THE N-POINT FUNCTIONS FOR INTERSECTION NUMBERS ON MODULI SPACES OF CURVES
|
|
- Suzanna Bryan
- 5 years ago
- Views:
Transcription
1 THE N-POINT FUNTIONS FOR INTERSETION NUMBERS ON MODULI SPAES OF URVES KEFENG LIU AND HAO XU Abstract. We derive from Witte s KdV equatio a simple formula of the -poit fuctios for itersectio umbers o moduli spaces of curves, geeralizig Dikgraaf s two-poit fuctio ad Zagier s three-poit fuctio. This formula ucovers may ew idetities about itegrals of ψ classes ad provides a elemetary ad more efficiet algorithm to compute itersectio umbers other tha the celebrated Witte-Kotsevich theorem.. Itroductio We deote by M g, the moduli space of stable -poited geus g complex algebraic curves. Let ψ i be the first her class of the lie budle whose fiber over each poited stable curve is the cotaget lie at the i-th marked poit. We adopt Witte s otatio i this paper, τ d τ d g : ψ d ψ d. M g, These itersectio umbers are the correlatio fuctios of two dimesioal topological quatum gravity. I the famous paper [7], Witte made the remarkable coecture proved by Kotsevich [4] that the geeratig fuctio of above itersectio umbers are govered by KdV hierarchy, which provides a recursive way to compute all these itersectio umbers. Witte s coecture was reformulated by Dikgraaf, Verlide, ad Verlide [DVV] i terms of the Virasoro algebra. Defiitio.. We call the followig geeratig fuctio F x,..., x τ d τ d g the -poit fuctio. P g0 d 3g 3+ The -poit fuctio ecodes all iformatio of the correlatio fuctios of two dimesioal topological quatum gravity. Okoukov [6] obtaied a aalytic expressio of the -poit fuctios usig -dimesioal error-fuctio-type itegrals. Brézi ad Hikami [] apply correlatio fuctios of GUE esemble to fid explicit formulae of -poit fuctios. The first key poit is to cosider the followig ormalized -poit fuctio Gx,..., x exp x3 F x,..., x. 4
2 KEFENG LIU AND HAO XU I particular, we have -poit fuctio Gx, Dikgraaf s -poit fuctio x Gx, y k k! xyx + y x + y k +! k 0 ad Zagier s 3-poit fuctio [8] which we leared from Faber, Gx, y, z r,s 0 r!s r x, y, z 4 r r +!! s 8 s r + s +!, where S r x, y, z ad are the homogeeous symmetric polyomials defied by S r x, y, z xyr x + y r+ + yz r y + z r+ + zx r z + x r+ Z[x, y, z], x + y + z x + y + z3 x, y, z x + yy + zz + x x3 + y 3 + z Although two ad three poit fuctios are foud i the early 990 s, it s ot obvious at all that clea explicit formulae of geeral -poit fuctios should exist. Recall that we oly have closed formula of itersectio umbers i geus zero ad oe. Now we state the mai theorem of this ote. Theorem.. For, Gx,..., x r,s 0 r + 3!! 4 s r + s +!! P rx,..., x x,..., x s, where P r ad are homogeeous symmetric polyomials defied by x,..., x x 3 x3, 3 P r x,..., x x x I J I J x i x i x i Gx I Gx J x i 3r+ 3 r G r x I G r r x J, where I, J, {,,..., } ad G g x I deotes the degree 3g + I 3 homogeeous compoet of the ormalized I -poit fuctio Gx k,..., x k I, where k I. Note that the degree 3r + 3 polyomial P r x,..., x Q[x,..., x ] is expressed by ormalized I -poit fuctios Gx I with I <. So we ca recursively obtai a explicit formula of the -poit fuctio F x,..., x exp x3 4 r 0 Gx,..., x, thus we have a elemetary algorithm to calculate all itersectio umbers of ψ classes other tha the celebrated Witte-Kotsevich s theorem [4, 7], which is the oly feasible way kow before to calculate all itersectio umbers of ψ classes.
3 THE N-POINT FUNTIONS FOR INTERSETION NUMBERS 3 Sice P 0 x, y x+y, P rx, y 0 for r > 0, we get Dikgraaf s -poit fuctio. From P r x, y, z r! r r +! xyr x + y r+ + yz r y + z r+ + zx r z + x r+, x + y + z we also easily recover Zagier s 3-poit fuctio obtaied more tha te years ago. There is aother slightly differet formula of -poit fuctios. Whe 3, this has also bee obtaied by Zagier [8]. Theorem.3. For, F x,..., x exp x 3 4 r,s 0 s P r x,..., x x,..., x s 8 s r + s + s! where P r ad are the same polyomials as defied i theorem.. Theorem.3 follows from Theorem. ad the followig lemma. Lemma.4. Let ad r, s 0. The the followig idetity holds, 3 s s 8 s r + s + s! Proof. Let p r + ad fp, s k0 s k0 k 8 k k! r + 3!! 4 s k r + s k +!! k k k!p + s k!!. We have s k p + s + s fp, s k k!p + s k +!! + k k k k!p + s k +!! k0 k0 s+ s p + s + fp, s + + fp, s s+ s +!p!! s s!p!!. So we have the followig idetity fp, s + s+ s+ p + s + s +!p 3!!, which is ust the idetity 3 if s + is replaced by s. I Sectio we give a proof of the mai theorem. Sectio 3 cotais may ew idetities of the itersectio umbers of the ψ classes derived from our formula of the -poit fuctios. I Sectio 4 we briefly discuss other applicatios of the -poit fuctios. Ackowledgemets. The authors would like to thak Professor Sergei Lado, Edward Witte ad Do Zagier for helpful commets ad their iterests i this work. We also wat to thak Professor arel Faber for his woderful Maple programm for calculatig Hodge itegrals ad for commuicatig Zagier s three-poit fuctio to us.
