.0, so
The mole fraction of solute, x s {\displaystyle x_{s}} , is 1 − x v {\displaystyle 1-x_{v}} , so ln ( x v ) {\displaystyle \ln(x_{v})} can be replaced with ln ( 1 − x s ) {\displaystyle \ln(1-x_{s})} , which, when x s {\displaystyle x_{s}} is small, can be approximated by − x s {\displaystyle -x_{s}} .
The mole fraction x s {\displaystyle x_{s}} is n s / ( n s + n v ) {\displaystyle n_{s}/(n_{s}+n_{v})} . When x s {\displaystyle x_{s}} is small, it may be approximated by x s = n s / n v {\displaystyle x_{s}=n_{s}/n_{v}} . Also, the molar volume V m {\displaystyle V_{m}} may be written as volume per mole, V m = V / n v {\displaystyle V_{m}=V/n_{v}} . Combining these gives the following.
For aqueous solutions of salts, ionisation must be taken into account. For example, 1 mole of NaCl ionises to 2 moles of ions. | https://en.wikipedia.org/wiki/Osmotic_pressure |
Osmotic shock or osmotic stress is physiologic dysfunction caused by a sudden change in the solute concentration around a cell , which causes a rapid change in the movement of water across its cell membrane . Under hypertonic conditions - conditions of high concentrations of either salts , substrates or any solute in the supernatant - water is drawn out of the cells through osmosis . This also inhibits the transport of substrates and cofactors into the cell thus “shocking” the cell. Alternatively, under hypotonic conditions - when concentrations of solutes are low - water enters the cell in large amounts, causing it to swell and either burst or undergo apoptosis . [ 1 ]
All organisms have mechanisms to respond to osmotic shock, with sensors and signal transduction networks providing information to the cell about the osmolarity of its surroundings; [ 2 ] these signals activate responses to deal with extreme conditions. [ 3 ] Cells that have a cell wall tend to be more resistant to osmotic shock because their cell wall enables them to maintain their shape. [ 4 ] Although single-celled organisms are more vulnerable to osmotic shock, since they are directly exposed to their environment, cells in large animals such as mammals still suffer these stresses under some conditions. [ 5 ] Current research also suggests that osmotic stress in cells and tissues may significantly contribute to many human diseases. [ 6 ]
In eukaryotes , calcium acts as one of the primary regulators of osmotic stress. Intracellular calcium levels rise during hypo-osmotic and hyper-osmotic stresses.
Calcium plays a large role in the recovery and tolerance for both hyper and hypo-osmotic stress situations. Under hyper-osmotic stress conditions, increased levels of intracellular calcium are exhibited. This may play a crucial role in the activation of second messenger pathways . [ 7 ]
One example of a calcium activated second messenger molecule is MAP Kinase Hog-1. It is activated under hyper-osmotic stress conditions [ 8 ] and is responsible for an increase in the production of glycerol within the cell succeeding osmotic stress. More specifically, it works by sending signals to the nucleus that activate genes responsible for glycerol production and uptake. [ 8 ]
Hypo-osmotic stress recovery is largely mediated by the influx and efflux of several ions and molecules. Cell recovery after hypo-osmotic stress has shown to be consistent with an influx of extracellular Calcium. [ 9 ] This influx of calcium may alter the cell's permeability. [ 9 ]
Additionally, in some organisms the efflux of amino acids associated with hypo-osmotic stress can be inhibited by phenothiazines . [ 9 ]
Hypo-osmotic stress is correlated with extracellular ATP release. ATP is used to activate purinergic receptors . [ 10 ] These receptors regulate sodium and potassium levels on either side of the cell membrane. | https://en.wikipedia.org/wiki/Osmotic_shock |
The osmotic stress technique is a method for measuring the effect of water on biological molecules, particularly enzymes . Just as the properties of molecules can depend on the presence of salts , pH , and temperature , they can depend significantly on the amount of water present. In the osmotic stress technique, flexible neutral polymers such as polyethylene glycol and dextran are added to the solution containing the molecule of interest, replacing a significant part of the water. The amount of water replaced is characterized by the chemical activity of water.
This biochemistry article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Osmotic_stress_technique |
Osram ne nsoromma is one of the Bono Adinkra symbols , which is interpreted to mean "Osram" Moon "Ne" and "Nsoromma" Star . This symbol signifies love, bonding and faithfulness in marriage. The symbol is represented by a half moon with a star slightly hanging within the circumference of the moon. Adinkra are symbols that carry a message or a concept. They are very much used by the Bono people of the Bonoman and the Gyaman , an Akan people of Ghana and Côte d'Ivoire . [ 1 ] Osram ne nsoromma symbols are incorporated into walls and other architectural features and quite recently has become common with tattoo designers. The most common ways through which the Adinkra messages are carried out or conveyed is having them printed extensively in fabrics and used in pottery.
Adinkra is an Akan name which means farewell or goodbye. [ 2 ] [ 3 ] [ 4 ] [ 5 ] [ 6 ] [ 7 ] Osram ne nsoromma are two different powerful objects of creation put together (Moon and Star). Both co exist in the sky to produce magnificent light or brightness at night. There are some wise sayings closely related to the Osram ne nsoromma symbol, often linked with proverbs
The Akans of Ghana use an Adinkra symbol to express proverbs and other philosophical ideas or traditional wisdom , aspects of life or the environment.
Some of the familiar proverbs are:
Awaree nye nsafufuo na waka ahwe , which means marriage is not palm-wine that you can decide to have a taste before you get served. It can also be interpreted to mean marriage is not a venture, committee or an organization that you can be a member today and withdraw your membership tomorrow. This proverb frowns upon break ups or separations in marriages or relationships.
Woreko awaree a bisa , This proverb talks about due diligence before marriage. it is wise to do a background check on your partner before marriage or the worse will happen. Making the wrong choice is sometimes costly. Similarly, most of the proverbs seem to put value on the object being addressed. This symbol signifies Love, bonding and faithfulness but one must carefully choose the right partner.
Ahwenepa nkasa, can be interpreted to mean "Ahwenepa" ( Good waist beads ) " nkasa " (makes no noise). This can be related to mean a good man or a good woman needs no introduction to be known. You can notice them when you find them. This proverb talks about the character of the would be partner. Good man or woman is hard to find but when you meet that special person, you will notice. [ 8 ] [ 9 ] | https://en.wikipedia.org/wiki/Osram_ne_nsoromma |
In the mathematical field of differential geometry , the Osserman–Xavier–Fujimoto theorem concerns the Gauss maps of minimal surfaces in the three-dimensional Euclidean space. It says that if a minimal surface is immersed and geodesically complete , then the image of the Gauss map either consists of a single point (so that the surface is a plane) or contains all of the sphere except for at most four points.
Bernstein's theorem says that a minimal graph in R 3 which is geodesically complete must be a plane. This can be rephrased to say that the Gauss map of a complete immersed minimal surface in R 3 is either constant or not contained within an open hemisphere. As conjectured by Louis Nirenberg and proved by Robert Osserman in 1959, in this form Bernstein's theorem can be generalized to say that the image of the Gauss map of a complete immersed minimal surface in R 3 either consists of a single point or is dense within the sphere. [ 1 ]
Osserman's theorem was improved by Frederico Xavier and Hirotaka Fujimoto in the 1980s. They proved that if the image of the Gauss map of a complete immersed minimal surface in R 3 omits more than four points of the sphere, then the surface is a plane. [ 2 ] This is optimal, since it was shown by Konrad Voss in the 1960s that for any subset A of the sphere whose complement consists of zero, one, two, three, or four points, there exists a complete immersed minimal surface in R 3 whose Gauss map has image A . [ 3 ] Particular examples include Riemann's minimal surface , whose Gauss map is surjective, the Enneper surface , whose Gauss map omits one point, the catenoid and helicoid , whose Gauss maps omit two points, and Scherk's first surface , whose Gauss map omits four points.
It is also possible to study the Gauss map of minimal surfaces of higher codimension in higher-dimensional Euclidean spaces. There are a number of variants of the results of Osserman, Xavier, and Fujimoto which can be studied in this setting. [ 4 ]
This geometry-related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Osserman–Xavier–Fujimoto_theorem |
Osteoimmunology (όστέον, osteon from Greek, "bone"; immunitas from Latin, "immunity"; and λόγος, logos , from Greek "study") is a field that emerged about 40 years ago that studies the interface between the skeletal system and the immune system , [ 1 ] [ 2 ] comprising the "osteo-immune system". [ 3 ] [ 4 ] Osteoimmunology also studies the shared components and mechanisms between the two systems in vertebrates, including ligands, receptors, signaling molecules and transcription factors. Over the past decade, osteoimmunology has been investigated clinically for the treatment of bone metastases , rheumatoid arthritis (RA), osteoporosis , osteopetrosis , and periodontitis . Studies in osteoimmunology reveal relationships between molecular communication among blood cells and structural pathologies in the body. [ 5 ]
The RANKL - RANK - OPG axis (OPG stands for osteoprotegerin) is an example of an important signaling system functioning both in bone [ 6 ] and immune cell communication. RANKL is expressed on osteoblasts and activated T cells , whereas RANK is expressed on osteoclasts , and dendritic cells (DCs), both of which can be derived from myeloid progenitor cells. Surface RANKL on osteoblasts as well as secreted RANKL provide necessary signals for osteoclast precursors to differentiate into osteoclasts. RANKL expression on activated T cells leads to DC activation through binding to RANK expressed on DCs. OPG, produced by DCs, is a soluble decoy receptor for RANKL that competitively inhibits RANKL binding to RANK. [ 7 ]
The bone marrow cavity is important for the proper development of the immune system, and houses important stem cells for maintenance of the immune system. Within this space, as well as outside of it, cytokines produced by immune cells also have important effects on regulating bone homeostasis. Some important cytokines that are produced by the immune system, including RANKL, M-CSF , TNFa , ILs , and IFNs , affect the differentiation and activity of osteoclasts and bone resorption . [ 8 ] [ 9 ] Such inflammatory osteoclastogenesis and osteoclast activation can be seen in ex vivo primary cultures of cells from the inflamed synovial fluid of patients with disease flare of the autoimmune disease rheumatoid arthritis. [ 9 ] [ 10 ]
Clinical osteoimmunology is a field that studies a treatment or prevention of the bone related diseases caused by disorders of the immune system. Aberrant and/or prolonged activation of immune system leads to derangement of bone modeling and remodeling. Common diseases caused by disorder of osteoimmune system is osteoporosis and bone destruction accompanied by RA characterized by high infiltration of CD4+ T cells in rheumatoid joints, in which two mechanisms are involved: One is an indirect effect on osteoclastogenesis from rheumatoid synovial cells in joints since synovial cells have osteoclast precursors and osteoclast supporting cells, synovial macrophages are highly differentiated into osteoclasts with help of RANKL released from osteoclast supporting cells. [ 11 ] [ 12 ] The second is an indirect effect on osteoclast differentiation and activity by the secretion of inflammatory cytokines such as IL-1 , IL-6 , TNFa , in synovium of RA, which increase RANKL signaling and finally bone destruction. A clinical approach to prevent bone related diseases caused by RA is OPG and RANKL treatment in arthritis. There is some evidence that infections (e.g. respiratory virus infection) can reduce the numbers of osteoblasts in bone, the key cells involved in bone formation. [ 13 ] | https://en.wikipedia.org/wiki/Osteoimmunology |
Osteophagy is the practice in which animals, usually herbivores , consume bones. Most vegetation around the world lacks sufficient amounts of phosphate . [ 1 ] Phosphorus is an essential mineral for all animals, as it plays a major role in the formation of the skeletal system , and is necessary for many biological processes including: energy metabolism, protein synthesis , cell signaling , and lactation . [ 2 ] Phosphate deficiencies can cause physiological side effects, especially pertaining to the reproductive system , [ 1 ] as well as side effects of delayed growth and failure to regenerate new bone . [ 2 ] The importance of having sufficient amounts of phosphorus further resides in the physiological importance of maintaining a proper phosphorus to calcium ratio. Having a Ca:P ratio of 2:1 is important for the absorption of these minerals, as deviations from this optimal ratio can inhibit their absorption. [ 3 ] Dietary calcium and phosphorus ratio, along with vitamin D , regulates bone mineralization and turnover by affecting calcium and phosphorus transport and absorption in the intestine . [ 4 ]
It has been suggested that osteophagy is an innate behavior that allows animals to supplement their phosphorus and calcium uptake in order to avoid the costly effects of deficiencies in these minerals. [ 1 ] Osteophagic behavior has been observed in pastoral and wild animals, most notably ungulates and other herbivores, for over two hundred years. [ 1 ] Osteophagy has been inferred from archaeological studies of dental wear in Pleistocene fossils dating back 780,000 years. [ 5 ] It has been seen in domestic animals, as well as red deer , camels , giraffes , wildebeest , antelopes , tortoises , and grizzly bears . [ 5 ] Due to differences in tooth structure, herbivores tend to chew old dry bones that are easier to break, while carnivores prefer to chew softer fresh bones. [ 6 ] Variations of the behavior have also been observed in humans. [ citation needed ]
While osteophagy has been regarded as a beneficial behavior to combat mineral deficiencies in animals, osteophagic practices have also been observed to be detrimental to the dentition of herbivores. It has been observed that the pattern of wear on the cheek teeth of herbivores is congruous to the manner in which herbivores hold and chew bones. [ 5 ] A major cost of osteophagy is therefore significant wear on teeth and dental breakage in herbivores, whose teeth did not evolve to enable the regular consumption of hard materials but rather for the grinding of vegetal fibers. [ 5 ]
Wolverines are observed finding large bones invisible in deep snow and are specialists at scavenging bones specifically to cache. Wolverine upper molars are rotated 90 degrees inward, which is the identifying dentition characteristic of the family Mustelidae (weasel family), of which the wolverine has the most mass, so they can crack the bones and eat the frozen marrow of large animals. This structural feature helps the wolverine be successful as a scavenger and adapt to a frozen habitat. [ 7 ]
Porcupine species including the largest, African porcupine and North American porcupine, are nocturnal bone collectors of thousands of bones, stored inside their den and in open piles in their vicinity. The bones do not satisfy seasonal nutritional deficiency, they prevent overgrown teeth but the shavings are ingested as the bulk of their diet. [ 8 ]
Osteophagy in desert tortoises has largely been observed in captivity, and more rarely in the wild where osteophagy observed above ground is quick and seldom, usually lasting only a few minutes. [ 9 ]
Desert plants are a major food source for desert tortoises ( Gopherus agassizii ), as they have a mainly herbivorous diet. [ 10 ] In addition to desert plants, desert tortoises also consume vulture feces (which contain bones), soil (layers contain calcium), mammal hairs, feathers, arthropods, stones, bones of conspecifics, as well as snake and lizard skin castings. [ 10 ] Desert tortoises have been observed to exhibit mounting behavior, aggressive biting, and repeated striking of carcasses when practicing osteophagy. [ 10 ]
Osteophagy in herbivores has been viewed to serve as a source for supplemental minerals. Desert plants grow in mineral-deficient soil, and may be a cause of mineral deficiency in desert tortoise diets, resulting in the intake of this supplemental material.
An observational study of tortoises near St. George, Utah, found that the tortoises exclusively consume the Mojave Desert's white stones, which are composed of calcite (mostly calcium carbonate), as opposed to the brown, grey, or other colored stones. [ 9 ] The ingestion of these white stones is attributed to the deliberate intake of additional calcium.
Furthermore, it is thought that these additional sources of food are sources of not only calcium, but also other nutrients including phosphorus, sodium , iron , copper , and selenium . [ 9 ]
It has also been hypothesized that osteophagy is a practice necessary for the maintenance of desert tortoise shells. [ 10 ] This parallels the phenomenon of osteophagy in birds, in which snail shells are ingested by egg-laying females to supplement the increased calcium needed for eggshell formation. Therefore, it would be expected that the increased physiological needs of juvenile and gravid female tortoises would also increase mineral demands and promote ingestion of bones, stones, and soil. Alternatively, the need to consume supplemental minerals may serve the purpose of detoxifying plant compounds, or may serve other purposes not related to nutrition such as to dislodge gut parasites. [ 10 ]
In the late 1800s, a then relatively unknown disease called botulism was seen in very high levels in South African cattle , especially those that grazed in pastures with low phosphorus levels. Researchers found that feeding the cattle sterile bonemeal , or maize with unnaturally high levels of phosphorus, nearly eliminated botulism. The simplest conclusion for this was that the botulism symptoms were caused by a lack of phosphorus. [ 11 ]
In the early 1900s, Sir Thomas Thieler revisited the issue, and began following herds of cattle to observe their behavior. [ 12 ] Incredibly, he found that the phosphorus-deficient cattle would eat the decomposing bones of dead cattle and other animals, and that this activity was highly correlated to botulism. Over the next several years, he was able to show that a bacterial strain living in the decomposing carcasses, Clostridium botulinum , was the true cause of the disease. [ 11 ] The cattle would eat the carcasses to replenish their phosphorus deficiency, and would contract the disease. [ citation needed ]
More recently, in 2005, it was found that cows experimentally depleted of phosphate through the extended provision of a low-phosphate diet exhibited a specific appetite for bones compared to controls who did not develop an interest in bones. After researchers increased blood plasma inorganic phosphate levels in the experimental group of cattle, the appetite for whole bones was suppressed. This experiment provided evidence for the causal link between osteophagy and phosphorus deficiency in cattle. [ 1 ]
Grizzly bears in the wild have been observed to gnaw on shed moose antlers, which can provide a valuable source of protein, calcium, and phosphorus. [ 3 ]
Grizzly bears are at the weakest point into their annual cycle following emergence from hibernation, in terms of lacking mineral and protein nutrition. Grizzly bears ( Ursus arctos ), after emerging from hibernation , may be experiencing a skewed phosphorus-to-calcium ratio due to the lack of consumption of animal resources during the period of hibernation. [ citation needed ]
In winter conditions, while grizzly bears may be able to continue to maintain calcium intake with the ingestion of plants and maintain levels of vitamin D from solar radiation, low protein availability results in phosphorus deficiency in grizzly bear diets. This lack of protein during winter conditions can be attributed to the scarcity of animal proteins, a phenomenon that occurs in many ecosystems prior to green-up, or the ending of winter conditions. Therefore, overall, bones can serve as a valuable source of minerals at times where animal protein availability is low. [ 3 ]
The resulting phosphorus deficiency in grizzly bear diets results in a skewed calcium to phosphorus ratio and creates an appetite for bone. Because this deficiency is associated with the cycle of the seasons, osteophagy in bears is likely to be a seasonal phenomenon rather than a constant dietary supplement. [ 3 ]
Giraffes rely solely on browsing to maintain their diet, which consists primarily of leafy material. [ 13 ] However, they are commonly observed supplementing their diet with bones. [ 13 ] [ 14 ] Although the exact purpose of this behavior is unknown, it is hypothesized that the ingestion of bones serves as an additional source of calcium and phosphorus . [ 13 ] While leaves usually serve as a sufficient source of these nutrients, calcium and phosphorus concentrations in the leaves vary seasonally with rainfall; the giraffes' osteophagic behavior has been observed to parallel this variance in mineral concentration. [ 13 ]
The benefits of this behavior remain unclear. Researchers have found that it is actually unlikely that the giraffes can sufficiently digest the bones to extract the calcium or phosphorus. [ 15 ] There is also evidence to suggest that osteophagy is associated with the development of kidney stones and medullary and cortical lesions in giraffes due to the nutritional imbalance in their diet. [ 13 ]
While the media often portrays domestic dogs chewing bones, this is slightly misleading. Dogs chew bones only to eat any residual meat and bone marrow left on them, so it is not truly a form of osteophagy. [ 16 ] Most modern toy "bones" for dogs are actually rawhide , which is simply dried animal skin, as animal bones are actually dangerous for dogs to chew. [ 17 ]
Osteophagic behavior has been frequently observed among several carnivorous bird species including hawks and owls , however the
motivations differ from those of the aforementioned herbivores. [ 18 ] Presumably, the bird's main purpose is to ingest the maximum amount of soft tissue from their prey as possible often resulting in the consumption of the prey's entire body. [ 18 ] The digestible materials are broken down while the indigestible material (i.e. bone) forms a pellet which is then regurgitated. [ 18 ] While the regurgitation of the bone is advantageous in that it frees space in the stomach for new prey, the behavior can be harmful in that the pellets are often larger than the digestive tract and could cause damage or obstruction. [ 18 ] In addition, the bearded vulture is a specialized bone-eater with bones making up 70–90% of its diet. [ 19 ]
Pica is the craving and consumption of non-nutrient substances that can cause health risks. [ 20 ] Osteophagy in humans would be considered a form of pica. Unlike calcium and phosphorus in most animals, pica is associated with iron deficiencies in humans. [ 21 ] Humans are unlikely to suffer from calcium and phosphorus deficiencies because the minerals are widely abundant in the foods they consume. [ 22 ]
Geophagy , the eating of earthen materials like clay , can be another form of pica that is more commonly observed than osteophagy. [ 21 ]
The Yanomami tribe live as nomads in the Brazilian and Venezuelan Amazon . [ 23 ] When a tribe member dies, it is a custom for their family to "set their spirit free" in a religious ritual. [ 23 ] During this ritual, the tribe grinds their bones to a fine ashen powder and mixes the powder into a plantain soup, which is eaten by the family of the deceased. [ 23 ] It is possible that this ritual originated as a way to increase phosphorus and other minerals in the tribe's diet, though it may just be a religious ritual without any other purpose. [ 23 ] | https://en.wikipedia.org/wiki/Osteophagy |
Ostrich oil is an oil derived from the fat of ostriches . Ostrich oil is composed of 36.51% of saturated fat , 46.75% of monounsaturated fat , and 18.24% of polyunsaturated fat . [ 1 ] Ostrich oil contains fatty acids , such as omega-3 , omega-6 , and omega-9 . [ 1 ] It also contains vitamins and minerals like vitamin E and selenium , which serve as natural antioxidants . [ 2 ] Emu oil in the USA has a similar composition to ostrich oil, but ostrich oil has a higher omega-3 content, containing 2.1% compared to 0.25% in emu oil. [ 3 ]
Ostrich oil has antibacterial properties, and is used for various skincare purposes, such as inflammation reduction. [ 4 ] Due to the moisturizing properties, ostrich oil is currently used in cosmetic formulations and food chemistry. [ 5 ] Ostrich oil is also used in the food industry as it has fatty acids and tocopherols , and a low cholesterol content. [ 6 ] | https://en.wikipedia.org/wiki/Ostrich_oil |
In mathematics , the Ostrowski–Hadamard gap theorem is a result about the analytic continuation of complex power series whose non-zero terms are of orders that have a suitable "gap" between them. Such a power series is "badly behaved" in the sense that it cannot be extended to be an analytic function anywhere on the boundary of its disc of convergence . The result is named after the mathematicians Alexander Ostrowski and Jacques Hadamard .
