Summation may refer to Summation , a mathematical notation Evaluating sums , evaluation of sums in closed form Summation Neurophysiology , a way of achieving action potential in a neuron In law, another name for the closing argument disambig ... more details
Orphan date October 2008 The Wolf summation , named after ????? Wolf , is a method for computing the electrostatic interactions of systems eg. crystal s . This method is generally more computationally efficient than the Ewald summation . Category Potential theory atomic physics stub ... more details
Orphan date October 2008 The summation generator , created in 1985, by Rainer Rueppel, was a cryptography and security front runner in the late 1980s. It operates by taking the output of two LFSR s through an adder with carry. The operation s strength is that it is nonlinear . However, through the early 1990s various attacks against the summation generator eventually led to its fall to a correlation attack. In 1995 Klapper and Goresky were able to determine the summation generator s sequence in only 2 sup 19 sup bits. An improved summation generator with 2 bit memory was then proposed by cryptographers Lee and Moon. In the new generator scheme an extra bit of memory is added to the nonlinear combining Function mathematics function . The objective in the modification was to make the summation generator immune to correlation attack . An attack against the improved summation generator was reported by Mex Perera and Shepherd in 2002 by exploting linear relations. Besides, in June 2005 an algebraic attack was developed. Using this attack a PC can calculate the initial state of the summation generator within 3 minutes even with 256 bit LFSR s References R. A. Rueppel, Correlation immunity and the Summation Generator, Advances in Cryptography EUROCRYPT 85 proceedings, Berlin Springer Verlag, 1986, pp. 260 272. W.Meier and O. Staffelbach, Correlation properties of Combiners with Memory in Stream Ciphers, Advances in Cryptography EUROCRYPT 90 proceedings, Berlin Springer Verlag, 1991, pp. 204 213. Bruce Schneir, Applied Cryptography, pg. 364, Summation Generator Mex Perera, J. C. and Shepherd, S. J. 2002. Cryptanalysis of a summation generator with 2 bit memory . Signal Process. 82, 12 Dec. 2002 , 2025 2028. An algebraic attack on the improved summation generator with 2 bit memory ... links http dsns.csie.nctu.edu.tw research crypto HTML PDF C85 260.PDF Correlation Immunity and the Summation ... Algebraic Attacks on Summation Generators Category Cryptographic algorithms ... more details
Summation notation may refer to Einstein notation , summation over like subscripted indices Capital sigma notation , mathematical symbol for summation mathdab Short pages monitor This long comment was added to the page to prevent it being listed on Special Shortpages. It and the accompanying monitoring template were generated via Template Longcomment. Please do not remove the monitor template without removing the comment as well. ... more details
In mathematics , a summation equation or discrete integral equation is an equation in which an unknown function mathematics function appears under a summation sign. The theories of summation equations and integral equation s can be unified as integral equations on time scales ref http web.maths.unsw.edu.au cct tis tomasia IJDE rev.pdf Volterra integral equations on time scales Basic qualitative and quantitative results with applications to initial value problems on unbounded domains , Tomasia Kulik, Christopher C. Tisdell, September 3, 2007 ref using time scale calculus . A summation equation compares to a difference equation as an integral equation compares to a differential equation . The Volterra summation equation is math x t f t sum i m n k t, s, x s math where x is the unknown function, and s, a, t are integers, and f, k are known functions. References references http scholar.google.com scholar?q 22discrete integral equations 22 OR 22summation equations 22 OR 22discrete integral equation 22 OR 22summation equation 22 Summation equations or discrete integral equations Category Integral equations ... more details
Retinal summation describes the relationship between different types of cells in the retina Cone cell cone photoreceptor cells , Bipolar cell of the retina bipolar cell s, and Retinal ganglion cell ganglion cell s. With high retinal summation, a large number of photoreceptor cells converge on a smaller number of bipolar cells in transferring their signals to ganglion cells. Zero summation occurs when each cone photoreceptor cell contacts a single ganglion cell via a single bipolar cell. ref name 2000Kay Cite journal last1 Kay first1 R.F. last2 Kirk first2 E.C. title Osteological evidence for the evolution of activity pattern and visual acuity in primates journal American Journal of Physical Anthropology year 2000 volume 113 pages 235 262 doi 10.1002 1096 8644 200010 113 2 235 AID AJPA7 3.0.CO 2 9 pmid 11002207 ref High summation increases sensitivity to light at the expense of visual acuity . Low retinal summation results in high visual acuity, with individual photoreceptor cells sending their own signals. High retinal summation yields high sensitivity to low light levels, where the signal is summed before reaching the brain presumably advantageous when the signals reaching individual photoreceptor cells are weak. ref name 2000Kay High retinal summation is an adaptation to low light levels, and low retinal summation to high light levels thus sharpening the images . ref name 2000Kay References Reflist Category eye ... more details
