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Sherman–Morrison formula-Alternative verification-Following is an alternate verification of the Sherman–Morrison formula using the easily verifiable identity (I+wvT)−1=I−wvT1+vTw .Let and A+uvT=A(I+wvT), then (A+uvT)−1=(I+wvT)−1A−1=(I−wvT1+vTw)A−1 .Substituting w=A−1u gives (A+uvT)−1=(I−A−1uvT1+vTA−1u)A−1=A−1−A−1uvTA... | milkshake721/2.1M-wiki-STEM |
Sherman–Morrison formula-Generalization (Woodbury matrix identity)-Given a square invertible n×n matrix A , an n×k matrix U , and a k×n matrix V , let B be an n×n matrix such that B=A+UV . Then, assuming (Ik+VA−1U) is invertible, we have B−1=A−1−A−1U(Ik+VA−1U)−1VA−1. | milkshake721/2.1M-wiki-STEM |
History of blogging-History of blogging-While the term "blog" was not coined until the late 1990s, the history of blogging starts with several digital precursors to it. Before "blogging" became popular, digital communities took many forms, including Usenet, commercial online services such as GEnie, BiX and the early Co... | milkshake721/2.1M-wiki-STEM |
History of blogging-1983–1993-Usenet was the primary serial medium included in the original definition of the Internet. It features the moderated newsgroup which allowed all posting in a newsgroup to be under the control of an individual or small group. Most such newsgroups were simply moderated discussion forums, howe... | milkshake721/2.1M-wiki-STEM |
History of blogging-1994–2001-The modern blog evolved from the online diary, where people would keep a running account of their personal lives. Most such writers called themselves diarists, journalists, or journalers. A few called themselves "escribitionists". The Open Pages webring included members of the online-journ... | milkshake721/2.1M-wiki-STEM |
History of blogging-1994–2001-Other forms of journals kept online also existed. A notable example was game programmer John Carmack's widely read journal, published via the finger protocol. Some of the very earliest bloggers, like Steve Gibson of sCary's Quakeholio (now Shacknews) and Stephen Heaslip of Blue's News (sti... | milkshake721/2.1M-wiki-STEM |
History of blogging-1994–2001-Another early example of an early online entry into the evolution of blogging was created by Dave Winer. Winer is considered a pioneer of Web syndication techniques and has been considered one of the “fathers” of blogging. As the editor of Scripting News claims that his site “bootstrapped ... | milkshake721/2.1M-wiki-STEM |
History of blogging-1994–2001-Early weblogs were simply manually updated components of common websites. However, the evolution of tools to facilitate the production and maintenance of web articles posted in reverse chronological order made the publishing process feasible to a much larger, less technical, population. Ul... | milkshake721/2.1M-wiki-STEM |
History of blogging-1994–2001-The term "weblog" was coined by Jorn Barger on 17 December 1997. The short form, "blog," was coined by Peter Merholz, who jokingly broke the word weblog into the phrase we blog in the sidebar of his blog Peterme.com in April or May 1999. Shortly thereafter, Evan Williams at Pyra Labs used ... | milkshake721/2.1M-wiki-STEM |
History of blogging-1994–2001-SlashDot, a still-popular blog for tech "nerds" launched in September 1997.
Brad Fitzpatrick, a well known blogger started LiveJournal in March 1999.