4 4 KEFENG LIU AND HAO XU. Proof of the mai theorem We ca derive from Witte s KdV equatio the followig coefficiet equatio see [3, 7], d + τ d τ 0 + {,...,}I J τ d 4 τ d τ0 4 τ d τ d τ 0 τ di τ0 3 τ di + τ d τ0 τ di τ0 τ di, which is equivalet to the followig differetial equatio of -poit fuctios F x,..., x, x + x x F x,..., x x 4 x 4 + x x F x,..., x + x x i x i 3 + x i x i F x I F x J. I J So i order to prove Theorem., we eed to check that Ex,..., x : satisfies the followig differetial equatio, 4 x x Gx,..., x x Ex,..., x + x + x3 x 4 x I J x + x x 4 x 3 Ex,..., x x i + x i x i Ex I Ex J. The verificatio is straightforward from the defiitio of Gx,..., x i Theorem.. We ow prove the followig iitial value coditio of Gx,..., x, thus coclude the proof of Theorem.. Gx,..., x, 0 x Gx,..., x. Let M r x,..., x : x i 3 x i r G r x I G r r x J. I J r 0
5 For the left had side, we have x LHS THE N-POINT FUNTIONS FOR INTERSETION NUMBERS 5 r +!! 4 s r + s +!! M r + r,s 0 r +!! 4 s r + s +!! M r s + r,s 0 r,s 0 r +!! 4 s r + s +!! + r+s r,s 0 kr x G r x,..., x x,..., x s r +!! 4 s r + s +!! p+qr k +!!r + 3!! 4 s r + s +!!k +!! where i the last equatio we have used chage of variables. While for the right had side, x RHS r + 3!! 4 s r + s +!! M r s. r,s 0 So we eed oly prove the followig combiatorial idetity p + 3!! 4 q r +!! M p q+s M r s, r+s r +!! 4 s r + s +!! + k +!!r + 3!! 4 s r + s +!!k +!! r + 3!! 4 s r + s +!! i.e. kr r+s r +!! r + 3!! + k +!! k +!! kr r + s +!! r + s +!! for all ad r, s 0. It follows easily from the followig idetity p +!! p!! + p +!! p +!! p + 3!! p +!!. It is typical that from the formula of -poit fuctios i Theorem., may assertios about itersectio umbers will be reduced to combiatorial idetities. 3. New properties of the -poit fuctios I this sectio we derive various ew idetities about the itersectio umbers of the ψ classes by usig our simple formula of the -poit fuctios. Lemma 3.. Let. We have the followig recursio relatio for ormalized -poit fuctios G g x,..., x g + P gx,..., x + x,..., x 4g + G g x,..., x. The followig idetity holds x,..., x x x + x x + x,..., x.
6 6 KEFENG LIU AND HAO XU Proof. We have G g x,..., x r+sg g + P gx,..., x + r + 3!! 4 s g +!! P rx,..., x x,..., x s r+sg r + 3!! 4 s+ g +!! P rx,..., x x,..., x s+ g + P gx,..., x + x,..., x 4g + G g x,..., x. The proof of is easy. Let xd, P x,..., x deotes the coefficiet of xd i a polyomial or formal power series P x,..., x. From the iductive structure i the defiitio of - poit fuctios, we have the followig basic properties of -poit fuctios, their proofs are purely combiatorial. First cosider the ormalized + -poit fuctio Gz, x,..., x. Here we use the variable z to distiguish oe poit. We have the followig theorem about the coefficiets of Gz, x,..., x. Theorem 3.. Let g + 0. If k > g +, d 0 ad d 3g + k, the z k z k, G gz, x,..., x, P gz, x,..., x 0, 0. Let d 0, d g ad a #{ d 0}. The z g + z g +, G gz, x,..., x, P gz, x,..., x 4 g d +!!, a 4 g d +!!. 3 Let d 0, d g +, a #{ d 0} ad b #{ d }. The z g 3+, G gz, x,..., x g + g a a 4 g d, +!! z g 3+, P gz, x,..., x ag + g g + a +5a + 3b 3 3b 4 g d. +!!
7 THE N-POINT FUNTIONS FOR INTERSETION NUMBERS 7 Proof. is obvious from theorem.. We ow prove iductively. z g +, P gz, x,..., x z g +, G gz, x,..., ˆx,..., x where a #{ d 0}. z g +, G g z, x,..., x r+sg g + g + r +!! 4 s g +!! P d g P d g P d r xd P 4 g d g d +!!. a 4 g d +!!, a xd 4 r d +!! a xd 4 g d +!! + x 4g + a xd 4 g d +!! + P d g The statemet 3 ca be proved similarly. x s P d g xd 4 g d +!! g + a xd 4 g d +!! Now cosider the ormalized special + -poit fuctio Gy, y, x,..., x. We have the followig theorem about the coefficiets of Gy, y, x,..., x. Theorem 3.3. Let g 0 ad. If k > g, d 0 ad d 3g + k, the y k, x i Gy, x I G y, x J 0, or equivaletly, I Jy + I J 0 x i y + k τ τ0 τ di g τ k τ0 τ di g g 0. If d ad d g +, the y g, + I Jy x i y + x i Gy, x I G y, x J g + +! 4 g g +! d!!.
