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wiki_5_chunk_159 | American Civil War | Civil War commemoration The American Civil War has been commemorated in many capacities ranging from the reenactment of battles to statues and memorial halls erected, to films being produced, to stamps and coins with Civil War themes being issued, all of which helped to shape public memory. This varied advent occurred ... | wikipedia |
wiki_5_chunk_160 | American Civil War | Hollywood's take on the war has been especially influential in shaping public memory, as seen in such film classics as The Birth of a Nation (1915), Gone with the Wind (1939), and Lincoln (2012). Ken Burns's PBS television series The Civil War (1990) is especially well-remembered, though criticized for its historical a... | wikipedia |
wiki_5_chunk_161 | American Civil War | Technological significance
Numerous technological innovations during the Civil War had a great impact on 19th-century science. The Civil War was one of the earliest examples of an "industrial war", in which technological might is used to achieve military supremacy in a war. New inventions, such as the train and telegra... | wikipedia |
wiki_5_chunk_162 | American Civil War | In works of culture and art
The Civil War is one of the most studied events in American history, and the collection of cultural works around it is enormous. This section gives an abbreviated overview of the most notable works. | wikipedia |
wiki_5_chunk_163 | American Civil War | Literature
When Lilacs Last in the Dooryard Bloom'd and O Captain! My Captain! (1865) by Walt Whitman, famous eulogies to Lincoln
Battle-Pieces and Aspects of the War (1866) poetry by Herman Melville
The Rise and Fall of the Confederate Government (1881) by Jefferson Davis
The Private History of a Campaign That Fai... | wikipedia |
wiki_5_chunk_164 | American Civil War | Film | wikipedia |
wiki_5_chunk_165 | American Civil War | The Birth of a Nation (1915, US)
The General (1926, US)
Operator 13 (1934, US)
Gone with the Wind (1939, US)
The Red Badge of Courage (1951, US)
The Horse Soldiers (1959, US)
Shenandoah (1965, US)
The Good, the Bad and the Ugly (1966, Italy-Spain-FRG)
The Beguiled (1971, US)
The Outlaw Josey Wales (1976, US)
... | wikipedia |
wiki_5_chunk_166 | American Civil War | Music Dixie
Battle Cry of Freedom
Battle Hymn of the Republic
The Bonnie Blue Flag
John Brown's Body
When Johnny Comes Marching Home
Marching Through Georgia
The Night They Drove Old Dixie Down Video games | wikipedia |
wiki_5_chunk_167 | American Civil War | North & South (1989, FR)
Sid Meier's Gettysburg! (1997, US)
Sid Meier's Antietam! (1999, US)
American Conqest: Divided Nation (2006, US)
Forge of Freedom: The American Civil War (2006, US)
The History Channel: Civil War – A Nation Divided (2006, US)
Ageod's American Civil War (2007, US/FR)
History Civil War: Sec... | wikipedia |
wiki_5_chunk_168 | American Civil War | See also General reference
American Civil War Corps Badges
List of American Civil War battles
List of costliest American Civil War land battles
List of weapons in the American Civil War
Second American Civil War Union
Presidency of Abraham Lincoln
Uniform of the Union Army Confederacy
Central Confederacy
Unifo... | wikipedia |
wiki_5_chunk_169 | American Civil War | Topical articles
Commemoration of the American Civil War
Commemoration of the American Civil War on postage stamps
Dorothea Dix
Education of freed people during the Civil War
Spies in the American Civil War
Infantry in the American Civil War
List of ships captured in the 19th century#American Civil War
Slaver... | wikipedia |
wiki_5_chunk_170 | American Civil War | Memorials
List of Confederate monuments and memorials
List of memorials and monuments at Arlington National Cemetery
List of memorials to Jefferson Davis
List of memorials to Robert E. Lee
List of memorials to Stonewall Jackson
List of monuments erected by the United Daughters of the Confederacy
List of monument... | wikipedia |
wiki_5_chunk_171 | American Civil War | Beringer, Richard E., Archer Jones, and Herman Hattaway, Why the South Lost the Civil War (1986), influential analysis of factors; an abridged version is The Elements of Confederate Defeat: Nationalism, War Aims, and Religion (1988)
