In other words, the afterlife is a direct product of life.
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Meddling in American and European politics is a direct product of that worldview.
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In other words, the hereafter is a direct product of the here and now.
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The Oversight Board is a direct product of Facebook's woes after the 1 election.
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Guy explains that the lyric is a direct product of his feelings after the Brexit vote.
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Colombia's then president, Juan Manuel Santos, blamed the mudslides caused by heavy rains, "as a direct product of climate change".
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There is an equal chance that the show's believability was a direct product of Bryan Fuller's involvement in the show.
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A young mutant herself, Laura's life is a direct product of everything Logan and Charles have worked for — and against.
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Everbank bolsters TIAA's retail services offering by adding direct product capabilities such as residential mortgages to provide to retail customers.
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If the WWE has effectively monopolized the popular idea of what wrestling is, Austin Theory is the direct product of that monoculture.
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This error, while only negligent, seems the direct product of Musk's failing to consult a lawyer (as his settlement with the SEC required).
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Thursday's ruling was, after all, a direct product of the 2014 and 2016 elections, which gave Republicans the Senate and the White House.
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The WSJ reports that the Commerce Department is drafting changes to the so-called foreign direct product rule to prevent Chinese semiconductor access.
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That type information is richly valued, because companies use it to ferret out trends in gaming habits, shape marketing strategies and direct product development.
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Not as something Trump dreamed up by accident, but as a direct product of the counter-jihad's influence on the president of the United States.
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These colors were the result of broadcasting US government NEXRAD, or Next Generation Weather Radar, information, a direct product of the NextGen Air Traffic Control system.
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The government agencies also have been considering changing the Foreign Direct Product Rule, which subjects foreign-made goods based on U.S. technology or software to U.S. oversight.
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It's going to be direct product impact of us understanding images more deeply but it also feeds back into our ability to build better algorithims in the future.
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The language in that AUMF pertained specifically and exclusively to operations in Iraq, and the US invasion of Iraq in 21625 was a direct product of this document.
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Accordingly, modern omnibus projects tend to reflect the ideas and ideals of the university (and often, as with the Very Short Introductions, to be a direct product of them).
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Besides the silliness of those remarks, much of the "entrenched partisanship" he bemoans is the direct product of his own intensely partisan leadership and open race-baiting over the past eight years.
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The striking opt-out rate of over 20 percent on state tests this spring — which puts New York at risk of federal penalties — was a direct product of his administration, he said.
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On the latter one, I suppose we're discussing it because it offers itself as the direct product of the piecing together, rather than blending, of genres, styles, references, and angles of social critique.
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While the plot unfurls naturally, somewhat inevitably towards shit being increasingly lost as the night passes, there's emotional weight to the game that's a direct product of how it presents its group of friends.
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The nostalgia associated with Native American crafts and with 19th- and early 20th-century images of Native peoples is a direct product of historical nostalgia—which, after all, grows out of unexamined and simplistic narratives.
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That isn't a targeted statement so much as a PSA (though it stings freshly of ABC's abominable Dirty Dancing), and a direct product of the fact that I was spoiled in 2006 by Omkara, an Indian film version of Othello.
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These higher bond yields are a boon for investors who may be buying US securities that have to be sold to finance America's ballooning deficit -- a direct product of the Trump tax cuts, narrower than hoped growth, among other factors.
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The singer's alias "Young Murphy the Kidd" might of been a direct product of his East Oakland roots, but his choice to stylize ymtk in lowercase letters draws from bell hooks' rhetoric to allow the work to speak for itself.
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Many Americans saw the shooting as emblematic of, and in many ways a direct product of, this larger issue, and so treated the shooting as not just an isolated death but as representative of a larger problem affecting millions of people.
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If the environment created by #MeToo is less than ideal in its standards of proof, that is a direct product of a deeply broken justice system where only an estimated seven out of a thousand rapes will result in a felony conviction.
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People familiar with the matter told the Journal that the Commerce Department is drafting changes to the foreign direct product rule that would require chip factories globally to obtain licenses if they want to use American equipment to create chips for Huawei.
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It can sometimes feel like a devolution, but this turn back toward words-as-words transforms both the nature of the game and the words themselves, and is a direct product of how Zynga, the publisher of Words With Friends, manages its word list.
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Trump's immigration policies are the direct product of a deliberate and sustained effort the group started in 1979 — and now critics on both left and right say FAIR has successfully hijacked the immigration debate and normalized its radical ideas, even as the American public grows more accepting of immigrants.
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The billionaire owners of Native-themed sports franchises in this nation continue to feel unbothered by the fact that their teams are a direct product of the historicizing and erasure of modern Native people, because to feel bothered and to act on that feeling would cost them money.
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Writing in a Nature News and Views article, Olshansky says this is simply not possible:Fixed genetic programs that directly cause ageing and death cannot exist as a direct product of evolution, because the end result would be death at an age beyond which almost every member of a species would ordinarily live.
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Also they're based in Mountain View, so they're closer to the product groups and they have more like 12 to 18 month research cycles, whereas we're more about algorithmic development and we tend to go for things that are two to three years long and don't necessarily have a direct product focus at the start.
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Goldman analysts believe making the jump to a direct product partnership would give Nike more control over how the brand is represented in the Amazon marketplace, as well as some seriously lucrative economic exposure — we're talking $300 million to $500 million more in revenue, a welcome sales jump for Nike, which announced nearly 1,400 layoffs just last week.
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Some authors draw a distinction between an internal direct product and an external direct product. If A, B \subset X and A \times B \cong X, then we say that X is an internal direct product of A and B, while if A and B are not subobjects then we say that this is an external direct product.
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Then is also isomorphic to Z2. But Z2 has only the trivial automorphism, so the only semi- direct product of N and is the direct product. Since Z4 is different from , we conclude that G is not a semi-direct product of N and .
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In mathematics, especially in the areas of abstract algebra known as universal algebra, group theory, ring theory, and module theory, a subdirect product is a subalgebra of a direct product that depends fully on all its factors without however necessarily being the whole direct product. The notion was introduced by Birkhoff in 1944 and has proved to be a powerful generalization of the notion of direct product.
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In particular, a finite chief factor is a direct product of isomorphic simple groups.
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Tatila Akufuna (son of Imbue) was a direct product of the Mbunda at Lukwakwa.
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This also explains the definition of the multiplication rule in . The direct product is a special case of the semidirect product. To see this, let be the trivial homomorphism (i.e., sending every element of to the identity automorphism of ) then is the direct product .
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Suppose is a semidirect product of the normal subgroup and the subgroup . If is also normal in , or equivalently, if there exists a homomorphism that is the identity on with kernel , then is the direct product of and . The direct product of two groups and can be thought of as the semidirect product of and with respect to for all in . Note that in a direct product, the order of the factors is not important, since is isomorphic to .
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Cyclic groups are Abelian, so the conjugate by every element of every element is the latter. Zmn / Zm \cong Zn. Zmn is the direct product of Zm and Zn if and only if m and n are coprime. Thus e.g. Z12 is the direct product of Z3 and Z4, but not of Z6 and Z2.
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For example, the Schur–Zassenhaus theorem implies the existence of a semi-direct product among groups of order 6; there are two such products, one of which is a direct product, and the other a dihedral group. In contrast, the Schur–Zassenhaus theorem does not say anything about groups of order 4 or groups of order 8 for instance.
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General Prohibition 3 applies to certain items that are produced outside of the U.S. and that are the "direct product" of U.S. technology or software, or they are developed from a plat which is the "direct product" of U.S. technology or software. Under General Prohibition 3, you may not, without a license or license exception, reexport any item that meets the direct product test to a destination in Country Group D:1, E:1, or E:2 (See supplement no.1 to part 740 of the EAR). Additionally, you may not, without a license or license exception, reexport or export from abroad any ECCN 0A919 commodities (foreign-made military commodities) that meet the direct product test to a destination in Country Group D:1, D:3, D:4, D:5, E:1, or E:2.
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In mathematics, specifically in group theory, the direct product is an operation that takes two groups and and constructs a new group, usually denoted . This operation is the group-theoretic analogue of the Cartesian product of sets and is one of several important notions of direct product in mathematics. In the context of abelian groups, the direct product is sometimes referred to as the direct sum, and is denoted G \oplus H. Direct sums play an important role in the classification of abelian groups: according to the fundamental theorem of finite abelian groups, every finite abelian group can be expressed as the direct sum of cyclic groups.
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The Baer–Specker group is the group B = ZN of all integer sequences with componentwise addition, that is, the direct product of countably many copies of Z.
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In mathematics, the Krull–Schmidt theorem states that a group subjected to certain finiteness conditions on chains of subgroups, can be uniquely written as a finite direct product of indecomposable subgroups.
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A metric on a Cartesian product of metric spaces, and a norm on a direct product of normed vector spaces, can be defined in various ways, see for example p-norm.
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In abstract algebra, the theory of fields lacks a direct product: the direct product of two fields, considered as a ring, is never itself a field. Nonetheless, it is often required to "join" two fields K and L, either in cases where K and L are given as subfields of a larger field M, or when K and L are both field extensions of a smaller field N (for example a prime field). The tensor product of fields is the best available construction on fields with which to discuss all the phenomena arising. As a ring, it is sometimes a field, and often a direct product of fields; it can, though, contain non-zero nilpotents (see radical of a ring).
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Given groups (with operation ) and (with operation ), the direct product is defined as follows: The resulting algebraic object satisfies the axioms for a group. Specifically: ;Associativity: The binary operation on is indeed associative. ;Identity: The direct product has an identity element, namely , where is the identity element of and is the identity element of . ;Inverses: The inverse of an element of is the pair , where is the inverse of in , and is the inverse of in .
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On July 30, 2009, Apple discontinued Shake. No direct product replacement was announced by Apple, but some features are now available in Final Cut Studio and Motion, such as the SmoothCam filter.
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A class of bands forms a variety if it is closed under formation of subsemigroups, homomorphic images and direct product. Each variety of bands can be defined by a single defining identity.
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The elements of the direct sum of modules \bigoplus M_i are sequences \alpha_i \in M_i where cofinitely many \alpha_i = 0. The analog (without requiring that cofinitely many are zero) is the direct product.
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The largest possible global symmetry group of a non-supersymmetric interacting field theory is a direct product of the conformal group with an internal group. Such theories are known as conformal field theories.
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For instance Dehn's algorithm does not solve the word problem for the fundamental group of the torus. However this group is the direct product of two infinite cyclic groups and so has a solvable word problem.
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For p ≠ 2, one is a semi-direct product of Cp×Cp with Cp, and the other is a semi- direct product of Cp2 with Cp. The first one can be described in other terms as group UT(3,p) of unitriangular matrices over finite field with p elements, also called the Heisenberg group mod p. For p = 2, both the semi-direct products mentioned above are isomorphic to the dihedral group Dih4 of order 8. The other non-abelian group of order 8 is the quaternion group Q8.