8 8 KEFENG LIU AND HAO XU or equivaletly, g I J 0 τ τ0 τ di g τ g τ0 τ di g g g + +! 4 g g +! d!!. 3 If d ad d g +, the y g, x i x i Gy, y, x I Gx J 0. or equivaletly, I J 0 I J 0 I J g τ τ g τ0 τ di g τ0 τ di g g 0. 4 If d 0 ad d g +, the g τ τ0 τ di τ g τ0 τ di + τ τ g τ0 τ di τ0 τ di g + + g 0 τ 0 τ τ g τ d τ d g 5 If k > g, d 0 ad d 3g + k, the y k y k, G gy, y, x,..., x, P gy, y, x,..., x 6 If d 0 ad d g +, the y g, P gy, y, x,..., x If moreover we have d, the y g g ++, G gy, y, x,..., x y g 0, 0., G gy, y, x,..., x g +! 4 g g +! d!!. Proof. We first show that ad imply the statemets is obvious, sice for d i, we have τ di , 5 ad the first idetity of 6 follow easily from Theorem.. τ 0.
9 THE N-POINT FUNTIONS FOR INTERSETION NUMBERS 9 Let d d d. We prove the secod idetity of 6 by iductig o d, the maximum idex. g τ g τ τ d τ d g 0 g 0 τ 0 τ g τ τ d +τ d τ d g g +! 4 g g +! d!!d + g +! 4 g g +! d!!, g k 0 k τ g τ τ d +τ d τ dk τ d g g +!d k 4 g g +! d!!d + where we have used 4. I fact the above idetity still holds if there is oly oe d 0. By explicitly writig dow the -poit fuctios, we give a proof of ad i the case, the geeral case ca be proved similarly. Note also that it is easy to prove Theorem 3.3 for g 0 sice G 0 x,..., x x + + x 3 see Lemma 4.3 ad orollary 4.4. It is easy to prove the followig idetity by iductig o g. 0 y g+, + I Jy x i y + g x i G g y, x I G g g y, x J g 0 r!! 4 s g +!! 4 r r +!! xr + x r x + x s 4 r r +!! 4 s s +!! xr x s. r+sg Because we have y g, + I Jy x i y + x i r!! 4 s g +!! 4 r r +!! r+sg g G g y, x I G g g y, x J g 0 sx r + x r x + x s x x + s + s x r + x r x + x s+ +s + r + x r+ + x r+ x + x s+ + r + r x r+ + x r+ x + x s + r + s + x r+ 4 r r +!! 4 s x s+ s + s x r s +!! x s+ r + r + 4 g g!! x + x g+, the proof of is also easy. x r+ x s It is easy to see that the statemets 5 ad 6 of Theorem 3.3 imply the followig idetities of itersectio umbers which we have aouced i [5]. They are related to Faber s itersectio umber coecture.
10 0 KEFENG LIU AND HAO XU orollary 3.4. Let d 0, #{ d 0} ad d g +. The g 0 τ g τ τ d τ d g g +! 4 g g +! d!!. If #{ d 0} ad a #{ d }, the the right had side becomes g +! 4 g g +! d!! g + a g + a. Let k > g, d 0 ad d 3g + k. The k τ k τ τ d τ d g 0. 0 We also have the followig geeralizatio of statemets ad of Theorem 3.3. The proof is similar. Theorem 3.5. Let g 0, ad r, s 0. If k > g + r + s, d 0 ad d 3g + r + s + k, the y k, x i +r Gy, x I G y, x J 0, or equivaletly, I Jy + k I J 0 τ τ +r 0 x i +s y + τ di g τ k τ +s 0 τ di g g 0. If d ad d g +, the y g+r+s, + I Jy x i +s y + x i +r Gy, x I G y, x J I J r g + + r + s +! 4 g g + r + s +! d!!. or equivaletly, g+r+s 0 τ τ +r 0 τ di g τ g+r+s τ +s 0 τ di g g r g + + r + s +! 4 g g + r + s +! d!!.
11 THE N-POINT FUNTIONS FOR INTERSETION NUMBERS 4. Other applicatios of -poit fuctios From the ew idetities i Sectio 3 ad their derivatios, we ca see that the simple formula of -poit fuctios may be used to prove the followig equivalet statemet of the Faber s itersectio umber coecture: oecture 4.. Let d 0 ad d g +. The τ d τ d τ g g τ d τ d τ d +g τ d+ τ d g I J g τ τ di g τ g τ di g g. 0 It is clear that our explicit formula of -poit fuctios should also shed light o the followig coectural idetity as stated i [5] where the case of has bee proved. oecture 4.. Let g, d ad d g. The g 3 +! g+ g 3! d!! τ d τ d τ g g τ d τ d τ d +g 3τ d+ τ d g + I J g 4 τ τ di g τ g 4 τ di g g If we have d 0, #{ d 0} ad a #{ d } i the above coecture, the the left had side becomes 0 g 3 +! g+ g 3! d!! g + + a g + 3 a. We will discuss the relatio of the above coectures with our simple formula of the - poit fuctios i a forthcomig paper. Here as the first step we oly prove two iterestig combiatorial idetities. Lemma 4.3. Let. 5 Assume that if I, the x i I. We have + {,...,}I Jx x i I x + x i J We have I J I,J x i I {,...,}I J x i J x x i I x i J
12 KEFENG LIU AND HAO XU Proof. Let xd 6 be ay moomial of {,...,}I Jx + x i I x + x i J. Sice d, so if d > 0, the their must exist some > such that d 0. The statemet meas that the polyomial 6 does ot cotai x, so we eed oly prove that after substitute x 0 i 6, the resultig polyomial does ot cotai x. {,..., }I J x x + {,..., }I Jx + x i I + x + x i I x + x i J + x + x i J. x i I x + x i J + So statemet follows by iductio. We prove statemet by iductio. Regard the LHS ad RHS of the idetity 5 as polyomials i x with degree, we eed to prove the equality of 5 whe substitute x x i for i.... It s sufficiet to check the case x x. LHS I J I J x + x i I 4 x + x x i I x + x i J + I J I,J x RHS. x i J + x i I + x i I x i J x i J x i I + x i J Note that if a term has power J, the J is assumed. Fially as a iterestig exercise we give a proof of the followig well-kow formula by usig our formula of the -poit fuctios. orollary 4.4. Let 3, d 0 ad d 3. The 3 τ d τ d 0. d,..., d
