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Korean Genome Project (Korea1K) is the largest genome sequencing project in Korea , first launched in 2015 as part of the Genome Korea in Ulsan . As of 2021, the project has sequenced over 10,000 human genomes and is the first large-scale data base for constructing a genetic map and diversity analysis of Koreans. [ 1 ] [ 2 ]
KGP was originated from the national initiative of sequencing the reference Korean and whole population genomes in 2006 by KOBIC , KRIBB and NCSRD , KRISS, Daejeon in Korea. From 2009, KGP was supported by the Genome Research Foundation and TheragenEtex to build the Variome of Koreans as well as the Korean Reference Genome ( KOREF ). Starting from KOREF, a consensus variome reference, providing information on millions of variants from 40 additional ethnically homogeneous genomes from the Korean Personal Genome Project was completed in 2017. [ 3 ] Updating the technology an improved version of KOREF was then constructed using long-read sequencing data produced by Oxford Nanopore PromethION and PacBio technologies has been released showcasing newer assembly technologies and techniques. [ 4 ] In 2022 a new chromosome-level haploid assembly of KOREF was published, assembled using Oxford Nanopore Technologies PromethION, Pacific Biosciences HiFi-CCS, and Hi-C technology. [ 5 ]
Since 2014, KGP has been supported by Ulsan National Institute of Science and Technology , Clinomics, and Ulsan City, Ulsan, Korea. [ citation needed ]
Korea1K) has been used in sequencing technologies such as MGI DNBSEQ-T7 and Illumina HiSeq2000, HiSeq2500, HiSeq4000, HiSeqX10, and NovaSeq6000 sequencing technologies. [ 6 ] The variome data has been a reference to study the origin and composition of Korean ethnicity when compared to ancient DNA sequences. [ 7 ]
Korea1K released 1,094 Korean whole genome sequences on 27 May 2020, published in Science Advances . [ 8 ]
In April 2024, Korea4K was published, making whole genome sequences of 4,157 Koreans publicly accessible alongside an imputation reference panel and 107 phenotypes derived from extensive health check-ups. [ 9 ]
This genetics article is a stub . You can help Wikipedia by expanding it .
This Korea -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Korean_Genome_Project |
Korean brining salt , also called Korean sea salt , is a variety of edible salt with a larger grain size compared to common kitchen salt . [ 1 ] [ 2 ] It is called gulgeun-sogeum ( 굵은소금 ; "coarse salt") or wang-sogeum ( 왕소금 ; "king/queen salt") in Korean . [ 3 ] [ 4 ] [ 5 ] The salt is used mainly for salting napa cabbages when making kimchi . Because it is minimally processed, there are microorganisms present in the salt, which serve to help develop flavours in fermented foods. [ 1 ]
This Korean cuisine –related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Korean_brining_salt |
The writing system of the Korean language is a syllabic alphabet of character parts ( jamo ) organized into character blocks ( 글자 ; geulja ) representing syllables . The character parts cannot be written from left to right on the computer, as in many Western languages. Every possible syllable in Korean would have to be rendered as syllable blocks by a font , or each character part would have to be encoded separately. Unicode has both options; the character parts ㅎ (h) and ㅏ (a), and the combined syllable 하 (ha), are encoded.
In RFC 1557 , a method known as ISO-2022-KR for seven-bit encoding of Korean characters in email was described. Where eight bits are allowed, EUC-KR encoding is preferred. These two encodings combine US-ASCII ( ISO 646 ) with the Korean standard KS X 1001 :1992 [ 1 ] (previously named KS C 5601:1987). Another character set, KPS 9566 (similar to KS X 1001), is used in North Korea .
The international Unicode standard contains special characters for the Korean language in the Hangul phonetic system. Unicode supports two methods. The method used by Microsoft Windows is to have each of the 11,172 syllable combinations as code and a preformed font character. The other method encodes letters ( jamos ) and lets the software combine them correctly. The Windows method requires more font memory but allows better shapes, since it is complicated to create stylistically correct combinations (preferable for documents).
Another possibility is stacking a sequence of medial (s) ( jungseong ) and a sequence of final (s) ( jongseong ) or a Middle Korean pitch mark (if needed) on top of the sequence of initial (s) ( choseong ) if the font has medial and final jamo with zero-width spacing inserted to the left of the cursor or caret, thus appearing in the right place below (or to the right of) the initial. If a syllable has a horizontal medial ( ㅗ , ㅛ , ㅜ , ㅠ or ㅡ ), the initial will probably appear further left in a complete syllable than in preformed syllables due to the space that must be reserved for a vertical medial, making aesthetically poor what may be the only way to display Middle Korean hangul text without resorting to images, romanization, replacement of obsolete jamo or non-standard encodings. However, most current fonts do not support this.
The Unicode standard also has attempted to create a unified CJK character set which can represent Chinese ( Hanzi ) and the Japanese ( Kanji ) and Korean ( Hanja ) derivatives of this script through Han unification , which does not discriminate by language or region in rendering Chinese characters if the typographic traditions have not resulted in major differences in what a character looks like. Han unification has been criticized.
While the first Korean typewriter (한글 타자기, Hangeul tajagi ) is unclear, the first moa-sseugi style (모아쓰기, the form of Hangul where consonants and vowels come together to form a letter; The standard form of Hangul used today) typewriter is thought to be first invented by Korean-American gyopo Lee Won-Ik (이원익) in 1914, where he modified a Smith Premier 10 typewriter's type into Hangul. [ 2 ] [ 3 ] Alongside Lee Won-ik's, Horace Grant Underwood 's 1913 US-patented Hangul type, the Underwood, and another Korean-American Kim Jun-Sung's Hangul type are also brought up when discussing the first Moa-Sugi type. [ 4 ]
In 1929, the first Dubeolsik typewriter was made by Song Ki-Ju, a student studying abroad in the US, gaining attention from the Donga ilbo , however, it no longer exists; In 1934 he showcased another type, which was a modification of the Underwood portable . [ 5 ] [ 6 ] Song's 1934 typewriter is stored in the Hangul museum as the oldest existing Korean typewriter. [ 7 ] The invention led to the development of other typewriters in 1945 by Kim Joon Sung and 1950 by Kong Byung Woo . [ 8 ]
In 1949, eye doctor Kong Byung-Woo made the first practical Hangul type able to write both in Moa-Sugi and horizontally. [ 9 ]
On a Korean computer keyboard, text is typically entered by pressing a key for the appropriate jamo ; the operating system creates each composite character on the fly. Depending on the Input method editor and keyboard layout, double consonants can be entered by holding the shift button. When all jamo making up a syllabic block has been entered, the user may initiate a conversion to hanja (or other special characters) using a keyboard shortcut or interface button; South Korean keyboards have a key for this. Subsequent semi-automated hanja conversion is supported in varying degrees by word processors.
When using a keyboard with another language, most operating systems require the user to type with an original Korean keyboard layout; the most common is Dubeolsik . In other languages, such as Japanese, text can be entered on non-native keyboards with romanization .
Operating systems such as Linux allow engine/hangul/hangul-keyboard='ro , resulting in a romaja keyboard; typing "seonggye" results in Korean : 성계 . [ 10 ] In this configuration, ㄲ is obtained by "gg" rather than ⇧ Shift + G . This allows keying "jasanGun" to obtain Korean : 자산군 , instead of keying "jasangun" (which would provide Korean : 자상운 ).
Korean text input is related to Korean typewriters ( 타자기 ) before computers. according to Jang Bong Seon, Horace Grant Underwood made a Korean typewriter during the first decade of the 20th century. [ 11 ] In 1927, Song Ki Joo invented the first Dubeolsik typewriter in Chicago.
South Korea originally had a Nebeolsik standard, but Dubeolsik became standard in 1985. [ 12 ]
Some Korean fonts do not include hanja , and word processors do not allow a user to specify which font to use as a fallback for any hanja in a text; each hanja sequence must be manually formatted for a desired font.
Vertical text is supported poorly (or not at all) by HTML and most word processors. This is not an issue for modern Korean, which is usually written horizontally; until the second half of the 20th century, however, Korean was often written vertically. Fifteenth-century texts written in hangul had pitch marks to the left of syllables which are included in Unicode, although current fonts do not support them.
Programs designed for Korean language-related use include:
Hangul letters are detailed in several parts of Unicode:
Pre-composed Hangul syllables in the Unicode Hangul Syllables block are algorithmically defined with the following formula:
To find the code point of "한" in Unicode:
Substituting these values in the formula above yields [(18 × 588) + (0 × 28) + 4] + 44032 = 54620. The Unicode value of 한 is 54620 in decimal, 한 in numeric character reference , and U+D55C in hexadecimal Unicode notation.
With the below module, calling e.g. hangul :: from_jamo ( 'ㅎ' , 'ㅏ' , Some ( 'ㄴ' )) will return Some ( '한' ) .
The Unicode Hangul Compatibility Jamo block has been allocated for compatibility with the KS X 1001 character set. It is usually used to represent hangul without distinguishing initials and finals.
The Hangul Jamo , Hangul Jamo Extended-A and Hangul Jamo Extended-B blocks contain initial, medial and final jamo, including obsolete jamo.
Hangul (word processor) shipped with fonts from Hanyang Information and Communication , which map obsolete Hangul characters with Unicode's Private Use Areas . Despite the use of PUAs instead of dedicated code points , Hanyang's mapping was the most popular way to represent obsolete Hangul in South Korea in 2007. With its Hangul 2010, however, Hancom deprecated Hanyang PUA code and began representing obsolete Hangul characters with Unicode Hangul jamo. | https://en.wikipedia.org/wiki/Korean_language_and_computers |
In mathematical analysis , Korn's inequality is an inequality concerning the gradient of a vector field that generalizes the following classical theorem: if the gradient of a vector field is skew-symmetric at every point, then the gradient must be equal to a constant skew-symmetric matrix. Korn's theorem is a quantitative version of this statement, which intuitively says that if the gradient of a vector field is on average not far from the space of skew-symmetric matrices, then the gradient must not be far from a particular skew-symmetric matrix. The statement that Korn's inequality generalizes thus arises as a special case of rigidity .
In ( linear ) elasticity theory , the symmetric part of the gradient is a measure of the strain that an elastic body experiences when it is deformed by a given vector-valued function. The inequality is therefore an important tool as an a priori estimate in linear elasticity theory.
Let Ω be an open , connected domain in n - dimensional Euclidean space R n , n ≥ 2 . Let H 1 (Ω) be the Sobolev space of all vector fields v = ( v 1 , ..., v n ) on Ω that, along with their (first) weak derivatives, lie in the Lebesgue space L 2 (Ω) . Denoting the partial derivative with respect to the i th component by ∂ i , the norm in H 1 (Ω) is given by
Then there is a (minimal) constant C ≥ 0 , known as the Korn constant of Ω , such that, for all v ∈ H 1 (Ω) ,
where e denotes the symmetrized gradient given by
Inequality (1) is known as Korn's inequality . | https://en.wikipedia.org/wiki/Korn's_inequality |
The Kornblum oxidation , named after Nathan Kornblum , is an organic oxidation reaction that converts alkyl halides and tosylates into carbonyl compounds. [ 1 ] [ 2 ] [ 3 ]
Similar to sulfonium-based oxidation of alcohols to aldehydes reactions, the Kornblum oxidation creates an alkoxysulphonium ion, which, in the presence of a base, such as triethylamine (Et 3 N), undergoes an elimination reaction to form the aldehyde or ketone .
The first step is an S N 2 reaction , so it is subject to the usual leaving group limitations of that reaction. While iodides work well, even bromides are often not reactive enough to be displaced by the DMSO. However, using an additive such as silver tetrafluoroborate allows the reaction to work on a wider range of substrates, as often seen for alkyl-halide substitutions, or they can be converted first to the corresponding alkyl tosylate. [ 4 ] [ 5 ] The reaction was initially limited to activated substrates, such as benzylic and α-halo ketones. To increase the range of viable substrates, Kornblum later added a preliminary conversion of the halide to a tosylate, which is a better leaving group, to the protocol, and using pyridine- N -oxide or similar reagents rather than DMSO. [ 5 ] The Ganem oxidation built on this latter modification, expanding on the use of various N -oxide reagents. | https://en.wikipedia.org/wiki/Kornblum_oxidation |
The Kornblum–DeLaMare rearrangement is a rearrangement reaction in organic chemistry in which a primary or secondary organic peroxide is converted to the corresponding ketone and alcohol under acid or base catalysis . The reaction is relevant as a tool in organic synthesis and is a key step in the biosynthesis of prostaglandins . [ 1 ]
The base can be a hydroxide such as potassium hydroxide or an amine such as triethylamine .
In the reaction mechanism for this organic reaction the base abstracts the acidic α-proton of the peroxide 1 to form the carbanion 4 as a reactive intermediate which rearranges to the ketone 2 with expulsion of the hydroxyl anion 3' . This intermediate gains a proton forming the alcohol 3 .
Deprotonation and rearrangement can also be a concerted reaction without formation of 4 .
An alternative reaction mechanism involving direct nucleophilic displacement on the peroxide link of the amine followed by an elimination reaction is considered unlikely based on the outcome of this model reaction: [ 2 ]
The peroxide 1 converts to the hydroxyketone 2 by action of triethylamine but the alternative route through hydroxylamine 3 by nucleophilic displacement with Lithium diisopropylamide and the ammonium salt 4 (by methylation with methyl trifluoromethanesulfonate ) fails.
The reaction, formally a rearrangement, ranks under the elimination reactions as already observed by the original authors. Not only alkoxides but any leaving group capable of carrying a negative charge will do for instance nitrate esters R–C(R)(H)–O–NO 2 .
The corresponding reaction involving an ether is the 1,2-Wittig rearrangement . The reaction course in this rearrangement is different because ether cleavage with carbanion formation is unfavorable. The Pummerer rearrangement in one of its reaction step contains a sulfur variation.
The original 1951 publication concerned the conversion of potassium t-butyl peroxide and 1-phenylethyl bromide to ultimately acetophenone and t-butanol with piperidine as the base:
The Kornblum–DeLaMare rearrangement can be carried out as an asymmetric reaction with a suitable chiral amine such as sparteine or a cinchona alkaloid: [ 3 ]
The first step in this one-pot reaction is 1,4-dioxygenation of 1,3-cycloheptadiene with singlet oxygen and a TPP catalyst. | https://en.wikipedia.org/wiki/Kornblum–DeLaMare_rearrangement |
In mathematics the Korovkin approximation is a convergence statement in which the approximation of a function is given by a certain sequence of functions. In practice a continuous function can be approximated by polynomials . With Korovkin approximations one comes a convergence for the whole approximation with examination of the convergence of the process at a finite number of functions. The Korovkin approximation is named after Pavel Korovkin . [ 1 ] [ 2 ]
This mathematical analysis –related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Korovkin_approximation |
Korps Commander , or "Korps Commander: The Road to Berlin" is a set of micro-armour Miniature wargaming rules designed by Bruce Rea Taylor and Andy Ashton and published in the UK by Tabletop games, August 1988.
These rules simulate actions in the second world war on the Western or Eastern fronts during the final two years of the war for forces up to Corps or Army Levels.
The basic units in the game are tank and mechanised infantry platoons, infantry companies, and artillery batteries. Aircraft may be in flights or squadrons. Logistics and engineers are fully covered.
Bruce Rea Taylor and Andy Ashton originally intended that Korps Commander would be the start of a family of rules, each covering a period and location, as can be seen by the quote from the Rules.
Road to Berlin is a set of rules in the Korps Commander Series, which will eventually cover all periods of warfare at a level which will allow large battles and campaigns to be fought. The series aim is to utilise the techniques of boardgaming whilst retaining the flavour of miniature gaming.
The Road to Berlin was the first and last of the Korps Commander publications. Bruce Rea Taylor [ 1 ] died shortly after the publication of these rules (1989), at the age of 40. I was unable to find any evidence Andy Ashton made any plans to continue this project.
Korps Commander used the Corps Commander : OMG (Operational Manoeuvre Group) game system, which was published in 1986. The differences between these two rules are summarized in the Introduction of Korps Commander.
For those who have played OMG the most significant difference is that the ground scale and time scales have been halved due to the shorter effective ranges of direct fire weapons, although the only noticeable difference is the range of artillery.
Bruce Rea Taylor gives a special mention to the Wallasey Wargames Club in the Introduction of the Korps Commander Rules.
We would like to acknowledge the help of the Wallasey Wargames Club in the preparation and playtesting of these rules.
The primary significance of these rules was it was one of the first game systems which used a scale of 1-2 base(s) per company. The reason why a scale of 1 base per company was not used was to allow for small formations, such as 1-2 SP-AA weapons allocated to a HQ and to ensure the record keeping for the standard company sized formations was not too difficult. However, apart from this, the rules attempted to use a 1 Base = 1 company scale, which allowed players to command one, or more, divisions. Compared with the most common rules used in this period, this was unusual.
In practice the detail and complexity of the rules precluded any possibility of playing a corps level game within a reasonable time frame. Most games would typically field 50 bases, or elements, per side. This represents a force of 25 companies per side. A game of this size which consisted of 24 game-turns, or one game day duration, could be completed within a 4-8 hour period.
Bruce Rea Taylor and Andy Ashton published Corps Commander in July 1986. This used the same game system as Korps Commander and can be considered the first of the Corps/Korps Commander system publications. In October 1986 "Digest #3, Engineering Equipment Data, Engineering Lists, Engineering Scenarios" was published, this contained additional material for Corps Commander. In June 1988 "Corps Commander: By Air & Sea" was published, containing yet more supporting material for Corps Commander. The final publication in this family was in August 1988 when "Korps Commander" was published.
While Bruce Rea Taylor published a number of additional books after August 1988, they were all dated after his death on 3 March 1989. [ 2 ] No additional Corp/Korps Commander books were published after August 1988. These post August 1988 publications are listed here:
There is strong evidence Bruce Rea Taylor built on his earlier rules, Challenger and Firefly, when developing the Corp/Korps Commander Rules. This is especially evident in the equipment specification. Players in the last 20 years have used these earlier rules to expand Corps/Korps Commander, such as can be seen in the "nikita - Corps Commander: OMG and Korps Kommander" site referenced earlier, and attempts to rewrite the rules as can be seen with Korps-Kommandeur. [ 3 ]
Ground Scale (1/300 scale) : Each centimetre represents 50 metres (1:5000)
Game-Turn Scale : Each daylight Turn represents 1/2 hour of actual time. Each night Turn represents 1 hour of actual time.
Base/Stand Scale - Each base represents a company, part of a company or a platoon. A Base can have a maximum of 9 Strength points. (If there are more than 9 vehicles or Gun's in a Company, the Company is split into two bases, each of which are similar in strength. This system allows a base to also represent a platoon, such as a support platoon attached to a Headquarters company.)
Base Size (1/300 scale) - A Base which has 1 to 5 strength points has a size of 20mm by 30mm. A Base with more than 5 strength points has a size of 30mm by 30mm.
For comparison purposes the earlier Corps Commander used a Ground scale of 10 cm = 1 km (1:10000) and a Daylight Game Turn = 1 hour. Lightning War - Red Storm uses a base size of 30 cm x 30 cm, a ground scale of 1 cm = 100, (1:10000), a Daylight Game Turn = 2 hours. Panzer Korps, while designed for 15mm/20mm, uses a Ground Scale of 1 cm = 50m (1:5000) for 1/300 scale.
The primary significance of these rules is the use of a 1 Company = 1-2 Base(s) scale. In 1986/1988 this was unusual. Examples of other similar scaled rules are, Great Battles of WWII Vol. 1 (1995), [ 4 ] Panzer Korps (2004). [ 5 ] and Lightning War - Red Storm (2001-2008).
The most common scale in the late 80's was 1 Vehicle = 1 Base, later rules used scales of 1 Platoon = 1 Base. The primary issue with the increase of the scale was related to the combat specifications of a base. When a base represents a common vehicle or gun, its speed and combat capabilities are based on the figure itself. When a base represents a mix of vehicles or guns, its speed and combat capabilities may differ from the figure on the base.
Example : A Base which contains 5 x Pz I and 2 x Pz II compared with another base which contains 5 x Pz I and 2 x Pz III. Both would be represented by a Pz I, but would differ in combat capabilities.
Players may be required to keep track of the combat capabilities of individual bases by the use of separate written records, or a value printed on the base. In this case the use of a Board-Game system may be preferable.
Corps/Korps Commander resolved this issue by giving each a Base a value, representing the number of common vehicles or guns. When there were different vehicles or guns in a company, the company would be represented by more than 1 base. Each Base would contain the same vehicle/guns and its value would be the number of vehicle/gun in each base. | https://en.wikipedia.org/wiki/Korps_Commander |
In the theory of stochastic processes , the Karhunen–Loève theorem (named after Kari Karhunen and Michel Loève ), also known as the Kosambi–Karhunen–Loève theorem [ 1 ] [ 2 ] states that a stochastic process can be represented as an infinite linear combination of orthogonal functions , analogous to a Fourier series representation of a function on a bounded interval. The transformation is also known as Hotelling transform and eigenvector transform, and is closely related to principal component analysis (PCA) technique widely used in image processing and in data analysis in many fields. [ 3 ]
There exist many such expansions of a stochastic process: if the process is indexed over [ a , b ] , any orthonormal basis of L 2 ([ a , b ]) yields an expansion thereof in that form. The importance of the Karhunen–Loève theorem is that it yields the best such basis in the sense that it minimizes the total mean squared error .
In contrast to a Fourier series where the coefficients are fixed numbers and the expansion basis consists of sinusoidal functions (that is, sine and cosine functions), the coefficients in the Karhunen–Loève theorem are random variables and the expansion basis depends on the process. In fact, the orthogonal basis functions used in this representation are determined by the covariance function of the process. One can think that the Karhunen–Loève transform adapts to the process in order to produce the best possible basis for its expansion.
In the case of a centered stochastic process { X t } t ∈ [ a , b ] ( centered means E [ X t ] = 0 for all t ∈ [ a , b ] ) satisfying a technical continuity condition, X admits a decomposition
where Z k are pairwise uncorrelated random variables and the functions e k are continuous real-valued functions on [ a , b ] that are pairwise orthogonal in L 2 ([ a , b ]) . It is therefore sometimes said that the expansion is bi-orthogonal since the random coefficients Z k are orthogonal in the probability space while the deterministic functions e k are orthogonal in the time domain. The general case of a process X t that is not centered can be brought back to the case of a centered process by considering X t − E [ X t ] which is a centered process.
Moreover, if the process is Gaussian , then the random variables Z k are Gaussian and stochastically independent . This result generalizes the Karhunen–Loève transform . An important example of a centered real stochastic process on [0, 1] is the Wiener process ; the Karhunen–Loève theorem can be used to provide a canonical orthogonal representation for it. In this case the expansion consists of sinusoidal functions.
The above expansion into uncorrelated random variables is also known as the Karhunen–Loève expansion or Karhunen–Loève decomposition . The empirical version (i.e., with the coefficients computed from a sample) is known as the Karhunen–Loève transform (KLT), principal component analysis , proper orthogonal decomposition (POD) , empirical orthogonal functions (a term used in meteorology and geophysics ), or the Hotelling transform .
The square-integrable condition E [ X t 2 ] < ∞ {\displaystyle \mathbf {E} [X_{t}^{2}]<\infty } is logically equivalent to K X ( s , t ) {\displaystyle K_{X}(s,t)} being finite for all s , t ∈ [ a , b ] {\displaystyle s,t\in [a,b]} . [ 4 ]
Theorem . Let X t be a zero-mean square-integrable stochastic process defined over a probability space (Ω, F , P ) and indexed over a closed and bounded interval [ a , b ], with continuous covariance function K X ( s , t ) .
Then K X ( s,t ) is a Mercer kernel and letting e k be an orthonormal basis on L 2 ([ a , b ]) formed by the eigenfunctions of T K X with respective eigenvalues λ k , X t admits the following representation
where the convergence is in L 2 , uniform in t and
Furthermore, the random variables Z k have zero-mean, are uncorrelated and have variance λ k
Note that by generalizations of Mercer's theorem we can replace the interval [ a , b ] with other compact spaces C and the Lebesgue measure on [ a , b ] with a Borel measure whose support is C .
Since the limit in the mean of jointly Gaussian random variables is jointly Gaussian, and jointly Gaussian random (centered) variables are independent if and only if they are orthogonal, we can also conclude:
Theorem . The variables Z i have a joint Gaussian distribution and are stochastically independent if the original process { X t } t is Gaussian.
In the Gaussian case, since the variables Z i are independent, we can say more:
almost surely.
This is a consequence of the independence of the Z k .
In the introduction, we mentioned that the truncated Karhunen–Loeve expansion was the best approximation of the original process in the sense that it reduces the total mean-square error resulting of its truncation. Because of this property, it is often said that the KL transform optimally compacts the energy.
More specifically, given any orthonormal basis { f k } of L 2 ([ a , b ]) , we may decompose the process X t as:
where
and we may approximate X t by the finite sum
for some integer N .
Claim . Of all such approximations, the KL approximation is the one that minimizes the total mean square error (provided we have arranged the eigenvalues in decreasing order).
Consider the error resulting from the truncation at the N -th term in the following orthonormal expansion:
The mean-square error ε N 2 ( t ) can be written as:
We then integrate this last equality over [ a , b ]. The orthonormality of the f k yields:
The problem of minimizing the total mean-square error thus comes down to minimizing the right hand side of this equality subject to the constraint that the f k be normalized. We hence introduce β k , the Lagrangian multipliers associated with these constraints, and aim at minimizing the following function:
Differentiating with respect to f i ( t ) (this is a functional derivative ) and setting the derivative to 0 yields:
which is satisfied in particular when
In other words, when the f k are chosen to be the eigenfunctions of T K X , hence resulting in the KL expansion.
An important observation is that since the random coefficients Z k of the KL expansion are uncorrelated, the Bienaymé formula asserts that the variance of X t is simply the sum of the variances of the individual components of the sum:
Integrating over [ a , b ] and using the orthonormality of the e k , we obtain that the total variance of the process is:
In particular, the total variance of the N -truncated approximation is
As a result, the N -truncated expansion explains
of the variance; and if we are content with an approximation that explains, say, 95% of the variance, then we just have to determine an N ∈ N {\displaystyle N\in \mathbb {N} } such that
Given a representation of X t = ∑ k = 1 ∞ W k φ k ( t ) {\displaystyle X_{t}=\sum _{k=1}^{\infty }W_{k}\varphi _{k}(t)} , for some orthonormal basis φ k ( t ) {\displaystyle \varphi _{k}(t)} and random W k {\displaystyle W_{k}} , we let p k = E [ | W k | 2 ] / E [ | X t | L 2 2 ] {\displaystyle p_{k}=\mathbb {E} [|W_{k}|^{2}]/\mathbb {E} [|X_{t}|_{L^{2}}^{2}]} , so that ∑ k = 1 ∞ p k = 1 {\displaystyle \sum _{k=1}^{\infty }p_{k}=1} . We may then define the representation entropy to be H ( { φ k } ) = − ∑ i p k log ( p k ) {\displaystyle H(\{\varphi _{k}\})=-\sum _{i}p_{k}\log(p_{k})} . Then we have H ( { φ k } ) ≥ H ( { e k } ) {\displaystyle H(\{\varphi _{k}\})\geq H(\{e_{k}\})} , for all choices of φ k {\displaystyle \varphi _{k}} . That is, the KL-expansion has minimal representation entropy.
Proof:
Denote the coefficients obtained for the basis e k ( t ) {\displaystyle e_{k}(t)} as p k {\displaystyle p_{k}} , and for φ k ( t ) {\displaystyle \varphi _{k}(t)} as q k {\displaystyle q_{k}} .
Choose N ≥ 1 {\displaystyle N\geq 1} . Note that since e k {\displaystyle e_{k}} minimizes the mean squared error, we have that
Expanding the right hand size, we get:
Using the orthonormality of φ k ( t ) {\displaystyle \varphi _{k}(t)} , and expanding X t {\displaystyle X_{t}} in the φ k ( t ) {\displaystyle \varphi _{k}(t)} basis, we get that the right hand size is equal to:
We may perform identical analysis for the e k ( t ) {\displaystyle e_{k}(t)} , and so rewrite the above inequality as:
Subtracting the common first term, and dividing by E [ | X t | L 2 2 ] {\displaystyle \mathbb {E} [|X_{t}|_{L^{2}}^{2}]} , we obtain that:
This implies that:
Consider a whole class of signals we want to approximate over the first M vectors of a basis. These signals are modeled as realizations of a random vector Y [ n ] of size N . To optimize the approximation we design a basis that minimizes the average approximation error. This section proves that optimal bases are Karhunen–Loeve bases that diagonalize the covariance matrix of Y . The random vector Y can be decomposed in an orthogonal basis
as follows:
where each
is a random variable. The approximation from the first M ≤ N vectors of the basis is
The energy conservation in an orthogonal basis implies
This error is related to the covariance of Y defined by
For any vector x [ n ] we denote by K the covariance operator represented by this matrix,
The error ε [ M ] is therefore a sum of the last N − M coefficients of the covariance operator
The covariance operator K is Hermitian and Positive and is thus diagonalized in an orthogonal basis called a Karhunen–Loève basis. The following theorem states that a Karhunen–Loève basis is optimal for linear approximations.
Theorem (Optimality of Karhunen–Loève basis). Let K be a covariance operator. For all M ≥ 1 , the approximation error
is minimum if and only if
is a Karhunen–Loeve basis ordered by decreasing eigenvalues.
Linear approximations project the signal on M vectors a priori. The approximation can be made more precise by choosing the M orthogonal vectors depending on the signal properties. This section analyzes the general performance of these non-linear approximations. A signal f ∈ H {\displaystyle f\in \mathrm {H} } is approximated with M vectors selected adaptively in an orthonormal basis for H {\displaystyle \mathrm {H} } [ definition needed ]
Let f M {\displaystyle f_{M}} be the projection of f over M vectors whose indices are in I M :
The approximation error is the sum of the remaining coefficients
To minimize this error, the indices in I M must correspond to the M vectors having the largest inner product amplitude
These are the vectors that best correlate f. They can thus be interpreted as the main features of f. The resulting error is necessarily smaller than the error of a linear approximation which selects the M approximation vectors independently of f. Let us sort
in decreasing order
The best non-linear approximation is
It can also be written as inner product thresholding:
with
The non-linear error is
this error goes quickly to zero as M increases, if the sorted values of | ⟨ f , g m k ⟩ | {\displaystyle \left|\left\langle f,g_{m_{k}}\right\rangle \right|} have a fast decay as k increases. This decay is quantified by computing the I P {\displaystyle \mathrm {I} ^{\mathrm {P} }} norm of the signal inner products in B:
The following theorem relates the decay of ε [ M ] to ‖ f ‖ B , p {\displaystyle \|f\|_{\mathrm {B} ,p}}
Theorem (decay of error). If ‖ f ‖ B , p < ∞ {\displaystyle \|f\|_{\mathrm {B} ,p}<\infty } with p < 2 then
and
Conversely, if ε [ M ] = o ( M 1 − 2 p ) {\displaystyle \varepsilon [M]=o\left(M^{1-{\frac {2}{p}}}\right)} then
‖ f ‖ B , q < ∞ {\displaystyle \|f\|_{\mathrm {B} ,q}<\infty } for any q > p .
To further illustrate the differences between linear and non-linear approximations, we study the decomposition of a simple non-Gaussian random vector in a Karhunen–Loève basis. Processes whose realizations have a random translation are stationary. The Karhunen–Loève basis is then a Fourier basis and we study its performance. To simplify the analysis, consider a random vector Y [ n ] of size N that is random shift modulo N of a deterministic signal f [ n ] of zero mean
The random shift P is uniformly distributed on [0, N − 1]:
Clearly
and
Hence
Since R Y is N periodic, Y is a circular stationary random vector. The covariance operator is a circular convolution with R Y and is therefore diagonalized in the discrete Fourier Karhunen–Loève basis
The power spectrum is Fourier transform of R Y :
Example: Consider an extreme case where f [ n ] = δ [ n ] − δ [ n − 1 ] {\displaystyle f[n]=\delta [n]-\delta [n-1]} . A theorem stated above guarantees that the Fourier Karhunen–Loève basis produces a smaller expected approximation error than a canonical basis of Diracs { g m [ n ] = δ [ n − m ] } 0 ≤ m < N {\displaystyle \left\{g_{m}[n]=\delta [n-m]\right\}_{0\leq m<N}} . Indeed, we do not know a priori the abscissa of the non-zero coefficients of Y , so there is no particular Dirac that is better adapted to perform the approximation. But the Fourier vectors cover the whole support of Y and thus absorb a part of the signal energy.
Selecting higher frequency Fourier coefficients yields a better mean-square approximation than choosing a priori a few Dirac vectors to perform the approximation. The situation is totally different for non-linear approximations. If f [ n ] = δ [ n ] − δ [ n − 1 ] {\displaystyle f[n]=\delta [n]-\delta [n-1]} then the discrete Fourier basis is extremely inefficient because f and hence Y have an energy that is almost uniformly spread among all Fourier vectors. In contrast, since f has only two non-zero coefficients in the Dirac basis, a non-linear approximation of Y with M ≥ 2 gives zero error. [ 5 ]
We have established the Karhunen–Loève theorem and derived a few properties thereof. We also noted that one hurdle in its application was the numerical cost of determining the eigenvalues and eigenfunctions of its covariance operator through the Fredholm integral equation of the second kind
However, when applied to a discrete and finite process ( X n ) n ∈ { 1 , … , N } {\displaystyle \left(X_{n}\right)_{n\in \{1,\ldots ,N\}}} , the problem takes a much simpler form and standard algebra can be used to carry out the calculations.
Note that a continuous process can also be sampled at N points in time in order to reduce the problem to a finite version.
We henceforth consider a random N -dimensional vector X = ( X 1 X 2 … X N ) T {\displaystyle X=\left(X_{1}~X_{2}~\ldots ~X_{N}\right)^{T}} . As mentioned above, X could contain N samples of a signal but it can hold many more representations depending on the field of application. For instance it could be the answers to a survey or economic data in an econometrics analysis.
As in the continuous version, we assume that X is centered, otherwise we can let X := X − μ X {\displaystyle X:=X-\mu _{X}} (where μ X {\displaystyle \mu _{X}} is the mean vector of X ) which is centered.
Let us adapt the procedure to the discrete case.
Recall that the main implication and difficulty of the KL transformation is computing the eigenvectors of the linear operator associated to the covariance function, which are given by the solutions to the integral equation written above.
Define Σ, the covariance matrix of X , as an N × N matrix whose elements are given by:
Rewriting the above integral equation to suit the discrete case, we observe that it turns into:
where e = ( e 1 e 2 … e N ) T {\displaystyle e=(e_{1}~e_{2}~\ldots ~e_{N})^{T}} is an N -dimensional vector.
The integral equation thus reduces to a simple matrix eigenvalue problem, which explains why the PCA has such a broad domain of applications.
Since Σ is a positive definite symmetric matrix, it possesses a set of orthonormal eigenvectors forming a basis of R N {\displaystyle \mathbb {R} ^{N}} , and we write { λ i , φ i } i ∈ { 1 , … , N } {\displaystyle \{\lambda _{i},\varphi _{i}\}_{i\in \{1,\ldots ,N\}}} this set of eigenvalues and corresponding eigenvectors, listed in decreasing values of λ i . Let also Φ be the orthonormal matrix consisting of these eigenvectors:
It remains to perform the actual KL transformation, called the principal component transform in this case. Recall that the transform was found by expanding the process with respect to the basis spanned by the eigenvectors of the covariance function. In this case, we hence have:
In a more compact form, the principal component transform of X is defined by:
The i -th component of Y is Y i = φ i T X {\displaystyle Y_{i}=\varphi _{i}^{T}X} , the projection of X on φ i {\displaystyle \varphi _{i}} and the inverse transform X = Φ Y yields the expansion of X on the space spanned by the φ i {\displaystyle \varphi _{i}} :
As in the continuous case, we may reduce the dimensionality of the problem by truncating the sum at some K ∈ { 1 , … , N } {\displaystyle K\in \{1,\ldots ,N\}} such that
where α is the explained variance threshold we wish to set.
We can also reduce the dimensionality through the use of multilevel dominant eigenvector estimation (MDEE). [ 6 ]
There are numerous equivalent characterizations of the Wiener process which is a mathematical formalization of Brownian motion . Here we regard it as the centered standard Gaussian process W t with covariance function
We restrict the time domain to [ a , b ]=[0,1] without loss of generality.
The eigenvectors of the covariance kernel are easily determined. These are
and the corresponding eigenvalues are
In order to find the eigenvalues and eigenvectors, we need to solve the integral equation:
differentiating once with respect to t yields:
a second differentiation produces the following differential equation:
The general solution of which has the form:
where A and B are two constants to be determined with the boundary conditions. Setting t = 0 in the initial integral equation gives e (0) = 0 which implies that B = 0 and similarly, setting t = 1 in the first differentiation yields e' (1) = 0, whence:
which in turn implies that eigenvalues of T K X are:
The corresponding eigenfunctions are thus of the form:
A is then chosen so as to normalize e k :
This gives the following representation of the Wiener process:
Theorem . There is a sequence { Z i } i of independent Gaussian random variables with mean zero and variance 1 such that
Note that this representation is only valid for t ∈ [ 0 , 1 ] . {\displaystyle t\in [0,1].} On larger intervals, the increments are not independent. As stated in the theorem, convergence is in the L 2 norm and uniform in t .
Similarly the Brownian bridge B t = W t − t W 1 {\displaystyle B_{t}=W_{t}-tW_{1}} which is a stochastic process with covariance function
can be represented as the series
Adaptive optics systems sometimes use K–L functions to reconstruct wave-front phase information (Dai 1996, JOSA A).
Karhunen–Loève expansion is closely related to the Singular Value Decomposition . The latter has myriad applications in image processing, radar, seismology, and the like. If one has independent vector observations from a vector valued stochastic process then the left singular vectors are maximum likelihood estimates of the ensemble KL expansion.
In communication, we usually have to decide whether a signal from a noisy channel contains valuable information. The following hypothesis testing is used for detecting continuous signal s ( t ) from channel output X ( t ), N ( t ) is the channel noise, which is usually assumed zero mean Gaussian process with correlation function R N ( t , s ) = E [ N ( t ) N ( s ) ] {\displaystyle R_{N}(t,s)=E[N(t)N(s)]}
When the channel noise is white, its correlation function is
and it has constant power spectrum density. In physically practical channel, the noise power is finite, so:
Then the noise correlation function is sinc function with zeros at n 2 ω , n ∈ Z . {\displaystyle {\frac {n}{2\omega }},n\in \mathbf {Z} .} Since are uncorrelated and gaussian, they are independent. Thus we can take samples from X ( t ) with time spacing
Let X i = X ( i Δ t ) {\displaystyle X_{i}=X(i\,\Delta t)} . We have a total of n = T Δ t = T ( 2 ω ) = 2 ω T {\displaystyle n={\frac {T}{\Delta t}}=T(2\omega )=2\omega T} i.i.d observations { X 1 , X 2 , … , X n } {\displaystyle \{X_{1},X_{2},\ldots ,X_{n}\}} to develop the likelihood-ratio test. Define signal S i = S ( i Δ t ) {\displaystyle S_{i}=S(i\,\Delta t)} , the problem becomes,
The log-likelihood ratio
As t → 0 , let:
Then G is the test statistics and the Neyman–Pearson optimum detector is
As G is Gaussian, we can characterize it by finding its mean and variances. Then we get
where
is the signal energy.
The false alarm error
And the probability of detection:
where Φ is the cdf of standard normal, or Gaussian, variable.
When N(t) is colored (correlated in time) Gaussian noise with zero mean and covariance function R N ( t , s ) = E [ N ( t ) N ( s ) ] , {\displaystyle R_{N}(t,s)=E[N(t)N(s)],} we cannot sample independent discrete observations by evenly spacing the time. Instead, we can use K–L expansion to decorrelate the noise process and get independent Gaussian observation 'samples'. The K–L expansion of N ( t ):
where N i = ∫ N ( t ) Φ i ( t ) d t {\displaystyle N_{i}=\int N(t)\Phi _{i}(t)\,dt} and the orthonormal bases { Φ i t } {\displaystyle \{\Phi _{i}{t}\}} are generated by kernel R N ( t , s ) {\displaystyle R_{N}(t,s)} , i.e., solution to
Do the expansion:
where S i = ∫ 0 T S ( t ) Φ i ( t ) d t {\displaystyle S_{i}=\int _{0}^{T}S(t)\Phi _{i}(t)\,dt} , then
under H and N i + S i {\displaystyle N_{i}+S_{i}} under K. Let X ¯ = { X 1 , X 2 , … } {\displaystyle {\overline {X}}=\{X_{1},X_{2},\dots \}} , we have
Hence, the log-LR is given by
and the optimum detector is
Define
then G = ∫ 0 T k ( t ) x ( t ) d t . {\displaystyle G=\int _{0}^{T}k(t)x(t)\,dt.}
Since
k(t) is the solution to
If N ( t )is wide-sense stationary,
which is known as the Wiener–Hopf equation . The equation can be solved by taking fourier transform, but not practically realizable since infinite spectrum needs spatial factorization. A special case which is easy to calculate k ( t ) is white Gaussian noise.
The corresponding impulse response is h ( t ) = k ( T − t ) = CS ( T − t ). Let C = 1, this is just the result we arrived at in previous section for detecting of signal in white noise.
Since X(t) is a Gaussian process,
is a Gaussian random variable that can be characterized by its mean and variance.
Hence, we obtain the distributions of H and K :
The false alarm error is
So the test threshold for the Neyman–Pearson optimum detector is
Its power of detection is
When the noise is white Gaussian process, the signal power is
For some type of colored noise, a typical practise is to add a prewhitening filter before the matched filter to transform the colored noise into white noise. For example, N(t) is a wide-sense stationary colored noise with correlation function
The transfer function of prewhitening filter is
When the signal we want to detect from the noisy channel is also random, for example, a white Gaussian process X ( t ), we can still implement K–L expansion to get independent sequence of observation. In this case, the detection problem is described as follows:
X ( t ) is a random process with correlation function R X ( t , s ) = E { X ( t ) X ( s ) } {\displaystyle R_{X}(t,s)=E\{X(t)X(s)\}}
The K–L expansion of X ( t ) is
where
and Φ i ( t ) {\displaystyle \Phi _{i}(t)} are solutions to
So X i {\displaystyle X_{i}} 's are independent sequence of r.v's with zero mean and variance λ i {\displaystyle \lambda _{i}} . Expanding Y ( t ) and N ( t ) by Φ i ( t ) {\displaystyle \Phi _{i}(t)} , we get
where
As N ( t ) is Gaussian white noise, N i {\displaystyle N_{i}} 's are i.i.d sequence of r.v with zero mean and variance 1 2 N 0 {\displaystyle {\tfrac {1}{2}}N_{0}} , then the problem is simplified as follows,
The Neyman–Pearson optimal test:
so the log-likelihood ratio is
Since
is just the minimum-mean-square estimate of X i {\displaystyle X_{i}} given Y i {\displaystyle Y_{i}} 's,
K–L expansion has the following property: If
where
then
So let
Noncausal filter Q ( t , s ) can be used to get the estimate through
By orthogonality principle , Q ( t , s ) satisfies
However, for practical reasons, it's necessary to further derive the causal filter h ( t , s ), where h ( t , s ) = 0 for s > t , to get estimate X ^ ( t ∣ t ) {\displaystyle {\widehat {X}}(t\mid t)} . Specifically, | https://en.wikipedia.org/wiki/Kosambi–Karhunen–Loève_theorem |
The Koschevnikov gland is a gland of the honeybee [ 1 ] located near the sting shaft . The gland produces an alarm pheromone that is released when a bee stings. The pheromone contains more than 40 different compounds, including pentylacetate , butyl acetate , 1-hexanol , n-butanol , 1-octanol , hexylacetate , octylacetate , and 2-nonanol . These components have a low molar mass and evaporate quickly. This collection of compounds is the least specific of all pheromones. The alarm pheromone is released when a honey bee stings another animal to attract other bees to attack, as well. The release of the alarm pheromone may entice more bees to sting at the same location. Smoking the bees can reduce the pheromone's efficacy.
This biochemistry article is a stub . You can help Wikipedia by expanding it .
This insect anatomy–related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Koschevnikov_gland |
Co-solvents (in water solvent ) are defined as kosmotropic (order-making) if they contribute to the stability and structure of water-water interactions. In contrast, chaotropic (disorder-making) agents have the opposite effect, disrupting water structure, increasing the solubility of nonpolar solvent particles, and destabilizing solute aggregates. [ 1 ] Kosmotropes cause water molecules to favorably interact, which in effect stabilizes intramolecular interactions in macromolecules such as proteins . [ 1 ]
Ionic kosmotropes tend to be small or have high charge density. Some ionic kosmotropes are CO 2− 3 , SO 2− 4 , HPO 2− 4 , Mg 2+ , Li + , Zn 2+ and Al 3+ . Large ions or ions with low charge density (such as Br − , I − , K + , Cs + ) instead act as chaotropes . [ 2 ] Kosmotropic anions are more polarizable and hydrate more strongly than kosmotropic cations of the same charge density. [ 3 ]
A scale can be established if one refers to the Hofmeister series or looks up the free energy of hydrogen bonding ( Δ G H B {\displaystyle \Delta G_{\rm {HB}}} ) of the salts, which quantifies the extent of hydrogen bonding of an ion in water. [ 4 ] For example, the kosmotropes CO 2− 3 and OH − have Δ G H B {\displaystyle \Delta G_{\rm {HB}}} between 0.1 and 0.4 J/mol , whereas the chaotrope SCN − has a Δ G H B {\displaystyle \Delta G_{\rm {HB}}} between −1.1 and −0.9. [ 4 ]
Recent simulation studies have shown that the variation in solvation energy between the ions and the surrounding water molecules underlies the mechanism of the Hofmeister series . [ 5 ] [ 6 ] Thus, ionic kosmotropes are characterized by strong solvation energy leading to an increase of the overall cohesiveness of the solution, which is also reflected by the increase of the viscosity and density of the solution. [ 6 ]
Ammonium sulfate is the traditional kosmotropic salt for the salting out of protein from an aqueous solution. Kosmotropes are used to induce protein aggregation in pharmaceutical preparation and at various stages of protein extraction and purification. [ 7 ] [ citation needed ]
Nonionic kosmotropes have no net charge but are very soluble and become very hydrated. Carbohydrates such as trehalose and glucose , as well as proline and tert -butanol , are kosmotropes. | https://en.wikipedia.org/wiki/Kosmotropic |
In Euclidean geometry , Kosnita's theorem is a property of certain circles associated with an arbitrary triangle .
Let A B C {\displaystyle ABC} be an arbitrary triangle, O {\displaystyle O} its circumcenter and O a , O b , O c {\displaystyle O_{a},O_{b},O_{c}} are the circumcenters of three triangles O B C {\displaystyle OBC} , O C A {\displaystyle OCA} , and O A B {\displaystyle OAB} respectively. The theorem claims that the three straight lines A O a {\displaystyle AO_{a}} , B O b {\displaystyle BO_{b}} , and C O c {\displaystyle CO_{c}} are concurrent. [ 1 ] This result was established by the Romanian mathematician Cezar Coşniţă (1910-1962). [ 2 ]
Their point of concurrence is known as the triangle's Kosnita point (named by Rigby in 1997). It is the isogonal conjugate of the nine-point center . [ 3 ] [ 4 ] It is triangle center X ( 54 ) {\displaystyle X(54)} in Clark Kimberling's list . [ 5 ] This theorem is a special case of Dao's theorem on six circumcenters associated with a cyclic hexagon in. [ 6 ] [ 7 ] [ 8 ] [ 9 ] [ 10 ] [ 11 ] [ 12 ]
This geometry-related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Kosnita's_theorem |
The Kostanecki acylation is a method used in organic synthesis to form chromones [ 1 ] or coumarins [ 2 ] by acylation of O -hydroxyaryl ketones with aliphatic acid anhydrides, followed by cyclization. [ 3 ] If benzoic anhydride (or benzoyl chloride ) is used, a particular type of chromone called a flavone is obtained. It was named after Polish chemist Stanisław Kostanecki .
The mechanism consists of three well-differentiated reactions: [ 4 ] [ 5 ] | https://en.wikipedia.org/wiki/Kostanecki_acylation |
In mathematics, Kostant's convexity theorem , introduced by Bertram Kostant ( 1973 ), can be used to derive Lie-theoretical extensions of the Golden–Thompson inequality and the Schur–Horn theorem for Hermitian matrices .
Konstant's convexity theorem states that the projection of every coadjoint orbit of a connected compact Lie group into the dual of a Cartan subalgebra is a convex set . It is a special case of a more general result for symmetric spaces . Kostant's theorem is a generalization of a result of Schur (1923) , Horn (1954) and Thompson (1972) for Hermitian matrices. They proved that the projection onto the diagonal matrices of the space of all n by n complex self-adjoint matrices with given eigenvalues Λ = (λ 1 , ..., λ n ) is the convex polytope with vertices all permutations of the coordinates of Λ.
Let K be a connected compact Lie group with maximal torus T and Weyl group W = N K ( T )/ T . Let their Lie algebras be k {\displaystyle {\mathfrak {k}}} and t {\displaystyle {\mathfrak {t}}} . Let P be the orthogonal projection of k {\displaystyle {\mathfrak {k}}} onto t {\displaystyle {\mathfrak {t}}} for some Ad-invariant inner product on k {\displaystyle {\mathfrak {k}}} . Then for X in t {\displaystyle {\mathfrak {t}}} , P (Ad( K )⋅ X ) is the convex polytope with vertices w ( X ) where w runs over the Weyl group.
Let G be a compact Lie group and σ an involution with K a compact subgroup fixed by σ and containing the identity component of the fixed point subgroup of σ. Thus G / K is a symmetric space of compact type. Let g {\displaystyle {\mathfrak {g}}} and k {\displaystyle {\mathfrak {k}}} be their Lie algebras and let σ also denote the corresponding involution of g {\displaystyle {\mathfrak {g}}} . Let p {\displaystyle {\mathfrak {p}}} be the −1 eigenspace of σ and let a {\displaystyle {\mathfrak {a}}} be a maximal Abelian subspace. Let Q be the orthogonal projection of p {\displaystyle {\mathfrak {p}}} onto a {\displaystyle {\mathfrak {a}}} for some Ad( K )-invariant inner product on p {\displaystyle {\mathfrak {p}}} . Then for X in a {\displaystyle {\mathfrak {a}}} , Q (Ad( K )⋅ X ) is the convex polytope with vertices the w ( X ) where w runs over the restricted Weyl group (the normalizer of a {\displaystyle {\mathfrak {a}}} in K modulo its centralizer).
The case of a compact Lie group is the special case where G = K × K , K is embedded diagonally and σ is the automorphism of G interchanging the two factors.
Kostant's proof for symmetric spaces is given in Helgason (1984) . There is an elementary proof just for compact Lie groups using similar ideas, due to Wildberger (1993) : it is based on a generalization of the Jacobi eigenvalue algorithm to compact Lie groups.
Let K be a connected compact Lie group with maximal torus T . For each positive root α there is a homomorphism of SU(2) into K . A simple calculation with 2 by 2 matrices shows that if Y is in k {\displaystyle {\mathfrak {k}}} and k varies in this image of SU(2), then P (Ad( k )⋅ Y ) traces a straight line between P ( Y ) and its reflection in the root α. In particular the component in the α root space—its "α off-diagonal coordinate"—can be sent to 0. In performing this latter operation, the distance from P ( Y ) to P (Ad( k )⋅ Y ) is bounded above by size of the α off-diagonal coordinate of Y . Let m be the number of positive roots, half the dimension of K / T . Starting from an arbitrary Y 1 take the largest off-diagonal coordinate and send it to zero to get Y 2 . Continue in this way, to get a sequence ( Y n ). Then
Thus P ⊥ ( Y n ) tends to 0 and
Hence X n = P ( Y n ) is a Cauchy sequence , so tends to X in t {\displaystyle {\mathfrak {t}}} . Since Y n = P ( Y n ) ⊕ P ⊥ ( Y n ), Y n tends to X . On the other hand, X n lies on the line segment joining X n +1 and its reflection in the root α. Thus X n lies in the Weyl group polytope defined by X n +1 . These convex polytopes are thus increasing as n increases and hence P ( Y ) lies in the polytope for X . This can be repeated for each Z in the K -orbit of X . The limit is necessarily in the Weyl group orbit of X and hence P (Ad( K )⋅ X ) is contained in the convex polytope defined by W ( X ).
To prove the opposite inclusion, take X to be a point in the positive Weyl chamber. Then all the other points Y in the convex hull of W ( X ) can be obtained by a series of paths in that intersection moving along the negative of a simple root. (This matches a familiar picture from representation theory: if by duality X corresponds to a dominant weight λ, the other weights in the Weyl group polytope defined by λ are those appearing in the irreducible representation of K with highest weight λ. An argument with lowering operators shows that each such weight is linked by a chain to λ obtained by successively subtracting simple roots from λ. [ 1 ] ) Each part of the path from X to Y can be obtained by the process described above for the copies of SU(2) corresponding to simple roots, so the whole convex polytope lies in P (Ad( K )⋅ X ).
Heckman (1982) gave another proof of the convexity theorem for compact Lie groups, also presented in Hilgert, Hofmann & Lawson (1989) . For compact groups, Atiyah (1982) and Guillemin & Sternberg (1982) showed that if M is a symplectic manifold with a Hamiltonian action of a torus T with Lie algebra t {\displaystyle {\mathfrak {t}}} , then the image of the moment map
is a convex polytope with vertices in the image of the fixed point set of T (the image is a finite set ). Taking for M a coadjoint orbit of K in k ∗ {\displaystyle {\mathfrak {k}}^{*}} , the moment map for T is the composition
Using the Ad-invariant inner product to identify k ∗ {\displaystyle {\mathfrak {k}}^{*}} and k {\displaystyle {\mathfrak {k}}} , the map becomes
the restriction of the orthogonal projection. Taking X in t {\displaystyle {\mathfrak {t}}} , the fixed points of T in the orbit Ad( K )⋅ X are just the orbit under the Weyl group, W ( X ). So the convexity properties of the moment map imply that the image is the convex polytope with these vertices. Ziegler (1992) gave a simplified direct version of the proof using moment maps.
Duistermaat (1983) showed that a generalization of the convexity properties of the moment map could be used to treat the more general case of symmetric spaces. Let τ be a smooth involution of M which takes the symplectic form ω to −ω and such that t ∘ τ = τ ∘ t −1 . Then M and the fixed point set of τ (assumed to be non-empty) have the same image under the moment map. To apply this, let T = exp a {\displaystyle {\mathfrak {a}}} , a torus in G . If X is in a {\displaystyle {\mathfrak {a}}} as before the moment map yields the projection map
Let τ be the map τ( Y ) = − σ( Y ). The map above has the same image as that of the fixed point set of τ, i.e. Ad( K )⋅ X . Its image is the convex polytope with vertices the image of the fixed point set of T on Ad( G )⋅ X , i.e. the points w ( X ) for w in W = N K ( T )/C K ( T ).
In Kostant (1973) the convexity theorem is deduced from a more general convexity theorem concerning the projection onto the component A in the Iwasawa decomposition G = KAN of a real semisimple Lie group G . The result discussed above for compact Lie groups K corresponds to the special case when G is the complexification of K : in this case the Lie algebra of A can be identified with i t {\displaystyle i{\mathfrak {t}}} . The more general version of Kostant's theorem has also been generalized to semisimple symmetric spaces by van den Ban (1986) . Kac & Peterson (1984) gave a generalization for infinite-dimensional groups. | https://en.wikipedia.org/wiki/Kostant's_convexity_theorem |
In mathematics , the Kostka number K λ μ {\displaystyle K_{\lambda \mu }} (depending on two integer partitions λ {\displaystyle \lambda } and μ {\displaystyle \mu } ) is a non-negative integer that is equal to the number of semistandard Young tableaux of shape λ {\displaystyle \lambda } and weight μ {\displaystyle \mu } . They were introduced by the mathematician Carl Kostka in his study of symmetric functions ( Kostka (1882) ). [ 1 ]
For example, if λ = ( 3 , 2 ) {\displaystyle \lambda =(3,2)} and μ = ( 1 , 1 , 2 , 1 ) {\displaystyle \mu =(1,1,2,1)} , the Kostka number K λ μ {\displaystyle K_{\lambda \mu }} counts the number of ways to fill a left-aligned collection of boxes with 3 in the first row and 2 in the second row with 1 copy of the number 1, 1 copy of the number 2, 2 copies of the number 3 and 1 copy of the number 4 such that the entries increase along columns and do not decrease along rows. The three such tableaux are shown at right, and K ( 3 , 2 ) ( 1 , 1 , 2 , 1 ) = 3 {\displaystyle K_{(3,2)(1,1,2,1)}=3} .
For any partition λ {\displaystyle \lambda } , the Kostka number K λ λ {\displaystyle K_{\lambda \lambda }} is equal to 1: the unique way to fill the Young diagram of shape λ = ( λ 1 , … , λ m ) {\displaystyle \lambda =(\lambda _{1},\dotsc ,\lambda _{m})} with λ 1 {\displaystyle \lambda _{1}} copies of 1, λ 2 {\displaystyle \lambda _{2}} copies of 2, and so on, so that the resulting tableau is weakly increasing along rows and strictly increasing along columns is if all the 1s are placed in the first row, all the 2s are placed in the second row, and so on. (This tableau is sometimes called the Yamanouchi tableau of shape λ {\displaystyle \lambda } .)