Expert subject Mathematics date November 2008 Ramanujan summation is a technique invented by the mathematician Srinivasa Ramanujan for assigning a sum to infinite divergent series . Although the Ramanujan summation of a divergent series is not a sum in the traditional sense, it has properties which make it mathematically useful in the study of divergent infinite series , for which conventional summation is undefined. Ramanujan summation essentially is a property of the partial sums, rather than a property of the entire sum, as that doesn t exist. If we take the Euler Maclaurin formula Euler Maclaurin summation formula together with the correction rule using Bernoulli number s, we see that math ... normally recover the usual summation for convergent series. For functions f x with no divergence at x ... summation and integration. Using standard extensions for known divergent series, he calculated Ramanujan summation of those. In particular, the sum of nowrap 1 2 3 4 is math 1 2 3 cdots frac 1 12 Re math where the notation math Re math indicates Ramanujan summation. This formula originally appeared in one of Ramanujan s notebooks, without any notation to indicate that it was a Ramanujan summation ... s summation, since then it can be assured that one series math sum k 1 infty f n math admits one and only one Ramanujan s summation, defined as the value in 1 of the only solution of the difference equation ..., http algo.inria.fr seminars sem01 02 delabaere2.pdf Ramanujan s Summation , Algorithms Seminar 2001 2002 , F. Chyzak ed. , INRIA, 2003 , pp. 83 88. ref This new definition of Ramanujan s summation denoted as math sum n ge 1 Re f n math does not coincide with the earlier defined Ramanujan s summation C 0 nor with the summation of convergent series, but it has interesting properties, such as If R ... using the Euler Maclaurin summation formula one can write math I n, Lambda frac n 2 I n 1, Lambda ... Field theories See also Borel summation Ces ro summation Ramanujan s sum Divergent series References ... more details
Euler summation is a summability method for convergent series convergent and divergent series . Given a series a sub n sub , if its Euler transform converges to a sum, then that sum is called the Euler sum of the original series. Euler summation can be generalized into a family of methods denoted E, q , where q 0. The E, 0 sum is the usual convergent sum, while E, 1 is the ordinary Euler sum. All of these methods are strictly weaker than Borel summation for q 0 they are incomparable with Abel summation . Definition Euler summation is particularly used to series acceleration accelerate the convergence of alternating series and allows evaluating divergent sums. math E y , sum j 0 infty a j sum i 0 infty frac 1 1 y i 1 sum j 0 i i choose j y j 1 a j . math To justify the approach notice that for interchanged sum, Euler s summation reduces to the initial series, because math y j 1 sum i j infty i choose j frac 1 1 y i 1 1. math This method itself cannot be improved by iterated application, as math E y 1 E y 2 sum , E frac y 1 y 2 1 y 1 y 2 sum. math Examples We have math sum j 0 infty x j P k j sum i 0 k frac x i 1 x i 1 sum j 0 i i choose j 1 i j P k j math , if math P k math is a polynomial of Degree of a polynomial degree k. Note that in this case Euler summation reduces an infinite series to a finite sum. The particular choice math P k j j 1 k math provides an explicit representation of the Bernoulli numbers , since math zeta k frac B k 1 k 1 math . Indeed, applying Euler summation to the zeta function yields math frac 1 1 2 k 1 sum i 0 k frac 1 2 i 1 sum j 0 i i choose j 1 j j 1 k math , which is polynomial for math k math a positive integer cf. Riemann zeta function Representations Riemann zeta function . math sum j 0 infty z j sum i 0 infty frac 1 1 y i 1 sum j 0 i ... of math y math this series converges to math frac 1 1 z math . See also Borel summation Ces ro summation Lambert summation Perron s formula Abelian and tauberian theorems Abel s summation formula Van ... more details
