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LK-99 [ 2 ] also called PCPOSOS , [ 3 ] is a gray–black, polycrystalline compound, identified as a copper - doped lead‒oxyapatite . A team from Korea University led by Lee Sukbae ( 이석배 ) and Kim Ji-Hoon ( 김지훈 ) began studying this material as a potential superconductor , and in July 2023 published preprints claiming that it acted as a room-temperature superconductor [ 4 ] : 8 at temperatures of up to 400 K (127 °C; 260 °F) at ambient pressure. [ 2 ] [ 5 ] [ 4 ] : 1
Many different researchers attempted to replicate the work, and were able to reach initial results within weeks, as the process of producing the material is relatively straightforward. [ 6 ] By mid-August 2023, the consensus [ 1 ] was that LK-99 is not a superconductor at room temperature, and is an insulator in pure form. [ 7 ] [ 8 ] [ 9 ]
As of 12 February 2024, no replications had gone through the peer review process of a journal, but some had been reviewed by a materials science lab. A number of replication attempts identified non-superconducting ferromagnetic and diamagnetic causes for observations that suggested superconductivity. A prominent cause was a copper sulfide impurity [ 10 ] occurring during the proposed synthesis, which can produce resistance drops, lambda transition in heat capacity , and magnetic response in small samples. [ 11 ] [ 12 ] [ 10 ] [ 13 ] [ 14 ] [ 15 ] [ 16 ]
After the initial preprints were published, Lee claimed they were incomplete, [ 17 ] and coauthor Kim Hyun-Tak ( 김현탁 ) said one of the papers contained flaws. [ 18 ]
The chemical composition of LK-99 is approximately Pb 9 Cu(PO 4 ) 6 O, in which— compared to pure lead-apatite (Pb 10 (PO 4 ) 6 O) [ 19 ] : 5 — approximately one quarter of Pb(II) ions in position 2 of the apatite structure are replaced by Cu(II) ions. [ 4 ] : 9
The structure is similar to that of apatite , space group P 6 3 / m (No. 176).
Lee et al . provide a method for chemical synthesis of LK-99 [ 19 ] : 2 in three steps. First they produce lanarkite from a 1:1 molar mixing of lead(II) oxide (PbO) and lead(II) sulfate (Pb(SO 4 )) powders, heated at 725 °C (1,000 K; 1,340 °F) for 24 hours:
Second, copper(I) phosphide (Cu 3 P) is produced by mixing copper (Cu) and phosphorus (P) powders in a 3:1 molar ratio in a sealed tube under a vacuum, and heated to 550 °C (820 K; 1,000 °F) for 48 hours: [ 19 ] : 3
Finally, lanarkite and copper phosphide crystals are ground into a powder, placed in a sealed tube under a vacuum, and heated to 925 °C (1,200 K; 1,700 °F) for between 5‒20 hours: [ 19 ] : 3
There were a number of problems with the above synthesis from the initial paper. The reaction is not balanced , and others reported the presence of copper(I) sulfide ( Cu 2 S ) as well. [ 20 ] [ 12 ] For x = 1 {\displaystyle x=1} a balanced reaction might be:
Many syntheses produced fragmentary results in different phases, where some of the resulting fragments were responsive to magnetic fields, and other fragments were not. [ 22 ] The first synthesis to produce pure crystals found them to be diamagnetic insulators. [ 23 ]
Some small LK-99 samples were reported to show strong diamagnetic properties, including a response confusingly [ 24 ] referred to as "partial levitation" over a magnet. [ 19 ] This is a sign of regular diamagnetism or ferromagnetism, however it was misinterpreted by some as a sign of superconductivity.
While initial preprints claimed the material was a room-temperature superconductor, [ 19 ] : 1 they did not report observing any definitive properties of superconductivity, such as zero resistance, the Meissner effect , flux pinning , AC magnetic susceptibility , the Josephson effect , a temperature-dependent critical field and current, or a sudden jump in specific heat around the critical temperature. [ 25 ]
Because it is common for a new material to spuriously seem like a potential candidate for high-temperature superconductivity , [ 14 ] thorough experimental reports normally demonstrate a number of these properties. None of these properties was ever observed by the original experiment or any replications. [ 26 ]
Partial replacement of Pb 2+ ions with smaller Cu 2+ ions is said to cause a 0.48% reduction in volume, creating internal stress in the material, [ 4 ] : 8 causing a heterojunction quantum well between the Pb(1) and oxygen within the phosphate ([PO 4 ] 3− ). Kim Hyun-Tak proposed that this quantum well could be superconducting [ 4 ] : 10 , in a 2021 paper [ 27 ] describing a novel and complicated theory combining ideas from a classical theory of metal-insulator transitions, [ 28 ] the standard Bardeen–Cooper–Schrieffer theory , and the theory of hole superconductivity [ 29 ] by J.E.Hirsch .
On 31 July 2023, Sinéad Griffin of Lawrence Berkeley National Laboratory analyzed LK-99 with density functional theory (DFT), showing that its structure might have correlated isolated flat bands , which might contribute to superconductivity. [ 30 ] However, while other researchers agreed with the DFT analysis, a number suggested that this was not compatible with superconductivity, and that a structure different from what was described in Lee, et al. would be necessary. [ 31 ] In August, a study by Alexandru Georgescu at Indiana University did not find flat bands at Fermi level, concluding that they related to an unfavored high-symmetry structure. [ 32 ]
Analyses by industrial and experimental physicists noted experimental and theoretical shortcomings of the published works. [ 33 ] Shortcomings included the lack of phase diagrams [ 29 ] spanning temperature, stoichiometry, [ 34 ] and stress; the lack of pathways for the very high T c of LK-99 compared to prior heavy fermion superconductors; the absence of flux pinning in any observations; the possibility of stochastic conductive artifacts [ 35 ] in conductivity measurements; the high resistance and low current capacity of the alleged superconducting state; and the lack of direct transmission electron microscopy (TEM) of the materials.
The name LK-99 comes from the initials of Lee and Kim, and the year they first started working with the material (1999). [ 2 ] The pair had worked with Tong-Seek Chair ( 최동식 ) at Korea University in the 1990s. [ 36 ] In 2008, they founded the Quantum Energy Research Centre (퀀텀 에너지연구소; also known as Q-Centre ) with other researchers from Korea University. [ 17 ] Lee would later become CEO of Q-Centre , and Kim would become director of research and development .
Lee stated that in 2020, an initial paper was submitted to Nature , but was rejected. [ 36 ] Similarly presented research on room-temperature superconductors (but a completely different chemical system) by Ranga P. Dias had been published in Nature earlier that year, and received with skepticism—Dias's paper would subsequently be retracted in 2022 after its data was questioned as having been falsified. [ 37 ]
In 2020, Lee and Kim Ji-Hoon filed a patent application. [ 38 ] A second patent application (additionally listing Young-Wan Kwon), was filed in 2021, which was published on 3 March 2023. [ 39 ] A World Intellectual Property Organization (WIPO) patent was also published on 2 March 2023. [ 40 ] On 4 April 2023, a Korean trademark application for "LK-99" was filed by the Q-Centre . [ 41 ]
A series of academic publications summarizing initial findings came out in 2023, with a total of seven authors across four publications.
Publications describing attempted replications began to be published as soon as the following week.
On 26 July 2023, Kim Hyun-Tak stated in an interview with the New Scientist that the first paper submitted by Kwon contained "many defects" and was submitted without his permission. [ 34 ] [ 42 ]
On 28 July 2023, Kwon presented the findings at a symposium held at Korea University. [ 45 ] [ 46 ] [ 47 ] That same day, Yonhap News Agency published an article quoting an official from Korea University as saying that Kwon was no longer in contact with the university. [ 17 ] The article also quoted Lee saying that Kwon had left the Q-Centre Research Institute four months previously. [ 17 ]
On the same day, Kim Hyun-Tak provided The New York Times with a new video presumably showing a sample displaying strong signs of diamagnetism. [ 2 ] The video appears to show a sample different to the one in the original preprint. On 4 August 2023, he told SBS News that high-quality LK-99 samples may exhibit diamagnetism over 5,000 times greater than graphite, which he claimed would be inexplicable unless the substance is a superconductor. [ 48 ]
Materials scientists and superconductor researchers responded with skepticism. [ 18 ] [ 49 ] The highest-temperature superconductors known at the time of publication had a critical temperature of 250 K (−23 °C; −10 °F) at pressures of over 170 gigapascals (1,680,000 atm; 24,700,000 psi). The highest-temperature superconductors at atmospheric pressure (1 atm) had a critical temperature of at most 150 K (−123 °C; −190 °F).
On 2 August 2023, The Korean Society of Superconductivity and Cryogenics established a verification committee as a response to the controversy and unverified claims of LK-99, in order to arrive at conclusions over these claims. The verification committee is headed by Kim Chang-Young of Seoul National University and consists of members of the university, Sungkyunkwan University and Pohang University of Science and Technology . Upon formation, the verification committee did not agree that the two 22 July arXiv papers by Lee et al. or the publicly available videos at the time supported the claim of LK-99 being a superconductor. [ 42 ] [ 50 ]
As of 15 August 2023, [update] the measured properties did not suggest that LK-99 is a superconductor. The published material does not explain how the LK-99's magnetisation can change, demonstrate its specific heat capacity, or demonstrate it crossing its transition temperature. [ 18 ] A more likely explanation for LK-99's magnetic response is a mix of ferromagnetism and non-superconductive diamagnetism . [ 42 ] [ 16 ] [ 51 ] A number of studies found that copper(I) sulfide contamination common to the synthesis process could closely replicate the observations that inspired the initial preprints. [ 10 ] [ 11 ]
The claims in the 22 July papers by Lee et al. went viral on social media platforms the following week. [ 6 ] [ 52 ] The viral nature of the claim resulted in posts from users using pseudonyms from Russia and China claiming to have replicated LK-99 on both Twitter and Zhihu . [ 53 ] Other viral videos described themselves as having replicated samples of LK-99 "partially levitating", most of which were found to be fake. [ 49 ]
Scientists interviewed by the press remained skeptical, [ 54 ] [ 55 ] because of the quality of both the original preprints, the lack of purity in the sample they reported, and the legitimacy of the claim after the failure of previous claims of room temperature superconductivity did not show legitimacy (such as the Ranga Dias affair ). [ 42 ] The Korean Society of Superconductivity and Cryogenics expressed concern on the social and economic impacts of the preliminary and unverified LK-99 research. [ 56 ]
A video from Huazhong University of Science and Technology uploaded on 1 August 2023 by a postdoctoral researcher on the team of Chang Haixin, [ 42 ] apparently showed a micrometre -sized sample of LK-99 partially levitating. This went viral on Chinese social media, becoming the most viewed video on Bilibili by the next day, [ 57 ] [ 42 ] and a prediction market briefly put the chance of successful replication at 60%. [ 58 ] A researcher from the Chinese Academy of Sciences refused to comment on the video for the press, dismissing the claim as "ridiculous". [ 57 ]
In early August, people began to create memes about "floating rocks", [ 59 ] and there was a brief surge in Korean and Chinese technology stocks, [ 60 ] [ 61 ] [ 62 ] despite warnings from the Korean stock exchange against speculative bets in light of the excitement around LK-99, [ 56 ] which eventually fell on August 8. [ 63 ] Following the publication of the Nature article on August 16 that proclaimed LK-99 is not a superconductor, [ 1 ] South Korean superconductor stocks fell further, as the interest about LK-99 from investors in previous weeks disappeared. [ 64 ]
After the July 2023 publication's release, independent groups reported that they had begun attempting to reproduce the synthesis, with initial results expected within weeks. [ 6 ] Some replication efforts gained global visibility, with the aid of online replication trackers that catalogued new announcements and status updates. [ 53 ] [ 26 ]
As of 15 August 2023, [update] no replication attempts had yet been peer-reviewed by a journal. Of the non-peer-reviewed attempts, over 15 notable labs published results that failed to observe any superconductivity, and a few observed magnetic response in small fragments that could be explained by normal diamagnetism or ferromagnetism. Some demonstrated and replicated alternate causes of the observations in the original papers: Copper-deficient copper (I) sulfide [ 10 ] has a known phase transition at 377 K (104 °C; 219 °F) from a low-temperature phase to a high-temperature superionic phase, with a sharp rise in resistivity [ 11 ] [ 10 ] and a λ-like-feature in the heat capacity. [ 10 ] Furthermore, Cu 2 S is diamagnetic.
Only one attempt observed anything that could have been a sign of superconductivity: Southeast University claimed to measure very low resistance in a flake of LK-99, in one of four synthesis attempts, below a temperature of 110 K (−163 °C; −262 °F). [ 2 ] [ 65 ] Doubts were expressed by experts in the field, as the study saw no dropoff to zero resistance, had large measurement artifacts, and used crude instruments that could not measure resistance below 10 μΩ (too high to distinguish superconductivity from less exotic low-temperature conductivity). [ 49 ] [ 66 ]
Results Key: # Success * Partial success ‡ Partial failure † Failure
Second attempt: strong diamagnetism in a fragment.
Analysis showed impurities of Iron and Cu 2 S, which could explain magnetic response rather than superconductivity.
2. The resistance of LK-99 material was measured, which is roughly equivalent to copper.
3. Observed strange metal phenomena
In the initial papers, the theoretical explanations for potential mechanisms of superconductivity in LK-99 were incomplete. Later analyses by other labs added simulations and theoretical evaluations of the material's electronic properties from first principles. An analysis by Georgescu, et al. comprehensively rebutted the claims of the original preprints, and was updated in early 2025 and published in the peer-reviewed Chemistry of Materials . [ 90 ]
Selected theoretical studies:
Media mentions: [ 92 ]
Similar work published the next day by Si & Held [ 31 ] and Kurleto, et al. [ 93 ]
Media mentions: [ 58 ] [ 59 ] | https://en.wikipedia.org/wiki/LK-99 |
LLM-as-a-Judge is a conceptual framework in natural language processing (NLP) that employs large language models (LLMs) as evaluators to assess the performance of other language-based systems or outputs.
Instead of relying solely on human annotators, the approach leverages the general language capabilities of advanced language models to serve at automated judges.
LLM-as-a-Judge may be more cost-effective and may be added to automated evaluation pipelines.
Unlike traditional automatic evaluation metrics such as ROUGE and BLEU —which rely on transparent, rule-based comparisons with surface-level n-grams—LLM-as-a-Judge relies on the opaque internal reasoning of large language models—offering evaluations that likely incorporate deeper semantic understanding, but at the cost of interpretability.
Typically, a more powerful LLM is employed to evaluate the outputs of smaller or less capable language models—for example, using GPT-4 to assess the performance of a 13-billion-parameter LLaMA model. [ 1 ]
This computing article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/LLM-as-a-Judge |
LLNL HRS (hot recycled solid) process is an above-ground shale oil extraction technology. It is classified as a hot recycled solids technology. [ 1 ]
The process was developed by the Lawrence Livermore National Laboratory . In 1984–1987, Lawrence Livermore National Laboratory operated a LLNL HRS process-based pilot pant at Parachute, Colorado , with capacity of one tonne of oil shale per day. In 1989, the pilot plant was upgraded to process four tonne of oil shale per day. The pilot plant was operated till 1993. [ 2 ] Later the process was modified and tested in the field of waste treatment and environmental cleanup for removing organic compounds and for decomposing sodium nitrate in contaminated soils. [ 3 ]
As a heat carrier, LLNL HRS process uses spent oil shale . Raw oil shale and spent oil shale are mixed in the fluidized bed mixer. The use of fluidized bed mixer results in better mixture, which in turn increases the mean quantity of oil yield and oil shale throughput. From the fluidized bed mixer oil shale moves downward to the packed-bed pyrolyzer. The heat is transferred from the heated spent oil shale to the raw oil shale causing pyrolysis . As a result, oil shale decomposes to shale oil vapors, oil shale gas and spent oil shale. Oil vapors are collected from the pyrolyzer. The spent oil shale, still including residual carbon ( char ), by the air pneumatic lift pipe to the delayed-fall combustor where it is combusted to heat the process. The delayed-fall combustor used in this process gives greater control over the combustion process as compared to a lift pipe combustor. From the delayed-fall combustor the oil shale ash and spent shale falls into a fluidized bed classifier where the finest parts of solids are removed and hot spent shale is forwarded to the fluidized bed mixer. [ 3 ] [ 4 ] | https://en.wikipedia.org/wiki/LLNL_HRS_process |
The LLNL RISE process was an experimental shale oil extraction technology developed by the Lawrence Livermore National Laboratory . The name comes from the abbreviation of the Lawrence Livermore National Laboratory and words 'rubble in situ extraction'. [ 1 ]
LLNL RISE is a modified in situ extraction technology originally proposed by Rio Blanco Oil Shale Co. and developed by the Lawrence Livermore National Laboratory. [ 1 ] It is classified as an internal combustion technology. [ 2 ] The process was described in 1975 by Lewis A. E. and A. J. Rothman. [ 3 ] [ 4 ]
In the LLNL RISE process a part of the oil shale deposit (roughly 20% of the total deposit) is removed by the conventional mining technique. The remaining deposit is then broken up with explosives to increase porosity of the deposit. As a result, a large underground retort chamber by 20 to 100 metres (66 to 328 ft) square and 100 to 300 metres (330 to 980 ft) high is created. The retort chamber is ignited at the top. The combustion zone moves downward as an oxygen gas provided, similar to the process developed by the Occidental Petroleum . The heat causes retorting process converting kerogen in oil shale to oil shale gas and shale oil vapors. Some oil is collected at the bottom of the retort, other collected at the surface as vapors. [ 1 ] [ 5 ]
The process was never used commercially. It was tested by using experimental simulated retort with capacity of 6 tonnes of oil shale per day. [ 5 ] | https://en.wikipedia.org/wiki/LLNL_RISE_process |
In mathematics, an LLT polynomial is one of a family of symmetric functions introduced as q -analogues of products of Schur functions . [ 1 ]
J. Haglund, M. Haiman , and N. Loehr showed how to expand Macdonald polynomials in terms of LLT polynomials. [ 2 ] Ian Grojnowski and Mark Haiman proved a positivity conjecture for LLT polynomials that combined with the previous result implies the Macdonald positivity conjecture for Macdonald polynomials , and extended the definition of LLT polynomials to arbitrary finite root systems. [ 3 ]
This polynomial -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/LLT_polynomial |
The LN-3 inertial navigation system is an inertial navigation system (INS) that was developed in the 1960s by Litton Industries . It equipped the Lockheed F-104 Starfighter versions used as strike aircraft in European forces. An inertial navigation system is a system which continually determines the position of a vehicle from measurements made entirely within the vehicle using sensitive instruments. These instruments are accelerometers which detect and measure vehicle accelerations, and gyroscopes which act to hold the accelerometers in proper orientation.
The development of Litton's first INS was the result of a collusive act by the engineer, Max Lipscomb of the Wright Air Force Base in Ohio and Dr. Henry E. Singleton , Head of the newly formed Guidance and Control Dept. of Litton Industries at Beverly Hills, California.
Lipscomb's department was not permitted to engage in development of navigation systems, but was permitted to engage in other aircraft avionics such as pitch, roll, and yaw indicators. Singleton proposed to provide a system that would provide highly accurate pitch, roll, and yaw indicators. The system would be a north seeking stable platform controlled by gyroscopes and accelerometers. Such a system would automatically provide velocities in the east–west and north–south direction. And later, by providing integrators for these two axes, one would then have a full-fledged Inertial Navigation System.
In about mid-1956 a contract for approximately $300,000 was awarded by Wright Air Force Base to Litton Industries for the development of such "Aircraft Attitude System." Singleton appointed Sidney Shapiro as Project Engineer for this program. The system was completed and ready for flight test by the end of 1958.
Mr. Shapiro selected Paul Mantz , a partner in Tallmantz Aviation, to supply the aircraft, principally because of Mantz's extensive experience with the movie industry. They had done their work on several Cinerama travelogs. Mantz's people had also recently finished work on the picture "North by Northwest" starring Cary Grant in which there was considerable stunt flying. Shapiro's idea was to photograph the ground periodically and at the same instant to photograph the Inertial Navigation System's output. In that way no possibility of finger pointing was possible since none of Shapiro's people were involved in the data taking. So the two extra integrators were installed and the system was ready for test by early 1959.
By 1959 things had gone well enough that Shapiro was able to obtain three successive flights in which the accuracies were substantially better than one mile an hour. On the basis of these results, Litton Industries was awarded a contract to provide 2000 systems for the F104 NATO Fighter Aircraft. [ 1 ]
The Cold War missile race spurred the development of smaller, lighter and more accurate inertial systems. Independent of its environment, the inertial system provides velocity and position information accurately and instantaneously for all manoeuvres, as well as being an accurate attitude and heading reference. The LN3-2A was the first inertial navigation system small and light and accurate enough to be fitted in a high performance fighter.
The early F-104's, model A through F, did not have an Inertial Navigator. It was the development of the F-104G, around 1959, for the European Air Forces with tactical bomber/strike capabilities, that brought the LN-3 into the aircraft. [ 2 ] The LN-3 gave the F-104G the capability to navigate at low level in adverse weather and to drop a nuclear weapon at a range of 1,000 km with the best possible precision; this being vital to the F-104G program.
The LN-3 is a full 3-degrees-of-freedom, 4-gimbal inertial navigator, covering the flight performance envelope of the F-104G which ranged from 0 to 70,000 feet altitude; 0 to Mach 2+ speed, and accelerations from −5 to +9 g.
The functional description of the LN3-2A requires some knowledge of some basic principles of inertial navigation to understand their application to the LN3-2A. The principal component of the system is the stable platform to which are mounted three accelerometers and two gyros. This stable platform is mounted in a system of platform gimbals. The acceleration of the airplane in any plane or direction is measured by the accelerometers and integrated in the computer to obtain velocity. Velocities in turn are integrated to obtain distance. With a known reference point representing initial position of the airplane with respect to Earth, this data can be converted to distance and heading traveled, and distance and bearing to destination.
The following characteristics of the platform are described: [ 3 ]
The LN3-2A computer controls the platform, computes navigational information and provides special AC and DC voltages required for equipment operation.
The functions of the computer are:
Before starting the Inertial navigator, the pilot has to enter the coordinates of the starting point in the "Align Control" panel in the right-hand console of the F-104G.
The first selection in the starting sequence is to rotate the mode selector switch of the "Inertial Navigation Control" Panel from Off to Standby .
In this mode the platform and component oven are brought up to operating temperature ; indicated by the "heat" light on the IN Control Panel, which takes several minutes depending on outside and system temperatures.
All at operating temperature the system may be switched to " Align ", allowing the machine to commence operation. The computer is powered up and nulls its velocity shafts; the gyros are powered by 115 V and 400 Hz and revving up; the platform is levelled in pitch, inner and outer roll relative to the aircraft using the gimbal synchrotransmitters; and the azimuth axis is driven to the grid north direction using the magnetic heading sensor. This phase of Alignment takes 1 minute and is called coarse align.
After this 1 minute the system switches to the fine align phase , during which the gyro spin motor power is brought down to 95 V and 375 Hz to avoid any magnetic interference with any other aircraft system using 400 Hz. The levelling of the platform is taken over by the X and Y accelerometers sensing even the smallest component of gravity which is an indication of not being precisely level. The levelling of the stable element is achieved by torquing the respective gyro torquers which makes the gimbal motors to follow up and level the stable element. The distance shafts are set to zero; the gyros are at operational speed and the computer is continuously feeding the gyros, and thereby the stable element, with corrections for local Earth rotation. This is called the levelling phase of fine align .
Leveling ends automatically when the computer decides that the platform stable element is exactly locally level, which may take a few minutes. If level, the final phase of alignment is switched on; gyrocompassing .
The stable element is exactly level and Schuler-tuned but the gyros are not yet aligned with the Earth rotation axis. Therefore, the stable element tends to turn off-level, which is sensed by the Y accelerometer which signal is fed to the gyro torquer to rotate the azimuth axis of the stable element. This process continues for a few minutes until the correction signal is getting smaller and can be kept almost zero for 50 seconds, which gives confidence that the system is level and aligned. This is visible for the pilot because the green Nav light flashes.
The system is now ready for use and the pilot selects " Nav " on the IN Control Panel, and all circuitry that was involved in the various alignment phases is switched to the navigate mode .
Other possible modes are Compass only which may be selected after a LN3 in-flight failure, and Alert Align to shorten the alignment phase. After the last flight but before shutting down aircraft power the precise heading of the running LN3 is stored and can be used at starting up the next time, if the aircraft is not moved.
Specified navigation accuracy for the LN-3 is a 50% circular-error probability (c.e.p.) of two nautical miles after one hour's operation, which is equivalent to a 98% c.e.p. of four nautical miles. Until the −9 version of the LN-3-2A came into service (~1963) results were outside these limits by a fair margin, but since then it has been greatly exceeded in a number of groups of flights.
During manufacturer's development flying at Palmdale, some 1167 flights were made up to October 1961, and the c.e.p. of the LN-3 and PHI-4 combined was a mile or so outside specification. From October 1961 to January 1962 a further 123 flights at Palmdale were assessed, following incorporation of the −9 modifications, and the c.e.p. came almost up to specification.
At Edwards AFB, during Category 2 testing, and at Palmdale during the "avionics marriage" period, mean time between failures of pre-9 systems was considerably below the 200 hr specified, but the target has been exceeded since then. [ 4 ]
In November 1965 a LN-3 system was installed in a prepared Flying Tigers Boeing 707 (the Pole Cat) to conduct a pole to pole 51 hours flight, and compare its performance with other means of navigation. The quoted error at the South pole was 2 miles.
Litton Systems Inc., or Litton Industries , the Guidance and Control Systems Division at Beverly Hills CA, were one of the major producers of inertial systems in the US in the 1950s and 1960s, and have made a series of systems for a number of American aircraft. [ 5 ]
The Genesis of inertial navigation systems is explained in the following reference.
The gimballed platform of the LN3-2A is the Litton P200 platform; the Gyro is the G200 Gyro; and the accelerometer is the A200 accelerometer. [ 19 ] (and Litton doc)
The G-200 Gyro is commonly used in the LN-2, LN-3 and the LN-12 systems. [ 20 ]
Manufacturers designation of the F-104G system is LN3-2A. Mark the difference in notation LN-3 and LN3-2A with the position of the dividing dash "-" .
The designation LN3-2A leaves room for a LN3-1, not known to author. Any additional information about the early Litton's is welcome!
The Litton LN-3 was one of the first inertial navigators on a production aircraft, but other systems, either inertial navigators or inertial measurement units, of other brands and for various applications with comparable technology existed.
The Autonetics Radar Enhanced Inertial Navigation System (REINS) of the North American A-5 Vigilante was more or less comparable to the LN-3/PHI-4. This system was derived from the XN-6 system developed for the SM-64 Navaho , the N5G system for the AGM-28 Hound Dog and the N2C/N2J/N3A/N3B system for the XB-70 , and was related to the N6A-1 navigation system used in the USS Nautilus (SSN-571) and the N10 inertial guidance system for the LGM-30 Minuteman . [ 21 ] Note that the Boeing history claims the REINS to be the first inertial navigation in a production airplane.
Nortronics had developed and produced Astro-inertial guidance /navigation systems for the SM-62 Snark . The system developed for the GAM-87 Skybolt was later adapted for use in the Lockheed SR-71 Blackbird and mostly referred to as NAS-14 and/or NAS-21.
The UGM-27 Polaris missile was equipped with a MIT-developed inertial system, which later evolved to the Delco produced IMU of the Apollo PGNCS .
The Saturn V was equipped with a MSFC-developed ST-124-M3 inertial platform which was a further development of the PGM-19 Jupiter 's ST-90.
The Convair B-58 Hustler was equipped by AN/ASQ-42 Dopler-inertial system, made by Sperry Corporation .
The LN-3 system was designed to constantly monitor critical parameters, and warn the pilot in case of a malfunction. Depending on the problem the pilot could switch-off the system, or continue in a dead reckoning mode. In case of serious self-diagnosed problems the system would auto shut-down.
Flight line maintenance of the LN-3, like systemchecks and fault isolation, was performed using specific test equipment :
At base (nav)shop level the platform, computer and adapter units were tested and repaired using the following test equipment :
For repairs beyond the capabilities of base level, the RNlAF Electronics Depot (DELM, at Rhenen) was equipped with specific test equipment and tooling to handle the (higher) depot level repairs of the LN-3 system.
The main test stations in use were:
The repair of the system's sensors, gyros and accelerometers, was performed by Litton. The RNlAF had its sensors repaired by Litton Canada, which also provided all necessary spare parts.
Other European users relied on German or Italian subsidiaries/licensees as LITEF at Freiburg and Hamburg. [ 22 ]
Exhibit of the LN3-2A system (without Alert Align Unit) in a vitrine. The platform gimbals can be rotated by the visitor with a remote control box.
Display of a complete system, running as new. On request explication and demonstration of the system is given. [ 23 ]
The LN-3 system was on display at RNlAF Air Force Day's, June 2019 Friday 14 and Saturday 15 at Volkel, Hangar 1. | https://en.wikipedia.org/wiki/LN-3_inertial_navigation_system |
LNAPL transmissivity is the discharge of light non-aqueous phase liquid (LNAPL) through a unit width of aquifer for a unit gradient.
Scholars Alex Mayer and S. Majid Hassanizadeh define LNAPL transmissivity as the "product of the porous medium permeability and the LNAPL relative permeability, which in turn is a function of saturation, and the thickness of the LNAPL". They wrote that once LNAPL is taken away, a lower recovery rate occurs because the "saturation and thickness of the mobile LNAPL fraction decreases". [ 1 ]
LNAPL transmissivity is a summary parameter that takes into account soil type and physical properties (e.g., porosity and permeability), LNAPL physical fluid properties(e.g., density and viscosity) and LNAPL saturation (i.e., amount of LNAPL present within the pore network). Consequently, LNAPL transmissivity is comparable across soil types, LNAPL types and
recoverable LNAPL volumes. More importantly, for LNAPL recovery from a given well, the soil and LNAPL physical properties do not change significantly through time. What changes, is the LNAPL saturation (amount of LNAPL present). As a result, LNAPL transmissivity decreases in direct proportion to the decrease in LNAPL saturation achievable through liquid recovery technology. LNAPL Transmissivity is not the only piece of data required when evaluating a site overall, because it requires a good LNAPL conceptual model in order to calculate. However, it is a superior summary metric to gauged LNAPL thickness to represent LNAPL recoverability and migration risk (e.g., on site maps) and direct remediation efforts. [ 2 ]
This chemistry -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/LNAPL_transmissivity |
A liquefied natural gas storage tank or LNG storage tank is a specialized type of storage tank used for the storage of Liquefied Natural Gas . LNG storage tanks can be found in ground, above ground or in LNG carriers . The common characteristic of LNG Storage tanks is the ability to store LNG at the very low temperature of -162 °C (-260 °F). LNG storage tanks have double containers, where the inner contains LNG and the outer container contains insulation materials. The most common tank type is the full containment tank. [ 1 ] Tanks vary greatly in size, depending on usage.
In-ground LNG tanks are also used; these are lined or unlined tanks beneath ground level. [ 2 ] The low temperature of the LNG freezes the soil and provides effective containment. The tank is sealed with an aluminium alloy roof at ground level. Historically there have been problems with some unlined tanks with the escape of LNG into fissures, the gradual expansion of extent of the frozen ground, and ice heave which have limited the operational capability of in-ground tanks. All piping connected to the LNG tanks, whether above ground or in-ground, are routed through the top of the vessel. [ 2 ] This mitigates against loss of containment in the event off a piping breach.
In LNG storage the pressure and temperature within the tank will continue to rise. LNG is a cryogen , and is kept in its liquid state at very low temperatures. The temperature within the tank will remain constant if the pressure is kept constant by allowing the boil off gas to escape from the tank. This is known as auto-refrigeration. [ 3 ]
The world's largest above-ground tank (delivered in 2000) is the 180 million liters full containment type for Osaka Gas Co., Ltd.
The world's largest tank (delivered in 2001) is the 200 million liters Membrane type for Toho Gas Co., Ltd. [ 4 ]
This article related to natural gas, petroleum or the petroleum industry is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/LNG_storage_tank |
A LO NO x burner is a type of burner that is typically used in utility boilers to produce steam and electricity .
Around 1986 John Joyce (of Bowin Cars fame), an influential Australian inventor, first learned about oxides of nitrogen ( NO x ) and their role in the production of smog and acid rain . His first introduction to the complexities of the subject was brought about by the work of Fred Barnes and Dr John Bromley from the state Energy Commission of Western Australia. [ 1 ]
The vast majority of the research and development stretching back over twenty years was about large scale industrial burners and complex mechanisms which, in the end, did not produce what one would consider low NO x (2 ng/J or ~ 4 ppm at 0% O 2 on dry basis ). [ 2 ]
In fact at that time, 15 ng/J NO 2 appears to have been considered low NO 2 . The one clear message that did flow through all the mass of information he studied, was the effect of temperature on the formation of NO x .
In the late 1980s, Health and Environment Authorities in Australia raised concerns about the indoor air quality and the extent that particularly older style unflued gas heaters were contributing to higher than acceptable levels of nitrogen dioxide (NO 2 ). Consequently in 1989 the New South Wales Department of School Education initiated an extensive investigation of nitrogen dioxide in schools throughout New South Wales. As an interim measure the Health Authorities advised that a level of 0.3 ppm NO 2 should become the upper limit for classrooms. [ 3 ] The Australian Gas Association in turn reduced the indoor emission rate of NO 2 for unflued gas heaters from 15 to 5 ng/J and this remains the current limit. [ 4 ] The New South Wales government, through the Public Works Department, also re-evaluated alternative methods of heating classrooms, to ensure a safe and healthy environment for students.
It was in this context, that John Joyce's company Bowin Technology embarked on a major research & development program aimed at minimising nitrogen dioxide emissions from unflued gas heaters. Bowin Technology set itself the task of solving the emission problem at its source: the gas burner . This was despite a generally long held belief by gas experts, that commercially warranted gas burner improvements could not deliver drastic nitrogen oxides (NO x ) reductions.
In 1989, an immediate call to reduce the indoor nitrogen dioxide (NO 2 ) level, was triggered by widely publicised articles and media coverage in New South Wales, highlighting the effect this chemical has on respiratory sensitive people, such as asthmatics and those with bronchial problems.
In the heat of the indoor air quality debate various State institutions in Australia were advised to switch to flued gas heaters and electric heating .
New South Wales in contrast, through combined action by Australian Gas Light Company , Health Authorities and the New South Wales Public Works Department, formulated initial indoor air quality guidelines. These guidelines formed the basis for Australian Gas Appliance Code restrictions for nitrogen dioxide NO 2 emissions from unflued heaters, now adopted Australia wide. [ 4 ]
John Joyce became aware that no other overseas regulatory body made a distinction between NO and NO 2 in their environmental guidelines or codes. Furthermore it appeared that total nitrogen oxides level requirements were in place irrespective of whether emissions were flued or not.
Consequently John Joyce learned that a 'harmless' part of NO x emissions, nitric oxide (NO), in the presence of hydrocarbons (such as household aerosol propellants , possible gas leaks and ingress of vehicle exhaust fume), converts to NO 2 . This was found to be the case in the New South Wales school investigation. [ 3 ] In a scientific sense it had become practice to calculate both NO + NO 2 , when measuring oxides of nitrogen levels in emissions. Hence the now commonly used reference to "total NO x ".
Natural gas by composition has a distinct advantage over other fossil fuels in terms of carbon dioxide , particulate and sulfur dioxides produced when converting to useful energy . In the early 1990s numerous countries were in the process of substituting oil and coal with natural gas for their energy and electric power needs.
To maintain this advantage as an "environmentally friendly" fuel, Australian gas utilities are effectively reducing gas losses ( methane emissions ) in their deliveries, and impose strict codes on appliance manufacturers and installers against gas leakage .
Nevertheless environmental experts see the production of oxides of nitrogen as a major menace in the formation of greenhouse gases and photochemical smog . The interaction of NO x with hydrocarbons from vehicle exhausts and sunlight can also form low level ozone . In the stratosphere (some 25 km up), ozone is helpful by absorbing the fiercer part of the ultraviolet radiation of the sun, but at ground level it damages materials and vegetation. It irritates throat, lungs and eyes, and strenuous exercise or work can become painful. Furthermore, the effectiveness of nitrous oxide as a greenhouse gas is magnified by its longer life than carbon dioxide , methane and CFC's.
In essence the rate at which low level ozone is formed is determined by hydrocarbons, whilst the availability of oxides of nitrogen influences the amount it produces. At this point the environmental debate takes a surprising turn as individual industries tend to blame each other's emission as a probable cause.
It is well established that conventional " blue flame " or bunsen gas burners produce oxides of nitrogen at levels of 30-50 nanograms per joule [ 5 ] [ 6 ] and are as such not considered to have potential for NO x reduction. Surface combustion burners or radiant tile burners in comparison produce nitrogen oxides' levels 60-70% less. [ 6 ] Therefore John Joyce's research into low NO x burners revolved primarily around surface combustion techniques. Another issue was the effect combustion temperatures have on the formation of NO x .
John Joyce's task became even more challenging when he decided not to direct his development towards radiant type surface combustion tiles. The use of radiant heating for most institutional purposes (other than spot heating) is considered impractical as is too hot close to the heater, while the loss of radiant heat over a distance to be reached is quite dramatic.
Investigations into numerous developments of other types of "low NO x " burners showed that so far such burners were either too complex in design or operation, too expensive or unsuitable. John Joyce's plan was to use high temperature steel mesh, and went on to produce scores of prototype burners until one showed "potential".
The scientific innovative nature of John Joyce's LO-NO x technologies are confirmed by full patent protection in Australia , United States , United Kingdom , Japan , Italy and France .
In 1993 John Joyce received an Australian Design Award and Powerhouse Museum Selection status for his "SLE" heater range, which incorporate LO-NO x burners.
The Australian Academy of Design selected the SLE unflued gas heater range to be featured in the Design Showcase during the "Innovation by Design" National Conference in October 1994
In the United States, John Joyce's LO-NO x water heater burners have successfully undergone a series of exhaustive tests to prove that these particular burners do not act as an ignition source in the presence of flammable vapours, resulting from accidental fuel spillage. There have also been extensive tests carried out to verify its reduction of NO 2 .
More tangible cost savings are defined when comparing the energy efficiencies of gas heaters with low NO x emissions with conventional flued types. Gas heaters with emission problems are flued and inherently lose substantial energies in the form of hot flue gases to the atmosphere. In addition, the choice of placement of flued heaters is greatly impaired due to flue installation restrictions.
In contrast, dedicated low emission gas heaters do not require a flue system. Furthermore, with the introduction of oxygen depletion sensors and thermostatic controls, they do not place critical reliance on ventilation as had been the case. These heaters can be positioned more conveniently and centralised to affect optimum warm air distribution. By definition unflued low NO x gas heaters are 100% efficient as all heat energy released from the flame is converted to useful heat. | https://en.wikipedia.org/wiki/LO-NOx_burner |
LOCOS , short for LOCal Oxidation of Silicon , is a microfabrication process where silicon dioxide is formed in selected areas on a silicon wafer having the Si-SiO 2 interface at a lower point than the rest of the silicon surface. As of 2008 it was largely superseded by shallow trench isolation .
This technology was developed to insulate MOS transistors from each other and limit transistor cross-talk. The main goal is to create a silicon oxide insulating structure that penetrates under the surface of the wafer, so that the Si-SiO 2 interface occurs at a lower point than the rest of the silicon surface. This cannot be easily achieved by etching field oxide. Thermal oxidation of selected regions surrounding transistors is used instead. The oxygen penetrates in depth of the wafer, reacts with silicon and transforms it into silicon oxide. In this way, an immersed structure is formed. For process design and analysis purposes, the oxidation of silicon surfaces can be modeled effectively using the Deal–Grove model . [ 1 ] | https://en.wikipedia.org/wiki/LOCOS |
The Large Observatory for X-ray Timing ( LOFT ) is a proposed ESA space mission originally slated to launch around 2022, and now proposed to launch around 2025. The mission will be devoted to the study of neutron stars , black holes and other compact objects by means of their very rapid X-ray variability. LOFT is supported by a large international collaboration, led by researchers spread over most of the European countries, including Italy, Switzerland , Germany, Denmark , United Kingdom, Greece , Ireland, the Netherlands, Poland, Czech Republic , Spain, and with contributions from Brazil , Canada, Israel , United States and Turkey . SRON Netherlands Institute for Space Research acts as principal investigator .
The mission was submitted to the ESA Cosmic Vision M3 call for proposals, [ 1 ] and was selected, together with other three missions, for an initial Assessment Phase. [ 2 ]
On February 19, 2014, the PLATO mission was selected in favour of the other candidates in the programme, including LOFT. [ 3 ] In spite of this, LOFT has been submitted to the Cosmic Vision M4
call for proposals for a planned launch date of 2025, if selected. [ 4 ]
The Large Observatory for X-ray Timing mission comprises two instruments.
The Large Area Detector (LAD) achieves an effective area of ~10 m 2 (more than an order of magnitude larger than current spaceborne X-ray detectors, e.g. RXTE ) in the 2-50 keV range, yet still fitting a conventional platform and small/medium-class launcher, thanks to the monolithic design of its large area Silicon Drift Detectors (SDD).
The Wide Field Monitor (WFM) is a coded mask X-ray monitor with a large field of view (observing about 50% of the sky available to the LAD at any time), and is also based on the Silicon Drift Detector technology. Its operating energy range is the same of the LAD, i.e. 2-50 keV.
The main scientific objectives of LOFT are:
Besides these topics, LOFT will in general study the X-ray spectra and variability for a wide range of astrophysical sources, e.g. magnetars , active galactic nuclei , cataclysmic variables , X-ray transients and gamma-ray bursts .
The unique capabilities of LOFT, make this instruments certainly able to provide new breakthroughs in a wide range of Astrophysical research areas. | https://en.wikipedia.org/wiki/LOFT |
The LOLI Database , an abbreviation of List Of LIsts , is an international chemical regulatory database developed and maintained by ChemADVISOR, Inc. [ 1 ]
The LOLI database is one of the primary sources of information for the creation of safety data sheets and other hazard communication documents. The database "lets users check individual or groups of chemical substances against any or all of the regulations noted within the database" in order to identify the regulations that apply to those substances. [ 2 ] The LOLI database can be accessed using either LOLI Desktop, LOLI in the Cloud or LOLI-On-Line. ChemADVISOR Navigator provides regulatory context. [ 3 ]
The database was first created in 1986. Since then, the database has grown to include over 5800 separate lists [ 4 ] from 128 countries. | https://en.wikipedia.org/wiki/LOLI_Database |
Long Range Kinematic ( LRK ) technology is a sophisticated kinematic method developed by Magellan (formerly Thales) Navigation that optimises the advantages of dual-frequency GPS operation. Other conventional methods use the dual-frequency only during initialisation . LRK makes solving ambiguities during initialisation easy and continuous dual-frequency kinematic operation possible at distances up to 40 kilometres .
Conventional dual-frequency kinematic operation is limited to about 10 kilometres, using a combined observation on GPS L1 and L2 frequencies to produce an initial wide lane solution, ambiguous to around 86 centimetres. During a second phase, the conventional kinematic method uses measurements from the L1 frequency only. This method only allows for kinematic operation as long as the de-correlation of atmospheric errors is compatible with a pure phase single-frequency solution.
Similar to the KART process, LRK is a simple and reliable method that allows any initialisation mode, from a static or fixed reference point, to On The Fly ambiguity resolution, when performing dual-frequency GPS positioning. LRK technology reduces initialisation times to a few seconds by efficiently using L2 measurements in every mode of operation. LRK maintains optimal real-time positioning accuracy to within a centimetre at a range up to 40-50 kilometres, even with a reduced number of visible satellites .
This technology-related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/LRK |
LSAT is the most common name for the inorganic compound lanthanum aluminate - strontium aluminium tantalate , which has the chemical formula (LaAlO 3 ) 0.3 (Sr 2 TaAlO 6 ) 0.7 or its less common alternative: (La 0.18 Sr 0.82 )(Al 0.59 Ta 0.41 )O 3 . LSAT is a hard, optically transparent oxide of the elements lanthanum , aluminium , strontium and tantalum . LSAT has the perovskite crystal structure , and its most common use is as a single crystal substrate for the growth of epitaxial thin films .
LSAT was originally developed as a substrate for the growth of high T c cuprate superconductors thin films, mostly of yttrium barium copper oxide (YBCO), for microwave device applications. The motivation for its development was to create a lattice-matched substrate with a similar thermal expansion coefficient and no structural phase transition over a wide temperature range, spanning from the high temperatures used for the growth of cuprates, to the cryogenic temperatures where they are superconducting. [ 1 ]
LSAT has a Mohs hardness of 6.5, placing it between quartz and the mineral feldspar . Its relative dielectric constant is ~22 and it has a thermal expansion coefficient of 8~10×10 −6 /K. The thermal conductivity of LSAT is 5.1 Wm −1 K −1 . [ 2 ] [ 3 ] LSAT's (cubic) lattice parameter of 3.868 Å makes it compatible for the growth of a wide range of perovskite oxides with a relatively low strain. [ citation needed ]
LSAT's melting temperature of 1,840C is lower compared to similar alternative substrates, such as LaAlO 3 . This property enables the growth of LSAT single crystals using the Czochralski process (CZ), which has commercial advantages. [ 4 ]
LSAT is primarily used in its single crystal form, typically as thin (≤1 mm ) wafers. These wafers are used as a common substrate for epitaxial growth of thin films . LSAT substrates are popular for epitaxial oxides and their heterostructures, often in the study of electron correlation phenomena. Typical materials grown on LSAT substrates include strontium titanate (SrTiO 3 ), cuprate superconductors (such as YBCO ), iron-based superconductors (iron-pnictides), rare-earth manganites , rare-earth nickelates and others. Semiconductors such as gallium nitride can also be grown on LSAT. [ 5 ]
LSAT's usefulness as a substrate for the growth of such films stems from its high chemical and thermal stability, and very low electrical conductivity. The growth conditions for such epitaxial layers can cause some substrates to form high densities of defects that can alter their properties. One example is the tendency of strontium titanate to form oxygen vacancy defects under high temperatures in high vacuum . These defects result in considerable variations of its properties, including the increase of electrical conductivity and optical opacity. LSAT on the other hand, is stable in both oxidizing and fairly reducing environments in high temperatures, thus enabling a larger window for the processing and growth conditions. [ citation needed ] | https://en.wikipedia.org/wiki/LSAT_(oxide) |
Lysergic acid diethylamide , commonly known as LSD (from German Lysergsäure-diethylamid ), is a potent psychedelic drug that intensifies thoughts, emotions, and sensory perception. [ 14 ] [ 9 ] [ 15 ] Often referred to as acid or lucy , LSD can cause mystical, spiritual, or religious experiences. [ 16 ] [ 17 ] At higher doses, it primarily induces visual and auditory hallucinations. [ 18 ] [ 19 ] LSD is not considered addictive, because it does not produce compulsive drug-seeking behavior. Using LSD can lead to adverse psychological reactions, such as anxiety, paranoia, and delusions. [ 7 ] Additionally, it may trigger "flashbacks," also known as hallucinogen persisting perception disorder (HPPD), where individuals experience persistent visual distortions after use. [ 20 ] [ 21 ]
The effects of LSD begin within 30 minutes of ingestion and can last up to 20 hours, with most trips averaging 8–12 hours. [ 22 ] [ 23 ] It is synthesized from lysergic acid and commonly administered via tabs of blotter paper . [ 24 ] LSD is mainly used recreationally or for spiritual purposes. [ 22 ] [ 25 ] As a serotonin receptor agonist , LSD's precise effects are not fully understood, but it is known to alter the brain’s default mode network , leading to its powerful psychedelic effects. [ 15 ] [ 26 ] [ 27 ]
The drug was first synthesized by Swiss chemist Albert Hofmann in 1938 and became widely studied in the 1950s and 1960s. [ 22 ] [ 20 ] It was used experimentally in psychiatry for treating alcoholism and schizophrenia . [ 28 ] However, its association with the counterculture movement of the 1960s led to its classification as a Schedule I drug in the U.S. in 1968. [ 29 ] It was also listed as a Schedule I controlled substance by the United Nations in 1971 and remains without approved medical uses. [ 22 ]
Despite its legal restrictions, LSD remains influential in scientific and cultural contexts. Its therapeutic potential has been explored, particularly in treating mental health disorders. [ 15 ] [ 30 ] As of 2017, about 10% of people in the U.S. had used LSD at some point, with 0.7% having used it in the past year. [ 31 ] Usage rates have risen, with a 56.4% increase in adult use in the U.S. from 2015 to 2018. [ 32 ]
LSD is commonly used as a recreational drug for its psychedelic effects. [ 33 ]
LSD can catalyze intense spiritual experiences and is thus considered an entheogen . Some users have reported out of body experiences. In 1966, Timothy Leary established the League for Spiritual Discovery with LSD as its sacrament . [ 34 ] [ 35 ] Stanislav Grof has written that religious and mystical experiences observed during LSD sessions appear similar to descriptions in sacred scriptures of great religions of the world and the texts of ancient civilizations . [ 36 ]
LSD currently has no approved uses in medicine . [ 37 ] [ 38 ] A meta analysis concluded that a single dose was shown to be effective at reducing alcohol consumption in people suffering from alcoholism . [ 39 ] LSD has also been studied in depression , anxiety , [ 40 ] [ 41 ] and drug dependence , with positive preliminary results. [ 42 ] [ 43 ]
LSD is an extraordinarily potent substance, [ 9 ] [ 44 ] [ 14 ] [ 45 ] and is one of the most potent psychoactive drugs known. [ 14 ] [ 45 ] This means that it produces its pharmacological effects at very small doses, with its dose range measured in micrograms (μg); that is, millionths of a gram. [ 9 ] [ 14 ] Noticeable effects can occur with doses of LSD as low as 20 μg, which is around 1/200th the mass of a grain of sand. [ 9 ] [ 44 ] [ 14 ] [ 22 ] LSD is approximately 200 times as potent as psilocybin and 5,000 times as potent as mescaline , meaning that it produces effects of similar magnitude at 1/200 and 1/5,000 times the respective doses. [ 9 ] [ 44 ] [ 46 ]
The usual dose range of LSD for psychedelic effects is 20 to 200 μg. [ 9 ] [ 44 ] The typical intermediate and "good effect" dose for a psychedelic experience is 100 μg (range 75–150 μg, while 20 to 50 μg is a low or "minidose" and 200 μg is a high or ego-dissolution dose. [ 9 ] [ 44 ] [ 7 ] A dose range as wide as 10 to 400 μg has been reported. [ 47 ] LSD may also be used in microdosing . [ 48 ] In this context, it may be used at subthreshold or microdoses of less than 10 μg. [ 9 ] [ 44 ]
The doses of LSD present in illicit LSD samples have decreased over time. In the mid-1960s, Owsley Stanley , the most important black market LSD manufacturer in the United States , distributed LSD at a standard concentration of 270 μg, [ 49 ] while street samples of the 1970s contained 30 to 300 μg. By the 1980s, the amount had reduced to between 100 and 125 μg, dropping more in the 1990s to the 20 to 80 μg range, [ 50 ] and even further in the 2000s. [ 49 ] [ 51 ]
LSD produces a variety of physical, psychological, and sensory effects. [ 9 ]
The primary immediate psychological effects of LSD are visual pseudo-hallucinations and altered thought, often referred to as "trips". These sensory alterations are considered pseudohallucinations because the subject does not perceive the patterns seen as being located in three-dimensional space outside the body. [ 52 ] LSD is not considered addictive. These effects typically begin within 20–30 minutes of oral ingestion, peak three to four hours after ingestion, and can last up to 20 hours, particularly with higher doses. An " afterglow " effect, characterized by an improved mood or perceived mental state, may persist for days or weeks following ingestion. [ 53 ] Positive experiences, or "good trips", are described as intensely pleasurable and can include feelings of joy, euphoria, an increased appreciation for life, decreased anxiety, a sense of spiritual enlightenment, and a feeling of interconnectedness with the universe. [ 54 ] [ 55 ]
Negative experiences, commonly known as " bad trips ", can induce feelings of fear, agitation, anxiety, panic, and paranoia. [ 7 ] [ 56 ] While the occurrence of a bad trip is unpredictable, factors such as mood, surroundings, sleep, hydration, and social setting, collectively referred to as " set and setting ", can influence the risk and are considered important in minimizing the likelihood of a negative experience. [ 57 ] [ 58 ]
LSD induces an animated sensory experience affecting senses, emotions, memories, time, and awareness. The effects range from subtle perceptual changes to profound cognitive shifts . Alterations in auditory and visual perception are common. [ 59 ] [ 60 ]
Users may experience enhanced visual phenomena, such as vibrant colors, objects appearing to morph, ripple or move, and geometric patterns on various surfaces. Changes in the perception of food's texture and taste are also noted, sometimes leading to aversion towards certain foods. [ 59 ] [ 61 ]
There are reports of inanimate objects appearing animated, with static objects seeming to move in additional spatial dimensions. [ 62 ] The auditory effects of LSD may include echo -like distortions of sounds, and an intensified experience of music. Basic visual effects often resemble phosphenes and can be influenced by concentration, thoughts, emotions, or music. [ 63 ] Higher doses can lead to more intense sensory perception alterations, including synesthesia, perception of additional dimensions, and temporary dissociation .
LSD can induce physical effects such as pupil dilation , decreased appetite , increased sweating , and wakefulness . The physical reactions to LSD vary greatly and some may be a result of its psychological effects. Commonly observed symptoms include increased body temperature , blood sugar , and heart rate , as well as goose bumps , jaw clenching , dry mouth , and hyperreflexia . In cases of adverse reactions, users may experience numbness , weakness , nausea , and tremors . [ 22 ]
The psychoactive effects of LSD last on average between 7 and 11 hours, with a possible range of 4 to 22 hours. [ 9 ] Higher doses tend to lead to a longer duration of action. [ 9 ] The onset of action when administered orally is 0.4 to 1.0 hours on average, with a possible range of 0.1 to 1.8 hours. [ 9 ]
LSD, a classical psychedelic, is deemed physiologically safe at standard dosages (50–200 μg) and its primary risks lie in psychological effects rather than physiological harm. [ 26 ] [ 68 ] A 2010 study by David Nutt ranked LSD as significantly less harmful than alcohol , placing it near the bottom of a list assessing the harm of 20 drugs. [ 69 ]
LSD can induce panic attacks or extreme anxiety, colloquially termed a " bad trip ". Despite lower rates of depression and substance abuse found in psychedelic drug users compared to controls, LSD presents heightened risks for individuals with severe mental illnesses like schizophrenia . [ 70 ] [ 71 ] These hallucinogens can catalyze psychiatric disorders in predisposed individuals, although they do not tend to induce illness in emotionally healthy people. [ 26 ]
While research from the 1960s indicated increased suggestibility under the influence of LSD among both mentally ill and healthy individuals, recent documents suggest that the CIA and Department of Defense have discontinued research into LSD as a means of mind control. [ 72 ] [ 73 ] [ 74 ] [ non-primary source needed ]
Flashbacks are psychological episodes where individuals re-experience some of LSD's subjective effects after the drug has worn off, persisting for days or months post- hallucinogen use. [ 75 ] [ 76 ] These experiences are associated with hallucinogen persisting perception disorder (HPPD), where flashbacks occur intermittently or chronically, causing distress or functional impairment. [ 21 ]
The etiology of flashbacks is varied. Some cases are attributed to somatic symptom disorder , where individuals fixate on normal somatic experiences previously unnoticed prior to drug consumption. [ 77 ] Other instances are linked to associative reactions to contextual cues, similar to responses observed in individuals with past trauma or emotional experiences. [ 78 ] The risk factors for flashbacks remain unclear, but pre-existing psychopathologies may be significant contributors. [ 79 ]
Estimating the prevalence of HPPD is challenging. It is considered rare, with occurrences ranging from 1 in 20 users experiencing the transient and less severe type 1 HPPD, to 1 in 50,000 for the more concerning type 2 HPPD. [ 21 ] Contrary to internet rumors, LSD is not stored long-term in the spinal cord or other body parts. Pharmacological evidence indicates LSD has a half-life of 175 minutes and is metabolized into water-soluble compounds like 2-oxo-3-hydroxy-LSD, eliminated through urine without evidence of long-term storage. [ 7 ] Clinical evidence also suggests that chronic use of SSRIs can potentiate LSD-induced flashbacks, even months after stopping LSD use. [ 80 ] : 145
LSD shows significant tachyphylaxis , with tolerance developing 24 hours after administration. The progression of tolerance at intervals shorter than 24 hours remains largely unknown. [ 81 ] Tolerance typically resets to baseline after 3–4 days of abstinence. [ 82 ] [ 83 ] Significant cross-tolerance occurs between LSD, mescaline and psilocybin . [ 84 ] [ 85 ] A slight cross-tolerance to DMT is observed in humans highly tolerant to LSD. [ 86 ] Tolerance to LSD also builds up with consistent use, [ 87 ] and is believed to result from serotonin 5-HT 2A receptor downregulation . [ 82 ] Researchers believe that tolerance returns to baseline after two weeks of not using psychedelics. [ 88 ]
LSD is widely considered to be non-addictive, despite its potential for abuse . [ 5 ] [ 26 ] [ 89 ] [ 90 ] Attempts to train laboratory animals to self-administer LSD have been largely unsuccessful. [ 26 ] Although tolerance to LSD builds up rapidly, a withdrawal syndrome does not appear, suggesting that a potential syndrome does not necessarily relate to the possibility of acquiring rapid tolerance to a substance. [ 91 ] A report examining substance use disorder for DSM-IV noted that almost no hallucinogens produced dependence, unlike psychoactive drugs of other classes such as stimulants and depressants . [ 92 ] [ 93 ]
The mutagenic potential of LSD is unclear. Overall, the evidence seems to point to limited or no effect at commonly used doses. [ 94 ] Studies showed no evidence of teratogenic or mutagenic effects. [ 7 ]
A potential risk of frequent repeated long-term use of LSD and other serotonergic psychedelics is cardiac fibrosis and valvulopathy due to serotonin 5-HT 2B receptor agonism . [ 95 ] [ 96 ] [ 97 ] [ 98 ] [ 99 ] This may also be the case with microdosing . [ 95 ] [ 96 ] [ 97 ] However, the risks are theoretical, and more research is needed to see if these complications can actually occur with psychedelics. [ 95 ] [ 98 ] A preliminary animal study found that chronic microdosing of LSD did not result in heart structure changes or valvulopathy in rodents. [ 100 ] Research appears to be mixed on whether LSD is a potent serotonin 5-HT 2B receptor agonist or not, with some studies finding it to be essentially inactive. [ 101 ]
Some psychedelics, including LSD, are metabolized by the cytochrome P450 enzyme CYP2D6 . Concurrent use of selective serotonin reuptake inhibitors (SSRIs), some of which are potent inhibitors of CYP2D6, with LSD may heighten the risk of serotonin syndrome . [ 80 ] : 145 Chronic usage of SSRIs, tricyclic antidepressants (TCAs), and monoamine oxidase inhibitors (MAOIs) is believed to diminish the subjective effects of psychedelics, likely due to 5-HT 2A receptor downregulation or desensitization induced by elevated serotonin levels. [ 7 ] [ 80 ] : 145 Contrary to the preceding notions however, a clinical study found that administration of LSD to people taking paroxetine , an SSRI and strong CYP2D6 inhibitor, increased LSD exposure by about 1.5-fold, was well-tolerated, and did not modify the pleasant subjective effects or physiological effects of LSD, whereas negative effects of LSD, including "bad drug effect", anxiety , and nausea , were reduced. [ 102 ] Similarly, a clinical study with LSD found that LSD levels were 75% higher in people with non-functional CYP2D6 ( poor metabolizers ) compared to those with functional CYP2D6. [ 103 ] [ 104 ] In contrast to certain other psychedelics, MAOIs do not inhibit the metabolism of or potentiate the effects of LSD and instead reduce its effects. [ 7 ] Interactions between psychedelics and antipsychotics or anticonvulsants are not well-documented; however, co-use with mood stabilizers like lithium may induce seizures and dissociative effects , particularly in individuals with bipolar disorder . [ 80 ] : 146 [ 105 ] [ 106 ] Lithium notably intensifies LSD reactions, potentially leading to acute comatose states when combined. [ 7 ]
There have been no documented fatal human overdoses from LSD, [ 7 ] [ 107 ] although there has been no "comprehensive review since the 1950s" and "almost no legal clinical research since the 1970s". [ 7 ] A 5 mg overdose of LSD (50 times the usual dose) produced severe nausea and vomiting along with behavioral and emotional disturbances. [ 108 ] [ 109 ] Eight individuals who had accidentally consumed an exceedingly high amount of LSD, mistaking it for cocaine , and had gastric levels of 1,000–7,000 μg/100 mL LSD tartrate and blood plasma levels up to 26 μg/mL, had suffered from comatose states , vomiting, respiratory problems, hyperthermia , and light gastrointestinal bleeding ; however, all of them survived without residual effects upon hospital intervention. [ 7 ] [ 110 ]
Individuals experiencing a bad trip after LSD intoxication may present with severe anxiety and tachycardia, often accompanied by phases of psychotic agitation and varying degrees of delusions. [ 68 ] Cases of death on a bad trip have been reported due to prone maximal restraint (commonly known as a hogtie) and positional asphyxia when the individuals were restrained by law enforcement personnel . [ 68 ]
Massive doses are largely managed by symptomatic treatments , and agitation can be addressed with benzodiazepines . [ 111 ] [ 112 ] Reassurance in a calm, safe environment is beneficial. [ 113 ] Antipsychotics such as haloperidol are not recommended as they may have adverse psychotomimetic effects . [ 111 ] Gastrointestinal decontamination with activated charcoal is of little use due to the rapid absorption of LSD, unless done within 30–60 minutes of ingesting exceedingly huge amounts. [ 111 ] Administration of anticoagulants , vasodilators , and sympatholytics may be useful for treating ergotism . [ 111 ]
The lethal oral dose of LSD in humans is estimated at 100 mg, based on LD 50 and lethal blood concentrations observed in rodent studies. [ 68 ]
Many novel psychoactive substances of 25-NB (NBOMe) series, such as 25I-NBOMe and 25B-NBOMe , are regularly sold as LSD in blotter papers. [ 114 ] [ 115 ] NBOMe compounds are often associated with life-threatening toxicity and death. [ 114 ] [ 116 ] Fatalities involved in NBOMe intoxication suggest that a significant number of individuals ingested the substance which they believed was LSD, [ 117 ] and researchers report that "users familiar with LSD may have a false sense of security when ingesting NBOMe inadvertently". [ 107 ] Researchers state that the alleged physiological toxicity of LSD is likely due to psychoactive substances other than LSD. [ 68 ]
NBOMe compounds are reported to have a bitter taste, [ 107 ] are not active orally, [ a ] and are usually taken sublingually. [ 119 ] When NBOMes are administered sublingually, numbness of the tongue and mouth followed by a metallic chemical taste was observed, and researchers describe this physical side effect as one of the main discriminants between NBOMe compounds and LSD. [ 120 ] [ 121 ] [ 122 ] Despite its high potency, recreational doses of LSD have only produced low incidents of acute toxicity, but NBOMe compounds have extremely different safety profiles. [ 107 ] [ 116 ] Testing with Ehrlich's reagent gives a positive result for LSD and a negative result for NBOMe compounds. [ 123 ] [ 124 ]
LSD is a serotonergic psychedelic and acts as a non-selective serotonin receptor modulator . [ 14 ] It binds with high affinity to most of the serotonin receptors . [ 9 ] The psychedelic effects of LSD are thought to be mediated specifically by activation of the serotonin 5-HT 2A receptor . [ 14 ] [ 9 ] However, a role of other serotonin receptors and targets in the effects of LSD cannot be ruled out and may be considered likely. [ 139 ] Uniquely among serotonergic psychedelics, LSD also shows potentially significant affinity for the dopamine receptors , albeit much lower than for most of the serotonin receptors. [ 9 ] [ 140 ]
LSD binds to most serotonin receptor subtypes except for the serotonin 5-HT 3 and 5-HT 4 receptors . [ 9 ] However, some of these serotonin receptors may not be affected at typical brain concentrations of LSD. [ 26 ] In humans, recreational doses of LSD may affect 5-HT 1A , 5-HT 2A , 5-HT 2B , 5-HT 2C , 5-HT 5A , and 5-HT 6 receptors . [ 141 ] Although not present in humans, 5-HT 5B receptors found in rodents also have a high affinity for LSD. [ 142 ] The psychedelic effects of LSD are attributed to activation of 5-HT 2A receptors. [ 143 ] Many but not all serotonin 5-HT 2A agonists are psychedelics and serotonin 5-HT 2A antagonists block the psychedelic activity of LSD. LSD exhibits functional selectivity at the serotonin 5-HT 2A and 5-HT 2C receptors in that it activates the signal transduction enzyme phospholipase A2 instead of activating the enzyme phospholipase C as the endogenous ligand serotonin does. [ 144 ]
Exactly how LSD produces its effects is unknown, but it is thought that it works by increasing glutamate release in the cerebral cortex [ 26 ] and therefore excitation in this area, specifically in layer V . [ 145 ] LSD, like many other drugs of recreational use, has been shown to activate DARPP-32 -related pathways. [ 146 ] The drug enhances dopamine D 2 receptor protomer recognition and signaling of D 2 –5-HT 2A receptor complexes, [ 147 ] which may contribute to its psychotropic effects. [ 147 ] LSD has been shown to have low affinity for H1 receptors , displaying antihistamine effects, although the significance of this at doses used in humans is unknown. [ 148 ] [ 149 ]
LSD is a biased agonist that induces a conformation in serotonin receptors that preferentially recruits β-arrestin over activating G proteins . [ 150 ] LSD also has an exceptionally long residence time when bound to serotonin receptors lasting hours, consistent with the long-lasting effects of LSD despite its relatively rapid clearance . [ 151 ] [ 150 ] A crystal structure of the serotonin 5-HT 2B receptor bound to LSD reveals an extracellular loop that forms a "lid" over the diethylamide end of the binding cavity and "traps" LSD in the binding pocket, which explains the slow rate of LSD unbinding from serotonin receptors. [ 151 ] [ 152 ] The related lysergamide lysergic acid amide (LSA) that lacks the diethylamide moiety is far less hallucinogenic in comparison. [ 152 ] Moreover, a specific residue in the binding pocket is partially responsible for the prolonged action of LSD, and this residue is found in the human protein but not in the receptors of rodents. [ 151 ]
LSD is an extraordinarily potent psychoactive drug and is among the most potent psychedelics known in humans. [ 14 ] [ 45 ] It is unclear why LSD is so potent. [ 45 ] [ 153 ] The affinity and activational potency of LSD at the human serotonin 5-HT 2A receptor in vitro is unremarkable compared to other psychedelics such as DOI and DOB . [ 45 ] [ 153 ] It appears that the N , N -diethylamide moiety of LSD fits into a sterically constrained region of the serotonin 5-HT 2A receptor that specifically accommodates this moiety. [ 45 ] [ 10 ] [ 153 ]
LSD, like other psychedelics, has been found to increase the expression of genes related to synaptic plasticity and hence to have psychoplastogenic effects. [ 154 ] This is in part due to binding to brain-derived neurotrophic factor (BDNF) receptor tropomyosin receptor kinase B (TrkB). [ 155 ]
There appears to be no significant acute tolerance to the subjective effects of LSD. [ 156 ] Hence, its duration appears to be dictated by pharmacokinetics rather than by pharmacodynamics . [ 156 ] This is in contrast to MDMA , which shows marked acute tolerance and a duration of effects that is shorter than its elimination half-life . [ 156 ]
The cryo-EM structures of the serotonin 5-HT 2A receptor with LSD, as well as with various other psychedelics and serotonin 5-HT 2A receptor agonists, have been solved and published by Bryan L. Roth and colleagues. [ 157 ] [ 158 ]
Neuroimaging studies using resting state fMRI recently suggested that LSD changes the cortical functional architecture. [ 160 ] These modifications spatially overlap with the distribution of serotoninergic receptors. In particular, increased connectivity and activity were observed in regions with high expression of 5-HT 2A receptor, while a decrease in activity and connectivity was observed in cortical areas that are dense with 5-HT 1A receptor. [ 161 ] Experimental data suggest that subcortical structures, particularly the thalamus, play a synergistic role with the cerebral cortex in mediating the psychedelic experience. LSD, through its binding to cortical 5-HT 2A receptor, may enhance excitatory neurotransmission along frontostriatal projections and, consequently, reduce thalamic filtering of sensory stimuli towards the cortex. [ 162 ] This phenomenon appears to selectively involve ventral, intralaminar, and pulvinar nuclei. [ 162 ]
The acute effects of LSD normally last between 6 and 12 hours depending on dosage, tolerance, and age. [ 163 ] [ 7 ] Aghajanian and Bing (1964) found LSD had an elimination half-life of only 175 minutes (about 3 hours); [ 141 ] however, using more accurate techniques, Papac and Foltz (1990) reported that 1 μg/kg oral LSD given to a single male volunteer had an apparent plasma half-life of 5.1 hours, with a peak plasma concentration of 5 ng/mL at 3 hours post-dose. [ 164 ]
The pharmacokinetics of LSD were not properly determined until 2015, which is not surprising for a drug with the kind of low-μg potency that LSD possesses. [ 8 ] [ 12 ] In a sample of 16 healthy subjects, a single mid-range 200 μg oral dose of LSD was found to produce mean maximal concentrations of 4.5 ng/mL at a median of 1.5 hours (range 0.5–4 hours) post-administration. [ 8 ] [ 12 ] Concentrations of LSD decreased following first-order kinetics with a half-life of 3.6±0.9 hours and a terminal half-life of 8.9±5.9 hours. [ 8 ] [ 12 ]
The effects of the dose of LSD given lasted for up to 12 hours and were closely correlated with the concentrations of LSD present in circulation over time, with no acute tolerance observed. [ 8 ] [ 12 ] Only 1% of the drug was eliminated in urine unchanged, whereas 13% was eliminated as the major metabolite 2-oxo-3-hydroxy-LSD (O-H-LSD) within 24 hours. [ 8 ] [ 12 ] O-H-LSD is formed by cytochrome P450 enzymes , although the specific enzymes involved are unknown, and it does not appear to be known whether O-H-LSD is pharmacologically active or not. [ 8 ] [ 12 ] The oral bioavailability of LSD was crudely estimated as approximately 71% using previous data on intravenous administration of LSD. [ 8 ] [ 12 ] The sample was equally divided between male and female subjects and there were no significant sex differences observed in the pharmacokinetics of LSD. [ 8 ] [ 12 ] In a subsequent, higher-quality study, the oral bioavailability of LSD was about 80%. [ 165 ]
A large meal before taking LSD has been found to result in circulating levels that were 50% lower than on an empty stomach. [ 7 ]
It has been said that there is a peculiar 40-minute lag before onset of the psychedelic effects of LSD when it is administered intravenously . [ 10 ] This has been said to be related to time-dependent interactions of LSD with the serotonin 5-HT 2A receptor. [ 10 ] However, contradicting the preceding claims, other sources have stated that intravenous injection of LSD results in onset of effects within a few minutes. [ 7 ] [ 11 ] In addition, intrathecal injection (intraspinal injection) is reported to have a virtually instantaneous onset of action. [ 7 ] [ 11 ] Doses of LSD are said to be similar by oral and injectable routes, with the exception of intrathecal injection in which the dosage is reduced to about one-third of usual. [ 11 ]
LSD is a chiral compound with two stereocenters at the carbon atoms C-5 and C-8, so that theoretically four different optical isomers of LSD could exist. LSD, also called (+)- d -LSD, [ 166 ] has the absolute configuration (5 R ,8 R ). 5 S stereoisomers of lysergamides do not exist in nature and are not formed during the synthesis from d -lysergic acid. Retrosynthetically , the C-5 stereocenter could be analysed as having the same configuration of the alpha carbon of the naturally occurring amino acid L- tryptophan , the precursor to all biosynthetic ergoline compounds.
