Background.

Internal market legislation of the European Union (EU) makes it mandatory for competent authorities in Member States to assist their counterparts abroad by providing them with information. Some legislation also requires communication between Member States and the European Commission (for example for the notification of national implementing measures of European Union law). IMI has been developed in order to facilitate this day-to-day exchange of information.  IMI was launched in February 2008. Development and maintenance has been funded by the programme Interoperability Solutions for European Public Administrations (ISA) since July 2010. ISA is the successor to the IDABC program, which initially funded IMI and came to an end on 31 December 2009.  IMI is one of the governance tools of the Single Market. Other such tools are Your Europe, Your Europe Advice, Solvit and the Points of Single Contact. IMI applies a "Privacy by Design" approach – integrating privacy and data protection compliance in all stages of the design of IMI – which has been developed in consultation with the European Data Protection Supervisor (EDPS). Administrative coordination is legally determinate and provided in a form of legal assistance between responsible institutions. In the case of the IMI system, administrative coordination encompasses some modes of administrative cooperation related to information provision (information exchange or data provision); shared administrative procedures execution (obligation to execute a single administrative decision/act by an institution from different Member States) and networking on a specific institutional structure (working of agencies or groups in formalised policy implementation processes)

Main Actors.

IMI has been rolled out in a decentralised way. Therefore, the practical implementation of IMI is the responsibility of the individual Member States. There are several actors that play a role in the IMI network.

Competent Authorities.

Competent authorities are the end users of IMI. They are public bodies that have been given the responsibility to deal with certain elements of application of internal market legislation. They can function on national, regional or local level.

IMI Coordinators.

There is one national IMI coordinator (NIMIC) per Member State, often located in a national ministry. Their task is to ensure the smooth operation of IMI in their country. IMI coordinators may delegate some of their responsibilities to additional coordinators who are in charge of, for example, one legislative area or a geographical region, depending on each Member State’s administrative structures.

European Commission.

The European Commission is responsible for maintenance and development of the tool, helpdesk services and training. It also manages and supports the network of IMI coordinators, promotes further expansion of IMI and reports on the functioning of the system.

IMI Workflows.

IMI offers a number of workflows to its users in order to facilitate different types of administrative cooperation across the Member states of the European Economic Area.

Information requests.

When a competent authority needs information from a counterpart abroad, it can send a request for information. This exchange mechanism uses lists of pre-translated questions and answers available in all EU languages. It is also possible to attach documents. Only the competent authorities that are directly involved in an information exchange have access to the content. A practical example of an information request is when a German teacher would like to continue his activities in Portugal. The Portuguese authority needs to verify the authenticity of his scanned diploma. It can then use IMI to send an information request to its partner authority in Germany. This authority can accept the request and send an answer back to the authority in Portugal. Due to the pre-translated question and answer sets, both authorities can communicate in their own language.

Notifications.

Notifications are based on one-to-many information exchanges where authorities can alert or notify one or more competent authorities and/or the European Commission. For example, the Services Directive requires that Member States alert each other of possible dangers to the health and safety of people or to the environment caused in the provision of services.

Information repositories.

IMI information repositories are databases storing specific information for certain policy areas. An example of such a repository is the directory of registers maintained by competent authorities throughout the European Economic Area. This directory is equipped with multilingual search functions. The content of a repository can be accessed either by a restricted group of competent authorities or by all IMI users.

Legal Framework.

IMI is used in all Member States of the European Economic Area for the administrative cooperation required by the Directive on the Recognition of Professional Qualifications (2005/36/EC), by the Directive on services in the internal market (2006/123/EC) and, on a pilot basis, by the Posting of Workers Directive. Since November 2012, it provides a repository for information on licence holders for the cross-border road transport of Euro cash and an IT platform for the problem solving network Solvit. IMI is being expanded to cover further legislative areas. For example, the Directive on Patients' Rights in Cross-border Healthcare. IMI aims to "become a flexible toolkit at the service of administrative cooperation, contributing to the improved governance of the Internal Market."  The IMI Regulation which came into force in December 2012 is an EU law establishing a comprehensive legal framework for IMI. It provides a complete set of rules for the processing of personal data in IMI and prescribes a method for future expansion of IMI to additional policy areas.

The Internal Market Information System (IMI) is an IT-based network that links public bodies in the European Economic Area. It was developed by the European Commission together with the Member States of the European Union to speed up cross-border administrative cooperation. IMI allows public administrations at national, regional and local level to identify their counterparts in other countries and to exchange information with them. Pre-translated questions and answers as well as machine translation make it possible for them to use their own language to communicate.

Systém pro výměnu informací o vnitřním trhu (IMI) je zabezpečená internetová aplikace, která spojuje veřejné orgány v Evropském hospodářském prostoru. Byla vyvinuta Evropskou komisí společně s členskými státy Evropské unie (odkaz na Wiki) za účelem urychlení administrativní spolupráce mezi jednotlivými zeměmi. IMI umožňuje orgánům ve veřejné správě na národní, regionální a obecní úrovni, aby rozpoznaly své protějšky v jiných zemích, a vyměňovaly si s nimi informace. Díky strojovému překladu a souboru předem přeložených standardizovaných otázek a odpovědí je možné pro komunikaci používat svůj vlastní jazyk.


Overview.

Upon returning from the Betty Ford Center in 1985 where he had undergone treatment for substance abuse, Osbourne was presented with a substantial quantity of music written by guitarist Jake E. Lee. After having been cheated out of his writing and publishing claims for Osbourne's previous album, 1983's "Bark at the Moon", Lee says this time he refused to contribute anything until he had a contract in front of him guaranteeing his writing credit and publishing rights. Much of this music would form the basis of the album. The album's lyrics were largely written by long-time Osbourne bassist and lyricist Bob Daisley. Daisley left the band prior to recording after having a disagreement with Osbourne, prompting the hiring of an unknown named Phil Soussan as his replacement. Daisley was not credited for his songwriting contributions on the initial 1986 pressing of the album, though this was corrected on subsequent pressings.  Future Y&T and Megadeth drummer Jimmy DeGrasso worked with Lee and Daisley on demos for the album, but this version of the band fell apart due to Osbourne's commitment to the Black Sabbath summer 1985 reunion concert at Live Aid; Castillo and Soussan ultimately replaced DeGrasso and Daisley once recording commenced.  The working title for the album was "Killer of Giants" after the album's song of the same name. Osbourne opted to change the title to "The Ultimate Sin" at the last minute. On 1 April 1986, a live performance promoting the album in Kansas City, Missouri, was filmed and released later that year as the home video "The Ultimate Ozzy". In 1987, after the various tours in support of the album had concluded, guitarist Lee was unexpectedly fired by Osbourne's wife and manager Sharon. The specific justification for Lee's dismissal remains unknown. Bassist Soussan also departed, with Bob Daisley once again entering the fold.

Reception.

At the time of its release, "The Ultimate Sin" was Osbourne's highest-charting studio album, as heavy metal was enjoying a surge in popularity during the mid-1980s. The RIAA awarded the album Platinum status on 14 May 1986, soon after its release; it was awarded Double Platinum status on 26 October 1994. The album sold over 2,000,000 copies worldwide. In the UK, it was the final of four Osbourne albums to attain Silver certification (60,000 units sold) by the British Phonographic Industry, achieving this in April 1986.  Despite its commercial success, Osbourne cites "The Ultimate Sin" as his least favorite solo album, saying "If there was ever an album I'd like to remix and do better, it would be "The Ultimate Sin"." Much of the vocalist's blame has been placed on the shoulders of producer Ron Nevison, saying "(He) didn't really do a great production job. The songs weren't bad, they were just put down weird. Everything felt and sounded the fucking same. There was no imagination."

Availability.

Despite being one of Osbourne's most successful solo albums, "The Ultimate Sin" was the only album among Osbourne's back catalog to not be reissued or remastered in 2002. Though no official reason was given, rumors circulated that an ongoing legal struggle with bassist/songwriter Phil Soussan over his song "Shot in the Dark" was responsible for the album's failure to be re-issued. Such reports are unfounded, as the only known legal proceeding between them was over an accounting issue that was believed to have been resolved in the early 1990s. The only available CD version of "The Ultimate Sin" remains the 1995 remastered version, which is missing four measures of music between the first chorus and second verse of "Shot in the Dark" found on the original CDs, vinyl copies, and cassettes.

The Ultimate Sin is the fourth studio album by British heavy metal vocalist Ozzy Osbourne. It was released on 22 February 1986, and was remastered and re-issued on compact disc on 22 August 1995. It is the second and last of Osbourne's albums to feature lead guitarist Jake E. Lee, and the first to feature drummer Randy Castillo and bassist Phil Soussan, who co-wrote the album's hit single "Shot in the Dark".

The Ultimate Sin je čtvrté sólové studiové album britského zpěváka Ozzyho Osbournea. Poprvé bylo vydáno v únoru 1986 pod hlavičkou vydavatelství Epic Records. Album produkoval Ron Nevison. V žebříčku Billboard 200 se album umístilo na šesté příčce.


Early life.

Conrad Bain was born in Lethbridge, Alberta, the son of Jean Agnes (née Young) and Stafford Harrison Bain, who was a wholesaler. He studied at the Banff School of Fine Arts before serving in the Canadian Army during World War II. He later studied in New York at the American Academy of Dramatic Arts, graduating in 1948; one of his classmates was comedian Don Rickles. Bain became a naturalized citizen of the United States in 1946.

