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Thue–Morse Sequence
In mathematics, the Thue–Morse sequence, or Prouhet–Thue–Morse sequence, is the binary sequence (an infinite sequence of 0s and 1s) obtained by starting with 0 and successively appending the Boolean complement of the sequence obtained thus far. The first few steps of this procedure yield the strings 0 then 01, 0110, 01101001, 0110100110010110, and so on, which are prefixes of the Thue–Morse sequence. The full sequence begins: :01101001100101101001011001101001.... The sequence is named after Axel Thue and Marston Morse. Definition There are several equivalent ways of defining the Thue–Morse sequence. Direct definition To compute the ''n''th element ''tn'', write the number ''n'' in binary. If the number of ones in this binary expansion is odd then ''tn'' = 1, if even then ''tn'' = 0. For this reason John H. Conway ''et al''. called numbers ''n'' satisfying ''tn'' = 1 ''odious'' (for ''odd'') numbers and numbers for which ''tn ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Recursion
Recursion (adjective: ''recursive'') occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics to logic. The most common application of recursion is in mathematics and computer science, where a function being defined is applied within its own definition. While this apparently defines an infinite number of instances (function values), it is often done in such a way that no infinite loop or infinite chain of references ("crock recursion") can occur. Formal definitions In mathematics and computer science, a class of objects or methods exhibits recursive behavior when it can be defined by two properties: * A simple ''base case'' (or cases) — a terminating scenario that does not use recursion to produce an answer * A ''recursive step'' — a set of rules that reduces all successive cases toward the base case. For example, the following is a recursive definition of a person's ''ancestor''. One's an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Formal Power Series
In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial sums, etc.). A formal power series is a special kind of formal series, whose terms are of the form a x^n where x^n is the nth power of a variable x (n is a non-negative integer), and a is called the coefficient. Hence, power series can be viewed as a generalization of polynomials, where the number of terms is allowed to be infinite, with no requirements of convergence. Thus, the series may no longer represent a function of its variable, merely a formal sequence of coefficients, in contrast to a power series, which defines a function by taking numerical values for the variable within a radius of convergence. In a formal power series, the x^n are used only as position-holders for the coefficients, so that the coefficient of x^5 is the fifth t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binary Field
(also denoted \mathbb F_2, or \mathbb Z/2\mathbb Z) is the finite field of two elements (GF is the initialism of ''Galois field'', another name for finite fields). Notations and \mathbb Z_2 may be encountered although they can be confused with the notation of p-adic integer, -adic integers. is the Field (mathematics), field with the smallest possible number of elements, and is unique if the additive identity and the multiplicative identity are denoted respectively and , as usual. The elements of may be identified with the two possible values of a bit and to the Boolean domain, boolean values ''true'' and ''false''. It follows that is fundamental and ubiquitous in computer science and its mathematical logic, logical foundations. Definition GF(2) is the unique field with two elements with its additive identity, additive and multiplicative identity, multiplicative identities respectively denoted and . Its addition is defined as the usual addition of integers but modulo 2 ... [...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-theo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Free Monoid
In abstract algebra, the free monoid on a set is the monoid whose elements are all the finite sequences (or strings) of zero or more elements from that set, with string concatenation as the monoid operation and with the unique sequence of zero elements, often called the empty string and denoted by ε or λ, as the identity element. The free monoid on a set ''A'' is usually denoted ''A''∗. The free semigroup on ''A'' is the subsemigroup of ''A''∗ containing all elements except the empty string. It is usually denoted ''A''+./ref> More generally, an abstract monoid (or semigroup) ''S'' is described as free if it is isomorphic to the free monoid (or semigroup) on some set. As the name implies, free monoids and semigroups are those objects which satisfy the usual universal property defining free objects, in the respective categories of monoids and semigroups. It follows that every monoid (or semigroup) arises as a homomorphic image of a free monoid (or semigroup). The st ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prolongable Morphism