For the song Cesaro Summability by the band Tool band Tool nima In mathematical analysis , Ces ro summation is an alternative means of assigning a sum to an Series mathematics infinite series . If the series Convergent series converges in the usual sense to a sum A , then the series is also Ces ro summable and has Ces ro sum A . The significance of Ces ro summation is that a series which does not converge may still have a well defined Ces ro sum. Ces ro summation is named for the Italian analyst Ernesto Ces ro 1859 1906 . Definition Let a sub n sub be a sequence , and let math s k a 1 cdots a k math be the k th partial sum of the series math sum n 1 infty a n. math The series s sub n sub is called Ces ro summable , with Ces ro sum math A in R math , if the average value of its partial sums tends to math A math math lim n to infty frac 1 n sum k 1 n s k A. math In other words, the Ces ro ... or repeat any obvious pattern. C, summation In 1890, Ernesto Ces ro stated a broader family of summation methods which have since been called C, n for non negative integers n . The C, 0 method is just ordinary summation, and C, 1 is Ces ro summation as described above. The higher order methods ... iterated application of the initial summation method and can be restated as math C, alpha sum j 0 infty ... . Then the C, sum of a sub n sub is defined as above. The existence of a C, summation implies every higher order summation, and also that a sub n sub o n sup sup if ... for all , and the value of the resulting limit is the same. See also Abel summation Abel s summation formula Borel summation Euler summation Lambert summation Ces ro mean Divergent ... theorem Summation by parts References citation last1 Shawyer first1 Bruce first2 Bruce last2 Watson ... 1986 publisher Chelsea Pub. Co. location New York, N.Y. . springer title Ces ro summation methods ... isbn 978 0521358859 edition 2nd . DEFAULTSORT Cesaro summation Category Summability methods es Sumaci n ... more details
can interact with each other when converting input to output. Summation , also known as frequency summation ... potential will be triggered by the summation of postsynaptic potential s. ref name temporal spatial cite web title temporal summation url http lecerveau.mcgill.ca flash capsules articles pdf temporal ... potential generation can be triggered by the summation of these individual units, each equivalent ... USA ref Types Image Temporal summation.JPG right thumb 300px A diagram of temporal summation. Main ... upon the spatial from multiple neurons and temporal summation from a single neuron of all inputs at that moment. In addition, the summation of excitatory and inhibitory influences will modulate the outcome ... summation. This is due to the fact that the postsynaptic potentials travel through the unmyelinated dendrites to the axon hillock, where they are combined in summation. Because dendrites are unmyelinated .... ref name temporal spatial Spatial summation Spatial summation is a way of achieving an action potential in a neuron which involves input from multiple presynaptic cells. Spatial summation is the algebraic summation of potentials from different areas of input, usually on the dendrites . Summation ... Neurophysiology year 2000 publisher W.B. Saunders Company location New York ref Temporal summation Temporal summation is another means of transmitting signals with increased frequency of impulse thus increasing the strength of signals in each fiber called Temporal summation . ref cite book title ... neuron as a way of achieving action potential . Summation occurs when the time constant Time constants ... stimulation can lead to EPSP summation and to surpassing the threshold potential. ref cite book ... of the postsynaptic cell, or IPSP. Summation with other IPSPs and contrasting EPSPs ... spatial summation and is complemented by temporal summation, wherein successive releases ..., and thereby allow summation of effect. thus, the EPSP differs from action potentials in a fundamental ... more details
In numerical analysis , pairwise summation , also called cascade summation , is a technique to sum a sequence ... Citation title The accuracy of floating point summation first1 Nicholas J. last1 Higham journal ... Although there are other techniques such as Kahan summation that typically have even smaller round off errors, pairwise summation is nearly as good differing only by a logarithmic factor while having ... the same number of arithmetic operations as naive summation. In particular, pairwise summation ... name Higham93 Kahan summation has a error bound worst case error of roughly O , independent of n ... to an average of math O varepsilon sqrt log n math for pairwise summation. ref name Tasche ... Boca Raton, FL CRC Press, 2000 . ref Precisely the recursive structure of pairwise summation is found ... In pseudocode , the pairwise summation algorithm for an Array data type array x of length n can be written s pairwise x 1&hellip n if n &le N base case naive summation for a sufficiently small ... sufficiently small N , this algorithm switches to a naive loop based summation as a base case , whose ... of a large base case for recursive summation is employed by high performance FFT implementations ..., the same as for naive summation, so if the recursion overhead is made negligible then pairwise summation has essentially the same computational cost as for naive summation. A variation on this idea ... with infinite precision . With pairwise summation for a base case N 1, one instead obtains ... error bound, the fraction &Sigma x sub i sub &Sigma x sub i sub is the condition number of the summation problem. Essentially, the condition number represents the intrinsic sensitivity of the summation ... Linear Algebra SIAM Philadelphia, 1997 . ref The relative error bound of every backwards stable summation ... to this condition number. ref name Higham93 An ill conditioned summation problem is one in which this ratio is large, and in this case even pairwise summation can have a large relative error. For example ... more details