The Kostka number K λ μ {\displaystyle K_{\lambda \mu }} is positive (i.e., there exist semistandard Young tableaux of shape λ {\displaystyle \lambda } and weight μ {\displaystyle \mu } ) if and only if λ {\displaystyle \lambda } and μ {\displaystyle \mu } are both partitions of the same integer n {\displaystyle n} and λ {\displaystyle \lambda } is larger than μ {\displaystyle \mu } in dominance order . [ 2 ]
In general, there are no nice formulas known for the Kostka numbers. However, some special cases are known. For example, if μ = ( 1 , 1 , … , 1 ) {\displaystyle \mu =(1,1,\dotsc ,1)} is the partition whose parts are all 1 then a semistandard Young tableau of weight μ {\displaystyle \mu } is a standard Young tableau; the number of standard Young tableaux of a given shape λ {\displaystyle \lambda } is given by the hook-length formula .
An important simple property of Kostka numbers is that K λ μ {\displaystyle K_{\lambda \mu }} does not depend on the order of entries of μ {\displaystyle \mu } . For example, K ( 3 , 2 ) ( 1 , 1 , 2 , 1 ) = K ( 3 , 2 ) ( 1 , 1 , 1 , 2 ) {\displaystyle K_{(3,2)(1,1,2,1)}=K_{(3,2)(1,1,1,2)}} . This is not immediately obvious from the definition but can be shown by establishing a bijection between the sets of semistandard Young tableaux of shape λ {\displaystyle \lambda } and weights μ {\displaystyle \mu } and μ ′ {\displaystyle \mu ^{\prime }} , where μ {\displaystyle \mu } and μ ′ {\displaystyle \mu ^{\prime }} differ only by swapping two entries. [ 3 ]
In addition to the purely combinatorial definition above, they can also be defined as the coefficients that arise when one expresses the Schur polynomial s λ {\displaystyle s_{\lambda }} as a linear combination of monomial symmetric functions m μ {\displaystyle m_{\mu }} :
where λ {\displaystyle \lambda } and μ {\displaystyle \mu } are both partitions of n {\displaystyle n} . Alternatively, Schur polynomials can also be expressed [ 4 ] as
where the sum is over all weak compositions α {\displaystyle \alpha } of n {\displaystyle n} and x α {\displaystyle x^{\alpha }} denotes the monomial x 1 α 1 x 2 α 2 … x n α n {\displaystyle x_{1}^{\alpha _{1}}x_{2}^{\alpha _{2}}\dotsc x_{n}^{\alpha _{n}}} .
On the level of representations of the symmetric group S n {\displaystyle S_{n}} , Kostka numbers express the decomposition of the permutation module M μ {\displaystyle M_{\mu }} in terms of the irreducible representations V λ {\displaystyle V_{\lambda }} where λ {\displaystyle \lambda } is a partition of n {\displaystyle n} , i.e.,
On the level of representations of the general linear group G L d ( C ) {\displaystyle \mathrm {GL} _{d}(\mathbb {C} )} , the Kostka number K λ μ {\displaystyle K_{\lambda \mu }} also counts the dimension of the weight space corresponding to μ {\displaystyle \mu } in the unitary irreducible representation U λ {\displaystyle U_{\lambda }} (where we require μ {\displaystyle \mu } and λ {\displaystyle \lambda } to have at most d {\displaystyle d} parts).
The Kostka numbers for partitions of size at most 3 are as follows:
These values are exactly the coefficients in the expansions of Schur functions in terms of monomial symmetric functions:
Kostka (1882 , pages 118-120) gave tables of these numbers for partitions of numbers up to 8.
Kostka numbers are special values of the 1 or 2 variable Kostka polynomials : | https://en.wikipedia.org/wiki/Kostka_number |
In mathematics , Kostka polynomials , named after the mathematician Carl Kostka , are families of polynomials that generalize the Kostka numbers . They are studied primarily in algebraic combinatorics and representation theory .
The two-variable Kostka polynomials K λμ ( q , t ) are known by several names including Kostka–Foulkes polynomials , Macdonald–Kostka polynomials or q , t -Kostka polynomials . Here the indices λ and μ are integer partitions and K λμ ( q , t ) is polynomial in the variables q and t . Sometimes one considers single-variable versions of these polynomials that arise by setting q = 0, i.e., by considering the polynomial K λμ ( t ) = K λμ (0, t ).
There are two slightly different versions of them, one called transformed Kostka polynomials . [ citation needed ]
The one-variable specializations of the Kostka polynomials can be used to relate Hall-Littlewood polynomials P μ to Schur polynomials s λ :
These polynomials were conjectured to have non-negative integer coefficients by Foulkes, and this was later proved in 1978 by Alain Lascoux and Marcel-Paul Schützenberger . [ 1 ] In fact, they show that
where the sum is taken over all semi-standard Young tableaux with shape λ and weight μ.
Here, charge is a certain combinatorial statistic on semi-standard Young tableaux.
The Macdonald–Kostka polynomials can be used to relate Macdonald polynomials (also denoted by P μ ) to Schur polynomials s λ :
where
Kostka numbers are special values of the one- or two-variable Kostka polynomials: | https://en.wikipedia.org/wiki/Kostka_polynomial |
In abstract algebra , a Koszul algebra R {\displaystyle R} is a graded k {\displaystyle k} - algebra over which the ground field k {\displaystyle k} has a linear minimal graded free resolution, i.e. , there exists an exact sequence :
for some nonnegative integers b i {\displaystyle b_{i}} . Here R ( − j ) {\displaystyle R(-j)} is the graded algebra R {\displaystyle R} with grading shifted up by j {\displaystyle j} , i.e. R ( − j ) i = R i − j {\displaystyle R(-j)_{i}=R_{i-j}} , and the exponent b i {\displaystyle b_{i}} refers to the b i {\displaystyle b_{i}} -fold direct sum. Choosing bases for the free modules in the resolution, the chain maps are given by matrices, and the definition requires the matrix entries to be zero or linear forms.
An example of a Koszul algebra is a polynomial ring over a field, for which the Koszul complex is the minimal graded free resolution of the ground field. There are Koszul algebras whose ground fields have infinite minimal graded free resolutions, e.g , R = k [ x , y ] / ( x y ) {\displaystyle R=k[x,y]/(xy)} .
The concept is named after the French mathematician Jean-Louis Koszul .
This algebra -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Koszul_algebra |
In mathematics , Koszul duality , named after the French mathematician Jean-Louis Koszul , is any of various kinds of dualities found in representation theory of Lie algebras , abstract algebras ( semisimple algebra ) [ 1 ] and topology (e.g., equivariant cohomology [ 2 ] ). The prototypical example of Koszul duality was introduced by Joseph Bernstein , Israel Gelfand , and Sergei Gelfand,. [ 3 ] It establishes a duality between the derived category of a symmetric algebra and that of an exterior algebra, as well as the BGG correspondence , which links the stable category of finite-dimensional graded modules over an exterior algebra to the bounded derived category of coherent sheaves on projective space. The importance of the notion rests on the suspicion that Koszul duality seems quite ubiquitous in nature. [ citation needed ]
The simplest, and in a sense prototypical case of Koszul duality arises as follows: for a 1-dimensional vector space V over a field k , with dual vector space V ∗ {\displaystyle V^{*}} , the exterior algebra of V has two non-trivial components, namely
This exterior algebra and the symmetric algebra of V ∗ {\displaystyle V^{*}} , Sym ( V ∗ ) {\displaystyle \operatorname {Sym} (V^{*})} , serve to build a two-step chain complex
whose differential is induced by natural evaluation map
Choosing a basis of V , Sym ( V ∗ ) {\displaystyle \operatorname {Sym} (V^{*})} can be identified with the polynomial ring in one variable, k [ t ] {\displaystyle k[t]} , and the previous chain complex becomes isomorphic to the complex
whose differential is multiplication by t . This computation shows that the cohomology of the above complex is 0 at the left hand term, and is k at the right hand term. In other words, k (regarded as a chain complex concentrated in a single degree) is quasi-isomorphic to the above complex, which provides a close link between the exterior algebra of V and the symmetric algebra of its dual.
Koszul duality, as treated by Alexander Beilinson , Victor Ginzburg , and Wolfgang Soergel [ 4 ] can be formulated using the notion of Koszul algebra . An example of such a Koszul algebra A is the symmetric algebra S ( V ) {\displaystyle S(V)} on a finite-dimensional vector space. More generally, any Koszul algebra can be shown to be a quadratic algebra , i.e., of the form
where T ( V ) {\displaystyle T(V)} is the tensor algebra on a finite-dimensional vector space, and R {\displaystyle R} is a submodule of T 2 ( V ) = V ⊗ V {\displaystyle T^{2}(V)=V\otimes V} . The Koszul dual then coincides with the quadratic dual
where V ∗ {\displaystyle V^{*}} is the ( k -linear) dual and R ′ ⊂ V ∗ ⊗ V ∗ {\displaystyle R'\subset V^{*}\otimes V^{*}} consists of those elements on which the elements of R (i.e., the relations in A ) vanish. The Koszul dual of A = S ( V ) {\displaystyle A=S(V)} is given by A ! = Λ ( V ∗ ) {\displaystyle A^{!}=\Lambda (V^{*})} , the exterior algebra on the dual of V . In general, the dual of a Koszul algebra is again a Koszul algebra. Its opposite ring is given by the graded ring of self- extensions of the underlying field k, thought of as an A -module:
If an algebra A {\displaystyle A} is Koszul, there is an equivalence between certain subcategories of the derived categories of graded A {\displaystyle A} - and A ! {\displaystyle A^{!}} -modules. These subcategories are defined by certain boundedness conditions on the grading vs. the cohomological degree of a complex.
As an alternative to passing to certain subcategories of the derived categories of A {\displaystyle A} and A ! {\displaystyle A^{!}} to obtain equivalences, it is possible instead to obtain equivalences between certain quotients of the homotopy categories. [ 5 ] Usually these quotients are larger than the derived category, as they are obtained by factoring out some subcategory of the category of acyclic complexes, but they have the advantage that every complex of modules determines some element of the category, without needing to impose boundedness conditions. A different reformulation gives an equivalence between the derived category of A {\displaystyle A} and the 'coderived' category of the coalgebra ( A ! ) ∗ {\displaystyle (A^{!})^{*}} .
An extension of Koszul duality to D -modules states a similar equivalence of derived categories between dg-modules over the dg-algebra Ω X {\displaystyle \Omega _{X}} of Kähler differentials on a smooth algebraic variety X and the D X {\displaystyle D_{X}} -modules. [ 6 ] [ 7 ] [ 8 ]
An extension of the above concept of Koszul duality was formulated by Ginzburg and Kapranov who introduced the notion of a quadratic operad and defined the quadratic dual of such an operad. [ 9 ] Very roughly, an operad is an algebraic structure consisting of an object of n -ary operations for all n . An algebra over an operad is an object on which these n -ary operations act. For example, there is an operad called the associative operad whose algebras are associative algebras, i.e., depending on the precise context, non-commutative rings (or, depending on the context, non-commutative graded rings, differential graded rings). Algebras over the so-called commutative operad are commutative algebras, i.e., commutative (possibly graded, differential graded) rings. Yet another example is the Lie operad whose algebras are Lie algebras . The quadratic duality mentioned above is such that the associative operad is self-dual, while the commutative and the Lie operad correspond to each other under this duality.
Koszul duality for operads states an equivalence between algebras over dual operads. The special case of associative algebras gives back the functor A ↦ A ! {\displaystyle A\mapsto A^{!}} mentioned above. | https://en.wikipedia.org/wiki/Koszul_duality |
A kotan ( Katakana : コタン) is a traditional settlement of the Ainu people . [ 1 ]
Due to the scarcity of primary source materials (as the Ainu did not have a system of writing ), all studies on the Ainu kotan (based on Russian , Japanese , and English works) will have different analyzations and opinions, varying largely depending on the researchers and the duration of their work.
The word kotan is often erroneously translated to as a " village "; the term generally applies to all human settlements, regardless of their size. For example, in the Ainu translation of the Bible , Rome and Jerusalem are referred to as yerusalem kotan and roma kotan , respectively.
Unlike other hunter gatherers , who did not settle in one place at any given time, the Ainu were highly dependent on fishing , therefore they settled by places that had good fishing (like river estuaries ) and built settlements there, though depending on the season the Ainu would move to a new fishing spot. For example, if the salmon spawning grounds differed along the same stretch of river, the Ainu would migrate along with the ground, leading to kotans being built at intervals of about 5 to 7 kilometers.
Average kotans were rather small and not very populous. A kotan can be made up of around five to seven houses, though there were larger settlements of ten or more. More than 20 households generally was the result of the forced Ainu labor mobilization under the place-contract system (場所請負制) established by the Matsumae clan during the Edo period, which cannot be called a traditional-style kotan anymore. In 1856, Takeshirō Matsuura (an explorer in Hokkaido ) reported the statistical kotan was inhabited by 10 families and 47 people in total. [ 2 ]
A kotan generally consisted of cise ( thatched-roof houses), hepereset (cages for keeping young bears usually for the iomante ceremony), an ashinru and/or menokol (lavatories for the males and females respectively), a pui (stilt warehouse for storing food), and various drying racks for wild plants, fish, and animal skins. There is usually an nusasan ( altar ) dedicated to the Inau god as well. In later years, chashis (Ainu fortifications) can be found around Ainu settlements. [ 3 ]
There was a common ground near the kotans called the iwolo , where kotan residents were free to cut down trees, hunt, fish, and forage for wild plants and cultivate them. Adjacent kotans were invited to share the hunting grounds, conduct the iomante ceremony together, and most often having one chief for several kotan. Such a collection of friendly kotan are called ekasi itokpa .
There is only one Ainu kotan still continually inhabited to the present day, the Lake Akan kotan in Kushiro. In 1959, there were still a scattering of Ainu kotans around Lake Akan, before Mitsuko Maeda of the Maeda Ippoen Foundation (an organization that helped in conserving Lake Akan) suggested the remaining Ainu to relocate to the new land purchased by him. As the Ainu relocated to the new land free of charge, the Lake Akan kotan was created. [ 4 ]
Each kotan had a unique place name, and as such, the suffix kotan can be found throughout Hokkaido, Sakhalin , and the Kuril islands . | https://en.wikipedia.org/wiki/Kotan_(village) |
Prof. Kotcherlakota Rangadhama Rao Memorial Lecture Award is given for the outstanding contributions in the subject of Spectroscopy in Physics . The award was established by the Indian National Science Academy of Calcutta in the year 1979. The honour is awarded to Indian citizens.
The Memorial Lecture Award was established in the year 1979 in the honour of Professor Kotcherlakota Rangadhama Rao by the students of Prof. K.Rangadhama Rao and Indian National Science Academy , formerly National Institute of Sciences of India, Calcutta . The lecture is awarded for outstanding contributions in the field of Spectroscopy. The award carries an honorarium of Rs. 25,000/- and a citation.
The below lists the recipients of the Memorial Award since its inception in the year 1979. [ 1 ]
Source: Indian National Science Academy | https://en.wikipedia.org/wiki/Kotcherlakota_Rangadhama_Rao_Memorial_Lecture_Award |
Kotoeri ( ことえり ) is a discontinued Japanese-language input method that came standard with OS X and earlier versions of Classic Mac OS until OS X Yosemite . Kotoeri (written ことえり or 言選り ) literally means "word selection".
The name "Kotoeri" comes from the chapter of Hahakigi [ ja ] in Tale of Genji : "Even though I write a letter, I must carefully choose my words ( 文を書けど、おほどかに言選りをし , fumi o kake do, ohodoka ni kotoeri o shi ) ".
In version 4, Kotoeri added support for Ainu , [ 1 ] colloquial language and Kansai dialect input, [ 2 ] the ability to search for kanji among related characters using keyboard shortcuts, and the option to use key bindings similar to Microsoft IME [ de ; jp ] . Additionally, it allowed converting accidentally typed kana into romanized letters by pressing the 英数 (alphanumeric) key twice and reverting confirmed characters to their original state by pressing the かな (kana) key twice.
Kotoeri supported key shortcuts including, for example, control + J to convert to Hiragana and control + K to convert to Katakana . [ 3 ] For users who are accustomed to the initial version of Kotoeri, the assigned shortcuts can still be used effectively. Specifically, ⌥ Option + Z shortcut can be used to convert text to Hiragana, [ 4 ] while ⌥ Option + X shortcut allows for conversion to Katakana. [ 5 ] These features ensure a seamless transition even for those familiar with the original version's shortcuts.
Starting from OS X Yosemite (OS X version 10.10), which was released on October 16, 2014, Kotoeri was entirely replaced with a different Japanese input method program [ ja ] . [ 6 ]
This article about Apple Inc. is a stub . You can help Wikipedia by expanding it .
This software article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Kotoeri |
Kounotori 6 ( こうのとり6号機 ) , also known as HTV-6 , was the sixth flight of the H-II Transfer Vehicle , an uncrewed cargo spacecraft launched to resupply the International Space Station . It was launched at 13:26:47 UTC on 9 December 2016 aboard H-IIB launch vehicle from Tanegashima Space Center .
Major changes from previous Kounotori include: [ 1 ] [ 2 ]
SFINKS (Solar Cell Film Array Sheet for Next Generation on Kounotori Six) will test thin film solar cells in space. [ 3 ]
KITE (Kounotori Integrated Tether Experiment) was an experimental electrodynamic tether (EDT). [ 4 ] [ 5 ] The tether was equipped with a 20 kg end-mass, and would have been 700 m long when deployed, [ 4 ] [ 6 ] [ 7 ] Unfortunately deployment failed, but useful data was still gathered from some of the instruments. [ 8 ] A maximum current of 10 mA was planned to run through the tether. [ 4 ] Kounotori's ISS rendezvous sensor would have been utilized to measure how the end-mass moves during the test. [ 4 ] The EDT experiment was scheduled following Kounotori 6's departure from the ISS, with a planned duration of one week. [ 4 ] After the experiment, the tether would have been separated before the spacecraft proceeds with the de-orbit maneuvers. The main objective of this experiment were the orbital demonstration of both extending an uncoated bare-tether , and driving electric currents through the EDT. [ 5 ] These two technologies will contribute to gaining capabilities to remove space debris . [ 5 ] [ 9 ] [ 10 ] [ 11 ]
Kounotori 6 carries about 5900 kg of cargo (including the support structure weight), consisting of 3900 kg in PLC (Pressurized Logistics Carrier) and 1900 kg in ULC (Unpressurized Logistics Carrier). [ 12 ]
Cargo in the pressurized compartment includes 30 bags filled with potable water (600 liters), [ 13 ] [ 14 ] food, crew commodities, CDRA Bed (Carbon Dioxide Removal Assembly), TPF (Two-Phase Flow) experiment unit, PS-TEPC (Position-Sensitive Tissue Equivalent Proportional Chamber) radiation measurement instrument, ExHAM (Exposed Experiment Handrail Attachment Mechanism), HDTV-EF2 high-definition and 4K camera, new J-SSOD (JEM Small Satellite Orbital Deployer), and CubeSats ( AOBA-Velox III , TuPOD which comprises two TubeSats ( Tancredo-1 and OSNSAT ), EGG , ITF-2 , STARS-C , FREEDOM , WASEDA-SAT3 ). [ 15 ] [ 16 ] [ 12 ] Cargo by NanoRacks includes TechEdSat-5 , [ 17 ] CubeRider , [ 18 ] RTcMISS , [ 19 ] NREP-P DM7, [ 20 ] [ 21 ] four Lemur-2 . [ 22 ] Additionally, the Blue SPHERES satellite of the MIT Space Systems Laboratory is being returned to the ISS for continued autonomous systems research. [ 23 ]
Cargo in the unpressurized compartment consists of six lithium-ion batteries and their associated adapter plates to replace existing nickel-hydrogen batteries of the International Space Station . Since each of the new lithium-ion battery has a capability equivalent to two of the current nickel-hydrogen batteries, the six new batteries will replace twelve old batteries, out of the 48 existing batteries of the ISS. [ 13 ]
On departure from the ISS, Kounotori 6 carries 9 out of the 12 replaced old batteries which will be disposed of by destructive reentry into Earth's atmosphere. The 3 remaining old batteries stay on the ISS. [ 14 ]
On 26 July 2016, the launch was scheduled for 30 September 2016, [ 24 ] but on 10 August 2016, postponement was announced due to the leak from piping. [ 25 ]
The H-IIB launch vehicle carrying Kounotori 6 lifted off at 13:26:47 UTC on 9 December 2016, and 15 minutes 11 seconds later, Kounotori 6 was released into initial 200 × 300 km orbit. [ 26 ] [ 27 ]
SFINKS experiment payload began the data collection at 14:16, on 9 December 2016, but it stopped unexpectedly after 509 seconds. [ 28 ]
After a series of orbital manoeuvres, Kounotori 6 arrived to the proximity of ISS and captured by SSRMS (Canadarm2) at 10:39 (10:37 according to NASA), on 13 December 2016. [ 29 ] [ 30 ] Kounotori was bolted to the CBM ( Common Berthing Mechanism ) of the Harmony nadir port by 13:48 UTC. [ 31 ]
Berthing operation completed at 18:24, on 13 December 2016 UTC, [ 32 ] and the hatch opened at 19:44 UTC. [ 33 ]
Since 07:44, 14 December 2016, Exposed Pallet (EP) was extracted from Unpressurised Logistics Carrier (ULC) of Kounotori 6 by SSRMS and transferred to Payload and ORU Accommodation (POA). [ 34 ] [ 35 ] After a combination of two Extra-Vehicular Activities and robotic operations, the lithium-ion battery units and adapter plates were installed. The Exposed Pallet carrying old Nickel-hydrogen battery units was returned to Kounotori 6's Unpresurised Logistics Carrier on 23 January 2017. [ 36 ]
SSRMS grappled and detached Kounotori 6 from the CBM of Harmony nadir port at 10:59, 27 January 2017, [ 37 ] and Kounotori 6 was released at 15:45, on 27 January 2017. [ 38 ]
Following the undocking after moving to a safe distance from ISS, the Kounotori 6 was to demonstrate the "Kounotori Integrated Tether Experiment" (KITE) using electrodynamic tether to demonstrate
space debris removal technology. [ 39 ] This experiment was planned for seven days before reentry to the Earth's atmosphere . [ 37 ] On 31 January 2017, media reported some problems in extending the tether, bringing to doubt the experiment's success. [ 40 ]
A series of deorbit manoeuvres were performed at 08:42, 10:12, and 14:42 UTC, on 5 February 2017. [ 41 ] [ 42 ] [ 43 ] Kounotori 6 reentered to Earth atmosphere over southern Pacific Ocean around 15:06 UTC, on 5 February 2017. | https://en.wikipedia.org/wiki/Kounotori_Integrated_Tether_Experiment |
The Koutecký–Levich equation models the measured electric current at an electrode from an electrochemical reaction in relation to the kinetic activity and the mass transport of reactants.
The Koutecký–Levich equation can be written as [ 1 ] :
1 i m = 1 i K + 1 i M T {\displaystyle {1 \over i_{m}}={1 \over i_{K}}+{1 \over i_{MT}}}
where
Note the similarity of this equation to the conductance of an electrical circuits in parallel .
The Koutecký–Levich equation is also commonly expressed as:
i m = i K i M T i K + i M T {\displaystyle i_{m}={i_{K}i_{MT} \over i_{K}+i_{MT}}}
The kinetic current ( i K ) can be modeled by the Butler-Volmer Equation and is characterized by being potential dependent. On the other hand, the mass transport current ( i MT ) depends on the particular electrochemical setup and amount of stirring.
In the case a rotating disk electrode setup is used and the electrode is flat and smooth, the i MT can modeled using the Levich equation . [ 1 ] [ 2 ] Inserted in the Koutecký–Levich equation, we get:
1 i m = 1 i K + 1 B L ω 0.5 {\displaystyle {1 \over i_{m}}={1 \over i_{K}}+{1 \over {B_{L}\omega ^{0.5}}}}
where:
From an experimental data set where the current is measured at different rotation rates, it is possible to extract the kinetic current from a so-called Koutecký–Levich plot. In a Koutecký–Levich plot the inverse measured current is plotted versus the inverse square root of the rotation rate. This will linearize the data set and the inverse of the kinetic current can be obtained by extrapolating the line to the ordinate . This y-intercept corresponds to taking the rotation rate up to infinity, where the reaction is not mass-transport limited. Koutecký–Levich analysis is therefore used to determine the kinetic constants of the reaction such as the kinetic constant k o {\displaystyle k^{o}} and the symmetry factor α {\displaystyle \alpha } .
This electrochemistry -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Koutecký–Levich_equation |
Kovács reagent is a biochemical reagent consisting of isoamyl alcohol , para-dimethylaminobenzaldehyde (DMAB), and concentrated hydrochloric acid . It is used for the diagnostical indole test , to determine the ability of the organism to split indole from the amino acid tryptophan . The indole produced yields a red complex with para-dimethylaminobenzaldehyde under the given conditions. [ 1 ] This was invented by the Hungarian physician Nicholas Kovács and was published in 1928. This reagent is used in the confirmation of E. coli and many other pathogenic microorganisms.
Ehrlich's reagent is similar but uses ethyl alcohol or 1-propyl alcohol .
2. Kovacs, N. (1928): Eine vereinfachte Methode zum Nachweis der Indolbildung durch Bakterien. Zeitschrift für Immunitätsforschung und Experimentelle Therapie, 55, 311.
This biochemistry article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Kovac's_reagent |
In statistical mechanics and condensed matter physics , the Kovacs effect is a kind of memory effect in glassy systems below the glass-transition temperature . A.J. Kovacs observed that a system’s state out of equilibrium is defined not only by its macro thermodynamical variables, but also by the inner parameters of the system. In the original effect, in response to a temperature change, under constant pressure, the isobaric volume and free energy of the system experienced a recovery characterized by non-monotonic departure from equilibrium, whereas all other thermodynamical variables were in their equilibrium values. It is considered a memory effect since the relaxation dynamics of the system depend on its thermal and mechanical history.
The effect was discovered by Kovacs in the 1960s in polyvinyl acetate . [ 1 ] [ 2 ] Since then, the Kovacs effect has been established as a very general phenomenon that comes about in a large variety of systems, model glasses, [ 3 ] [ 4 ] [ 5 ] [ 6 ] [ 7 ] [ 8 ] [ 9 ] [ 10 ] tapped dense granular matter, [ 11 ] spin-glasses, [ 12 ] molecular liquids, [ 13 ] granular gases, [ 14 ] active matter, [ 15 ] disordered mechanical systems, [ 16 ] protein molecules, [ 17 ] and more.
Kovacs’ experimental procedure on polyvinyl acetate consisted of two main stages. In the first step, the sample is instantaneously quenched from a high initial temperature T 0 {\displaystyle T_{0}} to a low reference temperature T r {\displaystyle T_{r}} , under constant pressure. The time-dependent volume of the system in T r {\displaystyle T_{r}} , V ( t ) | T r {\displaystyle V(t)|_{T_{r}}} , is recorded, until the time t e q {\displaystyle t_{eq}} when the system is considered to be at equilibrium. The volume at t e q {\displaystyle t_{eq}} is defined as the equilibrium volume of the system at temperature T r {\displaystyle T_{r}} :
In the second step, the sample is quenched again from T 0 {\displaystyle T_{0}} to a temperature T 1 {\displaystyle T_{1}} that is lower than T r {\displaystyle T_{r}} , so that T 0 > T r > T 1 {\displaystyle T_{0}>T_{r}>T_{1}} . But now, the system is held at temperature T 1 {\displaystyle T_{1}} only until the time t 1 {\displaystyle t_{1}} when its volume reaches the equilibrium value of T r {\displaystyle T_{r}} , meaning V ( t 1 ) | T 1 = V e q ( T r ) {\displaystyle V(t_{1})|_{T_{1}}=V_{eq}(T_{r})} .
Then, the temperature is raised instantaneously to T r {\displaystyle T_{r}} , so both the temperature and the volume agree with the same equilibrium state. Naively, one expects that nothing should happen when the system is at V = V e q ( T r ) {\displaystyle V=V_{eq}(T_{r})} and T = T r {\displaystyle T=T_{r}} . But instead, the volume of the system first increases and then relaxes back to V e q ( T r ) {\displaystyle V_{eq}(T_{r})} , while the temperature is held constant at T r {\displaystyle T_{r}} . This non-monotonic behavior in time of the volume V ( t ) {\displaystyle V(t)} after the jump in the temperature can be simply captured by:
where Δ V ≥ 0 {\displaystyle \Delta V\geq 0} , and Δ V ( t = t 1 ) = 0 , Δ V ( t → ∞ ) = 0 {\displaystyle \Delta V(t=t_{1})=0,\Delta V(t\rightarrow \infty )=0} . Δ V {\displaystyle \Delta V} is also referred as the “ Kovacs hump ”. Kovacs also found that the hump displayed some general features: Δ V ≥ 0 {\displaystyle \Delta V\geq 0} with only one maximum of height Δ V M {\displaystyle \Delta V_{M}} at a certain time t M {\displaystyle t_{M}} ; as the temperature T 1 {\displaystyle T_{1}} is lowered, the hump becomes larger, Δ V M {\displaystyle \Delta V_{M}} increases, and moves to shorter times, t M {\displaystyle t_{M}} decreases.
In the subsequent studies of the Kovacs hump in different systems, a similar protocol with two jumps in the temperature has been employed. The associated time evolution of a relevant physical quantity P {\displaystyle P} , often the energy, is monitored and displays the Kovacs hump. The physical relevance of this behavior is the same as in the Kovacs experiment: it shows that P {\displaystyle P} does not completely characterize the dynamical state of the system, and the necessity of incorporating additional variables to have the whole picture.
The Kovacs hump described above has been rationalized by employing linear response theory for molecular systems, in which the initial and final states are equilibrium ones. Therein, the "direct" relaxation function (with only one temperature jump, instead of two) is a superposition of positive exponentially decaying modes, as a consequence of the fluctuation-dissipation theorem. Linear response makes it possible to write the Kovacs hump in terms of the direct relaxation function. [ 18 ] Specifically, the positivity of the all the modes in the direct relaxation function ensures the "normal" character of the hump, i.e. the fact that Δ P ≥ 0 {\displaystyle \Delta P\geq 0} .
Recently, analogous experiments have been proposed for "athermal" systems, like granular systems or active matter, with the proper reinterpretation of the variables. For instance, in granular gases the relevant physical property P {\displaystyle P} is still the energy—although one usually employs the terminology "granular temperature" for the kinetic energy in this context—but it is the intensity of the external driving ξ {\displaystyle \xi } that plays the role of the temperature. The emergence of Kovacs-like humps highlights the relevance of non-Gaussianities to describe the physical state of granular gases.
"Anomalous" Kovacs humps have been reported in athermal systems, i.e. Δ P ≤ 0 {\displaystyle \Delta P\leq 0} , i.e. a minimum is observed instead of a maximum. [ 14 ] [ 15 ] Although the linear response connection between the Kovacs hump and the direct relaxation function can be extended to athermal systems, [ 15 ] [ 19 ] not all the modes are positive definite—the standard version of the fluctuation-dissipation theorem does not apply. This is the key that facilitates the emergence of anomalous behavior. [ 20 ] | https://en.wikipedia.org/wiki/Kovacs_effect |
Kovasznay flow corresponds to an exact solution of the Navier–Stokes equations and are interpreted to describe the flow behind a two-dimensional grid. The flow is named after Leslie Stephen George Kovasznay , who discovered this solution in 1948. [ 1 ] The solution is often used to validate numerical codes solving two-dimensional Navier-Stokes equations.
Let U {\displaystyle U} be the free stream velocity and let L {\displaystyle L} be the spacing between a two-dimensional grid. The velocity field ( u , v , 0 ) {\displaystyle (u,v,0)} of the Kovaszany flow, expressed in the Cartesian coordinate system is given by [ 2 ]
where λ {\displaystyle \lambda } is the root of the equation λ 2 − R e λ − 4 π 2 = 0 {\displaystyle \lambda ^{2}-Re\,\lambda -4\pi ^{2}=0} in which R e = U L / ν {\displaystyle Re=UL/\nu } represents the Reynolds number of the flow. The root that describes the flow behind the two-dimensional grid is found to be
The corresponding vorticity field ( 0 , 0 , ω ) {\displaystyle (0,0,\omega )} and the stream function ψ {\displaystyle \psi } are given by
Similar exact solutions, extending Kovasznay's, has been noted by Lin and Tobak [ 3 ] and C. Y. Wang. [ 4 ] [ 5 ] | https://en.wikipedia.org/wiki/Kovasznay_flow |
In gas chromatography , the Kovats retention index (shorter Kovats index , retention index ; plural retention indices ) is used to convert retention times into system-independent constants. The index is named after the Hungarian-born Swiss chemist Ervin Kováts (1927-2012), who outlined the concept in the 1950s while performing research into the composition of the essential oils . [ 1 ]
The retention index of a chemical compound is retention time interpolated between adjacent n - alkanes . While retention times vary with the individual chromatographic system (e.g. with regards to column length , film thickness, diameter and inlet pressure), the derived retention indices are quite independent of these parameters and allow comparing values measured by different analytical laboratories under varying conditions and analysis times from seconds to hours. Tables of retention indices are used to identify peaks by comparing measured retention indices with the tabulated values. [ 2 ] [ 3 ] [ 4 ]
The Kovats index applies to organic compounds . The method interpolates peaks between bracketing n -alkanes. The Kovats index of n-alkanes is 100 times their carbon number, e.g. the Kovats index of n - butane is 400. The Kovats index is dimensionless, unlike retention time or retention volume. For isothermal gas chromatography , the Kovats index is given by the equation:
where the variables used are:
The Kovats index also applies to packed columns with an equivalent equation:
Compounds elute in the carrier gas phase only. Compounds solved in the stationary phase stay put. The ratio of gas time t 0 {\displaystyle t_{0}} and residence time t i − t 0 {\displaystyle t_{i}-t_{0}} in the stationary liquid polymer phase is called the capacity factor k i {\displaystyle k_{i}} :
where the variables used are:
Capillary tubes with uniform coatings have this phase ratio β:
Capillary inner diameter d c {\displaystyle d_{c}} is well defined but film thickness d f {\displaystyle d_{f}} reduces by bleed and thermal breakdown that occur after column heating over time, depending on chemical bonding to the silica glass wall and polymer cross-linking of the stationary phase.
Above capacity factor k i {\displaystyle k_{i}} can be expressed explicit for retention time:
Retention time t i {\displaystyle t_{i}} is shorter by reduced d f {\displaystyle d_{f}} over column life time. Column length L {\displaystyle L} is introduced with average gas velocity u = L / t 0 {\displaystyle u=L/t_{0}} :
R {\displaystyle R} and temperature T {\displaystyle T} have a direct relation with t i {\displaystyle t_{i}} . However, warmer columns T {\displaystyle T} ↑ do not have longer t i {\displaystyle t_{i}} but shorter, following temperature programming experience. Pure liquid vapor pressure P i {\displaystyle P^{i}} rises exponentially with T {\displaystyle T} so that we do get shorter t i {\displaystyle t_{i}} warming the column T {\displaystyle T} ↑. Solubility of compounds S i {\displaystyle S_{i}} in the stationary phase may rise or fall with T {\displaystyle T} , but not exponentially. S i {\displaystyle S_{i}} is referred to as selectivity or polarity by gas chromatographers today.
Isothermal Kovats index in terms of the physical properties becomes:
Isothermal Kovats index is independent of R {\displaystyle R} , any GC dimension L {\displaystyle L} or ß or carrier gas velocity u {\displaystyle u} , which compares favorable to retention time t i {\displaystyle t_{i}} . Isothermal Kovats index is based on solubility S i {\displaystyle S_{i}} and vapor pressure P i {\displaystyle P^{i}} of compound i and n -Alkanes ( i = n {\displaystyle i=n} ). T {\displaystyle T} dependence depends on the compound compared to the n-alkanes. Kovats index of n-alkanes I n = 100 c {\displaystyle I_{n}=100c} is independent of T {\displaystyle T} .
Isothermal Kovats indices of hydrocarbon were measured by Axel Lubeck and Donald Sutton. [ 5 ]
IUPAC defines the temperature programmed chromatography Kovats index equation:
NOTE: TPGC index does depend on temperature program, gas velocity and the column used !
ASTM method D6730 defines the temperature programmed chromatography Kovats index equation:
Measured Kovats retention index values can be found in ASTM method D 6730 databases.
An extensive Kovats index database is compiled by NIST [1] .
The equations produce significant different Kovats indices.
Faster GC methods have shorter times but Kovats indices of the compounds may be conserved if proper method translation is applied.
Temperatures of the temperature program stay the same, but ramps and times change when using a smaller column or faster carrier gas.
If column dimensions Length×diameter×film are divided by 2 and gas velocity is doubled by using H2 in place of Helium , the hold times must be divided by 4 and the ramps must be multiplied by 4 to keep the same index and the same retention temperature for the same compound analyzed. Method translation rules are incorporated in some chromatography data systems. | https://en.wikipedia.org/wiki/Kovats_retention_index |
The Kowalski ester homologation is a chemical reaction for the homologation of esters . [ 1 ] [ 2 ]
This reaction was designed as a safer alternative to the Arndt–Eistert synthesis , avoiding the need for diazomethane . The Kowalski reaction is named after its inventor, Conrad J. Kowalski .
The mechanism is disputed. [ further explanation needed ]
By changing the reagent in the second step of the reaction, the Kowalski ester homologation can also be used for the preparation of silyl ynol ethers . [ citation needed ] | https://en.wikipedia.org/wiki/Kowalski_ester_homologation |
The Kozeny–Carman equation (or Carman–Kozeny equation or Kozeny equation ) is a relation used in the field of fluid dynamics to calculate the pressure drop of a fluid flowing through a packed bed of solids. It is named after Josef Kozeny and Philip C. Carman. The equation is only valid for creeping flow , i.e. in the slowest limit of laminar flow . The equation was derived by Kozeny (1927) [ 1 ] and Carman (1937, 1956) [ 2 ] [ 3 ] [ 4 ] from a starting point of (a) modelling fluid flow in a packed bed as laminar fluid flow in a collection of curving passages/tubes crossing the packed bed and (b) Poiseuille's law describing laminar fluid flow in straight, circular section pipes.
The equation is given as: [ 4 ] [ 5 ]
where:
This equation holds for flow through packed beds with particle Reynolds numbers up to approximately 1.0, after which point frequent shifting of flow channels in the bed causes considerable kinetic energy losses.
This equation is a particular case of Darcy's law , with a very specific permeability. Darcy's law states that " flow is proportional to the pressure gradient and inversely proportional to the fluid viscosity " and is given as:
Combining these equations gives the final Kozeny equation for absolute (single phase) permeability:
where:
The equation was first [ 8 ] proposed by Kozeny (1927) [ 1 ] and later modified by Carman (1937, 1956). [ 2 ] [ 3 ] A similar equation was derived independently by Fair and Hatch in 1933. [ 9 ] A comprehensive review of other equations has been published. [ 10 ] | https://en.wikipedia.org/wiki/Kozeny–Carman_equation |
Kozhikode Metro is a proposed rapid transit system for the city of Kozhikode , in India. In 2010, the State government explored the possibility of implementing a metro rail project for Kozhikode city and its suburbs. The proposal was to have a corridor connecting Meenchanda to the Kozhikode Medical College Hospital through the heart of the city. An inception report was submitted by a Bangalore -based consultant, Wilber Smith, on the detailed feasibility study on the prospect of implementing the Mass Rapid Transport System (MRTS) and Light Rail Transit System (LRTS) in the city. However, the project has been scrapped to be replaced by Kozhikode Monorail project.
The State Cabinet then decided to form a special purpose vehicle (SPV) to implement monorail projects in Kozhikode and Thiruvananthapuram, and administrative sanction was given in October 2012. [ 1 ] The state government issued orders entrusting the Thiruvananthapuram Monorail project to the KMCL on 26 November 2012. The government had handed over the Kozhikode Monorail project to the KMCL prior to that. [ 2 ] On 12 June 2013, the State Cabinet gave clearance for an agreement to be signed between KMCL and DMRC, that would make the latter the general consultant for the monorail projects in Kozhikode and Thiruvananthapuram. [ 3 ] The DMRC will receive a consultancy fee of 3.25% of the ₹ 55.81 billion ( ₹ 35.90 billion for Thiruvananthapuram and ₹ 19.91 billion for Kozhikode). [ 4 ] The agreement was signed on 19 June 2013. [ 5 ]
However, due to cost overrun and the cold response from the bidders the project was put on hold. Bombardier Transportation was the only bidder for the project. The project was later scrapped and Light metro was proposed. [ 6 ] [ 7 ] [ 8 ] [ 9 ]
In the 2025-2026 Kerala Budget, Finance minister KN Balagopal plans to develop the Kozhikode Metro project after the completion of the Thiruvananthapuram Metro. [ 10 ]
The Union Urban Development Ministry decided to consider the proposal for a Metro in Kozhikode after the success of the Delhi Metro and signed up for drawing the detailed project report (DPR) of the Rs. 27.71 billion Kozhikode metro transport project with Delhi Metro Rail Corporation as a feasibility study for the introduction of suburban services in Kozhikode city. The Ministry decided to bear 50% of the cost of the preparation of the DPR for the city that comes under the population cut-off bracket. The preliminary feasibility study had been carried out by the National Transportation Planning and Research Centre (NATPAC) in association with the Kerala Road Fund Board in December 2008. Based on this feasibility report, the Board entrusted Wilber Smith to conduct the study in June 2009. Already, the NATPAC has submitted a metro rail project covering a total distance of 32.6 km from Karipur to the Kozhikode Medical College . The cost of the project was estimated at Rs. 27.71 billion and was expected be completed within five years. The monorail project which replaced the metro rail project was estimated to cost Rs 1,991 crore has received a bid from the lone bidder Bombardier consortium, and was almost double of the estimate. The project was scrapped and the Light Metro has been approved. [ 6 ] [ 7 ] [ 8 ] [ 9 ]
As per the proposal for Metro, it would start from Karipur Airport , touching Ramanattukara , Meenchanda, Mini- Bypass, Arayadathupalam and culminate at the Medical College. An estimated 2,083,000 people would get the benefits of the new transportation system by 2031. The project, which can be partly finished within three years, will be economically and technically feasible. However the detailed project report prepared by Delhi Metro Rail Corporation, the alignment for Kozhikode Monorail is retained for the Light metro project.
The Union government was in favour of implementing the project with private participation, ruling out its own financial involvement. The Ministry of Urban Development and the Planning Commission were also against government investment in the project, and refused to accept it as a project in line with the Delhi Metro and Chennai Metro . The political rivalry between the earlier Left Front government in Kerala and the UPA government at the Centre was a major reason for such developments and the slow down in the project. The change in government in Kerala changed that scenario, making the Kozhikode Metro one of the top priorities of the UDF government. But later, not to affect the Kochi Metro project The Kerala cabinet under the Chief Ministership of Oommen Chandy decided to give clearance only for the Kozhikode Monorail project, replacing the Metro rail project.
The newly proposed Light Metro is proposed to be implemented as government initiative expecting a viability gap funding from the central and state government. Remaining fund is expected to be sourced internally and externally form competent agencies. [ 9 ]
The project was proposed to cover a distance of 14.2 km with 15 stations, from Medical College Hostel to Meenchanda. [ 11 ] The car depot was proposed to be located about 500 metres (1,600 ft) east of the Medical College Hostel station on 5.20 hectares (12.8 acres) of vacant land owned by the government.
The monorail was proposed to be built in two phases. The first from Medical College to Mananchira and the second from Mananchira to Meenchantha. [ 11 ] Approximately 10.65 ha (26.3 acres) of land was to be required for the project, [ 12 ] of which 80% is government-owned land.
Kozhikode monorail was proposed to have a total of 15 stations. [ 13 ]
The government had planned to extend the monorail to Civil Station and West Hill. It would have required ₹ 600 crore (US$71 million) for the 6-kilometre (3.7 mi) stretch connecting Malaparamba and Civil Station. [ 14 ]
Each train will be made up of 3 coaches on the formation - leading car / intermediate car / leading car. The length and width of the cars will be 18m and 2.8m respectively. The total length of train will be approximately 59.94 m.
Each train has a capacity of approximately 800 passengers. The metro is designed to carry 30,000 passengers per hour. | https://en.wikipedia.org/wiki/Kozhikode_Light_Metro |
Kqueue is a scalable event notification interface introduced in FreeBSD 4.1 in July 2000, [ 1 ] [ 2 ] also supported in NetBSD , OpenBSD , DragonFly BSD , and macOS . Kqueue was originally authored in 2000 by Jonathan Lemon, [ 1 ] [ 2 ] then involved with the FreeBSD Core Team . Kqueue makes it possible for software like nginx to solve the c10k problem . [ 3 ] [ 4 ] The term "kqueue" refers to its function as a "kernel event queue" [ 1 ] [ 2 ]
Kqueue provides efficient input and output event pipelines between the kernel and userland . Thus, it is possible to modify event filters as well as receive pending events while using only a single system call to kevent(2) per main event loop iteration. This contrasts with older traditional polling system calls such as poll(2) and select(2) which are less efficient, especially when polling for events on numerous file descriptors.
Kqueue not only handles file descriptor events but is also used for various other notifications such as file modification monitoring , signals , asynchronous I/O events (AIO), child process state change monitoring, and timers which support nanosecond resolution. Furthermore, kqueue provides a way to use user-defined events in addition to the ones provided by the kernel.
Some other operating systems which traditionally only supported select(2) and poll(2) also currently provide more efficient polling alternatives, such as epoll on Linux and I/O completion ports on Windows and Solaris .
libkqueue is a user space implementation of kqueue(2) , which translates calls to an operating system's native backend event mechanism. [ 5 ]
The function prototypes and types are found in sys/event.h . [ 6 ]
Creates a new kernel event queue and returns a descriptor.
Used to register events with the queue, then wait for and return any pending events to the user. In contrast to epoll , kqueue uses the same function to register and wait for events, and multiple event sources may be registered and modified using a single call. The changelist array can be used to pass modifications (changing the type of events to wait for, register new event sources, etc.) to the event queue, which are applied before waiting for events begins. nevents is the size of the user supplied eventlist array that is used to receive events from the event queue.
A macro that is used for convenient initialization of a struct kevent object.
OS-independent libraries with support for kqueue:
Kqueue equivalent for other platforms: | https://en.wikipedia.org/wiki/Kqueue |
Kr00k (also written as KrØØk ) is a security vulnerability that allows some WPA2 encrypted WiFi traffic to be decrypted. [ 1 ] The vulnerability was originally discovered by security company ESET in 2019 and assigned CVE - 2019-15126 on August 17th, 2019. [ 2 ] ESET estimates that this vulnerability affects over a billion devices. [ 3 ]
Kr00k was discovered by ESET Experimental Research and Detection Team, most prominently ESET security researcher Miloš Čermák . [ 1 ]
It was named Kr00k by Robert Lipovský and Štefan Svorenčík . It was discovered when trying variations of the KRACK attack. [ 4 ]
Initially found in chips made by Broadcom and Cypress , similar vulnerabilities have been found in other implementations, including those by Qualcomm and MediaTek . [ 5 ] [ 6 ]
The vulnerability is known to be patched in:
During their research, ESET confirmed over a dozen popular devices were vulnerable. [ 3 ]
Cisco has found several of their devices to be vulnerable and are working on patches. [ 7 ] They are tracking the issue with advisory id cisco-sa-20200226-wi-fi-info-disclosure. [ 8 ]
Known vulnerable devices include: | https://en.wikipedia.org/wiki/Kr00k |
Krypton difluoride , KrF 2 is a chemical compound of krypton and fluorine . It was the first compound of krypton discovered. [ 2 ] It is a volatile , colourless solid at room temperature. The structure of the KrF 2 molecule is linear, with Kr−F distances of 188.9 pm. It reacts with strong Lewis acids to form salts of the KrF + and Kr 2 F + 3 cations . [ 3 ]
The atomization energy of KrF 2 (KrF 2(g) → Kr (g) + 2 F (g) ) is 21.9 kcal/mol, giving an average Kr–F bond energy of only 11 kcal/mol, [ 4 ] the weakest of any isolable fluoride. In comparison, the dissociation of difluorine to atomic fluorine requires cleaving a F–F bond with a bond dissociation energy of 36 kcal/mol. Consequently, KrF 2 is a good source of the extremely reactive and oxidizing atomic fluorine. It is thermally unstable, with a decomposition rate of 10% per hour at room temperature. [ 5 ] The formation of krypton difluoride is endothermic, with a heat of formation (gas) of 14.4 ± 0.8 kcal/mol measured at 93 °C. [ 5 ]
Krypton difluoride can be synthesized using many different methods including electrical discharge, photoionization , hot wire, and proton bombardment. The product can be stored at −78 °C without decomposition. [ 6 ]
Electric discharge was the first method used to make krypton difluoride. It was also used in the only experiment ever reported to produce krypton tetrafluoride, although the identification of krypton tetrafluoride was later shown to be mistaken. The electrical discharge method involves having 1:1 to 2:1 mixtures of F 2 to Kr at a pressure of 40 to 60 torr and then arcing large amounts of energy between it. Rates of almost 0.25 g/h can be achieved. The problem with this method is that it is unreliable with respect to yield. [ 3 ] [ 7 ]
Using proton bombardment for the production of KrF 2 has a maximum production rate of about 1 g/h. This is achieved by bombarding mixtures of Kr and F 2 with a proton beam operating at an energy level of 10 MeV and at a temperature of about 133 K. It is a fast method of producing relatively large amounts of KrF 2 , but requires a source of high-energy protons, which usually would come from a cyclotron . [ 3 ] [ 8 ]
The successful photochemical synthesis of krypton difluoride was first reported by Lucia V. Streng in 1963. It was next reported in 1975 by J. Slivnik. [ 9 ] [ 10 ] [ 11 ] [ 3 ] The photochemical process for the production of KrF 2 involves the use of UV light and can produce under ideal circumstances 1.22 g/h. The ideal wavelengths to use are in the range of 303–313 nm. Harder UV radiation is detrimental to the production of KrF 2 . Using Pyrex glass, Vycor, or quartz will significantly increase yield because they all block harder UV light. In a series of experiments performed by S. A Kinkead et al., it was shown that a quartz insert (UV cut off of 170 nm) produced on average 158 mg/h, Vycor 7913 (UV cut off of 210 nm) produced on average 204 mg/h and Pyrex 7740 (UV cut off of 280 nm) produced on average 507 mg/h. It is clear from these results that higher-energy ultraviolet light reduces the yield significantly. The ideal circumstances for the production KrF 2 by a photochemical process appear to occur when krypton is a solid and fluorine is a liquid, which occur at 77 K. The biggest problem with this method is that it requires the handling of liquid F 2 and the potential of it being released if it becomes overpressurized. [ 3 ] [ 7 ]
The hot wire method for the production of KrF 2 uses krypton in a solid state with a hot wire running a few centimeters away from it as fluorine gas is then run past the wire. The wire has a large current, causing it to reach temperatures around 680 °C. This causes the fluorine gas to split into its radicals, which then can react with the solid krypton. Under ideal conditions, it has been known to reach a maximum yield of 6 g/h. In order to achieve optimal yields the gap between the wire and the solid krypton should be 1 cm, giving rise to a temperature gradient of about 900 °C/cm. A major downside to this method is the amount of electricity that has to be passed through the wire. It is dangerous if not properly set up. [ 3 ] [ 7 ]
Krypton difluoride can exist in one of two possible crystallographic morphologies: α-phase and β-phase. β-KrF 2 generally exists at above −80 °C, while α-KrF 2 is more stable at lower temperatures. [ 3 ] The unit cell of α-KrF 2 is body-centred tetragonal.
Krypton difluoride is primarily a powerful oxidising and fluorinating agent, more powerful even than elemental fluorine because Kr–F has less bond energy . It has a redox potential of +3.5 V for the KrF 2 /Kr couple, [ citation needed ] making it the most powerful known oxidising agent. However, the hypothetical KrF 4 could be even stronger [ 12 ] and nickel tetrafluoride comes close.
For example, krypton difluoride can oxidise gold to its highest-known oxidation state, +5:
KrF + AuF − 6 decomposes at 60 °C into gold(V) fluoride and krypton and fluorine gases: [ 13 ]
KrF 2 can also directly oxidise xenon to xenon hexafluoride : [ 12 ]
KrF 2 is used to synthesize the highly reactive BrF + 6 cation. [ 6 ] KrF 2 reacts with SbF 5 to form the salt KrF + SbF − 6 ; the KrF + cation is capable of oxidising both BrF 5 and ClF 5 to BrF + 6 and ClF + 6 , respectively. [ 14 ]
KrF 2 can also react with elemental silver to produce AgF 3 . [ 15 ] [ 16 ]
Irradiation of a crystal of KrF 2 at 77 K with γ-rays leads to the formation of the krypton monofluoride radical, KrF•, a violet-colored species that was identified by its ESR spectrum. The radical, trapped in the crystal lattice, is stable indefinitely at 77 K but decomposes at 120 K. [ 17 ] | https://en.wikipedia.org/wiki/KrF2 |
Krypton(IV) fluoride is a hypothetical inorganic chemical compound of krypton and fluorine with the chemical formula KrF 4 . At one time researchers thought they had synthesized it, but the claim was discredited. [ 1 ] The compound is predicted to be difficult to make and unstable if made. [ 2 ] However, it is predicted to become stable at pressures greater than 15 GPa . [ 3 ] Theoretical analysis indicates KrF 4 would have an approximately square planar molecular geometry . [ 2 ]
The claimed synthesis was by passing electric discharge through krypton-fluorine mixture: [ 4 ]
The claimed compound formed white crystalline solid. [ 5 ] Thermally, it is less stable than XeF 4 . [ 6 ]
This inorganic compound –related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/KrF4 |
Krypton hexafluoride is an inorganic chemical compound of krypton and fluorine with the chemical formula KrF 6 . It is still a hypothetical compound . [ 1 ] Calculations indicate it is unstable. [ 2 ]
In 1933, Linus Pauling predicted that the heavier noble gases would be able to form compounds with fluorine and oxygen. He also predicted the existence of krypton hexafluoride. [ 3 ] [ verification needed ] [ 4 ] Calculations suggest it would have octahedral molecular geometry . [ 1 ]
So far, out of all possible krypton fluorides, only krypton difluoride ( KrF 2 ) has actually been synthesized. [ citation needed ]
This inorganic compound –related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/KrF6 |
In colloidal chemistry , the Krafft temperature (or Krafft point , after German chemist Friedrich Krafft ) is defined as the minimum temperature at which the formation of micelles occurs in a solution of dissolved surfactant . It has been found that solubility at the Krafft point is nearly equal to critical micelle concentration (CMC). Below the Krafft temperature, the maximum solubility of the surfactant will be lower than the critical micelle concentration, meaning micelles will not form. The Krafft temperature is a point of phase change below which the surfactant remains in crystalline form, even in an aqueous solution. Visually the effect of going below the Krafft point is similar to that of going above the cloud point , with the solution becoming cloudy or opaque due to the surfactant molecules undergoing flocculation .
Surfactants in such a crystalline state will only solubilize and form micelles if another surfactant assists it in overcoming the forces that keep it crystallized, or if the temperature increases, thus causing entropy to increase and encouraging the crystalline structure to break apart.
Surfactants are usually composed of a hydrocarbon chain and a polar head group.
Increasing the length of the hydrocarbon chain increases the Krafft temperature because it improves Van der Waals forces.
Moreover, since Krafft point is related to solid-liquid transition, better-packed polar heads within surfactant crystals increase Krafft temperature. [ 1 ]
This physical chemistry -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Krafft_temperature |
The kraft process (also known as kraft pulping or sulfate process ) is a process for conversion of wood into wood pulp , which consists of almost pure cellulose fibres, the main component of paper . The kraft process involves treatment of wood chips with a hot mixture of water, sodium hydroxide (NaOH), and sodium sulfide (Na 2 S), known as white liquor , that breaks the bonds that link lignin , hemicellulose , and cellulose . The technology entails several steps, both mechanical and chemical. It is the dominant method for producing paper. In some situations, the process has been controversial because kraft plants can release odorous products and in some situations produce substantial liquid wastes . [ 1 ] [ 2 ] [ 3 ]
The process name is derived from the German word Kraft , meaning "strength" in this context, due to the strength of the kraft paper produced using this process. [ 4 ]
A precursor of the kraft process was used during the Napoleonic Wars in England. [ 5 ] The kraft process was invented by Carl F. Dahl in 1879 in Danzig , Prussia , Germany . U.S. patent 296,935 was issued in 1884, and a pulp mill using this technology began in Sweden in 1890. [ 6 ] The invention of the recovery boiler by G. H. Tomlinson in the early 1930s was a milestone in the advancement of the kraft process. [ 7 ] It enabled the recovery and reuse of the inorganic pulping chemicals such that a kraft mill is a nearly closed-cycle process with respect to inorganic chemicals, apart from those used in the bleaching process. For this reason, in the 1940s, the kraft process superseded the sulfite process as the dominant method for producing wood pulp. [ 6 ]
Common wood chips used in pulp production are 12–25 millimetres (0.47–0.98 in) long and 2–10 millimetres (0.079–0.394 in) thick. The chips are first wetted and preheated with steam . Cavities inside fresh wood chips are partly filled with liquid and partly with air. The steam treatment causes the air to expand and about 25% of the air to be expelled from the chips. The next step is to saturate the chips with black and white liquor . Air remaining in chips at the beginning of liquor impregnation is trapped within the chips. The impregnation can be done before or after the chips enter the digester and is normally done below 100 °C (212 °F). During impregnation, cooking liquors penetrate into the capillary structure of the chips and low temperature chemical reactions with the wood begin. A good impregnation is important to get a homogeneous cook and low rejects. About 40–60% of all alkali consumption, in the continuous process, occurs in the impregnation zone.