Career.

After a stint at the Stratford Festival in Canada, Bain had further success as a stage actor in the 1956 revival of Eugene O'Neill's "The Iceman Cometh". "The New York Times" reviewer noted that his role was "especially well acted." Bain's Broadway theatre credits include "Candide", "Advise and Consent", "An Enemy of the People", "Uncle Vanya", and "On Borrowed Time". Off Broadway, he appeared in the original run of "Steambath". While doing stage work in New York City, Bain also found work on television, appearing in the cult supernatural soap opera "Dark Shadows" as the town innkeeper, Mr. Wells, during seasons 1 and 2. His character was killed off by werewolf Chris Jennings (Don Briscoe).  In the early 1960s, Bain was one of the principal organizers of the Actors Federal Credit Union. In an article in the Credit Union Times, he recalled that "the effort grew out of the fact that, at least in the world of financial services, actors and other performers got no respect and, more importantly, no access to credit."  He wrote:  Bain served as the credit union's first president.  In the early 1970s, Bain appeared in New York–based films like "Lovers and Other Strangers" and Woody Allen's "Bananas" before achieving national recognition for his work in television. He is best known for his roles as Dr. Arthur Harmon, Bea Arthur's title character's conservative nemesis, who married her best friend, Vivian, in "Maude" (1972–78). He then starred as Park Avenue millionaire Phillip Drummond in "Diff'rent Strokes" (1978–86), in which he adopted two African-American orphaned boys from Harlem, Willis and Arnold, to live with him and his daughter, Kimberly, and housekeeper, Mrs. Garrett. In 1979, he played Phillip Drummond in an episode of "The Facts of Life". In 1996, Bain reprised his role of Phillip Drummond along with Gary Coleman as Arnold Jackson on the series finale of "The Fresh Prince of Bel-Air".

Personal life.

Bain had two sons and a daughter with Monica Sloan (1923-2009), to whom he was married in 1945 until her death in 2009. On an episode of "Tattletales", a question was asked of Monica as to whether she would like to live to 150 years old if in good health. Her answer was "No, 86 is my destiny". Bain's identical twin brother was actor Bonar Bain (1923-2005), who once played Arnold Harmon, the twin brother of Conrad's "Maude" character, Arthur Harmon. Two of Bain's fellow co-stars, Gary Coleman and Rue McClanahan died within a week of each other in 2010.

Death.

Bain died from complications following a stroke on January 14, 2013, in Livermore, California, three weeks before his 90th birthday. His body was cremated.

Conrad Stafford Bain (February 4, 1923 – January 14, 2013) was a Canadian/American actor and comedian and character actor. His television credits include a leading role as Phillip Drummond in the sitcom "Diff'rent Strokes" and as Arthur Harmon on "Maude".

Conrad Stafford Bain (4. února 1923, Lethbridge, Alberta, Kanada – 14. ledna 2013, Livemore, Kalifornie, USA) byl kanadsko-americký herec. Proslul zejména svojí rolí Phillipa Drummonda v situační komedii "Diff'rent Strokes" a jako Dr. Arthur Harmon v "Maude". Má bratra-dvojče jménem Bonar Bain.


Regional auditions.

Auditions were held in Bratislava, Košice, Prague, Brno in the spring of 2009.

Divadlo.

In Divadlo where 124 contestants. The contestants first emerged on stage in groups of 9 or 10 but performed solo unaccompanied, and those who did not impress the judges were cut after the group finished their individual performances. 50 made it to the next round Dlouhá cesta. 24 contestants made it to the Semi-final.

Semi-final.

24 semifinalists were revealed in September when the show premiered on screen. Six boys and six girls from both countries competed for a spot in the top 12. In three Semifinals the guys performed on Saturday and the girls on Sunday night. The following Monday the lowest vote getter from each gender and country got eliminated (the viewers could vote for contestants from both countries). It means that three Czech and Slovak boys and three Czech and Slovak girls would make the finals.

Finals.

Twelve contestants reached the finals. TOP 12 consisted of 3 Slovak boys, 3 Czech boys, 3 Slovak girls and 3 Czech girls. The first single recorded by TOP 12 is called "Příběh Nekončí / Príbeh Nekončí" (The Story Doesn't End) and it was composed by judge Pavol Habera (music) and Slovak poet Daniel Hevier. Every final night had a different theme. The audience could vote for contestants from the very beginning of the show, voting ended during the result show the following day. There were double eliminations in first two final rounds, with only one contestant being eliminated once the candidates became the TOP 8. All gender and nationality quotas are abolished in the finals.

Top 6 – Karel Gott and Duets.

Mentor: Karel Gott

Top 4 – Movie Songs and Michal David.

Mentor: Michal David

Top 3 – Richard Müller and Unplugged.

Mentor: Richard Müller

Česko Slovenská SuperStar (English: Czech&Slovak SuperStar) is the joint Czech-Slovak version of Idol series' "Pop Idol" merged from "Česko hledá SuperStar" and "Slovensko hľadá SuperStar" which previous to that had three individual seasons each. The first season premiered in September 2009 with castings held in Prague, Brno, Bratislava and Košice. It is broadcast on two channels: «TV Nova» (Czech Republic) and «Markíza» (Slovakia) which have also been the broadcast stations for the individual seasons. Also both hosts have been their hosts countries before as have been three out of the four judges. To legitimate a fair chance for each country's contestants to reach the final, twelve of the contestants will compete split into genders and nationalities in the semifinals, guaranteeing a 50% share for each country in the top 12.

První řada Česko Slovenská SuperStar měla premiéru 6. září 2009 na TV Nova v Česku a na TV Markíza na Slovensku. Vysílala se každou neděli a pondělí ve 20:00. Do prvního ročníku "Česko Slovenské SuperStar" mezi porotce zasedli za Slovensko Pavol Habera a zpěvačka Dara Rolins a za Česko Ondřej Hejma a zpěvačka Marta Jandová. Moderátorské pozice se ujali Leoš Mareš a Adela Banášová.


Formal definition.

A language "L" is in RP if and only if there exists a probabilistic Turing machine "M", such that  Alternatively, RP can be defined using only deterministic Turing machines. A language "L" is in RP if and only if there exists a polynomial "p" and deterministic Turing machine "M", such that In this definition, the string "y" corresponds to the output of the random coin flips that the probabilistic Turing machine would have made. For some applications this definition is preferable since it does not mention probabilistic Turing machines.

Related complexity classes.

The definition of RP says that a YES answer is always right and that a NO answer might be wrong (because a question with the YES answer can be sometimes answered NO). In other words, while NO questions are always answered NO, you cannot trust the NO answer, it may be a mistaken answer to a YES question. The complexity class co-RP is similarly defined, except that NO is always right and YES might be wrong. In other words, it accepts all YES instances but can either accept or reject NO instances. The class BPP describes algorithms that can give incorrect answers on both YES and NO instances, and thus contains both RP and co-RP. The intersection of the sets RP and co-RP is called ZPP. Just as RP may be called R, some authors use the name co-R rather than co-RP.

Connection to P and NP.

P is a subset of RP, which is a subset of NP. Similarly, P is a subset of co-RP which is a subset of co-NP. It is not known whether these inclusions are strict. However, if the commonly believed conjecture P = BPP is true, then RP, co-RP, and P collapse (are all equal). Assuming in addition that P ≠ NP, this then implies that RP is strictly contained in NP. It is not known whether RP = co-RP, or whether RP is a subset of the intersection of NP and co-NP, though this would be implied by P = BPP.  A natural example of a problem in co-RP currently not known to be in P is Polynomial Identity Testing, the problem of deciding whether a given multivariate arithmetic expression over the integers is the zero-polynomial. For instance, is the zero-polynomial while  An alternative characterization of RP that is sometimes easier to use is the set of problems recognizable by nondeterministic Turing machines where the machine accepts if and only if at least some constant fraction of the computation paths, independent of the input size, accept. NP on the other hand, needs only one accepting path, which could constitute an exponentially small fraction of the paths. This characterization makes the fact that RP is a subset of NP obvious.

In computational complexity theory, randomized polynomial time (RP) is the complexity class of problems for which a probabilistic Turing machine exists with these properties:

V teorii složitosti je RP jednou z významných tříd složitosti. Obsahuje všechny problémy řešitelné pomocí randomizovaného Turingova stroje v polynomiálním množství času s tím, že na kladné odpovědi stroje je možno se spolehnout.


Německo na Letních olympijských hrách 1992

Archery.

In Germany's fifth archery competition, only one of the six archers qualified for the individual elimination rounds. He lost his first match, as did both the men's and women's teams.  Women's Individual Competition:  Men's Individual Competition:  Women's Team Competition:  Men's Team Competition:

Athletics.

Men's competition.

Men's 400 metres  Men's 800 metres  Men's 1,500 metres  Men's 5,000 metres  Men's 10,000 metres  Men's 4 × 400 m Relay  Men's Marathon  Men's 110 metres Hurdles  Men's 400 metres Hurdles  Men's 3,000 metres Steeplechase  Men's 20 km Walk  Men's 50 km Walk  Men's High Jump  Men's Long Jump  Men's triple jump  Men's Javelin Throw  Men's Hammer Throw  Men's Discus Throw  Men's Shot Put  Men's Decathlon

Women's competition.