Andrew Smales created Pitas.com in July 1999 as an easier alternative to maintaining a "news page" on a website, followed by DiaryLand in Se... | milkshake721/2.1M-wiki-STEM |
History of blogging-1994–2001-In 2000, blogger Traciy Curry-Reyes started the blog Movies Based on True Stories Database. The site was the first to connect the real people with movies that were inspired by their cases. The information regarding her first blog in 1998 and based on the same subject has been lost. Traciy ... | milkshake721/2.1M-wiki-STEM |
History of blogging-1994–2001-Evan Williams and Meg Hourihan (Pyra Labs) launched blogger.com in August 1999 (purchased by Google in February 2003)Blogging combined the personal web page with tools to make linking to other pages easier — specifically permalinks, blogrolls and TrackBacks. This, together with weblog sear... | milkshake721/2.1M-wiki-STEM |
History of blogging-2001–2004-Several broadly popular American political blogs emerged in 2001: Ron Gunzburger's Politics1, Taegan Goddard's Political Wire, Glenn Reynolds' Instapundit, Charles Johnson's Little Green Footballs, and Jerome Armstrong's MyDD. Andrew Sullivan's AndrewSullivan.com — now entitled "The Daily ... | milkshake721/2.1M-wiki-STEM |
History of blogging-2001–2004-By 2001, blogging was enough of a phenomenon that how-to manuals began to appear, primarily focusing on technique. The importance of the blogging community (and its relationship to larger society) increased rapidly. Established schools of journalism began researching blogging and noting th... | milkshake721/2.1M-wiki-STEM |
History of blogging-2001–2004-Also in 2002, many blogs focused on comments by U.S. Senate Majority Leader Trent Lott. Senator Lott, at a party honoring U.S. Senator Strom Thurmond, praised Senator Thurmond by suggesting that the United States would have been better off had Thurmond been elected president. Lott's critic... | milkshake721/2.1M-wiki-STEM |
History of blogging-2001–2004-The impact of this story gave greater credibility to blogs as a medium of news dissemination. Though often seen as partisan gossips, bloggers sometimes lead the way in bringing key information to public light, with mainstream media having to follow their lead. More often, however, news blo... | milkshake721/2.1M-wiki-STEM |
History of blogging-2001–2004-Blogging was established by politicians and political candidates to express opinions on war and other issues and cemented blogs' role as a news source. (See Howard Dean and Wesley Clark.) Meanwhile, an increasing number of experts blogged, making blogs a source of in-depth analysis. (See D... | milkshake721/2.1M-wiki-STEM |
History of blogging-2001–2004-Bloggers began to provide nearly-instant commentary on televised events, creating a secondary meaning of the word "blogging": to simultaneously transcribe and editorialize speeches and events shown on television. (For example, "I am blogging Rice's testimony" means "I am posting my reactio... | milkshake721/2.1M-wiki-STEM |
History of blogging-2004–2013-In 2004, the role of blogs became increasingly mainstream, as political consultants, news services and candidates began using them as tools for outreach and opinion forming. Even politicians not actively campaigning, such as the UK's Labour Party's MP Tom Watson, began to blog to bond with... | milkshake721/2.1M-wiki-STEM |
History of blogging-2004–2013-Minnesota Public Radio broadcast a program by Christopher Lydon and Matt Stoller called "The blogging of the President," which covered a transformation in politics that blogging seemed to presage. The Columbia Journalism Review began regular coverage of blogs and blogging. Anthologies of b... | milkshake721/2.1M-wiki-STEM |
History of blogging-2004–2013-Some bloggers have moved over to other media. The following bloggers (and others) have appeared on radio and television: Duncan Black (known widely by his pseudonym, Atrios), Glenn Reynolds (Instapundit), Markos Moulitsas Zúniga (Daily Kos), Alex Steffen (Worldchanging) and Ana Marie Cox (... | milkshake721/2.1M-wiki-STEM |