The wood chips are then cooked in pressurized digesters. Some digesters operate in a batch manner and some in a continuous process. Digesters producing 1,000 tonnes or more of pulp per day are common, with the largest producing more than 3,500 tonnes per day. [ 8 ] Typically, delignification requires around two hours [ 9 ] at 170 to 176 °C (338 to 349 °F). Under digesting conditions, lignin and hemicellulose degrade to give fragments that are soluble in the strongly basic liquid. The solid pulp (about 50% by weight of the dry wood chips) is collected and washed. At this point the pulp is known as brown stock because of its color. The combined liquids, known as black liquor (because of its color), contain lignin fragments, carbohydrates from the breakdown of hemicellulose, sodium carbonate , sodium sulfate and other inorganic salts.
One of the main chemical reactions that underpin the kraft process is the scission of ether bonds by the nucleophilic sulfide (S 2− ) or bisulfide (HS − ) ions. [ 7 ]
The excess black liquor contains about 15% solids and is concentrated in a multiple effect evaporator . After the first step the black liquor has about 20–30% solids. At this concentration the rosin soap rises to the surface and is skimmed off. The collected soap is further processed to tall oil . Removal of the soap improves the evaporation operation of the later effects.
The weak black liquor is further evaporated to 65% or even 80% solids ("heavy black liquor" [ 10 ] ) and burned in the recovery boiler to recover the inorganic chemicals for reuse in the pulping process. Higher solids in the concentrated black liquor increases the energy and chemical efficiency of the recovery cycle, but also gives higher viscosity and precipitation of solids (plugging and fouling of equipment). [ 11 ] [ 12 ] During combustion, sodium sulfate is reduced to sodium sulfide by the organic carbon in the mixture:
This reaction is similar to thermochemical sulfate reduction in geochemistry.
The molten salts ("smelt") from the recovery boiler are dissolved in a process water known as "weak wash". This process water, also known as "weak white liquor" is composed of all liquors used to wash lime mud and green liquor precipitates. The resulting solution of sodium carbonate and sodium sulfide is known as "green liquor". The green liquor's eponymous green colour arises from the presence of colloidal iron sulfide. [ 13 ] This liquid is then mixed with calcium oxide , which becomes calcium hydroxide in solution, to regenerate the white liquor used in the pulping process through an equilibrium reaction (Na 2 S is shown since it is part of the green liquor, but does not participate in the reaction):
Calcium carbonate precipitates from the white liquor and is recovered and heated in a lime kiln where it is converted to calcium oxide (lime).
Calcium oxide (lime) is reacted with water to regenerate the calcium hydroxide used in Reaction 2:
The combination of reactions 1 through 4 form a closed cycle with respect to sodium, sulfur and calcium and is the main concept of the so-called recausticizing process where sodium carbonate is reacted to regenerate sodium hydroxide .
The recovery boiler also generates high pressure steam which is fed to turbogenerators, reducing the steam pressure for the mill use and generating electricity . A modern kraft pulp mill is more than self-sufficient in its electrical generation and normally will provide a net flow of energy which can be used by an associated paper mill or sold to neighboring industries or communities through to the local electrical grid. [ 14 ] Additionally, bark and wood residues are often burned in a separate power boiler to generate steam.
Although recovery boilers using G.H. Tomlinson's invention have been in general use since the early 1930s, attempts have been made to find a more efficient process for the recovery of cooking chemicals. Weyerhaeuser has operated a Chemrec first generation black liquor entrained flow gasifier successfully at its New Bern plant in North Carolina , while a second generation plant is run in pilot scale at Smurfit Kappa's plant in Piteå , Sweden . [ 15 ]
An additional technology is employed to lower the use of lime. In "partial borate autocausticizing" (PBAC), boric acid is added which produces sodium borate in place of sodium carbonate. [ 16 ]
The finished cooked wood chips are blown to a collection tank called a blow tank that operates at atmospheric pressure. This releases a lot of steam and volatiles. The volatiles are condensed and collected; in the case of northern softwoods this consists mainly of raw turpentine .
Screening of the pulp after pulping is a process whereby the pulp is separated from large shives , knots , dirt and other debris. The accept is the pulp. The material separated from the pulp is called reject .
The screening section consists of different types of sieves (screens) and centrifugal cleaning. The sieves are normally set up in a multistage cascade operation because considerable amounts of good fibres can go to the reject stream when trying to achieve maximum purity in the accept flow.
The fiber containing shives and knots are separated from the rest of the reject and reprocessed either in a refiner or is sent back to the digester. The content of knots is typically 0.5–3.0% of the digester output, while the shives content is about 0.1–1.0%.
The brownstock from the blowing goes to the washing stages where the used cooking liquors are separated from the cellulose fibers. Normally a pulp mill has 3-5 washing stages in series. Washing stages are also placed after oxygen delignification and between the bleaching stages as well. Pulp washers use countercurrent flow between the stages such that the pulp moves in the opposite direction to the flow of washing waters. Several processes are involved: thickening / dilution , displacement and diffusion . The dilution factor is the measure of the amount of water used in washing compared with the theoretical amount required to displace the liquor from the thickened pulp. Lower dilution factor reduces energy consumption, while higher dilution factor normally gives cleaner pulp. Thorough washing of the pulp reduces the chemical oxygen demand ( COD ).
Several types of washing equipment are in use:
In a modern mill, brownstock (cellulose fibers containing approximately 5% residual lignin) produced by the pulping is first washed to remove some of the dissolved organic material and then further delignified by a variety of bleaching stages. [ 17 ]
In the case of a plant designed to produce pulp to make brown sack paper or linerboard for boxes and packaging, the pulp does not always need to be bleached to a high brightness. Bleaching decreases the mass of pulp produced by about 5%, decreases the strength of the fibers and adds to the cost of manufacture.
Process chemicals are added to improve the production process:
Pulp produced by the kraft process is stronger than that made by other pulping processes and maintains a high effective sulfur ratio (sulfidity), an important determiner of the strength of the paper. Acidic sulfite processes degrade cellulose more than the kraft process, which leads to weaker fibers. Kraft pulping removes most of the lignin present originally in the wood whereas mechanical pulping processes leave most of the lignin in the fibers. The hydrophobic nature of lignin [ 20 ] interferes with the formation of the hydrogen bonds between cellulose (and hemicellulose) in the fibers needed for the strength of paper [ 6 ] (strength refers to tensile strength and resistance to tearing).
Kraft pulp is darker than other wood pulps, but it can be bleached to make very white pulp. Fully bleached kraft pulp is used to make high-quality paper where strength, whiteness, and resistance to yellowing are important.
The kraft process can use a wider range of fiber sources than most other pulping processes. All types of wood, including very resinous types like southern pine , [ 21 ] and non-wood species like bamboo and kenaf can be used in the kraft process.
The main byproducts of kraft pulping are crude sulfate turpentine and tall oil soap. The availability of these is strongly dependent on wood species, growth conditions, storage time of logs and chips, and the mill's process. [ 22 ] Pines are the most extractive-rich woods. The raw turpentine is volatile and is distilled off the digester, while the raw soap is separated from the spent black liquor by decantation of the soap layer formed on top of the liquor storage tanks. From pines the average yield of turpentine is 5–10 kg/t pulp and of crude tall oil is 30–50 kg/t pulp. [ 22 ]
Various byproducts containing hydrogen sulfide , methyl mercaptan , dimethyl sulfide , dimethyl disulfide , and other volatile sulfur compounds are the cause of the malodorous air emissions characteristic for pulp mills utilizing the kraft process. [ 23 ] [ 24 ] The sulfur dioxide emissions of kraft-pulp mills are much lower than those from sulfite mills. In the ambient air outside a typical modern kraft-pulp mill, the sulfur-dioxide odour is perceivable only during disturbance situations, for example when the mill is shut down for a maintenance break, or when an extended power outage occurs. Control of odours is achieved through the collection and burning of these odorous gases in the recovery boiler alongside the black liquor. In modern mills, where well-dried solids are burned in the recovery boiler, hardly any sulfur dioxide leaves the boiler. At high boiler temperatures, the sodium released from the black liquor droplets reacts with sulfur dioxide, thereby effectively scavenging it by forming odourless sodium sulfate crystals.
Pulp mills are almost always located near large bodies of water due to their substantial demand for water. Delignification of chemical pulps releases considerable amounts of organic material into the environment, particularly into rivers or lakes. The wastewater effluent can also be a major source of pollution, containing lignins from the trees, high biological oxygen demand (BOD) and dissolved organic carbon (DOC), along with alcohols , chlorates , heavy metals, and chelating agents. The process effluents can be treated in a biological effluent treatment plant , which can substantially reduce their toxicity. [ 25 ] [ 26 ] | https://en.wikipedia.org/wiki/Kraft_process |
In coding theory , the Kraft–McMillan inequality gives a necessary and sufficient condition for the existence of a prefix code [ 1 ] (in Leon G. Kraft's version) or a uniquely decodable code (in Brockway McMillan 's version) for a given set of codeword lengths. Its applications to prefix codes and trees often find use in computer science and information theory . The prefix code can contain either finitely many or infinitely many codewords.
Kraft's inequality was published in Kraft (1949) . However, Kraft's paper discusses only prefix codes, and attributes the analysis leading to the inequality to Raymond Redheffer . The result was independently discovered in McMillan (1956) . McMillan proves the result for the general case of uniquely decodable codes, and attributes the version for prefix codes to a spoken observation in 1955 by Joseph Leo Doob .
Kraft's inequality limits the lengths of codewords in a prefix code : if one takes an exponential of the length of each valid codeword, the resulting set of values must look like a probability mass function , that is, it must have total measure less than or equal to one. Kraft's inequality can be thought of in terms of a constrained budget to be spent on codewords, with shorter codewords being more expensive. Among the useful properties following from the inequality are the following statements:
Let each source symbol from the alphabet
be encoded into a uniquely decodable code over an alphabet of size r {\displaystyle r} with codeword lengths
Then
Conversely, for a given set of natural numbers ℓ 1 , ℓ 2 , … , ℓ n {\displaystyle \ell _{1},\ell _{2},\ldots ,\ell _{n}} satisfying the above inequality, there exists a uniquely decodable code over an alphabet of size r {\displaystyle r} with those codeword lengths.
Any binary tree can be viewed as defining a prefix code for the leaves of the tree. Kraft's inequality states that
Here the sum is taken over the leaves of the tree, i.e. the nodes without any children. The depth is the distance to the root node. In the tree to the right, this sum is
First, let us show that the Kraft inequality holds whenever the code for S {\displaystyle S} is a prefix code.
Suppose that ℓ 1 ⩽ ℓ 2 ⩽ ⋯ ⩽ ℓ n {\displaystyle \ell _{1}\leqslant \ell _{2}\leqslant \cdots \leqslant \ell _{n}} . Let A {\displaystyle A} be the full r {\displaystyle r} -ary tree of depth ℓ n {\displaystyle \ell _{n}} (thus, every node of A {\displaystyle A} at level < ℓ n {\displaystyle <\ell _{n}} has r {\displaystyle r} children, while the nodes at level ℓ n {\displaystyle \ell _{n}} are leaves). Every word of length ℓ ⩽ ℓ n {\displaystyle \ell \leqslant \ell _{n}} over an r {\displaystyle r} -ary alphabet corresponds to a node in this tree at depth ℓ {\displaystyle \ell } . The i {\displaystyle i} th word in the prefix code corresponds to a node v i {\displaystyle v_{i}} ; let A i {\displaystyle A_{i}} be the set of all leaf nodes (i.e. of nodes at depth ℓ n {\displaystyle \ell _{n}} ) in the subtree of A {\displaystyle A} rooted at v i {\displaystyle v_{i}} . That subtree being of height ℓ n − ℓ i {\displaystyle \ell _{n}-\ell _{i}} , we have
Since the code is a prefix code, those subtrees cannot share any leaves, which means that
Thus, given that the total number of nodes at depth ℓ n {\displaystyle \ell _{n}} is r ℓ n {\displaystyle r^{\ell _{n}}} , we have
from which the result follows.
Conversely, given any ordered sequence of n {\displaystyle n} natural numbers,
satisfying the Kraft inequality, one can construct a prefix code with codeword lengths equal to each ℓ i {\displaystyle \ell _{i}} by choosing a word of length ℓ i {\displaystyle \ell _{i}} arbitrarily, then ruling out all words of greater length that have it as a prefix. There again, we shall interpret this in terms of leaf nodes of an r {\displaystyle r} -ary tree of depth ℓ n {\displaystyle \ell _{n}} . First choose any node from the full tree at depth ℓ 1 {\displaystyle \ell _{1}} ; it corresponds to the first word of our new code. Since we are building a prefix code, all the descendants of this node (i.e., all words that have this first word as a prefix) become unsuitable for inclusion in the code. We consider the descendants at depth ℓ n {\displaystyle \ell _{n}} (i.e., the leaf nodes among the descendants); there are r ℓ n − ℓ 1 {\displaystyle r^{\ell _{n}-\ell _{1}}} such descendant nodes that are removed from consideration. The next iteration picks a (surviving) node at depth ℓ 2 {\displaystyle \ell _{2}} and removes r ℓ n − ℓ 2 {\displaystyle r^{\ell _{n}-\ell _{2}}} further leaf nodes, and so on. After n {\displaystyle n} iterations, we have removed a total of
nodes. The question is whether we need to remove more leaf nodes than we actually have available — r ℓ n {\displaystyle r^{\ell _{n}}} in all — in the process of building the code. Since the Kraft inequality holds, we have indeed
and thus a prefix code can be built. Note that as the choice of nodes at each step is largely arbitrary, many different suitable prefix codes can be built, in general.
Now we will prove that the Kraft inequality holds whenever S {\displaystyle S} is a uniquely decodable code. (The converse needs not be proven, since we have already proven it for prefix codes, which is a stronger claim.) The proof is by Jack I. Karush. [ 3 ] [ 4 ]
We need only prove it when there are finitely many codewords. If there are infinitely many codewords, then any finite subset of it is also uniquely decodable, so it satisfies the Kraft–McMillan inequality. Taking the limit, we have the inequality for the full code.
Denote C = ∑ i = 1 n r − l i {\displaystyle C=\sum _{i=1}^{n}r^{-l_{i}}} . The idea of the proof is to get an upper bound on C m {\displaystyle C^{m}} for m ∈ N {\displaystyle m\in \mathbb {N} } and show that it can only hold for all m {\displaystyle m} if C ≤ 1 {\displaystyle C\leq 1} . Rewrite C m {\displaystyle C^{m}} as
Consider all m -powers S m {\displaystyle S^{m}} , in the form of words s i 1 s i 2 … s i m {\displaystyle s_{i_{1}}s_{i_{2}}\dots s_{i_{m}}} , where i 1 , i 2 , … , i m {\displaystyle i_{1},i_{2},\dots ,i_{m}} are indices between 1 and n {\displaystyle n} . Note that, since S was assumed to uniquely decodable, s i 1 s i 2 … s i m = s j 1 s j 2 … s j m {\displaystyle s_{i_{1}}s_{i_{2}}\dots s_{i_{m}}=s_{j_{1}}s_{j_{2}}\dots s_{j_{m}}} implies i 1 = j 1 , i 2 = j 2 , … , i m = j m {\displaystyle i_{1}=j_{1},i_{2}=j_{2},\dots ,i_{m}=j_{m}} . This means that each summand corresponds to exactly one word in S m {\displaystyle S^{m}} . This allows us to rewrite the equation to
where q ℓ {\displaystyle q_{\ell }} is the number of codewords in S m {\displaystyle S^{m}} of length ℓ {\displaystyle \ell } and ℓ m a x {\displaystyle \ell _{max}} is the length of the longest codeword in S {\displaystyle S} . For an r {\displaystyle r} -letter alphabet there are only r ℓ {\displaystyle r^{\ell }} possible words of length ℓ {\displaystyle \ell } , so q ℓ ≤ r ℓ {\displaystyle q_{\ell }\leq r^{\ell }} . Using this, we upper bound C m {\displaystyle C^{m}} :
Taking the m {\displaystyle m} -th root, we get
This bound holds for any m ∈ N {\displaystyle m\in \mathbb {N} } . The right side is 1 asymptotically, so ∑ i = 1 n r − l i ≤ 1 {\displaystyle \sum _{i=1}^{n}r^{-l_{i}}\leq 1} must hold (otherwise the inequality would be broken for a large enough m {\displaystyle m} ).
Given a sequence of n {\displaystyle n} natural numbers,
satisfying the Kraft inequality, we can construct a prefix code as follows. Define the i th codeword, C i , to be the first ℓ i {\displaystyle \ell _{i}} digits after the radix point (e.g. decimal point) in the base r representation of
Note that by Kraft's inequality, this sum is never more than 1. Hence the codewords capture the entire value of the sum. Therefore, for j > i , the first ℓ i {\displaystyle \ell _{i}} digits of C j form a larger number than C i , so the code is prefix free.
The following generalization is found in. [ 5 ]
Theorem — If C , D {\textstyle C,D} are uniquely decodable, and every codeword in C {\textstyle C} is a concatenation of codewords in D {\textstyle D} , then ∑ c ∈ C r − | c | ≤ ∑ c ∈ D r − | c | {\displaystyle \sum _{c\in C}r^{-|c|}\leq \sum _{c\in D}r^{-|c|}}
The previous theorem is the special case when D = { a 1 , … , a r } {\displaystyle D=\{a_{1},\dots ,a_{r}\}} .
Let Q C ( x ) {\textstyle Q_{C}(x)} be the generating function for the code. That is, Q C ( x ) := ∑ c ∈ C x | c | {\displaystyle Q_{C}(x):=\sum _{c\in C}x^{|c|}}
By a counting argument, the k {\textstyle k} -th coefficient of Q C n {\textstyle Q_{C}^{n}} is the number of strings of length n {\textstyle n} with code length k {\textstyle k} . That is, Q C n ( x ) = ∑ k ≥ 0 x k # ( strings of length n with C -codes of length k ) {\displaystyle Q_{C}^{n}(x)=\sum _{k\geq 0}x^{k}\#({\text{strings of length }}n{\text{ with }}C{\text{-codes of length }}k)} Similarly, 1 1 − Q C ( x ) = 1 + Q C ( x ) + Q C ( x ) 2 + ⋯ = ∑ k ≥ 0 x k # ( strings with C -codes of length k ) {\displaystyle {\frac {1}{1-Q_{C}(x)}}=1+Q_{C}(x)+Q_{C}(x)^{2}+\cdots =\sum _{k\geq 0}x^{k}\#({\text{strings with }}C{\text{-codes of length }}k)}
Since the code is uniquely decodable, any power of Q C {\textstyle Q_{C}} is absolutely bounded by r | x | + r 2 | x | 2 + ⋯ = r | x | 1 − r | x | {\textstyle r|x|+r^{2}|x|^{2}+\cdots ={\frac {r|x|}{1-r|x|}}} , so each of Q C , Q C 2 , … {\textstyle Q_{C},Q_{C}^{2},\dots } and 1 1 − Q C ( x ) {\textstyle {\frac {1}{1-Q_{C}(x)}}} is analytic in the disk | x | < 1 / r {\textstyle |x|<1/r} .
We claim that for all x ∈ ( 0 , 1 / r ) {\textstyle x\in (0,1/r)} , Q C n ≤ Q D n + Q D n + 1 + ⋯ {\displaystyle Q_{C}^{n}\leq Q_{D}^{n}+Q_{D}^{n+1}+\cdots }
The left side is ∑ k ≥ 0 x k # ( strings of length n with C -codes of length k ) {\displaystyle \sum _{k\geq 0}x^{k}\#({\text{strings of length }}n{\text{ with }}C{\text{-codes of length }}k)} and the right side is
∑ k ≥ 0 x k # ( strings of length ≥ n with D -codes of length k ) {\displaystyle \sum _{k\geq 0}x^{k}\#({\text{strings of length}}\geq n{\text{ with }}D{\text{-codes of length }}k)}
Now, since every codeword in C {\textstyle C} is a concatenation of codewords in D {\textstyle D} , and D {\textstyle D} is uniquely decodable, each string of length n {\textstyle n} with C {\textstyle C} -code c 1 … c n {\textstyle c_{1}\dots c_{n}} of length k {\textstyle k} corresponds to a unique string s c 1 … s c n {\textstyle s_{c_{1}}\dots s_{c_{n}}} whose D {\textstyle D} -code is c 1 … c n {\textstyle c_{1}\dots c_{n}} . The string has length at least n {\textstyle n} .
Therefore, the coefficients on the left are less or equal to the coefficients on the right.
Thus, for all x ∈ ( 0 , 1 / r ) {\textstyle x\in (0,1/r)} , and all n = 1 , 2 , … {\textstyle n=1,2,\dots } , we have Q C ≤ Q D ( 1 − Q D ) 1 / n {\displaystyle Q_{C}\leq {\frac {Q_{D}}{(1-Q_{D})^{1/n}}}} Taking n → ∞ {\textstyle n\to \infty } limit, we have Q C ( x ) ≤ Q D ( x ) {\textstyle Q_{C}(x)\leq Q_{D}(x)} for all x ∈ ( 0 , 1 / r ) {\textstyle x\in (0,1/r)} .
Since Q C ( 1 / r ) {\textstyle Q_{C}(1/r)} and Q D ( 1 / r ) {\textstyle Q_{D}(1/r)} both converge, we have Q C ( 1 / r ) ≤ Q D ( 1 / r ) {\textstyle Q_{C}(1/r)\leq Q_{D}(1/r)} by taking the limit and applying Abel's theorem .
There is a generalization to quantum code . [ 6 ] | https://en.wikipedia.org/wiki/Kraft–McMillan_inequality |
Krakatoa is a modular explosive device used for explosive ordnance disposal (EOD) or demolitions developed by the British company Alford Technologies . The device is designed to fire a number of different projectiles, operates both in air and underwater, and can be used in a vertical or horizontal orientation. [ 1 ]
The device was featured during the second season of Discovery Channel 's television series Future Weapons , in which it was shown penetrating an inch of steel plate at 25 yards. The device's casing is made of plastic which is packed with plastique ( C4 ) and capped with an inverted copper cone.
The device itself is "no bigger than a standard can of coke."
It is named after the 1883 eruption of Krakatoa , which resulted in the loudest sound ever heard and was the second deadliest volcanic eruption in recorded history .
This explosive device was designed to play a role in covert operations, as a small but extremely powerful device that can disable tanks, vehicles, or even a warship. The device can be used underwater, at high altitudes, and in snow, hail, sleet, or any form of weather.
The device has copper cone with a driving charge of very fast high explosive behind it. The metal cone is the difference between a regular C4 'slappack' or hollow charge and a HEAT device.
When detonated, the copper cone is inverted into a narrow stream of copper and fired at extremely high velocity at the target, this can pierce certain thicknesses of steel armor or concrete.
The weapon looks like a small circular tube, no wider than a tea-cup plate and no taller than a soft-drink can. On one end, it is covered with plastic, but the other end houses the copper cone. Inside the actual device is C4, and this is used to form the copper into an explosively formed penetrator and turn it into a weapon. So far unused in war, it is tested standing on a small pair of bipod legs, and the copper cone is faced toward the target. There is a 'stand off distance' to allow correct formation of the penetrator stream. This is the reason for its legs.
For a special ops team, the weapon will already have the C4 placed inside, about two pounds of it, and being a versatile explosive, the weapon will not set off in any condition. One uses it by placing it on the stand, moving far away from it and then pressing the detonator. The whole weapon itself explodes, so it is not a handheld weapon or a device to stand near when operating or firing. | https://en.wikipedia.org/wiki/Krakatoa_(explosive) |
Kramers' law is a formula for the spectral distribution of X-rays produced by an electron hitting a solid target. The formula concerns only bremsstrahlung radiation, not the element specific characteristic radiation . It is named after its discoverer, the Dutch physicist Hendrik Anthony Kramers . [ 1 ]
The formula for Kramers' law is usually given as the distribution of intensity (photon count) I {\displaystyle I} against the wavelength λ {\displaystyle \lambda } of the emitted radiation: [ 2 ] I ( λ ) d λ = K ( λ λ min − 1 ) 1 λ 2 d λ {\displaystyle I(\lambda )d\lambda =K\left({\frac {\lambda }{\lambda _{\text{min}}}}-1\right){\frac {1}{\lambda ^{2}}}d\lambda }
The constant K is proportional to the atomic number of the target element, and λ min {\displaystyle \lambda _{\text{min}}} is the minimum wavelength given by the Duane–Hunt law . The maximum intensity is K 4 λ min 2 {\displaystyle {\frac {K}{4\lambda _{\text{min}}^{2}}}} at 2 λ min {\displaystyle 2\lambda _{\text{min}}} .
The intensity described above is a particle flux and not an energy flux as can be seen by the fact that the integral over values from λ m i n {\displaystyle \lambda _{min}} to ∞ {\displaystyle \infty } is infinite. However, the
integral of the energy flux is finite.
To obtain a simple expression for the energy flux, first change variables from λ {\displaystyle \lambda } (the wavelength) to ω {\displaystyle \omega } (the angular frequency) using λ = 2 π c / ω {\displaystyle \lambda =2\pi c/\omega } and also using I ~ ( ω ) = I ( λ ) − d λ d ω {\displaystyle {\tilde {I}}(\omega )=I(\lambda ){\frac {-d\lambda }{d\omega }}} . Now I ~ ( ω ) {\displaystyle {\tilde {I}}(\omega )} is that quantity which is integrated over ω {\displaystyle \omega } from 0 to ω max {\displaystyle \omega _{\text{max}}} to get the total number (still infinite) of photons, where ω max = 2 π c / λ min {\displaystyle \omega _{\text{max}}=2\pi c/\lambda _{\text{min}}} : I ~ ( ω ) = K 2 π c ( ω max ω − 1 ) {\displaystyle {\tilde {I}}(\omega )={\frac {K}{2\pi c}}\left({\frac {\omega _{\text{max}}}{\omega }}-1\right)}
The energy flux, which we will call ψ ( ω ) {\displaystyle \psi (\omega )} (but which may also be referred to as the "intensity" in conflict with the above name of I ( λ ) {\displaystyle I(\lambda )} ) is obtained by multiplying the above I ~ {\displaystyle {\tilde {I}}} by the energy ℏ ω {\displaystyle \hbar \omega } : ψ ( ω ) = K 2 π c ( ℏ ω max − ℏ ω ) {\displaystyle \psi (\omega )={\frac {K}{2\pi c}}(\hbar \omega _{\text{max}}-\hbar \omega )} for ω ≤ ω max {\displaystyle \omega \leq \omega _{\text{max}}} ψ ( ω ) = 0 {\displaystyle \psi (\omega )=0} for ω ≥ ω max {\displaystyle \omega \geq \omega _{\text{max}}} .
It is a linear function that is zero at the maximum energy ℏ ω max {\displaystyle \hbar \omega _{\text{max}}} .
This spectroscopy -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Kramers'_law |
The Kramers–Kronig relations , sometimes abbreviated as KK relations , are bidirectional mathematical relations, connecting the real and imaginary parts of any complex function that is analytic in the upper half-plane . The relations are often used to compute the real part from the imaginary part (or vice versa) of response functions in physical systems , because for stable systems, causality implies the condition of analyticity , and conversely, analyticity implies causality of the corresponding stable physical system. [ 1 ] The relation is named in honor of Ralph Kronig and Hans Kramers . [ 2 ] [ 3 ] In mathematics , these relations are known by the names Sokhotski–Plemelj theorem and Hilbert transform .
Let χ ( ω ) = χ 1 ( ω ) + i χ 2 ( ω ) {\displaystyle \chi (\omega )=\chi _{1}(\omega )+i\chi _{2}(\omega )} be a complex function of the complex variable ω {\displaystyle \omega } , where χ 1 ( ω ) {\displaystyle \chi _{1}(\omega )} and χ 2 ( ω ) {\displaystyle \chi _{2}(\omega )} are real . Suppose this function is analytic in the closed upper half-plane of ω {\displaystyle \omega } and tends to 0 {\displaystyle 0} as | ω | → ∞ {\displaystyle |\omega |\to \infty } . The Kramers–Kronig relations are given by χ 1 ( ω ) = 1 π P ∫ − ∞ ∞ χ 2 ( ω ′ ) ω ′ − ω d ω ′ {\displaystyle \chi _{1}(\omega )={\frac {1}{\pi }}{\mathcal {P}}\!\!\int _{-\infty }^{\infty }{\frac {\chi _{2}(\omega ')}{\omega '-\omega }}\,d\omega '} and χ 2 ( ω ) = − 1 π P ∫ − ∞ ∞ χ 1 ( ω ′ ) ω ′ − ω d ω ′ , {\displaystyle \chi _{2}(\omega )=-{\frac {1}{\pi }}{\mathcal {P}}\!\!\int _{-\infty }^{\infty }{\frac {\chi _{1}(\omega ')}{\omega '-\omega }}\,d\omega ',} where ω {\displaystyle \omega } is real and where P {\displaystyle {\mathcal {P}}} denotes the Cauchy principal value . The real and imaginary parts of such a function are not independent, allowing the full function to be reconstructed given just one of its parts.
The proof begins with an application of Cauchy's residue theorem for complex integration. Given any analytic function χ {\displaystyle \chi } in the closed upper half-plane, the function ω ′ ↦ χ ( ω ′ ) / ( ω ′ − ω ) {\displaystyle \omega '\mapsto \chi (\omega ')/(\omega '-\omega )} , where ω {\displaystyle \omega } is real, is analytic in the (open) upper half-plane. The residue theorem consequently states that ∮ χ ( ω ′ ) ω ′ − ω d ω ′ = 0 {\displaystyle \oint {\frac {\chi (\omega ')}{\omega '-\omega }}\,d\omega '=0} for any closed contour within this region. When the contour is chosen to trace the real axis, a hump over the pole at ω ′ = ω {\displaystyle \omega '=\omega } , and a large semicircle in the upper half-plane. This follows decomposition of the integral into its contributions along each of these three contour segments and pass them to limits. The length of the semicircular segment increases proportionally to | ω ′ | {\displaystyle |\omega '|} , but the integral over it vanishes in the limit because χ ( ω ′ ) ω ′ − ω {\displaystyle {\frac {\chi (\omega ')}{\omega '-\omega }}} vanishes faster than 1 / | ω ′ | {\displaystyle 1/|\omega '|} . We are left with the segments along the real axis and the half-circle around the pole. We pass the size of the half-circle to zero and obtain 0 = ∮ χ ( ω ′ ) ω ′ − ω d ω ′ = P ∫ − ∞ ∞ χ ( ω ′ ) ω ′ − ω d ω ′ − i π χ ( ω ) . {\displaystyle 0=\oint {\frac {\chi (\omega ')}{\omega '-\omega }}\,d\omega '={\mathcal {P}}\!\!\int _{-\infty }^{\infty }{\frac {\chi (\omega ')}{\omega '-\omega }}\,d\omega '-i\pi \chi (\omega ).}
The second term in the last expression is obtained using the theory of residues, [ 4 ] more specifically, the Sokhotski–Plemelj theorem . Rearranging, we arrive at the compact form of the Kramers–Kronig relations: χ ( ω ) = 1 i π P ∫ − ∞ ∞ χ ( ω ′ ) ω ′ − ω d ω ′ . {\displaystyle \chi (\omega )={\frac {1}{i\pi }}{\mathcal {P}}\!\!\int _{-\infty }^{\infty }{\frac {\chi (\omega ')}{\omega '-\omega }}\,d\omega '.}
The single i {\displaystyle i} in the denominator effectuates the connection between the real and imaginary components. Finally, split χ ( ω ) {\displaystyle \chi (\omega )} and the equation into their real and imaginary parts to obtain the forms quoted above.
The Kramers–Kronig formalism can be applied to response functions . In certain linear physical systems, or in engineering fields such as signal processing , the response function χ ( t − t ′ ) {\displaystyle \chi (t-t')} describes how some time-dependent property P ( t ) {\displaystyle P(t)} of a physical system responds to an impulse force F ( t ′ ) {\displaystyle F(t')} at time t ′ . {\displaystyle t'.} For example, P ( t ) {\displaystyle P(t)} could be the angle of a pendulum and F ( t ) {\displaystyle F(t)} the applied force of a motor driving the pendulum motion. The response χ ( t − t ′ ) {\displaystyle \chi (t-t')} must be zero for t < t ′ {\displaystyle t<t'} since a system cannot respond to a force before it is applied. It can be shown (for instance, by invoking Titchmarsh's theorem ) that this causality condition implies that the Fourier transform χ ( ω ) {\displaystyle \chi (\omega )} of χ ( t ) {\displaystyle \chi (t)} is analytic in the upper half plane. [ 5 ] Additionally, if the system is subjected to an oscillatory force with a frequency much higher than its highest resonant frequency, there will be almost no time for the system to respond before the forcing has switched direction, and so the frequency response χ ( ω ) {\displaystyle \chi (\omega )} will converge to zero as ω {\displaystyle \omega } becomes very large. From these physical considerations, it results that χ ( ω ) {\displaystyle \chi (\omega )} will typically satisfy the conditions needed for the Kramers–Kronig relations.
The imaginary part of a response function describes how a system dissipates energy , since it is in phase with the driving force . [ citation needed ] The Kramers–Kronig relations imply that observing the dissipative response of a system is sufficient to determine its out of phase (reactive) response, and vice versa.
The integrals run from − ∞ {\displaystyle -\infty } to ∞ {\displaystyle \infty } , implying we know the response at negative frequencies. Fortunately, in most physical systems, the positive frequency-response determines the negative-frequency response because χ ( ω ) {\displaystyle \chi (\omega )} is the Fourier transform of a real-valued response χ ( t ) {\displaystyle \chi (t)} . We will make this assumption henceforth.
As a consequence, χ ( − ω ) = χ ∗ ( ω ) {\displaystyle \chi (-\omega )=\chi ^{*}(\omega )} . This means χ 1 ( ω ) {\displaystyle \chi _{1}(\omega )} is an even function of frequency and χ 2 ( ω ) {\displaystyle \chi _{2}(\omega )} is odd .
Using these properties, we can collapse the integration ranges to [ 0 , ∞ ) {\displaystyle [0,\infty )} . Consider the first relation, which gives the real part χ 1 ( ω ) {\displaystyle \chi _{1}(\omega )} . We transform the integral into one of definite parity by multiplying the numerator and denominator of the integrand by ω ′ + ω {\displaystyle \omega '+\omega } and separating: χ 1 ( ω ) = 1 π P ∫ − ∞ ∞ ω ′ χ 2 ( ω ′ ) ω ′ 2 − ω 2 d ω ′ + ω π P ∫ − ∞ ∞ χ 2 ( ω ′ ) ω ′ 2 − ω 2 d ω ′ . {\displaystyle \chi _{1}(\omega )={1 \over \pi }{\mathcal {P}}\!\!\int _{-\infty }^{\infty }{\omega '\chi _{2}(\omega ') \over \omega '^{2}-\omega ^{2}}\,d\omega '+{\omega \over \pi }{\mathcal {P}}\!\!\int _{-\infty }^{\infty }{\chi _{2}(\omega ') \over \omega '^{2}-\omega ^{2}}\,d\omega '.}
Since χ 2 ( ω ) {\displaystyle \chi _{2}(\omega )} is odd, the second integral vanishes, and we are left with χ 1 ( ω ) = 2 π P ∫ 0 ∞ ω ′ χ 2 ( ω ′ ) ω ′ 2 − ω 2 d ω ′ . {\displaystyle \chi _{1}(\omega )={2 \over \pi }{\mathcal {P}}\!\!\int _{0}^{\infty }{\omega '\chi _{2}(\omega ') \over \omega '^{2}-\omega ^{2}}\,d\omega '.}
The same derivation for the imaginary part gives χ 2 ( ω ) = − 2 π P ∫ 0 ∞ ω χ 1 ( ω ′ ) ω ′ 2 − ω 2 d ω ′ = − 2 ω π P ∫ 0 ∞ χ 1 ( ω ′ ) ω ′ 2 − ω 2 d ω ′ . {\displaystyle \chi _{2}(\omega )=-{2 \over \pi }{\mathcal {P}}\!\!\int _{0}^{\infty }{\omega \chi _{1}(\omega ') \over \omega '^{2}-\omega ^{2}}\,d\omega '=-{2\omega \over \pi }{\mathcal {P}}\!\!\int _{0}^{\infty }{\chi _{1}(\omega ') \over \omega '^{2}-\omega ^{2}}\,d\omega '.}
These are the Kramers–Kronig relations in a form that is useful for physically realistic response functions.
Hu [ 6 ] and Hall and Heck [ 7 ] give a related and possibly more intuitive proof that avoids contour integration. It is based on the facts that:
Combining the formulas provided by these facts yields the Kramers–Kronig relations. This proof covers slightly different ground from the previous one in that it relates the real and imaginary parts in the frequency domain of any function that is causal in the time domain, offering an approach somewhat different from the condition of analyticity in the upper half plane of the frequency domain.
An article with an informal, pictorial version of this proof is also available. [ 8 ]
The conventional form of Kramers–Kronig above relates the real and imaginary part of a complex response function. A related goal is to find a relation between the magnitude and phase of a complex response function.
In general, unfortunately, the phase cannot be uniquely predicted from the magnitude. [ 9 ] A simple example of this is a pure time delay of time T , which has amplitude 1 at any frequency regardless of T , but has a phase dependent on T (specifically, phase = 2 π × T × frequency).
There is, however, a unique amplitude-vs-phase relation in the special case of a minimum phase system, [ 9 ] sometimes called the Bode gain–phase relation . The terms Bayard–Bode relations and Bayard–Bode theorem , after the works of Marcel Bayard (1936) and Hendrik Wade Bode (1945) are also used for either the Kramers–Kronig relations in general or the amplitude–phase relation in particular, particularly in the fields of telecommunication and control theory . [ 10 ] [ 11 ]
The Kramers–Kronig relations are used to relate the real and imaginary portions for the complex refractive index n ~ = n + i κ {\displaystyle {\tilde {n}}=n+i\kappa } of a medium, where κ {\displaystyle \kappa } is the extinction coefficient . [ 12 ] Hence, in effect, this also applies for the complex relative permittivity and electric susceptibility . [ 13 ]
The Sellmeier equation is directly connected to the Kramer-Kronig relations, and is used to approximate real and complex refractive index of materials far away from any resonances. [ 14 ] [ 15 ]
In optical rotation , the Kramers–Kronig relations establish a connection between optical rotary dispersion and circular dichroism .
Kramers–Kronig relations enable exact solutions of nontrivial scattering problems, which find applications in magneto-optics. [ 16 ]
In ellipsometry , Kramer-Kronig relations are applied to verify the measured values for the real and complex parts of the refractive index of thin films. [ 17 ]
In electron energy loss spectroscopy , Kramers–Kronig analysis allows one to calculate the energy dependence of both real and imaginary parts of a specimen's light optical permittivity , together with other optical properties such as the absorption coefficient and reflectivity . [ 18 ]
In short, by measuring the number of high energy (e.g. 200 keV) electrons which lose a given amount of energy in traversing a very thin specimen (single scattering approximation), one can calculate the imaginary part of permittivity at that energy. Using this data with Kramers–Kronig analysis, one can calculate the real part of permittivity (as a function of energy) as well.
This measurement is made with electrons, rather than with light, and can be done with very high spatial resolution. One might thereby, for example, look for ultraviolet (UV) absorption bands in a laboratory specimen of interstellar dust less than a 100 nm across, i.e. too small for UV spectroscopy. Although electron spectroscopy has poorer energy resolution than light spectroscopy , data on properties in visible, ultraviolet and soft x-ray spectral ranges may be recorded in the same experiment.
In angle resolved photoemission spectroscopy the Kramers–Kronig relations can be used to link the real and imaginary parts of the electrons self-energy . This is characteristic of the many body interaction the electron experiences in the material. Notable examples are in the high temperature superconductors , where kinks corresponding to the real part of the self-energy are observed in the band dispersion and changes in the MDC width are also observed corresponding to the imaginary part of the self-energy. [ 19 ]
The Kramers–Kronig relations are also used under the name "integral dispersion relations" with reference to hadronic scattering. [ 20 ] In this case, the function is the scattering amplitude. Through the use of the optical theorem the imaginary part of the scattering amplitude is then related to the total cross section , which is a physically measurable quantity.
Similarly to Hadronic scattering, the Kramers–Kronig relations are employed in high energy electron scattering . In particular, they enter the derivation of the Gerasimov–Drell–Hearn sum rule . [ 21 ]
For seismic wave propagation, the Kramer–Kronig relation helps to find right form for the quality factor in an attenuating medium. [ 22 ]
The Kramers-Kronig test is used in battery and fuel cell applications ( dielectric spectroscopy ) to test for linearity , causality and stationarity . Since, it is not possible in practice to obtain data in the whole frequency range, as the Kramers-Kronig formula requires, approximations are necessarily made.
At high frequencies (> 1 MHz) it is usually safe to assume, that the impedance is dominated by ohmic resistance of the electrolyte, although inductance artefacts are often observed.
At low frequencies, the KK test can be used to verify whether experimental data are reliable. In battery practice, data obtained with experiments of duration less than one minute usually fail the test for frequencies below 10 Hz. Therefore, care should be exercised, when interpreting such data. [ 23 ]
In electrochemistry practice, due to the finite frequency range of experimental data, Z-HIT relation is used instead of Kramers-Kronig relations. Unlike Kramers-Kronig (which is written for an infinite frequency range), Z-HIT integration requires only a finite frequency range. Furthermore, Z-HIT is more robust with respect to error in the Re and Im of impedance, since its accuracy depends mostly on the accuracy of the phase data. | https://en.wikipedia.org/wiki/Kramers–Kronig_relations |
In stochastic processes , Kramers–Moyal expansion refers to a Taylor series expansion of the master equation , named after Hans Kramers and José Enrique Moyal . [ 1 ] [ 2 ] [ 3 ] In many textbooks, the expansion is used only to derive the Fokker–Planck equation , and never used again. In general, continuous stochastic processes are essentially all Markovian, and so Fokker–Planck equations are sufficient for studying them. The higher-order Kramers–Moyal expansion only come into play when the process is jumpy . This usually means it is a Poisson-like process . [ 4 ] [ 5 ]
For a real stochastic process, one can compute its central moment functions from experimental data on the process, from which one can then compute its Kramers–Moyal coefficients, and thus empirically measure its Kolmogorov forward and backward equations. This is implemented as a python package [ 6 ]
Start with the integro-differential master equation
where p ( x , t | x 0 , t 0 ) {\displaystyle p(x,t|x_{0},t_{0})} is the transition probability function , and p ( x , t ) {\displaystyle p(x,t)} is the probability density at time t {\displaystyle t} . The Kramers–Moyal expansion transforms the above to an infinite order partial differential equation [ 7 ] [ 8 ] [ 9 ]
and also ∂ t p ( x , t | x 0 , t 0 ) = ∑ n = 1 ∞ ( − ∂ x ) n [ D n ( x , t ) p ( x , t | x 0 , t 0 ) ] {\displaystyle \partial _{t}p(x,t|x_{0},t_{0})=\sum _{n=1}^{\infty }(-\partial _{x})^{n}[D_{n}(x,t)p(x,t|x_{0},t_{0})]}
where D n ( x , t ) {\displaystyle D_{n}(x,t)} are the Kramers–Moyal coefficients, defined by D n ( x , t ) = 1 n ! lim τ → 0 1 τ μ n ( t | x , t − τ ) {\displaystyle D_{n}(x,t)={\frac {1}{n!}}\lim _{\tau \to 0}{\frac {1}{\tau }}\mu _{n}(t|x,t-\tau )} and μ n {\displaystyle \mu _{n}} are the central moment functions, defined by
The Fokker–Planck equation is obtained by keeping only the first two terms of the series in which D 1 {\displaystyle D_{1}} is the drift and D 2 {\displaystyle D_{2}} is the diffusion coefficient. [ 10 ]
Also, the moments, assuming they exist, evolves as [ 11 ]
∂ ∂ t ⟨ x n ⟩ = ∑ k = 1 n n ! ( n − k ) ! ⟨ x n − k D ( k ) ( x , t ) ⟩ {\displaystyle {\frac {\partial }{\partial t}}\left\langle x^{n}\right\rangle =\sum _{k=1}^{n}{\frac {n!}{(n-k)!}}\left\langle x^{n-k}D^{(k)}(x,t)\right\rangle } where angled brackets mean taking the expectation: ⟨ f ⟩ = ∫ f ( x ) p ( x , t ) d x {\displaystyle \left\langle f\right\rangle =\int f(x)p(x,t)dx} .
The above version is the one-dimensional version. It generalizes to n-dimensions. (Section 4.7 [ 9 ] )
In usual probability, where the probability density does not change, the moments of a probability density function determines the probability density itself by a Fourier transform (details may be found at the characteristic function page): p ( x ) = 1 2 π ∫ e − i k x p ~ ( k ) d k = ∑ n = 0 ∞ ( − 1 ) n n ! δ ( n ) ( x ) μ n {\displaystyle p(x)={\frac {1}{2\pi }}\int e^{-ikx}{\tilde {p}}(k)dk=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{n!}}\delta ^{(n)}(x)\mu _{n}} p ~ ( k ) = ∫ e i k x p ( x ) d x = ∑ n = 0 ∞ ( i k ) n n ! μ n {\displaystyle {\tilde {p}}(k)=\int e^{ikx}p(x)dx=\sum _{n=0}^{\infty }{\frac {(ik)^{n}}{n!}}\mu _{n}} Similarly, p ( x , t | x 0 , t 0 ) = ∑ n = 0 ∞ ( − 1 ) n n ! δ ( n ) ( x − x 0 ) μ n ( t | x 0 , t 0 ) {\displaystyle p(x,t|x_{0},t_{0})=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{n!}}\delta ^{(n)}(x-x_{0})\mu _{n}(t|x_{0},t_{0})} Now we need to integrate away the Dirac delta function. Fixing a small τ > 0 {\displaystyle \tau >0} , we have by the Chapman-Kolmogorov equation , p ( x , t ) = ∫ p ( x , t | x ′ , t − τ ) p ( x ′ , t − τ ) d x ′ = ∑ n = 0 ∞ ( − 1 ) n n ! ∫ p ( x ′ , t − τ ) δ ( n ) ( x − x ′ ) μ n ( t | x ′ , t − τ ) d x ′ = ∑ n = 0 ∞ ( − 1 ) n n ! ∂ x n ( p ( x , t − τ ) μ n ( t | x , t − τ ) ) {\displaystyle {\begin{aligned}p(x,t)&=\int p(x,t|x',t-\tau )p(x',t-\tau )dx'\\&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{n!}}\int p(x',t-\tau )\delta ^{(n)}(x-x')\mu _{n}(t|x',t-\tau )dx'\\&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{n!}}\partial _{x}^{n}(p(x,t-\tau )\mu _{n}(t|x,t-\tau ))\end{aligned}}} The n = 0 {\displaystyle n=0} term is just p ( x , t − τ ) {\displaystyle p(x,t-\tau )} , so taking derivative with respect to time, ∂ t p ( x , t ) = lim τ → 0 + 1 τ ∑ n = 1 ∞ ( − 1 ) n n ! ∂ x n ( p ( x , t − τ ) μ n ( t | x , t − τ ) ) = ∑ n = 1 ∞ ( − ∂ x ) n ( p ( x , t ) D n ( x , t ) ) {\displaystyle \partial _{t}p(x,t)=\lim _{\tau \to 0^{+}}{\frac {1}{\tau }}\sum _{n=1}^{\infty }{\frac {(-1)^{n}}{n!}}\partial _{x}^{n}(p(x,t-\tau )\mu _{n}(t|x,t-\tau ))=\sum _{n=1}^{\infty }(-\partial _{x})^{n}(p(x,t)D_{n}(x,t))}
The same computation with p ( x , t | x 0 , t 0 ) {\displaystyle p(x,t|x_{0},t_{0})} gives the other equation.
The equation can be recast into a linear operator form, using the idea of infinitesimal generator . Define the linear operator A f := ∑ n = 1 ∞ ( − ∂ x ) n [ D n ( x , t ) f ( x , t ) ] {\displaystyle {\mathcal {A}}f:=\sum _{n=1}^{\infty }(-\partial _{x})^{n}[D_{n}(x,t)f(x,t)]} then the equation above states ∂ t p ( x , t ) = A p ( x , t ) ∂ t p ( x , t | x 0 , t 0 ) = A p ( x , t | x 0 , t 0 ) {\displaystyle {\begin{aligned}\partial _{t}p(x,t)&={\mathcal {A}}p(x,t)\\\partial _{t}p(x,t|x_{0},t_{0})&={\mathcal {A}}p(x,t|x_{0},t_{0})\end{aligned}}} In this form, the equations are precisely in the form of a general Kolmogorov forward equation . The backward equation then states that ∂ t p ( x 1 , t 1 | x , t ) = − A † p ( x 1 , t 1 | x , t ) {\displaystyle \partial _{t}p(x_{1},t_{1}|x,t)=-{\mathcal {A}}^{\dagger }p(x_{1},t_{1}|x,t)} where A † f := ∑ n = 1 ∞ D n ( x , t ) ∂ x n [ f ( x , t ) ] {\displaystyle {\mathcal {A}}^{\dagger }f:=\sum _{n=1}^{\infty }D_{n}(x,t)\partial _{x}^{n}[f(x,t)]} is the Hermitian adjoint of A {\displaystyle {\mathcal {A}}} .
By definition, D n ( x , t ) = 1 n ! lim τ → 0 1 τ μ n ( t | x , t − τ ) {\displaystyle D_{n}(x,t)={\frac {1}{n!}}\lim _{\tau \to 0}{\frac {1}{\tau }}\mu _{n}(t|x,t-\tau )} This definition works because μ n ( t | x , t ) = 0 {\displaystyle \mu _{n}(t|x,t)=0} , as those are the central moments of the Dirac delta function. Since the even central moments are nonnegative, we have D 2 n ≥ 0 {\displaystyle D_{2n}\geq 0} for all n ≥ 1 {\displaystyle n\geq 1} . When the stochastic process is the Markov process d X = b d t + σ d W t {\displaystyle dX=bdt+\sigma dW_{t}} , we can directly solve for p ( x , t | x , t − τ ) {\displaystyle p(x,t|x,t-\tau )} as approximated by a normal distribution with mean x + b ( x ) τ {\displaystyle x+b(x)\tau } and variance σ 2 τ {\displaystyle \sigma ^{2}\tau } . This then allows us to compute the central moments, and so D 1 = b , D 2 = 1 2 σ 2 , D 3 = D 4 = ⋯ = 0 {\displaystyle D_{1}=b,\quad D_{2}={\frac {1}{2}}\sigma ^{2},\quad D_{3}=D_{4}=\cdots =0} This then gives us the 1-dimensional Fokker–Planck equation: ∂ t p = − ∂ x ( b p ) + 1 2 ∂ x 2 ( σ 2 p ) {\displaystyle \partial _{t}p=-\partial _{x}(bp)+{\frac {1}{2}}\partial _{x}^{2}(\sigma ^{2}p)}
Pawula theorem states that either the sequence D 1 , D 2 , D 3 , . . . {\displaystyle D_{1},D_{2},D_{3},...} becomes zero at the third term, or all its even terms are positive. [ 12 ] [ 13 ]
By Cauchy–Schwarz inequality , the central moment functions satisfy μ n + m 2 ≤ μ 2 n μ 2 m {\displaystyle \mu _{n+m}^{2}\leq \mu _{2n}\mu _{2m}} . So, taking the limit, we have D n + m 2 ≤ ( 2 n ) ! ( 2 m ) ! ( n + m ) ! 2 D 2 n D 2 m {\displaystyle D_{n+m}^{2}\leq {\frac {(2n)!(2m)!}{(n+m)!^{2}}}D_{2n}D_{2m}} . If some D 2 + n ≠ 0 {\displaystyle D_{2+n}\neq 0} for some n ≥ 1 {\displaystyle n\geq 1} , then D 2 D 2 + 2 n > 0 {\displaystyle D_{2}D_{2+2n}>0} . In particular, D 2 + n , D 2 + 2 n , D 2 + 4 n , . . . > 0 {\displaystyle D_{2+n},D_{2+2n},D_{2+4n},...>0} . So the existence of any nonzero coefficient of order ≥ 3 {\displaystyle \geq 3} implies the existence of nonzero coefficients of arbitrarily large order. Also, if D n ≠ 0 {\displaystyle D_{n}\neq 0} , then D 2 D 2 n − 2 > 0 , D 4 D 2 n − 4 > 0 , . . . {\displaystyle D_{2}D_{2n-2}>0,D_{4}D_{2n-4}>0,...} . So the existence of any nonzero coefficient of order n {\displaystyle n} implies all coefficients of order 2 , 4 , . . . , 2 n − 2 {\displaystyle 2,4,...,2n-2} are positive.
Let the operator A m {\displaystyle {\mathcal {A}}_{m}} be defined such A m f := ∑ n = 1 m ( − ∂ x ) n [ D n ( x , t ) f ( x , t ) ] {\displaystyle {\mathcal {A}}_{m}f:=\sum _{n=1}^{m}(-\partial _{x})^{n}[D_{n}(x,t)f(x,t)]} . The probability density evolves by ∂ t ρ ≈ A m ρ {\displaystyle \partial _{t}\rho \approx {\mathcal {A}}_{m}\rho } . Different order of m {\displaystyle m} gives different level of approximation.
Pawula theorem means that if truncating to the second term is not exact, that is, A 2 ≠ A {\displaystyle {\mathcal {A}}_{2}\neq {\mathcal {A}}} , then truncating to any term is still not exact. Usually, this means that for any truncation A m {\displaystyle {\mathcal {A}}_{m}} , there exists a probability density function ρ {\displaystyle \rho } that can become negative during its evolution ∂ t ρ ≈ A m ρ {\displaystyle \partial _{t}\rho \approx {\mathcal {A}}_{m}\rho } (and thus fail to be a probability density function). However, this doesn't mean that Kramers-Moyal expansions truncated at other choices of m {\displaystyle m} is useless. Though the solution must have negative values at least for sufficiently small times, the resulting approximation probability density may still be better than the m = 2 {\displaystyle m=2} approximation. | https://en.wikipedia.org/wiki/Kramers–Moyal_expansion |
The Kramers–Wannier duality is a symmetry in statistical physics . It relates the free energy of a two-dimensional square-lattice Ising model at a low temperature to that of another Ising model at a high temperature. It was discovered by Hendrik Kramers and Gregory Wannier in 1941. [ 1 ] With the aid of this duality Kramers and Wannier found the exact location of the critical point for the Ising model on the square lattice.
Similar dualities establish relations between free energies of other statistical models. For instance, in 3 dimensions the Ising model is dual to an Ising gauge model.
The 2-dimensional Ising model exists on a lattice, which is a collection of squares in a chessboard pattern. With the finite lattice, the edges can be connected to form a torus. In theories of this kind, one constructs an involutive transform . For instance, Lars Onsager suggested that the Star-Triangle transformation could be used for the triangular lattice. [ 2 ] Now the dual of the discrete torus is itself . Moreover, the dual of a highly disordered system (high temperature) is a well-ordered system (low temperature). This is because the Fourier transform takes a high bandwidth signal (more standard deviation ) to a low one (less standard deviation). So one has essentially the same theory with an inverse temperature.
When one raises the temperature in one theory, one lowers the temperature in the other. If there is only one phase transition , it will be at the point at which they cross, at which the temperatures are equal. Because the 2D Ising model goes from a disordered state to an ordered state, there is a near one-to-one mapping between the disordered and ordered phases.
The theory has been generalized, and is now blended with many other ideas. For instance, the square lattice is replaced by a circle, [ 3 ] random lattice, [ 4 ] nonhomogeneous torus, [ 5 ] triangular lattice, [ 6 ] labyrinth, [ 7 ] lattices with twisted boundaries, [ 8 ] chiral Potts model, [ 9 ] and many others.
One of the consequences of Kramers–Wannier duality is an exact correspondence in the spectrum of excitations on each side of the critical point. This was recently demonstrated via THz spectroscopy in Kitaev chains . [ 10 ]
We define first the variables. In the two-dimensional square lattice Ising model the number of horizontal and vertical links are taken to be equal. The couplings J , J ′ {\displaystyle J,J'} of the spins σ i {\displaystyle \sigma _{i}} in the two directions are different, and one sets K ∗ = β J {\displaystyle K^{*}=\beta J} and L ∗ = β J ′ {\displaystyle L^{*}=\beta J'} with β = 1 / k T {\displaystyle \beta =1/kT} .
The low temperature expansion of the N {\displaystyle N} spin partition function Z N {\displaystyle Z_{N}} for (K * ,L * )
obtained from the standard expansion
is
the factor 2 originating from a spin-flip symmetry for each P {\displaystyle P} .
Here the sum over P {\displaystyle P} stands for summation over closed polygons on the lattice resulting in the graphical correspondence from the sum over spins with values ± 1 {\displaystyle \pm 1} .
By using the following transformation to variables ( K , L ) {\displaystyle (K,L)} , i.e.
one obtains
where v = tanh K {\displaystyle v=\tanh K} and w = tanh L {\displaystyle w=\tanh L} . This yields a mapping relation between the low temperature expansion Z N ( K ∗ , L ∗ ) {\displaystyle Z_{N}(K^{*},L^{*})} and the high-temperature expansion Z N ( K , L ) {\displaystyle Z_{N}(K,L)} described as duality (here Kramers-Wannier duality). With the help of the relations
the above hyperbolic tangent relations defining K {\displaystyle K} and L {\displaystyle L} can be written more symmetrically as
With the free energy per site in the thermodynamic limit
the Kramers–Wannier duality gives
In the isotropic case where K = L , if there is a critical point at K = K c then there is another at K = K * c . Hence, in the case of there being a unique critical point, it would be located at K = K * = K * c , implying sinh 2K c = 1 , yielding
The result can also be written and is obtained below as
The Kramers-Wannier duality appears also in other contexts. [ 11 ] [ 12 ] [ 13 ] We consider here particularly the two-dimensional theory of a scalar field Φ . {\displaystyle \Phi .} [ 14 ] [ 15 ] In this case a more convenient variable than sinh ( 2 K ) {\displaystyle \sinh(2K)} is
With this expression one can construct the self-dual quantity
In field theory contexts the quantity ξ {\displaystyle \xi } is called correlation length . Next set
This function is the beta function of renormalization theory. Now suppose there is a value K ∗ {\displaystyle K^{*}} of K {\displaystyle K} for which β ( K ∗ ) = 0 {\displaystyle \beta (K^{*})=0} , i.e. s ( K ∗ ) = 1 {\displaystyle s(K^{*})=1} . The zero of the beta function is usually related to a symmetry - but only if the zero is unique. The solution of s ( K ∗ ) = + 1 {\displaystyle s(K^{*})=+1} yields (obtained with MAPLE)
Only the second solution is real and gives the critical value of Kramers and Wannier as | https://en.wikipedia.org/wiki/Kramers–Wannier_duality |
In probability theory , Kramkov's optional decomposition theorem (or just optional decomposition theorem ) is a mathematical theorem on the decomposition of a positive supermartingale V {\displaystyle V} with respect to a family of equivalent martingale measures into the form
where C {\displaystyle C} is an adapted (or optional ) process.