Gallagher, Gary W. (2011). The Union War. Cambridge, M... | wikipedia |
wiki_5_chunk_172 | American Civil War | Further reading Bibliography of the American Civil War
Bibliography of American Civil War naval history | wikipedia |
wiki_5_chunk_173 | American Civil War | External links
West Point Atlas of Civil War Battles
Civil War photos at the National Archives
View images from the Civil War Photographs Collection at the Library of Congress
American Battlefield Trust – A non-profit land preservation and educational organization with two divisions, the Civil War Trust and the Rev... | wikipedia |
wiki_5_chunk_174 | American Civil War | Rebellions against the United States
Conflicts in 1861
Conflicts in 1862
Conflicts in 1863
Conflicts in 1864
Conflicts in 1865
19th-century conflicts
Civil War
1860s in the United States
Wars of independence
Internal wars of the United States
1860s conflicts | wikipedia |
wiki_6_chunk_0 | Advanced Chemistry | Advanced Chemistry is a German hip hop group from Heidelberg, a scenic city in Baden-Württemberg, South Germany. Advanced Chemistry was founded in 1987 by Toni L, Linguist, Gee-One, DJ Mike MD (Mike Dippon) and MC Torch. Each member of the group holds German citizenship, and Toni L, Linguist, and Torch are of Italian, ... | wikipedia |
wiki_6_chunk_1 | Advanced Chemistry | Influenced by North American socially conscious rap and the Native tongues movement, Advanced Chemistry is regarded as one of the main pioneers in German hip hop. They were one of the first groups to rap in German (although their name is in English). Furthermore, their songs tackled controversial social and political i... | wikipedia |
wiki_6_chunk_2 | Advanced Chemistry | Career
Advanced Chemistry frequently rapped about their lives and experiences as children of immigrants, exposing the marginalization experienced by most ethnic minorities in Germany, and the feelings of frustration and resentment that being denied a German identity can cause. The song "Fremd im eigenen Land" (Foreign ... | wikipedia |
wiki_6_chunk_3 | Advanced Chemistry | This idea of national identity is important, as many rap artists in Germany have been of foreign origin. These so-called Gastarbeiter (guest workers) children saw breakdance, graffiti, rap music, and hip hop culture as a means of expressing themselves. Since the release of "Fremd im eigenen Land", many other German-lan... | wikipedia |
wiki_6_chunk_4 | Advanced Chemistry | Advanced Chemistry helped to found the German chapter of the Zulu nation. | wikipedia |
wiki_6_chunk_5 | Advanced Chemistry | The rivalry between Advanced Chemistry and Die Fantastischen Vier has served to highlight a dichotomy in the routes that hip hop has taken in becoming a part of the German soundscape. While Die Fantastischen Vier may be said to view hip hop primarily as an aesthetic art form, Advanced Chemistry understand hip hop as ... | wikipedia |
wiki_6_chunk_6 | Advanced Chemistry | With the release of the single “Fremd im eigenen Land”, Advanced Chemistry separated itself from the rest of the rap being produced in Germany. This single was the first of its kind to go beyond simply imitating US rap and addressed the current issues of the time. Fremd im eigenen Land which translates to “foreign in ... | wikipedia |
wiki_6_chunk_7 | Advanced Chemistry | After the reunification of Germany, an abundance of anti-immigrant sentiment emerged, as well as attacks on the homes of refugees in the early 1990s. Advanced Chemistry came to prominence in the wake of these actions because of their pro-multicultural society stance in their music. Advanced Chemistry's attitudes revo... | wikipedia |
wiki_6_chunk_8 | Advanced Chemistry | While Advanced Chemistry's use of the German language in their rap allows them to make claims to authenticity and true German heritage, bolstering pro-immigration sentiment, their style can also be problematic for immigrant notions of any real ethnic roots. Indeed, part of the Turkish ethnic minority of Frankfurt views... | wikipedia |
wiki_6_chunk_9 | Advanced Chemistry | Influences
Advanced Chemistry's work was rooted in German history and the country's specific political realities. However, they also drew inspiration from African-American hip-hop acts like A Tribe Called Quest and Public Enemy, who had helped bring a soulful sound and political consciousness to American hip-hop. One m... | wikipedia |
