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Microsoft Developer Network-Websites-Library MSDN Library is a library of official technical documentation intended for independent developers of software for Microsoft Windows. MSDN Library documents the APIs that ship with Microsoft products and also includes sample code, technical articles, and other programming inf... | milkshake721/2.1M-wiki-STEM |
Microsoft Developer Network-Websites-In Visual Studio 2010 MSDN Library is replaced with the new Help System, which is installed as a part of Visual Studio 2010 installation. Help Library Manager is used to install Help Content books covering selected topics.
In 2016, Microsoft introduced the new technical documentatio... | milkshake721/2.1M-wiki-STEM |
Microsoft Developer Network-Websites-Integration with Visual Studio Each edition of MSDN Library can only be accessed with one help viewer (Microsoft Document Explorer or other help viewer), which is integrated with the then current single version or sometimes two versions of Visual Studio. In addition, each new versio... | milkshake721/2.1M-wiki-STEM |
Microsoft Developer Network-Websites-Forums MSDN Forums are the web-based forums used by the community to discuss a wide variety of software development topics. MSDN Forums were migrated to an all-new platform during 2008 that provided new features designed to improve efficiency such as inline preview of threads, AJAX ... | milkshake721/2.1M-wiki-STEM |
Microsoft Developer Network-Websites-Blogs MSDN blogs is a series of blogs that were hosted under Microsoft's domain blogs.msdn.com. Some blogs are dedicated to a product – e.g. Visual Studio, Internet Explorer, PowerShell – or a version of a product – e.g. Windows 7, Windows 8 – while others belong to a Microsoft empl... | milkshake721/2.1M-wiki-STEM |
Microsoft Developer Network-Websites-Social bookmarking Social bookmarking on MSDN Social was first launched in 2008, built on a new web platform that has user-tagging and feeds at its core. The goal of the social bookmarking application is to provide a method whereby members of the developer community can: Contribute ... | milkshake721/2.1M-wiki-STEM |
Microsoft Developer Network-Websites-Find and follow experts' recommended sites. Each profile page includes a feed of the user's contributions. Users can be discovered through a drop-down menu on each bookmark.
Demonstrate their expertise through the links displayed in their profile.
Store their favorite links online.T... | milkshake721/2.1M-wiki-STEM |
Microsoft Developer Network-Software subscriptions-MSDN has historically offered a subscription package whereby developers have access and licenses to use nearly all Microsoft software that has ever been released to the public. Subscriptions are sold on an annual basis, and cost anywhere from US$1,000 to US$6,000 per y... | milkshake721/2.1M-wiki-STEM |
Microsoft Developer Network-Software subscriptions-Although in most cases the software itself functions exactly like the full product, the MSDN end-user license agreement prohibits use of the software in a business production environment. This is a legal restriction, not a technical one. An exception is made for Micros... | milkshake721/2.1M-wiki-STEM |
Microsoft Developer Network-MSDN Magazine-Microsoft provides the editorial content for MSDN Magazine, a monthly publication. The magazine was created as a merger between Microsoft Systems Journal (MSJ) and Microsoft Internet Developer (MIND) magazines in March 2000.|MSJ back issues are available online. MSDN Magazine w... | milkshake721/2.1M-wiki-STEM |
Microsoft Developer Network-MSDN Magazine-Microsoft Systems Journal Microsoft Systems Journal was a 1986-founded bi-monthly Microsoft magazine. | milkshake721/2.1M-wiki-STEM |
Microsoft Developer Network-History-MSDN was launched in September 1992 as a quarterly, CD-ROM-based compilation of technical articles, sample code, and software development kits. The first two MSDN CD releases (September 1992 and January 1993) were marked as pre-release discs (P1 and P2, respectively). Disc 3, release... | milkshake721/2.1M-wiki-STEM |
Microsoft Developer Network-History-MSDN2 was opened in November 2004 as a source for Visual Studio 2005 API information, with noteworthy differences being updated web site code, conforming better to web standards and thus giving a long-awaited improved support for alternative web browsers to Internet Explorer in the A... | milkshake721/2.1M-wiki-STEM |