Women's 100 metres  Women's 200 metres  Women's 400 metres  Women's 800 metres  Women's 1,500 metres  Women's 10,000 metres  Women's Marathon  Women's 100 metres Hurdles  Women's 400 metres Hurdles  Women's 4×100 metres Relay  Women's 4×400 metres Relay  Women's 10 km Walk  Women's High Jump  Women's Long Jump  Women's Shot Put  Women's Discus Throw  Women's Javelin Throw  Women's Heptathlon

Boxing.

Men's Light Flyweight (– 48 kg)

Cycling.

Seventeen cyclists, thirteen men and four women, represented Germany in 1992.

Diving.

Men's 3m Springboard  Men's 10m Platform  Women's 3m Springboard  Women's 10m Platform

Fencing.

20 fencers, 15 men and 5 women represented Germany in 1992.

Modern pentathlon.

Three male pentathletes represented Germany in 1992.

Sailing.

Men's Sailboard (Lechner A-390)  Women's 470 Class

Swimming.

Men's competition.

Men's 50 m Freestyle  Men's 100 m Freestyle  Men's 200 m Freestyle  Men's 400 m Freestyle  Men's 1500 m Freestyle  Men's 100 m Backstroke  Men's 200 m Backstroke  Men's 100 m Breaststroke  Men's 200 m Breaststroke  Men's 100 m Butterfly  Men's 200 m Butterfly  Men's 200 m Individual Medley  Men's 400 m Individual Medley  Men's 4 × 100 m Freestyle Relay  Men's 4 × 200 m Freestyle Relay  Men's 4 × 100 m Medley Relay

Women's competition.

Women's 50 m Freestyle  Women's 100 m Freestyle  Women's 200 m Freestyle  Women's 400 m Freestyle  Women's 800 m Freestyle  Women's 100 m Backstroke  Women's 200 m Backstroke  Women's 100 m Breaststroke  Women's 200 m Breaststroke  Women's 100 m Butterfly  Women's 200 m Butterfly  Women's 200 m Individual Medley  Women's 400 m Individual Medley  Women's 4 × 100 m Freestyle Relay  Women's 4 × 100 m Medley Relay

Synchronized swimming.

Two synchronized swimmers represented Germany in 1992.

Tennis.

Men's Singles Competition  Men's Doubles Competition  Women's Singles Competition

Germany competed at the 1992 Summer Olympics in Barcelona, Spain. It was the first time the European nation participated after German reunification in 1990. Previously, West Germany and East Germany had sent independent teams to the Games. 463 competitors, 300 men and 163 women, took part in 237 events in 26 sports.

Německo na Letních olympijských hrách v roce 1992 reprezentovala výprava 463 sportovců (300 mužů a 163 žen) v 26 sportech.


Introduction.

The first four hyperoperations are shown here, with tetration being considered the fourth in the series. The unary operation succession, defined as formula_5, is considered to be the zeroth operation.  Succession,, is the most basic operation; while addition () is a primary operation, for addition of natural numbers it can be thought of as a chained succession of successors of ; multiplication ) is also a primary operation, though for natural numbers it can analogously be thought of as a chained addition involving numbers of. Exponentiation can be thought of as a chained multiplication involving numbers of and tetration (formula_10) as a chained power involving numbers. Each of the operations above are defined by iterating the previous one; however, unlike the operations before it, tetration is not an elementary function.  The parameter is referred to as the base, while the parameter may be referred to as the height. In the original definition of tetration, the height parameter must be a natural number; for instance, it would be illogical to say "three raised to itself negative five times" or "four raised to itself one half of a time." However, just as addition, multiplication, and exponentiation can be defined in ways that allow for extensions to real and complex numbers, several attempts have been made to generalize tetration to negative numbers, real numbers, and complex numbers. One such way for doing so is using a recursive definition for tetration; for any positive real formula_11 and non-negative integer formula_12, we can define formula_13 recursively as: The recursive definition is equivalent to repeated exponentiation for natural heights; however, this definition allows for extensions to the other heights such as formula_15, formula_16, and formula_17 as well – many of these extensions are areas of active research.

Terminology.

There are many terms for tetration, each of which has some logic behind it, but some have not become commonly used for one reason or another. Here is a comparison of each term with its rationale and counter-rationale.  Owing in part to some shared terminology and similar notational symbolism, tetration is often confused with closely related functions and expressions. Here are a few related terms:  In the first two expressions is the "base", and the number of times appears is the "height" (add one for ). In the third expression, is the "height", but each of the bases is different.  Care must be taken when referring to iterated exponentials, as it is common to call expressions of this form iterated exponentiation, which is ambiguous, as this can either mean iterated powers or iterated exponentials.

Notation.

There are many different notation styles that can be used to express tetration. Some notations can also be used to describe other hyperoperations, while some are limited to tetration and have no immediate extension.  One notation above uses iterated exponential notation; this is defined in general as follows:  There are not as many notations for iterated exponentials, but here are a few:

Examples.

Because of the extremely fast growth of tetration, most values in the following table are too large to write in scientific notation. In these cases, iterated exponential notation is used to express them in base 10. The values containing a decimal point are approximate.

Properties.

Tetration has several properties that are similar to exponentiation, as well as properties that are specific to the operation and are lost or gained from exponentiation. Because exponentiation does not commute, the product and power rules do not have an analogue with tetration; the statements formula_20 and formula_21 are not necessarily true for all cases.  However, tetration does follow a different property, in which formula_22. This fact is most clearly shown using the recursive definition. From this property, a proof follows that formula_23, which allows for switching "b" and "c" in certain equations. The proof goes as follows:  This order is important because exponentiation is not associative, and evaluating the expression in the opposite order will lead to a different answer:  Evaluating the expression the left to right is considered less interesting; evaluating left to right, any expression formula_10 can be simplified to be formula_27. Because of this, the towers must be evaluated from right to left (or top to bottom). Computer programmers refer to this choice as right-associative.

Extensions.

Tetration can be extended in two different ways; in the equation formula_28, both the base and the height can be generalized using the definition and properties of tetration. Although the base and the height can be extended beyond the non-negative integers to different domains, including formula_29, complex functions such as formula_30, and heights of infinite, the more limited properties of tetration reduce the ability to extend tetration.

Extension of domain for bases.

Base zero.

The exponential formula_31 is not consistently defined. Thus, the tetrations formula_32 are not clearly defined by the formula given earlier. However, formula_33 is well defined, and exists:  Thus we could consistently define formula_35. This is analogous to defining formula_36.  Under this extension, formula_37, so the rule formula_38 from the original definition still holds.

Complex bases.

Since complex numbers can be raised to powers, tetration can be applied to "bases" of the form (where and are real). For example, in with, tetration is achieved by using the principal branch of the natural logarithm; using Euler's formula we get the relation:  This suggests a recursive definition for given any :  The following approximate values can be derived: Solving the inverse relation, as in the previous section, yields the expected and, with negative values of giving infinite results on the imaginary axis. Plotted in the complex plane, the entire sequence spirals to the limit, which could be interpreted as the value where is infinite.  Such tetration sequences have been studied since the time of Euler, but are poorly understood due to their chaotic behavior. Most published research historically has focused on the convergence of the infinitely iterated exponential function. Current research has greatly benefited by the advent of powerful computers with fractal and symbolic mathematics software. Much of what is known about tetration comes from general knowledge of complex dynamics and specific research of the exponential map.

Extensions of the domain for different heights.

Infinite heights.

Tetration can be extended to infinite heights; i.e., for certain and values in formula_41, there exists a well defined result for an infinite. This is because for bases within a certain interval, tetration converges to a finite value as the height tends to infinity. For example, formula_42 converges to 2, and can therefore be said to be equal to 2. The trend towards 2 can be seen by evaluating a small finite tower:  In general, the infinitely iterated exponential formula_44, defined as the limit of formula_45 as goes to infinity, converges for, roughly the interval from 0.066 to 1.44, a result shown by Leonhard Euler. The limit, should it exist, is a positive real solution of the equation. Thus,. The limit defining the infinite tetration of fails to converge for because the maximum of is.  This may be extended to complex numbers with the definition: where represents Lambert's W function.  As the limit (if existent, i.e. for ) must satisfy we see that is (the lower branch of) the inverse function of.

Negative heights.

We can use the recursive rule for tetration,  to prove formula_48:  Substituting −1 for gives  Smaller negative values cannot be well defined in this way. Substituting −2 for in the same equation gives  which is not well defined. They can, however, sometimes be considered sets.  For formula_52, any definition of formula_53 is consistent with the rule because

Real heights.

At this time there is no commonly accepted solution to the general problem of extending tetration to the real or complex values of. There have, however, been multiple approaches towards the issue, and different approaches are outlined below.  In general, the problem is finding — for any real — a "super-exponential function" formula_56 over real that satisfies  To find a more natural extension, one or more extra requirements are usually required. This is usually some collection of the following:  The fourth requirement differs from author to author, and between approaches. There are two main approaches to extending tetration to real heights; one is based on the "regularity" requirement, and one is based on the "differentiability" requirement. These two approaches seem to be so different that they may not be reconciled, as they produce results inconsistent with each other.  When formula_65 is defined for an interval of length one, the whole function easily follows for all.