History of blogging-2004–2013-Some blogs were an important news source during the December 2004 Tsunami such as Médecins Sans Frontières, which used SMS text messaging to report from affected areas in Sri Lanka and Southern India. Similarly, during Hurricane Katrina in August 2005 and the aftermath a few blogs which we... | milkshake721/2.1M-wiki-STEM |
History of blogging-2004–2013-In 2005, Global Voices Online, a site which "aggregates, curates, and amplifies the global conversation online – shining light on places and people other media often ignore" surfaced, bringing to light bloggers from around the world. Today, the site has a relationship with Reuters and is r... | milkshake721/2.1M-wiki-STEM |
History of blogging-2004–2013-In the United Kingdom, The Guardian newspaper launched a redesign in September 2005, which included a daily digest of blogs on page 2. Also in June 2006, BBC News launched a weblog for its editors, following other news companies.In January 2005, Fortune magazine listed eight bloggers that ... | milkshake721/2.1M-wiki-STEM |
History of blogging-2004–2013-In 2007, Tim O'Reilly proposed a Blogger's Code of Conduct. | milkshake721/2.1M-wiki-STEM |
History of blogging-2004–2013-In 2011, Tom Knighton, owner of Knighton Media, Inc, announced that his company was purchasing The Albany Journal. Knighton Media was formed to managed Knighton's blog, Laws-n-Sausages, and this was the first known time that a blog had purchased a newspaper anywhere in the world.In 2012, E... | milkshake721/2.1M-wiki-STEM |
History of blogging-2014 and After-The evolution of social media and the speed of people reacting to posted content led to increasing declarations of the death of the blog, even as it was acknowledged that what came after would contain a lot of the same DNA as the blog . | milkshake721/2.1M-wiki-STEM |
History of blogging-2014 and After-Even as the number of voices declaring blogs dead increased each year, others continued to see value, as for example in 2016 when the .blog domain name was launched. Depending on what one means by the word blog, blogging is alive and well - as of 2019, there are an estimated 500 milli... | milkshake721/2.1M-wiki-STEM |
Aleš Žiberna-Aleš Žiberna-Aleš Žiberna is a Slovene statistician, whose specialty is network analysis. His specific research interests include blockmodeling, multivariate analysis and computer intensive methods (e.g., computer simulations, resampling methods).Currently, he is employed at the Faculty of Social Sciences ... | milkshake721/2.1M-wiki-STEM |
Aleš Žiberna-Work-In 2007, he proposed a solution to the generalized valued blockmodeling by introducing homogeneity blockmodeling with the basic idea "that the inconsistency of an empirical block with its ideal block can be measured by within block variability of appropriate values". The newly-formed ideal blocks, whi... | milkshake721/2.1M-wiki-STEM |
Aleš Žiberna-Selected bibliography-ŽIBERNA, Aleš. Generalized blockmodeling of valued networks. Social Networks. [Print ed.]. Jan. 2007, vol. 29, no. 1, str. 105-126. ISSN 0378-8733. | milkshake721/2.1M-wiki-STEM |
Aleš Žiberna-Selected bibliography-ŽIBERNA, Aleš. Blockmodeling of multilevel networks. Social Networks. [Print ed.]. Oct. 2014, vol. 39, str. 46-61, ilustr. ISSN 0378-8733. DOI: 10.1016/j.socnet.2014.04.002. [COBISS.SI-ID 32697949] ŽIBERNA, Aleš. Generalized blockmodeling of sparse networks. Metodološki zvezki. [Tiska... | milkshake721/2.1M-wiki-STEM |
Brace (sailing)-Brace (sailing)-A brace on a square-rigged ship is a rope (line) used to rotate a yard around the mast, to allow the ship to sail at different angles to the wind. Braces are always used in pairs, one at each end of a yard (yardarm), termed port brace and starboard brace of a given yard or sail (e.g., th... | milkshake721/2.1M-wiki-STEM |
Brace (sailing)-Brace (sailing)-The braces are fixed to the outer ends of the yards, and are led to the deck as far aft as possible, to allow the crew to haul on them. The lower yards' braces can usually run directly to the deck, but to do so with those higher up would mean that most of the force was pulling downwards ... | milkshake721/2.1M-wiki-STEM |
