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Ehrenfeucht–Mycielski Sequence
The Ehrenfeucht–Mycielski sequence is a recursively defined sequence of binary digits with pseudorandom properties, defined by . Definition The sequence starts with the single bit 0; each successive digit is formed by finding the longest suffix of the sequence that also occurs earlier within the sequence, and complementing the bit following the most recent earlier occurrence of that suffix. (The empty string is a suffix and prefix of every string.) For example, the first few steps of this construction process are: #0: initial bit #01: the suffix "" of "0" occurs earlier, most-recently followed by a 0, so add 1 #010: the suffix "" of "01" occurs earlier, most-recently followed by a 1, so add 0 #0100: the suffix "0" of "010" occurs earlier, most-recently followed by a 1, so add 0 #01001: the suffix "0" of "0100" occurs earlier, most-recently followed by a 0, so add 1 #010011: the suffix "01" of "01001" occurs earlier, most-recently followed by a 0, so add 1 #0100110: the suffix "1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binary Digits
The bit is the most basic unit of information in computing and digital communications. The name is a portmanteau of binary digit. The bit represents a logical state with one of two possible values. These values are most commonly represented as either , but other representations such as ''true''/''false'', ''yes''/''no'', ''on''/''off'', or ''+''/''−'' are also commonly used. The relation between these values and the physical states of the underlying storage or device is a matter of convention, and different assignments may be used even within the same device or program. It may be physically implemented with a two-state device. The symbol for the binary digit is either "bit" per recommendation by the IEC 80000-13:2008 standard, or the lowercase character "b", as recommended by the IEEE 1541-2002 standard. A contiguous group of binary digits is commonly called a ''bit string'', a bit vector, or a single-dimensional (or multi-dimensional) ''bit array''. A group of eight bit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pseudorandom
A pseudorandom sequence of numbers is one that appears to be statistically random, despite having been produced by a completely deterministic Determinism is a philosophical view, where all events are determined completely by previously existing causes. Deterministic theories throughout the history of philosophy have developed from diverse and sometimes overlapping motives and consi ... and repeatable process. Background The generation of random numbers has many uses, such as for sampling (statistics), random sampling, Monte Carlo methods, board games, or gambling. In physics, however, most processes, such as gravitational acceleration, are deterministic, meaning that they always produce the same outcome from the same starting point. Some notable exceptions are radioactive decay and quantum measurement, which are both modeled as being truly random processes in the underlying physics. Since these processes are not practical sources of random numbers, people use pseudoran ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Suffix Tree
In computer science, a suffix tree (also called PAT tree or, in an earlier form, position tree) is a compressed trie containing all the suffixes of the given text as their keys and positions in the text as their values. Suffix trees allow particularly fast implementations of many important string operations. The construction of such a tree for the string S takes time and space linear in the length of S. Once constructed, several operations can be performed quickly, for instance locating a substring in S, locating a substring if a certain number of mistakes are allowed, locating matches for a regular expression pattern etc. Suffix trees also provide one of the first linear-time solutions for the longest common substring problem. These speedups come at a cost: storing a string's suffix tree typically requires significantly more space than storing the string itself. History The concept was first introduced by . Rather than the suffix S ..n/math>, Weiner stored in his trie the ''pref ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Constant Time
In computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations performed by the algorithm, supposing that each elementary operation takes a fixed amount of time to perform. Thus, the amount of time taken and the number of elementary operations performed by the algorithm are taken to be related by a constant factor. Since an algorithm's running time may vary among different inputs of the same size, one commonly considers the worst-case time complexity, which is the maximum amount of time required for inputs of a given size. Less common, and usually specified explicitly, is the average-case complexity, which is the average of the time taken on inputs of a given size (this makes sense because there are only a finite number of possible inputs of a given size). In both cases, the time complexity is generally expresse ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Disjunctive Sequence
A disjunctive sequence is an infinite sequence (over a finite alphabet of characters) in which every finite string appears as a substring. For instance, the binary Champernowne sequence :0\ 1\ 00\ 01\ 10\ 11\ 000\ 001 \ldots formed by concatenating all binary strings in shortlex order, clearly contains all the binary strings and so is disjunctive. (The spaces above are not significant and are present solely to make clear the boundaries between strings). The complexity function of a disjunctive sequence ''S'' over an alphabet of size ''k'' is ''p''''S''(''n'') = ''k''''n''.Bugeaud (2012) p.91 Any normal sequence (a sequence in which each string of equal length appears with equal frequency) is disjunctive, but the converse is not true. For example, letting 0''n'' denote the string of length ''n'' consisting of all 0s, consider the sequence :0\ 0^1\ 1\ 0^2\ 00\ 0^4\ 01\ 0^8\ 10\ 0^\ 11\ 0^\ 000\ 0^\ldots obtained by splicing exponentially long strings of 0s into the shortlex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ackermann Function
In computability theory, the Ackermann function, named after Wilhelm Ackermann, is one of the simplest and earliest-discovered examples of a total computable function that is not primitive recursive. All primitive recursive functions are total and computable, but the Ackermann function illustrates that not all total computable functions are primitive recursive. After Ackermann's publication of his function (which had three non-negative integer arguments), many authors modified it to suit various purposes, so that today "the Ackermann function" may refer to any of numerous variants of the original function. One common version, the two-argument Ackermann–Péter function is defined as follows for nonnegative integers ''m'' and ''n'': : \begin \operatorname(0, n) & = & n + 1 \\ \operatorname(m+1, 0) & = & \operatorname(m, 1) \\ \operatorname(m+1, n+1) & = & \operatorname(m, \operatorname(m+1, n)) \end Its value grows rapidly, even for small inputs. For example, is an integer o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