13 THE N-POINT FUNTIONS FOR INTERSETION NUMBERS 3 Proof. It s equivalet to prove that for 3 x 3 G 0 x,..., x This is ust the Lemma 4.3. x x I J I J x i x i G 0 x I G 0 x J x i I x i J. Refereces [] E. Brézi ad S. Hikami, Vertices from replica i a radom matrix theory, math-ph/ [] R. Dikgraaf, H. Verlide, ad E. Verlide, Topological strigs i d <, Nuclear Phys. B 35 99, [3]. Faber ad R. Padharipade, Logarithmic series ad Hodge itegrals i the tautological rig, with a appedix by Do Zagier, Michiga Math. J. Fulto volume , 5 5. [4] M. Kotsevich, Itersectio theory o the moduli space of curves ad the matrix Airy fuctio. omm. Math. Phys , o., 3. [5] K. Liu ad H. Xu, New properties of the itersectio umbers o moduli spaces of curves, math.ag/ [6] A. Okoukov, Geeratig fuctios for itersectio umbers o moduli spaces of curves, Iterat. Math. Res. Notices, 00, [7] E. Witte, Two-dimesioal gravity ad itersectio theory o moduli space, Surveys i Differetial Geometry, vol., [8] D. Zagier, The three-poit fuctio for M g, upublished. [9] H. Xu, A Maple program to compute itersectio idices by oecture A., available at eter of Mathematical Scieces, Zheiag Uiversity, Hagzhou, Zheiag 3007, hia; Departmet of Mathematics,Uiversity of aliforia at Los Ageles, Los Ageles, A , USA address: liu@math.ucla.edu, liu@cms.zu.edu.c eter of Mathematical Scieces, Zheiag Uiversity, Hagzhou, Zheiag 3007, hia address: haoxu@cms.zu.edu.c
CALCULATION OF FIBONACCI VECTORS
CALCULATION OF FIBONACCI VECTORS Stuart D. Aderso Departmet of Physics, Ithaca College 953 Daby Road, Ithaca NY 14850, USA email: saderso@ithaca.edu ad Dai Novak Departmet of Mathematics, Ithaca College
More informationMath 155 (Lecture 3)
Math 55 (Lecture 3) September 8, I this lecture, we ll cosider the aswer to oe of the most basic coutig problems i combiatorics Questio How may ways are there to choose a -elemet subset of the set {,,,
More informationMath 203A, Solution Set 8.
Math 20A, Solutio Set 8 Problem 1 Give four geeral lies i P, show that there are exactly 2 lies which itersect all four of them Aswer: Recall that the space of lies i P is parametrized by the Grassmaia
More informationThe r-generalized Fibonacci Numbers and Polynomial Coefficients
It. J. Cotemp. Math. Scieces, Vol. 3, 2008, o. 24, 1157-1163 The r-geeralized Fiboacci Numbers ad Polyomial Coefficiets Matthias Schork Camillo-Sitte-Weg 25 60488 Frakfurt, Germay mschork@member.ams.org,
More informationCHAPTER I: Vector Spaces
CHAPTER I: Vector Spaces Sectio 1: Itroductio ad Examples This first chapter is largely a review of topics you probably saw i your liear algebra course. So why cover it? (1) Not everyoe remembers everythig
More informationLinear Regression Demystified
Liear Regressio Demystified Liear regressio is a importat subject i statistics. I elemetary statistics courses, formulae related to liear regressio are ofte stated without derivatio. This ote iteds to
More informationSequences, Mathematical Induction, and Recursion. CSE 2353 Discrete Computational Structures Spring 2018
CSE 353 Discrete Computatioal Structures Sprig 08 Sequeces, Mathematical Iductio, ad Recursio (Chapter 5, Epp) Note: some course slides adopted from publisher-provided material Overview May mathematical
More informationCALCULATING FIBONACCI VECTORS
THE GENERALIZED BINET FORMULA FOR CALCULATING FIBONACCI VECTORS Stuart D Aderso Departmet of Physics, Ithaca College 953 Daby Road, Ithaca NY 14850, USA email: saderso@ithacaedu ad Dai Novak Departmet
More informationCourse : Algebraic Combinatorics
Course 18.312: Algebraic Combiatorics Lecture Notes # 18-19 Addedum by Gregg Musier March 18th - 20th, 2009 The followig material ca be foud i a umber of sources, icludig Sectios 7.3 7.5, 7.7, 7.10 7.11,
More informationLet us give one more example of MLE. Example 3. The uniform distribution U[0, θ] on the interval [0, θ] has p.d.f.
Lecture 5 Let us give oe more example of MLE. Example 3. The uiform distributio U[0, ] o the iterval [0, ] has p.d.f. { 1 f(x =, 0 x, 0, otherwise The likelihood fuctio ϕ( = f(x i = 1 I(X 1,..., X [0,
More information1 6 = 1 6 = + Factorials and Euler s Gamma function
Royal Holloway Uiversity of Lodo Departmet of Physics Factorials ad Euler s Gamma fuctio Itroductio The is a self-cotaied part of the course dealig, essetially, with the factorial fuctio ad its geeralizatio
More information1 Approximating Integrals using Taylor Polynomials
Seughee Ye Ma 8: Week 7 Nov Week 7 Summary This week, we will lear how we ca approximate itegrals usig Taylor series ad umerical methods. Topics Page Approximatig Itegrals usig Taylor Polyomials. Defiitios................................................
