A binary sequence BS is a sequence of math N math bits, math a j math for math j 0,1,...,N 1 math , i.e. math m math ones and math N m math zeros. A BS is pseudorandomness pseudo random PRBS if its autocorrelation function math C v sum j 0 N 1 a ja j v math has only two values math C v begin cases m, mbox if v equiv 0 mbox mod N mc, mbox otherwise end cases math where math c frac m 1 N 1 math is called the duty cycle of the PRBS. A PRBS is random in a sense that the value of an math a j math element is independent of the values of any of the other elements, similar to real random sequences. It is pseudo because it is deterministic and after math N math elements it starts to repeat itself, unlike real random sequences, such as sequences generated by radioactive decay or by white noise. The PRBS is more general than the n sequence , which is a special pseudo random binary sequence of n bits generated as the output of a linear shift register. An n sequence always has a 1 2 duty cycle and its number of elements math N 2 k 1 math . PRBS s are used in telecommunication , encryption , simulation , correlation technique and time of flight spectroscopy . Practical implementation Pseudorandom binary sequences can be generated using linear feedback shift register s. ref Paul H. Bardell, William H. McAnney, and Jacob Savir, Built In Test for VLSI Pseudorandom Techniques , John Wiley & Sons, New York, 1987. ref See also Pseudorandomnumber generator Gold code Complementary sequences Bit Error Rate Test References reflist External links http www.scriptwell.net correlation.htm Refimprove date January 2008 Category Pseudorandomness Category Binary sequences de Pseudo random bit stream pl PRBS ru uk ... more details
Rayleigh and Poisson distribution Poisson . See also Portal Mathematics List of pseudorandomnumber generators Pseudorandom binary sequence Quasi random Random number generator attack Randomness Notes ...Citation style date September 2009 A pseudorandomnumber generator PRNG , also known as a deterministic ... seed . Although sequences that are closer to truly random can be generated using hardware random number generator s, pseudorandom numbers are important in practice for their speed in number generation ... cipher. http eprint.iacr.org 2005 165.pdf Cryptographically secure pseudorandomnumber generators Main Cryptographically secure pseudorandomnumber generator A PRNG suitable for cryptography cryptographic ... criteria for quality of deterministic random number generators. They are summarized here K1 A sequence ... Attractors and TCP IP SequenceNumber Analysis an analysis of the strength of PRNGs used to create TCP IP sequencenumber s by various operating system s using strange attractor s. This is a good ... newtcp Strange Attractors and TCP IP SequenceNumber Analysis One Year Later a follow ... Security DEFAULTSORT PseudorandomNumber Generator Category Pseudorandomnumber generators cs ... Mar08 2007.pdf ref is an algorithm for generating a sequence of numbers that approximates the properties of random numbers. The sequence is not truly random in that it is completely determined ... . It will always produce the same sequence thereafter when initialized with that state. The maximum length of the sequence before it begins to repeat is determined by the size of the state, measured ... sequence. Mixes that are reversible permutations have periods of about 2 sup var n var 1 sup on average ... bit output. Most pseudorandom generator algorithms produce sequences which are uniform distribution ... quality PRNG from a truly random sequence without knowing the algorithm s used and the state with which ... random sequence. The simplest examples of this dependency are stream cipher s, which most often ... more details
sequence. This would be guaranteed if is a normal number , for example. However, this algorithm ... pseudorandomnumber generators, as their output is not completely determined by their initial state ... Test Suite for Random and PseudorandomNumber Generators , http csrc.nist.gov publications ... safe shrinking pseudorandomnumber generator. Interlang crypto navbox DEFAULTSORT Cryptographically Secure PseudorandomNumber Generator Categories Category Cryptographic algorithms Category Pseudorandomnumber generators Category Cryptographically secure pseudorandomnumber generators ...A cryptographically secure pseudo random number generator CSPRNG is a pseudo random number generator ... be a hardware random number generator or perhaps unpredictable system processes &mdash though ... bit test is as follows Given the first var k var bits of a random sequence, there is no polynomial ... output by computing bits of pi in sequence, starting from some unknown point in the binary expansion ... a 1, then encrypting a 2, etc. The counter can also be started at an arbitrary number ... doi isbn 978 0 7645 4975 5 ref Most stream cipher s work