Linear approximation for real heights.

A linear approximation (solution to the continuity requirement, approximation to the differentiability requirement) is given by:  hence: and so on. However, it is only piecewise differentiable; at integer values of the derivative is multiplied by formula_67. It is continuously differentiable for formula_68 if and only if formula_69. For example, using these methods formula_70 and formula_71  A main theorem in Hooshmand's paper states: Let formula_72. If formula_73 is continuous and satisfies the conditions:  then formula_75 is uniquely determined through the equation  where formula_80 denotes the fractional part of and formula_81 is the formula_82-iterated function of the function formula_83.  The proof is that the second through fourth conditions trivially imply that is a linear function on.  The linear approximation to natural tetration function formula_84 is continuously differentiable, but its second derivative does not exist at integer values of its argument. Hooshmand derived another uniqueness theorem for it which states:  If formula_85 is a continuous function that satisfies:  then formula_89. [Here formula_89 is Hooshmand's name for the linear approximation to the natural tetration function.]  The proof is much the same as before; the recursion equation ensures that formula_91 and then the convexity condition implies that formula_75 is linear on.  Therefore, the linear approximation to natural tetration is the only solution of the equation formula_93 and formula_94 which is convex on. All other sufficiently-differentiable solutions must have an inflection point on the interval.

Higher order approximations for real heights.

Beyond linear approximations, a quadratic approximation (to the differentiability requirement) is given by:  which is differentiable for all formula_62, but not twice differentiable. For example, formula_97 If formula_69 this is the same as the linear approximation.  Because of the way it is calculated, this function does not "cancel out", contrary to exponents, where formula_99. Namely,  Just as there is a quadratic approximation, cubic approximations and methods for generalizing to approximations of degree also exist, although they are much more unwieldy.

Complex heights.

It has now been proven that there exists a unique function which is a solution of the equation and satisfies the additional conditions that and approaches the fixed points of the logarithm (roughly ) as approaches and that is holomorphic in the whole complex -plane, except the part of the real axis at. This proof confirms a previous conjecture. The construction of such a function was originally demonstrated by Kneser in 1950. The complex map of this function is shown in the figure at right. The proof also works for other bases besides "e", as long as the base is bigger than formula_101. Subsequent work extended the construction to all complex bases. The complex double precision approximation of this function is available online. The requirement of the tetration being holomorphic is important for its uniqueness. Many functions can be constructed as  where and are real sequences which decay fast enough to provide the convergence of the series, at least at moderate values of.  The function satisfies the tetration equations,, and if and approach 0 fast enough it will be analytic on a neighborhood of the positive real axis. However, if some elements of or are not zero, then function has multitudes of additional singularities and cutlines in the complex plane, due to the exponential growth of sin and cos along the imaginary axis; the smaller the coefficients and are, the further away these singularities are from the real axis.  The extension of tetration into the complex plane is thus essential for the uniqueness; the real-analytic tetration is not unique.

Non-elementary recursiveness.

Tetration (restricted to formula_103) is not an elementary recursive function. One can prove by induction that for every elementary recursive function, there is a constant such that We denote the right hand side by formula_105. Suppose on the contrary that tetration is elementary recursive. formula_106 is also elementary recursive. By the above inequality, there is a constant such that formula_107. By letting formula_108, we have that formula_109, a contradiction.

Inverse operations.

Exponentiation has two inverse operations; roots and logarithms. Analogously, the inverses of tetration are often called the super-root, and the "super-logarithm" (In fact, all hyperoperations greater than or equal to 3 have analogous inverses); e.g., in the function formula_110, the two inverses are the cube super-root of and the super logarithm base of.

Super-root.

The super-root is the inverse operation of tetration with respect to the base: if formula_111, then is an th super root of (formula_112 or formula_113).  For example,  so 2 is the 4th super-root of 65,536.

Square super-root.

The "2nd-order super-root", "square super-root", or "super square root" has two equivalent notations, formula_115 and formula_116. It is the inverse of formula_117 and can be represented with the Lambert W function:  The function also illustrates the reflective nature of the root and logarithm functions as the equation below only holds true when formula_119:  Like square roots, the square super-root of may not have a single solution. Unlike square roots, determining the number of square super-roots of may be difficult. In general, if formula_121, then has two positive square super-roots between 0 and 1; and if formula_122, then has one positive square super-root greater than 1. If is positive and less than formula_123 it doesn't have any real square super-roots, but the formula given above yields countably infinitely many complex ones for any finite not equal to 1. The function has been used to determine the size of data clusters.  At formula_124 :  formula_125

Other super-roots.

For each integer, the function is defined and increasing for, and, so that the th super-root of, formula_112, exists for.  However, if the linear approximation above is used, then formula_127 if, so formula_128 cannot exist.  In the same way as the square super-root, terminology for other super roots can be based on the normal roots: "cube super-roots" can be expressed as formula_129; the "4th super-root" can be expressed as formula_130; and the "th super-root" is formula_112. Note that formula_112 may not be uniquely defined, because there may be more than one root. For example, has a single (real) super-root if is "odd", and up to two if is "even".  Just as with the extension of tetration to infinite heights, the super-root can be extended to, being well-defined if. Note that formula_133 and thus that formula_134. Therefore, when it is well defined, formula_135 and, unlike normal tetration, is an elementary function. For example, formula_136.  It follows from the Gelfond–Schneider theorem that super-root formula_137 for any positive integer is either integer or transcendental, and formula_138 is either integer or irrational. It is still an open question whether irrational super-roots are transcendental in the latter case.

Super-logarithm.

Once a continuous increasing (in ) definition of tetration,, is selected, the corresponding super-logarithm formula_139 or formula_140 is defined for all real numbers, and.  The function satisfies:

Open questions.

Other than the problems with the extensions of tetration, there are several open questions concerning tetration, particularly when concerning the relations between number systems such as integers and irrational numbers:

In mathematics, tetration (or hyper-4) is an operation based on iterated, or repeated, exponentiation. It is the next hyperoperation after exponentiation, but before pentation. The word was coined by Reuben Louis Goodstein from tetra- (four) and iteration.

Tetrace je matematická operace, která je jakýmsi rozšířením umocňování. Zatímco násobení je vlastně opakované sčítání a umocňování opakované násobení, tetrace je opakované umocňování.


Career.

Early career.

In his youth years on Curaçao, Anita started to play football at local team CVV Willemstad. After he and his family moved to the Netherlands in 1997, he started playing for amateur side VV Maarssen. Later he was scouted by Ajax and he joined their world-famous youth program in 1999.

Ajax.

As of the 2005–06 season he was promoted to the first team of Ajax and on 19 March 2006 he made his Eredivisie debut in a 3–2 away loss against FC Groningen. That year he also participated in the KNVB Cup semi-final against Roda JC. Ajax went on to win the KNVB Cup that year.  The following season he added another match to his Eredivisie total, bringing it to two appearances. He also made his European debut in the 3–0 away loss against Werder Bremen in the UEFA Cup. In the 2007–08 season Anita did not play any matches for the first team, but in the 2008–09 he returned to the first team squad. He played a total of 16 matches, mainly as a replacement for injured or suspended teammates.  In the 2009–10 season Anita became a first team regular under new head coach Martin Jol. Because of a suspension of Urby Emanuelson and a long-time injury of Thimothée Atouba, Anita was posted as a left back. Jol was satisfied with his performance and continued to let him play as a left back for most of the season. He also played in the KNVB Cup final against Feyenoord and won his second KNVB Cup. His good performances also led to his debut at the Dutch national team. However, he was not selected for the squad for 2010 FIFA World Cup by coach Bert van Marwijk.  The 2010–11 season started as the previous season ended with Anita playing most matches as a left back. Ajax managed to qualify for the Group stage of the UEFA Champions League for the first time since 2005–06. Because of the poor team performances during the first half of the season, Martin Jol left Ajax in December 2010 and was replaced by Frank de Boer. He believed Anita should play as a defensive midfielder rather than left back. As a consequence, Anita had to compete with other midfielders for his place in the starting line-up. He successfully did so and managed to play most matches. The season ended with two finals against FC Twente. Twente won the KNVB Cup final with 3–2, but Ajax secured the Eredivisie championship at the last matchday by winning 3–1. It was the first league title of Anita. In April 2011 it was announced that Anita signed a new contract at Ajax, binding him to the club until 2014.

Newcastle United.

After Ajax had rejected Newcastle's first bid, the clubs eventually came to an agreement after the second offer from the English club of €8.5 million (£6.7 million). He signed a five-year deal and took the number 8 shirt, last worn by Danny Guthrie. Anita made his debut on 18 August 2012, replacing Yohan Cabaye, as the Magpies defeated Tottenham Hotspur 2–1 on the opening day of the Premier League season. He made his first start for the club against Atromitos F.C. in the Europa League on 23 August, the game finished 1–1. He scored his first goal for Newcastle in a Europa League fixture against Club Brugge at the Jan Breydel Stadium on 8 November.  Anita scored his first league goal for Newcastle on 1 March 2014 in a comfortable 4–1 win away from home over Hull City, but the match was marred by controversy as manager Alan Pardew headbutted Hull player David Meyler. After Siem de Jong signed for Newcastle on 1 July 2014, the player revealed that both Anita and Tim Krul were very influential in persuading him to leave Ajax and join Tyneside.  Under new Manager Steve McClaren, Anita became a regular in centre midfield during the 2015–16 season. In the first half of the season he played eighteen of the first nineteen games before on 28 December 2015 he picked up a hamstring injury against West Bromwich Albion, which would keep him out for a lengthy period, the season saw Newcastle relegated to The Championship with Anita playing 28 games.  He was played mostly at right back during the 2016–17 season, playing 31 games in all competitions, which saw Anita win the EFL Championship 2016–17 season and gain promotion to the Premier League On 9 June 2017, Anita left Newcastle when his contract expired at the end of the 2016–17 season.