Brace (sailing)-Brace (sailing)-In many ways, braces are the equivalent of a modern yacht's sheets. However, where adjusting a sail on a yacht is a simple operation performed often, tacking or wearing ship using the braces usually requires the entire crew to be called to "bracing stations". This is because the braces c... | milkshake721/2.1M-wiki-STEM |
Brace (sailing)-Brace (sailing)-The sails on a tall ship's mast must all be turned together, because of all the gear that runs between them. The rate of turn is set by the course, the heaviest yard and hence the most difficult to move. The teams on the other braces for that mast must watch the course and keep their own... | milkshake721/2.1M-wiki-STEM |
Actoxumab-Actoxumab-Actoxumab is a human monoclonal antibody designed for the prevention of recurrence of Clostridium difficile infection.This drug, along with bezlotoxumab, was developed through Phase II efficacy trials by a partnership between Medarex Inc and MassBiologics of the University of Massachusetts Medical S... | milkshake721/2.1M-wiki-STEM |
Fly-whisk-Fly-whisk-A fly-whisk (or fly-swish) is a tool that is used to swat flies. A similar device is used as a hand fan in hot tropical climates, sometimes as part of regalia, and is called a chowrie, chāmara, or prakirnaka in South Asia and Tibet.In Indonesian art, a fly-whisk is one of the items that is associate... | milkshake721/2.1M-wiki-STEM |
Fly-whisk-Fly-whisk-Fly-whisks are in use in parts of the contemporary Middle East, such as Egypt, by some classes of society, e.g., outdoor merchants and shop keepers, especially in summer when flies become bothersome. Those have a wooden handle and plant fibers attached to them. The more expensive ones are made from ... | milkshake721/2.1M-wiki-STEM |
Fly-whisk-Fly-whisk-Fly-whisks appear frequently in the traditional regales of monarchs and nobility in many parts of the African continent. Fly whisks, called "ìrùkẹ̀rẹ̀" in Yoruba, were used by Yoruba monarchs and chiefs as a symbol of power and respect. This use has sometimes carried on into modern contexts: Kenyan ... | milkshake721/2.1M-wiki-STEM |
Fly-whisk-Fly-whisk-A fly-whisk forms part of the royal regalia of Thailand. It consists of the tail hairs of a white elephant.
Fly-whisks were also used in Polynesian culture as a ceremonial mark of authority. | milkshake721/2.1M-wiki-STEM |
Fly-whisk-Algeria incident-In 1827, the last Ottoman ruler of Algeria, Hussein Dey, struck the French consul, Pierre Deval in the face with a fly-whisk during a dispute over unpaid French debts to Algeria. That insult became a pretext for the French invasion of Algeria in 1830. | milkshake721/2.1M-wiki-STEM |
Difference density map-Difference density map-In X-ray crystallography, a difference density map shows the spatial distribution of the difference between the measured electron density of the crystal and the electron density explained by the current model.These coefficients are derived from the gradient of the likelihoo... | milkshake721/2.1M-wiki-STEM |
Difference density map-Display-Conventionally, they are displayed as isosurfaces with positive density—electron density where there's nothing in the model, usually corresponding to some constituent of the crystal that hasn't been modelled, for example a ligand or a crystallisation adjutant -- in green, and negative den... | milkshake721/2.1M-wiki-STEM |
Difference density map-Calculation-Difference density maps are usually calculated using Fourier coefficients which are the differences between the observed structure factor amplitudes from the X-ray diffraction experiment and the calculated structure factor amplitudes from the current model, using the phase from the mo... | milkshake721/2.1M-wiki-STEM |
Colored matroid-Colored matroid-In mathematics, a colored matroid is a matroid whose elements are labeled from a set of colors, which can be any set that suits the purpose, for instance the set of the first n positive integers, or the sign set {+, −}.