wiki_6_chunk_10 | Advanced Chemistry | Discography
1992 - "Fremd im eigenen Land" (12"/MCD, MZEE)
1993 - "Welcher Pfad führt zur Geschichte" (12"/MCD, MZEE)
1994 - "Operation § 3" (12"/MCD)
1994 - "Dir fehlt der Funk!" (12"/MCD)
1995 - Advanced Chemistry (2xLP/CD) External links
Official Website of MC Torch
Website of Toni L
Official Website of Lingui... | wikipedia |
wiki_6_chunk_11 | Advanced Chemistry | Bibliography
El-Tayeb, Fatima “‘If You Cannot Pronounce My Name, You Can Just Call Me
Pride.’ Afro-German Activism, Gender, and Hip Hop,” Gender & History15/3(2003):459-485. Felbert, Oliver von. “Die Unbestechlichen.” Spex (March 1993): 50–53. Weheliye, Alexander G. Phonographies:Grooves in Sonic Afro-Modernity, Duke ... | wikipedia |
wiki_7_chunk_0 | Alcohol (chemistry) | In chemistry, an alcohol is a type of organic compound that carries at least one hydroxyl functional group (−OH) bound to a saturated carbon atom. The term alcohol originally referred to the primary alcohol ethanol (ethyl alcohol), which is used as a drug and is the main alcohol present in alcoholic drinks. An importan... | wikipedia |
wiki_7_chunk_1 | Alcohol (chemistry) | The suffix -ol appears in the IUPAC chemical name of all substances where the hydroxyl group is the functional group with the highest priority. When a higher priority group is present in the compound, the prefix hydroxy- is used in its IUPAC name. The suffix -ol in non-IUPAC names (such as paracetamol or cholesterol) a... | wikipedia |
wiki_7_chunk_2 | Alcohol (chemistry) | History
The inflammable nature of the exhalations of wine was already known to ancient natural philosophers such as Aristotle (384–322 BCE), Theophrastus (c. 371–287 BCE), and Pliny the Elder (23/24–79 CE). However, this did not immediately lead to the isolation of alcohol, even despite the development of more advanced... | wikipedia |
wiki_7_chunk_3 | Alcohol (chemistry) | The works of Taddeo Alderotti (1223–1296) describe a method for concentrating alcohol involving repeated fractional distillation through a water-cooled still, by which an alcohol purity of 90% could be obtained. The medicinal properties of ethanol were studied by Arnald of Villanova (1240–1311 CE) and John of Rupesciss... | wikipedia |
wiki_7_chunk_4 | Alcohol (chemistry) | Nomenclature | wikipedia |
wiki_7_chunk_5 | Alcohol (chemistry) | Etymology
The word "alcohol" is from the Arabic kohl (), a powder used as an eyeliner. Al- is the Arabic definite article, equivalent to the in English. Alcohol was originally used for the very fine powder produced by the sublimation of the natural mineral stibnite to form antimony trisulfide . It was considered to be ... | wikipedia |
wiki_7_chunk_6 | Alcohol (chemistry) | Bartholomew Traheron, in his 1543 translation of John of Vigo, introduces the word as a term used by "barbarous" authors for "fine powder." Vigo wrote: "the barbarous auctours use alcohol, or (as I fynde it sometymes wryten) alcofoll, for moost fine poudre." | wikipedia |
wiki_7_chunk_7 | Alcohol (chemistry) | The 1657 Lexicon Chymicum, by William Johnson glosses the word as "antimonium sive stibium." By extension, the word came to refer to any fluid obtained by distillation, including "alcohol of wine," the distilled essence of wine. Libavius in Alchymia (1594) refers to "vini alcohol vel vinum alcalisatum". Johnson (1657) ... | wikipedia |
wiki_7_chunk_8 | Alcohol (chemistry) | The term ethanol was invented in 1892, blending "ethane" with the "-ol" ending of "alcohol", which was generalized as a libfix. | wikipedia |
wiki_7_chunk_9 | Alcohol (chemistry) | Systematic names
IUPAC nomenclature is used in scientific publications and where precise identification of the substance is important, especially in cases where the relative complexity of the molecule does not make such a systematic name unwieldy. In naming simple alcohols, the name of the alkane chain loses the termin... | wikipedia |
wiki_7_chunk_10 | Alcohol (chemistry) | In cases where the hydroxy group is bonded to an sp2 carbon on an aromatic ring, the molecule is classified separately as a phenol and is named using the IUPAC rules for naming phenols. Phenols have distinct properties and are not classified as alcohols. | wikipedia |