More informationFactors of sums and alternating sums involving binomial coefficients and powers of integers
Factors of sums ad alteratig sums ivolvig biomial coefficiets ad powers of itegers Victor J. W. Guo 1 ad Jiag Zeg 2 1 Departmet of Mathematics East Chia Normal Uiversity Shaghai 200062 People s Republic
More informationEnumerative & Asymptotic Combinatorics
C50 Eumerative & Asymptotic Combiatorics Notes 4 Sprig 2003 Much of the eumerative combiatorics of sets ad fuctios ca be geeralised i a maer which, at first sight, seems a bit umotivated I this chapter,
More informationSOME TRIGONOMETRIC IDENTITIES RELATED TO POWERS OF COSINE AND SINE FUNCTIONS
Folia Mathematica Vol. 5, No., pp. 4 6 Acta Uiversitatis Lodziesis c 008 for Uiversity of Lódź Press SOME TRIGONOMETRIC IDENTITIES RELATED TO POWERS OF COSINE AND SINE FUNCTIONS ROMAN WITU LA, DAMIAN S
More informationTopic 9: Sampling Distributions of Estimators
Topic 9: Samplig Distributios of Estimators Course 003, 2016 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be
More informationRecursive Algorithms. Recurrences. Recursive Algorithms Analysis
Recursive Algorithms Recurreces Computer Sciece & Egieerig 35: Discrete Mathematics Christopher M Bourke cbourke@cseuledu A recursive algorithm is oe i which objects are defied i terms of other objects
More informationConvergence of random variables. (telegram style notes) P.J.C. Spreij
Covergece of radom variables (telegram style otes).j.c. Spreij this versio: September 6, 2005 Itroductio As we kow, radom variables are by defiitio measurable fuctios o some uderlyig measurable space
More informationSequences of Definite Integrals, Factorials and Double Factorials
47 6 Joural of Iteger Sequeces, Vol. 8 (5), Article 5.4.6 Sequeces of Defiite Itegrals, Factorials ad Double Factorials Thierry Daa-Picard Departmet of Applied Mathematics Jerusalem College of Techology
More information#A51 INTEGERS 14 (2014) MULTI-POLY-BERNOULLI-STAR NUMBERS AND FINITE MULTIPLE ZETA-STAR VALUES
#A5 INTEGERS 4 (24) MULTI-POLY-BERNOULLI-STAR NUMBERS AND FINITE MULTIPLE ZETA-STAR VALUES Kohtaro Imatomi Graduate School of Mathematics, Kyushu Uiversity, Nishi-ku, Fukuoka, Japa k-imatomi@math.kyushu-u.ac.p
More informationSome remarks for codes and lattices over imaginary quadratic
Some remarks for codes ad lattices over imagiary quadratic fields Toy Shaska Oaklad Uiversity, Rochester, MI, USA. Caleb Shor Wester New Eglad Uiversity, Sprigfield, MA, USA. shaska@oaklad.edu Abstract
More informationDIVISIBILITY PROPERTIES OF GENERALIZED FIBONACCI POLYNOMIALS
DIVISIBILITY PROPERTIES OF GENERALIZED FIBONACCI POLYNOMIALS VERNER E. HOGGATT, JR. Sa Jose State Uiversity, Sa Jose, Califoria 95192 ad CALVIN T. LONG Washigto State Uiversity, Pullma, Washigto 99163
More informationBertrand s Postulate
Bertrad s Postulate Lola Thompso Ross Program July 3, 2009 Lola Thompso (Ross Program Bertrad s Postulate July 3, 2009 1 / 33 Bertrad s Postulate I ve said it oce ad I ll say it agai: There s always a
More informationChapter 8. Euler s Gamma function
Chapter 8 Euler s Gamma fuctio The Gamma fuctio plays a importat role i the fuctioal equatio for ζ(s that we will derive i the ext chapter. I the preset chapter we have collected some properties of the
More information62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 +
62. Power series Defiitio 16. (Power series) Give a sequece {c }, the series c x = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + is called a power series i the variable x. The umbers c are called the coefficiets of
More informationw (1) ˆx w (1) x (1) /ρ and w (2) ˆx w (2) x (2) /ρ.