The interest in colored matroids is through their invariants, especi... | milkshake721/2.1M-wiki-STEM |
Recombinant DNA-Recombinant DNA-Recombinant DNA (rDNA) molecules are DNA molecules formed by laboratory methods of genetic recombination (such as molecular cloning) that bring together genetic material from multiple sources, creating sequences that would not otherwise be found in the genome. | milkshake721/2.1M-wiki-STEM |
Recombinant DNA-Recombinant DNA-Recombinant DNA is the general name for a piece of DNA that has been created by combining two or more fragments from different sources. Recombinant DNA is possible because DNA molecules from all organisms share the same chemical structure, differing only in the nucleotide sequence. Recom... | milkshake721/2.1M-wiki-STEM |
Recombinant DNA-Recombinant DNA-The DNA sequences used in the construction of recombinant DNA molecules can originate from any species. For example, plant DNA can be joined to bacterial DNA, or human DNA can be joined with fungal DNA. In addition, DNA sequences that do not occur anywhere in nature can be created by the... | milkshake721/2.1M-wiki-STEM |
Recombinant DNA-Recombinant DNA-Proteins that can result from the expression of recombinant DNA within living cells are termed recombinant proteins. When recombinant DNA encoding a protein is introduced into a host organism, the recombinant protein is not necessarily produced. Expression of foreign proteins requires th... | milkshake721/2.1M-wiki-STEM |
Recombinant DNA-DNA creation-Molecular cloning is the laboratory process used to create recombinant DNA. It is one of two most widely used methods, along with polymerase chain reaction (PCR), used to direct the replication of any specific DNA sequence chosen by the experimentalist. There are two fundamental differences... | milkshake721/2.1M-wiki-STEM |
Recombinant DNA-DNA creation-Formation of recombinant DNA requires a cloning vector, a DNA molecule that replicates within a living cell. Vectors are generally derived from plasmids or viruses, and represent relatively small segments of DNA that contain necessary genetic signals for replication, as well as additional e... | milkshake721/2.1M-wiki-STEM |
Recombinant DNA-DNA creation-In standard cloning protocols, the cloning of any DNA fragment essentially involves seven steps: (1) Choice of host organism and cloning vector, (2) Preparation of vector DNA, (3) Preparation of DNA to be cloned, (4) Creation of recombinant DNA, (5) Introduction of recombinant DNA into the ... | milkshake721/2.1M-wiki-STEM |
Recombinant DNA-DNA expression-DNA expression requires the transfection of suitable host cells. Typically, either bacterial, yeast, insect, or mammalian cells (such as Human Embryonic Kidney cells or CHO cells) are used as host cells.Following transplantation into the host organism, the foreign DNA contained within the... | milkshake721/2.1M-wiki-STEM |
Recombinant DNA-Properties of organisms containing recombinant DNA-In most cases, organisms containing recombinant DNA have apparently normal phenotypes. That is, their appearance, behavior and metabolism are usually unchanged, and the only way to demonstrate the presence of recombinant sequences is to examine the DNA ... | milkshake721/2.1M-wiki-STEM |
Recombinant DNA-Properties of organisms containing recombinant DNA-If the rDNA sequences encode a gene that is expressed, then the presence of RNA and/or protein products of the recombinant gene can be detected, typically using RT-PCR or western hybridization methods. Gross phenotypic changes are not the norm, unless t... | milkshake721/2.1M-wiki-STEM |
Recombinant DNA-Properties of organisms containing recombinant DNA-In some cases, recombinant DNA can have deleterious effects even if it is not expressed. One mechanism by which this happens is insertional inactivation, in which the rDNA becomes inserted into a host cell's gene. In some cases, researchers use this phe... | milkshake721/2.1M-wiki-STEM |
Recombinant DNA-Applications of recombinant DNA-Recombinant DNA is widely used in biotechnology, medicine and research. Today, recombinant proteins and other products that result from the use of DNA technology are found in essentially every western pharmacy, physician or veterinarian office, medical testing laboratory... | milkshake721/2.1M-wiki-STEM |
Recombinant DNA-Applications of recombinant DNA-The most common application of recombinant DNA is in basic research, in which the technology is important to most current work in the biological and biomedical sciences. Recombinant DNA is used to identify, map and sequence genes, and to determine their function. rDNA pr... | milkshake721/2.1M-wiki-STEM |
Recombinant DNA-Applications of recombinant DNA-Recombinant chymosin Found in rennet, chymosin is an enzyme required to manufacture cheese. It was the first genetically engineered food additive used commercially. Traditionally, processors obtained chymosin from rennet, a preparation derived from the fourth stomach of ... | milkshake721/2.1M-wiki-STEM |