The theorem is of particular interest for financial mathematics , where the interpretation is: V {\displaystyle V} is the wealth process of a trader , ( H ⋅ X ) {\displaystyle (H\cdot X)} is the gain/loss and C {\displaystyle C} the consumption process.
The theorem was proven in 1994 by Russian mathematician Dmitry Kramkov . [ 1 ] The theorem is named after the Doob-Meyer decomposition but unlike there, the process C {\displaystyle C} is no longer predictable but only adapted (which, under the condition of the statement, is the same as dealing with an optional process ).
Let ( Ω , A , { F t } , P ) {\displaystyle (\Omega ,{\mathcal {A}},\{{\mathcal {F}}_{t}\},P)} be a filtered probability space with the filtration satisfying the usual conditions .
A d {\displaystyle d} -dimensional process X = ( X 1 , … , X d ) {\displaystyle X=(X^{1},\dots ,X^{d})} is locally bounded if there exist a sequence of stopping times ( τ n ) n ≥ 1 {\displaystyle (\tau _{n})_{n\geq 1}} such that τ n → ∞ {\displaystyle \tau _{n}\to \infty } almost surely if n → ∞ {\displaystyle n\to \infty } and | X t i | ≤ n {\displaystyle |X_{t}^{i}|\leq n} for 1 ≤ i ≤ d {\displaystyle 1\leq i\leq d} and t ≤ τ n {\displaystyle t\leq \tau _{n}} .
Let X = ( X 1 , … , X d ) {\displaystyle X=(X^{1},\dots ,X^{d})} be d {\displaystyle d} -dimensional càdlàg (or RCLL) process that is locally bounded. Let M ( X ) ≠ ∅ {\displaystyle M(X)\neq \emptyset } be the space of equivalent local martingale measures for X {\displaystyle X} and without loss of generality let us assume P ∈ M ( X ) {\displaystyle P\in M(X)} .
Let V {\displaystyle V} be a positive stochastic process then V {\displaystyle V} is a Q {\displaystyle Q} - supermartingale for each Q ∈ M ( X ) {\displaystyle Q\in M(X)} if and only if there exist an X {\displaystyle X} -integrable and predictable process H {\displaystyle H} and an adapted increasing process C {\displaystyle C} such that
The statement is still true under change of measure to an equivalent measure. | https://en.wikipedia.org/wiki/Kramkov's_optional_decomposition_theorem |
Krapcho decarboxylation is a chemical reaction used to manipulate certain organic esters . [ 1 ] This reaction applies to esters with a beta electron-withdrawing group (EWG).
The reaction proceeds by nucleophilic dealkylation of the ester by the halide followed by decarboxylation, followed by hydrolysis of the resulting stabilized carbanion. [ 2 ]
The reaction is carried in dipolar aprotic solvents such as dimethyl sulfoxide (DMSO) at high temperatures, often around 150 °C. [ 3 ] [ 4 ] [ 5 ]
A variety of salts assist in the reaction including NaCl , LiCl , KCN , and NaCN . [ 6 ] It is suggested that the salts were not necessary for reaction, but greatly accelerates the reaction when compared to the reaction with water alone.
The ester must contain an EWG in the beta position . The reaction works best with a methyl esters. [ 2 ] which are more susceptible to S N 2 reactions .
The mechanisms are still not fully uncovered. However, the following are suggested mechanisms for two different substituents:
α,α-Disubstituted ester
For an α,α-disubstituted ester, it is suggested that the anion in the salt attacks the R 3 in an S N 2 fashion, kicking off R 3 and leaving a negative charge on the oxygen. Then, decarboxylation occurs to produce a carbanion intermediate. The intermediate picks up a hydrogen from water to form the products. [ 2 ]
The byproducts of the reaction (X-R 3 and CO 2 ) are often lost as gases, which helps drive the reaction; entropy increases and Le Chatelier's principle takes place.
α-Monosubstituted ester
For an α-monosubstituted ester, it is speculated that the anion in the salt attacks the carbonyl group to form a negative charge on the oxygen, which then cleaves off the cyanoester. With the addition of water, the cyanoester is then hydrolyzed to form CO 2 and alcohol, and the carbanion intermediate is protonated. [ 7 ]
The byproduct of this reaction (CO 2 ) is also lost as gas, which helps drive the reaction; entropy increases and Le Chatelier's principle takes place.
The Krapcho decarboxylation is a comparatively simpler method to manipulate malonic esters because it cleaves only one ester group, without affecting the other ester group. [ 1 ] The conventional method involves saponification to form carboxylic acids, followed by decarboxylation to cleave the carboxylic acids, and an esterification step to regenerate the esters. [ 8 ] Additionally, Krapcho decarboxylation avoids harsh alkaline or acidic conditions. [ 9 ] | https://en.wikipedia.org/wiki/Krapcho_decarboxylation |
Kraton is the trade name given to a number of high-performance elastomers manufactured by Kraton Polymers, and used as synthetic replacements for rubber . Kraton polymers offer many of the properties of natural rubber, such as flexibility, high traction, and sealing abilities, but with increased resistance to heat, weathering, and chemicals.
The origin of Kraton polymers goes back to the synthetic rubber (GR-S) program funded by the U.S. government during World War II to develop and establish a domestic supply capability for synthetic styrene butadiene rubber (SBR) as an alternative to natural rubber. [ 1 ]
Shell Oil Company purchased the Torrance, California facility from the U.S. government that was built to make synthetic styrene butadiene rubber. [ 2 ] The company formed Elastomers Division that eventually became Kraton Corporation. Shell Oil Company broaden the product portfolio of elastomers in the 1950s, [ 3 ] under the technical leadership of Murray Luftglass and Norman R. Legge . [ 4 ]
As part of the divestment program that was announced by Shell in December 1998, the Kraton elastomers business was sold to a private equity firm Ripplewood Holdings in 2000. [ 5 ] [ 6 ] Kraton completed IPO on December 17, 2009, to became a separate publicly traded company. [ 7 ] In 2021 Kraton employees won an ASC Innovation Award for "Next Generation of Biobased Tackifiers REvolutionTM". [ 8 ] Kraton employees accept an ASC Innovation Award
Kraton polymers are styrenic block copolymer (SBC) consisting of polystyrene blocks and rubber blocks. The rubber blocks consist of polybutadiene , polyisoprene , or their hydrogenated equivalents. The tri-block with polystyrene blocks at both extremities linked together by a rubber block is the most important polymer structure observed in SBC. If the rubber block consists of polybutadiene, the corresponding triblock structure is: poly(styrene-block-butadiene-block-styrene) usually abbreviated as SBS. Kraton D (SBS and SIS) and their selectively hydrogenated versions Kraton G (SEBS and SEPS) are the major Kraton polymer structures. The microstructure of SBS consists of domains of polystyrene arranged regularly in a matrix of polybutadiene, as shown in the TEM micrograph. The picture was obtained on a thin film of polymer cast onto mercury from solution, and then stained with osmium tetroxide .
The glass transition temperature (T g ) of the polybutadiene blocks is typically −90 °C and T g of the polystyrene blocks is +100 °C. So, at any temperature between about −90 °C and +100 °C Kraton SBS will act as a physically crosslinked elastomer . If Kraton polymers are heated substantially above the T g of the styrene-derived blocks, that is, above about 100 °C, like 170 °C the physical cross-links change from rigid glassy regions to flowable melt regions and the entire material flows and therefore can be cast, molded, or extruded into any desired form. On cooling, this new form resumes its elastomeric character. This is the reason such a material is called a thermoplastic elastomer (TPE). The polystyrene blocks form domains of nanometre size in the microstructure, and they stabilize the form of the molded material. Depending on the rubber-to-polystyrene ratio in the material, the polystyrene domains can be spherical or form cylinders or lamellae . The hydrogenated Kraton polymers named Kraton G exhibit improved resistance to temperature (processing at 200–230 °C is common), to oxidation, and to UV. SEBS and SEPS due to their polyolefinic rubber nature present excellent compatibility with polyolefins and paraffinic oils.
Kraton polymers are always used in blends with various other ingredients like paraffinic oils, polyolefins, polystyrene, bitumen, tackifying resins, and fillers to provide a very large range of end-use products ranging from hot melt adhesives to impact-modified transparent polypropylene bins, from medical TPE compounds to modified bitumen roofing felts or from oil gel toys (including sex toys) to elastic attachments in diapers. [ 9 ]
It can make asphalt flexible, which is necessary if the asphalt is to be used to coat a surface that is below grade or for highly demanding paving applications like F1 racing tracks. [ 10 ]
Kraton-based compounds are also used in non-slip knife handles. [ 11 ] [ 12 ]
The earliest commercial components using Kraton G (thermoplastic rubber) in the automobile industry were in 1970s. [ 13 ] The implementation of U.S. requirements for automobile bumpers to absorb 5 mph (8 km/h) impacts with no damage to the car's safety equipment lead to the first successful commercial automotive application of specialized flexible polymers as fascia for the 1974 AMC Matador . [ 14 ]
American Motors Corporation (AMC) also used this polymer plastic on the AMC Eagle for the color matched flexible wheel arch flares that flowed into rocker panel extensions. [ 15 ] [ 16 ] This was needed because of the Eagle's 2-inch wider track compared to the AMC Concord platform on which the AWD cars were based on. [ 17 ] The Eagle's Kraton bodywork was lightweight, flexible, and did not crack in cold weather as is typical of fiberglass automobile body components. [ 18 ]
Some grades of Kraton can also be dissolved into hydrocarbon oils to create "shear thinning" grease-type products that are used in the manufacture of telecommunications cables containing optical fibers. | https://en.wikipedia.org/wiki/Kraton_(polymer) |
Kreft's dichromaticity index (DI) is a measure for quantification of dichromatism . It is defined as the difference in hue angle (Δh ab ) between the color of the sample at the dilution, where the chroma (color saturation) is maximal, and the color of four times more diluted (or thinner) and four times more concentrated (or thicker) sample. The two hue angle differences are called the dichromaticity index towards lighter (Kreft's DI L ) and
dichromaticity index towards darker (Kreft's DI D ) respectively. [ 1 ] Kreft's dichromaticity indexes DI L and DI D for pumpkin seed oil , which is one of the most dichromatic substances, are −9 and −44, respectively. This means, that pumpkin seed oil changes its color from green-yellow to orange-red (for 44 degrees in Lab color space ) when the thickness of the observed layer is increased from cca 0.5 mm to 2 mm; and it changes slightly towards green (for 9 degrees) if its thickness is reduced for four-fold.
The color of pumpkin oil at increasing thickness or concentration presented in CIELAB colorspace diagram. Straight lines are vectors showing hue (angle) and chroma (length) of the color at maximal chroma (toward the square mark), and the colors of four-fold less or more diluted or thick pumpkin oil (DI L and DI D ). Note that DI D is −44.1 degrees and DI L corresponds to −8.97 degrees.
Dichromaticity (DI L and DI D ) of selected substances, calculated from their VIS absorption spectra by the computer algorithm “Dichromaticity index calculator”:
Maximal chroma: chroma at concentration (thickness) where the color of the substance has maximal chroma (saturation). Angle at maximal chroma: the hue, which is represented by the angle of the vector to the color with maximal chroma in the CIELAB colorspace diagram. | https://en.wikipedia.org/wiki/Kreft's_dichromaticity_index |
In mathematical analysis , Krein's condition provides a necessary and sufficient condition for exponential sums
to be dense in a weighted L 2 space on the real line. It was discovered by Mark Krein in the 1940s. [ 1 ] A corollary, also called Krein's condition, provides a sufficient condition for the indeterminacy of the moment problem . [ 2 ] [ 3 ]
Let μ be an absolutely continuous measure on the real line, d μ ( x ) = f ( x ) d x . The exponential sums
are dense in L 2 ( μ ) if and only if
Let μ be as above; assume that all the moments
of μ are finite. If
holds, then the Hamburger moment problem for μ is indeterminate; that is, there exists another measure ν ≠ μ on R such that
This can be derived from the "only if" part of Krein's theorem above. [ 4 ]
Let
the measure d μ ( x ) = f ( x ) d x is called the Stieltjes–Wigert measure . Since
the Hamburger moment problem for μ is indeterminate. | https://en.wikipedia.org/wiki/Krein's_condition |
In mathematics, Krener's theorem is a result attributed to Arthur J. Krener in geometric control theory about the topological properties of attainable sets of finite-dimensional control systems. It states that any attainable set of a bracket-generating system has nonempty interior or, equivalently, that any attainable set has nonempty interior in the topology of the corresponding orbit . Heuristically, Krener's theorem prohibits attainable sets from being hairy .
Let q ˙ = f ( q , u ) {\displaystyle {\ }{\dot {q}}=f(q,u)} be a smooth control system, where q {\displaystyle {\ q}} belongs to a finite-dimensional manifold M {\displaystyle \ M} and u {\displaystyle \ u} belongs to a control set U {\displaystyle \ U} . Consider the family of vector fields F = { f ( ⋅ , u ) ∣ u ∈ U } {\displaystyle {\mathcal {F}}=\{f(\cdot ,u)\mid u\in U\}} .
Let L i e F {\displaystyle \ \mathrm {Lie} \,{\mathcal {F}}} be the Lie algebra generated by F {\displaystyle {\mathcal {F}}} with respect to the Lie bracket of vector fields .
Given q ∈ M {\displaystyle \ q\in M} , if the vector space L i e q F = { g ( q ) ∣ g ∈ L i e F } {\displaystyle \ \mathrm {Lie} _{q}\,{\mathcal {F}}=\{g(q)\mid g\in \mathrm {Lie} \,{\mathcal {F}}\}} is equal to T q M {\displaystyle \ T_{q}M} ,
then q {\displaystyle \ q} belongs to the closure of the interior of the attainable set from q {\displaystyle \ q} .
Even if L i e q F {\displaystyle \mathrm {Lie} _{q}\,{\mathcal {F}}} is different from T q M {\displaystyle \ T_{q}M} ,
the attainable set from q {\displaystyle \ q} has nonempty interior in the orbit topology,
as it follows from Krener's theorem applied to the control system restricted to the orbit through q {\displaystyle \ q} .
When all the vector fields in F {\displaystyle \ {\mathcal {F}}} are analytic, L i e q F = T q M {\displaystyle \ \mathrm {Lie} _{q}\,{\mathcal {F}}=T_{q}M} if and only if q {\displaystyle \ q} belongs to the closure of the interior of the attainable set from q {\displaystyle \ q} . This is a consequence of Krener's theorem and of the orbit theorem .
As a corollary of Krener's theorem one can prove that if the system is bracket-generating and if the attainable set from q ∈ M {\displaystyle \ q\in M} is dense in M {\displaystyle \ M} , then the attainable set from q {\displaystyle \ q} is actually equal to M {\displaystyle \ M} . | https://en.wikipedia.org/wiki/Krener's_theorem |
Kreyos was a consumer electronics company based in San Francisco, California. The company intended to develop digital wearable devices fitted for people with active lifestyles. The Kreyos Meteor, the company's first product, started as a crowd funding campaign from Indiegogo . Although the campaign was funded to a bit more than fifteen times its original goal, the resulting product was poorly received, many units were not delivered or were defective, and the company was closed in 2014. [ 2 ]
Initially a crowd funded project from Indiegogo, Kreyos raised $1.5 Million for the Meteor, surpassing the original goal of $100,000. The duration of bidding lasted for nearly three months - from June 23 to August 12, 2013. [ 3 ] Their next big update came on 10/08/2013 when they provided footage of an engineer test . The first delay was announced on 11/13/2013. Kreyos stated during their pilot production run they discovered the position of the speaker holes in the Meteor case muffled the speaker and it wasn't their idea of how to put out the best product they can develop. The solution they implement to solve this problem also increase the overall volume by 25%. As a result, this changed the delivery date from January 2014 to mid February 2014.
On November 22, 2013 Kreyos apologized for the first time about the lack of communication. On 12/08/2013 they announced their presence at CES 2014. On December 24, 2013 Kreyos sent out an update talking about implementing and testing the changes to improve the speaker, submission of the latest Meteor for official FCC, CE, and IC testing and providing dates for the next the production run 6 days later. On Jan 10th of 2014 a video was sent out showing the functionality of the watch with Siri in attempt to calm backers.
On January 22 the second delay came pushing back the delivery date to April. This time the delay was cause because of an “echoing” issue identified in the late stages of hardware testing. On Jan 28th they announced a solution to the problem along with a commitment to be more transparent to backers in regards to all things related to development and making sure they updated backers as often as possible moving forward. After Further testing on Feb 11th they announced the problem was solved and that the entire senior management team was overseeing activity related to this fix in our production facilities and were manufacturing the new molds to fix the existing hardware. In an attempt to thank those who didn't ask for a refund Kreyos offered a limited edition belt clip and a 40% discount on the Meteor 2 for all backers.
Kreyos' next update, on March 21 contained a video showing the watch interacting with Google Now on the Android platform along with images of the UI for the iPhone . In April they also announce the special color of the belt clip which backers voted on. The color was Grey. At the end of April came another delay, at not fault of Kreyos, which moved the delivery date back to the end of May. This time the delay was a result of a delay in the delivery of new speaker components. Then on May 16 the day after the next production run was supposed to start Kreyos announced a destructive storm hit their production facilities in Guangdong province and manufacturing has stalled due to flooding in the factory.
By the end of May, Kreyos said they were shipping the first 5,000 units to the United States to be distributed to backers. Later Kreyos said that the units would be sent to backer according to when they backed the Indiegogo. This meant that any backers before June or July 2013 had a chance at finally receiving the watch. They stated that the vessel should take 3 to 4 weeks to transport the unit from the manufacture to the United States. Kreyos addressed and apologized for their iOS app, which wasn't 100% ready, saying they submitted the app in advance since there's been some situations where the app isn't approved for a long period of time. Orders started to be received by backers from the beginning of August 2014.
As of August 17, 2014 the Android app and iOS app are still having issues - everything from disconnecting from the phone to not syncing at all. The Windows Phone app has yet to be released. This has been extremely bothersome to backers as all functionality other than basic watch functions are dependent on app for it to work properly.
On the 15th of September 2014, Steve Tan (owner of the company) issued a statement outlining the reasons for the project's failure and that the company would close. Tan describes Kreyos as having only marketing and sales experience, and that therefore the technical work on the project and most of the funding had been outsourced to an acquaintance in the ODM business. Said ODM then failed to meet project goals, and Kreyos had neither the experience nor finances to resolve the problems. Tan alleges that this was a deliberate effort to defraud Kreyos and use its resources to create a turnkey smartwatch solution for other companies. The statement explained that the company had no funds left to issue further refunds, that they were unable to persuade the ODM to address software and firmware issues, and that they would be selling whatever components and tooling they could recover from the ODM before permanently closing the business. The remaining software was released under an open source licence. [ 4 ]
The Kreyos Meteor is a smart watch created and developed by Kreyos and is considered as its flagship product.
The Meteor promised voice and gesture controls for sending messages and emails, making calls, and changing the music on your phone; however, none of these promised features materialized in the final product. It displays all options and information through an ultra low energy 1.26-inch memory LCD Screen . It has a backlight and a vibrating motor, along with an alarm that can be set through the device itself. Sensors include a three-axis gyrometer, falling short of the alleged six-axis accelerometer and a gyrometer. for connectivity purposes, the Meteor sometimes uses a Bluetooth 4.0 low energy and is compatible to Android, iOS and Windows Phone 8 devices. The watch also offers ANT+ connectivity, allowing it to connect to a cyclocomputer and a heart rate monitor , along with other such peripheral technology. Other missing features include an app-programmable gesture control system, Siri , Google Now and voice command support, a 'Find your Phone' option as well as cloud storage for keeping fitness information and other data.
The Meteor has a built-in microphone and low-quality speaker, specific tracking modes for cycling and running , and a number of other built-in activity trackers. [ 5 ] The unit is detachable from the watchband , and can be transferred to a lanyard or a belt clip. [ 6 ]
Pre-release discussions were positive. [ 7 ] However numerous delays, technical issues, and unmet goals meant that the project was very poorly received in practice.
The latest criticism on the Meteor is the recent delay in shipping due to an echoing issue that was discovered in a recent test. With the shipping date moved to April, there has been a collective sense of frustration among the early buyers and backers of the watch. Despite this, the team assures that they foresee no more delays in the future as recent testing suggested that the fix they identified does address the issue permanently. [ 8 ] The Meteor was then set to be released in April 2014. After a further delay, the Meteor was set to be delivered to the first 5,000 backers by the end of July. [ 9 ] Instead, the backers were once again made to wait, while pre-orders close to the distribution center were completed first.
Further criticism is the fact that Kreyos failed to release an App for the watch despite the fact that some devices had been delivered. They claimed to be wanting to release the app at the same time to ensure customers all had the same version, maintaining the App is currently with the App store just waiting for the go ahead from themselves. Social media is rife with doubt and unrest at Kreyos's lack of action. [ 10 ] [ 11 ]
Kreyos has also recently come under scrutiny from its backers for posting on its Facebook page that the watch was only waterproof up to 1 meter depth for 30 minutes, and also that the watch has a 3-axis gyroscope. This is contrary to the information originally given about the watch and also (as of July 27) still on their website that states the watch contains a 6-axis gyroscope as well as being waterproof up to 5 meters. Kreyos have also been under heavy fire for their lack of information and consistent vague statements on delays and deadlines. [ 12 ] Additionally, Kreyos has responded to negative feedback by deleting the comments and links to the Better Business Bureau left on their social networks. In a statement announcing the closure of the company, Steve Tan stated that the watch cannot be considered at all waterproof in its current state. | https://en.wikipedia.org/wiki/Kreyos |
In information theory , given an unknown stationary source π with alphabet A and a sample w from π , the Krichevsky–Trofimov (KT) estimator produces an estimate p i ( w ) of the probability of each symbol i ∈ A . This estimator is optimal in the sense that it minimizes the worst-case regret asymptotically.
For a binary alphabet and a string w with m zeroes and n ones, the KT estimator p i ( w ) is defined as: [ 1 ]
This corresponds to the posterior mean of a Beta-Bernoulli posterior distribution with prior 1 / 2 {\displaystyle 1/2} .
For the general case the estimate is made using a Dirichlet-Categorical distribution.
This probability -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Krichevsky–Trofimov_estimator |
Kripke semantics (also known as relational semantics or frame semantics , and often confused with possible world semantics ) [ 1 ] is a formal semantics for non-classical logic systems created in the late 1950s and early 1960s by Saul Kripke and André Joyal . It was first conceived for modal logics , and later adapted to intuitionistic logic and other non-classical systems. The development of Kripke semantics was a breakthrough in the theory of non-classical logics, because the model theory of such logics was almost non-existent before Kripke (algebraic semantics existed, but were considered 'syntax in disguise').
The language of propositional modal logic consists of a countably infinite set of propositional variables , a set of truth-functional connectives (in this article → {\displaystyle \to } and ¬ {\displaystyle \neg } ), and the modal operator ◻ {\displaystyle \Box } ("necessarily"). The modal operator ◊ {\displaystyle \Diamond } ("possibly") is (classically) the dual of ◻ {\displaystyle \Box } and may be defined in terms of necessity like so: ◊ A := ¬ ◻ ¬ A {\displaystyle \Diamond A:=\neg \Box \neg A} ("possibly A" is defined as equivalent to "not necessarily not A"). [ 2 ]
A Kripke frame or modal frame is a pair ⟨ W , R ⟩ {\displaystyle \langle W,R\rangle } , where W is a (possibly empty) set, and R is a binary relation on W . Elements
of W are called nodes or worlds , and R is known as the accessibility relation . [ 3 ]
A Kripke model is a triple ⟨ W , R , ⊩ ⟩ {\displaystyle \langle W,R,\Vdash \rangle } , [ 4 ] where ⟨ W , R ⟩ {\displaystyle \langle W,R\rangle } is a Kripke frame, and ⊩ {\displaystyle \Vdash } is a relation between nodes of W and modal formulas, such that for all w ∈ W and modal formulas A and B :
We read w ⊩ A {\displaystyle w\Vdash A} as “ w satisfies A ”, “ A is satisfied in w ”, or
“ w forces A ”. The relation ⊩ {\displaystyle \Vdash } is called the satisfaction relation , evaluation , or forcing relation .
The satisfaction relation is uniquely determined by its
value on propositional variables.
A formula A is valid in:
We define Thm( C ) to be the set of all formulas that are valid in C . Conversely, if X is a set of formulas, let Mod( X ) be the
class of all frames which validate every formula from X .
A modal logic (i.e., a set of formulas) L is sound with
respect to a class of frames C , if L ⊆ Thm( C ). L is complete wrt C if L ⊇ Thm( C ).
Semantics is useful for investigating a logic (i.e. a derivation system ) only if the semantic consequence relation reflects its syntactical counterpart, the syntactic consequence relation ( derivability ). [ 5 ] It is vital to know which modal logics are sound and complete with respect to a class of Kripke frames, and to determine also which class that is.
For any class C of Kripke frames, Thm( C ) is a normal modal logic (in particular, theorems of the minimal normal modal logic, K , are valid in every Kripke model). However, the converse does not hold in general: while most of the modal systems studied are complete of classes of frames described by simple conditions,
Kripke incomplete normal modal logics do exist. A natural example of such a system is Japaridze's polymodal logic .
A normal modal logic L corresponds to a class of frames C , if C = Mod( L ). In other words, C is the largest class of frames such that L is sound wrt C . It follows that L is Kripke complete if and only if it is complete of its corresponding class.
Consider the schema T : ◻ A → A {\displaystyle \Box A\to A} . T is valid in any reflexive frame ⟨ W , R ⟩ {\displaystyle \langle W,R\rangle } : if w ⊩ ◻ A {\displaystyle w\Vdash \Box A} , then w ⊩ A {\displaystyle w\Vdash A} since w R w . On the other hand, a frame which
validates T has to be reflexive: fix w ∈ W , and
define satisfaction of a propositional variable p as follows: u ⊩ p {\displaystyle u\Vdash p} if and only if w R u . Then w ⊩ ◻ p {\displaystyle w\Vdash \Box p} , thus w ⊩ p {\displaystyle w\Vdash p} by T , which means w R w using the definition of ⊩ {\displaystyle \Vdash } . T corresponds to the class of reflexive
Kripke frames.
It is often much easier to characterize the corresponding class of L than to prove its completeness, thus correspondence serves as a guide to completeness proofs. Correspondence is also used to show incompleteness of modal logics: suppose L 1 ⊆ L 2 are normal modal logics that correspond to the same class of frames, but L 1 does not prove all theorems of L 2 . Then L 1 is Kripke incomplete. For example, the schema ◻ ( A ↔ ◻ A ) → ◻ A {\displaystyle \Box (A\leftrightarrow \Box A)\to \Box A} generates an incomplete logic, as it
corresponds to the same class of frames as GL (viz. transitive and
converse well-founded frames), but does not prove the GL -tautology ◻ A → ◻ ◻ A {\displaystyle \Box A\to \Box \Box A} .
The following table lists common modal axioms together with their corresponding classes. The naming of the axioms often varies; Here, axiom K is named after Saul Kripke ; axiom T is named after the truth axiom in epistemic logic ; axiom D is named after deontic logic ; axiom B is named after L. E. J. Brouwer ; and axioms 4 and 5 are named based on C. I. Lewis 's numbering of symbolic logic systems .
Axiom K can also be rewritten as ◻ [ ( A → B ) ∧ A ] → ◻ B {\displaystyle \Box [(A\to B)\land A]\to \Box B} , which logically establishes modus ponens as a rule of inference in every possible world.
Note that for axiom D , ◊ A {\displaystyle \Diamond A} implicitly implies ◊ ⊤ {\displaystyle \Diamond \top } , which means that for every possible world in the model, there is always at least one possible world accessible from it (which could be itself). This implicit implication ◊ A → ◊ ⊤ {\displaystyle \Diamond A\rightarrow \Diamond \top } is similar to the implicit implication by existential quantifier on the range of quantification .
The following table lists several common normal modal systems. Frame conditions for some of the systems were simplified: the logics are sound and complete with respect to the frame classes given in the table, but they may correspond to a larger class of frames.
For any normal modal logic, L , a Kripke model (called the canonical model ) can be constructed that refutes precisely the non-theorems of L , by an adaptation of the standard technique of using maximal consistent sets as models. Canonical Kripke models play a
role similar to the Lindenbaum–Tarski algebra construction in algebraic
semantics.
A set of formulas is L - consistent if no contradiction can be derived from it using the theorems of L , and Modus Ponens. A maximal L-consistent set (an L - MCS for short) is an L -consistent set that has no proper L -consistent superset.
The canonical model of L is a Kripke model ⟨ W , R , ⊩ ⟩ {\displaystyle \langle W,R,\Vdash \rangle } , where W is the set of all L - MCS ,
and the relations R and ⊩ {\displaystyle \Vdash } are as follows:
The canonical model is a model of L , as every L - MCS contains
all theorems of L . By Zorn's lemma , each L -consistent set
is contained in an L - MCS , in particular every formula
unprovable in L has a counterexample in the canonical model.
The main application of canonical models are completeness proofs. Properties of the canonical model of K immediately imply completeness of K with respect to the class of all Kripke frames.
This argument does not work for arbitrary L , because there is no guarantee that the underlying frame of the canonical model satisfies the frame conditions of L .
We say that a formula or a set X of formulas is canonical with respect to a property P of Kripke frames, if
A union of canonical sets of formulas is itself canonical.
It follows from the preceding discussion that any logic axiomatized by
a canonical set of formulas is Kripke complete, and compact .
The axioms T, 4, D, B, 5, H, G (and thus
any combination of them) are canonical. GL and Grz are not
canonical, because they are not compact. The axiom M by itself is
not canonical (Goldblatt, 1991), but the combined logic S4.1 (in
fact, even K4.1 ) is canonical.
In general, it is undecidable whether a given axiom is
canonical. We know a nice sufficient condition: Henrik Sahlqvist identified a broad class of formulas (now called Sahlqvist formulas ) such that
This is a powerful criterion: for example, all axioms
listed above as canonical are (equivalent to) Sahlqvist formulas.
A logic has the finite model property (FMP) if it is complete
with respect to a class of finite frames. An application of this
notion is the decidability question: it
follows from Post's theorem that a recursively axiomatized modal logic L which has FMP is decidable, provided it is decidable whether a given
finite frame is a model of L . In particular, every finitely
axiomatizable logic with FMP is decidable.
There are various methods for establishing FMP for a given logic.
Refinements and extensions of the canonical model construction often
work, using tools such as filtration or unravelling . As another possibility,
completeness proofs based on cut-free sequent calculi usually produce finite models
directly.
Most of the modal systems used in practice (including all listed above) have FMP.
In some cases, we can use FMP to prove Kripke completeness of a logic:
every normal modal logic is complete with respect to a class of modal algebras , and a finite modal algebra can be transformed
into a Kripke frame. As an example, Robert Bull proved using this method
that every normal extension of S4.3 has FMP, and is Kripke
complete.
Kripke semantics has a straightforward generalization to logics with
more than one modality. A Kripke frame for a language with { ◻ i ∣ i ∈ I } {\displaystyle \{\Box _{i}\mid \,i\in I\}} as the set of its necessity operators
consists of a non-empty set W equipped with binary relations R i for each i ∈ I . The definition of a
satisfaction relation is modified as follows:
A simplified semantics, discovered by Tim Carlson, is often used for
polymodal provability logics . A Carlson model is a structure ⟨ W , R , { D i } i ∈ I , ⊩ ⟩ {\displaystyle \langle W,R,\{D_{i}\}_{i\in I},\Vdash \rangle } with a single accessibility relation R , and subsets D i ⊆ W for each modality. Satisfaction is
defined as
Carlson models are easier to visualize and to work with than usual
polymodal Kripke models; there are, however, Kripke complete polymodal
logics which are Carlson incomplete.
Kripke semantics for intuitionistic logic follows the same principles as the semantics of modal logic, but it uses a different definition of satisfaction. [ 8 ]
An intuitionistic Kripke model is a triple ⟨ W , ≤ , ⊩ ⟩ {\displaystyle \langle W,\leq ,\Vdash \rangle } , where ⟨ W , ≤ ⟩ {\displaystyle \langle W,\leq \rangle } is a preordered Kripke frame, and ⊩ {\displaystyle \Vdash } satisfies the following conditions: [ 9 ]
The negation of A , ¬ A , could be defined as an abbreviation for A → ⊥. If for all u such that w ≤ u , not u ⊩ A , then w ⊩ A → ⊥ is vacuously true , so w ⊩ ¬ A .
Intuitionistic logic is sound and complete with respect to its Kripke
semantics, and it has the finite model property .
Let L be a first-order language. A Kripke
model of L is a triple ⟨ W , ≤ , { M w } w ∈ W ⟩ {\displaystyle \langle W,\leq ,\{M_{w}\}_{w\in W}\rangle } , where ⟨ W , ≤ ⟩ {\displaystyle \langle W,\leq \rangle } is an intuitionistic Kripke frame, M w is a
(classical) L -structure for each node w ∈ W , and
the following compatibility conditions hold whenever u ≤ v :
Given an evaluation e of variables by elements of M w , we
define the satisfaction relation w ⊩ A [ e ] {\displaystyle w\Vdash A[e]} :
Here e ( x → a ) is the evaluation which gives x the
value a , and otherwise agrees with e . [ 10 ]
As part of the independent development of sheaf theory , it was realised around 1965 that Kripke semantics was intimately related to the treatment of existential quantification in topos theory . [ 11 ] That is, the 'local' aspect of existence for sections of a sheaf was a kind of logic of the 'possible'. Though this development was the work of a number of people, the name Kripke–Joyal semantics is often used in this connection.
As in classical model theory , there are methods for
constructing a new Kripke model from other models.
The natural homomorphisms in Kripke semantics are called p-morphisms (which is short for pseudo-epimorphism , but the
latter term is rarely used). A p-morphism of Kripke frames ⟨ W , R ⟩ {\displaystyle \langle W,R\rangle } and ⟨ W ′ , R ′ ⟩ {\displaystyle \langle W',R'\rangle } is a mapping f : W → W ′ {\displaystyle f\colon W\to W'} such that
A p-morphism of Kripke models ⟨ W , R , ⊩ ⟩ {\displaystyle \langle W,R,\Vdash \rangle } and ⟨ W ′ , R ′ , ⊩ ′ ⟩ {\displaystyle \langle W',R',\Vdash '\rangle } is a p-morphism of their
underlying frames f : W → W ′ {\displaystyle f\colon W\to W'} , which
satisfies
P-morphisms are a special kind of bisimulations . In general, a bisimulation between frames ⟨ W , R ⟩ {\displaystyle \langle W,R\rangle } and ⟨ W ′ , R ′ ⟩ {\displaystyle \langle W',R'\rangle } is a relation B ⊆ W × W’ , which satisfies
the following “zig-zag” property:
A bisimulation of models is additionally required to preserve forcing
of atomic formulas :
The key property which follows from this definition is that
bisimulations (hence also p-morphisms) of models preserve the
satisfaction of all formulas, not only propositional variables.
We can transform a Kripke model into a tree using unravelling . Given a model ⟨ W , R , ⊩ ⟩ {\displaystyle \langle W,R,\Vdash \rangle } and a fixed
node w 0 ∈ W , we define a model ⟨ W ′ , R ′ , ⊩ ′ ⟩ {\displaystyle \langle W',R',\Vdash '\rangle } , where W’ is the
set of all finite sequences s = ⟨ w 0 , w 1 , … , w n ⟩ {\displaystyle s=\langle w_{0},w_{1},\dots ,w_{n}\rangle } such
that w i R w i+1 for all i < n , and s ⊩ p {\displaystyle s\Vdash p} if and only if w n ⊩ p {\displaystyle w_{n}\Vdash p} for a propositional variable p . The definition of the accessibility relation R’ varies; in the simplest case we put
but many applications need the reflexive and/or transitive closure of
this relation, or similar modifications.
Filtration is a useful construction which one can use to prove FMP for many logics. Let X be a set of
formulas closed under taking subformulas. An X -filtration of a
model ⟨ W , R , ⊩ ⟩ {\displaystyle \langle W,R,\Vdash \rangle } is a mapping f from W to a model ⟨ W ′ , R ′ , ⊩ ′ ⟩ {\displaystyle \langle W',R',\Vdash '\rangle } such that
It follows that f preserves satisfaction of all formulas from X . In typical applications, we take f as the projection
onto the quotient of W over the relation
As in the case of unravelling, the definition of the accessibility
relation on the quotient varies.
The main defect of Kripke semantics is the existence of Kripke incomplete logics, and logics which are complete but not compact. It can be remedied by equipping Kripke frames with extra structure which restricts the set of possible valuations, using ideas from algebraic semantics. This gives rise to the general frame semantics.
Blackburn et al. (2001) point out that because a relational structure is simply a set together with a collection of relations on that set, it is unsurprising that relational structures are to be found just about everywhere. As an example from theoretical computer science , they give labeled transition systems , which model program execution . Blackburn et al. thus claim because of this connection that modal languages are ideally suited in providing "internal, local perspective on relational structures." (p. xii)
Similar work that predated Kripke's revolutionary semantic breakthroughs: [ 12 ] | https://en.wikipedia.org/wiki/Kripke_semantics |
The Krische allylation involves the enantioselective iridium-catalyzed addition of an allyl group to an aldehyde or an alcohol , resulting in the formation of a secondary homoallylic alcohol. [ 1 ] [ 2 ] The mechanism of the Krische allylation involves primary alcohol dehydrogenation or, when using aldehyde reactants, hydrogen transfer from 2-propanol. Unlike other allylation methods, the Krische allylation avoids the use of preformed allyl metal reagents and enables the direct conversion of primary alcohols to secondary homoallylic alcohols (precluding alcohol to aldehyde oxidation). [ 1 ] [ 3 ]
Enantioselective carbonyl allylations are frequently applied to the synthesis of polyketide natural products. [ 3 ] In 1978, Hoffmann reported the first asymmetric carbonyl allylation using a chiral allylmetal reagent, an allylborane derived from camphor . [ 4 ] [ 5 ] Subsequently, other chiral allylmetal reagents were developed by Kumada, Roush, Brown, Leighton, and others. [ 6 ] [ 7 ] [ 8 ] [ 9 ] [ 10 ] [ 11 ] These methods utilize preformed allyl metal reagents and generate stoichiometric quantities of metal byproducts.
In 1991, Yamamoto disclosed the first catalytic enantioselective method for carbonyl allylation , which employed a chiral boron Lewis acid-catalyst in combination with allyltrimethylsilane. [ 12 ] Numerous catalytic enantioselective methods for carbonyl allylation followed, including work by Umani-Ronchi [ 13 ] and Keck. [ 14 ] While these methods had a significant impact, they do not circumvent the use of preformed allylmetal reagents. Catalytic variants of the Nozaki-Hiyama-Kishi reaction represent an alternative method for asymmetric carbonyl allylation, but stoichiometric metallic reductants are required. [ 15 ]
Whereas the allylmetal reagents used in these first-generation technologies are often difficult to prepare and handle, the Krische allylation exploits highly tractable allylic acetates. Additionally, the Krische allylation avoids the use of preformed allyl metal reagents or metallic reductants and chiral auxiliaries , significantly reducing waste generation.
The Krische allylation involves “ transfer hydrogenative ” carbon-carbon bond formations. [ 16 ] In a series of papers published in the early 2000s, Krische and coworkers demonstrated that allenes , dienes , and allyl acetates could be converted to transient allylmetal nucleophiles via hydrogenation , transfer hydrogenation or hydrogen auto-transfer . [ 17 ] This strategy for enantioselective carbonyl allylation avoids preformed organometallic reagents or metallic reductants. A remarkable feature of these reactions is the ability to conduct carbonyl allylation from the alcohol oxidation state . Due to a kinetic preference for primary alcohol dehydrogenation, diols containing both primary and secondary alcohols undergo site-selective carbonyl allylation at the primary alcohol without the need for protecting groups . [ 18 ] Additionally, by using alcohol reactants, the use of chiral α-stereogenic aldehydes, which are prone to racemization , can be avoided. [ 19 ]
The excellent functional group compatibility of the Krische allylation combined with the tractability of the allyl acetate pronucleophiles enables the use of allyl donors bearing highly complex nitrogen-rich substituents. [ 20 ]
The figure below shows some of the different allyl donors that have been used in the Krische allylation. These methods are summarized in the review literature. [ 16 ] [ 17 ]
The active catalyst in the Krische allylation is a cyclometallated π-allyliridium C,O -benzoate complex. This complex can be generated in situ or can be isolated via precipitation or conventional chromatography on silica gel.
The mechanism of the Krische allylation has been corroborated by DFT calculations. [ 21 ] Entry into the catalytic cycle involves protonation of the cyclometallated π-allyliridium precatalyst to generate the iridium alkoxide I. β-Hydride elimination of alkoxide I generates the aldehyde , which dissociates to form the iridium hydride III. Deprotonation of the iridium hydride III provides an anionic iridium(I) species IV, which upon oxidative addition to the allyl donor forms the π-allyliridium complex V. Association of the aldehyde to the σ-allyliridium species VI triggers carbonyl addition by way of the six-centered transition structure VII to form the homoallylic alkoxide VIII. The homoallylic alkoxide VIII is stable with respect to beta-hydride elimination due to coordination of the double bond with the metal. Exchange with the primary alcohol reactant regenerates the iridium alkoxide I and releases the reaction product.
Iridium-catalyzed transfer-hydrogenative carbonyl allylation method has been applied to the synthesis of polyketide natural products. [ 3 ] Some examples are shown below. In every case, the target compound was prepared in significantly fewer steps than was previously achieved. For example, total syntheses of roxaticin, bryostatin and cryptocaryol were accomplished via double Krische allylation of 1,3-propane diol. [ 22 ] [ 23 ] [ 24 ] This method was also used in the synthesis of mandelalide A. [ 25 ]
The Krische bisallylation has been applied to the synthesis of psymberin in 17 LLS and 32 total steps. [ 26 ] Through the use of the Krische allylation, this synthesis was accomplished via a much shorter route than previous syntheses. The Krische allylation to his synthesis of callyspongiolide using the chiral SEGPHOS catalyst complex. [ 27 ] ] In 2018, Harran also prepared callyspongiolide using the Krische allylation as a convergent method for fragment union. [ 28 ] Double crotylation was used by Krische to prepare 6-deoxyerythronolide B and swinholide A. [ 29 ] [ 30 ] | https://en.wikipedia.org/wiki/Krische_allylation |
Krister Holmberg , born 1946, is a Swedish chemist .
Holmberg took a PhD in Organic Chemistry from Chalmers University of Technology in Gothenburg, Sweden in 1974. He then worked in industry for many years and he was R&D Director of Berol Nobel in Stenungsund, Sweden. During the period 1991-1998 he was Director of the Institute for Surface Chemistry in Stockholm. Since 1998 he has been Professor of Surface Chemistry at Chalmers University of Technology . His recent research has focussed on the behavior of surface active compounds in solution and at interfaces. He has been active in a number of industrial applications of surface and colloid chemistry.
Holmberg has been visiting professor at universities in the EU, the US and China: P. et M. Curie in Paris, the University of Florence, the University of California at Santa Barbara and the Chinese Academy of Sciences in Beijing. He is a member of the Royal Swedish Academy of Engineering Sciences (Kungliga Ingenjörsvetenskapsakademien) , [ 1 ] of the Royal Swedish Academy of Sciences (Kungliga Vetenskapsakademin) [ 2 ] and of the Royal Academy of Arts and Sciences in Gothenburg (Kungliga Vetenskaps- och Vitterhetssamhället i Göteborg) . [ 3 ] He was chairman of the latter academy during 2014.
In 2000, Holmberg was awarded the French National Order of Merit ( l'Ordre National du Mérite au grade de Chevalier ), [ 4 ] and in 2006 he won the Oscar Carlson medal from the Swedish Chemical Society . [ 5 ] He received the JCIS Life-Time Achievement Award [ 6 ] in 2008, the Kash Mittal Award in 2018 and the Quancheng Friendship Award by the city of Jinan, China in 2019. In 2021 he received the Friendship Award of China from the Chinese Government. | https://en.wikipedia.org/wiki/Krister_Holmberg |
Kristi Lynn Kiick is the Blue and Gold Distinguished Professor of Materials Science and Engineering at the University of Delaware . [ 1 ] She studies polymers, biomaterials and hydrogels for drug delivery and regenerative medicine. She is a Fellow of the American Chemical Society , the American Institute for Medical and Biological Engineering , and of the National Academy of Inventors . She served for nearly eight years as the deputy dean of the college of engineering at the University of Delaware .
Kiick first became interested in a career in the chemical sciences when she was at high school. [ 3 ] She studied chemistry at the University of Delaware , from which she graduated summa cum laude as a Eugene du Pont memorial distinguished scholar. [ 4 ] She was a Master's student at the University of Georgia , where she was awarded a National Science Foundation (NSF) predoctoral fellowship , and joined Kimberly-Clark as a research scientist in 1992. [ 4 ] Kiick returned to academia for a second master's degree in polymer science and engineering at the University of Massachusetts Amherst . [ 4 ] She completed her doctoral research at the California Institute of Technology , as a National Defense Science and Engineering Graduate (NDSEG) fellow. [ 4 ] She completed her PhD from the University of Massachusetts Amherst on templated macromolecular synthesis in 2001 under the supervision of David A. Tirrell , [ 5 ] prior to starting her faculty position at the University of Delaware in 2001.
Kiick designs polymer nanostructures for targeted therapies and hydrogel matrices for regenerative medicine . [ 6 ] She makes use of biomimetic self-assembly, bioconjugation and biosynthesis. [ 7 ] In particular, Kiick has worked on polymer-peptide macromolecular structures that can engage cellular targets. These include the use of polyethylene glycol (PEG) in click chemistry to form hydrogels that degrade selectively in response to molecules present in tissues and extracellular matrix . [ 7 ] Kiick has shown it is possible to selectively release small molecule cargo with a tuned release for applications in targeted drug-delivery and vascular grafts. [ 7 ] She has developed resilin -like polypeptides (RLP), elastomeric materials that can be cross-linked using small molecules, as well as hydrogels that contain nanoparticles for targeting tumors and inflammatory conditions. [ 6 ] Resilin is a primary elastomeric protein that is found in insects, and helps them to jump long distances and produce sound. [ 8 ]
She joined the faculty at the University of Delaware in 2001, and earned the rank of associate professor in 2007. In 2011 Kiick was promoted to the rank of professor of materials science and engineering and also named deputy dean of the University of Delaware ’s college of engineering. [ 9 ] In 2019-2020 she was awarded a Leverhulme Visiting Professorship from the Leverhulme Trust and a Fulbright Scholarship from the Fulbright Program to the University of Nottingham , to develop protocols for fabricating bioelastomeric materials. [ 10 ] [ 11 ]
Her awards and honours include:
Her publications include:
Kiick is married with two children. [ 3 ] | https://en.wikipedia.org/wiki/Kristi_Kiick |
Kristin Shrader-Frechette (born 1944) is O'Neill Family Professor, Department of Biological Sciences and Department of Philosophy, at the University of Notre Dame . She has previously held senior professorships at the University of California and the University of Florida . Most of Shrader-Frechette's research work analyzes the ethical problems in risk assessment , public health , or environmental justice - especially those related to radiological, ecological, and energy-related risks. [ 1 ]
Shrader-Frechette coined the phrase “ecological justice” more than 40 years ago, with the term changing to “ environmental justice ” over time. Among other things, "environmental injustice" references situations in which certain groups bear disproportionate environmental risks, have unequal access to goods like clean air or water, or have unequal voices in determining the imposition of environmental risks. Shrader-Frechette, who is considered one of the founders of the environmental justice movement, was also an early advocate of the concept of “intergenerational equity,” the idea that the environmental problems of future generations are also significant to the current generation.
Shrader-Frechette has received the Global Citizenship Award, and the Catholic Digest named her one of 12 "Heroes for the US and the World."
Kristin Shrader-Frechette studied physics at Xavier University and graduated (summa cum laude) in 1967. She received her Ph.D. in philosophy from the University of Notre Dame in 1972. Shrader-Frechette also did post-doctoral work relating to biology, economics, and hydrogeology. [ 2 ]
Shrader-Frechette has published more than 380 articles and 16 books/monographs, including Burying Uncertainty: Risk and the Case Against Geological Disposal of Nuclear Waste (1993); Method in Ecology (1993); The Ethics of Scientific Research (1994), Technology and Human Values (1996), Environmental Justice: Creating Equality, Reclaiming Democracy (2002), Taking Action, Saving Lives: Our Duties to Protect Environmental and Public Health (2007), and What Will Work: Fighting Climate Change with Renewable Energy, Not Nuclear Power (2011). Her books and articles have been translated into 13 languages. Shrader-Frechette is currently working on two new volumes: Risks of Risk Assessment and Philosophy of Science and Public Policy . [ 1 ]
Shrader-Frechette's 2011 book What Will Work says that nuclear power is not an economic or practical technology:
This book uses market data, scientific studies, and ethical analyses to show why we should pursue green energy and conservation, and not nuclear fission, to address global climate change. Chapter 6 uses classic scientific studies from Harvard, Princeton, and the US Department of Energy to show how improved conservation and energy efficiency—along with increased use of wind and solar-PV power—can supply all energy needs while costing less than either fossil fuels or nuclear fission. [ 3 ]
Shrader-Frechette has been a member of many boards and committees at the international level. She has been invited to address the National Academies of Science in three different countries. She has served as an advisor to numerous governments and international organizations, including the United Nations and the World Health Organization . Associate Editor of BioScience until 2002, Shrader-Frechette is Editor-in-Chief of the Oxford University Press monograph series on Environmental Ethics and Science Policy and spent two terms on the US EPA Science Advisory Board. She also serves on the editorial boards of 22 professional journals. [ 1 ] [ 2 ]
In 2004 Shrader-Frechette received the World Technology Award. [ 2 ] In 2007, Catholic Digest named her one of 12 "Heroes for the US and the World" because of her pro-bono environmental justice work with minority and poor communities. In 2011, Tufts University gave her the Jean Mayer Global Citizenship Award. [ 1 ] In 2023, she received the International Cosmos Prize , Japan for research and pro bono work on methods of quantitative risk assessment and stopping environmental injustice. | https://en.wikipedia.org/wiki/Kristin_Shrader-Frechette |
Krogh's principle states that "for such a large number of problems there will be some animal of choice, or a few such animals, on which it can be most conveniently studied." This concept is central to those disciplines of biology that rely on the comparative method , such as neuroethology , comparative physiology , and more recently functional genomics .
Krogh's principle is named after the Danish physiologist August Krogh , winner of the Nobel Prize in Physiology for his contributions to understanding the anatomy and physiology of the capillary system , who described it in The American Journal of Physiology in 1929. However, the principle was first elucidated nearly 60 years prior to this, and in almost the same words as Krogh, in 1865 by Claude Bernard , the French instigator of experimental medicine, on page 27 of his "Introduction à l'étude de la médecine expérimentale":
Dans l'investigation scientifique, les moindres procédés sont de la plus haute importance. Le choix heureux d'un animal , d'un instrument construit d'une certaine façon, l'emploi d'un réactif au lieu d'un autre, suffisent souvent pour résoudre les questions générales les plus élevées. ("In scientific research, the tiniest processes are of the greatest importance. The lucky choice of animal , of an instrument built in a particular way, the use of one reagent instead of another, often suffice to solve general questions of the highest order.")
Krogh wrote the following in his 1929 treatise on the then current 'status' of physiology (emphasis added):
...I want to emphasize that the route by which we can strive toward the ideal is by a study of the vital functions in all their aspects throughout the myriads of organisms . We may find out, nay, we will find out before very long the essential mechanisms of mammalian kidney function , but the general problem of excretion can be solved only when excretory organs are studied wherever we find them and in all their essential modifications. Such studies will be sure, moreover, to expand and deepen our insight into problems of the human kidney and will prove of value also from the narrowest utilitarian point of view. For such a large number of problems there will be some animal of choice or a few such animals on which it can be most conveniently studied . Many years ago when my teacher, Christian Bohr , was interested in the respiratory mechanism of the lung and devised the method of studying the exchange through each lung separately, he found that a certain kind of tortoise possessed a trachea dividing into the main bronchi high up in the neck, and we used to say as a laboratory joke that this animal had been created expressly for the purposes of respiration physiology . I have no doubt that there is quite a number of animals which are similarly "created" for special physiological purposes, but I am afraid that most of them are unknown to the men for whom they were "created," and we must apply to the zoologists to find them and lay our hands on them."
"Krogh's principle" was not utilized as a formal term until 1975 when the biochemist Hans Adolf Krebs (who initially described the Citric Acid Cycle ), first referred to it.
More recently, at the International Society for Neuroethology meeting in Nyborg , Denmark in 2004, Krogh's principle was cited as a central principle by the group at their 7th Congress. Krogh's principle has also been receiving attention in the area of functional genomics , where there has been increasing pressure and desire to expand genomics research to a more wide variety of organisms beyond the traditional scope of the field.
A central concept to Krogh's principle is evolutionary adaptation. Evolutionary theory maintains that organisms are suited to particular niches , some of which are highly specialized for solving particular biological problems. These adaptations are typically exploited by biologists in several ways: | https://en.wikipedia.org/wiki/Krogh's_principle |
Krogmann's salt is a linear chain compound consisting of stacks of tetracyanoplatinate. Sometimes described as molecular wires, Krogmann's salt exhibits highly anisotropic electrical conductivity. For this reason, Krogmann's salt and related materials are of some interest in nanotechnology . [ 2 ]
Krogmann's salt was first synthesized by Klaus Krogmann in the late 1960s. [ 3 ]
Krogmann's salt most commonly refers to a platinum metal complex of the formula K 2 [Pt(CN) 4 X 0.3 ] where X is usually bromine (or sometimes chlorine ). Many other non- stoichiometric metal salts containing the anionic complex [Pt(CN) 4 ] n− can also be characterized.
Krogmann's salt is a series of partially oxidized tetracyanoplatinate complexes linked by the platinum-platinum bonds on the top and bottom faces of the planar [Pt(CN) 4 ] n− anions. This salt forms infinite stacks in the solid state based on the overlap of the d z2 orbitals . [ 2 ]
Krogmann's salt has a tetragonal crystal structure with a Pt-Pt distance of 2.880 angstroms , which is much shorter than the metal-metal bond distances in other planar platinum complexes such as Ca[Pt(CN) 4 ]·5H 2 O (3.36 angstroms), Sr[Pt(CN) 4 ]·5H 2 O (3.58 angstroms), and Mg[Pt(CN) 4 ]·7H 2 O (3.16 angstroms). [ 3 ] [ 4 ] [ 1 ] The Pt-Pt distance in Krogmann's salt is only 0.1 angstroms longer than in platinum metal. [ 5 ]
Each unit cell contains a site for Cl − , corresponding to 0.5 Cl − per Pt. [ 1 ] However, this site is only filled 64% of the time, giving 0.32 Cl − per Pt in the actual compound. Because of this, the oxidation number of Pt does not rise above +2.32. [ 3 ]
Krogmann's salt has no recognizable phase range and is characterized by broad and intense intervalence bands in its electronic spectra. [ 6 ]
One of the most widely researched properties of Krogmann's salt is its unusual electric conductance. Because of its linear chain structure and overlap of the platinum d z 2 {\displaystyle d_{z^{2}}} orbitals, Krogmann's salt is an excellent conductor of electricity . [ 2 ] This property makes it an attractive material for nanotechnology. [ 7 ]
The usual preparation of Krogmann's salt involves the evaporation of a 5:1 molar ratio mixture of the salts K 2 [Pt(CN) 4 ] and K 2 [Pt(CN) 4 Br 2 ] in water to give copper-colored needles of K 2 [Pt(CN) 4 ]Br 0.32 ·2.6 H 2 O.
Because excess Pt II or Pt IV complex crystallizes out with the product when the reactant ratio is changed, the product is therefore well defined, although non-stoichiometric . [ 3 ]
Krogmann's salt nor any related material has found any commercial applications. | https://en.wikipedia.org/wiki/Krogmann's_salt |
The Kroll process is a pyrometallurgical industrial process used to produce metallic titanium from titanium tetrachloride . As of 2001 William Justin Kroll 's process replaced the Hunter process for almost all commercial production. [ 1 ]
In the Kroll process, titanium tetrachloride is reduced by liquid magnesium to give titanium metal:
The reduction is conducted at 800–850 °C in a stainless steel retort. [ 2 ] [ 3 ] Complications result from partial reduction of the TiCl 4 , giving to the lower chlorides TiCl 2 and TiCl 3 . The MgCl 2 can be further refined back to magnesium.