In telecommunication , the term summation check sum check has the following meanings A checksum based on the formation of the sum of the digits of a numeral. Note The sum of the individual digits is usually compared with a previously computed value. A comparison of checksums on the same data on different occasions or on different representations of the data in order to verify data integrity . References FS1037C Category Error detection and correction telecomm stub ... more details
quote box align right width 33 quote Borel, then an unknown young man, discovered that his summation method gave the right answer for many classical divergent series. He decided to make a pilgrimage to Stockholm to see G sta Mittag Leffler Mittag Leffler , who was the recognized lord of complex analysis. Mittag Leffler listened politely to what Borel had to say and then, placing his hand upon the complete works by Karl Weierstrass Weierstrass , his teacher, he said in Latin, The Master forbids it . source Mark Kac , quoted by harvtxt Reed Simon 1978 p 38 In mathematics, Borel summation is a summation method for divergent series , introduced by harvs txt first Emile last Borel year 1899 authorlink Emile Borel . There are several variations of this method that are also called Borel summation. Definition Let math y z sum k 0 infty y kz k math be a formal power series in z . Define the Borel transform math mathcal B y math of math y math by math mathcal B y t sum k 0 infty frac y k k t k. math Suppose that the Borel transform converges to an analytic function near 0 that can be analytic continuation analytically continued along the positive real axis to a function growing sufficiently slowly that the following integral is well defined as an improper integral . Then the Borel sum of y is given by math int 0 infty e t mathcal B y tz , dt. math A slightly weaker form of Borel s summation ... by the original divergent series. This is a typical example of the fact that Borel summation ... points and are orthogonal to lines joining the singular points to 0. Applications Borel summation ... be recovered from their perturbation series using Borel summation harv Glimm Jaffe 1987 p 461 . Some ... field theory harv Weinberg 2005 loc 20.7 . Notes references See also Divergent series Euler summation Ces ro summation Lambert summation Nachbin resummation Abelian and tauberian theorems Van Wijngaarden ... 521 55002 4 mr 2148467 year 2005 eom id B b017170 first A. A. last Zakharov title Borel summation method ... more details
refimprove date September 2011 In signal processing , any periodic function math f T math with period math T, math can be represented by a summation of an infinite number of instances of an aperiodic function, math f, math that are offset by integer multiples of math T. math This representation is called periodic summation math f T t sum n infty infty f t nT sum n infty infty f t nT . math When math f T math is alternatively represented as a complex Fourier series , the Fourier coefficients are proportional to the values or samples of the continuous Fourier transform of math f math at intervals of math scriptstyle 1 T. math ref The form of the transform assumed here is math F nu stackrel mathrm def int infty infty f t e i2 pi nu t , dt. math ref That identity is known as the Poisson summation formula . Quotient space as domain If a periodic function is represented using the Quotient space linear algebra quotient space Domain of a function domain math mathbb R T cdot mathbb Z math then one can write math varphi T mathbb R T cdot mathbb Z to mathbb R math math varphi T t sum tau in t f tau math instead. The arguments of math varphi T math are equivalence class es of real number s that share the same fractional part when divided by math T math . Notes reflist See also Dirac comb Circular convolution Discrete time Fourier transform Category Functions and mappings category of periodic function Category Signal processing category of a typical area of application ... more details