However, LSD and iso-LSD, the two C-8 isomers, rapidly interconvert in the presence of bases , as the alpha proton is acidic and can be deprotonated and reprotonated. Non-psychoactive iso-LSD which has formed during the synthesis can be separated by chromatography and can be isomerized to LSD.
Pure salts of LSD are triboluminescent , emitting small flashes of white light when shaken in the dark. [ 163 ] LSD is strongly fluorescent and will glow bluish-white under UV light .
LSD is an ergoline derivative. It is commonly synthesized by reacting diethylamine with an activated form of lysergic acid . Activating reagents include phosphoryl chloride [ 167 ] and peptide coupling reagents . [ 149 ] Lysergic acid is made by alkaline hydrolysis of lysergamides like ergotamine , a substance usually derived from the ergot fungus on agar plate . Lysergic acid can also be produced synthetically, although these processes are not used in clandestine manufacture due to their low yields and high complexity. [ 168 ] [ 169 ]
Albert Hofmann synthesized LSD in the following manner: (1) hydrazinolysis of ergotamine into D- and L-isolysergic acid hydrazide, (2) separation of the enantiomers with di-( p -toluyl)-D-tartaric acid to get D-isolysergic acid hydrazide, (3) enantiomerization into D-lysergic acid hydrazide, (4) substitution with HNO 2 to D-lysergic acid azide and (5) finally substitution with diethylamine to form D-lysergic acid diethylamide. [ 14 ]
The precursor for LSD, lysergic acid , has been produced by GMO baker's yeast . [ 170 ]
"LSD," writes the chemist Alexander Shulgin , "is an unusually fragile molecule ... As a salt, in water, cold, and free from air and light exposure, it is stable indefinitely." [ 163 ]
LSD has two labile protons at the tertiary stereogenic C5 and C8 positions, rendering these centers prone to epimerisation . The C8 proton is more labile due to the electron-withdrawing carboxamide attachment, but the removal of the chiral proton at the C5 position (which was once also an alpha proton of the parent molecule tryptophan ) is assisted by the inductively withdrawing nitrogen and pi electron delocalisation with the indole ring. [ citation needed ]
LSD also has enamine -type reactivity because of the electron-donating effects of the indole ring. Because of this, chlorine destroys LSD molecules on contact; even though chlorinated tap water contains only a slight amount of chlorine, the small quantity of compound typical to an LSD solution will likely be eliminated when dissolved in tap water. [ 163 ] The double bond between the 8-position and the aromatic ring , being conjugated with the indole ring, is susceptible to nucleophilic attacks by water or alcohol , especially in the presence of UV or other kinds of light. LSD often converts to "lumi-LSD," which is inactive in human beings. [ 163 ]
A controlled study was undertaken to determine the stability of LSD in pooled urine samples. [ 171 ]
The concentrations of LSD in urine samples were followed over time at various temperatures, in different types of storage containers, at various exposures to different wavelengths of light, and at varying pH values. These studies demonstrated no significant loss in LSD concentration at 25 °C for up to four weeks. After four weeks of incubation, a 30% loss in LSD concentration at 37 °C and up to a 40% at 45 °C were observed. Urine fortified with LSD and stored in amber glass or nontransparent polyethylene containers showed no change in concentration under any light conditions. The stability of LSD in transparent containers under light was dependent on the distance between the light source and the samples, the wavelength of light, exposure time, and the intensity of light. After prolonged exposure to heat in alkaline pH conditions, 10 to 15% of the parent LSD epimerized to iso-LSD. Under acidic conditions, less than 5% of the LSD was converted to iso-LSD. It was also demonstrated that trace amounts of metal ions in the buffer or urine could catalyze the decomposition of LSD and that this process can be avoided by the addition of EDTA .
LSD can be detected in concentrations larger than approximately 10% in a sample using Ehrlich's reagent and Hofmann's reagent . However, detecting LSD in human tissues is more challenging due to its active dosage being significantly lower (in micrograms ) compared to most other drugs (in milligrams ). [ 173 ]
LSD may be quantified in urine for drug testing programs, in plasma or serum to confirm poisoning in hospitalized victims, or in whole blood for forensic investigations. The parent drug and its major metabolite are unstable in biofluids when exposed to light, heat, or alkaline conditions, necessitating protection from light, low-temperature storage, and quick analysis to minimize losses. [ 174 ] Maximum plasma concentrations are typically observed 1.4 to 1.5 hours after oral administration of 100 μg and 200 μg, respectively, with a plasma half-life of approximately 2.6 hours (ranging from 2.2 to 3.4 hours among test subjects). [ 175 ]
Due to its potency in microgram quantities, LSD is often not included in standard pre-employment urine or hair analyses. [ 173 ] [ 176 ] However, advanced liquid chromatography–mass spectrometry methods can detect LSD in biological samples even after a single use. [ 176 ]
A variety of LSD analogues are known. [ 177 ] [ 178 ] [ 179 ] [ 180 ] [ 181 ] Many of them retain psychedelic effects similarly to LSD, although most have reduced potency and none are notably more potent than LSD. [ 177 ] [ 178 ] [ 179 ] [ 180 ] [ 182 ] Examples include ergine (lysergic acid amide; LSA), ergonovine (ergometrine), methylergonovine (methylergometrine), methysergide , ETH-LAD , AL-LAD , 1-methyl-LSD (MLD-41), and LA-SS-Az (LSZ), among many others. [ 177 ] [ 183 ] [ 184 ] Presumed or known prodrugs of LSD, including 1A-LSD (ALD-52), 1P-LSD , and 1V-LSD , have been developed or encountered. [ 185 ] [ 186 ] Some non- hallucinogenic LSD analogues, such as lisuride and 2-bromo-LSD (BOL-148), are known as well. [ 10 ] [ 187 ] [ 188 ] They are lower- efficacy serotonin 5-HT 2A receptor partial agonists and can notably act as hallucinogen antagonists against LSD. [ 187 ] [ 188 ] In addition to lysergamide derivatives , simplified or "partial" LSD analogues , such as NDTDI , N -DEAOP-NMT , and DEIMDHPCA , are known. [ 108 ] [ 189 ] [ 190 ] [ 191 ] A notable bioisostere of LSD is JRT , the isotryptamine analogue of LSD and a psychedelic and psychoplastogen under investigation to treat schizophrenia . [ 192 ] [ 193 ]
... affected by a remarkable restlessness, combined with a slight dizziness. At home I lay down and sank into a not unpleasant intoxicated-like condition, characterized by an extremely stimulated imagination. In a dreamlike state, with eyes closed (I found the daylight to be unpleasantly glaring), I perceived an uninterrupted stream of fantastic pictures, extraordinary shapes with intense, kaleidoscopic play of colors. After some two hours this condition faded away.
Swiss chemist Albert Hofmann first synthesized LSD in 1938 from lysergic acid , a chemical derived from the hydrolysis of ergotamine , an alkaloid found in ergot , a fungus that infects grain. [ 22 ] [ 20 ] LSD was the 25th of various lysergamides Hofmann synthesized from lysergic acid while trying to develop a new analeptic , hence the alternate name LSD-25. Hofmann discovered its effects in humans on April 16, in 1943, after unintentionally ingesting an unknown amount, possibly absorbing it through his skin. [ 195 ] [ 196 ] [ 197 ] On April 19, 1943, Hofmann intentionaly ingested 0.25 milligrams (250 micrograms) of LSD. [ 198 ] LSD was first published in the scientific literature by Hofmann and his colleague psychiatrist Werner Stoll in 1943 and the hallucinogenic effects of LSD were first published by Stoll in 1947. [ 199 ] [ 200 ] [ 201 ] [ 202 ] [ 203 ]
LSD was subject to exceptional interest within the field of psychiatry in the 1950s and early 1960s, with Sandoz distributing LSD to researchers under the trademark name Delysid in an attempt to find a marketable use for it. [ 196 ] During this period, LSD was controversially administered to hospitalised schizophrenic autistic children, with varying degrees of therapeutic success. [ 204 ] [ 205 ] [ 206 ] [ 207 ]
LSD-assisted psychotherapy was used in the 1950s and early 1960s by psychiatrists such as Humphry Osmond , who pioneered the application of LSD to the treatment of alcoholism , with promising results. [ 196 ] [ 208 ] [ 28 ] [ 39 ] Osmond coined the term "psychedelic" (mind manifesting) as a term for LSD and related hallucinogens , superseding the previously held " psychotomimetic " model in which LSD was believed to mimic schizophrenia . In contrast to schizophrenia, LSD can induce transcendent experiences, or mental states that transcend the experience of everyday consciousness, with lasting psychological benefit. [ 15 ] [ 196 ] During this time, the Central Intelligence Agency (CIA) began using LSD in the research project Project MKUltra , which used psychoactive substances to aid interrogation . The CIA administered LSD to unwitting test subjects to observe how they would react, the most well-known example of this being Operation Midnight Climax . [ 196 ] LSD was one of several psychoactive substances evaluated by the U.S. Army Chemical Corps as possible non-lethal incapacitants in the Edgewood Arsenal human experiments . [ 196 ]
According to declassified CIA documents, it's possible that the american agency spread LSD on civilians in Europe in the 50s. [ 209 ] [ 210 ]
In the 1960s, LSD and other psychedelics were adopted by and became synonymous with the counterculture movement due to their perceived ability to expand consciousness. This resulted in LSD being viewed as a cultural threat to American values and the Vietnam War effort, and it was designated as a Schedule I (illegal for medical as well as recreational use) substance in 1968. [ 211 ] It was listed as a Schedule I controlled substance by the United Nations in 1971 and currently has no approved medical uses. [ 22 ] As of 2017 [update] , about 10% of people in the United States have used LSD at some point in their lives, while 0.7% have used it in the last year. [ 31 ] It was most popular in the 1960s to 1980s. [ 22 ] The use of LSD among US adults increased by 56.4% from 2015 to 2018. [ 212 ]
LSD was first synthesized on November 16, 1938 [ 213 ] by Swiss chemist Albert Hofmann at the Sandoz Laboratories in Basel , Switzerland as part of a large research program searching for medically useful ergot alkaloid derivatives. The abbreviation "LSD" is from the German "Lysergsäurediethylamid". [ 214 ]
LSD's psychedelic properties were discovered 5 years later when Hofmann himself accidentally ingested an unknown quantity of the chemical. [ 215 ] The first intentional ingestion of LSD occurred on April 19, 1943, [ 194 ] when Hofmann ingested 250 μg of LSD. He said this would be a threshold dose based on the dosages of other ergot alkaloids. Hofmann found the effects to be much stronger than he anticipated. [ 216 ] Sandoz Laboratories introduced LSD as a psychiatric drug in 1947 and marketed LSD as a psychiatric panacea, hailing it "as a cure for everything from schizophrenia to criminal behavior, 'sexual perversions', and alcoholism." [ 217 ] Sandoz would send the drug for free to researchers investigating its effects. [ 195 ]
Beginning in the 1950s, the US Central Intelligence Agency (CIA) began a research program code-named Project MKUltra . [ 218 ] The CIA introduced LSD to the United States, purchasing the entire world's supply for $240,000 and propagating the LSD through CIA front organizations to American hospitals, clinics, prisons, and research centers. [ 219 ] Experiments included administering LSD to CIA employees, military personnel, doctors, other government agents, prostitutes, mentally ill patients, and members of the general public to study their reactions, usually without the subjects' knowledge. The project was revealed in the US congressional Rockefeller Commission report in 1975. However, the extent of the experiments conducted under Project MKUltra are still mostly unknown, as acting CIA director Richard Helms destroyed many of the key documents related to MKUltra in 1973. [ 220 ]
In 1963, the Sandoz patents on LSD expired [ 50 ] and the Czech company Spofa began to produce the substance. [ 195 ] Sandoz stopped the production and distribution in 1965. [ 195 ]
Several figures, including Aldous Huxley , Timothy Leary , and Al Hubbard , had begun to advocate the consumption of LSD. LSD became central to the counterculture of the 1960s. [ 221 ] In the early 1960s the use of LSD and other hallucinogens was advocated by new proponents of consciousness expansion such as Leary, Huxley, Alan Watts and Arthur Koestler , [ 222 ] [ 223 ] and according to L. R. Veysey they profoundly influenced the thinking of the new generation of youth. [ 224 ]
On October 24, 1968, possession of LSD was made illegal in the United States. [ 225 ] The last FDA approved study of LSD in patients ended in 1980, while a study in healthy volunteers was made in the late 1980s. Legally approved and regulated psychiatric use of LSD continued in Switzerland until 1993. [ 226 ]
In November 2020, Oregon became the first US state to decriminalize possession of small amounts of LSD after voters approved Ballot Measure 110 . [ 227 ]
By the mid-1960s, the youth countercultures in California, particularly in San Francisco, had widely adopted the use of hallucinogenic drugs, including LSD. The first major underground LSD factory was established by Owsley Stanley . [ 228 ] Around this time, the Merry Pranksters , associated with novelist Ken Kesey , organized the Acid Tests , events in San Francisco involving LSD consumption, accompanied by light shows and improvised music. [ 229 ] [ 230 ] Their activities, including cross-country trips in a psychedelically decorated bus and interactions with major figures of the beat movement, were later documented in Tom Wolfe 's The Electric Kool-Aid Acid Test (1968). [ 231 ]
In San Francisco's Haight-Ashbury neighborhood, the Psychedelic Shop was opened in January 1966 by brothers Ron and Jay Thelin to promote the safe use of LSD. This shop played a significant role in popularizing LSD in the area and establishing Haight-Ashbury as the epicenter of the hippie counterculture. The Thelins also organized the Love Pageant Rally in Golden Gate Park in October 1966, protesting against California's ban on LSD. [ 232 ] [ 233 ]
A similar movement developed in London, led by British academic Michael Hollingshead , who first tried LSD in America in 1961. After experiencing LSD and interacting with notable figures such as Aldous Huxley , Timothy Leary , and Richard Alpert , Hollingshead played a key role in the famous LSD research at Millbrook before moving to New York City for his experiments. In 1965, he returned to the UK and founded the World Psychedelic Center in Chelsea, London. [ 234 ]
Blotter art is an art form printed on perforated sheets of absorbent blotting paper infused with liquid LSD. The delivery method gained popularity following the banning of the hallucinogen LSD in the late 1960s. The use of graphics on blotter sheets originated as an underground art form in the early 1970s, sometimes to help identify the dosage, maker, or batch of LSD.
LSD art is any art or visual displays inspired by psychedelic experiences and hallucinations known to follow the ingestion of LSD, also known colloquially as acid). [ 235 ] Artists and scientists have been interested in the effect of LSD on drawing and painting since it first became available for legal use and general consumption. [ 236 ]
The influence of LSD in the realms of music and art became pronounced in the 1960s, especially through the Acid Tests and related events involving bands like the Grateful Dead , Jefferson Airplane , and Big Brother and the Holding Company . San Francisco-based artists such as Rick Griffin , Victor Moscoso , and Wes Wilson contributed to this movement through their psychedelic poster and album art. The Grateful Dead , in particular, became central to the culture of "Deadheads," with their music heavily influenced by LSD. [ 237 ]
In the United Kingdom, Michael Hollingshead, reputed for introducing LSD to various artists and musicians like Storm Thorgerson , Donovan , Keith Richards , and members of the Beatles , played a significant role in the drug's proliferation in the British art and music scene. Despite LSD's illegal status from 1966, it was widely used by groups including the Beatles , the Rolling Stones , and the Moody Blues . Their experiences influenced works such as the Beatles' Sgt. Pepper's Lonely Hearts Club Band and Cream's Disraeli Gears , featuring psychedelic-themed music and artwork. [ 238 ]
Psychedelic music of the 1960s often sought to replicate the LSD experience, incorporating exotic instrumentation, electric guitars with effects pedals, and elaborate studio techniques. Artists and bands utilized instruments like sitars and tablas, and employed studio effects such as backward tapes, panning, and phasing. [ 239 ] [ 240 ] Songs such as John Prine 's "Illegal Smile" and the Beatles' " Lucy in the Sky with Diamonds " have been associated with LSD, although the latter's authors denied such claims. [ 241 ] [ page needed ] [ 242 ]
Contemporary artists influenced by LSD include Keith Haring in the visual arts, [ 243 ] various electronic dance music creators, [ 244 ] and the jam band Phish . [ 245 ] The 2018 Leo Butler play All You Need is LSD is inspired by the author's interest in the history of LSD. [ 246 ]
The United Nations Convention on Psychotropic Substances of 1971 mandates that signing parties, including the United States, Australia, New Zealand, and most of Europe, prohibit LSD. Enforcement of these laws varies by country. The convention allows medical and scientific research with LSD. [ 247 ]
In Australia, LSD is classified as a Schedule 9 prohibited substance under the Poisons Standard (February 2017), indicating it may be abused or misused and its manufacture, possession, sale, or use should be prohibited except for approved research purposes. [ 248 ] In Western Australia, the Misuse of Drugs Act 1981 provides guidelines for possession and trafficking of substances like LSD. [ 249 ]
In Canada, LSD is listed under Schedule III of the Controlled Drugs and Substances Act. Unauthorized possession and trafficking of the substance can lead to significant legal penalties. [ 250 ]
In the United Kingdom, LSD is a Class A drug under the Misuse of Drugs Act 1971, making unauthorized possession and trafficking punishable by severe penalties. The Runciman Report and Transform Drug Policy Foundation have made recommendations and proposals regarding the legal regulation of LSD and other psychedelics. [ 251 ] [ 252 ]
In the United States, LSD is classified as a Schedule I controlled substance under the Controlled Substances Act of 1970, making its manufacture, possession, and distribution illegal without a DEA license. The law considers LSD to have a high potential for abuse, no legitimate medical use, and to be unsafe even under medical supervision. The US Supreme Court case Neal v. United States (1995) clarified the sentencing guidelines related to LSD possession. [ 253 ]
Oregon decriminalized personal possession of small amounts of drugs, including LSD, in February 2021, and California has seen legislative efforts to decriminalize psychedelics. [ 254 ]
Mexico decriminalized the possession of small amounts of drugs, including LSD, for personal use in 2009. The law specifies possession limits and establishes that possession is not a crime within designated quantities. [ 255 ]
In the Czech Republic, possession of "amount larger than small" of LSD is criminalized, while possession of smaller amounts is a misdemeanor. The definition of "amount larger than small" is determined by judicial practice and specific regulations. [ 256 ] [ 257 ]
An active dose of LSD is very minute, allowing a large number of doses to be synthesized from a comparatively small amount of raw material. Twenty-five kilograms of precursor ergotamine tartrate can produce 5–6 kg of pure crystalline LSD; this corresponds to around 50–60 million doses at 100 μg. Because the masses involved are so small, concealing and transporting illicit LSD is much easier than smuggling cocaine , cannabis , or other illegal drugs. [ 258 ]
Manufacturing LSD requires laboratory equipment and experience in the field of organic chemistry . It takes two to three days to produce 30 to 100 grams of pure compound. It is believed that LSD is not usually produced in large quantities, but rather in a series of small batches. This technique minimizes the loss of precursor chemicals in case a step does not work as expected. [ 258 ] Ali Altaft, the lead chemist at the University of Okara , in Punjab , Pakistan , performed the synthesis of LSD on video. [ 259 ]
LSD is produced in crystalline form and is then mixed with excipients or redissolved for production in ingestible forms. Liquid solution is either distributed in small vials or, more commonly, sprayed onto or soaked into a distribution medium. Historically, LSD solutions were first sold on sugar cubes, but practical considerations [ clarification needed ] forced a change to tablet form. Appearing in 1968 as an orange tablet measuring about 6 mm across, "Orange Sunshine" acid was the first largely available form of LSD after its possession was made illegal. Tim Scully , a prominent chemist, made some of these tablets, but said that most "Sunshine" in the USA came by way of Ronald Stark, who imported approximately thirty-five million doses from Europe. [ 260 ]
Over some time, tablet dimensions, weight, shape and concentration of LSD evolved from large (4.5–8.1 mm diameter), heavyweight (≥150 mg), round, high concentration (90–350 μg/tab) dosage units to small (2.0–3.5 mm diameter) lightweight (as low as 4.7 mg/tab), variously shaped, lower concentration (12–85 μg/tab, average range 30–40 μg/tab) dosage units. LSD tablet shapes have included cylinders, cones, stars, spacecraft, and heart shapes. The smallest tablets became known as "Microdots." [ 261 ]
After tablets came "computer acid" or "blotter paper LSD," typically made by dipping a preprinted sheet of blotting paper into an LSD/water/alcohol solution. [ 260 ] [ 261 ] More than 200 types of LSD tablets have been encountered since 1969 and more than 350 blotter paper designs have been observed since 1975. [ 261 ] About the same time as blotter paper LSD came "Windowpane" (AKA "Clearlight"), which contained LSD inside a thin gelatin square a quarter of an inch (6 mm) across. [ 260 ] LSD has been sold under a wide variety of often short-lived and regionally restricted street names including Acid, Trips, Uncle Sid, Blotter, Lucy , Alice and doses, as well as names that reflect the designs on the sheets of blotter paper. [ 54 ] [ 262 ] Authorities have encountered the drug in other forms—including powder or crystal, and capsule. [ 263 ]
Blotter art designs printed on blotter paper can serve to identify dosage strengths, different batches, or makers. [ 264 ]
On the other hand, blotters without art may be considered safer by some, since there is no guarantee that the printer ink used in clandestine production is edible or non-toxic for long-term exposure, and it is also possible for unscrupulous dealers to mimic reputable blotter art designs in order to boost sales.
LSD manufacturers and traffickers in the United States can be categorized into two groups: A few large-scale producers, and an equally limited number of small, clandestine chemists, consisting of independent producers who, operating on a comparatively limited scale, can be found throughout the country. [ 265 ] [ 266 ]
As a group, independent producers are of less concern to the Drug Enforcement Administration than the large-scale groups because their product reaches only local markets. [ 217 ]
Many LSD dealers and chemists describe a religious or humanitarian purpose that motivates their illicit activity. Nicholas Schou's book Orange Sunshine: The Brotherhood of Eternal Love and Its Quest to Spread Peace, Love, and Acid to the World describes one such group, the Brotherhood of Eternal Love . The group was a major American LSD trafficking group in the late 1960s and early 1970s. [ 267 ]
In the second half of the 20th century, dealers and chemists loosely associated with the Grateful Dead like Owsley Stanley , Nicholas Sand , Karen Horning, Sarah Maltzer, "Dealer McDope," and Leonard Pickard played an essential role in distributing LSD. [ 237 ]
Since 2005, law enforcement in the United States and elsewhere has seized several chemicals and combinations of chemicals in blotter paper which were sold as LSD mimics, including DOB , [ 268 ] [ 269 ] a mixture of DOC and DOI , [ 270 ] 25I-NBOMe , [ 271 ] and a mixture of DOC and DOB . [ 272 ] Many mimics are toxic in comparatively small doses, or have extremely different safety profiles. Many street users of LSD are often under the impression that blotter paper which is actively hallucinogenic can only be LSD because that is the only chemical with low enough doses to fit on a small square of blotter paper. While it is true that LSD requires lower doses than most other hallucinogens, blotter paper is capable of absorbing a much larger amount of material. The DEA performed a chromatographic analysis of blotter paper containing 2C-C which showed that the paper contained a much greater concentration of the active chemical than typical LSD doses, although the exact quantity was not determined. [ 273 ] Blotter LSD mimics can have relatively small dose squares; a sample of blotter paper containing DOC seized by Concord, California police had dose markings approximately 6 mm apart. [ 274 ] Several deaths have been attributed to 25I-NBOMe. [ 275 ] [ 276 ] [ 277 ] [ 278 ]
Some notable individuals have commented publicly on their experiences with LSD. [ 279 ] [ 280 ] Some of these comments date from the era when it was legally available in the US and Europe for non-medical uses, and others pertain to psychiatric treatment in the 1950s and 1960s. Still others describe experiences with illegal LSD, obtained for philosophic, artistic, therapeutic, spiritual, or recreational purposes.
In the United States, the earliest research began in the 1950s. Albert Kurland and his colleagues published research on LSD's therapeutic potential to treat schizophrenia. In Canada, Humphry Osmond and Abram Hoffer completed LSD studies as early as 1952. [ 302 ] By the 1960s, controversies surrounding "hippie" counterculture began to deplete institutional support for continued studies.
Currently, several organizations—including the Beckley Foundation , MAPS , Heffter Research Institute and the Albert Hofmann Foundation—exist to fund, encourage and coordinate research into the medicinal and spiritual uses of LSD and related psychedelics. [ 303 ] New clinical LSD experiments in humans started in 2009 for the first time in 35 years. [ 304 ] As it is illegal in many areas of the world, potential medical uses are difficult to study. [ 37 ]
In 2001 the United States Drug Enforcement Administration stated that LSD "produces no aphrodisiac effects, does not increase creativity, has no lasting positive effect in treating alcoholics or criminals, does not produce a " model psychosis ", and does not generate immediate personality change." [ 217 ] More recently, experimental uses of LSD have included the treatment of alcoholism, [ 305 ] pain and cluster headache relief, [ 7 ] [ 306 ] [ 307 ] and prospective studies on depression. [ 308 ]
A 2020 meta-review indicated possible positive effects of LSD in reducing psychiatric symptoms, mainly in cases of alcoholism. [ 309 ] There is evidence that psychedelics induce molecular and cellular adaptations related to neuroplasticity and that these could potentially underlie therapeutic benefits. [ 310 ] [ 311 ]
In the 1950s and 1960s, LSD was used in psychiatry to enhance psychotherapy, known as psychedelic therapy . Some psychiatrists, such as Ronald A. Sandison , who pioneered its use at Powick Hospital in England, believed LSD was especially useful at helping patients to "unblock" repressed subconscious material through other psychotherapeutic methods, [ 312 ] and also for treating alcoholism. [ 313 ] [ 314 ] One study concluded, "The root of the therapeutic value of the LSD experience is its potential for producing self-acceptance and self-surrender," [ 28 ] presumably by forcing the user to face issues and problems in that individual's psyche.
Two recent reviews concluded that conclusions drawn from most of these early trials are unreliable due to serious methodological flaws. These include the absence of adequate control groups , lack of follow-up, and vague criteria for therapeutic outcome. In many cases, studies failed to convincingly demonstrate whether the drug or the therapeutic interaction was responsible for any beneficial effects. [ 315 ] [ 316 ]
In recent years, organizations like the Multidisciplinary Association for Psychedelic Studies (MAPS) have renewed clinical research of LSD. [ 304 ]
It has been proposed that LSD be studied for use in the therapeutic setting, particularly in anxiety. [ 40 ] [ 41 ] [ 90 ] [ 317 ] In 2024, the FDA designated a form of LSD as a breakthrough therapy to treat generalized anxiety disorder which is being developed by MindMed . [ 318 ]
In the 1950s and 1960s, some psychiatrists (e.g., Oscar Janiger ) explored the potential effect of LSD on creativity. Experimental studies attempted to measure the effect of LSD on creative activity and aesthetic appreciation. [ 55 ] [ 319 ] [ 320 ] [ 321 ] In 1966 Dr. James Fadiman conducted a study with the central question "How can psychedelics be used to facilitate problem solving?" This study attempted to solve 44 different problems and had 40 satisfactory solutions when the FDA banned all research into psychedelics. LSD was a key component of this study. [ 322 ] [ 323 ]
Since 2008 there has been ongoing research into using LSD to alleviate anxiety for terminally ill cancer patients coping with their impending deaths. [ 40 ] [ 304 ] [ 324 ]
A 2012 meta-analysis found evidence that a single dose of LSD in conjunction with various alcoholism treatment programs was associated with a decrease in alcohol abuse, lasting for several months, but no effect was seen at one year. Adverse events included seizure, moderate confusion and agitation, nausea, vomiting , and acting in a bizarre fashion. [ 39 ]
LSD has been used as a treatment for cluster headaches with positive results in some small studies. [ 7 ]
LSD is a potent psychoplastogen , a compound capable of promoting rapid and sustained neural plasticity that may have wide-ranging therapeutic benefit. [ 325 ] LSD has been shown to increase markers of neuroplasticity in human brain organoids and improve memory performance in human subjects. [ 326 ]
LSD may have analgesic properties related to pain in terminally ill patients and phantom pain and may be useful for treating inflammatory diseases including rheumatoid arthritis. [ 327 ] | https://en.wikipedia.org/wiki/LSD |
Life Science Identifiers [ 1 ] [ 2 ] are a way to name and locate pieces of information on the web. Essentially, an LSID is a unique identifier for some data, and the LSID protocol specifies a standard way to locate the data (as well as a standard way of describing that data). They are a little like DOIs used by many publishers.
An LSID is represented as a uniform resource name (URN) with the following format:
The lsid: namespace, however, is not registered with the Internet Assigned Numbers Authority (IANA), and so these are not strictly URNs or URIs. [ 3 ]
LSIDs may be resolved in URLs, e.g. http://zoobank.org/urn:lsid:zoobank.org:pub:CDC8D258-8F57-41DC-B560-247E17D3DC8C
There has been a lot of interest in LSIDs in both the bioinformatics and the biodiversity communities, with the latter continuing to use them as a way of identifying species in global catalogues. [ 4 ] However, more recently, as understanding has increased of how HTTP URIs can perform a similar naming task, [ 5 ] [ 6 ] the use of LSIDs as identifiers has been criticized [ 7 ] as violating the Web Architecture good practice of reusing existing URI schemes. [ 8 ] Nevertheless, the explicit separation of data from metadata; specification of a method for discovering multiple locations for data-retrieval; and the ability to discover multiple independent sources of metadata for any identified thing were crucial parts of the LSID and its resolution specification that have not successfully been mimicked by an HTTP-only approach.
The World Wide Web provides a globally distributed communication framework that is essential for almost all scientific collaboration, including bioinformatics. However, several limits and inadequacies were thought to exist, one of which was the inability to programmatically identify locally named objects that may be widely distributed over the network. This perceived shortcoming would have limited our ability to integrate multiple knowledgebases, each of which gives partial information of a shared domain, as is commonly seen in bioinformatics. The Life Science Identifier (LSID) and LSID Resolution System (LSRS) were designed to provide simple and elegant solutions to this problem, consistent with next-generation Semantic Web and semantic grid , based on the extension of existing internet technologies. However, it has more recently been pointed out that some of these perceived shortcomings are not intrinsic to HTTP URIs, and much (though not all) of the functionality that LSIDs provide can be obtained using properly crafted HTTP URIs. [ 5 ]
Alternative identifiers have been proposed for organisms, e.g. the DOI system. NamesforLife (N4L), a private company, set up a system to apply DOIs to organisms. For example, doi:10.1601/nm.3093 is the DOI for Escherichia coli , and doi:10.1601/tx.3093 is the corresponding taxon. [ 9 ] | https://en.wikipedia.org/wiki/LSID |
LT50 is the median Lethal Time (time until death) [ 1 ] [ 2 ] after exposure of an organism to a toxic substance or stressful condition. LT50 is commonly used in toxicology studies to quantify amount of a stressor necessary to kill an organism. LT50 can be used in conjunction with EC50 (median Exposure Concentration) for even more precise quantification. [ 2 ]
This toxicology -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/LT50 |
LTE-M or LTE-MTC (" Long-Term Evolution Machine Type Communication ") is a type of low-power wide-area network radio communication technology standard developed by 3GPP for machine-to-machine and Internet of Things (IoT) applications. [ 1 ] [ 2 ] LTE-M includes eMTC ("enhanced Machine Type Communication"), also known as LTE Cat-M1 , whose specification was frozen in June 2016 as part of 3GPP Release 13 ( LTE Advanced Pro ), [ 3 ] as well as LTE Cat-M2 . [ 4 ]
Competing 3GPP IoT technologies include NB-IoT and EC-GSM-IoT . [ 5 ] The advantage of LTE-M over NB-IoT is its comparatively higher data rate, mobility, and voice over the network , but it requires more bandwidth, is more costly, and cannot be put into guard band portion of the frequency band for now. [ 6 ] Compared to LTE Release 12 Cat-0 modem, an LTE-M model is claimed to be 80% less expensive (in terms of the bill of materials), support up to 18 dB better coverage, and a battery lifetime that can last up to several years. [ 7 ] In March 2019, the Global Mobile Suppliers Association reported that over 100 operators had deployed/launched either NB-IoT or LTE-M networks. [ 8 ]
2 Mbit/s (EGPRS2B)
16.9 kbit/s (single-tone)
2 Mbit/s (EGPRS2B)
As of March 2019 the Global Mobile Suppliers Association had identified: [ 8 ]
As of February 2022, GSMA had listed LTE-M as being launched on 60 commercial networks. [ 11 ]
This computer networking article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/LTE-M |
LTE-WLAN aggregation ( LWA ) is a technology defined by the 3GPP . In LWA, a mobile handset supporting both LTE and Wi-Fi may be configured by the network to utilize both links simultaneously. It provides an alternative method of using LTE in unlicensed spectrum , which unlike LAA/LTE-U can be deployed without hardware changes to the network infrastructure equipment and mobile devices, while providing similar performance to that of LAA. Unlike other methods of using LTE and WLAN simultaneously (e.g. Multipath TCP ), LWA allows using both links for a single traffic flow and is generally more efficient, due to coordination at lower protocol stack layers.
For a user, LWA offers seamless usage of both LTE and Wi-Fi networks and substantially increased performance. For a cellular operator, LWA simplifies Wi-Fi deployment, improves system utilization and reduces network operation and management costs. LWA can be deployed in collocated manner, where the eNB and the Wi-Fi AP or AC are integrated into the same physical device or in non-collocated manner, where the eNB and the Wi-Fi AP or AC are connected via a standardized interface referred to as Xw. The latter deployment option is particularly suitable for the case when Wi-Fi needs to cover large areas and/or Wi-Fi services are provided by a 3rd party (e.g. a university campus), rather than a cellular operator.
LWA has been standardized by the 3GPP in Release-13. Release 14 Enhanced LWA (eLWA) adds support for 60 GHz band (802.11ad and 802.11ay aka WiGig) with 2.16 GHz bandwidth, uplink aggregation, mobility improvements and other enhancements.
Cellular networks have been designed for licensed spectrum. However, as usage patterns changed from voice-centric to data-centric and data usage surged, operators started looking into unlicensed spectrum opportunities. Using WLAN does not only allow operators to increase peak data rate and system capacity, but also to offer services for non-cellular devices, such as laptops.
To cater to operators demand, 3GPP have defined various methods for integrating WLAN access into operator's network deployments.
Based on how the WLAN access is integrated in the operator network there are two categories: 1) Core Network integration, in which the WLAN access is connected to the operator core network using either S2a or S2b interfaces) available in 3GPP networks since Release 8 and 2) RAN based integration in which the WLAN access is directly connected to RAN access nodes (eg. LWA or LWIP) available since Release 13. All of the above methods of integration assume a certain level of service continuity as well as the terminal devices being always under a licensed spectrum cellular coverage. When service continuity is not assumed the WLAN access it is said to be integrated through what is called Non-Seamless WLAN Offload (NSWO).
A terminal device may access either cellular or WLAN access or both. This procedure may be either network initiated or terminal initiated. When it is network initiated it may be either based on core network signaling (eg. NAS in the case of Network based IP flow mobility) or RAN based rules. Terminal based initiated procedures are based on either operator policies provided to the terminals (eg. through ANDSF), user based policies/preferences, etc. These policies may take into account various conditions (eg. time, location, network load, access load, radio conditions, etc.) in determining both the access selection as well as traffic switching from one access to another.
In LTE - WLAN Aggregation (eg. LWA or LWIP) the WLAN access is directly connected to RAN access nodes and the access selection and traffic steering/splitting is done entirely under the control of the Radio Access Network node (eg. eNB).
From the network perspective, there are two options that provide flexibility when looking at deploying LWA - collocated and non-collocated. In the former, the WLAN Access Point (AP) or Access Controller (AC) is physically integrated with the LTE eNB, whereas in the latter the WLAN network (i.e. APs and/or ACs) are connected to the LTE eNB via an external network interface (Xw).
LWA design primarily follows LTE Dual Connectivity (DC) architecture [3] as defined in 3GPP Release 12, which allows a UE to connect to multiple base stations simultaneously, with WLAN used instead of LTE Secondary eNB (SeNB).
In the user plane, LTE and WLAN are aggregated at the Packet Data Convergence Protocol (PDCP) level. In the downlink, the eNB may schedule PDCP PDUs of the same bearer to be delivered to the UE either via LTE or WLAN. This is possible as the PDCP layer can re-order packets received from both LTE and WLAN links, which in its turn results in substantial performance gains. In order to perform efficient scheduling and to assign packets to LTE and WLAN links in the most efficient manner, the eNB can receive radio information about both links, including flow control indication. In order to avoid changes to the WLAN MAC, LWA uses an EtherType allocated for this purpose, so that LWA traffic is transparent to WLAN AP.
In the control plane , Evolved Node B (eNB) is responsible for LWA activation, de-activation and the decision as to which bearers are offloaded to the WLAN. It does so using WLAN measurement information reported by the UE. Once LWA is activated, the eNB configures the UE with a list of WLAN identifiers (referred to as the WLAN Mobility Set) within which the UE can move without notifying the network. This is a tradeoff between fully network controlled mobility and fully UE controlled mobility.
Even though WLAN usage in LWA is controlled by cellular network, UE has the option to "opt out" in order to use home WLAN (in case UE does not support concurrent WLAN operation). Generally, the design tries to balance between LTE technology which is traditionally network-controlled and WLAN technology, which normally allows a lot of freedom for the terminal (e.g. in terms of network selection and traffic steering). With LWA design, the high level decision (e.g. LWA activation) is performed by the network, but there is sufficient level of UE freedom and flexibility (within the limits set by the network).
The first LWA version defined in Release-14 supported downlink aggregation only. This has been further enhanced in Release-15 with uplink aggregation and support for 60 GHz band (aka WiGig ).
Some contributions [ 1 ] use simulations of small cell deployments supporting both cellular and WLAN coverage using LWA. It shows that by using bearer split LWA improves the average as well as the cell edge user perceived throughput across all small cell users in the system when compared to the Rel-12/Rel-13 radio interworking schemes. In these schemes the WLAN is connected through the operator's core network instead of being anchored in the Radio access and the traffic may be switched from one access to another based on the radio conditions of the access including the load of the access. It is not clear though how much improvement LWA may bring when it is compared with other RAN based LTE-WLAN Aggregation solution (eg. LWIP).
On 19 August 2016 Singapore's M1 announced [ 2 ] Singapore’s first commercial HetNet ( Heterogeneous Network ) rollout, including LWA. Through leading edge LTE-WiFi Aggregation (LWA) technology, M1 expects to deliver peak download speeds of more than 1 Gbit/s by 2017.
Chunghwa Telecom (CHT) will open a commercial LTE/Wi-Fi Aggregation (LWA) network on February 23, [ 3 ] making it the world's first telecom operator to launch LWA, according to CHT. LWA technology standards were approved by 3GPP in June 2016. | https://en.wikipedia.org/wiki/LTE-WLAN_Aggregation |
The induction of NMDA receptor-dependent long-term potentiation (LTP) in chemical synapses in the brain occurs via a fairly straightforward mechanism. [ 1 ] [ 2 ] A substantial and rapid rise in calcium ion concentration inside the postsynaptic cell (or more specifically, within the dendritic spine ) is most possibly all that is required to induce LTP. But the mechanism of calcium delivery to the postsynaptic cell in inducing LTP is more complicated.
The AMPA receptor (AMPAR) is the engine that drives excitatory postsynaptic potentials (EPSPs). While some forms of the AMPAR can conduct calcium, most AMPARs found in the neocortex do not. The AMPAR, upon binding two glutamate molecules, undergoes a conformational change that resembles the opening of a clam shell. This conformational change opens an ion channel within the AMPAR protein structure that allows sodium ions to flow into the cell and potassium ions to flow out (i.e. it is a mixed cation-conducting channel). The Na + and K + permeabilities of the AMPAR channel are roughly equal, so when this channel is open the resulting change in membrane potential tends towards zero (a bit more than halfway between the equilibrium potentials E K and E Na ). This balance point is reached at around 0 mV (i.e. the reversal potential of the EPSP current is roughly 0 mV). However, the postsynaptic membrane potential will not change by more than a few millivolts from resting potential with a single presynaptic release of glutamate, because not many AMPAR channels open. The lifetime of the glutamate in the synaptic cleft is too short to allow more than a brief opening of the AMPAR channel, thus causing only a small depolarization . The open AMPAR channel is often considered to be non-calcium permeable, but this is only an approximation as AMPARs with certain subunit compositions will allow calcium through, albeit at different levels and frequency to NMDARs.
Historically, the most widely used experimental means of inducing LTP has been to deliver a tetanic stimulation to the presynaptic axon of a synapse or group of synapses. The frequency of this tetanus is typically 100 Hz, and the duration typically 1 s. A single AMPAR-mediated EPSP has a rise time-to-peak of approximately 2–5 ms and a duration of approximately 30 ms. If a synapse is being stimulated at 100 Hz, the presynaptic neuron will be attempting to release glutamate once every 10 ms. An EPSP occurring only 10 ms after a previous EPSP will arrive at a time when that previous EPSP is at its peak amplitude. Thus, during a 100 Hz stimulus train, each EPSP will add to the membrane depolarization caused by the previous EPSPs. This synaptic summation drives the membrane potential toward values that could not be reached with single synaptic stimuli. As the EPSPs summate, they will exceed the spike threshold.
The NMDA receptor (NMDAR) does not, in resting or near-resting membrane potential conditions, contribute significant current to the EPSP. Following the presynaptic release of the glutamate that binds to and opens the AMPAR, the NMDAR also binds this glutamate and opens. However, current does not flow through the NMDAR ion channel because it is instantaneously blocked by a magnesium ion (Mg 2+ ) that binds to a site "inside" the open pore of the NMDAR channel. Magnesium has access to this binding site only when the NMDAR channel is opened by glutamate binding, a so-called open channel block .
What makes this magnesium blockade of the NMDAR channel particularly significant in terms of LTP induction is that the block is membrane voltage-dependent. The basis of this voltage dependence is relatively straightforward. The NMDAR channel is a transmembrane protein ; that is, it spans the cell membrane. As such, it also spans the electric field generated by the membrane potential. The magnesium binding site within the NMDAR channel is physically located within this electric field. Magnesium ions carrying a double positive charge can be acted upon by the field. When the cell is hyperpolarized, magnesium is stabilized inside the channel (i.e. the two positive charges on the magnesium ion are attracted toward the negative pole of the electric field, which points toward the inside of the cell). As a cell is depolarized, the field effect on the magnesium ion weakens, and the dwell time of magnesium ions within the channel decreases. Thus, the kinetics of the binding reaction between magnesium and the NMDAR channel are such that magnesium periodically unbinds and leaves the channel, only to be replaced by another magnesium ion. During the (very brief) time that the magnesium is absent from the open channel, other ions (such as sodium and calcium) can flow through the channel. However, when the cell is more hyperpolarized, the bound state of magnesium is stabilized and it leaves the channel less often and for a shorter period of time (on average). When the cell is less hyperpolarized, the magnesium leaves the channel more often and stays away for longer (on average). Hence, the magnesium blockade of the open NMDAR channel is membrane voltage-dependent.
While the NMDAR channel itself displays little or no voltage dependence (its open channel I/V curve is more or less linear), the voltage dependence of the magnesium block effectively, if indirectly, confers voltage dependence to this channel. Thus, in effect, the NMDAR channel is both a ligand-gated and voltage-gated channel at the same time. [ 3 ] This fact is critical to the function of the NMDAR as a Hebbian coincidence detector. More strictly speaking, inward cationic current (sodium or calcium) through the open unblocked NMDAR does decrease with depolarization (because of the decreased electrochemical "driving force"), but the voltage-dependent unblocking seems to outweigh this decrease in driving force, so the calcium influx into the spine caused by a pair of appropriately timed pre- and postsynaptic spikes significantly exceeds the sum of the influxes due to the individual spikes alone. This extra, or "nonlinear", calcium entry triggers the strength change. | https://en.wikipedia.org/wiki/LTP_induction |
LUX or Phytoclock1 (PCL1) is a gene that codes for LUX ARRHYTHMO, a protein necessary for circadian rhythms in Arabidopsis thaliana . LUX protein associates with Early Flowering 3 (ELF3) and Early Flowering 4 (ELF4) to form the Evening Complex (EC), a core component of the Arabidopsis repressilator model of the plant circadian clock . [ 1 ] The LUX protein functions as a transcription factor that negatively regulates Pseudo-Response Regulator 9 ( PRR9 ), a core gene of the Midday Complex, another component of the Arabidopsis repressilator model. LUX is also associated with circadian control of hypocotyl growth factor genes PHYTOCHROME INTERACTING FACTOR 4 ( PIF4) and PHYTOCHROME INTERACTING FACTOR 5 ( PIF5) . [ 2 ]
In 2000, the LUX gene was first sequenced in Arabidopsis thaliana by a team at the Plant Gene Expression Center at UC Berkeley as a part of the Arabidopsis Genome Initiative. [ 3 ] In 2003, scientists from the Plant Gene Expression Center and the Genomic Analysis Laboratory at the Salk Institute for Biological Studies collaborated to identify expression of the LUX gene in Arabidopsis using cDNA arrays. [ 4 ] In 2005, scientists at the Center for Gene Research at Nagoya University and the Steve Kay lab at the Scripps Research Institute studied null mutations of LUX and the other Evening Complex genes to show that LUX was necessary for circadian rhythms in A. thaliana. [ 1 ] [ 5 ]
The LUX gene is located on the third chromosome of Arabidopsis thaliana and contains three exons . [ 6 ] Upstream of the LUX gene is a promoter containing a cis- regulatory element known as the "evening element" (EE) with the sequence AAAATATCT. It is overrepresented in evening-expressed genes in the Arabidopsis repressilator. The EE may be bound by Circadian Clock Associated 1 (CCA1) and Late Elongated Hypocotyl (LHY) proteins to suppress expression of LUX . [ 7 ] The LUX ARRHYTHMO protein has a length of 323 amino acids and contains a Myb -like GARP family transcription factor DNA-binding domain . [ 8 ] [ 9 ]
The LUX ARRHYTHMO protein encoded by the LUX gene participates in the regulation of the Arabidopsis thaliana circadian clock. Along with ELF3 and ELF4, it is a member of the Evening Complex, a component of the Arabidopsis repressilator model of gene regulation . This three- protein complex is expressed and assembled during the evening to repress transcription of the PRR9 gene, which codes for a component of the Midday Complex. LUX likely represses PRR9 via direct binding to a DNA sequence that has not yet been elucidated. PRR9 protein subsequently represses CCA1 and LHY , genes which express components of the Morning Complex. [ 1 ] [ 9 ] Although LUX and ELF4 are induced by low intensity, non-damaging UV -B radiation, the direct molecular mechanism of light input into the Arabidopsis circadian clock has yet to be elucidated. [ 7 ]
Additionally, as a part of the Arabidopsis thaliana repressilator, the LUX gene also represses its own transcription. [ 7 ]
The EC binds to promoters of Phytochrome Interacting Factor 4 ( PIF4 ) and Phytochrome Interacting Factor 5 ( PIF5 ), repressing their expression and subsequently inhibiting plant growth in the evening. PIF4 and PIF5 proteins are both basic helix-loop-helix (bHLH) domain transcription factors that are implicated in the induction of Flowering Locus T ( FT ), which expresses a florigen involved in promoting A. thaliana flowering. Mutants lacking functional LUX are unable to repress PIF4 and PIF5 , leading to early accumulation of PIF4 and PIF5 transcription factors and thus premature growth; consequently, LUX mutants often express an elongated hypocotyl phenotype due to excess growth during the night. [ 9 ]
The EC also plays a role in the detection and response to temperature. Despite variations in temperature which would normally reduce the expression of GI ( GIGANTEA ), LUX , PIF4 , PRR7 , and PRR9, these genes showed constitutively high expression in LUX (as well as ELF3 and ELF4 ) mutants. [ 9 ] This suggested that LUX mutants abolished the temperature-responsiveness of those clock genes. In addition, ELF3 association to LUX was found to be abolished at high temperatures, suggesting that temperature may play a role in recruiting EC components to their targeted promoters. [ 7 ]
Paralogs of LUX have been found to act in conjunction with LUX in Arabidopsis circadian clock regulation pathways. [ 7 ]
In the absence of LUX, ELF3 and ELF4 have also been found to form a complex with LUX paralog NOX (meaning “night” in Latin), also called BROTHER OF LUX ARRHYTHMO (BOA). NOX is a homologous Myb-like GARP transcription factor that binds to DNA sequences similar to LUX's binding, interacts directly with ELF4, and peaks in the late evening. [ 7 ]
Experiments involving artificial microRNA (amiRNA) methods have shown that both NOX and LUX are required to recruit the EC to the PIF4 and PIF5 promoters. There is evidence for NOX having an important role in the regulation of the plant circadian oscillator; overexpression of NOX has been found to have circadian phenotypes of long periods, as well as altered expressions of CCA1 , LHY , GI , and TOC1 . In particular, overexpression of NOX showed increased amplitudes of CCA1 expression. NOX likely regulates CCA1 through direct binding to the CCA1 promoter, and, conversely, CCA1 protein has been found to bind to the NOX promoter and inhibit NOX expression. [ 1 ]
In contrast to LUX , amiRNA knockouts of NOX have shown that NOX is not required for circadian rhythms, suggesting that the functionality of LUX and NOX are not completely redundant. RNAi experiments reducing NOX expression showed a continuation of circadian rhythms, whereas LUX null mutants are arrhythmic. Currently, more research must be done to determine how LUX and NOX differ in their contributions to the EC. [ 1 ] [ 7 ]
Specific studies of LUX (and ELF3 ) orthologous mutant alleles have identified variants in flowering and photoperiod -dependent growth. [ 7 ]
An ortholog for LUX named STERILE NODES (SN) was discovered in Pisum sativum . The name STERILE NODES came from the observation that photoperiod-responsive P. sativum lines formed more vegetative nodes before flowering compared to less photoperiod-responsive lines. The relationship of LUX and SN as orthologs was concluded based on the discovery of functionally and phenotypically similar mutations in SN and LUX , as well as apparent causal linkages between specific polymorphisms and SN mutant phenotypes. Like LUX , SN was found to be a major gene locus that controls regulation of circadian clock function and photoperiod-sensitive flowering. Also similar to LUX, SN protein is expressed rhythmically when exposed to light-dark cycles. [ 10 ]
Gene loci orthologous to Arabidopsis ELF3 , ELF4 , and GI have also been found in P. sativum , named HIGH RESPONSE TO PHOTOPERIOD ( HR ), DIE NEUTRALIS ( DNE ), and LATE BLOOMER1 ( LATE1 ) respectively. However, they have not yet been discovered to form a functional complex equivalent to the EC in Arabidopsis . [ 10 ]
HvLUX1 in Hordeum vulgare has been identified as an ortholog of LUX . The experiment leading to the discovery of HvLUX1 involved a mutation in the early maturity 10 ( eam10 ) locus in the H. vulgare genome. The mutation, called Bowman( eam10 ), abolished the circadian rhythm observed in H. vulgare flowering. Via high throughput sequencing , HvLUX1 has been identified as a candidate gene for this locus, though its specific mechanism of action in the circadian clock has yet to be demonstrated. [ 11 ]
Gene loci homologous to A. thaliana PRR genes ( PRR7 and PRR9 ), ELF3 and FT have also been found in H. vulgare , named PHOTOPERIOD 1 ( Ppd-H1 ), HvELF3 , and HvFT1 , respectively. [ 11 ]
Orthologs have been found for all three members of the Arabidopsis thaliana Evening Complex, but it is currently unknown if the EC is formed in species other than A. thaliana . Two orthologs of LUX , ROC15 and ROC75 , have been discovered in Chlamydomonas reinhardtii , but orthologs of ELF3 and ELF4 in C. reinhardtii have not yet been found. [ 7 ] | https://en.wikipedia.org/wiki/LUX |
LY-2365109 is a glycine reuptake inhibitor . It is able to inhibit the type 1 glycine transporter . [ 1 ] This inhibition increases extracellular levels of glycine. LY-2365109 has been shown to increase the seizure threshold in mice, meaning that this drug has potential as an anticonvulsant. [ 2 ]
This article about an organic compound is a stub . You can help Wikipedia by expanding it .
This pharmacology -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/LY-2365109 |
LZ4 is a lossless data compression algorithm that is focused on compression and decompression speed. It belongs to the LZ77 family of byte-oriented compression schemes.
The LZ4 algorithm aims to provide a good trade-off between speed and compression ratio. Typically, it has a smaller (i.e., worse) compression ratio than the similar LZO algorithm, which in turn is worse than algorithms like DEFLATE . However, LZ4 compression speed is similar to LZO and several times faster than DEFLATE, while decompression speed is significantly faster than LZO. [ 3 ]
LZ4 only uses a dictionary-matching stage (LZ77), and unlike other common compression algorithms does not combine it with an entropy coding stage (e.g. Huffman coding in DEFLATE). [ 4 ] [ 5 ]
The LZ4 algorithm represents the data as a series of sequences. Each sequence begins with a one-byte token that is broken into two 4-bit fields. The first field represents the number of literal bytes that are to be copied to the output. The second field represents the number of bytes to copy from the already decoded output buffer (with 0 representing the minimum match length of 4 bytes). A value of 15 in either of the bitfields indicates that the length is larger and there is an extra byte of data that is to be added to the length. A value of 255 in these extra bytes indicates that yet another byte is to be added. Hence arbitrary lengths are represented by a series of extra bytes containing the value 255. The string of literals comes after the token and any extra bytes needed to indicate string length. This is followed by an offset that indicates how far back in the output buffer to begin copying. The extra bytes (if any) of the match-length come at the end of the sequence. [ 6 ] [ 7 ]
Compression can be carried out in a stream or in blocks. Higher compression ratios can be achieved by investing more effort in finding the best matches. This results in both a smaller output and faster decompression.