The resulting porous metallic titanium sponge is purified by leaching or vacuum distillation . The sponge is crushed, and pressed before it is melted in a consumable electrode vacuum arc furnace , "backfilled with pure gettered argon of a pressure high enough to avoid a glow discharge". [ 4 ] The melted ingot is allowed to solidify under vacuum . It is often remelted to remove inclusions and ensure uniformity. These melting steps add to the cost of the product. Titanium is about six times as expensive as stainless steel: Potter noted in 2023 that "Titanium is just fundamentally difficult and expensive to deal with. Turning titanium ingots into bars and sheets is a challenge due to titanium’s reactivity: it readily absorbs impurities, requiring “frequent surface removal and trimming to eliminate surface defects” which are “costly and involve significant yield loss.”" The appurtenant processes that turn Kroll's sponge into useful metal have "changed little since the 1950s." [ 5 ]
Many methods had been applied to the production of titanium metal, beginning with a report in 1887 by Nilsen and Pettersen using sodium, which was optimized into the commercial Hunter process . In this process (which ceased to be commercial in the 1990s) TiCl 4 is reduced to the metal by sodium . [ 3 ]
In the 1920s Anton Eduard van Arkel working for Philips NV had described the thermal decomposition of titanium tetraiodide to give highly pure titanium.
Titanium tetrachloride was found to reduce with hydrogen at high temperatures to give hydrides that can be thermally processed to the pure metal.
With these three ideas as background, Kroll in Luxembourg developed both new reductants and new apparatus for the reduction of titanium tetrachloride. Its high reactivity toward trace amounts of water and other metal oxides presented challenges. Significant success came with the use of calcium as a reductant, but the resulting mixture still contained significant oxide impurities. [ 6 ] Major success using magnesium at 1000 °C using a molybdenum clad reactor, was reported by Kroll to the Electrochemical Society in Ottawa. [ 7 ] Kroll's titanium was highly ductile reflecting its high purity.
The Kroll process displaced the Hunter process and continues to be the dominant technology for the production of titanium metal, as well as driving the majority of the world's production of magnesium metal. [ citation needed ]
After moving to the United States, Kroll further developed the method for the production of zirconium at the Albany Research Center . [ 4 ] | https://en.wikipedia.org/wiki/Kroll_process |
The Krone BiG X is a self-propelled forage harvester manufactured by the agricultural machinery company Krone Agriculture from Spelle . [ 1 ] Since May 2000, this vehicle has been one of Krone's main revenue generators. [ 2 ] [ 3 ]
From May 2000, Maschinenfabrik Bernard Krone produced the first generation of the BiG X forage harvester as BiG X V8 and BiG X V12. [ 3 ] Krone initially marketed the engine power as 540 hp . According to the German agriculture magazine Agrarheute , the concept of a forage harvester the size of the BiG X, capable of reaching speeds of 40 km/h and boasting high engine power, was new at the turn of the millennium, and the company was concerned that the true engine power might be perceived as oversized. In reality, the installed V8 engine, a Mercedes-Benz OM 502, had a power output of 605 hp. [ 4 ] The BiG X V12 model was even more powerful, equipped with an OM 444 engine capable of up to 780 hp. [ 5 ]
In 2007, Krone introduced the second generation with the BiG X 800, which is powered by two different engines that can be synchronized when necessary. The second generation included the BiG X 1000, BiG X 650, and from 2008, the BiG X 500. [ 6 ] The BiG X 700, 850, and 1100 models were added in 2010. [ 7 ] Since 2007, Krone has equipped its forage harvesters with a photo-optical sensor as standard. Integrated into the corn header, this instrument detects the maturity of the plants through color matching, preserving the structure of the harvested crop and reducing the risk of secondary fermentation. [ 8 ]
At the 2013 Agritechnica agricultural trade fair in Hanover , Krone presented the small BiG X series, featuring the forage harvesters BiG X 480 and 580. [ 9 ] These new models replaced the previous BiG X 500 and distinguished themselves from their predecessor primarily due to a smaller chopping unit and overall more compact dimensions. [ 10 ]
In the same year, Krone introduced the BiG X 600 Edition 2013 featuring a V8 engine from MAN . [ 11 ] In 2015, Krone expanded its small range of forage harvesters with the BiG X 530 and 630 models. [ 12 ] In September 2018, Krone unveiled its new large forage harvester series with the BiG X 680, 780, 880, and 1180 models, which replaced the old series. [ 13 ] [ 14 ] According to Krone, the BiG X 1180 is considered the most powerful forage harvester in the world. [ 15 ] [ 16 ] [ 17 ] The vehicle reportedly has a harvesting capacity of 360 t/h according to agricultural technology magazine Profi . Apart from the engine, the BiG X 1180 is identical to the BiG X 680, 780, and 880 models. [ 18 ] Also in 2018, for the first time, Krone introduced a roller conditioner called OptiMaxx with a diameter of 305 mm as optional equipment for the BiG X 1180. [ 16 ] [ 19 ] [ 20 ]
In September 2020, Krone unveiled a new generation of the small BiG X series, including the 480, 530, 580, and 630 models, featuring a new cabin, modified crop flow, and transition to emissions stage 5. [ 21 ] The gap between the BiG X 880 and the BiG X 1180 in the larger series was filled in September 2022 with the introduction of the BiG X 980 and the BiG X 1080. Similar to the BiG X 1180, the largest forage harvester in the series, the two new models are equipped with V12 engines from Liebherr . [ 22 ]
Krone offers two main model ranges with the small BiG X series 480, 530, 580, and 630, as well as the large BiG X series 680, 780, 880, 980, 1080, and 1180. These ranges primarily differ in engine power and, if applicable, the diameter of the kernel processors. For the BiG X 1180, a 305 mm diameter kernel processor can be installed, unlike the other models which are shipped with a kernel processor diameter of 250 mm. [ 19 ] [ 23 ]
The following technical description primarily focuses on the BiG X 1180 model, which is largely identical to the BiG X 680, 780, 880, 980, and 1080 models, differing primarily in engine specifications. Reference will also be made to the smaller models in the current series where appropriate.
The BiG X 1180 is a self-propelled forage harvester powered by Bosch wheel motors with an independent wheel suspension for the crop feed and a 90° machine pass-through. The crop intake of the forage harvester consists of six hydraulically driven pre-compression rollers. The knives are arranged in pairs and taper V-shaped towards the front axle. The intake can be removed from the machine for maintenance purposes. The design allows access to the chopping drum which has a diameter of 660 mm and a channel width of 800 mm. It can be equipped with universal drums ( MaxFlow ) with configurations of 20, 28, and 36 knives and biogas drums with configurations of 40, and 48 knives. The chopping drum bottom is curved and directs the chopped material upwards to the conditioning rollers. [ 24 ] [ 25 ] The conditioner OptiMaxx 250 features a 5-degree angled, diagonally positioned sawtooth profile on the 250 mm large cracking rollers, which, with an additional shearing effect, also process the chopped material in the transverse direction. The OptiMaxx 305 , equipped with 305 mm large rollers, is available for the models BiG X 680 through 1180. Compared to the OptiMaxx 250 , it has a 20% higher peripheral speed, an 11% larger contact area, and temperature monitoring of the bearings. [ 18 ] The two conditioning rollers, each with a 5-degree helical tooth profile, are arranged in a counter-rotating pair and condition (or crack ) the chopped material through friction. Their speed difference is 30%, 40%, or 50%. [ 16 ] [ 24 ] Above the conditioning rollers is the accelerator, which has an adjustable throwing distance and transports the chopped material upward through the throwing channel from the forage harvester into the discharge chute. This chute has a swivel angle of 210° and a discharge height of 6000 mm. [ 26 ]
The BiG X 1180 is available with an automatic adjustment of the counter blade, allowing the driver to adjust it from the cab. With the camera-based 3D image analysis, the Easyload automatic loading system fills each transport vehicle alongside in parallel operation or to rear loading of the machine. Additionally, the BiG X features the Variloc system, which allows the drum speed to be adjusted in less than five minutes, enabling the forage harvester to be used for both short and long cuts. [ 27 ] [ 28 ] The automatic data management in the BiG X is based on Krone's Smart-Connect control unit with autologging function, which transfers the automatically recorded data to the operator's office and utilizes it for transparent billing. Alternatively, an order-based data management system is offered with an app, allowing the driver to navigate directly to the field entrance. [ 27 ]
The BiG X series 480, 530, 580 and 630 differs from the BiG X 1180 primarily in basic structure, with a narrower channel width of 630 mm instead of 800 mm. [ 12 ] The conditioner is technically identical to that of the larger models. [ 29 ] [ 30 ]
Four different attachments are available for the BiG X directly from the factory: an EasyFlow pick-up with optional working widths of 3000 or 3800 mm; an EasyCollect row-independent harvesting attachment with optional working widths of 4500, 6000, 7500, 9000, or 10500 mm; an XCollect row-independent harvesting attachment with optional working widths of 6000, 7500, or 9000 mm; and an XDisc direct cutting attachment with a working width of 6200 mm. The XCollect differs from a conventional corn header as it does not require forced cutting. The crop is cut using sickle knives and is structurally separated from the collectors that convey the crop. According to the German agricultural technology magazine Traction , this design ensures functionality even with increased wear. The sickle disc speed can be set to either 1000 or 3000 RPM . The XCollect can be folded for road transportation. [ 31 ]
For the BiG X series 480, 530, 580 and 630, the attachments are largely identical to those of the large series, apart from the row-independent corn harvesting attachments ( EasyCollect ), which are not offered with a 10500 mm working width for the small series. [ 32 ]
With the optional Krone Nir Control dual system, data on moisture and the content of harvested crops such as corn , grass , and whole plant silage are captured. [ 20 ] [ 33 ] [ 34 ] This data can be recorded in the machine terminal and assigned to the harvested area. The Krone Nir Control dual system is mounted on the discharge chute of the BiG X. [ 35 ] The Nir sensor is used both for determining content in Krone's harvesting technology and in the Zunhammer company's VanControl dual system to capture ingredients in organic fertilizers . [ 28 ] [ 33 ]
The forage harvester BiG X has two power paths, one mechanical and one hydraulic. All flow components (the chopping drum, conditioning rollers, and accelerator) are mechanically driven by a composite V-belt from the engine. The torque output to the pulley at the engine can be engaged and disengaged via a multi-plate clutch. The V-belt directly drives the chopping drum and accelerator via a pulley, while the conditioning rollers are driven on the opposite side by a second composite V-belt running over the accelerator. [ 36 ] Variloc, a two-stage planetary gear is integrated into the pulley of the chopping drum, enabling a chopping drum speed of either 1250 or 800 RPM. [ 37 ] A gearbox is flanged to the engine, to which the hydraulic pump for the propulsion drive and for driving the pre-compression rollers is flanged to. [ 38 ]
Unlike the large BiG X, the small series forage harvesters do not require a motor output gearbox for sending power to the wheels. Engagement of all flow components is done via a belt clutch. [ 39 ] [ 40 ]
As described, the harvester has hydraulic power transmission to the wheels; it features hub motors made by Bosch Rexroth , and planetary gear sets. The rear axle is a sprung double wishbone axle, which can also be equipped with hub motors upon request. [ 31 ] [ 41 ] There are two driving modes: a field driving mode with a range from 0 to 25 km/h and a road driving mode with a range from 0 to 40 km/h; the driving speed can be continuously adjusted. [ 31 ] From the factory, there are five different front tire combinations and four different rear tire combinations available, with front tire widths ranging from 680 to 900 mm and rim sizes of 32, 38, or 42 inches; the 500, 620, or 710 mm wide rear tires are always mounted on 30-inch rims. All tires are radial tires . [ 26 ] The cabin is optionally height-adjustable and can be raised by up to 700 mm. [ 31 ]
The chassis and cabin of the small series are identical to the large BiG X. The tire options range from 680 to 900 mm with rim sizes from 32 to 42 inches. [ 42 ] [ 43 ]
In the BiG X, conventional industrial diesel engines from various suppliers are installed, including Daimler-Chrysler , [ 44 ] MAN , MTU , and Liebherr . [ 45 ]
The BiG X 980/1080/1180 is equipped with a transversely mounted Liebherr D 9512 V12 diesel engine, [ 22 ] while the models BiG X 680/780/880 feature Liebherr D 9508 V8 diesels. With a cylinder bore of 128 mm and a piston stroke of 157 mm, this results in a displacement of 24.24 liters and 16.16 liters, respectively. The engine, with exhaust turbocharging and intercooling, has overhead valves, and does not have hydraulic tappets. [ 31 ]
According to information from the German agricultural magazine Profi , the BiG X 1180 engine has a continuous output of 850 kW (1156 hp) at 1400 to 1800 RPM. [ 18 ] The continuous output in XPower chopping mode is reported to be 818 kW (1112 hp), according to agricultural magazine Traction . In EcoPower mode, the power is electronically limited to 441 kW (600 hp). Maintenance intervals are set at 1000 operating hours. [ 31 ] The harvester complies with Tier 4 final emission standards; [ 28 ] an SCR -only strategy is used, which exclusively employs an SCR catalyst without an oxidation catalyst, exhaust gas recirculation, or diesel particulate filter. The BiG X 1180 is equipped with multiple fuel tanks, with a combined capacity of either 1100 or 1500 liters. [ 18 ] Additionally, there is a 150-liter auxiliary tank for AdBlue and a 230-liter auxiliary tank for silage additive, with an option for an additional 275-liter silage additive tank. [ 16 ] [ 31 ]
Sources: [ 5 ] [ 13 ] [ 15 ] [ 18 ] [ 20 ] [ 22 ] [ 46 ]
Sources: [ 5 ] [ 9 ] [ 12 ] [ 22 ] [ 47 ] | https://en.wikipedia.org/wiki/Krone_BiG_X |
In mathematics , Kronecker's congruence , introduced by Kronecker , states that
where p is a prime and Φ p ( x , y ) is the modular polynomial of order p , given by
for j the elliptic modular function and τ running through classes of imaginary quadratic integers of discriminant n .
This number theory -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Kronecker's_congruence |
In mathematics , Kronecker's lemma (see, e.g., Shiryaev (1996 , Lemma IV.3.2)) is a result about the relationship between convergence of infinite sums and convergence of sequences. The lemma is often used in the proofs of theorems concerning sums of independent random variables such as the strong Law of large numbers . The lemma is named after the German mathematician Leopold Kronecker .
If ( x n ) n = 1 ∞ {\displaystyle (x_{n})_{n=1}^{\infty }} is an infinite sequence of real numbers such that
exists and is finite, then we have for all 0 < b 1 ≤ b 2 ≤ b 3 ≤ … {\displaystyle 0<b_{1}\leq b_{2}\leq b_{3}\leq \ldots } and b n → ∞ {\displaystyle b_{n}\to \infty } that
Let S k {\displaystyle S_{k}} denote the partial sums of the x' s. Using summation by parts ,
Pick any ε > 0. Now choose N so that S k {\displaystyle S_{k}} is ε -close to s for k > N . This can be done as the sequence S k {\displaystyle S_{k}} converges to s . Then the right hand side is:
Now, let n go to infinity. The first term goes to s , which cancels with the third term. The second term goes to zero (as the sum is a fixed value). Since the b sequence is increasing, the last term is bounded by ϵ ( b n − b N ) / b n ≤ ϵ {\displaystyle \epsilon (b_{n}-b_{N})/b_{n}\leq \epsilon } .
This mathematical analysis –related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Kronecker's_lemma |
In mathematics, Kronecker coefficients g λ μν describe the decomposition of the tensor product (= Kronecker product ) of two irreducible representations of a symmetric group into irreducible representations. They play an important role in algebraic combinatorics and geometric complexity theory . They were introduced by Murnaghan in 1938.
Given a partition λ of n , write V λ for the Specht module associated to λ. Then the Kronecker coefficients g λ μν are given by the rule
One can interpret this on the level of symmetric functions , giving a formula for the Kronecker product of two Schur polynomials :
This is to be compared with Littlewood–Richardson coefficients , where one instead considers the induced representation
and the corresponding operation of symmetric functions is the usual product. Also note that the Littlewood–Richardson coefficients are the analogue of the Kronecker coefficients for representations of GL n , i.e. if we write W λ for the irreducible representation corresponding to λ (where λ has at most n parts), one gets that
Bürgisser & Ikenmeyer (2008) showed that computing Kronecker coefficients is #P-hard and contained in GapP . A recent work by Ikenmeyer, Mulmuley & Walter (2017) shows that deciding whether a given Kronecker coefficient is non-zero is NP-hard . [ 1 ] This recent interest in computational complexity of these coefficients arises from its relevance in the Geometric Complexity Theory program.
A major unsolved problem in representation theory and combinatorics is to give a combinatorial description of the Kronecker coefficients. It has been open since 1938, when Murnaghan asked for such a combinatorial description. [ 2 ] A combinatorial description would also imply that the problem is # P-complete in light of the above result.
The Kronecker coefficients can be computed as g ( λ , μ , ν ) = 1 n ! ∑ σ ∈ S n χ λ ( σ ) χ μ ( σ ) χ ν ( σ ) , {\displaystyle g(\lambda ,\mu ,\nu )={\frac {1}{n!}}\sum _{\sigma \in S_{n}}\chi ^{\lambda }(\sigma )\chi ^{\mu }(\sigma )\chi ^{\nu }(\sigma ),} where χ λ ( σ ) {\displaystyle \chi ^{\lambda }(\sigma )} is the character value of the irreducible representation corresponding to integer partition λ {\displaystyle \lambda } on a permutation σ ∈ S n {\displaystyle \sigma \in S_{n}} .
The Kronecker coefficients also appear in the generalized Cauchy identity ∑ λ , μ , ν g ( λ , μ , ν ) s λ ( x ) s μ ( y ) s ν ( z ) = ∏ i , j , k 1 1 − x i y j z k . {\displaystyle \sum _{\lambda ,\mu ,\nu }g(\lambda ,\mu ,\nu )s_{\lambda }(x)s_{\mu }(y)s_{\nu }(z)=\prod _{i,j,k}{\frac {1}{1-x_{i}y_{j}z_{k}}}.} | https://en.wikipedia.org/wiki/Kronecker_coefficient |
Kronecker substitution is a technique named after Leopold Kronecker for determining the coefficients of an unknown polynomial by evaluating it at a single value. If p ( x ) is a polynomial with integer coefficients, and x is chosen to be both a power of two and larger in magnitude than any of the coefficients of p , then the coefficients of each term of can be read directly out of the binary representation of p ( x ).
One application of this method is to reduce the computational problem of multiplying polynomials to the (potentially simpler) problem of multiplying integers.
If p ( x ) and q ( x ) are polynomials with known coefficients, then one can use these coefficients to determine a value of x that is a large enough power of two for the coefficients of the product pq ( x ) to be able to be read off from the binary representation of the number p ( x ) q ( x ). Since p ( x ) and q ( x ) are themselves straightforward to determine from the coefficients of p and q , this result shows that polynomial multiplication may be performed in the time of a single binary multiplication. [ 1 ]
This algebra -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Kronecker_substitution |
Krtin Nithiyanandam is a British medical researcher . He was awarded the Scientific American Innovator Award at the 2015 Google Science Fair for his work on developing a novel diagnosis test for early-onset Alzheimer's disease . [ 1 ] The award came with $25,000. [ 2 ] In 2017, Krtin's research on identifying a mechanism to make triple-negative breast cancer more treatable won the Intermediate Science stream at the national Big Bang Fair . [ 3 ] Recently, Nithiyanandam was the recipient of the U.K. Junior Water Prize for his project titled "A novel, photocatalytic, lead-sequestering bioplastic for sustainable water purification and environmental remediation ". He represented the U.K. at the international Stockholm Junior Water Prize . [ 4 ]
In 2017, Nithiyanandam was named as a Rising Star in Science by The Observer and as one of TIME 's 30 Most Influential Teens of 2017. [ 2 ] [ 5 ] Krtin currently attends Stanford University .
Krtin Nithiyanandam was born in Chennai, India and moved to Britain with his family. Nithiyanandam's interest in the medical sciences started after he suffered from hearing impairment as a child. He studied at Sutton Grammar School . [ 6 ] [ 7 ] Nithiyanandam has explained his research at TEDxLondon, TEDxGateway, WIRED : Next Generation, and the Royal Society of Medicine , and has advocated for increased student participation in scientific research. [ 8 ] [ 9 ] Krtin is a member of Stanford University's Class of 2022.
Nithiyanandam's work focused on oligomeric amyloid beta as a biomarker for Alzheimer's disease instead of amyloid beta plaques. [ 8 ] Nithiyanandam developed a bispecific antibody composed of two different Fab' fragments: one fragment from an anti-oligomeric amyloid beta IgG molecule and another fragment from an anti-transferrin receptor IgM molecule. [ 10 ] Nithiyanandam's bispecific antibody is conjugated to a quantum dot with MRI and fNIR detection capabilities. Nithiyanandam's in vitro studies suggest that the bispecific antibody quantum dot conjugate has little cross-reactivity and could potentially cross the blood-brain barrier . [ 7 ] [ 8 ] [ 10 ] He won the Scientific American Innovator Award at the Google Science Fair for this work. [ 1 ]
Nithiyanandam's research sought to develop a novel siRNA mechanism to decrease ID4 expression in aggressive triple-negative breast cancers. [ 5 ] [ 6 ] Nithiyanandam found that a knockdown in ID4 expression resulted in aggressive triple-negative breast cancers developing primitive oestrogen receptors on their surface, consequently making the cancer susceptible to existing breast-cancer treatments. [ 11 ] Moreover, Nithiyanandam found that increasing PTEN expression in several breast cancer cells lines, including MCF-7 , resulted in increased chemosensitivity to cisplatin . However, increased PTEN expression in "healthy cell line" MCF10A resulted in decreased chemosensitivity to cisplatin. [ 6 ] [ 11 ] He was the winner of the Intermediate stream of the Big Bang Fair for his work. [ 3 ]
Nithiyanandam developed a novel bioplastic capable of sequestering lead and purifying water through photocatalysis. [ 5 ] CIWEM , the organisation that awards the U.K. Junior Water Prize, commented that Nithiyanandam's project focused "on meeting global wastewater management challenges, and exhibits wastewater as an opportunity rather than a waste product". [ 4 ] | https://en.wikipedia.org/wiki/Krtin_Nithiyanandam |
In abstract algebra , Krull's separation lemma is a lemma in ring theory . It was proved by Wolfgang Krull in 1928. [ 1 ]
Let I {\displaystyle I} be an ideal and let M {\displaystyle M} be a multiplicative system ( i.e. M {\displaystyle M} is closed under multiplication) in a ring R {\displaystyle R} , and suppose I ∩ M = ∅ {\displaystyle I\cap M=\varnothing } . Then there exists a prime ideal P {\displaystyle P} satisfying I ⊆ P {\displaystyle I\subseteq P} and P ∩ M = ∅ {\displaystyle P\cap M=\varnothing } . [ 2 ]
This algebra -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Krull's_separation_lemma |
In mathematics , and more specifically in ring theory , Krull's theorem , named after Wolfgang Krull , asserts that a nonzero ring [ 1 ] has at least one maximal ideal . The theorem was proved in 1929 by Krull, who used transfinite induction . The theorem admits a simple proof using Zorn's lemma , and in fact is equivalent to Zorn's lemma , [ 2 ] which in turn is equivalent to the axiom of choice . | https://en.wikipedia.org/wiki/Krull's_theorem |
In commutative algebra , the Krull–Akizuki theorem states the following: Let A be a one-dimensional reduced noetherian ring , [ 1 ] K its total ring of fractions . Suppose L is a finite extension of K . [ 2 ] If A ⊂ B ⊂ L {\displaystyle A\subset B\subset L} and B is reduced,
then B is a noetherian ring of dimension at most one. Furthermore, for every nonzero ideal I {\displaystyle I} of B , B / I {\displaystyle B/I} is finite over A . [ 3 ] [ 4 ]
Note that the theorem does not say that B is finite over A . The theorem does not extend to higher dimension. One important consequence of the theorem is that the integral closure of a Dedekind domain A in a finite extension of the field of fractions of A is again a Dedekind domain. This consequence does generalize to a higher dimension: the Mori–Nagata theorem states that the integral closure of a noetherian domain is a Krull domain .
First observe that A ⊂ B ⊂ K B {\displaystyle A\subset B\subset KB} and KB is a finite extension of K , so we may assume without loss of generality that L = K B {\displaystyle L=KB} .
Then L = K x 1 + ⋯ + K x n {\displaystyle L=Kx_{1}+\cdots +Kx_{n}} for some x 1 , … , x n ∈ B {\displaystyle x_{1},\dots ,x_{n}\in B} .
Since each x i {\displaystyle x_{i}} is integral over K , there exists a i ∈ A {\displaystyle a_{i}\in A} such that a i x i {\displaystyle a_{i}x_{i}} is integral over A .
Let C = A [ a 1 x 1 , … , a n x n ] {\displaystyle C=A[a_{1}x_{1},\dots ,a_{n}x_{n}]} .
Then C is a one-dimensional noetherian ring, and C ⊂ B ⊂ Q ( C ) {\displaystyle C\subset B\subset Q(C)} , where Q ( C ) {\displaystyle Q(C)} denotes the total ring of fractions of C .
Thus we can substitute C for A and reduce to the case L = K {\displaystyle L=K} .
Let p i {\displaystyle {\mathfrak {p}}_{i}} be minimal prime ideals of A ; there are finitely many of them. Let K i {\displaystyle K_{i}} be the field of fractions of A / p i {\displaystyle A/{{\mathfrak {p}}_{i}}} and I i {\displaystyle I_{i}} the kernel of the natural map B → K → K i {\displaystyle B\to K\to K_{i}} . Then we have:
Now, if the theorem holds when A is a domain, then this implies that B is a one-dimensional noetherian domain since each B / I i {\displaystyle B/{I_{i}}} is and since B ≃ ∏ B / I i {\displaystyle B\simeq \prod B/{I_{i}}} . Hence, we reduced the proof to the case A is a domain. Let 0 ≠ I ⊂ B {\displaystyle 0\neq I\subset B} be an ideal and let a be a nonzero element in the nonzero ideal I ∩ A {\displaystyle I\cap A} . Set I n = a n B ∩ A + a A {\displaystyle I_{n}=a^{n}B\cap A+aA} . Since A / a A {\displaystyle A/aA} is a zero-dim noetherian ring; thus, artinian , there is an l {\displaystyle l} such that I n = I l {\displaystyle I_{n}=I_{l}} for all n ≥ l {\displaystyle n\geq l} . We claim
Since it suffices to establish the inclusion locally, we may assume A is a local ring with the maximal ideal m {\displaystyle {\mathfrak {m}}} . Let x be a nonzero element in B . Then, since A is noetherian, there is an n such that m n + 1 ⊂ x − 1 A {\displaystyle {\mathfrak {m}}^{n+1}\subset x^{-1}A} and so a n + 1 x ∈ a n + 1 B ∩ A ⊂ I n + 2 {\displaystyle a^{n+1}x\in a^{n+1}B\cap A\subset I_{n+2}} . Thus,
Now, assume n is a minimum integer such that n ≥ l {\displaystyle n\geq l} and the last inclusion holds. If n > l {\displaystyle n>l} , then we easily see that a n x ∈ I n + 1 {\displaystyle a^{n}x\in I_{n+1}} . But then the above inclusion holds for n − 1 {\displaystyle n-1} , contradiction. Hence, we have n = l {\displaystyle n=l} and this establishes the claim. It now follows:
Hence, B / a B {\displaystyle B/{aB}} has finite length as A -module. In particular, the image of I {\displaystyle I} there is finitely generated and so I {\displaystyle I} is finitely generated. The above shows that B / a B {\displaystyle B/{aB}} has dimension at most zero and so B has dimension at most one. Finally, the exact sequence B / a B → B / I → ( 0 ) {\displaystyle B/aB\to B/I\to (0)} of A -modules shows that B / I {\displaystyle B/I} is finite over A . ◻ {\displaystyle \square } | https://en.wikipedia.org/wiki/Krull–Akizuki_theorem |
In molecular genetics , the Krüppel-like family of transcription factors ( KLFs ) are a set of eukaryotic C2H2 zinc finger DNA-binding proteins that regulate gene expression . This family has been expanded to also include the Sp transcription factor and related proteins, forming the Sp/KLF family . [ 1 ]
The following human genes encode Kruppel-like factors: KLF1 , KLF2 , KLF3 , KLF4 , KLF5 , KLF6 , KLF7 , KLF8 , KLF9 , KLF10 , KLF11 , KLF12 , KLF13 , KLF14 , KLF15 , KLF16 , KLF17
The following genes are Sp factors: Sp1 , Sp2 , Sp3 , [ 2 ] Sp4 , Sp5 , Sp6 , Sp7 , Sp8 , and Sp9 .
Note that although KLF14 was an alias for Sp6 ( Q3SY56 ), it now refers to a protein ( Q8TD94 ) derived from KLF16 by a retrotransposon event. [ 3 ]
KLF/Sps are a family of transcription factors that contain three carboxyl-terminal ( C-terminal ) C2H2-type zinc finger structural motifs that bind to the GC-rich regions in DNA and regulate various cellular functions, such as proliferation , differentiation , and apoptosis , as well as the development and homeostasis of several types of tissue. The C-terminal end binds to the promoter and enhancer regions of a gene. Each KLF also has a unique amino-terminal ( N-terminal ) end that acts as the functional domain that allows it to bind specifically to a certain partner. KLFs share the similar function of transcription regulation via the recruitment of regulatory proteins . These transcription factors have a conserved structural homology between mammalian species, which allow for similar function due to similar protein interaction motifs at the N-terminal domains. The C-terminal end is also highly conserved with both the first and second zinc finger have 25 amino acids, while the third has 23 amino acids. Each of the three zinc fingers recognize three unique base pairs for their DNA-binding sites, which together make the general form NCR CRC CCN (where N is any base and R is a purine). There is some evidence that positively-charged amino acids within the three zinc fingers may contribute towards localizing the protein in the Nucleus. [ 5 ] The N-terminal end allows for the binding of various coactivators , corepressors , and modifiers . [ 4 ] All family members share the zinc finger signature KLF-DBD of CxxxxCxxxxxxxxxxxxHxxxHxxxxxxxCxxxxCxxxxxxxxxxxxHxxxHxxxxxxxCxxCxxxxxxxxxxxxHxxxH and use a 9aaTAD . [ 3 ]
KLFs are divided into three subgroups; Group 1 (KLF 3,8, and 12) are repressors via interaction with the C-terminal Binding Protein 1 and 2 ( CtBP1 and CtBP2 ). Group 2 (KLFs 1,2,4,5,6, and 7) are transcription activators . Group 3 (KLFs 9,10,11,13,12, and 16) have repressor activity via interaction with the common transcriptional co-repressor, Sin3A . KLFs 15 and 17 are distantly related without any defined protein interaction motifs. [ 4 ]
The Sp family members diverged from KLFs since Filozoa . They are typically divided into two groups of Sp1-4 and Sp5-9. One of the signatures is the "Btd box" CxCPxC preceding the KLF-DBD. [ 3 ]
The proliferation of KLF genes, presumably from an ancestral KLF, is also interesting. In some cases different family members are expressed in different tissues. The first KLF, KLF1 , originally known as Erythroid KLF (EKLF) is expressed only in red blood cells and megakaryocytes . It drives red blood cell differentiation and represses megakaryocyte formation. It appears that it has arisen as a KLF family member that has a particular role in these two blood lineages. [ 6 ] Other KLFs are more broadly expressed and there are interactions between family members. KLF3 for instance is driven by KLF1 as is KLF8 . [ 7 ] On the other hand, KLF3 represses KLF8 . Such cross-regulation occurs extensively in transcription factor families. Many transcription factor genes regulate their own promoters and when a gene duplicates during evolution then cross-regulation often occurs. The cross-regulation can ensure that the total amount of KLFs in the cell is monitored and controlled.
Finally, the biological roles of the KLFs are of wide interest. KLF1 is a very important factor in red blood cell biology. Naturally occurring human mutations in the KLF1 gene have been associated with de-repression of the fetal globin gene. [ 8 ] KLF2 (originally Lung KLF [ 9 ] ) also has a role in embryonic globin gene expression, [ 10 ] as does KLF3 (originally Basic KLF). KLF3 also has roles in adipocyte or fat formation, and in B lymphocytes. Recently, KLF3 was shown to be important in heart development. KLF4 (originally Gut KLF) is an important gene in the gut and skin but has more recently risen to prominence as one of the four genes that can reprogram body cells to become stem cells. [KLF4] is one of the so-called magic four transcription factors, KLF4 , Oct4, Sox2 and Myc. KLF5 , like KLF3 , is important in adipocytes [ 11 ] and KLF6 is an important tumour suppressor gene, that is often mutated in prostate cancers. [ 12 ]
KLF3 has a short motif in the N-terminus (of the form Proline-Isoleucine-Aspartate-Leucine-Serine or PIDLS) that recruits CtBP1 and 2. [ 13 ] CtBP in turn recruits histone modifying enzymes. It brings in histone deacetylases, histone demethylases and histone methylases, which are thought to
remove active chromatin marks and lay down repressive marks to eliminate gene expression.
Klf4, also known as gut-enriched Krüppel-like factor (GKLF), acts as a transcriptional activator or repressor depending on the promoter context and/or cooperation with other transcription factors. For example, Klf4 transactivates the iNOS promoter in cooperation with p65 ( RelA ), and the p21Cip1 /Waf1 promoter in cooperation with p53 , but it directly suppresses the p53 promoter and inhibits ornithine decarboxylase ( ODC ) promoter activity by competing with specificity protein-1 ( Sp-1 ). Klf4 also interacts with the p300/CBP transcription co-activators. Klf5, also known as intestinal enriched Krüppel-like factor (IKLF) or basic transcription element binding protein 2 (Bteb2), has been assigned purely transcriptional activation activity but, similar to Klf4, binds p300 which acetylates the first zinc finger conferring a trans-activating function. Importantly for Klf4 and Klf5, the amino acids that are predicted to interact with DNA are identical and the two compete for the same CACCC element or GC-rich sequence of the gene promoter region to regulate cell proliferation or differentiation-elated gene expression. Klf4 and Klf5 can act antagonistically during cellular proliferation, differentiation, and promoter activation, either via direct competition or via alterations in their own gene expression. The expression of Klf4 in terminally differentiated, post-mitotic intestinal epithelial cells as opposed to proliferating crypt cells which contain high levels of Klf5 is one example of such opposing effects. Klf4 inhibits proliferation through activation of p21Cip1/Waf1, and direct suppression of cyclin D1 and cyclin B1 gene expression. Both Klf4 & Klf5 proteins act on the Klf4 promoter where Klf4 increases expression and Klf5 decreases expression of Klf4 mRNA. The Wnt / APC signal pathway also plays an important role in the regulation of KLF4 expression. LOH , point mutations in the coding region and promoter hypermethylation are the main causes of klf4 gene silencing.
Klf4 is upregulated in vascular injury. It dramatically represses SRF/myocardin-induced activation of gene expression, and directly inhibits myocardin gene expression in vascular smooth muscle cells (VSMCs), therefore inhibiting the transition to a proliferative phenotype . Furthermore, Klf4 has been identified as an anti-proliferative shear stress-responsive gene, and forced over-expression of Klf4 in VSMCs induces growth arrest. Klf4 may therefore be an important protective factor in disease states affected by shear stress, such as thrombosis , restenosis and atherosclerosis . Klf4 also mediates the vascular response to nitric oxide (NO) by activating the promoters of inducible nitric oxide synthase (iNOS) in endothelial cells and cGMP-dependent protein kinase 1α/protein kinase G 1α ( PKG 1α ) in VSMCs. PKG 1α is activated by NO and mediates VSMC relaxation. This trans-activating effect of Klf4 on the PKG 1α promoter is inhibited by RhoA-induced actin polymerisation, possibly via G-actin regulation of a Klf4 co-activator or co-repressor. RhoA signalling pathways and RhoA activation are implicated in hypertension and increased vascular resistance which to some extent can be explained by this interaction with Klf4 and its effects on the response to NO. Klf5 has no effect on the PKG 1α promoter though the protein expression and nuclear localisation of Klf5 was similar to that of Klf4.
KLF-2 activation has been associated with laminar blood flow, a key protective force in the arterial walls that helps prevent atherosclerosis since it induces a protective phenotype in endothelial cells. In low-shear stress regions, KLF-2 inhibits a mechanosensory complex composed of platelet endothelial cell adhesion molecule (PECAM-1), vascular endothelial cadherin (VE-cadherin), and vascular endothelial growth factor receptor 2/3 (VEGFR2/3). [ 14 ]
Little is known of the Klfs in the myocardium. Klf5 activates the promoter of the hypertrophic agonist platelet derived growth factor ( PDGFA ) in cardiac fibroblasts a factor previously identified as being upregulated by ET-1, and Klf5+/- transgenic mice heterozygotes (described earlier) exhibited less cardiac fibrosis and hypertrophy when stimulated with angiotensin II compared with controls. [ 14 ] Klf5 is itself regulated by the immediate early gene egr-1 in VSMCs, which, if similarly regulated in the cardiomyocyte, places Klf5 potentially in a position to co-ordinate the acute response to external stress and tissue remodelling in the myocardium .
The understanding of the structure and function of KLFs has informed the design of artificial transcription factors. Artificial zinc fingers can be built to recognize chosen sites in DNA and artificial functional domains can be added to either activate or repress genes containing these sites. | https://en.wikipedia.org/wiki/Kruppel-like_factors |
The Krupp–Renn process was a direct reduction steelmaking process used from the 1930s to the 1970s. It used a rotary furnace and was one of the few technically and commercially successful direct reduction processes in the world, acting as an alternative to blast furnaces due to their coke consumption . The Krupp-Renn process consumed mainly hard coal and had the unique characteristic of partially melting the charge. This method is beneficial for processing low-quality or non-melting ores , as their waste material forms a protective layer that can be easily separated from the iron. It generates Luppen, nodules of pre-reduced iron ore, which can be easily melted down.
The first industrial furnaces emerged in the 1930s, firstly in Nazi Germany and then in the Japanese Empire . During the 1950s, new facilities were constructed, notably in Czechoslovakia and West Germany . The process was discontinued in the early 1970s, with a few nuances.
It was unproductive, intricate to master, and only pertinent to certain ores. In the beginning of the 21st century, Japan modernized the process to manufacture ferronickel , which is the sole surviving variant.
The direct reduction of iron ore principle was tested in the late 19th century using high-temperature stirring of ore powder mixed with coal and a small amount of limestone to adjust the ore's acidity . Carl Wilhelm Siemens ' [ 1 ] direct reduction process, which was sporadically employed in the United States and United Kingdom in the 1880s, is particularly noteworthy. This process is based on using a 3-meter in diameter and similarly lengthy drum with a horizontal axis for blowing gases preheated by two regenerators. [ 2 ]
The metallurgy industry underwent much research regarding the implementation of rotary tubular furnaces, inspired by similar equipment used in cement works. The Basset process, developed during the 1930s, is capable of even producing molten cast iron. [ 3 ] In the 1920s, German metallurgist Friedrich Johannsen [ fr ] , head of the metallurgy department at the Gruson plant [ fr ] and professor at the Clausthal University of Technology , [ 4 ] explored the metallurgical applications of this type of furnace. He filed a series of patents for removing volatile metals from steel raw materials. [ 5 ]
During the 1930s Johannsen initiated the development of direct-reduction iron production. The first installation underwent testing from 1931 to 1933 at the Gruson plant in Magdeburg . [ 6 ] Research on the Krupp-Renn process continued until 1939 at the Krupp facility in Essen-Borbeck . The process, named after the Krupp company that created it and the Rennfeuer , translating to " low furnace ," [ 7 ] displayed potential. As a result, Krupp procured patents overseas to safeguard the invention after 1932. [ 8 ]
In 1945 there were 38 furnaces worldwide, each with a capacity of 1 Mt/year. [ nb 1 ] [ 9 ] The process was favored in Germany due to the autarky policy of the Nazi regime, which prioritized the use of low-quality domestic iron ore. [ 10 ] The transfer of technology between Nazi Germany and Imperial Japan led to the Japanese Empire benefiting from this process. Furnaces were installed in the co-prosperity sphere and operated by Japanese technicians. By the eve of the Pacific War , the process was being used in four steelworks in Japan. [ 11 ]
After World War II all installations in Germany, China, and North Korea were dismantled, [ 12 ] with 29 furnaces sent to the USSR as war reparations. [ 13 ] Only the Japanese and Czechoslovakian plants remained functional. [ 12 ]
In the 1950s Krupp rebuilt several large furnaces in Spain, Greece , and Germany. [ 9 ] [ 12 ] The Czechoslovakians were the primary drivers, constructing 16 furnaces and increasing process efficiency. [ 14 ] The Great Soviet Encyclopedia reports that over 65 industrial plants, ranging from 60 to 110 meters in length and 3.6 to 4.6 meters in diameter, were constructed between 1930 and 1950. [ 6 ] By 1960, 50 furnaces were producing 2 million tons per year in several countries. [ 15 ]
The Soviet Union recovered 29 furnaces as war damage, but failed to gain significant profits from them. According to sources, the Red Army 's destructive techniques in dismantling German industrial plants proved inappropriate and wasted valuable resources. It was also challenging for Russians to reconstruct these factories within the Soviet Union. Travelers from Berlin to Moscow reported observing German machinery scattered, largely deteriorating, along every meter of track and shoulder, suffering from the harsh climatic conditions. [ 17 ] The Russian iron and steel industry did not heavily rely on technological input from the West. [ 18 ] Eventually, the Eastern Bloc only maintained this marginal technology to a limited extent in the recently sovietized European countries , [ 17 ] where it was eventually abandoned. [ 15 ]
Meanwhile large furnaces rebuilt in the 1950s in West Germany operated for approximately ten years before shutting down, due to the low cost of scrap and imported ore. [ nb 2 ] The process then vanished from West Germany, concurrently with Western Europe. [ 12 ]
In Japan furnaces also progressed towards increasingly bigger tools. However, the dwindling of local ferruginous sand deposits, along with the low cost of scrap and imported ores, eventually resulted in the gradual discontinuation of the process. The process was steadily improved by the Japanese, who developed it under various names for specialized products including ferroalloys [ 12 ] and the recycling of steelmaking by-products. [ 19 ] Currently, at the start of the 21st century, the Krupp-Renn process is exclusively used for ferronickel production in Japan. [ 20 ]
By 1972 most plants in Czechoslovakia, Japan, and West Germany had ceased operations. The process was widely considered obsolete and no longer garnered the attention of industrialists. [ 15 ]
The Krupp–Renn process is a direct reduction process that uses a long tubular furnace similar to those found in cement production. The most recent units constructed have a diameter of approximately 4.5 meters and a length of 110 meters. [ 9 ] The residence time of the product is influenced by the slope and speed of rotation of the rotary kiln, which is inclined at an angle of roughly 2.5 percent. [ 21 ]
Prior to usage, the iron ore is crushed to less than 6 mm in particle size . The iron ore is introduced into the furnace upstream and mixed with a small amount of fuel, typically hard coal. [ 21 ] After 6 to 8 hours, [ 22 ] it exits the furnace as pre-reduced iron ore at 1,000 °C. The amount of iron recovered ranges from 94% to 97.5% of the initial iron in the ore. [ 21 ]
A burner located at the lower end of the furnace provides heat, transforming it into a counter-current reactor. The fuel comprises finely pulverized coal, which, upon high-temperature combustion, generates reducing gas primarily consisting of CO . Once the furnace reaches an optimal temperature, the ore-coal mixture can serve as the primary fuel source. [ 21 ]
The fumes exiting the furnace's upper end attain temperatures ranging from 850 to 900 °C and are subsequently cooled and purged of dust by water injection before discharge through the chimney. [ 21 ]
The process is efficient in producing ferronickel due to the proximity of its constituent elements. At 800 °C, carbon easily reduces iron [ nb 3 ] and nickel oxides, while the gangue's other oxides are not significantly reduced . Specifically, iron(II) oxide (or wustite ), which is the stable iron oxide at 800 °C, has a reducibility similar to that of nickel(II) oxide , making it impossible to reduce one without reducing the other. [ 23 ]
The rotary kiln's maximum temperature ranges between 1,230 and 1,260 °C, which significantly exceeds the 1,000 to 1,050 °C threshold for iron oxide reduction. The main objective is to achieve a paste-like consistency of the ore gangue. [ 21 ] The reduced iron agglomerates into 3 to 8 mm metal nodules called Luppen . If the infusibility of the gangue is high, the temperature must be increased, up to 1,400 °C for a basic charge. [ 22 ] It is crucial to control the gangue's hot viscosity . [ 9 ] Among rotary drum direct reduction processes, it stands out for using high temperatures.
liq. ( cast iron )
Udy LARCO
SL/RN
Krupp
Renn
Another distinctive attribute of the procedure involves introducing powdered coal to the furnace outlet. Furthermore, the process has evolved to enable terminating the supply of coal and running exclusively on the coal dust or coke dust [ 25 ] introduced with the ore. [ 21 ] In this situation, solely combustion air is injected at the furnace outlet. Thermal efficiency is improved in shaft furnaces such as blast furnaces compared to rotary furnaces due to the air absorbing some of the Luppen heat. [ 14 ] However, the oxygen in the air partially re-oxidizes the product, meaning that the Luppen is still altered by contact with air at the end or after leaving the furnace, despite complete reduction of iron in the furnace. [ 7 ]
The hot assembly is discharged from the furnace and then rapidly cooled and crushed. The iron is separated from the slag via magnetic separation . Magnetically intermediate fines make up 5–15% of the charge. [ 22 ] While partial melting of the charge leads to the increased density of the products, it also requires significant energy consumption. [ 21 ]
The furnace comprises three distinct zones: [ 21 ] [ 22 ] [ 26 ]
Control of temperature is critical in regards to the ore's physicochemical characteristics. Overly high temperatures or unsuitable granulometry lead to the creation of rings of sintered material that accumulate on the walls of the furnace. Typically, a ring of iron-poor slag , known as slag, is formed at two-thirds of the distance along the furnace. Similarly, a metal ring usually forms around ten meters from the outlet. These rings disturb the flow of materials and gas, diminishing the furnace's useful capacity, sometimes completely obstructing it. The process's revival is hindered by the formation of a ring, particularly in China. In the early 21st century, industrialists abandoned its adoption after recognizing how critical and challenging managing this parameter was. [ 26 ]
While slag melting consumes energy, it enables us to govern the charge's behavior in the furnace. Additionally, we need a minimum of 800 to 1,000 kg of slag per ton of iron to prevent Luppen from growing too big. [ 27 ] Slag limits coal segregation as coal is much less dense than ore and would float to the surface of the mixture. It transforms into a paste that guards the metal against oxidation when heated and simplifies both Luppen processing and furnace cleaning during maintenance shutdowns through vitrification when it gets cold. [ 25 ]
The Krupp-Renn process is suitable for producing pre-reduced iron ore from highly siliceous and acidic ores (CaO/SiO2 basicity index of 0.1 to 0.4 [ 28 ] ), which begin generating a pasty slag at 1,200 °C. Additionally, due to the slag's acidity, it becomes vitreous, facilitating separation from the iron through easy crushing. [ 14 ] Furthermore, this process is also ideal for treating ores with high concentrations of titanium dioxide . Due to its ability to cause slag to become especially infusible and viscous, ores that contain this oxide cannot be used with blast furnaces as they must remove all their production in liquid form. [ 21 ] For this reason, the preferred ores for this technique are those that would become uneconomical if they had to be modified with basic additives, usually those with a low iron content (between 35 and 51%), and whose gangue needs to be neutralized. [ 7 ]
Integrated into a steelmaking complex, the Krupp-Renn process provides an alternative to sinter plants or beneficiation processes, [ 22 ] effectively eliminating waste rock and undesired elements like zinc , lead, and tin . In a blast furnace, these elements undergo vaporization-condensation cycles which progressively saturates the furnace. However, with the Krupp-Renn process, the high temperature of the fumes prevents condensation within the furnace, before they are retrieved by the dust-removal system. [ 28 ] The process recovers by-products or extracts specific metals. The Luppen is subsequently remelted in either the blast furnace or the cupola furnace , or the Martin-Siemens furnace , because it involves melting a pre-reduced, iron-rich charge. [ 22 ]
The process has been effective in treating ores abundant in nickel(II) oxide , vanadium , and other metals. [ 9 ] Additionally, the process is applicable in the production of ferronickel. [ 28 ] In this instance, saprolitic ores with a high magnesium [ 29 ] content are as infusible as highly acidic ores, distinguishing their relevance to the process. [ 11 ]
Direct reduction methods such as this one offer the flexibility of using any solid fuel and in this case, 240 to 300 kg of hard coal is needed to process one metric ton of iron ore that contains 30 to 40% iron. Assuming a consumption of 300 kg/ton of ore at 30%, the hard coal consumption is 800 kg per ton of iron. Additionally, 300 kg of coke is consumed during the smelting of Luppen in the blast furnace. When this ore is smelted entirely in the blast furnace, total fuel consumption remains the same. However, it only uses coke, which is a much more expensive fuel than hard coal. [ 22 ]
However, using slags with over 60% silica content, making them acidic, contradicts metal desulfurization that demands highly basic slags. [ 30 ] Consequently, 30% of the fuel's sulfur settles in the iron, entailing expensive after-treatments to eliminate it. [ 21 ] [ nb 4 ]
Depending on the ore and plant size, a furnace can daily output 250 to 800 tons of pre-reduced iron ore. [ 6 ] The biggest furnaces, up to 5 meters in diameter and 110 meters long, can process 950 to 1,000 tons of ore daily, excluding fuel. A properly operated plant typically runs for around 300 days per year. The internal refractory typically lasts 7 to 8 months in the most exposed part of the furnace and for 2 years elsewhere. In 1960, a Krupp-Renn furnace using low-grade ore yielded 100 kilotons of iron annually, [ 28 ] while a contemporaneous modern blast furnace produced ten times as much cast iron. [ 31 ]
Direct reduction processes employing rotary furnaces frequently face a significant challenge due to the localized formation of iron and slag rings, which sinter together and gradually obstruct the furnace. Understanding the mechanism of lining formation is a complex process involving mineralogy , chemical reactions, and ore preparation. The formation of the lining ring, which progressively grows and poisons the furnace, is caused by a few elements in minute quantities. To remedy this, increasing the supply of combustion air or interrupting the furnace charging process are effective solutions. [ 22 ] Otherwise, it may be necessary to adjust the grain size of the charged ore [ 32 ] or the chemical composition of the mineral blend. [ 33 ]
In 1958, Krupp constructed a plant that could generate 420,000 tons per year of pre-reduced iron ore (consisting of six furnaces) which had an estimated value of 90 million Deutsche Mark, [ 27 ] [ 28 ] or 21.4 million dollars. [ 34 ] By contrast, the plant erected in Salzgitter-Watenstedt in 1956–1957, which was well-integrated with an existing steelworks, only cost 33 million Deutsche Mark . [ 10 ] At that time, a Krupp-Renn plant presented itself as a feasible substitute to the established blast furnace process, considering its investment and operating costs: initial investment cost per ton produced was nearly half and operating costs were roughly two and a half times greater. [ 34 ]
The slag, a glassy silica, can be effortlessly employed as an additive for constructing road surfaces or concrete. [ 30 ] However, the method does not produce a recoverable gas similar to blast furnace gas , decreasing its profitability in most cases. Nevertheless, [ 22 ] it also solves the issue regarding gas recovery.
Unless otherwise specified, data are taken from ECSC (1960 [ 35 ] ), UNIDO (1963 [ 14 ] ), and Production étrangère de fer sans haut fourneau (Moscow, 1964 [ 36 ] ) publications.
(external or internal unknown)
Ząbkowice Śląskie (Poland)
Zakłady Górniczo-Hutnicze „Szklary”
2 × 275–300
1950–1953 [ 39 ]
1982 [ 39 ]
Garnierite processing (9% iron, 61% SiO 2 and 0.73% nickel).
4
Anshan I&S
8 × 300
1948 [ 42 ]
NC
2
1
1945-195
before 1964
NC
1,8
15
2
3,45
73
2
Hrudkovny Praha [ 47 ]
circa 1955 [ 47 ]
Ore containing 25–30% Fe.
2 [ nb 15 ]
4,6 [ nb 16 ]
70 [ nb 17 ]
NC
circa 1960
Technical failure: adoption of the LM process in 1963, followed by the Larco process in 1966 to transfer the pre-reduced smelting process to an electric furnace.
2
4,6
110
1957 and 1960
1
4,2
70
460 [ 25 ]
The difference in productivity between the first two furnaces is due to their inclination (2° and 3° respectively). [ 25 ]
1 [ 56 ]
1963–1973 [ 56 ]
1
1974 [ 59 ]
In view of its performance, the process seemed a suitable basis for the development of more efficient variants. Around 1940, the Japanese built several small reduction furnaces operating at lower temperatures: one at Tsukiji (1.8 m × 60 m), two at Hachinohe (2 furnaces of 2.8 m × 50 m), and three at Takasago (2 furnaces of 1.83 m × 27 m and 1 furnace of 1.25 m × 17 m). However, since they do not produce Luppen , they cannot be equated with the Krupp-Renn process. [ 40 ]
Although direct reduction in a rotary furnace has been the subject of numerous developments, the logical descendant of the Krupp-Renn process is the "Krupp-CODIR process". [ 62 ] Developed in the 1970s, it is based on the general principles of the Krupp-Renn process with a lower temperature reduction, typically between 950 and 1,050 °C, which saves fuel but is insufficient to achieve partial melting of the charge. The addition of basic corrective additives (generally limestone or dolomite ) mixed with the ore allows the removal of sulfur from the coal, although the thermolysis of these additives is highly endothermic . [ 21 ] This process has been adopted by three plants: ' Dunswart Iron & Steel Works' in South Africa in 1973, ' Sunflag Iron and Steel' in 1989, and ' Goldstar Steel & Alloy' in India in 1993. Although the industrial application is now well established, the process has not had the impact of its predecessor. [ 63 ]
Finally, there are many post-Krupp-Renn direct reduction processes based on a tubular rotary furnace. At the beginning of the 21st century, their combined output represented between 1% and 2% [ 64 ] of world steel production. [ 65 ] In 1935 and 1960, the output of the Krupp-Renn process (1 and 2 million tons respectively) represented just under 1% of world steel production. [ 9 ] [ 15 ]
The Krupp-Renn process, which specialized in the beneficiation of poor ores, was the logical basis for the development of recycling processes for ferrous by-products. In 1957, Krupp tested a furnace at Stürzelberg [ fr ] [ nb 20 ] for the treatment of roasted pyrites to extract iron (in the form of Luppen ) and zinc (vaporized in the flue gases). This process is therefore a hybrid of the Waelz and Krupp-Renn processes, which is why it is called the "Krupp-Waelz" (or "Renn-Waelz" [ 4 ] ) process. The trials were limited to a single 2.75 m × 40 m demonstrator capable of processing 70 to 80 t/day and were not followed up. [ 66 ]
The technical relationship between Krupp-Renn and Japanese direct reduction production processes is often cited. In the 1960s, Japanese steelmakers, sharing the observation that furnace plugging was difficult to control, developed their own low-temperature variants of the Krupp-Renn process. [ 11 ] Kawasaki Steel commissioned a direct-reduction furnace at its Eastern Japan steel plant [ fr ] (1968) and Western Japan steel plant [ fr ] (1975) plants, the most visible feature of which was a pelletizing unit for the site's steelmaking by-products (sludge and dust from the cleaning of converter and blast furnace gases). The "Kawasaki process" also incorporates other developments, such as the combustion of oil instead of pulverized coal [ nb 21 ] and the use of coke powder instead of coal mixed with ore... Almost identical to the Kawasaki process (with a more elaborate pelletizing unit), the "Koho process" was adopted by Nippon Steel , which commissioned a plant of this type at the Muroran steelworks [ fr ] in 1971. [ 19 ]
The production of ferronickel from laterites takes place in a context that is much more favorable to the Krupp-Renn process than to the steel industry. Lateritic ores in the form of saprolite are poor, very basic and contain iron. Production volumes are moderate, and the nickel chemistry is remarkably amenable to rotary kiln reduction. The process is therefore attractive, but regardless of the metal extracted, mastering all the physical and chemical transformations in a single reactor is a real challenge. [ 67 ] The failure of the Larco plant at Lárymna Greece, illustrates the risk involved in adopting this process: it was only when the ore was ready for industrial processing that it proved incompatible with the Krupp-Renn process. [ citation needed ]
As a result, lower-temperature reduction followed by electric furnace smelting allows each stage to have its own dedicated tool for greater simplicity and efficiency. Developed in 1950 at Koniambo [ fr ] in New Caledonia, this combination has proven to be both cost-effective and, above all, more robust. [ 26 ] Large rotating drums (5 m in diameter and 100 m or even 185 m long) are used to produce a dry powder from nickel ore concentrate. This powder contains 1.5 to 3% nickel. It leaves the drum at 800–900 °C and is immediately melted in electric furnaces. Only partial reduction takes place in the drums: a quarter of the nickel comes out in metallic form, the rest is still oxidized. Only 5% of the iron is reduced to metal, leaving unburned coal as fuel for the subsequent melting stage in the electric furnace. This proven process (also known as the RKEF process , for Rotary Kiln-Electric Furnace ) has become the norm: at the beginning of the 21st century, it accounted for almost all nickel laterite processing. [ 67 ]
In the early 21st century, however, the Nihon Yakin Kogyo foundry in Ōeyama, Japan, continued to use the Krupp-Renn process to produce intermediate grade ferronickel (23% nickel), sometimes called nickel pig iron. With a monthly output of 1,000 tons of Luppen [ 68 ] and a production capacity of 13 kt/year, the plant is operating at full capacity. [ 49 ] It is the only plant in the world using this process. It is also the only plant using a direct reduction process to extract nickel from laterite. [ 68 ] The process, which has been significantly upgraded, is called the "Ōeyama process". [ 69 ]
The Ōeyama process differs from the Krupp-Renn process in the use of limestone and the briquetting of the ore prior to charging. It retains its advantages, which are the concentration of all pyrometallurgical reactions in a single reactor and the use of standard (i.e. non-coking) coal, which covers 90% of the energy requirements of the process. Coal consumption is only 140 kg per ton of dry laterite , [ nb 22 ] and the quality of the ferronickel obtained is compatible with direct use by the steel industry. Although marginal, the Krupp-Renn process remains a modern, high-capacity process for the production of nickel pig iron. In this context, it remains a systematically studied alternative to the RKEF process and the "sinter plant-blast furnace" combination. [ 26 ] | https://en.wikipedia.org/wiki/Krupp–Renn_process |
Kruskal's algorithm [ 1 ] finds a minimum spanning forest of an undirected edge-weighted graph . If the graph is connected , it finds a minimum spanning tree . It is a greedy algorithm that in each step adds to the forest the lowest-weight edge that will not form a cycle . [ 2 ] The key steps of the algorithm are sorting and the use of a disjoint-set data structure to detect cycles. Its running time is dominated by the time to sort all of the graph edges by their weight.