Leeds United.

On 6 July 2017, Anita signed a three-year contract with Leeds United. On 6 August 2017, Anita made his Leeds United debut at left back coming on as a first half substitute for the injured Gaetano Berardi in the 3–2 victory against Bolton Wanderers. His first start came in the following game on 9 August 2017 in the League Cup 4–1 victory against Port Vale.  Due to injuries to regular fullbacks including Luke Ayling, Gaetano Berardi and Laurens De Bock during the season, Anita was used as right-back or left-back during the 2017–18 season, with Talksport's Billy Hawkins opining that Anita and De Bock "made appearances on the left of defence to little success".  On 26 July 2018, Anita was not given a shirt number for Leeds for the upcoming 2018–19 season.  On 2 September 2019, Anita's contract with Leeds was terminated by mutual consent.

Willem II loan.

On 31 August 2018, he joined Dutch club Willem II on a season-long loan. He made his debut on 1 September 2018 in a defeat to PSV Eindhoven. In total, he made 27 appearances for Willem II in Eredivisie.  After returning to Leeds after the end of his loan spell, Anita held talks with Super League Greece side Panathinaikos over a move.

International.

Anita performed in the 2005 FIFA U-17 World Championship in Peru for the Netherlands U-17, where the Dutch came third. He has also represented the Netherlands U-19 and Netherlands U-21 teams.  On 26 May 2010, Anita made his full international debut for the Netherlands, coming on as a substitute in a 2–1 friendly victory over Mexico. He made his first appearance in the starting eleven in a 1–1 friendly draw with Ukraine on 12 August 2010, being substituted at half-time by PSV Eindhoven defender Erik Pieters.

2010 World Cup.

Anita was included in the preliminary squad for the 2010 FIFA World Cup in South Africa. However, on 27 May 2010, Netherlands manager Bert van Marwijk announced that the player would not be part of the final squad of 23 participating in the competition.

Honours.

Club.

Ajax  Newcastle United

International.

Netherlands under-18

Vurnon San Benito Anita (born 4 April 1989) is a Dutch professional footballer who plays as a defensive midfielder or full-back. He has also represented Netherlands at international level.

Vurnon San Benito Anita (* 4. dubna 1989, Willemstad, Nizozemské Antily) je nizozemský fotbalový obránce původem z Curaçaa v Nizozemských Antilách, který hraje v anglickém klubu Newcastle United.


History.

The Soviet and American teams.

The Soviet Union entered the Lake Placid games as the heavy favorite, having won four consecutive gold medals dating back to the 1964 games. In the four Olympics following their 1960 bronze-medal finish at Squaw Valley, Soviet teams had gone 27–1–1 (wins-losses-ties) and outscored their opponents 175–44. In head-to-head matchups against the United States, the cumulative score over that period was 28–7. The Soviet team had not lost a game in Olympic play since 1968.  The Soviets were led by legendary players in world ice hockey, such as Boris Mikhailov (a top line right winger and team captain), Vladislav Tretiak (the consensus best goaltender in the world at the time), the speedy and skilled Valeri Kharlamov, and talented, dynamic players such as defenseman Viacheslav Fetisov and forwards Vladimir Krutov and Sergei Makarov. From that team, Tretiak, Kharlamov, Makarov, and Fetisov were eventually enshrined in the Hockey Hall of Fame. Many of the Soviet players had gained attention in the Summit Series eight years before and, in contrast to the American players, were "de facto" professionals with long histories of international play, employed by industrial firms or military organizations for the sole purpose of playing hockey on their organization's team. Western nations protested the Soviet Union's use of full-time athletes, as they were forced to use amateur (mainly college) players due to the International Olympic Committee's (IOC) amateur-only policy. The situation even led to Canadian withdrawal from the 1972 and 1976 Olympics, but the IOC did not change the rules until the late 1980s.  U.S. head coach Herb Brooks held tryouts in Colorado Springs in the summer of 1979. Of the 20 players who eventually made the final Olympic roster, Buzz Schneider was the only one returning from the 1976 Olympic team. Nine players had played under Brooks at the University of Minnesota (including Rob McClanahan, Mike Ramsey, and Phil Verchota), while four more were from Boston University (Dave Silk, Jack O'Callahan, goaltender Jim Craig, and team captain Mike Eruzione). As Boston University and Minnesota were perennial rivals in college hockey (for instance, they had faced off in a bitter NCAA national semifinal in 1976), Brooks' selection process was a 300-question psychological test that would give him insight on how every player would react under stress; anyone who refused to take the test would automatically fail. Brooks had to select from 68 players who started the tryout.  The average age of the U.S. team was 21 years old, making it the youngest team in U.S. history to play in the Olympics (in addition to being the youngest team in the 1980 Olympic tournament), but Brooks had selected carefully and he knew the limits of every player. As forward John Harrington said, "He knew exactly where to quit. He'd push you right to the limit where you were ready to say, 'I've had it, I'm throwing it in' — and then he'd back off." Brooks continued the organization by campaigning for the players' selection of Eruzione as the captain, and Craig had been the goalie for him in the 1979 World Championship tournament. Assistant coach Craig Patrick had played with Brooks on the 1967 U.S. national team.  The Soviet and American teams were natural rivals due to the decades-old Cold War. In addition, President Jimmy Carter was at the time considering a U.S. boycott of the 1980 Summer Olympics, to be held in Moscow, in protest of the December 1979 Soviet invasion of Afghanistan. On February 9, the same day the American and Soviet teams met in an exhibition game in New York City, U.S. Secretary of State Cyrus Vance denounced the impending Moscow games at an IOC meeting. President Carter eventually decided in favor of the boycott.

Exhibitions.

In exhibitions that year, Soviet club teams went 5–3–1 against National Hockey League (NHL) teams and, a year earlier, the Soviet national team had routed the NHL All-Stars 6–0 to win the Challenge Cup. In 1979–80, virtually all the top North American players were Canadians, although the number of U.S.-born professional players had been on the rise throughout the 1970s. The 1980 U.S. Olympic team featured several young players who were regarded as highly promising, and some had signed contracts to play in the NHL immediately after the tournament.  In September 1979, before the Olympics, the American team started exhibition play. They played a total of 61 games in five months against teams from Europe and the United States. Through these games, Brooks instilled a European style of play in his team, emphasizing wide-open play with sufficient body contact. He believed it would be the only way for the Americans to compete with the Soviets. From the start of the exhibition matches, he conducted the team through skating wind sprints consisting of end line to blue line and back, then end line to red line and back, then end line to far blue line and back, and finally end line all the way down and back. Some of the players took to calling these "Herbies". On September 17, the team played to a 3–3 tie in Norway. Believing the team wasn't putting up sufficient effort, an angry Brooks had them skate Herbies after the game and, after a while, arena custodians turned the lights off and the Herbies continued in the dark. The two teams played again the next night, with the US winning handily 9-0. Near the end of the exhibition season, Brooks, because of subpar play, threatened to cut Eruzione (the captain) from the team and replace Craig with Steve Janaszak as the starting goaltender, although he had supported them throughout.  In their last exhibition game, against the Soviets at Madison Square Garden on Saturday, February 9, the Americans were crushed 10–3. Soviet head coach Viktor Tikhonov later said that this victory "turned out to be a very big problem" by causing the Soviets to underestimate the American team. The game was also costly for the Americans off-ice, as defenseman Jack O'Callahan pulled a ligament in his knee; however, Brooks kept O'Callahan on the roster, which meant the U.S. was virtually playing with only 19 players throughout the tournament. O'Callahan eventually returned for the game against the Soviets, playing limited minutes.

Olympic group play.

In Olympic group play, the Americans surprised many observers with their physical, cohesive play. In their first game on February 12 against favored Sweden, Team USA earned a dramatic 2–2 draw by scoring with 27 seconds left after pulling goalie Jim Craig for an extra attacker. Then came a stunning 7–3 victory over Czechoslovakia, who were a favorite for the silver medal. With its two toughest games in the group phase out of the way, the U.S. team reeled off three more wins, beating Norway 5–1, Romania 7–2, and West Germany 4–2 to go 4–0–1 and advance to the medal round from its group, along with Sweden.  In the other group, the Soviets stormed through their opposition undefeated, often by grossly lopsided scores. They defeated Japan 16–0, the Netherlands 17–4, Poland 8–1, Finland 4–2, and Canada 6–4 to easily qualify for the next round, although both the Finns and the Canadians gave the Soviets tough games for two periods. In the end, the Soviet Union and Finland advanced from their group.

Game summary.