Recombinant DNA-Applications of recombinant DNA-Recombinant human insulin Almost completely replaced insulin obtained from animal sources (e.g. pigs and cattle) for the treatment of insulin-dependent diabetes. A variety of different recombinant insulin preparations are in widespread use. Recombinant insulin is synthes... | milkshake721/2.1M-wiki-STEM |
Recombinant DNA-Applications of recombinant DNA-Recombinant human growth hormone (HGH, somatotropin) Administered to patients whose pituitary glands generate insufficient quantities to support normal growth and development. Before recombinant HGH became available, HGH for therapeutic use was obtained from pituitary gl... | milkshake721/2.1M-wiki-STEM |
Recombinant DNA-Applications of recombinant DNA-Diagnosis of infection with HIV Each of the three widely used methods for diagnosing HIV infection has been developed using recombinant DNA. The antibody test (ELISA or western blot) uses a recombinant HIV protein to test for the presence of antibodies that the body has ... | milkshake721/2.1M-wiki-STEM |
Recombinant DNA-Applications of recombinant DNA-Herbicide-resistant crops Commercial varieties of important agricultural crops (including soy, maize/corn, sorghum, canola, alfalfa and cotton) have been developed that incorporate a recombinant gene that results in resistance to the herbicide glyphosate (trade name Roun... | milkshake721/2.1M-wiki-STEM |
Recombinant DNA-Applications of recombinant DNA-Insect-resistant crops Bacillus thuringeiensis is a bacterium that naturally produces a protein (Bt toxin) with insecticidal properties. The bacterium has been applied to crops as an insect-control strategy for many years, and this practice has been widely adopted in agr... | milkshake721/2.1M-wiki-STEM |
Recombinant DNA-History-The idea of recombinant DNA was first proposed by Peter Lobban, a graduate student of Prof. Dale Kaiser in the Biochemistry Department at Stanford University Medical School. The first publications describing the successful production and intracellular replication of recombinant DNA appeared in 1... | milkshake721/2.1M-wiki-STEM |
Recombinant DNA-History-Stanford University applied for a US patent on recombinant DNA in 1974, listing the inventors as Herbert W. Boyer (professor at the University of California, San Francisco) and Stanley N. Cohen (professor at Stanford University); this patent was awarded in 1980. The first licensed drug generated... | milkshake721/2.1M-wiki-STEM |
Recombinant DNA-Controversy-Scientists associated with the initial development of recombinant DNA methods recognized that the potential existed for organisms containing recombinant DNA to have undesirable or dangerous properties. At the 1975 Asilomar Conference on Recombinant DNA, these concerns were discussed and a vo... | milkshake721/2.1M-wiki-STEM |
Infusion-Infusion-Infusion is the process of extracting chemical compounds or flavors from plant material in a solvent such as water, oil or alcohol, by allowing the material to remain suspended in the solvent over time (a process often called steeping). An infusion is also the name for the resultant liquid. The proces... | milkshake721/2.1M-wiki-STEM |
Infusion-History-The first recorded use of essential oils was in the 10th or 11th century by the Persian polymath Avicenna, possibly in The Canon of Medicine.Tea is far older than this, dating back to the 10th century BC as the earliest recorded reference. | milkshake721/2.1M-wiki-STEM |
Infusion-Preparation techniques-Infusion is a chemical process that uses botanicals (typically dried herbs, flowers or berries) that are volatile and release their active ingredients readily in water, oil, or alcohol. In this process, a liquid is typically boiled (or brought to another appropriate temperature) and pour... | milkshake721/2.1M-wiki-STEM |
Infusion-Preparation techniques-The amount of time the herbs are left in the liquid depends on the kind of infusion. Infusion times can range from seconds (for some kinds of Chinese tea) to hours, days, or months (for liqueurs like Sloe Gin). There are several accessories and techniques for removing the steeped or lef... | milkshake721/2.1M-wiki-STEM |
Infusion-Examples-Tea is a common example of an infusion; most varieties of tea call for steeping the leaves in hot water, although some variants (e.g. Moroccan mint tea) call for decoction instead. Many herbal teas are prepared by infusion, as well; lemon, chamomile, senna, apple, ginger, rooibos, and many other plant... | milkshake721/2.1M-wiki-STEM |
Infusion-Examples-Flavored oils: Plants with desirable flavors may be steeped in an edible oil or vinegar for an extended period; the infused oil or vinegar is often sold still containing the plant and is then used as flavoring. Chilis, lemon, garlic, and many other plants may be used. There can be ambiguity in the lab... | milkshake721/2.1M-wiki-STEM |