by generating a pseudorandom stream of bits ... will return a new pseudorandom stream, possibly with a longer period. The cipher is only secure ... state must be kept secret. Number theoretic designs The Blum Blum Shub algorithm has a security ... logarithm problem but is also very inefficient. Special designs There are a number of practical PRNGs ... hac about chap5.pdf Chapter 5 Pseudorandom Bits and Sequences PDF ref Each time a random number is required it Obtains the current date time D to the maximum resolution possible. Computes ... links Wikibooks Cryptography Random number generation Portal Cryptography RFC 4086, Randomness Requirements ... class providing a cryptographically strong pseudo random number generator PRNG . http blogs.msdn.com michael howard archive 2005 01 14 353379.aspx Cryptographically Secure Random number on Windows ... more details
node1.html ref See also Maximum length sequencePseudorandomnumber generator Pseudorandomness White ...In cryptography , pseudorandom noise PRN ref cite web url http www.gps.gov technical icwg meetings 2011 09 13 WAS IS FINAL PRN Expansion 4May2011.pdf title Change Topic Pseudorandom Noise PRN Expansion publisher GPS.GOV accessdate 13 July 2011 ref ref cite web url http www.losangeles.af.mil library factsheets factsheet.asp?id 8618 title Pseudorandom Noise PRN Code Assignments publisher Los Angeles Air Force Base accessdate 13 July 2011 ref is a Signalling telecommunication signal similar to noise physics noise which satisfies one or more of the standard tests for statistical randomness . Although it seems to lack any definite pattern , pseudorandom noise consists of a deterministic sequence of Pulse ... cryptographic device s, the pseudorandom noise pattern is determined by a key cryptography key and the repetition period can be very long, even millions of years. Pseudorandom noise is used ... sequence has a very low correlation with any other sequence in the set, or with the same sequence .... Unlike random noise, it must be easy to generate exactly the same sequence at both the transmitter and the receiver, so the receiver s locally generated sequence has a very high correlation with the transmitted sequence. In a direct sequence spread spectrum system, each bit in the pseudorandom binary sequence is known as a chip and the inverse of its period as chip rate . Compare bit rate and baud . In a frequency hopping spread spectrum sequence, each value in the pseudorandomsequence is known as a channel number and the inverse of its period as the hop rate . FCC Part 15 mandates at least ... Code A pseudonoise code PN code is one that has a spectrum similar to a algorithmically random sequence random sequence of bits but is Deterministic computation deterministically generated. The most commonly used sequences in direct sequence spread spectrum systems are maximal length sequence s, Gold ... more details
of this term, see Pseudorandomnumber generator In theoretical computer science , a pseudorandom generator PRG is a deterministic algorithm deterministic procedure that produces a pseudorandom distribution ... sup , where s n , is a pseudorandom generator against F sub n sub with bias if for every f in F sub ... the seed length and the quantity n s is called the stretch of the pseudorandom generator. Functions from the class F sub n sub are sometimes called adversaries . A pseudorandom generator against a family of adversaries F F sub n sub with bias n is a collection of pseudorandom generators G sub n sub 0, 1 sup s n sup 0, 1 sup n sup , where G sub n sub is a pseudorandom generator against ... , and one is interested in desigining a pseudorandom generator that is computable in the same or some closely related model. Pseudorandom generators in cryptography In cryptography , the class F usually ... bit output, and one is interested in designing pseudorandom generators that are computable by a polynomial time algorithm and whose bias is negligible in the circuit size. These pseudorandom generators are sometimes called cryptographically secure pseudorandom generators CSPRGs . It is not known if cryptographic pseudorandom generators exist. Their existence would imply that P NP . However, the existence of cryptographic pseudorandom generators is widely believed to be true and their existence is necessary for many applications in cryptography . The existence of cryptographic pseudorandom generators is equivalent to the existence of one way function s see Pseudorandom generator theorem . Applications Pseudorandom generators have numerous applications in cryptography. For instance, pseudorandom ... of key length. Key length can be significantly reduced using a pseudorandom generator if perfect security is replaced by semantic security . Common constructions of stream cipher s are based on pseudorandom generators. Pseudorandom generators may also be used to construct symmetric key algorithm symmetric ... more details
