Set theory (music)

Musical set theory provides concepts for categorizing musical objects and describing their relationships. Many of the notions were first elaborated by Howard Hanson (1960) in connection with tonal music, and then mostly developed in connection with atonal music by theorists such as Allen Forte (1973), drawing on the work in twelve-tone theory of Milton Babbitt. The concepts of set theory are very general and can be applied to tonal and atonal styles in any equally tempered tuning system, and to some extent more generally than that.

One branch of musical set theory deals with collections (sets and permutations) of pitches and pitch classes (pitch-class set theory), which may be ordered or unordered, and can be related by musical operations such as transposition, inversion, and complementation. The methods of musical set theory are sometimes applied to the analysis of rhythm as well.

Mathematical set theory versus musical set theory

Although musical set theory is often thought to involve the application of mathematical set theory to music, there are numerous differences between the methods and terminology of the two. For example, musicians use the terms transposition and inversion where mathematicians would use translation and reflection. Furthermore, where musical set theory refers to ordered sets, mathematics would normally refer to tuples or sequences (though mathematics does speak of ordered sets, and although these can be seen to include the musical kind in some sense, they are far more involved).

2007 New Zealand police raids

The 2007 New Zealand raids were a series of armed police arrests conducted on Monday, 15 October 2007, in response to the discovery of an alleged paramilitary training camp in the Urewera mountain range near the town of Ruatoki in the eastern Bay of Plenty.

About 300 police, including members of the Armed Offenders Squad and Special Tactics Group, were involved in the arrests in which four guns and 230 rounds of ammunition were seized and 17 people arrested, all but one of them charged with firearms offences. According to police, the raids were a culmination of more than a year of surveillance that uncovered and monitored the training camps. Search warrants were executed under the Summary Proceedings Act to search for evidence relating to potential breaches of the Terrorism Suppression Act and the Arms Act.

On 29 October, police referred evidence gathered during the raids to the Solicitor-General to consider whether charges should be laid under the Terrorism Suppression Act. Authorisation for prosecutions under the Act is given by the Attorney-General though he has delegated this responsibility to Solicitor-General David Collins. On 8 November the Solicitor-General declined to press charges under the Terrorism Suppression Act, because of inadequacies of the legislation. According to Helen Clark, the Prime Minister at the time of the raids, one of the reasons police tried to lay charges under anti-terror legislation was because they could not use telephone interception evidence in prosecutions under the Arms Act.

Operation 40

Operation 40 was the code name for a Central Intelligence Agency-sponsored counterintelligence group composed of Cuban exiles. The group was formed to seize control of the Cuban government after the Bay of Pigs Invasion. Operation 40 continued to operate unofficially until it disbanded in 1970 due to allegations that an aircraft that was carrying cocaine and heroin in support of the group crashed in California.

It was approved by President Dwight D. Eisenhower in March 1960, after the January 1959 Cuban Revolution, and was presided over by Vice-president Richard Nixon. The group included Frank Sturgis (who would later become one of the Watergate burglars); Felix Rodriguez (a CIA officer who later was involved in the capture and summary execution of Che Guevara); Luis Posada Carriles (held in the US in 2010 on charges of illegal immigration, he is demanded by Venezuela for his key role in the execution of the 1976 Cubana Flight 455 bombing); Orlando Bosch (founder of the counterrevolutionary Coordination of United Revolutionary Organizations, that organized the 1976 murder of Chilean former minister Orlando Letelier); Rafael 'Chi Chi' Quintero; Virgilio Paz Romero; Pedro Luis Diaz Lanz; Bernard Barker; and Porter Goss.

Disaster (film)

Disaster is a 1948 American drama film directed by William H. Pine and written by Thomas Ahearn. The film stars Richard Denning, Trudy Marshall, Damian O'Flynn, Will Wright, James Millican and Jack Lambert. The film was released on December 3, 1948, by Paramount Pictures.

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Family Π

Family Π is a group of New Testament manuscripts. Belonging to the Byzantine text-type, it is one of the textual families of this group. The name of the family, "Π" (pronounced in English as "pie"), is drawn from the symbol used for the Codex Petropolitanus. One of the most distinct of the Byzantine sub-groups, it is very old and the third largest. The oldest Byzantine manuscripts belong to this family.

Hermann von Soden designated this group by the symbol "K^a". According to him, its text is not purely Byzantine.

Codices and manuscripts

Soden included the following in this group of codices: Cyprius (K), Petropolitanus (Π), 72, 114, 116, 178, 265, 389, 1008, 1009, 1079, 1154, 1200, 1219, 1346, and 1398. Lake added to this group of manuscripts: 489, 537, 652, 775, 796, 904, 1478, 1500, 1546, 1561, 1781, 1816. Soden also associated Codex Alexandrinus with this group. Wisse lists about 150 witnesses of the family, but the majority of them belong to this family only in some parts of their text. The Peshitta, in the Gospels, represents this family.

Index set

In mathematics, an index set is a set whose members label (or index) members of another set. For instance, if the elements of a set A may be indexed or labeled by means of a set J, then J is an index set. The indexing consists of a surjective function from J onto A and the indexed collection is typically called an (indexed) family, often written as (A_j)_j∈J.

Examples

An enumeration of a set S gives an index set , where f : J → S is the particular enumeration of S.

Any countably infinite set can be indexed by .

For , the indicator function on r is the function given by

The set of all the functions is an uncountable set indexed by .

Other uses

In computational complexity theory and cryptography, an index set is a set for which there exists an algorithm I that can sample the set efficiently; i.e., on input 1ⁿ, I can efficiently select a poly(n)-bit long element from the set.

See also

References

Analog Man (album)

Analog Man is the eleventh (and latest) studio solo album by the American singer-songwriter and multi-instrumentalist Joe Walsh, formerly of the James Gang and lead guitarist for the Eagles. The album was released in mid 2012, on the label Fantasy in the United States and the United Kingdom, It is his first studio solo album to be released since 1992's Songs for a Dying Planet, 20 years prior. The album features 10 new songs, and was co-produced by Jeff Lynne. The album also features contributions from the former Beatles drummer, Ringo Starr, the former Barnstorm members, Kenny Passarelli and Joe Vitale, former James Gang members, Jim Fox and Dale Peters, and also a duet with the infamous rock and roll legend, Little Richard.

The album peaked at No. 12 on the Billboard 200 chart, as well as No. 4 on the top rock albums chart.

Production and recording

After touring with the Eagles and struggling with alcohol and drug addictions for many years, Walsh decided that it was time to record a new album while being supported by his wife, Marjorie Bach (sister of Ringo Starr's wife, Barbara). To pursue making the album, she also gave Walsh Jeff Lynne's contact number. When Walsh was asked about his collaboration with co-producer Jeff Lynne, he said "Jeff and I met socially, and at one point he said, "Why don't you bring your tracks over sometime and we'll have a listen." That led to some comments and ideas that he had. Gradually, we worked on some stuff and checked out some of his stuff too. It ended up that he really helped me finish it up and ended up producing. He really put his stamp on my music and took it in a direction I never would have gone, and I'm really grateful to him."