The reference implementation in C by Yann Collet is licensed under a BSD license. There are ports and bindings in various languages including Java , C# , Rust , and Python . [ 8 ] The Apache Hadoop system uses this algorithm for fast compression. LZ4 was also implemented natively in the Linux kernel 3.11. [ 9 ] The FreeBSD, Illumos, ZFS on Linux, and ZFS-OSX implementations of the ZFS filesystem support the LZ4 algorithm for on-the-fly compression. [ 10 ] [ 11 ] [ 12 ] [ 13 ] Linux supports LZ4 for SquashFS since 3.19-rc1. [ 14 ] LZ4 is also supported by the newer zstd command line utility by Yann Collet, as well as a 7-Zip fork called 7-Zip-zstd. [ 15 ] | https://en.wikipedia.org/wiki/LZ4_(compression_algorithm) |
LZ77 and LZ78 are the two lossless data compression algorithms published in papers by Abraham Lempel and Jacob Ziv in 1977 [ 1 ] and 1978. [ 2 ] They are also known as Lempel-Ziv 1 (LZ1) and Lempel-Ziv 2 (LZ2) respectively. [ 3 ] These two algorithms form the basis for many variations including LZW , LZSS , LZMA and others. Besides their academic influence, these algorithms formed the basis of several ubiquitous compression schemes, including GIF and the DEFLATE algorithm used in PNG and ZIP .
They are both theoretically dictionary coders . LZ77 maintains a sliding window during compression. This was later shown to be equivalent to the explicit dictionary constructed by LZ78—however, they are only equivalent when the entire data is intended to be decompressed.
Since LZ77 encodes and decodes from a sliding window over previously seen characters, decompression must always start at the beginning of the input. Conceptually, LZ78 decompression could allow random access to the input if the entire dictionary were known in advance. However, in practice the dictionary is created during encoding and decoding by creating a new phrase whenever a token is output. [ 4 ]
The algorithms were named an IEEE Milestone in 2004. [ 5 ] In 2021 Jacob Ziv was awarded the IEEE Medal of Honor for his involvement in their development. [ 6 ]
In the second of the two papers that introduced these algorithms they are analyzed as encoders defined by finite-state machines. A measure analogous to information entropy is developed for individual sequences (as opposed to probabilistic ensembles). This measure gives a bound on the data compression ratio that can be achieved. It is then shown that there exists finite lossless encoders for every sequence that achieve this bound as the length of the sequence grows to infinity. In this sense an algorithm based on this scheme produces asymptotically optimal encodings. This result can be proven more directly, as for example in notes by Peter Shor . [ 7 ]
Formally, (Theorem 13.5.2 [ 8 ] ).
LZ78 is universal and entropic — If X {\textstyle X} is a binary source that is stationary and ergodic, then lim sup n 1 n l L Z 78 ( X 1 : n ) ≤ h ( X ) {\displaystyle \limsup _{n}{\frac {1}{n}}l_{LZ78}(X_{1:n})\leq h(X)} with probability 1. Here h ( X ) {\textstyle h(X)} is the entropy rate of the source.
Similar theorems apply to other versions of LZ algorithm.
LZ77 algorithms achieve compression by replacing repeated occurrences of data with references to a single copy of that data existing earlier in the uncompressed data stream. A match is encoded by a pair of numbers called a length-distance pair , which is equivalent to the statement "each of the next length characters is equal to the characters exactly distance characters behind it in the uncompressed stream". (The distance is sometimes called the offset instead.)
To spot matches, the encoder must keep track of some amount of the most recent data, such as the last 2 KB , 4 KB, or 32 KB. The structure in which this data is held is called a sliding window , which is why LZ77 is sometimes called sliding-window compression . The encoder needs to keep this data to look for matches, and the decoder needs to keep this data to interpret the matches the encoder refers to. The larger the sliding window is, the longer back the encoder may search for creating references.
It is not only acceptable but frequently useful to allow length-distance pairs to specify a length that actually exceeds the distance. As a copy command, this is puzzling: "Go back four characters and copy ten characters from that position into the current position". How can ten characters be copied over when only four of them are actually in the buffer? Tackling one byte at a time, there is no problem serving this request, because as a byte is copied over, it may be fed again as input to the copy command. When the copy-from position makes it to the initial destination position, it is consequently fed data that was pasted from the beginning of the copy-from position. The operation is thus equivalent to the statement "copy the data you were given and repetitively paste it until it fits". As this type of pair repeats a single copy of data multiple times, it can be used to incorporate a flexible and easy form of run-length encoding .
Another way to see things is as follows: While encoding, for the search pointer to continue finding matched pairs past the end of the search window, all characters from the first match at offset D and forward to the end of the search window must have matched input, and these are the (previously seen) characters that compose a single run unit of length L R , which must equal D . Then as the search pointer proceeds past the search window and forward, as far as the run pattern repeats in the input, the search and input pointers will be in sync and match characters until the run pattern is interrupted. Then L characters have been matched in total, L > D , and the code is [ D , L , c ].
Upon decoding [ D , L , c ], again, D = L R . When the first L R characters are read to the output, this corresponds to a single run unit appended to the output buffer. At this point, the read pointer could be thought of as only needing to return int( L / L R ) + (1 if L mod L R ≠ 0) times to the start of that single buffered run unit, read L R characters (or maybe fewer on the last return), and repeat until a total of L characters are read. But mirroring the encoding process, since the pattern is repetitive, the read pointer need only trail in sync with the write pointer by a fixed distance equal to the run length L R until L characters have been copied to output in total.
Considering the above, especially if the compression of data runs is expected to predominate, the window search should begin at the end of the window and proceed backwards, since run patterns, if they exist, will be found first and allow the search to terminate, absolutely if the current maximal matching sequence length is met, or judiciously, if a sufficient length is met, and finally for the simple possibility that the data is more recent and may correlate better with the next input.
The following pseudocode is a reproduction of the LZ77 compression algorithm sliding window.
Even though all LZ77 algorithms work by definition on the same basic principle, they can vary widely in how they encode their compressed data to vary the numerical ranges of a length–distance pair, alter the number of bits consumed for a length–distance pair, and distinguish their length–distance pairs from literals (raw data encoded as itself, rather than as part of a length–distance pair). A few examples:
The LZ78 algorithms compress sequential data by building a dictionary of token sequences from the input, and then replacing the second and subsequent occurrence of the sequence in the data stream with a reference to the dictionary entry. The observation is that the number of repeated sequences is a good measure of the non random nature of a sequence. The algorithms represent the dictionary as an n-ary tree where n is the number of tokens used to form token sequences. Each dictionary entry is of the form dictionary[...] = {index, token} , where index is the index to a dictionary entry representing a previously seen sequence, and token is the next token from the input that makes this entry unique in the dictionary. Note how the algorithm is greedy, and so nothing is added to the table until a unique making token is found. The algorithm is to initialize last matching index = 0 and next available index = 1 and then, for each token of the input stream, the dictionary searched for a match: {last matching index, token} . If a match is found, then last matching index is set to the index of the matching entry, nothing is output, and last matching index is left representing the input so far. Input is processed until a match is not found. Then a new dictionary entry is created, dictionary[next available index] = {last matching index, token} , and the algorithm outputs last matching index, followed by token, then resets last matching index = 0 and increments next available index. As an example consider the sequence of tokens AABBA which would assemble the dictionary;
and the output sequence of the compressed data would be 0A1B0B . Note that the last A is not represented yet as the algorithm cannot know what comes next. In practice an EOF marker is added to the input – AABBA$ for example. Note also that in this case the output 0A1B0B1$ is longer than the original input but compression ratio improves considerably as the dictionary grows, and in binary the indexes need not be represented by any more than the minimum number of bits. [ 11 ]
Decompression consists of rebuilding the dictionary from the compressed sequence. From the sequence 0A1B0B1$ the first entry is always the terminator 0 {...} , and the first from the sequence would be 1 {0,A} . The A is added to the output. The second pair from the input is 1B and results in entry number 2 in the dictionary, {1,B} . The token "B" is output, preceded by the sequence represented by dictionary entry 1. Entry 1 is an 'A' (followed by "entry 0" – nothing) so AB is added to the output. Next 0B is added to the dictionary as the next entry, 3 {0,B} , and B (preceded by nothing) is added to the output. Finally a dictionary entry for 1$ is created and A$ is output resulting in A AB B A$ or AABBA removing the spaces and EOF marker.
LZW is an LZ78-based algorithm that uses a dictionary pre-initialized with all possible characters (symbols) or emulation of a pre-initialized dictionary. The main improvement of LZW is that when a match is not found, the current input stream character is assumed to be the first character of an existing string in the dictionary (since the dictionary is initialized with all possible characters), so only the last matching index is output (which may be the pre-initialized dictionary index corresponding to the previous (or the initial) input character). Refer to the LZW article for implementation details.
BTLZ is an LZ78-based algorithm that was developed for use in real-time communications systems (originally modems) and standardized by CCITT/ITU as V.42bis . When the trie -structured dictionary is full, a simple re-use/recovery algorithm is used to ensure that the dictionary can keep adapting to changing data. A counter cycles through the dictionary. When a new entry is needed, the counter steps through the dictionary until a leaf node is found (a node with no dependents). This is deleted and the space re-used for the new entry. This is simpler to implement than LRU or LFU and achieves equivalent performance. | https://en.wikipedia.org/wiki/LZ77_and_LZ78 |
LZFSE ( Lempel–Ziv Finite State Entropy ) is an open source lossless data compression algorithm created by Apple Inc. It was released with a simpler algorithm called LZVN . [ 2 ]
The name is an acronym for Lempel–Ziv and finite-state entropy [ 3 ] (implementation of asymmetric numeral systems ).
LZFSE was introduced by Apple at its Worldwide Developer Conference 2015. It shipped with that year's iOS 9 and OS X 10.11 releases.
Apple claims that LZFSE compresses with a ratio comparable to that of zlib ( DEFLATE ) and decompresses two to three times faster while using fewer resources, therefore offering higher energy efficiency than zlib. It was aimed for scenarios where decompression speed and rate should be prioritised equally. [ 3 ] Part of this energy efficiency was achieved by optimising the algorithm for modern micro-architectures, specifically focusing on arm64 . [ 4 ] Third-party benchmarking confirms that LZFSE decompresses faster than zlib, but also suggests that many other modern compression algorithms may have more favorable compression algorithm performance characteristics such as density, compression speed and decompression speed by a significant margin. [ 5 ]
According to the Squash Benchmark, LZFSE is similar in speed to zstd (level 6), but has a slightly worse ratio. LZVN is similar in speed to LZ4 level 4, with a slightly worse ratio as well. [ 6 ] Neither LZFSE nor LZVN is tunable at runtime, although a few constants can be tweaked at compile time for the usual speed-ratio trade-off. [ 7 ]
A reference C library written by Eric Bainville was made available under the 3-clause BSD License after WWDC 2016. It includes an executable to compress and decompress LZFSE streams as well. There are no plans to expose an LZVN API. [ 1 ]
Apple's LZFSE implementation uses a simpler algorithm called LZVN when the input is smaller than LZFSE_ENCODE_LZVN_THRESHOLD (4096 bytes). This is a LZSS-type algorithm without entropy encoding but with three widths of REP (L,M,D) packets. In the open source reference implementation, Apple explains that LZFSE does not perform as well for small sizes, so LZVN is used instead. [ 7 ] This algorithm in libfastCompression.a was discovered earlier as the default kernelcache compression method in Mac OS X Yosemite Developer Preview 1 (2014), replacing the legacy lzss compression from Haruhiko Okumura . [ 8 ]
AppleFSCompression.framework (AFSC), the mechanism for quasi-transparent compression in HFS Plus and Apple File System , supports LZFSE and LZVN since OS X 10.9.
Apple's Disk Images framework has offered an LZFSE-based encoding called ULFO since Mac OS X 10.11 , [ 9 ] accessible via hdiutil(1) [ 10 ] and some third-party image utilities.
Apple introduced the Apple Archive format and its associated API in macOS High Sierra in 2017. [ 11 ] The extension name is .aar (since macOS Big Sur , used to be .yaa ). Encryption was introduced in macOS Monterey , when AA became the default Archive Utility format. Three command-line utilities are available in macOS to handle AA files. [ 12 ] [ 13 ] Of third-party programs, Keka is able to use the system APIs to handle AA files, but no independent implementations exist on other systems. [ 14 ] | https://en.wikipedia.org/wiki/LZFSE |
The Lempel–Ziv–Markov chain algorithm [ 1 ] ( LZMA ) is an algorithm used to perform lossless data compression . It has been used in the 7z format of the 7-Zip archiver since 2001. [ 2 ] This algorithm uses a dictionary compression scheme somewhat similar to the LZ77 algorithm published by Abraham Lempel and Jacob Ziv in 1977 and features a high compression ratio (generally higher than bzip2 ) [ 3 ] [ 4 ] and a variable compression-dictionary size (up to 4 GB ), [ 5 ] while still maintaining decompression speed similar to other commonly used compression algorithms. [ 6 ]
LZMA2 is a simple container format that can include both uncompressed data and LZMA data, possibly with multiple different LZMA encoding parameters. LZMA2 supports arbitrarily scalable multithreaded compression and decompression and efficient compression of data which is partially incompressible. [ 7 ]
LZMA uses a dictionary compression algorithm (a variant of LZ77 with huge dictionary sizes and special support for repeatedly used match distances), whose output is then encoded with a range encoder , using a complex model to make a probability prediction of each bit. The dictionary compressor finds matches using sophisticated dictionary data structures, and produces a stream of literal symbols and phrase references, which is encoded one bit at a time by the range encoder: many encodings are possible, and a dynamic programming algorithm is used to select an optimal one under certain approximations. [ 8 ]
Prior to LZMA, most encoder models were purely byte-based (i.e. they coded each bit using only a cascade of contexts to represent the dependencies on previous bits from the same byte). The main innovation of LZMA is that instead of a generic byte-based model, LZMA's model uses contexts specific to the bitfields in each representation of a literal or phrase: this is nearly as simple as a generic byte-based model, but gives much better compression because it avoids mixing unrelated bits together in the same context. Furthermore, compared to classic dictionary compression (such as the one used in zip and gzip formats), the dictionary sizes can be and usually are much larger, taking advantage of the large amount of memory available on modern systems. [ 8 ]
In LZMA compression, the compressed stream is a stream of bits, encoded using an adaptive binary range coder. The stream is divided into packets, each packet describing either a single byte, or an LZ77 sequence with its length and distance implicitly or explicitly encoded. Each part of each packet is modeled with independent contexts, so the probability predictions for each bit are correlated with the values of that bit (and related bits from the same field) in previous packets of the same type.
Both the lzip [ 9 ] and the LZMA SDK documentation describe this stream format. [ 8 ]
There are 7 types of packets: [ 9 ]
LONGREP[*] refers to LONGREP[0–3] packets, *REP refers to both LONGREP and SHORTREP, and *MATCH refers to both MATCH and *REP.
LONGREP[n] packets remove the distance used from the list of the most recent distances and reinsert it at the front, to avoid useless repeated entry, while MATCH just adds the distance to the front even if already present in the list and SHORTREP and LONGREP[0] don't alter the list.
The length is encoded as follows:
As in LZ77, the length is not limited by the distance, because copying from the dictionary is defined as if the copy was performed byte by byte, keeping the distance constant.
Distances are logically 32-bit and distance 0 points to the most recently added byte in the dictionary.
The distance encoding starts with a 6-bit "distance slot", which determines how many further bits are needed.
Distances are decoded as a binary concatenation of, from most to least significant, two bits depending on the distance slot, some bits encoded with fixed 0.5 probability, and some context encoded bits, according to the following table (distance slots 0−3 directly encode distances 0−3).
No complete natural language specification of the compressed format seems to exist, other than the one attempted in the following text.
The description below is based on the compact XZ Embedded decoder by Lasse Collin included in the Linux kernel source [ 10 ] from which the LZMA and LZMA2 algorithm details can be relatively easily deduced: thus, while citing source code as reference is not ideal, any programmer should be able to check the claims below with a few hours of work.
LZMA data is at the lowest level decoded one bit at a time by the range decoder, at the direction of the LZMA decoder.
Context-based range decoding is invoked by the LZMA algorithm passing it a reference to the "context", which consists of the unsigned 11-bit variable prob (typically implemented using a 16-bit data type) representing the predicted probability of the bit being 0, which is read and updated by the range decoder (and should be initialized to 2 10 {\displaystyle 2^{10}} , representing 0.5 probability).
Fixed probability range decoding instead assumes a 0.5 probability, but operates slightly differently from context-based range decoding.
The range decoder state consists of two unsigned 32-bit variables, range (representing the range size), and code (representing the encoded point within the range).
Initialization of the range decoder consists of setting range to 2 32 − 1 , and code to the 32-bit value starting at the second byte in the stream interpreted as big-endian; the first byte in the stream is completely ignored.
Normalization proceeds in this way:
Context-based range decoding of a bit using the prob probability variable proceeds in this way:
Fixed-probability range decoding of a bit proceeds in this way:
The Linux kernel implementation of fixed-probability decoding in rc_direct() , for performance reasons, does not include a conditional branch, but instead subtracts range from code unconditionally. The resulting sign bit is used to both decide the bit to return and to generate a mask that is combined with code and added to range .
Note that:
The range decoder also provides the bit-tree, reverse bit-tree and fixed probability integer decoding facilities, which are used to decode integers, and generalize the single-bit decoding described above.
To decode unsigned integers less than limit , an array of ( limit − 1) 11-bit probability variables is provided, which are conceptually arranged as the internal nodes of a complete binary tree with limit leaves.
Non-reverse bit-tree decoding works by keeping a pointer to the tree of variables, which starts at the root. As long as the pointer does not point to a leaf, a bit is decoded using the variable indicated by the pointer, and the pointer is moved to either the left or right children depending on whether the bit is 0 or 1; when the pointer points to a leaf, the number associated with the leaf is returned.
Non-reverse bit-tree decoding thus happens from most significant to least significant bit, stopping when only one value in the valid range is possible (this conceptually allows to have range sizes that are not powers of two, even though LZMA does not make use of this).
Reverse bit-tree decoding instead decodes from least significant bit to most significant bits, and thus only supports ranges that are powers of two, and always decodes the same number of bits. It is equivalent to performing non-reverse bittree decoding with a power of two limit , and reversing the last log 2 ( limit ) bits of the result.
In the rc_bittree function in the Linux kernel, integers are actually returned in the [ limit , 2 × limit ) range (with limit added to the conceptual value), and the variable at index 0 in the array is unused, while the one at index 1 is the root, and the left and right children indices are computed as 2 i and 2 i + 1. The rc_bittree_reverse function instead adds integers in the [0, limit ) range to a caller-provided variable, where limit is implicitly represented by its logarithm, and has its own independent implementation for efficiency reasons.
Fixed probability integer decoding simply performs fixed probability bit decoding repeatedly, reading bits from the most to the least significant.
The LZMA decoder is configured by an lclppb "properties" byte and a dictionary size. The value of the lclppb byte is lc + lp * 9 + pb * 9 * 5 , where:
In non-LZMA2 streams, lc must not be greater than 8, and lp and pb must not be greater than 4.
In LZMA2 streams, lc + lp and pb must not be greater than 4.
In the 7-zip LZMA file format, configuration is performed by a header containing the "properties" byte followed by the 32-bit little-endian dictionary size in bytes. In LZMA2, the properties byte can optionally be changed at the start of LZMA2 LZMA packets, while the dictionary size is specified in the LZMA2 header as later described.
The LZMA packet format has already been described, and this section specifies how LZMA statistically models the LZ-encoded streams, or in other words which probability variables are passed to the range decoder to decode each bit.
Those probability variables are implemented as multi-dimensional arrays; before introducing them, a few values that are used as indices in these multidimensional arrays are defined.
The state value is conceptually based on which of the patterns in the following table match the latest 2–4 packet types seen, and is implemented as a state machine state updated according to the transition table listed in the table every time a packet is output.
The initial state is 0, and thus packets before the beginning are assumed to be LIT packets.
The pos_state and literal_pos_state values consist of respectively the pb and lp (up to 4, from the LZMA header or LZMA2 properties packet) least significant bits of the dictionary position (the number of bytes coded since the last dictionary reset modulo the dictionary size). Note that the dictionary size is normally the multiple of a large power of 2, so these values are equivalently described as the least significant bits of the number of uncompressed bytes seen since the last dictionary reset.
The prev_byte_lc_msbs value is set to the lc (up to 4, from the LZMA header or LZMA2 properties packet) most significant bits of the previous uncompressed byte.
The is_REP value denotes whether a packet that includes a length is a LONGREP rather than a MATCH.
The match_byte value is the byte that would have been decoded if a SHORTREP packet had been used (in other words, the byte found at the dictionary at the last used distance); it is only used just after a *MATCH packet.
literal_bit_mode is an array of 8 values in the 0–2 range, one for each bit position in a byte, which are 1 or 2 if the previous packet was a *MATCH and it is either the most significant bit position or all the more significant bits in the literal to encode/decode are equal to the bits in the corresponding positions in match_byte , while otherwise it is 0; the choice between the 1 or 2 values depends on the value of the bit at the same position in match_byte .
The literal/Literal set of variables can be seen as a "pseudo-bit-tree" similar to a bit-tree but with 3 variables instead of 1 in every node, chosen depending on the literal_bit_mode value at the bit position of the next bit to decode after the bit-tree context denoted by the node.
The claim, found in some sources, that literals after a *MATCH are coded as the XOR of the byte value with match_byte is incorrect; they are instead coded simply as their byte value, but using the pseudo-bit-tree just described and the additional context listed in the table below.
The probability variable groups used in LZMA are those:
The LZMA2 container supports multiple runs of compressed LZMA data and uncompressed data. Each LZMA compressed run can have a different LZMA configuration and dictionary. This improves the compression of partially or completely incompressible files and allows multithreaded compression and multithreaded decompression by breaking the file into runs that can be compressed or decompressed independently in parallel.
Criticism of changes in LZMA2 over LZMA include header fields not being covered by CRCs,
and parallel decompression not being possible in practice. [ 7 ]
The LZMA2 header consists of a byte indicating the dictionary size:
LZMA2 data consists of packets starting with a control byte, with the following values:
Bits 5–6 for LZMA chunks can be:
LZMA state resets cause a reset of all LZMA state except the dictionary, and specifically:
Uncompressed chunks consist of:
LZMA chunks consist of:
The . xz format, which can contain LZMA2 data, is documented at tukaani.org , [ 11 ] while the .7z file format, which can contain either LZMA or LZMA2 data, is documented in the 7zformat.txt file contained in the LZMA SDK. [ 12 ]
The LZMA implementation extracted from 7-Zip is available as LZMA SDK. It was originally dual-licensed under both the GNU LGPL and Common Public License , [ 13 ] with an additional special exception for linked binaries, but was placed by Igor Pavlov in the public domain on December 2, 2008, with the release of version 4.62. [ 12 ]
LZMA2 compression, which is an improved version of LZMA, [ 14 ] is now the default compression method for the .7z format, starting with version 9.30 on October 26, 2012. [ 15 ]
The reference open source LZMA compression library was originally written in C++ but has been ported to ANSI C , C# , and Java . [ 12 ] There are also third-party Python bindings for the C++ library, [ 16 ] as well as ports of LZMA to Pascal , [ 17 ] Go [ 18 ] and Ada . [ 19 ]
The 7-Zip implementation uses several variants of hash chains , binary trees and Patricia trees as the basis for its dictionary search algorithm.
In addition to LZMA, the SDK and 7-Zip also implements multiple preprocessing filters intended to improve compression, ranging from simple delta encoding (for images) and BCJ for executable code. It also provides some other compression algorithms used in 7z.
Decompression-only code for LZMA generally compiles to around 5 KB, and the amount of RAM required during decompression is principally determined by the size of the sliding window used during compression. Small code size and relatively low memory overhead, particularly with smaller dictionary lengths, and free source code make the LZMA decompression algorithm well-suited to embedded applications.
In addition to the 7-Zip reference implementation, the following support the LZMA format. | https://en.wikipedia.org/wiki/LZMA |
Lempel–Ziv Ross Williams ( LZRW ) refers to variants of the LZ77 lossless data compression algorithms with an emphasis on improving compression speed through the use of hash tables and other techniques. This family was explored by Ross Williams , who published a series of algorithms [ 1 ] beginning with LZRW1 in 1991.
The variants are:
The LZJB algorithm used in ZFS is derived from LZRW1.
This computer science article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/LZRW |
LZWL is a syllable-based variant of the LZW (Lempel-Ziv-Welch) compression algorithm , designed to work with syllables derived from any syllable decomposition algorithm. This approach allows LZWL to efficiently process both syllables and words, offering a nuanced method for data compression.
The LZWL algorithm initializes by populating a dictionary with all characters from the alphabet. It then searches for the longest string, S , that exists in both the dictionary and as a prefix of the unencoded portion of the input. The algorithm outputs the identifier of S and augments the dictionary with a new phrase, which combines S with the subsequent character in the input. The input position advances by the length of S . During decoding, LZWL addresses scenarios where the received phrase identifier does not exist in the dictionary by constructing the missing phrase from the concatenation of the last added phrase and its initial character.
In its syllable-based adaptation, LZWL employs a list of syllables as its alphabet. The initialization step includes the empty syllable and integrates small, frequently occurring syllables into the dictionary. Identifying S and encoding its identifier mirrors the original algorithm, with the distinction that S represents a syllable string. If S is an empty syllable, the algorithm extracts a syllable K from the input and encodes K using methods for new syllables before adding K to the dictionary and advancing the input position accordingly.
A notable variation in the syllable-based LZWL involves dictionary expansion. When both S and the subsequent string S1 are non-empty syllables, a new phrase is added to the dictionary by concatenating S1 with S ’s initial syllable. This method prevents the formation of strings from syllables that appear only once and ensures the decoder does not encounter undefined phrase identifiers. | https://en.wikipedia.org/wiki/LZWL |
LZX is an LZ77 family compression algorithm , a slightly improved version of DEFLATE . [ 1 ] It is also the name of a file archiver with the same name. Both were invented by Jonathan Forbes and Tomi Poutanen in the 1990s.
LZX was publicly released as an Amiga file archiver in 1995, while the authors were studying at the University of Waterloo in Canada . The software was shareware , which was common for compression software at the time. The registered version contained fixes and improvements that were not available in the evaluation version. In 1997, the authors gave away a free keyfile, which allowed anyone to use the registered version, as they had stopped work on the archiver and stopped accepting registrations.
In 1996, Forbes went to work for Microsoft , [ 2 ] and Microsoft's cabinet archiver was enhanced to include the LZX compression method. Improvements included a variable search window size; Amiga LZX was fixed to 64 KB, and Microsoft LZX could range on powers of two between 32 and 2048 kilobytes (32,768 to 2,097,152 bytes). A special preprocessor was added to detect Intel 80x86 "CALL" instructions, converting their operands from relative addressing to absolute addressing, thus calls to the same location resulted in repeated strings that the compressor could match, improving compression of 80x86 binary code. (This technique is later generalized as Branch-Call-Jump [BCJ] filtering.)
When Microsoft introduced Microsoft Compressed HTML Help , the replacement for their classic Help file format, they chose to compress all of the HTML data with the LZX algorithm. However, in order to improve random access speed, the compressor was altered to reset itself after every 64 kilobyte (65,536 bytes) interval and re-align to a 16-bit boundary after every 32 kilobyte interval. Thus, the HTMLHelp software could immediately seek the nearest 64 kilobyte interval and start decoding from there, rather than decoding from the beginning of the compressed datastream at all times.
Microsoft LIT files for Microsoft Reader are simply an extension of the CHM file format, and thus also use LZX compression.
Windows Imaging Format , the installation/drive image file format of Windows Vista and Windows 7 , uses LZX as one of the compression methods. [ 3 ]
In Windows 10 , LZX compression from Windows Imaging Format is used for the new CompactOS NTFS file compression.
Microsoft uses LZX compression on Xbox Live Avatars to reduce their disk and bandwidth requirements. [ 4 ]
The unlzx program and XAD can unpack Amiga LZX archives. The cabextract program can unpack Microsoft cabinet files using the LZX method. [ 5 ] There are a multitude of cross-platform tools for decompiling or viewing CHM files, as stated in the CHM article. LIT files can be unpacked using the Convert LIT software. [ 6 ] | https://en.wikipedia.org/wiki/LZX |
In infrared astronomy , the L band is an atmospheric transmission window centred on 3.5 micrometres (in the mid-infrared ).
This astronomy -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/L_band_(infrared) |
The L-ring of the bacterial flagellum is the ring in the lipid outer cell membrane through which the axial filament (rod, hook, and flagellum) passes. [ 1 ] that l ring stands for lipopolysaccharide .
This bacteria -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/L_ring |
Lanthanum carbonate , La 2 ( C O 3 ) 3 , is the salt formed by lanthanum (III) cations and carbonate anions . It is an ore of lanthanum metal ( bastnäsite ), along with monazite .
Lanthanum carbonate is used as a starting material in lanthanum chemistry, particularly in forming mixed oxides, for example
Lanthanum carbonate is used in medicine as a phosphate binder . [ 1 ] As a medication it is sold under the trade name Fosrenol by the pharmaceutical company Shire Pharmaceuticals . Due to its large size (1000 mg tablet is 2.2 cm in diameter), it may be possible to choke on the tablet if it is not chewed. It is prescribed for the treatment of hyperphosphatemia , primarily in patients with chronic kidney disease . It is taken with meals and binds to dietary phosphate , preventing phosphate from being absorbed by the intestine . For cats with hyperphosphatemia it is available under the trade name Renalzin by Bayer Animal Health . [ 2 ]
However, when lanthanum carbonate is used for treating hyperphosphatemia, its side effects, namely myalgia , muscular cramping, and peripheral edema , should be clinically monitored. [ 3 ]
Lanthanum carbonate is also used for the tinting of glass, [ citation needed ] for water treatment, and as a catalyst for hydrocarbon cracking . | https://en.wikipedia.org/wiki/La2(CO3)3 |
Lanthanum(III) oxide , also known as lanthana , chemical formula La 2 O 3 , is an inorganic compound containing the rare earth element lanthanum and oxygen . It is used in some ferroelectric materials, as a component of optical materials, and is a feedstock for certain catalysts, among other uses.
Lanthanum oxide is a white solid that is insoluble in water, but dissolves in acidic solutions. La 2 O 3 absorbs moisture from air, converting to lanthanum hydroxide. [ 2 ] Lanthanum oxide has p-type semiconducting properties and a band gap of approximately 5.8 eV. [ 3 ] Its average room temperature resistivity is 10 kΩ·cm, which decreases with an increase in temperature. La 2 O 3 has the lowest lattice energy of the rare earth oxides, with very high dielectric constant ε = 27.
At low temperatures, La 2 O 3 has an A- M 2 O 3 hexagonal crystal structure. The La 3+ metal atoms are surrounded by a 7 coordinate group of O 2− atoms, the oxygen ions are in an octahedral shape around the metal atom and there is one oxygen ion above one of the octahedral faces. [ 4 ] On the other hand, at high temperatures lanthanum oxide converts to a C- M 2 O 3 cubic crystal structure. The La 3+ ion is surrounded by six O 2− ions in a hexagonal configuration. [ 5 ] [ 6 ]
Lanthanum oxide can crystallize in at least three polymorphs . [ 2 ]
Hexagonal La 2 O 3 has been produced by spray pyrolysis of lanthanum chloride. [ 7 ]
An alternative route to obtaining hexagonal La 2 O 3 involves precipitation of nominal La(OH) 3 from aqueous solution using a combination of 2.5% NH 3 and the surfactant sodium dodecyl sulfate followed by heating and stirring for 24 hours at 80 °C:
Other routes include:
Lanthanum oxide is used as an additive to develop certain ferroelectric materials, such as La-doped bismuth titanate ( Bi 4 Ti 3 O 12 - BLT).
Lanthanum oxide is used in optical materials; often the optical glasses are doped with La 2 O 3 to improve the glass' refractive index, chemical durability, and mechanical strength. [ 8 ]
The addition of the La 2 O 3 to the glass melt leads to a higher glass transition temperature from 658 °C to 679 °C. The addition also leads to a higher density, microhardness, and refractive index of the glass.
Lanthanum oxide is most useful as a precursor to other lanthanum compounds. [ 9 ] Neither the oxide nor any of the derived materials enjoys substantial commercial value, unlike some of the other lanthanides. Many reports describe efforts toward practical applications of La 2 O 3 , as described below.
La 2 O 3 forms glasses of high density, refractive index, and hardness. Together with oxides of tungsten , tantalum , and thorium , La 2 O 3 improves the resistance of the glass to attack by alkali. La 2 O 3 is an ingredient in some piezoelectric and thermoelectric materials.
La 2 O 3 has been examined for the oxidative coupling of methane . [ 10 ] | https://en.wikipedia.org/wiki/La2O3 |
Lanthanum(III) telluride is an inorganic compound, one of the tellurides of lanthanum , with the chemical formula La 2 Te 3 . Lanthanum(III) telluride is prepared by oxidizing a mixture of lanthanum , tellurium and sodium carbonate at 400 °C. [ 1 ] There are also literature reports on preparing lanthanum(III) telluride using lanthanum(III) chloride and tellurium as raw materials. In the reaction, tellurium is first oxidized to +4 oxidation state, and then hydrazine hydrate is reduced to −2. [ 2 ] Various phases such as CuLaTe 2 and Cu 4 La 2 Te 5 can be formed in the La 2 Te 3 -Cu 2 Te system. [ 3 ] | https://en.wikipedia.org/wiki/La2Te3 |
Lanthanum(III) bromide (LaBr 3 ) is an inorganic halide salt of lanthanum . When pure, it is a colorless white powder. The single crystals of LaBr 3 are hexagonal crystals with melting point of 783 °C. It is highly hygroscopic and water-soluble. There are several hydrates , La 3 Br·x H 2 O, of the salt also known. It is often used as a source of lanthanum in chemical synthesis and as a scintillation material in certain applications.
The scintillator material cerium activated lanthanum bromide (LaBr 3 :Ce) was first produced in 2001. [ 2 ] LaBr 3 :Ce-based radiation detectors offer improved energy resolution, fast emission and excellent temperature and linearity characteristics. Typical energy resolution at 662 keV is 3% as compared to sodium iodide detectors at 7%. [ 3 ] The improved resolution is due to a photoelectron yield that is 160% greater than is achieved with sodium iodide. Another advantage of LaBr 3 :Ce is the nearly flat photo emission over a 70 °C temperature range (~1% change in light output). [ citation needed ]
Today LaBr 3 detectors are offered with bialkali photomultiplier tubes (PMT) that can be two inches in diameter and 10 or more inches long. [ citation needed ] However, miniature packaging can be obtained by the use of a silicon drift detector (SDD) or a Silicon Photomultiplier (SiPM). [ 4 ] These UV enhanced diodes provide excellent wavelength matching to the 380 nm emission of LaBr 3 . The SDD is not as sensitive to temperature and bias drift as PMT. The reported spectroscopy performance of the SDD configuration resulted in a 2.8% energy resolution at 662 keV for the detector sizes considered.
LaBr 3 introduces an enhanced set of capabilities to a range of gamma spectroscopy radioisotope detection and identification systems used in the homeland security market. Isotope identification utilizes several techniques (known as algorithms) which rely on the detector's ability to discriminate peaks. The improvements in resolution allow more accurate peak discrimination in ranges where isotopes often have many overlapping peaks. This leads to better isotope classification. Screening of all types (pedestrians, cargo, conveyor belts, shipping containers, vehicles, etc.) often requires accurate isotopic identification to differentiate concerning materials from non-concerning materials ( medical isotopes in patients, naturally occurring radioactive materials, etc.) Heavy R&D and deployment of instruments utilizing LaBr 3 is expected in the upcoming years. | https://en.wikipedia.org/wiki/LaBr3 |
Lanthanum chloride is the inorganic compound with the formula La Cl 3 . It is a common salt of lanthanum which is mainly used in research. It is a white solid that is highly soluble in water and alcohols.
Anhydrous lanthanum(III) chloride can be produced by the ammonium chloride route . [ 3 ] [ 4 ] [ 5 ] In the first step, lanthanum oxide is heated with ammonium chloride to produce the ammonium salt of the pentachloride :
In the second step, the ammonium chloride salt is converted to the trichlorides by heating in a vacuum at 350-400 °C:
Lanthanum chloride is also used in biochemical research to block the activity of divalent cation channels , mainly calcium channels . Doped with cerium , it is used as a scintillator material. [ 6 ]
In organic synthesis , lanthanum trichloride functions as a mild Lewis acid for converting aldehydes to acetals . [ 7 ]
The compound has been identified as a catalyst for the high pressure oxidative chlorination of methane to chloromethane with hydrochloric acid and oxygen . [ 8 ] In Fluid Catalytic Cracking (FCC) , lanthanum chloride enhances catalysts to convert heavy crude into valuable fuels like gasoline and diesel. [ 9 ]
Also used in the field of geology as a very dilute solution, which when combined with the proper acids can help identify small >1% Strontium content in powdered rock samples.
Lanthanum chloride also contributes to environmental remediation by promoting- the catalytic transformation of pollutants, such as nitrogen oxides (NOx) , volatile organic compounds (VOCs) , sulfur dioxide (SO₂) , and phosphate ions, into less harmful products. [ 10 ] It can also be commercially viable and significantly outperform conventional materials such as ion exchange resins, activated carbons, and iron oxides in water and wastewater treatment , especially for the removal of anionic pollutants. [ 11 ] | https://en.wikipedia.org/wiki/LaCl3 |
Lanthanum trifluoride is a refractory ionic compound of lanthanum and fluorine . [ 4 ] The chemical formula is LaF 3 .
Bonding is ionic with lanthanum highly coordinated. The cation sits at the center of a trigonal prism . Nine fluorine atoms are close: three at the bottom corners of the trigonal prism, three in the faces of the trigonal prism, and three at top corners of the trigonal prism. There are also two fluorides a little further away above and below the prism. The cation can be considered 9-coordinate or 11-coordinate. [ 4 ] At 300 K, the structure allows the formation of Schottky defects with an activation energy of 0.07 eV, and free flow of fluoride ions with an activation energy of 0.45 eV, making the crystal unusually electrically conductive . [ 5 ] [ 6 ]
The larger sized rare earth elements ( lanthanides ), which are those with smaller atomic number, also form trifluorides with the LaF 3 structure. [ 4 ] Some actinides do as well.
This white salt is sometimes used as the "high-index" component in multilayer optical elements such as ultraviolet dichroic and narrowband mirrors. Fluorides are among the most commonly used compounds for UV optical coatings due to their relative inertness and transparency in the far ultraviolet (FUV) (100 nm < λ < 200 nm). Multilayer reflectors and antireflection coatings are typically composed of pairs of transparent materials, one with a low index of refraction , the other with a high index. LaF 3 is one of very few high-index materials in the far UV. [ 7 ] The material is also a component of multimetal fluoride glasses such as ZBLAN . [ 8 ] It is also doped with europium(II) fluoride in fluoride selective electrodes . [ 9 ]
LaF 3 occurs in the nature as the extremely rare mineral fluocerite-(La) . [ 10 ] [ 11 ] The suffix in the name is known as the Levinson modifier and, by showing the dominant element at a particular site in the structure, is used to differentiate from similar minerals (here: fluocerite-(Ce)). [ 12 ] | https://en.wikipedia.org/wiki/LaF3 |
Lanthanum decahydride is a polyhydride or superhydride compound of lanthanum and hydrogen (LaH 10 ) that has shown evidence of being a high-temperature superconductor . It was the first metal superhydride to be theoretically predicted, [ 2 ] [ 3 ] synthesized, [ 4 ] and experimentally confirmed [ 5 ] to superconduct at near room-temperatures. It has a superconducting transition temperature T C around 250 K (−23 °C; −10 °F) at a pressure of 150 gigapascals (22 × 10 ^ 6 psi), and its synthesis required pressures above approximately 160 gigapascals (23 × 10 ^ 6 psi). [ 6 ] [ 7 ]
Since its discovery in 2019, [ 5 ] the superconducting properties of LaH 10 and other lanthanum -based superhydrides have been experimentally confirmed in multiple independent experiments. [ 8 ] [ 9 ] [ 10 ] [ 11 ] The compound exhibits a Meissner effect below the superconducting transition temperature. [ 12 ] A cubic form can be synthesized at 1,000 K (730 °C; 1,340 °F), [ 6 ] and a hexagonal crystal structure can be formed at room temperature. [ 13 ] Further reports indicate T c is increased with nitrogen doping, [ 14 ] and decreased with the introduction of magnetic impurities. [ 15 ]
The cubic form has each lanthanum atom surrounded by 32 hydrogen atoms, which form the vertices of an 18-faced shape called a chamfered cube . [ 16 ]
A similar compound, lanthanum boron octahydride , was computationally predicted to be a superconductor at 126 K (−147 °C; −233 °F) and pressure 50 gigapascals (7.3 × 10 ^ 6 psi). [ 17 ] | https://en.wikipedia.org/wiki/LaH10 |
Lanthanum manganite is an inorganic compound with the formula LaMnO 3 , often abbreviated as LMO . Lanthanum manganite is formed in the perovskite structure , consisting of oxygen octahedra with a central Mn atom. The cubic perovskite structure is distorted into an orthorhombic structure by a strong Jahn–Teller distortion of the oxygen octahedra. [ 2 ]
LaMnO 3 often has lanthanum vacancies as evidenced by neutron scattering . For this reason, this material is usually referred as LaMnO 3+ẟ . These vacancies generate a structure with a rhombohedral unit cell in this perovskite. A temperatures below 140 K, this LaMnO 3+ẟ semiconductor exhibit a ferromagnetic order. [ 3 ]
Lanthanum manganite can be prepared via solid-state reactions at high temperatures, using their oxides or carbonates . [ 4 ] An alternative method is to use lanthanum nitrate and manganese nitrate as raw materials. The reaction occurs at high temperature after the solvents are vaporized. [ 5 ]
Lanthanum manganite is an electrical insulator and an A-type antiferromagnet . It is the parent compound of several important alloys, often termed rare-earth manganites or colossal magnetoresistance oxides. These families include lanthanum strontium manganite , lanthanum calcium manganite and others.
In lanthanum manganite, both the La and the Mn are in the +3 oxidation state. Substitution of some of the La atoms by divalent atoms such as Sr or Ca induces a similar amount of tetravalent Mn 4+ ions. Such substitution, or doping can induce various electronic effects, which form the basis of a rich and complex electron correlation phenomena that yield diverse electronic phase diagrams in these alloys. [ 6 ] | https://en.wikipedia.org/wiki/LaMnO3 |
Lanthanum(III) oxyfluoride is an inorganic compound of lanthanum , fluorine , and oxygen with the chemical formula LaOF. [ 1 ] [ 2 ]
Several methods of synthesizing LaOF are known:
The compound forms colorless crystals.
The compound is used to produce thin films. [ 3 ] [ 4 ] | https://en.wikipedia.org/wiki/LaOF |
Lanthanum phosphide is an inorganic compound of lanthanum and phosphorus with the chemical formula LaP .
Lanthanum phosphide can be made by heating lanthanum metal with excess phosphorus in a vacuum: [ 2 ]
Lanthanum phosphide forms black crystals of a cubic system , space group Fm 3 m, cell parameters a = 0.6025 nm, with number of formulas per unit cell Z = 4. [ 1 ]
The crystals are very unstable and decompose in the open air.
Lanthanum phosphide is an example of a strongly correlated material , [ 3 ] complicating theoretical prediction of its properties.
According to HSE06 calculations, lanthanum phosphide has been theoretically predicted to have an indirect band gap of 0.25 eV along the Γ-X direction . [ 4 ] According to HSE06 calculations with spin-orbit coupling, the band gap is predicted to be a direct gap of 0.72 eV at the X point. [ 5 ] Using EVGGA, the compound is predicted to have a band gap of 0.56 eV along the Γ-X direction. [ 6 ] FP-LAPW has predicted an indirect gap of 0.33 eV along the Γ-X direction. [ 3 ]
Lanthanum phosphide reacts with water, releasing highly toxic phosphine gas:
Lanthanum phosphide compound is a semiconductor used in high power, high frequency applications, and in laser diodes . [ 7 ] [ 8 ]
In addition to the simple phosphide, LaP, lanthanum and phosphorus can also form phosphorus-rich compounds such as LaP 2 [ 9 ] LaP 5 [ 10 ] and LaP 7 . [ 11 ] | https://en.wikipedia.org/wiki/LaP |
LaSalle's invariance principle (also known as the invariance principle , [ 1 ] Barbashin-Krasovskii-LaSalle principle , [ 2 ] or Krasovskii-LaSalle principle ) is a criterion for the asymptotic stability of an autonomous (possibly nonlinear) dynamical system .
Suppose a system is represented as
where x {\displaystyle \mathbf {x} } is the vector of variables, with
If a C 1 {\displaystyle C^{1}} (see Smoothness ) function V ( x ) {\displaystyle V(\mathbf {x} )} can be found such that
then the set of accumulation points of any trajectory [ clarification needed ] is contained in I {\displaystyle {\mathcal {I}}} where I {\displaystyle {\mathcal {I}}} is the union of complete trajectories contained entirely in the set { x : V ˙ ( x ) = 0 } {\displaystyle \{\mathbf {x} :{\dot {V}}(\mathbf {x} )=0\}} .
If we additionally have that the function V {\displaystyle V} is positive definite, i.e.
and if I {\displaystyle {\mathcal {I}}} contains no trajectory of the system except the trivial trajectory x ( t ) = 0 {\displaystyle \mathbf {x} (t)=\mathbf {0} } for t ≥ 0 {\displaystyle t\geq 0} , then the origin is asymptotically stable .
Furthermore, if V {\displaystyle V} is radially unbounded, i.e.
then the origin is globally asymptotically stable .
If
hold only for x {\displaystyle \mathbf {x} } in some neighborhood D {\displaystyle D} of the origin, and the set
does not contain any trajectories of the system besides the trajectory x ( t ) = 0 , t ≥ 0 {\displaystyle \mathbf {x} (t)=\mathbf {0} ,t\geq 0} , then the local version of the invariance principle states that the origin is locally asymptotically stable .
If V ˙ ( x ) {\displaystyle {\dot {V}}(\mathbf {x} )} is negative definite, then the global asymptotic stability of the origin is a consequence of Lyapunov's second theorem . The invariance principle gives a criterion for asymptotic stability in the case when V ˙ ( x ) {\displaystyle {\dot {V}}(\mathbf {x} )} is only negative semidefinite.
Example taken from "LaSalle's Invariance Principle, Lecture 23, Math 634", by Christopher Grant . [ 3 ]
Consider the vector field ( x ˙ , y ˙ ) = ( − y − x 3 , x 5 ) {\displaystyle ({\dot {x}},{\dot {y}})=(-y-x^{3},x^{5})} in the plane. The function V ( x , y ) = x 6 + 3 y 2 {\displaystyle V(x,y)=x^{6}+3y^{2}} satisfies V ˙ = − 6 x 8 {\displaystyle {\dot {V}}=-6x^{8}} , and is radially unbounded, showing that the origin is globally asymptotically stable.
This section will apply the invariance principle to establish the local asymptotic stability of a simple system, the pendulum with friction. This system can be modeled with the differential equation [ 4 ]
where θ {\displaystyle \theta } is the angle the pendulum makes with the vertical normal, m {\displaystyle m} is the mass of the pendulum, l {\displaystyle l} is the length of the pendulum, k {\displaystyle k} is the friction coefficient , and g is acceleration due to gravity.
This, in turn, can be written as the system of equations
Using the invariance principle, it can be shown that all trajectories that begin in a ball of certain size around the origin x 1 = x 2 = 0 {\displaystyle x_{1}=x_{2}=0} asymptotically converge to the origin. We define V ( x 1 , x 2 ) {\displaystyle V(x_{1},x_{2})} as
This V ( x 1 , x 2 ) {\displaystyle V(x_{1},x_{2})} is simply the scaled energy of the system. [ 4 ] Clearly, V ( x 1 , x 2 ) {\displaystyle V(x_{1},x_{2})} is positive definite in an open ball of radius π {\displaystyle \pi } around the origin. Computing the derivative,
Observe that V ( 0 ) = 0 {\displaystyle V(0)=0} and V ˙ ( 0 ) = 0 {\displaystyle {\dot {V}}(0)=0} . If it were true that V ˙ < 0 {\displaystyle {\dot {V}}<0} , we could conclude that every trajectory approaches the origin by Lyapunov's second theorem . Unfortunately, V ˙ ≤ 0 {\displaystyle {\dot {V}}\leq 0} and V ˙ {\displaystyle {\dot {V}}} is only negative semidefinite since x 1 {\displaystyle x_{1}} can be non-zero when V ˙ = 0 {\displaystyle {\dot {V}}=0} . However, the set
which is simply the set
does not contain any trajectory of the system, except the trivial trajectory x = 0 {\displaystyle x=0} . Indeed, if at some time t {\displaystyle t} , x 2 ( t ) = 0 {\displaystyle x_{2}(t)=0} , then because x 1 {\displaystyle x_{1}} must be less than π {\displaystyle \pi } away from the origin, sin x 1 ≠ 0 {\displaystyle \sin x_{1}\neq 0} and x ˙ 2 ( t ) ≠ 0 {\displaystyle {\dot {x}}_{2}(t)\neq 0} . As a result, the trajectory will not stay in the set S {\displaystyle S} .
All the conditions of the local version of the invariance principle are satisfied, and we can conclude that every trajectory that begins in some neighborhood of the origin will converge to the origin as t → ∞ {\displaystyle t\rightarrow \infty } . [ 5 ]
The general result was independently discovered by J.P. LaSalle (then at RIAS ) and N.N. Krasovskii , who published in 1960 and 1959 respectively. While LaSalle was the first author in the West to publish the general theorem in 1960, a special case of the theorem was communicated in 1952 by Barbashin and Krasovskii , followed by a publication of the general result in 1959 by Krasovskii . [ 6 ] | https://en.wikipedia.org/wiki/LaSalle's_invariance_principle |
La Géométrie ( French pronunciation: [la ʒeɔmetʁi] ) was published in 1637 as an appendix to Discours de la méthode ( Discourse on the Method ), written by René Descartes . In the Discourse , Descartes presents his method for obtaining clarity on any subject. La Géométrie and two other appendices, also by Descartes, La Dioptrique ( Optics ) and Les Météores ( Meteorology ), were published with the Discourse to give examples of the kinds of successes he had achieved following his method [ 1 ] (as well as, perhaps, considering the contemporary European social climate of intellectual competitiveness, to show off a bit to a wider audience).
The work was the first to propose the idea of uniting algebra and geometry into a single subject [ 2 ] and invented an algebraic geometry called analytic geometry , which involves reducing geometry to a form of arithmetic and algebra and translating geometric shapes into algebraic equations . For its time this was ground-breaking. It also contributed to the mathematical ideas of Leibniz and Newton and was thus important in the development of calculus.
This appendix is divided into three "books". [ 3 ]
Book I is titled Problems Which Can Be Constructed by Means of Circles and Straight Lines Only. In this book he introduces algebraic notation that is still in use today. The letters at the end of the alphabet, viz., x , y , z , etc. are to denote unknown variables, while those at the start of the alphabet, a , b , c , etc. denote constants. He introduces modern exponential notation for powers (except for squares, where he kept the older tradition of writing repeated letters, such as, aa ). He also breaks with the Greek tradition of associating powers with geometric referents, a 2 with an area, a 3 with a volume and so on, and treats them all as possible lengths of line segments. These notational devices permit him to describe an association of numbers to lengths of line segments that could be constructed with straightedge and compass . The bulk of the remainder of this book is occupied by Descartes's solution to "the locus problems of Pappus ." [ 4 ] According to Pappus, given three or four lines in a plane, the problem is to find the locus of a point that moves so that the product of the distances from two of the fixed lines (along specified directions) is proportional to the square of the distance to the third line (in the three line case) or proportional to the product of the distances to the other two lines (in the four line case). In solving these problems and their generalizations, Descartes takes two line segments as unknown and designates them x and y . Known line segments are designated a , b , c , etc. The germinal idea of a Cartesian coordinate system can be traced back to this work.
In the second book, called On the Nature of Curved Lines , Descartes described two kinds of curves, called by him geometrical and mechanical . Geometrical curves are those which are now described by algebraic equations in two variables, however, Descartes described them kinematically and an essential feature was that all of their points could be obtained by construction from lower order curves. This represented an expansion beyond what was permitted by straightedge and compass constructions. [ 5 ] Other curves like the quadratrix and spiral , where only some of whose points could be constructed, were termed mechanical and were not considered suitable for mathematical study. Descartes also devised an algebraic method for finding the normal at any point of a curve whose equation is known. The construction of the tangents to the curve then easily follows and Descartes applied this algebraic procedure for finding tangents to several curves.
The third book, On the Construction of Solid and Supersolid Problems , is more properly algebraic than geometric and concerns the nature of equations and how they may be solved. He recommends that all terms of an equation be placed on one side and set equal to 0 to facilitate solution. He points out the factor theorem for polynomials and gives an intuitive proof that a polynomial of degree n has n roots. He systematically discussed negative and imaginary roots [ 6 ] of equations and explicitly used what is now known as Descartes' rule of signs .
Descartes wrote La Géométrie in French rather than the language used for most scholarly publication at the time, Latin. His exposition style was far from clear, the material was not arranged in a systematic manner and he generally only gave indications of proofs, leaving many of the details to the reader. [ 7 ] His attitude toward writing is indicated by statements such as "I did not undertake to say everything," or "It already wearies me to write so much about it," that occur frequently. Descartes justifies his omissions and obscurities with the remark that much was deliberately omitted "in order to give others the pleasure of discovering [it] for themselves."
Descartes is often credited with inventing the coordinate plane because he had the relevant concepts in his book, [ 8 ] however, nowhere in La Géométrie does the modern rectangular coordinate system appear. This and other improvements were added by mathematicians who took it upon themselves to clarify and explain Descartes' work.
This enhancement of Descartes' work was primarily carried out by Frans van Schooten , a professor of mathematics at Leiden and his students. Van Schooten published a Latin version of La Géométrie in 1649 and this was followed by three other editions in 1659−1661, 1683 and 1693. The 1659−1661 edition was a two volume work more than twice the length of the original filled with explanations and examples provided by van Schooten and his students. One of these students, Johannes Hudde provided a convenient method for determining double roots of a polynomial, known as Hudde's rule , that had been a difficult procedure in Descartes's method of tangents. These editions established analytic geometry in the seventeenth century. [ 9 ] | https://en.wikipedia.org/wiki/La_Géométrie |
La Sociedad Interamericana de Astronomía en la Cultura (SIAC) is an organization dedicated to bringing together professionals in the field of cultural astronomy in the Americas. [ 1 ]
SIAC was founded during the Symposium on Ethno- and Archaeoastronomy of the International Congress of Americanists (Simposio de etno y arqueoastronomía del Congreso Internacional de Americanistas) in Santiago, Chile, in 2003. During this meeting, present researchers discussed the necessity of having a professional organization in Latin America dedicated to cultural astronomy and founded the organization. In 2011, the organization applied for and received grant funding and began to organize more meetings. [ 2 ]
The main objective of SIAC is to promote the exchange and development of interdisciplinary research concerning astronomical knowledge and practices in their cultural context, as an important contribution to understanding the relationships between human societies and the environment in which they are located.
SIAC seeks to promote professional meetings (congresses, workshops, schools) and activities in cultural astronomy in the Americas. As of 2024, the Society has members from 13 countries, most of them in Latin America. Together with SEAC (Société Européenne pour l'Astronomie dans la Culture) and ISSAC (International Society for Archaeoastronomy and Astronomy in Culture), SIAC is one of the three international associations of cultural astronomy.
SIAC also keeps a catalog of publications that includes books, magazines, and newsletters. [ 1 ] | https://en.wikipedia.org/wiki/La_Sociedad_Interamericana_de_Astronomía_en_la_Cultura |
A lab-on-a-chip ( LOC ) is a device that integrates one or several laboratory functions on a single integrated circuit (commonly called a "chip") of only millimeters to a few square centimeters to achieve automation and high-throughput screening . [ 1 ] LOCs can handle extremely small fluid volumes down to less than pico-liters . Lab-on-a-chip devices are a subset of microelectromechanical systems (MEMS) devices and sometimes called "micro total analysis systems " (μTAS). LOCs may use microfluidics , the physics, manipulation and study of minute amounts of fluids. However, strictly regarded "lab-on-a-chip" indicates generally the scaling of single or multiple lab processes down to chip-format, whereas "μTAS" is dedicated to the integration of the total sequence of lab processes to perform chemical analysis.
After the invention of microtechnology (≈1954) for realizing integrated semiconductor structures for microelectronic chips, these lithography -based technologies were soon applied in pressure sensor manufacturing (1966) as well. Due to further development of these usually CMOS -compatibility limited processes, a tool box became available to create micrometre or sub-micrometre sized mechanical structures in silicon wafers as well: the microelectromechanical systems (MEMS) era had started.
Next to pressure sensors, airbag sensors and other mechanically movable structures, fluid handling devices were developed. Examples are: channels (capillary connections), mixers, valves, pumps and dosing devices. The first LOC analysis system was a gas chromatograph , developed in 1979 by S.C. Terry at Stanford University. [ 2 ] [ 3 ] However, only at the end of the 1980s and beginning of the 1990s did the LOC research start to seriously grow as a few research groups in Europe developed micropumps, flowsensors and the concepts for integrated fluid treatments for analysis systems. [ 4 ] These μTAS concepts demonstrated that integration of pre-treatment steps, usually done at lab-scale, could extend the simple sensor functionality towards a complete laboratory analysis, including additional cleaning and separation steps.
A big boost in research and commercial interest came in the mid-1990s, when μTAS technologies turned out to provide interesting tooling for genomics applications, like capillary electrophoresis and DNA microarrays . A big boost in research support also came from the military, especially from DARPA (Defense Advanced Research Projects Agency), for their interest in portable systems to aid in the detection of biological and chemical warfare agents. The added value was not only limited to integration of lab processes for analysis but also the characteristic possibilities of individual components and the application to other, non-analysis, lab processes. Hence the term "lab-on-a-chip" was introduced.
Although the application of LOCs is still novel and modest, a growing interest of companies and applied research groups is observed in different fields such as chemical analysis, environmental monitoring, medical diagnostics and cellomics , but also in synthetic chemistry such as rapid screening and microreactors for pharmaceutics. Besides further application developments, research in LOC systems is expected to extend towards downscaling of fluid handling structures as well, by using nanotechnology . Sub-micrometre and nano-sized channels, DNA labyrinths, single cell detection and analysis, [ 5 ] and nano-sensors, might become feasible, allowing new ways of interaction with biological species and large molecules. Many books have been written that cover various aspects of these devices, including the fluid transport, [ 6 ] [ 7 ] [ 8 ] system properties, [ 9 ] sensing techniques, [ 10 ] and bioanalytical applications. [ 11 ] [ 12 ]
The size of the global lab on chip market was estimated at US$5,698 million in 2021 and is projected to increase to US$14,772 million by 2030, at a CAGR of 11.5% from 2022 to 2030 [ 13 ]
The basis for most LOC fabrication processes is photolithography . Initially most processes were in silicon, as these well-developed technologies were directly derived from semiconductor fabrication. Because of demands for e.g. specific optical characteristics, bio- or chemical compatibility, lower production costs and faster prototyping, new processes have been developed such as glass, ceramics and metal etching , deposition and bonding, polydimethylsiloxane (PDMS) processing (e.g., soft lithography ), Off-stoichiometry thiol-ene polymers (OSTEmer) processing, thick-film- and stereolithography -based 3D printing [ 14 ] as well as fast replication methods via electroplating , injection molding and embossing . The demand for cheap and easy LOC prototyping resulted in a simple methodology for the fabrication of PDMS microfluidic devices: ESCARGOT (Embedded SCAffold RemovinG Open Technology). [ 15 ] This technique allows for the creation of microfluidic channels, in a single block of PDMS, via a dissolvable scaffold (made by e.g. 3D printing ). [ 16 ] Furthermore, the LOC field more and more exceeds the borders between lithography-based microsystem technology, nanotechnology and precision engineering. Printing is considered as a well-established yet maturing method for rapid prototyping in chip fabrication. [ 17 ]
The development of LOC devices using printed circuit board (PCB) substrates is an interesting alternative due to these differentiating characteristics: commercially available substrates with integrated electronics, sensors and actuators; disposable devices at low cost, and very high potential of commercialization. [ 18 ] These devices are known as Lab-on-PCBs (LOPCBs). [ 19 ] The following are some of the advantages of PCB technology:
a) PCB-based circuit design offers great flexibility and can be tailored to specific demands. [ 20 ] b) PCB technology enables the integration of electronic and sensing modules on the same platform, reducing device size while maintaining accuracy of detection.
c) The standardized and established PCB manufacturing process allows for cost-effective large-scale production of PCB-based detection devices.
d) The growth of flexible PCB technology has driven the development of wearable detection devices. As a result, over the past decade, there have been numerous reports on the application of Lab-on-PCB to various biomedical fields, including the fastest SARS-CoV-2 molecular diagnostic test. [ 21 ] e) PCBs are compatible with wet deposition methods, to allow for the fabrication of sensors using novel nanomaterials (e.g. graphene). [ 22 ] [ 23 ]
LOCs may provide advantages, which are specific to their application. Typical advantages [ 10 ] are:
The most prominent disadvantages [ 26 ] of labs-on-chip are:
Lab-on-a-chip technology may soon become an important part of efforts to improve global health , [ 29 ] particularly through the development of point-of-care testing devices. [ 30 ] In countries with few healthcare resources, infectious diseases that would be treatable in a developed nation are often deadly. In some cases, poor healthcare clinics have the drugs to treat a certain illness but lack the diagnostic tools to identify patients who should receive the drugs. Many researchers believe that LOC technology may be the key to powerful new diagnostic instruments. The goal of these researchers is to create microfluidic chips that will allow healthcare providers in poorly equipped clinics to perform diagnostic tests such as microbiological culture assays , immunoassays and nucleic acid assays with no laboratory support.
For the chips to be used in areas with limited resources, many challenges must be overcome. In developed nations, the most highly valued traits for diagnostic tools include speed, sensitivity, and specificity; but in countries where the healthcare infrastructure is less well developed, attributes such as ease of use and shelf life must also be considered. The reagents that come with the chip, for example, must be designed so that they remain effective for months even if the chip is not kept in a climate controlled environment. Chip designers must also keep cost , scalability , and recyclability in mind as they choose what materials and fabrication techniques to use.