In cryptography , the term pseudorandom permutation , abbreviated PRP , refers to a function that cannot be distinguished from a random permutation that is, a permutation selected at random with uniform probability, from the family of all permutations on the function s domain with practical effort. A pseudorandom permutation family is a collection of pseudorandom permutations, where a specific permutation may be chosen using a key. The idealized abstraction of a block cipher is a truly random permutation. If a distinguishing algorithm exists that achieves significant advantage cryptography advantage with less effort than specified by the block cipher s security parameter this usually means the effort required should be about the same as a brute force search through the cipher s key space , then the cipher is considered broken at least in a certificational sense, even if such a break doesn t immediately lead to a practical security failure. See also Block cipher pseudorandom permutation families operating on fixed size blocks of bits Format Preserving Encryption pseudorandom permutation families operating on arbitrary finite sets Pseudorandom function Permutation box References cite book author Mihir Bellare , Phillip Rogaway date 2005 09 20 title Introduction to Modern Cryptography url http www cse.ucsd.edu mihir cse207 classnotes.html chapter Chapter 3 Pseudorandom functions accessdate 2007 09 30 Category Theory of cryptography crypto stub ... more details
Orphan date February 2009 Notability Products date August 2011 A pseudorandom encoder is an electro mechanical device used for measuring distance and finding position. It is different from the conventional absolute and incremental encoder in its disk pattern, which gives it aspects of the other two designs. The disk of a pseudorandom encoder has an evenly spaced notch design just as an incremental encoder to measure distance. A second track of code is used to find the position just like an absolute encoder however, an absolute encoder tells position from reading from multiple parallel tracks while a pseudorandom encoder uses one single track. This aspect allows pseudorandom rotary encoders to have smaller diameters and larger through holes. To know its position, the encoder must first move a certain distance to read the code. This distance is generally small, about 1 2 degrees for rotary encoders and 1 mm for linear encoders, though this idiosyncrasy does hinder its usefulness in some applications such as trackball track balls since position is not immediately known when the system starts. ref cite web url http www.virtualabsolute.com VAoverview.htm title A New Kind of Encoder virtualabsolute.com ref See also Rotary encoder References reflist Category Dimensional instruments ... more details
Let math U U n n in mathbb N math be a uniform ensemble and math X X n n in mathbb N math be an distribution ensemble ensemble . The ensemble math X math is called pseudorandom if math X math and math U math are Computationally indistinguishable indistinguishable in polynomial time . References Goldreich, Oded 2001 . Foundations of Cryptography Volume 1, Basic Tools . Cambridge University Press. ISBN 0 521 79172 3. Fragments available at the http www.wisdom.weizmann.ac.il oded frag.html author s web site . Category Algorithmic information theory Category Pseudorandomness Category Cryptography crypto stub ... more details
sequences of elements indexed by natural number s. It may be convenient to have the sequence start ... sequence Farey sequence Fibonacci number Fibonacci sequence Geometric progression Look and say sequence ...Other uses In mathematics , a sequence is an ordered list of objects or events . Like a Set mathematics set , it contains Element mathematics members also called elements or terms , and the number of terms possibly infinite is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence. A sequence is a Discrete mathematics discrete function mathematics function . For example, C, R, Y is a sequence of letters ..., or Infinite set infinite , such as the sequence of all even and odd numbers even ... notions of sequence, but may be excluded depending on the context. Image Cauchy sequence illustration2.svg right thumb 350px An infinite sequence of real numbers in blue . This sequence is neither increasing, nor decreasing, nor convergent, nor Cauchy sequence Cauchy . It is, however, bounded ... of which e.g. , exact sequence are not covered by the notations introduced below. In addition to identifying the elements of a sequence by their position, such as the 3rd element , elements may be given names for convenient referencing. For example a sequence might be written as a sub 1 sub , a sub ... definition of a finite sequence with terms in a set S is a function mathematics function from 1, 2, ..., n to S for some n 0. An infinite sequence in S is a function from 1, 2, ... to S . For example, the sequence of prime numbers 2,3,5,7,11, is the function 1 2 , 2 3 , 3 5 , 4 7 , 5 11 , . A sequence of a finite length n is also called an n tuple n tuple . Finite sequences include the empty sequence ... sequence or two way infinite sequence . An example is the bi infinite sequence of all even integers , 4, 2, 0, 2, 4, 6, 8 . Multiplicative Let A a sequence defined by a function f 1, 2, 3, ... 1, 2 ... more details