Microsoft Developer Network-History-Dr GUI and the MSDN Writers Team In 1996, Bob Gunderson began writing a column in Microsoft Developer Network News, edited by Andrew Himes, using the pseudonym "Dr.GUI". The column provided answers to questions submitted by MSDN subscribers. The caricature of Dr. GUI was based on a p... | milkshake721/2.1M-wiki-STEM |
DS-1 (drug)-DS-1 (drug)-DS-1 is a drug from the imidazopyridine family, which is the first drug developed that acts as a GABAA receptor positive allosteric modulator (PAM) selective for the α4β3δ subtype, which is not targeted by other GABAA receptor PAMs such as the benzodiazepines or other nonbenzodiazepine drugs. No... | milkshake721/2.1M-wiki-STEM |
Glare (vision)-Glare (vision)-Glare is difficulty of seeing in the presence of bright light such as direct or reflected sunlight or artificial light such as car headlamps at night. Because of this, some cars include mirrors with automatic anti-glare functions and in buildings, blinds or louvers are often used to protec... | milkshake721/2.1M-wiki-STEM |
Glare (vision)-Discomfort and disability-Glare can be generally divided into two types, discomfort glare and disability glare. Discomfort glare is a psychological sensation caused by high brightness (or brightness contrast) within the field of view, which does not necessarily impair vision. In buildings, discomfort gla... | milkshake721/2.1M-wiki-STEM |
Glare (vision)-Reducing factors-Glare can reduce visibility by: Reduction of brightness of the rest of the scene by constriction of the pupils Reduction in contrast of the rest of the scene by scattering of the bright light within the eye.
Reduction in contrast by scattering light in particles in the air, as when the h... | milkshake721/2.1M-wiki-STEM |
Glare (vision)-Reducing factors-bloom surrounding objects in front of glareSunglasses are often worn to reduce glare; polarized sunglasses are designed to reduce glare caused by light reflected from non-metallic surfaces such as water, glossy printed matter or painted surfaces. An anti-reflective treatment on eyeglasse... | milkshake721/2.1M-wiki-STEM |
Glare (vision)-Reducing factors-Light field measurements can be taken to reduce glare with digital post-processing. | milkshake721/2.1M-wiki-STEM |
Glare (vision)-Measurement-Methods Discomfort glare has often been studied using psychophysics experiments, where the common methods have been the luminance adjustment and category rating procedures. Studies conducted by Petherbridge and Hopkinson and Luckiesh and Guth. were amongst the first to compared subjective ass... | milkshake721/2.1M-wiki-STEM |
Glare (vision)-Measurement-Biases A comprehensive review of the methods used to measure glare showed that there are biases associated with its measurement. Luminance adjustments are sensitive to anchoring (cognitive bias) effects caused when the initial starting luminance viewed influences the final assessment of visua... | milkshake721/2.1M-wiki-STEM |
Glare (vision)-Measurement-Prediction models Glare from artificial lights is typically measured with luminance meters. From daylit windows, cameras are used to convert the pixels into luminance. Both of which are able to determine the luminance of objects within small solid angles. The glare of a scene i.e. visual fiel... | milkshake721/2.1M-wiki-STEM |
Glare (vision)-Measurement-Unified glare rating The unified glare rating (UGR) is a measure of the glare in a given environment, proposed by Sorensen in 1987 and adopted by the International Commission on Illumination (CIE). It is basically the logarithm of the glare of all visible lamps, divided by the background lumi... | milkshake721/2.1M-wiki-STEM |
Tensoba-Tensoba-Tensoba, or tempura soba, is a Japanese dish of soba noodles and tempura. | milkshake721/2.1M-wiki-STEM |
Tensoba-Overview-There are two varieties of tensoba: one is served with a hot broth of dashi and soy sauce; the other is served with cooled soba and dipped in tsukejiru (lit. 'dipping sauce'), either chilled or hot and usually strongly flavored. The dipping variety is also called tenzaru-soba or ten-seiro, depending on... | milkshake721/2.1M-wiki-STEM |
Tensoba-History-Tensoba originated during the mid-Edo-period. It was first eaten as a hot broth soba with kakiage, using the adductor muscles of surf clams. At that time, shrimp-tempura soba was more expensive than other ingredients. So, shrimp-tempura-soba is also called jo-tempura soba (lit. 'upper class tempura-soba... | milkshake721/2.1M-wiki-STEM |
