Wolfram Research Mathworld, Repository hosted at UIUC
(13.662 recursos)
Wolfram Research's MathWorld is a comprehensive and interactive mathematics encyclopedia intended for students, educators, math enthusiasts, and researchers. It has been assembled by internet encyclopedist Eric W.ÊWeisstein with assistance from the mathematics and internet communities. Like the discipline of mathematics, this site is continuously updated to include new material and incorporate new discoveries. This is a free service for the mathematical community provided by Wolfram Research, makers of Mathematica, with additional support from the National Science Foundation.
Mostrando recursos 1 - 20 de 164
1.
Ackermann Function -- from MathWorld - Weisstein, Eric W.
The Ackermann function is the simplest example of a well defined total function which is computable but not primitive recursive, providing a counterexample to the belief in the early 1900s that every computable function was also primitive recursive (Dötzel 1991). It grows faster than an exponential function, or even a multiple exponential function. The Ackermann function A(x,y) is defined for integer x and y by A(x,y)\equiv\cases{ y+1 & if x=0\cr A(x-1, 1) & if y=0\cr A(x-1, A(x,...
2.
Ackermann Number -- from MathWorld - Weisstein, Eric W.
A number of the form n\underbrace{\uparrow\cdots\uparrow}_n n, where arrow notation has been used. The first few Ackermann numbers are 1\uparrow 1=1, 2\uparrow\uparrow 2=4, and 3\uparrow\uparrow\uparrow 3=\underbrace{{3}^{{3}^{\cdot^{\cdot^{\cdot^{3}}}}}\!\!}_{7{,}625{,}597{,}484{,}987}. See also: Ackermann Function, Arrow Notation, Power Tower
3.
Algorithm -- from MathWorld - Weisstein, Eric W.
A specific set of instructions for carrying out a procedure or solving a problem, usually with the requirement that the procedure terminate at some point. Specific algorithms sometimes also go by the name method, procedure, or technique. The word "algorithm" is a distortion of al-Khwarizmi, a Persian mathematician who wrote an influential treatise about algebraic methods. The process of applying an algorithm to an input to obtain an output is called a computation. See also:...
4.
Archimedes Algorithm -- from MathWorld - Weisstein, Eric W.
Successive application of Archimedes' recurrence formula gives the Archimedes algorithm, which can be used to provide successive approximations to \pi (pi). The algorithm is also called the Borchardt-Pfaff algorithm. Archimedes obtained the first rigorous approximation of \pi by circumscribing and inscribing n=6\cdot 2^k-gons on a circle. From Archimedes' recurrence formula, the circumferences a and b of the circumscribed and inscribed polygons are a(n) = 2n\tan\left({\pi\over n}\right) b(n) =...
5.
Archimedes' Recurrence Formula -- from MathWorld - Weisstein, Eric W.
Let a_n and b_n be the perimeters of the circumscribed and inscribed n-gon and a_{2n} and b_{2n} the perimeters of the circumscribed and inscribed 2n-gon. Then a_{2n} = {2a_nb_n\over a_n+b_n} b_{2n} = \sqrt{a_{2n}b_n}\,. The first follows from the fact that side lengths of the polygons on a circle of radius r = 1 are s_R = 2\tan\left({\pi\over n}\right) s_r = 2\sin\left({\pi\over n}\right), so a_n = 2n\tan\left({\pi\over n}\right) b_n = 2n\sin\left({\pi\over n}\right). But {2a_nb_n\over...
6.
Automata Theory -- from MathWorld - Weisstein, Eric W.
The mathematical study of abstract computing machines (especially Turing machines) and the analysis of algorithms used by such machines. A connection between automata theory and number theory was provided by Christol et al. (1980), who showed that a sequence \{a_n\} is generated by a p-automaton iff the formal power series with coefficients a_n is algebraic on the field of rational elements A(X)/Q(X), where A(X) and Q(X) are polynomials with coefficients in the finite field F_p. See also:...
7.
Backward Stability -- from MathWorld - Weisstein, Eric W.