wiki_7_chunk_11 | Alcohol (chemistry) | Common names
In other less formal contexts, an alcohol is often called with the name of the corresponding alkyl group followed by the word "alcohol", e.g., methyl alcohol, ethyl alcohol. Propyl alcohol may be n-propyl alcohol or isopropyl alcohol, depending on whether the hydroxyl group is bonded to the end or middle c... | wikipedia |
wiki_7_chunk_12 | Alcohol (chemistry) | Alcohols are then classified into primary, secondary (sec-, s-), and tertiary (tert-, t-), based upon the number of carbon atoms connected to the carbon atom that bears the hydroxyl functional group. (The respective numeric shorthands 1°, 2°, and 3° are also sometimes used in informal settings.) The primary alcohols ha... | wikipedia |
wiki_7_chunk_13 | Alcohol (chemistry) | In archaic nomenclature, alcohols can be named as derivatives of methanol using "-carbinol" as the ending. For instance, can be named trimethylcarbinol. Applications | wikipedia |
wiki_7_chunk_14 | Alcohol (chemistry) | Alcohols have a long history of myriad uses. For simple mono-alcohols, which is the focus on this article, the following are most important industrial alcohols:
methanol, mainly for the production of formaldehyde and as a fuel additive
ethanol, mainly for alcoholic beverages, fuel additive, solvent
1-propanol, 1-butano... | wikipedia |
wiki_7_chunk_15 | Alcohol (chemistry) | Toxicity With respect to acute toxicity, simple alcohols have low acute toxicities. Doses of several milliliters are tolerated. For pentanols, hexanols, octanols and longer alcohols, LD50 range from 2–5 g/kg (rats, oral). Methanol and ethanol are less acutely toxic. All alcohols are mild skin irritants. | wikipedia |
wiki_7_chunk_16 | Alcohol (chemistry) | The metabolism of methanol (and ethylene glycol) is affected by the presence of ethanol, which has a higher affinity for liver alcohol dehydrogenase. In this way methanol will be excreted intact in urine. | wikipedia |
wiki_7_chunk_17 | Alcohol (chemistry) | Physical properties
In general, the hydroxyl group makes alcohols polar. Those groups can form hydrogen bonds to one another and to most other compounds. Owing to the presence of the polar OH alcohols are more water-soluble than simple hydrocarbons. Methanol, ethanol, and propanol are miscible in water. Butanol, with a... | wikipedia |
wiki_7_chunk_18 | Alcohol (chemistry) | Because of hydrogen bonding, alcohols tend to have higher boiling points than comparable hydrocarbons and ethers. The boiling point of the alcohol ethanol is 78.29 °C, compared to 69 °C for the hydrocarbon hexane, and 34.6 °C for diethyl ether. | wikipedia |
wiki_7_chunk_19 | Alcohol (chemistry) | Occurrence in nature
Simple alcohols are found widely in nature. Ethanol is the most prominent because it is the product of fermentation, a major energy-producing pathway. Other simple alcohols, chiefly fusel alcohols, are formed in only trace amounts. More complex alcohols however are pervasive, as manifested in sugar... | wikipedia |
wiki_7_chunk_20 | Alcohol (chemistry) | The process generates a range of alcohols that are separated by distillation. Many higher alcohols are produced by hydroformylation of alkenes followed by hydrogenation. When applied to a terminal alkene, as is common, one typically obtains a linear alcohol: Such processes give fatty alcohols, which are useful for dete... | wikipedia |
wiki_7_chunk_21 | Alcohol (chemistry) | Hydration reactions
Some low molecular weight alcohols of industrial importance are produced by the addition of water to alkenes. Ethanol, isopropanol, 2-butanol, and tert-butanol are produced by this general method. Two implementations are employed, the direct and indirect methods. The direct method avoids the formati... | wikipedia |
wiki_7_chunk_22 | Alcohol (chemistry) | Hydration is also used industrially to produce the diol ethylene glycol from ethylene oxide. Biological routes
Ethanol is obtained by fermentation using glucose produced from sugar from the hydrolysis of starch, in the presence of yeast and temperature of less than 37 °C to produce ethanol. For instance, such a process... | wikipedia |
