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Cobham–Semenov Theorem
In mathematics and theoretical computer science, an automatic sequence (also called a ''k''-automatic sequence or a ''k''-recognizable sequence when one wants to indicate that the base of the numerals used is ''k'') is an infinite sequence of terms characterized by a finite automaton. The ''n''-th term of an automatic sequence ''a''(''n'') is a mapping of the final state reached in a finite automaton accepting the digits of the number ''n'' in some fixed base ''k''.Allouche & Shallit (2003) p. 152Berstel et al (2009) p. 78 An automatic set is a set of non-negative integers ''S'' for which the sequence of values of its characteristic function χ''S'' is an automatic sequence; that is, ''S'' is ''k''-automatic if χ''S''(''n'') is ''k''-automatic, where χ''S''(''n'') = 1 if ''n'' \in ''S'' and 0 otherwise. Definition Automatic sequences may be defined in a number of ways, all of which are equivalent. Four common definitions are as follows. Automata-theo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Julius Richard Büchi
Julius Richard Büchi (1924–1984) was a Swiss logician and mathematician. He received his Dr. sc. nat. in 1950 at ETH Zurich under the supervision of Paul Bernays and Ferdinand Gonseth. Shortly afterwards he went to Purdue University in Lafayette, Indiana. He and his first student Lawrence Landweber had a major influence on the development of theoretical computer science. Together with his friend Saunders Mac Lane, a student of Paul Bernays as well, Büchi published numerous celebrated works. He invented what is now known as the Büchi automaton, a finite-state machine accepting certain sets of infinite sequences of characters known as omega-regular languages. The "''n'' squares' problem", known also as Büchi's problem, is an open problem from number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constru ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unary Numeral System
The unary numeral system is the simplest numeral system to represent natural numbers: to represent a number ''N'', a symbol representing 1 is repeated ''N'' times. In the unary system, the number 0 (zero) is represented by the empty string, that is, the absence of a symbol. Numbers 1, 2, 3, 4, 5, 6, ... are represented in unary as 1, 11, 111, 1111, 11111, 111111, ... Unary is a bijective numeral system. However, although it has sometimes been described as "base 1", it differs in some important ways from positional notations, in which the value of a digit depends on its position within a number. For instance, the unary form of a number can be exponentially longer than its representation in other bases. The use of tally marks in counting is an application of the unary numeral system. For example, using the tally mark (𝍷), the number 3 is represented as . In East Asian cultures, the number 3 is represented as 三, a character drawn with three strokes. (One and two are repres ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pumping Lemma For Regular Languages
In the theory of formal languages, the pumping lemma for regular languages is a Lemma (mathematics), lemma that describes an essential property of all regular languages. Informally, it says that all sufficiently long string (computer science), strings in a regular language may be ''pumped''—that is, have a middle section of the string repeated an arbitrary number of times—to produce a new string that is also part of the language. The pumping lemma is useful for proving that a specific language is not a regular language, by showing that the language does not have the property. Specifically, the pumping lemma says that for any regular language L, there exists a constant p such that any string w in L with length at least p can be split into three substrings x, y and z (w = xyz, with y being non-empty), such that the strings xz, xyz, xyyz, xyyyz, ... are also in L. The process of repeating y zero or more times is known as "pumping". Moreover, the pumping lemma guarantees that the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multiplicative Independence
In number theory, two positive integers ''a'' and ''b'' are said to be multiplicatively independent if their only common integer power is 1. That is, for integers ''n'' and ''m'', a^n=b^m implies n=m=0. Two integers which are not multiplicatively independent are said to be multiplicatively dependent. As examples, 36 and 216 are multiplicatively dependent since 36^3=(6^2)^3=(6^3)^2=216^2, whereas 2 and 3 are multiplicatively independent. Properties Being multiplicatively independent admits some other characterizations. ''a'' and ''b'' are multiplicatively independent if and only if \log(a)/\log(b) is irrational. This property holds independently of the base of the logarithm. Let a = p_1^p_2^ \cdots p_k^ and b = q_1^q_2^ \cdots q_l^ be the canonical representations of ''a'' and ''b''. The integers ''a'' and ''b'' are multiplicatively dependent if and only if ''k'' = ''l'', p_i=q_i and \frac=\frac for all ''i'' and ''j''. Applications Büchi arithmeti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Academic Press
Academic Press (AP) is an academic book publisher founded in 1941. It launched a British division in the 1950s. Academic Press was acquired by Harcourt, Brace & World in 1969. Reed Elsevier said in 2000 it would buy Harcourt, a deal completed the next year, after a regulatory review. Thus, Academic Press is now an imprint of Elsevier. Academic Press publishes reference books, serials and online products in the subject areas of: * Communications engineering * Economics * Environmental science * Finance * Food science and nutrition * Geophysics * Life sciences * Mathematics and statistics * Neuroscience * Physical sciences * Psychology Psychology is the scientific study of mind and behavior. Its subject matter includes the behavior of humans and nonhumans, both consciousness, conscious and Unconscious mind, unconscious phenomena, and mental processes such as thoughts, feel ... Well-known products include the '' Methods in Enzymology'' series and encyclopedias such ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Morphic Word