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A ring is left coherent if and only if every direct product of flat right modules is flat , . Compare this to: A ring is left Noetherian if and only if every direct sum of injective left modules is injective.
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The Weyl group of E7 is of order 2903040: it is the direct product of the cyclic group of order 2 and the unique simple group of order 1451520 (which can be described as PSp6(2) or PSΩ7(2)).
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The automorphism group of the Tutte 12-cage is of order and is a semi-direct product of the projective special unitary group PSU(3,3) with the cyclic group Z/2Z.Geoffrey Exoo & Robert Jajcay, Dynamic cage survey, Electr. J. Combin. 15 (2008).
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Society and culture among the Kipsigis was a direct product of its heritage. This thus means the culture of the Kipsigis was stifled and stringent, hardly changing. The Kipsigis have a circular view of life. They were a superstitious as well as a spiritual community.
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In mathematics, it is possible to combine several rings into one large product ring. This is done by giving the Cartesian product of a (possibly infinite) family of rings coordinatewise addition and multiplication. The resulting ring is called a direct product of the original rings.
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In some classical texts, the notion of direct sum of algebras over a field is also introduced. This construction, however, does not provide a coproduct in the category of algebras, but a direct product (see note below and the remark on direct sums of rings).
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Non-trivial semidirect products do not arise in abelian categories, such as the category of modules. In this case, the splitting lemma shows that every semidirect product is a direct product. Thus the existence of semidirect products reflects a failure of the category to be abelian.
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Since the finite direct product is the same as the finite direct sum of groups, it follows that the unrestricted A WrΩ H and the restricted wreath product A wrΩ H agree if the H-set Ω is finite. In particular this is true when Ω = H is finite.
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A Henselian local ring is called strictly Henselian if its residue field is separably closed. A field with valuation is said to be Henselian if its valuation ring is Henselian. A ring is called Henselian if it is a direct product of a finite number of Henselian local rings.
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Let us split Y into the direct product Y= Y_1 \oplus Y_2 , where \dim Y_2 < \infty . Let Q be the projection operator onto Y_1 . Consider also the direct product X= X_1 \oplus X_2 . Applying the operators Q and I-Q to the original equation, one obtains the equivalent system : Qf(x,\lambda)=0 \, : (I-Q)f(x,\lambda)=0 \, Let x_1\in X_1 and x_2 \in X_2 , then the first equation : Qf(x_1+x_2,\lambda)=0 \, can be solved with respect to x_2 by applying the implicit function theorem to the operator : Qf(x_1+x_2,\lambda): \quad X_2\times(X_1\times\Lambda)\to Y_1 \, (now the conditions of the implicit function theorem are fulfilled).
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It was also seen as a key step towards a new legal agreement on climate change, the Paris Agreement, which was adopted by the COP21 in Paris in December 2015 and became effective in November 2016. Another direct product of the Climate Summit 2014 was the New York Declaration on Forests.
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Structures whose axioms unavoidably include nonidentities are among the most important ones in mathematics, e.g., fields and division rings. Structures with nonidentities present challenges varieties do not. For example, the direct product of two fields is not a field, because (1,0)\cdot(0,1)=(0,0), but fields do not have zero divisors.
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More generally, biproducts exist in the category of modules over a ring. On the other hand, biproducts do not exist in the category of groups.Borceux, 7 Here, the product is the direct product, but the coproduct is the free product. Also, biproducts do not exist in the category of sets.
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New York: McGraw-Hill, 1969. Print. Operant conditioning has to do with rewards and punishments and how they can either strengthen or weaken certain behaviours.Schaefer, Halmuth H., and Patrick L. Martin. Behavioral Therapy, 20-24. New York: McGraw-Hill, 1969. Print. Contingency management programs are a direct product of research from operant conditioning.
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A spherical triangle The area of a spherical triangle on the unit sphere is . The isometry group of the unit sphere in is the orthogonal group , with the rotation group as the subgroup of isometries preserving orientation. It is the direct product of with the antipodal map, sending to ., Chapter II, Spherical geometry.
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Co2 is one of the 26 sporadic groups and was discovered by as the group of automorphisms of the Leech lattice Λ fixing a lattice vector of type 2. It is thus a subgroup of Co0. It is isomorphic to a subgroup of Co1. The direct product 2×Co2 is maximal in Co0.
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A choice function (selector, selection) is a mathematical function f that is defined on some collection X of nonempty sets and assigns to each set S in that collection some element f(S) of S. In other words, f is a choice function for X if and only if it belongs to the direct product of X.
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Complements generalize both the direct product (where the subgroups H and K are normal in G), and the semidirect product (where one of H or K is normal in G). The product corresponding to a general complement is called the internal Zappa–Szép product. When H and K are nontrivial, complement subgroups factor a group into smaller pieces.
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The remaining two electrons are used to generate a proton motive force and reduce NAD(P) through reverse electron transport. Recent results, however, show that HAO does not produce nitrite as a direct product of catalysis. This enzyme instead produces nitric oxide and three electrons. Nitric oxide can then be oxidized by other enzymes (or oxygen) to nitrite.
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The ultraproduct is a mathematical construction that appears mainly in abstract algebra and mathematical logic, in particular in model theory and set theory. An ultraproduct is a quotient of the direct product of a family of structures. All factors need to have the same signature. The ultrapower is the special case of this construction in which all factors are equal.
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A second aspect relating to the infringement of European patents is prescribed in . The EPC requires that national courts must consider the "direct product of a patented process" to be an infringement. reads: :If the subject-matter of the European patent is a process, the protection conferred by the patent shall extend to the products directly obtained by such process.
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The above map to has a section: we can view as the subgroup of that are diagonal with in the upper left corner and on the rest of the diagonal. Therefore is a semi-direct product of with . The unitary group is not abelian for . The center of is the set of scalar matrices with ; this follows from Schur's lemma.
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Wordsmith is Ai's platform for natural language generation. It is "an artificial intelligence system that uses mounds of data, quantitative analysis and some rules about style and good writing" to produce stories. Wordsmith is sold as both a direct product and service to clients. In October 2015, the Wordsmith platform was updated to allow users to create their own narratives through online software.
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Goursat's lemma, named after the French mathematician Édouard Goursat, is an algebraic theorem about subgroups of the direct product of two groups. It can be stated more generally in a Goursat variety (and consequently it also holds in any Maltsev variety), from which one recovers a more general version of Zassenhaus' butterfly lemma. In this form, Goursat's theorem also implies the snake lemma.
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However, if a short exact sequence of groups is right split (2.), then it need not be left split or a direct sum (neither 1. nor 3. follows): the problem is that the image of the right splitting need not be normal. What is true in this case is that is a semidirect product, though not in general a direct product.
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The full automorphism group of the diamond graph is a group of order 4 isomorphic to the Klein four-group, the direct product of the cyclic group Z/2Z with itself. The characteristic polynomial of the diamond graph is x(x+1)(x^2-x-4). It is the only graph with this characteristic polynomial, making it a graph determined by its spectrum.
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The infrared spectrum of HCl gas There are many types of coupled transition such as are observed in vibration-rotation spectra. The excited-state wave function is the product of two wave functions such as vibrational and rotational. The general principle is that the symmetry of the excited state is obtained as the direct product of the symmetries of the component wave functions.Harris & Berolucci, p.
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Let C and D be categories. The collection of all functors from C to D forms the objects of a category: the functor category. Morphisms in this category are natural transformations between functors. Functors are often defined by universal properties; examples are the tensor product, the direct sum and direct product of groups or vector spaces, construction of free groups and modules, direct and inverse limits.
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The space splits as a semi-direct product, :LG = \Omega G \rtimes G. We may also think of as the loop space on . From this point of view, is an H-space with respect to concatenation of loops. On the face of it, this seems to provide with two very different product maps. However, it can be shown that concatenation and pointwise multiplication are homotopic.
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The external central product is the quotient of the direct product H × K by the normal subgroup N = { ( h, k ) : h in H1, k in K1, and θ(h)⋅k = 1 }, . Sometimes the stricter requirement that H1 = Z(H) and K1 = Z(K) is imposed, as in . An internal central product is isomorphic to an external central product with H1 = K1 = H ∩ K and θ the identity.
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In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one talks about the product in category theory, which formalizes these notions. Examples are the product of sets, groups (described below), rings, and other algebraic structures.
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The proof of validity is the same as above, replacing the direct product of the fields Fq[x]/fi by the direct product of their subfields with q elements. The complexity is decomposed in O(n^2\log(r)\log(q)) for the algorithm itself, O(n^2(\log(q)+n)) for the computation of the matrix of the linear map (which may be already computed in the square-free factorization) and O(n3) for computing its kernel. It may be noted that this algorithm works also if the factors have not the same degree (in this case the number r of factors, needed for stopping the while loop, is found as the dimension of the kernel). Nevertheless, the complexity is slightly better if square-free factorization is done before using this algorithm (as n may decrease with square-free factorization, this reduces the complexity of the critical steps).
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6 (glide-reflections, translations and rotations) is generated by a glide reflection and a rotation about a point on the line of reflection. It is isomorphic to a semi-direct product of Z and C2. Example pattern with this symmetry group: :File:Frieze example p2mg.png A typical example of glide reflection in everyday life would be the track of footprints left in the sand by a person walking on a beach.
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The symmetry group of special relativity is not entirely simple, due to translations. The Lorentz group is the set of the transformations that keep the origin fixed, but translations are not included. The full Poincaré group is the semi-direct product of translations with the Lorentz group. If translations are to be similar to elements of the Lorentz group, then as boosts are non-commutative, translations would also be non-commutative.
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In mathematics, especially in the field of group theory, the central product is one way of producing a group from two smaller groups. The central product is similar to the direct product, but in the central product two isomorphic central subgroups of the smaller groups are merged into a single central subgroup of the product. Central products are an important construction and can be used for instance to classify extraspecial groups.
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One may consider the ring of integers mod n, where n is squarefree. By the Chinese remainder theorem, this ring factors into the direct product of rings of integers mod p. Now each of these factors is a field, so it is clear that the factor's only idempotents will be 0 and 1. That is, each factor has two idempotents. So if there are m factors, there will be 2m idempotents.
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In category theory, the product of two (or more) objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as the Cartesian product of sets, the direct product of groups or rings, and the product of topological spaces. Essentially, the product of a family of objects is the "most general" object which admits a morphism to each of the given objects.
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If we do not impose the coprime condition, the theorem is not true: consider for example the cyclic group C_4 and its normal subgroup C_2. Then if C_4 were a semidirect product of C_2 and C_4 / C_2 \cong C_2 then C_4 would have to contain two elements of order 2, but it only contains one. Another way to explain this impossibility of splitting C_4 (i.e. expressing it as a semidirect product) is to observe that the automorphisms of C_2 are the trivial group, so the only possible [semi]direct product of C_2 with itself is a direct product (which gives rise to the Klein four-group, a group that is non-isomorphic with C_4). An example where the Schur–Zassenhaus theorem does apply is the symmetric group on 3 symbols, S_3, which has a normal subgroup of order 3 (isomorphic with C_3) which in turn has index 2 in S_3 (in agreement with the theorem of Lagrange), so S_3 / C_3 \cong C_2.