Infusion-Examples-Cucumber water, a mix of sliced cucumber with citrus slices and herbs such as mint, is a popular infusion. It is sometimes known as "spa water" due to it being commonly served in day spas and similar personal care establishments. | milkshake721/2.1M-wiki-STEM |
Triangular decomposition-Triangular decomposition-In computer algebra, a triangular decomposition of a polynomial system S is a set of simpler polynomial systems S1, ..., Se such that a point is a solution of S if and only if it is a solution of one of the systems S1, ..., Se. | milkshake721/2.1M-wiki-STEM |
Triangular decomposition-Triangular decomposition-When the purpose is to describe the solution set of S in the algebraic closure of its coefficient field, those simpler systems are regular chains. If the coefficients of the polynomial systems S1, ..., Se are real numbers, then the real solutions of S can be obtained by... | milkshake721/2.1M-wiki-STEM |
Triangular decomposition-History-The Characteristic Set Method is the first factorization-free algorithm, which was proposed for decomposing an algebraic variety into equidimensional components. Moreover, the Author, Wen-Tsun Wu, realized an implementation of this method and reported experimental data in his 1987 pione... | milkshake721/2.1M-wiki-STEM |
Triangular decomposition-History-Let K be an algebraically closed field and k be a subfield of K. A subset V ⊂ Kn is an (affine) algebraic variety over k if there exists a polynomial set F ⊂ k[x1, ..., xn] such that the zero set V(F) ⊂ Kn of F equals V.
Recall that V is said irreducible if for all algebraic varieties V... | milkshake721/2.1M-wiki-STEM |
Triangular decomposition-History-Moreover, if Vi ⊈ Vj holds for 1 ≤ i < j ≤ e then the set {V1, ..., Ve} is unique and forms the irreducible decomposition of V.The varieties V1, ..., Ve in the above Theorem are called the irreducible components of V and can be regarded as a natural output for a decomposition algorithm,... | milkshake721/2.1M-wiki-STEM |
Triangular decomposition-History-In order to lead to a computer program, this algorithm specification should prescribe how irreducible components are represented. Such an encoding is introduced by Joseph Ritt through the following result. | milkshake721/2.1M-wiki-STEM |
Triangular decomposition-History-Theorem (Ritt). If V ⊂ Kn is a non-empty and irreducible variety then one can compute a reduced triangular set C contained in the ideal ⟨F⟩ generated by F in k[x1, ..., xn] and such that all polynomials g in ⟨F⟩ reduces to zero by pseudo-division w.r.t C.We call the set C in Ritt's Th... | milkshake721/2.1M-wiki-STEM |
Triangular decomposition-History-Joseph Ritt described a method for solving polynomial systems based on polynomial factorization over field extensions and computation of characteristic sets of prime ideals. | milkshake721/2.1M-wiki-STEM |
Triangular decomposition-History-Deriving a practical implementation of this method, however, was and remains a difficult problem. In the 1980s, when the Characteristic set Method was introduced, polynomial factorization was an active research area and certain fundamental questions on this subject were solved recentlyN... | milkshake721/2.1M-wiki-STEM |
Triangular decomposition-History-The Characteristic Set Method relies on the following variant of Ritt's Theorem. | milkshake721/2.1M-wiki-STEM |
Triangular decomposition-History-Theorem (Wen-Tsun Wu). For any finite polynomial set F ⊂ k[x1, ..., xn], one can compute a reduced triangular set C⊂⟨F⟩ such that all polynomial g in F reduces to zero by pseudo-division w.r.t C.Different concepts and algorithms extended the work of Wen-Tsun Wu. In the early 1990s, the... | milkshake721/2.1M-wiki-STEM |
Triangular decomposition-History-In Kalkbrener's vision, regular chains are used to represent the generic zeros of the irreducible components of an algebraic variety. In the original work of Yang and Zhang, they are used to decide whether a hypersurface intersects a quasi-variety (given by a regular chain). Regular cha... | milkshake721/2.1M-wiki-STEM |
Triangular decomposition-History-Regular chains have been investigated in many papers.The abundant literature on the subject can be explained by the many equivalent definitions of a regular chain. Actually, the original formulation of Kalkbrener is quite different from that of Yang and Zhang. A bridge between these two... | milkshake721/2.1M-wiki-STEM |
Triangular decomposition-Formal definitions-Let k be a field and x1 < ... < xn be ordered variables. We denote by R = k[x1, ..., xn] the corresponding polynomial ring. For F ⊂ R, regarded as a system of polynomial equations, there are two notions of a triangular decomposition over the algebraic closure of k. The first ... | milkshake721/2.1M-wiki-STEM |
Triangular decomposition-Formal definitions-(F)=⋂i=1esat(Ti).