In mathematics and computer science, a morphic word or substitutive word is an infinite sequence of symbols which is constructed from a particular class of endomorphism of a free monoid. Every automatic sequence is morphic. Definition Let ''f'' be an endomorphism of the free monoid ''A''∗ on an alphabet ''A'' with the property that there is a letter ''a'' such that ''f''(''a'') = ''as'' for a non-empty string ''s'': we say that ''f'' is prolongable at ''a''. The word : a s f(s) f(f(s)) \cdots f^(s) \cdots \ is a pure morphic or pure substitutive word. Note that it is the limit of the sequence ''a'', ''f''(''a''), ''f''(''f''(''a'')), ''f''(''f''(''f''(''a''))), ... It is clearly a fixed point of the endomorphism ''f'': the unique such sequence beginning with the letter ''a''.Lothaire (2011) p. 10Honkala (2010) p.505 In general, a morphic word is the image of a pure morphic word under a coding, that is, a morphism that maps letter to letter. If a morphic word is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Paperfolding Sequence
In mathematics the regular paperfolding sequence, also known as the dragon curve sequence, is an infinite sequence of 0s and 1s. It is obtained from the repeating partial sequence by filling in the question marks by another copy of the whole sequence. The first few terms of the resulting sequence are: If a strip of paper is folded repeatedly in half in the same direction, i times, it will get 2^i-1 folds, whose direction (left or right) is given by the pattern of 0's and 1's in the first 2^i-1 terms of the regular paperfolding sequence. Opening out each fold to create a right-angled corner (or, equivalently, making a sequence of left and right turns through a regular grid, following the pattern of the paperfolding sequence) produces a sequence of polygonal chains that approaches the dragon curve fractal: Properties The value of any given term t_n in the regular paperfolding sequence, starting with n=1, can be found recursively as follows. Divide n by two, as many times as possi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Baum–Sweet Sequence
In mathematics the Baum–Sweet sequence is an infinite automatic sequence of 0s and 1s defined by the rule: :''b''''n'' = 1 if the binary representation of ''n'' contains no block of consecutive 0s of odd length; :''b''''n'' = 0 otherwise; for ''n'' ≥ 0. For example, ''b''4 = 1 because the binary representation of 4 is 100, which only contains one block of consecutive 0s of length 2; whereas ''b''5 = 0 because the binary representation of 5 is 101, which contains a block of consecutive 0s of length 1. Starting at ''n'' = 0, the first few terms of the Baum–Sweet sequence are: :1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1 ... Historical motivation The properties of the sequence were first studied by Leonard E. Baum and Melvin M. Sweet in 1976. In 1949, Khinchin conjectured that there does not exist a non-quadratic algebraic real number having bounded partial quotients in its continued fraction expansion. A counterexample to this conjecture is still not known. Baum an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rudin–Shapiro Sequence
In mathematics, the Rudin–Shapiro sequence, also known as the Golay–Rudin–Shapiro sequence, is an infinite 2- automatic sequence named after Marcel Golay, Harold S. Shapiro, and Walter Rudin, who investigated its properties. Definition Each term of the Rudin–Shapiro sequence is either 1 or -1. If the binary expansion of n is given by : n = \sum_ \epsilon_k(n) 2^k, then let : u_n = \sum_ \epsilon_k(n)\epsilon_(n). (So u_n is the number of times the block 11 appears in the binary expansion of n.) The Rudin–Shapiro sequence (r_n)_ is then defined by : r_n = (-1)^. Thus r_n = 1 if u_n is even and r_n = -1 if u_n is odd. The sequence u_n is known as the complete Rudin–Shapiro sequence, and starting at n = 0, its first few terms are: : 0, 0, 0, 1, 0, 0, 1, 2, 0, 0, 0, 1, 1, 1, 2, 3, ... and the corresponding terms r_n of the Rudin–Shapiro sequence are: : +1, +1, +1, −1, +1, +1, −1, +1, +1, +1, +1, −1, −1, −1, +1, −1, ... For exam ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modulo Operation
In computing and mathematics, the modulo operation returns the remainder or signed remainder of a Division (mathematics), division, after one number is divided by another, the latter being called the ''modular arithmetic, modulus'' of the operation. Given two positive numbers and , modulo (often abbreviated as ) is the remainder of the Euclidean division of by , where is the Division (mathematics), dividend and is the divisor. For example, the expression "5 mod 2" evaluates to 1, because 5 divided by 2 has a quotient of 2 and a remainder of 1, while "9 mod 3" would evaluate to 0, because 9 divided by 3 has a quotient of 3 and a remainder of 0. Although typically performed with and both being integers, many computing systems now allow other types of numeric operands. The range of values for an integer modulo operation of is 0 to . mod 1 is always 0. When exactly one of or is negative, the basic definition breaks down, and programming languages differ in how these valu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