Let 0 < p 1 < p 2 < ... be a sequence of integers such that, for some λ > 1 and all j ∈ N ,
Let ( α j ) j ∈ N be a sequence of complex numbers such that the power series
has radius of convergence 1. Then no point z with | z | = 1 is a regular point for f ; i.e. f cannot be analytically extended from the open unit disc D to any larger open set—not even to a single point on the boundary of D .
This mathematical analysis –related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Ostrowski–Hadamard_gap_theorem |
In materials science , Ostwald's rule or Ostwald's step rule , conceived by Wilhelm Ostwald , [ 1 ] describes the formation of polymorphs . The rule states that usually the less stable polymorph crystallizes first. [ 2 ] Ostwald's rule is not a universal law but a common tendency observed in nature. [ 3 ]
This can be explained on the basis of irreversible thermodynamics, structural relationships, or a combined consideration of statistical thermodynamics and structural variation with temperature. Unstable polymorphs more closely resemble the state in solution, and thus are kinetically advantaged .
For example, out of hot water, metastable , fibrous crystals of benzamide appear first, only later to spontaneously convert to the more stable rhombic polymorph. A dramatic example is phosphorus , which upon sublimation first forms the less stable white phosphorus, which only slowly polymerizes to the red allotrope . This is notably the case for the anatase polymorph of titanium dioxide , which having a lower surface energy is commonly the first phase to form by crystallisation from amorphous precursors or solutions despite being metastable, with rutile being the equilibrium phase at all temperatures and pressures. [ 4 ]
This mineralogy article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Ostwald's_rule |
The Ostwald process is a chemical process used for making nitric acid (HNO 3 ). [ 1 ] The Ostwald process is a mainstay of the modern chemical industry , and it provides the main raw material for the most common type of fertilizer production. [ 2 ] Historically and practically, the Ostwald process is closely associated with the Haber process , which provides the requisite raw material, ammonia (NH 3 ). This method is preferred over other methods of nitric acid production, in that it is less expensive and more efficient. [ 3 ]
Ammonia is converted to nitric acid in 2 stages.
The Ostwald process begins with burning ammonia . Ammonia burns in oxygen at temperature about 900 °C (1,650 °F) and pressure up to 8 standard atmospheres (810 kPa) [ 4 ] in the presence of a catalyst such as platinum gauze, alloyed with 10% rhodium to increase its strength and nitric oxide yield, platinum metal on fused silica wool, copper or nickel to form nitric oxide (nitrogen(II) oxide) and water (as steam). This reaction is strongly exothermic , making it a useful heat source once initiated: [ 5 ]
A number of side reactions compete with the formation of nitric oxide. Some reactions convert the ammonia to N 2 , such as:
This is a secondary reaction that is minimised by reducing the time the gas mixtures are in contact with the catalyst. [ 6 ] Another side reaction produces nitrous oxide :
The platinum and rhodium catalyst is frequently replaced due to decomposition as a result of the extreme conditions which it operates under, leading to a form of degradation called cauliflowering . [ 7 ] The exact mechanism of this process is unknown, the main theories being physical degradation by hydrogen atoms penetrating the platinum-rhodium lattice, or by metal atom transport from the centre of the metal to the surface. [ 7 ]
The nitric oxide (NO) formed in the prior catalysed reaction is then cooled down from around 900˚C to roughly 250˚C to be further oxidised to nitrogen dioxide (NO 2 ) [ 8 ] by the reaction:
2NO + O 2 → 2NO 2 (Δ H = -114.2 kJ/mol) [ 9 ]
The reaction:
2NO 2 → N 2 O 4 (Δ H = -57.2 kJ/mol) [ 10 ]
also occurs once the nitrogen dioxide has formed. [ 11 ]
Stage two encompasses the absorption of nitrous oxides in water and is carried out in an absorption apparatus, a plate column containing water. [ citation needed ] This gas is then readily absorbed by the water, yielding the desired product (nitric acid in a dilute form), while reducing a portion of it back to nitric oxide: [ 5 ]
The NO is recycled, and the acid is concentrated to the required strength by distillation .
This is only one of over 40 absorption reactions of nitrous oxides recorded, [ 11 ] with other common reactions including:
And, if the last step is carried out in air:
The overall reaction is the sum of the first equation, 3 times the second equation, and 2 times the last equation; all divided by 2:
Alternatively, if the last step is carried out in the air, the overall reaction is the sum of equation 1, 2 times equation 2, and equation 4; all divided by 2.
Without considering the state of the water,
Wilhelm Ostwald developed the process, and he patented it in 1902. [ 12 ] [ 13 ] | https://en.wikipedia.org/wiki/Ostwald_process |
Ostwald ripening is a phenomenon observed in solid solutions and liquid sols that involves the change of an inhomogeneous structure over time, in that small crystals or sol particles first dissolve and then redeposit onto larger crystals or sol particles. [ 3 ]
Dissolution of small crystals or sol particles and the redeposition of the dissolved species on the surfaces of larger crystals or sol particles was first described by Wilhelm Ostwald in 1896. [ 4 ] [ 5 ] For colloidal systems, Ostwald ripening is also found in water-in-oil emulsions , while flocculation is found in oil-in-water emulsions. [ 6 ]
This thermodynamically -driven spontaneous process occurs because larger particles are more energetically favored than smaller particles. [ 7 ] This stems from the fact that molecules on the surface of a particle are energetically less stable than the ones in the interior.
Consider a cubic crystal of atoms: all the atoms inside are bonded to 6 neighbours and are quite stable, but atoms on the surface are only bonded to 5 neighbors or fewer, which makes these surface atoms less stable. Large particles are more energetically favorable since, continuing with this example, more atoms are bonded to 6 neighbors and fewer atoms are at the unfavorable surface. As the system tries to lower its overall energy, molecules on the surface of a small particle (energetically unfavorable, with only 3 or 4 or 5 bonded neighbors) will tend to detach from the particle and diffuse into the solution.
Kelvin's equation describes the relationship between the radius of curvature and the chemical potential between the surface and the inner volume:
where μ {\displaystyle \mu } corresponds to the chemical potential , σ {\displaystyle \sigma } to the surface tension , ν a t {\displaystyle \nu _{\mathrm {at} }} to the atomic volume and r {\displaystyle r} to the radius of the particle.
The chemical potential of an ideal solution can also be expressed as a function of the solute’s concentration if liquid and solid phases are in equilibrium.
where k B {\displaystyle k_{\mathrm {B} }} corresponds to the Boltzmann constant , T {\displaystyle T} to the temperature and C e q {\displaystyle C_{\mathrm {eq} }} to the solute concentration in a solution in which the solid and the liquid phase are in equilibrium.
Combining both expressions the following equation is obtained:
Thus, the equilibrium concentration, C e q {\displaystyle C_{eq}} , is lower around bigger particles than it is around smaller particles.
where r {\displaystyle r} and R {\displaystyle R} are the particles radius, and r < R {\displaystyle r<R} .
Inferring from Fick’s first law of diffusion , the particles will move from big concentrations, corresponding to areas surrounding small particles, to small concentrations, corresponding to areas surrounding large nanoparticles. Thus, the small particles will tend to shrink while the big particles will grow. As a result, the average size of the nanoparticles in the solution will grow, and the dispersion of sizes will decrease.
Therefore, if a solution is left for a long time, in the extreme case of t → ∞ {\displaystyle t\rightarrow \infty } , its particles would evolve until they would finally form a single huge spherical particle to minimize the total surface area.
The history of research progress in quantitatively modeling Ostwald ripening is long, with many derivations. [ 8 ] In 1958, Lifshitz and Slyozov [ 9 ] performed a mathematical investigation of Ostwald ripening in the case where diffusion of material is the slowest process. They began by stating how a single particle grows in a solution. This equation describes where the boundary is between small, shrinking particles and large, growing particles. They finally conclude that the average radius of the particles ⟨R⟩, grows as follows:
where
Note that the quantity ⟨R⟩ 3 is different from ⟨R 3 ⟩ , and that the statement that ⟨R⟩ goes as t 1/3 relies on ⟨R⟩ 0 being zero; but because nucleation is a separate process from growth, this places ⟨R⟩ 0 outside the bounds of validity of the equation. In contexts where the actual value of ⟨R⟩ 0 is irrelevant, an approach that respects the meanings of all terms is to take the time derivative of the equation to eliminate ⟨R⟩ 0 and t . Another such approach is to change the ⟨R⟩ 0 to ⟨R⟩ i with the initial time i having a positive value. [ citation needed ]
Also contained in the Lifshitz and Slyozov derivation is an equation for the size distribution function f(R, t) of particles. For convenience, the radius of particles is divided by the average radius to form a new variable, ρ = R(⟨R⟩) −1 .
Three years after that Lifshitz and Slyozov published their findings (in Russian, 1958), Carl Wagner performed his own mathematical investigation of Ostwald ripening, [ 10 ] examining both systems where diffusion was slow and also where attachment and detachment at the particle surface was slow. Although his calculations and approach were different, Wagner came to the same conclusions as Lifshitz and Slyozov for slow-diffusion systems. This duplicate derivation went unnoticed for years because the two scientific papers were published on opposite sides of the Iron Curtain in 1961. [ citation needed ] It was not until 1975 that Kahlweit addressed the fact that the theories were identical [ 11 ] and combined them into the Lifshitz-Slyozov-Wagner or LSW theory of Ostwald ripening. Many experiments and simulations have shown LSW theory to be robust and accurate. Even some systems that undergo spinodal decomposition have been shown to quantitatively obey LSW theory after initial stages of growth. [ 12 ]
Wagner derived that when attachment and detachment of molecules is slower than diffusion, then the growth rate becomes
where k s is the reaction rate constant of attachment with units of length per time. Since the average radius is usually something that can be measured in experiments, it is fairly easy to tell if a system is obeying the slow-diffusion equation or the slow-attachment equation. If the experimental data obeys neither equation, then it is likely that another mechanism is taking place and Ostwald ripening is not occurring.
Although LSW theory and Ostwald ripening were intended for solids ripening in a fluid, Ostwald ripening is also observed in liquid-liquid systems, for example, in an oil-in-water emulsion polymerization . [ 6 ] In this case, Ostwald ripening causes the diffusion of monomers (i.e. individual molecules or atoms) from smaller droplets to larger droplets due to greater solubility of the single monomer molecules in the larger monomer droplets. The rate of this diffusion process is linked to the solubility of the monomer in the continuous (water) phase of the emulsion. This can lead to the destabilization of emulsions (for example, by creaming and sedimentation). [ 13 ]
Inhibition of sulfathiazole crystal growth by polyvinylpyrrolidone. The polymer forms a noncondensed netlike film over the sulfathiazole crystal, allowing the crystal to grow out only through the openings of the net. The growth is thus controlled by the pore size of the polymer network at the crystal surface. The smaller the pore size, the higher is the supersaturation of the solution required for the crystals to grow. [ 14 ]
One example of Ostwald ripening is the re-crystallization of water within ice cream which gives old ice cream a gritty, crunchy texture. Larger ice crystals grow at the expense of smaller ones within the ice cream, creating a coarser texture. [ 15 ]
Another gastronomical example is the ouzo effect , where the droplets in the cloudy microemulsion grow by Ostwald ripening.
In geology , it is the textural coarsening, aging or growth of phenocrysts and crystals in solid rock which is below the solidus temperature. It is often ascribed as a process in the formation of orthoclase megacrysts , [ 16 ] as an alternative to the physical processes governing crystal growth from nucleation and growth rate thermochemical limitations.
In aqueous solution chemistry and precipitates ageing, the term refers to the growth of larger crystals from those of smaller size which have a higher solubility than the larger ones. In the process, many small crystals formed initially ( nuclei ) slowly disappear, except for a few that grow larger, at the expense of the small crystals ( crystal growth ). The smaller crystals act as fuel for the growth of bigger crystals. Limiting Ostwald ripening is fundamental in modern technology for the solution synthesis of quantum dots . [ 17 ] Ostwald ripening is also the key process in the digestion and aging of precipitates, an important step in gravimetric analysis . The digested precipitate is generally purer, and easier to wash and filter.
Ostwald ripening can also occur in emulsion systems, with molecules diffusing from small droplets to large ones through the continuous phase. When a miniemulsion is desired, an extremely hydrophobic compound is added to stop this process from taking place. [ 18 ]
Diffusional growth of larger drops in liquid water clouds in the atmosphere at the expense of smaller drops is also characterized as Ostwald ripening. [ 19 ] | https://en.wikipedia.org/wiki/Ostwald_ripening |
The Ostwald–Freundlich equation governs boundaries between two phases ; specifically, it relates the surface tension of the boundary to its curvature , the ambient temperature, and the vapor pressure or chemical potential in the two phases.
The Ostwald–Freundlich equation for a droplet or particle with radius R {\displaystyle R} is:
One consequence of this relation is that small liquid droplets (i.e., particles with a high surface curvature) exhibit a higher effective vapor pressure , since the surface is larger in comparison to the volume.
Another notable example of this relation is Ostwald ripening , in which surface tension causes small precipitates to dissolve and larger ones to grow. Ostwald ripening is thought to occur in the formation of orthoclase megacrysts in granites as a consequence of subsolidus growth. See rock microstructure for more.
In 1871, Lord Kelvin ( William Thomson ) obtained the following relation governing a liquid-vapor interface: [ 1 ]
where:
In his dissertation of 1885, Robert von Helmholtz (son of the German physicist Hermann von Helmholtz ) derived the Ostwald–Freundlich equation and showed that Kelvin's equation could be transformed into the Ostwald–Freundlich equation. [ 2 ] [ 3 ] The German physical chemist Wilhelm Ostwald derived the equation apparently independently in 1900; [ 4 ] however, his derivation contained a minor error which the German chemist Herbert Freundlich corrected in 1909. [ 5 ]
According to Lord Kelvin's equation of 1871, [ 6 ] [ 7 ]
If the particle is assumed to be spherical, then r = r 1 = r 2 {\displaystyle r=r_{1}=r_{2}} ; hence,
Note: Kelvin defined the surface tension γ {\displaystyle \gamma } as the work that was performed per unit area by the interface rather than on the interface; hence his term containing γ {\displaystyle \gamma } has a minus sign. In what follows, the surface tension will be defined so that the term containing γ {\displaystyle \gamma } has a plus sign.
Since ρ l i q u i d ≫ ρ v a p o r {\displaystyle \rho \,_{\rm {liquid}}\gg \rho \,_{\rm {vapor}}} , then ρ l i q u i d − ρ v a p o r ≈ ρ l i q u i d {\displaystyle \rho \,_{\rm {liquid}}-\rho \,_{\rm {vapor}}\approx \rho \,_{\rm {liquid}}} ; hence,
Assuming that the vapor obeys the ideal gas law , then
where:
Since M W N A {\displaystyle {\frac {MW}{N_{\rm {A}}}}} is the mass of one molecule of vapor or liquid, then
Hence
Thus
Since
then
Since p ( r ) ≈ P {\displaystyle p(r)\approx P} , then P − p ( r ) P ≪ 1 {\displaystyle {\frac {P-p(r)}{P}}\ll 1} . If x ≪ 1 {\displaystyle x\ll 1} , then log ( 1 − x ) ≈ − x {\displaystyle \log \left(1-x\right)\approx -x} . Hence
Therefore
which is the Ostwald–Freundlich equation. | https://en.wikipedia.org/wiki/Ostwald–Freundlich_equation |
Defunct
Defunct
Oswald Arnold Gottfried Spengler [ a ] (29 May 1880 – 8 May 1936) was a German polymath whose areas of interest included history , philosophy , mathematics , science , and art , as well as their relation to his organic theory of history. He is best known for his two-volume work The Decline of the West ( Der Untergang des Abendlandes ), published in 1918 and 1922, covering human history . Spengler's model of history postulates that human cultures and civilizations are akin to biological entities, each with a limited, predictable, and deterministic lifespan.
Spengler predicted that about the year 2000, Western civilization would enter the period of pre‑death emergency which would lead to 200 years of Caesarism (extra-constitutional omnipotence of the executive branch of government) before Western civilization's final collapse. [ 11 ]
Spengler is regarded as a German nationalist and a critic of republicanism , and he was a prominent member of the Weimar -era Conservative Revolution . [ 4 ] : 3–30, 63 [ 5 ] [ 3 ] While the Nazis had viewed his writings as a means to provide a "respectable pedigree" to their ideology, [ 12 ] Spengler later criticized Nazism for what he considered to be excessive racialist elements. He saw Benito Mussolini , and entrepreneurial types, like the mining magnate Cecil Rhodes , [ 13 ] as examples of the impending Caesars of Western culture —later showcasing his disappointment in Mussolini's colonialist adventures. [ 14 ]
Oswald Arnold Gottfried Spengler was born on 29 May 1880 in Blankenburg , Duchy of Brunswick , German Empire , the oldest surviving child of Bernhard Spengler (1844–1901) and Pauline Spengler (1840–1910), née Grantzow, the descendant of an artistic family. [ 15 ] [ 16 ] Oswald's elder brother was born prematurely in 1879, when his mother tried to move a heavy laundry basket, and died at the age of three weeks. Oswald was born ten months after his brother's death. [ 17 ] His younger sisters were Adele (1881–1917), Gertrud (1882–1957), and Hildegard (1885–1942). [ 15 ] Oswald's paternal grandfather, Theodor Spengler (1806–1876), was a metallurgical inspector ( Hütteninspektor ) in Altenbrak . [ 18 ]
Spengler's maternal great-grandfather, Friedrich Wilhelm Grantzow, a tailor's apprentice in Berlin, had three children out of wedlock with a Jewish woman named Bräunchen Moses ( c. 1769–1849) whom he later married, on 26 May 1799. [ 19 ] Shortly before the wedding, Moses was baptized as Johanna Elisabeth Anspachin; the surname was chosen after her birthplace— Anspach . [ 20 ] Her parents, Abraham and Reile Moses, were both deceased by then. The couple had another five children, [ 19 ] one of whom was Spengler's maternal grandfather, Gustav Adolf Grantzow (1811–1883)—a solo dancer and ballet master in Berlin, who in 1837 married Katharina Kirchner (1813–1873), a solo dancer from a Munich Catholic family; [ 20 ] the second of their four daughters was Oswald Spengler's mother Pauline Grantzow. [ 21 ] Like the Grantzows in general, Pauline was of a Bohemian disposition, and, before marrying Bernhard Spengler, accompanied her dancer sisters on tours. In appearance, she was plump. Her temperament, which Oswald inherited, was moody, irritable, and morose. [ 22 ]
When Oswald was ten years of age, his family moved to the university city of Halle . Here he received a classical education at the local Gymnasium (academically oriented secondary school), studying Greek, Latin, mathematics and sciences. Here, too, he developed his propensity for the arts—especially poetry, drama, and music—and came under the influence of the ideas of Johann Wolfgang von Goethe and Friedrich Nietzsche . [ 1 ] At 17, he wrote a drama titled Montezuma . [ 16 ]
After his father's death in 1901, Spengler attended several universities ( Munich , Berlin , and Halle ) as a private scholar, taking courses in a wide range of subjects. His studies were undirected. In 1903, he failed his doctoral thesis on Heraclitus —titled Der metaphysische Grundgedanke der heraklitischen Philosophie ( The Fundamental Metaphysical Thought of the Heraclitean Philosophy ) and conducted under the direction of Alois Riehl —because of insufficient references. He took the doctoral oral exam again and received his PhD from Halle on 6 April 1904. In December 1904, he began to write the secondary dissertation ( Staatsexamensarbeit ) necessary to qualify as a high school teacher. This became The Development of the Organ of Sight in the Higher Realms of the Animal Kingdom ( Die Entwicklung des Sehorgans bei den Hauptstufen des Tierreiches ), a text now lost. [ 23 ] It was approved and he received his teaching certificate. In 1905, Spengler suffered a nervous breakdown .