A minimum spanning tree of a connected weighted graph is a connected subgraph, without cycles, for which the sum of the weights of all the edges in the subgraph is minimal. For a disconnected graph, a minimum spanning forest is composed of a minimum spanning tree for each connected component .
This algorithm was first published by Joseph Kruskal in 1956, [ 3 ] and was rediscovered soon afterward by Loberman & Weinberger (1957) . [ 4 ] Other algorithms for this problem include Prim's algorithm , Borůvka's algorithm , and the reverse-delete algorithm .
The algorithm performs the following steps:
At the termination of the algorithm, the forest forms a minimum spanning forest of the graph. If the graph is connected, the forest has a single component and forms a minimum spanning tree.
The following code is implemented with a disjoint-set data structure . It represents the forest F as a set of undirected edges, and uses the disjoint-set data structure to efficiently determine whether two vertices are part of the same tree.
For a graph with E edges and V vertices, Kruskal's algorithm can be shown to run in time O ( E log E ) time, with simple data structures. This time bound is often written instead as O ( E log V ) , which is equivalent for graphs with no isolated vertices, because for these graphs V /2 ≤ E < V 2 and the logarithms of V and E are again within a constant factor of each other.
To achieve this bound, first sort the edges by weight using a comparison sort in O ( E log E ) time. Once sorted, it is possible to loop through the edges in sorted order in constant time per edge. Next, use a disjoint-set data structure , with a set of vertices for each component, to keep track of which vertices are in which components. Creating this structure, with a separate set for each vertex, takes V operations and O ( V ) time. The final iteration through all edges performs two find operations and possibly one union operation per edge. These operations take amortized time O ( α ( V )) time per operation, giving worst-case total time O ( E α ( V )) for this loop, where α is the extremely slowly growing inverse Ackermann function . This part of the time bound is much smaller than the time for the sorting step, so the total time for the algorithm can be simplified to the time for the sorting step.
In cases where the edges are already sorted, or where they have small enough integer weight to allow integer sorting algorithms such as counting sort or radix sort to sort them in linear time, the disjoint set operations are the slowest remaining part of the algorithm and the total time is O ( E α ( V )) .
The proof consists of two parts. First, it is proved that the algorithm produces a spanning tree . Second, it is proved that the constructed spanning tree is of minimal weight.
Let G {\displaystyle G} be a connected, weighted graph and let Y {\displaystyle Y} be the subgraph of G {\displaystyle G} produced by the algorithm. Y {\displaystyle Y} cannot have a cycle, as by definition an edge is not added if it results in a cycle. Y {\displaystyle Y} cannot be disconnected, since the first encountered edge that joins two components of Y {\displaystyle Y} would have been added by the algorithm. Thus, Y {\displaystyle Y} is a spanning tree of G {\displaystyle G} .
We show that the following proposition P is true by induction : If F is the set of edges chosen at any stage of the algorithm, then there is some minimum spanning tree that contains F and none of the edges rejected by the algorithm.
Kruskal's algorithm is inherently sequential and hard to parallelize. It is, however, possible to perform the initial sorting of the edges in parallel or, alternatively, to use a parallel implementation of a binary heap to extract the minimum-weight edge in every iteration. [ 5 ] As parallel sorting is possible in time O ( n ) {\displaystyle O(n)} on O ( log n ) {\displaystyle O(\log n)} processors, [ 6 ] the runtime of Kruskal's algorithm can be reduced to O ( E α( V )), where α again is the inverse of the single-valued Ackermann function .
A variant of Kruskal's algorithm, named Filter-Kruskal, has been described by Osipov et al. [ 7 ] and is better suited for parallelization. The basic idea behind Filter-Kruskal is to partition the edges in a similar way to quicksort and filter out edges that connect vertices of the same tree to reduce the cost of sorting. The following pseudocode demonstrates this.
Filter-Kruskal lends itself better to parallelization as sorting, filtering, and partitioning can easily be performed in parallel by distributing the edges between the processors. [ 7 ]
Finally, other variants of a parallel implementation of Kruskal's algorithm have been explored. Examples include a scheme that uses helper threads to remove edges that are definitely not part of the MST in the background, [ 8 ] and a variant which runs the sequential algorithm on p subgraphs, then merges those subgraphs until only one, the final MST, remains. [ 9 ] | https://en.wikipedia.org/wiki/Kruskal's_algorithm |
In mathematics , Kruskal's tree theorem states that the set of finite trees over a well-quasi-ordered set of labels is itself well-quasi-ordered under homeomorphic embedding.
A finitary application of the theorem gives the existence of the fast-growing TREE function . TREE(3) is largely accepted to be one of the largest simply defined finite numbers, dwarfing other large numbers such as Graham's number and googolplex . [ 1 ]
The theorem was conjectured by Andrew Vázsonyi and proved by Joseph Kruskal ( 1960 ); a short proof was given by Crispin Nash-Williams ( 1963 ). It has since become a prominent example in reverse mathematics as a statement that cannot be proved in ATR 0 (a second-order arithmetic theory with a form of arithmetical transfinite recursion ).
In 2004, the result was generalized from trees to graphs as the Robertson–Seymour theorem , a result that has also proved important in reverse mathematics and leads to the even-faster-growing SSCG function , which dwarfs TREE ( 3 ) {\displaystyle {\text{TREE}}(3)} .
The version given here is that proven by Nash-Williams; Kruskal's formulation is somewhat stronger. All trees we consider are finite.
Given a tree T with a root , and given vertices v , w , call w a successor of v if the unique path from the root to w contains v , and call w an immediate successor of v if additionally the path from v to w contains no other vertex.
Take X to be a partially ordered set . If T 1 , T 2 are rooted trees with vertices labeled in X , we say that T 1 is inf-embeddable in T 2 and write T 1 ≤ T 2 {\displaystyle T_{1}\leq T_{2}} if there is an injective map F from the vertices of T 1 to the vertices of T 2 such that:
Kruskal's tree theorem then states:
If X is well-quasi-ordered , then the set of rooted trees with labels in X is well-quasi-ordered under the inf-embeddable order defined above. (That is to say, given any infinite sequence T 1 , T 2 , … of rooted trees labeled in X , there is some i < j {\displaystyle i<j} so that T i ≤ T j {\displaystyle T_{i}\leq T_{j}} .)
For a countable label set X , Kruskal's tree theorem can be expressed and proven using second-order arithmetic . However, like Goodstein's theorem or the Paris–Harrington theorem , some special cases and variants of the theorem can be expressed in subsystems of second-order arithmetic much weaker than the subsystems where they can be proved. This was first observed by Harvey Friedman in the early 1980s, an early success of the then-nascent field of reverse mathematics. In the case where the trees above are taken to be unlabeled (that is, in the case where X has size one), Friedman found that the result was unprovable in ATR 0 , [ 2 ] thus giving the first example of a predicative result with a provably impredicative proof. [ 3 ] This case of the theorem is still provable by Π 1 1 -CA 0 , but by adding a "gap condition" [ 4 ] to the definition of the order on trees above, he found a natural variation of the theorem unprovable in this system. [ 5 ] [ 6 ] Much later, the Robertson–Seymour theorem would give another theorem unprovable by Π 1 1 -CA 0 .
Ordinal analysis confirms the strength of Kruskal's theorem, with the proof-theoretic ordinal of the theorem equaling the small Veblen ordinal (sometimes confused with the smaller Ackermann ordinal ). [ 7 ]
Suppose that P ( n ) {\displaystyle P(n)} is the statement:
All the statements P ( n ) {\displaystyle P(n)} are true as a consequence of Kruskal's theorem and Kőnig's lemma . For each n , Peano arithmetic can prove that P ( n ) {\displaystyle P(n)} is true, but Peano arithmetic cannot prove the statement " P ( n ) {\displaystyle P(n)} is true for all n ". [ 8 ] Moreover, the length of the shortest proof of P ( n ) {\displaystyle P(n)} in Peano arithmetic grows phenomenally fast as a function of n , far faster than any primitive recursive function or the Ackermann function , for example. [ citation needed ] The least m for which P ( n ) {\displaystyle P(n)} holds similarly grows extremely quickly with n .
Define tree ( n ) {\displaystyle {\text{tree}}(n)} , the weak tree function, as the largest m so that we have the following:
It is known that tree ( 1 ) = 2 {\displaystyle {\text{tree}}(1)=2} , tree ( 2 ) = 5 {\displaystyle {\text{tree}}(2)=5} , tree ( 3 ) ≥ 844 , 424 , 930 , 131 , 960 {\displaystyle {\text{tree}}(3)\geq 844,424,930,131,960} (about 844 trillion), tree ( 4 ) ≫ g 64 {\displaystyle {\text{tree}}(4)\gg g_{64}} (where g 64 {\displaystyle g_{64}} is Graham's number ), and TREE ( 3 ) {\displaystyle {\text{TREE}}(3)} (where the argument specifies the number of labels ; see below ) is larger than t r e e t r e e t r e e t r e e t r e e 8 ( 7 ) ( 7 ) ( 7 ) ( 7 ) ( 7 ) . {\displaystyle \mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{8}(7)}(7)}(7)}(7)}(7).}
To differentiate the two functions, "TREE" (with all caps) is the big TREE function, and "tree" (with all letters in lowercase) is the weak tree function.
By incorporating labels, Friedman defined a far faster-growing function. [ 9 ] For a positive integer n , take TREE ( n ) {\displaystyle {\text{TREE}}(n)} [a] to be the largest m so that we have the following:
The TREE sequence begins TREE ( 1 ) = 1 {\displaystyle {\text{TREE}}(1)=1} , TREE ( 2 ) = 3 {\displaystyle {\text{TREE}}(2)=3} , before TREE ( 3 ) {\displaystyle {\text{TREE}}(3)} suddenly explodes to a value so large that many other "large" combinatorial constants, such as Friedman's n ( 4 ) {\displaystyle n(4)} , n n ( 5 ) ( 5 ) {\displaystyle n^{n(5)}(5)} , and Graham's number , [b] are extremely small by comparison. A lower bound for n ( 4 ) {\displaystyle n(4)} , and, hence, an extremely weak lower bound for TREE ( 3 ) {\displaystyle {\text{TREE}}(3)} , is A A ( 187196 ) ( 1 ) {\displaystyle A^{A(187196)}(1)} . [c] [ 10 ] Graham's number, for example, is much smaller than the lower bound A A ( 187196 ) ( 1 ) {\displaystyle A^{A(187196)}(1)} , which is approximately g 3 ↑ 187196 3 {\displaystyle g_{3\uparrow ^{187196}3}} , where g x {\displaystyle g_{x}} is Graham's function .
Citations
Bibliography | https://en.wikipedia.org/wiki/Kruskal's_tree_theorem |
The Krylov–Bogolyubov averaging method ( Krylov–Bogolyubov method of averaging ) is a mathematical method for approximate analysis of oscillating processes in non-linear mechanics. [ 1 ] The method is based on the averaging principle when the exact differential equation of the motion is replaced by its averaged version. The method is named after Nikolay Krylov and Nikolay Bogoliubov .
Various averaging schemes for studying problems of celestial mechanics were used since works of Carl Friederich Gauss , Pierre Fatou , Boris Delone and George William Hill . The importance of the contribution of Krylov and Bogoliubov is that they developed a general averaging approach and proved that the solution of the averaged system approximates the exact dynamics. [ 2 ] [ 3 ] [ 4 ]
Krylov–Bogoliubov averaging can be used to approximate oscillatory problems when a classical perturbation expansion fails. That is singular perturbation problems of oscillatory type, for example Einstein's correction to the perihelion precession of Mercury . [ 5 ]
The method deals with differential equations in the form
for a smooth function f along with appropriate initial conditions. The parameter ε is assumed to satisfy
If ε = 0 then the equation becomes that of the simple harmonic oscillator with constant forcing, and the general solution is
where A and B are chosen to match the initial conditions. The solution to the perturbed equation (when ε ≠ 0) is assumed to take the same
form, but now A and B are allowed to vary with t (and ε ). If it is also assumed that
then it can be shown that A and B satisfy the differential equation: [ 5 ]
where ϕ = k t + B {\displaystyle \phi =kt+B} . Note that this equation is still exact — no approximation has been made as yet. The method of Krylov and Bogolyubov is to note that the functions A and B vary slowly
with time (in proportion to ε), so their dependence on ϕ {\displaystyle \phi } can be (approximately) removed by averaging on the right hand side of the previous equation:
where A 0 {\displaystyle A_{0}} and B 0 {\displaystyle B_{0}} are held fixed during the integration. After solving this (possibly) simpler set of differential equations, the Krylov–Bogolyubov averaged approximation for the original function is then given by
This approximation has been shown to satisfy [ 6 ]
where t satisfies
for some constants C 1 {\displaystyle C_{1}} and C 2 {\displaystyle C_{2}} , independent of ε. | https://en.wikipedia.org/wiki/Krylov–Bogoliubov_averaging_method |
In mathematics , the Krylov–Bogolyubov theorem (also known as the existence of invariant measures theorem ) may refer to either of the two related fundamental theorems within the theory of dynamical systems . The theorems guarantee the existence of invariant measures for certain "nice" maps defined on "nice" spaces and were named after Russian - Ukrainian mathematicians and theoretical physicists Nikolay Krylov and Nikolay Bogolyubov who proved the theorems. [ 1 ]
Theorem (Krylov–Bogolyubov) . Let X be a compact , metrizable topological space and F : X → X a continuous map . Then F admits an invariant Borel probability measure .
That is, if Borel( X ) denotes the Borel σ-algebra generated by the collection T of open subsets of X , then there exists a probability measure μ : Borel( X ) → [0, 1] such that for any subset A ∈ Borel( X ),
In terms of the push forward , this states that
Let X be a Polish space and let P t , t ≥ 0 , {\displaystyle P_{t},t\geq 0,} be the transition probabilities for a time-homogeneous Markov semigroup on X , i.e.
Theorem (Krylov–Bogolyubov) . If there exists a point x ∈ X {\displaystyle x\in X} for which the family of probability measures { P t ( x , ·) | t > 0 } is uniformly tight and the semigroup ( P t ) satisfies the Feller property , then there exists at least one invariant measure for ( P t ), i.e. a probability measure μ on X such that
This article incorporates material from Krylov-Bogolubov theorem on PlanetMath , which is licensed under the Creative Commons Attribution/Share-Alike License . | https://en.wikipedia.org/wiki/Krylov–Bogolyubov_theorem |
KryoFlux is a hardware and software solution for preserving software on floppy disks . [ 2 ] [ 3 ] It was developed by the Software Preservation Society.
KryoFlux consists of a small hardware device, [ 4 ] [ 5 ] which is a software-programmable FDC system that runs on small ARM -based devices that connects to a floppy disk drive and a host PC over USB, and software for accessing the device. KryoFlux reads "flux transitions" from floppy disks at a very fine resolution. [ 6 ] It can also read disks originally written with different bit cell widths and drive speeds, with a normal fixed-speed drive. [ 7 ] The software is available for Microsoft Windows , [ 8 ] Mac OS and Linux . The KryoFlux controller plugs into a standard USB port, and allows normal PC floppy disk drives to be plugged into it.
Because the device operates on data bits at the lowest possible level with very precise timing resolution, it allows modern PCs to read, decode and write floppy disks that use practically any data format or method of copy protection to aid in digital preservation . [ 9 ] It has been tested successfully with many generations of floppy disk drive including 8", 5.25", 3.5" and 3" mechanisms, and dozens of disk formats including numerous schemes originally designed to prevent software piracy , allowing the preservation (typically to an image file stored on hard disk or other modern media) of programs and data that will inevitably succumb to data degradation as the original physical media deteriorates and becomes unreadable over time. [ 3 ] The image files produced may be rewritten to fresh disk media or, more commonly, used with software emulations of the original systems.
When reading old disks (especially those stored in non-climate controlled environments for long periods) there are a number of problems that can arise, including weakening of the magnetic field storing the data, deterioration of the binder holding the metal particles to the plastic disk surface, friction issues preventing the disk rotating freely in its outer protective sleeve, and issues caused by physical misalignment of the drive that originally wrote the disk or the one being used to read it. Users have detailed [ 10 ] various techniques to aid in the recovery of data stored on such marginal disks.
This software article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/KryoFlux |
Krypton-85 ( 85 Kr ) is a radioisotope of krypton .
Krypton-85 has a half-life of 10.756 years and a maximum decay energy of 687 keV . [ 1 ] It decays into stable rubidium -85. Its most common decay (99.57%) is by beta particle emission with a maximum energy of 687 keV and an average energy of 251 keV. The second most common decay (0.43%) is by beta particle emission (maximum energy of 173 keV) followed by gamma ray emission (energy of 514 keV). [ 2 ] Other decay modes have very small probabilities and emit less energetic gamma rays. [ 1 ] [ 3 ] Krypton-85 is mostly synthetic , though it is produced naturally in trace quantities by cosmic ray spallation .
In terms of radiotoxicity , 440 Bq of 85 Kr is equivalent to 1 Bq of radon-222 , without considering the rest of the radon decay chain .
Krypton-85 is produced in small quantities by the interaction of cosmic rays with stable krypton-84 in the atmosphere. Natural sources maintain an equilibrium inventory of about 0.09 PBq in the atmosphere. [ 4 ]
As of 2009, the total amount in the atmosphere is estimated at 5500 PBq due to anthropogenic sources. [ 5 ] At the end of the year 2000, it was estimated to be 4800 PBq, [ 4 ] and in 1973, an estimated 1961 PBq (53 megacuries). [ 6 ] The most important of these human sources is nuclear fuel reprocessing , as krypton-85 is one of the seven common medium-lived fission products . [ 4 ] [ 5 ] [ 6 ] Nuclear fission produces about three atoms of krypton-85 for every 1000 fissions (i.e., it has a fission yield of 0.3%). [ 7 ] Most or all of this krypton-85 is retained in the spent nuclear fuel rods; spent fuel on discharge from a reactor contains between 0.13–1.8 PBq/Mg of krypton-85. [ 4 ] Some of this spent fuel is reprocessed . Current nuclear reprocessing releases the gaseous 85 Kr into the atmosphere when the spent fuel is dissolved. It would be possible in principle to capture and store this krypton gas as nuclear waste or for use. The cumulative global amount of krypton-85 released from reprocessing activity has been estimated as 10,600 PBq as of 2000. [ 4 ] The global inventory noted above is smaller than this amount due to radioactive decay; a smaller fraction is dissolved into the deep oceans. [ 4 ]
Other man-made sources are small contributors to the total. Atmospheric nuclear weapons tests released an estimated 111–185 PBq. [ 4 ] The 1979 accident at the Three Mile Island nuclear power plant released about 1.6 PBq (43 kCi). [ 8 ] The Chernobyl accident released about 35 PBq, [ 4 ] [ 5 ] and the Fukushima Daiichi accident released an estimated 44–84 PBq. [ 9 ]
The average atmospheric concentration of krypton-85 was approximately 0.6 Bq/m 3 in 1976, and has increased to approximately 1.3 Bq/m 3 as of 2005. [ 4 ] [ 10 ] These are approximate global average values; concentrations are higher locally around nuclear reprocessing facilities, and are generally higher in the northern hemisphere than in the southern hemisphere.
For wide-area atmospheric monitoring, krypton-85 is the best indicator for clandestine plutonium separations. [ 11 ]
Krypton-85 releases increase the electrical conductivity of atmospheric air. Meteorological effects are expected to be stronger closer to the source of the emissions. [ 12 ]
Krypton-85 is used in arc discharge lamps commonly used in the entertainment industry for large HMI film lights as well as high-intensity discharge lamps . [ 13 ] [ 14 ] [ 15 ] [ 16 ] [ 17 ] The presence of krypton-85 in discharge tube of the lamps can make the lamps easy to ignite. [ 14 ] Early experimental krypton-85 lighting developments included a railroad signal light designed in 1957 [ 18 ] and an illuminated highway sign erected in Arizona in 1969. [ 19 ] A 60 μCi (2.22 MBq) capsule of krypton-85 was used by the random number server HotBits (an allusion to the radioactive element being a quantum mechanical source of entropy), but was replaced with a 5 μCi (185 kBq) Cs-137 source in 1998. [ 20 ] [ 21 ]
Krypton-85 is also used to inspect aircraft components for small defects. Krypton-85 is allowed to penetrate small cracks, and then its presence is detected by autoradiography . The method is called "krypton gas penetrant imaging". [ 22 ] The gas penetrates smaller openings than the liquids used in dye penetrant inspection and fluorescent penetrant inspection . [ 23 ]
Krypton-85 was used in cold-cathode voltage regulator electron tubes, such as the type 5651. [ 24 ]
Krypton-85 is also used for Industrial Process Control mainly for thickness and density measurements as an alternative to Sr-90 or Cs-137 . [ 25 ] [ 26 ]
Krypton-85 is also used as a charge neutralizer in aerosol sampling systems. [ 27 ] | https://en.wikipedia.org/wiki/Krypton-85 |
Krypton hexafluoride is an inorganic chemical compound of krypton and fluorine with the chemical formula KrF 6 . It is still a hypothetical compound . [ 1 ] Calculations indicate it is unstable. [ 2 ]
In 1933, Linus Pauling predicted that the heavier noble gases would be able to form compounds with fluorine and oxygen. He also predicted the existence of krypton hexafluoride. [ 3 ] [ verification needed ] [ 4 ] Calculations suggest it would have octahedral molecular geometry . [ 1 ]
So far, out of all possible krypton fluorides, only krypton difluoride ( KrF 2 ) has actually been synthesized. [ citation needed ]
This inorganic compound –related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Krypton_hexafluoride |
Krypton(IV) fluoride is a hypothetical inorganic chemical compound of krypton and fluorine with the chemical formula KrF 4 . At one time researchers thought they had synthesized it, but the claim was discredited. [ 1 ] The compound is predicted to be difficult to make and unstable if made. [ 2 ] However, it is predicted to become stable at pressures greater than 15 GPa . [ 3 ] Theoretical analysis indicates KrF 4 would have an approximately square planar molecular geometry . [ 2 ]
The claimed synthesis was by passing electric discharge through krypton-fluorine mixture: [ 4 ]
The claimed compound formed white crystalline solid. [ 5 ] Thermally, it is less stable than XeF 4 . [ 6 ]
This inorganic compound –related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Krypton_tetrafluoride |
In Chemistry , a kryptoracemic compound or kryptoracemate (sometimes false conglomerate ) is a racemic compound crystallizing in a Sohncke space group .
In most of the cases, racemic compounds crystallize in centrosymmetric crystal structures. In a kryptoracemic compound the chemical composition of the crystal is racemic although the crystal belongs to space groups in which all enantiomerically pure molecules have to crystallize.
Crystallographically, in kryptoracemic compounds, the number of independent molecules in the asymmetric unit (Z′) is necessarily greater than 1 and should take an even value (to respect the racemic composition). By extension, the scalemic compounds (or unbalanced compounds), i.e. crystal with non - stoichiometric ratio of enantiomer, crystallizing in Sohncke space group are sometimes included in kryptoracemic compounds although they are not strito-sensu kryptoracemic.
The term (kryptoracemate) was coined by Ivan Bernal who employed this term during a meeting of the American Crystallographic Association in 1995. [ 1 ]
The name is made of krypto (from Ancient Greek : κρυπτός, romanized : kryptos "the hidden one") and racemic . It comes from the fact that the racemic composition is "hidden" in a Sohncke space group (usually enantiomerically pure).
There is no space group restriction for the crystallization of racemic compound crystallizing either in centrosymmetric or in non-centrosymmetric space group (SG). The frequency of organic racemic compounds in the Cambridge Structural Database is summarized in the following table: [ 2 ]
(Non-centrosymmetric Chiral SGs or Sohncke SGs)
Kryptoracemic compounds are thus very rare and represent circa 1% of the racemic compounds. [ 2 ] The frequency of kryptoracemic compounds in the whole organic Cambridge Structural Database was estimated to circa 0.4% to 0.8%. [ 2 ] [ 3 ] [ 4 ] [ 5 ]
A review covering organometallic compounds with a stereogenic metal atom sorted a list of 26 possible kryptoracemic compounds. [ 6 ]
This stereochemistry article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Kryptoracemic_compounds |
Kröger–Vink notation is a set of conventions that are used to describe electric charges and lattice positions of point defect species in crystals . It is primarily used for ionic crystals and is particularly useful for describing various defect reactions. It was proposed by Ferdinand Anne Kröger [ fr ] and Hendrik Jan Vink [ nl ] . [ 1 ] [ 2 ]
The notation follows the scheme:
When using Kröger–Vink notation for both intrinsic and extrinsic defects, it is imperative to keep all masses, sites, and charges balanced in each reaction. If any piece is unbalanced, the reactants and the products do not equal the same entity and therefore all quantities are not conserved as they should be. The first step in this process is determining the correct type of defect and reaction that comes along with it; Schottky and Frenkel defects begin with a null reactant (∅) and produce either cation and anion vacancies (Schottky) or cation/anion vacancies and interstitials (Frenkel). Otherwise, a compound is broken down into its respective cation and anion parts for the process to begin on each lattice. From here, depending on the required steps for the desired outcome, several possibilities occur. For example, the defect may result in an ion on its own ion site or a vacancy on the cation site. To complete the reactions, the proper number of each ion must be present (mass balance), an equal number of sites must exist (site balance), and the sums of the charges of the reactants and products must also be equal (charge balance).
Assume that the cation C has +1 charge and anion A has −1 charge.
The following oxidation–reduction tree for a simple ionic compound, AX, where A is a cation and X is an anion, summarizes the various ways in which intrinsic defects can form. Depending on the cation-to-anion ratio, the species can either be reduced and therefore classified as n-type , or if the converse is true, the ionic species is classified as p-type . Below, the tree is shown for a further explanation of the pathways and results of each breakdown of the substance.
From the chart above, there are total of four possible chemical reactions using Kröger–Vink Notation depending on the intrinsic deficiency of atoms within the material. Assume the chemical composition is AX, with A being the cation and X being the anion. (The following assumes that X is a diatomic gas such as oxygen and therefore cation A has a +2 charge. Note that materials with this defect structure are often used in oxygen sensors .)
Using the law of mass action , a defect's concentration can be related to its Gibbs free energy of formation, and the energy terms ( enthalpy of formation ) can be calculated given the defect concentration or vice versa.
For a Schottky reaction in MgO , the Kröger–Vink defect reaction can be written as follows:
Note that the vacancy on the Mg sublattice site has a −2 effective charge, and the vacancy on the oxygen sublattice site has a +2 effective charge. Using the law of mass action , the reaction equilibrium constant can be written as ( square brackets indicating concentration):
Based on the above reaction, the stoichiometric relation is as follows:
Also, the equilibrium constant can be related to the Gibbs free energy of formation Δ f G according to the following relations,
Relating equations 2 and 4 , we get:
Using equation 5 , the formula can be simplified into the following form where the enthalpy of formation can be directly calculated:
Therefore, given a temperature and the formation energy of Schottky defect, the intrinsic Schottky defect concentration can be calculated from the above equation. | https://en.wikipedia.org/wiki/Kröger–Vink_notation |
Kröhnke pyridine synthesis is a reaction in organic synthesis that occurs between α-pyridinium methyl ketone salts and α, β-unsaturated carbonyl compounds and is used to generate highly functionalized pyridines . Pyridines occur widely in natural and synthetic products, so there is wide interest in routes for their synthesis. The method is named after Fritz Kröhnke.
In his work at the University of Giessen , Kröhnke observed condensation of α-pyridinium methyl ketone salts 1 with α,β-unsaturated carbonyl compounds 2 via a Michael reaction when treated with ammonium acetate to give 2,4,6-trisubstituted pyridines in high yields under mild reaction conditions. [ 1 ] The proposed intermediates, 1, 5-dicarbonyl compound 3 , have not been isolated. [ 2 ] Since its discovery, the Kröhnke synthesis has enjoyed broad applicability to the preparation of di-,tri- and tetrapyridine derivatives, demonstrating a number of advantages over related reactions such as the Hantzsch pyridine synthesis .
The mechanism of the Kröhnke pyridine synthesis begins with enolization of α-pyridinium methyl ketone 4 followed by 1,4-addition to the α, β-unsaturated ketone 5 to form the Michael adduct 6 , which immediately tautomerizes to the 1,5-dicarbonyl 7 . Addition of ammonia to 7 followed by dehydration via 8 generates the imine intermediate 9 ., [ 3 ] [ 4 ] The imine intermediate is then deprotonated to enamine 10 and cyclizes with the carbonyl to generate intermediate 11 . The pyridinium cation is then eliminated to form hydroxy-dienamine 12 . Aromatization of 12 via subsequent loss of water generates the desired pyridine heterocycle 13 .
The starting materials for the Kröhnke synthesis are often trivial to prepare. The α-pyridinium methyl ketone salts can be easily prepared by treatment of the corresponding bromomethyl ketone with pyridine. The α,β-unsaturated ketones are often available commercially. Additionally, Mannich bases can also be utilized as the Michael acceptor for the scheme, further diversifying the scope of starting materials that can be incorporated into the Kröhnke scheme. [ 5 ]
The reaction conditions for the Kröhnke synthesis are generally facile and the reactions often proceed in high yields with reaction temperatures generally not exceeding 140 °C. [ 6 ] The Kröhnke synthesis is generally performed in either glacial acetic acid or methanol as solvent, but it can also be done under aqueous conditions and under solvent-free conditions.
1,3-Dicarbonyl compounds have also been shown to be viable starting materials in place of the α-pyridinium methyl ketone salts. [ 7 ] For example, treatment of 1,3-diketone 14 with base in ethanol followed by ammonium acetate, acetic acid, the corresponding enone and a Lewis acid yields 3-acyltriarylpyridines of the form 15 . These acyl pyridine are attractive intermediates because they have an electrophilic handle that allows for additional functionality to be incorporated into the molecule. This allows for straightforward construction of complex polyaryl systems, an attractive method for library synthesis of drug targets containing functionalized pyridine moieties.
UUnlike the Hantzsch synthesis, [ 8 ] the Kröhnke method does not require oxidation to generate the desired product since the α-pyridinium methyl ketone already possesses the correct oxidation state.
Another advantage of the Kröhnke synthesis is its high atom economy . For example, the Chichibabin synthesis requires 2 equivalents of unsaturated starting material. [ 9 ] Additionally, the byproducts of the Kröhnke synthesis is water and pyridine, which allow for easy workup and purification. Unlike comparable methods for pyridine synthesis, the Kröhkne synthesis benefits from being a high-yielding one pot synthesis, which ultimately allows for abbreviation of synthetic pathways and further simplifies combinatorial library cataloging.
The broad scope of the Kröhnke pyridine synthesis has made it particularly useful for the synthesis of poly aryl systems including pyridyl, thienyl , and furanyl moieties as well. The method tolerates a broad array of aryl substitiuents on both the α-pyridinium methyl ketone fragment and the α, β-unsaturated carbonyl compounds and can thus be used to generate a wide catalog of poly-aryl systems. Additionally, electron-withdrawing groups and electron-donating groups on the incoming aryl substituents are both well tolerated. The Kröhnke synthesis can also employ alkyl and vinyl substituents giving moderated to good yields as well. [ 10 ] Due to its broad scope, the Kröhnke method has seen wide applicability to for the synthesis of bipyridines ( 16 ), terpyridines ( 17 ), quaterpyridines ( 18 ) and even up to septipyridines ( 19 ) as shown below. [ 11 ]
The Kröhnke method is featured in a solvent-free synthesis of triarylpyridines that proceeds via a homo-coupling of two diaryl substituted α, β-unsaturated carbonyl compounds. [ 12 ] This strategy offers a facile means for preparation of pyridnyl aryl systems that are important fragments of many useful drug scaffolds.
In 1992, Robinson and co-workers developed a similar pyridine synthesis using enamino nitriles as one of the three-carbon fragments in place of an α-pyridinium methyl ketone. [ 13 ] This improvement increases the reactivity of the system and allows for formation of fully substituted pyridines whereas use of an α-pyridinium methyl ketone requires that the 3- or 5- position on the resulting pyridine be unsubstituted. Kröhnke condensation of enamino nitrile 20 with enone 21 yielded fused pyridine 22 .
The mechanism of this Kröhnke-type reaction likely proceeds via a vinylogous cyanamide 23 which undergoes elimination of hydrocyanic acid , deprotonation to form enamine 24 and cyclization to form intermediate 25 , which is then dehydrated to form the desired pyridine product.
A clean one-pot Kröhnke method in aqueous media generates 4’-aryl-2,2’:6’, 2’’-terpyridines. [ 14 ] Reaction of aryl aldehyde 26 with two equivalents of 2-acetylpyridine ( 27 ) yielded terpyridines of the form 28 .
In addition to variations on the original method, a number of combinatorial studies using the Kröhnke synthesis and its variations have been employed to synthesize vast libraries of highly functionalized pyridines. Janda and co-workers utilized the general Kröhnke reaction scheme to generate a 220 compound library. [ 15 ] Various methyl ketones 29 and aldehydes 30 were coupled via aldol condensation to give enones of the form 31 . These compounds were then reacted with various α-pyridinium methyl ketones 32 to give the desired tri-substituted pyridine 33 .
In 2009, Tu and coworkers developed a 3 fragment, one-pot combinatorial strategy for developing 3-cyanoterpyridines 34 and 1-amino-2-acylterpyridines 35 . [ 16 ] These combinatorial variations of the Kröhnke reaction provide an efficient synthetic strategy to poly arylpyridine scaffolds. This methodology would also be advantageous for biological assays and screening experiments.
The Kröhnke methodology has also been utilized to generate a number of interesting metal-binding ligands since polypyridyl complexes such as bipyridine (bipy) have been used extensively as ligands. The Kröhnke synthesis was used to prepare a family of tetrahydroquinoline-based N, S-type ligands. [ 17 ] 2-thiophenylacetophenone ( 36 ) was reacted with iodine gas and pyridine in quantitative yield to generate acylmethylpyridinium iodide 37 . Reaction with a chiral cyclic α, β-unsaturated ketone derived from 2-(+)-carene yielded the desired N, S-type ligand 38 .
Novel, chiral P, N-ligands have been prepared using the Kröhnke method. [ 18 ] α-pyridinium acyl ketone salt 39 was cyclized with pinocarvone derivative 40 to generate pyridine 41 . The benzylic position of 41 was methylated and subsequent SnAr reaction with potassium diphenylphosphide to generate ligand 42 .
The Kröhnke reaction has also enjoyed applicability to the synthesis of a number of biologically active compounds in addition to ones cataloged in combinatorial studies. Kelly and co-workers developed a route to cyclo-2,2′:4′,4′′:2′′,2′′′:4′′′,4′′′′:2′′′′,2′′′′′:4′′′′′,4-sexipyridine utilizing the Kröhnke reactions as the key macrocyclization step. [ 19 ] Polypyridine complex 43 was treated with N-Bromosuccinimide in wet tetrahydrofuran followed by pyridine to generate the acylmethylpyridinium salt 44 which can then undergo the macrocyclization under standard conditions to yield the desired product 45 . The Kröhnke method in this synthesis was crucial due to the failure of other cyclization techniques such as the Glaser coupling or Ullmann coupling .
Another use of the Kröhnke pyridine synthesis was the generation of a number of 2,4,6-trisubstituted pyridines that were investigated as potential topoisomerase 1 inhibitors. [ 20 ] 2-acetylthiophene ( 46 ) was treated with iodine and pyridine to generate α-pyridinium acyl ketone 47 . Reaction with Michael acceptor 48 under standard conditions yielded functionalized pyridine 49 in 60% overall yield.
Ultimately, the Kröhnke pyridine synthesis offers a facile and straightforward approach to the synthesis of a wide breadth of functionalized pyridines and poly aryl systems. The Kröhnke methodology has been applied to a number of strategies towards interesting ligands and biologically relevant molecules. Additionally, the Kröhnke reaction and its variations offer a number of advantages than alternative methods to pyridine synthesis ranging from one-pot, organic solvent-free variations to high atom economy. | https://en.wikipedia.org/wiki/Kröhnke_pyridine_synthesis |
The Krüppel associated box ( KRAB ) domain is a category of transcriptional repression domains present in approximately 400 human zinc finger protein -based transcription factors (KRAB zinc finger proteins). [ 1 ] The KRAB domain typically consists of about 75 amino acid residues, while the minimal repression module is approximately 45 amino acid residues. [ 2 ] It is predicted to function through protein-protein interactions via two amphipathic helices. The most prominent interacting protein is called TRIM28 initially visualized as SMP1, [ 3 ] cloned as KAP1 [ 4 ] and TIF1-beta. [ 5 ] Substitutions for the conserved residues abolish repression.
Over 10 independently encoded KRAB domains have been shown to be effective repressors of transcription, suggesting this activity to be a common property of the domain. KRAB domains can be fused with dCas9 CRISPR tools to form even stronger repressors. [ 6 ]
The KRAB domain had initially been identified in 1988 as a periodic array of leucine residues separated by six amino acids 5’ to the zinc finger region of KOX1/ ZNF10 [ 7 ] coined heptad repeat of leucines (also known as a leucine zipper ). [ 8 ] Later, this domain was named in association with the C2H2-Zinc finger proteins Krüppel associated box (KRAB). [ 9 ] [ 10 ] The KRAB domain is confined to genomes from tetrapod organisms. The KRAB containing C2H2-ZNF genes constitute the largest sub-family of zinc finger genes. More than half of the C2H2-ZNF genes are associated with a KRAB domain in the human genome. They are more prone to clustering and are found in large clusters on the human genome. [ 11 ]
The KRAB domain presents one of the strongest repressors in the human genome. [ 2 ] Once the KRAB domain was fused to the tetracycline repressor (TetR), the TetR-KRAB fusion proteins were the first engineered drug-inducible repressor that worked in mammalian cells. [ 3 ] Two distinct types of KRAB A domains can be structurally and functionally distinguished. Ancestral KRAB A domains present in human PDRM9 proteins are even evolutionary conserved in mussel genomes. Modern KRAB A domain sequences are found in coelacanth latimeria chalumnae and in Lungfish genomes. [ 12 ]
Human genes encoding KRAB-ZFPs include KOX1/ ZNF10 , KOX8/ZNF708, ZNF43 , ZNF184 , ZNF91 , HPF4 , HTF10 and HTF34 .
This biochemistry article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Krüppel_associated_box |
In medicine, Kt/V is a number used to quantify hemodialysis and peritoneal dialysis treatment adequacy.
In the context of hemodialysis, Kt/V is a pseudo-dimensionless number; it is dependent on the pre- and post-dialysis concentration (see below). It is not the product of K and t divided by V , as would be the case in a true dimensionless number . [ 1 ] In peritoneal dialysis, it isn't dimensionless at all.
It was developed by Frank Gotch and John Sargent as a way for measuring the dose of dialysis when they analyzed the data from the National Cooperative Dialysis Study. [ 2 ] In hemodialysis the US National Kidney Foundation Kt / V target is ≥ 1.3, so that one can be sure that the delivered dose is at least 1.2. [ 3 ] In peritoneal dialysis the target is ≥ 1.7/week. [ 3 ]
Despite the name, Kt / V is quite different from standardized Kt/V .
K (clearance) multiplied by t (time) is a volume (since mL/min × min = mL, or L/h × h = L), and ( K × t ) can be thought of as the mL or L of fluid (blood in this case) cleared of urea (or any other solute) during the course of a single treatment. V also is a volume, expressed in mL or L. So the ratio of K × t / V is a so-called "dimensionless ratio" and can be thought of as a multiple of the volume of plasma cleared of urea divided by the distribution volume of urea. When Kt/V = 1.0, a volume of blood equal to the distribution volume of urea has been completely cleared of urea.
The relationship between Kt / V and the concentration of urea C at the end of dialysis can be derived from the first-order differential equation that describes exponential decay and models the clearance of any substance from the body where the concentration of that substance decreases in an exponential fashion:
where
From the above definitions it follows that d C d t {\displaystyle {\frac {dC}{dt}}} is the first derivative of concentration with respect to time, i.e. the change in concentration with time.
This equation is separable and can be integrated (assuming K and V are constant) as follows:
After integration,
where
If one takes the antilog of equation 2b the result is:
where
By integer exponentiation this can be written as:
where
The above equation can also be written as [ 2 ]
Normally we measure postdialysis serum urea nitrogen concentration C and compare this with the initial or predialysis level C 0 . The session length or time is t and this is measured by the clock. The dialyzer clearance K is usually estimated, based on the urea transfer ability of the dialyzer (a function of its size and membrane permeability), the blood flow rate, and the dialysate flow rate. [ 4 ] In some dialysis machines, the urea clearance during dialysis is estimated by testing the ability of the dialyzer to remove a small salt load that is added to the dialysate during dialysis.
The URR or Urea reduction ratio is simply the fractional reduction of urea during dialysis. So by definition, URR = 1 − C / C 0 . So 1−URR = C / C 0 . So by algebra, substituting into equation ( 4 ) above, since ln C / C 0 = − ln C 0 / C , we get:
Patient has a mass of 70 kg (154 lb) and gets a hemodialysis treatment that lasts 4 hours where the urea clearance is 215 mL/min.
Therefore:
This means that if you dialyze a patient to a Kt/V of 1.23, and measure the postdialysis and predialysis urea nitrogen levels in the blood, then calculate the URR, then −ln(1−URR) should be about 1.23.
The math does not quite work out, and more complicated relationships have been worked-out to account for the fluid removal (ultrafiltration) during dialysis as well as urea generation (see urea reduction ratio ). Nevertheless, the URR and Kt/V are so closely related mathematically, that their predictive power has been shown to be no different in terms of prediction of patient outcomes in observational studies.
The above analysis assumes that urea is removed from a single compartment during dialysis. In fact, this Kt/V is usually called the "single-pool" Kt/V. Due to the multiple compartments in the human body, a significant concentration rebound occurs following hemodialysis. Usually rebound lowers the Kt/V by about 15%. The amount of rebound depends on the rate of dialysis (K) in relation to the size of the patient (V). Equations have been devised to predict the amount of rebound based on the ratio of K/V, but usually this is not necessary in clinical practice. One can use such equations to calculate an "equilibrated Kt/V" or a "double-pool Kt/V", and some think that this should be used as a measure of dialysis adequacy, but this is not widely done in the United States, and the KDOQI guidelines (see below) recommend using the regular single pool Kt/V for simplicity.
Kt / V (in the context of peritoneal dialysis) was developed by Michael J. Lysaght in a series of articles on peritoneal dialysis. [ 5 ] [ 6 ]
The steady-state solution of a simplified mass transfer equation that is used to describe the mass exchange over a semi-permeable membrane and models peritoneal dialysis is
where
This can also be written as:
The mass generation (of urea), in steady state , can be expressed as the mass (of urea) in the effluent per time:
where
Lysaght, motivated by equations 6b and 6c , defined the value K D :
Lysaght uses "ml/min" for the clearance. In order to convert the above clearance (which is in m 3 /s) to ml/min one has to multiply by 60 × 1000 × 1000.
Once K D is defined the following equation is used to calculate Kt / V :
where
The 7/3 is used to adjust the Kt / V value so it can be compared to the Kt / V for hemodialysis, which is typically done thrice weekly in the USA.
To calculate the weekly Kt/V (for peritoneal dialysis) K D has to be in litres/day.
Weekly Kt / V is defined by the following equation:
Assume:
Then by equation 6d , K D is: 1.3334 × 10 −07 m 3 /s or 8.00 mL/min or 11.52 L/d.
Kt/V and the weekly Kt/V by equations 7a and 7b respectively are thus: 0.45978 and 1.9863.
On a practical level, in peritoneal dialysis the calculation of Kt/V is often relatively easy because the fluid drained is usually close to 100% saturated with urea, [ citation needed ] i.e. the dialysate has equilibriated with the body. Therefore, the daily amount of plasma cleared is simply the drain volume divided by an estimate of the patient's volume of distribution.
As an example, if someone is infusing four 2 liter exchanges a day, and drains out a total of 9 liters per day, then they drain 9 × 7 = 63 liters per week. If the patient has an estimated total body water volume V of about 35 liters, then the weekly Kt/V would be 63/35, or about 1.8.
The above calculation is limited by the fact that the serum concentration of urea is changing during dialysis. So ideally this should not be used as it has not taken in account the urea level in dialysate or serum...so it cannot be labelled as urea clearance
In automated PD this change cannot be ignored; thus, blood samples are usually measured at some time point in the day and assumed to be representative of an average value. The clearance is then calculated using this measurement.
Kt / V has been widely adopted because it was correlated with survival. Before Kt / V nephrologists measured the serum urea concentration (specifically the time-averaged concentration of urea (TAC of urea)), which was found not to be correlated with survival (due to its strong dependence on protein intake) and thus deemed an unreliable marker of dialysis adequacy.
Kt / V has been criticized because quite high levels can be achieved, particularly in smaller patients, during relatively short dialysis sessions. This is especially true for small people, where "adequate" levels of Kt/V often can be achieved over 2 to 2.5 hours. One important part of dialysis adequacy has to do with adequate removal of salt and water, and also of solutes other than urea, especially larger molecular weight substances and phosphorus. Phosphorus and similar molecular weights remain elusive to filtration of any degree. A number of studies suggest that a longer amount of time on dialysis, or more frequent dialysis sessions, lead to better results. There have been various alternative methods of measuring dialysis adequacy, most of which have proposed some number based on Kt/V and number of dialysis sessions per week, e.g., the standardized Kt/V , or simply number of dialysis sessions per week squared multiplied by the hours on dialysis per session; e.g. the hemodialysis product by Scribner and Oreopoulos [ 9 ] It is not practical to give long dialysis sessions (greater than 4.5 hours) thrice a week in a dialysis center during the day. Longer sessions can be practically delivered if dialysis is done at home. Most experience has been gained with such long dialysis sessions given at night. Some centers are offering every-other-night or thrice a week nocturnal dialysis. The benefits of giving more frequent dialysis sessions is also an area of active study, and new easy-to-use machines are permitting easier use of home dialysis, where 2–3+ hour sessions can be given 4–7 days per week.
One question in terms of Kt/V is, how much is enough? The answer has been based on observational studies, and the NIH-funded HEMO trial done in the United States, and also, on kinetic analysis.
For a US perspective, see the KDOQI clinical practice guidelines [ 10 ] and for a United Kingdom perspective see: U.K. Renal Association clinical practice guidelines [ 11 ] According to the US guidelines, for thrice a week dialysis a Kt / V (without rebound) should be 1.2 at a minimum with a target value of 1.4 (15% above the minimum values). However, there is suggestive evidence that larger amounts may need to be given to women, smaller patients, malnourished patients, and patients with clinical problems. The recommended minimum Kt/V value changes depending on how many sessions per week are given, and is reduced for patients who have a substantial degree of residual renal function.
For a US perspective, see: [ 12 ] For the United States, the minimum weekly Kt/V target used to be 2.0. This was lowered to 1.7 in view of the results of a large randomized trial done in Mexico, the ADEMEX trial, [ 13 ] and also from reanalysis of previous observational study results from the perspective of residual kidney function.
For a United Kingdom perspective see: [ 14 ] This is still in draft form. | https://en.wikipedia.org/wiki/Kt/V |
Kuber is a smokeless tobacco product, known for its highly addictive properties and its unique presentation disguised as a mouth freshener. It originated in India and has gained attention for its widespread use and impact on public health in various countries, including Uganda .
The product is typically sold in small sachets , with contents resembling tea leaves . This deceptive packaging has contributed to its accessibility and misuse. Kuber is known for its high nicotine content, making it more potent and addictive than traditional cigarettes . [ 1 ] Users commonly add it to tea or consume it directly by placing a pinch under the lower lip. [ 2 ]
Kuber's high nicotine content raises significant health concerns, including: [ 3 ]
Withdrawal from kuber can result in cravings and changes in mood and appetite. [ 3 ]
The legal status of kuber varies by region. In some countries , its sale and distribution, especially under the guise of a mouth freshener, have led to legal scrutiny and regulatory measures. For instance, in Uganda , the government has taken steps to ban the use of kuber due to its impact on public health , particularly among youth. [ 4 ] The governments of Malawi and Tanzania also banned the manufacture, import, sale, and consumption of kuber. [ 5 ] Despite efforts to ban kuber in Kenya , [ 6 ] it remains popular, particularly in Nairobi and Mombasa counties. [ 7 ]
The widespread use of kuber, especially among young people in high schools and colleges, has raised societal concerns. Its addictive nature and the ease of access have led to a rise in nicotine addiction among adolescents, with implications for long-term public health and social dynamics . [ 5 ] | https://en.wikipedia.org/wiki/Kuber_(tobacco) |
In mathematics, the Kubilius model relies on a clarification and extension of a finite probability space on which the behaviour of additive arithmetic functions can be modeled by sum of independent random variables . [ 1 ]
The method was introduced in Jonas Kubilius 's monograph Tikimybiniai metodai skaičių teorijoje (published in Lithuanian in 1959) [ 2 ] / Probabilistic Methods in the Theory of Numbers (published in English in 1964) . [ 3 ]
Eugenijus Manstavičius and Fritz Schweiger wrote about Kubilius's work in 1992, "the most impressive work has been done on the statistical theory of arithmetic functions which almost created a new research area called Probabilistic Number Theory. A monograph ( Probabilistic Methods in the Theory of Numbers ) devoted to this topic was translated into English in 1964 and became very influential." [ 4 ] : xi | https://en.wikipedia.org/wiki/Kubilius_model |
The Kubo formula , named for Ryogo Kubo who first presented the formula in 1957, [ 1 ] [ 2 ] is an equation which expresses the linear response of an observable quantity due to a time-dependent perturbation .
Among numerous applications of the Kubo formula, one can calculate the charge and spin susceptibilities of systems of electrons in response to applied electric and magnetic fields. Responses to external mechanical forces and vibrations can be calculated as well.
Consider a quantum system described by the (time independent) Hamiltonian H 0 {\displaystyle H_{0}} . The expectation value of a physical quantity at equilibrium temperature T {\displaystyle T} , described by the operator A ^ {\displaystyle {\hat {A}}} , can be evaluated as:
where β = 1 / k B T {\displaystyle \beta =1/k_{\rm {B}}T} is the thermodynamic beta , ρ ^ 0 {\displaystyle {\hat {\rho }}_{0}} is density operator, given by
and Z 0 = Tr [ ρ ^ 0 ] {\displaystyle Z_{0}=\operatorname {Tr} \,\left[{\hat {\rho }}_{0}\right]} is the partition function .
Suppose now that just after some time t = t 0 {\displaystyle t=t_{0}} an external perturbation is applied to the system. The perturbation is described by an additional time dependence in the Hamiltonian:
where θ ( t ) {\displaystyle \theta (t)} is the Heaviside function (1 for positive times, 0 otherwise) and V ^ ( t ) {\displaystyle {\hat {V}}(t)} is hermitian and defined for all t , so that H ^ ( t ) {\displaystyle {\hat {H}}(t)} has for positive t − t 0 {\displaystyle t-t_{0}} again a complete set of real eigenvalues E n ( t ) . {\displaystyle E_{n}(t).} But these eigenvalues may change with time.
However, one can again find the time evolution of the density matrix ρ ^ ( t ) {\displaystyle {\hat {\rho }}(t)} rsp. of the partition function Z ( t ) = Tr [ ρ ^ ( t ) ] , {\displaystyle Z(t)=\operatorname {Tr} \,\left[{\hat {\rho }}(t)\right],} to evaluate the expectation value of
The time dependence of the states | n ( t ) ⟩ {\displaystyle |n(t)\rangle } is governed by the Schrödinger equation
which thus determines everything, corresponding of course to the Schrödinger picture . But since V ^ ( t ) {\displaystyle {\hat {V}}(t)} is to be regarded as a small perturbation, it is convenient to now use instead the interaction picture representation, | n ^ ( t ) ⟩ , {\displaystyle \left|{\hat {n}}(t)\right\rangle ,} in lowest nontrivial order. The time dependence in this representation is given by | n ( t ) ⟩ = e − i H ^ 0 t / ℏ | n ^ ( t ) ⟩ = e − i H ^ 0 t / ℏ U ^ ( t , t 0 ) | n ^ ( t 0 ) ⟩ , {\displaystyle |n(t)\rangle =e^{-i{\hat {H}}_{0}t/\hbar }\left|{\hat {n}}(t)\right\rangle =e^{-i{\hat {H}}_{0}t/\hbar }{\hat {U}}(t,t_{0})\left|{\hat {n}}(t_{0})\right\rangle ,} where by definition for all t and t 0 {\displaystyle t_{0}} it is: | n ^ ( t 0 ) ⟩ = e i H ^ 0 t 0 / ℏ | n ( t 0 ) ⟩ {\displaystyle \left|{\hat {n}}(t_{0})\right\rangle =e^{i{\hat {H}}_{0}t_{0}/\hbar }|n(t_{0})\rangle }
To linear order in V ^ ( t ) {\displaystyle {\hat {V}}(t)} , we have
Thus one obtains the expectation value of A ^ ( t ) {\displaystyle {\hat {A}}(t)} up to linear order in the perturbation:
thus [ 3 ]
⟨ A ^ ( t ) ⟩ = ⟨ A ^ ⟩ 0 − i ℏ ∫ t 0 t d t ′ ⟨ [ A ^ ( t ) , V ^ ( t ′ ) ] ⟩ 0 {\displaystyle \langle {\hat {A}}(t)\rangle =\left\langle {\hat {A}}\right\rangle _{0}-{\frac {i}{\hbar }}\int _{t_{0}}^{t}dt'\left\langle \left[{\hat {A}}(t),{\hat {V}}{\mathord {\left(t'\right)}}\right]\right\rangle _{0}}
The brackets ⟨ ⟩ 0 {\displaystyle \langle \rangle _{0}} mean an equilibrium average with respect to the Hamiltonian H 0 . {\displaystyle H_{0}.} Therefore, although the result is of first order in the perturbation, it involves only the zeroth-order eigenfunctions, which is usually the case in perturbation theory and moves away all complications which otherwise might arise for t > t 0 {\displaystyle t>t_{0}} .
The above expression is true for any kind of operators. (see also Second quantization ) [ 4 ] | https://en.wikipedia.org/wiki/Kubo_formula |
Note:
The Kuching Declaration ( Iban : Jaku penetap Kuching ; Malay : Perisytiharan Kuching ) was adopted by the three component parties of the Pakatan Rakyat (i.e. People's Justice Party (PKR) signed by Anwar Ibrahim and Baru Bian , Democratic Action Party (DAP) signed by Lim Kit Siang and Wong Ho Leng , Pan-Malaysian Islamic Party (PAS) signed by Abdul Hadi Awang and Adam Ahid ) on 16 September 2012. The signing was held at Chonglin Park, Kuching , Sarawak , coinciding with Malaysia Day celebrations . The declaration, which was written in English, pledges and promises to honor the spirit of the Malaysia Agreement of 1963 to the nations and the peoples of the states of Sarawak and Sabah [ 1 ] when they form the next government of Malaysia. [ 2 ]
Lest we forget, and lest all the peoples of our great Nation of Malaysia forget, we the undersigned do once again firmly, resolutely and unequivocally pledge and promise before the whole Nation of Malaysia as our witness, on this historically day 16 September 2012, in the City of Kuching, and on behalf of our respective parties and Pakatan Rakyat will honour all its pledges and promises to the peoples of Malaysia.
We will honourably execute all the policies set forth in the Buku Jingga so that Malaysia will once again be a great Nation, her peoples prosperous, her future secure and peaceful, and her name celebrated by all the nations of the world.
We will honour the spirit of the Malaysia Agreement of 1963 which our founding fathers put their hands to, and as a sign of our deep commitment to the peoples of Sarawak and Sabah , [ 1 ] consistent with democratic principles and justice for all Malaysians, in particular:
We, the undersigned, make this declaration as an incontrovertible contract between the Pakatan Rakyat and the peoples of Malaysia, this historic day of 16 September 2012 on Malaysia Day, so that it may ring out resoundingly from Malaysia's high forest hills down to the open sea; so that freedom may ever reign; and our peoples live in unity!
In witness whereof the undersigned, being duly authorised thereto, have signed this Declaration, and all the peoples of Malaysia being witnesses thereof.
Done at Kuching, this 16th day of September, 2012, in six copies of which one shall be deposited with each of the signatories. [ 3 ]
Sebedau kami nyau enda ingat agi, lalu sebedau bala orang di menua Malaysia tu enda ingat agi, kami, ti ngelabuhka sainjari ditu, sekali agi bersemaya ba mua menua Malaysia tu jadika saksi kami, kena sehari tu 16hb Sembilan, 2012, di negeri Kuching, lalu mega ngarika semua parti politik enggau Pakatan Rakyat, iya nya, lebuh kami numbuhka perintah baru di menua Malaysia, Pakatan Rakyat deka bebasa sereta ngamatika semua danji enggau semaya iya ngagai peranak menua Malaysia.