Prior to the Friday game, ABC requested that it be rescheduled from 5:00 p.m. to 8:00 p.m. EST, so that it could be broadcast live in primetime. However, the IIHF declined the request after the Soviets complained that it would cause the game to air at 4 a.m. Moscow Time, as opposed to 1 a.m. As a result, ABC decided not to broadcast the game live for the U.S. audience and tape delayed it for broadcast during its primetime block of Olympics coverage. Before the game aired, ABC's Olympics host Jim McKay openly stated that the game had already occurred but that they had promised not to spoil its results. In order to accommodate coverage of the men's slalom competition, portions of the game were also edited for time. ABC's 8 to 8:30 p.m. timeslot was instead, devoted to the animated special "". To this day, some who watched the game on television still believe that it was live.  With a capacity of 8,500, the Field House was packed. The home crowd waved U.S. flags and sang patriotic songs such as "God Bless America". Before the game, Brooks read his players a statement he had written out on a piece of paper, telling them that "You were born to be a player. You were meant to be here. This moment is yours." Brooks believed they could win and later said, "The Russians were ready to cut their own throats. But we had to get to the point to be ready to pick up the knife and hand it to them. So the morning of the game I called the team together and told them, 'It's meant to be. This is your moment and it's going to happen.' It's kind of corny and I could see them thinking, 'Here goes Herb again...' But I believed it."

First period.

As in several previous games, the U.S. team fell behind early. Vladimir Krutov deflected a slap shot by Alexei Kasatonov past U.S. goaltender Jim Craig at the 9:12 mark to give the Soviets a 1–0 lead. At the 14:03 mark, Buzz Schneider scored for the United States on a 50-foot shot from the left boards to tie the game. The Soviets struck again when Sergei Makarov scored with 17:34 gone. With his team down 2–1, Craig improved his play, turning away many Soviet shots before the U.S. team had another shot on goal.  In the waning seconds of the first period, Dave Christian fired a slap shot on Tretiak from away. The Soviet goalie saved the shot but misplayed the rebound, which bounced out some in front of him. Mark Johnson sliced between the two defenders, found the loose puck, and fired it past a diving Tretiak to tie the score with one second left in the period. The first period ended with the game tied 2–2.

Second period.

Tikhonov replaced Tretiak with backup goaltender Vladimir Myshkin immediately after Johnson's tying goal, a move that shocked players on both teams. Tikhonov later identified this as the "turning point of the game" and called it "the biggest mistake of my career". Years later, when Johnson asked Viacheslav Fetisov, now an NHL teammate, about the move, Fetisov responded with "Coach crazy." Myshkin allowed no goals in the second period. The Soviets dominated play in the second period, outshooting the Americans 12–2, but scored only once, on a power play goal by Aleksandr Maltsev 2:18 into play. After two periods the Soviet Union led, 3–2.

Third period.

Vladimir Krutov was sent to the penalty box at the 6:47 mark of the third period for high-sticking. The Americans, who had managed only two shots on Myshkin in 27 minutes, had a power play and a rare offensive opportunity. Myshkin stopped a Mike Ramsey shot, then U.S. team captain Mike Eruzione fired a shot wide. Late in the power play, Dave Silk was advancing into the Soviet zone when Valeri Vasiliev knocked him to the ice. The puck slid to Mark Johnson. Johnson fired off a shot that went under Myshkin and into the net at the 8:39 mark, as the power play was ending, tying the game at 3. Only a couple of shifts later, Mark Pavelich passed to Eruzione, who was left undefended in the high slot. Eruzione, who had just come onto the ice, fired a shot past Myshkin, who was screened by Vasili Pervukhin. This goal gave Team USA a 4–3 lead, its first of the game, with exactly 10 minutes remaining to play.  The Soviets, trailing for the first time in the game, attacked ferociously. Moments after Eruzione's goal, Maltsev fired a shot which ricocheted off the right goal post. As the minutes wound down, Brooks kept repeating to his players, "Play your game. Play your game." Instead of going into a defensive crouch, the United States continued to play offense, even getting off a few more shots on goal. The Soviets began to shoot wildly, and Sergei Starikov admitted that "we were panicking." As the clock ticked down below a minute, the Soviets got the puck back into the American zone, and Mikhailov passed to Vladimir Petrov, who shot wide. The Americans fully expected Tikhonov to pull the goalie in the waning seconds. To their surprise, Myshkin stayed in the game. Starikov later explained that "We never did six-on-five," not even in practice, because "Tikhonov just didn't believe in it." Craig kicked away a Petrov slap shot with 33 seconds left. Kharlamov fired the puck back in as the clock ticked below 20 seconds. A wild scramble for the puck ensued, ending when Johnson found it and passed it to Ken Morrow. As the U.S. team tried to clear the zone (move the puck over the blue line, which they did with seven seconds remaining), the crowd began to count down the seconds left. Sportscaster Al Michaels, who was calling the game on ABC along with former Montreal Canadiens goaltender Ken Dryden, picked up on the countdown in his broadcast, and delivered his famous call:  As his team ran all over the ice in celebration, Herb Brooks sprinted back to the locker room and cried. In the locker room afterwards, players spontaneously broke into a chorus of "God Bless America". After the gold medal-securing victory over Finland, they received a congratulatory phone call from President Jimmy Carter.  During the broadcast wrap-up after the game, ABC Olympic sports anchor Jim McKay compared the American victory over the Soviet professionals to a group of Canadian college football players defeating the Pittsburgh Steelers (the recent Super Bowl champions and at the height of their dynasty).

American aftermath.

Gold medal.

The United States did not win the gold medal by defeating the USSR. In 1980, the medal round was a round-robin, not a single elimination format as it is today. Under Olympic rules at the time, the group game with Sweden was counted along with the medal round games versus the Soviet Union and Finland, so it was mathematically possible for the United States to finish anywhere from first to third.  Needing to win to secure the gold medal, Team USA came back from a 2–1 third-period deficit to defeat Finland 4–2. According to Mike Eruzione, coming into the dressing room for the second intermission, Brooks turned to his players, looked at them, and said, "If you lose this game, you'll take it to your fucking graves." He then walked towards the locker room door, paused, looked over his shoulder, and said to them again, "Your fucking graves."  At the time, the players ascended a podium to receive their medals and then lined up on the ice for the playing of the national anthem, as the podium was only meant to accommodate one person. Only the team captains remained on the podium for the duration. After the completion of the anthem, Eruzione motioned for his teammates to join him on the podium. Today, podiums are not used for ice hockey; the teams line up on their respective blue lines after the final game.  The cover of the March 3, 1980, issue of "Sports Illustrated" was a photograph by Heinz Kluetmeier of the American players celebrating and waving an American flag; it did not feature any explanatory captions or headlines, because, as Kluetmeier put it, "It didn't need it. Everyone in America knew what happened." The U.S. team also received the magazine's "Sportsmen of the Year" award, and were also named Athlete of the Year by the Associated Press and ABC's "Wide World of Sports". In 2004, ESPN, as part of its 25th anniversary, declared the Miracle on Ice to be the top sports headline moment and game of the period 1979–2004. The victory was voted the greatest sports moment of the 20th century by "Sports Illustrated".

After the 1980 Winter Olympics.

At the 1981 Canada Cup, the United States, with seven players from their 1980 Olympic team, again faced the Soviet Union. The Soviets took the opening round encounter 4–1 in Edmonton. At the 1982 World Championship in Finland, with Mike Ramsey, Mark Johnson, Buzz Schneider, and John Harrington, the Americans again met the Soviets, but once again the U.S. lost, 8–4.

Later careers.

Of the 20 players on Team USA, 13 eventually played in the NHL. Five of them played over 500 NHL games, and three of them played over 1,000 NHL games.

Soviet aftermath.

Silver medal.

In the Soviet locker room, Tikhonov singled out first-line players Tretiak, Kharlamov, Petrov, and Mikhailov, and told each of them, "This is your loss!" Two days after the Miracle on Ice, the Soviet team defeated Sweden 9–2, winning the silver medal. The Soviet players were so upset at their loss that they did not turn in their silver medals to get their names inscribed on them, as is custom. The result stunned the Soviet Union and its news media.

After the 1980 Winter Olympics.

Despite the loss, the USSR remained the pre-eminent power in Olympic hockey until its 1991 break-up. The Soviet team did not lose a World Championship game until 1985 and did not lose to the United States again until 1991. Throughout the 1980s, NHL teams continued to draft Soviet players in hopes of enticing them to eventually play in North America. Soviet emigrant Victor Nechayev made a brief appearance with the Los Angeles Kings in the 1982–83 season and, during the 1988–89 season, the Soviet Ice Hockey Federation agreed to let veteran Sergei Pryakhin join the Calgary Flames.

NHL careers.

In the 1989-90 season, Soviet authorities permitted six more 1980 Olympians – Helmuts Balderis, Viacheslav Fetisov, Alexei Kasatonov, Vladimir Krutov, Sergei Makarov, and Sergei Starikov – to join NHL clubs, but only after they agreed to play in their final World Championship (where they won gold). Makarov won the Calder Memorial Trophy as NHL Rookie of the Year in 1989–90, becoming the oldest player to win that award. Fetisov was a teammate of Mike Ramsey on the 1995 Detroit Red Wings team that lost the Stanley Cup Final to Neal Broten and the New Jersey Devils. Fetisov completed his career by winning Cups with the Red Wings in 1997 and 1998; the first Cup win also made Fetisov a member of the Triple Gold Club, consisting of individuals who have won a Stanley Cup plus gold medals at the Olympics and World Championships.