Tensoba-Regional variety-There are some regional varieties for tensoba toppings. In Kanto and Kyushu, the soba shops often use satsuma age (fried fish cake) or chikuwa for tempura. These two fish cakes are sometimes batter-fried. | milkshake721/2.1M-wiki-STEM |
Eskimo Rescue-Eskimo Rescue-An Eskimo rescue, bow rescue or T-rescue is a kayaking technique performed to recover a kayaker from a capsize without them having to leave their boat or perform a self-rescue such as a kayak roll. The advantages of this manoeuvre are that the kayaker does not have to get out of the kayak an... | milkshake721/2.1M-wiki-STEM |
Eskimo Rescue-Technique-After drawing attention to the capsize by banging on the bottom of their boat, the kayaker who capsized waits upside down underwater until another kayak arrives to help. The capsized kayaker finds the other kayak, usually the bow, with their hand and uses this for support while they perform a hi... | milkshake721/2.1M-wiki-STEM |
Eskimo Rescue-Naming-An eskimo rescue is often used synonymously with T rescue and bow rescue, these names come from the shape the boats make (a T) and the part of the boat that is presented to the capsized kayaker respectively. However, an eskimo rescue is really the general term for any rescue in which a capsized kay... | milkshake721/2.1M-wiki-STEM |
Biscuits and gravy-Biscuits and gravy-Biscuits and gravy is a popular breakfast dish in the United States, especially in the South. The dish consists of soft dough biscuits covered in white gravy (sawmill gravy), made from the drippings of cooked pork sausage, flour, milk, and often (but not always) bits of sausage, ba... | milkshake721/2.1M-wiki-STEM |
Biscuits and gravy-History-The meal emerged as a distinct regional dish after the American Revolutionary War (1775–1783), when stocks of foodstuffs were in short supply. Breakfast was necessarily the most substantial meal of the day for a person facing a day of work on the plantations in the American South. In addition... | milkshake721/2.1M-wiki-STEM |
Biscuits and gravy-Variations-Tomato gravy is white gravy mixed with crushed or diced tomatoes. | milkshake721/2.1M-wiki-STEM |
Computerised Pilot Selection System-Computerised Pilot Selection System-The Computerised Pilot Selection System is used for screening candidates into the flight branch of the Indian Air Force. It replaced the earlier pilot selection test named Pilot Aptitude Battery Test (PABT). It was originally conceived by then Scie... | milkshake721/2.1M-wiki-STEM |
Nucleoside oxidase-Nucleoside oxidase-Nucleoside oxidase (EC 1.1.3.28) is an enzyme with systematic name nucleoside:oxygen 5'-oxidoreductase. This enzyme catalyses the following chemical reaction inosine + O2 ⇌ 9-riburonosylhypoxanthine + H2O (1a) 2 inosine + O2 ⇌ 2 5'-dehydroinosine + 2 H2O (1b) 2 5'-dehydroinosine ... | milkshake721/2.1M-wiki-STEM |
Libwww-Libwww-Libwww is an early World Wide Web software library providing core functions for web browsers, implementing HTML, HTTP, and other technologies. Tim Berners-Lee, at the European Organization for Nuclear Research (CERN), released libwww (then also called the Common Library) in late 1992, comprising reusable ... | milkshake721/2.1M-wiki-STEM |
Libwww-Libwww-Libwww was relied upon by the then popular browser Mosaic. By 1997, interest in libwww declined, and the World Wide Web Consortium (W3C), which took over from CERN, reduced its commitment to the project. Later, the purpose of libwww was redefined to be "a testbed for protocol experiments"; in that role it... | milkshake721/2.1M-wiki-STEM |
Libwww-History-In 1991 and 1992, Tim Berners-Lee and a student at CERN named Jean-François Groff rewrote various components of the original WorldWideWeb browser for the NeXTstep operating system in portable C code, in order to demonstrate the potential of the World Wide Web. In the beginning, libwww was referred to as ... | milkshake721/2.1M-wiki-STEM |
Libwww-Features-In 2003, Kahan claimed that "libwww is the only library that has a full implementation of the HTTP specification, including caching and pipelining."Libwww supports following protocols: file FTP Gopher HTTP 1.1 with a Persistent Cache Manager, pipelining NNTP Telnet WAISOther features include: TLS and SS... | milkshake721/2.1M-wiki-STEM |
Libwww-Applications using libwww-It has been used for applications of varying sizes, including web browsers, editors, Internet bots, and batch tools. Pluggable modules provided with libwww add support for HTTP/1.1 with caching, pipelining, POST, Digest Authentication, and deflate.