The second is to describe explicitly all the points of V(F) in the so-called sense of in Lazard and Wen-Tsun Wu.
V(F)=⋃i=1eW(Ti).
In both cases T1, ..., Te are finitely many regular chains of R and sat(Ti) denotes the radical of the saturated ideal of Ti wh... | milkshake721/2.1M-wiki-STEM |
Triangular decomposition-Formal definitions-Assume from now on that k is a real closed field. Consider S a semi-algebraic system with polynomials in R. There exist finitely many regular semi-algebraic systems S1, ..., Se such that we have Zk(S)=Zk(S1)∪⋯∪Zk(Se) where Zk(S) denotes the points of kn which solve S. The reg... | milkshake721/2.1M-wiki-STEM |
Triangular decomposition-Examples-Denote Q the rational number field. In Q[x,y,z] with variable ordering x>y>z , consider the following polynomial system: S={x2+y+z=1x+y2+z=1x+y+z2=1 According to the Maple code: One possible triangular decompositions of the solution set of S with using RegularChains library is: {z=0... | milkshake721/2.1M-wiki-STEM |
Cross of Merit-Cross of Merit-A Cross of Merit is a personal decoration which is generally above the rank of medal and below that of knight, and may refer to: | milkshake721/2.1M-wiki-STEM |
Cross of Merit-The Netherlands-Cross of Merit of the Order of the House of Orange Cross of Merit (Netherlands), a decoration for bravery Cross of Merit of the Netherlands Red Cross | milkshake721/2.1M-wiki-STEM |
Cross of Merit-Poland-Cross of Merit (Poland) Cross of Merit with Swords (Poland) Cross of Merit for Bravery (Poland) Cross of Merit (Polish Scouting and Guiding Association) | milkshake721/2.1M-wiki-STEM |
Cross of Merit-Ukraine-Cross of Merit (Ukrainian Insurgent Army) | milkshake721/2.1M-wiki-STEM |
Cross of Merit-Other-Cross of Merit (EOHSJ) Cross of the Order pro Merito Melitensi, Sovereign Military Order of Malta | milkshake721/2.1M-wiki-STEM |
Bus garage-Bus garage-A bus garage, also known as a bus depot, bus base or bus barn, is a facility where buses are stored and maintained. In many conurbations, bus garages are on the site of former car barns or tram sheds, where trams (streetcars) were stored, and the operation transferred to buses. In other areas, gar... | milkshake721/2.1M-wiki-STEM |
Bus garage-Description-Most bus garages will contain the following elements: Internal parking External parking Fueling point Fuel storage tanks Engineering section Inspection pits Bus wash Brake test lane Staff canteen/break room Administration officeSmaller garages may contain the minimum engineering facilities, restr... | milkshake721/2.1M-wiki-STEM |
Bus garage-Description-Overnight, the more valuable or regularly in-service buses will usually be stored in the interior of the garage, with less used or older service vehicles, and vehicles withdrawn for storage or awaiting disposal, stored externally. During the day, internal and external areas will see a variety of ... | milkshake721/2.1M-wiki-STEM |
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