Notable rematches.

The U.S. and the Soviet Union next met at the Winter Olympics in 1988. As in 1980, the Soviets were represented by their star-studded veterans, while the Americans fielded a team of college players. The Soviets won the encounter 7–5 and won the gold medal, while the U.S. placed seventh.  The two teams met again at the 1992 Olympics in a semi-final match. There, the Unified Team (the successor to the Soviet Union) won 5–2. While some stars had left the Soviet Union to play in the NHL, the Unified Team still boasted many veterans from their domestic professional league, while the Americans were represented primarily by college players. The Unified Team eventually won the gold medal, while the U.S. placed fourth.  The U.S., coached by Herb Brooks, and Russia, coached by Viacheslav Fetisov, met twice in the 2002 Winter Olympics in Salt Lake City, which included a 2–2 round-robin draw and a 3–2 semi-final win for the Americans. The semi-final match was played 22 years to the day after the "Miracle on Ice" game. The U.S. eventually won silver, while Russia won bronze.  The U.S. and Russia played each other in a round-robin game at the 2014 Winter Olympics in Sochi. The game was tied 2–2 after overtime before the Americans prevailed in an eight-round shootout, with T.J. Oshie scoring on 4 of 6 attempts for the United States. The match has been dubbed by some as the "Marathon on Ice" due to its length. Both teams, however, failed to medal; the Americans finished fourth (losing to Canada in the semifinals), while the Russians placed fifth (losing to Finland in the quarterfinals).

Popular culture.

A made-for-TV movie "Miracle on Ice", starring Karl Malden as Brooks and Steve Guttenberg as Craig, aired on ABC television in 1981. It incorporated actual game footage and original commentary from the 1980 Winter Games.  The final episode of "People's Century", "Fast Forward", featured the Miracle on Ice as illustrating the Cold War.  The documentary film "Do You Believe in Miracles?", narrated by Liev Schreiber, premiered on HBO in 2001 and was subsequently released on home video.  In 2004, Walt Disney Pictures released the film "Miracle", directed by Gavin O'Connor and starring Kurt Russell as Brooks. Al Michaels recreated his commentary for most of the games. The final ten seconds, however, and his "Do you believe in miracles? YES!" call, were from the original broadcast and used in the film since the filmmakers felt that they could not ask him to recreate the emotion he felt at that moment. The film was dedicated to Brooks, who died shortly after principal photography completed.  The documentary "Of Miracles And Men", which was directed by Jonathan Hock, premiered on ESPN in 2015 as part of the channel's "30 for 30" series. The story of the 1980 matchup is told from the Soviet perspective.

Team rosters.

Soviet Union.

<nowiki>* Starting line up</nowiki

The "Miracle on Ice" was a medal-round game during the men's ice hockey tournament at the 1980 Winter Olympics in Lake Placid, New York, played between the hosting United States and the four-time defending gold medalists, the Soviet Union.

Zázrak na ledě () je termín dodnes používaný pro utkání, které se odehrálo dne 22. února 1980 v rámci Zimních olympijských her v Lake Placid a v němž se utkaly hokejové reprezentace USA a Sovětského svazu.


History.

Corrugated (also called pleated) paper was patented in England in 1856, and used as a liner for tall hats, but corrugated boxboard was not patented and used as a shipping material until 20 December 1871. The patent was issued to Albert Jones of New York City for single-sided (single-face) corrugated board. Jones used the corrugated board for wrapping bottles and glass lantern chimneys. The first machine for producing large quantities of corrugated board was built in 1874 by G. Smyth, and in the same year Oliver Long improved upon Jones' design by inventing corrugated board with liner sheets on both sides, thereby inventing corrugated board as it came to be known in modern times.  Scottish-born Robert Gair invented the pre-cut paperboard box in 1890 – flat pieces manufactured in bulk that folded into boxes. Gair's invention resulted from an accident. He was a Brooklyn printer and paper-bag maker during the 1870s. While he was printing seed bags, a metal ruler used to crease bags shifted in position and cut them. Gair discovered that by cutting and creasing in one operation he could make prefabricated paperboard boxes. Applying this idea to corrugated boxboard was a straightforward development when the material became available in the early 20th century.  The corrugated box was first used for packaging glass and pottery containers. In the mid-1950s, the case enabled fruit and produce to be shipped from farm to retailer without bruising, improving the return to producers and opening export markets.

Properties.

Several properties and characteristics can be measured for corrugated board. Some of these include:  Corrugated fiberboard is anisotropic; many of its properties are highly directional. For example, edge crush, bending stiffness, tensile, and surface characteristics are different, depending on the orientation to the flutes and the machine direction of manufacture.

Manufacturing.

Corrugated board is manufactured on large high-precision machinery lines called corrugators, usually running at about or more. These machines, over time, have become very complex with the objective of avoiding some common problems in corrugated board production, such as warp and washboarding.  The key raw material in corrugating is paper, different grades for each layer making up the corrugated box. Due to supply chain and scale considerations, paper is produced in separate plants called paper mills. Most corrugating plants keep an inventory of paper reels.  In the classical corrugator, the paper is softened with high-pressure steam. After the board is formed it is dried in the so-called dry-end. Here the newly formed corrugated board is heated from the bottom by hot plates. On the top, various pressures are applied by a load system on the belt.  The corrugated medium is often basis weight in the US; in the UK, a fluting paper is common. At the single-facer, it is heated, moistened, and formed into a fluted pattern on geared wheels. This is joined to a flat linerboard with a starch based adhesive to form single face board. At the double-backer, a second flat linerboard is adhered to the other side of the fluted medium to form single wall corrugated board. Linerboards are test liners (recycled paper) or kraft paperboard (of various grades). The liner may be bleached white, mottled white, colored, or preprinted.  Common flute sizes are "A", "B", "C", "E" and "F" or microflute. The letter designation relates to the order that the flutes were invented, not the relative sizes. Flute size refers to the number of flutes per linear foot, although the actual flute dimensions for different corrugator manufacturers may vary slightly. Measuring the number of flutes per linear foot is a more reliable method of identifying flute size than measuring board thickness, which can vary due to manufacturing conditions. The most common flute size in corrugated boxes is "C" flute.  Corrugated fiberboard can be specified by the construction (single face, singlewall, doublewall, etc.), flute size, burst strength, edge crush strength, flat crush, basis weights of components (pounds per thousand square feet, grams per square meter, etc.), surface treatments and coatings, etc. TAPPI and ASTM test methods for these are standardized.  The choice of corrugated medium, flute size, combining adhesive, and linerboards can be varied to engineer a corrugated board with specific properties to match a wide variety of potential uses. Double and triple-wall corrugated board is also produced for high stacking strength and puncture resistance.  Most corrugators are "two knife" corrugators, which means that they can produce two different sheet lengths side-by-side. This leads to an optimisation problem, known as the cutting stock problem.

Box manufacture process.

Box design.

Packaging engineers design corrugated boxes to meet the particular needs of the product being shipped, the hazards of the shipping environment, (shock, vibration, compression, moisture, etc.), and the marketing needs of retailers and consumers.  The most common box style is the Regular Slotted Container (RSC). All flaps are the same length from the score to the edge. Typically, the longer major flaps meet in the middle and the minor flaps do not. However, with the rise of E-commerce and a more competitive retail environment, box designs are becoming more dynamic and functional, straying away from typical shipping box designs.  The manufacturer's joint is most often joined with adhesive but may also be taped or stitched. The box is shipped flat (knocked down) to the packager who sets up the box, fills it, and closes it for shipment. Box closure may be by tape, adhesive, staples, strapping, etc.  The size of a box can be measured for either internal (for product fit) or external (for handling machinery or palletizing) dimensions. Boxes are usually specified and ordered by the internal dimensions.

Box maker's certificate.

A box maker's certificate (also known as box certificate, or box certificate of maker) is a seal printed on an outside surface, typically the bottom of the box, that includes some information about how strong the box is. The certificate is not required, but if it is used that implies compliance with regulations relating to the certificate. Significant information includes: 1) Bursting Test or Edge Crush Test; 2) Size Limit (the maximum outside dimensions of a finished box when the length, width and depth of the box are added together); 3) Gross Weight Limit.

Manufacturing.

Boxes can be formed in the same plant as the corrugator. Such plants are known as "integrated plants". Part of the scoring and cutting takes place in-line on the corrugator. Alternatively, sheets of corrugated board may be sent to a different manufacturing facility for box fabrication; these are sometimes called "sheet plants" or "converters".  The corrugated board is creased or scored to provide controlled bending of the board. Most often, slots are cut to provide flaps on the box. Scoring and slotting can also be accomplished by die-cutting. Along with the print being produced on the box in these plants. These conversions are produce on different machines, depending on the conversion being made: "Flexo Folder Gluers (FFG)", "Rotary Die Cutters (RDC), "Printer Slotters" or " Printing Presses".

Single-face laminate.