wiki_7_chunk_23 | Alcohol (chemistry) | Several species of the benign bacteria in the intestine use fermentation as a form of anaerobic metabolism. This metabolic reaction produces ethanol as a waste product. Thus, human bodies contain some quantity of alcohol endogenously produced by these bacteria. In rare cases, this can be sufficient to cause "auto-brewe... | wikipedia |
wiki_7_chunk_24 | Alcohol (chemistry) | Like ethanol, butanol can be produced by fermentation processes. Saccharomyces yeast are known to produce these higher alcohols at temperatures above . The bacterium Clostridium acetobutylicum can feed on cellulose to produce butanol on an industrial scale. | wikipedia |
wiki_7_chunk_25 | Alcohol (chemistry) | Substitution
Primary alkyl halides react with aqueous NaOH or KOH mainly to primary alcohols in nucleophilic aliphatic substitution. (Secondary and especially tertiary alkyl halides will give the elimination (alkene) product instead). Grignard reagents react with carbonyl groups to secondary and tertiary alcohols. Rela... | wikipedia |
wiki_7_chunk_26 | Alcohol (chemistry) | Reduction
Aldehydes or ketones are reduced with sodium borohydride or lithium aluminium hydride (after an acidic workup). Another reduction by aluminiumisopropylates is the Meerwein-Ponndorf-Verley reduction. Noyori asymmetric hydrogenation is the asymmetric reduction of β-keto-esters. | wikipedia |
wiki_7_chunk_27 | Alcohol (chemistry) | Hydrolysis
Alkenes engage in an acid catalysed hydration reaction using concentrated sulfuric acid as a catalyst that gives usually secondary or tertiary alcohols. The hydroboration-oxidation and oxymercuration-reduction of alkenes are more reliable in organic synthesis. Alkenes react with NBS and water in halohydrin f... | wikipedia |
wiki_7_chunk_28 | Alcohol (chemistry) | Deprotonation
With aqueous pKa values of around 16–19, they are, in general, slightly weaker acids than water. With strong bases such as sodium hydride or sodium they form salts called alkoxides, with the general formula RO− M+. | wikipedia |
wiki_7_chunk_29 | Alcohol (chemistry) | The acidity of alcohols is strongly affected by solvation. In the gas phase, alcohols are more acidic than in water. In DMSO, alcohols (and water) have a pKa of around 29–32. As a consequence, alkoxides (and hydroxide) are powerful bases and nucleophiles (e.g., for the Williamson ether synthesis) in this solvent. In... | wikipedia |
wiki_7_chunk_30 | Alcohol (chemistry) | Nucleophilic substitution
The OH group is not a good leaving group in nucleophilic substitution reactions, so neutral alcohols do not react in such reactions. However, if the oxygen is first protonated to give , the leaving group (water) is much more stable, and the nucleophilic substitution can take place. For instanc... | wikipedia |
wiki_7_chunk_31 | Alcohol (chemistry) | Alcohols may, likewise, be converted to alkyl bromides using hydrobromic acid or phosphorus tribromide, for example: In the Barton-McCombie deoxygenation an alcohol is deoxygenated to an alkane with tributyltin hydride or a trimethylborane-water complex in a radical substitution reaction. Dehydration
Meanwhile, the oxy... | wikipedia |
wiki_7_chunk_32 | Alcohol (chemistry) | Upon treatment with strong acids, alcohols undergo the E1 elimination reaction to produce alkenes. The reaction, in general, obeys Zaitsev's Rule, which states that the most stable (usually the most substituted) alkene is formed. Tertiary alcohols eliminate easily at just above room temperature, but primary alcohols re... | wikipedia |
wiki_7_chunk_33 | Alcohol (chemistry) | Protonolysis
Tertiary alcohols react with strong acids to generate carbocations. The reaction is related to their dehydration, e.g. isobutylene from tert-butyl alcohol. A special kind of dehydration reaction involves triphenylmethanol and especially its amine-substituted derivatives. When treated with acid, these alcoh... | wikipedia |
wiki_7_chunk_34 | Alcohol (chemistry) | Esterification
Alcohol and carboxylic acids react in the so-called Fischer esterification. The reaction usually requires a catalyst, such as concentrated sulfuric acid:
Other types of ester are prepared in a similar manner for example, tosyl (tosylate) esters are made by reaction of the alcohol with p-toluenesulfonyl... | wikipedia |