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A coproduct in the category of algebras is a free product of algebras.) Direct products are commutative and associative (up to isomorphism), meaning that it doesn't matter in which order one forms the direct product. If Ai is an ideal of Ri for each i in I, then is an ideal of R. If I is finite, then the converse is true, i.e., every ideal of R is of this form. However, if I is infinite and the rings Ri are non-zero, then the converse is false: the set of elements with all but finitely many nonzero coordinates forms an ideal which is not a direct product of ideals of the Ri. The ideal A is a prime ideal in R if all but one of the Ai are equal to Ri and the remaining Ai is a prime ideal in Ri. However, the converse is not true when I is infinite.
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Examples of CDRH-regulated devices include cellular phones, airport baggage screening equipment, television receivers, microwave ovens, tanning booths, and laser products. CDRH regulatory powers include the authority to require certain technical reports from the manufacturers or importers of regulated products, to require that radiation-emitting products meet mandatory safety performance standards, to declare regulated products defective, and to order the recall of defective or noncompliant products. CDRH also conducts limited amounts of direct product testing.
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It can also be of interest to go beyond prime order components to prime-power order. Consider an elementary abelian group G to be of type (p,p,...,p) for some prime p. A homocyclic group (of rank n) is an abelian group of type (m,m,...,m) i.e. the direct product of n isomorphic cyclic groups of order m, of which groups of type (pk,pk,...,pk) are a special case.
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Note that is of order , so the subgroup of generated by contains elements and thus is a proper subset of , because includes all elements of this subgroup and the center which does not contain but at least elements. Hence the order of is strictly larger than , therefore = , therefore is an element of the center of . Hence is abelian and in fact isomorphic to the direct product of two cyclic groups each of order .
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Main Event Championship Wrestling (MECW) is a privately owned professional wrestling promotion founded by Jason Daniel. The company is active in the integrated-media industry, broadcasting its events on television and the Internet. MECW also gains revenue from live events, product licensing, and direct product sales. Jason Daniel is also the President of the company and head booker for MECW events, but only currently works as a part-time wrestler on camera.
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Similarly, n5 must divide 3, and n5 must equal 1 (mod 5); thus it must also have a single normal subgroup of order 5. Since 3 and 5 are coprime, the intersection of these two subgroups is trivial, and so G must be the internal direct product of groups of order 3 and 5, that is the cyclic group of order 15. Thus, there is only one group of order 15 (up to isomorphism).
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Co0 has 4 conjugacy classes of elements of order 3. In M24 an element of shape 38 generates a group normal in a copy of S3, which commutes with a simple subgroup of order 168. A direct product in M24 permutes the octads of a trio and permutes 14 dodecad diagonal matrices in the monomial subgroup. In Co0 this monomial normalizer is expanded to a maximal subgroup of the form , where 2.
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Monday, March 16, 2020 "BTL", or "Below The Line", suggests that the advertising is going to target a specific group of potential consumers. BTL advertising agencies will be hired to help companies to develop ads and promotion strategies in a creative way, directed to certain groups of people, using tools like direct emailing, or direct product demonstrations for a specific group of people, like giving away vitamin samples at the door of a famous gym.
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The principal rings constructed in Example 5. above are always Artinian rings; in particular they are isomorphic to a finite direct product of principal Artinian local rings. A local Artinian principal ring is called a special principal ring and has an extremely simple ideal structure: there are only finitely many ideals, each of which is a power of the maximal ideal. For this reason, special principal rings are examples of uniserial rings.
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Notational abuse to be found below includes for the exponential map given an argument, writing for the element in a direct product ( is the identity in ), and analogously for Lie algebra direct sums (where also and are used interchangeably). Likewise for semidirect products and semidirect sums. Canonical injections (both for groups and Lie algebras) are used for implicit identifications. Furthermore, if , , ..., are groups, then the default names for elements of , , ..., are , , ..., and their Lie algebras are , , ... .
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Proving existence is relatively straightforward: let be the set of all normal subgroups that can not be written as a product of indecomposable subgroups. Moreover, any indecomposable subgroup is (trivially) the one-term direct product of itself, hence decomposable. If Krull-Schmidt fails, then contains ; so we may iteratively construct a descending series of direct factors; this contradicts the DCC. One can then invert the construction to show that all direct factors of appear in this way.
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This subset does indeed form a group, and for a finite set of groups {Hi} the external direct sum is equal to the direct product. If G = ∑Hi, then G is isomorphic to ∑E{Hi}. Thus, in a sense, the direct sum is an "internal" external direct sum. For each element g in G, there is a unique finite set S and a unique set {hi ∈ Hi : i ∈ S} such that g = ∏ {hi : i in S}.
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In algebra, more specifically group theory, a p-elementary group is a direct product of a finite cyclic group of order relatively prime to p and a p-group. A finite group is an elementary group if it is p-elementary for some prime number p. An elementary group is nilpotent. Brauer's theorem on induced characters states that a character on a finite group is a linear combination with integer coefficients of characters induced from elementary subgroups.
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The automorphism group of the Nauru graph is a group of order 144.Royle, G. F024A data It is isomorphic to the direct product of the symmetric groups S4 and S3 and acts transitively on the vertices, on the edges and on the arcs of the graph. Therefore, the Nauru graph is a symmetric graph (though not distance transitive). It has automorphisms that take any vertex to any other vertex and any edge to any other edge.
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Every infinite cyclic group is isomorphic to the additive group of Z, the integers. Every finite cyclic group of order n is isomorphic to the additive group of Z/nZ, the integers modulo n. Every cyclic group is an abelian group (meaning that its group operation is commutative), and every finitely generated abelian group is a direct product of cyclic groups. Every cyclic group of prime order is a simple group which cannot be broken down into smaller groups.
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Industrialized nations more often face malnutrition in the form of over- nutrition from excess calories and non-nutritious carbohydrates, which has contributed greatly to the public health epidemic of obesity. Disparities, according to gender, geographic location and socio-economic position, both within and between countries, represent the biggest threat to child nutrition in industrialized countries. These disparities are a direct product of social inequalities and social inequalities are rising throughout the industrialized world, particularly in Europe.
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Crimpshrine is featured in the 2017 documentary Turn It Around: The Story of East Bay Punk. Although closely associated with the Gilman Street Project, Cometbus was quick to note in a 1988 interview with Flipside magazine that they were not a direct product of Gilman, having been established as a band years before the launch of the club. "We helped them as much as they helped us because we are a part of it just as much," Cometbus said.
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In mathematics, in the field of group theory, a group is said to be characteristically simple if it has no proper nontrivial characteristic subgroups. Characteristically simple groups are sometimes also termed elementary groups. Characteristically simple is a weaker condition than being a simple group, as simple groups must not have any proper nontrivial normal subgroups, which include characteristic subgroups. A finite group is characteristically simple if and only if it is the direct product of isomorphic simple groups.
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The zero scheme of Q in H/L defines a smooth quadric surface X in the associated projective 3-space over k. Over an algebraic closure of k, X is a product of two projective lines, so by a descent argument X is the Weil restriction to k of the projective line over a quadratic étale algebra K. Since Q is not split over k, an auxiliary argument with special orthogonal groups over k forces K to be a field (rather than a product of two copies of k). The natural S6-action on everything in sight defines a map from S6 to the k-automorphism group of X, which is the semi- direct product G of PGL2(K) = PGL2(9) against the Galois involution. This map carries the simple group A6 nontrivially into (hence onto) the subgroup PSL2(9) of index 4 in the semi-direct product G, so S6 is thereby identified as an index-2 subgroup of G (namely, the subgroup of G generated by PSL2(9) and the Galois involution).
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In other words, these works were programmatic rather than abstract.Kennedy, 711. The form was a direct product of Romanticism which encouraged literary, pictorial and dramatic associations in music. It developed into an important form of programme music in the second half of the 19th century.Spencer, P., 1233 The first 12 symphonic poems were composed in the decade 1848–58 (though some use material conceived earlier); one other, Von der Wiege bis zum Grabe (From the Cradle to the Grave), followed in 1882.
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Robert Remak was born in Berlin. He studied at Humboldt University of Berlin under Ferdinand Georg Frobenius and received his doctorate in 1911. His dissertation, Über die Zerlegung der endlichen Gruppen in indirekte unzerlegbare Faktoren ("On the decomposition of a finite group into indirect indecomposable factors") established that any two decompositions of a finite group into a direct product are related by a central automorphism. A weaker form of this statement, uniqueness, was first proved by Joseph Wedderburn in 1909.
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The bio-mechanical model predicts that morphological variation in torus size is the direct product of differential tension caused by mastication, as indicated by an increase in load/lever ratio and broad craniofacial angle.Oyen and Russell, 1984, p. 368-369 Research done on this model has largely been based on earlier work of Endo. By applying pressure similar to the type associated with chewing, he carried out an analysis of the structural function of the supraorbital region on dry human and gorilla skulls.
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It was assumed that youth violence was the direct product of gangs involvement almost in every youth violence incident. Some gang members were interviewed and it was learned that many did not classify themselves as gangs or gang members. Researchers with the help of gang and patrol officer identify the areas of operation pertaining to each gang or information was also acquired from gang members. Each area was highlighted on a printed map; this facilitated the identification of gang- controlled territory.
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The kernel of is the centralizer of in , and so is at least a semidirect product, , but the action of on is trivial, and so the product is direct. This proof is somewhat interesting since the original exact sequence is reversed during the proof. This can be restated in terms of elements and internal conditions: If is a normal, complete subgroup of a group, , then is a direct product. The proof follows directly from the definition: is centerless giving is trivial.
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LIM functions around the premise that marketing or advertising agencies aim to appeal to companies' target demographic. Avenues such as sponsorship or direct product placement and sampling are explored in turn. Unlike traditional event marketing, LIM suggests that end-users can sample the product or service in a comfortable and relaxed atmosphere. The theory posits that the end-user will have as positive as possible an interaction with the given brand, thereby leading to word-of- mouth communication and potential future purchases.
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In abstract algebra, the direct sum is a construction which combines several modules into a new, larger module. The direct sum of modules is the smallest module which contains the given modules as submodules with no "unnecessary" constraints, making it an example of a coproduct. Contrast with the direct product, which is the dual notion. The most familiar examples of this construction occur when considering vector spaces (modules over a field) and abelian groups (modules over the ring Z of integers).
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The Hoffman graph is not a vertex-transitive graph and its full automorphism group is a group of order 48 isomorphic to the direct product of the symmetric group S4 and the cyclic group Z/2Z. The characteristic polynomial of the Hoffman graph is equal to :(x-4) (x-2)^4 x^6 (x+2)^4 (x+4) making it an integral graph—a graph whose spectrum consists entirely of integers. It is the same spectrum as the hypercube Q4.
