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Sesquipower
In mathematics, a sesquipower or Zimin word is a string over an alphabet with identical prefix and suffix. Sesquipowers are unavoidable patterns, in the sense that all sufficiently long strings contain one. Formal definition Formally, let ''A'' be an alphabet and ''A''∗ be the free monoid of finite strings over ''A''. Every non-empty word ''w'' in ''A''+ is a sesquipower of order 1. If ''u'' is a sesquipower of order ''n'' then any word ''w'' = ''uvu'' is a sesquipower of order ''n'' + 1.Lothaire (2011) p. 135 The ''degree'' of a non-empty word ''w'' is the largest integer ''d'' such that ''w'' is a sesquipower of order ''d''.Lothaire (2011) p. 136 Bi-ideal sequence A bi-ideal sequence is a sequence of words ''f''''i'' where ''f''1 is in ''A''+ and :f_ = f_i g_i f_i \ for some ''g''''i'' in ''A''∗ and ''i'' ≥ 1. The degree of a word ''w'' is thus the length of the longest bi-ideal sequence ending in ''w''. Unavoidabl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unavoidable Pattern
In mathematics and theoretical computer science, a pattern is an unavoidable pattern if it is unavoidable on any finite alphabet. Definitions Pattern Like a word, a pattern (also called ''term'') is a sequence of symbols over some alphabet. The minimum multiplicity of the pattern ''p'' is ''m(p)=\min(\mathrm(x):x \in p)'' where ''\mathrm(x)'' is the number of occurrence of symbol ''x'' in pattern ''p''. In other words, it is the number of occurrences in ''p'' of the least frequently occurring symbol in ''p''. Instance Given finite alphabets \Sigma and \Delta, a word x\in\Sigma^* is an instance of the pattern p\in\Delta^* if there exists a non-erasing semigroup morphism f:\Delta^*\rightarrow\Sigma^* such that f(p)=x, where \Sigma^* denotes the Kleene star of \Sigma. Non-erasing means that f(a)\neq\varepsilon for all a\in\Delta, where \varepsilon denotes the empty string. Avoidance / Matching A word w is said to ''match'', or ''encounter'', a pattern p if a factor (also c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monoid Factorisation
In mathematics, a factorisation of a free monoid is a sequence of subsets of words with the property that every word in the free monoid can be written as a concatenation of elements drawn from the subsets. The Chen–Fox–Lyndon theorem states that the Lyndon words furnish a factorisation. The Schützenberger theorem relates the definition in terms of a multiplicative property to an additive property. Let ''A''* be the free monoid on an alphabet ''A''. Let ''X''''i'' be a sequence of subsets of ''A''* indexed by a totally ordered index set ''I''. A factorisation of a word ''w'' in ''A''* is an expression :w = x_ x_ \cdots x_ \ with x_ \in X_ and i_1 \ge i_2 \ge \ldots \ge i_n. Some authors reverse the order of the inequalities. Chen–Fox–Lyndon theorem A Lyndon word over a totally ordered alphabet ''A'' is a word that is lexicographically less than all its rotations.Lothaire (1997) p.64 The Chen–Fox–Lyndon theorem states that every string may be formed in a unique ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
ABACABA Pattern
The ABACABA pattern is a recursive fractal pattern that shows up in many places in the real world (such as in geometry, art, music, poetry, number systems, literature and higher dimensions). Patterns often show a DABACABA type subset. ''AA'', ''ABBA'', and ''ABAABA'' type forms are also considered.Halter-Koch, Franz and Tichy, Robert F.; eds. (2000). ''Algebraic Number Theory and Diophantine Analysis'', p.478. W. de Gruyter. . Generating the pattern In order to generate the next sequence, first take the previous pattern, add the next letter from the alphabet, and then repeat the previous pattern. The first few steps are listed here. A generator can be founhere ABACABA is a "quickly growing word", often described as ''chiastic'' or "symmetrically organized around a central axis" (see: Chiastic structure and Χ). The number of members in each iteration is , the Mersenne numbers (). Gallery See also * Arch form * Farey sequence * Rondo * Sesquipower In mathematics, a s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
String (computer Science)
In computer programming, a string is traditionally a sequence of characters, either as a literal constant or as some kind of variable. The latter may allow its elements to be mutated and the length changed, or it may be fixed (after creation). A string is generally considered as a data type and is often implemented as an array data structure of bytes (or words) that stores a sequence of elements, typically characters, using some character encoding. ''String'' may also denote more general arrays or other sequence (or list) data types and structures. Depending on the programming language and precise data type used, a variable declared to be a string may either cause storage in memory to be statically allocated for a predetermined maximum length or employ dynamic allocation to allow it to hold a variable number of elements. When a string appears literally in source code, it is known as a string literal or an anonymous string. In formal languages, which are used in mathematical ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prefix (computer Science)