The W3C created the Arena web browser ... | milkshake721/2.1M-wiki-STEM |
Libwww-Criticism-The developers of libcurl have criticised libwww as being not as portable, not thread-safe and lacking several HTTP authentication types. Neither libcurl nor libwww are lightweight enough for some projects. | milkshake721/2.1M-wiki-STEM |
(S)-tetrahydroprotoberberine N-methyltransferase-(S)-tetrahydroprotoberberine N-methyltransferase-In enzymology, a (S)-tetrahydroprotoberberine N-methyltransferase (EC 2.1.1.122) is an enzyme that catalyzes the chemical reaction S-adenosyl-L-methionine + (S)-7,8,13,14-tetrahydroprotoberberine ⇌ S-adenosyl-L-homocystei... | milkshake721/2.1M-wiki-STEM |
Double penetration-Double penetration-Double penetration (sometimes called DP for short) is a term that usually refers to a vaginal and anal sex act involving one penis penetrating a woman's vagina while another penetrates her anus. | milkshake721/2.1M-wiki-STEM |
Double penetration-Practice-Double penetration usually involves the insertion and thrusting of two erect penises into a woman's vagina and anus simultaneously. It is a common practice in pornography. The term can also describe the insertion and thrusting of two erect penises into a single vagina or anus. This is known ... | milkshake721/2.1M-wiki-STEM |
Double penetration-Practice-Double penetration can be carried out not only with penises, but also with different parts of the body (hands, fingers) or with specific sex toys.The sexual act may be pleasurable for the penetrated partner due to the simultaneous stimulation of the G-spot and the anterior fornix, or the pro... | milkshake721/2.1M-wiki-STEM |
Double penetration-History-Representations of double penetration have been depicted in many Roman erotic objects, as well as in the Kama Sutra. The first filmed double penetration in history appeared in 1970, in the movie "Delphia the Greek", by director Lasse Braun.The feminist Bernadette Barton has argued that double... | milkshake721/2.1M-wiki-STEM |
Double penetration-Spit-roast-The spit-roast is a variation of double penetration whereby a person is penetrated in the rear by one penis (either in the vagina or anus) and performs oral sex on another penis. This sexual act combines both the doggy style position with fellatio; the "spit-roast" is frequently depicted w... | milkshake721/2.1M-wiki-STEM |
Elliptic-curve Diffie–Hellman-Elliptic-curve Diffie–Hellman-Elliptic-curve Diffie–Hellman (ECDH) is a key agreement protocol that allows two parties, each having an elliptic-curve public–private key pair, to establish a shared secret over an insecure channel. This shared secret may be directly used as a key, or to deri... | milkshake721/2.1M-wiki-STEM |
Elliptic-curve Diffie–Hellman-Key establishment protocol-The following example illustrates how a shared key is established. Suppose Alice wants to establish a shared key with Bob, but the only channel available for them may be eavesdropped by a third party. Initially, the domain parameters (that is, (p,a,b,G,n,h) in t... | milkshake721/2.1M-wiki-STEM |
Elliptic-curve Diffie–Hellman-Key establishment protocol-Alice computes point (xk,yk)=dA⋅QB . Bob computes point (xk,yk)=dB⋅QA . The shared secret is xk (the x coordinate of the point). Most standardized protocols based on ECDH derive a symmetric key from xk using some hash-based key derivation function. | milkshake721/2.1M-wiki-STEM |
Elliptic-curve Diffie–Hellman-Key establishment protocol-The shared secret calculated by both parties is equal, because dA⋅QB=dA⋅dB⋅G=dB⋅dA⋅G=dB⋅QA The only information about her key that Alice initially exposes is her public key. So, no party except Alice can determine Alice's private key (Alice of course knows it by... | milkshake721/2.1M-wiki-STEM |
