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K-regular Sequence
In mathematics and theoretical computer science, a ''k''-regular sequence is a sequence satisfying linear recurrence equations that reflect the base-''k'' representations of the integers. The class of ''k''-regular sequences generalizes the class of ''k''-automatic sequences to alphabets of infinite size. Definition There exist several characterizations of ''k''-regular sequences, all of which are equivalent. Some common characterizations are as follows. For each, we take ''R''′ to be a commutative Noetherian ring and we take ''R'' to be a ring containing ''R''′. ''k''-kernel Let ''k'' ≥ 2. The ''k-kernel'' of the sequence s(n)_ is the set of subsequences :K_(s) = \. The sequence s(n)_ is (''R''′, ''k'')-regular (often shortened to just "''k''-regular") if the R'-module generated by ''K''''k''(''s'') is a finitely-generated ''R''′-module.Allouche and Shallit (1992), Definition 2.1. In the special case when R' = R = \mathbb, the sequence s(n)_ is k-regular i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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''&n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Automata (computation)
An automaton (; plural: automata or automatons) is a relatively self-operating machine, or control mechanism designed to automatically follow a sequence of operations, or respond to predetermined instructions.Automaton – Definition and More from the Free Merriam-Webster Dictionary http://www.merriam-webster.com/dictionary/automaton Some automata, such as bellstrikers in mechanical clocks, are designed to give the illusion to the casual observer that they are operating under their own power. Since long ago, the term is commonly associated with automated puppets that resemble moving humans or animals, built to impress and/or to entertain people. Animatronics are a modern type of automata with electronics, often used for the portrayal of characters in films and in theme park attractions. Etymology The word "automaton" is the latinization of the Ancient Greek , , (neuter) "acting of one's own will". This word was first used by Homer to describe an automatic door opening, or au ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combinatorics On Words
Combinatorics on words is a fairly new field of mathematics, branching from combinatorics, which focuses on the study of words and formal languages. The subject looks at letters or symbols, and the sequences they form. Combinatorics on words affects various areas of mathematical study, including algebra and computer science. There have been a wide range of contributions to the field. Some of the first work was on square-free words by Axel Thue in the early 1900s. He and colleagues observed patterns within words and tried to explain them. As time went on, combinatorics on words became useful in the study of algorithms and coding. It led to developments in abstract algebra and answering open questions. Definition Combinatorics is an area of discrete mathematics. Discrete mathematics is the study of countable structures. These objects have a definite beginning and end. The study of enumerable objects is the opposite of disciplines such as analysis, where calculus and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press A university press is an academic publishing house specializing in monographs and scholarly journals. Most are nonprofit organizations and an integral component of a large research university. They publish work that has been reviewed by schola ... in the world. It is also the King's Printer. Cambridge University Press is a department of the University of Cambridge and is both an academic and educational publisher. It became part of Cambridge University Press & Assessment, following a merger with Cambridge Assessment in 2021. With a global sales presence, publishing hubs, and offices in more than 40 Country, countries, it publishes over 50,000 titles by authors from over 100 countries. Its publishing includes more than 380 academic journals, monographs, reference works, school and uni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convolution
In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution'' refers to both the result function and to the process of computing it. It is defined as the integral of the product of the two functions after one is reflected about the y-axis and shifted. The choice of which function is reflected and shifted before the integral does not change the integral result (see #Properties, commutativity). The integral is evaluated for all values of shift, producing the convolution function. Some features of convolution are similar to cross-correlation: for real-valued functions, of a continuous or discrete variable, convolution (f*g) differs from cross-correlation (f \star g) only in that either or is reflected about the y-axis in convolution; thus it is a cross-c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
K-synchronized Sequence
In mathematics and theoretical computer science, a ''k''-synchronized sequence is an infinite sequence of terms ''s''(''n'') characterized by a finite automaton taking as input two strings ''m'' and ''n'', each expressed in some fixed base ''k'', and accepting if ''m'' = ''s''(''n''). The class of ''k''-synchronized sequences lies between the classes of ''k''-automatic sequences and ''k''-regular sequences. Definitions As relations Let Σ be an alphabet of ''k'' symbols where ''k'' ≥ 2, and let 'n''sub>''k'' denote the base-''k'' representation of some number ''n''. Given ''r'' ≥ 2, a subset ''R'' of \mathbb^ is ''k''-synchronized if the relation is a right-synchronized rational relation over Σ∗ × ... × Σ∗, where (''n''1, ..., ''n''''r'') \in ''R''.Carpi & Maggi (2010) Language-theoretic Let ''n'' ≥ 0 be a natural number and let ''f'': \mathbb \rightarrow \mathbb be a map, where both ''n'' a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gould's Sequence