Kami deka enggau basa ti tinggi berjalaika semua polisi ti udah ditetepka kami dalam Bup Jingga ngambika menua Malaysia ulih nyadi sebengkah menua ti tampak, bala peranak menua iya tau bertambah kaya, dudi hari iya tau likun sereta selamat, lalu nama iya deka dikemesaika sereta dipelandikka menua bukai di dunya tu.
Kami deka bebasaka semengat dalam Sempekat Malaysia tahun 1963 ti ditanggam bala tuai-tuai menua kami, ti laluadika kelai penaluk kami ke orang Sarawak enggau Sabah , [ 1 ] enda begilik penemu dalam atur demokrasi lalu enda bebelah bagi ba semua peranak Malaysia, kelebih agi:
: PEKARA KESATU: SEKUNSI :Kami deka mulaika semengat dalam Sempekat Malaysia lalu berakupka penuduk Sarawak enggau Sabah dalam Serakup Malaysia enggau chara meri pulai kuasa merintah diri (autonomi) ke Sarawak enggau Sabah [ 1 ] belalauka Tampuk Adat Perintah Besai.
: PEKARA KEDUA: BEPENGARI SAMA RATA :Kami deka ngemanahka agi kaul entara Sarawak , Sabah [ 1 ] enggau Semenanjung Malaysia nengah atur bekunsi kuasa ti nyeridi semengat Sempekat Malaysia .
: PEKARA KETIGA: PERANAK MENUA :Kami deka numbuhka Komisyen Di-Raja dikena mutarka penanggul peranak menua luar ke enda sah nyadi peranak menua Malaysia, pia mega penanggul tu ke udah nuntung Sarawak enggau Sabah.
: PEKARA KEEMPAT: MERI PULAI TANAH BANSA ASAL (NCR) :Kami deka ngamatka sereta ngemendarka kuasa ti udah ditetapka dalam Adat Menua Sarawak enggau Sabah dikena numbuhka Komisyen Tanah dikena mansik, mutarka penyarut, nyikapka atur nyukat lalu meri pulai hak Tanah Bansa Asal ba Tanah Bansa Asal.
: PEKARA KELIMA: PENGELANDIK GAWA PERANAK MENUA SARAWAK ENGGAU SABAH :Kami deka ngamat sereta ngemendarka padu mangku pengawa ke semua peranak menua Sarawak enggau Sabah nyadi kepala dalam Opis-Opis Perintah di menua sida empu lalu ngena kuasa-kuasa Sekretari-Sekretari Menua ke dua buah menua nya lalu pia mega ngenuluka padu mangku pengawa ke peranak menua Sarawak enggau Sabah ba ikas Perintah Besai ti bekereja di menua Sarawak enggau Sabah.
: PEKARA KEENAM: ASIL MINYAK TI PATUT ::Kami deka nikika bayar minyak petroleum enggau asil hidrokarbon ke menua Sarawak enggau Sabah ngagai ikas 20% ari penyampau 5% ke ngelama tu.
: PEKARA KETUJUH: PEMANSANG SAMA RATA :Kami deka nikika ikas pemansang infrastrukta di menua Sarawak enggau Sabah sama bela sebaka enggau ikas di Semenanjung Malaysia.
kami, jadika ti ngengkahka sainjari, ngaga penetap ti jadika semaya entara Pakatan Rakyat enggau semua peranak Malaysia, kena seharitu 16hb September 2012, beserimbai enggau Hari Malaysia, ngambika auh iya didinga ari tuchung munggu lalu nelusur ngili batang sungai nyentuk ke tasik besai; awakka kitai dilepaska; lalu peranak raban bansa kitai ulih idup dalam penyerakup !
Disaksika ngelabuhka sainjari, ti diberi kuasa, lalu semua peranak Malaysia jadika saksi ditu.
Disain di nengeri Kuching, kena seharitu 16hb Sembilan, 2012, dalam enam salin ti endur genap salin deka disimpan genap iku orang ti ngelabuhka sainjari.
Jangan sampai kita lupa, dan jangan sampai masyarakat Malaysia yang terbilang ini betah terlupa bahawasanya kami, yang bertandatangan di bawah, sekali lagi dengan penuh tekad dan iltizam sepenuh jiwa dan raga berikrar dan berjanji bersaksikan seluruh warganegara Malaysia, pada hari ini yakni pada tarikh bersejarah 16 September 2012, di bandaraya Kuching, bagi pihak parti-parti yang berkenaan dan Pakatan Rakyat seluruhnya bila mana kami mengijtirafkan kerajaan baru buat Malaysia, Pakatan Rakyat akan melunasi setiap ikrar dan janji kami pada setiap lapisan masyarakat dan rakyat Malaysia.
Bahawasanya kami akan memelihara dan menjunjung setia dasar-dasar yang diterapkan didalam Buku Jingga supaya Malaysia bakal bangkit semula sebagai negara yang makmur, agar terwujud rakyat yang cemerlang, bersatu padu dan berkembang maju, dengan masa depan yang selamat dan aman damai, agar namanya harum di serata pelusuk dunia.
Bahawa sesungguhnya kami berpegang teguh dengan semangat Perjanjian Malaysia 1963 yang telah dicetuskan dan dikendalikan oleh pemimpin-pemimpin kami yang terdahulu, dan sebagai bukti dan tanda utuhnya kesetiaan kami terhadap rakyat Serawak dan Sabah , [ 1 ] berteraskan prinsip-prinsip demokrasi dan keadilan untuk rakyat Malaysia seluruhnya, dengan pengkhususan istimewa terhadap perkara-perkara berikut:
: PERKARA 1: KERJASAMA BERSEKUTU :Kami akan memulihara intipati Perjanjian Malaysia dan memperkasakan kedudukan Sarawak dan Sabah [ 1 ] setanding dengan Semenanjung Malaysia dengan mengembalikan autonomi Sarawak dan Sabah mengikut lunas Perlembagaan Persekutuan.
: PERKARA 2: PERWAKILAN YANG ADIL :Kami akan mempertingkatkan integrasi nasional di antara Sarawak , Sabah [ 1 ] dan Semenanjung Malaysia melalui penggemblengan tenaga dan perkongsian kuasa secara adil yang menjunjung semangat Perjanjian Malaysia .
: PERKARA 3: KEWARGANEGARAAN :Kami akan menubuhkan Suruhanjaya DiRaja bagi menyelesaikan masalah nasional berkaitan pendatang tanpa izin dan isu-isu kerakyatan, khususnya di Sarawak dan Sabah.
: PERKARA 4: PENGIKTIRAFAN HAK KE ATAS TANAH ADAT :Kami akan menghormati dan mengiktiraf hak anak watan ke atas tanah adat sepertimana yang tertakluk dan termaktub dalam Undang-undang Negeri Sarawak dan Sabah dengan menubuhkan Suruhanjaya Tanah untuk menyiasat, menyelesaikan pertikaian, memperbaharui, menyelidik dan memulihkan Hak Adat Peribumi ke atas pemilikan Tanah Adat.
: PERKARA 5: KECEKAPAN DAN KEMANDIRIAN SARAWAK DAN SABAH :Kami akan mengiktiraf perlantikan rakyat Sarawak dan Sabah untuk memimpin dan memegang jawatan di Jabatan Kerajaan di negeri masing-masing, dikuatkuasakan oleh Setiausaha-setiausaha kerajaan kedua-dua negeri dan memberikan keutamaan kepada rakyat Sarawak dan Sabah di peringkat Kerajaan Persekutuan untuk menjalankan tugas di negeri masing-masing.
: PERKARA 6: KEADILAN DALAM ISU MINYAK :Kami akan menaikkan bayaran royalti ke atas petroleum dan sumber sumber hidrokarbon kepada Sarawak dan Sabah dari 5% ke 20%.
: PERKARA 7: PEMBANGUNAN SEKATA :Kami akan membawa tahap kemajuan infrastruktur di Sarawak dan Sabah setanding dengan perkembangan di Semenanjung Malaysia.
Dengan ini, kami yang bertanda tangan seperti tertera di bawah, mengiktiraf perisytiharan ini sebagai satu perjanjian yang tidak boleh disangkal dan dipersoalkan lagi, termaktub di antara Pakatan Rakyat dan segenap lapisan masyarakat Malaysia, pada hari ini, tanggal 16 September 2012, bersempena dengan sambutan Hari Malaysia, agar tersohorlah gegak gempitanya dari puncak tertinggi di bumi Malaysia tercinta, dan akan tersebar seluas samudera yang terbentang supaya kebebasan dapat kita capai dan kita semua boleh hidup bersatu padu dalam suasana aman dan damai.
Sebagai saksi padanya, penandatangan yang diberi kuasa dalam hal ini, telah menandatangani Perisytiharan ini, dan semua rakyat Malaysia adalah saksi kepadanya.
Dimeterai di Kuching, pada 16 haribulan September tahun 2012, dalam enam salinan di mana setiap satu darinya akan disimpan oleh setiap penandatangan. | https://en.wikipedia.org/wiki/Kuching_Declaration |
A Kugelrohr ( German for "ball tube") is a short-path vacuum distillation apparatus [ 1 ] : 150 typically used to distill relatively small amounts of compounds with high boiling points (usually greater than 300 °C) under greatly reduced pressure.
" Short path " refers to the short distance that the vapors of the distillate need to travel, which helps reduce loss and speed up collection of the distillate. This type of distillation is generally performed under vacuum to prevent the compound from charring due to atmospheric oxygen , as well as to allow the distillation to proceed at a lower temperature. [ 2 ] [ 3 ]
The apparatus consists of a tube furnace or other electric heater controlled by a thermostat, and two or more bulbs connected with ground glass joints . The compound to be distilled is placed in the terminal bulb. The other bulbs can be used to collect the distillates sequentially, when the desired fraction is being collected the bulb is cooled with water or ice to aid condensation. A motor drive is often used to rotate the string of bulbs to reduce bumping , give even heating, and increase the surface area for evaporation. | https://en.wikipedia.org/wiki/Kugelrohr |
Kugel–Khomskii coupling describes a coupling between the spin and orbital degrees of freedom in a solid; it is named after the Russian physicists Kliment I. Kugel (Климент Ильич Кугель) and Daniel I. Khomskii (Daniil I. Khomskii, Даниил Ильич Хомский). The Hamiltonian used is:
This condensed matter physics -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Kugel–Khomskii_coupling |
In game theory , Kuhn's theorem relates perfect recall , mixed and unmixed strategies and their expected payoffs. It was formalized by Harold W. Kuhn in 1953. [ 1 ]
The theorem states that in a game where players may remember all of their previous moves/states of the game available to them, for every mixed strategy there is a behavioral strategy that has an equivalent payoff (i.e., the strategies are equivalent). The theorem does not specify what this strategy is, only that it exists. It is valid both for finite games, as well as infinite games (i.e., games with continuous choices, or iterated infinitely). [ 2 ]
This game theory article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Kuhn's_theorem |
The Kuhn length is a theoretical treatment, developed by Werner Kuhn , in which a real polymer chain is considered as a collection of N {\displaystyle N} Kuhn segments each with a Kuhn length b {\displaystyle b} . Each Kuhn segment can be thought of as if they are freely jointed with each other. [ 1 ] [ 2 ] [ 3 ] [ 4 ] Each segment in a freely jointed chain can randomly orient in any direction without the influence of any forces, independent of the directions taken by other segments. Instead of considering a real chain consisting of n {\displaystyle n} bonds and with fixed bond angles, torsion angles, and bond lengths, Kuhn considered an equivalent ideal chain with N {\displaystyle N} connected segments, now called Kuhn segments, that can orient in any random direction.
The length of a fully stretched chain is L = N b {\displaystyle L=Nb} for the Kuhn segment chain. [ 5 ] In the simplest treatment, such a chain follows the random walk model, where each step taken in a random direction is independent of the directions taken in the previous steps, forming a random coil . The mean square end-to-end distance for a chain satisfying the random walk model is ⟨ R 2 ⟩ = N b 2 {\displaystyle \langle R^{2}\rangle =Nb^{2}} .
Since the space occupied by a segment in the polymer chain cannot be taken by another segment, a self-avoiding random walk model can also be used. The Kuhn segment construction is useful in that it allows complicated polymers to be treated with simplified models as either a random walk or a self-avoiding walk , which can simplify the treatment considerably.
For an actual homopolymer chain (consists of the same repeat units) with bond length l {\displaystyle l} and bond angle θ with a dihedral angle energy potential, [ clarification needed ] the mean square end-to-end distance can be obtained as
The fully stretched length L = n l cos ( θ / 2 ) {\displaystyle L=nl\,\cos(\theta /2)} . By equating the two expressions for ⟨ R 2 ⟩ {\displaystyle \langle R^{2}\rangle } and the two expressions for L {\displaystyle L} from the actual chain and the equivalent chain with Kuhn segments, the number of Kuhn segments N {\displaystyle N} and the Kuhn segment length b {\displaystyle b} can be obtained.
For worm-like chain , Kuhn length equals two times the persistence length . [ 6 ] | https://en.wikipedia.org/wiki/Kuhn_length |
In mathematics , Kuiper's theorem (after Nicolaas Kuiper ) is a result on the topology of operators on an infinite-dimensional, complex Hilbert space H . It states that the space GL( H ) of invertible bounded endomorphisms of H is such that all maps from any finite complex Y to GL( H ) are homotopic to a constant, for the norm topology on operators.
A significant corollary, also referred to as Kuiper's theorem , is that this group is weakly contractible , ie. all its homotopy groups are trivial. This result has important uses in topological K-theory .
For finite dimensional H , this group would be a complex general linear group and not at all contractible. In fact it is homotopy equivalent to its maximal compact subgroup , the unitary group U of H . The proof that the complex general linear group and unitary group have the same homotopy type is by the Gram-Schmidt process , or through the matrix polar decomposition , and carries over to the infinite-dimensional case of separable Hilbert space , basically because the space of upper triangular matrices is contractible as can be seen quite explicitly. The underlying phenomenon is that passing to infinitely many dimensions causes much of the topological complexity of the unitary groups to vanish; but see the section on Bott's unitary group, where the passage to infinity is more constrained, and the resulting group has non-trivial homotopy groups.
It is a surprising fact that the unit sphere , sometimes denoted S ∞ , in infinite-dimensional Hilbert space H is a contractible space , while no finite-dimensional spheres are contractible. This result, certainly known decades before Kuiper's, may have the status of mathematical folklore , but it is quite often cited. [ 1 ] [ 2 ] In fact more is true: S ∞ is diffeomorphic to H , which is certainly contractible by its convexity. [ 3 ] One consequence is that there are smooth counterexamples to an extension of the Brouwer fixed-point theorem to the unit ball in H . [ 4 ] The existence of such counter-examples that are homeomorphisms was shown in 1943 by Shizuo Kakutani , who may have first written down a proof of the contractibility of the unit sphere. [ 5 ] But the result was anyway essentially known (in 1935 Andrey Nikolayevich Tychonoff showed that the unit sphere was a retract of the unit ball). [ 6 ]
The result on the group of bounded operators was proved by the Dutch mathematician Nicolaas Kuiper , for the case of a separable Hilbert space; the restriction of separability was later lifted. [ 7 ] The same result, but for the strong operator topology rather than the norm topology, was published in 1963 by Jacques Dixmier and Adrien Douady . [ 8 ] The geometric relationship of the sphere and group of operators is that the unit sphere is a homogeneous space for the unitary group U . The stabiliser of a single vector v of the unit sphere is the unitary group of the orthogonal complement of v ; therefore the homotopy long exact sequence predicts that all the homotopy groups of the unit sphere will be trivial. This shows the close topological relationship, but is not in itself quite enough, since the inclusion of a point will be a weak homotopy equivalence only, and that implies contractibility directly only for a CW complex . In a paper published two years after Kuiper's, [ 9 ]
There is another infinite-dimensional unitary group, of major significance in homotopy theory , that to which the Bott periodicity theorem applies. It is certainly not contractible. The difference from Kuiper's group can be explained: Bott's group is the subgroup in which a given operator acts non-trivially only on a subspace spanned by the first N of a fixed orthonormal basis { e i }, for some N , being the identity on the remaining basis vectors.
An immediate consequence, given the general theory of fibre bundles , is that every Hilbert bundle is a trivial bundle . [ 10 ]
The result on the contractibility of S ∞ gives a geometric construction of classifying spaces for certain groups that act freely on it, such as the cyclic group with two elements and the circle group . The unitary group U in Bott's sense has a classifying space BU for complex vector bundles (see Classifying space for U(n) ). A deeper application coming from Kuiper's theorem is the proof of the Atiyah–Jänich theorem (after Klaus Jänich and Michael Atiyah ), stating that the space of Fredholm operators on H , with the norm topology, represents the functor K (.) of topological (complex) K-theory, in the sense of homotopy theory. This is given by Atiyah. [ 11 ]
The same question may be posed about invertible operators on any Banach space of infinite dimension. Here there are only partial results. Some classical sequence spaces have the same property, namely that the group of invertible operators is contractible. On the other hand, there are examples known where it fails to be a connected space . [ 12 ] Where all homotopy groups are known to be trivial, the contractibility in some cases may remain unknown. | https://en.wikipedia.org/wiki/Kuiper's_theorem |
Kukulkanins are chalcones isolated from Mexican Mimosa . [ 1 ] [ 2 ]
This organic chemistry article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Kukulkanin |
The Kulinkovich reaction describes the organic synthesis of substituted cyclopropanols through reaction of esters with dialkyldialkoxytitanium reagents, which are generated in situ from Grignard reagents containing a hydrogen in beta-position and titanium(IV) alkoxides such as titanium isopropoxide . [ 1 ] This reaction was first reported by Oleg Kulinkovich and coworkers in 1989. [ 2 ]
Titanium catalysts are ClTi(OiPr) 3 or Ti(OiPr) 4 , ClTi(OtBu) 3 or Ti(OtBu) 4 , Grignard reagents are EtMgX, PrMgX or BuMgX. Solvents can be Et 2 O, THF, toluene. Tolerated Functional Groups : Ethers R–O–R, R–S–R, Imines RN=CHR. Amides , primary and secondary amines . Carbamates typically do not tolerate the reaction conditions, but tert-butyl carbamates (N-Boc derivatives) survive the transformation.
An asymmetric version of this reaction is also known with a TADDOL -based catalyst. [ 1 ]
The generally accepted reaction mechanism initially utilizes two successive stages of transmetallation of the committed Grignard reagent , leading to an intermediate dialkyldiisopropoxytitanium complex. This complex undergoes a dismutation to give an alkane molecule and a titanacyclopropane 1 . The insertion of the carbonyl group of the ester in the weakest carbon-titanium bond, leads to an oxatitanacyclopentane 2 being rearranged to ketone 3 . Lastly, the insertion of the carbonyl group of 3 in the residual carbon-titanium connection forms a cyclopropane ring. In the transition state of this elementary stage, which is the limiting stage of the reaction, an agostic interaction stabilizing between the beta hydrogen and the R2 group and the titanium atom was called upon to explain the diastereoselectivity observed. Complex 4 obtained is a tetraalkyloxytitanium compound able to play a part similar to that of the starting tetraisopropyloxytitanate, which closes the catalytic cycle . At the end of the reaction, the product is mainly in the shape of the magnesium alcoholate 5 , giving the cyclopropanol after hydrolysis by the reaction medium.
The step leading to the titanacyclopropane has been scrutinized computationally. Although the dialkyldiisopropoxytitanium complex has been proposed to undergo β hydrogen elimination followed by C–H reductive elimination to give the alkane and 1 , it was found that β hydrogen abstraction by the alkyl group, leading directly to products without the intermediate titanium hydride, is a more favorable process. [ 3 ]
In broad strokes, and in a formal retrosynthetic sense, titanacyclopropane 1 behaves like a 1,2-dianion which adds into the ester twice: after the first addition into the ester, the resultant tetrahedral intermediate 2 collapses to give β-titanio ketone 3 , which undergoes a second intramolecular addition to give the titanium salt of the cyclopropanol ( 4 ). (This species then undergoes transmetalation with Grignard reagent to regenerate 1 and close the catalytic cycle and give the product in the form of the magnesium salt ( 5 ).)
The reaction mechanism has been the subject of theoretical analysis. [ 4 ] Certain points remain nevertheless obscure. Intermediate titanium complexes of the ate type have been proposed by Kulinkovich. [ 5 ]
In 1993, the team of Kulinkovich highlighted the aptitude of the titanacyclopropanes to undergo ligand exchange with olefins. [ 6 ] This discovery was important, because it gave access to cyclopropanols more functionalized by making economic use of the Grignard of which normally at least two equivalents should have been engaged to obtain good yields. Cha and its team introduced the use of cyclic Grignard reagents, particularly adapted for these reactions. [ 7 ]
The methodology has been extended to intramolecular reactions [ 8 ]
With amides instead of esters the reaction product is an aminocyclopropane in the De Meijere variation [ 9 ] [ 10 ]
The intramolecular reaction is also known: [ 11 ] [ 12 ] [ 13 ] [ 14 ] [ 15 ] [ 16 ] [ 17 ] [ 18 ] [ 19 ] [ 20 ]
In the Bertus-Szymoniak variation the substrate is a nitrile and the reaction product a cyclopropane with a primary amine group. [ 21 ] [ 22 ]
The reaction mechanism is akin the Kulinkovich reaction: | https://en.wikipedia.org/wiki/Kulinkovich_reaction |
Karlsruhe Accurate Arithmetic ( KAA ), or Karlsruhe Accurate Arithmetic Approach ( KAAA ), augments conventional floating-point arithmetic with good error behaviour with new operations to calculate scalar products with a single rounding error. [ 1 ]
The foundations for KAA were developed at the University of Karlsruhe starting in the late 1960s. [ 2 ] [ 3 ] [ 4 ]
This computing article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Kulisch_arithmetic |
In the mathematical field of differential geometry , the Kulkarni–Nomizu product (named for Ravindra Shripad Kulkarni and Katsumi Nomizu ) is defined for two (0, 2) - tensors and gives as a result a (0, 4) -tensor.
If h and k are symmetric (0, 2) -tensors, then the product is defined via: [ 1 ]
where the X j are tangent vectors and | ⋅ | {\displaystyle |\cdot |} is the matrix determinant . Note that h ∧ ◯ k = k ∧ ◯ h {\displaystyle h{~\wedge \!\!\!\!\!\!\!\!\;\bigcirc ~}k=k{~\wedge \!\!\!\!\!\!\!\!\;\bigcirc ~}h} , as it is clear from the second expression.
With respect to a basis { ∂ i } {\displaystyle \{\partial _{i}\}} of the tangent space, it takes the compact form
where [ … ] {\displaystyle [\dots ]} denotes the total antisymmetrisation symbol .
The Kulkarni–Nomizu product is a special case of the product in the graded algebra
where, on simple elements,
( ⊙ {\displaystyle \odot } denotes the symmetric product ).
The Kulkarni–Nomizu product of a pair of symmetric tensors has the algebraic symmetries of the Riemann tensor . [ 2 ] For instance, on space forms (i.e. spaces of constant sectional curvature ) and two-dimensional smooth Riemannian manifolds, the Riemann curvature tensor has a simple expression in terms of the Kulkarni–Nomizu product of the metric g = g i j d x i ⊗ d x j {\displaystyle g=g_{ij}dx^{i}\otimes dx^{j}} with itself; namely, if we denote by
the (1, 3) -curvature tensor and by
the Riemann curvature tensor with R i j k l = g i m R m j k l {\displaystyle R_{ijkl}=g_{im}{R^{m}}_{jkl}} , then
where Scal = tr g Ric = R i i {\displaystyle \operatorname {Scal} =\operatorname {tr} _{g}\operatorname {Ric} ={R^{i}}_{i}} is the scalar curvature and
is the Ricci tensor , which in components reads R i j = R k i k j {\displaystyle R_{ij}={R^{k}}_{ikj}} .
Expanding the Kulkarni–Nomizu product g ∧ ◯ g {\displaystyle g~\wedge \!\!\!\!\!\!\!\!\;\bigcirc ~g} using the definition from above, one obtains
This is the same expression as stated in the article on the Riemann curvature tensor .
For this very reason, it is commonly used to express the contribution that the Ricci curvature (or rather, the Schouten tensor ) and the Weyl tensor each makes to the curvature of a Riemannian manifold . This so-called Ricci decomposition is useful in differential geometry .
When there is a metric tensor g , the Kulkarni–Nomizu product of g with itself is the identity endomorphism of the space of 2-forms, Ω 2 ( M ), under the identification (using the metric) of the endomorphism ring End(Ω 2 ( M )) with the tensor product Ω 2 ( M ) ⊗ Ω 2 ( M ).
A Riemannian manifold has constant sectional curvature k if and only if the Riemann tensor has the form
where g is the metric tensor . | https://en.wikipedia.org/wiki/Kulkarni–Nomizu_product |
In information theory and statistics , Kullback's inequality is a lower bound on the Kullback–Leibler divergence expressed in terms of the large deviations rate function . [ 1 ] If P and Q are probability distributions on the real line, such that P is absolutely continuous with respect to Q , i.e. P << Q , and whose first moments exist, then D K L ( P ∥ Q ) ≥ Ψ Q ∗ ( μ 1 ′ ( P ) ) , {\displaystyle D_{KL}(P\parallel Q)\geq \Psi _{Q}^{*}(\mu '_{1}(P)),} where Ψ Q ∗ {\displaystyle \Psi _{Q}^{*}} is the rate function, i.e. the convex conjugate of the cumulant -generating function, of Q {\displaystyle Q} , and μ 1 ′ ( P ) {\displaystyle \mu '_{1}(P)} is the first moment of P . {\displaystyle P.}
The Cramér–Rao bound is a corollary of this result.
Let P and Q be probability distributions (measures) on the real line, whose first moments exist, and such that P << Q . Consider the natural exponential family of Q given by Q θ ( A ) = ∫ A e θ x Q ( d x ) ∫ − ∞ ∞ e θ x Q ( d x ) = 1 M Q ( θ ) ∫ A e θ x Q ( d x ) {\displaystyle Q_{\theta }(A)={\frac {\int _{A}e^{\theta x}Q(dx)}{\int _{-\infty }^{\infty }e^{\theta x}Q(dx)}}={\frac {1}{M_{Q}(\theta )}}\int _{A}e^{\theta x}Q(dx)} for every measurable set A , where M Q {\displaystyle M_{Q}} is the moment-generating function of Q . (Note that Q 0 = Q .) Then D K L ( P ∥ Q ) = D K L ( P ∥ Q θ ) + ∫ supp P ( log d Q θ d Q ) d P . {\displaystyle D_{KL}(P\parallel Q)=D_{KL}(P\parallel Q_{\theta })+\int _{\operatorname {supp} P}\left(\log {\frac {\mathrm {d} Q_{\theta }}{\mathrm {d} Q}}\right)\mathrm {d} P.} By Gibbs' inequality we have D K L ( P ∥ Q θ ) ≥ 0 {\displaystyle D_{KL}(P\parallel Q_{\theta })\geq 0} so that D K L ( P ∥ Q ) ≥ ∫ supp P ( log d Q θ d Q ) d P = ∫ supp P ( log e θ x M Q ( θ ) ) P ( d x ) {\displaystyle D_{KL}(P\parallel Q)\geq \int _{\operatorname {supp} P}\left(\log {\frac {\mathrm {d} Q_{\theta }}{\mathrm {d} Q}}\right)\mathrm {d} P=\int _{\operatorname {supp} P}\left(\log {\frac {e^{\theta x}}{M_{Q}(\theta )}}\right)P(dx)} Simplifying the right side, we have, for every real θ where M Q ( θ ) < ∞ : {\displaystyle M_{Q}(\theta )<\infty :} D K L ( P ∥ Q ) ≥ μ 1 ′ ( P ) θ − Ψ Q ( θ ) , {\displaystyle D_{KL}(P\parallel Q)\geq \mu '_{1}(P)\theta -\Psi _{Q}(\theta ),} where μ 1 ′ ( P ) {\displaystyle \mu '_{1}(P)} is the first moment, or mean, of P , and Ψ Q = log M Q {\displaystyle \Psi _{Q}=\log M_{Q}} is called the cumulant-generating function . Taking the supremum completes the process of convex conjugation and yields the rate function : D K L ( P ∥ Q ) ≥ sup θ { μ 1 ′ ( P ) θ − Ψ Q ( θ ) } = Ψ Q ∗ ( μ 1 ′ ( P ) ) . {\displaystyle D_{KL}(P\parallel Q)\geq \sup _{\theta }\left\{\mu '_{1}(P)\theta -\Psi _{Q}(\theta )\right\}=\Psi _{Q}^{*}(\mu '_{1}(P)).}
Let X θ be a family of probability distributions on the real line indexed by the real parameter θ, and satisfying certain regularity conditions . Then lim h → 0 D K L ( X θ + h ∥ X θ ) h 2 ≥ lim h → 0 Ψ θ ∗ ( μ θ + h ) h 2 , {\displaystyle \lim _{h\to 0}{\frac {D_{KL}(X_{\theta +h}\parallel X_{\theta })}{h^{2}}}\geq \lim _{h\to 0}{\frac {\Psi _{\theta }^{*}(\mu _{\theta +h})}{h^{2}}},}
where Ψ θ ∗ {\displaystyle \Psi _{\theta }^{*}} is the convex conjugate of the cumulant-generating function of X θ {\displaystyle X_{\theta }} and μ θ + h {\displaystyle \mu _{\theta +h}} is the first moment of X θ + h . {\displaystyle X_{\theta +h}.}
The left side of this inequality can be simplified as follows: lim h → 0 D K L ( X θ + h ∥ X θ ) h 2 = lim h → 0 1 h 2 ∫ − ∞ ∞ log ( d X θ + h d X θ ) d X θ + h = − lim h → 0 1 h 2 ∫ − ∞ ∞ log ( d X θ d X θ + h ) d X θ + h = − lim h → 0 1 h 2 ∫ − ∞ ∞ log ( 1 − ( 1 − d X θ d X θ + h ) ) d X θ + h = lim h → 0 1 h 2 ∫ − ∞ ∞ [ ( 1 − d X θ d X θ + h ) + 1 2 ( 1 − d X θ d X θ + h ) 2 + o ( ( 1 − d X θ d X θ + h ) 2 ) ] d X θ + h Taylor series for log ( 1 − t ) = lim h → 0 1 h 2 ∫ − ∞ ∞ [ 1 2 ( 1 − d X θ d X θ + h ) 2 ] d X θ + h = lim h → 0 1 h 2 ∫ − ∞ ∞ [ 1 2 ( d X θ + h − d X θ d X θ + h ) 2 ] d X θ + h = 1 2 I X ( θ ) {\displaystyle {\begin{aligned}\lim _{h\to 0}{\frac {D_{KL}(X_{\theta +h}\parallel X_{\theta })}{h^{2}}}&=\lim _{h\to 0}{\frac {1}{h^{2}}}\int _{-\infty }^{\infty }\log \left({\frac {\mathrm {d} X_{\theta +h}}{\mathrm {d} X_{\theta }}}\right)\mathrm {d} X_{\theta +h}\\&=-\lim _{h\to 0}{\frac {1}{h^{2}}}\int _{-\infty }^{\infty }\log \left({\frac {\mathrm {d} X_{\theta }}{\mathrm {d} X_{\theta +h}}}\right)\mathrm {d} X_{\theta +h}\\&=-\lim _{h\to 0}{\frac {1}{h^{2}}}\int _{-\infty }^{\infty }\log \left(1-\left(1-{\frac {\mathrm {d} X_{\theta }}{\mathrm {d} X_{\theta +h}}}\right)\right)\mathrm {d} X_{\theta +h}\\&=\lim _{h\to 0}{\frac {1}{h^{2}}}\int _{-\infty }^{\infty }\left[\left(1-{\frac {\mathrm {d} X_{\theta }}{\mathrm {d} X_{\theta +h}}}\right)+{\frac {1}{2}}\left(1-{\frac {\mathrm {d} X_{\theta }}{\mathrm {d} X_{\theta +h}}}\right)^{2}+o\left(\left(1-{\frac {\mathrm {d} X_{\theta }}{\mathrm {d} X_{\theta +h}}}\right)^{2}\right)\right]\mathrm {d} X_{\theta +h}&&{\text{Taylor series for }}\log(1-t)\\&=\lim _{h\to 0}{\frac {1}{h^{2}}}\int _{-\infty }^{\infty }\left[{\frac {1}{2}}\left(1-{\frac {\mathrm {d} X_{\theta }}{\mathrm {d} X_{\theta +h}}}\right)^{2}\right]\mathrm {d} X_{\theta +h}\\&=\lim _{h\to 0}{\frac {1}{h^{2}}}\int _{-\infty }^{\infty }\left[{\frac {1}{2}}\left({\frac {\mathrm {d} X_{\theta +h}-\mathrm {d} X_{\theta }}{\mathrm {d} X_{\theta +h}}}\right)^{2}\right]\mathrm {d} X_{\theta +h}\\&={\frac {1}{2}}{\mathcal {I}}_{X}(\theta )\end{aligned}}} which is half the Fisher information of the parameter θ .
The right side of the inequality can be developed as follows: lim h → 0 Ψ θ ∗ ( μ θ + h ) h 2 = lim h → 0 1 h 2 sup t { μ θ + h t − Ψ θ ( t ) } . {\displaystyle \lim _{h\to 0}{\frac {\Psi _{\theta }^{*}(\mu _{\theta +h})}{h^{2}}}=\lim _{h\to 0}{\frac {1}{h^{2}}}{\sup _{t}\{\mu _{\theta +h}t-\Psi _{\theta }(t)\}}.} This supremum is attained at a value of t =τ where the first derivative of the cumulant-generating function is Ψ θ ′ ( τ ) = μ θ + h , {\displaystyle \Psi '_{\theta }(\tau )=\mu _{\theta +h},} but we have Ψ θ ′ ( 0 ) = μ θ , {\displaystyle \Psi '_{\theta }(0)=\mu _{\theta },} so that Ψ θ ″ ( 0 ) = d μ θ d θ lim h → 0 h τ . {\displaystyle \Psi ''_{\theta }(0)={\frac {d\mu _{\theta }}{d\theta }}\lim _{h\to 0}{\frac {h}{\tau }}.} Moreover, lim h → 0 Ψ θ ∗ ( μ θ + h ) h 2 = 1 2 Ψ θ ″ ( 0 ) ( d μ θ d θ ) 2 = 1 2 Var ( X θ ) ( d μ θ d θ ) 2 . {\displaystyle \lim _{h\to 0}{\frac {\Psi _{\theta }^{*}(\mu _{\theta +h})}{h^{2}}}={\frac {1}{2\Psi ''_{\theta }(0)}}\left({\frac {d\mu _{\theta }}{d\theta }}\right)^{2}={\frac {1}{2\operatorname {Var} (X_{\theta })}}\left({\frac {d\mu _{\theta }}{d\theta }}\right)^{2}.}
We have: 1 2 I X ( θ ) ≥ 1 2 Var ( X θ ) ( d μ θ d θ ) 2 , {\displaystyle {\frac {1}{2}}{\mathcal {I}}_{X}(\theta )\geq {\frac {1}{2\operatorname {Var} (X_{\theta })}}\left({\frac {d\mu _{\theta }}{d\theta }}\right)^{2},} which can be rearranged as: Var ( X θ ) ≥ ( d μ θ / d θ ) 2 I X ( θ ) . {\displaystyle \operatorname {Var} (X_{\theta })\geq {\frac {(d\mu _{\theta }/d\theta )^{2}}{{\mathcal {I}}_{X}(\theta )}}.} | https://en.wikipedia.org/wiki/Kullback's_inequality |
In mathematical statistics , the Kullback–Leibler ( KL ) divergence (also called relative entropy and I-divergence [ 1 ] ), denoted D KL ( P ∥ Q ) {\displaystyle D_{\text{KL}}(P\parallel Q)} , is a type of statistical distance : a measure of how much a model probability distribution Q is different from a true probability distribution P . [ 2 ] [ 3 ] Mathematically, it is defined as
D KL ( P ∥ Q ) = ∑ x ∈ X P ( x ) log P ( x ) Q ( x ) . {\displaystyle D_{\text{KL}}(P\parallel Q)=\sum _{x\in {\mathcal {X}}}P(x)\,\log {\frac {P(x)}{Q(x)}}.}
A simple interpretation of the KL divergence of P from Q is the expected excess surprise from using Q as a model instead of P when the actual distribution is P . While it is a measure of how different two distributions are and is thus a distance in some sense, it is not actually a metric , which is the most familiar and formal type of distance. In particular, it is not symmetric in the two distributions (in contrast to variation of information ), and does not satisfy the triangle inequality . Instead, in terms of information geometry , it is a type of divergence , [ 4 ] a generalization of squared distance , and for certain classes of distributions (notably an exponential family ), it satisfies a generalized Pythagorean theorem (which applies to squared distances). [ 5 ]
Relative entropy is always a non-negative real number , with value 0 if and only if the two distributions in question are identical. It has diverse applications, both theoretical, such as characterizing the relative (Shannon) entropy in information systems, randomness in continuous time-series , and information gain when comparing statistical models of inference ; and practical, such as applied statistics, fluid mechanics , neuroscience , bioinformatics , and machine learning .
Consider two probability distributions P and Q . Usually, P represents the data, the observations, or a measured probability distribution. Distribution Q represents instead a theory, a model, a description or an approximation of P . The Kullback–Leibler divergence D KL ( P ∥ Q ) {\displaystyle D_{\text{KL}}(P\parallel Q)} is then interpreted as the average difference of the number of bits required for encoding samples of P using a code optimized for Q rather than one optimized for P . Note that the roles of P and Q can be reversed in some situations where that is easier to compute, such as with the expectation–maximization algorithm (EM) and evidence lower bound (ELBO) computations.
The relative entropy was introduced by Solomon Kullback and Richard Leibler in Kullback & Leibler (1951) as "the mean information for discrimination between H 1 {\displaystyle H_{1}} and H 2 {\displaystyle H_{2}} per observation from μ 1 {\displaystyle \mu _{1}} ", [ 6 ] where one is comparing two probability measures μ 1 , μ 2 {\displaystyle \mu _{1},\mu _{2}} , and H 1 , H 2 {\displaystyle H_{1},H_{2}} are the hypotheses that one is selecting from measure μ 1 , μ 2 {\displaystyle \mu _{1},\mu _{2}} (respectively). They denoted this by I ( 1 : 2 ) {\displaystyle I(1:2)} , and defined the "'divergence' between μ 1 {\displaystyle \mu _{1}} and μ 2 {\displaystyle \mu _{2}} " as the symmetrized quantity J ( 1 , 2 ) = I ( 1 : 2 ) + I ( 2 : 1 ) {\displaystyle J(1,2)=I(1:2)+I(2:1)} , which had already been defined and used by Harold Jeffreys in 1948. [ 7 ] In Kullback (1959) , the symmetrized form is again referred to as the "divergence", and the relative entropies in each direction are referred to as a "directed divergences" between two distributions; [ 8 ] Kullback preferred the term discrimination information . [ 9 ] The term "divergence" is in contrast to a distance (metric), since the symmetrized divergence does not satisfy the triangle inequality. [ 10 ] Numerous references to earlier uses of the symmetrized divergence and to other statistical distances are given in Kullback (1959 , pp. 6–7, §1.3 Divergence). The asymmetric "directed divergence" has come to be known as the Kullback–Leibler divergence, while the symmetrized "divergence" is now referred to as the Jeffreys divergence .
For discrete probability distributions P and Q defined on the same sample space , X {\displaystyle {\mathcal {X}}} , the relative entropy from Q to P is defined [ 11 ] to be
D KL ( P ∥ Q ) = ∑ x ∈ X P ( x ) log P ( x ) Q ( x ) , {\displaystyle D_{\text{KL}}(P\parallel Q)=\sum _{x\in {\mathcal {X}}}P(x)\,\log {\frac {P(x)}{Q(x)}}\,,}
which is equivalent to
D KL ( P ∥ Q ) = − ∑ x ∈ X P ( x ) log Q ( x ) P ( x ) . {\displaystyle D_{\text{KL}}(P\parallel Q)=-\sum _{x\in {\mathcal {X}}}P(x)\,\log {\frac {Q(x)}{P(x)}}\,.}
In other words, it is the expectation of the logarithmic difference between the probabilities P and Q , where the expectation is taken using the probabilities P .
Relative entropy is only defined in this way if, for all x , Q ( x ) = 0 {\displaystyle Q(x)=0} implies P ( x ) = 0 {\displaystyle P(x)=0} ( absolute continuity ). Otherwise, it is often defined as + ∞ {\displaystyle +\infty } , [ 1 ] but the value + ∞ {\displaystyle \ +\infty \ } is possible even if Q ( x ) ≠ 0 {\displaystyle Q(x)\neq 0} everywhere, [ 12 ] [ 13 ] provided that X {\displaystyle {\mathcal {X}}} is infinite in extent. Analogous comments apply to the continuous and general measure cases defined below.
Whenever P ( x ) {\displaystyle P(x)} is zero the contribution of the corresponding term is interpreted as zero because
lim x → 0 + x log ( x ) = 0 . {\displaystyle \lim _{x\to 0^{+}}x\,\log(x)=0\,.}
For distributions P and Q of a continuous random variable , relative entropy is defined to be the integral [ 14 ]
D KL ( P ∥ Q ) = ∫ − ∞ ∞ p ( x ) log p ( x ) q ( x ) d x , {\displaystyle D_{\text{KL}}(P\parallel Q)=\int _{-\infty }^{\infty }p(x)\,\log {\frac {p(x)}{q(x)}}\,dx\,,}
where p and q denote the probability densities of P and Q .
More generally, if P and Q are probability measures on a measurable space X , {\displaystyle {\mathcal {X}}\,,} and P is absolutely continuous with respect to Q , then the relative entropy from Q to P is defined as
D KL ( P ∥ Q ) = ∫ x ∈ X log P ( d x ) Q ( d x ) P ( d x ) , {\displaystyle D_{\text{KL}}(P\parallel Q)=\int _{x\in {\mathcal {X}}}\log {\frac {P(dx)}{Q(dx)}}\,P(dx)\,,}
where P ( d x ) Q ( d x ) {\displaystyle {\frac {P(dx)}{Q(dx)}}} is the Radon–Nikodym derivative of P with respect to Q , i.e. the unique Q almost everywhere defined function r on X {\displaystyle {\mathcal {X}}} such that P ( d x ) = r ( x ) Q ( d x ) {\displaystyle P(dx)=r(x)Q(dx)} which exists because P is absolutely continuous with respect to Q . Also we assume the expression on the right-hand side exists. Equivalently (by the chain rule ), this can be written as
D KL ( P ∥ Q ) = ∫ x ∈ X P ( d x ) Q ( d x ) log P ( d x ) Q ( d x ) Q ( d x ) , {\displaystyle D_{\text{KL}}(P\parallel Q)=\int _{x\in {\mathcal {X}}}{\frac {P(dx)}{Q(dx)}}\ \log {\frac {P(dx)}{Q(dx)}}\ Q(dx)\,,}
which is the entropy of P relative to Q . Continuing in this case, if μ {\displaystyle \mu } is any measure on X {\displaystyle {\mathcal {X}}} for which densities p and q with P ( d x ) = p ( x ) μ ( d x ) {\displaystyle P(dx)=p(x)\mu (dx)} and Q ( d x ) = q ( x ) μ ( d x ) {\displaystyle Q(dx)=q(x)\mu (dx)} exist (meaning that P and Q are both absolutely continuous with respect to μ {\displaystyle \mu } ), then the relative entropy from Q to P is given as
D KL ( P ∥ Q ) = ∫ x ∈ X p ( x ) log p ( x ) q ( x ) μ ( d x ) . {\displaystyle D_{\text{KL}}(P\parallel Q)=\int _{x\in {\mathcal {X}}}p(x)\,\log {\frac {p(x)}{q(x)}}\ \mu (dx)\,.}
Note that such a measure μ {\displaystyle \mu } for which densities can be defined always exists, since one can take μ = 1 2 ( P + Q ) {\textstyle \mu ={\frac {1}{2}}\left(P+Q\right)} although in practice it will usually be one that applies in the context like counting measure for discrete distributions, or Lebesgue measure or a convenient variant thereof like Gaussian measure or the uniform measure on the sphere , Haar measure on a Lie group etc. for continuous distributions.
The logarithms in these formulae are usually taken to base 2 if information is measured in units of bits , or to base e if information is measured in nats . Most formulas involving relative entropy hold regardless of the base of the logarithm.
Various conventions exist for referring to D KL ( P ∥ Q ) {\displaystyle D_{\text{KL}}(P\parallel Q)} in words. Often it is referred to as the divergence between P and Q , but this fails to convey the fundamental asymmetry in the relation. Sometimes, as in this article, it may be described as the divergence of P from Q or as the divergence from Q to P . This reflects the asymmetry in Bayesian inference , which starts from a prior Q and updates to the posterior P . Another common way to refer to D KL ( P ∥ Q ) {\displaystyle D_{\text{KL}}(P\parallel Q)} is as the relative entropy of P with respect to Q or the information gain from P over Q .
Kullback [ 3 ] gives the following example (Table 2.1, Example 2.1). Let P and Q be the distributions shown in the table and figure. P is the distribution on the left side of the figure, a binomial distribution with N = 2 {\displaystyle N=2} and p = 0.4 {\displaystyle p=0.4} . Q is the distribution on the right side of the figure, a discrete uniform distribution with the three possible outcomes x = 0 , 1 , 2 (i.e. X = { 0 , 1 , 2 } {\displaystyle {\mathcal {X}}=\{0,1,2\}} ), each with probability p = 1 / 3 {\displaystyle p=1/3} .
Relative entropies D KL ( P ∥ Q ) {\displaystyle D_{\text{KL}}(P\parallel Q)} and D KL ( Q ∥ P ) {\displaystyle D_{\text{KL}}(Q\parallel P)} are calculated as follows. This example uses the natural log with base e , designated ln to get results in nats (see units of information ):
D KL ( P ∥ Q ) = ∑ x ∈ X P ( x ) ln P ( x ) Q ( x ) = 9 25 ln 9 / 25 1 / 3 + 12 25 ln 12 / 25 1 / 3 + 4 25 ln 4 / 25 1 / 3 = 1 25 ( 32 ln 2 + 55 ln 3 − 50 ln 5 ) ≈ 0.0852996 , {\displaystyle {\begin{aligned}D_{\text{KL}}(P\parallel Q)&=\sum _{x\in {\mathcal {X}}}P(x)\,\ln {\frac {P(x)}{Q(x)}}\\&={\frac {9}{25}}\ln {\frac {9/25}{1/3}}+{\frac {12}{25}}\ln {\frac {12/25}{1/3}}+{\frac {4}{25}}\ln {\frac {4/25}{1/3}}\\&={\frac {1}{25}}\left(32\ln 2+55\ln 3-50\ln 5\right)\\&\approx 0.0852996,\end{aligned}}}
D KL ( Q ∥ P ) = ∑ x ∈ X Q ( x ) ln Q ( x ) P ( x ) = 1 3 ln 1 / 3 9 / 25 + 1 3 ln 1 / 3 12 / 25 + 1 3 ln 1 / 3 4 / 25 = 1 3 ( − 4 ln 2 − 6 ln 3 + 6 ln 5 ) ≈ 0.097455. {\displaystyle {\begin{aligned}D_{\text{KL}}(Q\parallel P)&=\sum _{x\in {\mathcal {X}}}Q(x)\,\ln {\frac {Q(x)}{P(x)}}\\&={\frac {1}{3}}\,\ln {\frac {1/3}{9/25}}+{\frac {1}{3}}\,\ln {\frac {1/3}{12/25}}+{\frac {1}{3}}\,\ln {\frac {1/3}{4/25}}\\&={\frac {1}{3}}\left(-4\ln 2-6\ln 3+6\ln 5\right)\\&\approx 0.097455.\end{aligned}}}
In the field of statistics, the Neyman–Pearson lemma states that the most powerful way to distinguish between the two distributions P and Q based on an observation Y (drawn from one of them) is through the log of the ratio of their likelihoods: log P ( Y ) − log Q ( Y ) {\displaystyle \log P(Y)-\log Q(Y)} . The KL divergence is the expected value of this statistic if Y is actually drawn from P . Kullback motivated the statistic as an expected log likelihood ratio. [ 15 ]
In the context of coding theory , D KL ( P ∥ Q ) {\displaystyle D_{\text{KL}}(P\parallel Q)} can be constructed by measuring the expected number of extra bits required to code samples from P using a code optimized for Q rather than the code optimized for P .
In the context of machine learning , D KL ( P ∥ Q ) {\displaystyle D_{\text{KL}}(P\parallel Q)} is often called the information gain achieved if P would be used instead of Q which is currently used. By analogy with information theory, it is called the relative entropy of P with respect to Q .
Expressed in the language of Bayesian inference , D KL ( P ∥ Q ) {\displaystyle D_{\text{KL}}(P\parallel Q)} is a measure of the information gained by revising one's beliefs from the prior probability distribution Q to the posterior probability distribution P . In other words, it is the amount of information lost when Q is used to approximate P . [ 16 ]
In applications, P typically represents the "true" distribution of data, observations, or a precisely calculated theoretical distribution, while Q typically represents a theory, model, description, or approximation of P . In order to find a distribution Q that is closest to P , we can minimize the KL divergence and compute an information projection .
While it is a statistical distance , it is not a metric , the most familiar type of distance, but instead it is a divergence . [ 4 ] While metrics are symmetric and generalize linear distance, satisfying the triangle inequality , divergences are asymmetric and generalize squared distance, in some cases satisfying a generalized Pythagorean theorem . In general D KL ( P ∥ Q ) {\displaystyle D_{\text{KL}}(P\parallel Q)} does not equal D KL ( Q ∥ P ) {\displaystyle D_{\text{KL}}(Q\parallel P)} , and the asymmetry is an important part of the geometry. [ 4 ] The infinitesimal form of relative entropy, specifically its Hessian , gives a metric tensor that equals the Fisher information metric ; see § Fisher information metric . Fisher information metric on the certain probability distribution let determine the natural gradient for information-geometric optimization algorithms. [ 17 ] Its quantum version is Fubini-study metric. [ 18 ] Relative entropy satisfies a generalized Pythagorean theorem for exponential families (geometrically interpreted as dually flat manifolds ), and this allows one to minimize relative entropy by geometric means, for example by information projection and in maximum likelihood estimation . [ 5 ]
The relative entropy is the Bregman divergence generated by the negative entropy, but it is also of the form of an f -divergence . For probabilities over a finite alphabet , it is unique in being a member of both of these classes of statistical divergences . The application of Bregman divergence can be found in mirror descent. [ 19 ]
Consider a growth-optimizing investor in a fair game with mutually exclusive outcomes
(e.g. a “horse race” in which the official odds add up to one).
The rate of return expected by such an investor is equal to the relative entropy
between the investor's believed probabilities and the official odds. [ 20 ] This is a special case of a much more general connection between financial returns and divergence measures. [ 21 ]
Financial risks are connected to D KL {\displaystyle D_{\text{KL}}} via information geometry. [ 22 ] Investors' views, the prevailing market view, and risky scenarios form triangles on the relevant manifold of probability distributions. The shape of the triangles determines key financial risks (both qualitatively and quantitatively). For instance, obtuse triangles in which investors' views and risk scenarios appear on “opposite sides” relative to the market describe negative risks, acute triangles describe positive exposure, and the right-angled situation in the middle corresponds to zero risk. Extending this concept, relative entropy can be hypothetically utilised to identify the behaviour of informed investors, if one takes this to be represented by the magnitude and deviations away from the prior expectations of fund flows, for example. [ 23 ]
In information theory, the Kraft–McMillan theorem establishes that any directly decodable coding scheme for coding a message to identify one value x i {\displaystyle x_{i}} out of a set of possibilities X can be seen as representing an implicit probability distribution q ( x i ) = 2 − ℓ i {\displaystyle q(x_{i})=2^{-\ell _{i}}} over X , where ℓ i {\displaystyle \ell _{i}} is the length of the code for x i {\displaystyle x_{i}} in bits. Therefore, relative entropy can be interpreted as the expected extra message-length per datum that must be communicated if a code that is optimal for a given (wrong) distribution Q is used, compared to using a code based on the true distribution P : it is the excess entropy.
D KL ( P ∥ Q ) = ∑ x ∈ X p ( x ) log 1 q ( x ) − ∑ x ∈ X p ( x ) log 1 p ( x ) = H ( P , Q ) − H ( P ) {\displaystyle {\begin{aligned}D_{\text{KL}}(P\parallel Q)&=\sum _{x\in {\mathcal {X}}}p(x)\log {\frac {1}{q(x)}}-\sum _{x\in {\mathcal {X}}}p(x)\log {\frac {1}{p(x)}}\\[5pt]&=\mathrm {H} (P,Q)-\mathrm {H} (P)\end{aligned}}}
where H ( P , Q ) {\displaystyle \mathrm {H} (P,Q)} is the cross entropy of Q relative to P and H ( P ) {\displaystyle \mathrm {H} (P)} is the entropy of P (which is the same as the cross-entropy of P with itself).
The relative entropy D KL ( P ∥ Q ) {\displaystyle D_{\text{KL}}(P\parallel Q)} can be thought of geometrically as a statistical distance , a measure of how far the distribution Q is from the distribution P . Geometrically it is a divergence : an asymmetric, generalized form of squared distance. The cross-entropy H ( P , Q ) {\displaystyle H(P,Q)} is itself such a measurement (formally a loss function ), but it cannot be thought of as a distance, since H ( P , P ) =: H ( P ) {\displaystyle H(P,P)=:H(P)} is not zero. This can be fixed by subtracting H ( P ) {\displaystyle H(P)} to make D KL ( P ∥ Q ) {\displaystyle D_{\text{KL}}(P\parallel Q)} agree more closely with our notion of distance, as the excess loss. The resulting function is asymmetric, and while this can be symmetrized (see § Symmetrised divergence ), the asymmetric form is more useful. See § Interpretations for more on the geometric interpretation.
Relative entropy relates to " rate function " in the theory of large deviations . [ 24 ] [ 25 ]
Arthur Hobson proved that relative entropy is the only measure of difference between probability distributions that satisfies some desired properties, which are the canonical extension to those appearing in a commonly used characterization of entropy . [ 26 ] Consequently, mutual information is the only measure of mutual dependence that obeys certain related conditions, since it can be defined in terms of Kullback–Leibler divergence .
In particular, if P ( d x ) = p ( x ) μ ( d x ) {\displaystyle P(dx)=p(x)\mu (dx)} and Q ( d x ) = q ( x ) μ ( d x ) {\displaystyle Q(dx)=q(x)\mu (dx)} , then p ( x ) = q ( x ) {\displaystyle p(x)=q(x)} μ {\displaystyle \mu } - almost everywhere . The entropy H ( P ) {\displaystyle \mathrm {H} (P)} thus sets a minimum value for the cross-entropy H ( P , Q ) {\displaystyle \mathrm {H} (P,Q)} , the expected number of bits required when using a code based on Q rather than P ; and the Kullback–Leibler divergence therefore represents the expected number of extra bits that must be transmitted to identify a value x drawn from X , if a code is used corresponding to the probability distribution Q , rather than the "true" distribution P .