Elliptic-curve Diffie–Hellman-Key establishment protocol-The public keys are either static (and trusted, say via a certificate) or ephemeral (also known as ECDHE, where final 'E' stands for "ephemeral"). Ephemeral keys are temporary and not necessarily authenticated, so if authentication is desired, authenticity assura... | milkshake721/2.1M-wiki-STEM |
Elliptic-curve Diffie–Hellman-Key establishment protocol-If Alice maliciously chooses invalid curve points for her key and Bob does not validate that Alice's points are part of the selected group, she can collect enough residues of Bob's key to derive his private key. Several TLS libraries were found to be vulnerable t... | milkshake721/2.1M-wiki-STEM |
Elliptic-curve Diffie–Hellman-Software-Curve25519 is a popular set of elliptic curve parameters and reference implementation by Daniel J. Bernstein in C. Bindings and alternative implementations are also available.
LINE messenger app has used the ECDH protocol for its "Letter Sealing" end-to-end encryption of all messa... | milkshake721/2.1M-wiki-STEM |
Kinamycin-Kinamycin-Kinamycins are a group of bacterial polyketide secondary metabolites containing a diazo group. Kinamycins are known for their cytotoxicity and are considered of interest for potential use in anti-cancer therapies. | milkshake721/2.1M-wiki-STEM |
Kinamycin-Synthesis-In 2006 and 2007 the means to totally and enantioselectively synthesize kinamycins C, F, and J were discovered. In 2010 a method was found to allow easier synthesis of these compounds in fewer steps, making research into their properties more feasible. | milkshake721/2.1M-wiki-STEM |
Tuberculosis hut-Tuberculosis hut-A tuberculosis hut or TB hut is a small wooden building that was used, mostly in the early twentieth century, by tuberculosis patients to recover in solitude. | milkshake721/2.1M-wiki-STEM |
Tuberculosis hut-Introduction-By the end of the 19th century, one out of four deaths in Europe was related to tuberculosis. The disease was often associated with bad hygienical circumstances and air pollution in the cities. As a result of improvements in housing and healthcare in the beginning of the 20th century, ther... | milkshake721/2.1M-wiki-STEM |
Tuberculosis hut-Locations-In the United Kingdom and the Netherlands, the houses could be found in groups near hospitals or sanatoria. In the Netherlands, these could also be found near health associations, on farmground just outside the town center or in gardens of individuals. Once a day a nurse visited the patient f... | milkshake721/2.1M-wiki-STEM |
Tuberculosis hut-Design-Tuberculosis huts existed in various forms, but in general they were simple premanufactured wooden buildings, that could be put together on the spot. They were white or green, with a lot of glass to allow the entering of as much sunlight as possible. On the front side, the houses were either ful... | milkshake721/2.1M-wiki-STEM |
Tuberculosis hut-Use-The original purpose of the hut was that the patient could recover by resting in solitude. The patient was supposed to stay in the hut night and day, and this stay could take months or even years. The huts were acquired for other purposes too, like a summerhouse or gazebo. At least the Irish playwr... | milkshake721/2.1M-wiki-STEM |
Sensor (character)-Sensor (character)-This page discusses the post-Zero Hour reboot version of the character. For the other versions, see Princess Projectra.
Jeka Wynzorr, codenamed Sensor, is a fictional character, a superheroine in the future of the DC Comics universe, and a member of the Legion of Super-Heroes.