Spengler briefly served as a teacher in Saarbrücken then in Düsseldorf . From 1908 to 1911 he worked at a grammar school ( Realgymnasium ) in Hamburg , where he taught science, German history, and mathematics. Biographers report that his life as a teacher was uneventful. [ 24 ]
In 1911, following his mother's death, he moved to Munich , where he lived for the rest of his life. While there, he was a cloistered scholar, supported by his modest inheritance. Spengler survived on very limited means and was marked by loneliness. He owned no books, and took work as a tutor and wrote for magazines to earn additional income. Due to a severe heart problem, Spengler was exempted from military service. [ 16 ] During the war, his inheritance was useless because it was invested overseas; thus, he lived in genuine poverty for this period. [ citation needed ]
He began work on the first volume of The Decline of the West intending to focus on Germany within Europe. However, the Agadir Crisis of 1911 affected him deeply, so he widened the scope of his study. According to Spengler the book was completed in 1914, but the first edition was published in the summer of 1918, shortly before the end of World War I . [ 25 ] Spengler wrote about the years immediately prior to World War I in Decline :
At that time the World-War appeared to me both as imminent and also as the inevitable outward manifestation of the historical crisis, and my endeavor was to comprehend it from an examination of the spirit of the preceding centuries—not years. ... Thereafter I saw the present—the approaching World-War—in a quite other light. It was no longer a momentary constellation of casual facts due to national sentiments, personal influences, or economic tendencies endowed with an appearance of unity and necessity by some historian's scheme of political or social cause-and-effect, but the type of historical change of phase occurring within a great historical organism of definable compass at the point preordained for it hundreds of years ago. [ 26 ]
When the first volume of The Decline of the West was published, it was a wild success. [ b ] Spengler became an instant celebrity. [ 25 ] The national humiliation of the Treaty of Versailles (1919), followed by economic depression in 1923 and hyperinflation , seemed to prove Spengler right. Decline comforted Germans because it could be used as a rationale for their diminished pre-eminence, i.e. due to larger world-historical processes. The book met with wide success outside of Germany as well, and by 1919 had been translated into several other languages. [ citation needed ]
The second volume of Decline was published in 1922. In it, Spengler argued that German socialism differed from Marxism ; instead, he said it was more compatible with traditional German conservatism. Spengler declined an appointment as Professor of Philosophy at the University of Göttingen , saying he needed time to focus on writing. [ citation needed ]
The book was widely discussed, even by those who had not read it. Historians took umbrage at his unapologetically non-scientific approach. Novelist Thomas Mann compared reading Spengler's book to reading Arthur Schopenhauer 's works for the first time. Academics gave it a mixed reception. Sociologist Max Weber described Spengler as a "very ingenious and learned dilettante", while philosopher Karl Popper called the thesis "pointless". The first volume of Decline was published in English by Alfred A. Knopf in 1926, the second in 1928. [ 27 ]
In 1924, following the social-economic upheaval and hyperinflation , Spengler entered politics in an effort to bring Reichswehr General Hans von Seeckt to power as the country's leader . The attempt failed and Spengler proved ineffective in practical politics. [ citation needed ]
A 1928 Time review of the second volume of Decline described the immense influence and controversy Spengler's ideas enjoyed during the 1920s: "When the first volume of The Decline of the West appeared in Germany a few years ago, thousands of copies were sold. Cultivated European discourse quickly became Spengler-saturated. Spenglerism spurted from the pens of countless disciples. It was imperative to read Spengler, to sympathize or revolt. It still remains so". [ 28 ]
In 1931, he published Man and Technics , which warned against the dangers of technology and industrialism to culture. He especially pointed to the tendency of Western technology to spread to hostile "Colored races" which would then use the weapons against the West. [ 29 ] It was poorly received because of its anti-industrialism. [ citation needed ] This book contains the well-known Spengler quote "Optimism is cowardice". [ 30 ]
Despite voting for Hitler over Hindenburg in 1932, Spengler found the Führer vulgar. He met Hitler in 1933 and after a lengthy discussion remained unimpressed, saying that Germany did not need a heroic tenor but a real hero ". He quarreled publicly with Alfred Rosenberg , and his pessimism and remarks about the Führer resulted in isolation and public silence. He further rejected offers from Joseph Goebbels to give public speeches. However, Spengler did become a member of the German Academy that year. [ citation needed ]
The Hour of Decision , published in 1934, was a bestseller, but was later banned for its critique of National Socialism . Spengler's criticisms of liberalism [ 31 ] were welcomed by the Nazis, but Spengler disagreed with their biological ideology and anti-Semitism . [ 32 ] While racial mysticism played a key role in his own worldview, Spengler had always been an outspoken critic of the racial theories professed by the Nazis and many others in his time, and was not inclined to change his views during and after Hitler's rise to power. [ 33 ] Although a German nationalist, Spengler viewed the Nazis as too narrowly German, and not occidental enough to lead the fight against other peoples. The book also warned of a coming world war in which Western Civilization risked being destroyed, and was widely distributed abroad before eventually being banned by the National Socialist German Workers Party in Germany. A Time review of The Hour of Decision noted Spengler's international popularity as a polemicist, observing that "When Oswald Spengler speaks, many a Western Worldling stops to listen". The review recommended the book for "readers who enjoy vigorous writing", who "will be glad to be rubbed the wrong way by Spengler's harsh aphorisms" and his pessimistic predictions. [ 34 ]
On 13 October 1933, Spengler became one of the hundred senators of the German Academy . [ 35 ] [ 36 ]
Spengler spent his final years in Munich, listening to Beethoven , reading Molière and Shakespeare , buying several thousand books, and collecting ancient Turkish , Persian and Indian weapons. He made occasional trips to the Harz mountains and to Italy.
Spengler died of a heart attack on 8 May 1936, in Munich, at age 55. [ 37 ] He was buried in the Nordfriedhof in Munich .
In the introduction to The Decline of the West , Spengler cites Johann W. von Goethe and Friedrich Nietzsche as his major influences. Goethe's vitalism and Nietzsche's cultural criticism, in particular, are highlighted in his works. [ 38 ]
I feel urged to name once more those to whom I owe practically everything: Goethe and Nietzsche. Goethe gave me method, Nietzsche the questioning faculty… [ 39 ]
Spengler was also influenced by the universal and cyclical vision of world history proposed by the German historian Eduard Meyer . [ 38 ] The belief in the progression of civilizations through an evolutionary process comparable with living beings can be traced back to classical antiquity, although it is difficult to assess the extent of the influence those thinkers had on Spengler: Cato the Elder , Cicero , Seneca , Florus , Ammianus Marcellinus , and later, Francis Bacon , who compared different empires with each other with the help of biological analogies. [ 40 ]
The concept of historical philosophy developed by Spengler is founded upon two assumptions:
Spengler enumerates nine Cultures: Ancient Egyptian , Babylonian , Indian, Chinese, Greco-Roman or 'Apollonian', 'Magian' or 'Arabic' (including early and Byzantine Christianity and Islam), Mexican, Western or 'Faustian', and Russian. They interacted with each other in time and space but were distinctive due to 'internal' attributes. According to Spengler, "Cultures are organisms, and world-history is their collective biography." [ 41 ]
'Mankind'… has no aim, no idea, no plan, any more than the family of butterflies or orchids. 'Mankind' is a zoological expression, or an empty word. … I see, in place of that empty figment of one linear history which can only be kept up by shutting one’s eyes to the overwhelming multitude of the facts, the drama of a number of mighty Cultures, each springing with primitive strength from the soil of a mother region to which it remains firmly bound throughout its whole life-cycle; each stamping its material, its mankind, in its own image; each having its own idea, its own passions, its own life, will and feeling, its own death. [ 42 ]
Spengler also compares the evolution of Cultures to the different ages of human life, "Every Culture passes through the age-phases of the individual man. Each has its childhood, youth, manhood and old age." When a Culture enters its late stage, Spengler argues, it becomes a 'Civilization' ( Zivilisation ), a petrified body characterized in the modern age by technology, imperialism, and mass society, which he expected to fossilize and decline from the 2000s onward. [ 43 ] The first-millennium Near East was, in his view, not a transition between Classical Antiquity , Western Christianity , and Islam , but rather an emerging new Culture he named 'Arabian' or 'Magian', explaining messianic Judaism, early Christianity , Gnosticism , Mandaeism , Zoroastrianism , and Islam as different expressions of a single Culture sharing a unique worldview. [ 44 ]
The great historian of antiquity Eduard Meyer thought highly of Spengler, although he also had some criticisms of him. Spengler's obscurity, intuitiveness, and mysticism were easy targets, especially for the positivists and neo-Kantians who rejected the possibility that there was meaning in world history. The critic and aesthete Count Harry Kessler thought him unoriginal and rather inane, especially in regard to his opinion on Nietzsche . Philosopher Ludwig Wittgenstein , however, shared Spengler's cultural pessimism. Spengler's work became an important foundation for social cycle theory . [ 45 ]
Prussianism and Socialism ( German : Preußentum und Sozialismus [ˈpʁɔʏsn̩tuːm ʔʊnt zotsi̯aˈlɪsmʊs] ) is a 1919 book by Oswald Spengler originally based on notes intended for the second volume of The Decline of the West , in which he argues for " Prussian " socialism, characterized by an emphasis on social roles rather than capital, in contrast to mainstream socialism, which he refers to as "English" socialism. [ 46 ] [ 47 ]
Spengler responded to the claim that socialism's rise in Germany had not begun with the German revolution of 1918–1919 , but rather in 1914 when Germany waged war, uniting the German nation in a national struggle that he claimed was based on socialistic Prussian characteristics, including creativity, discipline, concern for the greater good, productivity, and self-sacrifice. [ 48 ] Spengler claimed that these socialistic Prussian qualities were present across Germany and stated that the merger of German nationalism with this form of socialism while resisting Marxist and internationalist socialism would be in the interests of Germany. [ 49 ]
Spengler claimed that socialistic Prussian characteristics existed across Germany that included creativity, discipline, concern for the greater good, productivity, and self-sacrifice. [ 48 ] He described socialism outside of a class conflict perspective and said "The meaning of socialism is that life is controlled not by the opposition between rich and poor, but by the rank that achievement and talent bestow. That is our freedom, freedom from the economic despotism of the individual." [ 50 ] Spengler addressed the need of Germans to accept Prussian socialism to free themselves from foreign forms of government:
Prussiandom and socialism stand together against the inner England , against the world-view that infuses our entire life as a people, crippling it and stealing its soul…The working class must liberate itself from the illusions of Marxism. Marx is dead. As a form of existence, socialism is just beginning, but the socialism of the German proletariat is at an end. For the worker, there is only Prussian socialism or nothing ... For conservatives, there is only conscious socialism or destruction. But we need liberation from the forms of Anglo-French democracy. We have our own. [ 50 ]
Spengler went further to demonstrate the difference between priorities held in England and Prussia:
English society is founded on the distinction between rich and poor, Prussian society on the distinction between command and obedience...Democracy in England means the possibility for everyone to become rich , in Prussia the possibility of attaining to every existing rank . [ 51 ]
Spengler denounced Marxism for having developed socialism from an English perspective, while not understanding Germans' socialist nature. [ 51 ] In the pamphlet, a central argument is that the corrupt forces promoting English socialism in his country comprised an "invisible English army, which Napoleon had left behind on German soil after the Battle of Jena ." [ 52 ]
Spengler accused Marxism of following the British tradition in which the poor envy the rich:
The socialism of a Fichte would accuse [those who don't work] of sloth, it would brand them as irresponsible, dispensable shirkers and parasites. But Marxian instinct envies them. They are too well-off, and therefore they should be revolted against. Marx has inoculated his proletariat with a contempt for work. [ 51 ]
He claimed that Marxism sought to train the proletariat to "expropriate the expropriator", the capitalist, so that the proletariat could live a life of leisure on this expropriation. [ 51 ] In summary, Spengler concluded that "Marxism is the capitalism of the working class" and not true socialism. [ 51 ]
In contrast to Marxism, Spengler claimed that "true socialism" in its German form "does not mean nationalization through expropriation or robbery." [ 51 ] Spengler justified this claim by saying:
In general, it is a question not of nominal possession but of the technique of administration. For a slogan’s sake to buy up enterprises immoderately and purposelessly and to turn them over to public administration in the place of the initiative and responsibility of their owners, who must eventually lose all power of supervision—that means the destruction of socialism. The old Prussian idea was to bring under legislative control the formal structure of the whole national productive force, at the same time carefully preserving the right of property and inheritance, and leaving scope for the kind of personal enterprise, talent, energy, and intellect displayed by an experienced chess player, playing within the rules of the game and enjoying that sort of freedom which the very sway of the rule affords…. Socialization means the slow transformation—taking centuries to complete—of the worker into an economic functionary, and the employer into a responsible supervisory official. [ 53 ]
True socialism according to Spengler would take the form of a corporatism in which "local corporate bodies organized according to the importance of each occupation to the people as a whole; higher representation in stages up to a supreme council of the state; mandates revocable at any time; no organized parties, no professional politicians, no periodic elections." [ 51 ]
Spengler was an important influence on Nazi ideology. He "provided skeletal Nazi ideas" to the early Nazi movement "and gave them a respectable pedigree". [ 12 ] Key parts of his writings were incorporated into Nazi Party ideology. [ 12 ]
Spengler's criticism of the Nazi Party was taken seriously by Hitler, and Carl Deher credited him for inspiring Hitler to carry out the Night of the Long Knives in which Ernst Röhm and other leaders of the Sturmabteilung (SA) were executed. [ 12 ] In 1934, Spengler pronounced the funeral oration for one of the victims of the Night of the Long Knives and retired in 1935 from the board of the highly influential Nietzsche Archive which was viewed as opposition to the regime. [ 33 ]
Spengler considered Judaism to be a "disintegrating element" (zersetzendes Element) that acts destructively "wherever it intervenes" (wo es auch eingreift). In his view, Jews are characterized by a "cynical intelligence" (zynische Intelligenz) and their "money thinking" (Gelddenken). [ 55 ] Therefore, they were incapable of adapting to Western culture and represented a foreign body in Europe. He also clarifies in The Decline of the West that this is a pattern shared in all civilizations: He mentions how the ancient Jew would have seen the cynical, atheistic Romans of the late Roman empire the same way Westerners today see Jews. Alexander Bein argues that with these characterizations Spengler contributed significantly to the enforcement of Jewish stereotypes in pre-WW2 German circles. [ 56 ]
Spengler viewed Nazi anti-Semitism as self-defeating, and personally took an ethnological view of race and culture. [ 12 ] In his private papers, he remarked upon "how much envy of the capability of other people in view of one's lack of it lies hidden in anti-Semitism!", and arguing that "when one would rather destroy business and scholarship than see Jews in them, one is an ideologue, i.e., a danger for the nation. Idiotic." [ 32 ]
Spengler, however, regarded the transformation of ultra-capitalist mass democracies into dictatorial regimes as inevitable, and he had expressed acknowledgement for Benito Mussolini and the Italian Fascist movement as a first symptom of this development. [ 33 ]
Spengler influenced other academics, including historians Arnold J. Toynbee , [ 57 ] Carroll Quigley , and Samuel P. Huntington . [ citation needed ] Others include fascist ideologues Francis Parker Yockey and Oswald Mosley . [ 58 ] John Calvert notes that Oswald Spengler's criticism of Western civilisation remains popular among Islamists . [ 59 ] | https://en.wikipedia.org/wiki/Oswald_Spengler |
The Oswald Veblen Prize in Geometry is an award granted by the American Mathematical Society for notable research in geometry or topology . It was funded in 1961 in memory of Oswald Veblen and first issued in 1964. The Veblen Prize is now worth US$5000, and is awarded every three years.
The first seven prize winners were awarded for works in topology. James Harris Simons and William Thurston were the first ones to receive it for works in geometry. [ 1 ] As of 2022, there have been thirty-seven prize recipients. | https://en.wikipedia.org/wiki/Oswald_Veblen_Prize_in_Geometry |
The Oswald efficiency , similar to the span efficiency , is a correction factor that represents the change in drag with lift of a three-dimensional wing or airplane, as compared with an ideal wing having the same aspect ratio and an elliptical lift distribution. [ 1 ]
The Oswald efficiency is defined for the cases where the overall coefficient of drag of the wing or airplane has a constant+quadratic dependence on the aircraft lift coefficient
where
For conventional fixed-wing aircraft with moderate aspect ratio and sweep, Oswald efficiency number with wing flaps retracted is typically between 0.7 and 0.85. At supersonic speeds, Oswald efficiency number decreases substantially. For example, at Mach 1.2 Oswald efficiency number is likely to be between 0.3 and 0.5. [ 1 ]
It is frequently assumed that Oswald efficiency number is the same as the span efficiency factor which appears in lifting-line theory , and in fact the same symbol e is typically used for both. But this assumes that the profile drag coefficient is independent of C L {\displaystyle C_{L}} , which is certainly not true in general. Assuming that the profile drag itself has a constant+quadratic dependence on C L {\displaystyle C_{L}} ,
an alternative drag coefficient breakdown can be given by [ citation needed ]
where
Equating the two C D {\displaystyle C_{D}} expressions gives the relation between the Oswald efficiency number e 0 and the lifting-line span efficiency e .
For the typical situation c d 2 > 0 {\displaystyle c_{d_{2}}>0} , we have e 0 < e {\displaystyle e_{0}<e} .
This engineering-related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Oswald_efficiency_number |
Othala ( ᛟ ), also known as ēðel and odal , is a rune that represents the o and œ phonemes in the Elder Futhark and the Anglo-Saxon Futhorc writing systems respectively. Its name is derived from the reconstructed Proto-Germanic * ōþala- "heritage; inheritance, inherited estate". As it does not occur in Younger Futhark , it disappears from the Scandinavian record around the 8th century, but its usage continued in England into the 11th century, where it was sometimes further used in manuscripts as a shorthand for the word ēðel ("homeland"), similarly to how other runes were sometimes used at the time.
As with other symbols used historically in Europe such as the swastika and Celtic cross , othala has been appropriated by far-right groups such as the Nazi party and neo-Nazis , who have used it to represent ideas like Aryan heritage, a usage that is wholly modern and not attested in any ancient or medieval source. The rune also continues to be used in non-racist contexts, both in Heathenry and in wider popular culture such as the works of J.R.R. Tolkien and video games.
The sole attested name of the rune is Old English : ēþel , meaning "homeland". Based on this, and cognates in other Germanic languages such as Old Norse : óðal and Old Frisian : ēthel , the Proto-Germanic : * ōþalą can be reconstructed, meaning "ancestral land", "the land owned by one's kin", and by extension "property" or "inheritance". * ōþalą is in turn derived from Proto-Germanic : * aþalą , meaning "nobility" and "disposition". [ citation needed ]
Terms derived from * ōþalą are formative elements in some Germanic names , notably Ulrich . [ citation needed ]
The term "odal" ( Old Norse : óðal ) refers to Scandinavian laws of inheritance which established land rights for families that had owned that parcel of land over a number of generations, restricting its sale to others. Among other aspects, this protected the inheritance rights of daughters against males from outside the immediate family. [ 1 ] Some of these laws remain in effect today in Norway as the Odelsrett ( allodial right ). The tradition of Udal law found in Shetland , Orkney , and the Isle of Man , is from the same origin. [ citation needed ]
The o -rune is attested early, in inscriptions from the 3rd century, such as the Thorsberg chape ( DR7 ) and the Vimose planer ( Vimose-Høvelen , DR 206 ). [ citation needed ] The corresponding Gothic letter is 𐍉 (derived from Greek Ω ), which had the name oþal . [ citation needed ] The othala rune is found in some transitional inscriptions of the 6th or 7th century, such as the Gummarp , Björketorp and Stentoften runestones, but it disappears from the Scandinavian record by the 8th century. The Old Norse o phoneme at this time becomes written in Younger Futhark in the same way as the u phoneme, with the Ur rune . [ citation needed ]
It has been suggested that the othala rune on the Ring of Pietroassa is used to represent the word "*oþal", referencing the ring as hereditary treasure. [ 2 ] Similarly, Wolfgang Krause speculated that the o rune is used as an ideograph denoting possession in the Thorsberg chape inscription, reading the inscription owlþuþewaz as O[þila] - W[u]lþu-þewaz "inherited property - the servant of Wulþuz ". [ 3 ] [ 4 ] [ 5 ] [ 6 ]
The Anglo-Saxon runes preserve the full set of 24 Elder Futhark runes (as well as introducing innovations), but in some cases these runes are given new sound values due to Anglo-Frisian sound changes. The othala rune is such a case: the o sound in the Anglo-Saxon system is now expressed by ōs ᚩ, a derivation of the old Ansuz rune ; the othala rune is known in Old English as ēðel (with umlaut due to the form ōþila- ) and is used to express an œ sound, but is attested only rarely in epigraphy (outside of simply appearing in a futhark row). [ citation needed ] In some runic inscriptions, such as on the Seax of Beagnoth , and more commonly in manuscripts, othala is written with a single vertical line instead of the two diagonal legs, perhaps due to its simpler form. [ 7 ]
The rune is also used as a shorthand for the word ēþel or œþel ("ancestral property or land") in texts such as Beowulf , Waldere and the Old English translation of Orosius ' Historiae adversus paganos . [ 8 ] [ 9 ] This is similar to wider practices of the time, in which runes such as ᛞ , ᚹ and ᛗ were also used as shorthands to write the name of the rune. [ 9 ]
Epigraphical attestations include:
The Anglo-Saxon rune poem preserves the meaning "an inherited estate" for the rune name:
ᛟ bẏþ oferleof æghƿẏlcum men, gif he mot ðær rihtes and gerẏsena on brucan on bolde bleadum oftast.
[An estate] is very dear to every man, if he can enjoy there in his house whatever is right and proper in constant prosperity.
The othala rune, like some other runes, was adopted as an occult symbol by German Nazi occultists and thereof in the 1930s, later being adopted by the German Schutzstaffel (SS) as an SS-rune to symbolise kinship, family and blood ties within the Aryan race . The SS modified the symbol with serifs , also called "feet" or "wings", subsequently being nicknamed "Winged Othala" and thereof in modern times. It was subsequently used by various military divisions within the German Army during World War II and also became the badge of the SS Race and Settlement Main Office , which was responsible for maintaining the racial purity of the SS. [ 10 ]
After World War II, this symbol has seen continued by Neo-Nazis and similar far-right collectives. White supremacists who use the rune often claim it symbolises the heritage or land of " white " or " Aryan " people which should be free from foreigners. Usages such as these are not attested in any source from before the modern period, being invented by members of these groups. [ 11 ]
Othala is widely used in popular culture, including by J.R.R. Tolkien along with other historical runes in The Hobbit , as seen on Thror's map of Erebor . These further form the base for the dwarvish Cirth writing systems used in The Lord of the Rings and described in Tolkien's Legendarium . [ 12 ] [ 13 ] The rune is also used as the symbol for the "Lore" resource in Northgard , released in 2018, [ 14 ] and in Stargate SG-1 , Othala is a world in the Ida Galaxy where the Asgard had lived. [ citation needed ] The Anti-Defamation League notes that because it is part of the runic alphabet, the othala rune is often used in non-racist manners, such as these, and should be interpreted in conjunction with its context. [ 15 ]
Othala, along with other runes more widely, often feature prominently in the practices of Heathens , [ 16 ] [ 17 ] [ 18 ] and are commonly used to decorate items and in tattoos. [ 19 ] The use of runes such as othala by far-right groups has been strongly condemned by some Heathen groups, including Asatru UK which released a public statement that "[it] is categorically opposed to fascist movements, or any movements, using the symbols of our faith for hate". [ 20 ] | https://en.wikipedia.org/wiki/Othala |
The Otto-Naegeli-Preis is a Swiss award for medical research that is awarded every two years. [ 1 ] It is one of the most prestigious Swiss medical awards and is given with an award sum of 200,000 Swiss Francs . It was established in 1960 and is named after Otto Naegeli , a former professor of internal medicine at the University of Zurich . [ 1 ]
The Awardees of the prize are the following: [ 2 ] | https://en.wikipedia.org/wiki/Otto-Naegeli-Preis |
Ludwig Otto Hesse (22 April 1811 – 4 August 1874) was a German mathematician . Hesse was born in Königsberg , Prussia , and died in Munich , Bavaria . He worked mainly on algebraic invariants , and geometry . The Hessian matrix , the Hesse normal form , the Hesse configuration , the Hessian group , Hessian pairs , Hesse's theorem , Hesse pencil , and the Hesse transfer principle [ 2 ] are named after him. Many of Hesse's research findings were presented for the first time in Crelle's Journal or Hesse's textbooks. [ 3 ]
Hesse was born in Königsberg (today Kaliningrad ) as the son of Johann Gottlieb Hesse, a businessman and brewery owner and his wife Anna Karoline Reiter (1788–1865). He studied in his hometown at the Albertina under Carl Gustav Jacob Jacobi . Among his teachers were count Friedrich Wilhelm Bessel and Friedrich Julius Richelot . He earned his doctorate in 1840 at the University of Königsberg with the dissertation De octo punctis intersectionis trium superficium secundi ordinis . In 1841, Hesse completed his habilitation thesis. In the same year he married Sophie Marie Emilie Dulk, the daughter of pharmacists and chemistry professor Friedrich Philipp Dulk (1788–1852). The couple had a son and five daughters. Hesse taught for some time physics and chemistry at the Vocational School in Königsberg and lectured at the Albertina. In 1845 he was appointed associate professor in Königsberg. In 1855 he moved to Halle and in 1856 to Heidelberg until 1868, when he finally moved to Munich to the newly established Polytechnic School . In 1869 he joined the Bavarian Academy of Sciences .