Denote f ( α ) := D KL ( ( 1 − α ) Q + α P ∥ Q ) {\displaystyle f(\alpha ):=D_{\text{KL}}((1-\alpha )Q+\alpha P\parallel Q)} and note that D KL ( P ∥ Q ) = f ( 1 ) {\displaystyle D_{\text{KL}}(P\parallel Q)=f(1)} . The first derivative of f {\displaystyle f} may be derived and evaluated as follows f ′ ( α ) = ∑ x ∈ X ( P ( x ) − Q ( x ) ) ( log ( ( 1 − α ) Q ( x ) + α P ( x ) Q ( x ) ) + 1 ) = ∑ x ∈ X ( P ( x ) − Q ( x ) ) log ( ( 1 − α ) Q ( x ) + α P ( x ) Q ( x ) ) f ′ ( 0 ) = 0 {\displaystyle {\begin{aligned}f'(\alpha )&=\sum _{x\in {\mathcal {X}}}(P(x)-Q(x))\left(\log \left({\frac {(1-\alpha )Q(x)+\alpha P(x)}{Q(x)}}\right)+1\right)\\&=\sum _{x\in {\mathcal {X}}}(P(x)-Q(x))\log \left({\frac {(1-\alpha )Q(x)+\alpha P(x)}{Q(x)}}\right)\\f'(0)&=0\end{aligned}}} Further derivatives may be derived and evaluated as follows f ″ ( α ) = ∑ x ∈ X ( P ( x ) − Q ( x ) ) 2 ( 1 − α ) Q ( x ) + α P ( x ) f ″ ( 0 ) = ∑ x ∈ X ( P ( x ) − Q ( x ) ) 2 Q ( x ) f ( n ) ( α ) = ( − 1 ) n ( n − 2 ) ! ∑ x ∈ X ( P ( x ) − Q ( x ) ) n ( ( 1 − α ) Q ( x ) + α P ( x ) ) n − 1 f ( n ) ( 0 ) = ( − 1 ) n ( n − 2 ) ! ∑ x ∈ X ( P ( x ) − Q ( x ) ) n Q ( x ) n − 1 {\displaystyle {\begin{aligned}f''(\alpha )&=\sum _{x\in {\mathcal {X}}}{\frac {(P(x)-Q(x))^{2}}{(1-\alpha )Q(x)+\alpha P(x)}}\\f''(0)&=\sum _{x\in {\mathcal {X}}}{\frac {(P(x)-Q(x))^{2}}{Q(x)}}\\f^{(n)}(\alpha )&=(-1)^{n}(n-2)!\sum _{x\in {\mathcal {X}}}{\frac {(P(x)-Q(x))^{n}}{\left((1-\alpha )Q(x)+\alpha P(x)\right)^{n-1}}}\\f^{(n)}(0)&=(-1)^{n}(n-2)!\sum _{x\in {\mathcal {X}}}{\frac {(P(x)-Q(x))^{n}}{Q(x)^{n-1}}}\end{aligned}}} Hence solving for D KL ( P ∥ Q ) {\displaystyle D_{\text{KL}}(P\parallel Q)} via the Taylor expansion of f {\displaystyle f} about 0 {\displaystyle 0} evaluated at α = 1 {\displaystyle \alpha =1} yields D KL ( P ∥ Q ) = ∑ n = 0 ∞ f ( n ) ( 0 ) n ! = ∑ n = 2 ∞ 1 n ( n − 1 ) ∑ x ∈ X ( Q ( x ) − P ( x ) ) n Q ( x ) n − 1 {\displaystyle {\begin{aligned}D_{\text{KL}}(P\parallel Q)&=\sum _{n=0}^{\infty }{\frac {f^{(n)}(0)}{n!}}\\&=\sum _{n=2}^{\infty }{\frac {1}{n(n-1)}}\sum _{x\in {\mathcal {X}}}{\frac {(Q(x)-P(x))^{n}}{Q(x)^{n-1}}}\end{aligned}}} P ≤ 2 Q {\displaystyle P\leq 2Q} a.s. is a sufficient condition for convergence of the series by the following absolute convergence argument ∑ n = 2 ∞ | 1 n ( n − 1 ) ∑ x ∈ X ( Q ( x ) − P ( x ) ) n Q ( x ) n − 1 | = ∑ n = 2 ∞ 1 n ( n − 1 ) ∑ x ∈ X | Q ( x ) − P ( x ) | | 1 − P ( x ) Q ( x ) | n − 1 ≤ ∑ n = 2 ∞ 1 n ( n − 1 ) ∑ x ∈ X | Q ( x ) − P ( x ) | ≤ ∑ n = 2 ∞ 1 n ( n − 1 ) = 1 {\displaystyle {\begin{aligned}\sum _{n=2}^{\infty }\left\vert {\frac {1}{n(n-1)}}\sum _{x\in {\mathcal {X}}}{\frac {(Q(x)-P(x))^{n}}{Q(x)^{n-1}}}\right\vert &=\sum _{n=2}^{\infty }{\frac {1}{n(n-1)}}\sum _{x\in {\mathcal {X}}}\left\vert Q(x)-P(x)\right\vert \left\vert 1-{\frac {P(x)}{Q(x)}}\right\vert ^{n-1}\\&\leq \sum _{n=2}^{\infty }{\frac {1}{n(n-1)}}\sum _{x\in {\mathcal {X}}}\left\vert Q(x)-P(x)\right\vert \\&\leq \sum _{n=2}^{\infty }{\frac {1}{n(n-1)}}\\&=1\end{aligned}}} P ≤ 2 Q {\displaystyle P\leq 2Q} a.s. is also a necessary condition for convergence of the series by the following proof by contradiction. Assume that P > 2 Q {\displaystyle P>2Q} with measure strictly greater than 0 {\displaystyle 0} . It then follows that there must exist some values ε > 0 {\displaystyle \varepsilon >0} , ρ > 0 {\displaystyle \rho >0} , and U < ∞ {\displaystyle U<\infty } such that P ≥ 2 Q + ε {\displaystyle P\geq 2Q+\varepsilon } and Q ≤ U {\displaystyle Q\leq U} with measure ρ {\displaystyle \rho } . The previous proof of sufficiency demonstrated that the measure 1 − ρ {\displaystyle 1-\rho } component of the series where P ≤ 2 Q {\displaystyle P\leq 2Q} is bounded, so we need only concern ourselves with the behavior of the measure ρ {\displaystyle \rho } component of the series where P ≥ 2 Q + ε {\displaystyle P\geq 2Q+\varepsilon } . The absolute value of the n {\displaystyle n} th term of this component of the series is then lower bounded by 1 n ( n − 1 ) ρ ( 1 + ε U ) n {\displaystyle {\frac {1}{n(n-1)}}\rho \left(1+{\frac {\varepsilon }{U}}\right)^{n}} , which is unbounded as n → ∞ {\displaystyle n\to \infty } , so the series diverges.
The following result, due to Donsker and Varadhan, [ 29 ] is known as Donsker and Varadhan's variational formula .
Theorem [Duality Formula for Variational Inference] — Let Θ {\displaystyle \Theta } be a set endowed with an appropriate σ {\displaystyle \sigma } -field F {\displaystyle {\mathcal {F}}} , and two probability measures P and Q , which formulate two probability spaces ( Θ , F , P ) {\displaystyle (\Theta ,{\mathcal {F}},P)} and ( Θ , F , Q ) {\displaystyle (\Theta ,{\mathcal {F}},Q)} , with Q ≪ P {\displaystyle Q\ll P} . ( Q ≪ P {\displaystyle Q\ll P} indicates that Q is absolutely continuous with respect to P .) Let h be a real-valued integrable random variable on ( Θ , F , P ) {\displaystyle (\Theta ,{\mathcal {F}},P)} . Then the following equality holds
log E P [ exp h ] = sup Q ≪ P { E Q [ h ] − D KL ( Q ∥ P ) } . {\displaystyle \log E_{P}[\exp h]=\operatorname {sup} _{Q\ll P}\{E_{Q}[h]-D_{\text{KL}}(Q\parallel P)\}.}
Further, the supremum on the right-hand side is attained if and only if it holds
Q ( d θ ) P ( d θ ) = exp h ( θ ) E P [ exp h ] , {\displaystyle {\frac {Q(d\theta )}{P(d\theta )}}={\frac {\exp h(\theta )}{E_{P}[\exp h]}},}
almost surely with respect to probability measure P , where Q ( d θ ) P ( d θ ) {\displaystyle {\frac {Q(d\theta )}{P(d\theta )}}} denotes the Radon-Nikodym derivative of Q with respect to P .
For a short proof assuming integrability of exp ( h ) {\displaystyle \exp(h)} with respect to P , let Q ∗ {\displaystyle Q^{*}} have P -density exp h ( θ ) E P [ exp h ] {\displaystyle {\frac {\exp h(\theta )}{E_{P}[\exp h]}}} , i.e. Q ∗ ( d θ ) = exp h ( θ ) E P [ exp h ] P ( d θ ) {\displaystyle Q^{*}(d\theta )={\frac {\exp h(\theta )}{E_{P}[\exp h]}}P(d\theta )} Then
D KL ( Q ∥ Q ∗ ) − D KL ( Q ∥ P ) = − E Q [ h ] + log E P [ exp h ] . {\displaystyle D_{\text{KL}}(Q\parallel Q^{*})-D_{\text{KL}}(Q\parallel P)=-E_{Q}[h]+\log E_{P}[\exp h].}
Therefore,
E Q [ h ] − D KL ( Q ∥ P ) = log E P [ exp h ] − D KL ( Q ∥ Q ∗ ) ≤ log E P [ exp h ] , {\displaystyle E_{Q}[h]-D_{\text{KL}}(Q\parallel P)=\log E_{P}[\exp h]-D_{\text{KL}}(Q\parallel Q^{*})\leq \log E_{P}[\exp h],}
where the last inequality follows from D KL ( Q ∥ Q ∗ ) ≥ 0 {\displaystyle D_{\text{KL}}(Q\parallel Q^{*})\geq 0} , for which equality occurs if and only if Q = Q ∗ {\displaystyle Q=Q^{*}} . The conclusion follows.
Suppose that we have two multivariate normal distributions , with means μ 0 , μ 1 {\displaystyle \mu _{0},\mu _{1}} and with (non-singular) covariance matrices Σ 0 , Σ 1 . {\displaystyle \Sigma _{0},\Sigma _{1}.} If the two distributions have the same dimension, k , then the relative entropy between the distributions is as follows: [ 30 ]
D KL ( N 0 ∥ N 1 ) = 1 2 [ tr ( Σ 1 − 1 Σ 0 ) − k + ( μ 1 − μ 0 ) T Σ 1 − 1 ( μ 1 − μ 0 ) + ln det Σ 1 det Σ 0 ] . {\displaystyle D_{\text{KL}}\left({\mathcal {N}}_{0}\parallel {\mathcal {N}}_{1}\right)={\frac {1}{2}}\left[\operatorname {tr} \left(\Sigma _{1}^{-1}\Sigma _{0}\right)-k+\left(\mu _{1}-\mu _{0}\right)^{\mathsf {T}}\Sigma _{1}^{-1}\left(\mu _{1}-\mu _{0}\right)+\ln {\frac {\det \Sigma _{1}}{\det \Sigma _{0}}}\right].}
The logarithm in the last term must be taken to base e since all terms apart from the last are base- e logarithms of expressions that are either factors of the density function or otherwise arise naturally. The equation therefore gives a result measured in nats . Dividing the entire expression above by ln ( 2 ) {\displaystyle \ln(2)} yields the divergence in bits .
In a numerical implementation, it is helpful to express the result in terms of the Cholesky decompositions L 0 , L 1 {\displaystyle L_{0},L_{1}} such that Σ 0 = L 0 L 0 T {\displaystyle \Sigma _{0}=L_{0}L_{0}^{T}} and Σ 1 = L 1 L 1 T {\displaystyle \Sigma _{1}=L_{1}L_{1}^{T}} . Then with M and y solutions to the triangular linear systems L 1 M = L 0 {\displaystyle L_{1}M=L_{0}} , and L 1 y = μ 1 − μ 0 {\displaystyle L_{1}y=\mu _{1}-\mu _{0}} ,
D KL ( N 0 ∥ N 1 ) = 1 2 ( ∑ i , j = 1 k ( M i j ) 2 − k + | y | 2 + 2 ∑ i = 1 k ln ( L 1 ) i i ( L 0 ) i i ) . {\displaystyle D_{\text{KL}}\left({\mathcal {N}}_{0}\parallel {\mathcal {N}}_{1}\right)={\frac {1}{2}}\left(\sum _{i,j=1}^{k}{\left(M_{ij}\right)}^{2}-k+|y|^{2}+2\sum _{i=1}^{k}\ln {\frac {(L_{1})_{ii}}{(L_{0})_{ii}}}\right).}
A special case, and a common quantity in variational inference , is the relative entropy between a diagonal multivariate normal, and a standard normal distribution (with zero mean and unit variance):
D KL ( N ( ( μ 1 , … , μ k ) T , diag ( σ 1 2 , … , σ k 2 ) ) ∥ N ( 0 , I ) ) = 1 2 ∑ i = 1 k [ σ i 2 + μ i 2 − 1 − ln ( σ i 2 ) ] . {\displaystyle D_{\text{KL}}\left({\mathcal {N}}\left(\left(\mu _{1},\ldots ,\mu _{k}\right)^{\mathsf {T}},\operatorname {diag} \left(\sigma _{1}^{2},\ldots ,\sigma _{k}^{2}\right)\right)\parallel {\mathcal {N}}\left(\mathbf {0} ,\mathbf {I} \right)\right)={\frac {1}{2}}\sum _{i=1}^{k}\left[\sigma _{i}^{2}+\mu _{i}^{2}-1-\ln \left(\sigma _{i}^{2}\right)\right].}
For two univariate normal distributions p and q the above simplifies to [ 31 ] D KL ( p ∥ q ) = log σ 1 σ 0 + σ 0 2 + ( μ 0 − μ 1 ) 2 2 σ 1 2 − 1 2 {\displaystyle D_{\text{KL}}\left({\mathcal {p}}\parallel {\mathcal {q}}\right)=\log {\frac {\sigma _{1}}{\sigma _{0}}}+{\frac {\sigma _{0}^{2}+{\left(\mu _{0}-\mu _{1}\right)}^{2}}{2\sigma _{1}^{2}}}-{\frac {1}{2}}}
In the case of co-centered normal distributions with k = σ 1 / σ 0 {\displaystyle k=\sigma _{1}/\sigma _{0}} , this simplifies [ 32 ] to:
D KL ( p ∥ q ) = log 2 k + ( k − 2 − 1 ) / 2 / ln ( 2 ) b i t s {\displaystyle D_{\text{KL}}\left({\mathcal {p}}\parallel {\mathcal {q}}\right)=\log _{2}k+(k^{-2}-1)/2/\ln(2)\mathrm {bits} }
Consider two uniform distributions, with the support of p = [ A , B ] {\displaystyle p=[A,B]} enclosed within q = [ C , D ] {\displaystyle q=[C,D]} ( C ≤ A < B ≤ D {\displaystyle C\leq A<B\leq D} ). Then the information gain is:
D KL ( p ∥ q ) = log D − C B − A {\displaystyle D_{\text{KL}}\left({\mathcal {p}}\parallel {\mathcal {q}}\right)=\log {\frac {D-C}{B-A}}}
Intuitively, [ 32 ] the information gain to a k times narrower uniform distribution contains log 2 k {\displaystyle \log _{2}k} bits. This connects with the use of bits in computing, where log 2 k {\displaystyle \log _{2}k} bits would be needed to identify one element of a k long stream.
The exponential family of distribution is given by
p X ( x | θ ) = h ( x ) exp ( θ T T ( x ) − A ( θ ) ) {\displaystyle p_{X}(x|\theta )=h(x)\exp \left(\theta ^{\mathsf {T}}T(x)-A(\theta )\right)}
where h ( x ) {\displaystyle h(x)} is reference measure, T ( x ) {\displaystyle T(x)} is sufficient statistics, θ {\displaystyle \theta } is canonical natural parameters, and A ( θ ) {\displaystyle A(\theta )} is the log-partition function.
The KL divergence between two distributions p ( x | θ 1 ) {\displaystyle p(x|\theta _{1})} and p ( x | θ 2 ) {\displaystyle p(x|\theta _{2})} is given by [ 33 ]
D KL ( θ 1 ∥ θ 2 ) = ( θ 1 − θ 2 ) T μ 1 − A ( θ 1 ) + A ( θ 2 ) {\displaystyle D_{\text{KL}}(\theta _{1}\parallel \theta _{2})={\left(\theta _{1}-\theta _{2}\right)}^{\mathsf {T}}\mu _{1}-A(\theta _{1})+A(\theta _{2})}
where μ 1 = E θ 1 [ T ( X ) ] = ∇ A ( θ 1 ) {\displaystyle \mu _{1}=E_{\theta _{1}}[T(X)]=\nabla A(\theta _{1})} is the mean parameter of p ( x | θ 1 ) {\displaystyle p(x|\theta _{1})} .
For example, for the Poisson distribution with mean λ {\displaystyle \lambda } , the sufficient statistics T ( x ) = x {\displaystyle T(x)=x} , the natural parameter θ = log λ {\displaystyle \theta =\log \lambda } , and log partition function A ( θ ) = e θ {\displaystyle A(\theta )=e^{\theta }} . As such, the divergence between two Poisson distributions with means λ 1 {\displaystyle \lambda _{1}} and λ 2 {\displaystyle \lambda _{2}} is
D KL ( λ 1 ∥ λ 2 ) = λ 1 log λ 1 λ 2 − λ 1 + λ 2 . {\displaystyle D_{\text{KL}}(\lambda _{1}\parallel \lambda _{2})=\lambda _{1}\log {\frac {\lambda _{1}}{\lambda _{2}}}-\lambda _{1}+\lambda _{2}.}
As another example, for a normal distribution with unit variance N ( μ , 1 ) {\displaystyle N(\mu ,1)} , the sufficient statistics T ( x ) = x {\displaystyle T(x)=x} , the natural parameter θ = μ {\displaystyle \theta =\mu } , and log partition function A ( θ ) = μ 2 / 2 {\displaystyle A(\theta )=\mu ^{2}/2} . Thus, the divergence between two normal distributions N ( μ 1 , 1 ) {\displaystyle N(\mu _{1},1)} and N ( μ 2 , 1 ) {\displaystyle N(\mu _{2},1)} is
D KL ( μ 1 ∥ μ 2 ) = ( μ 1 − μ 2 ) μ 1 − μ 1 2 2 + μ 2 2 2 = ( μ 2 − μ 1 ) 2 2 . {\displaystyle D_{\text{KL}}(\mu _{1}\parallel \mu _{2})=\left(\mu _{1}-\mu _{2}\right)\mu _{1}-{\frac {\mu _{1}^{2}}{2}}+{\frac {\mu _{2}^{2}}{2}}={\frac {{\left(\mu _{2}-\mu _{1}\right)}^{2}}{2}}.}
As final example, the divergence between a normal distribution with unit variance N ( μ , 1 ) {\displaystyle N(\mu ,1)} and a Poisson distribution with mean λ {\displaystyle \lambda } is
D KL ( μ ∥ λ ) = ( μ − log λ ) μ − μ 2 2 + λ . {\displaystyle D_{\text{KL}}(\mu \parallel \lambda )=(\mu -\log \lambda )\mu -{\frac {\mu ^{2}}{2}}+\lambda .}
While relative entropy is a statistical distance , it is not a metric on the space of probability distributions, but instead it is a divergence . [ 4 ] While metrics are symmetric and generalize linear distance, satisfying the triangle inequality , divergences are asymmetric in general and generalize squared distance, in some cases satisfying a generalized Pythagorean theorem . In general D KL ( P ∥ Q ) {\displaystyle D_{\text{KL}}(P\parallel Q)} does not equal D KL ( Q ∥ P ) {\displaystyle D_{\text{KL}}(Q\parallel P)} , and while this can be symmetrized (see § Symmetrised divergence ), the asymmetry is an important part of the geometry. [ 4 ]
It generates a topology on the space of probability distributions . More concretely, if { P 1 , P 2 , … } {\displaystyle \{P_{1},P_{2},\ldots \}} is a sequence of distributions such that
lim n → ∞ D KL ( P n ∥ Q ) = 0 , {\displaystyle \lim _{n\to \infty }D_{\text{KL}}(P_{n}\parallel Q)=0,}
then it is said that
P n → D Q . {\displaystyle P_{n}\xrightarrow {D} \,Q.}
Pinsker's inequality entails that
P n → D P ⇒ P n → T V P , {\displaystyle P_{n}\xrightarrow {D} P\Rightarrow P_{n}\xrightarrow {TV} P,}
where the latter stands for the usual convergence in total variation .
Relative entropy is directly related to the Fisher information metric . This can be made explicit as follows. Assume that the probability distributions P and Q are both parameterized by some (possibly multi-dimensional) parameter θ {\displaystyle \theta } . Consider then two close by values of P = P ( θ ) {\displaystyle P=P(\theta )} and Q = P ( θ 0 ) {\displaystyle Q=P(\theta _{0})} so that the parameter θ {\displaystyle \theta } differs by only a small amount from the parameter value θ 0 {\displaystyle \theta _{0}} . Specifically, up to first order one has (using the Einstein summation convention ) P ( θ ) = P ( θ 0 ) + Δ θ j P j ( θ 0 ) + ⋯ {\displaystyle P(\theta )=P(\theta _{0})+\Delta \theta _{j}\,P_{j}(\theta _{0})+\cdots }
with Δ θ j = ( θ − θ 0 ) j {\displaystyle \Delta \theta _{j}=(\theta -\theta _{0})_{j}} a small change of θ {\displaystyle \theta } in the j direction, and P j ( θ 0 ) = ∂ P ∂ θ j ( θ 0 ) {\displaystyle P_{j}\left(\theta _{0}\right)={\frac {\partial P}{\partial \theta _{j}}}(\theta _{0})} the corresponding rate of change in the probability distribution. Since relative entropy has an absolute minimum 0 for P = Q {\displaystyle P=Q} , i.e. θ = θ 0 {\displaystyle \theta =\theta _{0}} , it changes only to second order in the small parameters Δ θ j {\displaystyle \Delta \theta _{j}} . More formally, as for any minimum, the first derivatives of the divergence vanish
∂ ∂ θ j | θ = θ 0 D KL ( P ( θ ) ∥ P ( θ 0 ) ) = 0 , {\displaystyle \left.{\frac {\partial }{\partial \theta _{j}}}\right|_{\theta =\theta _{0}}D_{\text{KL}}(P(\theta )\parallel P(\theta _{0}))=0,}
and by the Taylor expansion one has up to second order
D KL ( P ( θ ) ∥ P ( θ 0 ) ) = 1 2 Δ θ j Δ θ k g j k ( θ 0 ) + ⋯ {\displaystyle D_{\text{KL}}(P(\theta )\parallel P(\theta _{0}))={\frac {1}{2}}\,\Delta \theta _{j}\,\Delta \theta _{k}\,g_{jk}(\theta _{0})+\cdots }
where the Hessian matrix of the divergence
g j k ( θ 0 ) = ∂ 2 ∂ θ j ∂ θ k | θ = θ 0 D KL ( P ( θ ) ∥ P ( θ 0 ) ) {\displaystyle g_{jk}(\theta _{0})=\left.{\frac {\partial ^{2}}{\partial \theta _{j}\,\partial \theta _{k}}}\right|_{\theta =\theta _{0}}D_{\text{KL}}(P(\theta )\parallel P(\theta _{0}))}
must be positive semidefinite . Letting θ 0 {\displaystyle \theta _{0}} vary (and dropping the subindex 0) the Hessian g j k ( θ ) {\displaystyle g_{jk}(\theta )} defines a (possibly degenerate) Riemannian metric on the θ parameter space, called the Fisher information metric.
When p ( x , ρ ) {\displaystyle p_{(x,\rho )}} satisfies the following regularity conditions:
∂ log ( p ) ∂ ρ , ∂ 2 log ( p ) ∂ ρ 2 , ∂ 3 log ( p ) ∂ ρ 3 {\displaystyle {\frac {\partial \log(p)}{\partial \rho }},{\frac {\partial ^{2}\log(p)}{\partial \rho ^{2}}},{\frac {\partial ^{3}\log(p)}{\partial \rho ^{3}}}} exist, | ∂ p ∂ ρ | < F ( x ) : ∫ x = 0 ∞ F ( x ) d x < ∞ , | ∂ 2 p ∂ ρ 2 | < G ( x ) : ∫ x = 0 ∞ G ( x ) d x < ∞ | ∂ 3 log ( p ) ∂ ρ 3 | < H ( x ) : ∫ x = 0 ∞ p ( x , 0 ) H ( x ) d x < ξ < ∞ {\displaystyle {\begin{aligned}\left|{\frac {\partial p}{\partial \rho }}\right|&<F(x):\int _{x=0}^{\infty }F(x)\,dx<\infty ,\\\left|{\frac {\partial ^{2}p}{\partial \rho ^{2}}}\right|&<G(x):\int _{x=0}^{\infty }G(x)\,dx<\infty \\\left|{\frac {\partial ^{3}\log(p)}{\partial \rho ^{3}}}\right|&<H(x):\int _{x=0}^{\infty }p(x,0)H(x)\,dx<\xi <\infty \end{aligned}}}
where ξ is independent of ρ ∫ x = 0 ∞ ∂ p ( x , ρ ) ∂ ρ | ρ = 0 d x = ∫ x = 0 ∞ ∂ 2 p ( x , ρ ) ∂ ρ 2 | ρ = 0 d x = 0 {\displaystyle \left.\int _{x=0}^{\infty }{\frac {\partial p(x,\rho )}{\partial \rho }}\right|_{\rho =0}\,dx=\left.\int _{x=0}^{\infty }{\frac {\partial ^{2}p(x,\rho )}{\partial \rho ^{2}}}\right|_{\rho =0}\,dx=0}
then: D ( p ( x , 0 ) ∥ p ( x , ρ ) ) = c ρ 2 2 + O ( ρ 3 ) as ρ → 0. {\displaystyle {\mathcal {D}}(p(x,0)\parallel p(x,\rho ))={\frac {c\rho ^{2}}{2}}+{\mathcal {O}}\left(\rho ^{3}\right){\text{ as }}\rho \to 0.}
Another information-theoretic metric is variation of information , which is roughly a symmetrization of conditional entropy . It is a metric on the set of partitions of a discrete probability space .
MAUVE is a measure of the statistical gap between two text distributions, such as the difference between text generated by a model and human-written text. This measure is computed using Kullback–Leibler divergences between the two distributions in a quantized embedding space of a foundation model.
Many of the other quantities of information theory can be interpreted as applications of relative entropy to specific cases.
The self-information , also known as the information content of a signal, random variable, or event is defined as the negative logarithm of the probability of the given outcome occurring.
When applied to a discrete random variable , the self-information can be represented as [ citation needed ]
I ( m ) = D KL ( δ im ∥ { p i } ) , {\displaystyle \operatorname {\operatorname {I} } (m)=D_{\text{KL}}\left(\delta _{\text{im}}\parallel \{p_{i}\}\right),}
is the relative entropy of the probability distribution P ( i ) {\displaystyle P(i)} from a Kronecker delta representing certainty that i = m {\displaystyle i=m} — i.e. the number of extra bits that must be transmitted to identify i if only the probability distribution P ( i ) {\displaystyle P(i)} is available to the receiver, not the fact that i = m {\displaystyle i=m} .
The mutual information ,
I ( X ; Y ) = D KL ( P ( X , Y ) ∥ P ( X ) P ( Y ) ) = E X { D KL ( P ( Y ∣ X ) ∥ P ( Y ) ) } = E Y { D KL ( P ( X ∣ Y ) ∥ P ( X ) ) } {\displaystyle {\begin{aligned}\operatorname {I} (X;Y)&=D_{\text{KL}}(P(X,Y)\parallel P(X)P(Y))\\[5pt]&=\operatorname {E} _{X}\{D_{\text{KL}}(P(Y\mid X)\parallel P(Y))\}\\[5pt]&=\operatorname {E} _{Y}\{D_{\text{KL}}(P(X\mid Y)\parallel P(X))\}\end{aligned}}}
is the relative entropy of the joint probability distribution P ( X , Y ) {\displaystyle P(X,Y)} from the product P ( X ) P ( Y ) {\displaystyle P(X)P(Y)} of the two marginal probability distributions — i.e. the expected number of extra bits that must be transmitted to identify X and Y if they are coded using only their marginal distributions instead of the joint distribution. Equivalently, if the joint probability P ( X , Y ) {\displaystyle P(X,Y)} is known, it is the expected number of extra bits that must on average be sent to identify Y if the value of X is not already known to the receiver.
The Shannon entropy ,
H ( X ) = E [ I X ( x ) ] = log N − D KL ( p X ( x ) ∥ P U ( X ) ) {\displaystyle {\begin{aligned}\mathrm {H} (X)&=\operatorname {E} \left[\operatorname {I} _{X}(x)\right]\\&=\log N-D_{\text{KL}}{\left(p_{X}(x)\parallel P_{U}(X)\right)}\end{aligned}}}
is the number of bits which would have to be transmitted to identify X from N equally likely possibilities, less the relative entropy of the uniform distribution on the random variates of X , P U ( X ) {\displaystyle P_{U}(X)} , from the true distribution P ( X ) {\displaystyle P(X)} — i.e. less the expected number of bits saved, which would have had to be sent if the value of X were coded according to the uniform distribution P U ( X ) {\displaystyle P_{U}(X)} rather than the true distribution P ( X ) {\displaystyle P(X)} . This definition of Shannon entropy forms the basis of E.T. Jaynes 's alternative generalization to continuous distributions, the limiting density of discrete points (as opposed to the usual differential entropy ), which defines the continuous entropy as lim N → ∞ H N ( X ) = log N − ∫ p ( x ) log p ( x ) m ( x ) d x , {\displaystyle \lim _{N\to \infty }H_{N}(X)=\log N-\int p(x)\log {\frac {p(x)}{m(x)}}\,dx,} which is equivalent to: log ( N ) − D KL ( p ( x ) | | m ( x ) ) {\displaystyle \log(N)-D_{\text{KL}}(p(x)||m(x))}
The conditional entropy [ 34 ] ,
H ( X ∣ Y ) = log N − D KL ( P ( X , Y ) ∥ P U ( X ) P ( Y ) ) = log N − D KL ( P ( X , Y ) ∥ P ( X ) P ( Y ) ) − D KL ( P ( X ) ∥ P U ( X ) ) = H ( X ) − I ( X ; Y ) = log N − E Y [ D KL ( P ( X ∣ Y ) ∥ P U ( X ) ) ] {\displaystyle {\begin{aligned}\mathrm {H} (X\mid Y)&=\log N-D_{\text{KL}}(P(X,Y)\parallel P_{U}(X)P(Y))\\[5pt]&=\log N-D_{\text{KL}}(P(X,Y)\parallel P(X)P(Y))-D_{\text{KL}}(P(X)\parallel P_{U}(X))\\[5pt]&=\mathrm {H} (X)-\operatorname {I} (X;Y)\\[5pt]&=\log N-\operatorname {E} _{Y}\left[D_{\text{KL}}\left(P\left(X\mid Y\right)\parallel P_{U}(X)\right)\right]\end{aligned}}}
is the number of bits which would have to be transmitted to identify X from N equally likely possibilities, less the relative entropy of the product distribution P U ( X ) P ( Y ) {\displaystyle P_{U}(X)P(Y)} from the true joint distribution P ( X , Y ) {\displaystyle P(X,Y)} — i.e. less the expected number of bits saved which would have had to be sent if the value of X were coded according to the uniform distribution P U ( X ) {\displaystyle P_{U}(X)} rather than the conditional distribution P ( X | Y ) {\displaystyle P(X|Y)} of X given Y .
When we have a set of possible events, coming from the distribution p , we can encode them (with a lossless data compression ) using entropy encoding . This compresses the data by replacing each fixed-length input symbol with a corresponding unique, variable-length, prefix-free code (e.g.: the events (A, B, C) with probabilities p = (1/2, 1/4, 1/4) can be encoded as the bits (0, 10, 11)). If we know the distribution p in advance, we can devise an encoding that would be optimal (e.g.: using Huffman coding ). Meaning the messages we encode will have the shortest length on average (assuming the encoded events are sampled from p ), which will be equal to Shannon's Entropy of p (denoted as H ( p ) {\displaystyle \mathrm {H} (p)} ). However, if we use a different probability distribution ( q ) when creating the entropy encoding scheme, then a larger number of bits will be used (on average) to identify an event from a set of possibilities. This new (larger) number is measured by the cross entropy between p and q .
The cross entropy between two probability distributions ( p and q ) measures the average number of bits needed to identify an event from a set of possibilities, if a coding scheme is used based on a given probability distribution q , rather than the "true" distribution p . The cross entropy for two distributions p and q over the same probability space is thus defined as follows.
H ( p , q ) = E p [ − log q ] = H ( p ) + D KL ( p ∥ q ) . {\displaystyle \mathrm {H} (p,q)=\operatorname {E} _{p}[-\log q]=\mathrm {H} (p)+D_{\text{KL}}(p\parallel q).}
For explicit derivation of this, see the Motivation section above.
Under this scenario, relative entropies (kl-divergence) can be interpreted as the extra number of bits, on average, that are needed (beyond H ( p ) {\displaystyle \mathrm {H} (p)} ) for encoding the events because of using q for constructing the encoding scheme instead of p .
In Bayesian statistics , relative entropy can be used as a measure of the information gain in moving from a prior distribution to a posterior distribution : p ( x ) → p ( x ∣ I ) {\displaystyle p(x)\to p(x\mid I)} . If some new fact Y = y {\displaystyle Y=y} is discovered, it can be used to update the posterior distribution for X from p ( x ∣ I ) {\displaystyle p(x\mid I)} to a new posterior distribution p ( x ∣ y , I ) {\displaystyle p(x\mid y,I)} using Bayes' theorem :
p ( x ∣ y , I ) = p ( y ∣ x , I ) p ( x ∣ I ) p ( y ∣ I ) {\displaystyle p(x\mid y,I)={\frac {p(y\mid x,I)p(x\mid I)}{p(y\mid I)}}}
This distribution has a new entropy :
H ( p ( x ∣ y , I ) ) = − ∑ x p ( x ∣ y , I ) log p ( x ∣ y , I ) , {\displaystyle \mathrm {H} {\big (}p(x\mid y,I){\big )}=-\sum _{x}p(x\mid y,I)\log p(x\mid y,I),}
which may be less than or greater than the original entropy H ( p ( x ∣ I ) ) {\displaystyle \mathrm {H} (p(x\mid I))} . However, from the standpoint of the new probability distribution one can estimate that to have used the original code based on p ( x ∣ I ) {\displaystyle p(x\mid I)} instead of a new code based on p ( x ∣ y , I ) {\displaystyle p(x\mid y,I)} would have added an expected number of bits:
D KL ( p ( x ∣ y , I ) ∥ p ( x ∣ I ) ) = ∑ x p ( x ∣ y , I ) log p ( x ∣ y , I ) p ( x ∣ I ) {\displaystyle D_{\text{KL}}{\big (}p(x\mid y,I)\parallel p(x\mid I){\big )}=\sum _{x}p(x\mid y,I)\log {\frac {p(x\mid y,I)}{p(x\mid I)}}}
to the message length. This therefore represents the amount of useful information, or information gain, about X , that has been learned by discovering Y = y {\displaystyle Y=y} .
If a further piece of data, Y 2 = y 2 {\displaystyle Y_{2}=y_{2}} , subsequently comes in, the probability distribution for x can be updated further, to give a new best guess p ( x ∣ y 1 , y 2 , I ) {\displaystyle p(x\mid y_{1},y_{2},I)} . If one reinvestigates the information gain for using p ( x ∣ y 1 , I ) {\displaystyle p(x\mid y_{1},I)} rather than p ( x ∣ I ) {\displaystyle p(x\mid I)} , it turns out that it may be either greater or less than previously estimated:
∑ x p ( x ∣ y 1 , y 2 , I ) log p ( x ∣ y 1 , y 2 , I ) p ( x ∣ I ) {\displaystyle \sum _{x}p(x\mid y_{1},y_{2},I)\log {\frac {p(x\mid y_{1},y_{2},I)}{p(x\mid I)}}} may be ≤ or > than ∑ x p ( x ∣ y 1 , I ) log p ( x ∣ y 1 , I ) p ( x ∣ I ) {\textstyle \sum _{x}p(x\mid y_{1},I)\log {\frac {p(x\mid y_{1},I)}{p(x\mid I)}}}
and so the combined information gain does not obey the triangle inequality:
D KL ( p ( x ∣ y 1 , y 2 , I ) ∥ p ( x ∣ I ) ) {\displaystyle D_{\text{KL}}{\big (}p(x\mid y_{1},y_{2},I)\parallel p(x\mid I){\big )}} may be <, = or > than D KL ( p ( x ∣ y 1 , y 2 , I ) ∥ p ( x ∣ y 1 , I ) ) + D KL ( p ( x ∣ y 1 , I ) ∥ p ( x ∣ I ) ) {\displaystyle D_{\text{KL}}{\big (}p(x\mid y_{1},y_{2},I)\parallel p(x\mid y_{1},I){\big )}+D_{\text{KL}}{\big (}p(x\mid y_{1},I)\parallel p(x\mid I){\big )}}
All one can say is that on average , averaging using p ( y 2 ∣ y 1 , x , I ) {\displaystyle p(y_{2}\mid y_{1},x,I)} , the two sides will average out.
A common goal in Bayesian experimental design is to maximise the expected relative entropy between the prior and the posterior. [ 35 ] When posteriors are approximated to be Gaussian distributions, a design maximising the expected relative entropy is called Bayes d-optimal .
Relative entropy D KL ( p ( x ∣ H 1 ) ∥ p ( x ∣ H 0 ) ) {\textstyle D_{\text{KL}}{\bigl (}p(x\mid H_{1})\parallel p(x\mid H_{0}){\bigr )}} can also be interpreted as the expected discrimination information for H 1 {\displaystyle H_{1}} over H 0 {\displaystyle H_{0}} : the mean information per sample for discriminating in favor of a hypothesis H 1 {\displaystyle H_{1}} against a hypothesis H 0 {\displaystyle H_{0}} , when hypothesis H 1 {\displaystyle H_{1}} is true. [ 36 ] Another name for this quantity, given to it by I. J. Good , is the expected weight of evidence for H 1 {\displaystyle H_{1}} over H 0 {\displaystyle H_{0}} to be expected from each sample.
The expected weight of evidence for H 1 {\displaystyle H_{1}} over H 0 {\displaystyle H_{0}} is not the same as the information gain expected per sample about the probability distribution p ( H ) {\displaystyle p(H)} of the hypotheses,
D KL ( p ( x ∣ H 1 ) ∥ p ( x ∣ H 0 ) ) ≠ I G = D KL ( p ( H ∣ x ) ∥ p ( H ∣ I ) ) . {\displaystyle D_{\text{KL}}(p(x\mid H_{1})\parallel p(x\mid H_{0}))\neq IG=D_{\text{KL}}(p(H\mid x)\parallel p(H\mid I)).}
Either of the two quantities can be used as a utility function in Bayesian experimental design, to choose an optimal next question to investigate: but they will in general lead to rather different experimental strategies.
On the entropy scale of information gain there is very little difference between near certainty and absolute certainty—coding according to a near certainty requires hardly any more bits than coding according to an absolute certainty. On the other hand, on the logit scale implied by weight of evidence, the difference between the two is enormous – infinite perhaps; this might reflect the difference between being almost sure (on a probabilistic level) that, say, the Riemann hypothesis is correct, compared to being certain that it is correct because one has a mathematical proof. These two different scales of loss function for uncertainty are both useful, according to how well each reflects the particular circumstances of the problem in question.
The idea of relative entropy as discrimination information led Kullback to propose the Principle of Minimum Discrimination Information ( MDI ): given new facts, a new distribution f should be chosen which is as hard to discriminate from the original distribution f 0 {\displaystyle f_{0}} as possible; so that the new data produces as small an information gain D KL ( f ∥ f 0 ) {\displaystyle D_{\text{KL}}(f\parallel f_{0})} as possible.
For example, if one had a prior distribution p ( x , a ) {\displaystyle p(x,a)} over x and a , and subsequently learnt the true distribution of a was u ( a ) {\displaystyle u(a)} , then the relative entropy between the new joint distribution for x and a , q ( x ∣ a ) u ( a ) {\displaystyle q(x\mid a)u(a)} , and the earlier prior distribution would be:
D KL ( q ( x ∣ a ) u ( a ) ∥ p ( x , a ) ) = E u ( a ) { D KL ( q ( x ∣ a ) ∥ p ( x ∣ a ) ) } + D KL ( u ( a ) ∥ p ( a ) ) , {\displaystyle D_{\text{KL}}(q(x\mid a)u(a)\parallel p(x,a))=\operatorname {E} _{u(a)}\left\{D_{\text{KL}}(q(x\mid a)\parallel p(x\mid a))\right\}+D_{\text{KL}}(u(a)\parallel p(a)),}
i.e. the sum of the relative entropy of p ( a ) {\displaystyle p(a)} the prior distribution for a from the updated distribution u ( a ) {\displaystyle u(a)} , plus the expected value (using the probability distribution u ( a ) {\displaystyle u(a)} ) of the relative entropy of the prior conditional distribution p ( x ∣ a ) {\displaystyle p(x\mid a)} from the new conditional distribution q ( x ∣ a ) {\displaystyle q(x\mid a)} . (Note that often the later expected value is called the conditional relative entropy (or conditional Kullback–Leibler divergence ) and denoted by D KL ( q ( x ∣ a ) ∥ p ( x ∣ a ) ) {\displaystyle D_{\text{KL}}(q(x\mid a)\parallel p(x\mid a))} [ 3 ] [ 34 ] ) This is minimized if q ( x ∣ a ) = p ( x ∣ a ) {\displaystyle q(x\mid a)=p(x\mid a)} over the whole support of u ( a ) {\displaystyle u(a)} ; and we note that this result incorporates Bayes' theorem, if the new distribution u ( a ) {\displaystyle u(a)} is in fact a δ function representing certainty that a has one particular value.
MDI can be seen as an extension of Laplace 's Principle of Insufficient Reason , and the Principle of Maximum Entropy of E.T. Jaynes . In particular, it is the natural extension of the principle of maximum entropy from discrete to continuous distributions, for which Shannon entropy ceases to be so useful (see differential entropy ), but the relative entropy continues to be just as relevant.
In the engineering literature, MDI is sometimes called the Principle of Minimum Cross-Entropy (MCE) or Minxent for short. Minimising relative entropy from m to p with respect to m is equivalent to minimizing the cross-entropy of p and m , since
H ( p , m ) = H ( p ) + D KL ( p ∥ m ) , {\displaystyle \mathrm {H} (p,m)=\mathrm {H} (p)+D_{\text{KL}}(p\parallel m),}
which is appropriate if one is trying to choose an adequate approximation to p . However, this is just as often not the task one is trying to achieve. Instead, just as often it is m that is some fixed prior reference measure, and p that one is attempting to optimise by minimising D KL ( p ∥ m ) {\displaystyle D_{\text{KL}}(p\parallel m)} subject to some constraint. This has led to some ambiguity in the literature, with some authors attempting to resolve the inconsistency by redefining cross-entropy to be D KL ( p ∥ m ) {\displaystyle D_{\text{KL}}(p\parallel m)} , rather than H ( p , m ) {\displaystyle \mathrm {H} (p,m)} [ citation needed ] .
Surprisals [ 37 ] add where probabilities multiply. The surprisal for an event of probability p is defined as s = − k ln p {\displaystyle s=-k\ln p} . If k is { 1 , 1 / ln 2 , 1.38 × 10 − 23 } {\displaystyle \left\{1,1/\ln 2,1.38\times 10^{-23}\right\}} then surprisal is in { {\displaystyle \{} nats, bits, or J / K } {\displaystyle J/K\}} so that, for instance, there are N bits of surprisal for landing all "heads" on a toss of N coins.
Best-guess states (e.g. for atoms in a gas) are inferred by maximizing the average surprisal S ( entropy ) for a given set of control parameters (like pressure P or volume V ). This constrained entropy maximization , both classically [ 38 ] and quantum mechanically, [ 39 ] minimizes Gibbs availability in entropy units [ 40 ] A ≡ − k ln Z {\displaystyle A\equiv -k\ln Z} where Z is a constrained multiplicity or partition function .
When temperature T is fixed, free energy ( T × A {\displaystyle T\times A} ) is also minimized. Thus if T , V {\displaystyle T,V} and number of molecules N are constant, the Helmholtz free energy F ≡ U − T S {\displaystyle F\equiv U-TS} (where U is energy and S is entropy) is minimized as a system "equilibrates." If T and P are held constant (say during processes in your body), the Gibbs free energy G = U + P V − T S {\displaystyle G=U+PV-TS} is minimized instead. The change in free energy under these conditions is a measure of available work that might be done in the process. Thus available work for an ideal gas at constant temperature T o {\displaystyle T_{o}} and pressure P o {\displaystyle P_{o}} is W = Δ G = N k T o Θ ( V / V o ) {\displaystyle W=\Delta G=NkT_{o}\Theta (V/V_{o})} where V o = N k T o / P o {\displaystyle V_{o}=NkT_{o}/P_{o}} and Θ ( x ) = x − 1 − ln x ≥ 0 {\displaystyle \Theta (x)=x-1-\ln x\geq 0} (see also Gibbs inequality ).
More generally [ 41 ] the work available relative to some ambient is obtained by multiplying ambient temperature T o {\displaystyle T_{o}} by relative entropy or net surprisal Δ I ≥ 0 , {\displaystyle \Delta I\geq 0,} defined as the average value of k ln ( p / p o ) {\displaystyle k\ln(p/p_{o})} where p o {\displaystyle p_{o}} is the probability of a given state under ambient conditions. For instance, the work available in equilibrating a monatomic ideal gas to ambient values of V o {\displaystyle V_{o}} and T o {\displaystyle T_{o}} is thus W = T o Δ I {\displaystyle W=T_{o}\Delta I} , where relative entropy
Δ I = N k [ Θ ( V V o ) + 3 2 Θ ( T T o ) ] . {\displaystyle \Delta I=Nk\left[\Theta {\left({\frac {V}{V_{o}}}\right)}+{\frac {3}{2}}\Theta {\left({\frac {T}{T_{o}}}\right)}\right].}
The resulting contours of constant relative entropy, shown at right for a mole of Argon at standard temperature and pressure, for example put limits on the conversion of hot to cold as in flame-powered air-conditioning or in the unpowered device to convert boiling-water to ice-water discussed here. [ 42 ] Thus relative entropy measures thermodynamic availability in bits.
For density matrices P and Q on a Hilbert space , the quantum relative entropy from Q to P is defined to be
D KL ( P ∥ Q ) = Tr ( P ( log P − log Q ) ) . {\displaystyle D_{\text{KL}}(P\parallel Q)=\operatorname {Tr} (P(\log P-\log Q)).}
In quantum information science the minimum of D KL ( P ∥ Q ) {\displaystyle D_{\text{KL}}(P\parallel Q)} over all separable states Q can also be used as a measure of entanglement in the state P .
Just as relative entropy of "actual from ambient" measures thermodynamic availability, relative entropy of "reality from a model" is also useful even if the only clues we have about reality are some experimental measurements. In the former case relative entropy describes distance to equilibrium or (when multiplied by ambient temperature) the amount of available work , while in the latter case it tells you about surprises that reality has up its sleeve or, in other words, how much the model has yet to learn .
Although this tool for evaluating models against systems that are accessible experimentally may be applied in any field, its application to selecting a statistical model via Akaike information criterion are particularly well described in papers [ 43 ] and a book [ 44 ] by Burnham and Anderson. In a nutshell the relative entropy of reality from a model may be estimated, to within a constant additive term, by a function of the deviations observed between data and the model's predictions (like the mean squared deviation ) . Estimates of such divergence for models that share the same additive term can in turn be used to select among models.
When trying to fit parametrized models to data there are various estimators which attempt to minimize relative entropy, such as maximum likelihood and maximum spacing estimators. [ citation needed ]
Kullback & Leibler (1951) also considered the symmetrized function: [ 6 ]
D KL ( P ∥ Q ) + D KL ( Q ∥ P ) {\displaystyle D_{\text{KL}}(P\parallel Q)+D_{\text{KL}}(Q\parallel P)}
which they referred to as the "divergence", though today the "KL divergence" refers to the asymmetric function (see § Etymology for the evolution of the term). This function is symmetric and nonnegative, and had already been defined and used by Harold Jeffreys in 1948; [ 7 ] it is accordingly called the Jeffreys divergence .
This quantity has sometimes been used for feature selection in classification problems, where P and Q are the conditional pdfs of a feature under two different classes. In the Banking and Finance industries, this quantity is referred to as Population Stability Index ( PSI ), and is used to assess distributional shifts in model features through time.
An alternative is given via the λ {\displaystyle \lambda } -divergence,
D λ ( P ∥ Q ) = λ D KL ( P ∥ λ P + ( 1 − λ ) Q ) + ( 1 − λ ) D KL ( Q ∥ λ P + ( 1 − λ ) Q ) , {\displaystyle D_{\lambda }(P\parallel Q)=\lambda D_{\text{KL}}(P\parallel \lambda P+(1-\lambda )Q)+(1-\lambda )D_{\text{KL}}(Q\parallel \lambda P+(1-\lambda )Q),}
which can be interpreted as the expected information gain about X from discovering which probability distribution X is drawn from, P or Q , if they currently have probabilities λ {\displaystyle \lambda } and 1 − λ {\displaystyle 1-\lambda } respectively. [ clarification needed ] [ citation needed ]
The value λ = 0.5 {\displaystyle \lambda =0.5} gives the Jensen–Shannon divergence , defined by
D JS = 1 2 D KL ( P ∥ M ) + 1 2 D KL ( Q ∥ M ) {\displaystyle D_{\text{JS}}={\tfrac {1}{2}}D_{\text{KL}}(P\parallel M)+{\tfrac {1}{2}}D_{\text{KL}}(Q\parallel M)}
where M is the average of the two distributions,
M = 1 2 ( P + Q ) . {\displaystyle M={\tfrac {1}{2}}\left(P+Q\right).}
We can also interpret D JS {\displaystyle D_{\text{JS}}} as the capacity of a noisy information channel with two inputs giving the output distributions P and Q . The Jensen–Shannon divergence, like all f -divergences, is locally proportional to the Fisher information metric . It is similar to the Hellinger metric (in the sense that it induces the same affine connection on a statistical manifold ).
Furthermore, the Jensen–Shannon divergence can be generalized using abstract statistical M-mixtures relying on an abstract mean M. [ 45 ] [ 46 ]
There are many other important measures of probability distance . Some of these are particularly connected with relative entropy. For example:
Other notable measures of distance include the Hellinger distance , histogram intersection , Chi-squared statistic , quadratic form distance , match distance , Kolmogorov–Smirnov distance , and earth mover's distance . [ 49 ]
Just as absolute entropy serves as theoretical background for data compression , relative entropy serves as theoretical background for data differencing – the absolute entropy of a set of data in this sense being the data required to reconstruct it (minimum compressed size), while the relative entropy of a target set of data, given a source set of data, is the data required to reconstruct the target given the source (minimum size of a patch ). | https://en.wikipedia.org/wiki/Kullback–Leibler_divergence |
In organic chemistry , the Kumada coupling is a type of cross coupling reaction , useful for generating carbon–carbon bonds by the reaction of a Grignard reagent and an organic halide. The procedure uses transition metal catalysts , typically nickel or palladium, to couple a combination of two alkyl , aryl or vinyl groups . The groups of Robert Corriu and Makoto Kumada reported the reaction independently in 1972. [ 1 ] [ 2 ]
The reaction is notable for being among the first reported catalytic cross-coupling methods. Despite the subsequent development of alternative reactions ( Suzuki , Sonogashira , Stille , Hiyama , Negishi ), the Kumada coupling continues to be employed in many synthetic applications, including the industrial-scale production of aliskiren , a hypertension medication, and polythiophenes , useful in organic electronic devices.
The first investigations into the catalytic coupling of Grignard reagents with organic halides date back to the 1941 study of cobalt catalysts by Morris S. Kharasch and E. K. Fields. [ 3 ] In 1971, Tamura and Kochi elaborated on this work in a series of publications demonstrating the viability of catalysts based on silver, [ 4 ] copper [ 5 ] and iron. [ 6 ] However, these early approaches produced poor yields due to substantial formation of homocoupling products, where two identical species are coupled.
These efforts culminated in 1972, when the Corriu and Kumada groups concurrently reported the use of nickel-containing catalysts. With the introduction of palladium catalysts in 1975 by the Murahashi group, the scope of the reaction was further broadened. [ 7 ] Subsequently, many additional coupling techniques have been developed, culminating in the 2010 Nobel Prize in Chemistry recognized Ei-ichi Negishi , Akira Suzuki and Richard F. Heck for their contributions to the field.
According to the widely accepted mechanism, the palladium-catalyzed Kumada coupling is understood to be analogous to palladium's role in other cross coupling reactions. The proposed catalytic cycle involves both palladium(0) and palladium(II) oxidation states. Initially, the electron-rich Pd(0) catalyst ( 1 ) inserts into the R–X bond of the organic halide. This oxidative addition forms an organo-Pd(II)-complex ( 2 ). Subsequent transmetalation with the Grignard reagent forms a hetero-organometallic complex ( 3 ). Before the next step, isomerization is necessary to bring the organic ligands next to each other into mutually cis positions. Finally, reductive elimination of ( 4 ) forms a carbon–carbon bond and releases the cross coupled product while regenerating the Pd(0) catalyst ( 1 ). [ 8 ] For palladium catalysts, the frequently rate-determining oxidative addition occurs more slowly than with nickel catalyst systems. [ 8 ]
Current understanding of the mechanism for the nickel-catalyzed coupling is limited. Indeed, the reaction mechanism is believed to proceed differently under different reaction conditions and when using different nickel ligands. [ 9 ] In general the mechanism can still be described as analogous to the palladium scheme (right). Under certain reaction conditions, however, the mechanism fails to explain all observations. Examination by Vicic and coworkers using tridentate terpyridine ligand identified intermediates of a Ni(II)-Ni(I)-Ni(III) catalytic cycle, [ 10 ] suggesting a more complicated scheme. Additionally, with the addition of butadiene, the reaction is believed to involve a Ni(IV) intermediate. [ 11 ]
The Kumada coupling has been successfully demonstrated for a variety of aryl or vinyl halides. In place of the halide reagent pseudohalides can also be used, and the coupling has been shown to be quite effective using tosylate [ 12 ] and triflate [ 13 ] species in variety of conditions.
Despite broad success with aryl and vinyl couplings, the use of alkyl halides is less general due to several complicating factors. Having no π-electrons, alkyl halides require different oxidative addition mechanisms than aryl or vinyl groups, and these processes are currently poorly understood. [ 9 ] Additionally, the presence of β-hydrogens makes alkyl halides susceptible to competitive elimination processes. [ 14 ]
These issues have been circumvented by the presence of an activating group, such as the carbonyl in α-bromoketones, that drives the reaction forward. However, Kumada couplings have also been performed with non-activated alkyl chains, often through the use of additional catalysts or reagents. For instance, with the addition of 1,3-butadienes Kambe and coworkers demonstrated nickel catalyzed alkyl–alkyl couplings that would otherwise be unreactive. [ 15 ]
Though poorly understood, the mechanism of this reaction is proposed to involve the formation of an octadienyl nickel complex. This catalyst is proposed to undergo transmetalation with a Grignard reagent first, prior to the reductive elimination of the halide, reducing the risk of β-hydride elimination. However, the presence of a Ni(IV) intermediate is contrary to mechanisms proposed for aryl or vinyl halide couplings. [ 11 ]
Couplings involving aryl and vinyl Grignard reagents were reported in the original publications by Kumada and Corriu. [ 2 ] Alkyl Grignard reagents can also be used without difficulty, as they do not suffer from β-hydride elimination processes. Although the Grignard reagent inherently has poor functional group tolerance, low-temperature syntheses have been prepared with highly functionalized aryl groups. [ 16 ]
Kumada couplings can be performed with a variety of nickel(II) or palladium(II) catalysts. The structures of the catalytic precursors can be generally formulated as ML 2 X 2 , where L is a phosphine ligand. [ 17 ] Common choices for L 2 include bidentate diphosphine ligands such as dppe and dppp among others.
Work by Alois Fürstner and coworkers on iron-based catalysts have shown reasonable yields. The catalytic species in these reactions is proposed to be an "inorganic Grignard reagent" consisting of Fe(MgX) 2 . [ 18 ]
The reaction typically is carried out in tetrahydrofuran or diethyl ether as solvent. Such ethereal solvents are convenient because these are typical solvents for generating the Grignard reagent. [ 2 ] Due to the high reactivity of the Grignard reagent, Kumada couplings have limited functional group tolerance which can be problematic in large syntheses. In particular, Grignard reagents are sensitive to protonolysis from even mildly acidic groups such as alcohols . They also add to carbonyls and other oxidative groups.
As in many coupling reactions, the transition metal palladium catalyst is often air-sensitive, requiring an inert Argon or nitrogen reaction environment.
A sample synthetic preparation is available at the Organic Syntheses website .
Both cis- and trans- olefin halides promote the overall retention of geometric configuration when coupled with alkyl Grignards. This observation is independent of other factors, including the choice of catalyst ligands and vinylic substituents. [ 17 ]
Conversely, a Kumada coupling using vinylic Grignard reagents proceeds without stereospecificity to form a mixture of cis- and trans- alkenes. The degree of isomerization is dependent on a variety of factors including reagent ratios and the identity of the halide group. According to Kumada, this loss of stereochemistry is attributable to side-reactions between two equivalents of the allylic Grignard reagent. [ 17 ]
Asymmetric Kumada couplings can be effected through the use of chiral ligands. Using planar chiral ferrocene ligands, enantiomeric excesses (ee) upward of 95% have been observed in aryl couplings. [ 19 ] More recently, Gregory Fu and co-workers have demonstrated enantioconvergent couplings of α-bromoketones using catalysts based on bis-oxazoline ligands, wherein the chiral catalyst converts a racemic mixture of starting material to one enantiomer of product with up to 95% ee. [ 20 ] The latter reaction is also significant for involving a traditionally inaccessible alkyl halide coupling.
Grignard reagents do not typically couple with chlorinated arenes. This low reactivity is the basis for chemoselectivity for nickel insertion into the C–Br bond of bromochlorobenzene using a NiCl 2 -based catalyst. [ 21 ]
The Kumada coupling is suitable for large-scale, industrial processes, such as drug synthesis. The reaction is used to construct the carbon skeleton of aliskiren (trade name Tekturna), a treatment for hypertension . [ 22 ]
The Kumada coupling also shows promise in the synthesis of conjugated polymers , polymers such as polyalkylthiophenes (PAT), which have a variety of potential applications in organic solar cells and light-emitting diodes . [ 23 ] In 1992, McCollough and Lowe developed the first synthesis of regioregular polyalkylthiophenes by utilizing the Kumada coupling scheme pictured below, which requires subzero temperatures. [ 24 ]
Since this initial preparation, the synthesis has been improved to obtain higher yields and operate at room temperature. [ 25 ] | https://en.wikipedia.org/wiki/Kumada_coupling |
Kumeyaay astronomy or cosmology ( Kumeyaay : My Uuyow , "sky knowledge") comprises the astronomical knowledge of the Kumeyaay people , a Native American group whose traditional homeland occupies what is now Southern California in the United States and adjacent parts of northern Baja California in Mexico. [ 1 ] A deeply rooted cosmological belief system was developed and followed by the Kumeyaay civilization based on this knowledge including the computing of time (Kumeyaay Mat’taam). [ 2 ]
The first evidence of astronomical observations and visual registration was discovered in the El Vallecito archeological zone. The "Men in a square" rupestric painting located at El Diablito area of El Vallecito depicted a square that aligns with sunlight on the Fall equinox . These paintings were made by the Kumeyaay people, possibly during nomadic travels. [ 3 ] [ 4 ] Kumeyaay sand paintings and rock art modeled the passage of the sun, moon, and constellations. [ 5 ]
Observation areas were made by the Kumeyaay to watch and register astronomical events. However many were destroyed by vandals before protection measures were instituted. [ 6 ]
Constellations: [ 7 ] | https://en.wikipedia.org/wiki/Kumeyaay_astronomy |
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