Infobox musical artist See Wikipedia WikiProject Musicians name The Sequence image caption image size Only for images narrower than 220 pixels background group or band alias origin Columbia, South Carolina Columbia , South Carolina , United States U.S. genre Old school hip hop br Funk years active 1979 1985 label Sugar Hill Records rap Sugar Hill associated acts Spoonie Gee website past members Angie Stone Angie Brown Stone Angie B. br Cheryl Cook Cheryl The Pearl br Gwendolyn Chisolm Blondie The Sequence is a former female old school hip hop trio signed to the Sugar Hill Records rap Sugar Hill label in the early 1980s. The group consisted of Cheryl Cook Cheryl The Pearl , Gwendolyn Chisolm Blondie , and lead singer rapper Angie Stone Angie Brown Stone Angie B. . The group originated from Columbia, South Carolina Columbia , South Carolina as a group of high school cheerleader s. Their most notable single was Funk You Up 1979 , which was the first rap record released by a female group and the second single released by Sugar Hill Records rap Sugar Hill Records . ref name Greenberg1999 Greenberg, Steve Light, Alan ed. 1999 . The VIBE History of Hip Hop . Three Rivers Press. p. 28. ISBN 0609805037 ref Elements of Funk You Up were later used by Dr. Dre for his 1995 single Keep Their Heads Ringin . ref Ego Trip s Book of Rap Lists Book of Rap Lists . 1999. Macmillan Publishers Macmillan ... Hold and later a solo artist. Discography Albums Sugarhill Presents the Sequence 1980 , Sugar Hill Records rap Sugar Hill The Sequence 1982 , Sugar Hill 51 Black Albums The Sequence Party 1983 , Sugar Hill Compilations Funky Sound 1995 , P Vine The Best of the Sequence 1996 , Deep Beats Monster ... The Sequence . Allmusic . External links http www.discogs.com artist Sequence, The Discography DEFAULTSORT Sequence, The Category African American musical groups Category African American rappers Category Female hip hop groups Category Female rappers Category Musical trios Hiphop band stub no The Sequence ... more details
In cryptography , a pseudorandom function family , abbreviated PRF , is a collection of efficiently computable Function computer science functions which emulate a random oracle in the following way no efficient algorithm can distinguish with significant Advantage cryptography advantage between a function chosen randomly from the PRF family and a random oracle a function whose outputs are fixed completely at random . Pseudorandom functions are vital tools in the construction of cryptographic primitive s, especially secure encryption encryption schemes . Pseudorandom functions are not to be confused with pseudorandomnumber generator pseudorandom generators PRGs . The guarantee of a PRG is that a single output appears random if the input was chosen at random. On the other hand, the guarantee of a PRF is that all its outputs appear random, regardless of how the corresponding inputs were chosen, as long as the function was drawn at random from the PRF family. A pseudorandom function family can be constructed from any pseudorandom generator, using, for example, the construction given by Goldreich, Shafi Goldwasser Goldwasser , and Micali. ref Oded Goldreich , Shafi Goldwasser , Silvio Micali 1986 How to Construct Random Functions , Journal of the ACM , vol.33, no.4, p.792 807. doi 10.1145 6490.6503 http theory.lcs.mit.edu cis pubs shafi 1986 jacm.pdf preprint http www.math.weizmann.ac.il oded ggm.html web page and preprint ref See also Pseudorandom permutation References references Category Theory of cryptography Category Cryptographic primitives Category Pseudorandomness he ... more details
between two of the earlier results. This attacker wants to predict the next sequence element. Such an attack ... 