One of the most prominent and well known LOC devices to reach the market is the at home pregnancy test kit, a device that utilizes paper-based microfluidics technology.
Another active area of LOC research involves ways to diagnose and manage common infectious diseases caused by bacteria , e.g. bacteriuria , or viruses , e.g. influenza . A gold standard for diagnosing bacteriuria ( urinary tract infections ) is microbial culture . A recent study based on lab-on-a-chip technology, Digital Dipstick, [ 31 ] miniaturised microbiological culture into a dipstick format and enabled it to be used at the point-of-care . Lab-on-a-chip technology can also be useful for the diagnosis and management of viral infections. In 2023, researchers developed a working prototype of an RT-LAMP lab-on-a-chip system called LoCKAmp, which provided results for SARS-CoV-2 tests within three minutes. [ 32 ] [ 33 ] Managing HIV infections is another area where lab-on-a-chips may be useful. Around 36.9 million people are infected with HIV in the world today, and 59% of these people receive anti-retroviral treatment. Only 75% of people living with HIV knew their status. [ 34 ] Measuring the number of CD4+ T lymphocytes in a person's blood is an accurate way to determine if a person has HIV and to track the progress of an HIV infection. [ citation needed ] At the moment, flow cytometry is the gold standard for obtaining CD4 counts, but flow cytometry is a complicated technique that is not available in most developing areas because it requires trained technicians and expensive equipment. Recently such a cytometer was developed for just $5. [ 35 ] Another active area of LOC research is for controlled separation and mixing. In such devices it is possible to quickly diagnose and potentially treat diseases. As mentioned above, a big motivation for development of these is that they can potentially be manufactured at very low cost. [ 24 ] One more area of research that is being looked into with regards to LOC is with home security. Automated monitoring of volatile organic compounds (VOCs) is a desired functionality for LOC. If this application becomes reliable, these micro-devices could be installed on a global scale and notify homeowners of potentially dangerous compounds. [ 36 ]
Lab-on-a-chip devices could be used to characterize pollen tube guidance in Arabidopsis thaliana . Specifically, plant on a chip is a miniaturized device in which pollen tissues and ovules could be incubated for plant sciences studies. [ 37 ] | https://en.wikipedia.org/wiki/Lab-on-a-chip |
LabKey Server is a software suite available for scientists to integrate , analyze , and share biomedical research data. The platform provides a secure data repository that allows web-based querying, reporting, and collaborating across a range of data sources. Specific scientific applications and workflows can be added on top of the basic platform and leverage a data processing pipeline .
LabKey licenses LabKey Server and its documentation for free under the Apache License . [ 2 ]
The base platform is written in Java . It can be extended through the addition of Java-based modules or simple, file-based modules written in HTML , XML and JavaScript . [ 3 ] The platform can also be extended using LabKey Server's Java , JavaScript , R , Python , Perl and SAS client libraries. [ 4 ]
LabKey Server, originally known as the Computational Proteomics Analysis System (CPAS), was developed at the Fred Hutchinson Cancer Research Center to manage high volumes of data generated at the Fred Hutch Computational Proteomics Lab. In 2005, a small team spun out of the Hutch and began operating independently as LabKey Software after contributors realized that the software could be beneficial to the broader scientific community. [ 5 ] [ 6 ] [ 7 ]
LabKey Server provides a secure data repository for all types of biomedical data, including mass spectrometry , flow cytometry , microarray , microplate , ELISpot , ELISA , NAb and observational study information. A customizable data processing pipeline allows the upload and processing of the large data files common in biomedical research.
The platform also provides domain-specific support for several areas of research, including:
In 2016, LabKey and Professor Dave O'Connor of the University of Wisconsin–Madison launched the Zika Open Research Portal [1] using LabKey Server. The portal provides direct access to experiment data being produced by members of the Zika Experimental Science Team (ZEST). The portal received attention from the scientific community for being the first platform of its kind to share real-time research data. [ 8 ] [ 9 ]
Labkey is licensed in a variety of manners. Source-code is provided for a core set of features with the Community Edition, and there are also Premium Editions available. [ 10 ]
Users range from individual labs to large research consortia. In 2017, the program's users included the following: [ 11 ] | https://en.wikipedia.org/wiki/LabKey_Server |
LabLynx, Inc. is a privately held American laboratory informatics company that develops, supports, and markets laboratory information management system (LIMS) solutions. Its primary offerings over the years have included webLIMS and ELab. [ 3 ] The company’s primary clients include laboratories in the agriculture, clinical, environmental, forensics , health care, and manufacturing industries, including government agencies. [ 4 ] [ 5 ] [ 6 ] [ 7 ] The company is known for introducing one of the first browser-based LIMS products in 1997 [ 8 ] [ 9 ] and being in the laboratory informatics industry for decades. [ 10 ] [ 11 ]
Before LabLynx was a company, it was a LIMS product offered by Atlanta Systems Consultants, Inc. (ASC). [ 12 ] [ 13 ] Formed in 1992, [ 10 ] ASC’s LabLynx division later began work on a laboratory information management system designed specifically for a web browser . ASC demonstrated its new Internet Explorer -based LabLynx LIMS at Pittcon in 1997, among the first browser-based LIMS to appear at the time. [ 8 ] [ 9 ] The company again showcased LabLynx at Pittcon in 1998 [ 14 ] and soon after picked up a major LIMS-based contract with the U.S. Customs Service . [ 15 ]
By July 2000, the LabLynx division of ASC separated to become its own incorporated entity. [ 1 ] And while ASC eventually ceased to exist in 2005, [ 10 ] LabLynx, Inc. went on to diversify its offerings. The LabLynx’s browser-based LIMS previously demonstrated at Pittcon in 1997 expanded to become ELab, which in 2001 took on an application service provider (ASP) model of distribution. [ 16 ] In 2004 LabLynx released a browser-based tool called openLIMS, which gave consultants and end-users the ability "to build custom LIMS solutions that are geared to the exact operational needs of many different laboratories." [ 17 ]
On June 19, 2006, LabLynx established the Laboratory Informatics Institute, an open membership group with the purpose of advancing the field of laboratory informatics and shaping the standards associated with it. [ 18 ]
In 2011, LabLynx was involved in an initiative to standardize and structure the transmission of laboratory data that first originates in a LIMS or LIS and then moves to a person's or population of people's electronic health records . [ 19 ] This laboratory results interface (LRI) pilot began in August 2011 and included collaborations with the supported open source project mdDigest and the U.S. Office of the National Coordinator for Health Information Technology (ONC). [ 20 ]
In February 2015, LabLynx released HealthCloudPOL, a cloud-based laboratory information system (LIS) for the physician office laboratory (POL). [ 21 ]
In July 2019, LabLynx released its CannaQA LIMS software for the cannabis testing industry. [ 22 ] The solution's name changed to "ELab LIMS for Cannabis Testing" by 2022.
In April 2020, LabLynx announced it had modified its existing ELab LIMS platform to effectively manage COVID-19 workflows. [ 23 ]
Since transitioning from Atlanta Systems Consultants, Inc. to LabLynx, Inc. in 2000, LabLynx has become increasingly active in the laboratory informatics community. Projects that LabLynx has started or been involved in within the community (past and present) include:
LabLynx products include:
Prior products include: | https://en.wikipedia.org/wiki/LabLynx |
LabVantage Solutions, Inc. is a laboratory information management system (LIMS) provider based in Somerset , New Jersey . Founded in 1981, [ 1 ] LabVantage is the third largest LIMS provider in the world. [ 2 ]
Laboratory MicroSystems was founded by Mark Chudzicki and Michael Boskin in 1981. [ 3 ] Chudzicki started the company when he was attending graduate school at Rensselaer Polytechnic Institute . [ 4 ] The company began making money several years after it was founded but needed a $100,000 loan from New York state's Corporation for Innovation Development. [ 4 ] [ 5 ] Laboratory MicroSystems received financing from 100 Capital District shareholders in 1985. [ 6 ] The company in 1986 was based in Hendrik Hudson Hotel , a downtown Troy hotel that was refurbished into an office building, employed 15 people, and had annual sales of $1 million. [ 4 ]
It was named to the Inc. 500 in 1987 after a 532% increase in sales in its first five years in business. [ 7 ] The company in 1988 primarily served Fortune 500 companies including General Electric , Dow Corning , Pennzoil , DuPont , Monsanto , and Exxon . [ 6 ] It set up software that cost between $10,000 and $70,000 in 1990. [ 8 ] Chudzicki and Boskin in 1990 sold the company, which had 12 employees at the time, to Instron . [ 3 ] The offer was for about $2.5 million, half to be paid in Instron stock and half to be paid in cash. [ 8 ] The acquisition was finalized at $2.42 million to be distributed among Laboratory MicroSystems' 100 shareholders. [ 9 ]
In 1997, Instron sold Laboratory MicroSystems to Axiom Systems, a subsidiary of Purnendu Chatterjee 's The Chatterjee Group, which renamed the company to LabVantage. [ 3 ] [ 10 ] Strategic Directions International said in a 2000 report, "by the late 1990s, the company appeared to have lost its way" because the numerous products LabVantage had to maintain caused it to be "overwhelmed" which hurt its client contentment. [ 10 ] The report further noted that the company's sales failed to increase meaningfully in the past few years. [ 10 ]
In 2005, about half of the company's employees work in India , while 60 employees are in North America and 20 are in Europe . [ 11 ]
LabVantage's customers in the United States include Aventis , Pfizer , and Unilever's Best Foods (now called Hellmann's and Best Foods ). [ 2 ] In India, LabVantage provides services for GAIL , Indian Oil Corporation , and Reliance Industries . [ 12 ] | https://en.wikipedia.org/wiki/LabVantage |
Lab on a Chip is a peer-reviewed scientific journal which publishes original (primary) research and review articles on any aspect of miniaturisation at the micro and nano scale. Lab on a Chip is published twice monthly by the Royal Society of Chemistry (RSC) and the editor-in-chief is Aaron Wheeler ( University of Toronto ). The journal was established in 2001 and hosts other RSC publications: Highlights in Chemical Technology and Highlights in Chemical Biology . According to the Journal Citation Reports , the journal has a 2023 impact factor of 6.1. [ 1 ] [ 2 ]
Lab on a Chip publishes research at the micro- and nano-scale across a variety of disciplines including chemistry , biology , bioengineering , physics , electronics , clinical science , chemical engineering , and materials science focusing on lab on a chip technologies.
Lab on a Chip publishes full research papers, urgent communications, critical and tutorial reviews.
Chips & Tips is an online resource launched in 2006. It is moderated by David Beebe ( University of Wisconsin–Madison ) and Glenn Walker ( North Carolina State University and University of North Carolina at Chapel Hill ). Chips & Tips pages are brief practical tips for the miniaturisation community, including pictures, videos, and schematics. | https://en.wikipedia.org/wiki/Lab_on_a_Chip_(journal) |
Labcorp Holdings Inc. , operating under the brand name Labcorp , headquartered in Burlington, North Carolina , provides laboratory services used for diagnosis and healthcare decisions. [ 1 ] It operates one of the largest clinical laboratory networks in the world and has operations in over 100 countries; although its operations are primarily in the U.S. [ 1 ]
Its Diagnostics Laboratories segment operates 2,000 patient service centers with more than 6,000 in-office phlebotomists in the United States. In addition to healthcare testing such as oncology testing, human immunodeficiency virus (HIV) genotyping and phenotyping , it provides testing for: employment, DNA testing to determine parentage and to determine immigration eligibility, environmental issues, wellness, toxicology , pain management , and medical drug monitoring. It also provides 50 tests that patients can complete at home. It processes over 160 million tests per year. Approximately 10% of this segment’s revenue are from the U.S. Medicare health insurance program. [ 1 ]
Its Biopharma Laboratory Services segment provides drug development , medical device and diagnostic development services to pharmaceutical, biotechnology, medical device, and diagnostic companies. [ 1 ] In 2023, this division provided support to 84% of the new drugs and therapeutic products approved by the Food and Drug Administration . [ 1 ]
Labcorp performs its largest volume of specialty testing at its Center for Esoteric Testing in Burlington, North Carolina. [ 2 ]
Labcorp was an early pioneer of genomic testing using polymerase chain reaction (PCR) technology at its Center for Molecular Biology and Pathology in Research Triangle Park , North Carolina , where it also performs other molecular diagnostics. Labcorp operates the National Genetics Institute, Inc. (NGI), in Los Angeles, California , which develops PCR testing methods. [ 3 ]
In 1971, Revlon acquired DCL BioMedical, a clinical laboratory business founded in 1968. In 1974, it changed its name to National Health Laboratories Incorporated. By 1977, it operated clinical testing laboratories in 13 cities and maintained auxiliary service centers and satellite laboratories in 15 other cities. In 1978, it acquired American Biomedical Corporation, giving it operations in the Southwestern United States and data processing technology. In 1985, Revlon was acquired by Ronald Perelman . Revlon divested its other businesses and a major stake in the company was acquired by MacAndrews & Forbes . [ 4 ]
In 1988, National Health Laboratories became a public company via an initial public offering on the NASDAQ exchange. [ 5 ]
In 1989, the company generated revenue of about US$400 million, with about US$70 million in earnings. [ 6 ]
In the early 1990s, worries about malpractice lawsuits led doctors to conduct more clinical testing before diagnosing, which increased business for the company. [ 4 ]
In 1990, the company's revenues reached US$500 million, with over US$70 million in earnings. [ 6 ]
In 1991, National Health Laboratories moved its listing from the NASDAQ to the New York Stock Exchange . [ 6 ]
In June 1992, the company offered to acquire Damon Corporation for $260 million in cash and stock. [ 7 ] However, the company was outbid by Corning Inc. , which acquired Damon for $370 million. [ 8 ]
By 1993, the company had 22 major laboratories. [ 9 ]
On March 8, 1994, National Health Laboratories Inc. reorganized as a holding company , National Health Laboratories Holdings Inc.
In 1994, National Health Laboratories acquired Allied Clinical Laboratories. [ 10 ] The acquisition price was reduced to $204 million after federal officials issued subpoenas in an investigation of Medicare billing practices. [ 11 ] [ 12 ] [ 13 ]
In April 1995, Hoffmann-La Roche , a division of Roche , contributed Roche Biomedical Laboratories, Inc. and US$186.7 million in cash to National Health Laboratories Holdings, in exchange for 49.9% of the combined company. [ 14 ] Perelman received about US$100 million from the deal, which made the new company the largest blood-testing company in the United States. [ 9 ] [ 10 ] [ 15 ] [ 16 ] The company changed its name to Laboratory Corporation of America Holdings and relocated its headquarters to Burlington, North Carolina. [ 9 ]
In July 1998, Labcorp acquired the Michigan-based laboratory division of Universal Standard Healthcare (UHCI) and made an equity investment in the company. Labcorp also became UHCI's clinical laboratory long-term testing provider [ 17 ] but terminated this agreement in March 1999. [ 18 ]
In June 2000, Labcorp acquired the laboratory testing business of Pathology Medical Laboratories. [ 19 ] [ 20 ]
In May 2001, Labcorp acquired Path Lab Holdings, the largest regional laboratory in New England . [ 21 ]
In June 2001, it acquired ViroMed, which specialized on clinical diagnostic testing in virology , molecular biology , serology , microbiology , mycology and mycobacteriology, as well as in tissue and eye bank testing. [ 22 ] In 2013, it closed the Viromed facility in Minnetonka, Minnesota and laid off 79 workers. [ 23 ]
In December 2001, Labcorp became the exclusive marketer for genomics and proteomics tests for breast cancer , colon cancer , melanoma , and hypertension made by Myriad Genetics . [ 24 ]
In March 2002, Roche sold its remaining interest in the company. [ 25 ]
In May 2002, Labcorp acquired Dynacare , a Canadian medical laboratory services company, for $480 million. [ 26 ] [ 27 ]
In January 2003, Labcorp acquired Dianon, a provider of oncology and genomic diagnostic testing services, for $598 million in cash. [ 28 ] [ 29 ] [ 30 ]
In February 2005, the company acquired US Pathology Labs Inc., a provider of anatomical pathology and oncology testing services, for $155 million. [ 20 ] [ 31 ]
In March 2005, Labcorp acquired Esoterix, a provider of specialty reference testing, for approximately $150 million in cash from Behrman Capital. [ 32 ] [ 33 ]
In November 2006, Labcorp acquired Litholink, a kidney stone analysis laboratory. [ 33 ]
In January 2008, Labcorp acquired Tandem Labs, a contract research organization specializing in advanced mass spectrometry , immunoanalytical support, pharmacokinetics , and pharmacodynamics . [ 34 ] [ 33 ]
In June 2009, Labcorp acquired Monogram Biosciences , a diagnostic lab specializing in HIV resistance testing, for approximately $155 million including debt. [ 35 ]
In September 2010, Labcorp acquired Genzyme Genetics, formerly a division of Genzyme Corporation , with 9 testing laboratories and approximately 1,900 employees, for $925 million in cash. [ 36 ]
In February 2011, Labcorp acquired the assets of bankrupt Westcliff Medical Laboratory. The FTC challenged the acquisition but lost in court. [ 37 ] [ 38 ] [ 39 ]
In June 2011, Labcorp acquired Canadian central labs partner Clearstone from Czura Thornton, adding laboratories in China, France, Singapore and Canada, and the central laboratory protocol management system APOLLO CLPM. [ 40 ]
In November 2011, Labcorp acquired more than 90% of the shares of DNA testing company Orchid Cellmark for $85 million. To receive approval of the transaction from the Federal Trade Commission , Labcorp sold parts of Orchid's DNA paternity testing business to DNA Diagnostics Center. [ 41 ] [ 42 ]
In May 2012, Labcorp Clinical Trials sold its European biological sampling kit building operation located in Hamburg to Marken. [ 43 ]
In August 2012, Labcorp acquired testing lab Medtox Scientific for $241 million. [ 44 ]
In September 2013, Labcorp acquired MuirLab, the clinical laboratory outreach business of John Muir Health . [ 45 ]
In November 2014, Labcorp acquired LipoScience, a developer of diagnostic tests based on nuclear magnetic resonance spectroscopy measuring heart disease risk, for $85 million. [ 46 ] [ 47 ] [ 48 ]
In December 2014, Labcorp acquired Bode Technology Group, a provider of forensic DNA analysis, DNA collection products, and relationship testing and the largest DNA forensic testing company in the U.S., from SolutionPoint International. [ 49 ] [ 50 ] [ 51 ]
In February 2015, Labcorp acquired Covance for $6.1 billion. [ 52 ] [ 53 ]
In October 2015, Labcorp acquired Safe Foods International Holdings and its two operating companies, International Food Network and The National Food Laboratory, expanding its capabilities in food and beverage product-development and product-integrity. [ 54 ]
In March 2016, Labcorp acquired Pathology Inc., a provider of reproductive donor testing as well as anatomic, molecular and digital pathology services focused on women's health . [ 55 ]
In September 2016, Labcorp acquired Sequenom for $371 million including debt, expanding its operations in Europe and Asia. [ 56 ] [ 57 ]
In October 2016, Labcorp acquired ClearPath Diagnostics, a provider of laboratory diagnostic services in the Northeastern United States , from Shore Capital Partners. [ 58 ] [ 59 ]
In May 2017, Labcorp acquired Pathology Associates Medical Laboratories from Providence Health & Services and Catholic Health Initiatives. [ 60 ]
In September 2017, Labcorp acquired Chiltern, a contract research organization (CRO), for $1.2 billion, and contributed it to Covance. [ 61 ] [ 62 ]
In 2019, Labcorp acquired Wellness Corporate Solutions , a workplace wellness company. [ 63 ]
During the COVID-19 pandemic , the company was one of the major developers and processors of COVID-19 testing . In March 2020, Labcorp received emergency use authorization from the FDA for a test for SARS-CoV-2 . [ 64 ] In April 2020, the company developed the first COVID-19 test in which people were able to collect a sample at home. [ 65 ] In July 2020, the company was processing 165,000 COVID-19 tests per day. [ 66 ]
In December 2021, the company acquired Toxikon, a contract research organization developing non-clinical testing services. [ 67 ] [ 68 ] [ 69 ]
In February 2022, the company acquired Personal Genome Diagnostics and its liquid biopsy and tissue-based genomic product, for $450 million in cash plus a possible earn-out of an additional $125 million. [ 70 ] [ 71 ] [ 72 ]
Also in February 2022, Labcorp entered into agreements with Ascension, one of the nation’s largest Catholic and nonprofit health systems, to manage Ascension's hospital-based laboratories in ten states and purchase assets of the health system's outreach laboratory business. [ 73 ] [ 74 ] [ 75 ]
In June 2023, Labcorp completed the corporate spin-off of Fortrea . [ 76 ] [ 77 ]
In August 2024, Labcorp acquired the assets of bankrupt Invitae for $234 million. [ 78 ]
In 1992, the company was one of the first to be prosecuted as part of Operation Labscam , a nationwide crackdown on fraud in the healthcare system initiated by the U.S. Attorney's Office in San Diego, California . The company and others were accused of routinely submitting false claims to Medicare and Medicaid for unnecessary tests on blood samples that physicians had never ordered. In addition, charges for the tests billed to the government were significantly higher than what the company charged private insurers. [ 79 ] That year, the company agreed to pay $111 million to settle claims. [ 80 ] In November 1996, the company agreed to pay a total of $182 million as a result of the investigation. [ 81 ]
In February 2023, the company paid $19 million to resolve allegations that it violated the False Claims Act of 1863 by its submission of false claims to Medicare. [ 82 ]
In 2012, Labcorp was criticized for its practice of paying the salaries of genetic counselors in hospitals and doctors' offices, which is perceived to be a possible conflict of interest . [ 83 ]
In July 2018, Labcorp's servers were affected by a variant of the SamSam ransomware, with the goal of locking or disabling computers and servers to prevent access to data. The attack resulted in delays, but no data was believed to have been stolen. The malware was blocked on the network by midnight on July 14, 2018. [ 84 ] [ 85 ] [ 86 ]
Labcorp has been criticized over faulty paternity tests, many of which have resulted in lawsuits. The most notable case was the 2005 false accusation of Washington hairdresser Andre Chreky, who spent $200,000 and years in court proving, despite a false-positive test, that he was not the father. In the trial, it was determined that only five people at Labcorp reviewed data and made paternity determinations, including during 10 hour shifts that required making determinations every 4 minutes. [ 87 ]
In 2024, Labcorp faced a class action lawsuit for allowing Google to collect confidential patient information such as appointments made and website login details. [ 88 ]
Employees have criticized the company's productivity goals, in which they say volume and speed of work done is prioritized over quality. Labs face staffing shortages and employees say they are overworked. [ 89 ]
Labcorp has been criticized for its process of having all employees terminated upon hospital laboratory management acquisitions, and making them reapply with LabCorp, thereby losing accrued employee benefits , including employee eligibility for Public Service Loan Forgiveness (PSLF) in the case of non-profit hospitals. [ 90 ]
In 2024, Labcorp stated that its goal is to "operate the company in a non-union environment." [ 91 ] After Labcorp became the manager of laboratories of Legacy Health, outsourced laboratory personnel unionized with the Oregon Federation of Nurses and Health Professionals. [ 92 ]
Labcorp ( ICAO : SKQ , call sign : SKYLAB ) utilizes a fleet of 12 Pilatus PC-12 and a single Pilatus PC-24 aircraft on nightly runs from Burlington for use on the East Coast. [ 93 ] Prior to the acquisition of PC-12 aircraft Labcorp utilized seven PA-31-350s . | https://en.wikipedia.org/wiki/Labcorp |
Label dispenser and label applicator are machine built to simplify the process of removing a label from its liner or backing tape. Some are bench-top for dispensing the labels while others include the application of the label to the item (such as a package). Unlike label printer applicators , they dispense preprinted labels.
Label dispensers are generally intended to dispense a label to an operator who manually applies the label to the package. They are designed with varying sizes and features which are often specific to the type of label they can dispense and to the degree of automation desired.
Label applicators are usually part of a larger packaging line. They receive the package from a previous automation stage, apply the label, and feed the package to the next stage in the packaging line.
Many everyday items have stick-on labels, which were either applied by a machine or by hand, and most likely were peeled from the backing paper using some sort of label dispenser. Some of the more common applications are bulk mailing , manufacturing , packaging , food and beverage , fast food , and photo labs.
Patents on manual label dispensers go back to the 1920s. They are designed for light-duty use. They are operated by hand and are not automated, but still assist in the process of removing labels from their liners. Many manual label dispensers can dispense multiple rolls at once. Operation is performed by pulling the liner/backing paper around a plate or bar which causes the label to peel away from the backing paper. This happens because the backing paper is usually thinner than the label itself and is also underneath. When the liner is forced around a tight radius the label lifts away and protrudes through the front or top of the dispenser. [ 1 ]
Electric semi-automatic label dispensers were first patented in the early 1970s. They were originally designed for multiple-row address labels for bulk mailing houses. On average a good mailing house employee could apply approximately 500 labels per hour to envelopes. The label dispenser increased this to over 2,000 per hour. These dispensers advance individual or multiple-row labels and remove them from their lining similar to a manual dispenser, but instead of manually pulling on the liner, label advancement occurs when a trigger on the dispenser detects the absence of a label, such as when the operator removes the label. The sensor then closes the circuit and engages the motor, dispensing the next label until the sensor once again detects the label which opens the circuit.
The first electric dispenser was designed with the limit switch on the left of a 16" wide machine. 4-up multiple-row labels were loaded into the machine and once activated would advance one row of labels. The operator would take the labels from right to left, so that when the leftmost label was taken, the next row advanced, automatically providing a constant supply of labels to apply. The labels are also peeled without the natural curl that will happen when pulled from the backing paper with fingers. Also, only one hand was needed to take the label, the other hand could be used to move the material the label was being applied to.
Label applicators are fully automated and can range from simple slower speed models to large machines capable of applying hundreds or even thousands of labels per minute. [ 2 ]
Applicators advance the label stock over the peeler plate until a portion of the label, called the "flag," is extended into the path of the oncoming package. When the package engages the label flag, the label web is advanced to match the speed of the package and label is either tamped or wiped on to assure adhesion.
Proper alignment of the label on the package depends on sensors that sense the location/orientation of the package and label sensors that detect the location of the label edge. Package sensors can be a variety of Position sensors , often optical sensors or Ultrasonic sensors . Label sensors are usually Photoelectric sensors because they are inexpensive. But clear labels cannot be detected by photoelectric sensors. Capacitive and ultrasonic technologies are used for clear label detection. [ 3 ] [ 4 ] | https://en.wikipedia.org/wiki/Label_dispenser |
A label printer is a computer printer that prints on self- adhesive label material and/or card-stock (tags). A label printer with built-in keyboard and display for stand-alone use (not connected to a separate computer) is often called a label maker . Label printers are different from ordinary printers because they need to have special feed mechanisms to handle rolled stock, or tear sheet (fanfold) stock. Common connectivity for label printers include RS-232 serial , Universal Serial Bus (USB), parallel , Ethernet and various kinds of wireless . Label printers have a wide variety of applications, including supply chain management , retail price marking, packaging labels, blood and laboratory specimen marking, and fixed assets management . [ 1 ]
Label printers use a wide range of label materials, including paper and synthetic polymer ("plastic") materials. Several types of print mechanisms are also used, including laser and impact, but thermal printer mechanisms are perhaps the most common.
There are two common types of thermal printer. [ 2 ]
There are three grades of ribbon for use with thermal transfer printers. Wax is the most popular with some smudge resistance, and is suitable for matte and semi-gloss paper labels. Wax/resin is smudge resistant, suitable for semi-gloss paper and some synthetic labels. Resin alone is scratch and chemical resistant, suitable for coated synthetic labels.
When printing on continuous label stock, there is a tendency for the print location to shift slightly from label to label. To ensure registration of the print area with the target media, many label printers use a sensor that detects a gap, notch, line or perforation between labels. This allows the printer to adjust the intake of label stock so that the print aligns correctly with the media.
Label printer capabilities vary between home, corporate and industrial-oriented models. | https://en.wikipedia.org/wiki/Label_printer |
A label printer applicator is a basic robot that can automatically print and apply pressure-sensitive labels to various products. Some types of labeling include shipping labeling, content labeling, graphic images, and labeling to comply with specific standards such as those of GS1 and Universal Product Code U.P.C. A pressure-sensitive label consists of a label substrate and adhesive.
First developed in the late 1970s, today there are over 70 manufacturers of these types of machines worldwide.
Basic label printer applicators consist of three primary parts: a printer, or print engine, an applicator and a method to handle label and ribbons, referred to as media. Computing power also has the potential to increase the efficiency of label printer applicators.
The print engine can be taken from an industrial table top printer, it can be a specifically designed module that can be "bolted" onto an applicator or it can be a proprietary element constructed by the printer applicator manufacturer. A print engine’s primary function is to accept data from a computer and print the data onto a label for application. This printing can be accomplished using either the direct thermal method or the thermal transfer method. Both methods heat up very fine elements (up to 600 per inch) on a print head. Direct thermal burns the image onto the face of specially designed label stock. This is the preferred method for shipping labels and is also very popular in Europe. The thermal transfer process utilizes a ribbon coated with wax, resin, or a hybrid of the two. It is then heated and melted onto the surface of the label substrate. Thermal transfer is the most popular method in the United States. The printer knows what to print via data communication from an outside software package, much like common inkjet printers. The software delivers data formatted in a specific layout and the printer reads the format based on its own driver.
The applicator section delivers the label to the product. This can be accomplished by several methods. Typically application is achieved with a pneumatic or electric cylinder with a specially designed label pad. The cylinder will extend out and touch (tamp) the adhesive side of the label to a product. Variations of this method will extend the cylinder and then use air to blow the label to the product surface (tamp-blow). Another popular method is a blow-on system that will use a burst of air to deliver the label from the pad to the product surface without the use of a cylinder. Other methods can be used to wipe a label onto a surface, or even place two duplicate or unique labels on different sides of a product.
Media handling controls how the label stock is delivered to the print engine. It also performs the separation of the label from its backing and rewinds the waste label backing that remains after label application. This process can be difficult since consistent tension must be maintained for the label to peel off the liner and onto the applicator. Too much tension can cause the liner to break, which requires the machine to be rethreaded.
Today, a fourth element to label printer applicators is emerging: computing power. Recently label printer applicators have been introduced which have the power to store large amounts of data. These machines can also control and harvest data from other input devices such as barcode scanners and weighing scales. These printer applicators can be programmed with special languages such as Fingerprint designed by Intermec for Intermec print engines or MCL (Macro Command Language), a Datamax programming language. Now label printer applicators can communicate directly with an array of devices and hosts on the production line without the aid of a computer. | https://en.wikipedia.org/wiki/Label_printer_applicator |
In combinatorial mathematics, the labelled enumeration theorem is the counterpart of the Pólya enumeration theorem for the labelled case, where we have a set of labelled objects given by an exponential generating function (EGF) g ( z ) which are being distributed into n slots and a permutation group G which permutes the slots, thus creating equivalence classes of configurations. There is a special re-labelling operation that re-labels the objects in the slots, assigning labels from 1 to k , where k is the total number of nodes, i.e. the sum of the number of nodes of the individual objects. The EGF f n ( z ) {\displaystyle f_{n}(z)} of the number of different configurations under this re-labelling process is given by
In particular, if G is the symmetric group of order n (hence, | G | = n !), the functions f n ( z ) {\displaystyle f_{n}(z)} can be further combined into a single generating function :
which is exponential w.r.t. the variable z and ordinary w.r.t. the variable t .
We assume that an object ω {\displaystyle \omega } of size | ω | {\displaystyle |\omega |} represented by z | ω | / | ω | ! {\displaystyle z^{|\omega |}/|\omega |!} contains | ω | = m {\displaystyle |\omega |=m} labelled internal nodes, with the labels going from 1 to m . The action of G on the slots is greatly simplified compared to the unlabelled case, because the labels distinguish the objects in the slots, and the orbits under G all have the same size | G | {\displaystyle |G|} . (The EGF g ( z ) may not include objects of size zero. This is because they are not distinguished by labels and therefore the presence of two or more of such objects creates orbits whose size is less than | G | {\displaystyle |G|} .) As mentioned, the nodes of the objects are re-labelled when they are distributed into the slots. Say an object of size r 1 {\displaystyle r_{1}} goes into the first slot, an object of size r 2 {\displaystyle r_{2}} into the second slot, and so on, and the total size of the configuration is k , so that
The re-labelling process works as follows: choose one of
partitions of the set of k labels into subsets of size r 1 , r 2 , … r n . {\displaystyle r_{1},r_{2},\ldots r_{n}.} Now re-label the internal nodes of each object using the labels from the respective subset, preserving the order of the labels. E.g. if the first object contains four nodes labelled from 1 to 4 and the set of labels chosen for this object is {2, 5, 6, 10}, then node 1 receives the label 2, node 2, the label 5, node 3, the label 6 and node 4, the label 10. In this way the labels on the objects induce a unique labelling using the labels from the subset of [ k ] {\displaystyle [k]} chosen for the object.
It follows from the re-labelling construction that there are
or
different configurations of total size k . The formula evaluates to an integer because [ z k ] g ( z ) n {\displaystyle [z^{k}]g(z)^{n}} is zero for k < n (remember that g does not include objects of size zero) and when k ≥ n {\displaystyle k\geq n} we have n ! | k ! {\displaystyle n!|k!} and the order | G | {\displaystyle |G|} of G divides the order of S n {\displaystyle S_{n}} , which is n ! {\displaystyle n!} , by Lagrange's theorem . The conclusion is that the EGF of the labelled configurations is given by
This formula could also be obtained by enumerating sequences, i.e. the case when the slots are not being permuted, and by using the above argument without the 1 / | G | {\displaystyle 1/|G|} -factor to show that their generating function under re-labelling is given by g ( z ) n {\displaystyle g(z)^{n}} . Finally note that every sequence belongs to an orbit of size | G | {\displaystyle |G|} , hence the generating function of the orbits is given by g ( z ) n / | G | . {\displaystyle g(z)^{n}/|G|.} | https://en.wikipedia.org/wiki/Labelled_enumeration_theorem |
In cellular biology , labile cells are cells that continuously multiply and divide throughout life . [ 1 ] [ 2 ] Labile cells replace the cells that are lost from the body. [ 1 ] When injured, labile cells are repaired rapidly due to an aggressive TR response. [ 1 ] This continual division of labile cells allows them to reproduce new stem cells and replace functional cells that are lost in the body. [ 1 ] Functional cells may be lost through necrosis , which is the premature death of cells caused by environmental disturbances, such as diseases or injuries. [ 3 ] Functional cells may also need to be replaced after undergoing apoptosis, which is the programmed death of cells that occurs normally as part of an organism's development. [ 3 ] Labile cells continually regenerate by undergoing mitosis and are one of three types of cells that are involved in cell division, classified by their regenerative capacity. [ citation needed ] The other two cell types include stable cells and permanent cells. Each of these three cell types respond to injuries to their corresponding tissues differently. Stable cells, unlike labile cells, are typically not dividing and only do so when an injury occurs. [ 4 ] Permanent cells are not capable of division after maturing. [ 5 ]
Some examples of labile cells, which act as stem cells, include skin cells, such as the epidermis , the epithelia of ducts, hematopoietic stem cells, [ 1 ] cells within the gastrointestinal tract , and some cells found within bone marrow . [ 6 ]
Labile cells exhibit a very short G1 phase and never enter G0 phase (the resting phase), as they are continually proliferating throughout their life. [ 6 ]
Cells that are constantly dividing have a higher risk of dividing uncontrollably and becoming malignant, or cancerous. [ 7 ] Muscle tissue does not consist of constantly dividing cells, which is likely why cancer of the muscle is not nearly as common as, for example, cancer of the skin. [ 8 ]
In addition, cytotoxic drugs used in chemotherapy target dividing cells and inhibit their proliferation. [ 9 ] The cytotoxic drugs aim to target the dividing cells which are malignant in the body; however, these drugs target all dividing cells and are not capable of only selecting the cancerous ones. [ 10 ] Healthy cells, that are normally dividing in the body, are targeted and affected as well. [ 10 ] For this reason, adverse effects are often produced from chemotherapy. [ 11 ] The labile cells within epithelial tissue and bone marrow, for example, may be targeted, resulting in possible hair loss or bone marrow suppression. [ citation needed ]
This cell biology article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Labile_cell |
Lability refers to the degree that something is likely to undergo change. It is the opposite ( antonym ) of stability.
In reference to biochemistry , this is an important concept as far as kinetics is concerned in metalloproteins . This can allow for the rapid synthesis and degradation of substrates in biological systems.
Labile cells refer to cells that constantly divide by entering and remaining in the cell cycle . [ 1 ] These are contrasted with "stable cells" and "permanent cells".
An important example of this is in the epithelium of the cornea , where cells divide at the basal level and move upwards, and the topmost cells die and fall off .
In medicine, the term "labile" means susceptible to alteration or destruction. For example, a heat-labile protein is one that can be changed or destroyed at high temperatures.
The opposite of labile in this context is "stable". [ 2 ]
Compounds or materials that are easily transformed (often by biological activity ) are termed labile. For example, labile phosphate is that fraction of soil phosphate that is readily transformed into soluble or plant-available phosphate. [ 3 ] Labile organic matter is the soil organic matter that is easily decomposed by microorganisms . [ 4 ]
The term is used to describe a transient chemical species . As a general example, if a molecule exists in a particular conformation for a short lifetime, before adopting a lower energy conformation (structural arrangement), the former molecular structure is said to have 'high lability' (such as C 25 , a 25-carbon fullerene spheroid). The term is sometimes also used in reference to reactivity – for example, a complex that quickly reaches equilibrium in solution is said to be labile (with respect to that solution). Another common example is the cis effect in organometallic chemistry, which is the labilization of CO ligands in the cis position of octahedral transition metal complexes.
This biochemistry article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Lability |
Labor burden is the actual cost of a company to have an employee, in addition to wages that the employee earns. Labor burden costs include benefits that a company pays for employees that are included on their payroll, including payroll taxes , pension costs, workers compensation , health and dental insurance , and the cost of any other benefits that a company provides an employee. [ 1 ]
Fully-burdened costs for individual employees can be expressed as a yearly total to provide an estimate of how much the company will spend that year on an employee. It can also be expressed as an hourly cost by dividing the total yearly cost by the number of hours the employee will work. [ 2 ]
This labor -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Labor_burden |
A laboratory-acquired infection or LAI is an infection that is acquired in a laboratory, usually as part of a medical research facility or hospital. [ 1 ] [ 2 ] [ 3 ]
There are various microbes, viruses, fungi, and parasites that can infect a host via several routes of transmission. [ 4 ]
Laboratory facilities handling microbes, viruses and/or parasites adhere to various biosecurity measures in order to prevent biosecurity accidents and incidents . [ 5 ] [ 6 ]
In 2001, experts from OECD countries created a consensus report called, calling upon "national governments to undertake actions to bring the BRC concept into being in concert with the international scientific community". The report details "Biological Resource Centres" (BRCs) as "repositories and providers of high-quality biological materials and information". [ 7 ]
The first laboratory-acquired infection was reported at the time of Pasteur and Koch in 1890. [ 3 ]
Prior to 1950, few reports were made on laboratory-acquired infections, due to the lower level of awareness concerning the problem. In 1951, a paper from Sulkin and Pike presented data on viral infections contracted in laboratories, which advised caution on handling viruses in laboratory environments and brought public awareness to the issue. Soon after, the American Public Health Association formed a standing committee on Laboratory Infections and Accidents and created a file to document cases of laboratory-acquired infections reported by the public and through private communications. [ 8 ] | https://en.wikipedia.org/wiki/Laboratory-acquired_infection |
Laboratory B ( Russian : Лаборатория Б ), also known as Object B ( Объект Б ) or Object 2011 during its period of operation, [ 1 ] [ 2 ] [ 3 ] [ 4 ] was a former Soviet nuclear research site constructed in 1946 by Lake Sungul [ ru ] in Chelyabinsk Oblast in Russia . [ citation needed ] Operated under the 9th Chief Directorate of the Soviet Ministry of Internal Affairs , it was a major site for the Soviet program of nuclear weapons that works on handling, treatment, and the use of the radioactive products generated in reactors, as well as radiation biology, dosimetry, and radiochemistry. It had two divisions: radiochemistry and radiobiophysics ; the latter was headed by N. V. Timofeev-Resovskij .
Laboratory B was run as a sharashka —a secret facility run as a prison, with at least ten of its German staff classified as prisoners of war from World War II . For two years, the German chemist Nikolaus Riehl was the scientific director.
It was closed in 1955, and has since been abandoned and left as a ruin.
From early in 1945, Colonel General A. P. Zavenyagin , as head of the 9th Chief Directorate of the NKVD ( MVD after 1946), was responsible for the acquisition of German scientists, equipment, materiel, and intellectual property, under the Russian Alsos , to help Russia with the Soviet atomic bomb project . The issue of Decree No. 9877 from the Council of Ministers on 20 August 1945 created a special committee of which Zavenyagin was a member, [ a ] Zavenyagin was responsible for establishing, building, managing, and providing security for facilities supporting the atomic bomb project. Zavenyagin's purview also included the resources of the Gulag ; some of the facilities to which the German scientists were assigned were run as a sharashka . German scientists were available for recruitment from the Soviet occupation zone in Germany. Also, immediately after World War II and extending into 1949, the Russians also had a large pool of German PoW scientists and highly skilled specialists from which to recruit; the main camp was at Krasnogorsk . [ 6 ] [ 7 ] [ 8 ] [ 9 ] [ 10 ] [ 11 ]
Facilities to which the German scientists were assigned were under the under authority of the 9th Chief Directorate and included the following (with annotations of prominent Germans at the facilities): [ 12 ] [ 13 ] [ 14 ] [ 15 ]
Laboratory B had two scientific divisions, a radiobiophysics division headed by the geneticist N. V. Timofeev-Resovskij (prisoner), and a radiochemistry division headed by Sergej Aleksandrovich Voznesenskij (prisoner). [ 18 ]
In 1925, as the Russian part of a collaborative effort between Russia and Germany, the Russians sent Timofeev-Resovskij, and his colleague Sergei Romanovich Tsarapkin, to Germany. There, they worked with Oskar Vogt , director of the Kaiser-Wilhelm Institut für Hirnforschung (KWIH, Kaiser Wilhelm Institute for Brain Research), [ 19 ] to establish the Abteilung für Experimentelle Genetik (Department of Experimental Genetics) and Timofeev-Resovskij became its director. Timofeev-Resovskij stayed in Germany through World War II, and built his department to world-renowned status. On the basis of false denunciations, Timofeev-Resovskij and Tsarapkin were arrested by the NKVD in September 1945, returned to Russia, and both sentenced to 10 years in the Gulag . They ended up in the Karaganda prison camp in northern Kazakhstan , one of the most terrible camps in the Gulag; the harsh conditions of Timofeev-Resovskij's transportation and incarceration in the labor camp contributed to a significant decline in his health, including the degradation of his vision brought on by malnutrition. Colonel General Zavenyagin, who had intended to utilize Timofeev-Resovskij's talents in the Soviet atomic bomb project, had Timofeev-Resovskij and Tsarapkin sent to Laboratory B in 1947. Timofeev-Resovskij's wife Elena Aleksandrovna, after receipt of a letter in his handwriting, left Berlin in 1948, with their son Andrew, to join him in Sungul'. The house occupied by the three Timofeev-Resovskijs was every bit as nice as that planned for the German scientists working at the Sungul' institute. [ 20 ] [ 21 ] [ 22 ] [ 23 ] [ 24 ] (In 1992, Timofeev-Resovskij was rehabilitated, 11 years after his death! [ 25 ] [ 26 ] )
Born, Catsch, and Zimmer, who had worked for Timofeev-Resovskij in Berlin and who were sent to Laboratory B by Riehl in December 1947, were able to conduct work similar to that which they had done in Germany, and all three became section heads in Timofeev-Resovskij's department. Born examined fission products, developed methods of separating plutonium from fission products created in a nuclear reactor, and investigated and developed radiation health and safety measures. Catsch began his work on developing methods to extract radionucleotides from various organs, which he would continue when he left Russia. [ 27 ] [ 28 ] [ 29 ]
The radiobiophysics division under Timofeev-Resovskij had four sections which conducted experimental studies in four basic directions: [ 18 ]
The agrobiological and hydrobiological experiments were united on the general basis of the biogeochemical analysis of the experimentally created elementary biogeocenosis [ 30 ] and the introduction of special factor radioactive materials into it. [ 18 ]
On the basis of a false denunciation, Sergej Aleksandrovich Voznesenskij was arrested in June 1941; in April 1942, he was sentenced to 10 years in the Gulag. From March 1943 to 1947, he led a research group in the 4th Special Department of the NKVD in Moscow; the 4th Special Department provided military research and development by utilizing specialist prisoners, i.e., scientists. In December 1947, he was transferred to Laboratory B to head up the radiochemistry division. [ 31 ] With the liquidation of Laboratory B and its merger into NII-1011 in 1955, Voznesenskij was transferred to the Ural Polytechnical Institute to head up the Department of Radiochemistry, and was simultaneously appointed as a scientific consultant at Combine No. 817 on problems of radioactive waste cleanup. (Voznesenskij had been fully rehabilitate in May 1953.) [ 32 ] [ 33 ]
The radiochemistry division had four sections and conducted research and development in the following areas: [ 18 ] [ 32 ]
Owing to its proximity to the radiochemical plutonium facility Combine No. 817, [ 34 ] the scientists at the institute had access to high-dose radioactive materials. [ 27 ] [ 18 ]
The scientific staff at Laboratory B – a Sharashka – was both Soviet and German, the former being mostly political prisoners or exiles, although some of the service staff were criminals – one had been convicted of murder. In 1955, the institute had 451 staff members; in 1946 there had been 95. The institute had a maximum of 26 German scientists, and more than 10 of them initially were classified as PoWs. The German contingent left the institute in 1953. The institute had two departments: radiobiophysics (No. 1) and radiochemistry (No. 2). In 1955, the institute was merged into the newly created second [ 35 ] nuclear weapons design institute Nauchno-Issledovatel'skij Institut-1011 (NII-1011). [ 36 ] During the merger, the radiopathology section of the radiochemistry department was transferred to Combine No. 817 ( Ozersk ) and a section of the radiobiophysics department was transferred to the Ural Branch of the USSR Academy of Sciences . [ 8 ] [ 27 ] [ 18 ] [ 28 ] [ 37 ]
Accomplishments of Laboratory B include the development of technology for the isolation of fission by-products such as strontium-90, caesium-137, zirconium-65, and the technology to remove these isotopes from chemical compounds. [ 27 ]
The first director of Laboratory B, starting in 1946, was MVD Colonel Alexander Konstantinovich Uralets, who had previously worked on the Soviet atomic bomb project . He received the Order of Lenin for his management of Laboratory B. [ 38 ]
From 26 December 1952 to 14 June 1955, the director was the chemist Gleb Arkad'evich Sereda. [ 18 ] [ 39 ]
Nikolaus Riehl was the scientific director of Laboratory B from September 1950 to early autumn in 1952. [ 40 ]
Riehl, scientific director of the Auergesellschaft , was sent by the Russians, in 1945, to head a group at Plant No. 12 in Ehlektrostal' to develop an industrial process for production of reactor-grade uranium. Other Germans sent to work there included A. Baroni (PoW), Werner Kirst, Henry E. Ortmann (chemist from Auergesellschaft ), Przybilla, Herbert Schmitz (PoW), Herbert Thieme, Tobein, and Günter Wirths (chemist from Auergesellschaft ). When Riehl learned that professional colleagues from the Kaiser-Wilhelm Institut für Hirnforschung (Kaiser Wilhelm Institute for Brain Research) in Berlin, Hans-Joachim Born and Karl Zimmer , were being held in Krasnogorsk , in the main PoW camp for Germans with scientific degrees, Riehl arranged though Zavenyagin to have them sent to Ehlektrostal'. Alexander Catsch was also sent there. At Ehlektrostal', Riehl had a hard time incorporating Born, Catsch, and Zimmer into his tasking on uranium production, as Born was a radiochemist, Catsch was a physician and radiation biologist, and Zimmer was a physicist and radiation biologist; in December 1947, Riehl sent all three to Laboratory B to work with Timofeev-Resovskij. [ 41 ] [ 42 ] [ 43 ]
After the detonation of the Russian uranium bomb, uranium production was going smoothly and Riehl's oversight was no longer necessary at Plant No. 12. Riehl then went, in 1950, to be the scientific director of Laboratory B, where he stayed until 1952. Essentially the remaining personnel in his Ehlektrostal' group were assigned elsewhere, with the exception of Henry E. Ortmann, A. Baroni (PoW), and Herbert Schmitz (PoW), who went with Riehl to Sungul'. [ 44 ] [ 45 ]
Besides those already mentioned, other Germans at Laboratory were Rinatia von Ardenne (sister of Manfred von Ardenne, director of Institute A, in Sukhumi) Wilhelm Menke (botanist), Willi Lange (who married the widow of Karl-Heinrich Riewe , who had been at Heinz Pose's Laboratory V, in Obninsk ), Joachim Pani, and K. K. Rintelen. Until Riehl's return to Germany in June 1955, which Riehl had to request and negotiate, he was quarantined in Agudzery (Agudseri) starting in 1952; Augudzery, was the location of Institute G. [ 46 ] [ 47 ] | https://en.wikipedia.org/wiki/Laboratory_B |
Laboratory animal allergy (LAA) is an occupational disease of laboratory animal technicians and scientists. [ 1 ] [ 2 ] It manifests as an allergic response to animal urine, specifically the major urinary proteins (Mups) of rodents, and can lead to the development of asthma . [ 3 ] A study of 5641 workers in Japan who were exposed to laboratory animals found 23.1% had one or more allergic symptoms; globally the prevalence among at risk workers is estimated between 11 and 30% [ 4 ] According to the National Institutes of Health , prevention of animal allergy depends on the control of allergens in the work environment. This involves a combination of measures to eliminate or control allergen exposure, including engineering controls , administrative controls , and personal protective equipment . [ 5 ]
The protein product of the mouse Mup17 gene, [ 6 ] known as Mus m 1 , Ag1 or MA1 , [ 7 ] accounts for much of the allergenic properties of mouse urine. Similarly, the product of the rat Mup13 gene, Rat n 1 , is also a potent human allergen . [ 3 ] One study found that two-thirds of laboratory workers who had developed asthmatic reactions to animals had antibodies to Rat n 1. [ 8 ] | https://en.wikipedia.org/wiki/Laboratory_animal_allergy |
Animals used by laboratories for testing purposes are largely supplied by dealers who specialize in selling them to universities, medical and veterinary schools, and companies that provide contract animal-testing services. It is comparatively rare that animals are procured from sources other than specialized dealers, [ 1 ] as this poses the threat of introducing disease into a colony and confounding any data collected. [ 2 ] However, suppliers of laboratory animals may include breeders who supply purpose-bred animals, businesses that trade in wild animals, and dealers who supply animals sourced from pounds, auctions, and newspaper ads. Animal shelters may also supply the laboratories directly. Some animal dealers, termed Class B dealers, have been reported to engage in kidnapping pets from residences or illegally trapping strays, a practice dubbed as bunching . [ 3 ] [ 4 ] [ 5 ]
All laboratories using vertebrate lab animals in the United States are required by law to have a licensed veterinarian on staff and to adhere to the NIH Guide for the Use and Care of Laboratory Animals, which further stipulates that all protocols, including the sources for obtaining the animals, must be reviewed by an independent committee . [ 6 ]
Class A breeders are licensed by the U.S. Department of Agriculture (USDA) to sell animals bred specifically for research. [ 7 ] In July 2004, there were 4,117 licensed Class A dealers in the United States. [ 1 ]
Class B dealers are licensed by the USDA to buy animals from "random sources". This refers to animals who were not purpose-bred or raised on the dealers' property. [ 5 ] Animals from "random sources" come from auctions, pounds, newspaper ads (including "free-to-home" ads), and some may be stolen pets or illegally trapped strays. [ 3 ] As of February 2013, there were only seven active Class B dealers remaining in the United States. However, these sources round up "thousands" of cats and dogs each year for sale. [ 5 ]
Animals are also sold directly to laboratories by shelters. According to the American Society for the Prevention of Cruelty to Animals (ASPCA), Iowa, Minnesota, Oklahoma, South Dakota, and Utah require publicly funded shelters to surrender animals to any Class B dealer who asks for them. [ 8 ] Fourteen states prohibit the practice, [ 9 ] and the remainder either have no relevant legislation, or permit the practice in certain circumstances. [ 8 ] [ 10 ] [ 11 ]
According to a paper presented to the American Society of Criminology in 2006, an illegal economy in the theft of pets, mostly dogs, has emerged in the U.S. in recent years, with the thieves known as "bunchers". The bunchers sell the animals to Class B animal dealers, who pay $25 per animal. The dealers then sell the animals to universities, medical and veterinary schools, and companies providing animal-testing services. Lawrence Salinger and Patricia Teddlie of Arkansas State University told the conference that these institutions pay up to $500 for a stolen animal, who is often accompanied by forged documents and fake health certificates. Salinger and Teddlie argue that the stolen animals may affect research results, because they come from unknown backgrounds and have an uncertain health profile. [ 3 ] Conversely the Foundation for Biomedical Research claim that pets being stolen for animal research is largely an urban myth and that the majority of stolen dogs are most likely used for dog fighting . [ 12 ]
The largest Class B dealer in dogs in the U.S. was investigated for bunching by the U.S. Department of Agriculture (USDA) in 2005. Chester C. Baird, of Martin Creek Kennels and Pat's Pine Tree Farms in Willifore, Arkansas , lost his licence after being convicted of 100 counts of animal abuse and neglect, and of stealing pets for laboratories and forging documentation. The criminal charges were filed after an eight-year investigation by an animal protection group, Last Chance for Animals . The group filmed over 72 hours of undercover video at Martin Creek Kennels, which included footage of dogs being shot. [ 13 ] [ 14 ] In 2006, HBO produced Dealing Dogs , a documentary film based on this footage. [ 15 ] [ 16 ]
Baird's customers included the University of Missouri , University of Colorado Health Sciences Center , and Oregon State University . According to the Humane Society of the United States, Missouri was experiencing such a high rate of pet theft that animal protection groups had dubbed it the "Steal Me State". In a 2008 article, Last Chance for Animals estimated that around two million pets are stolen in the U.S. each year. [ 14 ]
Animal dealers in the European Union (EU) are governed by Council Directive 86/609/EEC . [ 17 ] This directive sets forth specific requirements regulating the supply and breeding of animals intended for use by testing facilities within the EU. The directive defines 'breeding establishment' as a facility engaged in breeding animals for their use in experiments, and 'supplying establishment' as a facility other than a breeding establishment, which supplies animals for experiments.
Article 15 of the directive requires supplying establishments to obtain animals only from approved breeding or other supplying establishments, "unless the animal has been lawfully imported and is not a feral or stray animal." Nonetheless, the directive allows
exemptions from this sourcing requirement "under arrangements determined by the authority." [ 17 ]
Animal rights supporters have raised concerns that these rules allow strays and pets to be used for experimentation, either by exemptions or by importing animals from non-EU countries, where the rules may be more lax. [ 18 ]
In 2010, a new EU directive was published on the protection of animals used for scientific purposes, repealing the old directive 86/609/EEC on January 1, 2013, with the exception of Article 13 (statistical information on the use of animals in experiments) which has been repealed on May 10, 2013. [ 19 ] | https://en.wikipedia.org/wiki/Laboratory_animal_sources |
Laboratory automation is a multi-disciplinary strategy to research, develop, optimize and capitalize on technologies in the laboratory that enable new and improved processes. Laboratory automation professionals are academic, commercial and government researchers, scientists and engineers who conduct research and develop new technologies to increase productivity, elevate experimental data quality, reduce lab process cycle times, or enable experimentation that otherwise would be impossible.
The most widely known application of laboratory automation technology is laboratory robotics . More generally, the field of laboratory automation comprises many different automated laboratory instruments , devices (the most common being autosamplers ), software algorithms, and methodologies used to enable, expedite and increase the efficiency and effectiveness of scientific research in laboratories.
The application of technology in today's laboratories is required to achieve timely progress and remain competitive. Laboratories devoted to activities such as high-throughput screening , combinatorial chemistry , automated clinical and analytical testing, diagnostics, large-scale biorepositories, and many others, would not exist without advancements in laboratory automation.
Some universities offer entire programs that focus on lab technologies. For example, Indiana University-Purdue University at Indianapolis offers a graduate program devoted to Laboratory Informatics. Also, the Keck Graduate Institute in California offers a graduate degree with an emphasis on development of assays, instrumentation and data analysis tools required for clinical diagnostics, high-throughput screening , genotyping , microarray technologies, proteomics , imaging and other applications.
At least since 1875 there have been reports of automated devices for scientific investigation. [ 1 ] These first devices were mostly built by scientists themselves in order to solve problems in the laboratory. After the second world war, companies started to provide automated equipment with greater and greater complexity.
Automation steadily spread in laboratories through the 20th century, but then a revolution took place: in the early 1980s, the first fully automated laboratory was opened by Dr. Masahide Sasaki . [ 2 ] [ 3 ] In 1993, Dr. Rod Markin at the University of Nebraska Medical Center created one of the world's first clinical automated laboratory management systems. [ 4 ] In the mid-1990s, he chaired a standards group called the Clinical Testing Automation Standards Steering Committee (CTASSC) of the American Association for Clinical Chemistry , [ 5 ] [ 6 ] which later evolved into an area committee of the Clinical and Laboratory Standards Institute . [ 7 ] In 2004, the National Institutes of Health (NIH) and more than 300 nationally recognized leaders in academia, industry, government, and the public completed the NIH Roadmap to accelerate medical discovery to improve health. The NIH Roadmap clearly identifies technology development as a mission critical factor in the Molecular Libraries and Imaging Implementation Group (see the first theme – New Pathways to Discovery – at https://web.archive.org/web/20100611171315/http://nihroadmap.nih.gov/ ).
Despite the success of Dr. Sasaki laboratory and others of the kind, the multi-million dollar cost of such laboratories has prevented adoption by smaller groups. [ 8 ] This is all more difficult because devices made by different manufactures often cannot communicate with each other. However, recent advances based on the use of scripting languages like Autoit have made possible the integration of equipment from different manufacturers. [ 9 ] Using this approach, many low-cost electronic devices, including open-source devices, [ 10 ] become compatible with common laboratory instruments.
Some startups such as Emerald Cloud Lab and Strateos provide on-demand and remote laboratory access on a commercial scale. A 2017 study indicates that these commercial-scale, fully integrated automated laboratories can improve reproducibility and transparency in basic biomedical experiments, and that over nine in ten biomedical papers use methods currently available through these groups. [ 11 ]
A large obstacle to the implementation of automation in laboratories has been its high cost. Many laboratory instruments are very expensive. This is justifiable in many cases, as such equipment can perform very specific tasks employing cutting-edge technology. However, there are devices employed in the laboratory that are not highly technological but still are very expensive. This is the case of many automated devices, which perform tasks that could easily be done by simple and low-cost devices like simple robotic arms , [ 12 ] [ 13 ] [ 14 ] universal (open-source) electronic modules, [ 15 ] [ 16 ] [ 17 ] [ 18 ] [ 19 ] Lego Mindstorms , [ 20 ] or 3D printers .
So far, using such low-cost devices together with laboratory equipment was considered to be very difficult. However, it has been demonstrated that such low-cost devices can substitute without problems the standard machines used in laboratory. [ 12 ] [ 21 ] [ 22 ] It can be anticipated that more laboratories will take advantage of this new reality as low-cost automation is very attractive for laboratories.
A technology that enables the integration of any machine regardless of their brand is scripting, more specifically, scripting involving the control of mouse clicks and keyboard entries, like AutoIt . By timing clicks and keyboard inputs, different software interfaces controlling different devices can be perfectly synchronized. [ 9 ] [ 23 ] | https://en.wikipedia.org/wiki/Laboratory_automation |
A laboratory centrifuge is a piece of laboratory equipment , driven by a motor, which spins liquid samples at high speed.
There are various types of centrifuges, depending on the size and the sample capacity. [ 1 ]
Like all other centrifuges , laboratory centrifuges work by the sedimentation principle , where the centripetal acceleration is used to separate substances of greater and lesser density.
There are various types of centrifugation:
There are different types of laboratory centrifuges:
Because of the heat generated by air friction (even in ultracentrifuges, where the rotor operates in a good vacuum), and the frequent necessity of maintaining samples at a given temperature, many types of laboratory centrifuges are refrigerated and temperature regulated.
Centrifuge tubes are precision-made, high-strength tubes of glass or plastic made to fit exactly in rotor cavities. They may vary in capacity from 50 mL down to much smaller capacities used in microcentrifuges used extensively in molecular biology laboratories. Microcentrifuges typically accommodate disposable plastic microcentrifuge tubes with capacities from 250 μL to 2.0 mL .
Glass centrifuge tubes can be used with most solvents, but tend to be more expensive. They can be cleaned like other laboratory glassware , and can be sterilized by autoclaving . Small scratches from careless handling can cause failure under the strong forces imposed during a run. Glass tubes are inserted into soft rubber sleeves to cushion them during runs. Plastic centrifuge tubes, especially tend to be less expensive and, with care, can be just as durable as glass. Water is preferred when plastic centrifuge tubes are used. They are more difficult to clean thoroughly, and are usually inexpensive enough to be considered disposable.
Disposable plastic "microlitre tubes" of 0.5ml to 2ml are commonly used in microcentrifuges. They are molded from a flexible transparent plastic similar to polythene , are semi-conical in shape, with integral, hinged sealing caps.
Larger samples are spun using centrifuge bottles, which range in capacity from 250 to 1000 millilitres. Although some are made of heavy glass, centrifuge bottles are usually made of shatterproof plastics such as polypropylene or polycarbonate. Sealing closures may be used for added leak-proof assurance.
The load in a laboratory centrifuge must be carefully balanced. This is achieved by using a combination of samples and balance tubes which all have the same weight or by using various balancing patterns without balance tubes. [ 2 ] It is an interesting mathematical problem to solve the balance pattern given n slots and k tubes with the same weight. It is known that the solution exists if and only if both k and n-k can be expressed as a sum of prime factors of n. [ 3 ] Small differences in mass of the load can result in a large force imbalance when the rotor is at high speed. This force imbalance strains the spindle and may result in damage to the centrifuge or personal injury. Some centrifuges have an automatic rotor imbalance detection feature that immediately discontinues the run when an imbalance is detected.
Before starting a centrifuge, an accurate check of the rotor and lid locking mechanisms is mandatory. A spinning rotor can cause serious injury if touched. Modern centrifuges generally have features that prevent accidental contact with a moving rotor as the main lid is locked during the run.
