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Double Exponential Function
A double exponential function is a constant raised to the power of an exponential function. The general formula is f(x) = a^=a^ (where ''a''>1 and ''b''>1), which grows much more quickly than an exponential function. For example, if ''a'' = ''b'' = 10: *''f''(x) = 1010x *''f''(0) = 10 *''f''(1) = 1010 *''f''(2) = 10100 = googol *''f''(3) = 101000 *''f''(100) = 1010100 = googolplex. Factorials grow faster than exponential functions, but much more slowly than double exponential functions. However, tetration and the Ackermann function grow faster. See Big O notation for a comparison of the rate of growth of various functions. The inverse of the double exponential function is the double logarithm log(log(''x'')). The complex double exponential function is entire, because it is the composition of two entire functions f(x)=a^x=e^ and g(x)=b^x=e^. Double exponential sequences A sequence of positive integers (or real numbers) is said to have ''double exponential rate of growth ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mills' Constant
In number theory, Mills' constant is defined as the smallest positive real number ''A'' such that the floor function of the double exponential function : \left\lfloor A^ \right\rfloor is a prime number for all positive natural numbers ''n''. This constant is named after William Harold Mills who proved in 1947 the existence of ''A'' based on results of Guido Hoheisel and Albert Ingham on the prime gaps. Its value is unproven, but if the Riemann hypothesis is true, it is approximately 1.3063778838630806904686144926... . Mills primes The primes generated by Mills' constant are known as Mills primes; if the Riemann hypothesis is true, the sequence begins :2, 11, 1361, 2521008887, 16022236204009818131831320183, :4113101149215104800030529537915953170486139623539759933135949994882770404074832568499, \ldots . If ''ai'' denotes the ''i'' th prime in this sequence, then ''ai'' can be calculated as the smallest prime number larger than a_^3. In order to ensure that rounding A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chan's Algorithm
In computational geometry, Chan's algorithm, named after Timothy M. Chan, is an optimal output-sensitive algorithm to compute the convex hull of a set P of n points, in 2- or 3-dimensional space. The algorithm takes O(n \log h) time, where h is the number of vertices of the output (the convex hull). In the planar case, the algorithm combines an O(n \log n) algorithm ( Graham scan, for example) with Jarvis march (O(nh)), in order to obtain an optimal O(n \log h) time. Chan's algorithm is notable because it is much simpler than the Kirkpatrick–Seidel algorithm, and it naturally extends to 3-dimensional space. This paradigm has been independently developed by Frank Nielsen in his Ph.D. thesis. Algorithm Overview A single pass of the algorithm requires a parameter m which is between 0 and n (number of points of our set P). Ideally, m = h but h, the number of vertices in the output convex hull, is not known at the start. Multiple passes with increasing values of m are done whi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Michael O
Michael may refer to: People * Michael (given name), a given name * Michael (surname), including a list of people with the surname Michael Given name * Michael (bishop elect), English 13th-century Bishop of Hereford elect * Michael (Khoroshy) (1885–1977), cleric of the Ukrainian Orthodox Church of Canada * Michael Donnellan (fashion designer), Michael Donnellan (1915–1985), Irish-born London fashion designer, often referred to simply as "Michael" * Michael (footballer, born 1982), Brazilian footballer * Michael (footballer, born 1983), Brazilian footballer * Michael (footballer, born 1993), Brazilian footballer * Michael (footballer, born February 1996), Brazilian footballer * Michael (footballer, born March 1996), Brazilian footballer * Michael (footballer, born 1999), Brazilian footballer Rulers Byzantine emperors *Michael I Rangabe (d. 844), married the daughter of Emperor Nikephoros I *Michael II (770–829), called "the Stammerer" and "the Amorian" *Michael III ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Michael J
Michael may refer to: People * Michael (given name), a given name * he He ..., a given name * Michael (surname), including a list of people with the surname Michael Given name * Michael (bishop elect)">Michael (surname)">he He ..., a given name * Michael (surname), including a list of people with the surname Michael Given name * Michael (bishop elect), English 13th-century Bishop of Hereford elect * Michael (Khoroshy) (1885–1977), cleric of the Ukrainian Orthodox Church of Canada * Michael Donnellan (fashion designer), Michael Donnellan (1915–1985), Irish-born London fashion designer, often referred to simply as "Michael" * Michael (footballer, born 1982), Brazilian footballer * Michael (footballer, born 1983), Brazilian footballer * Michael (footballer, born 1993), Brazilian footballer * Michael (footballer, born February 1996), Brazilian footballer * Michael (footballer, born March 1996), Brazilian footballer * Michael (footballer, born 1999), Brazilian football ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Presburger Arithmetic
Presburger arithmetic is the first-order theory of the natural numbers with addition, named in honor of Mojżesz Presburger, who introduced it in 1929. The signature of Presburger arithmetic contains only the addition operation and equality, omitting the multiplication operation entirely. The theory is computably axiomatizable; the axioms include a schema of induction. Presburger arithmetic is much weaker than Peano arithmetic, which includes both addition and multiplication operations. Unlike Peano arithmetic, Presburger arithmetic is a decidable theory. This means it is possible to algorithmically determine, for any sentence in the language of Presburger arithmetic, whether that sentence is provable from the axioms of Presburger arithmetic. The asymptotic running-time computational complexity of this algorithm is at least doubly exponential, however, as shown by . Overview The language of Presburger arithmetic contains constants 0 and 1 and a binary function +, interprete ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Christos Papadimitriou