1 1 4 1 4 math . Then, the attacker wants to predict the next sequence element of this function, math ... f a 1 math and math f a 5 math . There are other attacks that would be very bad for a PseudorandomNumber Generator the user expects to get random numbers from the output, so of course the stream should ... Complexity One natural measure of how useful a sequence may be for cryptography cryptographic purposes is the size of its linear complexity . The linear complexity of an n element sequence W x , x ... sub math mathcal R math , which is satisfied by this sequence. For some math gamma math 0,n 1 math gamma ... of the sequence math f a x math ,0 x 2 sup n 1 sup , denoted by math L a math satisfies math L a geqslant ... F p math , then each vector a defines a Sequence Finite and infinite finite sequence in the subgroup ... elliptic curve sequence is defined as math u k X f a k mbox where X P mbox is the abscissa of P in E ... generator Generalized inversive congruential pseudorandom numbers Notes Reflist refs ref name NaorReingold Naor, M., Reingold, O. Number theoretic constructions of efficient pseudo random functions ... complexity of the Naor Reingold sequence with elliptic curves, Finite Fields and Their Applications ... Proceedings of the Third International Symposium on Algorithmic Number Theory,1998,48 63. ref References citation last Shparlinski first Igor title Cryptographic Applications of Analytic Number Theory ... and Pseudorandomness year 1998 edition first publisher Springer isbn 978 3540647669 Category Pseudorandomnumber generators Category Cryptography ... more details
number i 1 . Now, assume that G sub l sub is not a pseudorandom distribution that is, there exists ...In computational complexity theory and cryptography , the existence of pseudorandom generator s is related to the existence of one way function s through a number of theorems, collectively referred to as the pseudorandom generator theorem . Introduction Pseudorandomness A distribution is considered pseudorandom ... D sub n sub is pseudorandom if for any polynomial size circuit C , and any inversely polynomial ... . Pseudorandom generators A function G sub l sub 0,1 sup l sup 0,1 sup m sup , where l m is a pseudorandom generator if G sub l sub can be computed in time polynomial in l G sub l sub x is pseudorandom, when x is uniformly random. One additional pseudorandom bit implies polynomially more pseudorandom bits It can be shown that if there is a pseudorandom generator G sub l sub 0,1 sup l sup 0,1 sup l 1 sup , i.e. a generator that adds only one pseudorandom bit, then for any m poly l , there is a pseudorandom generator G sub l sub 0,1 sup l sup 0,1 sup m sup ... and used as the seed to the first instance of G sub l sub , which is known to be a pseudorandom generator ... of the output. Repeating this process for m times yields an output of m pseudorandom bits. It can be shown that such G sub l sub , that consists of m instances of G sub l sub , is indeed a pseudorandom ... l sub or a drawn from uniform distribution. Let s reuse the approach of building large pseudorandom generators out of instances of G sub l sub and construct a string of pseudorandom bits of length m&minus ... is not a pseudorandom generator, which is a contradiciton to the hypothesis. Q.E.D. Now, let s illustrate ... to circuit C for distinguishing between G sub l sub and U sub l 1 sub . Existence of pseudorandom generators The existence of pseudorandom generators is related to the existence of one way function s and hard core predicate s. Formally, pseudorandom generators exist if and only if one way functions ... more details
orphan date February 2010 In theoretical computer science a pseudorandom generator for low degree polynomials is an efficiently computable function whose output is indistinguishable from the uniform distribution by evaluation of low degree polynomials in the following sense. Definition A pseudorandom generator math G mathbb F s rightarrow mathbb F n math for polynomials of degree math d math over a Finite field F is an efficient procedure that stretches math s n math field elements into math n math field elements and fools any polynomial of degree d in n variables over F For every such polynomial p, the Total variation Total variation distance in probability theory statistical distance between the distributions math p U n math , for uniform math U n math in math mathbb F n math , and math p G U s math , for uniform math U s math in math mathbb F s math , is at most a small math epsilon math . Construction The case of linear polynomials is solved by Epsilon Biased Sample Spaces small bias spaces which give constructions with seed length math s O log n log 1 epsilon math this is optimal up to constant factors . Following the sequence of papers http www.ccs.neu.edu home viola papers gen.pdf , http shachar.lovett.googlepages.com prg poly.pdf it was established in http www.ccs.neu.edu home viola papers d.pdf that a sum of math d math small bias spaces fools degree math d math polynomials. This gives a construction with seed length math s O log n 2 d log 1 epsilon math . References http www.ccs.neu.edu home viola papers gen.pdf The paper proposed taking a sum of independent small bias spaces for fooling low degree polynomials . http shachar.lovett.googlepages.com prg poly.pdf The paper gave the first unconditional result showing that sum of math 2 d math small bias spaces fools low degree polynomials . http www.ccs.neu.edu home viola papers d.pdf The paper shows that sum of math d math small bias spaces fools low degree polynomials . Category Pseudorandomness ... more details