Centrifuge rotors have tremendous kinetic energy during high speed rotation. Rotor failure, caused by mechanical stress from the high forces imparted by the motor, can occur due to manufacturing defects, routine wear and tear, or improper use and maintenance. Such a failure can be catastrophic failure , especially with larger centrifuges, and generally results in total destruction of the centrifuge. While centrifuges generally have safety shielding to contain these failures, such shielding may be inadequate, especially in older models, or the entire centrifuge unit may be propelled from its position, resulting in damage to nearby personnel and equipment. Uncontained rotor failures have shattered laboratory windows and destroyed refrigerators and cabinetry. To reduce the risk of rotor failures, centrifuge manufacturers specify operating and maintenance procedures to ensure that rotors are regularly inspected and removed from service or derated (only operated at lower speeds) when they are past their expected lifetime. [ 4 ]
Another potential hazard is the aerosolization of hazardous samples during centrifugation. To prevent contamination of the laboratory, rotor lids with special aerosol-tight gaskets are available. The rotor can be loaded with the samples within a hood and the rotor lid fixed on the rotor. Afterwards, the aerosol-tight system of rotor and lid is transferred to the centrifuge. The rotor can then be fixed within the centrifuge without opening the lid. After the run, the entire rotor assembly, including the lid, is removed from the centrifuge to the hood for further steps, maintaining the samples within a closed system. | https://en.wikipedia.org/wiki/Laboratory_centrifuge |
Laboratory developed test ( LDT ) is a term used to refer to a certain class of in vitro diagnostics (IVDs) that, in the U.S., were traditionally regulated under the Clinical Laboratory Improvement Amendments program. [ 1 ]
Laboratory-developed tests (LDTs) are a class of in vitro diagnostics (IVDs) designed, manufactured, and used within a single laboratory. They are employed for various medical diagnoses and research applications, offering advantages in flexibility and fostering innovation in the diagnostics field. [ 2 ]
In the United States, the Food and Drug Administration (FDA) has determined that while such tests qualify as medical devices , these products could enter the market without prior approval from the agency. In 2014, the FDA announced that it would start regulating some LDTs. [ 3 ] [ 4 ] In general, however, it has not done so, as of April 2019. [ 5 ]
As LDTs do not require FDA 510(k) clearance required by other diagnostic tests, they have been viewed as a regulatory loophole by opponents. [ 6 ] [ 7 ]
Direct-to-consumer tests are regulated as medical devices, although they are not necessarily reviewed by the FDA. [ 8 ]
23andMe direct-to-consumer genetic tests were originally offered as LDTs, but the FDA challenged that and forced the company to submit the test for approval as a class II medical device. [ 9 ] [ 10 ]
Several companies offer lab-developed tests. [ 3 ]
Several prominent companies are at the forefront of developing innovative Laboratory Developed Tests solutions, including Adaptive Biotechnologies Corporation, Quest Diagnostics, Roche, and Illumina [ 2 ]
The global market for laboratory-developed testing (LDT) is experiencing significant growth, with a projected value of US$ 4582.6 million by 2030 from US$ 3518.7 million in 2023 (CAGR of 3.8%) [ 2 ] This growth is driven by advancements in genetic testing, the increasing demand for personalized medicine, and the ongoing expansion of the healthcare and diagnostics sectors [ 2 ]
This medical diagnostic article is a stub . You can help Wikipedia by expanding it .
This article related to medical technology is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Laboratory_developed_test |
In the diagnostic laboratory, virus infections can be confirmed by a myriad of methods. Diagnostic virology has changed rapidly due to the advent of molecular techniques and increased clinical sensitivity of serological assays. [ 1 ]
A wide variety of samples can be used for virological testing. The type of sample sent to the laboratory often depends on the type of viral infection being diagnosed and the test required. Proper sampling technique is essential to avoid potential pre-analytical errors. For example, different types of samples must be collected in appropriate tubes to maintain the integrity of the sample and stored at appropriate temperatures (usually 4 °C) to preserve the virus and prevent bacterial or fungal growth. Sometimes multiple sites may also be sampled. [ citation needed ]
Types of samples include the following:
For example, a nasal mucus test may be done to diagnose rhinovirus. [ 2 ]
Viruses are often isolated from the initial patient sample. This allows the virus sample to be grown into larger quantities and allows a larger number of tests to be run on them. This is particularly important for samples that contain new or rare viruses for which diagnostic tests are not yet developed. [ citation needed ]
Many viruses can be grown in cell culture in the lab. To do this, the virus sample is mixed with cells, a process called adsorption , after which the cells become infected and produce more copies of the virus. [ 3 ] Although different viruses often only grow in certain types of cells, there are cells that support the growth of a large variety of viruses and are a good starting point, for example, the African monkey kidney cell line ( Vero cells ), human lung fibroblasts ( MRC-5 ), and human epidermoid carcinoma cells ( HEp-2 ). One means of determining whether the cells are successfully replicating the virus is to check for a change in cell morphology or for the presence of cell death using a microscope.
Other viruses may require alternative methods for growth such as the inoculation of embryonated chicken eggs (e.g. avian influenza viruses [ 4 ] ) or the intracranial inoculation of virus using newborn mice (e.g. lyssaviruses [ 5 ] ).
Molecular techniques are the most specific and sensitive diagnostic tests. [ 6 ] They are capable of detecting either the whole viral genome or parts of the viral genome. In the past nucleic acid tests have mainly been used as a secondary test to confirm positive serological results. [ 3 ] However, as they become cheaper and more automated, they are increasingly becoming the primary tool for diagnostics and can also be use for monitoring of treatment of viral infected individuals t. [ 3 ]
Detection of viral RNA and DNA genomes can be performed using polymerase chain reaction . This technique makes many copies of the virus genome using virus-specific probes. Variations of PCR such as nested reverse transcriptase PCR and real time PCR can also be used to determine viral loads in patient serum. This is often used to monitor treatment success in HIV cases. [ citation needed ]
Sequencing is the only diagnostic method that will provide the full sequence of a virus genome. Hence, it provides the most information about very small differences between two viruses that would look the same using other diagnostic tests. Currently it is only used when this depth of information is required. For example, sequencing is useful when specific mutations in the patient are tested for in order to determine antiviral therapy and susceptibility to infection. However, as the tests are getting cheaper, faster and more automated, sequencing will likely become the primary diagnostic tool in the future. [ citation needed ]
Immunofluorescence or immunoperoxidase assays are commonly used to detect whether a virus is present in a tissue sample. These tests are based on the principle that if the tissue is infected with a virus, an antibody specific to that virus will be able to bind to it. To do this, antibodies that are specific to different types of viruses are mixed with the tissue sample. After the tissue is exposed to a specific wavelength of light or a chemical that allows the antibody to be visualized. [ citation needed ]
These tests require specialized antibodies that are produced and purchased from commercial companies. These commercial antibodies are usually well characterized and are known to bind to only one specific type of virus. They are also conjugated to a special kind of tag that allows the antibody to be visualized in the lab, i.e.so that it will emit fluorescence or a color. Hence, immunofluorescence refers
to the detection of a fluorescent antibody (immuno) and immunoperoxidase refers to the detection of a colored antibody (peroxidase produces a dark brown color). [ citation needed ]
Electron microscopy is a method that can take a picture of a whole virus and can reveal its shape and structure. It is not typically used as a routine diagnostic test as it requires a highly specialized type of sample preparation, microscope and technical expertise. However, electron microscopy is highly versatile due to its ability to analyze any type of sample and identify any type of virus. Therefore, it remains the gold standard for identifying viruses that do not show up on routine diagnostic tests or for which routine tests present conflicting results. [ 7 ]
A person who has recently been infected by a virus will produce antibodies in their bloodstream that specifically recognize that virus. This is called humoral immunity . Two types of antibodies are important. The first called IgM is highly effective at neutralizing viruses but is only produced by the cells of the immune system for a few weeks. The second, called, IgG is produced indefinitely. Therefore, the presence of IgM in the blood of the host is used to test for acute infection, whereas IgG indicates an infection sometime in the past. [ 8 ] Both types of antibodies are measured when tests for immunity are carried out. [ 9 ]
Antibody testing has become widely available. It can be done for individual viruses (e.g. using an ELISA assay) but automated panels that can screen for many viruses at once are becoming increasingly common. [ citation needed ]
Some viruses attach to molecules present on the surface of red blood cells, for example, influenza virus. [ 10 ] A consequence of this is that – at certain concentrations – a viral suspension may bind together ( agglutinate ) the red blood cells thus preventing them from settling out of suspension. [ citation needed ] | https://en.wikipedia.org/wiki/Laboratory_diagnosis_of_viral_infections |
A laboratory information management system ( LIMS ), sometimes referred to as a laboratory information system ( LIS ) or laboratory management system ( LMS ), is a software -based solution with features that support a modern laboratory 's operations. Key features include—but are not limited to— workflow and data tracking support, flexible architecture, and data exchange interfaces, which fully "support its use in regulated environments". The features and uses of a LIMS have evolved over the years from simple sample tracking to an enterprise resource planning tool that manages multiple aspects of laboratory informatics .
There is no useful definition of the term "LIMS" as it is used to encompass a number of different laboratory informatics components. The spread and depth of these components is highly dependent on the LIMS implementation itself. All LIMSs have a workflow component and some summary data management facilities but beyond that there are significant differences in functionality.
Historically the LIMyS, LIS, and process development execution system (PDES) have all performed similar functions. The term "LIMS" has tended to refer to informatics systems targeted for environmental, research, or commercial analysis such as pharmaceutical or petrochemical work. "LIS" has tended to refer to laboratory informatics systems in the forensics and clinical markets, which often required special case management tools. "PDES" has generally applied to a wider scope, including, for example, virtual manufacturing techniques, while not necessarily integrating with laboratory equipment .
In recent times LIMS functionality has spread even further beyond its original purpose of sample management. Assay data management, data mining , data analysis, and electronic laboratory notebook (ELN) integration have been added to many LIMS, enabling the realization of translational medicine completely within a single software solution. Additionally, the distinction between LIMS and LIS has blurred, as many LIMS now also fully support comprehensive case-centric clinical data.
Up until the late 1970s, the management of laboratory samples and the associated analysis and reporting were time-consuming manual processes often riddled with transcription errors. This gave some organizations impetus to streamline the collection of data and how it was reported. Custom in-house solutions were developed by a few individual laboratories, while some enterprising entities sought to develop commercial reporting solutions in the form of special instrument-based systems. [ 1 ]
In 1982 the first generation of LIMS was introduced in the form of a centralized minicomputer, which offered automated reporting tools. As the interest in these early LIMS grew, industry leaders like Gerst Gibbon of the Federal Energy Technology Center in Pittsburgh began planting the seeds through LIMS-related conferences. By 1988 the second-generation commercial offerings were tapping into relational databases to expand LIMS into more application-specific territory, and International LIMS Conferences were in full swing. As personal computers became more powerful and prominent, a third generation of LIMS emerged in the early 1990s. These new LIMS took advantage of client/server architecture , allowing laboratories to implement better data processing and exchanges. [ 1 ]
By 1995 the client/server tools allowed the processing of data anywhere on the network. Web-enabled LIMS were introduced the following year, enabling researchers to extend operations outside the laboratory. From 1996 to 2002 additional functionality was included, from wireless networking and georeferencing of samples, to the adoption of XML standards and Internet purchasing. [ 1 ]
As of 2012, some LIMS have added additional characteristics such as clinical functionality, electronic laboratory notebook (ELN) functionality, as well a rise in the software as a service (SaaS) distribution model.
The LIMS is an evolving concept, with new features and functionality being added often. As laboratory demands change and technological progress continues, the functions of a LIMS will likely also change. Despite these changes, a LIMS tends to have a base set of functionality that defines it. That functionality can roughly be divided into five laboratory processing phases, with numerous software functions falling under each: [ 2 ] (1) the reception and log in of a sample and its associated customer data,
(2) the assignment, scheduling, and tracking of the sample and the associated analytical workload,
(3) the processing and quality control associated with the sample and the utilized equipment and inventory,
(4) the storage of data associated with the sample analysis,
(5) the inspection, approval, and compilation of the sample data for reporting and/or further analysis.
There are several pieces of core functionality associated with these laboratory processing phases that tend to appear in most LIMS:
The core function of LIMS has traditionally been the management of samples. This typically is initiated when a sample is received in the laboratory, at which point the sample will be registered in the LIMS. Some LIMS will allow the customer to place an "order" for a sample directly to the LIMS at which point the sample is generated in an "unreceived" state. The processing could then include a step where the sample container is registered and sent to the customer for the sample to be taken and then returned to the lab. The registration process may involve accessioning the sample and producing barcodes to affix to the sample container. Various other parameters such as clinical or phenotypic information corresponding with the sample are also often recorded. The LIMS then tracks chain of custody as well as sample location. Location tracking usually involves assigning the sample to a particular freezer location, often down to the granular level of shelf, rack, box, row, and column. Other event tracking such as freeze and thaw cycles that a sample undergoes in the laboratory may be required.
Modern LIMS have implemented extensive configurability as each laboratory's needs for tracking additional data points can vary widely. LIMS vendors cannot typically make assumptions about what these data tracking needs are, and therefore vendors must create LIMS that are adaptable to individual environments. LIMS users may also have regulatory concerns to comply with such as CLIA , HIPAA , GLP , and FDA specifications, affecting certain aspects of sample management in a LIMS solution. One key to compliance with many of these standards is audit logging of all changes to LIMS data, and in some cases a full electronic signature system is required for rigorous tracking of field-level changes to LIMS data.
Modern LIMS offer an increasing amount of integration with laboratory instruments and applications. A LIMS may create control files that are "fed" into the instrument and direct its operation on some physical item such as a sample tube or sample plate. The LIMS may then import instrument results files to extract data for quality control assessment of the operation on the sample. Access to the instrument data can sometimes be regulated based on chain of custody assignments or other security features if need be.
Modern LIMS products now also allow for the import and management of raw assay data results. [ 3 ] Modern targeted assays such as qPCR and deep sequencing can produce tens of thousands of data points per sample. Furthermore, in the case of drug and diagnostic development as many as 12 or more assays may be run for each sample. In order to track this data, a LIMS solution needs to be adaptable to many different assay formats at both the data layer and import creation layer, while maintaining a high level of overall performance. Some LIMS products address this by simply attaching assay data as BLOBs to samples, but this limits the utility of that data in data mining and downstream analysis.
The exponentially growing volume of data created in laboratories, coupled with increased business demands and focus on profitability, have pushed LIMS vendors to increase attention to how their LIMS handles electronic data exchanges . Attention must be paid to how an instrument's input and output data is managed, how remote sample collection data is imported and exported, and how mobile technology integrates with the LIMS. The successful transfer of data files in spreadsheets and other formats is a pivotal aspect of the modern LIMS. In fact, the transition "from proprietary databases to standardized database management systems such as MySQL " has arguably had one of the biggest impacts on how data is managed and exchanged in laboratories. In addition to mobile and database electronic data exchange, many LIMS support real-time data exchange with Electronic Health Records used in core hospital or clinic operations.
Aside from the key functions of sample management, instrument and application integration, and electronic data exchange, there are numerous additional operations that can be managed in a LIMS. This includes but is not limited to:
A LIMS has utilized many architectures and distribution models over the years. As technology has changed, how a LIMS is installed, managed, and utilized has also changed with it. The following represents architectures which have been utilized at one point or another.
A thick-client LIMS is a more traditional client/server architecture, with some of the system residing on the computer or workstation of the user ( the client ) and the rest on the server. The LIMS software is installed on the client computer, which does all of the data processing. Later it passes information to the server, which has the primary purpose of data storage. Most changes, upgrades, and other modifications will happen on the client side.
This was one of the first architectures implemented into a LIMS, having the advantage of providing higher processing speeds (because processing is done on the client and not the server). Additionally, thick-client systems have also provided more interactivity and customization, though often at a greater learning curve. The disadvantages of client-side LIMS include the need for more robust client computers and more time-consuming upgrades, as well as a lack of base functionality through a web browser . The thick-client LIMS can become web-enabled through an add-on component. [ 4 ]
Although there is a claim of improved security through the use of a thick-client LIMS, [ 4 ] this is based on the misconception that "only users with the client application installed on their PC can access server side information". This secrecy-of-design reliance is known as security through obscurity and ignores an adversary's ability to mimic client-server interaction through, for example, reverse engineering , network traffic interception , or simply purchasing a thick-client license. Such a view is in contradiction of the "Open Design" principle of the National Institute of Standards and Technology 's Guide to General Server Security which states that "system security should not depend on the secrecy of the implementation or its components", [ 5 ] which can be considered as a reiteration of Kerckhoffs's principle .
A thin-client LIMS is a more modern architecture which offers full application functionality accessed through a device's web browser. The actual LIMS software resides on a server (host) which feeds and processes information without saving it to the user's hard disk. Any necessary changes, upgrades, and other modifications are handled by the entity hosting the server-side LIMS software, meaning all end-users see all changes made. To this end, a true thin-client LIMS will leave no "footprint" on the client's computer, and only the integrity of the web browser need be maintained by the user. The advantages of this system include significantly lower cost of ownership and fewer network and client-side maintenance expenses. However, this architecture has the disadvantage of requiring real-time server access, a need for increased network throughput, and slightly less functionality. A sort of hybrid architecture that incorporates the features of thin-client browser usage with a thick client installation exists in the form of a web-based LIMS.
Some LIMS vendors are beginning to rent hosted, thin-client solutions as " software as a service " (SaaS). These solutions tend to be less configurable than on-premises solutions and are therefore considered for less demanding implementations such as laboratories with few users and limited sample processing volumes.
Another implementation of the thin client architecture is the maintenance, warranty , and support (MSW) agreement. Pricing levels are typically based on a percentage of the license fee, with a standard level of service for 10 concurrent users being approximately 10 hours of support and additional customer service, at a roughly $200 per hour rate. Though some may choose to opt out of an MSW after the first year, it is often more economical to continue the plan in order to receive updates to the LIMS, giving it a longer life span in the laboratory.
A web-enabled LIMS architecture is essentially a thick-client architecture with an added web browser component. In this setup, the client-side software has additional functionality that allows users to interface with the software through their device's browser. This functionality is typically limited only to certain functions of the web client. The primary advantage of a web-enabled LIMS is the end-user can access data both on the client side and the server side of the configuration. As in a thick-client architecture, updates in the software must be propagated to every client machine. However, the added disadvantages of requiring always-on access to the host server and the need for cross-platform functionality mean that additional overhead costs may arise.
A web-based LIMS architecture is a hybrid of the thick- and thin-client architectures. While much of the client-side work is done through a web browser, the LIMS may also require the support of desktop software installed on the client device. The end result is a process that is apparent to the end-user through a web browser, but perhaps not so apparent as it runs thick-client-like processing in the background. In this case, web-based architecture has the advantage of providing more functionality through a more friendly web interface. The disadvantages of this setup are more sunk costs in system administration and reduced functionality on mobile platforms.
LIMS implementations are notorious for often being lengthy and costly. This is partly due to the diversity of requirements within each lab, but also to the inflexible nature of most LIMS products for adapting to these widely varying requirements. Newer LIMS solutions are beginning to emerge that take advantage of modern techniques in software design that are inherently more configurable and adaptable — particularly at the data layer — than prior solutions. This means not only that implementations are much faster, but also that the costs are lower and the risk of obsolescence is minimized.
Until recently, the LIMS and Laboratory Information System (LIS) have exhibited a few key differences, making them noticeably separate entities.
A LIMS traditionally has been designed to process and report data related to batches of samples from biology labs, water treatment facilities , drug trials , and other entities that handle complex batches of data. A LIS has been designed primarily for processing and reporting data related to individual patients in a clinical setting.
A LIMS may need to satisfy good manufacturing practice (GMP) and meet the reporting and audit needs of the regulatory bodies and research scientists in many different industries. A LIS, however, must satisfy the reporting and auditing needs of health service agencies e.g. the hospital accreditation agency, HIPAA in the US, or other clinical medical practitioners.
A LIMS is most competitive in group-centric settings (dealing with "batches" and "samples") that often deal with mostly anonymous research-specific laboratory data, whereas a LIS is usually most competitive in patient-centric settings (dealing with "subjects" and "specimens") and clinical labs. An LIS is regulated as a medical device by the FDA, and the companies that produce the software are therefore liable for defects. Due to this, a LIS can not be customized by the client.
A LIMS covers standards such as 21 CFR Part 11 from the Food and Drug Administration (United States) , ISO/IEC 17025 , ISO 15189 , ISO 20387, Good Clinical Practice (GCP), Good Laboratory Practice (GLP), Good Manufacturing Practice (GMP), FDA Food Safety Modernization Act (FSMA), HACCP , and ISBER Best Practices. | https://en.wikipedia.org/wiki/Laboratory_information_management_system |
A water bath is laboratory equipment made from a container filled with heated water. It is used to incubate samples in water at a constant temperature over a long period of time. Most water baths have a digital or an analogue interface to allow users to set a desired temperature, but some water baths have their temperature controlled by a current passing through a reader.
Uses include warming of reagents , melting of substrates , determination of boiling point , or incubation of cell cultures. It is also used to enable certain chemical reactions to occur at high temperature.
Water baths are preferred heat sources for heating flammable chemicals, as their lack of open flame prevents ignition . [ 1 ] Different types of water baths are used depending on application. For all water baths, it can be used up to 99.9 °C. [ 2 ] [ 3 ]
When the required temperature is above 100 °C, alternative methods such as oil bath, silicone oil bath or sand bath may be used. [ 4 ]
It is not recommended to use water bath with moisture sensitive or pyrophoric reactions. [ 5 ] Do not heat a bath fluid above its flash point. [ 5 ] [ 6 ] Water level should be regularly monitored, and filled with distilled water only. [ 7 ] [ 8 ] This is required to prevent salts from depositing on the heater. [ 8 ] Disinfectants can be added to prevent growth of organisms. [ 6 ] [ 7 ] If application involves liquids that give off fumes , it is recommended to operate water bath in fume hood or in a well ventilated area. [ 9 ] The cover is closed to prevent evaporation and to help reaching high temperatures. [ 9 ]
Circulating water baths (also called stirrers [ 10 ] ) are ideal for applications when temperature uniformity and consistency are critical, such as enzymatic and serologic experiments. Water is thoroughly circulated throughout the bath resulting in a more uniform temperature.
This type of water bath relies primarily on convection instead of water being uniformly heated. Therefore, it is less accurate in terms of temperature control. In addition, there are add-ons that provide stirring to non-circulating water baths to create more uniform heat transfer. [ 4 ]
This type of water bath has extra control for shaking, which moves liquids around. This shaking feature can be turned on or off. In microbiological practices, constant shaking allows liquid-grown cell cultures grown to constantly mix with the air.
Some key benefits of shaking water bath are user-friendly operation via keypad, convenient bath drains, adjustable shaking frequencies, bright LED-display, optional lift-up bath cover, power switch integrated in keypad and warning and cut-off protection for low/high temperature.
The bath is a fundamental product in any laboratory. Over the years, water baths have evolved from basic analog tools to advanced digital machines capable of sophisticated and programmable controls, functions, and capabilities.
Key features in water baths often include: | https://en.wikipedia.org/wiki/Laboratory_water_bath |
The lacUV5 promoter is a mutated promoter from the Escherichia coli lac operon which is used in molecular biology to drive gene expression on a plasmid . lacUV5 is very similar to the classical lac promoter, containing just 2 base pair mutations in the -10 hexamer region, compared to the lac promoter. [ 1 ] LacUV5 is among the most commonly used promoters in molecular biology because it requires no additional activators and it drives high levels of gene expression. [ 2 ]
The lacUV5 promoter sequence conforms more closely to the consensus sequence recognized by bacterial sigma factors than the traditional lac promoter does. [ 3 ] : 4b Due to this, lacUV5 recruits RNA Polymerase more effectively, thus leading to higher transcription of target genes. Additionally, unlike the lac promoter, lacUV5 works independently of activator proteins or other cis regulatory elements (apart from the -10 and -35 promoter regions). [ 2 ] While no activators are required, lacUV5 promoter expression can be regulated by the LacI repressor and can be induced with IPTG , which is an effective inducer of protein expression when used in the concentration range of 100 μM to 1.5 mM. Due to this control, the lacUV5 promoter is commonly found on expression plasmids and is used when controllable but high levels of a product are desired. [ citation needed ]
The lacUV5 mutation was first identified in 1970 in a study of lac promoter mutants that produce higher yields. Some of them, including UV5, has lost catabolite repression at the CAP site. [ 4 ] Development into cloning vectors is known since 1982, when a UV5-carrying phage known as "λ h80 lac UV5 cI857" has its genome spliced with the HaeIII restriction enzyme to make plasmids carrying the fragment with UV5. [ 5 ]
Modern lacUV5 is seen in the BL21(DE3) strain, which carries both a lac operon with the standard promoter and a lacUV5 operon split by the DE3 prophage (and as a result driving the T7 RNA polymerase instead). [ 1 ] The two important mutations are underlined.
This genetics article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/LacUV5 |
The lactose operon ( lac operon) is an operon required for the transport and metabolism of lactose in E. coli and many other enteric bacteria . Although glucose is the preferred carbon source for most enteric bacteria, the lac operon allows for the effective digestion of lactose when glucose is not available through the activity of β-galactosidase . [ 1 ] Gene regulation of the lac operon was the first genetic regulatory mechanism to be understood clearly, so it has become a foremost example of prokaryotic gene regulation . It is often discussed in introductory molecular and cellular biology classes for this reason. This lactose metabolism system was used by François Jacob and Jacques Monod to determine how a biological cell knows which enzyme to synthesize. Their work on the lac operon won them the Nobel Prize in Physiology in 1965. [ 1 ]
Most bacterial cells including E. coli lack introns in their genome. They also lack a nuclear membrane . Hence the gene regulation by lac operon occurs at the transcriptional level, by controlling transcription of DNA .
Bacterial operons are polycistronic transcripts that are able to produce multiple proteins from one mRNA transcript. In this case, when lactose is required as a sugar source for the bacterium, the three genes of the lac operon can be transcribed and their subsequent proteins translated: lacZ , lacY , and lacA . The gene product of lacZ is β-galactosidase which cleaves lactose, a disaccharide , into glucose and galactose . lacY encodes β-galactoside permease , a membrane protein which becomes embedded in the Plasma membrane to enable the cellular transport of lactose into the cell. Finally, lacA encodes β-galactoside transacetylase .
Note that the number of base pairs in diagram given above are not to scale. There are in fact over 5300 base pairs in the lac operon. [ 2 ]
It would be wasteful to produce enzymes when no lactose is available or if a preferable energy source such as glucose were available. The lac operon uses a two-part control mechanism to ensure that the cell expends energy producing the enzymes encoded by the lac operon only when necessary. [ 3 ]
In the absence of lactose, the lac repressor , encoded by lacI, halts production of the enzymes and transport proteins encoded by the lac operon. [ 4 ] It does so by blocking the DNA dependent RNA polymerase . This blocking/ halting is not perfect, and a minimal amount of gene expression does take place all the time. The repressor protein is always expressed, but the lac operon (i.e. enzymes and transport proteins) are almost completely repressed, allowing for a small level of background expression. If this weren't the case, there would be no lacY transporter protein in the cellular membrane; consequently, the lac operon would not be able to detect the presence of lactose.
When lactose is available but not glucose, then some lactose enters the cell using pre-existing transport protein encoded by lacY. This lactose then combines with the repressor and inactivates it, hence allowing the lac operon to be expressed. Then more β-galactoside permease is synthesized allowing even more lactose to enter and the enzymes encoded by lacZ and lacA can digest it.
However, in the presence of glucose, regardless of the presence of lactose, the operon will be repressed. This is because the catabolite activator protein (CAP), required for production of the enzymes, remains inactive, and EIIA Glc shuts down lactose permease to prevent transport of lactose into the cell. This dual control mechanism causes the sequential utilization of glucose and lactose in two distinct growth phases, known as diauxie .
Only lacZ and lacY appear to be necessary for lactose catabolic pathway .
By numbers, lacI has 1100 bps, lacZ has 3000 bps, lacY has 800 bps, lacA has 800 bps, with 3 bps corresponding to 1 amino acid. [ 5 ]
Three-letter abbreviations are used to describe phenotypes in bacteria including E. coli .
Examples include:
In the case of Lac, wild type cells are Lac + and are able to use lactose as a carbon and energy source, while Lac − mutant derivatives cannot use lactose. The same three letters are typically used (lower-case, italicized) to label the genes involved in a particular phenotype, where each different gene is additionally distinguished by an extra letter. The lac genes encoding enzymes are lacZ , lacY , and lacA . The fourth lac gene is lacI , encoding the lactose repressor—"I" stands for inducibility .
One may distinguish between structural genes encoding enzymes, and regulatory genes encoding proteins that affect gene expression. Current usage expands the phenotypic nomenclature to apply to proteins: thus, LacZ is the protein product of the lacZ gene, β-galactosidase. Various short sequences that are not genes also affect gene expression, including the lac promoter, lac p , and the lac operator, lac o . Although it is not strictly standard usage, mutations affecting lac o are referred to as lac o c , for historical reasons.
Specific control of the lac genes depends on the availability of the substrate lactose to the bacterium. The proteins are not produced by the bacterium when lactose is unavailable as a carbon source.
The lac genes are organized into an operon ; that is, they are oriented in the same direction immediately adjacent on the chromosome and are co-transcribed into a single polycistronic mRNA molecule. Transcription of all genes starts with the binding of the enzyme RNA polymerase (RNAP), a DNA-binding protein , which binds to a specific DNA binding site, the promoter , immediately upstream of the genes. Binding of RNA polymerase to the promoter is aided by the cAMP -bound catabolite activator protein (CAP, also known as the cAMP receptor protein). [ 6 ] However, the lacI gene (regulatory gene for lac operon) produces a protein that blocks RNAP from binding to the operator of the operon. This protein can only be removed when allolactose binds to it, and inactivates it. The protein that is formed by the lacI gene is known as the lac repressor. The type of regulation that the lac operon undergoes is referred to as negative inducible, meaning that the gene is turned off by the regulatory factor ( lac repressor) unless some molecule (lactose) is added. Once the repressor is removed, RNAP then proceeds to transcribe all three genes ( lacZYA ) into mRNA. Each of the three genes on the mRNA strand has its own Shine-Dalgarno sequence , so the genes are independently translated. [ 7 ] The DNA sequence of the E. coli lac operon , the lacZYA mRNA , and the lacI genes are available from GenBank (view) .
The first control mechanism is the regulatory response to lactose, which uses an intracellular regulatory protein called the lactose repressor to hinder production of β-galactosidase in the absence of lactose. The lacI gene coding for the repressor lies nearby the lac operon and is always expressed ( constitutive ). If lactose is missing from the growth medium, the repressor binds very tightly to a short DNA sequence just downstream of the promoter near the beginning of lacZ called the lac operator . The repressor binding to the operator interferes with binding of RNAP to the promoter, and therefore mRNA encoding LacZ and LacY is only made at very low levels. When cells are grown in the presence of lactose, however, a lactose metabolite called allolactose, made from lactose by the product of the lacZ gene, binds to the repressor, causing an allosteric shift. Thus altered, the repressor is unable to bind to the operator, allowing RNAP to transcribe the lac genes and thereby leading to higher levels of the encoded proteins.
The second control mechanism is a response to glucose, which uses the catabolite activator protein (CAP) homodimer to greatly increase production of β-galactosidase in the absence of glucose. Cyclic adenosine monophosphate (cAMP) is a signal molecule whose prevalence is inversely proportional to that of glucose. It binds to the CAP, which in turn allows the CAP to bind to the CAP binding site (a 16 bp DNA sequence upstream of the promoter on the left in the diagram below, about 60 bp upstream of the transcription start site), [ 8 ] which assists the RNAP in binding to the DNA. In the absence of glucose, the cAMP concentration is high and binding of CAP-cAMP to the DNA significantly increases the production of β-galactosidase, enabling the cell to hydrolyse lactose and release galactose and glucose.
More recently inducer exclusion was shown to block expression of the lac operon when glucose is present. Glucose is transported into the cell by the PEP-dependent phosphotransferase system . The phosphate group of phosphoenolpyruvate is transferred via a phosphorylation cascade consisting of the general PTS (phosphotransferase system) proteins HPr and EIA and the glucose-specific PTS proteins EIIA Glc and EIIB Glc , the cytoplasmic domain of the EII glucose transporter. Transport of glucose is accompanied by its phosphorylation by EIIB Glc , draining the phosphate group from the other PTS proteins, including EIIA Glc . The unphosphorylated form of EIIA Glc binds to the lac permease and prevents it from bringing lactose into the cell. Therefore, if both glucose and lactose are present, the transport of glucose blocks the transport of the inducer of the lac operon. [ 9 ]
The lac repressor is a four-part protein, a tetramer, with identical subunits. Each subunit contains a helix-turn-helix (HTH) motif capable of binding to DNA. The operator site where repressor binds is a DNA sequence with inverted repeat symmetry. The two DNA half-sites of the operator together bind to two of the subunits of the repressor. Although the other two subunits of repressor are not doing anything in this model, this property was not understood for many years.
Eventually it was discovered that two additional operators are involved in lac regulation. [ 10 ] One (O 3 ) lies about −90 bp upstream of O 1 in the end of the lacI gene, and the other (O 2 ) is about +410 bp downstream of O 1 in the early part of lacZ . These two sites were not found in the early work because they have redundant functions and individual mutations do not affect repression very much. Single mutations to either O 2 or O 3 have only 2 to 3-fold effects. However, their importance is demonstrated by the fact that a double mutant defective in both O 2 and O 3 is dramatically de-repressed (by about 70-fold).
In the current model, lac repressor is bound simultaneously to both the main operator O 1 and to either O 2 or O 3 . The intervening DNA loops out from the complex. The redundant nature of the two minor operators suggests that it is not a specific looped complex that is important. One idea is that the system works through tethering; if bound repressor releases from O 1 momentarily, binding to a minor operator keeps it in the vicinity, so that it may rebind quickly. This would increase the affinity of repressor for O 1 .
The repressor is an allosteric protein , i.e. it can assume either one of two slightly different shapes, which are in equilibrium with each other. In one form the repressor will bind to the operator DNA with high specificity, and in the other form it has lost its specificity. According to the classical model of induction, binding of the inducer, either allolactose or IPTG, to the repressor affects the distribution of repressor between the two shapes. Thus, repressor with inducer bound is stabilized in the non-DNA-binding conformation. However, this simple model cannot be the whole story, because repressor is bound quite stably to DNA, yet it is released rapidly by addition of inducer. Therefore, it seems clear that an inducer can also bind to the repressor when the repressor is already bound to DNA. It is still not entirely known what the exact mechanism of binding is. [ 11 ]
Non-specific binding of the repressor to DNA plays a crucial role in the repression and induction of the Lac-operon. The specific binding site for the Lac-repressor protein is the operator. The non-specific interaction is mediated mainly by charge-charge interactions while binding to the operator is reinforced by hydrophobic interactions. Additionally, there is an abundance of non-specific DNA sequences to which the repressor can bind. Essentially, any sequence that is not the operator, is considered non-specific. Studies have shown, that without the presence of non-specific binding, induction (or unrepression) of the Lac-operon could not occur even with saturated levels of inducer. It had been demonstrated that, without non-specific binding, the basal level of induction is ten thousand times smaller than observed normally. This is because the non-specific DNA acts as sort of a "sink" for the repressor proteins, distracting them from the operator. The non-specific sequences decrease the amount of available repressor in the cell. This in turn reduces the amount of inducer required to unrepress the system. [ 12 ]
A number of lactose derivatives or analogs have been described that are useful for work with the lac operon. These compounds are mainly substituted galactosides, where the glucose moiety of lactose is replaced by another chemical group.
The experimental microorganism used by François Jacob and Jacques Monod was the common laboratory bacterium, E. coli , but many of the basic regulatory concepts that were discovered by Jacob and Monod are fundamental to cellular regulation in all organisms. [ 17 ] The key idea is that proteins are not synthesized when they are not needed— E. coli conserves cellular resources and energy by not making the three Lac proteins when there is no need to metabolize lactose, such as when other sugars like glucose are available. The following section discusses how E. coli controls certain genes in response to metabolic needs.
During World War II , Monod was testing the effects of combinations of sugars as nutrient sources for E. coli and B. subtilis . Monod was following up on similar studies that had been conducted by other scientists with bacteria and yeast. He found that bacteria grown with two different sugars often displayed two phases of growth. For example, if glucose and lactose were both provided, glucose was metabolized first (growth phase I, see Figure 2) and then lactose (growth phase II). Lactose was not metabolized during the first part of the diauxic growth curve because β-galactosidase was not made when both glucose and lactose were present in the medium. Monod named this phenomenon diauxie . [ 18 ]
Monod then focused his attention on the induction of β-galactosidase formation that occurred when lactose was the sole sugar in the culture medium. [ 19 ]
A conceptual breakthrough of Jacob and Monod [ 20 ] was to recognize the distinction between regulatory substances and sites where they act to change gene expression. A former soldier, Jacob used the analogy of a bomber that would release its lethal cargo upon receipt of a special radio transmission or signal. A working system requires both a ground transmitter and a receiver in the airplane. Now, suppose that the usual transmitter is broken. This system can be made to work by introduction of a second, functional transmitter. In contrast, he said, consider a bomber with a defective receiver. The behavior of this bomber cannot be changed by introduction of a second, functional aeroplane.
To analyze regulatory mutants of the lac operon, Jacob developed a system by which a second copy of the lac genes ( lacI with its promoter, and lacZYA with promoter and operator) could be introduced into a single cell. A culture of such bacteria, which are diploid for the lac genes but otherwise normal, is then tested for the regulatory phenotype. In particular, it is determined whether LacZ and LacY are made even in the absence of IPTG (due to the lactose repressor produced by the mutant gene being non-functional). This experiment, in which genes or gene clusters are tested pairwise, is called a complementation test .
This test is illustrated in the figure ( lacA is omitted for simplicity). First, certain haploid states are shown (i.e. the cell carries only a single copy of the lac genes). Panel (a) shows repression, (b) shows induction by IPTG, and (c) and (d) show the effect of a mutation to the lacI gene or to the operator, respectively. In panel (e) the complementation test for repressor is shown. If one copy of the lac genes carries a mutation in lacI , but the second copy is wild type for lacI , the resulting phenotype is normal—but lacZ is expressed when exposed to inducer IPTG. Mutations affecting repressor are said to be recessive to wild type (and that wild type is dominant ), and this is explained by the fact that repressor is a small protein which can diffuse in the cell. The copy of the lac operon adjacent to the defective lacI gene is effectively shut off by protein produced from the second copy of lacI .
If the same experiment is carried out using an operator mutation, a different result is obtained (panel (f)). The phenotype of a cell carrying one mutant and one wild type operator site is that LacZ and LacY are produced even in the absence of the inducer IPTG; because the damaged operator site, does not permit binding of the repressor to inhibit transcription of the structural genes. The operator mutation is dominant. When the operator site where repressor must bind is damaged by mutation, the presence of a second functional site in the same cell makes no difference to expression of genes controlled by the mutant site.
A more sophisticated version of this experiment uses marked operons to distinguish between the two copies of the lac genes and show that the unregulated structural gene(s) is(are) the one(s) next to the mutant operator (panel (g). For example, suppose that one copy is marked by a mutation inactivating lacZ so that it can only produce the LacY protein, while the second copy carries a mutation affecting lacY and can only produce LacZ. In this version, only the copy of the lac operon that is adjacent to the mutant operator is expressed without IPTG. We say that the operator mutation is cis-dominant , it is dominant to wild type but affects only the copy of the operon which is immediately adjacent to it.
This explanation is misleading in an important sense, because it proceeds from a description of the experiment and then explains the results in terms of a model. But in fact, it is often true that the model comes first, and an experiment is fashioned specifically to test the model. Jacob and Monod first imagined that there must be a site in DNA with the properties of the operator, and then designed their complementation tests to show this.
The dominance of operator mutants also suggests a procedure to select them specifically. If regulatory mutants are selected from a culture of wild type using phenyl-Gal, as described above, operator mutations are rare compared to repressor mutants because the target-size is so small. But if instead we start with a strain which carries two copies of the whole lac region (that is diploid for lac ), the repressor mutations (which still occur) are not recovered because complementation by the second, wild type lacI gene confers a wild type phenotype. In contrast, mutation of one copy of the operator confers a mutant phenotype because it is dominant to the second, wild type copy.
Source: [ 21 ]
Explanation of diauxie depended on the characterization of additional mutations affecting the lac genes other than those explained by the classical model. Two other genes, cya and crp , subsequently were identified that mapped far from lac , and that, when mutated, result in a decreased level of expression in the presence of IPTG and even in strains of the bacterium lacking the repressor or operator. The discovery of cAMP in E. coli led to the demonstration that mutants defective the cya gene but not the crp gene could be restored to full activity by the addition of cAMP to the medium.
The cya gene encodes adenylate cyclase, which produces cAMP. In a cya mutant, the absence of cAMP makes the expression of the lacZYA genes more than ten times lower than normal. Addition of cAMP corrects the low Lac expression characteristic of cya mutants. The second gene, crp , encodes a protein called catabolite activator protein (CAP) or cAMP receptor protein (CRP). [ 22 ]
However the lactose metabolism enzymes are made in small quantities in the presence of both glucose and lactose (sometimes called leaky expression) due to the fact that the RNAP can still sometimes bind and initiate transcription even in the absence of CAP. Leaky expression is necessary in order to allow for metabolism of some lactose after the glucose source is expended, but before lac expression is fully activated.
In summary:
The delay between growth phases reflects the time needed to produce sufficient quantities of lactose-metabolizing enzymes. First, the CAP regulatory protein has to assemble on the lac promoter, resulting in an increase in the production of lac mRNA . More available copies of the lac mRNA results in the production (see translation ) of significantly more copies of LacZ (β-galactosidase, for lactose metabolism) and LacY (lactose permease to transport lactose into the cell). After a delay needed to increase the level of the lactose metabolizing enzymes, the bacteria enter into a new rapid phase of cell growth .
Two puzzles of catabolite repression relate to how cAMP levels are coupled to the presence of glucose, and secondly, why the cells should even bother. After lactose is cleaved it actually forms glucose and galactose (easily converted to glucose). In metabolic terms, lactose is just as good a carbon and energy source as glucose. The cAMP level is related not to intracellular glucose concentration but to the rate of glucose transport, which influences the activity of adenylate cyclase. (In addition, glucose transport also leads to direct inhibition of the lactose permease.) As to why E. coli works this way, one can only speculate. All enteric bacteria ferment glucose, which suggests they encounter it frequently. It is possible that a small difference in efficiency of transport or metabolism of glucose v. lactose makes it advantageous for cells to regulate the lac operon in this way. [ 23 ]
The lac gene and its derivatives are amenable to use as a reporter gene in a number of bacterial-based selection techniques such as two hybrid analysis, in which the successful binding of a transcriptional activator to a specific promoter sequence must be determined. [ 16 ] In LB plates containing X-gal , the colour change from white colonies to a shade of blue corresponds to about 20–100 β-galactosidase units, while tetrazolium lactose and MacConkey lactose media have a range of 100–1000 units, being most sensitive in the high and low parts of this range respectively. [ 16 ] Since MacConkey lactose and tetrazolium lactose media both rely on the products of lactose breakdown, they require the presence of both lacZ and lacY genes. The many lac fusion techniques which include only the lacZ gene are thus suited to X-gal plates [ 16 ] or ONPG liquid broths. [ 24 ] | https://en.wikipedia.org/wiki/Lac_operon |
The lac repressor (LacI) is a DNA-binding protein that inhibits the expression of genes coding for proteins involved in the metabolism of lactose in bacteria. These genes are repressed when lactose is not available to the cell, ensuring that the bacterium only invests energy in the production of machinery necessary for uptake and utilization of lactose when lactose is present. When lactose becomes available, it is firstly converted into allolactose by β-Galactosidase ( lacZ ) in bacteria. The DNA binding ability of lac repressor bound with allolactose is inhibited due to allosteric regulation , thereby genes coding for proteins involved in lactose uptake and utilization can be expressed.
The lac repressor (LacI) operates by a helix-turn-helix motif in its DNA-binding domain , binding base-specifically to the major groove of the operator region of the lac operon , with base contacts also made by residues of symmetry-related alpha helices, the "hinge" helices, which bind deeply in the minor groove. [ 1 ] This bound repressor can reduce transcription of the Lac proteins by occluding the RNA polymerase binding site or by prompting DNA looping. [ 2 ] When lactose is present, allolactose binds to the lac repressor, causing an allosteric change in its shape. In its changed state, the lac repressor is unable to bind tightly to its cognate operator. Thus, the gene is mostly off in the absence of inducer and mostly on in the presence of inducer, although the degree of gene expression depends on the number of repressors in the cell and on the repressor's DNA-binding affinity. [ 3 ] Isopropyl β-D-1-thiogalactopyranoside (IPTG) is a commonly used allolactose mimic which can be used to induce transcription of genes being regulated by lac repressor.
Structurally, the lac repressor protein is a homotetramer . More precisely, the tetramer contains two DNA-binding subunits composed of two monomers each (a dimer of dimers). Each monomer consists of four distinct regions: [ 4 ] [ 5 ] [ 6 ]
DNA binding occurs via an N-terminal helix-turn-helix structural motif and is targeted to one of several operator DNA sequences (known as O 1 , O 2 and O 3 ). The O 1 operator sequence slightly overlaps with the promoter, which increases the affinity of RNA polymerase for the promoter sequence such that it cannot enter elongation and remains in abortive initiation . Additionally, because each tetramer contains two DNA-binding subunits, binding of multiple operator sequences by a single tetramer induces DNA looping. [ 7 ]
Each monomer has 360 amino acids, so it has 1440 amino acids in total, and 154,520 Dalton of atomic mass. [ 8 ]
LacI finds its target operator DNA surprisingly fast. In vitro the search is 10-100 times faster than the theoretical upper limit for two particles searching for each other via diffusion in three dimensions (3D). [ 9 ] To explain the fast search, it was hypothesized that LacI and other transcription factors (TFs) find their binding sites by facilitated diffusion, a combination of free diffusion in 3D and 1D-sliding on the DNA. [ 10 ] During sliding the repressor is in contact with the DNA helix, sliding around and tracking its major groove, which speeds up the search process by extending the target length when the TF slides in onto the operator from the side. In vivo single-molecule experiments with E. coli cells have now tested and verified the facilitated diffusion model, and shown that the TF scans on average 45 bp during each sliding event, before the TF detaches spontaneously and resumes exploring the genome in 3D. [ 11 ] These experiments also suggest that LacI slides over the O 1 operator several times before binding, meaning that different DNA sequences can have different probabilities to be recognized at each encounter with the TF. This implies a trade-off between fast search on nonspecific sequences and binding to specific sequences. [ 11 ] In vivo and in vitro experiments have shown that it is this probability to recognize the operator that changes with DNA sequence, while the time the TF remains in the bound conformation on the operator changes less with sequence. [ 12 ] The TF often leaves the sequence it is intended to regulate, but at a strong target site, it almost always make a very short journey before finding the way back again. On the macroscopic scale, this looks like a stable interaction. This binding mechanism explains how DNA binding proteins manage to quickly search through the genome of the cell without getting stuck too long at sequences that resemble the true target.
An all-atom molecular dynamics simulation suggests that the transcription factor encounters a barrier of 1 k B T during sliding and 12 k B T for dissociation, implying that the repressor will slide over 8 bp on average before dissociating. [ 13 ] The in vivo search model for the lac repressor includes intersegment transfer and hopping as well as crowding by other proteins which make the genome in E. coli cells less accessible for the repressor. [ 14 ] The existence of hopping, where the protein slips out of the major groove of DNA to land in another nearby groove along the DNA chain, has been proven more directly in vitro , where the lac repressor has been observed to bypass operators, flip orientation, and rotate with a longer pitch than the 10.5 bp period of DNA while moving along it. [ 15 ]
The lac repressor was first isolated by Walter Gilbert and Benno Müller-Hill in 1966. [ 16 ] They showed that in vitro the protein bound to DNA containing the lac operon, and it released the DNA when IPTG (an analog of allolactose) was added. | https://en.wikipedia.org/wiki/Lac_repressor |
Lachryphagy is the practice of feeding on tears and other eye secretions. [ 1 ] Certain bees , butterflies , and flies have been observed feeding on the tears of reptiles , birds , and mammals , including humans. Lachryphagous insects gather nutrients, especially sodium and proteins , from the tears. Lachryphagous feeding can be unbothersome or painful, with some feeding insects damaging the eye and introducing pathogens to the host. Lachryphagy has been studied as a form of commensalism and puddling .
It is best known as a behavior of butterflies and moths ( Lepidoptera ). Flies, stingless bees, and other insects such as cockroaches and lice have also been recorded feeding on tears, and there is evidence that some stingless bee colonies in Southeast Asia have specialized tear collectors as a division of labor.
Lachryphagy is best known as a behavior of adult butterflies and moths ( Lepidoptera ). [ 1 ] [ 2 ] It was first recorded in moths in 1852. [ 2 ] Tear-drinking and eye-frequenting behavior has been observed in butterflies and moths throughout tropical and subtropical regions of Africa and Asia, particularly in their " savanna belts and monsoon regions ... where the dry season covers a period of at least three to four months, and where rainfall and humidity are too low for rainforests." [ 3 ] Wilhelm Büttiker and J. D. Bezuidenhout recorded the first accounts of lepidopteran lachryphagy in southwest Africa in 1974, of three species [ a ] in the genus Arcyophora . [ 3 ] They found these eye-frequenting moths drinking from the eyes of domestic cows and goats. [ 3 ] It is also documented in South and Central American moth species, and a single account of moth lachryphagy (from a horse's eye) in the United States was recorded in 1972. [ 4 ] [ 2 ]
Lachryphagy in lepidopterans is thought to have evolved from mud-puddling behavior, in contrast to hematophagous Lepidoptera whose behavior is thought to derive from fruit-piercing feeding. [ 2 ] Most moths observed drinking tears are male, suggesting that male lachryphagous moths collect sodium (which is essential to moth reproduction) to transfer to females during mating. [ 2 ] Tears form the largest part of the diets of most eye-frequenting moths. [ 4 ] Only two species, Lobocraspis griseifusa and Arcyophora sylvatica , are known to be eulachryphagous (feeding exclusively on tears). [ 2 ] [ 5 ] L. griseifusa secrete gut proteases not present in other lachryphagous moths, which allow them to digest proteins in tears. [ 2 ] [ 6 ] Lachryphagous behavior among moths seems to be limited to nocturnal species, likely because hosts are less alert at night. [ 7 ]
The wild and domesticated mammalian hosts (besides humans) of moths include antelope , elephants , horses , mules , pigs , sambar deer , sheep , tapirs , water buffalo , and cattle . [ 8 ] [ 5 ] Butterflies have been observed drinking the tears of reptiles, including turtles and crocodiles. [ 9 ] Lachryphagic moths generally do not specialize in any host species, [ 4 ] [ 5 ] and generally prefer large hosts, who are less reactive and sensitive to the moths. [ 2 ] [ 5 ] Lachryphagy on large hosts (such as crocodiles ) is generally observed during the day, while lachryphagy on small hosts is more commonly practiced at night. [ 10 ]
The first report of moths feeding on the tears of birds was published in 2007. The Malagasy moth Hemiceratoides hieroglyphica was found to probe its sharp proboscis into the closed eyes of sleeping birds at night to drink their tears. H. hieroglyphica 's proboscis is unique among those of lachryphagous moths, [ 2 ] being armored with "hooks, barbs, and spines" and "shaped like an ancient harpoon". [ 11 ] [ 4 ] The proboscis is similar in anatomy to those of fruit-eating and blood-sucking calpine moths, to which H. hieroglyphica is more closely related than it is to other lachryphagous moths. The researchers who observed it noted that H. hieroglyphica does not use its harpoon-like proboscis to pierce the bird's eyelid; they theorize that its barbs are used to secure the proboscis in place between the bird's eyelids while the moth feeds. This species is specialized to drinking the tears of birds because of the lack of large mammals in Madagascar, whose most common large-eyed mammals, such as lemurs, are nocturnal. [ 4 ] David Plotkin considers it likely that H. hieroglyphica is the "most competent vector for a disease agent" among lachryphagous moths. [ 2 ]
Lachryphagy of human tears by moths was first observed in 1966. According to Bänziger's 1972 study of lachryphagous moths in Southeast Asia , the people most exposed to tear-feeding moths are outdoor-sleeping herders, woodcutters, porters, and caravan travelers who walk forest paths during the monsoon season to transport goods such as betel nuts , opium , and cattle. [ 5 ] No moth is known to specialize in human tears. [ 2 ] Experiments by Bänziger (wearing unwashed clothes, deliberately strengthening his body odor, smearing his face with banteng tears) along with his observation that most incidents of human tear-feeding occurred when subjects were within 10 meters of a larger animal host suggest that humans are not attractive hosts—"certainly less so than water buffalo". [ 5 ] According to Bänziger, Thai villagers are often reluctant to associate eye diseases with moth lachryphagy "due to their respect for nature and fear of cosmic punishment." Many locals believe that illnesses of this kind are caused by forest spirits or are punishment for trespassing into sacred areas of jungle , and are hesitant in discussing their interactions with insects. They refrain from catching or harming lachryphagous moths, and do not try to swat them away even if they land on them, instead "suffer[ing] the irritation in silence unless the annoyance becomes unbearable." [ 5 ]
Lachryphagous moths drink tears directly from the eye as well as from tear-stained cheeks. [ 5 ] Most moths circle the head before landing quickly on the face, probing for any wetness (in the nose, mouth, eyes, or ears) before settling at the eye . [ 5 ] [ 8 ] By contrast, L. griseifusa flies directly to the eye, extending its proboscis to drink after "quivering for a few seconds" upon landing. [ 8 ] Both L. griefusa and A. sylvatica —the two known moths whose entire diet consists of tears—drink only directly from the eye, never from the cheek. [ 5 ] Crambids like Filodes fluvidorsalis rest with their wings open, and feed from afar with the proboscis extended, while lachryphagous Geometridae , whose wings are elevated and folded at rest, feed very close to the eye. The very large hawkmoth Rhagastis olivacea , whose spiny legs would likely cause discomfort to a host, hovers in the air with its proboscis extended to drink from a human eye "with minimal interference". Bänziger described the experience as painless and less irritating than other species, like a "cool, smooth foreign body moving between the lower lid and the cornea ." [ 12 ] This suggests a correlation between morphology and feeding position in lachryphagous moths, as more obtrusively-shaped moths must avoid causing irritation to the host in order to continue their drinking in peace. [ 2 ]
Groups of moths are commonly seen drinking together, and as many as 13 moths have been observed feeding on a single banteng's eye. Feeding groups are often mixed-species, and individual moths do not competitively drive each other away. Newcoming moths attempting to join groups of 8–10 may struggle to land and be driven away by the wingbeats of already-feeding moths as they attempt to wedge themselves into the group. Individual animals with heavy tear flow attract more moths. [ 5 ] Bänziger and his team noted that "certain persons were visited relatively often, while others never were." [ 8 ] Lachryphagous moths excrete excess water during and after tear-feeding. [ 5 ]
Hans Bänziger 's study of eye-frequenting moths in Northern Thailand describes mechanical damage to the eye caused by the moths' proboscises. [ 8 ] In a later study, Bänziger described Chaeopsestis ludovicae scratching the conjunctiva with its sharp tarsal claws , causing a sensation like "a grain of sand being rubbed between eye and lid" even as its proboscis produced no irritation. [ 2 ] Bänziger describes the sensation of Pionea aureolalis 's proboscis probing his eyeball as uncomfortable, inducing a flow of tears, but not painful. [ 8 ] Though Bänziger proposed that the probing around the eye was a deliberate maneuver to induce lachrymal flow, David Plotkin notes that this hypothesis is untested, and the probing behavior may simply be a search for an optimal feeding site. [ 2 ] Other moths, however, elicited "stinging pain". After 30 minutes of allowing a Lobocraspis griseifusa to drink from his closed eye, Bänziger was so irritated that he had to stop, and for the rest of the day his eye was red and inflamed and hard to keep open. [ 8 ] Plotkin suggested that the difference in sensation of being fed on with one's eye open (less irritating) and closed (very painful) may be caused by contact between the host's eyelid and triangular spines on the proboscis. Plotkin suggests that this may be strategic, so as to induce the host to keep its eye open, increasing available eye surface for drinking to allow for multiple moths to imbibe. [ 2 ] Rain generally reduces tear-feeding activity, though a brief increase is often seen at the start of a light shower. Wind strongly inhibits feeding, while temperature has little effect—moths are active in both cool and warm conditions. Activity typically rises after long dry periods or dry days following rain. Smoky fires lit by Thai farmers to keep biting insects from cattle were found to repel lachryphagous moths. Some lachryphagous moths also seemed to avoid bright lights, rarely landing on eyes illuminated by researchers' flashlights. [ 5 ]
According to Plotkin, "there is no known threat of disease from Lepidoptera that feed on human body fluids." [ 2 ] No instance of pathogenic transmission from adult moth lachryphagy has ever been documented; if it were to happen, the pathogen would likely move from the moth's salivary glands to the host's eye via the moth's proboscis. Pathogens could also be transmitted directly from the proboscis to the host's eye. Bänziger and Büttiker posit that a lachryphagous moth that feeds from one animal's wound and then from another animal's eye could transmit pathogens from the wound to the eye, and that scratching of the conjunctiva by the moth's legs (as was observed in Chaeopsestis ludovicae ) could transmit pathogens from the leg into the eye via the scratches. [ 8 ] [ 2 ] Studies of conjunctivitis in African mammals have implicated lachryphagous moths in the spread of eye infections on the continent, especially keratoconjunctivitis . [ 3 ] A 1995 study found some bacteria found in cattle with opthalmia in the proboscises of lachryphagous moths, implying inconclusively that lachryphagous moths can transmit harmful bacteria to cattle. [ 2 ] [ 13 ]
Many flies visit human and animal eyes for tears, especially those in the families Chloropidae , Cryptochetidae , Drosophilidae , and Muscidae . [ 14 ] Human lachryphagy was first recorded in 1921, when John Russell Malloch described the Steganin fly Amiota minor persistently trying to land on his hands and face, attracted to his sweat and tears. [ 1 ] Most Amiota are lachryphagous, attracted to the tears of humans and other animals. [ 1 ] The fly Phortica variegata infects humans and other animals with the larvae of the nematode Thelazia callipaeda , transmitting thelaziasis with its lachryphagous behavior. [ 1 ] Lachryphagous drosophilids tend to approach human faces in vertical zig-zagging motion. [ 14 ]
Lachryphagy is widespread among stingless bees , who have been observed collecting tears from "humans, zebu, dogs, cats, rabbits, chickens, and yellow tortoises ", with chickens being the least sensitive and reactive to the bees. [ 14 ] An uncertain observation of a stingless bee "molesting a goliath frog on its eye" has also been documented. [ 14 ] Bees of the genus Lisotrigona have evolved specialized abilities—likely involving both sight and smell—to "detect and recognize" the differently shaped eyes of various vertebrates in order to harvest their tears. [ 14 ]
Worker bees of the Chinese species Ebaiotrigona carpenteri [ b ] occasionally switch from their typical sudophagy (sweat-sucking) to lachryphagy in spring and summer, [ 15 ] possibly due to the relatively richer protein content in tears as compared to sweat (200 times greater), and as an adaptation to times of decreased availability of nectar and pollen. [ 15 ] [ 14 ] It has also been proposed that lachryphagy and sudophagy in Lisotrigona represents a state of evolutionary transition from carnivory to highly specialized nectarivorous and palynivorous feeding behavior. [ 15 ] Other theories posit that stingless bee lachryphagy evolved in connection with contact between humans and bees; that it evolved from pursuit of moisture in the eyes of birds during a dry period 30–70 million years ago, or from dinosaurs in the late Cretaceous , when flowering plants were uncommon. [ 14 ] Lachryphagy in stingless bees may have developed, been lost, and redeveloped multiple times. [ 14 ]
Hans Bänziger proposes that Lisotrigona colonies include specialized tear collectors as a division of labor, storing collected tears in cells alongside honey and pollen for use by other bees in the colony. [ 16 ] [ 14 ] He also proposed that tears may be used to dilute honey or to produce food for larvae. [ 14 ] According to Bänziger, Lisotrigona likely regurgitate the tears they collect—either sharing them via trophallaxis or storing them in nest containers—rather than excreting them, since the dissolved proteins in tears would be digested and used as nutrition. Though pollen is richer in protein overall, tear proteins are more readily digestible and include lysozyme , which may help prevent spoilage. Tear-harvesting is energetically efficient, can yield larger loads than pollen, and offers a continuous, year-round protein source—especially valuable for small bees like Lisotrigona with limited foraging ranges and intermittent access to flowers. [ 14 ] Tears may also be regurgitated onto the surface of the nest and fanned to cool it during periods of life-threatening heat. [ 14 ] Stingless bee lachryphagy rarely occurs near the bee's nest. [ 17 ] The maximum observed distance a host was recorded traveling before a Lisotrigona tear-collector flew back to the nest is 680 meters—a great distance for such a small bee. [ 14 ]
Bänziger observed marked Lisotrigona cacciae and Lisotrigona furva workers repeatedly returning to human eyes to collect tears over spans that can last from hours to multiple days. The individual visits averaged around 2 minutes long, with the same worker bee visiting human eyes up to 148 times in one day. None of the bees observed drinking human tears carried pollen loads. [ 14 ] The same was observed in the lachryphagous Pariotrigona klossi . [ 17 ] Lisotrigona tear-collectors notably were not observed collecting salt from other sources such as sweat, which is more available and easily collected. This suggests that tear collection by Lisotrigona is driven by pursuit of proteins, not salt. [ 14 ] Lisotrigona also often collects tears from animals which do not sweat, showing that their lachryphagy is not incidental or secondary to sudophagy. [ 14 ] Lisotrigona generally approach human eyes in horizontal zig-zagging motion, landing gently at the corner of the eye to suck tears from between the eyelid and eyeball with their tiny proboscises . They crawl on the face with their claws retracted, making contact with the skin using only their soft tarsal pads. Bänziger describes the sensation as subtle and not bothersome— P. klossi , he notes, "gently sipped his tears in peaceful congregations"—though the effect can become irritating when large numbers of bees are involved. [ 14 ] [ 16 ] Lisotrigona scent-mark eyes once they finish collecting tears, making the hosts easy to locate for future visits. Exponential increases in the number of pursuant bees following the initial visit suggests that the tear-collectors recruit fellows from their colonies to imbibe in certain hosts' tears. [ 14 ] It is unlikely that stingless bees can transmit pathogens to the eyes of their hosts. [ 14 ]
Cockroaches in the Amazon have been observed feeding on the tears of anoles , which are natural predators of cockroaches. [ 10 ] Such lachryphagy, lasting several minutes, is practiced at night, while the anole is resting, to mitigate the risk of predation. [ 10 ] The scientists who observed this behavior, the first report of cockroach lachryphagy, linked it with the presence of nutrients essential for reproduction, particularly uric acid , in the tears of anoles. [ 10 ]
Lachryphagy was first observed in chewing lice in 1734, and has been documented across 14 species belonging to 9 genera. Chewing lice have been observed feeding on the tears of both living and "freshly dead" hosts belonging to 10 families. [ 18 ] [ 19 ] | https://en.wikipedia.org/wiki/Lachryphagy |
Lack's principle , proposed by the British ornithologist David Lack in 1954, states that "the clutch size of each species of bird has been adapted by natural selection to correspond with the largest number of young for which the parents can, on average, provide enough food". [ 1 ] As a biological rule , the principle can be formalised and generalised to apply to reproducing organisms in general, including animals and plants. Work based on Lack's principle by George C. Williams and others has led to an improved mathematical understanding of population biology .