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In mathematics, the Kronecker product, sometimes denoted by ⊗, is an operation on two matrices of arbitrary size resulting in a block matrix. It is a generalization of the outer product (which is denoted by the same symbol) from vectors to matrices, and gives the matrix of the tensor product with respect to a standard choice of basis. The Kronecker product is to be distinguished from the usual matrix multiplication, which is an entirely different operation. The Kronecker product is also sometimes called matrix direct product.
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These symmetries lead, for s>2, to 'too many' conservation laws that trivialise scattering so that S=1. Another well-known result is the Coleman-Mandula theorem. that, under certain assumptions, states that any symmetry group of S-matrix is necessarily locally isomorphic to the direct product of an internal symmetry group and the Poincaré group. This means that there cannot be any symmetry generators transforming as tensors of the Lorentz group - S-matrix cannot have symmetries that would be associated with higher spin charges.
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One can similarly define the Cartesian product of n sets, also known as an n-fold Cartesian product, which can be represented by an n-dimensional array, where each element is an n-tuple. An ordered pair is a 2-tuple or couple. More generally still, one can define the Cartesian product of an indexed family of sets. The Cartesian product is named after René Descartes, whose formulation of analytic geometry gave rise to the concept, which is further generalized in terms of direct product.
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Assume that a group, , is a group extension given as a short exact sequence of groups : with kernel, , and quotient, . If the kernel, , is a complete group then the extension splits: is isomorphic to the direct product, . A proof using homomorphisms and exact sequences can be given in a natural way: The action of (by conjugation) on the normal subgroup, gives rise to a group homomorphism, . Since and has trivial center the homomorphism is surjective and has an obvious section given by the inclusion of in .
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The ASP hotspot was originally located beneath Australia and a chain of seamounts connecting it to the southern end of the Ninety East Ridge, i.e. the ASP hotspot track, indicate it probably contributed to the formation of the Ninety East Ridge before the SEIR opened. The opening of the Southern Ocean began west of Australia around 100 Ma from where it propagated eastward at about 2 cm/yr. This rifting was not the direct product of hotspot interaction as it occurred over a cooler than normal mantle.
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A quasigroup is semisymmetric if the following equivalent identities hold: :xy = y / x, :yx = x \ y, :x = (yx)y, :x = y(xy). Although this class may seem special, every quasigroup Q induces a semisymmetric quasigroup QΔ on the direct product cube Q3 via the following operation: :(x_1, x_2, x_3) \cdot (y_1, y_2, y_3) = (y_3/x_2, y_1 \backslash x_3 , x_1y_2) = (x_2//y_3, x_3 \backslash \backslash y_1, x_1y_2) , where "//" and "\\\" are the conjugate division operations given by y // x = x / y and y \backslash\backslash x = x \backslash y.
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In 1933 he got magister degree at Jan Kazimierz University of Lwów, and in 1939 he got master's degree at Columbia University. Supervised by Francis Joseph Murray, he got doctorate degree in 1942 for the thesis "On the Direct Product of Banach Spaces". Shortly after being appointed to a junior professorship, he joined the United States army where during training he suffered a back injury which affected him for the remainder of his life. In 1943 he was appointed to an assistant professorship at University of Vermont.
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Various raw meats Animals are used as food either directly or indirectly by the products they produce. Meat is an example of a direct product taken from an animal, which comes from muscle systems or from organs (offal). Food products produced by animals include milk produced by mammary glands, which in many cultures is drunk or processed into dairy products (cheese, butter, etc.). In addition, birds and other animals lay eggs, which are often eaten, and bees produce honey, a reduced nectar from flowers, which is a popular sweetener in many cultures.
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While it might seem as though a natural law was at work, it happened each and every time only because God willed it to happen—the event was "a direct product of divine intervention as any more attention grabbing miracle". Averroes, by contrast insisted while God created the natural law, humans "could more usefully say that fire caused cotton to burn—because creation had a pattern that they could discern." For al-Ghazali's argument see The Incoherence of the Philosophers. Translated by Michael E. Marmura. 2nd ed, Provo Utah, 2000, pp.116-7.
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In glycogenolysis, it is the direct product of the reaction in which glycogen phosphorylase cleaves off a molecule of glucose from a greater glycogen structure. A deficiency of muscle glycogen phosphorylase is known as glycogen storage disease type V (McArdle Disease). To be utilized in cellular catabolism it must first be converted to glucose 6-phosphate by the enzyme phosphoglucomutase. One reason that cells form glucose 1-phosphate instead of glucose during glycogen breakdown is that the very polar phosphorylated glucose cannot leave the cell membrane and so is marked for intracellular catabolism.
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"Dieselwall" in Berlin Starting in 1991, Diesel has been known for producing ads that invoke surreal images in lieu of direct product details, in partnership with Swedish ad agency Paradiset DDB, Stockholm. These included 1997 ads portraying life in Communist North Korea (shot in Hong Kong). Another ad campaign imitated automobile crashes.Bruzzi and Gibson, Pg. 143 Campaigns have also used social consciousness as a theme, ironic plays on global issues (such as their Global Warming Ready campaign featuring post-global warming backdrops in global locations), as well as anti-establishment messages.
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A subdirect product is a subalgebra (in the sense of universal algebra) A of a direct product ΠiAi such that every induced projection (the composite pjs: A → Aj of a projection pj: ΠiAi → Aj with the subalgebra inclusion s: A → ΠiAi) is surjective. A direct (subdirect) representation of an algebra A is a direct (subdirect) product isomorphic to A. An algebra is called subdirectly irreducible if it is not subdirectly representable by "simpler" algebras. Subdirect irreducibles are to subdirect product of algebras roughly as primes are to multiplication of integers.
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Miyako island, Okinawa Habushu, a version of awamori bottled with habu vipers Awamori (泡盛, Okinawan: , 'āmui) is an alcoholic beverage indigenous and unique to Okinawa, Japan. It is made from long grain indica rice, and is not a direct product of brewing (like sake) but of distillation (like shōchū). All awamori made today is from indica rice imported from Thailand, the local production being insufficient to meet domestic demand. Awamori is typically 60–86 proof (30–43% alcohol), although "export" brands (including brands shipped to mainland Japan) are increasingly 50 proof (25% alcohol).
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In the Calvin cycle, the energy from the electron carriers is used in carbon fixation, the conversion of carbon dioxide and water into carbohydrates. RPIA is essential in the cycle, as Ru5P generated from R5P is subsequently converted to ribulose-1,5-bisphosphate (RuBP), the acceptor of carbon dioxide in the first dark reaction of photosynthesis (Figure 3). The direct product of RuBP carboxylase reaction is glyceraldehyde-3-phosphate; these are subsequently used to make larger carbohydrates. Glyceraldehyde-3-phosphate is converted to glucose which is later converted by the plant to storage forms (e.g.
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Universal properties may be used in other areas of mathematics implicitly, but the abstract and more precise definition of it can be studied in category theory. This article gives a general treatment of universal properties. To understand the concept, it is useful to study several examples first, of which there are many: all free objects, direct product and direct sum, free group, free lattice, Grothendieck group, Dedekind–MacNeille completion, product topology, Stone–Čech compactification, tensor product, inverse limit and direct limit, kernel and cokernel, pullback, pushout and equalizer.
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In Euclidean plane geometry terms, being a parallelogram is affine since affine transformations always take one parallelogram to another one. Being a circle is not affine since an affine shear will take a circle into an ellipse. To explain accurately the relationship between affine and Euclidean geometry, we now need to pin down the group of Euclidean geometry within the affine group. The Euclidean group is in fact (using the previous description of the affine group) the semi- direct product of the orthogonal (rotation and reflection) group with the translations.
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In particular, a finite solvable group is characteristically simple if and only if it is an elementary abelian group. This does not hold in general for infinite groups; for example, the rational numbers form a characteristically simple group that is not a direct product of simple groups. A minimal normal subgroup of a group G is a nontrivial normal subgroup N of G such that the only proper subgroup of N that is normal in G is the trivial subgroup. Every minimal normal subgroup of a group is characteristically simple.
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Pricesearcher uses PriceBot, its custom web crawler, to search the web for prices, and it allows direct product feeds from retailers at no cost. The search engine's rapid growth has been attributed to its enabling technology: a retailer can upload their product feed in any format, without the need for further development. Pricesearcher processes 1.5 billion prices every day and uses Amazon Web Services (AWS), to which it migrated in December 2016, to enable the high volume of data processing required. The rest of the business uses algorithms, NLP, Machine learning, data science and artificial intelligence to organise all the data.
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Electroconvulsive therapy has been depicted in fiction, including fictional works partly based on true experiences. These include Sylvia Plath's autobiographical novel, The Bell Jar, Ken Loach's film Family Life, and Ken Kesey's novel One Flew Over the Cuckoo's Nest; Kesey's novel is a direct product of his time working the graveyard shift as an orderly at a mental health facility in Menlo Park, California. In the 2000 film Requiem for a Dream, Sarah Goldfarb receives "unmodified" electroconvulsive therapy after experiencing severe amphetamine psychosis following prolonged stimulant abuse. Unlike typical ECT treatment, she is given no anesthetic or medication before.
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Let D be the set of diagonal matrices in the matrix ring Mn(R), that is the set of the matrices such that every nonzero entry, if any, is on the main diagonal. Then D is closed under matrix addition and matrix multiplication, and contains the identity matrix, so it is a subalgebra of Mn(R). As an algebra over R, D is isomorphic to the direct product of n copies of R. It is a free R-module of dimension n. The idempotent elements of D are the diagonal matrices such that the diagonal entries are themselves idempotent.
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The natural sum of α and β is often denoted by α⊕β or α#β, and the natural product by α⊗β or α⨳β. The natural operations come up in the theory of well partial orders; given two well partial orders S and T, of types (maximum linearizations) o(S) and o(T), the type of the disjoint union is o(S)⊕o(T), while the type of the direct product is o(S)⊗o(T).D. H. J. De Jongh and R. Parikh, Well-partial orderings and hierarchies, Indag. Math. 39 (1977), 195–206.
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Available here One may take this relation as a definition of the natural operations by choosing S and T to be ordinals α and β; so α⊕β is the maximum order type of a total order extending the disjoint union (as a partial order) of α and β; while α⊗β is the maximum order type of a total order extending the direct product (as a partial order) of α and β.Philip W. Carruth, Arithmetic of ordinals with applications to the theory of ordered Abelian groups, Bull. Amer. Math. Soc. 48 (1942), 262–271. See Theorem 1.
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Any product of (even infinitely many) injective modules is injective; conversely, if a direct product of modules is injective, then each module is injective . Every direct sum of finitely many injective modules is injective. In general, submodules, factor modules, or infinite direct sums of injective modules need not be injective. Every submodule of every injective module is injective if and only if the ring is Artinian semisimple ; every factor module of every injective module is injective if and only if the ring is hereditary, ; every infinite direct sum of injective modules is injective if and only if the ring is Noetherian, .