An M sequence may refer to Regular sequence algebra Regular sequence , which is an important topic in commutative algebra. A maximum length sequence , which is a type of pseudorandom binary sequence. disambig ... more details
multiple image direction vertical width footer 256 points from a pseudorandomnumber source top compared with the first 256 points from the 2,3 Halton sequence below . The Halton sequence covers the space more evenly. image1 Pseudorandomsequence 2D.svg alt1 caption1 image2 Halton sequence 2D.svg alt2 caption2 In statistics , Halton sequences Who s Halton? are sequence s used to generate points in space for numerical methods such as Monte Carlo simulations . Although these sequences are Deterministic system mathematics deterministic they are of Low discrepancy sequence low discrepancy , that is, appear to be random for many purposes. They were first introduced in 1960 and are an example of a quasi random numbersequence. They generalise the one dimensional van der Corput sequence s. Example of Halton sequence used to generate points in 0, 1 × 0, 1 in R sup 2 sup The Halton sequence is constructed according to a deterministic method that uses a prime number as its base. As a simple example, let s take one dimension of the Halton sequence to be based on 2 and the other on 3. To generate the sequence for 2, we start by dividing the interval 0,1 in half, then in fourths, eighths, etc, which generates frac 2 , frac 4 , frac 3 4 , frac 8 , frac 5 8 , frac 3 8 , frac 7 8 , frac 16 , frac 9 16 ,... and to generate the sequence for 3, we divide the interval 0,1 in thirds, then ninths ... , frac 2 9 , frac 5 9 , frac 8 9 , frac 27 ,... When we pair them up, we get a sequence of points in a unit ... to drop the first 20 entries, or some other predetermined number depending on the primes ... prominent solutions is the scrambled Halton sequence, which uses permutations of the coefficients used in the construction of the standard sequence. Implementation in Pseudo Code FUNCTION index, base ... number generation and quasi Monte Carlo methods publisher Society for Industrial and Applied Mathematics ... Radical inverse quasi random point sequence publisher ACM date 1964 doi 10.1145 355588.365104 page ... more details
In mathematics, a Sturmian sequence may refer to A Sturmian word a sequence with minimal complexity function A sequence used to determine the number of distinct real roots of a polynomial by Sturm s theorem mathdab ... more details
number s Baum Sweet sequence Bell number s Binomial coefficient s Carmichael number s Catalan number s Composite number s Deficient number s Euler number s Even and odd numbers Factorial numbers Fibonacci number s Fibonacci word Figurate numbers Golomb sequence Happy number s Highly totient number s Highly composite number s Home prime s Hyperperfect number s Juggler sequence Kolakoski sequence Lucky number s Lucas number s Padovan sequence Padovan number s Partition number s Perfect number s Pseudoperfect number s Prime number s Pseudoprime numbers Regular paperfolding sequence Rudin Shapiro sequence Semiperfect number s Semiprime numbers Superperfect number s Thue Morse sequence Ulam numbers Weird number s div Computable and definable sequences An integer sequence is a Recursion theory computable sequence , if there exists an algorithm which given n , calculates a sub n sub , for all n > 0. An integer sequence is a definable set definable sequence , if there exists some statement P x which is true for that integer sequence x and false for all other integer sequences. The set of computable ...In mathematics , an integer sequence is a sequence i.e., an ordered list of integer s. An integer sequence ... between its terms. For example, the sequence 0, 1, 1, 2, 3, 5, 8, 13, the Fibonacci numbers Fibonacci sequence is formed by starting with 0 and 1 and then adding any two consecutive terms to obtain the next one an implicit description. The sequence 0, 3, 8, 15, is formed according ..., an integer sequence may be defined by a property which members of the sequence possess and other integers do not possess. For example, we can determine whether a given integer is a perfect number , even though we do not have a formula for the n th perfect number. Examples Integer sequences .... Complete sequences An integer sequence is called a complete sequence if every positive integer can be expressed as a sum of values in the sequence, using each value at most once. See also On Line Encyclopedia ... more details
In mathematics , the term normal sequence has multiple meanings, depending on the area of specialty. In general, it is a sequence with nice properties. In set theory , a normal function normal sequence is one that is continuous function set theory continuous and strictly increasing . In probability theory , normal sequence may be used synonymously with normal number . References Thomas Jech . Set Theory , 3rd millennium ed., 2002, Springer Monographs in Mathematics,Springer, ISBN 3 540 44085 2 disambig ... more details