Lack's principle implies that birds that happen to lay more eggs than the optimum will most likely have fewer fledglings (young that successfully fly from the nest) because the parent birds will be unable to collect enough food for them all. [ 1 ] Evolutionary biologist George C. Williams notes that the argument applies also to organisms other than birds, both animals and plants, giving the example of the production of ovules by seed plants as an equivalent case. Williams formalised the argument to create a mathematical theory of evolutionary decision-making , based on the framework outlined in 1930 by R. A. Fisher , namely that the effort spent on reproduction must be worth the cost, compared to the long-term reproductive fitness of the individual. [ 2 ] Williams noted that this would contribute to the discussion on whether (as Lack argued) an organism's reproductive processes are tuned to serve its own reproductive interest ( natural selection ), or as V.C. Wynne-Edwards proposed, [ 3 ] to increase the chances of survival of the species to which the individual belonged ( group selection ). The zoologist J.L. Cloudsley-Thompson argued that a large bird would be able to produce more young than a small bird. [ 4 ] Williams replied that this would be a bad reproductive strategy, as large birds have lower mortality and therefore a higher residual reproductive value over their whole lives (so taking a large short-term risk is unjustified). [ 5 ] Williams' reply "is one of the most cited papers in life history evolution because it ... made it conceptually possible to find the optimal life history strategies in age-structured populations". [ 6 ] | https://en.wikipedia.org/wiki/Lack's_principle |
Lacquer is a type of hard and usually shiny coating or finish applied to materials such as wood or metal. It is most often made from resin extracted from trees and waxes and has been in use since antiquity. [ 1 ]
Asian lacquerware , which may be called "true lacquer", are objects coated with the treated, dyed and dried sap of Toxicodendron vernicifluum or related trees, applied in several coats to a base that is usually wood. This dries to a very hard and smooth surface layer which is durable, waterproof, and attractive in feel and look. Asian lacquer is sometimes painted with pictures, inlaid with shell and other materials, or carved , as well as dusted with gold and given other further decorative treatments.
In modern techniques, lacquer means a range of clear or pigmented coatings that dry by solvent evaporation to produce a hard, durable finish. The finish can be of any sheen level from ultra matte to high gloss , and it can be further polished as required. Lacquer finishes are usually harder and more brittle than oil-based or latex paints and are typically used on hard and smooth surfaces. [ citation needed ]
In terms of modern finishing products, finishes based on shellac dissolved in alcohol are often called shellac or lac to distinguish them from synthetic lacquer, often called simply lacquer , which consists of synthetic polymers (such as nitrocellulose , cellulose acetate butyrate ("CAB"), or acrylic resin ) dissolved in lacquer thinner , a mixture of various organic solvents . [ 2 ] Although synthetic lacquer is more durable than shellac, traditional shellac finishes are nevertheless often preferred for their aesthetic characteristics, as with French polish , as well as their "all-natural" and generally food-safe ingredients.
The English lacquer is from the archaic French word lacre , "a kind of sealing wax", from Portuguese lacre , itself an unexplained variant of Medieval Latin lacca "resinous substance," from Arabic lakk ( لك ), from Persian lāk ( لاک ), from Hindi lākh ( लाख ); Prakrit lakkha , 𑀮𑀓𑁆𑀔 ), [ 3 ] [ 4 ] [ 5 ] [ 6 ] itself from the Sanskrit word lākshā ( लाक्षा ) for lac bug , representing the number one hundred thousand (100,000), used as wood finish in ancient India and neighbouring areas. [ 7 ]
Lacquer sheen is a measurement of the shine for a given lacquer. [ 8 ] Different manufacturers have their own names and standards for their sheen. [ 8 ] The most common names from least shiny to most shiny are: flat, matte, eggshell, satin, semi-gloss, and gloss (high).
In India shellac derived from insect lac was used since ancient times. Shellac is the secretion of the lac bug ( Tachardia lacca Kerr. or Laccifer lacca ). It is used for wood finish, lacquerware, skin cosmetic, ornaments, dye for textiles, production of different grades of shellac for surface coating. [ 7 ] [ 9 ] [ 10 ]
Urushiol -based lacquers differ from most others, being slow-drying, and set by oxidation and polymerization , rather than by evaporation alone. The active ingredient of the resin is urushiol, a mixture of various phenols suspended in water, plus a few proteins. In order for it to set properly it requires a humid and warm environment. The phenols oxidize and polymerize under the action of laccase enzymes, yielding a substrate that, upon proper evaporation of its water content, is hard. These lacquers produce very hard, durable finishes that are both beautiful and very resistant to damage by water, acid, alkali or abrasion. The resin is derived from trees indigenous to East Asia, like lacquer tree Toxicodendron vernicifluum , and wax tree Toxicodendron succedaneum . [ 11 ] The fresh resin from the T. vernicifluum trees causes urushiol-induced contact dermatitis and great care is therefore required in its use. The Chinese treated the allergic reaction with crushed shellfish, which supposedly prevents lacquer from drying properly. [ 12 ] Lacquer skills became very highly developed in Asia, and many highly decorated pieces were produced.
It has been confirmed that the lacquer tree has existed in Japan since nearly 12,600 years ago in the incipient Jōmon period . This was confirmed by radioactive carbon dating of the lacquer tree found at the Torihama shell mound and is the oldest lacquer tree in the world found as of 2011. [ 13 ] Lacquer was used in Japan as early as 7000 BCE, during the Jōmon period. Evidence for the earliest lacquerware was discovered at the Kakinoshima "B" Excavation Site in Hokkaido . The ornaments woven with lacquered red thread were discovered in a pit grave dating from the first half of the Initial Jōmon period. Also, at Kakinoshima "A" Excavation Site, earthenware with a spout painted with vermilion lacquer, which was made 3200 years ago, was found almost completely intact. [ 14 ] [ 15 ] [ 13 ]
During the Shang dynasty (1600–1046 BC), the sophisticated techniques used in the lacquer process were first developed and it became a highly artistic craft, [ 16 ] although various prehistoric lacquerwares have been unearthed in China dating back to the Neolithic period. [ 16 ] The earliest extant Chinese lacquer object, a red wooden bowl, [ 17 ] was unearthed at a Hemudu culture (5000–4500 BC) site in China. [ 18 ] By the Han dynasty (206 BC – 220 AD), many centres of lacquer production became firmly established. [ 16 ] The knowledge of the Chinese methods of the lacquer process spread from China during the Han , Tang and Song dynasties. Eventually it was introduced to Korea and Japan. [ 19 ]
Trade of lacquer objects travelled through various routes to the Middle East. Known applications of lacquer in China included coffins, music instruments, furniture, and various household items. [ 16 ] Lacquer mixed with powdered cinnabar is used to produce the traditional red lacquerware from China.
From the 16th century to the 17th century, lacquer was introduced to Europe on a large scale for the first time through trade with Japanese . Until the 19th century, lacquerware was one of Japan's major exports, and European royalty, aristocrats and religious people represented by Marie-Antoinette , Maria Theresa and The Society of Jesus collected Japanese lacquerware luxuriously decorated with maki-e . [ 20 ] [ 21 ] The terms related to lacquer such as " Japanning ", "Urushiol" and " maque " which means lacquer in Mexican Spanish, are derived from Japanese. [ 22 ] [ 23 ]
The trees must be at least ten years old before cutting to bleed the resin. It sets by a process called "aqua-polymerization", absorbing oxygen to set; placing in a humid environment allows it to absorb more oxygen from the evaporation of the water.
Lacquer-yielding trees in Thailand , Vietnam , Burma and Taiwan , called Thitsi , are slightly different; they do not contain urushiol, but similar substances called laccol or thitsiol. The result is similar but softer than the Chinese or Japanese lacquer. Burmese lacquer sets slower, and is painted by craftsmen's hands without using brushes.
Raw lacquer can be "coloured" by the addition of small amounts of iron oxides , giving red or black depending on the oxide. There is some evidence that its use is even older than 8,000 years from archaeological digs in Japan and China. Later, pigments were added to make colours. It is used not only as a finish, because if mixed with ground fired and unfired clays applied to a mould with layers of hemp cloth, it can produce objects without need for another core like wood. The process is called "kanshitsu" in Japan. In the lacquering of the Chinese musical instrument, the guqin , the lacquer is mixed with deer horn powder (or ceramic powder) to give it more strength so it can stand up to the fingering.
There are a number of forms of urushiol. They vary by the length of the R chain, which depends on the species of plant producing the urushiol. Urushiol can also vary in the degree of saturation in the carbon chain. Urushiol can be drawn as follows: , where:
R = (CH 2 ) 14 CH 3 or R = (CH 2 ) 7 CH=CH(CH 2 ) 5 CH 3 or R = (CH 2 ) 7 CH=CHCH 2 CH=CH(CH 2 ) 2 CH 3 or R = (CH 2 ) 7 CH=CHCH 2 CH=CHCH=CHCH 3 or R = (CH 2 ) 7 CH=CHCH 2 CH=CHCH 2 CH=CH 2
Types of lacquer vary from place to place but they can be divided into unprocessed and processed categories.
The basic unprocessed lacquer is called raw lacquer (生漆: ki-urushi in Japanese, shengqi in Chinese). This is directly from the tree itself with some impurities filtered out. Raw lacquer has a water content of around 25% and appears in a light brown colour. This comes in a standard grade made from Chinese lacquer, which is generally used for ground layers by mixing with a powder, and a high-quality grade made from Japanese lacquer called kijomi-urushi (生正味漆) which is used for the last finishing layers.
The processed form (in which the lacquer is stirred continuously until much of the water content has evaporated) is called guangqi (光漆) in Chinese but comes under many different Japanese names depending on the variation, for example, kijiro-urushi (木地呂漆) is standard transparent lacquer sometimes used with pigments and kuroroiro-urushi (黒呂色漆) is the same but pre-mixed with iron hydroxide to produce a black coloured lacquer. Nashiji-urushi (梨子地漆) is the transparent lacquer but mixed with gamboge to create a yellow-tinged lacquer and is especially used for the sprinkled-gold technique. These lacquers are generally used for the middle layers. Japanese lacquers of this type are generally used for the top layers and are prefixed by the word jo- (上) which means 'top (layer)'.
Processed lacquers can have oil added to them to make them glossy, for example, shuai-urushi (朱合漆) is mixed with linseed oil. Other specialist lacquers include ikkake-urushi (釦漆) which is thick and used mainly for applying gold or silver leaf.
Solvent-based dipping lacquers that contain nitrocellulose , a resin obtained from the nitration of cotton and other cellulosic materials, debuted in the 19th century along with nitrocellulose's other commercial applications. They were used, for example, on brass items such as musical instruments. Faster-drying and more durable versions of these lacquers were developed in the early 1920s, when the end of the WWI caused a massive overcapacity of nitrocellulose production, and soon greatly displaced much use of the slower-drying paints and lacquers that preceded them; they were extensively used in the automotive industry and others for the next 30 years until further chemical advancements replaced them. Prior to their introduction, mass-produced automotive finishes were limited in colour, damaged easily, and took a long time to dry, [ 24 ] : 295–301 with Japan black being the fastest drying and thus the most economical to use.
The problem with using nitrocellulose in lacquers was its high viscosity, which necessitated dilution of the product with large amounts of thinner for application, leaving only a very thin film of finish not durable enough for outdoor use. This problem was overcome by decreasing the viscosity of the polymer (the term actually post-dates the empirical solution, with Staudinger 's modern structural theory explaining polymer solution viscosity by length of molecular chains not yet experimentally proven in 1920s) with heat treatments, either with 2% of mineral acid or in an autoclave at considerable pressure. [ 25 ]
The first practical nitrocellulose enamel Glossy White S.2567, still for interior use, was introduced in 1919 in the UK by Nobel Explosives . [ 26 ] In 1923, General Motors' Oakland brand automobile was the first to introduce one of the new fast-drying nitrocellulose lacquers, a bright blue, produced by DuPont under their Duco tradename. [ 24 ] : 295–301 In 1924 the other GM makes followed suit, and by 1925 nitrocellulose lacquers were thoroughly disrupting the traditional paint business for automobiles, appliances, furniture, musical instruments, caskets, and other products. [ 24 ] : 295–301 Henry Ford and, in the UK, Herbert Austin were introducing nitrocellulose lacquers at the same time, and soon the market flourished.
Nitrocellulose lacquers are also used to make firework fuses waterproof. The nitrocellulose and other resins and plasticizers are dissolved in the solvent, and each coat of lacquer dissolves some of the previous coat. These lacquers were a huge improvement over earlier automobile and furniture finishes, both in ease of application and in colour retention. The preferred method of applying quick-drying lacquers is by spraying, and the development of nitrocellulose lacquers led to the first extensive use of spray guns. Nitrocellulose lacquers produce a hard yet flexible, durable finish that can be polished to a high sheen. Drawbacks of these lacquers include the hazardous nature of the solvent, which is flammable and toxic, and the hazards of nitrocellulose in the manufacturing process. The lacquer grade of soluble nitrocellulose is closely related to the more highly nitrated form which is used to make explosives. They become relatively non-toxic after approximately a month since, at this point, the lacquer has evaporated most of the solvents used in its production.
Lacquers using acrylic resin , a synthetic polymer, were developed in the 1950s. Acrylic resin is colourless, transparent thermoplastic , obtained by the polymerization of derivatives of acrylic acid . Acrylic is also used in enamel paints , which have the advantage of not needing to be buffed to obtain a shine. Enamels, however, are slow drying. The advantage of acrylic lacquer is its exceptionally fast drying time. The use of lacquers in automobile finishes was discontinued when tougher, more durable, weather- and chemical-resistant two-component polyurethane coatings were developed. The system usually consists of a primer, colour coat and clear topcoat, commonly known as clear coat finishes.
Due to health risks and environmental considerations involved in the use of solvent-based lacquers, much work has gone into the development of water-based lacquers. Such lacquers are considerably less toxic, more environmentally friendly, and, in many cases, produce acceptable results. While water-based lacquer's fumes are considerably less hazardous, and it does not have the combustibility issues of solvent-based lacquers, the product still dries fairly quickly. Even though its odor is weaker, water-based lacquers can still produce airborne particulates that can get into the lungs, so proper protective wear still needs to be worn. More and more water-based colored lacquers are replacing solvent-based clear and colored lacquers in under-hood and interior applications in the automobile and other similar industrial applications. Water-based lacquers are used extensively in wood furniture finishing as well.
One drawback of water-based lacquer is that it has a tendency to be highly reactive to other fresh finishes such as quick-dry primer (excluding waterborne lacquer primers), caulking and even some paints that have a paint/primer aspect. Tannin bleed-through can also be an issue, depending on the brand of lacquer used. Once it happens, there is no easy fix as the lacquer is so reactive to other products.
Water-based lacquer used for wood finishing is also not rated for exterior wear, unless otherwise specified.
Just as china is a common name for porcelain , japanning is an old name to describe the European technique to imitate Asian lacquerware . [ 27 ] As Asian lacquer work became popular in England, France, the Netherlands, and Spain in the 17th century, the Europeans developed imitation techniques. The European technique, which is used on furniture and other objects, uses finishes that have a resin base similar to shellac. The technique, which became known as japanning, involves applying several coats of varnish which are each heat-dried and polished. In the 18th century, japanning gained a large popular following. Although traditionally a pottery and wood coating, japanning was the popular (mostly black) coating of the accelerating metalware industry. By the twentieth century, the term was freely applied to coatings based on various varnishes and lacquers besides the traditional shellac. | https://en.wikipedia.org/wiki/Lacquer |
Lactacystin is an organic compound naturally synthesized by bacteria of the genus Streptomyces first identified as an inducer of neuritogenesis in neuroblastoma cells in 1991. [ 1 ] The target of lactacystin was subsequently found to be the proteasome on the basis of its affinity for certain catalytic subunits of the proteasome by Fenteany and co-workers in 1995. [ 2 ] The proteasome is a protein complex responsible for the bulk of proteolysis in the cell, as well as proteolytic activation of certain protein substrates. Lactacystin was the first non-peptidic proteasome inhibitor discovered and is widely used as a research tool in biochemistry and cell biology. The transformation product of lactacystin clasto-lactacystin β-lactone (also known as omuralide) covalently modifies the amino-terminal threonine of specific catalytic subunits of the proteasome, a discovery that helped to establish the proteasome as a mechanistically novel class of protease: an amino-terminal threonine protease . The molecule is commonly used in biochemistry and cell biology laboratories as a selective inhibitor of the proteasome . [ 2 ] [ 3 ] The first total synthesis of lactacystin was developed in 1992 by Corey and Reichard, [ 4 ] and a number of other syntheses of this molecule have also been published. There are more than 1,660 entries for lactacystin in PubMed as of January 2019. | https://en.wikipedia.org/wiki/Lactacystin |
A lactam is a cyclic amide , formally derived from an amino alkanoic acid through cyclization reactions. The term is a portmanteau of the words lactone + amide .
Greek prefixes in alphabetical order indicate ring size.
This ring-size nomenclature stems from the fact that hydrolysis of an α-lactam gives an α- amino acid and that of a β-Lactam gives a β-amino acid, and so on.
General synthetic methods are used for the organic synthesis of lactams.
Lactams form by the acid-catalyzed rearrangement of oximes in the Beckmann rearrangement .
Lactams form from cyclic ketones and hydrazoic acid in the Schmidt reaction . Cyclohexanone with hydrazoic acid, forms ε - Caprolactum , which upon treatment with excess acid forms Cardiazole , a heart stimulant.
Lactams can be formed from cyclisation of amino acids via the coupling between an amine and a carboxylic acid within the same molecule.
Lactamization is most efficient in this way if the product is a γ-lactam. For example, Fmoc-Dab(Mtt)-OH, although its side-chain amine is sterically protected by extremely bulky 4-Methyltrityl (Mtt) group, the amine can still intramolecularly couple with the carboxylic acid to form a γ-lactam. This reaction almost finished within 5 minutes with many coupling reagents (e.g. HATU and PyAOP ). [ 1 ]
Lactams form from intramolecular attack of linear acyl derivatives from the nucleophilic abstraction reaction.
An iminium ion reacts with a halonium ion formed in situ by reaction of an alkene with iodine . [ 2 ]
Lactams form by copper-catalyzed 1,3-dipolar cycloaddition of alkynes and nitrones in the Kinugasa reaction
Diels-Alder reaction between cyclopentadiene and chlorosulfonyl isocyanate (CSI) can be utilized to obtain both β- as well as γ-lactam. At lower temp (−78 °C), β-lactam is the preferred product. At optimum temperatures, a highly useful γ-lactam known as Vince Lactam [ 3 ] is obtained. [ 4 ]
A lactim is a cyclic imidic acid compound characterized by an endocyclic carbon-nitrogen double bond . They are formed when lactams undergo tautomerization . | https://en.wikipedia.org/wiki/Lactam |
Lactate dehydrogenase ( LDH or LD ) is an enzyme found in nearly all living cells. LDH catalyzes the conversion of pyruvate to lactate and back, as it converts NAD + to NADH and back. A dehydrogenase is an enzyme that transfers a hydride from one molecule to another.
LDH exists in four distinct enzyme classes. This article is specifically about the NAD(P) -dependent L -lactate dehydrogenase. Other LDHs act on D -lactate and/or are dependent on cytochrome c : D-lactate dehydrogenase (cytochrome) and L-lactate dehydrogenase (cytochrome) .
LDH is expressed extensively in body tissues, such as blood cells and heart muscle. Because it is released during tissue damage, it is a marker of common injuries and disease such as heart failure.
Lactate dehydrogenase catalyzes the interconversion of pyruvate and lactate with concomitant interconversion of NADH and NAD + . It converts pyruvate, the final product of glycolysis , to lactate when oxygen is absent or in short supply, and it performs the reverse reaction during the Cori cycle in the liver . At high concentrations of lactate, the enzyme exhibits feedback inhibition, and the rate of conversion of pyruvate to lactate is decreased. It also catalyzes the dehydrogenation of 2-hydroxybutyrate , but this is a much poorer substrate than lactate.
LDH in humans uses His (193) as the proton acceptor, and works in unison with the coenzyme ( Arg 99 and Asn 138), and substrate (Arg106; Arg169; Thr 248) binding residues. [ 1 ] [ 2 ] The His(193) active site, is not only found in the human form of LDH, but is found in many different animals, showing the convergent evolution of LDH. The two different subunits of LDH (LDHA, also known as the M subunit of LDH, and LDHB, also known as the H subunit of LDH) both retain the same active site and the same amino acids participating in the reaction. The noticeable difference between the two subunits that make up LDH's tertiary structure is the replacement of alanine (in the M chain) with a glutamine (in the H chain). This tiny but notable change is believed to be the reason the H subunit can bind NAD faster, and the M subunit's catalytic activity isn't reduced in the presence of acetylpyridine adenine dinucleotide, whereas the H subunit's activity is reduced fivefold. [ 3 ]
Enzymatically active lactate dehydrogenase is consisting of four subunits (tetramer). The two most common subunits are the LDH-M and LDH-H peptides, named for their discovery in muscle and heart tissue, and encoded by the LDHA and LDHB genes, respectively. These two subunits can form five possible tetramers (isoenzymes): LDH-1 (4H), LDH-5 (4M), and the three mixed tetramers (LDH-2/3H1M, LDH-3/2H2M, LDH-4/1H3M). These five isoforms are enzymatically similar but show different tissue distribution.
LDH-2 is usually the predominant form in the serum . An LDH-1 level higher than the LDH-2 level (a "flipped pattern") suggests myocardial infarction (damage to heart tissues releases heart LDH, which is rich in LDH-1, into the bloodstream). The use of this phenomenon to diagnose infarction has been largely superseded by the use of Troponin I or T measurement. [ citation needed ]
There are two more mammalian LDH subunits that can be included in LDH tetramers: LDHC and LDHBx. LDHC is a testes-specific LDH protein, that is encoded by the LDHC gene. LDHBx is a peroxisome -specific LDH protein. LDHBx is the readthrough-form of LDHB. LDHBx is generated by translation of the LDHB mRNA , but the stop codon is interpreted as an amino acid -encoding codon . In consequence, translation continues to the next stop codon. This leads to the addition of seven amino acid residues to the normal LDH-H protein. The extension contains a peroxisomal targeting signal , so that LDHBx is imported into the peroxisome. [ 6 ]
The family also contains L-lactate dehydrogenases that catalyse the conversion of pyruvate to L-lactate , the last step in anaerobic glycolysis. Malate dehydrogenases that catalyse the interconversion of malate to oxaloacetate and participate in the citric acid cycle, and L-2-hydroxyisocaproate dehydrogenases are also members of the family. The N-terminus is a Rossmann NAD-binding fold and the C-terminus is an unusual alpha+beta fold. [ 7 ] [ 8 ]
Click on genes, proteins and metabolites below to link to respective articles. [ § 1 ]
This protein may use the morpheein model of allosteric regulation . [ 9 ]
Ethanol is dehydrogenated to acetaldehyde by alcohol dehydrogenase , and further into acetyl CoA by acetaldehyde dehydrogenase . During this reaction 2 NADH are produced. If large amounts of ethanol are present, then large amounts of NADH are produced, leading to a depletion of NAD + . Thus, the conversion of pyruvate to lactate is increased due to the associated regeneration of NAD + . Therefore, anion-gap metabolic acidosis ( lactic acidosis ) may ensue in ethanol poisoning .
The increased NADH/NAD+ ratio also can cause hypoglycemia in an (otherwise) fasting individual who has been drinking and is dependent on gluconeogenesis to maintain blood glucose levels. Alanine and lactate are major gluconeogenic precursors that enter gluconeogenesis as pyruvate. The high NADH/NAD+ ratio shifts the lactate dehydrogenase equilibrium to lactate, so that less pyruvate can be formed and, therefore, gluconeogenesis is impaired.
LDH is also regulated by the relative concentrations of its substrates. LDH becomes more active under periods of extreme muscular output due to an increase in substrates for the LDH reaction. When skeletal muscles are pushed to produce high levels of power, the demand for ATP in regards to aerobic ATP supply leads to an accumulation of free ADP, AMP, and Pi. The subsequent glycolytic flux, specifically production of pyruvate, exceeds the capacity for pyruvate dehydrogenase and other shuttle enzymes to metabolize pyruvate. The flux through LDH increases in response to increased levels of pyruvate and NADH to metabolize pyruvate into lactate. [ 10 ]
LDH undergoes transcriptional regulation by PGC-1α. PGC-1α regulates LDH by decreasing LDH A mRNA transcription and the enzymatic activity of pyruvate to lactate conversion. [ 11 ]
The M and H subunits are encoded by two different genes :
Mutations of the M subunit have been linked to the rare disease exertional myoglobinuria (see OMIM article), and mutations of the H subunit have been described but do not appear to lead to disease.
In rare cases, a mutation in the genes controlling the production of lactate dehydrogenase will lead to a medical condition known as lactate dehydrogenase deficiency. Depending on which gene carries the mutation, one of two types will occur: either lactate dehydrogenase-A deficiency (also known as glycogen storage disease XI) or lactate dehydrogenase-B deficiency. Both of these conditions affect how the body breaks down sugars, primarily in certain muscle cells. Lactate dehydrogenase-A deficiency is caused by a mutation to the LDHA gene, while lactate dehydrogenase-B deficiency is caused by a mutation to the LDHB gene. [ 12 ]
This condition is inherited in an autosomal recessive pattern, meaning that both parents must contribute a mutated gene in order for this condition to be expressed. [ 13 ]
A complete lactate dehydrogenase enzyme consists of four protein subunits . [ 14 ] Since the two most common subunits found in lactate dehydrogenase are encoded by the LDHA and LDHB genes, either variation of this disease causes abnormalities in many of the lactate dehydrogenase enzymes found in the body. In the case of lactate dehydrogenase-A deficiency, mutations to the LDHA gene result in the production of an abnormal lactate dehydrogenase-A subunit that cannot bind to the other subunits to form the complete enzyme . This lack of a functional subunit reduces the amount of enzyme formed, leading to an overall decrease in activity. During the anaerobic phase of glycolysis (the Cori cycle ), the mutated enzyme is unable to convert pyruvate into lactate to produce the extra energy the cells need. Since this subunit has the highest concentration in the LDH enzymes found in the skeletal muscles (which are the primary muscles responsible for movement), high-intensity physical activity will lead to an insufficient amount of energy being produced during this anaerobic phase. [ 15 ] This in turn will cause the muscle tissue to weaken and eventually break down, a condition known as rhabdomyolysis . The process of rhabdomyolysis also releases myoglobin into the blood, which will eventually end up in the urine and cause it to become red or brown: another condition known as myoglobinuria . [ 16 ] Some other common symptoms are exercise intolerance, which consists of fatigue, muscle pain, and cramps during exercise, and skin rashes. [ 17 ] [ 18 ] In severe cases, myoglobinuria can damage the kidneys and lead to life-threatening kidney failure. [ 19 ] In order to obtain a definitive diagnosis, a muscle biopsy may be performed to confirm low or absent LDH activity. There is currently no specific treatment for this condition.
In the case of lactate dehydrogenase-B deficiency, mutations to the LDHB gene result in the production of an abnormal lactate dehydrogenase-B subunit that cannot bind to the other subunits to form the complete enzyme. As with lactate dehydrogenase-A deficiency, this mutation reduces the overall effectiveness in the enzyme. [ 20 ] However, there are some major differences between these two cases. The first is the location where the condition manifests itself. With lactate dehydrogenase-B deficiency, the highest concentration of B subunits can be found within the cardiac muscle, or the heart. Within the heart, lactate dehydrogenase plays the role of converting lactate back into pyruvate so that the pyruvate can be used again to create more energy. [ 21 ] With the mutated enzyme, the overall rate of this conversion is decreased. However, unlike lactate dehydrogenase-A deficiency, this mutation does not appear to cause any symptoms or health problems linked to this condition. [ 18 ] [ 22 ] At the present moment, it is unclear why this is the case. Affected individuals are usually discovered only when routine blood tests indicate low LDH levels present within the blood.
The onset of acidosis during periods of intense exercise is commonly attributed to accumulation of hydrogens that are dissociated from lactate. Previously, lactic acid was thought to cause fatigue. From this reasoning, the idea of lactate production being a primary cause of muscle fatigue during exercise was widely adopted. A closer, mechanistic analysis of lactate production under “anaerobic” conditions shows that there is no biochemical evidence for the production of lactate through LDH contributing to acidosis. While LDH activity is correlated to muscle fatigue, [ 23 ] the production of lactate by means of the LDH complex works as a system to delay the onset of muscle fatigue. George Brooks and Colleagues at UC Berkeley where the lactate shuttle was discovered showed that lactate was actually a metabolic fuel not a waste product or the cause of fatigue.
LDH works to prevent muscular failure and fatigue in multiple ways. The lactate-forming reaction generates cytosolic NAD+, which feeds into the glyceraldehyde 3-phosphate dehydrogenase reaction to help maintain cytosolic redox potential and promote substrate flux through the second phase of glycolysis to promote ATP generation. This, in effect, provides more energy to contracting muscles under heavy workloads. The production and removal of lactate from the cell also ejects a proton consumed in the LDH reaction- the removal of excess protons produced in the wake of this fermentation reaction serves to act as a buffer system for muscle acidosis. [ citation needed ] Once proton accumulation exceeds the rate of uptake in lactate production and removal through the LDH symport, [ 24 ] muscular acidosis occurs.
On blood tests , an elevated level of lactate dehydrogenase usually indicates tissue damage, which has multiple potential causes, reflecting its widespread tissue distribution:
Low and normal levels of LDH do not usually indicate any pathology. [ 28 ] Low levels may be caused by large intake of vitamin C .
LDH is a protein that normally appears throughout the body in small amounts .
Many cancers can raise LDH levels, so LDH may be used as a tumor marker , but at the same time, it is not useful in identifying a specific kind of cancer. Measuring LDH levels can be helpful in monitoring treatment for cancer. Noncancerous conditions that can raise LDH levels include heart failure, hypothyroidism, anemia, pre-eclampsia, meningitis, encephalitis, acute pancreatitis, HIV and lung or liver disease. [ 30 ]
Tissue breakdown releases LDH, and therefore, LDH can be measured as a surrogate for tissue breakdown (e.g., hemolysis ). LDH is measured by the lactate dehydrogenase (LDH) test (also known as the LDH test or lactic acid dehydrogenase test). Comparison of the measured LDH values with the normal range help guide diagnosis. [ 31 ]
In medicine , LDH is often used as a marker of tissue breakdown as LDH is abundant in red blood cells and can function as a marker for hemolysis . A blood sample that has been handled incorrectly can show false-positively high levels of LDH due to erythrocyte damage.
It can also be used as a marker of myocardial infarction . Following a myocardial infarction, levels of LDH peak at 3–4 days and remain elevated for up to 10 days. In this way, elevated levels of LDH (where the level of LDH1 is higher than that of LDH2, i.e. the LDH Flip, as normally, in serum, LDH2 is higher than LDH1) can be useful for determining whether a patient has had a myocardial infarction if they come to doctors several days after an episode of chest pain.
Other uses are assessment of tissue breakdown in general; this is possible when there are no other indicators of hemolysis . It is used to follow up cancer (especially lymphoma ) patients, as cancer cells have a high rate of turnover, with destroyed cells leading to an elevated LDH activity.
LDH is often measured in HIV patients as a non-specific marker for pneumonia due to Pneumocystis jirovecii (PCP). Elevated LDH in the setting of upper respiratory symptoms in a HIV patient suggests, but is not diagnostic for, PCP. However, in HIV-positive patients with respiratory symptoms, a very high LDH level (>600 IU/L) indicated histoplasmosis (9.33 times more likely) in a study of 120 PCP and 30 histoplasmosis patients. [ 32 ]
Measuring LDH in fluid aspirated from a pleural effusion (or pericardial effusion ) can help in the distinction between exudates (actively secreted fluid, e.g., due to inflammation ) or transudates (passively secreted fluid, due to a high hydrostatic pressure or a low oncotic pressure ). The usual criterion (included in Light's criteria ) is that a ratio of pleural LDH to serum LDH greater than 0.6 [ 33 ] or 2 ⁄ 3 the upper limit of the normal laboratory value for serum LDH [ 34 ] indicates an exudate, while a ratio of less indicates a transudate. Different laboratories have different values for the upper limit of serum LDH, but examples include 200 [ 35 ] and 300 [ 35 ] IU /L. [ 36 ] In empyema , the LDH levels, in general, will exceed 1000 IU/L.
High levels of lactate dehydrogenase in cerebrospinal fluid are often associated with bacterial meningitis . [ 37 ] In the case of viral meningitis, high LDH, in general, indicates the presence of encephalitis and poor prognosis .
LDH is involved in tumor initiation and metabolism. Cancer cells rely on increased glycolysis resulting in increased lactate production in addition to aerobic respiration in the mitochondria, even under oxygen-sufficient conditions (a process known as the Warburg effect [ 38 ] ). This state of fermentative glycolysis is catalyzed by the A form of LDH. This mechanism allows tumorous cells to convert the majority of their glucose stores into lactate regardless of oxygen availability, shifting use of glucose metabolites from simple energy production to the promotion of accelerated cell growth and replication.
LDH A and the possibility of inhibiting its activity has been identified as a promising target in cancer treatments focused on preventing carcinogenic cells from proliferating. Chemical inhibition of LDH A has demonstrated marked changes in metabolic processes and overall survival of carcinoma cells. Oxamate is a cytosolic inhibitor of LDH A that significantly decreases ATP production in tumorous cells as well as increasing production of reactive oxygen species (ROS). These ROS drive cancer cell proliferation by activating kinases that drive cell cycle progression growth factors at low concentrations, [ 39 ] but can damage DNA through oxidative stress at higher concentrations. Secondary lipid oxidation products can also inactivate LDH and impact its ability to regenerate NADH, [ 40 ] directly disrupting the enzymes ability to convert lactate to pyruvate.
While recent studies have shown that LDH activity is not necessarily an indicator of metastatic risk, [ 41 ] LDH expression can act as a general marker in the prognosis of cancers. Expression of LDH5 and VEGF in tumors and the stroma has been found to be a strong prognostic factor for diffuse or mixed-type gastric cancers. [ 42 ]
A cap- membrane -binding domain is found in prokaryotic lactate dehydrogenase. This consists of a large seven-stranded antiparallel beta-sheet flanked on both sides by alpha-helices . It allows for membrane association. [ 43 ] | https://en.wikipedia.org/wiki/Lactate_dehydrogenase |
The lactate shuttle hypothesis describes the movement of lactate intracellularly (within a cell) and intercellularly (between cells). The hypothesis is based on the observation that lactate is formed and utilized continuously in diverse cells under both anaerobic and aerobic conditions. [ 1 ] Further, lactate produced at sites with high rates of glycolysis and glycogenolysis can be shuttled to adjacent or remote sites including heart or skeletal muscles where the lactate can be used as a gluconeogenic precursor or substrate for oxidation. [ 2 ] [ 3 ] The hypothesis was proposed in 1985 by George Brooks of the University of California at Berkeley . [ 3 ] [ 4 ]
In addition to its role as a fuel source predominantly in the muscles, heart, brain, and liver, the lactate shuttle hypothesis also relates the role of lactate in redox signalling , gene expression , and lipolytic control. These additional roles of lactate have given rise to the term "lactormone", pertaining to the role of lactate as a signalling hormone. [ 5 ]
Prior to the formation of the lactate shuttle hypothesis, lactate had long been considered a byproduct resulting from glucose breakdown through glycolysis in times of anaerobic metabolism. [ 6 ] [ 3 ] As a means of regenerating oxidized NAD + , lactate dehydrogenase catalyzes the conversion of pyruvate to lactate in the cytosol , oxidizing NADH to NAD + , regenerating the necessary substrate needed to continue glycolysis. Lactate is then transported from the peripheral tissues to the liver by means of the Cori Cycle where it is reformed into pyruvate through the reverse reaction using lactate dehydrogenase . By this logic, lactate was traditionally considered a toxic metabolic byproduct that could give rise to fatigue and muscle pain during times of anaerobic respiration. [ 7 ] Lactate was essentially payment for ‘ oxygen debt ’ defined by Hill and Lupton as the ‘total amount of oxygen used, after cessation of exercise in recovery therefrom’. [ 8 ]
In addition to Cori Cycle, the lactate shuttle hypothesis proposes complementary functions of lactate in multiple tissues. Contrary to the long-held belief that lactate is formed as a result of oxygen-limited metabolism, substantial evidence exists that suggests lactate is formed under both aerobic and anaerobic conditions, as a result of substrate supply and equilibrium dynamics. [ 9 ]
During physical exertion or moderate intensity exercise lactate released from working muscle and other tissue beds is the primary fuel source for the heart, exiting the muscles through monocarboxylate transport protein (MCT). [ 10 ] This evidence is supported by an increased amount of MCT shuttle proteins in the heart and muscle in direct proportion to exertion as measured through muscular contraction. [ 11 ]
Furthermore, both neurons and astrocytes have been shown to express MCT proteins, suggesting that the lactate shuttle may be involved in brain metabolism. Astrocytes express MCT4, a low affinity transporter for lactate (Km = 35mM), suggesting its function is to export lactate produced by glycolysis. Conversely, neurons express MCT2, a high affinity transporter for lactate (Km = 0.7mM) [ clarification needed ] . Thus, it is hypothesized that the astrocytes produce lactate which is then taken up by the adjacent neurons and oxidized for fuel.
The lactate shuttle hypothesis also explains the balance of lactate production in the cytosol, via glycolysis or glycogenolysis , and lactate oxidation in the mitochondria (described below).
MCT2 transporters within the peroxisome function to transport pyruvate into the peroxisome where it is reduced by peroxisomal LDH (pLDH) to lactate. In turn, NADH is converted to NAD + , regenerating this necessary component for subsequent β-oxidation . Lactate is then shuttled out of the peroxisome via MCT2, where it is oxidized by cytoplasmic LDH (cLDH) to pyruvate, generating NADH for energy use and completing the cycle (see figure). [ 12 ]
While the cytosolic fermentation pathway of lactate is well established, a novel feature of the lactate shuttle hypothesis is the oxidation of lactate in the mitochondria. Baba and Sherma (1971) were the first to identify the enzyme lactate dehydrogenase (LDH) in the mitochondrial inner membrane and matrix of rat skeletal and cardiac muscle. [ 13 ] Subsequently, LDH was found in the rat liver, kidney, and heart mitochondria. [ 14 ] It was also found that lactate could be oxidized as quickly as pyruvate in rat liver mitochondria. Because lactate can either be oxidized in the mitochondria (back to pyruvate for entry into the Krebs cycle , generating NADH in the process), or serve as a gluconeogenic precursor, the intracellular lactate shuttle has been proposed to account for the majority of lactate turnover in the human body (as evidenced by the slight increases in arterial lactate concentration). Brooks et al. confirmed this in 1999, when they found that lactate oxidation exceeded that of pyruvate by 10-40% in rat liver, skeletal, and cardiac muscle.
In 1990, Roth and Brooks found evidence for the facilitated transporter of lactate, monocarboxylate transport protein (MCT), in the sarcolemma vesicles of rat skeletal muscle. Later, MCT1 was the first of the MCT super family to be identified. [ 15 ] The first four MCT isoforms are responsible for pyruvate/lactate transport. MCT1 was found to be the predominant isoform in many tissues including skeletal muscle, neurons, erythrocytes , and sperm. [ 16 ] In skeletal muscle, MCT1 is found in the membranes of the sarcolemma, [ 15 ] peroxisome, [ 12 ] and mitochondria. [ 6 ] Because of the mitochondrial localization of MCT (to transport lactate into the mitochondria), LDH (to oxidize the lactate back to pyruvate), and COX ( cytochrome c oxidase , the terminal element of the electron transport chain ), Brooks et al. proposed the possibility of a mitochondrial lactate oxidation complex in 2006. This is supported by the observation that the ability of muscle cells to oxidize lactate was related to the density of mitochondria. [ 17 ] Furthermore, it was shown that training increases MCT1 protein levels in skeletal muscle mitochondria, and that corresponded with an increase in the ability of muscle to clear lactate from the body during exercise. [ 18 ] The affinity of MCT for pyruvate is greater than lactate, however two reactions will ensure that lactate will be present in concentrations that are orders of magnitude greater than pyruvate: first, the equilibrium constant of LDH (3.6 × 10 4 ) greatly favors the formation of lactate. Secondly, the immediate removal of pyruvate from the mitochondria (either via the Krebs cycle or gluconeogenesis) ensures that pyruvate is not present in great concentrations within the cell.
LDH isoenzyme expression is tissue-dependent. It was found that in rats, LDH-1 was the predominant form in the mitochondria of myocardium , but LDH-5 was predominant in the liver mitochondria. [ 6 ] It is suspected that this difference in isoenzyme is due to the predominant pathway the lactate will take – in liver it is more likely to be gluconeogenesis, whereas in the myocardium it is more likely to be oxidation. Despite these differences, it is thought that the redox state of the mitochondria dictates the ability of the tissues to oxidize lactate, not the particular LDH isoform.
As illustrated by the peroxisomal intracellular lactate shuttle described above, the interconversion of lactate and pyruvate between cellular compartments plays a key role in the oxidative state of the cell. Specifically, the interconversion of NAD + and NADH between compartments has been hypothesized to occur in the mitochondria. However, the evidence for this is lacking, as both lactate and pyruvate are quickly metabolized inside the mitochondria. However, the existence of the peroxisomal lactate shuttle suggests that this redox shuttle could exist for other organelles . [ 12 ]
Increased intracellular levels of lactate can act as a signalling hormone, inducing changes in gene expression that will upregulate genes involved in lactate removal. [ 19 ] These genes include MCT1, cytochrome c oxidase (COX), and other enzymes involved in the lactate oxidation complex. Additionally, lactate will increase levels of peroxisome proliferator activated receptor gamma coactivator 1-alpha (PGC1-α), suggesting that lactate stimulates mitochondrial biogenesis. [ 1 ]
In addition to the role of the lactate shuttle in supplying NAD + substrate for β-oxidation in the peroxisomes, the shuttle also regulates FFA mobilization by controlling plasma lactate levels. Research has demonstrated that lactate functions to inhibit lipolysis in fat cells through activation of an orphan G-protein couple receptor ( GPR81 ) that acts as a lactate sensor, inhibiting lipolysis in response to lactate . [ 20 ]
As found by Brooks, et al., while lactate is disposed of mainly through oxidation and only a minor fraction supports gluconeogenesis , lactate is the main gluconeogenic precursor during sustained exercise. [ 1 ]
Brooks demonstrated in his earlier studies that little difference in lactate production rates were seen in trained and untrained subjects at equivalent power outputs. What was seen, however, was more efficient clearance rates of lactate in the trained subjects suggesting an upregulation of MCT protein. [ 1 ]
Local lactate use depends on exercise exertion. During rest, approximately 50% of lactate disposal take place through lactate oxidation whereas in time of strenuous exercise (50-75% VO 2 max ) approximately 75-80% of lactate is used by the active cell, indicating lactate's role as a major contributor to energy conversion during increased exercise exertion.
Highly malignant tumors rely heavily on anaerobic glycolysis (metabolism of glucose to lactic acid even under ample tissue oxygen; Warburg effect ) and thus need to efflux lactic acid via MCTs to the tumor micro-environment to maintain a robust glycolytic flux and to prevent the tumor from being "pickled to death". [ 21 ] The MCTs have been successfully targeted in pre-clinical studies using RNAi [ 22 ] and a small-molecule inhibitor alpha-cyano-4-hydroxycinnamic acid (ACCA; CHC) to show that inhibiting lactic acid efflux is a very effective therapeutic strategy against highly glycolytic malignant tumors. [ 23 ] [ 24 ] [ 25 ]
In some tumor types, growth and metabolism relies on the exchange of lactate between glycolytic and rapidly respiring cells. This is of particular importance during tumor cell development when cells often undergo anaerobic metabolism, as described by the Warburg effect . Other cells in the same tumor may have access to or recruit sources of oxygen (via angiogenesis ), allowing it to undergo aerobic oxidation. The lactate shuttle could occur as the hypoxic cells anaerobically metabolize glucose and shuttle the lactate via MCT to the adjacent cells capable of using the lactate as a substrate for oxidation. Investigation into how MCT-mediated lactate exchange in targeted tumor cells can be inhibited, therefore depriving cells of key energy sources, could lead to promising new chemotherapeutics . [ 26 ]
Additionally, lactate has been shown to be a key factor in tumor angiogenesis . Lactate promotes angiogenesis by upregulating HIF-1 in endothelial cells . Thus a promising target of cancer therapy is the inhibition of lactate export, through MCT-1 blockers, depriving developing tumors of an oxygen source. [ 27 ] | https://en.wikipedia.org/wiki/Lactate_shuttle_hypothesis |
Lactate inflection point ( LIP ) is the exercise intensity at which the blood concentration of lactate and/or lactic acid begins to increase rapidly. [ 1 ] It is often expressed as 85% of maximum heart rate or 75% of maximum oxygen intake. [ 2 ] When exercising at or below the lactate threshold, any lactate produced by the muscles is removed by the body without it building up. [ 3 ]
The onset of blood lactate accumulation (OBLA) is often confused with the lactate threshold. With an exercise intensity higher than the threshold the lactate production exceeds the rate at which it can be broken down. The blood lactate concentration will show an increase equal to 4.0 mM; it then accumulates in the muscle and then moves to the bloodstream. [ 2 ]
Regular endurance exercise leads to adaptations in skeletal muscle which raises the threshold at which lactate levels will rise. This is mediated via activation of the protein receptor PGC-1α , which alters the isoenzyme composition of the lactate dehydrogenase (LDH) complex and decreases the activity of lactate dehydrogenase A (LDHA), while increasing the activity of lactate dehydrogenase B (LDHB). [ 4 ]
The lactate threshold is a useful measure for deciding exercise intensity for training and racing in endurance sports (e.g., long distance running, cycling, rowing, long distance swimming and cross country skiing), but varies between individuals and can be increased with training. [ 2 ]
Interval training alternates work and rest periods allowing the body to temporarily exceed the lactate threshold at a high intensity, and then recover (reduce blood-lactate). [ 2 ] This type of training uses the ATP-PC and the lactic acid system while exercising, which provides the most energy when there are short bursts of high intensity exercise followed by a recovery period. [ 5 ] Interval training can take the form of many different types of exercise and should closely replicate the movements found in the sport being trained for. [ 2 ] Interval training can be adjusted to the individual, however it is important to consider the intensity of each interval, duration or distance of each interval, length of rest/recovery, number of repetitions, frequency of training and recovery type. [ 2 ]
Fartlek and interval training are similar, the main difference being the structure of the exercise. Fartlek is a Swedish word, meaning speed play. [ 2 ] This type of training is a combination of continuous (generally aerobic) and interval training (generally anaerobic), involving consistent changes of pace/intensity throughout the session. [ 2 ]
Aerobic training will not increase lactic acid tolerance, however, it will increase the lactate threshold. [ 2 ] Anaerobic training will increase tolerance of the effects of lactic acid over time, allowing the muscles’ ability to work in the presence of increased lactic acid. Training at or slightly above the lactate threshold improves the lactic acid tolerance. [ 3 ]
Muscles are producing lactate even at rest, with resting blood lactate levels in the 1–2 mmol/L range. [ 6 ] Although the lactate threshold is defined as the point when lactic acid starts to accumulate, some testers approximate this by crossing the lactate threshold and using the point at which lactate reaches a concentration of 4 mmol/ L of lactate. [ 3 ] Accurately measuring the lactate blood concentration involves taking blood samples (normally a pinprick to the finger, earlobe or thumb) during a ramp test where the exercise intensity is progressively increased. [ 7 ]
The chemistry behind the blood lactate meter takes place in the single-use strips. These strips are coated with a layer of the enzyme lactate oxidase. When the blood sample hits this layer in the presence of atmospheric oxygen, the enzyme converts the lactate to pyruvate. The byproduct of this reaction is hydrogen peroxide. The hydrogen peroxide is then oxidized to hydrogen ions, oxygen, and electrons. This is done by applying a small voltage to the strip to initiate the reaction, The electrodes in the test strip then measure the current produced in the reaction. and convert it to voltage. The detector/computer analysis is done by the device to magnify the voltage reading and correct for temperature before converting back to a blood lactate reading (mmol or mg/dL) on the screen. [ 8 ]
Blood samples are a popular way of measuring the lactate blood concentration, however there are many factors that may affect the sample. Every individual has a different health status, thus the results from the blood lactate response can vary from factors prior to exercise such as the glycogen status of the participant and ambient temperature. [ 9 ] “Furthermore, the lactate concentration measured may vary depending on the sampling site sweat contamination, and the accuracy of the lactate analyser.” [ 9 ] There are many factors that may give this test a false reading; it is important that an individual takes these into consideration, to receive an accurate test. [ 7 ]
The aerobic threshold (AeT or AerT) is sometimes defined equivalently to the lactate threshold (LT); as the exercise intensity at which blood lactate concentrations rise above resting levels. [ 9 ] In contrast, at the anaerobic threshold (AnT) the exercise is at an intensity beyond which blood lactate concentration is linearly related to exercise intensity, but increases with both exercise intensity and duration. [ 3 ] The blood lactate concentration at the anaerobic threshold is called the "maximum steady-state lactate concentration" (MLSS). [ 9 ]
AeT is the exercise intensity at which anaerobic energy pathways start to operate, considered to be around 65-85% of an individual's maximum heart rate. [ 2 ] Some have suggested this is where blood lactate reaches a concentration of 2 mmol /litre (at rest it is around 1). [ 3 ] The anaerobic energy system increases the ability to produce blood lactate during maximal exercise, resulting from an increased amount of glycogen stores [ clarification needed ] and glycolytic enzymes. [ 2 ]
In zone-based polarized training methodologies, LT1 is commonly used to designate the linear inflection point, often observed around blood lactate levels of 2.0 mmol/L, while LT2 is commonly used to designate the non-linear inflection point, often observed around blood lactate levels of 4.0 mmol/L. | https://en.wikipedia.org/wiki/Lactate_threshold |
Lactic acid fermentation is a metabolic process by which glucose or other six-carbon sugars (also, disaccharides of six-carbon sugars, e.g. sucrose or lactose ) are converted into cellular energy and the metabolite lactate , which is lactic acid in solution. It is an anaerobic fermentation reaction that occurs in some bacteria and animal cells , such as muscle cells . [ 1 ] [ 2 ] [ 3 ] [ page needed ]
If oxygen is present in the cell, many organisms will bypass fermentation and undergo cellular respiration ; however, facultative anaerobic organisms will both ferment and undergo respiration in the presence of oxygen. [ 3 ] Sometimes even when oxygen is present and aerobic metabolism is happening in the mitochondria , if pyruvate is building up faster than it can be metabolized, the fermentation will happen anyway.
Lactate dehydrogenase catalyzes the interconversion of pyruvate and lactate with concomitant interconversion of NADH and NAD + .
In homolactic fermentation , one molecule of glucose is ultimately converted to two molecules of lactic acid. Heterolactic fermentation , by contrast, yields carbon dioxide and ethanol in addition to lactic acid, in a process called the phosphoketolase pathway. [ 1 ]
Chemical analysis of archaeological finds show that milk fermentation had been used since prehistory; its first applications were probably a part of the Neolithic Revolution . Since milk naturally contains lactic acid bacteria , the discovery of the fermentation process was quite evident, since it happens spontaneously at an adequate temperature. The problem of these first farmers was that fresh milk is nearly indigestible by adults, so they had an interest to discover this mechanism. In fact, lactic acid bacteria contain the needed enzymes to digest lactose, and their populations multiply strongly during the fermentation. Therefore, milk fermented even a short time contains enough enzymes to digest the lactose molecules, after the milk is in the human body, which allows adults to consume it. Even safer was a longer fermentation, which was practiced for cheesemaking . This process was also discovered a very long time ago, which is proven by recipes for cheese production on Cuneiform scripts , the first written documents that exist, and later in Babylonian and Egyptian texts. There is a theory of competitive advantage related to fermented milk products. This theory suggests that the women of these first settled agricultural civilisations could shorten the time between two children thanks to the additional lactose uptake from milk consumption. This factor may have given them an important advantage to out-compete the hunter-gatherer societies. [ 4 ]
With the increasing consumption of milk products these societies developed a lactase persistence by epigenetic inheritance, which means that the milk-digesting enzyme lactase was present in their bodies during the whole lifetime, so they could drink unfermented milk as adults too. This early habituation to lactose consumption in the first settler societies can still be observed today in regional differences of this mutation's concentration. It is estimated that about 65% of world population still lacks it. [ 5 ] Since these first societies came from regions around eastern Turkey to central Europe , the gene appears more frequently there and in North America, as it was settled by Europeans. It is because of the dominance of this mutation that Western cultures believe it is unusual to have a lactose intolerance , when it is in fact more common than the mutation . On the contrary, lactose intolerance is much more present in Asian countries. [ citation needed ]
Milk products and their fermentation have had an important influence on some cultures' development. This is the case in Mongolia , where people often practice a pastoral form of agriculture . The milk that they produce and consume in these cultures is mainly mare milk and has a long tradition. But not every part or product of the fresh milk has the same meaning. For instance, the fattier part on the top, the "deež", is seen as the most valuable part and is therefore often used to honor guests.
Very important with often a traditional meaning as well are fermentation products of mare milk, like for example the slightly-alcoholic yogurt kumis . Consumption of these peaks during cultural festivities such as the Mongolian lunar new year (in spring). The time of this celebration is called the "white month", which indicates that milk products (called "white food" together with starchy vegetables, in comparison to meat products, called "black food") are a central part of this tradition. The purpose of these festivities is to "close" the past year – clean the house or the yurt , honor the animals for having provided their food, and prepare everything for the coming summer season – to be ready to "open" the new year. Consuming white food in this festive context is a way to connect to the past and to a national identity, which is the Mongol Empire personified by Genghis Khan . During the time of this empire, the fermented mare milk was the drink to honor and thank warriors and leading persons, it was not meant for everybody. Although it eventually became a drink for normal people, it has kept its honorable meaning. Like many other traditions, this one feels the influence of globalization . Other products, like industrial yogurt , coming mainly from China and western countries, have tended to replace it more and more, mainly in urban areas. However, in rural and poorer regions it is still of great importance. [ 6 ]
Although this chemical process had been used in food production for thousand of years, microbial lactic acid fermentation was not properly described before much later. During the 19th century, several chemists discovered some fundamental concepts of organic chemistry . One of these was the French chemist Joseph Louis Gay-Lussac , who was especially interested in fermentation processes, and he passed this fascination to one of his best students, Justus von Liebig . With a difference of some years, each of them described, together with colleagues, the chemical structure of the lactic acid molecule as we know it today. They had a purely chemical understanding of the fermentation process; it could not be observed using a microscope , and could only be optimized by chemical catalyzers . In 1857, the French chemist Louis Pasteur first described lactic acid as the product of a microbial fermentation. During this time, he worked at the University of Lille , where a local distillery asked him for advice concerning some fermentation problems. Per chance and with the badly equipped laboratory he had at that time, he was able to discover that in this distillery, two fermentations were taking place, a lactic acid one and an alcoholic one, both induced by microorganisms . He then continued the research on these discoveries in Paris, where he also published his theories that presented a stable contradiction to the purely chemical version represented by Liebig and his followers. Even though Pasteur described some concepts that are still accepted today, Liebig refused to accept them. But even Pasteur himself wrote that he was "driven" to a completely new understanding of this chemical phenomenon. Although Pasteur didn't find every detail of this process, he still discovered the main mechanism of how microbial lactic acid fermentation works. He was the first to describe fermentation as a "form of life without air". [ 7 ] [ 8 ]
Homofermentative bacteria convert glucose to two molecules of lactate and use this reaction to perform substrate-level phosphorylation to make two molecules of ATP :
Heterofermentative bacteria produce less lactate and less ATP, but produce several other end products:
Examples include Leuconostoc mesenteroides , Lactobacillus bifermentous , and Leuconostoc lactis .
Bifidobacterium bifidum utilizes a lactic acid fermentation pathway that produces more ATP than either homolactic fermentation or heterolactic fermentation:
Some major bacterial strains identified as being able to ferment lactose are in the genera Escherichia, Citrobacter, Enterobacter and Klebsiella . All four of these groups fall underneath the family of Enterobacteriaceae . These four genera are able to be separated from each other by using biochemical testing, and simple biological tests are readily available. Apart from whole-sequence genomics , common tests include H2S production, motility and citrate use, indole , methyl red and Voges-Proskauer tests . [ 9 ]
Lactic acid fermentation is used in many areas of the world to produce foods that cannot be produced through other methods. [ 10 ] [ 11 ] The most commercially important genus of lactic acid-fermenting bacteria is Lactobacillus , though other bacteria and even yeast are sometimes used. [ 10 ] Two of the most common applications of lactic acid fermentation are in the production of yogurt and sauerkraut.
Pickling in brine is a very common way to use lactic acid fermentation to aid in the preservation of food. Lactic acid bacteria (LAB) already exists as part of the natural flora in most vegetables, so by creating a selective environment of oxygen-poor brine, LAB will dominate in growth and convert sugars to lactic acid.
Silage fermentation uses the same principle of creating an anaerobic environment. Different types of LAB will produce different types of silage fermentation. [ 12 ]
Kimchi also uses lactic acid fermentation. [ 13 ]
Lactic acid fermentation is also used in the production of sauerkraut . The main type of bacteria used in the production of sauerkraut is of the genus Leuconostoc . [ 1 ] [ 14 ]
As in yogurt, when the acidity rises due to lactic acid-fermenting organisms, many other pathogenic microorganisms are killed. The bacteria produce lactic acid, as well as simple alcohols and other hydrocarbons . These may then combine to form esters , contributing to the unique flavor of sauerkraut. [ 1 ]
In some Asian cuisines, fish is traditionally fermented with rice to produce lactic acid that preserves the fish. Examples of these dishes include burong isda of the Philippines ; narezushi of Japan ; and pla ra of Thailand . The same process is also used for shrimp in the Philippines in the dish known as balao-balao . [ 15 ] [ 16 ] [ 17 ]
Lactic acid is a component in the production of sour beers , including Lambics and Berliner Weisses . [ 18 ]
The main method of producing yogurt is through the lactic acid fermentation of milk with harmless bacteria. [ 10 ] [ 19 ] The primary bacteria used are typically Lactobacillus bulgaricus and Streptococcus thermophilus , and United States as well as European law requires all yogurts to contain these two cultures (though others may be added as probiotic cultures). [ 19 ] These bacteria produce lactic acid in the milk culture, decreasing its pH and causing it to congeal. The bacteria also produce compounds that give yogurt its distinctive flavor. An additional effect of the lowered pH is the incompatibility of the acidic environment with many other types of harmful bacteria. [ 10 ] [ 19 ]
For a probiotic yogurt, additional types of bacteria such as Lactobacillus acidophilus are also added to the culture. [ 19 ]
Lactobacillus fermentation and accompanying production of acid provides a protective vaginal microbiome that protects against the proliferation of pathogenic organisms. [ 20 ]
The vaginal environment is heavily influenced by lactic acid producing bacteria. Lactobacilli spp. that live in the vaginal canal assist in pH control. If the pH in the vagina becomes too basic, more lactic acid will be produced to lower the pH back to a more acidic level. Lactic acid producing bacteria also act as a protective barrier against possible pathogens such as bacterial vaginosis and vaginitis species, different fungi, and protozoa through the production of hydrogen peroxide, and antibacterial compounds. It is unclear if further use of lactic acid, through fermentation, in the vaginal canal is present. [ citation needed ]
Human (and other eukaryote) cells can produce ATP from glucose without oxygen in a process called glycolysis . This is not as efficient as respiration, but provides a high instantaneous output, and is hence used by some muscle cells. Glycolysis consumes ADP, Pi, glucose, and NAD+ to produce ATP, pyruvate, and NADH. Through lactate fermentation, pyruvate and NADH are turned into lactate and NAD+, thereby regenerating the NAD+ required for more glycolysis.
During the 1990s, the lactic acid hypothesis was created to explain why people experienced burning or muscle cramps that occurred during and after intense exercise. The hypothesis proposes that a lack of oxygen in muscle cells results in a switch from cellular respiration to fermentation. Lactic acid created as a byproduct of fermentation of pyruvate from glycolysis accumulates in muscles causing a burning sensation and cramps.
Research from 2006 has suggested that acidosis isn't the main cause of muscle cramps. Instead cramps may be due to a lack of potassium in muscles, leading to contractions under high stress.
Animals, in fact, do not produce lactic acid during fermentation. Despite the common use of the term lactic acid in the literature, the byproduct of fermentation in animal cells is lactate. [ 21 ]
Another change to the lactic acid hypothesis is that when sodium lactate is inside of the body, there is a higher period of exhaustion in the host after a period of exercise. [ 22 ]
In small amounts, lactic acid is good for the human body by providing energy and substrates while it moves through the cycle. In lactose intolerant people, the fermentation of lactose to lactic acid has been shown in small studies to help lactose intolerant people. The process of fermentation limits the amount of lactose available. With the amount of lactose lowered, there is less build up inside of the body, reducing bloating. Success of lactic fermentation was most evident in yogurt cultures. Further studies are being conducted on other milk products like acidophilus milk. [ 23 ] | https://en.wikipedia.org/wiki/Lactic_acid_fermentation |
A Lactofuchsin mount (also spelled Lacto-fuchsin or Lacto-Fuchsin ) is a technique used for mounting fungi with hyphae on a microscope slide for examination. [ 1 ] The main advantage of a lactofuchsin mount is that if performed correctly, it preserves the structure and arrangement of any hyphae that are present.
To examine the hyphae of fungi under a microscope, a wet mount is essential. While this is possible to do with a water based mount, a better result can be obtained with lactofuchsin mounting fluid, which both sticks to the cell walls and colours the cell walls red in the process. Lactofuchsin, a 1% solution of basic fuchsine in lactic acid , [ 2 ] dries much slower than water, so the slide may be preserved for a longer period, particularly if the edges of the finished slide are sealed, for example with clear nail polish. [ 3 ] In addition, the refractive index of the fluid is significantly different to that of the cell walls, which provides a stronger visual contrast of the cell walls against the background.