2 5. Weighted umber of late jobs 5.1. Release dates ad due dates: maximimizig the weight of o-time jobs Oce we add release dates, miimizig the umber of late jobs becomes a sigificatly harder problem. For
More informationIntroduction to Extreme Value Theory Laurens de Haan, ISM Japan, Erasmus University Rotterdam, NL University of Lisbon, PT
Itroductio to Extreme Value Theory Laures de Haa, ISM Japa, 202 Itroductio to Extreme Value Theory Laures de Haa Erasmus Uiversity Rotterdam, NL Uiversity of Lisbo, PT Itroductio to Extreme Value Theory
More informationThe Arakawa-Kaneko Zeta Function
The Arakawa-Kaeko Zeta Fuctio Marc-Atoie Coppo ad Berard Cadelpergher Nice Sophia Atipolis Uiversity Laboratoire Jea Alexadre Dieudoé Parc Valrose F-0608 Nice Cedex 2 FRANCE Marc-Atoie.COPPO@uice.fr Berard.CANDELPERGHER@uice.fr
More informationarxiv: v1 [math.fa] 3 Apr 2016
Aticommutator Norm Formula for Proectio Operators arxiv:164.699v1 math.fa] 3 Apr 16 Sam Walters Uiversity of Norther British Columbia ABSTRACT. We prove that for ay two proectio operators f, g o Hilbert
More informationLecture 7: Properties of Random Samples
Lecture 7: Properties of Radom Samples 1 Cotiued From Last Class Theorem 1.1. Let X 1, X,...X be a radom sample from a populatio with mea µ ad variace σ
More informationNICK DUFRESNE. 1 1 p(x). To determine some formulas for the generating function of the Schröder numbers, r(x) = a(x) =
AN INTRODUCTION TO SCHRÖDER AND UNKNOWN NUMBERS NICK DUFRESNE Abstract. I this article we will itroduce two types of lattice paths, Schröder paths ad Ukow paths. We will examie differet properties of each,
More informationInfinite Sequences and Series
Chapter 6 Ifiite Sequeces ad Series 6.1 Ifiite Sequeces 6.1.1 Elemetary Cocepts Simply speakig, a sequece is a ordered list of umbers writte: {a 1, a 2, a 3,...a, a +1,...} where the elemets a i represet
More informationKinetics of Complex Reactions
Kietics of Complex Reactios by Flick Colema Departmet of Chemistry Wellesley College Wellesley MA 28 wcolema@wellesley.edu Copyright Flick Colema 996. All rights reserved. You are welcome to use this documet
More informationCourse : Algebraic Combinatorics
Course 8.32: Algebraic Combiatorics Lecture Notes # Addedum by Gregg Musier February 4th - 6th, 2009 Recurrece Relatios ad Geeratig Fuctios Give a ifiite sequece of umbers, a geeratig fuctio is a compact
More informationP. Z. Chinn Department of Mathematics, Humboldt State University, Arcata, CA
RISES, LEVELS, DROPS AND + SIGNS IN COMPOSITIONS: EXTENSIONS OF A PAPER BY ALLADI AND HOGGATT S. Heubach Departmet of Mathematics, Califoria State Uiversity Los Ageles 55 State Uiversity Drive, Los Ageles,
More informationA GENERALIZATION OF THE SYMMETRY BETWEEN COMPLETE AND ELEMENTARY SYMMETRIC FUNCTIONS. Mircea Merca
Idia J Pure Appl Math 45): 75-89 February 204 c Idia Natioal Sciece Academy A GENERALIZATION OF THE SYMMETRY BETWEEN COMPLETE AND ELEMENTARY SYMMETRIC FUNCTIONS Mircea Merca Departmet of Mathematics Uiversity
More informationComparison Study of Series Approximation. and Convergence between Chebyshev. and Legendre Series
Applied Mathematical Scieces, Vol. 7, 03, o. 6, 3-337 HIKARI Ltd, www.m-hikari.com http://d.doi.org/0.988/ams.03.3430 Compariso Study of Series Approimatio ad Covergece betwee Chebyshev ad Legedre Series
More informationTopic 9: Sampling Distributions of Estimators
Topic 9: Samplig Distributios of Estimators Course 003, 2018 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be
More informationACO Comprehensive Exam 9 October 2007 Student code A. 1. Graph Theory
1. Graph Theory Prove that there exist o simple plaar triagulatio T ad two distict adjacet vertices x, y V (T ) such that x ad y are the oly vertices of T of odd degree. Do ot use the Four-Color Theorem.
More informationOn the Inverse of a Certain Matrix Involving Binomial Coefficients
It. J. Cotemp. Math. Scieces, Vol. 3, 008, o. 3, 5-56 O the Iverse of a Certai Matrix Ivolvig Biomial Coefficiets Yoshiari Iaba Kitakuwada Seior High School Keihokushimoyuge, Ukyo-ku, Kyoto, 60-0534, Japa
More informationAsymptotic distribution of products of sums of independent random variables
Proc. Idia Acad. Sci. Math. Sci. Vol. 3, No., May 03, pp. 83 9. c Idia Academy of Scieces Asymptotic distributio of products of sums of idepedet radom variables YANLING WANG, SUXIA YAO ad HONGXIA DU ollege
More informationThe picture in figure 1.1 helps us to see that the area represents the distance traveled. Figure 1: Area represents distance travelled
1 Lecture : Area Area ad distace traveled Approximatig area by rectagles Summatio The area uder a parabola 1.1 Area ad distace Suppose we have the followig iformatio about the velocity of a particle, how
More informationGeneralization of Samuelson s inequality and location of eigenvalues
Proc. Idia Acad. Sci. Math. Sci.) Vol. 5, No., February 05, pp. 03. c Idia Academy of Scieces Geeralizatio of Samuelso s iequality ad locatio of eigevalues R SHARMA ad R SAINI Departmet of Mathematics,
More informationFLOOR AND ROOF FUNCTION ANALOGS OF THE BELL NUMBERS. H. W. Gould Department of Mathematics, West Virginia University, Morgantown, WV 26506, USA
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A58 FLOOR AND ROOF FUNCTION ANALOGS OF THE BELL NUMBERS H. W. Gould Departmet of Mathematics, West Virgiia Uiversity, Morgatow, WV
More information1 Introduction to reducing variance in Monte Carlo simulations
Copyright c 010 by Karl Sigma 1 Itroductio to reducig variace i Mote Carlo simulatios 11 Review of cofidece itervals for estimatig a mea I statistics, we estimate a ukow mea µ = E(X) of a distributio by
More informationA solid Foundation for q-appell Polynomials
Advaces i Dyamical Systems ad Applicatios ISSN 0973-5321, Volume 10, Number 1, pp. 27 35 2015) http://campus.mst.edu/adsa A solid Foudatio for -Appell Polyomials Thomas Erst Uppsala Uiversity Departmet
More informationSeunghee Ye Ma 8: Week 5 Oct 28
Week 5 Summary I Sectio, we go over the Mea Value Theorem ad its applicatios. I Sectio 2, we will recap what we have covered so far this term. Topics Page Mea Value Theorem. Applicatios of the Mea Value
More informationMDIV. Multiple divisor functions
MDIV. Multiple divisor fuctios The fuctios τ k For k, defie τ k ( to be the umber of (ordered factorisatios of ito k factors, i other words, the umber of ordered k-tuples (j, j 2,..., j k with j j 2...