A limitation of common corrugated material has been the difficulty in applying fine graphic print for informative and marketing purposes. The reasons for this stem from the fact that prefabricated corrugated sheets are relatively thick and spongy, compared to the thin and incompressible nature of solid fibre paper such as paperboard. Due to these characteristics of corrugated, it has been mainly printed using a flexographic process, which is by nature a coarse application with loose registration properties.  A more recent development popular in usage is a hybrid product featuring the structural benefits of corrugated combined with the high-graphics print of lithography previously restricted to paperboard folding cartons. This application, generally referred to as 'Single-Face Laminate', begins its process as a traditional fluted medium adhered to a single linerboard (single-face), but in place of a second long-fibered liner, a pre-printed sheet of paperboard such as SBS (solid bleached sulfate) is laminated to the outer facing. The sheet can then be converted with the same processes used for other corrugated manufacturing into any desired form.  Specialized equipment is necessary for the material construction of SFL, so users may expect to pay a premium for these products. However, this cost is often offset by the savings over a separate paperboard sleeve and the labor necessary to assemble the completed package.

Recycling.

Old corrugated containers are an excellent source of fiber for recycling. They can be compressed and baled for cost effective transport. The baled boxes are put in a hydropulper, which is a large vat of warm water for cleaning and processing. The pulp slurry is then used to make new paper and fiber products.  Mill and corrugator scrap, or broke, is the cleanest source for recycling. The high rates of post-consumer recycling reflect the efficiency of recycling mills to clean and process the incoming materials. Several technologies are available to sort, screen, filter, and chemically treat the recycled paper.  Many extraneous materials are readily removed. Twine, strapping, etc. are removed from the hydropulper by a "ragger". Metal straps and staples can be screened out or removed by a magnet. Film-backed pressure-sensitive tape stays intact: the PSA adhesive and the backing are both removed together.  Materials which are more difficult to remove include wax coatings on corrugated boxes and "stickies", soft rubbery particles which can clog the paper maker and contaminate the recycled paper. Stickies can originate from book bindings, hot melt adhesives, pressure-sensitive adhesives from paper labels, laminating adhesives of reinforced gummed tapes, etc.  Corrugated fiberboard shredders are now available which convert post-consumer corrugated board into packing/cushioning materials by means of a specialized shredding process.  and others.

Corrugated fiberboard is a material consisting of a fluted corrugated sheet and one or two flat linerboards. It is made on "flute lamination machines" or "corrugators" and is used for making cardboard boxes. The corrugated medium sheet and the linerboard(s) are made of kraft containerboard, a paperboard material usually over thick. Corrugated fiberboard is sometimes called "corrugated cardboard", although cardboard might be any heavy paper-pulp based board.

Vlnitá lepenka je papírový materiál skládající se z vlnité vrstvy a jedné nebo dvou hladkých vrstev papíru. Je široce využívána především při výrobě kartonových krabic a přepravních kontejnerů. Vlnitá i hladká část lepenky jsou vyrobeny z bednové lepenky, materiál podobný papíru s tloušťkou více než 0,25 mm. Vlnitá lepenka je někdy nazývána jako vlnitý karton, ačkoli kartonem může být označován každý těžší papír s vyšším obsahem buničiny.


History.

In January 1850 Bauer, a cavalryman during the German-Danish War, designed "Brandtaucher" as a way to end the Danish naval blockade of Germany. Bauer's early sketch attracted the attention of the Minister of Marine, who allowed him to construct a model. The model was demonstrated in Kiel harbour in front of naval dignitaries. Its satisfactory performance led to the construction of a full-scale model, which was funded by contributions from army personnel and local civilians. Due to the inadequate funding, the scale of the boat had to be downgraded and the design altered and simplified; resulting in a reduced diving depth from 30 m to 9.5 m. This redesign included eliminating the use of enclosed ballast tanks to contain the water being taken into and expelled from the submarine. Instead, the water was allowed to pool inside the bottom of the hull, below the main floor, and was able to move relatively unobstructed within this area when the ship changed orientation. The resulting instability was probably a significant contributing factor to the loss of the vessel.  As built, "Brandtaucher" was 8.07 m long and 2.02 m at maximum beam, with a draught of 2.63 m. It was propelled by a crew of three turning large tread wheels connected to a propeller. The boat could reach a speed of three knots, but this could not be maintained for long periods of time.  On 1 February 1851, "Brandtaucher" sank after a diving accident during acceptance trials in Kiel Harbour. The submarine experienced equipment failure, and sank to the bottom of a 60-foot hole at the bottom of Kiel Harbour. Bauer escaped by letting in water, thus increasing the air pressure, which allowed Bauer and his two companions to open the hatch and swim to the surface. This was the first submarine escape to be witnessed and reported.  In 1887 the wreck was discovered, and it was raised on 5 July 1887. "Brandtaucher" was first placed on display at the Naval Academy in Kiel and then in 1906 it was moved to the "Museum für Meereskunde" in Berlin. From 1963 to 1965 it was restored in the DDR at Rostock, and placed on display at the Nationale Volksarmee Museum in Potsdam. The boat can now be viewed at the "Militärhistorisches Museum der Bundeswehr" (German Armed Forces Museum of Military History), in Dresden.

Brandtaucher (German for Fire-diver) was a submersible designed by the Bavarian inventor and engineer Wilhelm Bauer and built by Schweffel & Howaldt in Kiel for Schleswig-Holstein's Flotilla (part of the "Reichsflotte") in 1850. The "Brandtaucher" is the oldest known surviving submarine in the world.

Brandtaucher (z něm. "Brand" = požár/žár/topivo a "Taucher" = potápěč), v literatuře také někdy označován jako ""Železný tuleň" " nebo "Plongeur Marin" je ponorka, kterou navrhl a zkonstruoval německý vynálezce Wilhelm Bauer v roce 1850. On sám však ve svých plánech ponorku nijak zvlášť nenazýval, označovaná byla jen jako "ponorný člun" nebo "podmořská loď". Název tedy není možné považovat za oficiální. Je nejstarší zachovanou ponorkou na světě.


Biography.

Hempel studied mathematics, physics and philosophy at the University of Göttingen and subsequently at the University of Berlin and the Heidelberg University. In Göttingen, he encountered David Hilbert and was impressed by his program attempting to base all mathematics on solid logical foundations derived from a limited number of axioms.  After moving to Berlin, Hempel participated in a congress on scientific philosophy in 1929 where he met Rudolf Carnap and became involved in the Berlin Circle of philosophers associated with the Vienna Circle. In 1934, he received his doctoral degree from the University of Berlin with a dissertation on probability theory, titled " Beiträge zur logischen Analyse des Wahrscheinlichkeitsbegriffs" ("Contributions to the Logical Analysis of the Concept of Probability"). Hans Reichenbach was Hempel's main doctoral supervisor, but after Reichenbach lost his philosophy chair in Berlin in 1933, Wolfgang Köhler and Nicolai Hartmann became the official supervisors.  Within a year of completing his doctorate, the increasingly repressive and anti-semitic Nazi regime in Germany had prompted Hempel to emigratehis wife was of Jewish ancestryto Belgium. In this, he was aided by the scientist Paul Oppenheim, with whom he co-authored the book "Der Typusbegriff im Lichte der neuen Logik" on typology and logic in 1936. In 1937, Hempel emigrated to the United States, where he accepted a position as Carnap's assistant at the University of Chicago. He later held positions at the City College of New York (1939–1948), Yale University (1948–1955) and Princeton University, where he taught alongside Thomas Kuhn and remained until made emeritus in 1973. Between 1974 and 1976, he was an emeritus at the Hebrew University in Jerusalem before becoming University Professor of Philosophy at the University of Pittsburgh in 1977 and teaching there until 1985. In 1989 the Department of Philosophy at Princeton University renamed its Three Lecture Series the 'Carl G. Hempel Lectures' in his honor.  Hempel never embraced the term "logical positivism" as an accurate description of the Vienna Circle and Berlin Group, preferring to describe those philosophersand himselfas "logical empiricists". He believed that the term "positivism", with its roots in Auguste Comte, invoked a materialist metaphysics that empiricists need not embrace. He regarded Ludwig Wittgenstein as a philosopher with a genius for stating philosophical insights in striking and memorable language, but believed that he (or, at least, the Wittgenstein of the "Tractatus") made claims that could only be supported by recourse to metaphysics. To Hempel, metaphysics involved claims to know things which were not knowable; that is, metaphysical hypotheses were incapable of confirmation or disconfirmation by evidence.

Legacy.

In 2005, the City of Oranienburg, Hempel's birthplace, renamed one of its streets "Carl-Gustav-Hempel-Straße" in his memory.

Carl Gustav "Peter" Hempel (January 8, 1905 – November 9, 1997) was a German writer and philosopher. He was a major figure in logical empiricism, a 20th-century movement in the philosophy of science. He is especially well known for his articulation of the deductive-nomological model of scientific explanation, which was considered the "standard model" of scientific explanation during the 1950s and 1960s. He is also known for the raven paradox (also known as "Hempel's paradox").

Carl Gustav Hempel (8. ledna 1905, Oranienburg – 9. listopadu 1997, Princeton) byl německo-americký filozof, představitel analytické filozofie a novopozitivismu, měl blízko k tzv. Vídeňskému kroužku. Věnoval se též filozofii vědy. Vytvořil tzv. deduktivně-nomologický model. Věnoval se problematice indukce, definoval v této souvislosti tzv. paradox havrana (známý též jako Hempelův paradox).