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 c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fixed Point (mathematics)
A fixed point (sometimes shortened to fixpoint, also known as an invariant point) is a value that does not change under a given transformation. Specifically, in mathematics, a fixed point of a function is an element that is mapped to itself by the function. In physics, the term fixed point can refer to a temperature that can be used as a reproducible reference point, usually defined by a phase change or triple point. Fixed point of a function Formally, is a fixed point of a function if belongs to both the domain and the codomain of , and . For example, if is defined on the real numbers by f(x) = x^2 - 3 x + 4, then 2 is a fixed point of , because . Not all functions have fixed points: for example, , has no fixed points, since is never equal to for any real number. In graphical terms, a fixed point means the point is on the line , or in other words the graph of has a point in common with that line. Fixed-point iteration In numerical analysis, ''fixed-point i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monoid Morphism
In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoids are semigroups with identity. Such algebraic structures occur in several branches of mathematics. The functions from a set into itself form a monoid with respect to function composition. More generally, in category theory, the morphisms of an object to itself form a monoid, and, conversely, a monoid may be viewed as a category with a single object. In computer science and computer programming, the set of strings built from a given set of characters is a free monoid. Transition monoids and syntactic monoids are used in describing finite-state machines. Trace monoids and history monoids provide a foundation for process calculi and concurrent computing. In theoretical computer science, the study of monoids is fundamental for automa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Squarefree Word
In combinatorics, a squarefree word is a word (a sequence of symbols) that does not contain any squares. A square is a word of the form , where is not empty. Thus, a squarefree word can also be defined as a word that avoids the pattern . Finite squarefree words Binary alphabet Over a binary alphabet \, the only squarefree words are the empty word \epsilon,0,1,01,10,010, and 101. Ternary alphabet Over a ternary alphabet ''\'', there are infinitely many squarefree words. It is possible to count the number c(n) of ternary squarefree words of length . This number is bounded by c(n) = \Theta(\alpha^n) , where 1.3017597 in \Sigma_ uniformly at random set \chi_w to ''w /math>'' followed by all other letters of \Sigma_ in increasing order set the number of iterations to 0 while , w, to the end of update \chi_w shifting the first elements to the right and setting \chi_w = a increment by if ends with a square of rank then ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Palindromic Number
A palindromic number (also known as a numeral palindrome or a numeric palindrome) is a number (such as 16461) that remains the same when its digits are reversed. In other words, it has reflectional symmetry across a vertical axis. The term ''palindromic'' is derived from palindrome, which refers to a word (such as ''rotor'' or ''racecar'') whose spelling is unchanged when its letters are reversed. The first 30 palindromic numbers (in decimal) are: : 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 101, 111, 121, 131, 141, 151, 161, 171, 181, 191, 202, … . Palindromic numbers receive most attention in the realm of recreational mathematics. A typical problem asks for numbers that possess a certain property ''and'' are palindromic. For instance: * The palindromic primes are 2, 3, 5, 7, 11, 101, 131, 151, ... . * The palindromic square numbers are 0, 1, 4, 9, 121, 484, 676, 10201, 12321, ... . It is obvious that in any base there are infinitely many palindr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Periodic Sequence
In mathematics, a periodic sequence (sometimes called a cycle) is a sequence for which the same terms are repeated over and over: :''a''1, ''a''2, ..., ''a''''p'', ''a''1, ''a''2, ..., ''a''''p'', ''a''1, ''a''2, ..., ''a''''p'', ... The number ''p'' of repeated terms is called the period (period). Definition A (purely) periodic sequence (with period ''p''), or a ''p-''periodic sequence, is a sequence ''a''1, ''a''2, ''a''3, ... satisfying :''a''''n''+''p'' = ''a''''n'' for all values of ''n''. If a sequence is regarded as a function whose domain is the set of natural numbers, then a periodic sequence is simply a special type of periodic function. The smallest ''p'' for which a periodic sequence is ''p''-periodic is called its least period or exact period. Examples Every constant function is 1-periodic. The sequence 1,2,1,2,1,2\dots is periodic with least period 2. The sequence of digits in the decimal expansion of 1/7 is periodic with period 6: :\fr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