Ewald summation , named after Paul Peter Ewald , is a method for computing the interaction energies of periodic systems e.g. crystals , particularly electrostatic energies. Ewald summation is a special case of the Poisson summation formula , replacing the summation of interaction energies in real space with an equivalent summation in Fourier space. The advantage of this approach is the rapid convergence of the Fourier space summation compared to its real space equivalent when the real space interactions are long range. Because electrostatic energies consist of both short and long range interactions ... summed in real space and a long range component summed in Fourier space. Derivation Ewald summation ... becomes the summation of the long range term. Due to the use of the Fourier sum, the method implicitly ... 1 m 2 mathbf b 2 m 3 mathbf b 3 math in the final summation. This is the essential result. Once math tilde rho uc mathbf k math is calculated, the summation integration over math mathbf k math is straightforward ... Ewald summation was developed as a method of theoretical physics , long before the advent of computer ... galaxies and molecules . As in normal Ewald summation, a generic interaction potential is separated ... mesh Ewald summation is to replace the direct summation of interaction energies between point ... sr mathbf r j mathbf r i math that is the particle part of particle mesh Ewald and a summation in Fourier ... that s the mesh part . Due to the periodicity assumption implicit in Ewald summation, applications ... convergent , i.e., depends on the order of the summation. For example, if the dipole ... falls like math frac 1 R 3 math and 3 the mathematical summation math sum n 1 infty frac 1 ... mathbf r math , which points towards the charge, not away from it. History The Ewald summation ... and, hence, the Madelung constant of ionic crystals. Scaling Generally different Ewald summation methods ... Madelung constant Poisson summation formula Molecular modeling References Reflist Ewald P. 1921 Die ... more details
Binocular summation is the process by which the brain combines the information it receives from the left and the right eyes. ref Foundations of binocular vision a clinical perspective by Scott B. Steinman, Barbara A. Steinman, Ralph Philip Garzia 2000 ISBN 0838526705 pages 153 160 http books.google.com books?id XUtkHMO0C5cC&pg PA153&dq Binocular summation&hl en&ei hmFjTrbtJaiM4gTN fW Cg&sa X&oi book result&ct result&resnum 3&ved 0CDsQ6AEwAg v onepage&q Binocular 20summation&f false ref ref Normal Binocular Vision Theory, Investigation and Practical Aspects by David Stidwill, Robert Fletcher 2010 ISBN 140519250X pages 29 35 http books.google.com books?id cGTz58hxLqEC&pg PA29&dq Binocular summation&hl en&ei hmFjTrbtJaiM4gTN fW Cg&sa X&oi book result&ct result&resnum 1&ved 0CDEQ6AEwAA v onepage&q Binocular 20summation&f false ref See also Binocular disparity Binocular rivalry Binoviewer References Reflist Category Vision optics stub ... more details
In mathematical analysis , Lambert summation is a summability method for a class of divergent series . Definition A series math sum a n math is Lambert summable to A , written math sum a n A , mathrm L math , if math lim r rightarrow 1 1 r sum n 1 infty frac n a n r n 1 r n A . , math If a series is convergent to A then it is Lambert summable to A an Abelian theorem . Examples math sum n 1 infty frac mu n n 0 mathrm L math , where &mu is the M bius function . Hence if this series converges at all, it converges to zero. See also Lambert series Abelian and tauberian theorems References cite book author Jacob Korevaar title Tauberian theory. A century of developments series Grundlehren der Mathematischen Wissenschaften volume 329 publisher Springer Verlag year 2004 isbn 3 540 21058 X pages 18 cite book author Hugh L. Montgomery authorlink Hugh Montgomery mathematician coauthors Robert Charles Vaughan mathematician Robert C. Vaughan title Multiplicative number theory I. Classical theory series Cambridge tracts in advanced mathematics volume 97 year 2007 isbn 0 521 84903 9 pages 159 160 publisher Cambridge Univ. Press location Cambridge cite journal author Norbert Wiener authorlink Norbert Wiener title Tauberian theorems journal Ann. Of Math. year 1932 volume 33 pages 1 100 doi 10.2307 1968102 issue 1 publisher The Annals of Mathematics, Vol. 33, No. 1 jstor 1968102 Mathanalysis stub Category Mathematical series Category Summability methods ... more details
Correlate summation analysis is a data mining method. It is designed to find the Variable mathematics variables that are most covariance covariant with all of the other variables being studied, relative to Data clustering clustering . Aggregate correlate summation is the product of the totaled negative logarithm of the p value s for all of the correlation s to a given variable and its normalized standard deviation to mean quotient. Discrete correlate summation is the product of the totaled absolute value of the logarithm of the p value ratios between two groups correlations to a given variable and its absolute value of the logarithm of the group mean ratios. Correlate summation template This zipped Excel template performs a correlate summation analysis for up to 100 variables for 4 groups of 15 subjects http sites.google.com site correlatesummationtemplate