A significant disadvantage of Lactofuchsin is its cost; prices are over US$100 for a small 20mL bottle. Only a few drops are used for each mount. Lactofuchsin is poisonous .
This microbiology -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Lactofuchsin_mount |
In organic chemistry , a lactol is a functional group which is the cyclic equivalent of a hemiacetal ( −CH(OH)O− ) or a hemiketal ( >C(OH)O− ).
The compound is formed by the intramolecular , nucleophilic addition of a hydroxyl group ( −OH ) to the carbonyl group ( C=O ) of an aldehyde ( −CH=O ) or a ketone ( >C=O ). [ 1 ]
A lactol is often found as an equilibrium mixture with the corresponding hydroxyaldehyde. The equilibrium can favor either direction depending on ring size and other conformational effects.
The lactol functional group is prevalent in nature as component of aldose sugars.
Lactols can participate in a variety of chemical reactions including: [ 2 ] | https://en.wikipedia.org/wiki/Lactol |
A hydrometer or lactometer is an instrument used for measuring density or relative density of liquids based on the concept of buoyancy . They are typically calibrated and graduated with one or more scales such as specific gravity .
A hydrometer usually consists of a sealed hollow glass tube with a wider bottom portion for buoyancy , a ballast such as lead or mercury for stability, and a narrow stem with graduations for measuring. The liquid to test is poured into a tall container, often a graduated cylinder , and the hydrometer is gently lowered into the liquid until it floats freely. The point at which the surface of the liquid touches the stem of the hydrometer correlates to relative density. Hydrometers can contain any number of scales along the stem corresponding to properties correlating to the density.
Hydrometers are calibrated for different uses, such as a lactometer for measuring the density (creaminess) of milk, a saccharometer for measuring the density of sugar in a liquid, or an alcoholometer for measuring higher levels of alcohol in spirits .
The hydrometer makes use of Archimedes' principle : a solid suspended in a fluid is buoyed by a force equal to the weight of the fluid displaced by the submerged part of the suspended solid. The lower the density of the fluid, the deeper a hydrometer of a given weight sinks; the stem is calibrated to give a numerical reading.
The hydrometer probably dates back to the Greek philosopher Archimedes (3rd century BC) who used its principles to find the density of various liquids. [ 1 ] [ 2 ] An early description of a hydrometer comes from a Latin poem, written in the 2nd century AD by Remnius, who compared the use of a hydrometer to the method of fluid displacement used by Archimedes to determine the gold content of Hiero II's crown. [ 3 ]
Hypatia of Alexandria (b. c. 350– 370; d. 415 CE), an important female Greek mathematician, is the first person traditionally associated with the hydrometer. [ 3 ] In a letter, Synesius of Cyrene asks Hypatia, his teacher, to make a hydrometer for him:
The instrument in question is a cylindrical tube, which has the shape of a flute and is about the same size. It has notches in a perpendicular line, by means of which we are able to test the weight of the waters. A cone forms a lid at one of the extremities, closely fitted to the tube. The cone and the tube have one base only. This is called the baryllium. Whenever you place the tube in water, it remains erect. You can then count the notches at your ease, and in this way ascertain the weight of the water. [ 4 ]
According to the Encyclopedia of the History of Arabic Science , it was used by Abū Rayhān al-Bīrūnī in the 11th century and described by Al-Khazini in the 12th century. [ 5 ] It was rediscovered in 1612 by Galileo and his circle of friends, and used in experiments especially at the Accademia del Cimento. [ 6 ] It appeared again in the 1675 work of Robert Boyle (who coined the name "hydrometer" ), [ 3 ] with types devised by Antoine Baumé (the Baumé scale ), William Nicholson , and Jacques Alexandre César Charles in the late 18th century, [ 7 ] more or less contemporarily with Benjamin Sikes ' discovery of the device by which the alcoholic content of a liquid can be automatically determined. The use of the Sikes device was made obligatory by British law in 1818. [ 8 ]
The hydrometer sinks deeper in low-density liquids such as kerosene , gasoline , and alcohol , and less deep in high-density liquids such as brine , milk , and acids . It is usual for hydrometers to be used with dense liquids to have the mark 1.000 (for water) near the top of the stem, and those for use with lighter liquids to have 1.000 near the bottom. In many industries a set of hydrometers is used (1.0–0.95, 0.95–.) to have instruments covering the range of specific gravities that may be encountered.
Modern hydrometers usually measure specific gravity but different scales were (and sometimes still are) used in certain industries. Examples include:
Specialized hydrometers are frequently named for their use: a lactometer, for example, is a hydrometer designed especially for use with dairy products. They are sometimes referred to by this specific name, sometimes as hydrometers.
An alcoholmeter is a hydrometer that indicates the alcoholic strength of liquids which are essentially a mixture of alcohol and water. It is also known as a proof and Tralles hydrometer (after Johann Georg Tralles , but commonly misspelled as traille and tralle ). It measures the density of the fluid. Where no sugar or other dissolved substances are present, the specific gravity of a solution of ethanol in water can be directly correlated to the concentration of alcohol. Saccharometers for measuring sugar-water mixtures measure densities greater than water. Many have scales marked with volume percents of "potential alcohol", based on a pre-calculated specific gravity. A higher "potential alcohol" reading on this scale is caused by a greater specific gravity, assumed to be caused by the introduction of dissolved sugars or carbohydrate based material. A reading is taken before and after fermentation and approximate alcohol content is determined by subtracting the post fermentation reading from the pre-fermentation reading. [ 10 ]
These were important instruments for determining tax, and specific maker's instruments could be specified. Bartholomew Sikes had a monopoly in the UK and Mary Dicas and her family enjoyed a similar monopoly in the US. [ 11 ]
A lactometer is used to check purity of cow's milk. The specific gravity of milk does not give a conclusive indication of its composition since milk contains a variety of substances that are either heavier or lighter than water. Additional tests for fat content are necessary to determine overall composition. The instrument is graduated into a hundred parts. Milk is poured in and allowed to stand until the cream has formed, then the depth of the cream deposit in degrees determines the quality of the milk. If the milk sample is pure, the lactometer floats higher than if it is adulterated or impure. [ 12 ] [ 13 ]
A saccharometer is a type of hydrometer used for determining the amount of sugar in a solution, invented by Thomas Thomson . [ 14 ] It is used primarily by winemakers and brewers , [ 15 ] and it can also be used in making sorbets and ice-creams. [ 16 ] The first brewers' saccharometer was constructed by Benjamin Martin (with distillation in mind), and initially used for brewing by James Baverstock Sr in 1770. [ 17 ] Henry Thrale adopted its use and it was later popularized by John Richardson in 1784. [ 18 ]
It consists of a large weighted glass bulb with a thin stem rising from the top with calibrated markings. The sugar level can be determined by reading the value where the surface of the liquid crosses the scale. The higher the sugar content, the denser the solution, and thus the higher the bulb will float.
A thermohydrometer is a hydrometer that has a thermometer enclosed in the float section. For measuring the density of petroleum products, such as fuel oils, the specimen is usually heated in a temperature jacket with a thermometer placed behind it since density is dependent on temperature. Light oils are placed in cooling jackets, typically at 15 °C.
Very light oils with many volatile components are measured in a variable volume container using a floating piston sampling device to minimize light end losses. [ 19 ]
The state of charge of a lead-acid battery can be estimated from the density of the sulfuric acid solution used as electrolyte . A hydrometer calibrated to read specific gravity relative to water at 60 °F (16 °C) is a standard tool for servicing automobile batteries . Tables are used to correct the reading to the standard temperature. Hydrometers are also used for maintenance of wet-cell nickel-cadmium batteries to ensure the electrolyte is of the proper strength for the application; for this battery chemistry the specific gravity of the electrolyte is not related to the state of charge of the battery.
A battery hydrometer with thermometer (thermohydrometer) measures the temperature-compensated specific gravity and electrolyte temperature.
Another automotive use of hydrometers is testing the quality of the antifreeze solution used for engine cooling. The degree of freeze protection can be related to the density (and so concentration) of the antifreeze; different types of antifreeze have different relations between measured density and freezing point.
An acidometer, or acidimeter, is a hydrometer used to measure the specific gravity of an acid . [ 20 ]
A barkometer is calibrated to test the strength of tanning liquors used in tanning leather . [ 21 ]
A salinometer is a hydrometer used to measure the salt content of the feed water to a marine steam boiler.
A urinometer is a medical hydrometer designed for urinalysis . As urine's specific gravity is dictated by its ratio of solutes (wastes) to water, a urinometer makes it possible to quickly assess a patient's overall level of hydration.
A hydrometer analysis is the process by which fine-grained soils, silts and clays , are graded. Hydrometer analysis is performed if the grain sizes are too small for sieve analysis . The basis for this test is Stokes' Law for falling spheres in a viscous fluid in which the terminal velocity of fall depends on the grain diameter and the densities of the grain in suspension and of the fluid. The grain diameter thus can be calculated from a knowledge of the distance and time of fall. The hydrometer also determines the specific gravity (or density) of the suspension, and this enables the fraction of particles of a certain equivalent particle diameter to be calculated. [ 22 ] | https://en.wikipedia.org/wiki/Lactometer |
Lactonase (EC 3.1.1.81, acyl-homoserine lactonase ; systematic name N -acyl- L -homoserine-lactone lactonohydrolase ) is a metalloenzyme , produced by certain species of bacteria, which targets and inactivates acylated homoserine lactones (AHLs). It catalyzes the reaction
Many species of α -, β -, and γ-proteobacteria produce acylated homoserine lactones, small hormone-like molecules commonly used as communication signals between bacterial cells in a population to regulate certain gene expression and phenotypic behaviours. [ 1 ] This type of gene regulation is known as quorum sensing .
Other names for these types of enzymes are Quorum-quenching N -acyl-homoserine lactonase, acyl homoserine degrading enzyme, acyl-homoserine lactone acylase, AHL lactonase, AHL-degrading enzyme, AHL-inactivating enzyme, AHLase, AhlD, AhlK, AiiA, AiiA lactonase, AiiA-like protein, AiiB, AiiC, AttM, delactonase, lactonase-like enzyme, N -acyl homoserine lactonase, N -acyl homoserine lactone hydrolase, N -acyl-homoserine lactone lactonase, N -acyl- L -homoserine lactone hydrolase, quorum-quenching lactonase, quorum-quenching N -acyl homoserine lactone hydrolase . [ 2 ] [ 3 ] [ 4 ] [ 5 ] [ 6 ] [ 7 ] [ 8 ] [ 9 ] [ 10 ] [ 11 ] [ 12 ]
Lactonase hydrolyzes the ester bond of the homoserine lactone ring of acylated homoserine lactones. In hydrolysing the lactone bond, lactonase prevents these signaling molecules from binding to their target transcriptional regulators, thus inhibiting quorum sensing . [ 13 ]
A dinuclear zinc binding site is conserved in all known lactonases and essential for enzyme activity and protein folding. [ 14 ]
Zn1 is tetracoordinated by His104, His106, His169, and the bridging hydroxide ion. Zn2 has five ligands, including Asp191, His235, His109, Asp108, and the bridging hydroxide ion. The metal ions assist in polarizing the lactone bond, increasing the electrophilicity of the lactone ring’s carbonyl carbon. Isotopic labeling studies indicated that the ring opening occurs via an addition elimination reaction with water addition shown below. [ 15 ]
Lactonases are able to interfere with AHL-mediated quorum sensing. Some examples of these lactonases are AiiA produced by Bacillus species, AttM and AiiB produced by Agrobacterium tumefaciens , and QIcA produced by Hyphomicrobiales species. [ 16 ]
Lactonases have been reported for Bacillus, Agrobacterium, Rhodococcus, Streptomyces, Arthrobacter, Pseudomonas , and Klebsiella . [ 17 ] The Bacillus cereus group (consisting of B. cereus, B. thuringiensis, B. mycoides , and B. anthracis ) was found to contain nine genes homologous to the AiiA gene that encode AHL-inactivating enzymes, with the catalytic zinc-binding motif conserved in all cases. [ 18 ]
In the phytopathogen A. tumefaciens , AiiB lactonase acts as a fine modulator that essentially delays the release of lactone OC8-HSL and the resultant number of tumors produced by the pathogen. AttM lactonase mediates the degradation of the lactone OC8-HSL in wounded plant tissues. [ 19 ]
The primary activity of the anti- atherosclerotic paraoxonase (PON) enzymes is as lactonase. [ 20 ] Oxidized polyunsaturated fatty acids (notably in oxidized low-density lipoprotein ) form lactone -like structures that are PON substrates. [ 20 ]
It is still unclear the ecological effects of lactonase but it has been proposed that since bacteria mostly coexist with other microorganisms in the environment, some bacteria strains could have evolved its feeding strategies and utilize AHLs as their main resource for energy and nitrogen. [ 21 ]
Understanding the mechanisms and purposes of lactonase activity could lead to potential applied roles for these lactonases to control bacterial infections by inhibiting quorum-sensing activity and bring about profound effects on human health and the environment. However, in both the chemical and enzymatic lactonolysis, the reaction is reversible, complicating direct therapeutic application of lactonases. [ 22 ]
Pseudomonas aeruginosa , is an AHL-producing bacteria an opportunistic pathogen that infects immuno-compromised patients, [ 23 ] and is found in lung infections of cystic fibrosis patients. P. aeruginosa relies on quorum sensing via production of lactones N -butanoyl- L -homoserine (C4-HSL) and N -(3-oxododecanoyl)-l-HSL (3-oxo-C12-HSL) to regulate swarming, toxin and protease production, and proper biofilm formation. The absence of one or more components of the quorum-sensing system results in a significant reduction in virulence of the pathogen. [ 24 ]
Erwinia carotovora is a plant pathogen that causes soft rot in a number of crops such as potatoes and carrots [ 25 ] by using N-hexanoyl-l-HSL (C6-HSL) quorum sensing to evade the plant's defense systems and coordinate its production of pectate lyase during the infection process. [ 26 ]
Plants expressing AHL-Lactonase were shown to demonstrate enhanced resistance to infection from the pathogen Erwinia carotovora. Expression of virulence genes in E. Carotovora is regulated by N -(3-oxohexanoyl)- L -homoserine lactone (OHHL). Presumably, OHHL-hydrolysis via lactonase reduced OHHL levels, inhibiting the quorum-sensing systems driving virulence gene expression. [ 18 ] | https://en.wikipedia.org/wiki/Lactonase |
Lactones are cyclic carboxylic esters . They are derived from the corresponding hydroxycarboxylic acids by esterification . They can be saturated or unsaturated. [ 1 ]
Lactones are formed by lactonization , the intramolecular esterification of the corresponding hydroxycarboxylic acids. [ 2 ]
Greek prefixes in alphabetical order indicate ring size.
Lactones are usually named according to the precursor acid molecule ( aceto = 2 carbon atoms, propio = 3, butyro = 4, valero = 5, capro = 6, etc.), with a -lactone suffix and a Greek letter prefix that specifies the number of carbon atoms in the heterocycle — that is, the distance between the relevant -OH and the -COOH groups along said backbone. The first carbon atom after the carbon in the -COOH group on the parent compound is labelled α, the second will be labeled β, and so forth. Therefore, the prefixes also indicate the size of the lactone ring: α-lactone = 3-membered ring, β-lactone = 4-membered, γ-lactone = 5-membered, δ-lactone = 6-membered, etc. Macrocyclic lactones are known as macrolactones . [ 3 ]
The other suffix used to denote a lactone is -olide , used in substance class names like butenolide , macrolide , cardenolide or bufadienolide .
To obtain the preferred IUPAC names , lactones are named as heterocyclic pseudoketones by adding the suffix 'one', 'dione', 'thione', etc. and the appropriate multiplicative prefixes to the name of the heterocyclic parent hydride. [ 4 ]
The name lactone derives from the ring compound called lactide , which is formed from the dehydration of 2-hydroxypropanoic acid ( lactic acid ) CH 3 -CH(OH)-COOH. Lactic acid, in turn, derives its name from its original isolation from soured milk (Latin: lac, lactis). The name was coined in 1844 by the French chemist Théophile-Jules Pelouze , who first obtained it as a derivative of lactic acid. [ 5 ] An internal dehydration reaction within the same molecule of lactic acid would have produced alpha-propiolactone , a lactone with a 3-membered ring.
In 1880 the German chemist Wilhelm Rudolph Fittig extended the name "lactone" to all intramolecular carboxylic esters. [ 6 ]
Lactone rings occur widely as building blocks in nature, such as in ascorbic acid , kavain , nepetalactone , gluconolactone , hormones ( spironolactone , mevalonolactone ), enzymes ( lactonase ), neurotransmitters ( butyrolactone , avermectins ), antibiotics ( macrolides like erythromycin ; amphotericin B ), anticancer drugs ( vernolepin , epothilones ), phytoestrogens ( resorcylic acid lactones, cardiac glycosides ).
5-Membered γ-lactones and 6-membered δ-lactones are prevalent. β-lactones appear in a number of natural products. [ 7 ] α‑Lactones can be detected as transient species in mass spectrometry experiments. [ 8 ]
Macrocyclic lactones are also important natural products. [ 9 ] Lactones are present in oak wood, and they contribute to the flavour profile of barrel-aged beers . [ 10 ]
Many methods in ester synthesis can also be applied to that of lactones. Lactonization competes with polymerization for longer hydroxy acids, or the strained β‑lactones. γ‑Lactones, on the other hand, are so stable that 4-hydroxy acids (R-CH(OH)-(CH 2 ) 2 -CO 2 H) spontaneously cyclize.
In one industrial synthesis of oxandrolone the key lactone-forming step is an organic reaction – esterification. [ 11 ] [ 12 ]
In halolactonization , an alkene is attacked by a halogen via electrophilic addition with the cationic intermediate captured intramolecularly by an adjacent carboxylic acid . [ 13 ]
Specific methods include Yamaguchi esterification , Shiina macrolactonization , Corey-Nicolaou macrolactonization , Baeyer–Villiger oxidation and nucleophilic abstraction .
An alternative radical reaction yielding γ-lactones is the manganese-mediated coupling .
Lactones exhibit the reactions characteristic of esters.
Heating a lactone with a base ( sodium hydroxide ) will hydrolyse the lactone to its parent compound, the straight chained bifunctional compound. Like straight-chained esters, the hydrolysis-condensation reaction of lactones is a reversible reaction , with an equilibrium . However, the equilibrium constant of the hydrolysis reaction of the lactone is lower than that of the straight-chained ester i.e. the products (hydroxyacids) are less favored in the case of the lactones. This is because although the enthalpies of the hydrolysis of esters and lactones are about the same, the entropy of the hydrolysis of lactones is less than the entropy of straight-chained esters. Straight-chained esters give two products upon hydrolysis, making the entropy change more favorable than in the case of lactones which gives only a single product.
Lactones also react with amines to give the ring-opened alcohol and amide.
Lactones can be reduced to diols using lithium aluminium hydride . For instance, gamma-lactones is reduced to butane-1,4-diol, (CH 2 (OH)-(CH 2 ) 2 -CH 2 (OH).
Some lactones convert to polyesters: [ 14 ] [ 15 ] For example the double lactone called lactide polymerizes to polylactic acid (polylactide). The resulting polylactic acid has been heavily investigated for commercial applications. [ 16 ] [ 17 ]
Lactones contribute significantly to the flavor of fruit, and of unfermented and fermented dairy products, [ 18 ] and are therefore used as flavors and fragrances. [ 9 ] Some examples are γ-decalactone (4-decanolide), which has a characteristic peach flavor; [ 18 ] δ-decalactone (5-decanolide), which has a creamy coconut/peach flavour; γ-dodecalactone (4-dodecanolide), which also has a coconut/fruity flavor, [ 18 ] a description which also fits γ-octalactone (4-octanolide), [ 19 ] although it also has a herbaceous character; [ 18 ] γ-nonalactone , which has an intense coconut flavor of this series, despite not occurring in coconut, [ 20 ] and γ-undecalactone .
Macrocyclic lactones ( cyclopentadecanolide , 15-pentadec-11/12-enolide ) have odors similar to macrocyclic ketones of animal origin ( muscone , civetone ). [ 9 ]
Polycaprolactone is an important plastic. Its formation has even been considered in the context of the origin of life . [ 21 ] | https://en.wikipedia.org/wiki/Lactone |
The lactoscope is an instrument for estimating the amount of cream in milk, based on its relative opacity. The higher the opacity, the greater the amount of cream present. Credit for the development of this instrument is given to Alfred Donné in 1843. [ 1 ] [ 2 ] The instrument was also used to measure the fat content of milk, but it gave inaccurate results. [ 3 ] | https://en.wikipedia.org/wiki/Lactoscope |
Lactulose is a non-absorbable sugar used in the treatment of constipation and hepatic encephalopathy . [ 3 ] [ 4 ] It is administered orally for constipation, and either orally or rectally for hepatic encephalopathy . [ 3 ] It generally begins working after 8–12 hours, but may take up to 2 days to improve constipation. [ 1 ] [ 2 ]
Common side effects include abdominal bloating and cramps. [ 3 ] A potential exists for electrolyte problems as a result of the diarrhea it produces. [ 3 ] No evidence of harm to the fetus has been found when used during pregnancy . [ 3 ] It is generally regarded as safe during breastfeeding. [ 5 ] It is classified as an osmotic laxative . [ 6 ]
Lactulose was first made in 1929, and has been used medically since the 1950s. [ 7 ] [ 8 ] Lactulose is made from the milk sugar lactose , which is composed of two simple sugars , galactose and glucose . [ 9 ] [ 3 ] It is on the World Health Organization's List of Essential Medicines . [ 10 ] It is available as a generic medication . [ 4 ] In 2022, it was the 267th most commonly prescribed medication in the United States, with more than 900,000 prescriptions. [ 11 ] [ 12 ]
Lactulose is used in the treatment of chronic constipation in patients of all ages as a long-term treatment. [ 13 ] The dosage of lactulose for chronic idiopathic constipation is adjusted depending on the constipation severity and desired effect, from a mild stool softener to causing diarrhea. Lactulose is contraindicated in case of galactosemia , as most preparations contain the monosaccharide galactose due to its synthesis process. [ 14 ] [ 15 ]
Lactulose may be used to counter the constipating effects of opioids , and in the symptomatic treatment of hemorrhoids as a stool softener. [ medical citation needed ]
Lactulose is commonly prescribed for children who develop fear of their bowel movements and are withholders. This is because lactulose, when dosed in the proper amount, causes a bowel movement that is impossible to retain for very long. Lactulose is also used for the elderly because of its gentle and consistent results. [ medical citation needed ]
Lactulose is useful in treating hyperammonemia (high blood ammonia), which can lead to hepatic encephalopathy . Lactulose helps trap the ammonia (NH 3 ) in the colon and bind to it. [ 16 ] It does this by using gut flora to acidify the colon, transforming the freely diffusible ammonia into ammonium ions ( NH + 4 ), which can no longer diffuse back into the blood. [ 17 ] It is also useful for preventing hyperammonemia caused as a side effect of administration of valproic acid . [ 18 ]
Lactulose is used as a test of small intestine bacterial overgrowth (SIBO). Recently, the reliability of it for diagnosing SIBO has been seriously questioned. [ 19 ] [ 20 ] [ 21 ] [ 22 ] A large amount of it is given with subsequent testing of molecular hydrogen gas in the breath . The test is positive if an increase in exhaled hydrogen occurs before that which would be expected by normal digestion by the normal gut flora in the colon. An earlier result has been hypothesized to indicate digestion occurring within the small intestine. An alternate explanation for differences in results is the variance in small bowel transit time among tested subjects. [ 22 ]
No evidence of harm to the fetus has been found when used during pregnancy . [ 3 ] It is generally regarded as safe during breastfeeding. [ 5 ]
Common side effects of lactulose are abdominal cramping, borborygmus , and flatulence . In normal individuals, overdose is considered uncomfortable, but not life-threatening. [ 23 ] Uncommon side effects are nausea and vomiting . In sensitive individuals, such as the elderly or people with reduced kidney function, excess lactulose dosage can result in dehydration and electrolyte disturbances such as low magnesium levels . Ingestion of lactulose does not cause a weight gain because it is not digestible, with no nutritional value. Although lactulose is less likely to cause dental caries than sucrose, as a sugar, a potential for this exists, which is relevant when taken by people with a high susceptibility to this condition. [ citation needed ]
Lactulose is not absorbed in the small intestine nor broken down by human enzymes, thus stays in the digestive bolus through most of its course, causing retention of water through osmosis leading to softer, easier-to-pass stool. It has a secondary laxative effect in the colon, where it is fermented by the gut flora , producing metabolites which have osmotic powers and peristalsis -stimulating effects (such as acetate ), but also methane associated with flatulence . [ citation needed ]
Lactulose is metabolized in the colon by bacterial flora into short-chain fatty acids, including lactic acid and acetic acid . These partially dissociate, acidifying the colonic contents (increasing the H + concentration in the gut). [ 17 ] This favors the formation of the nonabsorbable NH + 4 from NH 3 , trapping NH 3 in the colon and effectively reducing plasma NH 3 concentrations. Lactulose is therefore effective in treating hepatic encephalopathy. [ 24 ] Specifically, it is effective as secondary prevention of hepatic encephalopathy in people with cirrhosis. [ 25 ] Moreover, research showed improved cognitive functions and health-related quality of life in people with cirrhosis with minimal hepatic encephalopathy treated with lactulose. [ 26 ]
Lactulose is a disaccharide formed from one molecule each of the simple sugars ( monosaccharides ) glucose and galactose . Lactulose is not normally present in raw milk, but is a product of heat processes: [ 27 ] the greater the heat, the greater amount of this substance (from 3.5 mg/L in low-temperature pasteurized milk to 744 mg/L in in-container sterilized milk). [ 28 ]
Lactulose is produced commercially by isomerization of lactose . A variety of reaction conditions and catalysts can be used. [ 14 ]
Lactulose is its international nonproprietary name (INN). [ 29 ] It is sold under various brand names. [ citation needed ]
Lactulose is available as a generic medication . [ 4 ] It is available without prescription in most countries, but a prescription is required in the United States, [ 3 ] Philippines, and Austria. [ citation needed ]
In some countries where lactulose may be obtained without a prescription, lactulose is commonly used as a food additive to improve taste and promote intestinal transit. [ citation needed ]
Lactulose is used in veterinary medicine for constipation [ 30 ] and hepatic encephalopathy—the same indications as human lactulose. [ 31 ] | https://en.wikipedia.org/wiki/Lactulose |
Lacunarity , from the Latin lacuna , meaning "gap" or "lake", is a specialized term in geometry referring to a measure of how patterns, especially fractals , fill space, where patterns having more or larger gaps generally have higher lacunarity. Beyond being an intuitive measure of gappiness, lacunarity can quantify additional features of patterns such as "rotational invariance" and more generally, heterogeneity. [ 1 ] [ 2 ] [ 3 ] This is illustrated in Figure 1 showing three fractal patterns. When rotated 90°, the first two fairly homogeneous patterns do not appear to change, but the third more heterogeneous figure does change and has correspondingly higher lacunarity. The earliest reference to the term in geometry is usually attributed to Benoit Mandelbrot , who, in 1983 or perhaps as early as 1977, introduced it as, in essence, an adjunct to fractal analysis . [ 4 ] Lacunarity analysis is now used to characterize patterns in a wide variety of fields and has application in multifractal analysis [ 5 ] [ 6 ] in particular (see Applications ).
In many patterns or data sets, lacunarity is not readily perceivable or quantifiable, so computer-aided methods have been developed to calculate it. As a measurable quantity, lacunarity is often denoted in scientific literature by the Greek letters Λ {\displaystyle \Lambda } or λ {\displaystyle \lambda } but it is important to note that there is no single standard and several different methods exist to assess and interpret lacunarity.
One well-known method of determining lacunarity for patterns extracted from digital images uses box counting , the same essential algorithm typically used for some types of fractal analysis . [ 1 ] [ 4 ] Similar to looking at a slide through a microscope with changing levels of magnification, box counting algorithms look at a digital image from many levels of resolution to examine how certain features change with the size of the element used to inspect the image. Basically, the arrangement of pixels is measured using traditionally square (i.e., box-shaped) elements from an arbitrary set of E {\displaystyle \mathrm {E} } sizes, conventionally denoted ε {\displaystyle \varepsilon } s. For each ε {\displaystyle \varepsilon } , a box of size ε {\displaystyle \varepsilon } is placed successively on the image, in the end covering it completely, and each time it is laid down, the number of pixels that fall within the box is recorded. [ note 1 ] In standard box counting , the box for each ε {\displaystyle \varepsilon } in E {\displaystyle \mathrm {E} } is placed as though it were part of a grid overlaid on the image so that the box does not overlap itself, but in sliding box algorithms the box is slid over the image so that it overlaps itself and the "Sliding Box Lacunarity" or SLac is calculated. [ 3 ] [ 7 ] Figure 2 illustrates both types of box counting.
The data gathered for each ε {\displaystyle \varepsilon } are manipulated to calculate lacunarity. One measure, denoted here as λ ε {\displaystyle \lambda _{\varepsilon }} , is found from the coefficient of variation ( C V {\displaystyle {\mathit {CV}}} ), calculated as the standard deviation ( σ {\displaystyle \sigma } ) divided by the mean ( μ {\displaystyle \mu } ), for pixels per box. [ 1 ] [ 3 ] [ 6 ] Because the way an image is sampled will depend on the arbitrary starting location, for any image sampled at any ε {\displaystyle \varepsilon } there will be some number ( G {\displaystyle {\mathit {G}}} ) of possible orientations, each denoted here by g {\displaystyle {\mathit {g}}} , that the data can be gathered over, which can have varying effects on the measured distribution of pixels. [ 5 ] [ note 2 ] Equation 1 shows the basic method of calculating λ ε , g {\displaystyle \lambda _{\varepsilon ,g}} :
Alternatively, some methods sort the numbers of pixels counted into a probability distribution having B {\displaystyle B} bins, and use the bin sizes (masses, m {\displaystyle m} ) and their corresponding probabilities ( p {\displaystyle p} ) to calculate λ ε , g {\displaystyle \lambda _{\varepsilon ,g}} according to Equations 2 through 5 :
Lacunarity based on λ ε , g {\displaystyle \lambda _{\varepsilon ,g}} has been assessed in several ways including by using the variation in or the average value of λ ε , g {\displaystyle \lambda _{\varepsilon ,g}} for each ε {\displaystyle \varepsilon } (see Equation 6 ) and by using the variation in or average over all grids (see Equation 7 ). [ 1 ] [ 5 ] [ 7 ] [ 8 ]
Lacunarity analyses using the types of values discussed above have shown that data sets extracted from dense fractals, from patterns that change little when rotated, or from patterns that are homogeneous, have low lacunarity, but as these features increase, [ clarification needed ] so generally does lacunarity. In some instances, it has been demonstrated that fractal dimensions and values of lacunarity were correlated, [ 1 ] but more recent research has shown that this relationship does not hold for all types of patterns and measures of lacunarity. [ 5 ] Indeed, as Mandelbrot originally proposed, lacunarity has been shown to be useful in discerning amongst patterns (e.g., fractals, textures, etc.) that share or have similar fractal dimensions in a variety of scientific fields including neuroscience. [ 8 ]
Other methods of assessing lacunarity from box counting data use the relationship between values of lacunarity (e.g., λ ε , g {\displaystyle \lambda _{\varepsilon ,g}} ) and ε {\displaystyle \varepsilon } in different ways from the ones noted above. One such method looks at the ln {\displaystyle \ln } vs ln {\displaystyle \ln } plot of these values. According to this method, the curve itself can be analyzed visually, or the slope at g {\displaystyle {\mathit {g}}} can be calculated from the ln {\displaystyle \ln } vs ln {\displaystyle \ln } regression line. [ 3 ] [ 7 ] Because they tend to behave in certain ways for respectively mono-, multi-, and non-fractal patterns, ln {\displaystyle \ln } vs ln {\displaystyle \ln } lacunarity plots have been used to supplement methods of classifying such patterns. [ 5 ] [ 8 ]
To make the plots for this type of analysis, the data from box counting first have to be transformed as in Equation 9 :
This transformation avoids undefined values, which is important because homogeneous images will have σ {\displaystyle \sigma } at some ε {\displaystyle \varepsilon } equal to 0 so that the slope of the ln {\displaystyle \ln } vs ln {\displaystyle \ln } regression line would be impossible to find. With f λ ε , g {\displaystyle f\lambda _{\varepsilon ,g}} , homogeneous images have a slope of 0, corresponding intuitively to the idea of no rotational or translational invariance and no gaps. [ 9 ]
One box counting technique using a "gliding" box calculates lacunarity according to:
S i {\displaystyle S_{i}} is the number of filled data points in the box and Q ( S i , r ) {\displaystyle Q(S_{i},r)} the normalized frequency distribution of S i {\displaystyle S_{i}} for different box sizes.
Another proposed way of assessing lacunarity using box counting, the Prefactor method, is based on the value obtained from box counting for the fractal dimension ( D B {\displaystyle D_{B}} ). This statistic uses the variable A {\displaystyle A} from the scaling rule N = A ε − D B {\displaystyle N=A\varepsilon ^{-D_{B}}} , where A {\displaystyle A} is calculated from the y-intercept ( y {\displaystyle {\mathit {y}}} ) of the ln-ln regression line for ε {\displaystyle \varepsilon } and either the count ( N {\displaystyle N} ) of boxes that had any pixels at all in them or else m {\displaystyle m} at g {\displaystyle g} . A {\displaystyle A} is particularly affected by image size and the way data are gathered, especially by the lower limit of ε {\displaystyle \varepsilon } s used. The final measure is calculated as shown in Equations 11 through 13 : [ 1 ] [ 4 ]
Below is a list of some fields where lacunarity plays an important role, along with links to relevant research illustrating practical uses of lacunarity. | https://en.wikipedia.org/wiki/Lacunarity |
In chemistry , a ladder polymer is a type of double stranded polymer with the connectivity of a ladder. In a typical one-dimensional polymer, e.g. polyethylene and polysiloxanes, the monomers form two bonds, giving a chain. In a ladder polymer the monomers are interconnected by four bonds. Inorganic ladder polymers are found in synthetic and natural settings. Ladder polymers are a special case of cross-linked polymers because the crosslinks exist only with pairs of chains. [ 1 ]
According to one definition, a ladder polymer, adjacent rings have two or more atoms in common. [ 2 ]
Organic ladder polymers are of interest because they can exhibit exceptional thermal stabilities and the conformation of the subunits is constrained. Because they are less flexible, their processing can be challenging. An early example was derived from condensation of the 1,2,4,5-tetraaminobenzene with naphthalenetetracarboxylic dianhydride . [ 3 ] [ 4 ]
Poly(benzimidazobenzophenanthroline) (BBL) is a conjugated ladder polymer. [ 5 ] Its backbone is composed of aromatic rings and the ladder structures enable the uninterrupted polymer chains with periodic linkages. However, conjugated ladder polymers additionally contain pi conjugation via strong pi-pi stacking interactions and charge transport. [ 6 ] Traditionally, p-typed doped poly(3,4-ethylenedioxythiophene):poly(styrene sulfonate) (PEDOT:PSS) is used as conductive polymers, but BBL doped with poly(ethyleneimine) (PEI) can provide a n-type doped conductive properties for fabricating high-performance organic electronic devices. [ 7 ] BBL's glass transition temperature (Tg) is estimated to be around 500 C based on differential scanning calorimetry (DSC) measurements. [ 8 ] [ 9 ] BBL is stable at higher temperatures. In addition to this, the stress-strain curves of BBL fibers were observed to be very high compared to other semiconductor fibers with a value around 105.8 MPa with the highest BBL polymer concentration. [ 10 ]
Some polysilicates are ladder polymers. One example is provided by the mineral tremolite .
In the area of coordination chemistry , the ladder structure is seen in some coordination polymers . Illustrative is the polymer [CuI( 2-picoline ] n . When the 2-picoline is replaced by a tertiary phosphine , it forms a tetrameric cubane-type cluster , [CuI([[PR 3 ]] 4 (R = organic group_. In both cases, the Cu(I) centers adopt tetrahedral molecular geometry . [ 11 ] [ 12 ] | https://en.wikipedia.org/wiki/Ladder_polymer |
Ladeana Hillier is a biomedical engineer and computational biologist from the University of Washington. [ 1 ] [ 2 ] She was one of the earliest scientists involved in the Human Genome Project [ 3 ] and is noted for her work in various branches of DNA sequencing , [ 4 ] [ 5 ] [ 6 ] [ 7 ] [ 8 ] as well as for having co-developed Phred , [ 9 ] a widely used DNA trace analyzer. [ 10 ] [ 11 ] Her multidisciplinary approach integrates Computational biology and Gene in her work, where she created a software to speed up sequencing. [ 12 ] LaDeana W. Hillier performs integrative study on Gene expression and Regulation of gene expression. [ 13 ] She also worked with the Children's Hospital Oakland Research Institute & Wellcome Trust Sanger Institute. [ 14 ]
This article about a biologist is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Ladeana_Hillier |
Ladislav Svante Rieger (1916–1963) [ 1 ] was a Czechoslovak mathematician who worked in the areas of algebra , mathematical logic , and axiomatic set theory . He is considered to be the founder of mathematical logic in Czechoslovakia, having begun his work around 1957. [ 2 ] [ 3 ]
This article about a European mathematician is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Ladislav_Rieger |
In metallurgy , a ladle is a bucket-shaped container or vessel used to transport and pour out molten metals. [ 1 ] Ladles are often used in foundries and range in size from small hand-carried vessels that resemble a kitchen ladle and hold 20 kilograms (44 lb) to large steelmill ladles that hold up to 300 tonnes (295 long tons; 331 short tons). Many non-ferrous foundries also use ceramic crucibles for transporting and pouring molten metal and will also refer to these as ladles.
The basic term is often prefixed to define the actual purpose of the ladle. The basic ladle design can therefore include many variations that improve the usage of the ladle for specific tasks. For example:
Unless the ladle is to be used with alloys that have very low temperature melting point, the ladle is also fitted with a refractory lining. It is the refractory lining that stops the steel vessel from suffering damage when the ladle is used to transport metals with high melting temperatures that, if the molten metal came in direct contact with the ladle shell, would rapidly melt through the shell. Refractory lining materials come in many forms and the right choice very much depends on each foundry's working practices. Traditionally ladles used to be lined using pre-cast firebricks however refractory concretes have tended to supersede these in many countries.
Foundry ladles are normally rated by their working capacity rather than by their physical size. Hand-held ladles are typically known as handshank ladles and are fitted with a long handle to keep the heat of the metal away from the person holding it. Their capacity is limited to what a man can safely handle. Larger ladles are usually referred to as geared crane ladles. Their capacity is usually determined by the ladle function. Small hand-held ladles might also be crucibles that are fitted with carrying devices. However, in most foundries, the foundry ladle refers to a steel vessel that has a lifting bail fitted so that the vessel can be carried by an overhead crane or monorail system and is also fitted with a mechanical means for rotating the vessel, usually in the form of a gearbox. The gearbox can either be manually operated or powered operation. (See the paragraph below for further details).
For the transportation of very large volumes of molten metal, such as in steel mills , the ladle can run on wheels, a purpose-built ladle transfer car or be slung from an overhead crane and will be tilted using a second overhead lifting device.
The most common shape for a ladle is a vertical cone, but other shapes are possible. Having a tapered cone as the shell adds strength and rigidity to the shell. Having the taper also helps when it comes time to remove the refractory lining. However straight sided shells are also fabricated as are other shapes.
The most common of these other shapes is known as a drum ladle and is shaped as a horizontal cylinder suspended between two bogies . Large versions, often having capacities in excess of 100 tonnes (98 long tons; 110 short tons) are used in steel mills are often referred to as torpedo ladles. [ citation needed ] Torpedo ladles are commonly used to transport liquid iron from a blast furnace to another part of the steel mill. Some versions are even adapted so that they can be carried on special bogies that can be transported by either road or rail.
Ladles can be "lip pour" design, "teapot spout" design, "lip-axis design" or "bottom pour" design:
Ladles can be either open-topped or covered. Covered ladles have a (sometimes removable) dome-shaped lid to contain radiant heat ; they lose heat slower than open-topped ladles. Small ladles do not commonly have covers, although a ceramic blanket may be used instead (where available).
Medium and large ladles which are suspended from a crane have a bail which holds the ladle on shafts, called trunnions . To tilt the ladle a gearbox is used and this is typically a worm gear . The gear mechanism may be hand operated with a large wheel or may be operated by an electric motor or pneumatic motor. Powered rotation allows the ladle operator to be moved to a safe distance and control the rotation of the ladle via a pendant or radio remote control. Powered rotation also allows the ladle to have a number of rotation speeds which may be beneficial to the overall casting process. Powered rotation obviously also reduces the effort required by the ladle operator and allows high volumes of molten metal to be transferred and poured for long periods without operator fatigue. Where the ladle is fitted with a manually operated gearbox, the type of gearbox most commonly used is the worm and wheel design because in most practical circumstances, and when correctly maintained it can be considered as "self-locking" and does not need an internal friction brake to regulate the tilting speed of the ladle. Other types of gear system can also be used but they have to be fitted with an additional braking system that can hold the ladle if the operator takes his hand off the hand-wheel.
Lip-axis ladles may also use hydraulic rams to tilt the ladle. The largest ladles are un-geared and are typically poured using a special, two-winch crane, where the main winch carries the ladle while the second winch engages a lug at the bottom of the ladle. Raising the second winch then rotates the ladle on its trunnions.
Ladles are often designed for special purposes such as adding alloys to the molten metal. Ladles may also have porous plugs inserted into the base, so inert gases can be bubbled through the ladle to enhance alloying or metallic treatment practices. | https://en.wikipedia.org/wiki/Ladle_(metallurgy) |
In mathematics , Ladyzhenskaya's inequality is any of a number of related functional inequalities named after the Soviet Russian mathematician Olga Aleksandrovna Ladyzhenskaya . The original such inequality, for functions of two real variables, was introduced by Ladyzhenskaya in 1958 to prove the existence and uniqueness of long-time solutions to the Navier–Stokes equations in two spatial dimensions (for smooth enough initial data). There is an analogous inequality for functions of three real variables, but the exponents are slightly different; much of the difficulty in establishing existence and uniqueness of solutions to the three-dimensional Navier–Stokes equations stems from these different exponents. Ladyzhenskaya's inequality is one member of a broad class of inequalities known as interpolation inequalities .
Let Ω {\displaystyle \Omega } be a Lipschitz domain in R n {\displaystyle \mathbb {R} ^{n}} for n = 2 or 3 {\displaystyle n=2{\text{ or }}3} and let u : Ω → R {\displaystyle u:\Omega \rightarrow \mathbb {R} } be a weakly differentiable function that vanishes on the boundary of Ω {\displaystyle \Omega } in the sense of trace (that is, u {\displaystyle u} is a limit in the Sobolev space H 1 ( Ω ) {\displaystyle H^{1}(\Omega )} of a sequence of smooth functions that are compactly supported in Ω {\displaystyle \Omega } ). Then there exists a constant C {\displaystyle C} depending only on Ω {\displaystyle \Omega } such that, in the case n = 2 {\displaystyle n=2} :
and in the case n = 3 {\displaystyle n=3} : | https://en.wikipedia.org/wiki/Ladyzhenskaya's_inequality |
A lagg , also called a moat , is the very wet zone on the perimeter of peatland or a bog where water from the adjacent upland collects and flows slowly around the main peat mass. [ 1 ]
A lagg is an area of wetland, especially at the edge of raised bogs , in which water collects. [ 2 ] It is often markedly different from the terrain either side and may consist of a morass of shrubs and murky water. [ 1 ]
In addition to water gathered from surrounding uplands, the lagg also picks up water flowing down from the domed centre of a raised bog through small channels - soaks or water tracks - to the steeply sloping shoulder or rand of the bog. At the foot of the rand , the water collects and meets the water of the surrounding area on the boundary between the peat soil and mineral soil.
This article related to topography is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Lagg_(landform) |
Lagrange's four-square theorem , also known as Bachet's conjecture , states that every nonnegative integer can be represented as a sum of four non-negative integer squares . [ 1 ] That is, the squares form an additive basis of order four: p = a 2 + b 2 + c 2 + d 2 , {\displaystyle p=a^{2}+b^{2}+c^{2}+d^{2},} where the four numbers a , b , c , d {\displaystyle a,b,c,d} are integers. For illustration, 3, 31, and 310 can be represented as the sum of four squares as follows: 3 = 1 2 + 1 2 + 1 2 + 0 2 31 = 5 2 + 2 2 + 1 2 + 1 2 310 = 17 2 + 4 2 + 2 2 + 1 2 = 16 2 + 7 2 + 2 2 + 1 2 = 15 2 + 9 2 + 2 2 + 0 2 = 12 2 + 11 2 + 6 2 + 3 2 . {\displaystyle {\begin{aligned}3&=1^{2}+1^{2}+1^{2}+0^{2}\\[3pt]31&=5^{2}+2^{2}+1^{2}+1^{2}\\[3pt]310&=17^{2}+4^{2}+2^{2}+1^{2}\\[3pt]&=16^{2}+7^{2}+2^{2}+1^{2}\\[3pt]&=15^{2}+9^{2}+2^{2}+0^{2}\\[3pt]&=12^{2}+11^{2}+6^{2}+3^{2}.\end{aligned}}}
This theorem was proven by Joseph Louis Lagrange in 1770. It is a special case of the Fermat polygonal number theorem .
From examples given in the Arithmetica , it is clear that Diophantus was aware of the theorem. This book was translated in 1621 into Latin by Bachet (Claude Gaspard Bachet de Méziriac) , who stated the theorem in the notes of his translation. But the theorem was not proved until 1770 by Lagrange. [ 2 ]
Adrien-Marie Legendre extended the theorem in 1797–8 with his three-square theorem , by proving that a positive integer can be expressed as the sum of three squares if and only if it is not of the form 4 k ( 8 m + 7 ) {\displaystyle 4^{k}(8m+7)} for integers k and m . Later, in 1834, Carl Gustav Jakob Jacobi discovered a simple formula for the number of representations of an integer as the sum of four squares with his own four-square theorem .
The formula is also linked to Descartes' theorem of four "kissing circles", which involves the sum of the squares of the curvatures of four circles. This is also linked to Apollonian gaskets , which were more recently related to the Ramanujan–Petersson conjecture . [ 3 ]
Several very similar modern versions [ 4 ] [ 5 ] [ 6 ] of Lagrange's proof exist. The proof below is a slightly simplified version, in which the cases for which m is even or odd do not require separate arguments.
It is sufficient to prove the theorem for every odd prime number p . This immediately follows from Euler's four-square identity (and from the fact that the theorem is true for the numbers 1 and 2).
The residues of a 2 modulo p are distinct for every a between 0 and ( p − 1)/2 (inclusive).
To see this, take some a and define c as a 2 mod p . a is a root of the polynomial x 2 − c over the field Z/ p Z .
So is p − a (which is different from a ).
In a field K , any polynomial of degree n has at most n distinct roots ( Lagrange's theorem (number theory) ),
so there are no other a with this property, in particular not among 0 to ( p − 1)/2 .
Similarly, for b taking integral values between 0 and ( p − 1)/2 (inclusive), the − b 2 − 1 are distinct.
By the pigeonhole principle , there are a and b in this range, for which a 2 and − b 2 − 1 are congruent modulo p , that is for which a 2 + b 2 + 1 2 + 0 2 = n p . {\displaystyle a^{2}+b^{2}+1^{2}+0^{2}=np.}
Now let m be the smallest positive integer such that mp is the sum of four squares, x 1 2 + x 2 2 + x 3 2 + x 4 2 (we have just shown that there is some m (namely n ) with this property, so there is a least one m , and it is smaller than p ). We show by contradiction that m equals 1: supposing it is not the case, we prove the existence of a positive integer r less than m , for which rp is also the sum of four squares (this is in the spirit of the infinite descent [ 7 ] method of Fermat).
For this purpose, we consider for each x i the y i which is in the same residue class modulo m and between (– m + 1)/2 and m /2 (possibly included). It follows that y 1 2 + y 2 2 + y 3 2 + y 4 2 = mr , for some strictly positive integer r less than m .
Finally, another appeal to Euler's four-square identity shows that mpmr = z 1 2 + z 2 2 + z 3 2 + z 4 2 . But the fact that each x i is congruent to its corresponding y i implies that all of the z i are divisible by m . Indeed, { z 1 = x 1 y 1 + x 2 y 2 + x 3 y 3 + x 4 y 4 ≡ x 1 2 + x 2 2 + x 3 2 + x 4 2 = m p ≡ 0 ( mod m ) , z 2 = x 1 y 2 − x 2 y 1 + x 3 y 4 − x 4 y 3 ≡ x 1 x 2 − x 2 x 1 + x 3 x 4 − x 4 x 3 = 0 ( mod m ) , z 3 = x 1 y 3 − x 2 y 4 − x 3 y 1 + x 4 y 2 ≡ x 1 x 3 − x 2 x 4 − x 3 x 1 + x 4 x 2 = 0 ( mod m ) , z 4 = x 1 y 4 + x 2 y 3 − x 3 y 2 − x 4 y 1 ≡ x 1 x 4 + x 2 x 3 − x 3 x 2 − x 4 x 1 = 0 ( mod m ) . {\displaystyle {\begin{cases}z_{1}&=x_{1}y_{1}+x_{2}y_{2}+x_{3}y_{3}+x_{4}y_{4}&\equiv x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}&=mp\equiv 0&{\pmod {m}},\\z_{2}&=x_{1}y_{2}-x_{2}y_{1}+x_{3}y_{4}-x_{4}y_{3}&\equiv x_{1}x_{2}-x_{2}x_{1}+x_{3}x_{4}-x_{4}x_{3}&=0&{\pmod {m}},\\z_{3}&=x_{1}y_{3}-x_{2}y_{4}-x_{3}y_{1}+x_{4}y_{2}&\equiv x_{1}x_{3}-x_{2}x_{4}-x_{3}x_{1}+x_{4}x_{2}&=0&{\pmod {m}},\\z_{4}&=x_{1}y_{4}+x_{2}y_{3}-x_{3}y_{2}-x_{4}y_{1}&\equiv x_{1}x_{4}+x_{2}x_{3}-x_{3}x_{2}-x_{4}x_{1}&=0&{\pmod {m}}.\end{cases}}}
It follows that, for w i = z i / m , w 1 2 + w 2 2 + w 3 2 + w 4 2 = rp , and this is in contradiction with the minimality of m .
In the descent above, we must rule out both the case y 1 = y 2 = y 3 = y 4 = m /2 (which would give r = m and no descent), and also the case y 1 = y 2 = y 3 = y 4 = 0 (which would give r = 0 rather than strictly positive). For both of those cases, one can check that mp = x 1 2 + x 2 2 + x 3 2 + x 4 2 would be a multiple of m 2 , contradicting the fact that p is a prime greater than m .
Another way to prove the theorem relies on Hurwitz quaternions , which are the analog of integers for quaternions . [ 8 ]
The Hurwitz quaternions consist of all quaternions with integer components and all quaternions with half-integer components. These two sets can be combined into a single formula α = 1 2 E 0 ( 1 + i + j + k ) + E 1 i + E 2 j + E 3 k = a 0 + a 1 i + a 2 j + a 3 k {\displaystyle \alpha ={\frac {1}{2}}E_{0}(1+\mathbf {i} +\mathbf {j} +\mathbf {k} )+E_{1}\mathbf {i} +E_{2}\mathbf {j} +E_{3}\mathbf {k} =a_{0}+a_{1}\mathbf {i} +a_{2}\mathbf {j} +a_{3}\mathbf {k} } where E 0 , E 1 , E 2 , E 3 {\displaystyle E_{0},E_{1},E_{2},E_{3}} are integers. Thus, the quaternion components a 0 , a 1 , a 2 , a 3 {\displaystyle a_{0},a_{1},a_{2},a_{3}} are either all integers or all half-integers, depending on whether E 0 {\displaystyle E_{0}} is even or odd, respectively. The set of Hurwitz quaternions forms a ring ; that is to say, the sum or product of any two Hurwitz quaternions is likewise a Hurwitz quaternion.
The (arithmetic, or field) norm N ( α ) {\displaystyle \mathrm {N} (\alpha )} of a rational quaternion α {\displaystyle \alpha } is the nonnegative rational number N ( α ) = α α ¯ = a 0 2 + a 1 2 + a 2 2 + a 3 2 {\displaystyle \mathrm {N} (\alpha )=\alpha {\bar {\alpha }}=a_{0}^{2}+a_{1}^{2}+a_{2}^{2}+a_{3}^{2}} where α ¯ = a 0 − a 1 i − a 2 j − a 3 k {\displaystyle {\bar {\alpha }}=a_{0}-a_{1}\mathbf {i} -a_{2}\mathbf {j} -a_{3}\mathbf {k} } is the conjugate of α {\displaystyle \alpha } . Note that the norm of a Hurwitz quaternion is always an integer. (If the coefficients are half-integers, then their squares are of the form 1 4 + n : n ∈ Z {\displaystyle {\tfrac {1}{4}}+n:n\in \mathbb {Z} } , and the sum of four such numbers is an integer.)
Since quaternion multiplication is associative, and real numbers commute with other quaternions, the norm of a product of quaternions equals the product of the norms: N ( α β ) = α β ( α β ¯ ) = α β β ¯ α ¯ = α N ( β ) α ¯ = α α ¯ N ( β ) = N ( α ) N ( β ) . {\displaystyle \mathrm {N} (\alpha \beta )=\alpha \beta ({\overline {\alpha \beta }})=\alpha \beta {\bar {\beta }}{\bar {\alpha }}=\alpha \mathrm {N} (\beta ){\bar {\alpha }}=\alpha {\bar {\alpha }}\mathrm {N} (\beta )=\mathrm {N} (\alpha )\mathrm {N} (\beta ).}
For any α ≠ 0 {\displaystyle \alpha \neq 0} , α − 1 = α ¯ N ( α ) − 1 {\displaystyle \alpha ^{-1}={\bar {\alpha }}\mathrm {N} (\alpha )^{-1}} . It follows easily that α {\displaystyle \alpha } is a unit in the ring of Hurwitz quaternions if and only if N ( α ) = 1 {\displaystyle \mathrm {N} (\alpha )=1} .
The proof of the main theorem begins by reduction to the case of prime numbers. Euler's four-square identity implies that if Lagrange's four-square theorem holds for two numbers, it holds for the product of the two numbers. Since any natural number can be factored into powers of primes, it suffices to prove the theorem for prime numbers. It is true for 2 = 1 2 + 1 2 + 0 2 + 0 2 {\displaystyle 2=1^{2}+1^{2}+0^{2}+0^{2}} . To show this for an odd prime integer p , represent it as a quaternion ( p , 0 , 0 , 0 ) {\displaystyle (p,0,0,0)} and assume for now (as we shall show later) that it is not a Hurwitz irreducible ; that is, it can be factored into two non-unit Hurwitz quaternions p = α β . {\displaystyle p=\alpha \beta .}
The norms of p , α , β {\displaystyle p,\alpha ,\beta } are integers such that N ( p ) = p 2 = N ( α β ) = N ( α ) N ( β ) {\displaystyle \mathrm {N} (p)=p^{2}=\mathrm {N} (\alpha \beta )=\mathrm {N} (\alpha )\mathrm {N} (\beta )} and N ( α ) , N ( β ) > 1 {\displaystyle \mathrm {N} (\alpha ),\mathrm {N} (\beta )>1} . This shows that both N ( α ) {\displaystyle \mathrm {N} (\alpha )} and N ( β ) {\displaystyle \mathrm {N} (\beta )} are equal to p (since they are integers), and p is the sum of four squares p = N ( α ) = a 0 2 + a 1 2 + a 2 2 + a 3 2 . {\displaystyle p=\mathrm {N} (\alpha )=a_{0}^{2}+a_{1}^{2}+a_{2}^{2}+a_{3}^{2}.}
If it happens that the α {\displaystyle \alpha } chosen has half-integer coefficients, it can be replaced by another Hurwitz quaternion. Choose ω = ( ± 1 ± i ± j ± k ) / 2 {\displaystyle \omega =(\pm 1\pm \mathbf {i} \pm \mathbf {j} \pm \mathbf {k} )/2} in such a way that γ ≡ ω + α {\displaystyle \gamma \equiv \omega +\alpha } has even integer coefficients. Then p = ( γ ¯ − ω ¯ ) ω ω ¯ ( γ − ω ) = ( γ ¯ ω − 1 ) ( ω ¯ γ − 1 ) . {\displaystyle p=({\bar {\gamma }}-{\bar {\omega }})\omega {\bar {\omega }}(\gamma -\omega )=({\bar {\gamma }}\omega -1)({\bar {\omega }}\gamma -1).}
Since γ {\displaystyle \gamma } has even integer coefficients, ( ω ¯ γ − 1 ) {\displaystyle ({\bar {\omega }}\gamma -1)} will have integer coefficients and can be used instead of the original α {\displaystyle \alpha } to give a representation of p as the sum of four squares.
As for showing that p is not a Hurwitz irreducible, Lagrange proved that any odd prime p divides at least one number of the form u = 1 + l 2 + m 2 {\displaystyle u=1+l^{2}+m^{2}} , where l and m are integers. [ 8 ] This can be seen as follows: since p is prime, a 2 ≡ b 2 ( mod p ) {\displaystyle a^{2}\equiv b^{2}{\pmod {p}}} can hold for integers a , b {\displaystyle a,b} , only when a ≡ ± b ( mod p ) {\displaystyle a\equiv \pm b{\pmod {p}}} . Thus, the set X = { 0 2 , 1 2 , … , ( ( p − 1 ) / 2 ) 2 } {\displaystyle X=\{0^{2},1^{2},\dots ,((p-1)/2)^{2}\}} of squares contains ( p + 1 ) / 2 {\displaystyle (p+1)/2} distinct residues modulo p . Likewise, Y = { − ( 1 + x ) : x ∈ X } {\displaystyle Y=\{-(1+x):x\in X\}} contains ( p + 1 ) / 2 {\displaystyle (p+1)/2} residues. Since there are only p residues in total, and | X | + | Y | = p + 1 > p {\displaystyle |X|+|Y|=p+1>p} , the sets X and Y must intersect.
The number u can be factored in Hurwitz quaternions: 1 + l 2 + m 2 = ( 1 + l i + m j ) ( 1 − l i − m j ) . {\displaystyle 1+l^{2}+m^{2}=(1+l\;\mathbf {i} +m\;\mathbf {j} )(1-l\;\mathbf {i} -m\;\mathbf {j} ).}
The norm on Hurwitz quaternions satisfies a form of the Euclidean property: for any quaternion α = a 0 + a 1 i + a 2 j + a 3 k {\displaystyle \alpha =a_{0}+a_{1}\mathbf {i} +a_{2}\mathbf {j} +a_{3}\mathbf {k} } with rational coefficients we can choose a Hurwitz quaternion β = b 0 + b 1 i + b 2 j + b 3 k {\displaystyle \beta =b_{0}+b_{1}\mathbf {i} +b_{2}\mathbf {j} +b_{3}\mathbf {k} } so that N ( α − β ) < 1 {\displaystyle \mathrm {N} (\alpha -\beta )<1} by first choosing b 0 {\displaystyle b_{0}} so that | a 0 − b 0 | ≤ 1 / 4 {\displaystyle |a_{0}-b_{0}|\leq 1/4} and then b 1 , b 2 , b 3 {\displaystyle b_{1},b_{2},b_{3}} so that | a i − b i | ≤ 1 / 2 {\displaystyle |a_{i}-b_{i}|\leq 1/2} for i = 1 , 2 , 3 {\displaystyle i=1,2,3} . Then we obtain N ( α − β ) = ( a 0 − b 0 ) 2 + ( a 1 − b 1 ) 2 + ( a 2 − b 2 ) 2 + ( a 3 − b 3 ) 2 ≤ ( 1 4 ) 2 + ( 1 2 ) 2 + ( 1 2 ) 2 + ( 1 2 ) 2 = 13 16 < 1. {\displaystyle {\begin{aligned}\mathrm {N} (\alpha -\beta )&=(a_{0}-b_{0})^{2}+(a_{1}-b_{1})^{2}+(a_{2}-b_{2})^{2}+(a_{3}-b_{3})^{2}\\&\leq \left({\frac {1}{4}}\right)^{2}+\left({\frac {1}{2}}\right)^{2}+\left({\frac {1}{2}}\right)^{2}+\left({\frac {1}{2}}\right)^{2}={\frac {13}{16}}<1.\end{aligned}}}
It follows that for any Hurwitz quaternions α , β {\displaystyle \alpha ,\beta } with α ≠ 0 {\displaystyle \alpha \neq 0} , there exists a Hurwitz quaternion γ {\displaystyle \gamma } such that N ( β − α γ ) < N ( α ) . {\displaystyle \mathrm {N} (\beta -\alpha \gamma )<\mathrm {N} (\alpha ).}
The ring H of Hurwitz quaternions is not commutative, hence it is not an actual Euclidean domain, and it does not have unique factorization in the usual sense. Nevertheless, the property above implies that every right ideal is principal . Thus, there is a Hurwitz quaternion α {\displaystyle \alpha } such that α H = p H + ( 1 − l i − m j ) H . {\displaystyle \alpha H=pH+(1-l\;\mathbf {i} -m\;\mathbf {j} )H.}
In particular, p = α β {\displaystyle p=\alpha \beta } for some Hurwitz quaternion β {\displaystyle \beta } . If β {\displaystyle \beta } were a unit, 1 − l i − m j {\displaystyle 1-l\;\mathbf {i} -m\;\mathbf {j} } would be a multiple of p , however this is impossible as 1 / p − l / p i − m / p j {\displaystyle 1/p-l/p\;\mathbf {i} -m/p\;\mathbf {j} } is not a Hurwitz quaternion for p > 2 {\displaystyle p>2} . Similarly, if α {\displaystyle \alpha } were a unit, we would have ( 1 + l i + m j ) H = ( 1 + l i + m j ) p H + ( 1 + l i + m j ) ( 1 − l i − m j ) H ⊆ p H {\displaystyle (1+l\;\mathbf {i} +m\;\mathbf {j} )H=(1+l\;\mathbf {i} +m\;\mathbf {j} )pH+(1+l\;\mathbf {i} +m\;\mathbf {j} )(1-l\;\mathbf {i} -m\;\mathbf {j} )H\subseteq pH} so p divides 1 + l i + m j {\displaystyle 1+l\;\mathbf {i} +m\;\mathbf {j} } , which again contradicts the fact that 1 / p − l / p i − m / p j {\displaystyle 1/p-l/p\;\mathbf {i} -m/p\;\mathbf {j} } is not a Hurwitz quaternion. Thus, p is not Hurwitz irreducible, as claimed.