She ... | milkshake721/2.1M-wiki-STEM |
Sensor (character)-Fictional character biography-Jeka Wynzorr was the princess of the planet Orando, a world populated by a ruling class of large, sentient, snakes and an underclass of (similarly sentient) small, raccoon-like mammals. She renounced this heritage to travel the galaxy, using her illusion-casting powers t... | milkshake721/2.1M-wiki-STEM |
Sensor (character)-Fictional character biography-Eventually, she arrived on Earth, and chose to join in the Legion tryouts. She was accepted, alongside Magno and Umbra, and chose the codename Sensor (a homage to "Sensor Girl", Princess Projectra's later codename in the pre-Zero Hour Legion) because of her powers as a m... | milkshake721/2.1M-wiki-STEM |
Sensor (character)-Fictional character biography-After most of the Lost members returned and the team formally reconstituted, she continued to serve with the team, until she was struck by a bolt of "Hypertaxis energy" while on Xanthu. This caused her to mutate out of control until Ra's al Ghul managed to stabilize her ... | milkshake721/2.1M-wiki-STEM |
Sensor (character)-Fictional character biography-In the new Legion continuity launched in 2004, Sensor has disappeared and in her place is a new version of the original, humanoid Princess Projectra. | milkshake721/2.1M-wiki-STEM |
Sensor (character)-Fictional character biography-Final Crisis In Final Crisis: Legion of 3 Worlds #2, the reboot Legion and the "threeboot" Legion are summoned to the 31st Century of New Earth. Sensor is among the Legionnaires rescued from limbo, as Princess Projectra is a member of the threeboot Legion, and Sensor Gir... | milkshake721/2.1M-wiki-STEM |
Sensor (character)-Name-Sensor's first name, Jeka, is a reference to her preboot counterpart's nickname "Jeckie" (short for "Projectra"). Her surname, Wynzorr, is a reference to the House of Windsor, the current British royal family. In keeping with this, her father is King Charlz, and she has a brother named Willum, a... | milkshake721/2.1M-wiki-STEM |
Sensor (character)-In other media-Sensor appears in Adventures in the DC Universe #10. | milkshake721/2.1M-wiki-STEM |
Drawing (manufacturing)-Drawing (manufacturing)-Drawing is a metalworking process that uses tensile forces to elongate metal, glass, or plastic. As the material is drawn (pulled), it stretches and becomes thinner, achieving a desired shape and thickness. Drawing is classified into two types: sheet metal drawing and wir... | milkshake721/2.1M-wiki-STEM |
Drawing (manufacturing)-Metal-Successful drawing depends on the flow and stretch of the material. Steels, copper alloys, and aluminium alloys are commonly drawn metals.In sheet metal drawing, as a die forms a shape from a flat sheet of metal (the "blank"), the material is forced to move and conform to the die. The flow... | milkshake721/2.1M-wiki-STEM |
Drawing (manufacturing)-Metal-Sheet metal drawing becomes deep drawing when the workpiece is longer than its diameter. It is common that the workpiece is also processed using other forming processes, such as piercing, ironing, necking, rolling, and beading. In shallow drawing, the depth of drawing is less than the smal... | milkshake721/2.1M-wiki-STEM |
Drawing (manufacturing)-Metal-Bar, tube, and wire drawing all work upon the same principle: the starting stock is drawn through a die to reduce its diameter and increase its length. Usually, the die is mounted on a draw bench. The starting end of the workpiece is narrowed or pointed to get the end through the die. The ... | milkshake721/2.1M-wiki-STEM |
Drawing (manufacturing)-Metal-The cold drawing process for steel bars and wire is as follows: Tube lubrication: The surface of the bar or tube is coated with a drawing lubricant such as phosphate or oil to aid cold drawing.
Push Pointing: Several inches of the lead ends of the bar or tube are reduced in size by swaging... | milkshake721/2.1M-wiki-STEM |
Drawing (manufacturing)-Metal-Cold drawing, process drawing: In this process, the material is drawn at room temperature. The reduced end of the bar or coil, which is smaller than the die opening, is passed through the die where it enters a gripping device of the drawing machine. The drawing machine pulls ("draws") the ... | milkshake721/2.1M-wiki-STEM |
Drawing (manufacturing)-Metal-Finished product: The drawn product, which is referred to as "cold drawn" or "cold finished", exhibits a bright or polished finish, increased mechanical properties, improved machining characteristics, and precise and uniform dimensional tolerances.
