Flat, straight-edged shapes (like triangles) or 3D ones (like pyramids) have only a finite number of symmetries. In contrast, the affine symmetric group is a way to mathematically describe all the symmetries possible when an infinitely large flat surface is covered by triangular tiles. As with many subjects in mathematics, it can also be thought of in a number ways: for example, it also describes the symmetries of the infinitely long number line, or the possible arrangements of all integers (..., −2, −1, 0, 1, 2, ...) with certain repetitive patterns. As a result, studying the affine symmetric group extends the study of symmetries of polyhedra or of groups of permutations to the infinite case. It also connects several topics in mathematics that were originally studied for independent reasons, ranging from complex reflection groups to juggling sequences.
LZ77 and LZ78
LZ77 and LZ78 are two different types of lossless data compression algorithms. They were both published in papers by Abraham Lempel and Jacob Ziv in 1977 and 1978. Both of these types of algorithm, LZ77 and LZ78, are mainly designed to compress data without destroying the contents of the data when decompressed. Like all data compression algorithms, LZ77 and LZ78 both use a process called "data compression", or just "compression". Additionally, LZ77 and LZ78 both have their own processes for decompression (a type of process which reverses compression).
LZW is an improved version of LZ78.
LZ78 is an algorithm that has two variants of itself, LZ78 compression and LZ78 decompression.
In LZ78 compression, this algorithm does compression by replacing repeated occurrences of data with references to a dictionary. The dictionary used for this algorithm is built based on the input data stream. The dictionary stores tuples that represent a set of characters that were part of the input data stream. Each index of a dictionary contains a tuple and its respective character. A tuple stored in an index of the dictionary contains a certain index of the dictionary and a character. Each dictionary item with its respective tuple will be represented as codice_1. Within this type of tuple, the index refers to a point in the dictionary where there is an occurrence of a duplicated part of the dictionary.
If the algorithm finds a unique character not yet found in the dictionary, then a new dictionary entry is added and the index for that new dictionary entry is set to 0. Otherwise, if there is already a dictionary entry, the algorithm just lists a new dictionary entry with an index equal to an existing dictionary entry's index and the character becomes whatever the following character is. Afterwards, a value representing each recorded tuple is stored into a file. The entire process continues until all characters in the input data stream are read.
Suppose the string in the input data stream is "abbaaaabbbbccabcbaaa".
Whenever doing LZ78 compression, the dictionary always starts out empty. Any new entries will start at the first index. The "empty" index is actually the 0th index.
Again, a dictionary contains a set of indexes. Its indexes each contain a tuple that represents a certain value. After LZ78 compression is done, the dictionary should end up like this:
After each time a certain section of the input data stream has been read, a representation of the currently recorded tuple will be stored into the file. The information stored into the file should end up looking like this:
In LZ78 decompression, the process is reversed. Instead of taking a string of many characters, LZ78 decompression scans a file containing a series of stored values. These stored values represent tuples that were featured in LZ78 compression. Like LZ78 compression, LZ78 decompression uses a dictionary to record values. In LZ78 decompression, it will take information that was stored in a file and then record these values into a dictionary. Once all values are added into the dictionary, the program will refer to the dictionary. From there, the program will convert each line of the file into a certain value. That value is determined from converting each line of the file into a tuple. This tuple is stored into the dictionary and will contain a dictionary index and a letter, just as it does in LZ78 compression.
Untamed (1955 movie)
Untamed is a 1955 American romantic adventure movie directed by Henry King and was based on the 1950 novel by Helga Moray. It stars Tyrone Power, Susan Hayward, Richard Egan, Agnes Moorehead, Henry O'Neill, John Justin, Rita Moreno, Brad Dexter, Hope Emerson and was distributed by 20th Century Fox.
Box 25
Box 25 () is a 2015 Panamanian American documentary movie directed by Mercedes Arias and Delfina Vidal about the men who built the Panama Canal.
The Four Musketeers (movie)
The Four Musketeers is a 1974 Spanish Panamanian British American adventure movie directed by Richard Lester and was based on the 1844 novel by Alexandre Dumas and is the sequel of the 1973 movie "The Three Musketeers". It stars Michael York, Oliver Reed, Frank Finlay, Richard Chamberlain, Jean-Pierre Cassel, Geraldine Chaplin, Charlton Heston, Faye Dunaway, Christopher Lee, Raquel Welch, Roy Kinnear and was distributed by 20th Century Fox. It was nominated for an Academy Award in 1975.
Ruben Blades Is Not My Name
Ruben Blades Is Not My Name () is a 2018 Panamanian Argentine Colombian documentary movie directed by Abner Benaim.
Hogfish
The hogfish (known as boquinete, doncella de pluma or pez perro in Mexico) is a type of fish which lives in the Western Atlantic Ocean. The hogfish lives in the area from Nova Scotia, Canada, to northern South America, including the Gulf of Mexico. It is classified as a wrasse.
Beth din
A Beit din (plural: Battei dinim) is a Jewish court of law.
There are several types of Battei dinim. The minimum and usual number for a Beit din is 3 Rabbi's. There are also battei dinim of 23 and a beit din of 70.
Zaal Udumashvili
Zaal (Zaliko) Udumashvili (born 27 February 1971) is a Georgian journalist and politician. Member of Parliament of Georgia.
Pfaffing, Austria
Pfaffing is a municipality of the district of Vöcklabruck in the Austrian state of Upper Austria.
Lignières-Châtelain
Lignières-Châtelain is a commune. It is in Hauts-de-France in the Somme department in north France.
Springfield, Virginia
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Lignières-en-Vimeu
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Morcourt, Somme
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Blood and Roses
Blood and Roses () is a 1960 French Italian horror movie directed by Roger Vadim and is based on the 1872 novella "Carmilla" by Sheridan Le Fanu. It stars Mel Ferrer, Elsa Martinelli, Annette Vadim, René-Jean Chauffard, Marc Allégret, Alberto Bonucci, Serge Marquand, Gabriella Farinon and was distributed by Paramount Pictures.
2013 Campeonato Amapaense
The 2013 Campeonato Amapaense de Futebol is the 68th edition of the Amapá's top professional football league. The competition began on July, and ended on October 12.
On the first stage all teams play against each other once. The two best teams play against each other to find the round's winner, and the six best advance to the second stage.
On the second stage, all six teams from the first stage play against each other once. The two best teams play against each other to find the second round's winner.
On the final stage, the winner of the first stage play against the winner of the second stage.
The champion qualifies to the 2014 Copa do Brasil and the 2014 Campeonato Brasileiro Série D.
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