His doctoral students include Olaus Henrici , Gustav Kirchhoff , Jacob Lüroth , Adolph Mayer , Carl Neumann , Max Noether , Ernst Schröder , and Heinrich Martin Weber . [ 1 ]
His collected works were published in 1897 by Bavarian Academy of Sciences and Humanities . | https://en.wikipedia.org/wiki/Otto_Hesse |
The Otto Laporte Award (1972–2003) was an annual award by the American Physical Society (APS) to "recognize outstanding contributions to fluid dynamics " and to honour Otto Laporte (1902–1971). It was established as the Otto Laporte Memorial Lectureship by the APS Division of Fluid Dynamics in 1972, and became an APS award in 1985. The Otto Laporte Award was merged into the Fluid Dynamics Prize in 2004, in order to obtain one major prize in fluid dynamics by the APS.
This fluid dynamics –related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Otto_Laporte_Award |
Otto Neubauer (8 April 1874 – 24 November 1957) was a Bohemia-born physician and biochemist who was responsible for several clinical diagnostic innovations including the Neubauer-Fischer test to evaluate kidney function and the Neubauer counting chamber.
Neubauer was born in Karlsbad (then in Bohemia) to physician Wolfgang and Hedwig Arnstein née Sadler. In 1892 he passed the examination for qualifying admission to a university after studying at the humanistic gymnasium of Chomutov. He then went to the German University in Prague he received a medical degree in 1898 and became interested in physiological chemistry through the influence of Karl H. Huppert . He then joined as an assistant to Friedrich von Müller at Basel. He moved to Munich in 1902. In 1908 he joined the University of Munich and served in a reserve hospital during World War I . His major work in this period was on amino acid metabolism in human health and disease. [ 1 ] [ 2 ] Neubauer and Konrad Fromherz examined the role of pyruvic acid in fermentation. [ 3 ] He innovated several clinical diagnostics including tests of peptolytic activity. Gastric juice incubated with glycyl-tryptophan for twenty four hours tested with bromine to see if free tryptophan causes a rose-violet colour was used as an indication of stomach carcinoma. In 1918 he became head physician at Schwabinger Hospital, working there until his dismissal by the Nazi government in June 1933 as a person of Jewish ancestry. In 1920 he developed a blood pressure measuring device and still later a measuring slide (known as a Neubauer slide or Neubauer counting chamber [ 4 ] ) for counting cells under a microscope. With assistance and support from The Society for the Protection of Science and Learning, he emigrated to England in 1939 along with his wife Lilly Caroline (1876-1962, who was married to composer Fritz Cassirer until his death) and worked in Oxford for the remainder of his life. His contributions included studies on arsenic [ 5 ] and other chemicals [ 6 ] as carcinogens. [ 7 ] [ 8 ]
Neubauer's students included Siegfried Thannhauser, Rudolf Schindler, and Konrad Dobriner. [ 7 ] [ 9 ] | https://en.wikipedia.org/wiki/Otto_Neubauer |
Otto Perutz (27 July 1847 – 18 January 1922, Munich ) was an Austrian - German chemist .
Otto Perutz was born on 27 July 1847 in Teplice , Bohemia , Austrian Empire . From 1872 to 1876, Perutz was director of Bayerische Aktiengesellschaft für chemische und landwirtschaftlich-chemische Fabrikate (Bavarian Corporation for Chemical and Agrochemical Products Inc., later Süd-Chemie AG) in Munich-Heufeld. [ 1 ]
On 13 April 1880, he purchased the Chemische und pharmacheuische Produktenhandlung Dr. F. Snitter & Co. , a merchant in photochemicals in Munich , [ 1 ] and founded his own firm, Otto Perutz Trockenplattenfabrik . He developed a method for the industrial production of Eosin-Silver-Plates which had been invented by Hermann Wilhelm Vogel and Johann Baptist Obernetter, and was licensed by Vogel to produce dry plates. [ 1 ] This was crucial for the development of colour photography. [ citation needed ] They were an immediate success when they were introduced in August 1887. [ 1 ] In 1896, Perutz-Plates were used for radiography for the first time. [ 1 ]
Perutz sold his firm on 1 July 1897. [ 1 ] He was a member of the supervisory board of the Bayerische Aktiengesellschaft für chemische und landwirtschaftlich-chemische Fabrikate from 1902 until his death in 1922. [ 1 ] He died on 18 January 1922 in Munich. The Perutz-Photowerke became part of Agfa in 1964. [ 1 ] | https://en.wikipedia.org/wiki/Otto_Perutz |
Otto Scherzer (9 March 1909 – 15 November 1982) was a German theoretical physicist who made contributions to electron microscopy , including the Scherzer's theorem for the spherical aberration of electronic lenses.
Scherzer studied physics at the Munich Technical University [ 1 ] and the Ludwig Maximilian University of Munich (LMU) from 1927 to 1931. At LMU his thesis advisor was Arnold Sommerfeld , [ failed verification ] and he was granted his doctorate in 1931. His thesis was on the quantum theory of Bremsstrahlung , titled " Über die Ausstrahlung bei der Bremsung von Protonen und schnellen Elektronen ". [ 2 ] [ 3 ] From 1932 to 1933, Scherzer was an assistant to Carl Ramsauer at the Allgemeine Elektrizitäts-Gesellschaft , an electric combine with headquarters in Berlin and Frankfurt-on-Main . There, he did research on electron optics. [ 4 ] He completed his Habilitation in 1934, and he then became a Privatdozent at LMU and an assistant to Sommerfeld. [ 5 ] [ 6 ]
In 1935, Scherzer moved to the Technische Hochschule Darmstadt . [ 7 ] In 1936, he became an extraordinarius professor and director of the theoretical physics department. [ 8 ] In a landmark 1936 paper, Scherzer proved that the spherical and chromatic aberrations of a rotationally symmetric, static, space-charge-free, dioptric lens for electron beams cannot be eliminated by skillful design, in contrast to the case for glass lenses. [ 9 ] This was later called Scherzer's theorem and is the only named and well-established theorem in the field of charged particle optics. [ 10 ] In 1947, Scherzer published a sequel to this paper proposing various corrected lenses, dependent upon abandoning one or other requirements as set forth in the 1936 paper. [ 11 ] Scherzer’s derivations contributed to the development of electron microscopy.
From 1939 to 1945, Scherzer worked on radar at the communications research headquarters of the Kriegsmarine . [ 8 ] In a communication with Sommerfeld, dated 2 December 1944, Scherzer reported war damage in Darmstadt and commented on his work on radar . [ 12 ] From 1944 to 1945, Scherzer was head of radar finding research ( Arbeitsbereich Funkmesstechnik ) for the Reich Research Council ( Reichsforschungsrat ), [ 13 ] which was the coordinating agency in the Reich Education Ministry ( German : Reichsziehungsministerium ) for the centralized planning of basic and applied research. [ 14 ]
In 1954, Scherzer became ordinarius professor at the Technische Hochschule Darmstadt , where he helped found the Society for Heavy Ion Research. [ 8 ] A literature citation places Scherzer at Darmstadt as late as 1978. [ 15 ] Scherzer died in Darmstadt . | https://en.wikipedia.org/wiki/Otto_Scherzer |
Otto Franz Georg Schilling (3 November 1911 – 20 June 1973) was a German-American mathematician known as one of the leading algebraists of his time. [ 1 ]
He was born in Apolda and studied in the 1930s at the Universität Jena and the Universität Göttingen under Emmy Noether . After Noether was forced to leave Germany by the Nazis, he found a new advisor in Helmut Hasse , [ 2 ] and obtained his Ph.D. from Marburg University in 1934 on the thesis Über gewisse Beziehungen zwischen der Arithmetik hyperkomplexer Zahlsysteme und algebraischer Zahlkörper . [ 3 ] He then was post doc at Trinity College, Cambridge before moving to Institute for Advanced Study 1935–37 [ 4 ] and the Johns Hopkins University 1937–39. He became an instructor with the University of Chicago in 1939, [ 2 ] promoted to assistant professor 1943, associate 1945 and full professor in 1958. In 1961 he moved to Purdue University . He died in Highland Park, Illinois . His students were, among others, the game theorist Anatol Rapoport and the mathematician Harley Flanders . [ 3 ] | https://en.wikipedia.org/wiki/Otto_Schilling |
Friedrich Otto Schott (1851–1935) was a German chemist , glass technologist, and the inventor of borosilicate glass . Schott systematically investigated the relationship between the chemical composition of the glass and its properties. In this way, he solved fundamental problems in glass properties, identifying compositions with optical properties that approach the theoretical limit. Schott's findings were a major advance in the optics for microscopy and optical astronomy . [ 1 ] His work has been described as "a watershed in the history of glass composition". [ 2 ]
Schott was the son of a window glass maker, Simon Schott. His mother was Karoline Schott. [ 3 ] From 1870 to 1873 Schott studied chemical technology at the technical college in Aachen and at the University of Würzburg and at the University of Leipzig . He earned a doctorate in chemistry at Friedrich Schiller University of Jena , specializing in glass science. His doctoral thesis was entitled “Contributions to the Theory and Practice of Glass Fabrication” (1875). [ 4 ] [ 5 ] [ 6 ]
In 1879, Schott developed a new lithium -based glass that possessed novel optical properties. Schott shared this discovery with Ernst Abbe , a professor of physics at Jena University whose comments on glass had stimulated Schott's interest in the subject. [ 2 ] [ 7 ]
I recently produced a glass, in which a considerable amount of lithium was introduced, and the specific gravity of which was relatively low. I suspect that such a glass will exhibit excellent optical properties, and I therefore wanted to inquire whether you or one of your colleagues might be willing to test it for refractive index and dispersion to determine whether my above supposition is correct.
Not long after Schott had completed his formal university training, he had become aware that Abbe had articulated the deficiencies in glass that was available at the time. The deficiencies were particularly acute in scientific instruments for which optical performance of the glass in lenses such as for telescopes and microscopes is paramount. Scientifically, as the magnification power of the lenses were increased, chromatic aberration became large. Chromatic aberration causes the optical quality of the visual image to become dependent on the color of the light, resulting in a significant limitation of the scientific instrument. [ 1 ] [ 8 ]
In response to Abbe's scientific provocation, Schott began a systematic investigation of the properties of glass as the properties varied with the chemical composition. Schott substituted one element for another, such as borate and phosphate for a portion of the silica in the glass and substituting fluoride for oxygen. [ 2 ]
Schott's 1879 letter to Abbe was the beginning of a long collaboration between the two scientists. [ 2 ] Abbe was already working with Carl Zeiss , an instrument-maker, on the making of glass for microscopes. Zeiss participated in the three-way collaboration by testing improved glass compositions that Schott and Abbe identified in actual optical instruments, such as telescopes. [ 2 ] In 1882, Schott moved to Jena, where he could work more closely with Abbe and Zeiss. [ 2 ]
They created types of glass and examined their properties using silica, soda, potash, lime, lead oxide and 28 other elements. Lacking a theoretical basis for the work, they relied on careful and systematic observation and measurement. The addition of elements that had no direct effect on optical properties might help to correct other properties of a glass such as the occurrence of surface staining when exposed to air. [ 2 ]
By 1886, Schott had completed thorough investigations of structure-property relationships in glass compositions. Through these investigations, Schott discovered that the refractive index of a glass (important to its ability to function as a magnifying lens) could be disconnected from its chromatic aberration. In this way, Schott settled on a lithium-containing glass that could perform close to its theoretical limit in scientific instruments, which was a significant advance in optical instrumentation such as for microscopy and astronomy. [ 1 ]
By mastering the process of small-scale melt-stirring, Schott was able to create a homogeneous product, whose refractive index and dispersion could be exactly measured and characterized. Through systematic experiment, he applied this to the creation of an array of different glass types. Based on his experiments, Schott worked with A. Winkelmann to develop the first composition-property model for the calculation of glass properties . [ 2 ]
Schott systematized the chemical composition of a significant range of glass compositions. Representative examples are summarized in the table.
In 1884, in association with Dr. Ernst Abbe and Carl Zeiss , Otto founded Glastechnische Laboratorium Schott & Genossen (Schott & Associates Glass Technology Laboratory) in Jena. It was here, during the period 1887 through to 1893, that Schott developed borosilicate glass. Borosilicate glass is distinguished for its high tolerance to heat and a substantial resistance to thermal shock resulting from sudden temperature changes and resistance to degradation when exposed to corrosive chemicals. This type of glass initially became known under the brand name Duran . Their business enterprise also commercialized apochromatic lenses that had low chromatic aberration and was based on Schott's systematic investigations of the composition and properties of glass. [ 1 ] [ 9 ] [ 10 ]
Schott used borosilicate glass to make laboratory and medical supplies, including thermometers, glassware for laboratory use, medicine vials and pharmaceutical tubing. Schott produced domestic glassware under the brandname "Jenaer Glas". He also produced heat resistant lamp cylinders for use in gas lighting . Carl Auer 's incandescent gas lamps were first sold in 1894 and became a lucrative source of income for Schott's glassworks. [ 11 ] In late 1890s he was also involved in the electrification of the industry in Jena. [ 12 ] Schott's business enterprise held a near monopoly on global optical glass from its inception until the start of World War I . [ 2 ]
In 1919, Schott & Associates became wholly owned by the Carl Zeiss Foundation, although Schott & Associates is known in the early 21st century as Schott AG . [ 13 ] The Schott Company's brand became associated with high quality and specialty optics. [ 3 ]
As of 2020, vials made of glass from Schott AG were being used in vaccination efforts against COVID-19 disease . [ 14 ]
In 1917, Otto Schott's eldest son, Rolf Schott, was killed in World War I . Shortly thereafter, Otto's son Erich Schott joined Schott & Gen. [ 15 ] In 1926, Otto Schott retired from active work at Schott & Gen. Shortly thereafter, Erich Schott took over Otto Schott's responsibilities in managing the company. [ 16 ]
In 1909, Schott received the Liebig Medal from the Association of German Chemists. [ 17 ] Otto-Schott-Straße in Jena, Germany, the location of Schott's home, was renamed in Schott's honor. The Schott Glass Museum is on the same premises. Both can be visited. The Schott Glass Museum displays developments in glass science beginning with the innovations of Otto Schott. [ 18 ]
Since 1991, the Otto Schott Research Award has been presented every two years to meritorious researchers in the field of glass science and ceramics science. The award is organized and funded by the Abbe Fund of the Carl Zeiss Foundation . [ 19 ] | https://en.wikipedia.org/wiki/Otto_Schott |
The Otto Warburg Medal is awarded annually by the German Society for Biochemistry and Molecular Biology (German: Gesellschaft für Biochemie und Molekularbiologie or GBM ) to honour scientists who have contributed important work in the field of biological chemistry. It is named after Otto Warburg , a renowned German physiologist and Nobel Prize laureate. It was first awarded on his 80th birthday on 8 October 1963. [ 1 ]
Up to 2025, ten Warburg Medal recipients have also been awarded the Nobel Prize.
Source: GBM | https://en.wikipedia.org/wiki/Otto_Warburg_Medal |
Ottobock SE & Co. KGaA , formerly Otto Bock , is an international company based in Duderstadt Germany, that operates in the field of orthopedic technology. It is considered the world market leader in the field of prosthetics and one of the leading suppliers in orthotics , wheelchairs and exoskeletons . [ 2 ]
Näder Holding GmbH & Co. KG is entirely owned by the Näder family, direct descendants of the company's founder, Otto Bock. Näder Holding controls 80% of the shares in Ottobock SE & Co. KGaA. The remaining 20% of the shares were previously held by the Swedish financial investor EQT. However, in March 2024, it was announced that Näder Holding had repurchased these shares from EQT for EUR 1.1 billion. [ 3 ]
In 2022, the Ottobock Group as a whole generated sales of €1,3 billion. [ 4 ] As of 2019 [update] they had 8,367 employees worldwide. [ 5 ]
The company was founded on January 13, 1919 [ 6 ] under the appellation Orthopädische Industrie GmbH, headquartered in Berlin . Initiated by a group surrounding a manufacturer named Otto Bock, who hailed from Krefeld , [ 7 ] its objective was to supply prostheses and orthopedic products to the many thousands of war invalids of World War I . [ 8 ] Bock acted as production manager during this phase.
In 1920, production operations were relocated to Königsee in Thuringia , where at times, a workforce of up to 600 people was employed. Faced with high demand that could not be met with traditional handicraft or artisanal methods, Otto Bock began the mass-production of prosthetic components, thus laying the foundation for the orthopedic industry. Bock moved into the management of the company in 1924 and finally took over as sole managing director in 1927. [ 7 ] With the evolution of the industry, new materials began to be used in production, notably aluminum , which found early application in prosthetic construction during the 1930s.
In May 1933, Bock joined the NSDAP . During the 1930s he became a supporting member of the SS. He paid a monthly contribution of six Reichmarks, according to his own account, until 1938. At the end of 1933, Bock had Orthopädische Industrie GmbH liquidated and paid out the remaining shareholders. The company was renamed Orthopedic Industry Otto Bock in Königsee . [ 8 ] [ 7 ]
Max Näder , after completing his secondary education in 1935, commenced his professional journey by undertaking training as an orthopedic mechanic and industrial clerk at Otto Bock. During his later studies in Berlin, he became part of the National Socialist German Student Association (NSDStB). During the African campaign, Näder was awarded the Iron Cross II Class. [ 7 ] In 1943, while on leave, he married Maria Bock, the daughter of Otto Bock.
During the tumult of World War II , the company resorted to the utilisation of forced laborers to sustain its operational endeavors. [ 9 ] A company chronicle quotes former employees as saying that from 1942 onwards, around 100 Russian women aged between 18 and 22 were employed in the bandage, sewing and timber departments. Letters from Marie Bock also suggest that the entrepreneurial family used forced labourers not only in the company but also in the private household. [ 7 ]
After World War II, when all the family's private assets as well as the factory in Königsee had been confiscated by the Soviet occupiers, [ 9 ] the company settled in Duderstadt in southern Lower Saxony in 1946. [ 10 ] In 1950, plastics were introduced into production for the first time. The invention of a braking knee joint with high stability, called the Jüpa knee, brought the economic breakthrough after 1949. [ 11 ] Together with a newly developed balance device and two other apparatuses for prosthetic alignment, it was also in demand on the American market. In 1955, Ottobock exported the first 500 Jüpa knees to the U.S. [ 12 ] [ 13 ] The establishment of an American branch in Minneapolis in 1958 marked the beginning of the company's international sales structure.
In 1965, Max Näder introduced myoelectric arm prostheses to the market. For the first time, light and fragile as well as heavy objects could be grasped with them. In myoelectrics, weak electrical voltages control the prosthesis. [ 14 ] [ 15 ] Another development was a fitting solution for modular leg prostheses. The pyramid adapter, patented in 1969, connects the prosthetic foot, knee joint and stem and allows static corrections as well as the exchange of the modules. It remains an integrative element of innovative joints to this day. [ 16 ]
After reunification , Hans Georg Näder took over the management of the family company from his father Max Näder, the son-in-law of company founder Otto Bock, in 1990. In the same year, the company was able to reacquire the old Ottobock site in Königsee. Today, manual wheelchairs, power wheelchairs, rehabilitation products for children and seat shell bases are produced at the former headquarters .
After a five-year development period, the world's first microprocessor -controlled knee joint, the C-Leg, was presented at the World Prosthetics Congress in Nuremberg in 1997. [ 17 ] The company's 90th anniversary was also marked by the launch of the C-Leg.