Gould's sequence is an integer sequence named after Henry W. Gould that counts how many odd numbers are in each row of Pascal's triangle. It consists only of power of two, powers of two, and begins:. :1, 2, 2, 4, 2, 4, 4, 8, 2, 4, 4, 8, 4, 8, 8, 16, 2, 4, ... For instance, the sixth number in the sequence is 4, because there are four odd numbers in the sixth row of Pascal's triangle (the four bold numbers in the sequence 1, 5, 10, 10, 5, 1). Additional interpretations The th value in the sequence (starting from ) gives the highest power of 2 that divides the central binomial coefficient \tbinom, and it gives the numerator of 2^n/n! (expressed as a fraction in lowest terms). Gould's sequence also gives the number of live cells in the th generation of the Rule 90 cellular automaton starting from a single live cell.. It has a characteristic growing sawtooth wave, sawtooth shape that can be used to recognize physical processes that behave similarly to Rule 90.. Related sequences ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer-valued Polynomial
In mathematics, an integer-valued polynomial (also known as a numerical polynomial) P(t) is a polynomial whose value P(n) is an integer for every integer ''n''. Every polynomial with integer coefficients is integer-valued, but the converse is not true. For example, the polynomial : \frac t^2 + \frac t=\fract(t+1) takes on integer values whenever ''t'' is an integer. That is because one of ''t'' and t + 1 must be an even number. (The values this polynomial takes are the triangular numbers.) Integer-valued polynomials are objects of study in their own right in algebra, and frequently appear in algebraic topology.. See in particular pp. 213–214. Classification The class of integer-valued polynomials was described fully by . Inside the polynomial ring \Q /math> of polynomials with rational number coefficients, the subring of integer-valued polynomials is a free abelian group. It has as basis the polynomials :P_k(t) = t(t-1)\cdots (t-k+1)/k! for k = 0,1,2, \dots, i.e., th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sorting Number
In mathematics and computer science, the sorting numbers are a sequence of numbers introduced in 1950 by Hugo Steinhaus for the analysis of comparison sort algorithms. These numbers give the worst-case number of comparisons used by both binary insertion sort and merge sort. However there are other algorithms that use fewer comparisons. Formula and examples The nth sorting number is given by the formula :n\,S(n)-2^+1 where :S(n)=\lfloor 1 + \log_2 n \rfloor. The sequence of numbers given by this formula (starting with n=1) is :0, 1, 3, 5, 8, 11, 14, 17, 21, 25, 29, 33, 37, 41, ... . The same sequence of numbers can also be obtained from the recurrence relation :A(n) = A\bigl(\lfloor n/2\rfloor\bigr)+A\bigl(\lceil n/2\rceil\bigr)+n-1. It is an example of a 2-regular sequence. Asymptotically, the value of the nth sorting number fluctuates between approximately n\log_2 n-n and n\log_2 n-0.915n depending on the ratio between n and the nearest power of two. Application to sorting In ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Merge Sort
In computer science, merge sort (also commonly spelled as mergesort) is an efficient, general-purpose, and comparison-based sorting algorithm. Most implementations produce a stable sort, which means that the order of equal elements is the same in the input and output. Merge sort is a divide-and-conquer algorithm that was invented by John von Neumann in 1945. A detailed description and analysis of bottom-up merge sort appeared in a report by Goldstine and von Neumann as early as 1948. Algorithm Conceptually, a merge sort works as follows: #Divide the unsorted list into ''n'' sublists, each containing one element (a list of one element is considered sorted). #Repeatedly merge sublists to produce new sorted sublists until there is only one sublist remaining. This will be the sorted list. Top-down implementation Example C-like code using indices for top-down merge sort algorithm that recursively splits the list (called ''runs'' in this example) into sublists until sublist size i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stanley Sequence
In mathematics, a Stanley sequence is an integer sequence generated by a greedy algorithm that chooses the sequence members to avoid arithmetic progressions. If S is a finite set of non-negative integers on which no three elements form an arithmetic progression (that is, a Salem–Spencer set), then the Stanley sequence generated from S starts from the elements of S, in sorted order, and then repeatedly chooses each successive element of the sequence to be a number that is larger than the already-chosen numbers and does not form any three-term arithmetic progression with them. These sequences are named after Richard P. Stanley. Binary–ternary sequence The Stanley sequence starting from the empty set consists of those numbers whose ternary representations have only the digits 0 and 1. That is, when written in ternary, they look like binary numbers. These numbers are :0, 1, 3, 4, 9, 10, 12, 13, 27, 28, 30, 31, 36, 37, 39, 40, ... By their construction as a Stanley sequence, this s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