wiki_7_chunk_35 | Alcohol (chemistry) | Primary alcohols () can be oxidized either to aldehydes (R-CHO) or to carboxylic acids (). The oxidation of secondary alcohols (R1R2CH-OH) normally terminates at the ketone (R1R2C=O) stage. Tertiary alcohols (R1R2R3C-OH) are resistant to oxidation. | wikipedia |
wiki_7_chunk_36 | Alcohol (chemistry) | The direct oxidation of primary alcohols to carboxylic acids normally proceeds via the corresponding aldehyde, which is transformed via an aldehyde hydrate () by reaction with water before it can be further oxidized to the carboxylic acid. | wikipedia |
wiki_7_chunk_37 | Alcohol (chemistry) | Reagents useful for the transformation of primary alcohols to aldehydes are normally also suitable for the oxidation of secondary alcohols to ketones. These include Collins reagent and Dess-Martin periodinane. The direct oxidation of primary alcohols to carboxylic acids can be carried out using potassium permanganate o... | wikipedia |
wiki_8_chunk_0 | Algebraically closed field | In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . | wikipedia |
wiki_8_chunk_1 | Algebraically closed field | Examples
As an example, the field of real numbers is not algebraically closed, because the polynomial equation x2 + 1 = 0 has no solution in real numbers, even though all its coefficients (1 and 0) are real. The same argument proves that no subfield of the real field is algebraically closed; in particular, the field ... | wikipedia |
wiki_8_chunk_2 | Algebraically closed field | Equivalent properties
Given a field F, the assertion "F is algebraically closed" is equivalent to other assertions: The only irreducible polynomials are those of degree one
The field F is algebraically closed if and only if the only irreducible polynomials in the polynomial ring F[x] are those of degree one. | wikipedia |
wiki_8_chunk_3 | Algebraically closed field | The assertion "the polynomials of degree one are irreducible" is trivially true for any field. If F is algebraically closed and p(x) is an irreducible polynomial of F[x], then it has some root a and therefore p(x) is a multiple of x − a. Since p(x) is irreducible, this means that p(x) = k(x − a), for some k ∈ F \ {0}. ... | wikipedia |
wiki_8_chunk_4 | Algebraically closed field | Every polynomial is a product of first degree polynomials
The field F is algebraically closed if and only if every polynomial p(x) of degree n ≥ 1, with coefficients in F, splits into linear factors. In other words, there are elements k, x1, x2, ..., xn of the field F such that p(x) = k(x − x1)(x − x2) ⋯ (x − xn). | wikipedia |
wiki_8_chunk_5 | Algebraically closed field | If F has this property, then clearly every non-constant polynomial in F[x] has some root in F; in other words, F is algebraically closed. On the other hand, that the property stated here holds for F if F is algebraically closed follows from the previous property together with the fact that, for any field K, any polynom... | wikipedia |
wiki_8_chunk_6 | Algebraically closed field | Polynomials of prime degree have roots
If every polynomial over F of prime degree has a root in F, then every non-constant polynomial has a root in F. It follows that a field is algebraically closed if and only if every polynomial over F of prime degree has a root in F. The field has no proper algebraic extension
The f... | wikipedia |
wiki_8_chunk_7 | Algebraically closed field | If F has no proper algebraic extension, let p(x) be some irreducible polynomial in F[x]. Then the quotient of F[x] modulo the ideal generated by p(x) is an algebraic extension of F whose degree is equal to the degree of p(x). Since it is not a proper extension, its degree is 1 and therefore the degree of p(x) is 1. On ... | wikipedia |
wiki_8_chunk_8 | Algebraically closed field | The field has no proper finite extension
The field F is algebraically closed if and only if it has no proper finite extension because if, within the previous proof, the term "algebraic extension" is replaced by the term "finite extension", then the proof is still valid. (Note that finite extensions are necessarily alge... | wikipedia |