To mark the company's 90th anniversary, the newly built Science Center Medical Technology was inaugurated in Berlin in June 2009. [ 6 ] Until 2019, this building near Potsdamer Platz served both as a venue for the public exhibition Begreifen, was uns bewegt , and as a venue for congresses and seminars . [ 18 ] On January 1, 2009, the subsidiary Otto Bock Mobility Solutions GmbH based in Königsee emerged from the HealthCare division. At the end of 2011, the old logo with the original signature of Otto Bock was replaced by a new international logo. [ 19 ]
Advancements in electronic knee joint components and mechatronic prosthetic feet, led to enhanced individual fitting and personalised care for recipients. In 2011, these technological improvements enabled prosthetic users to walk backwards safely, overcome obstacles, or climb stairs in alternating steps for the first time. [ 20 ]
In 2016, Ottobock was banned from operating in parts of Bosnia following an investigation by the Centre for Investigative Reporting that revealed the company was implicated in a scandal involving the misuse of public health funds in which prosthetic limb users were forced to buy Ottoboc's products. [ 21 ] [ 22 ]
In February 2017, Ottobock purchased the myoelectric arm or hand prostheses developed under the product name BeBionic from the British medical technology company Steeper . Since May 2017, the prostheses have been part of Ottobock's product range. [ 23 ] [ 24 ] In April 2017, Ottobock acquired Boston-based BionX Medical Technologies that manufactured a prosthetic foot and ankle product that utilises robotics technology. [ 25 ] In June 2017, Swedish venture capitalist , EQT , acquired a 20 percent stake in Ottobock. [ 26 ]
In 2018, Ottobock expanded its presence in the orthopaedic technology market, acquiring a 51 percent stake in Pohlig GmbH, a medium-sized orthopedic company based in Traunstein , Bavaria , and one of the most important orthopedic technology companies in Germany. [ 27 ] In the same year, Pohlig GmbH became a wholly owned subsidiary of Ottobock. [ 28 ]
During the period from 2012 to 2018, Hans Georg Näder withdrew substantial sums from Ottobock, exceeding the company's generated profits. [ 29 ] [ 30 ] This financial practice led to a significant decline in Ottobock's equity ratio, which dropeed from 50% in 2011 [ 31 ] to 16% by 2021. [ 32 ]
In late 2018, Ottobock's subsidiary, Sycor, planned a merger with the IT service provider Allgeier Enterprise Services . [ 33 ] However, Ottobock cancelled the merger at the beginning of 2019. [ 34 ] Following a series of acquisitions, Ottobock reported in 2019 that for the first time in its history the company's sales exceeded €1 billion. [ 35 ] [ 36 ]
In November 2019, Ottobock was compelled to sell the U.S.-based prothesis manufacturer Freedom Innovations LLC and divest all assets acquired via its purchase of the industry competitor in 2017. [ 37 ] This sale was mandated after the U.S. Federal Trade Commission (FTC) filed an anti-competitive complain against Ottobock for breaking competition laws, incurring a damage of €78.1 million to Ottobock. [ 38 ] [ 39 ] [ 37 ] The shares of Freedom Innovations were subsequently acquired by the French prosthesis manufacturer Proteor. [ 40 ] In December 2019, the European Investment Bank (EIB) announced that it will provide up to €100 million to Ottobock to support the company's development of new products. [ 41 ]
In 2018, Ottobock's new generation of orthoses incorporated sensor technology to regulate the stance and swing phases of the leg throughout the gait cycle, enabling an almost natural walking pattern. [ 42 ] Additionally, the company introduced an exoskeleton, the first product of the new Ottobock Bionic Exoskeletons business unit, designed to reduce strain during overhead work. [ 43 ] [ 44 ] Ottobock expanded its exoskeletons business after acquiring US-based exoskeleton startup SuitX, a spinoff from Berkeley Robotics and Human Engineering Laboratory , in November 2021. [ 45 ]
At the end of 2021, Ottobock announced plans for an initial public offering (IPO) slated for 2022. [ 46 ] [ 47 ] However, this IPO was repeatedly postponed throughout the following year, accompanied by significant changes in the company's executive leadership. [ 48 ] By the end of 2022, Handelsblatt reported that the IPO had been abandoned due to unfavourable market conditions and that the financial investor EQT was considering a direct sale of its shares. [ 49 ] [ 50 ]
In May 2020, an Ottobock subsidiary based in Russia was fined by Russian anti-monopoly authorities for suspected cartel collusion which gave Ottobock and its co-conspirators a monopoly over state tenders for prosthetics, worth 168.1 million Russian Roubles. [ 51 ] [ 52 ]
In June 2023, it was announced that EQT, with the assistance of JP Morgan , had initiated the sale of its 20% stake in Ottobock. Additionally, a 10% stake held by Hans Georg Näder was included as part of the planned transaction. [ 53 ] In December 2023, Näder Holding declared its intention to repurchase all of its shares. This buyback was completed in March 2024, [ 54 ] [ 55 ] with Näder securing €1.1 billion in credit funds for the transaction. [ 56 ] Prior to the buyback, Hans Georg Näder sold his company Sycor, which had acquired Näder Ventures GmbH from Ottobock at the beginning of 2021. [ 57 ]
In March 2023, Ottobock expanded its operations by acquiring the Brillinger chain of medical supply stores. [ 58 ] That same month, Cranial Technologies filed a patent infringement lawsuit against Ottobock and Active Life, alleging that the companies used a patented process for 3D printing certain components of infant cranial helmets without authorisation. [ 59 ]
Despite Russian invasion of Ukraine , Ottobock continues its operations in Russia, including maintaining a manufacturing site in Tolyatti. [ 60 ]
The largest shareholder of Ottobock SE & Co. KGaA is Näder Holding GmbH & Co. KG, which is headquartered in Duderstadt. It is 100 percent owned by the owner family Näder, the direct descendants of the company founder Otto Bock. A further 20 percent is held by the Swedish financial investor EQT . [ 61 ]
Hans Georg Näder has publicly stated that he intends to float Ottobock via an initial public offering (IPO) scheduled for 2022, despite previously announcing the intention to take Ottobock public in 2015. [ 62 ] [ 63 ] In February 2022, the company delayed the IPO to September 2022. [ 64 ] According to Reuters , Ottobock announced in May 2022 that it would not pursue the IPO due to market conditions, while company insiders claimed the company is unlikely to reach the target valuation of five to six billion euros. [ 65 ]
The Board of Directors manages the business of Ottobock SE & Co. KGaA and determines the basic guidelines and strategic direction of the company. It consists of four non-executive directors and currently two of the four executive directors (CEO/CSO and CFO). The Chairman of the Board of Directors is Hans Georg Näder. [ 66 ]
The company's Supervisory Board is European co-determined and consists of six shareholder representatives and four employee representatives. It monitors the activities of the board of directors. The supervisory board is chaired by Bernd Bohr, long-time head of the automotive division of the Bosch Group . [ 67 ] Other members include Gesche Joost and Michael Kaschke. [ 68 ] [ 69 ]
Since July 2022, the company has been managed operationally by four executive directors : Oliver Jakobi, chief executive officer (CEO) and chief revenue officer (CRO), [ 70 ] Arne Kreitz, chief financial officer (CFO), [ 71 ] Arne Jörn chief operating officer (COO) and chief technology officer (CTO) and Martin Böhm chief experience officer . [ 72 ] [ 73 ]
This leadership transition followed a significant restructuring by Hans Georg Näder, who ousted three of the four previous managing directors from the company over a period of three days following the intervention by Hans Georg Näder, who opposed with the plan to take the company public in 2022. [ 74 ] These were namely Philipp Schulte-Noelle, Kathrin Dahnke, and Andreas Goppelt. Philipp Schulte-Noelle is a former senior executive of German healthcare public company Fresenius , who was appointed as the CEO of Ottobock in 2019 amid the plan to take Ottobock public. [ 75 ] Kathrin Dahnke was hired by Ottobock in July 2021 after she left her position as CFO at German electric lights manufacturer Osram . [ 76 ] Kathrin Dahnke told reporters just days before her departure that Ottobock still intends to go public. [ 77 ]
By February 2022, the company had expanded its operations to a total of almost 52 sites distributed across North and South America , Europe , Asia , Africa and Australia . Ottobock SE & Co. KGaA is the global market leader in technical orthopedics/prosthetics, with sales and service locations in more than 50 countries. [ 78 ]
At the end of 2022, Ottobock employed over 9,000 people worldwide. The company's corporate headquarters are located in Duderstadt, Germany, with additional German locations in Königsee , Hanover , Traunstein , and Berlin . A competence center and research and development workshop are situated in Göttingen . [ 79 ] Ottobock maintains research and development facilities in other locations, including Duderstadt, Salt Lake City, and Vienna. [ 80 ]
Since its inception, Ottobock has concentrated on developing prosthetic devices, [ 81 ] and it has emerged as a global leader in the field of exo-prosthetics. [ 82 ] [ 83 ] Another focus area is orthotics, specifically designing devices that help individuals with partial leg paralysis regain mobility. [ 84 ]
Ottobock's NeuroMobility division focuses on neuro-orthotics solutions alongside its rehabilitation and wheelchair business segments. Since 2018, the development of high-tech wheelchairs has been undertaken at the company's facility in Berlin. [ 85 ] Before production begins, these wheelchairs undergo rigorous testing on a specialised test track and in an integrated workshop to ensure quality and functionalist. The production of the wheelchairs takes place in Königsee, Thuringia, where Ottobock maintains its manufacturing operations. [ 86 ] [ 87 ]
Ottobock operates more than 340 care centres worldwide. [ 88 ] Additionally, the company continuously optimises processes within its orthopaedic workshops to enhance the quality and effectiveness of its treatments, orthotics and prosethic products.
In 2018, Ottobock established a new business division focusing on biomechanics, specifically through its Ottobock Bionic Exoskeletons unit. This division specialises in developing and marketing exoskeletons designed for use in industrial work settings to support people in physically demanding work, [ 89 ] such as the automotive sector and smartphone manufacturing. [ 90 ] The exoskeletons relieve strain on muscles and joints, for example during overhead work or heavy lifting activities.
In October 2021, Ottobock completed the acquisition of the US company SuitX. SuitX is a spin-off from the Robotics and Human Engineering Lab at the University of California, Berkeley, and focuses on the research and development of exoskeletons for both professional and medical applications. The acquisition aims to enhance Ottobock's development and distribution efforts in the exoskeleton technology space. [ 91 ] [ 92 ]
In 2009, Ottobock reestablished its presence in Berlin by opening the Science Center at Potsdamer Platz, marking a return to the city where the company was originally founded in Kreuzberg in 1919. [ 93 ] The Science Center served as Ottobock's representative office and showroom in the capital for nine years. [ 94 ] During its operation, it attracted over one million visitors from around the globe to its interactive exhibition, "Begreifen, was uns bewegt" ("Understanding What Moves Us"). [ 95 ] In the summer of 2018, Ottobock relocated to the renovated former Bötzow Brewery buildings, leading to the closure of the Science Center Berlin to the public. [ 96 ] [ 97 ]
Ottobock established a digital think tank known as the Ottobock Future Lab at the Bötzow Brewery, once Berlin's largest private brewery. [ 98 ] After acquiring the site in 2010, Ottobock initiated a revitalization project based on a master plan designed by architect David Chipperfield . [ 99 ] [ 100 ] This redevelopment blended modern working environments with the brewery's historic brick and industrial architecture. [ 101 ] The Future Lab serves as a hub where new products, technologies, and supply solutions are developed and tested by cross-functional teams. The site hosts a variety of digital start-ups and also houses employees from departments including IT , Human Resources, Marketing, Corporate Strategy, Corporate Communications, and Public Affairs.
Ottobock is an official global partner to the International Paralympic Committee (IPC) [ 102 ] since 2005, [ 103 ] and has been providing technical services at the Paralympic Games since the Summer Games in Seoul, 1988. [ 104 ]
As an official technical service partner at the Paralympic Games, Ottobock provides support to athletes by offering services free of charge. Many athletes rely heavily on technical aids that undergo extreme stresses, especially wheelchairs in contact sports, which often suffer damage. [ 105 ] To address this, Ottobock deploys a technical team on-site during the games and establishes workshops near the Paralympic village, and at key training and competition venues. [ 106 ] The team performs repairs and maintenance on equipment, servicing athletes' regardless of nationality or the brand of aids used. [ 107 ] [ 108 ]
Advertising partners in this area include paralympians Johannes Floors , [ 109 ] Léon Schäfer, Anna Schaffelhuber [ 110 ] and Heinrich Popow . [ 111 ]
The 2016 Paralympic Games in Rio de Janeiro marked the 13th games at which Ottobock provided technical services. [ 112 ] This involved shipping 18 tonnes (18 long tons; 20 short tons) of equipment, including 15,000 spare parts, 1,100 wheelchair tyres, 70 running blades and 300 prosthetic feet, 300 kilometres (190 mi) from Duderstadt to the port at Bremerhaven , 10,100 kilometres (6,300 mi) by sea to Santos , and then 500 kilometres (310 mi) by road to Rio de Janeiro. [ 113 ] At Seoul in 1988, four Ottobock technicians carried out 350 repairs; [ 104 ] in Rio de Janeiro in 2016, 100 technicians from 29 countries speaking 26 languages carried out 3,361 repairs for 1,162 athletes, including 2,745 repairs to wheelchairs, 438 to prosthetics, and 178 to orthotics. [ 113 ]
In Rio on 10 September, the IPC's president, Sir Philip Craven , announced that Ottobock had agreed to extend its world-wide partnership to the end of 2020, encompassing the 2020 Paralympic Games in Tokyo. [ 114 ] | https://en.wikipedia.org/wiki/Ottobock |
Ouabain / w ɑː ˈ b ɑː ɪ n / [ 1 ] or / ˈ w ɑː b eɪ n , ˈ w æ -/ (from Somali waabaayo , "arrow poison" through French ouabaïo ) also known as g-strophanthin , is a plant derived toxic substance that was traditionally used as an arrow poison in eastern Africa for both hunting and warfare. Ouabain is a cardiac glycoside and, in lower doses, can be used medically to treat hypotension and some arrhythmias. It acts by inhibiting the Na/K-ATPase , also known as the sodium–potassium ion pump. [ 2 ] However, adaptations to the alpha-subunit of the Na + /K + -ATPase via amino acid substitutions, have been observed in certain species, namely some herbivore insect species, that have resulted in toxin resistance. [ 3 ]
It is classified as an extremely hazardous substance in the United States as defined in Section 302 of the U.S. Emergency Planning and Community Right-to-Know Act (42 U.S.C. 11002), and is subject to strict reporting requirements by facilities which produce, store, or use it in significant quantities. [ 4 ]
Ouabain can be found in the roots, stems, leaves, and seeds of the Acokanthera schimperi and Strophanthus gratus plants, both of which are native to eastern Africa. [ 5 ]
Ouabain is a cardiac glycoside that acts by non-selectively inhibiting the Na + /K + -ATPase sodium–potassium ion pump. [ 2 ] Once ouabain binds to this enzyme, the enzyme ceases to function, leading to an increase of intracellular sodium. This increase in intracellular sodium reduces the activity of the sodium–calcium exchanger (NCX), which pumps one calcium ion out of the cell and three sodium ions into the cell down their concentration gradient. Therefore, the decrease in the concentration gradient of sodium into the cell which occurs when the Na/K-ATPase is inhibited reduces the ability of the NCX to function. This in turn elevates intracellular calcium. [ 6 ] This results in higher cardiac contractility and an increase in cardiac vagal tone . The change in ionic gradients caused by ouabain can also affect the membrane voltage of the cell and result in cardiac arrhythmias.
An overdose of ouabain can be detected by the presence of the following symptoms: rapid twitching of the neck and chest musculature, respiratory distress, increased and irregular heartbeat, rise in blood pressure, convulsions, wheezing, clicking, and gasping rattling. Death is caused by cardiac arrest. [ 5 ]
Ouabain is a highly toxic compound, however, it has a low bioavailability [ 2 ] and is absorbed poorly from the alimentary tract as so much of the oral dose is destroyed. Intravenous administration results in greater available concentrations. After intravenous administration, the onset of action occurs within 2–10 minutes in humans with the maximum effect enduring for 1.5 hours.
Ouabain is eliminated by renal excretion, largely unchanged. [ 2 ]
In 1991, a specific high affinity sodium pump inhibitor indistinguishable from ouabain was first discovered in the human circulation [ 7 ] and proposed as one of the potential mediators of long term blood pressure and the enhanced salt excretion following salt and volume loading. [ 8 ] This agent was an inhibitor of the sodium pump that acted similarly to digitalis . A number of analytical techniques led to the conclusion that this circulating molecule was ouabain and that humans were producing it as an endogenous hormone. [ 8 ] A large portion of the scientific community agreed that this inhibitor was endogenous ouabain and that there was strong evidence to indicate that it was synthesized in the adrenal gland . [ 8 ] One early speculative interpretation of the analytical data led to the proposal that endogenous ouabain may have been the 11 epimer, i.e., an isomer of plant ouabain. [ 9 ] However, this possibility was excluded by various methods including the synthesis of the 11 epimer and the demonstration that it has different chromatographic behavior from ouabain. Critically, the primary observations concerning the identification of ouabain in mammals were repeated and confirmed using a variety of tissue sources on three different continents with advanced analytical methods as summarized elsewhere. [ 10 ]
Despite widespread analytical confirmation, some questioned whether or not this endogenous substance is ouabain. The arguments were based less upon rigorous analytical data but more on the fact that immunoassays are neither entirely specific nor reliable. Hence, it was suggested that some assays for endogenous ouabain detected other compounds or failed to detect ouabain at all. [ 11 ] Additionally, it was suggested [ 11 ] that rhamnose, the L-sugar component of ouabain, could not be synthesized within the body despite published data to the contrary. [ 12 ] Yet another argument against the existence of endogenous ouabain was the lack of effect of rostafuroxin (a first generation ouabain receptor antagonist) on blood pressure in an unselected population of hypertensive patients. [ 13 ]
Ouabain is no longer approved for use in the USA. In France and Germany, however, intravenous ouabain has a long history in the treatment of heart failure, and some continue to advocate its use intravenously and orally in angina pectoris and myocardial infarction despite its poor and variable absorption. The positive properties of ouabain regarding the prophylaxis and treatment of these two indications are documented by several studies. [ 14 ] [ 15 ]
The African crested rat ( Lophiomys imhausi ) has a broad, white-bordered strip of hairs covering an area of glandular skin on the flank. When the animal is threatened or excited, the mane on its back erects and this flank strip parts, exposing the glandular area. The hairs in this flank area are highly specialised; at the tips they are like ordinary hairs, but are otherwise spongy, fibrous, and absorbent. The rat is known to deliberately chew the roots and bark of the Poison-arrow tree ( Acokanthera schimperi ), which contains ouabain. After the rat has chewed the tree, instead of swallowing the poison it slathers the resulting masticate onto its specialised flank hairs which are adapted to absorb the poisonous mixture. It thereby creates a defense mechanism that can sicken or even kill predators which attempt to bite it. [ 16 ] [ 17 ] [ 18 ]
The total synthesis of ouabain was achieved in 2008 by Deslongchamps laboratory in Canada. [ 19 ] It was synthesized under the hypothesis that a polyanionic cyclization (double Michael addition followed by aldol condensation ) would allow access to a tetracyclic intermediate with the desired functionality. [ 19 ] The figure below shows the key steps in the synthesis of ouabain.
In their synthesis, Zhang et al. from the Deslongchamps laboratory condensed cyclohexenone A with Nazarov substitute B in a double Michael addition to produce tricycle C. At the indicated position, C was reduced to the aldehyde and the alcohol group was protected with p-methoxybenzyl ether (PMB) to form the aldol precursor needed to produce D. After several steps, intermediate E was produced. E contained all the required functionalities and stereochemistry needed to produce ouabain. The structure of E was confirmed by comparison against the degradation product of ouabain. Methylation of E, catalyzed by rhodium, produced F. The dehydroxylation and selective oxidation of the secondary hydroxy group of F produced G. G reacted with triphenyl phosphoranylidene ketene and the ester bonds in G were hydrolyzed to produce ouabagenin, a precursor to ouabain. The glycosylation of ouabagenin with rhamnose produced ouabain.
Poisons derived from Acokanthera plants are known to have been used in Africa as far back as the 3rd century BC when Theophrastus reported a toxic substance that the Ethiopians would smear on their arrows. [ 5 ] [ 21 ] The poisons derived from this genus of plants were used throughout eastern Africa, typically as arrow poisons for hunting and warfare. Acokanthera schimperi , in particular, exhibits a very large amount of ouabain, which the Kenyans, Tanzanians, Rwandans, Ethiopians, and Somalis would use as an arrow poison. [ 5 ]
The poison was extracted from the branches and leaves of the plant by boiling them over a fire. Arrows would then be dipped into the concentrated black tar-like juice that formed. [ 5 ] Often, certain magical additives were also mixed in with the ouabain extract in order to make the poison work according to the hunter's wishes. In Kenya, the Giriama and Langulu poison makers would add an elephant shrew to the poison mixture in order to facilitate the pursuit of their prey. [ 5 ] They had observed that an elephant shrew would always run straight ahead or follow a direct path and thought that these properties would be transferred to the poison. A poisonous arrow made with this shrew was thought to cause the hunted animal to behave like the shrew and run in a straight path. In Rwanda members of the Nyambo tribe, also known poison arrow makers, harvest the Aconkathera plants according to how many dead insects are found under it - more dead insects under a shrub indicating a higher potency of poison. [ 5 ]
Although ouabain was used as an arrow poison primarily for hunting, it was also used during battle. One example of this occurred during a battle against the Portuguese, who had stormed Mombasa in 1505. Portuguese records indicated that they had suffered a great deal from the poisoned arrows. [ 5 ]
European imperial expansion and exploration into Africa overlapped with the rise of the European pharmaceutical industry towards the end of the nineteenth century. [ 22 ] British troops were the target of arrows poisoned with the extracts of various Strophanthus species. [ 22 ] They were familiar with the deadly properties of these plants and brought samples back to Europe. Around this time, interest in the plant grew. It was known that ouabain was a cardiac poison, but there was some speculation about its potential medical uses. [ 5 ] [ 22 ]
In 1882, ouabain was first isolated from the plant by the French chemist Léon-Albert Arnaud as an amorphous substance, which he identified as a glycoside . [ 5 ] Ouabain was seen as a possible treatment for certain cardiac conditions. | https://en.wikipedia.org/wiki/Ouabain |
Ouchterlony double immunodiffusion (also known as passive double immunodiffusion ) is an immunological technique used in the detection, identification and quantification of antibodies and antigens , such as immunoglobulins and extractable nuclear antigens . The technique is named after Örjan Ouchterlony , the Swedish physician who developed the test in 1948 to evaluate the production of diphtheria toxins from isolated bacteria. [ 1 ]
A gel plate is cut to form a series of holes ("wells") in an agar or agarose gel. A sample extract of interest (for example human cells harvested from tonsil tissue) is placed in one well, sera or purified antibodies are placed in another well and the plate left for 48 hours to develop. During this time the antigens in the sample extract and the antibodies each diffuse out of their respective wells. Where the two diffusion fronts meet, if any of the antibodies recognize any of the antigens, they will bind to the antigens and form an immune complex . The immune complex precipitates in the gel to give a thin white line (precipitin line), which is a visual signature of antigen recognition. [ citation needed ] [ 2 ] [ 3 ]
The method can be conducted in parallel with multiple wells filled with different antigen mixtures and multiple wells with different antibodies or mixtures of antibodies, and antigen-antibody reactivity can be seen by observing between which wells the precipitate is observed. When more than one well is used there are many possible outcomes based on the reactivity of the antigen and antibody selected. The zone of equivalence lines may give a full identity (i.e. a continuous line), partial identity (i.e. a continuous line with a spur at one end), or a non-identity (i.e. the two lines cross completely). [ citation needed ]
The sensitivity of the assay can be increased by using a stain such as Coomassie brilliant blue , this is done by repeated staining and destaining of the assay until the precipitin lines are at maximum visibility. [ 1 ]
Precipitation occurs with most antigens because the antigen is multivalent (i.e. has several antigenic determinants per molecule to which antibodies can bind). Antibodies have at least two antigen binding sites (and in the case of immunoglobulin M there is a multimeric complex with up to 10 antigen binding sites), thus large aggregates or gel-like lattices of antigen and antibody are formed. Experimentally, an increasing amount of antigen is added to a constant amount of antibody in solution. Initially at low antigen concentration, all of the antibody is contained in the precipitate. This is called the antibody-excess zone (i.e. prozone phenomenon). As more antigen is added, the amount of protein precipitated increases until the antigen/antibody molecules are at an optimal ratio. This is known as the zone of equivalence or equivalence point. When the amount of antigen in solution exceeds the amount of antibody, the amount of precipitation will decrease. This is known as the antigen excess zone. | https://en.wikipedia.org/wiki/Ouchterlony_double_immunodiffusion |
Our Final Invention: Artificial Intelligence and the End of the Human Era is a 2013 non-fiction book by the American author James Barrat . The book discusses the potential benefits and possible risks of human-level ( AGI ) or super-human ( ASI ) artificial intelligence . [ 1 ] Those supposed risks include extermination of the human race . [ 2 ]
James Barrat weaves together explanations of AI concepts, AI history, and interviews with prominent AI researchers including Eliezer Yudkowsky and Ray Kurzweil . The book starts with an account of how an artificial general intelligence could become an artificial super-intelligence through recursive self-improvement. In subsequent chapters, the book covers the history of AI , including an account of the work done by I. J. Good , up to the work and ideas of researchers in the field today.