Multi-pass drawing: The cold drawing of c... | milkshake721/2.1M-wiki-STEM |
Drawing (manufacturing)-Glass-Similar drawing processes are applied in glassblowing and in making glass and plastic optical fiber. | milkshake721/2.1M-wiki-STEM |
Drawing (manufacturing)-Plastics-Plastic drawing, sometimes referred to as cold drawing, is the same process as used on metal bars, applied to plastics. Plastic drawing is primarily used in manufacturing plastic fibers. The process was discovered by Julian W. Hill in 1930 while trying to make fibers from an early polye... | milkshake721/2.1M-wiki-STEM |
Μ operator-Μ operator-In computability theory, the μ-operator, minimization operator, or unbounded search operator searches for the least natural number with a given property. Adding the μ-operator to the primitive recursive functions makes it possible to define all computable functions. | milkshake721/2.1M-wiki-STEM |
Μ operator-Definition-Suppose that R(y, x1, ..., xk) is a fixed (k+1)-ary relation on the natural numbers. The μ-operator "μy", in either the unbounded or bounded form, is a "number theoretic function" defined from the natural numbers to the natural numbers. However, "μy" contains a predicate over the natural numbers, ... | milkshake721/2.1M-wiki-STEM |
Μ operator-Definition-The bounded μ-operator appears earlier in Kleene (1952) Chapter IX Primitive Recursive Functions, §45 Predicates, prime factor representation as: The least such that if otherwise ,z. | milkshake721/2.1M-wiki-STEM |
Μ operator-Definition-" (p. 225)Stephen Kleene notes that any of the six inequality restrictions on the range of the variable y is permitted, i.e. y < z, y ≤ z, w < y < z, w < y ≤ z, w ≤ y < z and w ≤ y ≤ z. "When the indicated range contains no y such that R(y) [is "true"], the value of the "μy" expression is the card... | milkshake721/2.1M-wiki-STEM |
Μ operator-Definition-In Chapter XI §57 General Recursive Functions, Kleene defines the unbounded μ-operator over the variable y in the following manner, the least (natural number) such that R(y)} " (p. 279, where " (∃y) " means "there exists a y such that...")In this instance R itself, or its representing function, de... | milkshake721/2.1M-wiki-STEM |
Μ operator-Definition-For a given R(y) the unbounded μ-operator μyR(y) (note no requirement for "(Ey)" ) is a partial function. Kleene makes it as a total function instead (cf. p. 317): the least such that if otherwise .
The total version of the unbounded μ-operator is studied in higher-order reverse mathematics (Ko... | milkshake721/2.1M-wiki-STEM |
Μ operator-Properties-(i) In context of the primitive recursive functions, where the search variable y of the μ-operator is bounded, e.g. y < z in the formula below, if the predicate R is primitive recursive (Kleene Proof #E p. 228), then μyy<zR(y, x1, ..., xn) is a primitive recursive function.(ii) In the context of t... | milkshake721/2.1M-wiki-STEM |
Μ operator-Properties-Here (xi) means "for all xi" and Ey means "there exists at least one value of y such that..." (cf Kleene (1952) p. 279.)then the five primitive recursive operators plus the unbounded-but-total μ-operator give rise to what Kleene called the "general" recursive functions (i.e. total functions define... | milkshake721/2.1M-wiki-STEM |
Μ operator-Properties-(iii) In the context of the partial recursive functions: Suppose that the relation R holds if and only if a partial recursive function converges to zero. And suppose that that partial recursive function converges (to something, not necessarily zero) whenever μyR(y, x1, ..., xk) is defined and y is... | milkshake721/2.1M-wiki-STEM |
Μ operator-Properties-The μ-operator is used in the characterization of the computable functions as the μ recursive functions.