wiki_8_chunk_9 | Algebraically closed field | An endomorphism of Fn has an eigenvector if and only if its characteristic polynomial has some root. Therefore, when F is algebraically closed, every endomorphism of Fn has some eigenvector. On the other hand, if every endomorphism of Fn has an eigenvector, let p(x) be an element of F[x]. Dividing by its leading coeffi... | wikipedia |
wiki_8_chunk_10 | Algebraically closed field | Decomposition of rational expressions
The field F is algebraically closed if and only if every rational function in one variable x, with coefficients in F, can be written as the sum of a polynomial function with rational functions of the form a/(x − b)n, where n is a natural number, and a and b are elements of F. If F ... | wikipedia |
wiki_8_chunk_11 | Algebraically closed field | On the other hand, suppose that the property stated above holds for the field F. Let p(x) be an irreducible element in F[x]. Then the rational function 1/p can be written as the sum of a polynomial function q with rational functions of the form a/(x – b)n. Therefore, the rational expression can be written as a quotient... | wikipedia |
wiki_8_chunk_12 | Algebraically closed field | Relatively prime polynomials and roots
For any field F, if two polynomials p(x),q(x) ∈ F[x] are relatively prime then they do not have a common root, for if a ∈ F was a common root, then p(x) and q(x) would both be multiples of x − a and therefore they would not be relatively prime. The fields for which the reverse im... | wikipedia |
wiki_8_chunk_13 | Algebraically closed field | If the field F is algebraically closed, let p(x) and q(x) be two polynomials which are not relatively prime and let r(x) be their greatest common divisor. Then, since r(x) is not constant, it will have some root a, which will be then a common root of p(x) and q(x). | wikipedia |
wiki_8_chunk_14 | Algebraically closed field | If F is not algebraically closed, let p(x) be a polynomial whose degree is at least 1 without roots. Then p(x) and p(x) are not relatively prime, but they have no common roots (since none of them has roots). | wikipedia |
wiki_8_chunk_15 | Algebraically closed field | Other properties
If F is an algebraically closed field and n is a natural number, then F contains all nth roots of unity, because these are (by definition) the n (not necessarily distinct) zeroes of the polynomial xn − 1. A field extension that is contained in an extension generated by the roots of unity is a cyclotomi... | wikipedia |
wiki_8_chunk_16 | Algebraically closed field | If a proposition which can be expressed in the language of first-order logic is true for an algebraically closed field, then it is true for every algebraically closed field with the same characteristic. Furthermore, if such a proposition is valid for an algebraically closed field with characteristic 0, then not only is... | wikipedia |
wiki_8_chunk_17 | Algebraically closed field | Every field F has some extension which is algebraically closed. Such an extension is called an algebraically closed extension. Among all such extensions there is one and only one (up to isomorphism, but not unique isomorphism) which is an algebraic extension of F; it is called the algebraic closure of F. The theory of ... | wikipedia |
wiki_9_chunk_0 | Algorithms for calculating variance | Algorithms for calculating variance play a major role in computational statistics. A key difficulty in the design of good algorithms for this problem is that formulas for the variance may involve sums of squares, which can lead to numerical instability as well as to arithmetic overflow when dealing with large values. N... | wikipedia |
wiki_9_chunk_1 | Algorithms for calculating variance | This algorithm can easily be adapted to compute the variance of a finite population: simply divide by N instead of n − 1 on the last line. Because and can be very similar numbers, cancellation can lead to the precision of the result to be much less than the inherent precision of the floating-point arithmetic used to ... | wikipedia |
wiki_9_chunk_2 | Algorithms for calculating variance | The variance is invariant with respect to changes in a location parameter, a property which can be used to avoid the catastrophic cancellation in this formula. with any constant, which leads to the new formula the closer is to the mean value the more accurate the result will be, but just choosing a value inside the