Throughout the book, Barrat takes a cautionary tone, focusing on the threats artificial super-intelligence poses to human existence. Barrat emphasizes how difficult it would be to control or even to predict the actions of something that may become orders of magnitude more intelligent than the most intelligent humans.
On 13 December 2013, journalist Matt Miller interviewed Barrat for his podcast , "This... is interesting". The interview and related matters to Barrat's book, Our Final Invention , were then captured in Miller's weekly opinion piece for The Washington Post . [ 3 ]
Seth Baum , executive director of the Global Catastrophic Risk Institute and one of the people cited by Barrat in his book, reviewed the book favorably on Scientific American's "invited guest" blog, calling it a welcome counterpoint to the vision articulated by Ray Kurzweil in his book The Singularity is Near . [ 4 ]
Gary Marcus questions Barrat's argument "that tendencies toward self-preservation and resource acquisition are inherent in any sufficiently complex, goal-driven system", noting that present-day AI does not have such drives, but Marcus concedes "that the goals of machines could change as they get smarter", and he feels that "Barrat is right to ask" about these important issues. [ 5 ]
Our Final Invention was a Huffington Post Definitive Tech Book of 2013. [ 6 ] | https://en.wikipedia.org/wiki/Our_Final_Invention |
Our Mathematical Universe: My Quest for the Ultimate Nature of Reality is a 2014 non-fiction book by the Swedish-American cosmologist Max Tegmark . Written in popular science format, the book interweaves what a New York Times reviewer called "an informative survey of exciting recent developments in astrophysics and quantum theory" with Tegmark's mathematical universe hypothesis , which posits that reality is a mathematical structure. [ 1 ] This mathematical nature of the universe, Tegmark argues, has important consequences for the way researchers should approach many questions of physics.
Tegmark, whose background and scientific research have been in the fields of theoretical astrophysics and cosmology, mixes autobiography and humor into his analysis of the universe. The book begins with an account of a bicycle accident in Stockholm in which Tegmark was killed—in some theoretical parallel universes , though not in our own. [ 2 ]
The rest of the book is divided into three parts. [ 3 ] Part one, "Zooming Out," deals with locating ourselves in the cosmos and/or multiverse . Part two, "Zooming In," looks for added perspective from quantum mechanics and particle physics. Part three, "Stepping Back," interweaves a scientific viewpoint with Tegmark's speculative ideas about the mathematical nature of reality. By the end of the book, Tegmark has hypothesized four different levels of multiverse.
According to Andrew Liddle, reviewing the book for Nature : [ 4 ]
The culmination that Tegmark seeks to lead us to is the "Level IV multiverse". This level contends that the Universe is not just well described by mathematics, but, in fact, is mathematics. All possible mathematical structures have a physical existence, and collectively, give a multiverse that subsumes all others. Here, Tegmark is taking us well beyond accepted viewpoints, advocating his personal vision for explaining the Universe.
Reviews of the book have generally praised Tegmark's writing and exposition of established physics, while often criticizing the content and speculativeness of his new "mathematical universe" hypothesis.
In a very positive review, Clive Cookson in The Financial Times wrote that "physics could do with more characters like Tegmark" and that his book "should engage any reader interested in the infinite variety of nature." [ 5 ] Giles Whitsell in The Times described the book as "mind-bending." [ 6 ] Peter Forbes in The Independent praised the last chapter of the book, on the risks of extinction humanity faces, as "wise and bracing". [ 7 ]
Brian Rotman , writing for The Guardian , was unconvinced by Tegmark's conclusions but also wrote that the book is "at the cutting edge of cosmology and quantum theory in friendly and relaxed prose, full of entertaining anecdotes and down-to-earth analogies." [ 8 ] Similarly, cosmologist Andrew Liddle , in Nature , summarized: [ 4 ]
This is a valuable book, written in a deceptively simple style but not afraid to make significant demands on its readers, especially once the multiverse level gets turned up to four. It is impressive how far Tegmark can carry you until, like a cartoon character running off a cliff, you wonder whether there is anything holding you up.
Mathematical physicist Edward Frenkel , writing for The New York Times , alleged that the meaning of Tegmark's hypothesis "is a big question, which is never fully answered" and said that parts of the book "[pretend] to stay in the realm of science" while actually espousing "science fiction and mysticism." [ 9 ] In a positive review, cosmologist Andreas Albrecht , writing for SIAM Review , criticized Tegmark's proposed test of the "mathematical universe" hypothesis (the hypothetical identification of physical phenomena which cannot be described mathematically) as meaningless. [ 10 ] In a review written for The Wall Street Journal , physicist Peter Woit said that the problem with Tegmark's proposal is "not that it's wrong but that it's empty" and "radically untestable." [ 11 ] In Physics Today , Francis Sullivan particularly praised Tegmark's explanation of the theory of inflation but criticized his purportedly physical application of Emile Borel 's theorem on normal numbers , and regarded his overall argument as circular. [ 12 ] In New Scientist , Mark Buchanan contrasted what he saw as the "uninhibited speculation" in parts of Tegmark's book with his earlier "hard, empirical" work which established him as a physicist. [ 13 ]
In The New York Times , science writer Amir Alexander concluded that the book is "brilliantly argued and beautifully written" and "never less than thought-provoking," although Tegmark's hypothesis is "simply too far removed from the frontiers of today's mainstream science" to judge its legitimacy. [ 2 ] | https://en.wikipedia.org/wiki/Our_Mathematical_Universe |
Out in Science, Technology, Engineering, and Mathematics, Inc. , abbreviated oSTEM , is a 501(c)(3) non-profit professional society dedicated to LGBTQ+ individuals within the science, technology, engineering, and mathematics (STEM) community.
In October of 2005. IBM sponsored a focus group where students from across the United States convened at the Human Rights Campaign headquarters in Washington, D.C. These students discussed topics relevant to LGBTQ+ communities at their colleges and universities. They debated how to structure an organization that serves students in science, technology, engineering, and mathematics. [ 1 ]
Founded in 2009, the organization was granted 501(c)(3) status in 2010. oSTEM currently consists of more than 100 chapters across the United States and the United Kingdom .
oSTEM strives to identify, address, and advocate for the needs of LGBTQ+ students and professionals within the STEM fields. oSTEM fulfills these needs by providing networking opportunities, mentorship connections, strategic collaborations, and professional/leadership development , as well as an annual global conference. [ 2 ] [ 3 ]
oSTEM hosts annual conferences [ 4 ] [ 5 ] that discuss LGBTQ+ topics in STEM as well as intelligence fields . [ 6 ] Topics discussed include inclusion, outreach, and diversity within the workplace. [ 7 ] [ 8 ] The goal of workshops, talks, and networking events for LGBTQ+ people is to help them integrate and move up in their fields. [ 9 ] The fourth annual conference was hosted jointly with the National Organization of Gay and Lesbian Scientists and Technical Professionals ' Out to Innovate in Atlanta in 2014. [ 10 ]
On July 5, 2018, oSTEM along with Pride in STEM , [ 11 ] House of STEM, [ 12 ] and InterEngineering [ 13 ] created international awareness for LGBTQ+ people in Science, Technology, Engineering, and Math . [ 14 ]
oSTEM presents a variety of awards annually to individuals and organizations that demonstrate a strong dedication to advancing and empowering LGBTQ+ in STEM fields. [ 15 ]
According to the oSTEM website, "This award recognizes volunteers who have gone above and beyond to bring oSTEM to new heights in the last year. The Executive Board recognizes volunteers who have demonstrated their drive to achieve great things and push our organization to grow. In the past, awardees have been recognized for creating our scholarship program, spearheading the pivot to an online format for our cornerstone event during COVID, cultivating and expanding institutional support for professional members and collegiate members, and stepping into leadership roles with grace and professionalism. oSTEM is powered by the hard work and commitment of our volunteers." [ 16 ]
Previous awardees include:
The oSTEM Global STEM Service Award is given to present and past oSTEM members who show strong dedication to inclusion, diversity, and equality for LGBTQ+ and other marginalized individuals in STEM fields. [ 17 ]
Awardees are:
The oSTEM Strategic Alliance Award is presented to a current sponsoring organization, community partner, or grant provider of oSTEM who demonstrates strong dedication, engagement, and support to oSTEM and its values.
Awardees are:
The oSTEM Partner Excellence Award is presented to individuals associated with oSTEM accomplished in their much academic or professional lives who regularly advocate for the full inclusion of people of all marginalized identities.
Awardees are:
The Overall Student Chapter of the Year is given to oSTEM chapters that educate, empower, and engage a diverse community. These chapters contrihelp a lot with finding LGBTQ students in the STEM community, helping them, and speaking up for themselves are:
The Rookie Student Chapter of the Year celebrates achievements by oSTEM chapters that have been founded within two years of application submission.
Awardees are:
There are over 100 chapters affiliated with the parent organization. Chapters are organized into six geographic regions (A–F) and a region that encompasses all chapters dedicated specifically to graduate students.
The six regions are:
The first professional chapter is currently being tested in the Boston metropolitan area. In 2020, there was a shift to a virtual professional chapter with members in the United States and United Kingdom, with a number of smaller in-person events occurring in those two regions. | https://en.wikipedia.org/wiki/Out_in_Science,_Technology,_Engineering,_and_Mathematics |
Out of autoclave composite manufacturing is an alternative to the traditional high pressure autoclave (industrial) curing process commonly used by the aerospace manufacturers for manufacturing composite material . Out of autoclave (OOA) is a process that achieves the same quality as an autoclave but through a different process. [ 1 ] OOA curing achieves the desired fiber content and elimination of voids by placing the layup within a closed mold and applying vacuum, pressure, and heat by means other than an autoclave. A resin transfer molding (RTM) press is the typical method of applying heat and pressure to the closed mold. There are several out of autoclave technologies in current use including RTM, same qualified resin transfer molding (SQRTM), vacuum-assisted resin transfer molding (VARTM), and balanced pressure fluid molding. The most advanced of these processes can produce high-tech net shape aircraft components.
Resin transfer molding (RTM) is a method of fabricating high-tech composite structures. The RTM process is capable of consistently producing composite parts with high strength, complex geometries, tight dimensional tolerances, and part quality typically required of aerospace applications. RTM uses a closed mold commonly made of aluminum. A fiber "layup" such as graphite is placed into the mold. The mold is closed, sealed, heated, and placed under vacuum. Heated resin is injected into the mold to impregnate the fiber layup. Having the mold heated and under vacuum, as in vacuum assisted resin transfer molding (VARTM) assists the resin flow. The mold is then held at a temperature sufficient to cure the resin. Current RTM technology produces lightweight parts with excellent mechanical properties. With these qualities, composite materials are gaining wide use in a variety of structural and non-structural applications common in aerospace and aviation. RTM is one method of fabricating these composite structures. [ 1 ] [ 2 ]
Same qualified resin transfer molding (SQRTM) is a closed mold composites manufacturing method similar to RTM. "Same qualified" refers to this method injecting the same resin as that used in the prepreg layup. The attributes of "same qualified" are significant to a manufacturer because those who adopt this process need not re-qualify resin materials for their production process. What sets SQRTM apart from standard resin transfer molding is the substitution of a prepreg layup rather than a dry fiber preform. [ 3 ]
SQRTM is an RTM process adapted to prepreg technology. The prepreg is placed in a closed mold and during the cure cycle, a small amount of resin is injected into the cavity through ports positioned around the part. This resin does not go into the laminate, but only presses up against the edge of the laminate in order to establish hydrostatic pressure on the prepreg, similar to the goal of autoclave curing. This pressure is similar to the autoclave, on the order of 6-7 bars (90-100 psi). Hydrostatic pressure minimizes voids by keeping dissolved air, water and resin monomers in solution in the resin. The tool can either be self-clamped and self-heated or heated and clamped by a press. The equipment is composed of a tool, a press, an injector, and a vacuum pump. [ 4 ]
The key factors in the SQRTM process include precision machined closed mold tooling, high pressure presses, a high vacuum applied to the tool interior, and precise control of heating platens, injected resin volume, heat, and pressure. [ citation needed ]
The advantages of the SQRTM process include a high level of integration, tight tolerances and the use of qualified prepregs. Its disadvantages include higher tool costs and a lower level of flexibility to design changes. [ 5 ]
Vacuum assisted resin transfer molding (VARTM) differs from pre-preg processing in that fiber reinforcements and core materials are laid up on a one-sided mold and vacuum bagged. Liquid resin is introduced through ports in the mold and vacuum-drawn through the reinforcements by way of designed-in channels and infusion media that facilitate fiber wetout. Subsequent curing does not require high heat or high pressure, unlike the autoclave. The process's comparatively low-cost tooling allows inexpensive production of large, complex parts in one shot, [ 1 ] such as the tail of the Mitsubishi SpaceJet . [ 6 ]
Balanced pressure molding using fluid as the heat transfer is commercially practiced as the 'quickstep' process. This process allows for the curing, partial curing, and joining of composite materials . The process involves a fluid-filled, pressure balanced, heated floating mould technology. The heated floating mold technology used within the process works by rapidly applying heat to the laminate which is trapped between a free floating rigid or semi-rigid mold that floats in, and is surrounded by, a heat transfer fluid (HTF). The rapid heating can lead to significantly lower resin viscosities, and this in turn allows achieving full laminate consolidation using pressures lower than those used in autoclave. The mold and laminate become separated from the circulating HTF by a flexible membrane. The part, typically under full vacuum, is subject to pressures as high as 250kPa fluid pressure and can be rapidly heated to the desired cure temperature without risk of catastrophic exothermic reaction, as the HTF can draw excess heat as desired. The air is then removed under vacuum and the laminate is compacted and heated until the part is cured.
A flexible membrane beneath the mold is bonded into a pressure chamber creating the lower half of a 'clamshell' or 'chamber' like mold set. A second flexible membrane is bonded to a second pressure chamber creating the upper half of the clamshell. These pressure chambers are clamped together during processing, permitting the laminate to be compressed while reducing stress to the mold as it is floating in a balanced pressure environment within the HTF.
The process can use thermosetting , thermoplastic prepregs (pre-impregnated composite fibers), and wet resin with dry fiber to produce superior composite parts. This out of autoclave process can achieve aerospace grade void contents of less than 2%, with extremely fast cycle times, and at significantly lower pressures and lower labor costs than many alternative autoclave production systems using many typical autoclave qualified prepregs. The quickstep out of autoclave system is unique in that it uses fully immersed balanced pressure fluid curing and it allows the user to stop the composite cure reaction at any point in the cure cycle, and thus can halt processing on all or part of the laminate and either return to it at a later to complete cure or to co-cure, join and bond other composites to it to create larger parts.
The use of fluid to control temperature, as opposed to the gas generally used within methods such as autoclave and oven curing equates to lower energy consumption , faster cycle times and extremely accurate part temperature control.
Another out of autoclave method for achieving external compression on prepreg based composite parts is through the use of heat shrink tape. This method, however, does not achieve the high quality of RTM or autoclave processes because without the autoclave or a closed mold, the part must be cured in a non-pressurized oven. These compression tapes are typically made from polyester (PET) film. Heat shrink tape is applied to a composite part prior to the heating, or curing cycle. When heated, the tape will shrink in the linear (machine direction). Heat shrink tape works best on parts that are cylindrical or semi-circular in cross section, as this allows the tape to exert even compaction forces on the part surface. Examples would be composite tubes for aerospace, wind energy, consumer sporting goods, etc. Heat shrink tape allows these parts to be processed without the need to cure with the heat and pressure of an autoclave. | https://en.wikipedia.org/wiki/Out_of_autoclave_composite_manufacturing |
An out-of-danger species is an animal or plant species formerly categorized as Rare , Vulnerable , or Endangered that has since been removed from these lists because the species' survival has been relatively secured, [ 1 ] e.g. Ginkgo biloba . Often known as a delisted species , these animals have been moved out of the Rare, Vulnerable, or Endangered categories through conservation efforts and government policymaking to ensure their survival and population growth. [ 2 ] The International Union for Conservation of Nature ( IUCN ) established its list of endangered species in 1964, subsequently becoming a global authority on wildlife conservation. [ 3 ] The following year, the United States created the U.S. Fish and Wildlife Service to act as a federal authority on endangered species. [ 4 ] Currently, both international and domestic organizations implement recovery efforts and track species' population growth, delisting when necessary. [ 5 ] [ 6 ] Removing a species from the endangered species list is generally a slow process; most organizations and governments require long periods of observation both before and after delisting. [ 7 ] There have been numerous efforts to delist endangered species, with both international and country-wide recovery plans being regularly implemented. [ 8 ] These programs have led to the recovery of dozens of species, but their overall effectiveness remains contested. [ 9 ] [ 10 ]
The first wildlife conservation law passed in the United States was the Lacey Act of 1900 , which required the secretary of agriculture to "preserve, introduce, distribute, and restore" wild and game birds. In the 1960s, the Department of Interior formed a Committee on Rare and Endangered Wildlife Species to identify species in danger of extinction. The first official document listing the species that the federal government declared in danger of extinction was published as the 'Redbook on Rare and Endangered Fish and Wildlife of the United States in 1964. [ 11 ] The Bureau of Sport Fisheries and Wildlife (renamed U.S. Fish and Wildlife Service in 1974) was created in 1965 by The Fish and Wildlife Act and is the authority on the official federal list of endangered species today. [ 12 ] The first species to be delisted from the Endangered Species list due to recovery (as opposed to extinction or listing error) was the Brown Pelican in 1985. [ 4 ] Beyond domestic classifications within the U.S., international non-governmental organizations ( NGOs ) have developed separate classification and prevention systems. The International Union for Conservation of Nature (IUCN), which established its list of endangered species in 1964, is the global authority on species conservation and recovery. [ 3 ] Many nations have implemented laws that protect endangered species by, for example, banning hunting or creating protected areas. More extensive measures such as captive breeding and habitat restoration have also been undertaken, especially by nations that rely on revenue from tourism. [ 8 ]
The IUCN Red List is the world's most comprehensive inventory of the conservation status of biological species . It serves as a global indicator for biodiversity and provides information about population size, habitat, and threats to the population that help inform conservation decisions. Species are reassessed each time a new version of the list is published, and some are downlisted or delisted if certain criteria are met. Species are examined for a multitude of factors, including if their main threats remain prevalent and whether conservation measures have engendered enough of an improvement to warrant a change in threat category or complete removal from the list. The IUCN relies on global scientific research to refine its assessments and accurately assess whether a species is improving or deteriorating. [ 5 ]
Country-wide efforts take on many forms, as each nation develops different strategies to shorten the endangered species list. Some, like Australia and the United States, use recovery plans enacted by the national government to guide conservation, while others rely more heavily on captive breeding programs. [ 13 ] [ 14 ] The effectiveness of these efforts also differ, with ninety percent of North/Central American countries and seventy percent of African countries being classified as above-average performers on a Megafauna Conservation Index (MCI) developed by researchers in Global Ecology and Conservation. Conversely, approximately twenty-five percent of Asian countries and twenty percent of European countries were found to be under-performers; some argue that these disparities are due to disparate levels of reliance on wildlife tourism . [ 8 ]
Currently, the delisting of out-of-danger species in the United States is governed by the Endangered Species Act of 1973 (ESA). The law was enacted to prevent endangered species from becoming extinct and is jointly administered by the U.S Department of the Interior , the U.S Department of Commerce , and the U.S Department of Agriculture . Federal policy differentiates between an " endangered species ," which is at risk of extinction throughout most or all of its population, and a " threatened species ," a less severe classification referring to a species that is likely to become endangered in the foreseeable future. [ 6 ] The delisting of a species, which can be formally defined as the removal of species from the Federal Lists of Endangered and Threatened Wildlife and Plants, is governed by section 4 of the ESA. The process occurs when a species is determined to no longer be at risk; this assessment is based on factors such as population size, habitat quality, and elimination of threats. After being delisted, the species must be monitored for at least five years to ensure that recovery remains stable. There is a similar process governed by the ESA known as downlisting. While it is close to delisting, it deals with the downgrade of a species from endangered to threatened as opposed to their complete removal from the list. [ 2 ]
In the United States, recovery is defined as the process of restoring endangered and threatened species to the point where they no longer require the safeguards of the ESA. Recovery plans are developed by departments like National Oceanic and Atmospheric Administration (NOAA) [ 15 ] and the U.S Fish and Wildlife Services [ 16 ] to outline a strategy to restore self-sufficient wild populations of engendered species. They are non-regulatory documents developed in conjunction with interested parties in federal, state, local, and tribal governments; successful implementation often results in downlisting or delisting, and the removal of ESA protections. [ 15 ] Species are tracked over time while these agencies implement individual recovery actions. [ 16 ] For example, ten federal agencies formed the Columbia River Basin Federal Caucus to promote recovery of native fish and wildlife listed under the Endangered Species Act in the Columbia River Basin , including the Middle Columbia River Steelhead. [ 17 ]
In July 2021, the IUCN implemented a new metric for assessing species recovery. Known as the "green status," it ranges from 0 to 100 and is calculated using the population of a species prior to human interference. It also tracks the impact of previous conservation efforts; the hypothetical effect of stopping current conservation efforts; as well as future potential species recovery. [ 18 ]
Despite these efforts, listings of endangered species tend to outpace delistings. [ 19 ] Some hold the view that most species can expect an extended, if not permanent residence on the endangered species list. They argue that a lack of protections against important causes of species decline results in most species remaining on the list forever, and warn that the detriments of a mistaken delisting generally outweigh those of extended retention on the protected list. The slow speed of delisting is not always been seen as negative, and has been cited as demonstrative of the importance of the ESA. [ 19 ] Occasionally, advocacy groups have filed lawsuits to challenge the Fish and Wildlife Service's delisting of a species, as was the case with both Yellowstone grizzly bear and Sonoran Desert bald eagle . Controversy is not uncommon in these decisions, as some worry about the consequences engendered by a species' loss of ESA protections. [ 10 ] Critics of the ESA argue that recovery efforts focus on charismatic species to the detriment of others, especially plants. Proposed improvements to current recovery policy come in many forms, including strengthening partnerships with states and corporations, a higher level of species monitoring, and the use of climate-smart conservation strategies. [ 20 ]
