1.
Tetration
–
In mathematics, tetration is the next hyperoperation after exponentiation, and is defined as iterated exponentiation. The word was coined by Reuben Louis Goodstein, from tetra-, Tetration is used for the notation of very large numbers. The notation n a means a a ⋅ ⋅ a, the application of exponentiation n −1 times. Shown here are the first four hyperoperations, with tetration as the fourth, multiplication a × n = a + a + ⋯ + a ⏟ n n copies of a combined by addition. Exponentiation a n = a × a × ⋯ × a ⏟ n n copies of a combined by multiplication, Tetration n a = a a ⋅ ⋅ a ⏟ n n copies of a combined by exponentiation, right-to-left. The above example is read as the nth tetration of a, each operation is defined by iterating the previous one. Tetration is not an elementary function, analogously, tetration can be thought of as a chained power involving n numbers a. The parameter a may be called the base-parameter in the following, computer programmers refer to this choice as right-associative. When a and 10 are coprime, we can compute the last m decimal digits of n a using Eulers theorem, 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, the term tetration, introduced by Goodstein in his 1947 paper Transfinite Ordinals in Recursive Number Theory, has gained dominance. It was also popularized in Rudy Ruckers Infinity and the Mind, the term superexponentiation was published by Bromer in his paper Superexponentiation in 1987. It was used earlier by Ed Nelson in his book Predicative Arithmetic, the term hyperpower is a natural combination of hyper and power, which aptly describes tetration. The problem lies in the meaning of hyper with respect to the hyperoperation sequence, when considering hyperoperations, the term hyper refers to all ranks, and the term super refers to rank 4, or tetration. So under these considerations hyperpower is misleading, since it is referring to tetration. The term power tower is used, in the form the power tower of order n for a a ⋅ ⋅ a ⏟ n. This is a misnomer, however, because tetration cannot be expressed with iterated power functions, the term snap is occasionally used in informal contexts, in the form a snap n for a a ⋅ ⋅ a ⏟ n. This term is not yet accepted, although it is used within select communities. It is believed to be a reference to Jounce, the derivative of position in physics

2.
Turing machine
–
Despite the models simplicity, given any computer algorithm, a Turing machine can be constructed that is capable of simulating that algorithms logic. The machine operates on an infinite memory tape divided into discrete cells, the machine positions its head over a cell and reads the symbol there. The Turing machine was invented in 1936 by Alan Turing, who called it an a-machine, thus, Turing machines prove fundamental limitations on the power of mechanical computation. Turing completeness is the ability for a system of instructions to simulate a Turing machine, a Turing machine is a general example of a CPU that controls all data manipulation done by a computer, with the canonical machine using sequential memory to store data. More specifically, it is a capable of enumerating some arbitrary subset of valid strings of an alphabet. Assuming a black box, the Turing machine cannot know whether it will eventually enumerate any one specific string of the subset with a given program and this is due to the fact that the halting problem is unsolvable, which has major implications for the theoretical limits of computing. The Turing machine is capable of processing an unrestricted grammar, which implies that it is capable of robustly evaluating first-order logic in an infinite number of ways. This is famously demonstrated through lambda calculus, a Turing machine that is able to simulate any other Turing machine is called a universal Turing machine. The thesis states that Turing machines indeed capture the notion of effective methods in logic and mathematics. Studying their abstract properties yields many insights into computer science and complexity theory, at any moment there is one symbol in the machine, it is called the scanned symbol. The machine can alter the scanned symbol, and its behavior is in part determined by that symbol, however, the tape can be moved back and forth through the machine, this being one of the elementary operations of the machine. Any symbol on the tape may therefore eventually have an innings, the Turing machine mathematically models a machine that mechanically operates on a tape. On this tape are symbols, which the machine can read and write, one at a time, in the original article, Turing imagines not a mechanism, but a person whom he calls the computer, who executes these deterministic mechanical rules slavishly. If δ is not defined on the current state and the current tape symbol, Q0 ∈ Q is the initial state F ⊆ Q is the set of final or accepting states. The initial tape contents is said to be accepted by M if it eventually halts in a state from F, Anything that operates according to these specifications is a Turing machine. The 7-tuple for the 3-state busy beaver looks like this, Q = Γ = b =0 Σ = q 0 = A F = δ = see state-table below Initially all tape cells are marked with 0. In the words of van Emde Boas, p.6, The set-theoretical object provides only partial information on how the machine will behave and what its computations will look like. For instance, There will need to be many decisions on what the symbols actually look like, and a failproof way of reading and writing symbols indefinitely

3.
International Standard Book Number
–
The International Standard Book Number is a unique numeric commercial book identifier. An ISBN is assigned to each edition and variation of a book, for example, an e-book, a paperback and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, the method of assigning an ISBN is nation-based and varies from country to country, often depending on how large the publishing industry is within a country. The initial ISBN configuration of recognition was generated in 1967 based upon the 9-digit Standard Book Numbering created in 1966, the 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108. Occasionally, a book may appear without a printed ISBN if it is printed privately or the author does not follow the usual ISBN procedure, however, this can be rectified later. Another identifier, the International Standard Serial Number, identifies periodical publications such as magazines, the ISBN configuration of recognition was generated in 1967 in the United Kingdom by David Whitaker and in 1968 in the US by Emery Koltay. The 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108, the United Kingdom continued to use the 9-digit SBN code until 1974. The ISO on-line facility only refers back to 1978, an SBN may be converted to an ISBN by prefixing the digit 0. For example, the edition of Mr. J. G. Reeder Returns, published by Hodder in 1965, has SBN340013818 -340 indicating the publisher,01381 their serial number. This can be converted to ISBN 0-340-01381-8, the check digit does not need to be re-calculated, since 1 January 2007, ISBNs have contained 13 digits, a format that is compatible with Bookland European Article Number EAN-13s. An ISBN is assigned to each edition and variation of a book, for example, an ebook, a paperback, and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, a 13-digit ISBN can be separated into its parts, and when this is done it is customary to separate the parts with hyphens or spaces. Separating the parts of a 10-digit ISBN is also done with either hyphens or spaces, figuring out how to correctly separate a given ISBN number is complicated, because most of the parts do not use a fixed number of digits. ISBN issuance is country-specific, in that ISBNs are issued by the ISBN registration agency that is responsible for country or territory regardless of the publication language. Some ISBN registration agencies are based in national libraries or within ministries of culture, in other cases, the ISBN registration service is provided by organisations such as bibliographic data providers that are not government funded. In Canada, ISBNs are issued at no cost with the purpose of encouraging Canadian culture. In the United Kingdom, United States, and some countries, where the service is provided by non-government-funded organisations. Australia, ISBNs are issued by the library services agency Thorpe-Bowker