Home correlate summation template correlate.zip?attredirects 0 The paper ref name rat cite journal author Westwood, B coauthors Chappell, M. year 2006 title Proceedings of the 1st international workshop on Text mining in bioinformatics TMBIO 06 publisher TMBIO 06 ACM pages 21 26 doi 10.1145 1183535.1183542 chapter Application of correlate summation to data clustering in the estrogen and salt sensitive female mRen2.Lewis rat isbn 1595935266 ref describing the method is embedded in the spreadsheet. Discrete correlate summation Given ... summation array. As in the log correlation ratio log sub cr sub , the log mean ratio log sub mr sub for the two groups means was acquired for each variable. The correlate summation was then multiplied by the log mean ratio, to yield the discrete mean correlate summation DC sub x sub ref name rat . Aggregate correlate summation As in the discrete correlate summation, a linear probability matrix ... values the columns were totaled to give the aggregate correlate summation AC array. The standard ... summation AC sub x sub ref name rat . Non linear modeling A linear correlation between variables ... more details
In numerical analysis , the Kahan summation algorithm also known as compensated summation ref Strictly, there exist other variants of compensated summation as well see cite book first Nicholas last Higham ... walk . ref name Higham93 Citation title The accuracy of floating point summation first1 Nicholas J ... 0914050 year 1993 ref With compensated summation, the worst case error bound is independent of n ..., which rounds to 10005.9. With a plain summation, each incoming value would be aligned with sum and many .... However, with compensated summation, we get the correct rounded result of 10005.9. Assume ... 10005.9 Exact result is 10005.85987, this is correctly rounded to 6 digits. So the summation is performed ... , to nudge the low order part of sum the next time around. Thus the summation proceeds with guard digits ... in compensated summation is needed to appreciate its accuracy characteristics. While it is more accurate than naive summation, it can still give large relative errors for ill conditioned sums. Suppose ... i math computed with infinite precision With compensated summation, one instead obtains math S n E ... sub i sub is the condition number of the summation problem. Essentially, the condition number represents the intrinsic sensitivity of the summation problem to errors, regardless of how it is computed ... error bound of every backwards stable summation method by a fixed algorithm in fixed precision i.e. ... summation problem is one in which this ratio is large, and in this case even compensated summation ... number is 1. Given a condition number, the relative error of compensated summation is effectively ... condition number, the errors of compensated summation are effectively O , independent of n . In comparison, the relative error bound for naive summation simply adding the numbers in sequence ... summation has a root mean square relative error that grows as math O varepsilon sqrt n math multiplied ... much worse than compensated summation, however. Note, however, that if the sum can be performed in twice ... more details
In mathematics , the Poisson summation formula is an equation that relates the Fourier series coefficients of the periodic summation of a function mathematics function to values of the function s continuous Fourier transform . Consequently, the periodic summation of a function is completely defined by discrete samples of the original function s Fourier transform. And conversely, the periodic summation .... The Poisson summation formula was discovered by Sim on Denis Poisson and is sometimes called Poisson resummation . Forms of the equation For appropriate functions , the Poisson summation formula ... F s t x hat s nu cdot e i 2 pi nu t , math EquationNote Eq.2 becomes a periodic summation ... 2 pi frac k P t . math EquationRef Eq.3 Similarly, the periodic summation of a function s Fourier transform .... At those values the summation diverges at a rate that is independent of k . The summation can ... such as Ces ro summation Ces ro summability . When interpreting convergence in this way EquationNote ... on the interval 0, P . Applications In partial differential equations , the Poisson summation ..., and that of a rectangle is determined by taking the periodization. The Poisson summation formula ... dimensions harv Grafakos 2004 . In signal processing, the Poisson summation formula ... . Computationally, the Poisson summation formula is useful since a slowly converging summation in real space is guaranteed to be converted into a quickly converging equivalent summation in Fourier ... space and vice versa. This is the essential idea behind Ewald summation . The Poisson summation ... hat f math both have compact support then math f 0 , math harv Pinsky 2002 . Poisson summation ... may then interpret the summation formula as the formula NumBlk math begin align underbrace sum n infty ... Eq.3 holds in the sense of distributions. Generalizations A version of the Poisson summation ... D over lattice points is exactly the question, so that the Sides of an equation LHS of the summation ... more details