In formal language theory and computer science, a substring is a contiguous sequence of characters within a string. For instance, "''the best of''" is a substring of "''It was the best of times''". In contrast, "''Itwastimes''" is a subsequence of "''It was the best of times''", but not a substring. Prefixes and suffixes are special cases of substrings. A prefix of a string S is a substring of S that occurs at the beginning of S; likewise, a suffix of a string S is a substring that occurs at the end of S. The substrings of the string "''apple''" would be: "''a''", "''ap''", "''app''", "''appl''", "''apple''", "''p''", "''pp''", "''ppl''", "''pple''", "''pl''", "''ple''", "''l''", "''le''" "''e''", "" (note the empty string at the end). Substring A string u is a substring (or factor) of a string t if there exists two strings p and s such that t = pus. In particular, the empty string is a substring of every string. Example: The string u=ana is equal to substrings (and subse ... [...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 study ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Recurrent Word
In mathematics, a recurrent word or sequence is an infinite word over a finite alphabet in which every factor occurs infinitely many times.Lothaire (2011) p. 30Allouche & Shallit (2003) p.325Pytheas Fogg (2002) p.2 An infinite word is recurrent if and only if it is a sesquipower.Lothaire (2011) p. 141Berstel et al (2009) p.133 A uniformly recurrent word is a recurrent word in which for any given factor ''X'' in the sequence, there is some length ''n''''X'' (often much longer than the length of ''X'') such that ''X'' appears in ''every'' block of length ''n''''X''.Berthé & Rigo (2010) p.7Allouche & Shallit (2003) p.328 The terms minimal sequencePytheas Fogg (2002) p.6 and almost periodic sequence (Muchnik, Semenov, Ushakov 2003) are also used. Examples * The easiest way to make a recurrent sequence is to form a periodic sequence, one where the sequence repeats entirely after a given number ''m'' of steps. Such a sequence is then uniformly recurrent and ''n''''X'' can be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Total Order
In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( reflexive). # If a \leq b and b \leq c then a \leq c ( transitive). # If a \leq b and b \leq a then a = b ( antisymmetric). # a \leq b or b \leq a (strongly connected, formerly called total). Total orders are sometimes also called simple, connex, or full orders. A set equipped with a total order is a totally ordered set; the terms simply ordered set, linearly ordered set, and loset are also used. The term ''chain'' is sometimes defined as a synonym of ''totally ordered set'', but refers generally to some sort of totally ordered subsets of a given partially ordered set. An extension of a given partial order to a total order is called a linear extension of that partial order. Strict and non-strict total orders A on a set X is a strict partial ord ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lyndon Word
In mathematics, in the areas of combinatorics and computer science, a Lyndon word is a nonempty string that is strictly smaller in lexicographic order than all of its rotations. Lyndon words are named after mathematician Roger Lyndon, who investigated them in 1954, calling them standard lexicographic sequences. Anatoly Shirshov introduced Lyndon words in 1953 calling them regular words. Lyndon words are a special case of Hall words; almost all properties of Lyndon words are shared by Hall words. Definitions Several equivalent definitions exist. A k-ary Lyndon word of length n > 0 is an n-character string over an alphabet of size k, and which is the unique minimum element in the lexicographical ordering in the multiset of all its rotations. Being the singularly smallest rotation implies that a Lyndon word differs from any of its non-trivial rotations, and is therefore aperiodic.; . Alternately, a word w is a Lyndon word if and only if it is nonempty and lexicographically stri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs. The society is one of the four parts of the Joint Policy Board for Mathematics and a member of the Conference Board of the Mathematical Sciences. History The AMS was founded in 1888 as the New York Mathematical Society, the brainchild of Thomas Fiske, who was impressed by the London Mathematical Society on a visit to England. John Howard Van Amringe was the first president and Fiske became secretary. The society soon decided to publish a journal, but ran into some resistance, due to concerns about competing with the American Journal of Mathematics. The result was the ''Bulletin of the American Mathematical Society'', with Fiske as editor-in-chief. The de facto journal, as intended, was influential in in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