De Bruijn Sequence
In combinatorial mathematics, a de Bruijn sequence of order ''n'' on a size-''k'' alphabet ''A'' is a cyclic sequence in which every possible length-''n'' string on ''A'' occurs exactly once as a substring (i.e., as a ''contiguous'' subsequence). Such a sequence is denoted by and has length , which is also the number of distinct strings of length ''n'' on ''A''. Each of these distinct strings, when taken as a substring of , must start at a different position, because substrings starting at the same position are not distinct. Therefore, must have ''at least'' symbols. And since has ''exactly'' symbols, De Bruijn sequences are optimally short with respect to the property of containing every string of length ''n'' at least once. The number of distinct de Bruijn sequences is :\dfrac. The sequences are named after the Dutch mathematician Nicolaas Govert de Bruijn, who wrote about them in 1946. As he later wrote, the existence of de Bruijn sequences for each order together ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convergence Rate
In numerical analysis, the order of convergence and the rate of convergence of a convergent sequence are quantities that represent how quickly the sequence approaches its limit. A sequence (x_n) that converges to x^* is said to have ''order of convergence'' q \geq 1 and ''rate of convergence'' \mu if : \lim _ \frac=\mu. The rate of convergence \mu is also called the ''asymptotic error constant''. Note that this terminology is not standardized and some authors will use ''rate'' where this article uses ''order'' (e.g., ). In practice, the rate and order of convergence provide useful insights when using iterative methods for calculating numerical approximations. If the order of convergence is higher, then typically fewer iterations are necessary to yield a useful approximation. Strictly speaking, however, the asymptotic behavior of a sequence does not give conclusive information about any finite part of the sequence. Similar concepts are used for discretization methods. The solutio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Random Sequence
The concept of a random sequence is essential in probability theory and statistics. The concept generally relies on the notion of a sequence of random variables and many statistical discussions begin with the words "let ''X''1,...,''Xn'' be independent random variables...". Yet as D. H. Lehmer stated in 1951: "A random sequence is a vague notion... in which each term is unpredictable to the uninitiated and whose digits pass a certain number of tests traditional with statisticians". Axiomatic probability theory ''deliberately'' avoids a definition of a random sequence. Traditional probability theory does not state if a specific sequence is random, but generally proceeds to discuss the properties of random variables and stochastic sequences assuming some definition of randomness. The Bourbaki school considered the statement "let us consider a random sequence" an abuse of language. Early history Émile Borel was one of the first mathematicians to formally address randomness in 190 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Law Of The Iterated Logarithm
In probability theory, the law of the iterated logarithm describes the magnitude of the fluctuations of a random walk. The original statement of the law of the iterated logarithm is due to A. Ya. Khinchin (1924). Another statement was given by A. N. Kolmogorov in 1929. Statement Let be independent, identically distributed random variables with means zero and unit variances. Let ''S''''n'' = ''Y''1 + ... + ''Y''''n''. Then : \limsup_ \frac = 1 \quad \text, where “log” is the natural logarithm, “lim sup” denotes the limit superior, and “a.s.” stands for “almost surely”. Discussion The law of iterated logarithms operates “in between” the law of large numbers and the central limit theorem. There are two versions of the law of large numbers — the weak and the strong — and they both state that the sums ''S''''n'', scaled by ''n''−1, converge to zero, respectively in probability and almost surely: : \frac \ \xrightarrow\ 0, \qquad \frac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normal Number
In mathematics, a real number is said to be simply normal in an integer base b if its infinite sequence of digits is distributed uniformly in the sense that each of the b digit values has the same natural density 1/b. A number is said to be normal in base b if, for every positive integer n, all possible strings n digits long have density b−''n''. Intuitively, a number being simply normal means that no digit occurs more frequently than any other. If a number is normal, no finite combination of digits of a given length occurs more frequently than any other combination of the same length. A normal number can be thought of as an infinite sequence of coin flips ( binary) or rolls of a die (base 6). Even though there ''will'' be sequences such as 10, 100, or more consecutive tails (binary) or fives (base 6) or even 10, 100, or more repetitions of a sequence such as tail-head (two consecutive coin flips) or 6-1 (two consecutive rolls of a die), there will also be equally ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Mathematical Monthly
''The American Mathematical Monthly'' is a mathematical journal founded by Benjamin Finkel in 1894. It is published ten times each year by Taylor & Francis for the Mathematical Association of America. The ''American Mathematical Monthly'' is an expository journal intended for a wide audience of mathematicians, from undergraduate students to research professionals. Articles are chosen on the basis of their broad interest and reviewed and edited for quality of exposition as well as content. In this the ''American Mathematical Monthly'' fulfills a different role from that of typical mathematical research journals. The ''American Mathematical Monthly'' is the most widely read mathematics journal in the world according to records on JSTOR. Tables of contents with article abstracts from 1997–2010 are availablonline The MAA gives the Lester R. Ford Awards annually to "authors of articles of expository excellence" published in the ''American Mathematical Monthly''. Editors *2022– ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