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The idea is generalized in universal algebra: A congruence relation on an algebra A is a subset of the direct product A × A that is both an equivalence relation on A and a subalgebra of A × A. The kernel of a homomorphism is always a congruence. Indeed, every congruence arises as a kernel. For a given congruence ~ on A, the set A/~ of equivalence classes can be given the structure of an algebra in a natural fashion, the quotient algebra. The function that maps every element of A to its equivalence class is a homomorphism, and the kernel of this homomorphism is ~.
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The bridge was a direct product of the City Beautiful Movement in Philadelphia in the early years of the 20th century. Seeking to provide community harmony and cooperation through improved public spaces, the bridge was viewed as an achievement that could unite the communities and cultures of Roxborough and Germantown in addition to inspiring a greater civic engagement. It was also believed that more beautiful construction techniques could help to reform a corrupt political system within the city. The Philadelphia community members rallied around the construction of the bridge and the opening was highly anticipated by all ages alike.
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A reducible pair can be decomposed into a direct product of irreducible ones, and for many purposes, it is enough to restrict one's attention to the irreducible case. Several classes of reductive dual pairs had appeared earlier in the work of André Weil. Roger Howe proved a classification theorem, which states that in the irreducible case, those pairs exhaust all possibilities. An irreducible reductive dual pair (G, G′) in Sp(W) is said to be of type II if there is a lagrangian subspace X in W that is invariant under both G and G′, and of type I otherwise.
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The automorphism group of the Horton graph is of order 96 and is isomorphic to Z/2Z×Z/2Z×S4, the direct product of the Klein four-group and the symmetric group S4. The characteristic polynomial of the Horton graph is : (x-3) (x-1)^{14} x^4 (x+1)^{14} (x+3) (x^2-5)^3 (x^2-3)^{11}(x^2-x-3) (x^2+x-3) (x^{10}-23 x^8+188 x^6-644 x^4+803 x^2-101)^2 (x^{10}-20 x^8+143 x^6-437 x^4+500 x^2-59).
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Without the worry and potential of falling victim to Belial's demonic sway, the Qumran people would never feel impelled to craft a curse. This very fact illuminates the power Belial was believed to hold over mortals, and the fact that sin proved to be a temptation that must stem from an impure origin. In Jubilees 1:20, Belial's appearance continues to support the notion that sin is a direct product of his influence. Moreover, Belial's presence acts as a placeholder for all negative influences or those that would potentially interfere with God's will and a pious existence.
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Dedekind and Baer have shown (in the finite and respectively infinite order case) that every Hamiltonian group is a direct product of the form , where B is an elementary abelian 2-group, and D is a periodic abelian group with all elements of odd order. Dedekind groups are named after Richard Dedekind, who investigated them in , proving a form of the above structure theorem (for finite groups). He named the non-abelian ones after William Rowan Hamilton, the discoverer of quaternions. In 1898 George Miller delineated the structure of a Hamiltonian group in terms of its order and that of its subgroups.
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Not all neutrons are emitted as a direct product of fission; some are instead due to the radioactive decay of some of the fission fragments. The neutrons that occur directly from fission are called "prompt neutrons," and the ones that are a result of radioactive decay of fission fragments are called "delayed neutrons". The fraction of neutrons that are delayed is called β, and this fraction is typically less than 1% of all the neutrons in the chain reaction. The delayed neutrons allow a nuclear reactor to respond several orders of magnitude more slowly than just prompt neutrons would alone.
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It can be shown that there is, up to sign, a unique choice of transverse vector field ξ for which the two conditions that and are both satisfied. These two special transverse vector fields are called affine normal vector fields, or sometimes called Blaschke normal fields. From its dependence on volume forms for its definition we see that the affine normal vector field is invariant under volume preserving affine transformations. These transformations are given by where SL(n+1,R) denotes the special linear group of matrices with real entries and determinant 1, and ⋉ denotes the semi-direct product.
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Believing that the catastrophe was a direct product of their neglect of Jerusalem, the surviving members of the perushim community in Safed decided that the only hope for their future in the Land of Israel would be to reestablish themselves in Jerusalem. However, entrance to the Jerusalem could only be gained once the decree against Ashkenazim had been annulled. The perushim could then reclaim ownership of the Hurva Synagogue and its surrounding courtyard and homes, sites that were historically Ashkenazi property. The refugees succeeded in renewing the Ashkenazi presence in Jerusalem, after nearly a hundred years of banishment by the local Arabs.
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Internalized racism as a phenomenon is a direct product of a racial classification system, and is found across different racial groups and regions around the world where race exists as a social construct. In these places, internalized racism can have adverse effects on those who experience it. For example, high internalized racism scores have been linked to poor health outcomes among Caribbean black women, higher propensity for violence among African American young males, and increased domestic violence among Native American populations in the US. Responses to internalized racism have been varied. Many of the approaches focus on dispelling false narratives learned from racial oppression.
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The cyclotomic fields are examples. A cyclotomic extension, under either definition, is always abelian. If a field K contains a primitive n-th root of unity and the n-th root of an element of K is adjoined, the resulting Kummer extension is an abelian extension (if K has characteristic p we should say that p doesn't divide n, since otherwise this can fail even to be a separable extension). In general, however, the Galois groups of n-th roots of elements operate both on the n-th roots and on the roots of unity, giving a non-abelian Galois group as semi-direct product.
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One Flew Over the Cuckoo's Nest was written in 1959 and published in 1962 in the midst of the Civil Rights Movement and deep changes to the way psychology and psychiatry were being approached in America. The 1960s began the controversial movement towards deinstitutionalization, an act that would have affected the characters in Kesey's novel. The novel is a direct product of Kesey's time working the graveyard shift as an orderly at a mental health facility in Menlo Park, California. Not only did he speak to the patients and witness the workings of the institution; he also voluntarily took psychoactive drugs, including mescaline and LSD, as part of Project MKUltra.
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In algebra, the Mori–Nagata theorem introduced by and , states the following: let A be a noetherian reduced commutative ring with the total ring of fractions K. Then the integral closure of A in K is a direct product of r Krull domains, where r is the number of minimal prime ideals of A. The theorem is a partial generalization of the Krull–Akizuki theorem, which concerns a one-dimensional noetherian domain. A consequence of the theorem is that if R is a Nagata ring, then every R-subalgebra of finite type is again a Nagata ring . The Mori–Nagata theorem follows from Matijevic's theorem.
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Conversely, humans that remained in warmer climates are more physiologically comfortable simply due to temperature, and so have less incentive to work to increase their comfort levels. Therefore, according to Parker GDP is a direct product of the natural compensation of humans to their climate. Political geographers have used climatic determinism ideology to attempt to predict and rationalize the history of civilization, as well as to explain existing or perceived social and cultural divides between peoples. Some argue that one of the first attempts geographers made to define the development of human geography across the globe was to relate a country's climate to human development.
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The album came at the tail-end of the group's commercial success with their Top 40 debut, Ultimate Spinach, a direct product of the marketing campaign known as the Bosstown Sound. However, the Sound's advertising had begun to have an adverse effect on the Boston bands that spearheaded the movement, leading to Ultimate Spinach's popularity to go on the decline by the time the group initiated recording sessions. Lead vocalist, multi-instrumentalist, and songwriter Ian Bruce-Douglas, again, fronted Ultimate Spinach throughout the recording process. For the album, female vocalist Barbara Jean Hudson was reserved to backing vocals while guest singer Carol Lee Brit is featured on "Where You're At".
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In physics, the S-matrix or scattering matrix relates the initial state and the final state of a physical system undergoing a scattering process. It is used in quantum mechanics, scattering theory and quantum field theory (QFT). More formally, in the context of QFT, the S-matrix is defined as the unitary matrix connecting sets of asymptotically free particle states (the in-states and the out-states) in the Hilbert space of physical states. A multi-particle state is said to be free (non-interacting) if it transforms under Lorentz transformations as a tensor product, or direct product in physics parlance, of one-particle states as prescribed by equation below.
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Consider the group (the group of order 12 that is the direct product of the symmetric group of order 6 and a cyclic group of order 2). The center of is its second factor . Note that the first factor, , contains subgroups isomorphic to , for instance ; let be the morphism mapping onto the indicated subgroup. Then the composition of the projection of onto its second factor , followed by , followed by the inclusion of into as its first factor, provides an endomorphism of under which the image of the center, , is not contained in the center, so here the center is not a fully characteristic subgroup of .
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The MIOX enzyme has been the object of intense metabolic engineering efforts to produce glucaric acid through biosynthetic pathways. In 2004, the U.S. Department of Energy released a list of the top value-added chemicals from biomass which included glucaric acid—the direct product of the oxidation of glucuronic acid. The first biosynthetic production of glucaric acid was achieved in 2009 with use of the uronate dehydrogenase (UDH) enzyme. Since then, the MIOX enzyme has been engineered for improved glucaric acid production through numerous strategies including appendage of an N-terminal SUMO-tag, directed evolution and also the use of modular, synthetic scaffolds to increase its effective local concentration.
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The properties of the dihedral groups with depend on whether is even or odd. For example, the center of consists only of the identity if n is odd, but if n is even the center has two elements, namely the identity and the element rn/2 (with Dn as a subgroup of O(2), this is inversion; since it is scalar multiplication by −1, it is clear that it commutes with any linear transformation). In the case of 2D isometries, this corresponds to adding inversion, giving rotations and mirrors in between the existing ones. For n twice an odd number, the abstract group is isomorphic with the direct product of and .
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The different ways of constructing new representations from given ones can be used for compact groups as well, except for the dual representation with which we will deal later. The direct sum and the tensor product with a finite number of summands/factors are defined in exactly the same way as for finite groups. This is also the case for the symmetric and alternating square. However, we need a Haar measure on the direct product of compact groups in order to extend the theorem saying that the irreducible representations of the product of two groups are (up to isomorphism) exactly the tensor product of the irreducible representations of the factor groups.
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A group is called virtually cyclic if it contains a cyclic subgroup of finite index (the number of cosets that the subgroup has). In other words, any element in a virtually cyclic group can be arrived at by applying a member of the cyclic subgroup to a member in a certain finite set. Every cyclic group is virtually cyclic, as is every finite group. An infinite group is virtually cyclic if and only if it is finitely generated and has exactly two ends; an example of such a group is the direct product of Z/nZ and Z, in which the factor Z has finite index n.
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In mathematics, the Alperin–Brauer–Gorenstein theorem characterizes the finite simple groups with quasidihedral or wreathedA 2-group is wreathed if it is a nonabelian semidirect product of a maximal subgroup that is a direct product of two cyclic groups of the same order, that is, if it is the wreath product of a cyclic 2-group with the symmetric group on 2 points. Sylow 2-subgroups. These are isomorphic either to three-dimensional projective special linear groups or projective special unitary groups over a finite field of odd order, depending on a certain congruence, or to the Mathieu group M_{11}. proved this in the course of 261 pages.