Christos Charilaos Papadimitriou (; born August 16, 1949) is a Greek-American theoretical computer scientist and the Donovan Family Professor of Computer Science at Columbia University. Education Papadimitriou studied at the National Technical University of Athens, where in 1972 he received his Bachelor of Arts degree in electrical engineering. He then pursued graduate studies at Princeton University, where he received his Ph.D. in electrical engineering and computer science in 1976 after completing a doctoral dissertation titled "The complexity of combinatorial optimization problems." Career Papadimitriou has taught at Harvard, MIT, the National Technical University of Athens, Stanford, UCSD, University of California, Berkeley and is currently the Donovan Family Professor of Computer Science at Columbia University. Papadimitriou co-authored a paper on pancake sorting with Bill Gates, then a Harvard undergraduate. Papadimitriou recalled "Two years later, I called to tell him ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
EXPSPACE
In computational complexity theory, is the set of all decision problems solvable by a deterministic Turing machine in exponential space, i.e., in O(2^) space, where p(n) is a polynomial function of n. Some authors restrict p(n) to be a linear function, but most authors instead call the resulting class . If we use a nondeterministic machine instead, we get the class , which is equal to by Savitch's theorem. A decision problem is if it is in , and every problem in has a polynomial-time many-one reduction to it. In other words, there is a polynomial-time algorithm that transforms instances of one to instances of the other with the same answer. problems might be thought of as the hardest problems in . is a strict superset of , , and . It contains and is believed to strictly contain it, but this is unproven. Formal definition In terms of and , :\mathsf = \bigcup_ \mathsf\left(2^\right) = \bigcup_ \mathsf\left(2^\right) Examples of problems Formal languages An examp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alternating Turing Machine
In computational complexity theory, an alternating Turing machine (ATM) is a non-deterministic Turing machine (NTM) with a rule for accepting computations that generalizes the rules used in the definition of the complexity classes NP and co-NP. The concept of an ATM was set forth by Chandra and Stockmeyer and independently by Kozen in 1976, with a joint journal publication in 1981. Definitions Informal description The definition of NP uses the ''existential mode'' of computation: if ''any'' choice leads to an accepting state, then the whole computation accepts. The definition of co-NP uses the ''universal mode'' of computation: only if ''all'' choices lead to an accepting state does the whole computation accept. An alternating Turing machine (or to be more precise, the definition of acceptance for such a machine) alternates between these modes. An alternating Turing machine is a non-deterministic Turing machine whose states are divided into two sets: existential state ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
2-EXPTIME
In computational complexity theory, the complexity class 2-EXPTIME (sometimes called 2-EXP, sometimes also written 2EXPTIME) is the set of all decision problems solvable by a deterministic Turing machine in O(22''p''(''n'')) time, where ''p''(''n'') is a polynomial function of ''n''. In terms of DTIME, : \mathsf = \bigcup_ \mathsf \left( 2^ \right) . Comparison with other complexity classes We know : P ⊆ NP ⊆ PSPACE ⊆ EXPTIME ⊆ NEXPTIME ⊆ EXPSPACE ⊆ 2-EXPTIME ⊆ ELEMENTARY. 2-EXPTIME can also be reformulated as the space class AEXPSPACE, the problems that can be solved by an alternating Turing machine in exponential space. This is one way to see that EXPSPACE ⊆ 2-EXPTIME, since an alternating Turing machine is at least as powerful as a deterministic Turing machine. 2-EXPTIME is one class in a hierarchy of complexity classes with increasingly higher time bounds. The class 3-EXPTIME is defined similarly to 2-EXPTIME but with a triply exponential time bound 2^. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computational Complexity Theory
In theoretical computer science and mathematics, computational complexity theory focuses on classifying computational problems according to their resource usage, and explores the relationships between these classifications. A computational problem is a task solved by a computer. A computation problem is solvable by mechanical application of mathematical steps, such as an algorithm. A problem is regarded as inherently difficult if its solution requires significant resources, whatever the algorithm used. The theory formalizes this intuition, by introducing mathematical models of computation to study these problems and quantifying their computational complexity, i.e., the amount of resources needed to solve them, such as time and storage. Other measures of complexity are also used, such as the amount of communication (used in communication complexity), the number of logic gate, gates in a circuit (used in circuit complexity) and the number of processors (used in parallel computing). O ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fibonacci Quarterly
The ''Fibonacci Quarterly'' is a scientific journal on mathematical topics related to the Fibonacci numbers, published four times per year. It is the primary publication of The Fibonacci Association, which has published it since 1963. Its founding editors were Verner Emil Hoggatt Jr. and Alfred Brousseau; by Clark Kimberling the present editor is Professor Curtis Cooper of the Mathematics Department of the University of Central Missouri. The ''Fibonacci Quarterly'' has an editorial board of nineteen members and is overseen by the nine-me ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
