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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] |
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] |
Automatic Sequence
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-the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Leonard E
Leonard or ''Leo'' is a common English masculine given name and a surname. The given name and surname originate from the Old High German '' Leonhard'' containing the prefix ''levon'' ("lion") from the Greek Λέων ("lion") through the Latin '' Leo,'' and the suffix ''hardu'' ("brave" or "hardy"). The name has come to mean "lion strength", "lion-strong", or "lion-hearted". Leonard was the name of a Saint in the Middle Ages period, known as the patron saint of prisoners. Leonard is also an Irish origin surname, from the Gaelic ''O'Leannain'' also found as O'Leonard, but often was anglicised to just Leonard, consisting of the prefix ''O'' ("descendant of") and the suffix ''Leannan'' ("lover"). The oldest public records of the surname appear in 1272 in Huntingdonshire, England, and in 1479 in Ulm, Germany. Variations The name has variants in other languages: * Anard/Nardu/Lewnardu/Leunardu (Maltese) * Leen, Leendert, Lenard (Dutch) * Lehnertz, Lehnert (Luxembourgish) * Len ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Aleksandr Khinchin
Aleksandr Yakovlevich Khinchin (, ), July 19, 1894 – November 18, 1959, was a Soviet mathematician and one of the most significant contributors to the Soviet school of probability theory. Due to romanization conventions, his name is sometimes written as "Khinchin" and other times as "Khintchine". Life and career He was born in the village of Kondrovo, Kaluga Governorate, Russian Empire. While studying at Moscow State University, he became one of the first followers of the famous Luzin school. Khinchin graduated from the university in 1916 and six years later he became a full professor there, retaining that position until his death. Khinchin's early works focused on real analysis. Later he applied methods from the metric theory of functions to problems in probability theory and number theory. He became one of the founders of modern probability theory, discovering the law of the iterated logarithm in 1924, achieving important results in the field of limit theorems, giving ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Power Series
In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''a_n'' represents the coefficient of the ''n''th term and ''c'' is a constant called the ''center'' of the series. Power series are useful in mathematical analysis, where they arise as Taylor series of infinitely differentiable functions. In fact, Borel's theorem implies that every power series is the Taylor series of some smooth function. In many situations, the center ''c'' is equal to zero, for instance for Maclaurin series. In such cases, the power series takes the simpler form \sum_^\infty a_n x^n = a_0 + a_1 x + a_2 x^2 + \dots. The partial sums of a power series are polynomials, the partial sums of the Taylor series of an analytic function are a sequence of converging polynomial approximations to the function at the center, and a converging power series can be seen as a kind of generalized polynom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hensel's Lemma
In mathematics, Hensel's lemma, also known as Hensel's lifting lemma, named after Kurt Hensel, is a result in modular arithmetic, stating that if a univariate polynomial has a simple root modulo a prime number , then this root can be ''lifted'' to a unique root modulo any higher power of . More generally, if a polynomial factors modulo into two coprime polynomials, this factorization can be lifted to a factorization modulo any higher power of (the case of roots corresponds to the case of degree for one of the factors). By passing to the "limit" (in fact this is an inverse limit) when the power of tends to infinity, it follows that a root or a factorization modulo can be lifted to a root or a factorization over the p-adic integer, -adic integers. These results have been widely generalized, under the same name, to the case of polynomials over an arbitrary commutative ring, where is replaced by an ideal (ring theory), ideal, and "coprime polynomials" means "polynomials that gene ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite-state Machine
A finite-state machine (FSM) or finite-state automaton (FSA, plural: ''automata''), finite automaton, or simply a state machine, is a mathematical model of computation. It is an abstract machine that can be in exactly one of a finite number of ''State (computer science), states'' at any given time. The FSM can change from one state to another in response to some Input (computer science), inputs; the change from one state to another is called a ''transition''. An FSM is defined by a list of its states, its initial state, and the inputs that trigger each transition. Finite-state machines are of two types—Deterministic finite automaton, deterministic finite-state machines and Nondeterministic finite automaton, non-deterministic finite-state machines. For any non-deterministic finite-state machine, an equivalent deterministic one can be constructed. The behavior of state machines can be observed in many devices in modern society that perform a predetermined sequence of actions d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
String Substitution
In computer science, in the area of formal language theory, frequent use is made of a variety of string functions; however, the notation used is different from that used for computer programming, and some commonly used functions in the theoretical realm are rarely used when programming. This article defines some of these basic terms. Strings and languages A string is a finite sequence of characters. The empty string is denoted by \varepsilon. The concatenation of two string s and t is denoted by s \cdot t, or shorter by s t. Concatenating with the empty string makes no difference: s \cdot \varepsilon = s = \varepsilon \cdot s. Concatenation of strings is associative: s \cdot (t \cdot u) = (s \cdot t) \cdot u. For example, (\langle b \rangle \cdot \langle l \rangle) \cdot (\varepsilon \cdot \langle ah \rangle) = \langle bl \rangle \cdot \langle ah \rangle = \langle blah \rangle. A language is a finite or infinite set of strings. Besides the usual set operations like union, inters ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Automatic Sequence
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-the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cambridge University Press
Cambridge University Press was the university press of the University of Cambridge. Granted a letters patent by King Henry VIII in 1534, it was the oldest university press in the world. Cambridge University Press merged with Cambridge Assessment to form Cambridge University Press and Assessment under Queen Elizabeth II's approval in August 2021. With a global sales presence, publishing hubs, and offices in more than 40 countries, it published over 50,000 titles by authors from over 100 countries. Its publications include more than 420 academic journals, monographs, reference works, school and university textbooks, and English language teaching and learning publications. It also published Bibles, runs a bookshop in Cambridge, sells through Amazon, and has a conference venues business in Cambridge at the Pitt Building and the Sir Geoffrey Cass Sports and Social Centre. It also served as the King's Printer. Cambridge University Press, as part of the University of Cambridge, was a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