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The non-triviality of the (additional) conjugacy conclusion can be illustrated with the Klein four-group V as the non-example. Any of the three proper subgroups of V (all of which have order 2) is normal in V; fixing one of these subgroups, any of the other two remaining (proper) subgroups complements it in V, but none of these three subgroups of V is a conjugate of any other one, because V is Abelian. The quaternion group has normal subgroups of order 4 and 2 but is not a [semi]direct product. Schur's papers at the beginning of the 20th century introduced the notion of central extension to address examples such as C_4 and the quaternions.
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Every elementary abelian p-group is a vector space over the prime field with p elements, and conversely every such vector space is an elementary abelian group. By the classification of finitely generated abelian groups, or by the fact that every vector space has a basis, every finite elementary abelian group must be of the form (Z/pZ)n for n a non-negative integer (sometimes called the group's rank). Here, Z/pZ denotes the cyclic group of order p (or equivalently the integers mod p), and the superscript notation means the n-fold direct product of groups. In general, a (possibly infinite) elementary abelian p-group is a direct sum of cyclic groups of order p.
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If d is a divisor of n, then the number of elements in Z/nZ which have order d is φ(d), and the number of elements whose order divides d is exactly d. If G is a finite group in which, for each , G contains at most n elements of order dividing n, then G must be cyclic. The order of an element m in Z/nZ is n/gcd(n,m). If n and m are coprime, then the direct product of two cyclic groups Z/nZ and Z/mZ is isomorphic to the cyclic group Z/nmZ, and the converse also holds: this is one form of the Chinese remainder theorem.
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Shelah, S., Whitehead groups may not be free even assuming CH I, Israel Journal of Mathematics (28) 1972Shelah, S., Whitehead groups may not be free even assuming CH II, Israel Journal of Mathematics (350 1980 Consider the ring A = R[x,y,z] of polynomials in three variables over the real numbers and its field of fractions M = R(x,y,z). The projective dimension of M as A-module is either 2 or 3, but it is independent of ZFC whether it is equal to 2; it is equal to 2 if and only if CH holds. A direct product of countably many fields has global dimension 2 if and only if the continuum hypothesis holds.
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A categorical product can be characterized by a universal construction. For concreteness, one may consider the Cartesian product in Set, the direct product in Grp, or the product topology in Top, where products exist. Let X and Y be objects of a category D with finite products. The product of X and Y is an object X × Y together with two morphisms :\pi_1 : X \times Y \to X :\pi_2 : X \times Y \to Y such that for any other object Z of D and morphisms f: Z \to X and g: Z \to Y there exists a unique morphism h: Z \to X \times Y such that f = \pi_1 \circ h and g = \pi_2 \circ h.
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"The bootlegging and gang killings...are not the by-product but the direct product of the Volstead Act, and the supporters of this crime breeding legislation must claim the new cult of American criminals entirely as their own."Davis, Marni, "Jews And Booze: Becoming American In The Age Of Prohibition," New York University Press, 2012, p. 191, As a leading Democrat he chaired the powerful House Rules Committee after 1937. He was an ineffective chairman, with a small weak staff, who proved unable to lead his committee, was frequently at odds with the House leadership, and was inclined to write the President little letters "informing" on House Speakers William B. Bankhead and Sam Rayburn.
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This is false for some noncommutative rings, and a counterexample can be constructed using the Eilenberg swindle as follows. Let X be an abelian group such that X ≅ X ⊕ X (for example the direct sum of an infinite number of copies of any nonzero abelian group), and let R be the ring of endomorphisms of X. Then the left R-module R is isomorphic to the left R-module R ⊕ R. Example: If A and B are any groups then the Eilenberg swindle can be used to construct a ring R such that the group rings R[A] and R[B] are isomorphic rings: take R to be the group ring of the restricted direct product of infinitely many copies of A ⨯ B.
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To describe the above properties in the case where G is the direct sum of an infinite (perhaps uncountable) set of subgroups, more care is needed. If g is an element of the cartesian product ∏{Hi} of a set of groups, let gi be the ith element of g in the product. The external direct sum of a set of groups {Hi} (written as ∑E{Hi}) is the subset of ∏{Hi}, where, for each element g of ∑E{Hi}, gi is the identity e_{H_i} for all but a finite number of gi (equivalently, only a finite number of gi are not the identity). The group operation in the external direct sum is pointwise multiplication, as in the usual direct product.
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In 1885 he discarded the work by Georg Ludwig Carius, according to which thionyl chloride formed by the action of phosphorus pentachloride on inorganic sulfites was regarded as a direct product of the reaction and which formed the only experimental evidence in favour of the symmetrical constitution of the sulfites. Divers demonstrated that thionyl chloride was instead produced by a secondary reaction between sulphur dioxide and phosphorus pentachloride. It was in the course of this work, on 24 November 1884, that Divers lost vision in his right eye as he was badly cut by pieces of glass resulting from the sudden bursting of the bottle with phosphorus oxychloride. Chemistry of sulfonated nitrogen compounds was the subject of most attention for Divers while staying in Japan.
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There are several related but distinct notions of central product. Similarly to the direct product, there are both internal and external characterizations, and additionally there are variations on how strictly the intersection of the factors is controlled. A group G is an internal central product of two subgroups H, K if (1) G is generated by H and K and (2) every element of H commutes with every element of K . Sometimes the stricter requirement that H ∩ K is exactly equal to the center is imposed, as in . The subgroups H and K are then called central factors of G. The external central product is constructed from two groups H and K, two subgroups H1 ≤ Z(H), K1 ≤ Z(K), and a group isomorphism θ:H1 → K1.
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Intervals can be associated with points of the plane, and hence regions of intervals can be associated with regions of the plane. Generally, an interval in mathematics corresponds to an ordered pair (x,y) taken from the direct product R × R of real numbers with itself, where it is often assumed that y > x. For purposes of mathematical structure, this restriction is discarded,Kaj Madsen (1979) Review of "Interval analysis in the extended interval space" by Edgar Kaucher from Mathematical Reviews and "reversed intervals" where y − x < 0 are allowed. Then, the collection of all intervals [x,y] can be identified with the topological ring formed by the direct sum of R with itself, where addition and multiplication are defined component-wise.
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Much of the structure of a finite group is carried in the structure of its so-called local subgroups, the normalizers of non-identity p-subgroups. The large elementary abelian subgroups of a finite group exert control over the group that was used in the proof of the Feit–Thompson theorem. Certain central extensions of elementary abelian groups called extraspecial groups help describe the structure of groups as acting on symplectic vector spaces. Richard Brauer classified all groups whose Sylow 2-subgroups are the direct product of two cyclic groups of order 4, and John Walter, Daniel Gorenstein, Helmut Bender, Michio Suzuki, George Glauberman, and others classified those simple groups whose Sylow 2-subgroups were abelian, dihedral, semidihedral, or quaternion.
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In graph theory, a branch of mathematics, a crown graph on 2n vertices is an undirected graph with two sets of vertices {u1, u2, ..., un} and {v1, v2, ..., vn} and with an edge from ui to vj whenever i ≠ j. The crown graph can be viewed as a complete bipartite graph from which the edges of a perfect matching have been removed, as the bipartite double cover of a complete graph, as the tensor product Kn × K2, as the complement of the Cartesian direct product of Kn and K2, or as a bipartite Kneser graph Hn,1 representing the 1-item and (n − 1)-item subsets of an n-item set, with an edge between two subsets whenever one is contained in the other.
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Consumer products include direct product sale of chemicals such as soaps, detergents, and cosmetics. Typical growth rates are 0.8 to 1.0 times GDP. Consumers rarely if ever come into contact with basic chemicals but polymers and speciality chemicals are the materials that they will encounter everywhere in their everyday lives, such as in plastics, cleaning materials, cosmetics, paints & coatings, electronic gadgets, automobiles and the materials used to construct their homes. These speciality products are marketed by chemical companies to the downstream manufacturing industries as pesticides, speciality polymers, electronic chemicals, surfactants, construction chemicals, Industrial Cleaners, flavours and fragrances, speciality coatings, printing inks, water-soluble polymers, food additives, paper chemicals, oil field chemicals, plastic adhesives, adhesives and sealants, cosmetic chemicals, water management chemicals, catalysts, textile chemicals.
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SO(3) is a subgroup of E+(3), which consists of direct isometries, i.e., isometries preserving orientation; it contains those that leave the origin fixed. O(3) is the direct product of SO(3) and the group generated by inversion (denoted by its matrix −I): :O(3) = SO(3) × { I , −I } Thus there is a 1-to-1 correspondence between all direct isometries and all indirect isometries, through inversion. Also there is a 1-to-1 correspondence between all groups of direct isometries H in O(3) and all groups K of isometries in O(3) that contain inversion: :K = H × { I , −I } :H = K ∩ SO(3) For instance, if H is C2, then K is C2h, or if H is C3, then K is S6.
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This result can also be stated as "any subgroup of index 2 is normal", and in this form it applies also to infinite groups. Furthermore, if p is the smallest prime number dividing the order of a finite group, G, then if has order p, H must be a normal subgroup of G. Given G and a normal subgroup N, then G is a group extension of by N. One could ask whether this extension is trivial or split; in other words, one could ask whether G is a direct product or semidirect product of N and . This is a special case of the extension problem. An example where the extension is not split is as follows: Let G = Z4 = {0, 1, 2, 3}, and N = {0, 2}, which is isomorphic to Z2.
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This is based on a notion referred to as phenotypic plasticity, essentially phenotypes under the EE approach can respond to varying environmental conditions. What is important to gather is,that by applying an EE analysis to anthropological and archaeological phenomena, it allows researchers to employ phenotypic plasticity to the explanations of human behavior. By doing so, this explanatory framework gives humans the cognitive abilities to “adapt to change quicker that they could through natural selection acting on genetic variation” (Boone and Smith, 1998) Evolutionary ecology assumes that “behavioral variation itself is not the direct product of natural selection, rather, selection enters the explanation only indirectly, as the process that designed the behaving organism (or in fact its ancestors) to respond facultatively and adaptively to particular environmental conditions” (Boone and Smith, 1998).
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This is called a unification of space and time because the Lorentz group is simple, while the Galilean group is a semi-direct product of rotations and Galilean boosts. This means that the Lorentz group mixes up space and time such that they cannot be disentangled, while the Galilean group treats time as a parameter with different units of measurement than space. An analogous thing can be made to happen with the ordinary rotation group in three dimensions. If you imagine a nearly flat world, one in which pancake-like creatures wander around on a pancake flat world, their conventional unit of height might be the micrometre (μm), since that is how high typical structures are in their world, while their unit of distance could be the metre, because that is their body's horizontal extent.
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Yet, the selfish determinants of collective action are, according to Popkin, a direct product of the inherent instability of peasant life. The goal of a laborer, for example, will be to move to a tenant position, then smallholder, then landlord; where there is less variance and more income. Voluntarism is thus non-existent in such communities. Popkin singles out four variables that impact individual participation: # Contribution to the expenditure of resources: collective action has a cost in terms of contribution, and especially if it fails (an important consideration with regards to rebellion) # Rewards : the direct (more income) and indirect (less oppressive central state) rewards for collective action # Marginal impact of the peasant's contribution to the success of collective action # Leadership "viability and trust" : to what extent the resources pooled will be effectively used.