More informationSome families of generating functions for the multiple orthogonal polynomials associated with modified Bessel K-functions
J. Math. Aal. Appl. 297 2004 186 193 www.elsevier.com/locate/jmaa Some families of geeratig fuctios for the multiple orthogoal polyomials associated with modified Bessel K-fuctios M.A. Özarsla, A. Altı
More informationDe la Vallée Poussin Summability, the Combinatorial Sum 2n 1
J o u r a l of Mathematics ad Applicatios JMA No 40, pp 5-20 (2017 De la Vallée Poussi Summability, the Combiatorial Sum 1 ( 2 ad the de la Vallée Poussi Meas Expasio Ziad S. Ali Abstract: I this paper
More informationA Note on the Symmetric Powers of the Standard Representation of S n
A Note o the Symmetric Powers of the Stadard Represetatio of S David Savitt 1 Departmet of Mathematics, Harvard Uiversity Cambridge, MA 0138, USA dsavitt@mathharvardedu Richard P Staley Departmet of Mathematics,
More informationGENERALIZED HARMONIC NUMBER IDENTITIES AND A RELATED MATRIX REPRESENTATION
J Korea Math Soc 44 (2007), No 2, pp 487 498 GENERALIZED HARMONIC NUMBER IDENTITIES AND A RELATED MATRIX REPRESENTATION Gi-Sag Cheo ad Moawwad E A El-Miawy Reprited from the Joural of the Korea Mathematical
More informationApplication to Random Graphs
A Applicatio to Radom Graphs Brachig processes have a umber of iterestig ad importat applicatios. We shall cosider oe of the most famous of them, the Erdős-Réyi radom graph theory. 1 Defiitio A.1. Let
More informationStat 421-SP2012 Interval Estimation Section
Stat 41-SP01 Iterval Estimatio Sectio 11.1-11. We ow uderstad (Chapter 10) how to fid poit estimators of a ukow parameter. o However, a poit estimate does ot provide ay iformatio about the ucertaity (possible
More informationHoggatt and King [lo] defined a complete sequence of natural numbers
REPRESENTATIONS OF N AS A SUM OF DISTINCT ELEMENTS FROM SPECIAL SEQUENCES DAVID A. KLARNER, Uiversity of Alberta, Edmoto, Caada 1. INTRODUCTION Let a, I deote a sequece of atural umbers which satisfies
More informationMost text will write ordinary derivatives using either Leibniz notation 2 3. y + 5y= e and y y. xx tt t
Itroductio to Differetial Equatios Defiitios ad Termiolog Differetial Equatio: A equatio cotaiig the derivatives of oe or more depedet variables, with respect to oe or more idepedet variables, is said
More information7.1 Convergence of sequences of random variables
Chapter 7 Limit Theorems Throughout this sectio we will assume a probability space (, F, P), i which is defied a ifiite sequece of radom variables (X ) ad a radom variable X. The fact that for every ifiite
More informationNotes on the Combinatorial Nullstellensatz
Notes o the Combiatorial Nullstellesatz Costructive ad Nocostructive Methods i Combiatorics ad TCS U. Chicago, Sprig 2018 Istructor: Adrew Drucker Scribe: Roberto Ferádez For the followig theorems ad examples
More informationIntroducing a Novel Bivariate Generalized Skew-Symmetric Normal Distribution
Joural of mathematics ad computer Sciece 7 (03) 66-7 Article history: Received April 03 Accepted May 03 Available olie Jue 03 Itroducig a Novel Bivariate Geeralized Skew-Symmetric Normal Distributio Behrouz
More informationarxiv: v1 [cs.sc] 2 Jan 2018
Computig the Iverse Melli Trasform of Holoomic Sequeces usig Kovacic s Algorithm arxiv:8.9v [cs.sc] 2 Ja 28 Research Istitute for Symbolic Computatio RISC) Johaes Kepler Uiversity Liz, Alteberger Straße
More informationThe standard deviation of the mean
Physics 6C Fall 20 The stadard deviatio of the mea These otes provide some clarificatio o the distictio betwee the stadard deviatio ad the stadard deviatio of the mea.. The sample mea ad variace Cosider
More informationMATH 112: HOMEWORK 6 SOLUTIONS. Problem 1: Rudin, Chapter 3, Problem s k < s k < 2 + s k+1
MATH 2: HOMEWORK 6 SOLUTIONS CA PRO JIRADILOK Problem. If s = 2, ad Problem : Rudi, Chapter 3, Problem 3. s + = 2 + s ( =, 2, 3,... ), prove that {s } coverges, ad that s < 2 for =, 2, 3,.... Proof. The
More informationTopic 9: Sampling Distributions of Estimators
Topic 9: Samplig Distributios of Estimators Course 003, 2018 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be
More informationMath 2784 (or 2794W) University of Connecticut
ORDERS OF GROWTH PAT SMITH Math 2784 (or 2794W) Uiversity of Coecticut Date: Mar. 2, 22. ORDERS OF GROWTH. Itroductio Gaiig a ituitive feel for the relative growth of fuctios is importat if you really
More informationA symbolic approach to multiple zeta values at the negative integers
A symbolic approach to multiple zeta values at the egative itegers Victor H. Moll a, Li Jiu a Christophe Vigat a,b a Departmet of Mathematics, Tulae Uiversity, New Orleas, USA Correspodig author b LSS/Supelec,
More informationLecture 23: Minimal sufficiency
Lecture 23: Miimal sufficiecy Maximal reductio without loss of iformatio There are may sufficiet statistics for a give problem. I fact, X (the whole data set) is sufficiet. If T is a sufficiet statistic