Lagrange's four-square theorem is a special case of the Fermat polygonal number theorem and Waring's problem . Another possible generalization is the following problem: Given natural numbers a , b , c , d {\displaystyle a,b,c,d} , can we solve
n = a x 1 2 + b x 2 2 + c x 3 2 + d x 4 2 {\displaystyle n=ax_{1}^{2}+bx_{2}^{2}+cx_{3}^{2}+dx_{4}^{2}}
for all positive integers n in integers x 1 , x 2 , x 3 , x 4 {\displaystyle x_{1},x_{2},x_{3},x_{4}} ? The case a = b = c = d = 1 {\displaystyle a=b=c=d=1} is answered in the positive by Lagrange's four-square theorem. The general solution was given by Ramanujan . [ 9 ] He proved that if we assume, without loss of generality, that a ≤ b ≤ c ≤ d {\displaystyle a\leq b\leq c\leq d} then there are exactly 54 possible choices for a , b , c , d {\displaystyle a,b,c,d} such that the problem is solvable in integers x 1 , x 2 , x 3 , x 4 {\displaystyle x_{1},x_{2},x_{3},x_{4}} for all n . (Ramanujan listed a 55th possibility a = 1 , b = 2 , c = 5 , d = 5 {\displaystyle a=1,b=2,c=5,d=5} , but in this case the problem is not solvable if n = 15 {\displaystyle n=15} . [ 10 ] )
In 1986, Michael O. Rabin and Jeffrey Shallit [ 11 ] proposed randomized polynomial-time algorithms for computing a single representation n = x 1 2 + x 2 2 + x 3 2 + x 4 2 {\displaystyle n=x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}} for a given integer n , in expected running time O ( log ( n ) 2 ) {\displaystyle \mathrm {O} (\log(n)^{2})} . It was further improved to O ( log ( n ) 2 log ( log ( n ) ) − 1 ) {\displaystyle \mathrm {O} (\log(n)^{2}\log(\log(n))^{-1})} by Paul Pollack and Enrique Treviño in 2018. [ 12 ]
The number of representations of a natural number n as the sum of four squares of integers is denoted by r 4 ( n ). Jacobi's four-square theorem states that this is eight times the sum of the divisors of n if n is odd and 24 times the sum of the odd divisors of n if n is even (see divisor function ), i.e.
r 4 ( n ) = { 8 ∑ m ∣ n m if n is odd 24 ∑ m | n m odd m if n is even . {\displaystyle r_{4}(n)={\begin{cases}8\sum \limits _{m\mid n}m&{\text{if }}n{\text{ is odd}}\\[12pt]24\sum \limits _{\begin{smallmatrix}m|n\\m{\text{ odd}}\end{smallmatrix}}m&{\text{if }}n{\text{ is even}}.\end{cases}}}
Equivalently, it is eight times the sum of all its divisors which are not divisible by 4, i.e.
r 4 ( n ) = 8 ∑ m : 4 ∤ m ∣ n m . {\displaystyle r_{4}(n)=8\sum _{m\,:\,4\nmid m\mid n}m.}
We may also write this as r 4 ( n ) = 8 σ ( n ) − 32 σ ( n / 4 ) , {\displaystyle r_{4}(n)=8\sigma (n)-32\sigma (n/4)\ ,} where the second term is to be taken as zero if n is not divisible by 4. In particular, for a prime number p we have the explicit formula r 4 ( p ) = 8( p + 1) . [ 13 ]
Some values of r 4 ( n ) occur infinitely often as r 4 ( n ) = r 4 (2 m n ) whenever n is even. The values of r 4 ( n )/ n can be arbitrarily large: indeed, r 4 ( n )/ n is infinitely often larger than 8 √ log n . [ 13 ]
The sequence of positive integers which have only one representation as a sum of four squares of non-negative integers (up to order) is:
These integers consist of the seven odd numbers 1, 3, 5, 7, 11, 15, 23 and all numbers of the form 2 ( 4 k ) , 6 ( 4 k ) {\displaystyle 2(4^{k}),6(4^{k})} or 14 ( 4 k ) {\displaystyle 14(4^{k})} .
The sequence of positive integers which cannot be represented as a sum of four non-zero squares is:
These integers consist of the eight odd numbers 1, 3, 5, 9, 11, 17, 29, 41 and all numbers of the form 2 ( 4 k ) , 6 ( 4 k ) {\displaystyle 2(4^{k}),6(4^{k})} or 14 ( 4 k ) {\displaystyle 14(4^{k})} .
Lagrange's four-square theorem can be refined in various ways. For example, Zhi-Wei Sun [ 14 ] proved that each natural number can be written as a sum of four squares with some requirements on the choice of these four numbers.
One may also wonder whether it is necessary to use the entire set of square integers to write each natural as the sum of four squares. Eduard Wirsing proved that there exists a set of squares S with | S | = O ( n 1 / 4 log 1 / 4 n ) {\displaystyle |S|=O(n^{1/4}\log ^{1/4}n)} such that every positive integer smaller than or equal to n can be written as a sum of at most 4 elements of S . [ 15 ] | https://en.wikipedia.org/wiki/Lagrange's_four-square_theorem |
In the algebra , Lagrange's identity , named after Joseph Louis Lagrange , is: [ 1 ] [ 2 ] ( ∑ k = 1 n a k 2 ) ( ∑ k = 1 n b k 2 ) − ( ∑ k = 1 n a k b k ) 2 = ∑ i = 1 n − 1 ∑ j = i + 1 n ( a i b j − a j b i ) 2 ( = 1 2 ∑ i = 1 n ∑ j = 1 , j ≠ i n ( a i b j − a j b i ) 2 ) , {\displaystyle {\begin{aligned}\left(\sum _{k=1}^{n}a_{k}^{2}\right)\left(\sum _{k=1}^{n}b_{k}^{2}\right)-\left(\sum _{k=1}^{n}a_{k}b_{k}\right)^{2}&=\sum _{i=1}^{n-1}\sum _{j=i+1}^{n}\left(a_{i}b_{j}-a_{j}b_{i}\right)^{2}\\&\left(={\frac {1}{2}}\sum _{i=1}^{n}\sum _{j=1,j\neq i}^{n}(a_{i}b_{j}-a_{j}b_{i})^{2}\right),\end{aligned}}} which applies to any two sets { a 1 , a 2 , ..., a n } and { b 1 , b 2 , ..., b n } of real or complex numbers (or more generally, elements of a commutative ring ). This identity is a generalisation of the Brahmagupta–Fibonacci identity and a special form of the Binet–Cauchy identity .
In a more compact vector notation, Lagrange's identity is expressed as: [ 3 ] ‖ a ‖ 2 ‖ b ‖ 2 − ( a ⋅ b ) 2 = ∑ 1 ≤ i < j ≤ n ( a i b j − a j b i ) 2 , {\displaystyle \left\|\mathbf {a} \right\|^{2}\left\|\mathbf {b} \right\|^{2}-(\mathbf {a} \cdot \mathbf {b} )^{2}=\sum _{1\leq i<j\leq n}\left(a_{i}b_{j}-a_{j}b_{i}\right)^{2}\,,} where a and b are n -dimensional vectors with components that are real numbers. The extension to complex numbers requires the interpretation of the dot product as an inner product or Hermitian dot product. Explicitly, for complex numbers, Lagrange's identity can be written in the form: [ 4 ] ( ∑ k = 1 n | a k | 2 ) ( ∑ k = 1 n | b k | 2 ) − | ∑ k = 1 n a k b k | 2 = ∑ i = 1 n − 1 ∑ j = i + 1 n | a i b ¯ j − a j b ¯ i | 2 {\displaystyle \left(\sum _{k=1}^{n}|a_{k}|^{2}\right)\left(\sum _{k=1}^{n}|b_{k}|^{2}\right)-\left|\sum _{k=1}^{n}a_{k}b_{k}\right|^{2}=\sum _{i=1}^{n-1}\sum _{j=i+1}^{n}\left|a_{i}{\overline {b}}_{j}-a_{j}{\overline {b}}_{i}\right|^{2}} involving the absolute value . [ 5 ] [ 6 ]
Since the right-hand side of the identity is clearly non-negative, it implies Cauchy's inequality in the finite-dimensional real coordinate space R n and its complex counterpart C n .
Geometrically, the identity asserts that the square of the volume of the parallelepiped spanned by a set of vectors is the Gram determinant of the vectors.
In terms of the wedge product , Lagrange's identity can be written ( a ⋅ a ) ( b ⋅ b ) − ( a ⋅ b ) 2 = ( a ∧ b ) ⋅ ( a ∧ b ) . {\displaystyle (a\cdot a)(b\cdot b)-(a\cdot b)^{2}=(a\wedge b)\cdot (a\wedge b).}
Hence, it can be seen as a formula which gives the length of the wedge product of two vectors, which is the area of the parallelogram they define, in terms of the dot products of the two vectors, as ‖ a ∧ b ‖ = ( a ⋅ a ) ( b ⋅ b ) − ( a ⋅ b ) 2 = ‖ a ‖ 2 ‖ b ‖ 2 − ( a ⋅ b ) 2 . {\displaystyle \|a\wedge b\|={\sqrt {(a\cdot a)(b\cdot b)-(a\cdot b)^{2}}}={\sqrt {\|a\|^{2}\|b\|^{2}-(a\cdot b)^{2}}}.}
In three dimensions, Lagrange's identity asserts that if a and b are vectors in R 3 with lengths | a | and | b |, then Lagrange's identity can be written in terms of the cross product and dot product : [ 7 ] [ 8 ] | a | 2 | b | 2 − ( a ⋅ b ) 2 = | a × b | 2 {\displaystyle |\mathbf {a} |^{2}|\mathbf {b} |^{2}-(\mathbf {a} \cdot \mathbf {b} )^{2}=|\mathbf {a} \times \mathbf {b} |^{2}}
Using the definition of angle based upon the dot product (see also Cauchy–Schwarz inequality ), the left-hand side is | a | 2 | b | 2 ( 1 − cos 2 θ ) = | a | 2 | b | 2 sin 2 θ {\displaystyle \left|\mathbf {a} \right|^{2}\left|\mathbf {b} \right|^{2}\left(1-\cos ^{2}\theta \right)=\left|\mathbf {a} \right|^{2}\left|\mathbf {b} \right|^{2}\sin ^{2}\theta } where θ is the angle formed by the vectors a and b . The area of a parallelogram with sides | a | and | b | and angle θ is known in elementary geometry to be | a | | b | | sin θ | , {\displaystyle \left|\mathbf {a} \right|\left|\mathbf {b} \right|\left|\sin \theta \right|,} so the left-hand side of Lagrange's identity is the squared area of the parallelogram. The cross product appearing on the right-hand side is defined by a × b = ( a 2 b 3 − a 3 b 2 ) i + ( a 3 b 1 − a 1 b 3 ) j + ( a 1 b 2 − a 2 b 1 ) k {\displaystyle \mathbf {a} \times \mathbf {b} =\left(a_{2}b_{3}-a_{3}b_{2}\right)\mathbf {i} +\left(a_{3}b_{1}-a_{1}b_{3}\right)\mathbf {j} +\left(a_{1}b_{2}-a_{2}b_{1}\right)\mathbf {k} } which is a vector whose components are equal in magnitude to the areas of the projections of the parallelogram onto the yz , zx , and xy planes, respectively.
For a and b as vectors in R 7 , Lagrange's identity takes on the same form as in the case of R 3 [ 9 ] | a | 2 | b | 2 − | a ⋅ b | 2 = | a × b | 2 , {\displaystyle |\mathbf {a} |^{2}|\mathbf {b} |^{2}-|\mathbf {a} \cdot \mathbf {b} |^{2}=|\mathbf {a} \times \mathbf {b} |^{2}\ ,}
However, the cross product in 7 dimensions does not share all the properties of the cross product in 3 dimensions. For example, the direction of a × b in 7-dimensions may be the same as c × d even though c and d are linearly independent of a and b . Also the seven-dimensional cross product is not compatible with the Jacobi identity . [ 9 ]
A quaternion p is defined as the sum of a scalar t and a vector v : p = t + v = t + x i + y j + z k . {\displaystyle p=t+\mathbf {v} =t+x\ \mathbf {i} +y\ \mathbf {j} +z\ \mathbf {k} .}
The product of two quaternions p = t + v and q = s + w is defined by p q = ( s t − v ⋅ w ) + s v + t w + v × w . {\displaystyle pq=(st-\mathbf {v} \cdot \mathbf {w} )+s\ \mathbf {v} +t\ \mathbf {w} +\mathbf {v} \times \mathbf {w} .}
The quaternionic conjugate of q is defined by q ¯ = t − v , {\displaystyle {\overline {q}}=t-\mathbf {v} ,} and the norm squared is | q | 2 = q q ¯ = t 2 + x 2 + y 2 + z 2 . {\displaystyle |q|^{2}=q{\overline {q}}=t^{2}\ +\ x^{2}+\ y^{2}\ +\ z^{2}.}
The multiplicativity of the norm in the quaternion algebra provides, for quaternions p and q : [ 10 ] | p q | = | p | | q | . {\displaystyle \left|pq\right|=\left|p\right|\left|q\right|.}
The quaternions p and q are called imaginary if their scalar part is zero; equivalently, if p = v , q = w . {\displaystyle p=\mathbf {v} ,\quad q=\mathbf {w} .}
Lagrange's identity is just the multiplicativity of the norm of imaginary quaternions, | v w | 2 = | v | 2 | w | 2 , {\displaystyle |\mathbf {v} \mathbf {w} |^{2}=|\mathbf {v} |^{2}|\mathbf {w} |^{2},} since, by definition, | v w | 2 = ( v ⋅ w ) 2 + | v × w | 2 . {\displaystyle |\mathbf {v} \mathbf {w} |^{2}=(\mathbf {v} \cdot \mathbf {w} )^{2}+|\mathbf {v} \times \mathbf {w} |^{2}.}
The vector form follows from the Binet-Cauchy identity by setting c i = a i and d i = b i . The second version follows by letting c i and d i denote the complex conjugates of a i and b i , respectively,
Here is also a direct proof. [ 11 ] The expansion of the first term on the left side is:
which means that the product of a column of a s and a row of b s yields (a sum of elements of) a square of ab s , which can be broken up into a diagonal and a pair of triangles on either side of the diagonal.
The second term on the left side of Lagrange's identity can be expanded as:
which means that a symmetric square can be broken up into its diagonal and a pair of equal triangles on either side of the diagonal.
To expand the summation on the right side of Lagrange's identity, first expand the square within the summation: ∑ i = 1 n − 1 ∑ j = i + 1 n ( a i b j − a j b i ) 2 = ∑ i = 1 n − 1 ∑ j = i + 1 n ( a i 2 b j 2 + a j 2 b i 2 − 2 a i b j a j b i ) . {\displaystyle \sum _{i=1}^{n-1}\sum _{j=i+1}^{n}(a_{i}b_{j}-a_{j}b_{i})^{2}=\sum _{i=1}^{n-1}\sum _{j=i+1}^{n}\left(a_{i}^{2}b_{j}^{2}+a_{j}^{2}b_{i}^{2}-2a_{i}b_{j}a_{j}b_{i}\right).}
Distribute the summation on the right side, ∑ i = 1 n − 1 ∑ j = i + 1 n ( a i b j − a j b i ) 2 = ∑ i = 1 n − 1 ∑ j = i + 1 n a i 2 b j 2 + ∑ i = 1 n − 1 ∑ j = i + 1 n a j 2 b i 2 − 2 ∑ i = 1 n − 1 ∑ j = i + 1 n a i b j a j b i . {\displaystyle \sum _{i=1}^{n-1}\sum _{j=i+1}^{n}(a_{i}b_{j}-a_{j}b_{i})^{2}=\sum _{i=1}^{n-1}\sum _{j=i+1}^{n}a_{i}^{2}b_{j}^{2}+\sum _{i=1}^{n-1}\sum _{j=i+1}^{n}a_{j}^{2}b_{i}^{2}-2\sum _{i=1}^{n-1}\sum _{j=i+1}^{n}a_{i}b_{j}a_{j}b_{i}.}
Now exchange the indices i and j of the second term on the right side, and permute the b factors of the third term, yielding:
Back to the left side of Lagrange's identity: it has two terms, given in expanded form by Equations ( 1 ) and ( 2 ). The first term on the right side of Equation ( 2 ) ends up canceling out the first term on the right side of Equation ( 1 ), yielding
which is the same as Equation ( 3 ), so Lagrange's identity is indeed an identity, Q.E.D.
Normed division algebras require that the norm of the product is equal to the product of the norms. Lagrange's identity exhibits this equality.
The product identity used as a starting point here, is a consequence of the norm of the product equality with the product of the norm for scator algebras. This proposal, originally presented in the context of a deformed Lorentz metric, is based on a transformation stemming from the product operation and magnitude definition in hyperbolic scator algebra. [ 12 ] Lagrange's identity can be proved in a variety of ways. [ 4 ]
Let a i , b i ∈ C {\displaystyle a_{i},b_{i}\in \mathbb {C} } be complex numbers and the overbar represents complex conjugate.
The product identity ∏ i = 1 n ( 1 − a i a ¯ i − b i b ¯ i + a i a ¯ i b i b ¯ i ) = ∏ i = 1 n ( 1 − a i a ¯ i ) ∏ i = 1 n ( 1 − b i b ¯ i ) {\displaystyle \prod _{i=1}^{n}\left(1-a_{i}{\bar {a}}_{i}-b_{i}{\bar {b}}_{i}+a_{i}{\bar {a}}_{i}b_{i}{\bar {b}}_{i}\right)=\prod _{i=1}^{n}\left(1-a_{i}{\bar {a}}_{i}\right)\prod _{i=1}^{n}\left(1-b_{i}{\bar {b}}_{i}\right)} reduces to the complex Lagrange's identity when fourth order terms, in a series expansion, are considered.
In order to prove it, expand the product on the LHS of the product identity in terms of
series up to fourth order. To this end, recall that products of the form ( 1 + x i ) {\displaystyle \left(1+x_{i}\right)} can be expanded in terms of sums as ∏ i = 1 n ( 1 + x i ) = 1 + ∑ i = 1 n x i + ∑ i < j n x i x j + O 3 + ( x ) , {\displaystyle \prod _{i=1}^{n}\left(1+x_{i}\right)=1+\sum _{i=1}^{n}x_{i}+\sum _{i<j}^{n}x_{i}x_{j}+{\mathcal {O}}^{3+}(x),} where O 3 + ( x ) {\displaystyle {\mathcal {O}}^{3+}(x)} means terms with order three or higher in x {\displaystyle x} .
∏ i = 1 n ( 1 − a i a ¯ i − b i b ¯ i + a i a ¯ i b i b ¯ i ) = 1 − ∑ i = 1 n ( a i a ¯ i + b i b ¯ i ) + ∑ i = 1 n a i a ¯ i b i b ¯ i + ∑ i < j n ( a i a ¯ i a j a ¯ j + b i b ¯ i b j b ¯ j ) + ∑ i < j n ( a i a ¯ i b j b ¯ j + a j a ¯ j b i b ¯ i ) + O 5 + . {\displaystyle \prod _{i=1}^{n}\left(1-a_{i}{\bar {a}}_{i}-b_{i}{\bar {b}}_{i}+a_{i}{\bar {a}}_{i}b_{i}{\bar {b}}_{i}\right)=1-\sum _{i=1}^{n}\left(a_{i}{\bar {a}}_{i}+b_{i}{\bar {b}}_{i}\right)+\sum _{i=1}^{n}a_{i}{\bar {a}}_{i}b_{i}{\bar {b}}_{i}+\sum _{i<j}^{n}\left(a_{i}{\bar {a}}_{i}a_{j}{\bar {a}}_{j}+b_{i}{\bar {b}}_{i}b_{j}{\bar {b}}_{j}\right)+\sum _{i<j}^{n}\left(a_{i}{\bar {a}}_{i}b_{j}{\bar {b}}_{j}+a_{j}{\bar {a}}_{j}b_{i}{\bar {b}}_{i}\right)+{\mathcal {O}}^{5+}.}
The two factors on the RHS are also written in terms of series ∏ i = 1 n ( 1 − a i a ¯ i ) ∏ i = 1 n ( 1 − b i b ¯ i ) = ( 1 − ∑ i = 1 n a i a ¯ i + ∑ i < j n a i a ¯ i a j a ¯ j + O 5 + ) ( 1 − ∑ i = 1 n b i b ¯ i + ∑ i < j n b i b ¯ i b j b ¯ j + O 5 + ) . {\displaystyle \prod _{i=1}^{n}\left(1-a_{i}{\bar {a}}_{i}\right)\prod _{i=1}^{n}\left(1-b_{i}{\bar {b}}_{i}\right)=\left(1-\sum _{i=1}^{n}a_{i}{\bar {a}}_{i}+\sum _{i<j}^{n}a_{i}{\bar {a}}_{i}a_{j}{\bar {a}}_{j}+{\mathcal {O}}^{5+}\right)\left(1-\sum _{i=1}^{n}b_{i}{\bar {b}}_{i}+\sum _{i<j}^{n}b_{i}{\bar {b}}_{i}b_{j}{\bar {b}}_{j}+{\mathcal {O}}^{5+}\right).}
The product of this expression up to fourth order is ∏ i = 1 n ( 1 − a i a ¯ i ) ∏ i = 1 n ( 1 − b i b ¯ i ) = 1 − ∑ i = 1 n ( a i a ¯ i + b i b ¯ i ) + ( ∑ i = 1 n a i a ¯ i ) ( ∑ i = 1 n b i b ¯ i ) + ∑ i < j n ( a i a ¯ i a j a ¯ j + b i b ¯ i b j b ¯ j ) + O 5 + . {\displaystyle \prod _{i=1}^{n}\left(1-a_{i}{\bar {a}}_{i}\right)\prod _{i=1}^{n}\left(1-b_{i}{\bar {b}}_{i}\right)=1-\sum _{i=1}^{n}\left(a_{i}{\bar {a}}_{i}+b_{i}{\bar {b}}_{i}\right)+\left(\sum _{i=1}^{n}a_{i}{\bar {a}}_{i}\right)\left(\sum _{i=1}^{n}b_{i}{\bar {b}}_{i}\right)+\sum _{i<j}^{n}\left(a_{i}{\bar {a}}_{i}a_{j}{\bar {a}}_{j}+b_{i}{\bar {b}}_{i}b_{j}{\bar {b}}_{j}\right)+{\mathcal {O}}^{5+}.} Substitution of these two results in the product identity give ∑ i = 1 n a i a ¯ i b i b ¯ i + ∑ i < j n ( a i a ¯ i b j b ¯ j + a j a ¯ j b i b ¯ i ) = ( ∑ i = 1 n a i a ¯ i ) ( ∑ i = 1 n b i b ¯ i ) . {\displaystyle \sum _{i=1}^{n}a_{i}{\bar {a}}_{i}b_{i}{\bar {b}}_{i}+\sum _{i<j}^{n}\left(a_{i}{\bar {a}}_{i}b_{j}{\bar {b}}_{j}+a_{j}{\bar {a}}_{j}b_{i}{\bar {b}}_{i}\right)=\left(\sum _{i=1}^{n}a_{i}{\bar {a}}_{i}\right)\left(\sum _{i=1}^{n}b_{i}{\bar {b}}_{i}\right).}
The product of two conjugates series can be expressed as series involving the product of conjugate terms. The conjugate series product is ( ∑ i = 1 n x i ) ( ∑ i = 1 n x ¯ i ) = ∑ i = 1 n x i x ¯ i + ∑ i < j n ( x i x ¯ j + x ¯ i x j ) , {\displaystyle \left(\sum _{i=1}^{n}x_{i}\right)\left(\sum _{i=1}^{n}{\bar {x}}_{i}\right)=\sum _{i=1}^{n}x_{i}{\bar {x}}_{i}+\sum _{i<j}^{n}\left(x_{i}{\bar {x}}_{j}+{\bar {x}}_{i}x_{j}\right),} thus ( ∑ i = 1 n a i b i ) ( ∑ i = 1 n a i b i ¯ ) − ∑ i < j n ( a i b i a ¯ j b ¯ j + a ¯ i b ¯ i a j b j ) + ∑ i < j n ( a i a ¯ i b j b ¯ j + a j a ¯ j b i b ¯ i ) = ( ∑ i = 1 n a i a ¯ i ) ( ∑ i = 1 n b i b ¯ i ) . {\displaystyle \left(\sum _{i=1}^{n}a_{i}b_{i}\right)\left(\sum _{i=1}^{n}{\overline {a_{i}b_{i}}}\right)-\sum _{i<j}^{n}\left(a_{i}b_{i}{\bar {a}}_{j}{\bar {b}}_{j}+{\bar {a}}_{i}{\bar {b}}_{i}a_{j}b_{j}\right)+\sum _{i<j}^{n}\left(a_{i}{\bar {a}}_{i}b_{j}{\bar {b}}_{j}+a_{j}{\bar {a}}_{j}b_{i}{\bar {b}}_{i}\right)=\left(\sum _{i=1}^{n}a_{i}{\bar {a}}_{i}\right)\left(\sum _{i=1}^{n}b_{i}{\bar {b}}_{i}\right).}
The terms of the last two series on the LHS are grouped as a i a ¯ i b j b ¯ j + a j a ¯ j b i b ¯ i − a i b i a ¯ j b ¯ j − a ¯ i b ¯ i a j b j = ( a i b ¯ j − a j b ¯ i ) ( a ¯ i b j − a ¯ j b i ) , {\displaystyle a_{i}{\bar {a}}_{i}b_{j}{\bar {b}}_{j}+a_{j}{\bar {a}}_{j}b_{i}{\bar {b}}_{i}-a_{i}b_{i}{\bar {a}}_{j}{\bar {b}}_{j}-{\bar {a}}_{i}{\bar {b}}_{i}a_{j}b_{j}=\left(a_{i}{\bar {b}}_{j}-a_{j}{\bar {b}}_{i}\right)\left({\bar {a}}_{i}b_{j}-{\bar {a}}_{j}b_{i}\right),} in order to obtain the complex Lagrange's identity: ( ∑ i = 1 n a i b i ) ( ∑ i = 1 n a i b i ¯ ) + ∑ i < j n ( a i b ¯ j − a j b ¯ i ) ( a i b ¯ j − a j b ¯ i ¯ ) = ( ∑ i = 1 n a i a ¯ i ) ( ∑ i = 1 n b i b ¯ i ) . {\displaystyle \left(\sum _{i=1}^{n}a_{i}b_{i}\right)\left(\sum _{i=1}^{n}{\overline {a_{i}b_{i}}}\right)+\sum _{i<j}^{n}\left(a_{i}{\bar {b}}_{j}-a_{j}{\bar {b}}_{i}\right)\left({\overline {a_{i}{\bar {b}}_{j}-a_{j}{\bar {b}}_{i}}}\right)=\left(\sum _{i=1}^{n}a_{i}{\bar {a}}_{i}\right)\left(\sum _{i=1}^{n}b_{i}{\bar {b}}_{i}\right).}
In terms of the moduli, | ∑ i = 1 n a i b i | 2 + ∑ i < j n | a i b ¯ j − a j b ¯ i | 2 = ( ∑ i = 1 n | a i | 2 ) ( ∑ i = 1 n | b i | 2 ) . {\displaystyle \left|\sum _{i=1}^{n}a_{i}b_{i}\right|^{2}+\sum _{i<j}^{n}\left|a_{i}{\bar {b}}_{j}-a_{j}{\bar {b}}_{i}\right|^{2}=\left(\sum _{i=1}^{n}\left|a_{i}\right|^{2}\right)\left(\sum _{i=1}^{n}\left|b_{i}\right|^{2}\right).}
Lagrange's identity for complex numbers has been obtained from a straightforward
product identity. A derivation for the reals is obviously even more succinct. Since the Cauchy–Schwarz inequality is a particular case of Lagrange's identity, [ 4 ] this
proof is yet another way to obtain the CS inequality. Higher order terms in the series produce novel identities. | https://en.wikipedia.org/wiki/Lagrange's_identity |
In the study of ordinary differential equations and their associated boundary value problems in mathematics, Lagrange's identity , named after Joseph Louis Lagrange , gives the boundary terms arising from integration by parts of a self-adjoint linear differential operator . Lagrange's identity is fundamental in Sturm–Liouville theory . In more than one independent variable, Lagrange's identity is generalized by Green's second identity .
In general terms, Lagrange's identity for any pair of functions u and v in function space C 2 (that is, twice differentiable) in n dimensions is: [ 1 ] v L [ u ] − u L ∗ [ v ] = ∇ ⋅ M , {\displaystyle vL[u]-uL^{*}[v]=\nabla \cdot {\boldsymbol {M}},} where: M i = ∑ j = 1 n a i j ( v ∂ u ∂ x j − u ∂ v ∂ x j ) + u v ( b i − ∑ j = 1 n ∂ a i j ∂ x j ) , {\displaystyle M_{i}=\sum _{j=1}^{n}a_{ij}\left(v{\frac {\partial u}{\partial x_{j}}}-u{\frac {\partial v}{\partial x_{j}}}\right)+uv\left(b_{i}-\sum _{j=1}^{n}{\frac {\partial a_{ij}}{\partial x_{j}}}\right),} and ∇ ⋅ M = ∑ i = 1 n ∂ ∂ x i M i , {\displaystyle \nabla \cdot {\boldsymbol {M}}=\sum _{i=1}^{n}{\frac {\partial }{\partial x_{i}}}M_{i},}
The operator L and its adjoint operator L * are given by: L [ u ] = ∑ i , j = 1 n a i , j ∂ 2 u ∂ x i ∂ x j + ∑ i = 1 n b i ∂ u ∂ x i + c u {\displaystyle L[u]=\sum _{i,\ j=1}^{n}a_{i,j}{\frac {\partial ^{2}u}{\partial x_{i}\partial x_{j}}}+\sum _{i=1}^{n}b_{i}{\frac {\partial u}{\partial x_{i}}}+cu} and L ∗ [ v ] = ∑ i , j = 1 n ∂ 2 ( a i , j v ) ∂ x i ∂ x j − ∑ i = 1 n ∂ ( b i v ) ∂ x i + c v . {\displaystyle L^{*}[v]=\sum _{i,\ j=1}^{n}{\frac {\partial ^{2}(a_{i,j}v)}{\partial x_{i}\partial x_{j}}}-\sum _{i=1}^{n}{\frac {\partial (b_{i}v)}{\partial x_{i}}}+cv.}
If Lagrange's identity is integrated over a bounded region, then the divergence theorem can be used to form Green's second identity in the form: ∫ Ω v L [ u ] d Ω = ∫ Ω u L ∗ [ v ] d Ω + ∫ S M ⋅ n d S , {\displaystyle \int _{\Omega }vL[u]\,d\Omega =\int _{\Omega }uL^{*}[v]\ d\Omega +\int _{S}{\boldsymbol {M\cdot n}}\,dS,}
where S is the surface bounding the volume Ω and n is the unit outward normal to the surface S .
Any second order ordinary differential equation of the form: a ( x ) d 2 y d x 2 + b ( x ) d y d x + c ( x ) y + λ w ( x ) y = 0 , {\displaystyle a(x){\frac {d^{2}y}{dx^{2}}}+b(x){\frac {dy}{dx}}+c(x)y+\lambda w(x)y=0,} can be put in the form: [ 2 ] d d x ( p ( x ) d y d x ) + ( q ( x ) + λ w ( x ) ) y ( x ) = 0. {\displaystyle {\frac {d}{dx}}\left(p(x){\frac {dy}{dx}}\right)+\left(q(x)+\lambda w(x)\right)y(x)=0.}
This general form motivates introduction of the Sturm–Liouville operator L , defined as an operation upon a function f such that: L f = d d x ( p ( x ) d f d x ) + q ( x ) f . {\displaystyle Lf={\frac {d}{dx}}\left(p(x){\frac {df}{dx}}\right)+q(x)f.}
It can be shown that for any u and v for which the various derivatives exist, Lagrange's identity for ordinary differential equations holds: [ 2 ] u L v − v L u = − d d x [ p ( x ) ( v d u d x − u d v d x ) ] . {\displaystyle uLv-vLu=-{\frac {d}{dx}}\left[p(x)\left(v{\frac {du}{dx}}-u{\frac {dv}{dx}}\right)\right].}
For ordinary differential equations defined in the interval [0, 1], Lagrange's identity can be integrated to obtain an integral form (also known as Green's formula): [ 3 ] [ 4 ] [ 5 ] [ 6 ] ∫ 0 1 d x ( u L v − v L u ) = [ p ( x ) ( u d v d x − v d u d x ) ] 0 1 , {\displaystyle \int _{0}^{1}dx\ (uLv-vLu)=\left[p(x)\left(u{\frac {dv}{dx}}-v{\frac {du}{dx}}\right)\right]_{0}^{1},}
where p = P ( x ) {\displaystyle p=P(x)} , q = Q ( x ) {\displaystyle q=Q(x)} , u = U ( x ) {\displaystyle u=U(x)} and v = V ( x ) {\displaystyle v=V(x)} are functions of x {\displaystyle x} . u {\displaystyle u} and v {\displaystyle v} having continuous second derivatives on the interval [ 0 , 1 ] {\displaystyle [0,1]} .
We have: u L v = u [ d d x ( p ( x ) d v d x ) + q ( x ) v ] , {\displaystyle uLv=u\left[{\frac {d}{dx}}\left(p(x){\frac {dv}{dx}}\right)+q(x)v\right],} and v L u = v [ d d x ( p ( x ) d u d x ) + q ( x ) u ] . {\displaystyle vLu=v\left[{\frac {d}{dx}}\left(p(x){\frac {du}{dx}}\right)+q(x)u\right].}
Subtracting: u L v − v L u = u d d x ( p ( x ) d v d x ) − v d d x ( p ( x ) d u d x ) . {\displaystyle uLv-vLu=u{\frac {d}{dx}}\left(p(x){\frac {dv}{dx}}\right)-v{\frac {d}{dx}}\left(p(x){\frac {du}{dx}}\right).}
The leading multiplied u and v can be moved inside the differentiation, because the extra differentiated terms in u and v are the same in the two subtracted terms and simply cancel each other. Thus, u L v − v L u = d d x ( p ( x ) u d v d x ) − d d x ( v p ( x ) d u d x ) , = d d x [ p ( x ) ( u d v d x − v d u d x ) ] , {\displaystyle {\begin{aligned}uLv-vLu&={\frac {d}{dx}}\left(p(x)u{\frac {dv}{dx}}\right)-{\frac {d}{dx}}\left(vp(x){\frac {du}{dx}}\right),\\&={\frac {d}{dx}}\left[p(x)\left(u{\frac {dv}{dx}}-v{\frac {du}{dx}}\right)\right],\end{aligned}}} which is Lagrange's identity. Integrating from zero to one: ∫ 0 1 d x ( u L v − v L u ) = [ p ( x ) ( u d v d x − v d u d x ) ] 0 1 , {\displaystyle \int _{0}^{1}dx\ (uLv-vLu)=\left[p(x)\left(u{\frac {dv}{dx}}-v{\frac {du}{dx}}\right)\right]_{0}^{1},} as was to be shown. | https://en.wikipedia.org/wiki/Lagrange's_identity_(boundary_value_problem) |
In the mathematical field of group theory , Lagrange's theorem states that if H is a subgroup of any finite group G , then | H | {\displaystyle |H|} is a divisor of | G | {\displaystyle |G|} , i.e. the order (number of elements) of every subgroup H divides the order of group G.
The theorem is named after Joseph-Louis Lagrange . The following variant states that for a subgroup H {\displaystyle H} of a finite group G {\displaystyle G} , not only is | G | / | H | {\displaystyle |G|/|H|} an integer, but its value is the index [ G : H ] {\displaystyle [G:H]} , defined as the number of left cosets of H {\displaystyle H} in G {\displaystyle G} .
Lagrange's theorem — If H is a subgroup of a group G , then | G | = [ G : H ] ⋅ | H | . {\displaystyle \left|G\right|=\left[G:H\right]\cdot \left|H\right|.}
This variant holds even if G {\displaystyle G} is infinite, provided that | G | {\displaystyle |G|} , | H | {\displaystyle |H|} , and [ G : H ] {\displaystyle [G:H]} are interpreted as cardinal numbers .
The left cosets of H in G are the equivalence classes of a certain equivalence relation on G : specifically, call x and y in G equivalent if there exists h in H such that x = yh .
Therefore, the set of left cosets forms a partition of G .
Each left coset aH has the same cardinality as H because x ↦ a x {\displaystyle x\mapsto ax} defines a bijection H → a H {\displaystyle H\to aH} (the inverse is y ↦ a − 1 y {\displaystyle y\mapsto a^{-1}y} ).
The number of left cosets is the index [ G : H ] .
By the previous three sentences,
Lagrange's theorem can be extended to the equation of indexes between three subgroups of G . [ 1 ]
Extension of Lagrange's theorem — If H is a subgroup of G and K is a subgroup of H , then
Let S be a set of coset representatives for K in H ,
so H = ⨆ s ∈ S s K {\displaystyle H=\bigsqcup _{s\in S}sK} (disjoint union), and | S | = [ H : K ] {\displaystyle |S|=[H:K]} .
For any a ∈ G {\displaystyle a\in G} , left-multiplication-by- a is a bijection G → G {\displaystyle G\to G} ,
so a H = ⨆ s ∈ S a s K {\displaystyle aH=\bigsqcup _{s\in S}asK} .
Thus each left coset of H decomposes into [ H : K ] {\displaystyle [H:K]} left cosets of K .
Since G decomposes into [ G : H ] {\displaystyle [G:H]} left cosets of H ,
each of which decomposes into [ H : K ] {\displaystyle [H:K]} left cosets of K ,
the total number [ G : K ] {\displaystyle [G:K]} of left cosets of K in G is [ G : H ] [ H : K ] {\displaystyle [G:H][H:K]} .
If we take K = { e } ( e is the identity element of G ), then [ G : { e }] = | G | and [ H : { e }] = | H | . Therefore, we can recover the original equation | G | = [ G : H ] | H | .
A consequence of the theorem is that the order of any element a of a finite group (i.e. the smallest positive integer number k with a k = e , where e is the identity element of the group) divides the order of that group, since the order of a is equal to the order of the cyclic subgroup generated by a . If the group has n elements, it follows
This can be used to prove Fermat's little theorem and its generalization, Euler's theorem . These special cases were known long before the general theorem was proved.
The theorem also shows that any group of prime order is cyclic and simple , since the subgroup generated by any non-identity element must be the whole group itself.
Lagrange's theorem can also be used to show that there are infinitely many primes : suppose there were a largest prime p {\displaystyle p} . Any prime divisor q {\displaystyle q} of the Mersenne number 2 p − 1 {\displaystyle 2^{p}-1} satisfies 2 p ≡ 1 ( mod q ) {\displaystyle 2^{p}\equiv 1{\pmod {q}}} (see modular arithmetic ), meaning that the order of 2 {\displaystyle 2} in the multiplicative group ( Z / q Z ) ∗ {\displaystyle (\mathbb {Z} /q\mathbb {Z} )^{*}} is p {\displaystyle p} . By Lagrange's theorem, the order of 2 {\displaystyle 2} must divide the order of ( Z / q Z ) ∗ {\displaystyle (\mathbb {Z} /q\mathbb {Z} )^{*}} , which is q − 1 {\displaystyle q-1} . So p {\displaystyle p} divides q − 1 {\displaystyle q-1} , giving p < q {\displaystyle p<q} , contradicting the assumption that p {\displaystyle p} is the largest prime. [ 2 ]
Lagrange's theorem raises the converse question as to whether every divisor of the order of a group is the order of some subgroup. This does not hold in general: given a finite group G and a divisor d of | G |, there does not necessarily exist a subgroup of G with order d . The smallest example is A 4 (the alternating group of degree 4), which has 12 elements but no subgroup of order 6.
A "Converse of Lagrange's Theorem" (CLT) group is a finite group with the property that for every divisor of the order of the group, there is a subgroup of that order. It is known that a CLT group must be solvable and that every supersolvable group is a CLT group. However, there exist solvable groups that are not CLT (for example, A 4 ) and CLT groups that are not supersolvable (for example, S 4 , the symmetric group of degree 4).
There are partial converses to Lagrange's theorem. For general groups, Cauchy's theorem guarantees the existence of an element, and hence of a cyclic subgroup, of order any prime dividing the group order. Sylow's theorem extends this to the existence of a subgroup of order equal to the maximal power of any prime dividing the group order. For solvable groups, Hall's theorems assert the existence of a subgroup of order equal to any unitary divisor of the group order (that is, a divisor coprime to its cofactor).
The converse of Lagrange's theorem states that if d is a divisor of the order of a group G , then there exists a subgroup H where | H | = d .
We will examine the alternating group A 4 , the set of even permutations as the subgroup of the Symmetric group S 4 .
| A 4 | = 12 so the divisors are 1, 2, 3, 4, 6, 12 . Assume to the contrary that there exists a subgroup H in A 4 with | H | = 6 .
Let V be the non-cyclic subgroup of A 4 called the Klein four-group .
Let K = H ⋂ V . Since both H and V are subgroups of A 4 , K is also a subgroup of A 4 .
From Lagrange's theorem, the order of K must divide both 6 and 4 , the orders of H and V respectively. The only two positive integers that divide both 6 and 4 are 1 and 2 . So | K | = 1 or 2 .
Assume | K | = 1 , then K = { e } . If H does not share any elements with V , then the 5 elements in H besides the Identity element e must be of the form ( a b c ) where a, b, c are distinct elements in {1, 2, 3, 4} .
Since any element of the form ( a b c ) squared is ( a c b ) , and ( a b c )( a c b ) = e , any element of H in the form ( a b c ) must be paired with its inverse. Specifically, the remaining 5 elements of H must come from distinct pairs of elements in A 4 that are not in V . This is impossible since pairs of elements must be even and cannot total up to 5 elements. Thus, the assumptions that | K | = 1 is wrong, so | K | = 2 .
Then, K = { e , v } where v ∈ V , v must be in the form ( a b )( c d ) where a, b, c, d are distinct elements of {1, 2, 3, 4} . The other four elements in H are cycles of length 3.
Note that the cosets generated by a subgroup of a group form a partition of the group. The cosets generated by a specific subgroup are either identical to each other or disjoint . The index of a subgroup in a group [ A 4 : H ] = | A 4 |/| H | is the number of cosets generated by that subgroup. Since | A 4 | = 12 and | H | = 6 , H will generate two left cosets, one that is equal to H and another, gH , that is of length 6 and includes all the elements in A 4 not in H .
Since there are only 2 distinct cosets generated by H , then H must be normal. Because of that, H = gHg −1 (∀ g ∈ A 4 ) . In particular, this is true for g = ( a b c ) ∈ A 4 . Since H = gHg −1 , gvg −1 ∈ H .
Without loss of generality, assume that a = 1 , b = 2 , c = 3 , d = 4 . Then g = (1 2 3) , v = (1 2)(3 4) , g −1 = (1 3 2) , gv = (1 3 4) , gvg −1 = (1 4)(2 3) . Transforming back, we get gvg −1 = ( a d )( b c ) . Because V contains all disjoint transpositions in A 4 , gvg −1 ∈ V . Hence, gvg −1 ∈ H ⋂ V = K .
Since gvg −1 ≠ v , we have demonstrated that there is a third element in K . But earlier we assumed that | K | = 2 , so we have a contradiction.
Therefore, our original assumption that there is a subgroup of order 6 is not true and consequently there is no subgroup of order 6 in A 4 and the converse of Lagrange's theorem is not necessarily true. Q.E.D.
Lagrange himself did not prove the theorem in its general form. He stated, in his article Réflexions sur la résolution algébrique des équations , [ 3 ] that if a polynomial in n variables has its variables permuted in all n ! ways, the number of different polynomials that are obtained is always a factor of n ! . (For example, if the variables x , y , and z are permuted in all 6 possible ways in the polynomial x + y − z then we get a total of 3 different polynomials: x + y − z , x + z − y , and y + z − x . Note that 3 is a factor of 6.) The number of such polynomials is the index in the symmetric group S n of the subgroup H of permutations that preserve the polynomial. (For the example of x + y − z , the subgroup H in S 3 contains the identity and the transposition ( x y ) .) So the size of H divides n ! . With the later development of abstract groups, this result of Lagrange on polynomials was recognized to extend to the general theorem about finite groups which now bears his name.
In his Disquisitiones Arithmeticae in 1801, Carl Friedrich Gauss proved Lagrange's theorem for the special case of ( Z / p Z ) ∗ {\displaystyle (\mathbb {Z} /p\mathbb {Z} )^{*}} , the multiplicative group of nonzero integers modulo p , where p is a prime. [ 4 ] In 1844, Augustin-Louis Cauchy proved Lagrange's theorem for the symmetric group S n . [ 5 ]
Camille Jordan finally proved Lagrange's theorem for the case of any permutation group in 1861. [ 6 ] | https://en.wikipedia.org/wiki/Lagrange's_theorem_(group_theory) |
In numerical analysis , the Lagrange interpolating polynomial is the unique polynomial of lowest degree that interpolates a given set of data.
Given a data set of coordinate pairs ( x j , y j ) {\displaystyle (x_{j},y_{j})} with 0 ≤ j ≤ k , {\displaystyle 0\leq j\leq k,} the x j {\displaystyle x_{j}} are called nodes and the y j {\displaystyle y_{j}} are called values . The Lagrange polynomial L ( x ) {\displaystyle L(x)} has degree ≤ k {\textstyle \leq k} and assumes each value at the corresponding node, L ( x j ) = y j . {\displaystyle L(x_{j})=y_{j}.}
Although named after Joseph-Louis Lagrange , who published it in 1795, [ 1 ] the method was first discovered in 1779 by Edward Waring . [ 2 ] It is also an easy consequence of a formula published in 1783 by Leonhard Euler . [ 3 ]
Uses of Lagrange polynomials include the Newton–Cotes method of numerical integration , Shamir's secret sharing scheme in cryptography , and Reed–Solomon error correction in coding theory .
For equispaced nodes, Lagrange interpolation is susceptible to Runge's phenomenon of large oscillation.
Given a set of k + 1 {\textstyle k+1} nodes { x 0 , x 1 , … , x k } {\displaystyle \{x_{0},x_{1},\ldots ,x_{k}\}} , which must all be distinct, x j ≠ x m {\displaystyle x_{j}\neq x_{m}} for indices j ≠ m {\displaystyle j\neq m} , the Lagrange basis for polynomials of degree ≤ k {\textstyle \leq k} for those nodes is the set of polynomials { ℓ 0 ( x ) , ℓ 1 ( x ) , … , ℓ k ( x ) } {\textstyle \{\ell _{0}(x),\ell _{1}(x),\ldots ,\ell _{k}(x)\}} each of degree k {\textstyle k} which take values ℓ j ( x m ) = 0 {\textstyle \ell _{j}(x_{m})=0} if m ≠ j {\textstyle m\neq j} and ℓ j ( x j ) = 1 {\textstyle \ell _{j}(x_{j})=1} . Using the Kronecker delta this can be written ℓ j ( x m ) = δ j m . {\textstyle \ell _{j}(x_{m})=\delta _{jm}.} Each basis polynomial can be explicitly described by the product:
ℓ j ( x ) = ( x − x 0 ) ( x j − x 0 ) ⋯ ( x − x j − 1 ) ( x j − x j − 1 ) ( x − x j + 1 ) ( x j − x j + 1 ) ⋯ ( x − x k ) ( x j − x k ) = ∏ 0 ≤ m ≤ k m ≠ j x − x m x j − x m | . {\displaystyle {\begin{aligned}\ell _{j}(x)&={\frac {(x-x_{0})}{(x_{j}-x_{0})}}\cdots {\frac {(x-x_{j-1})}{(x_{j}-x_{j-1})}}{\frac {(x-x_{j+1})}{(x_{j}-x_{j+1})}}\cdots {\frac {(x-x_{k})}{(x_{j}-x_{k})}}\\[8mu]&=\prod _{\begin{smallmatrix}0\leq m\leq k\\m\neq j\end{smallmatrix}}{\frac {x-x_{m}}{x_{j}-x_{m}}}{\vphantom {\Bigg |}}.\end{aligned}}}
Notice that the numerator ∏ m ≠ j ( x − x m ) {\textstyle \prod _{m\neq j}(x-x_{m})} has k {\textstyle k} roots at the nodes { x m } m ≠ j {\textstyle \{x_{m}\}_{m\neq j}} while the denominator ∏ m ≠ j ( x j − x m ) {\textstyle \prod _{m\neq j}(x_{j}-x_{m})} scales the resulting polynomial so that ℓ j ( x j ) = 1. {\textstyle \ell _{j}(x_{j})=1.}
The Lagrange interpolating polynomial for those nodes through the corresponding values { y 0 , y 1 , … , y k } {\displaystyle \{y_{0},y_{1},\ldots ,y_{k}\}} is the linear combination:
L ( x ) = ∑ j = 0 k y j ℓ j ( x ) . {\displaystyle L(x)=\sum _{j=0}^{k}y_{j}\ell _{j}(x).}
Each basis polynomial has degree k {\textstyle k} , so the sum L ( x ) {\textstyle L(x)} has degree ≤ k {\textstyle \leq k} , and it interpolates the data because L ( x m ) = ∑ j = 0 k y j ℓ j ( x m ) = ∑ j = 0 k y j δ m j = y m . {\textstyle L(x_{m})=\sum _{j=0}^{k}y_{j}\ell _{j}(x_{m})=\sum _{j=0}^{k}y_{j}\delta _{mj}=y_{m}.}
The interpolating polynomial is unique. Proof: assume the polynomial M ( x ) {\textstyle M(x)} of degree ≤ k {\textstyle \leq k} interpolates the data. Then the difference M ( x ) − L ( x ) {\textstyle M(x)-L(x)} is zero at k + 1 {\textstyle k+1} distinct nodes { x 0 , x 1 , … , x k } . {\textstyle \{x_{0},x_{1},\ldots ,x_{k}\}.} But the only polynomial of degree ≤ k {\textstyle \leq k} with more than k {\textstyle k} roots is the constant zero function, so M ( x ) − L ( x ) = 0 , {\textstyle M(x)-L(x)=0,} or M ( x ) = L ( x ) . {\textstyle M(x)=L(x).}
Each Lagrange basis polynomial ℓ j ( x ) {\textstyle \ell _{j}(x)} can be rewritten as the product of three parts, a function ℓ ( x ) = ∏ m ( x − x m ) {\textstyle \ell (x)=\prod _{m}(x-x_{m})} common to every basis polynomial, a node-specific constant w j = ∏ m ≠ j ( x j − x m ) − 1 {\textstyle w_{j}=\prod _{m\neq j}(x_{j}-x_{m})^{-1}} (called the barycentric weight ), and a part representing the displacement from x j {\textstyle x_{j}} to x {\textstyle x} : [ 4 ]
ℓ j ( x ) = ℓ ( x ) w j x − x j {\displaystyle \ell _{j}(x)=\ell (x){\dfrac {w_{j}}{x-x_{j}}}}
By factoring ℓ ( x ) {\textstyle \ell (x)} out from the sum, we can write the Lagrange polynomial in the so-called first barycentric form :
If the weights w j {\displaystyle w_{j}} have been pre-computed, this requires only O ( k ) {\displaystyle {\mathcal {O}}(k)} operations compared to O ( k 2 ) {\displaystyle {\mathcal {O}}(k^{2})} for evaluating each Lagrange basis polynomial ℓ j ( x ) {\displaystyle \ell _{j}(x)} individually.
The barycentric interpolation formula can also easily be updated to incorporate a new node x k + 1 {\displaystyle x_{k+1}} by dividing each of the w j {\displaystyle w_{j}} , j = 0 … k {\displaystyle j=0\dots k} by ( x j − x k + 1 ) {\displaystyle (x_{j}-x_{k+1})} and constructing the new w k + 1 {\displaystyle w_{k+1}} as above.
For any x , {\textstyle x,} ∑ j = 0 k ℓ j ( x ) = 1 {\textstyle \sum _{j=0}^{k}\ell _{j}(x)=1} because the constant function g ( x ) = 1 {\textstyle g(x)=1} is the unique polynomial of degree ≤ k {\displaystyle \leq k} interpolating the data { ( x 0 , 1 ) , ( x 1 , 1 ) , … , ( x k , 1 ) } . {\textstyle \{(x_{0},1),(x_{1},1),\ldots ,(x_{k},1)\}.} We can thus further simplify the barycentric formula by dividing L ( x ) = L ( x ) / g ( x ) : {\displaystyle L(x)=L(x)/g(x)\colon }
This is called the second form or true form of the barycentric interpolation formula.
This second form has advantages in computation cost and accuracy: it avoids evaluation of ℓ ( x ) {\displaystyle \ell (x)} ; the work to compute each term in the denominator w j / ( x − x j ) {\displaystyle w_{j}/(x-x_{j})} has already been done in computing ( w j / ( x − x j ) ) y j {\displaystyle {\bigl (}w_{j}/(x-x_{j}){\bigr )}y_{j}} and so computing the sum in the denominator costs only k {\textstyle k} addition operations; for evaluation points x {\textstyle x} which are close to one of the nodes x j {\textstyle x_{j}} , catastrophic cancelation would ordinarily be a problem for the value ( x − x j ) {\textstyle (x-x_{j})} , however this quantity appears in both numerator and denominator and the two cancel leaving good relative accuracy in the final result.
Using this formula to evaluate L ( x ) {\displaystyle L(x)} at one of the nodes x j {\displaystyle x_{j}} will result in the indeterminate ∞ y j / ∞ {\displaystyle \infty y_{j}/\infty } ; computer implementations must replace such results by L ( x j ) = y j . {\displaystyle L(x_{j})=y_{j}.}
Each Lagrange basis polynomial can also be written in barycentric form:
Solving an interpolation problem leads to a problem in linear algebra amounting to inversion of a matrix. Using a standard monomial basis for our interpolation polynomial L ( x ) = ∑ j = 0 k x j m j {\textstyle L(x)=\sum _{j=0}^{k}x^{j}m_{j}} , we must invert the Vandermonde matrix ( x i ) j {\displaystyle (x_{i})^{j}} to solve L ( x i ) = y i {\displaystyle L(x_{i})=y_{i}} for the coefficients m j {\displaystyle m_{j}} of L ( x ) {\displaystyle L(x)} . By choosing a better basis, the Lagrange basis, L ( x ) = ∑ j = 0 k l j ( x ) y j {\textstyle L(x)=\sum _{j=0}^{k}l_{j}(x)y_{j}} , we merely get the identity matrix , δ i j {\displaystyle \delta _{ij}} , which is its own inverse: the Lagrange basis automatically inverts the analog of the Vandermonde matrix.
This construction is analogous to the Chinese remainder theorem . Instead of checking for remainders of integers modulo prime numbers, we are checking for remainders of polynomials when divided by linears.
Furthermore, when the order is large, Fast Fourier transformation can be used to solve for the coefficients of the interpolated polynomial.
We wish to interpolate f ( x ) = x 2 {\displaystyle f(x)=x^{2}} over the domain 1 ≤ x ≤ 3 {\displaystyle 1\leq x\leq 3} at the three nodes { 1 , 2 , 3 } {\displaystyle \{1,\,2,\,3\}} :
The node polynomial ℓ {\displaystyle \ell } is
The barycentric weights are
The Lagrange basis polynomials are
The Lagrange interpolating polynomial is:
In (second) barycentric form,
The Lagrange form of the interpolation polynomial shows the linear character of polynomial interpolation and the uniqueness of the interpolation polynomial. Therefore, it is preferred in proofs and theoretical arguments. Uniqueness can also be seen from the invertibility of the Vandermonde matrix, due to the non-vanishing of the Vandermonde determinant .
But, as can be seen from the construction, each time a node x k changes, all Lagrange basis polynomials have to be recalculated. A better form of the interpolation polynomial for practical (or computational) purposes is the barycentric form of the Lagrange interpolation (see below) or Newton polynomials .
Lagrange and other interpolation at equally spaced points, as in the example above, yield a polynomial oscillating above and below the true function. This behaviour tends to grow with the number of points, leading to a divergence known as Runge's phenomenon ; the problem may be eliminated by choosing interpolation points at Chebyshev nodes . [ 5 ]
The Lagrange basis polynomials can be used in numerical integration to derive the Newton–Cotes formulas .
When interpolating a given function f by a polynomial of degree k at the nodes x 0 , . . . , x k {\displaystyle x_{0},...,x_{k}} we get the remainder R ( x ) = f ( x ) − L ( x ) {\displaystyle R(x)=f(x)-L(x)} which can be expressed as [ 6 ]
where f [ x 0 , … , x k , x ] {\displaystyle f[x_{0},\ldots ,x_{k},x]} is the notation for divided differences . Alternatively, the remainder can be expressed as a contour integral in complex domain as
The remainder can be bound as
Clearly, R ( x ) {\displaystyle R(x)} is zero at nodes. To find R ( x ) {\displaystyle R(x)} at a point x p {\displaystyle x_{p}} , define a new function F ( x ) = R ( x ) − R ~ ( x ) = f ( x ) − L ( x ) − R ~ ( x ) {\displaystyle F(x)=R(x)-{\tilde {R}}(x)=f(x)-L(x)-{\tilde {R}}(x)} and choose R ~ ( x ) = C ⋅ ∏ i = 0 k ( x − x i ) {\textstyle {\tilde {R}}(x)=C\cdot \prod _{i=0}^{k}(x-x_{i})} where C {\displaystyle C} is the constant we are required to determine for a given x p {\displaystyle x_{p}} . We choose C {\displaystyle C} so that F ( x ) {\displaystyle F(x)} has k + 2 {\displaystyle k+2} zeroes (at all nodes and x p {\displaystyle x_{p}} ) between x 0 {\displaystyle x_{0}} and x k {\displaystyle x_{k}} (including endpoints). Assuming that f ( x ) {\displaystyle f(x)} is k + 1 {\displaystyle k+1} -times differentiable, since L ( x ) {\displaystyle L(x)} and R ~ ( x ) {\displaystyle {\tilde {R}}(x)} are polynomials, and therefore, are infinitely differentiable, F ( x ) {\displaystyle F(x)} will be k + 1 {\displaystyle k+1} -times differentiable. By Rolle's theorem , F ( 1 ) ( x ) {\displaystyle F^{(1)}(x)} has k + 1 {\displaystyle k+1} zeroes, F ( 2 ) ( x ) {\displaystyle F^{(2)}(x)} has k {\displaystyle k} zeroes... F ( k + 1 ) {\displaystyle F^{(k+1)}} has 1 zero, say ξ , x 0 < ξ < x k {\displaystyle \xi ,\,x_{0}<\xi <x_{k}} . Explicitly writing F ( k + 1 ) ( ξ ) {\displaystyle F^{(k+1)}(\xi )} :
The equation can be rearranged as [ 7 ]
Since F ( x p ) = 0 {\displaystyle F(x_{p})=0} we have R ( x p ) = R ~ ( x p ) = f k + 1 ( ξ ) ( k + 1 ) ! ∏ i = 0 k ( x p − x i ) {\displaystyle R(x_{p})={\tilde {R}}(x_{p})={\frac {f^{k+1}(\xi )}{(k+1)!}}\prod _{i=0}^{k}(x_{p}-x_{i})}
The d th derivative of a Lagrange interpolating polynomial can be written in terms of the derivatives of the basis polynomials,
Recall (see § Definition above) that each Lagrange basis polynomial is
ℓ j ( x ) = ∏ m = 0 m ≠ j k x − x m x j − x m . {\displaystyle {\begin{aligned}\ell _{j}(x)&=\prod _{\begin{smallmatrix}m=0\\m\neq j\end{smallmatrix}}^{k}{\frac {x-x_{m}}{x_{j}-x_{m}}}.\end{aligned}}}
The first derivative can be found using the product rule :
The second derivative is
The third derivative is
and likewise for higher derivatives.
Note that all of these formulas for derivatives are invalid at or near a node. A method of evaluating all orders of derivatives of a Lagrange polynomial efficiently at all points of the domain, including the nodes, is converting the Lagrange polynomial to power basis form and then evaluating the derivatives.
The Lagrange polynomial can also be computed in finite fields . This has applications in cryptography , such as in Shamir's Secret Sharing scheme. | https://en.wikipedia.org/wiki/Lagrange_polynomial |
In mathematics , the Lagrange reversion theorem gives series or formal power series expansions of certain implicitly defined functions ; indeed, of compositions with such functions.
Let v be a function of x and y in terms of another function f such that
Then for any function g , for small enough y :
If g is the identity, this becomes
In which case the equation can be derived using perturbation theory .
In 1770, Joseph Louis Lagrange (1736–1813) published his power series solution of the implicit equation for v mentioned above. However, his solution used cumbersome series expansions of logarithms. [ 1 ] [ 2 ] In 1780, Pierre-Simon Laplace (1749–1827) published a simpler proof of the theorem, which was based on relations between partial derivatives with respect to the variable x and the parameter y. [ 3 ] [ 4 ] [ 5 ] Charles Hermite (1822–1901) presented the most straightforward proof of the theorem by using contour integration. [ 6 ] [ 7 ] [ 8 ]
Lagrange's reversion theorem is used to obtain numerical solutions to Kepler's equation .
We start by writing:
Writing the delta-function as an integral we have:
The integral over k then gives δ ( x − z ) {\displaystyle \delta (x-z)} and we have:
Rearranging the sum and cancelling then gives the result: | https://en.wikipedia.org/wiki/Lagrange_reversion_theorem |
Lagrange stability is a concept in the stability theory of dynamical systems , named after Joseph-Louis Lagrange .
For any point in the state space, x ∈ M {\displaystyle x\in M} in a real continuous dynamical system ( T , M , Φ ) {\displaystyle (T,M,\Phi )} , where T {\displaystyle T} is R {\displaystyle \mathbb {R} } , the motion Φ ( t , x ) {\displaystyle \Phi (t,x)} is said to be positively Lagrange stable if the positive semi-orbit γ x + {\displaystyle \gamma _{x}^{+}} is compact . If the negative semi-orbit γ x − {\displaystyle \gamma _{x}^{-}} is compact , then the motion is said to be negatively Lagrange stable . The motion through x {\displaystyle x} is said to be Lagrange stable if it is both positively and negatively Lagrange stable. If the state space M {\displaystyle M} is the Euclidean space R n {\displaystyle \mathbb {R} ^{n}} , then the above definitions are equivalent to γ x + , γ x − {\displaystyle \gamma _{x}^{+},\gamma _{x}^{-}} and γ x {\displaystyle \gamma _{x}} being bounded , respectively.
A dynamical system is said to be positively-/negatively-/Lagrange stable if for each x ∈ M {\displaystyle x\in M} , the motion Φ ( t , x ) {\displaystyle \Phi (t,x)} is positively-/negatively-/Lagrange stable, respectively.
This mathematical analysis –related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Lagrange_stability |
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