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A direct product of neuropsychopharmacological research is the knowledge base required to develop drugs which act on very specific receptors within a neurotransmitter system. These "hyperselective-action" drugs would allow the direct targeting of specific sites of relevant neural activity, thereby maximizing the efficacy (or technically the potency) of the drug within the clinical target and minimizing adverse effects. However, there are some cases when some degree of pharmacological promiscuity is tolerable and even desirable, producing more desirable results than a more selective agent would. An example of this is Vortioxetine, a drug which is not particularly selective as a serotonin reuptake inhibitor, having a significant degree of serotonin modulatory activity, but which has demonstrated reduced discontinuation symptoms (and reduced likelihood of relapse) and greatly reduced incidence of sexual dysfunction, without loss in antidepressant efficacy.
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The heavy cruiser was a direct product of the Washington Naval Treaty of 1922, which limited cruisers to a standard displacement of no more than 10,000 tons, with main guns not exceeding 8 inches (203 mm) caliber. There were also important technical differences between the heavy cruiser and the armored cruiser, some of which reflected the generational gap between them. Heavy cruisers were typically powered by oil-fired superheated steam boilers and steam turbine engines, and were capable of far faster speeds than armored cruisers (propelled by coal-fired reciprocating steam engines of their era) ever had been. Countries withdrawing from the Washington Treaty and the London Naval Treaty of 1930 and Second London Naval Treaty 1936 eventually rendered all limitations on heavy cruisers moot, although the only supersized or large cruisers actually built were the two members of the Alaska class.
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In the context of group theory, the socle of a group G, denoted soc(G), is the subgroup generated by the minimal normal subgroups of G. It can happen that a group has no minimal non-trivial normal subgroup (that is, every non-trivial normal subgroup properly contains another such subgroup) and in that case the socle is defined to be the subgroup generated by the identity. The socle is a direct product of minimal normal subgroups. As an example, consider the cyclic group Z12 with generator u, which has two minimal normal subgroups, one generated by u4 (which gives a normal subgroup with 3 elements) and the other by u6 (which gives a normal subgroup with 2 elements). Thus the socle of Z12 is the group generated by u4 and u6, which is just the group generated by u2.
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All the national American home shopping channels are broadcast live, 24 hours a day, so mistakes and errors cannot be edited out. This has led to some memorable on-air home shopping bloopers, such as a QVC demonstrator falling from a ladder and getting injured, or a Shop at Home host insisting that a moth in a photograph is actually a horse. Many of the guests who accompany the home shopping host on-air are not particularly TV-savvy or experienced, so the host must help guests better define the attributes and value of the items being showcased. Because of the large influx of guests and products, hosts often lack the time to study the product beforehand, which increases the need for an effective host to possess sufficient verbal skills to mask this lack of direct product knowledge.
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In a speech on the one-year anniversary of the CARD Act, then-CFPB Special Adviser Elizabeth Warren said that "much of the [credit card] industry has gone further than the law requires in curbing re-pricing and overlimit fees." However, she said there was still much work to be done, that the Consumer Financial Protection Bureau's "next challenges will be about further clarifying price and risks and making it easier for consumers to make direct product comparisons." In 2012, many stay-at-home spouses complained that because they have no individual income, the act prevented them from acquiring credit cards without their husbands' permission. On April 29, 2013, the CFPB amended regulations to allow credit card issuers to consider third-party income for applicants who are 21 or older, if the applicant has a reasonable expectation of access to it.
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Piaget believed in two basic principles relating to character education: that children develop moral ideas in stages and that children create their conceptions of the world. According to Piaget, "the child is someone who constructs his own moral world view, who forms ideas about right and wrong, and fair and unfair, that are not the direct product of adult teaching and that are often maintained in the face of adult wishes to the contrary" (Gallagher, 1978, p. 26). Piaget believed that children made moral judgments based on their own observations of the world. Piaget's theory of morality was radical when his book The Moral Judgment of the Child was published in 1932 for two reasons: his use of philosophical criteria to define morality (as universalizable, generalizable, and obligatory) and his rejection of equating cultural norms with moral norms.
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In particular, if S and T intersect only in the identity, then every element of ST has a unique expression as a product st with s in S and t in T. If S and T also commute, then ST is a group, and is called a Zappa–Szép product. Even further, if S or T is normal in ST, then ST coincides with the semidirect product of S and T. Finally, if both S and T are normal in ST, then ST coincides with the direct product of S and T. If S and T are subgroups whose intersection is the trivial subgroup (identity element) and additionally ST = G, then S is called a complement of T and vice versa. By a (locally unambiguous) abuse of terminology, two subgroups that intersect only on the (otherwise obligatory) identity are sometimes called disjoint.
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Prior to the Haag–Łopuszański–Sohnius theorem, the Coleman–Mandula theorem was the strongest of a series of no-go theorems, stating that the symmetry group of a consistent 4-dimensional quantum field theory is the direct product of the internal symmetry group and the Poincaré group. In 1971 Yuri Golfand and E. P. Likhtman published the first paper on four-dimensional supersymmetry which presented (in modern notation) N=1 superalgebra and N=1 super-QED with charged matter and a mass term for the photon field. They proved that the conserved supercharges can exist in four dimensions by allowing both commuting and anticommuting symmetry generators, thus providing a nontrivial extension of the Poincaré algebra, namely the supersymmetry algebra. In 1975, Rudolf Haag, Jan Łopuszański, and Martin Sohnius further generalized superalgebras by analyzing extended supersymmetries (e.g.
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County Farm Park has a wide variety of different gardens, including an octagonal perennial flower garden, a "native grasses display" on the Outer Loop trail, and a "Kids & Trees" nursery near the children's playground. It also includes the Project Grow Community Gardens, which consist of 84 total plots in two sections available for rental by individuals. Project Grow's gardens at County Farm Park constitute the oldest and largest part of its network, which in total consists of 15 individual sites and over 350 separate plots throughout the Ann Arbor area. The Project Grow gardens at County Farm Park have been in existence since 1972, when they emerged as a direct product of the environmental and community gardening movements of the 1970s, although they were also substantially influenced by the world war-era tradition of "victory gardens".
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A quasivariety defined logically as the class of models of a universal Horn theory can equivalently be defined algebraically as a class of structures closed under isomorphisms, subalgebras, and reduced products. Since the notion of reduced product is more intricate than that of direct product, it is sometimes useful to blend the logical and algebraic characterizations in terms of pseudoelementary classes. One such blended definition characterizes a quasivariety as a pseudoelementary class closed under isomorphisms, subalgebras, and direct products (the pseudoelementary property allows "reduced" to be simplified to "direct"). A corollary of this characterization is that one can (nonconstructively) prove the existence of a universal Horn axiomatization of a class by first axiomatizing some expansion of the structure with auxiliary sorts and relations and then showing that the pseudoelementary class obtained by dropping the auxiliary constructs is closed under subalgebras and direct products.
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Infringement is remitted entirely to national law and to national courts. In one of its very few substantive interventions into national law, the EPC requires that national courts must consider the "direct product of a patented process" to be an infringement. The "extent of the protection" conferred by a European patent is determined primarily by reference to the claims of the European patent (rather than by the disclosure of the specification and drawings, as in some older patent systems), though the description and drawings are to be used as interpretive aids in determining the meaning of the claims. A "Protocol on the Interpretation of Article 69 EPC" provides further guidance, that claims are to be construed using a "fair" middle position, neither "strict, literal" nor as mere guidelines to considering the description and drawings, though of course even the protocol is subject to national interpretation.E.g.
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While an accidental collision with a sperm whale at night accounted for sinking of the Union in 1807,Report of the Commissioner By United States Commission of Fish and Fisheries, p11 the Essex incident some 30 years beforehand was the only other documented case of a whale deliberately attacking, holing, and sinking a ship. However, these two incidents are probably not as much of a freak occurrence as they appear to be. Observations of aggression in males of the cetacean species suggest that head-butting during male–male aggression is a basal behavior, and that the enlarged melon or spermaceti organ is a direct product of sexual dimorphism, having evolved as a battering ram to injure an opponent in such attacks. The ability of the sperm whale to aggressively attack and destroy ships some 3–5 times its body mass in this manner is therefore hardly surprising.
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The concept of avant-garde refers primarily to artists, writers, composers and thinkers whose work is opposed to mainstream cultural values and often has a trenchant social or political edge. Many writers, critics and theorists made assertions about vanguard culture during the formative years of modernism, although the initial definitive statement on the avant-garde was the essay Avant-Garde and Kitsch by New York art critic Clement Greenberg, published in Partisan Review in 1939. Greenberg argued that vanguard culture has historically been opposed to "high" or "mainstream" culture, and that it has also rejected the artificially synthesized mass culture that has been produced by industrialization. Each of these media is a direct product of Capitalism—they are all now substantial industries—and as such they are driven by the same profit-fixated motives of other sectors of manufacturing, not the ideals of true art.
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Haumea is the largest member of its collisional family, a group of astronomical objects with similar physical and orbital characteristics thought to have formed when a larger progenitor was shattered by an impact. This family is the first to be identified among TNOs and includes—beside Haumea and its moons— (≈364 km), (≈174 km), (≈200 km), (≈230 km), and (≈252 km). Brown and colleagues proposed that the family were a direct product of the impact that removed Haumea's ice mantle, but a second proposal suggests a more complicated origin: that the material ejected in the initial collision instead coalesced into a large moon of Haumea, which was later shattered in a second collision, dispersing its shards outwards. This second scenario appears to produce a dispersion of velocities for the fragments that is more closely matched to the measured velocity dispersion of the family members.
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The Burnside ring of a finite group G is constructed from the category of finite G-sets as a Grothendieck group. More precisely, let M(G) be the commutative monoid of isomorphism classes of finite G-sets, with addition the disjoint union of G-sets and identity element the empty set (which is a G-set in a unique way). Then A(G), the Grothendieck group of M(G), is an abelian group. It is in fact a free abelian group with basis elements represented by the G-sets G/H, where H varies over the subgroups of G. (Note that H is not assumed here to be a normal subgroup of G, for while G/H is not a group in this case, it is still a G-set.) The ring structure on A(G) is induced by the direct product of G-sets; the multiplicative identity is the (isomorphism class of any) one-point set, which becomes a G-set in a unique way.
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Let G, H be two groups and let (\pi,V) and (\rho,W) be representations of G and H, respectively. Then we can let the direct product group G\times H act on the tensor product space V\otimes W by the formula :(g, h) \cdot (v \otimes w) = \pi(g) v \otimes \rho(h) w. Even if G=H, we can still perform this construction, so that the tensor product of two representations of G could, alternatively, be viewed as a representation of G\times G rather than a representation of G. It is therefore important to clarify whether the tensor product of two representations of G is being viewed as a representation of G or as a representation of G\times G. In contrast to the Clebsch–Gordan problem discussed above, the tensor product of two irreducible representations of G is irreducible when viewed as a representation of the product group G\times G.