In mathematics, a polyphase sequence is a sequence whose terms are complex number complex root of unity roots of unity math a n e i frac 2 pi q x n , math where x sub n sub is an integer . Polyphase sequences is an important class of sequences and play important roles in synchronizing sequence design. References Pingzhi Fan and Michael Darnell, Sequence Design for Communications Applications , 1996 Category Sequences and series ... more details
In mathematics , an interleave sequence is obtained by merging together two sequence s. Let math S math be a Set mathematics set , and let math x i math and math y i math , math i 0,1,2,..., math be two sequence s in math S. math The interleave sequence is defined to be the sequence math x 0, y 0, x 1, y 1, dots. math Formally, it is the sequence math z i , i 0,1,2,... math given by math z i left begin matrix x k & mbox if i 2k mbox is even, y k & mbox if i 2k 1 mbox is odd. end matrix right. math Properties The interleave sequence math z i math is limit of a sequence convergent if and only if the sequences math x i math and math y i math are convergent and have the same limit. Consider two real number s a and b greater than zero and smaller than 1. One can interleave the sequences of digits of a and b , which will determine a third number c , also greater than zero and smaller than 1. In this way one obtains an Injective function injection from the square 0, 1 × 0, 1 to the interval mathematics interval 0, 1 . planetmath id 449 title Interleave sequence Category Real analysis Category Sequences and series ... more details
for sequence transformations include the binomial transform , M bius transform , Stirling transform and others. Definitions For a given sequence math S s n n in N , , math the transformed sequence is math mathbf T S S s n n in N , , math where the members of the transformed sequence are usually computed from some finite number of members of the original sequence, i.e. math s n T s n,s n 1 , dots ...In mathematics , a sequence transformation is an Operator mathematics operator acting on a given space of sequence s. Sequence transformations include linear mappings such as convolution with another sequence, and resummation of a sequence and, more generally, are commonly used for series acceleration , that is, for improving the rate of convergence of a slowly convergent sequence or series mathematics series . Sequence transformations are also commonly used to compute the antilimit of a divergent ... . In the simplest case, the math s n math and the math s n math are real number real or complex number ... of convergence, the transformed sequence is said to converge faster than the original sequence ... , assumed to be convergent. In this case, convergence acceleration is obtained. If the original sequence is divergent , the sequence transformation acts as extrapolation method to the antilimit math ... 0 k c m s n m math for some constants math c 0, dots,c k math which may depend on n , the sequence transformation math mathbf T math is called a linear sequence transformation . Sequence transformations that are not linear are called nonlinear sequence transformation s. Examples Simplest examples of linear sequence transformations include shifting all elements, math s n s n k math resp. 0 if n k 0 for a fixed k , and scalar multiplication of the sequence. A little less trivial ... sequence. The binomial transform is another linear transformation of a still more general type. An example of a nonlinear sequence transformation is Aitken s delta squared process , used to improve ... more details
Unreferenced date December 2009 In mathematics , an ergodic sequence is a certain type of integer sequence , having certain equidistribution properties. Definition Let math A a j math be an infinite, strictly increasing sequence of positive integers. Then, given an integer q , this sequence is said to be ergodic mod q if, for all integers math 1 leq k leq q math , one has math lim t to infty frac N A,t,k,q N A,t frac 1 q math where math N A,t mbox card a j in A a j leq t math and cardinality card is the count the number of elements of a set, so that math N A,t math is the number of elements in the sequence A that are less than or equal to t , and math N A,t,k,q mbox card a j in A a j leq t, , a j mod q k math so math N A,t,k,q math is the number of elements in the sequence A , less than t , that are equivalent to k modulo q . That is, a sequence is an ergodic sequence if it becomes uniformly distributed mod q as the sequence is taken to infinity. An equivalent definition is that the sum math lim t to infty frac 1 N A,t sum j a j leq t exp frac 2 pi ika j q 0 math vanish for