One event that contributed to the recovery efforts of several bird species in the United States was the banning of the chemical commonly known as DDT (dichlorodiphenyltrichloroethane). This chemical became a well-known synthetic pesticide after its use during WWII to prevent insect-carried diseases from affecting American troops. After the war, DDT became a popular agricultural insecticide. [ citation needed ] DDT's harmful environmental effects largely affected bird populations, as the chemical caused the bird's eggshells to become dangerously brittle and reduced the reproductive abilities of bird populations. [ 21 ] In 1962, American biologist Rachel Carson published the book Silent Spring , which raised public awareness about the harmful effects of DDT and questioned the widespread release of the chemical into the environment. On July 14, 1972, the Administrator of the Environmental Protection Agency canceled nearly all remaining federal registrations of DDT products. [ citation needed ] After the federal ban of DDT, several bird populations that had become endangered due to the widespread use of the chemical were able to recover and were removed from the Endangered Species list. These populations include the Bald Eagle , Peregrine Falcon , Osprey , and Brown Pelican . [ 22 ]
The American alligator ( Alligator mississippiensis ) is a member of the order Crocodilia . They are apex predators who help to control the number of rodents and other animals that might otherwise overtax marshland vegetation. The species saw a dramatic population decline in the mid-20th century. [ 23 ] When its population reached an all-time low in 1967, it was officially recognized as an endangered species. The U.S. government, in conjunction with the southern states, cracked down on the hunting of alligators and heavily monitored population growth. They were officially delisted in 1987 after U.S. Fish and Wildlife Services announced they had made a full recovery. [ 9 ] Moreover, the IUCN now considers them of lowest risk/least concern and has commented on their positive response to intervention and rapid recovery. Sustainable management programs have operated in Louisiana , Florida , Texas , and other southeastern states for years. The prevalence and availability of healthy populations has led to numerous investigations of alligator biology. [ 24 ] [ 23 ]
The Arabian oryx ( Oryx leucoryx ) is a species of antelope native to the Arabian Peninsula and is locally referred to as Al Maha. Despite the last wild Oryx being shot in 1972, captive breeding and subsequent reintroduction efforts downlisted the species from endangered to vulnerable on the IUCN Red List, a three-category improvement. [ 13 ] This reintroduction process was heavily supported by the local Harasis bedu people that established safe grazing areas for the reintroduced herds. The reintroduction process faced many challenges and setbacks due to poachers and an oil pipeline being built on Oman 's Arabian Oryx Sanctuary in 2007. [ 25 ] However, as of 2011, the peninsula hosted over 1,000 wild specimens, with another 7,000 living in captivity. [ 9 ] The Arabian Oryx's shift from endangered to vulnerable in 2011 was the first time the IUCN had reclassified a species as vulnerable after it had been extinct in the wild. [ 25 ] [ 13 ]
The bald eagle ( Haliaeetus leucocephalus ) is a large, carnivorous bird of prey that can often be found near bodies of water. [ 26 ] The population saw a decrease in the mid-20th century as they lost habitat due to urbanization , hunting, and the widespread use of DDT . People at that time believed that the eagles were responsible for the abduction of smaller cattle, and in some cases, attacking children. However, this was disproven, and in 1978, the Endangered Species Technical Bulletin attributed most eagle deaths to preventative killings caused by these beliefs. [ 27 ] Along with direct attacks on the species, the pesticide DDT played a key role in the population diminishing through biomagnification . DDT did not affect grown adult eagles, but it did alter their calcium metabolism to make them either sterile or incapable of producing healthy eggs. Researchers found that eggs that were produced were often too brittle to withstand the weight of a brooding adult eagle. Subsequently, regulations and protections have been implemented, including permit requirements to hunt eagles. The U.S. government removed the species from its endangered species list in 1995, and it was removed from the endangered and threatened category in 2007. While they are out of danger, the bald eagle is still under the protection of the Bald and Golden Eagle Protection Act and the Migratory Bird Treaty Act . [ 27 ] [ 28 ]
The gray wolf ( Canis lupus ) is one of the most recent species to be categorized as out-of-danger in the U.S. The population of gray wolves declined in America through the 1960s as productivity in the agriculture industry increased and gray wolves were seen as threats to cattle. At its low in 1985, their population reached about 300 in total and their habitat was reduced to just northern Michigan and Minnesota , as well as Wisconsin . [ 29 ] Thanks to being put under the protection of the Endangered Species Act of 1973, conservationists were able to promote the species' safety and the population began to slowly rise and return to its normal numbers. [ 30 ] By 2020, the population in America reached over 6,000, and the wolves' geographic territory had expended, which exceeded the expectations of conservationists. The gray wolf has now been removed from the list of endangered and threatened species. Now that the numbers have returned to a sustainable state, the U.S. Fish and Wildlife Service will continue to monitor the species to ensure that both it and the cattle to which they are considered a danger remain safe. [ 29 ] [ 31 ]
The giant panda ( Ailuropoda melanoleuca) is native to south-central China . The species mainly reside in temperate forests high in the mountains and subsist almost entirely on bamboo. These bears must eat large quantities of bamboo every day and are excellent tree climbers, despite their size. [ 32 ] In the 1960s, the species experienced near extinction due to a diminishing habitat and a pelt that was considered valuable to humans, which led to widespread hunting and confinement. The wild population continued to decline until conservation groups and government agencies stepped in. Since then, the population has dramatically increased relative to its almost non-existence. In 2016, they were moved from the list of Endangered Species to Vulnerable Species according to the International Union for Conservation of Nature (IUCN) . At their current rate of growth, they may soon exit that category as well. There are currently 67 panda reserves that protect 66% of wild pandas. [ 30 ] [ 32 ]
Ginkgo biloba , commonly known as the maidenhair tree, is the last of the division Ginkgophyta and has fossils dating back to the Jurassic Age , making it one of the oldest living tree species in the world. [ 33 ] The rest of its division is believed to have gone extinct around the same time as the dinosaurs . Populations of this plant in the wild are still considered endangered by the IUCN. However, it is widely cultivated worldwide, especially in China and Japan where it is native and is popular for both medicinal uses as well as its use in culinary practices. In the medical world, it is believed to be helpful in treating Alzheimer's , dementia , and vertigo . Its fruit is often used in cooking and is considered a delicacy in China . [ 34 ] [ 35 ]
The Potentilla robbinsiana or Robbins's cinquefoil is a dwarf alpine plant located in New Hampshire ’s White Mountains and Franconia Ridge . [ 9 ] It is a small, yellow-flowered perennial member of the rose family. [ 36 ] After a large hiking trail was built through Monroe Flats—home to more than 95 percent of the world's Robbins' cinquefoil—a combination of harvesting and foot traffic pushed the species to the brink of extinction. After being placed on the endangered species list, the Monroe Flats trail was rerouted, and work began to germinate new satellite colonies of the species. These efforts caused the population to grow from 1,801 to 4,831 between 1973 and 2006, and the species was officially delisted in 2002. [ 37 ] New colonies are especially prevalent in the Mount Washington area, where the species survives best on rocky sites similar to its natural habitat. [ 38 ] [ 4 ] | https://en.wikipedia.org/wiki/Out_of_danger_species |
An out-of-the-box feature or functionality (also called OOTB or off the shelf ), particularly in software , is a native feature or built-in functionality of a product that comes directly from the vendor and works immediately when the product is placed in service. [ 1 ] [ 2 ] In the context of software, out-of-the-box features and functionality are available for all users by default and do not require customization, modification, configuration, scripting, add-ons, modules, third-party tools, or additional fees in order to be used. [ 3 ] [ 4 ] [ 5 ]
This computing article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Out_of_the_box_(feature) |
Out to Innovate , previously known as the National Organization of Gay and Lesbian Scientists and Technical Professionals (NOGLSTP), is a professional society for professionals in science, technology, mathematics, and engineering. [ 1 ] [ 2 ] Each year, Out to Innovate gives the Walt Westman Award to members who helped make significant contributions to the association's mission.
The organization was organized along the lines of earlier organizations of gay scientists in Los Angeles and the Research Triangle area of North Carolina, and arose out of a session at the 1980 American Association for the Advancement of Science (AAAS) meeting. It was formally organized in 1983 and incorporated in California in 1991. The foundation of the organization was in response to issues such as gay scientists not being able to get visas to immigrate to the United States or security clearances to work in government laboratories, the lack of research on LGBT health issues, and loss of productivity due to the stress of stigmatization. Much of the organization's early work related to increasing the visibility of LGBT scientists and opposing homophobia . In the 1990s, it focused on encouraging corporations to adopt nondiscrimination policies and assisted in a 1995 Government Accounting Office report that recommended that LGBT status should not be considered a vulnerability to blackmail in security clearance investigations. In the 2000s and 2010s, awards for LGBT scientists, engineers, and STEM educators were established.
Out to Innovate supports regional groups and caucuses who choose to affiliate with Out to Innovate. Out to Innovate affiliates and partners with other national STEM organizations, including AAAS. Out to Innovate also organizes a mentoring network, a scholarship program for students, and a biannual career summit. [ 1 ]
In July 2019, Out to Innovate partnered with the Out Astronaut Project, a nonprofit initiative aimed at sending the first out LGBTQIA+ astronaut into space. [ 3 ] The goal of the partnership, according to a press release from OAP, is to "provide opportunities for LGBTQ persons to become actively involved in space-related research." [ 4 ] The goals of OAP, beyond sending the first LGBTQIA+ astronaut into space includes providing a robust presence in STEM fields for LGBTQIA+ individuals "by highlighting the contributions of LGBTQ members currently working in science and space while providing grants to promising LGBTQ students." [ 4 ] On September 24, 2019, the OAP announced via Facebook that they had found the winner of the first phase of their project. [ 5 ]
Out to Innovate has a number of other partnerships and affiliations. [ 6 ] They include: The American Association for the Advancement of Science , the National Postdoctoral Association , and the American Chemical Society .
Out to Innovate recognizes an LGBTQ+ Scientist, Engineer, and Educator each year "who has made outstanding contributions to their field". [ 7 ] In addition, they give the Walt Westman Award to recognize Out to Innovate members who have significantly advanced Out to Innovate's mission.
Awardees are: | https://en.wikipedia.org/wiki/Out_to_Innovate |
In Information theory , outage probability of a communication channel is the probability that a given information rate is not supported, because of variable channel capacity . Outage probability is defined as the probability that information rate is less than the required threshold information rate. It is the probability that an outage will occur within a specified time period. [ 1 ]
For example, the channel capacity for slow- fading channel is C = log 2 (1 + h 2 SNR), where h is the fading coefficient and SNR is a signal to noise ratio without fading. As C is random , no constant rate is available. There may be a chance that information rate may go below to required threshold level. For slow fading channel, outage probability = P( C < r ) = P(log 2 (1 + h 2 SNR) < r ), where r is the required threshold information rate.
This mathematics -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Outage_probability |
In game theory , the outcome of a game is the ultimate result of a strategic interaction with one or more people, dependant on the choices made by all participants in a certain exchange. It represents the final payoff resulting from a set of actions that individuals can take within the context of the game. Outcomes are pivotal in determining the payoffs and expected utility for parties involved. [ 1 ] Game theorists commonly study how the outcome of a game is determined and what factors affect it.
In game theory, a strategy is a set of actions that a player can take in response to the actions of others. Each player’s strategy is based on their expectation of what the other players are likely to do, often explained in terms of probability. [ 2 ] Outcomes are dependent on the combination of strategies chosen by involved players and can be represented in a number of ways; one common way is a payoff matrix showing the individual payoffs for each players with a combination of strategies, as seen in the payoff matrix example below. Outcomes can be expressed in terms of monetary value or utility to a specific person. Additionally, a game tree can be used to deduce the actions leading to an outcome by displaying possible sequences of actions and the outcomes associated. [ 3 ]
Strategies of Player A
A commonly used theorem in relation to outcomes is the Nash equilibrium . This theorem is a combination of strategies in which no player can improve their payoff or outcome by changing their strategy, given the strategies of the other players. In other words, a Nash equilibrium is a set of strategies in which each player is doing the best possible, assuming what the others are doing to receive the most optimal outcome for themselves. [ 4 ] Not all games have a unique nash equilibrium and if they do, it may not be the most desirable outcome. [ 5 ] Additionally, the desired outcomes is greatly affected by individuals chosen strategies, and their beliefs on what they believe other players will do under the assumption that players will make the most rational decision for themselves. [ 6 ] A common example of the nash equilibrium and undesirable outcomes is the Prisoner’s Dilemma game. [ 7 ]
Many different concepts exist to express how players might interact. An optimal interaction may be one in which no player's payoff can be made greater, without making any other player's payoff lesser. Such a payoff is described as Pareto efficient , and the set of such payoffs is called the Pareto frontier.
Many economists study the ways in which payoffs are in some sort of economic equilibrium . One example of such an equilibrium is the Nash equilibrium , where each player plays a strategy such that their payoff is maximized given the strategy of the other players.
Players are persons who make logical economic decisions. It is assumed that human people make all of their economic decisions based only on the idea that they are irrational. A player's rewards (utilities, profits, income, or subjective advantages) are assumed to be maximised. [ 8 ] The purpose of game-theoretic analysis, when applied to a rational approach, is to provide recommendations on how to make choices against other rational players. First, it reduces the possible outcomes; logical action is more predictable than irrational. Second, it provides a criterion for assessing an economic system's efficiency.
In a Prisoner's Dilemma game between two players, player one and player two can choose the utilities that are the best response to maximise their outcomes. "A best response to a coplayer’s strategy is a strategy that yields the highest payoff against that particular strategy". [ 9 ] A matrix is used to present the payoff of both players in the game. For example, the best response of player one is the highest payoff for player one’s move, and vice versa. For player one, they will pick the payoffs from the column strategies. For player two, they will choose their moves based on the two row strategies. Assuming both players do not know the opponents strategies. [ 10 ] It is a dominant strategy for the first player to choose a payoff of 5 rather than a payoff of 3 because strategy D is a better response than strategy C.
Outcome optimisation in game theory has many real world applications that can help predict actions and economic behaviours by other players. [ 11 ] Examples of this include stock trades and investments , cost of goods in business, corporate behaviour and even social sciences. [ citation needed ]
Equilibria are not always Pareto efficient, and a number of game theorists design ways to enforce Pareto efficient play, or play that satisfies some other sort of social optimality. The theory of this is called implementation theory . | https://en.wikipedia.org/wiki/Outcome_(game_theory) |
' Outcome primacy ' is a psychological phenomenon that describes lasting effects on a subject's behavior based on the outcome of first experiences with a given task or decision. [ 1 ] It was found that this outcome primacy can account for much of the underweighting of rare events in experience based decisions, [ 2 ] where participants apparently underestimate small probabilities (in contrast to prospect theory where people tend to overestimate low probabilities, when lotteries are described).
Behaviour in this task can be modelled using a standard, model-free reinforcement learning algorithm. In this model, the values of the different actions are learned over time and are used to determine the next action according to a predefined action-selection rule. It was shown that a substantial effect of first experience on behaviour is consistent with the reinforcement learning model if one assumes that the outcome of first experience resets the values of the experienced actions, but not if symmetric initial conditions are assumed. Moreover, the predictive power of the resetting model outperforms previously published [ 3 ] models regarding the aggregate choice behaviour.
These findings suggest that first experience has a disproportionately large effect on subsequent actions, similar to primacy effects in other fields of cognitive psychology, such as in the application of the serial position effect . The mechanism of resetting of the initial conditions that underlies outcome primacy may thus also account for other forms of primacy. | https://en.wikipedia.org/wiki/Outcome_primacy |
Outcomes theory provides the conceptual basis for thinking about, and working with outcomes systems of any type. An outcomes system is any system that: identifies; prioritizes; measures; attributes; or hold parties to account for outcomes of any type in any area.
Outcomes systems go under various names such as: strategic plans; management by results; results-based management systems; outcomes-focused management systems; accountability systems; evidence-based practice systems; and best-practice systems. In addition, outcomes issues are dealt with in traditional areas such as: strategic planning; business planning and risk management.
Outcomes theory theorizes a sub-set of topics covered in diverse ways in other disciplines such as: performance management, organizational development, program evaluation, policy analysis, economics and the other social sciences. The different treatment of outcomes issues in different technical languages in these different disciplines means that it is hard for those building outcomes systems to gain quick access to a generic body of principles about how to set up outcomes systems and fix issues with existing outcomes systems.
Outcomes theory was developed by Paul Duignan. [ 1 ] [ 2 ] [ 3 ]
Outcomes theory is made up of several key conceptual frameworks and a set of principles. The most important framework is Duignan's Outcomes System Diagram. [ 1 ] This diagram identifies seven different building-blocks of outcomes systems. These building-blocks are analogous to the building-blocks that make up accounting systems (e.g. general ledger, assets register). In the case of an outcomes system they are a different set of building-blocks which are necessary for outcomes systems to function properly. The building blocks are:
An example of a principle within outcomes theory is the Impact evaluation only option for high-level outcome attribution if no controllable indicators at top of outcomes model principle . [ clarification needed ] This is the principle that in an instance where building-block two (controllable indicators) does not connect to building block one of the outcomes model, then building-block five (impact evaluation) offers the only way to obtain more information about whether changes in high-level indicators can be attributed to an intervention.
Williams and Hummelbrunner (2009) summarize some of the uses of outcomes theory: "Outcomes theory intends to improve outcomes system architecture, that is, related systems that deal in one way or another with outcomes, by providing a clear common technical language, thus helping to avoid unnecessary duplication and identify gaps to be filled. Outcomes theory also specifies the structural features and the key principles of well-constructed outcomes systems. ... This helps people without significant background in outcomes thinking to construct sound and sustainable outcomes systems." [ 4 ]
Duignan's Outcomes-Focused Visual Strategic Planning is an applied implementation of outcomes theory. It is based on building a visual strategic plan and then using it for: prioritization; performance management; and assessing organizational impact. | https://en.wikipedia.org/wiki/Outcomes_theory |
Outer billiards is a dynamical system based on a convex shape in the plane. Classically, this system is defined for the Euclidean plane [ 1 ] but one can also consider the system in the hyperbolic plane [ 2 ] or in other spaces that suitably generalize the plane. Outer billiards differs from a usual dynamical billiard in that it deals with a discrete sequence of moves outside the shape rather than inside of it.
Let P be a convex shape in the plane.
Given a point x0 outside P, there is typically a unique
point x1 (also outside P) so that the line segment connecting x0 to x1 is tangent to P at its midpoint and
a person walking from x0 to x1 would see P on the right. (See Figure.) The map
F: x0 -> x1 is called the outer billiards map .
The inverse (or backwards) outer billiards map is also defined, as the map x1 -> x0.
One gets the inverse map simply by replacing the word right by the word left in the definition given above.
The figure shows the situation in the Euclidean plane , but the definition in the hyperbolic plane is essentially the same.
An outer billiards orbit is the set of all iterations of the point, namely ... x0 ↔ x1 ↔ x2 ↔ x3 ... That is, start at x0 and
iteratively apply both the outer billiards map and the backwards outer billiards map.
When P is a strictly convex shape, such as an ellipse ,
every point in the exterior of P has a well defined orbit. When P
is a polygon , some points might not have well-defined orbits, on account of the
potential ambiguity of choosing the midpoint of the relevant tangent line. Nevertheless, in
the polygonal case, almost every point has a well-defined orbit.
Defining an outer billiards system in a higher-dimensional space is beyond the scope of this article. Unlike the case of ordinary billiards , the definition is not straightforward. One natural setting for the map is a complex vector space . In this case, there is a natural choice of line tangent to a convex body at each point. One obtains these tangents by starting with the normals and using the complex structure to rotate 90 degrees. These distinguished tangent lines can be used
to define the outer billiards map roughly as above. [ 1 ]
Most people attribute the introduction of outer billiards to Bernhard Neumann in the late 1950s, [ 3 ] though it seems that a few people cite an earlier construction in 1945, due to M. Day. Jürgen Moser popularized the system in the 1970s as a toy model for celestial mechanics . [ 4 ] [ 5 ] This system has been studied classically in the Euclidean plane , and more recently in
the hyperbolic plane . One can also consider higher-dimensional spaces, though no serious study has yet been made. Bernhard Neumann informally posed the question as to whether or not one can
have unbounded orbits in an outer billiards system, and Moser put it in writing in 1973. [ 4 ] Sometimes this basic question has been called the Moser-Neumann question . This question, originally posed for shapes in the Euclidean plane and solved only recently, has been a guiding problem in the field.
In the 70's, Jürgen Moser sketched a proof, based on K.A.M. theory , that outer
billiards relative to a
6-times- differentiable shape of positive curvature has all orbits bounded.
In 1982, Raphael Douady gave the full proof of this result. [ 6 ] A big advance in the polygonal case came over a period of several years when
three teams of authors, Vivaldi-Shaidenko, [ 7 ] Kolodziej, [ 8 ] and Gutkin-Simanyi, [ 9 ] each
using different methods, showed that outer billiards relative to a quasirational polygon has all orbits bounded. The notion of quasirational is technical
(see references) but it includes the class of regular polygons and convex rational polygons ,
namely those convex polygons whose vertices have rational coordinates. In the case of rational polygons, all the orbits are
periodic. In 1995, Sergei Tabachnikov showed that outer billiards for the regular pentagon has some aperiodic orbits,
thus clarifying the distinction between the dynamics in the rational and regular cases. [ 1 ] In 1996, Philip Boyland showed that outer billiards relative to some shapes can have orbits which accumulate on
the shape. [ 10 ] In 2005, Daniel Genin showed that all orbits are bounded when the shape is a trapezoid , thus
showing that quasirationality is not a necessary condition for the system to have all orbits bounded. [ 11 ] (Not all trapezoids are quasirational.)
In 2007, Richard Schwartz showed that outer billiards has some unbounded orbits when defined
relative to the Penrose Kite, thus answering the original Moser-Neumann question in the affirmative. [ 12 ] The Penrose kite is the convex quadrilateral from the kites-and-darts Penrose tilings . Subsequently, Schwartz showed that outer billiards has unbounded orbits when defined relative
to any irrational kite. [ 13 ] An irrational kite is a quadrilateral with the following property:
One of the diagonals of the quadrilateral divides the region into two triangles of equal area
and the other diagonal divides the region into two triangles whose areas are not rational multiples
of each other. In 2008, Dmitry Dolgopyat and Bassam Fayad showed that outer billiards defined relative to the semidisk has
unbounded orbits. [ 14 ] The semidisk is the region one gets by cutting a disk in half.