dablink Another concept sometimes known by this name is summation by parts . In mathematics , Abel s summation formula , introduced by Niels Henrik Abel , is intensively used in number theory to compute series mathematics series .. Identity Let math a n , math be a sequence of real number real or complex number s and math phi x , math a function mathematics function of class math mathcal C 1 , math . Then math sum 1 le n le x a n phi n A x phi x int 1 x A u phi u , mathrm d u , math where math A x sum 0 n le x a n ,. math Indeed, this is integration by parts for a Riemann Stieltjes integral . More generally, we have math sum x n le y a n phi n A y phi y A x phi x int x y A u phi u , mathrm d u ,. math Examples Euler Mascheroni constant If math a n 1 , math and math phi x frac 1 x ,, math then math A x lfloor x rfloor , math and math sum 1 x frac 1 n frac lfloor x rfloor x int 1 x frac lfloor u rfloor u 2 , mathrm d u math which is a method to represent the Euler Mascheroni constant . Representation of Riemann s zeta function If math a n 1 , math and math phi x frac 1 x s ,, math then math A x lfloor x rfloor , math and math sum 1 infty frac 1 n s s int 1 infty frac lfloor u rfloor u 1 s mathrm d u ,. math The formula holds for math Re s 1 ,. math It may be used to derive Dirichlet s theorem, that is, math zeta s , math has a simple pole complex analysis pole with residue complex analysis residue 1 in s 1. Riemann zeta function If math a n mu n , math is the M bius function and math phi x frac 1 x s ,, math then math A x M u sum n le x mu x , math is Mertens function and math sum 1 infty frac mu n n s s int 1 infty frac M u u 1 s mathrm d u ,. math This formula holds for math Re s 1 ,. math See also Summation by parts References citation first Tom last Apostol authorlink Tom Apostol title Introduction to Analytic Number Theory publisher Springer Verlag series Undergraduate Texts in Mathematics year 1976 . Category Number theory Category Summability ... more details
fractional C, a methods. ref Smail p.131 ref Abel sum Abel summation is similar to Euler s attempted ... summation. For Grandi s series one has math A sum n 0 infty 1 n lim x rightarrow 1 sum n 0 infty ... summation Borel sum of Grandi s series is again sup 1 sup sub 2 sub , since math 1 x frac x 2 ... publisher Stanford M.S. theses div Category Divergent series Grandi s series, Summation of pt Soma da ... more details
Abel s sum formula may refer to Abel s summation formula , a formula used in number theory to compute series Summation by parts , a transformation of the summation of products of sequences into other summations disambiguation ... more details
In mathematics, Borel transform may refer to A transform used in Borel summation A generalization of this in Nachbin s theorem mathdab ... more details
in three sections 1 Summation of Sura chapter numbers and Ayah verse numbers 2 Split into even and odd 3 Extended analysis of odd and even summation numbers. Summation of chapter numbers and verse ... and verse numbers. It is easy to see that the result will be equal to the summation of chapter numbers 6555 and summation of verse numbers 6236 giving 12791. The next step the researcher did was to divide ... chapter of the quran gives a summation of chapter number and verse number of 8 which is even so this will be classified ... the 114 summation numbers the researcher found that there is an exact distribution of 57 even and 57 ... distribution found by Abdullah Jalghoom. He then added the 57 even summation numbers and 57 odd summation numbers. The addition of the 57 even summation numbers give 6236 and the addition of the 57 odd summation numbers give 6555. This was a marvelous symmetry that the researcher discovered as the addition of even summation numbers 6236 is equal to the summation of verse numbers, and the addition of odd summation numbers 6555 is equal to the summation of chapter numbers. This occurrence is very rare considering probabilities. The researcher further continued and analyzed the 57 odd summation number and the 57 even summation numbers The sum being the addition of verse number and chapter number Extended analysis of odd and even summation numbers This extended analysis by the researcher ... the even summation numbers and odd summation numbers The sum being the addition of verse number and chapter number as before . He created two number sets for the even summation numbers, one listing ... listing the original verse number. He then repeated this again for the odd summation numbers. After ... of 2933. This finding of Abdullah is remarkable. The summation of chapter numbers 6555 is different from the summation of verse numbers 6236, but splitting the chapters numbers into two groups and verse ... more details
Encyclopedia
top