every integer k with math k mod q ne 0 math . If a sequence is ergodic for all q , then it is sometimes said to be ergodic for periodic systems . Examples The sequence of positive integers is ergodic for all q . Almost all Bernoulli sequence s, that is, sequences associated with a Bernoulli process , are ergodic for all q . That is, let math Omega,Pr math be a probability space of random variable s over two letters math 0,1 math . Then, given math omega in Omega math , the random variable math X j omega math is 1 with some probability p and is zero with some probability 1 p this is the definition of a Bernoulli process. Associated with each math omega math is the sequence of integers math mathbb Z omega n in mathbb Z X n omega 1 math Then almost every sequence math mathbb Z omega math is ergodic. See ... Sequence Category Ergodic theory Category Sequences and series ... more details
Unreferenced date July 2009 In mathematics , a periodic sequence is a sequence for which the same terms are repeated over and over a sub 1 sub , a sub 2 sub , ..., a sub p sub , a sub 1 sub , a sub 2 sub , ..., a sub p sub , a sub 1 sub , a sub 2 sub , ..., a sub p sub , ... The number p of repeated terms is called the period . Definition A periodic sequence is a sequence a sub 1 sub , a sub 2 sub , a sub 3 sub , ... satisfying a sub n p sub a sub n sub for all values of n . If we regard a sequence as a function mathematics function whose domain is the set of natural number s, then a periodic sequence is simply a special type of periodic function . Examples The sequence of digits in the decimal expansion of 1 7 is periodic with period six 1 7 0 . 1 4 2 8 5 7 1 4 2 8 5 7 1 4 2 8 5 7 ... More generally, the sequence of digits in the decimal expansion of any rational number is eventually periodic see below . The sequence of powers of &minus 1 is periodic with period two &minus 1, 1, &minus 1, 1, &minus 1, 1, ... More generally, the sequence ... math is a periodic sequence. Periodic points are important in the theory of dynamical systems . Generalizations A sequence is eventually periodic if it can be made periodic by dropping some finite number of terms from the beginning. For example, the sequence of digits in the decimal expansion of 1 56 ... 7 1 4 2 ... A sequence is asymptotically periodic if its terms approach those of a periodic sequence. That is, the sequence x sub 1 sub , x sub 2 sub , x sub 3 sub , ... is asymptotically periodic if there exists a periodic sequence a sub 1 sub , a sub 2 sub , a sub 3 sub , ... for which math lim n rightarrow infty x n a n 0. math For example, the sequence ... periodic, since its terms approach those of the periodic sequence 0, 1, 0, 1, 0, 1, .... DEFAULTSORT Periodic Sequence Category Sequences and series fr Suite p riodique ... more details
Notability date March 2011 Refimprove date March 2011 Sequence , a board game board and card game , was invented by Douglas Reuter in Minneapolis, Minnesota over a two year period in the 1970 s. In June, 1981, Mr. Reuter entered into a license agreement with Jax Ltd., Inc. which authorizes it to manufacture and distribute Sequence and was first sold in a retail store in 1982. It involves an average sized playing board, on which are depicted two decks of playing cards minus the Jokers and Jacks , two full standard decks, 35 red, 50 blue and 50 green playing chips which total 135 ref name rules Rules from official site http www.jaxgames.com seq.htm ref . The game is played by two players, two teams, three players or three teams. Players are dealt a certain number of cards 3 7 depending on the number ... teams or three players only 1 sequence needs to be formed in order to win. The four corners of the grid ... as your own color chip, that is any series of 4 chips including a wild square forms a sequence. The jacks ... an opposing player s token from anywhere. Sequence employs some distinct hard and fast rules that include ... of the board. No other number or suit is eligible to become a foot , only the 9 of Spades. If a chip ... the Sequence. Variations Sequence comes in several versions including Sequence States & Capitals, Sequence Numbers, Sequence 25th Anniversary Edition, Jumbo Sequence, Travel Sequence, Sequence Deluxe Edition, and Sequence for Kids. There is also another variation, called Sequence Dice that uses ... one and numbers and dice are used instead of cards. The object of Sequence dice is to be the first person to connect a sequence of five chips in a row on the board, just as in the original game. Another major difference is that only one sequence is required instead of the two sometimes ... Jack misc jack.html . ref References Reflist External links http www.jaxgames.com seq.htm Sequence at jax, Ltd bgg 2375 Sequence Category Abstract strategy games Category Miscellaneous card games ... more details
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