The proof of Dolgopyat-Fayad is robust, and also works for regions obtained by cutting a disk nearly in half, when the word nearly is suitably interpreted.
In 2003, Filiz Doǧru and Sergei Tabachnikov showed that all orbits are unbounded for a certain class of convex polygons in the hyperbolic plane . [ 15 ] The authors call such polygons large .
(See the reference for the definition.) Filiz Doǧru and Samuel Otten then extended this work in 2011 by specifying the conditions under which a regular polygonal table in the hyperbolic plane have all orbits unbounded, that is, are large. [ 16 ]
In ordinary polygonal billiards , the existence of periodic
orbits is a major unsolved problem. For instance, it is unknown if every
triangular shaped table has a periodic billiard path. More progress has
been made for outer billiards, though the situation is far from well understood.
As mentioned above, all the orbits are periodic when the system is defined
relative to a convex rational polygon in the Euclidean plane . Moreover, it is a
recent theorem of Chris Culter (written up by Sergei Tabachnikov) that outer
billiards relative to any convex polygon has periodic orbits—in fact a
periodic orbit outside of any given bounded region. [ 17 ]
Outer billiards is a subject still in its beginning phase. Most problems are still unsolved.
Here are some open problems in the area. | https://en.wikipedia.org/wiki/Outer_billiards |
The extracellular polysaccharide colanic acid is produced by species of the family Enterobacteriaceae . In Escherichia coli strain K12 the colanic acid cluster comprises 19 genes . The wzx gene encodes a protein with multiple transmembrane segments that may function in export of the colanic acid repeat unit from the cytoplasm into the periplasm in a process analogous to O-unit export . The colanic acid gene clusters may be involved in the export of polysaccharide from the cell . [ 1 ]
This biochemistry article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Outer_membrane_polysaccharide_transporter |
Outer sphere refers to an electron transfer (ET) event that occurs between chemical species that remain separate and intact before, during, and after the ET event. [ 1 ] In contrast, for inner sphere electron transfer the participating redox sites undergoing ET become connected by a chemical bridge . Because the ET in outer sphere electron transfer occurs between two non-connected species, the electron is forced to move through space from one redox center to the other.
The main theory describing the rates of outer sphere electron transfer was developed by Rudolph A. Marcus in the 1950s, for which he was awarded the Nobel Prize in Chemistry in 1992. [ 2 ] A major aspect of Marcus theory is the dependence of the electron transfer rate on the thermodynamic driving force (difference in the redox potentials of the electron-exchanging sites). For most reactions, the rates increase with increased driving force. A second aspect is that the rate of outer sphere electron-transfer depends inversely on the "reorganizational energy." Reorganization energy describes the changes in bond lengths and angles that are required for the oxidant and reductant to switch their oxidation states . This energy is assessed by measurements of the self-exchange rates (see below).
Outer sphere electron transfer is the most common type of electron transfer, especially in biochemistry , where redox centers are separated by several (up to about 11) angstroms by intervening protein . In biochemistry, there are two main types of outer sphere ET: ET between two separate biological molecules or fixed distance electron transfer, in which the electron transfers within a single biomolecule (e.g., intraprotein). [ 3 ]
Outer sphere electron transfer can occur between chemical species that are identical except for their oxidation state . [ 4 ] This process is termed self-exchange. An example is the degenerate reaction between the tetrahedral ions permanganate and manganate :
For octahedral metal complexes , the rate constant for self-exchange reactions correlates with changes in the population of the e g orbitals , the population of which most strongly affects the length of metal-ligand bonds:
Outer sphere ET is the basis of the biological function of the iron-sulfur proteins . The Fe centers are typically further coordinated by cysteinyl ligands . The [Fe 4 S 4 ] electron-transfer proteins ([Fe 4 S 4 ] ferredoxins ) may be further subdivided into low-potential (bacterial-type) and high-potential (HiPIP) ferredoxins . Low- and high-potential ferredoxins are related by the following redox scheme:
Because of the small structural differences between the individual redox states, ET is rapid between these clusters. | https://en.wikipedia.org/wiki/Outer_sphere_electron_transfer |
In cladistics or phylogenetics , an outgroup [ 1 ] is a more distantly related group of organisms that serves as a reference group when determining the evolutionary relationships of the ingroup, the set of organisms under study, and is distinct from sociological outgroups . Character states present in the ingroup but absent in the outgroup are (often) synapomorphies that provide empirical support for the inferred monophyly of the ingroup; character states that are present in the outgroup and some members of the ingroup are symplesiomorphies , and their complementary synapomorphies shared among some members of the ingroup provide hypotheses of relationship within the ingroup clade. The outgroup is used as a point of comparison for the ingroup and specifically allows for the phylogeny to be rooted. Because the polarity (direction) of character change can be determined only on a rooted phylogeny, the choice of outgroup is essential for understanding the evolution of traits along a phylogeny. [ 2 ]
Although the concept of outgroups has been in use from the earliest days of cladistics, the term "outgroup" is thought to have been coined in the early 1970s at the American Museum of Natural History . [ 3 ] Prior to the advent of the term, various other terms were used by evolutionary biologists, including "exgroup", "related group", and "outside groups". [ 3 ]
The chosen outgroup is hypothesized to be less closely related to the ingroup than the ingroup is related to itself. The evolutionary assumption from these relationships is that the outgroup species has a common ancestor with the ingroup that is older than the common ancestor of the ingroup. Choice of outgroup can change the topology of a phylogeny. [ 4 ] Therefore, phylogeneticists typically use more than one outgroup in cladistic analysis. The use of multiple outgroups is preferable because it provides a more robust phylogeny, buffering against poor outgroup candidates. If one outgroup taxon is designated as the root and the others are included simply as additional taxa in the analysis, this provides a test of the ingroup's hypothesized monophyly. [ 3 ] [ 5 ] [ 6 ]
To qualify as an outgroup, a taxon must satisfy the following two characteristics:
Therefore, an appropriate outgroup must be unambiguously outside the clade of interest in the phylogenetic study. An outgroup that is nested within the ingroup will, when used to root the phylogeny, result in incorrect inference of phylogenetic relationships and trait evolution. [ 7 ] However, the optimal level of relatedness of the outgroup to the ingroup depends on the depth of phylogenetic analysis. Choosing a closely related outgroup relative to the ingroup is more useful when looking at subtle differences, while choosing an unduly distant outgroup can result in mistaking convergent evolution for a direct evolutionary relationship due to a common ancestor . [ 8 ] [ 9 ] For shallow phylogenetics—for example, resolving the evolutionary relationships of a clade within a genus—an appropriate outgroup would be a member of the sister clade. [ 10 ] However, for deeper phylogenetic analysis, less closely related taxa can be used. For example, Jarvis et al. (2014) used humans and crocodiles as outgroups while resolving the early branches of the avian phylogeny. [ 11 ] In molecular phylogenetics , satisfying the second requirement typically means that DNA or protein sequences from the outgroup can be successfully aligned to sequences from the ingroup. Although there are algorithmic approaches to identify the outgroups with maximum global parsimony, they are often limited by failing to reflect the continuous, quantitative nature of certain character states. [ 12 ] Character states are traits, either ancestral or derived, that affect the construction of branching patterns in a phylogenetic tree. [ 13 ]
In each example, a phylogeny of organisms in the ingroup may be rooted by scoring the same character states for one or more members of the outgroup. | https://en.wikipedia.org/wiki/Outgroup_(cladistics) |
The following outline is provided as an overview of and topical guide to Gottfried Wilhelm Leibniz:
Gottfried Wilhelm (von) Leibniz (1 July 1646 [O.S. 21 June] – 14 November 1716); German polymath, philosopher logician, mathematician. [ 1 ] Developed differential and integral calculus at about the same time and independently of Isaac Newton. Leibniz earned his keep as a lawyer, diplomat, librarian, and genealogist for the House of Hanover, and contributed to diverse areas. His impact continues to reverberate, especially his original contributions in logic and binary representations. [ 2 ]
Maria Rosa Antognazza 's 2009 Leibniz biography is a major recent resource. [ 4 ] | https://en.wikipedia.org/wiki/Outline_of_Gottfried_Wilhelm_Leibniz |
The following outline is provided as an overview of and topical guide to actuarial science:
Actuarial science – discipline that applies mathematical and statistical methods to assess risk in the insurance and finance industries.
Actuarial science can be described as all of the following:
Actuarial science
History of actuarial science
Reinsurance
Financial mathematics
Pensions | https://en.wikipedia.org/wiki/Outline_of_actuarial_science |
The following outline is provided as an overview of and topical guide to air pollution dispersion:
In environmental science , air pollution dispersion is the distribution of air pollution into the atmosphere . Air pollution is the introduction of particulates , biological molecules, or other harmful materials into Earth's atmosphere, causing disease , death to humans, damage to other living organisms such as food crops, and the natural or built environment . Air pollution may come from anthropogenic or natural sources. Dispersion refers to what happens to the pollution during and after its introduction; understanding this may help in identifying and controlling it.
Air pollution dispersion has become the focus of environmental conservationists and governmental environmental protection agencies (local, state, province and national) of many countries (which have adopted and used much of the terminology of this field in their laws and regulations) regarding air pollution control .
Air pollution emission plume – flow of pollutant in the form of vapor or smoke released into the air. Plumes are of considerable importance in the atmospheric dispersion modelling of air pollution. There are three primary types of air pollution emission plumes :
There are five types of air pollution dispersion models, as well as some hybrids of the five types: [ 1 ]
Effect of turbulence on dispersion – turbulence increases the entrainment and mixing of unpolluted air into the plume and thereby acts to reduce the concentration of pollutants in the plume (i.e., enhances the plume dispersion). It is therefore important to categorize the amount of atmospheric turbulence present at any given time. This type of dispersion is scale dependent. [ 10 ] Such that, for flows where the cloud of pollutant is smaller than the largest eddies present, there will be mixing. There is no limit on the size on mixing motions in the atmosphere and therefore bigger clouds will experience larger and stronger mixing motions. And hence, this type of dispersion is scale dependent.
Pasquill atmospheric stability classes – oldest and, for a great many years, the most commonly used method of categorizing the amount of atmospheric turbulence present was the method developed by Pasquill in 1961. [ 11 ] He categorized the atmospheric turbulence into six stability classes named A, B, C, D, E and F with class A being the most unstable or most turbulent class, and class F the most stable or least turbulent class.
Table 1: The Pasquill stability classes
Table 2: Meteorological conditions that define the Pasquill stability classes
Incoming solar radiation is based on the following: strong (> 700 W m −2 ), moderate (350–700 W m −2 ), slight (< 350 W m −2 ) [ 13 ]
The stability class can be defined also by using the
Advanced air pollution dispersion models – they do not categorize atmospheric turbulence by using the simple meteorological parameters commonly used in defining the six Pasquill classes as shown in Table 2 above. The more advanced models use some form of Monin–Obukhov similarity theory . Some examples include: | https://en.wikipedia.org/wiki/Outline_of_air_pollution_dispersion |
Algebra is one of the main branches of mathematics , covering the study of structure , relation and quantity . Algebra studies the effects of adding and multiplying numbers , variables , and polynomials , along with their factorization and determining their roots . In addition to working directly with numbers, algebra also covers symbols , variables, and set elements . Addition and multiplication are general operations , but their precise definitions lead to structures such as groups , rings , and fields .
An algebraic equation is an equation involving only algebraic expressions in the unknowns. These are further classified by degree . | https://en.wikipedia.org/wiki/Outline_of_algebra |
In mathematics , many types of algebraic structures are studied. Abstract algebra is primarily the study of specific algebraic structures and their properties. Algebraic structures may be viewed in different ways, however the common starting point of algebra texts is that an algebraic object incorporates one or more sets with one or more binary operations or unary operations satisfying a collection of axioms .
Another branch of mathematics known as universal algebra studies algebraic structures in general. From the universal algebra viewpoint, most structures can be divided into varieties and quasivarieties depending on the axioms used. Some axiomatic formal systems that are neither varieties nor quasivarieties, called nonvarieties , are sometimes included among the algebraic structures by tradition.
Concrete examples of each structure will be found in the articles listed.
Algebraic structures are so numerous today that this article will inevitably be incomplete. In addition to this, there are sometimes multiple names for the same structure, and sometimes one name will be defined by disagreeing axioms by different authors. Most structures appearing on this page will be common ones which most authors agree on. Other web lists of algebraic structures, organized more or less alphabetically, include Jipsen and PlanetMath. These lists mention many structures not included below, and may present more information about some structures than is presented here.
Algebraic structures appear in most branches of mathematics, and one can encounter them in many different ways.
In full generality, an algebraic structure may use any number of sets and any number of axioms in its definition. The most commonly studied structures, however, usually involve only one or two sets and one or two binary operations . The structures below are organized by how many sets are involved, and how many binary operations are used. Increased indentation is meant to indicate a more exotic structure, and the least indented levels are the most basic.
The following group-like structures consist of a set with a binary operation. The binary operation can be indicated by any symbol, or with no symbol (juxtaposition). The most common structure is that of a group . Other structures involve weakening or strengthening the axioms for groups, and may additionally use unary operations.
The main types of structures with one set having two binary operations are ring-like or ringoids and lattice-like or simply lattices . Ringoids and lattices can be clearly distinguished despite both having two defining binary operations. In the case of ringoids, the two operations are linked by the distributive law ; in the case of lattices, they are linked by the absorption law . Ringoids also tend to have numerical models , while lattices tend to have set-theoretic models.
In ring-like structures or ringoids, the two binary operations are often called addition and multiplication , with multiplication linked to addition by the distributive law .
Lattice-like structures have two binary operations called meet and join , connected by the absorption law .
The following module-like structures have the common feature of having two sets, A and B , so that there is a binary operation from A × A into A and another operation from A × B into A . Modules, counting the ring operations, have at least three binary operations.
These structures are defined over two sets, a ring R and an R -module M equipped with an operation called multiplication. This can be viewed as a system with five binary operations: two operations on R , two on M and one involving both R and M . Many of these structures are hybrid structures of the previously mentioned ones.
There are many examples of mathematical structures where algebraic structure exists alongside non-algebraic structure.
Some algebraic structures find uses in disciplines outside of abstract algebra. The following is meant to demonstrate some specific applications in other fields.
In Physics :
In Mathematical logic :
In Computer science :
A monograph available free online: | https://en.wikipedia.org/wiki/Outline_of_algebraic_structures |
The following outline is provided as an overview of, and topical guide to, applied physics:
Applied physics – physics intended for a particular technological or practical use. [ 1 ] It is usually considered as a bridge or a connection between "pure" physics and engineering . [ 2 ]
Applied Physics – is the proper name of a journal founded and edited by Helmut K.V. Lotsch in 1972 and published by Springer-Verlag Berlin Heidelberg New York from 1973 on [ 3 ]
Topics in Applied Physics – is the proper name of a series of quasi-monographs founded by Helmut K.V. Lotsch and published by Springer-Verlag Berlin Heidelberg New York [ 4 ]
Applied physics can be described as all of the following:
Fields and areas of research include: | https://en.wikipedia.org/wiki/Outline_of_applied_physics |
Arithmetic is an elementary branch of mathematics that is widely used for tasks ranging from simple day-to-day counting to advanced science and business calculations. | https://en.wikipedia.org/wiki/Outline_of_arithmetic |
The following outline is provided as an overview of and topical guide to astronomy:
Astronomy – studies the universe beyond Earth, including its formation and development , and the evolution, physics , chemistry , meteorology , and motion of celestial objects (such as galaxies , planets , etc.) and phenomena that originate outside the atmosphere of Earth (such as the cosmic background radiation ). Astronomy also intersects with biology, as astrobiology , studying potential life throughout the universe.
Astronomy can be described as all the following:
History of astronomy
Astronomical object
Sun
Small Solar System body
Variable star
Supernova
Black hole
Obsolete constellations including Ptolemy's Argo Navis Anser
Space agencies
1 Preceded by the Soviet space program | https://en.wikipedia.org/wiki/Outline_of_astronomy |
The following outline is provided as an overview of and topical guide to automation:
Automation – use of control systems and information technologies to reduce the need for human work in the production of goods and services. In the scope of industrialization , automation is a step beyond mechanization .
Automation-related social movement – a movement that advocates semi- or fully automatic systems to provide for human needs globally. For example, automation of farming and food distribution throughout the world so that no one will go hungry. One goal is to automate all mundane labor, to free humans to engage in more creative activities (or less work).
James S. Albus – a US government engineer and a prolific pioneering inventor of intelligent systems, automation and robotics. Also wrote extensively on the impact of automation on jobs, incomes, well-being and prosperity | https://en.wikipedia.org/wiki/Outline_of_automation |
The following outline is provided as an overview of and topical guide to biochemistry:
Biochemistry – study of chemical processes in living organisms , including living matter. Biochemistry governs all living organisms and living processes.
Biotechnology , Bioluminescence , Molecular chemistry , Enzymatic chemistry , Genetic engineering , Pharmaceuticals , Endocrinology , Neurochemistry , Hematology , Nutrition , Photosynthesis , Environmental , Toxicology | https://en.wikipedia.org/wiki/Outline_of_biochemistry |
The following outline is provided as an overview of and topical guide to biophysics:
Biophysics – interdisciplinary science that uses the methods of physics to study biological systems. [ 1 ]
Biophysical techniques – methods used for gaining information about biological systems on an atomic or molecular level. They overlap with methods from many other branches of science. | https://en.wikipedia.org/wiki/Outline_of_biophysics |
The following outline is provided as an overview of and topical guide to biotechnology:
Biotechnology – field of applied biology that involves the use of living organisms and bioprocesses in engineering, technology, medicine and other fields requiring bioproducts . Biotechnology also utilizes these products for manufacturing purposes.
History of biotechnology | https://en.wikipedia.org/wiki/Outline_of_biotechnology |
The following outline is provided as an overview of and topical guide to black holes:
Black hole – mathematically defined region of spacetime exhibiting such a strong gravitational pull that no particle or electromagnetic radiation can escape from inside it. The theory of general relativity predicts that a sufficiently compact mass can deform spacetime to form a black hole. The boundary of the region from which no escape is possible is called the event horizon . Although crossing the event horizon has enormous effect on the fate of the object crossing it, it appears to have no locally detectable features. In many ways a black hole acts like an ideal black body , as it reflects no light. Moreover, quantum field theory in curved spacetime predicts that event horizons emit Hawking radiation , with the same spectrum as a black body of a temperature inversely proportional to its mass. This temperature is on the order of billionths of a kelvin for black holes of stellar mass, making it essentially impossible to observe.
A black hole can be described as all of the following:
History of black holes | https://en.wikipedia.org/wiki/Outline_of_black_holes |
The following outline is an overview of and topical guide to botany , the biological academic discipline involving the study of plants .
Branches of botany
In alphabetical order by surname: | https://en.wikipedia.org/wiki/Outline_of_botany |
The following outline is provided as an overview of and topical guide to brain mapping:
Brain mapping – set of neuroscience techniques predicated on the mapping of (biological) quantities or properties onto spatial representations of the (human or non-human) brain resulting in maps. Brain mapping is further defined as the study of the anatomy and function of the brain and spinal cord through the use of imaging (including intra-operative, microscopic, endoscopic and multi-modality imaging), immunohistochemistry, molecular and optogenetics, stem cell and cellular biology, engineering (material, electrical and biomedical), neurophysiology and nanotechnology.
This section covers imaging and recording systems. The general section covers history, neuroimaging, and techniques for mapping specific neural connections. The specific systems section covers the various specific technologies, including experimental and widely deployed imaging and recording systems. | https://en.wikipedia.org/wiki/Outline_of_brain_mapping |
Calculus is a branch of mathematics focused on limits , functions , derivatives , integrals , and infinite series . This subject constitutes a major part of contemporary mathematics education . Calculus has widespread applications in science , economics , and engineering and can solve many problems for which algebra alone is insufficient. | https://en.wikipedia.org/wiki/Outline_of_calculus |
The following outline is provided as an overview of and guide to category theory , the area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows (also called morphisms , although this term also has a specific, non category-theoretical sense), where these collections satisfy certain basic conditions. Many significant areas of mathematics can be formalised as categories, and the use of category theory allows many intricate and subtle mathematical results in these fields to be stated, and proved, in a much simpler way than without the use of categories. | https://en.wikipedia.org/wiki/Outline_of_category_theory |
The following outline is provided as an overview of and topical guide to cell biology:
Cell biology – A branch of biology that includes study of cells regarding their physiological properties, structure, and function; the organelles they contain; interactions with their environment; and their life cycle , division , and death . This is done both on a microscopic and molecular level. Cell biology research extends to both the great diversities of single-celled organisms like bacteria and the complex specialized cells in multicellular organisms like humans . Formerly, the field was called cytology (from Greek κύτος, kytos , "a hollow;" and -λογία, -logia ).
Cell biology can be described as all of the following:
History of cell biology – is intertwined with the history of biochemistry and the history of molecular biology . Other articles pertaining to the history of cell biology include: | https://en.wikipedia.org/wiki/Outline_of_cell_biology |
The following outline is provided as an overview of and topical guide to chemical engineering:
Chemical engineering – deals with the application of physical science (e.g., chemistry and physics ), and life sciences (e.g., biology , microbiology and biochemistry ) with mathematics and economics , to the process of converting raw materials or chemicals into more useful or valuable forms. In addition to producing useful materials, modern chemical engineering is also concerned with pioneering valuable new materials and techniques – such as nanotechnology , fuel cells and biomedical engineering . [ 1 ]
History of chemical engineering | https://en.wikipedia.org/wiki/Outline_of_chemical_engineering |
The following outline acts as an overview of and topical guide to chemistry:
Chemistry is the science of atomic matter (matter that is composed of chemical elements ), especially its chemical reactions , but also including its properties, structure, composition, behavior, and changes as they relate to the chemical reactions. [ 1 ] [ 2 ] Chemistry is centrally concerned with atoms and their interactions with other atoms, and particularly with the properties of chemical bonds .
Chemistry can be described as all of the following:
Other
History of chemistry
Atomic theory
Thermochemistry | https://en.wikipedia.org/wiki/Outline_of_chemistry |
Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures .
History of combinatorics | https://en.wikipedia.org/wiki/Outline_of_combinatorics |
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