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There are various connections between the lower central series (LCS) and upper central series (UCS) , particularly for nilpotent groups. Most simply, a group is abelian if and only if the LCS terminates at the first step (the commutator subgroup is trivial) if and only if the UCS stabilizes at the first step (the center is the entire group). More generally, for a nilpotent group, the length of the LCS and the length of the UCS agree (and is called the nilpotency class of the group). However, the LCS and UCS of a nilpotent group may not necessarily have the same terms. For example, while the UCS and LCS agree for the cyclic group C2 and quaternion group Q8 (which are C2 ⊵ {e} and Q8 ⊵ {1, -1} ⊵ {1} respectively), the UCS and LCS of their direct product C2 × Q8 do not: its lower central series is C2 × Q8 ⊵ {e} × {-1, 1} ⊵ {e} × {1}, while the upper central series is C2 × Q8 ⊵ C2 × {-1, 1} ⊵ {e} × {1}.
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If the finite group G is not perfect, then its Schur covering groups (all such C of maximal order) are only isoclinic. It is also called more briefly a universal central extension, but note that there is no largest central extension, as the direct product of G and an abelian group form a central extension of G of arbitrary size. Stem extensions have the nice property that any lift of a generating set of G is a generating set of C. If the group G is presented in terms of a free group F on a set of generators, and a normal subgroup R generated by a set of relations on the generators, so that G \cong F/R, then the covering group itself can be presented in terms of F but with a smaller normal subgroup S, that is, C\cong F/S. Since the relations of G specify elements of K when considered as part of C, one must have S \le [F,R].
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As noted by Chris Allen, the EDL is nevertheless "not a direct product of the traditional far-right milieu" in Britain, differing from other groups in its willingness to reach out to communities that the far-right historically discriminates against, namely Jews, people of colour, and LGBT people. The criminologists James Treadwell and Jon Garland suggested that the EDL reflected "both a continuation of and a departure from traditional far-right activity", while Paul Jackson—a historian of the far-right—referred to it as part of the "new far right", a movement that presents itself as being more moderate than older far-right groups. Ideologically, the EDL was not wholly clear; it had no specific policies, goals, or manifesto, and no intellectual vanguard to lead it.The political scientist Julian Richards suggested that one of the reasons that the EDL should be categorised as far-right was because of how many of its members acted, in contrast to what the group officially stated in its public pronouncements.
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The EAR have very broad application. With only the exceptions noted below, the EAR apply to the following categories of things: # All items in the United States, including in a U.S. Foreign Trade Zone or moving intransit through the United States from one foreign country to another; # All U.S. origin items wherever located; # Foreign-made commodities that incorporate controlled U.S.-origin commodities, foreign-made commodities that are ‘bundled’ with controlled U.S.-origin software, foreign-made software that is commingled with controlled U.S.-origin software, and foreign-made technology that is commingled with controlled U.S.-origin technology in certain quantities (see the "de minimis" rules found at 15 CFR part 734); # Certain foreign-made direct products of U.S. origin technology or software, as described in 15 CFR §736.2(b)(3); and # Certain commodities produced by any plant or major component of a plant located outside the United States that is a direct product of U.S.-origin technology or software, as described in §736.2(b)(3) of the EAR.
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As an adaptation strategy to improve income of fishermen and to help them cope up with adverse climatic events, the ICAR-Central Marine Fisheries Research Institute(CMFRI) developed a multivendor E-commerce website and associated android app (marinefishsales) through the National Innovations on Climate Resilient Agriculture (NICRA) project. The website and mobile app is aimed at helping fish farmers and fishermen to sell their farmed fish and marine catch directly to the customers online and to fetch better income without depending middlemen. Various fishermen SHGs can register as vendors (fishers and farmers) based on their fish products and update their stock availability under pre-approved categories and products, which shall be displayed in the website and associated mobile app. Customers visiting the website or app could place the order and subsequently the registered fisher/farmer shall be notified through email and SMS, upon which the quality products within the pre-assigned time frame shall be delivered, enabling direct product sale between customers and SHGs.
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Karagwe, Nkore, and Buhaya formed small neighboring states to the major kingdoms of Bunyoro and Buganda in the Great Lakes region. Karagwe and Nkore were individual polities, while Buhaya refers to an area along the western side of Lake Victoria in which seven small states were recognized: Kiamutwara, Kiziba, Ihangiro, Kihanja, Bugabo, Maruku, and Missenye. Although this entry only deals with the period up to the end of the eighteenth century, it is essential to recognize that the earlier histories of these polities and the detail with which they have been recorded are a direct product of nineteenth- and twentieth-century history and the circumstances which befell them. Nkore (Ankole in colonial times) found itself within the British Protectorate of Uganda and became a cornerstone of Protectorate policy, being one of the four main kingdoms and enjoying a considerably enlarged territorial status under the Protectorate than it had done in precolonial times.
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In mathematics, the Klein four-group is a group with four elements, in which each element is self-inverse (composing it with itself produces the identity) and in which composing any two of the three non-identity elements produces the third one. It can be described as the symmetry group of a non-square rectangle (with the three non-identity elements being horizontal and vertical reflection and 180-degree rotation), as the group of bitwise exclusive or operations on two-bit binary values, or more abstractly as , the direct product of two copies of the cyclic group of order 2. It was named Vierergruppe (meaning four-group) by Felix Klein in 1884.Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade (Lectures on the icosahedron and the solution of equations of the fifth degree) It is also called the Klein group, and is often symbolized by the letter V or as K4.
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Over time, Land Rover grew into its own brand (and for a while also a company), encompassing a consistently growing range of four-wheel drive, off-road capable models. Starting with the much more upmarket 1970 Range Rover, and subsequent introductions of the mid-range Discovery and entry-level Freelander line (in 1989 and 1997), as well as the 1990 Land Rover Defender refresh, the marque today includes two models of Discovery, four distinct models of Range Rover, and after a three-year hiatus, a second generation of Defenders have gone into production for the 2020 model year—in short or long wheelbase, as before. For half a century (from the original 1948 model, through 1997, when the Freelander was introduced), Land Rovers and Range Rovers exclusively relied on their trademark boxed-section vehicle frames. Land Rover used boxed frames in a direct product bloodline until the termination of the original Defender in 2016; and their last body-on-frame model was replaced by a monocoque with the third generation Discovery in 2017.
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More recently Fraser Hunter has reassessed the context of the Scottish examples and some of the unstamped Welsh examples and argues that they could in fact be Iron Age in date or at least reflect native rather than Roman copper working (Hunter 1999, 338-40.). Although ingots of any sort are not common in the British Iron Age, planoconvex or bun-shaped ingots are not unknown - e.g. a tin ingot which was discovered within the Iron Age hillfort at Chun Castle, Cornwall (Tylecote 1987, 204.). ;Composition and Structure of Roman Ingots The Roman Bun Ingots are less pure than the earlier LBA examples and Tylecote suggests that they may be a direct product of smelting (1987, 24.) Although theoretically such an ingot could be formed in the base of the furnace this is problematic in the case of the stamped examples as this would require the furnace to be dismantled or else have a short shaft to allow access for stamping (Merkel 1986, Tylecote 1986, 22.) As a solution the furnace could have been tapped into a mould at the completion of smelting.
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For example, Z/12Z is isomorphic to the direct product Z/3Z × Z/4Z under the isomorphism (k mod 12) → (k mod 3, k mod 4); but it is not isomorphic to Z/6Z × Z/2Z, in which every element has order at most 6. If p is a prime number, then any group with p elements is isomorphic to the simple group Z/pZ. A number n is called a cyclic number if Z/nZ is the only group of order n, which is true exactly when .. The cyclic numbers include all primes, but some are composite such as 15\. However, all cyclic numbers are odd except 2. The cyclic numbers are: :1, 2, 3, 5, 7, 11, 13, 15, 17, 19, 23, 29, 31, 33, 35, 37, 41, 43, 47, 51, 53, 59, 61, 65, 67, 69, 71, 73, 77, 79, 83, 85, 87, 89, 91, 95, 97, 101, 103, 107, 109, 113, 115, 119, 123, 127, 131, 133, 137, 139, 141, 143, ... The definition immediately implies that cyclic groups have group presentation and for finite n..
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Since commutators in each simple group form a generating set, pairs of commutators form a generating set of the direct product. More generally, a quasisimple group (a perfect central extension of a simple group) which is a non-trivial extension (and therefore not a simple group itself) is perfect but not simple; this includes all the insoluble non- simple finite special linear groups SL(n,q) as extensions of the projective special linear group PSL(n,q) (SL(2,5) is an extension of PSL(2,5), which is isomorphic to A5). Similarly, the special linear group over the real and complex numbers is perfect, but the general linear group GL is never perfect (except when trivial or over F2, where it equals the special linear group), as the determinant gives a non-trivial abelianization and indeed the commutator subgroup is SL. A non-trivial perfect group, however, is necessarily not solvable; and 4 divides its order (if finite), moreover, if 8 does not divide the order, then 3 does. Every acyclic group is perfect, but the converse is not true: A5 is perfect but not acyclic (in fact, not even superperfect), see .
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The subdirect representation theorem of universal algebra states that every algebra is subdirectly representable by its subdirectly irreducible quotients. An equivalent definition of "subdirect irreducible" therefore is any algebra A that is not subdirectly representable by those of its quotients not isomorphic to A. (This is not quite the same thing as "by its proper quotients" because a proper quotient of A may be isomorphic to A, for example the quotient of the semilattice (Z, min) obtained by identifying just the two elements 3 and 4.) An immediate corollary is that any variety, as a class closed under homomorphisms, subalgebras, and direct products, is determined by its subdirectly irreducible members, since every algebra A in the variety can be constructed as a subalgebra of a suitable direct product of the subdirectly irreducible quotients of A, all of which belong to the variety because A does. For this reason one often studies not the variety itself but just its subdirect irreducibles. An algebra A is subdirectly irreducible if and only if it contains two elements that are identified by every proper quotient, equivalently, if and only if its lattice Con A of congruences has a least nonidentity element.
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Portrait of Pope Julius II by Raphael, 1511–12, recognized since 1970 as the prime version In the case of official portraits the later creation of replica versions was likely to have been anticipated from the start, as in other types of art such as printmaking, but in other types of painting, especially history painting, the normal presumption, in the past as today, was that each work was a unique creation. This was a matter of some importance to the owner, and there was evidently in some periods a general understanding that a work should not be replicated or copied without the permission of the owner of the prime version, which needed to be asked for carefully and was not always given. In many periods "replicas were the direct product of collecting, as collectors have always preferred recognizable masterpieces to what is offbeat".Christiansen (2001), 21 This was true of 17th century Rome, where artists such as Orazio Gentileschi and Bernardo Strozzi routinely made replicas, and others such as Guercino and Guido Reni sometimes did,Christiansen (2001), 20-27 as of Victorian London, where artists such as William Powell Frith often painted one or more replica versions of their successes.
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