The property of certain algorithms that accurate answers are returned for well-conditioned problems, and the inaccuracy of the answers returned for ill-conditioned problems is proportional to the sensitivity.
9.
Baud -- from MathWorld - Weisstein, Eric W.
One baud is defined as the state of a signal in a communication channel changing once per second. See also: Baud Rate, Bit Rate
10.
Baud Rate -- from MathWorld - Weisstein, Eric W.
Baud rate is a measure of the number of times per second a signal in a communications channel changes state. The state is usually voltage level, frequency, or phase angle. See also: Baud, Bit Rate
11.
Binary Search -- from MathWorld - Weisstein, Eric W.
A searching algorithm which works on a sorted table by testing the middle of an interval, eliminating the half of the table in which the key cannot lie, and then repeating the procedure iteratively. See also: Searching
13.
Birthday Attack -- from MathWorld - Weisstein, Eric W.
Birthday attacks are a class of brute-force techniques used in an attempt to solve a class of cryptographic hash function problems. These methods take advantage of functions which, when supplied with a random input, return one of k equally likely values. By repeatedly evaluating the function for different inputs, the same output is expected to be obtained after about 1.2\sqrt{k} evaluations. See also: Birthday Problem, Cryptographic Hash Function
14.
Bit -- from MathWorld - Weisstein, Eric W.
The smallest unit of binary information, equal to a single 0 or 1. Two bits are called a crumb, four bits are called a nibble, and eight bits are called 1 byte. See also: Binary, Bit Rate, Byte, Crumb, Least Significant Bit, Nibble, Quantum Bit
15.
Bit Complexity -- from MathWorld - Weisstein, Eric W.
The number of single operations (of addition, subtraction, and multiplication) required to complete an algorithm. See also: Strassen Formulas
16.
Bit Length -- from MathWorld - Weisstein, Eric W.
The number of binary bits necessary to represent a number, given explicitly by {\it BL}(n)=\left\lceil{\lg n}\right\rceil , where \left\lceil{x}\right\rceil is the ceiling function and \lg n is lg, the logarithm to base 2. For n = 0, 1, 2, ..., the first few values are 0, 1, 2, 2, 3, 3, 3, 3, 4, 4, ... (Sloane's A036377). The function is given by the Mathematica version 4.0 function Developer`BitLength[n].
17.
Bit Rate -- from MathWorld - Weisstein, Eric W.
Bit rate (i.e., bits per second, abbreviated bps) is a measure of the number of data bits (digital 0s and 1s) transmitted each second in a digital communications channel. If simple frequency shift key modulation (FSK) is used, each baud causes the transmission of just one bit, so the baud rate is equal to the bit rate. However, for a channel that uses four bits per baud (e.g., CCITT V.22), the baud rate is 1/4 of the bit rate. See also: Baud, Baud Rate, Bit
18.
Bloch Sphere -- from MathWorld - Weisstein, Eric W.
The qubit \left\vert{\psi}\right\rangle = a\left\vert{0}\right\rangle + b\left\vert{1}\right\rangle can be represented as a point (\theta,\phi) on a unit sphere called the Bloch sphere. Define the angles \theta and \phi by letting a=\cos(\theta/2) and b = e^{i\phi} \sin(\theta/2). Here, a is taken to be real, which can always be made real by multiplying \left\vert{\psi}\right\rangle by an overall phase factor (that is unobservable). Then \left\vert{\psi}\right\rangle is represented by...
19.
Bloch Vector -- from MathWorld - Weisstein, Eric W.
A Bloch vector is a unit vector (\cos\phi\sin\theta, \sin\phi\sin\theta, \cos\theta) used to represent points on a Bloch sphere. See also: Bloch Sphere, Qubit
20.
Blum's Speed-Up Theorem -- from MathWorld - Weisstein, Eric W.
There exists a total computable predicate P such that for any algorithm computing P(x) with running time T(x), there exists another algorithm computing P(x) with computation time O(\ln T(x)). This means that there is no algorithm for the predicate P that is even nearly optimal. See also: Computation Time
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