s... | wikipedia |
wiki_9_chunk_3 | Algorithms for calculating variance | If just the first sample is taken as the algorithm can be written in Python programming language as | wikipedia |
wiki_9_chunk_4 | Algorithms for calculating variance | def shifted_data_variance(data):
if len(data) < 2:
return 0.0
K = data[0]
n = Ex = Ex2 = 0.0
for x in data:
n = n + 1
Ex += x - K
Ex2 += (x - K) * (x - K)
variance = (Ex2 - (Ex * Ex) / n) / (n - 1)
# use n instead of (n-1) if want to compute the exact variance of ... | wikipedia |
wiki_9_chunk_5 | Algorithms for calculating variance | This formula also facilitates the incremental computation that can be expressed as
K = n = Ex = Ex2 = 0.0 def add_variable(x):
global K, n, Ex, Ex2
if n == 0:
K = x
n += 1
Ex += x - K
Ex2 += (x - K) * (x - K) def remove_variable(x):
global K, n, Ex, Ex2
n -= 1
Ex -= x - K
Ex2... | wikipedia |
wiki_9_chunk_6 | Algorithms for calculating variance | def get_mean():
global K, n, Ex
return K + Ex / n def get_variance():
global n, Ex, Ex2
return (Ex2 - (Ex * Ex) / n) / (n - 1) Two-pass algorithm
An alternative approach, using a different formula for the variance, first computes the sample mean, and then computes the sum of the squares of the differenc... | wikipedia |
wiki_9_chunk_7 | Algorithms for calculating variance | for x in data:
n += 1
sum1 += x mean = sum1 / n for x in data:
sum2 += (x - mean) * (x - mean) variance = sum2 / (n - 1)
return variance | wikipedia |
wiki_9_chunk_8 | Algorithms for calculating variance | This algorithm is numerically stable if n is small. However, the results of both of these simple algorithms ("naïve" and "two-pass") can depend inordinately on the ordering of the data and can give poor results for very large data sets due to repeated roundoff error in the accumulation of the sums. Techniques such as c... | wikipedia |
wiki_9_chunk_9 | Algorithms for calculating variance | Welford's online algorithm
It is often useful to be able to compute the variance in a single pass, inspecting each value only once; for example, when the data is being collected without enough storage to keep all the values, or when costs of memory access dominate those of computation. For such an online algorithm, a... | wikipedia |
wiki_9_chunk_10 | Algorithms for calculating variance | The following formulas can be used to update the mean and (estimated) variance of the sequence, for an additional element xn. Here, denotes the sample mean of the first n samples , their biased sample variance, and their unbiased sample variance. These formulas suffer from numerical instability, as they repeatedly s... | wikipedia |
wiki_9_chunk_11 | Algorithms for calculating variance | An example Python implementation for Welford's algorithm is given below. | wikipedia |
wiki_9_chunk_12 | Algorithms for calculating variance | # For a new value newValue, compute the new count, new mean, the new M2.
# mean accumulates the mean of the entire dataset
# M2 aggregates the squared distance from the mean
# count aggregates the number of samples seen so far
def update(existingAggregate, newValue):
(count, mean, M2) = existingAggregate
count ... | wikipedia |
wiki_9_chunk_13 | Algorithms for calculating variance | # Retrieve the mean, variance and sample variance from an aggregate
def finalize(existingAggregate):
(count, mean, M2) = existingAggregate
if count < 2:
return float("nan")
else:
(mean, variance, sampleVariance) = (mean, M2 / count, M2 / (count - 1))
return (mean, variance, sampleVar... | wikipedia |
wiki_9_chunk_14 | Algorithms for calculating variance | This algorithm is much less prone to loss of precision due to catastrophic cancellation, but might not be as efficient because of the division operation inside the loop. For a particularly robust two-pass algorithm for computing the variance, one can first compute and subtract an estimate of the mean, and then use thi... | wikipedia |
wiki_9_chunk_15 | Algorithms for calculating variance | def weighted_incremental_variance(data_weight_pairs):
w_sum = w_sum2 = mean = S = 0 for x, w in data_weight_pairs:
w_sum = w_sum + w
w_sum2 = w_sum2 + w * w
mean_old = mean
mean = mean_old + (w / w_sum) * (x - mean_old)
S = S + w * (x - mean_old) * (x - mean) | wikipedia |
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