In constructive mathematics, the unbounded search operator is related to Markov's principle. | milkshake721/2.1M-wiki-STEM |
Μ operator-Examples-Example 1: The bounded μ-operator is a primitive recursive function In the following x represents the string xi, ..., xn.The bounded μ-operator can be expressed rather simply in terms of two primitive recursive functions (hereafter "prf") that also are used to define the CASE function—the product-of... | milkshake721/2.1M-wiki-STEM |
Μ operator-Examples-In a footnote Minsky does allow his operator to terminate when the function inside produces a match to the parameter "k"; this example is also useful because it shows another author's format: "For μt[φ(t) = k]" (p. 210)The reason for zero is that the unbounded operator μy will be defined in terms of... | milkshake721/2.1M-wiki-STEM |
Μ operator-Examples-The function μy produces as "output" a single natural number y = {0, 1, 2, 3, ...}. However, inside the operator one of a couple "situations" can appear: (a) a "number-theoretic function" χ that produces a single natural number, or (b) a "predicate" R that produces either {t = true, f = false}. (And... | milkshake721/2.1M-wiki-STEM |
Μ operator-Examples-Kleene's proof is informal and uses an example similar to the first example, but first he casts the μ-operator into a different form that uses the "product-of-terms" Π operating on function χ that yields a natural number n, which can be any natural number, and 0 in the instance when the u-operator's... | milkshake721/2.1M-wiki-STEM |
Μ operator-Examples-The definition recast with the Π-function: μyy<zχ(y) = (i): π(x, y) = Πs<yχ(x, s) (ii): φ(x) = τ(π(x, y), π(x, y' ), y) (iii): τ(z' , 0, y) = y ;τ(u, v, w) is undefined for u = 0 or v > 0.This is subtle. At first glance the equations seem to be using primitive recursion. But Kleene has not provided ... | milkshake721/2.1M-wiki-STEM |
Μ operator-Examples-τ(π(x, y), π(x, y' ), y), i.e.: τ(π(x, 0), π(x, 1), 0), τ(π(x, 1), π(x, 2), 1) τ(π(x, 2), π(x, 3), 2) τ(π(x, 3), π(x, 4), 3) ... until a match occurs at y=n and then: τ(z' , 0, y) = τ(z' , 0, n) = n and the μ-operator's search is done.For the example Kleene "...consider[s] any fixed values of (xi, .... | milkshake721/2.1M-wiki-STEM |
Μ operator-Examples-The following demonstration follows Minsky without the "peculiarity" mentioned in the footnote. The demonstration will use a "successor" counter machine model closely related to the Peano Axioms and the primitive recursive functions. The model consists of (i) a finite state machine with a TABLE of i... | milkshake721/2.1M-wiki-STEM |
Μ operator-Examples-Note †: The unbounded μ-operator will continue this attempt-to-match process ad infinitum or until a match occurs. Thus the "y" register must be unbounded -- it must be able to "hold" a number of arbitrary size. Unlike a "real" computer model, abstract machine models allow this. In the case of a bou... | milkshake721/2.1M-wiki-STEM |
Μ operator-Footnotes-Total function demonstration What is mandatory if the function is to be a total function is a demonstration by some other method (e.g. induction) that for each and every combination of values of its parameters xi some natural number y will satisfy the μ-operator so that the algorithm that represent... | milkshake721/2.1M-wiki-STEM |
Pulmonary artery catheter-Pulmonary artery catheter-A pulmonary artery catheter (PAC), also known as a Swan-Ganz catheter or right heart catheter, is a balloon-tipped catheter that is inserted into a pulmonary artery in a procedure known as pulmonary artery catheterization or right heart catheterization. Pulmonary arte... | milkshake721/2.1M-wiki-STEM |
Pulmonary artery catheter-Pulmonary artery catheter-The pulmonary artery catheter allows direct, simultaneous measurement of pressures in the right atrium, right ventricle, pulmonary artery, and the filling pressure (pulmonary wedge pressure) of the left atrium. The pulmonary artery catheter is frequently referred to a... | milkshake721/2.1M-wiki-STEM |
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