More informationMath 475, Problem Set #12: Answers
Math 475, Problem Set #12: Aswers A. Chapter 8, problem 12, parts (b) ad (d). (b) S # (, 2) = 2 2, sice, from amog the 2 ways of puttig elemets ito 2 distiguishable boxes, exactly 2 of them result i oe
More informationREGULARIZATION OF CERTAIN DIVERGENT SERIES OF POLYNOMIALS
REGULARIZATION OF CERTAIN DIVERGENT SERIES OF POLYNOMIALS LIVIU I. NICOLAESCU ABSTRACT. We ivestigate the geeralized covergece ad sums of series of the form P at P (x, where P R[x], a R,, ad T : R[x] R[x]
More informationChapter 6 Infinite Series
Chapter 6 Ifiite Series I the previous chapter we cosidered itegrals which were improper i the sese that the iterval of itegratio was ubouded. I this chapter we are goig to discuss a topic which is somewhat
More informationOn forward improvement iteration for stopping problems
O forward improvemet iteratio for stoppig problems Mathematical Istitute, Uiversity of Kiel, Ludewig-Mey-Str. 4, D-24098 Kiel, Germay irle@math.ui-iel.de Albrecht Irle Abstract. We cosider the optimal
More informationMatrix representations of Fibonacci-like sequences
NTMSCI 6, No. 4, 03-0 08 03 New Treds i Mathematical Scieces http://dx.doi.org/0.085/tmsci.09.33 Matrix represetatios of Fiboacci-like sequeces Yasemi Tasyurdu Departmet of Mathematics, Faculty of Sciece
More informationAppendix to Quicksort Asymptotics
Appedix to Quicksort Asymptotics James Alle Fill Departmet of Mathematical Scieces The Johs Hopkis Uiversity jimfill@jhu.edu ad http://www.mts.jhu.edu/~fill/ ad Svate Jaso Departmet of Mathematics Uppsala
More informationAppendix: The Laplace Transform
Appedix: The Laplace Trasform The Laplace trasform is a powerful method that ca be used to solve differetial equatio, ad other mathematical problems. Its stregth lies i the fact that it allows the trasformatio
More information4. Partial Sums and the Central Limit Theorem
1 of 10 7/16/2009 6:05 AM Virtual Laboratories > 6. Radom Samples > 1 2 3 4 5 6 7 4. Partial Sums ad the Cetral Limit Theorem The cetral limit theorem ad the law of large umbers are the two fudametal theorems
More information1. By using truth tables prove that, for all statements P and Q, the statement
Author: Satiago Salazar Problems I: Mathematical Statemets ad Proofs. By usig truth tables prove that, for all statemets P ad Q, the statemet P Q ad its cotrapositive ot Q (ot P) are equivalet. I example.2.3
More information(VII.A) Review of Orthogonality
VII.A Review of Orthogoality At the begiig of our study of liear trasformatios i we briefly discussed projectios, rotatios ad projectios. I III.A, projectios were treated i the abstract ad without regard
More informationPhysics 324, Fall Dirac Notation. These notes were produced by David Kaplan for Phys. 324 in Autumn 2001.
Physics 324, Fall 2002 Dirac Notatio These otes were produced by David Kapla for Phys. 324 i Autum 2001. 1 Vectors 1.1 Ier product Recall from liear algebra: we ca represet a vector V as a colum vector;
More informationA Simplified Binet Formula for k-generalized Fibonacci Numbers
A Simplified Biet Formula for k-geeralized Fiboacci Numbers Gregory P. B. Dresde Departmet of Mathematics Washigto ad Lee Uiversity Lexigto, VA 440 dresdeg@wlu.edu Zhaohui Du Shaghai, Chia zhao.hui.du@gmail.com
More informationRandom Models. Tusheng Zhang. February 14, 2013
Radom Models Tusheg Zhag February 14, 013 1 Radom Walks Let me describe the model. Radom walks are used to describe the motio of a movig particle (object). Suppose that a particle (object) moves alog the
More information1 Generating functions for balls in boxes
Math 566 Fall 05 Some otes o geeratig fuctios Give a sequece a 0, a, a,..., a,..., a geeratig fuctio some way of represetig the sequece as a fuctio. There are may ways to do this, with the most commo ways
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 19 11/17/2008 LAWS OF LARGE NUMBERS II THE STRONG LAW OF LARGE NUMBERS
MASSACHUSTTS INSTITUT OF TCHNOLOGY 6.436J/5.085J Fall 2008 Lecture 9 /7/2008 LAWS OF LARG NUMBRS II Cotets. The strog law of large umbers 2. The Cheroff boud TH STRONG LAW OF LARG NUMBRS While the weak
More informationThe Random Walk For Dummies
The Radom Walk For Dummies Richard A Mote Abstract We look at the priciples goverig the oe-dimesioal discrete radom walk First we review five basic cocepts of probability theory The we cosider the Beroulli
More information7.1 Convergence of sequences of random variables
Chapter 7 Limit theorems Throughout this sectio we will assume a probability space (Ω, F, P), i which is defied a ifiite sequece of radom variables (X ) ad a radom variable X. The fact that for every ifiite
More informationRecurrence Relations
Recurrece Relatios Aalysis of recursive algorithms, such as: it factorial (it ) { if (==0) retur ; else retur ( * factorial(-)); } Let t be the umber of multiplicatios eeded to calculate factorial(). The
More informationCHAPTER 5. Theory and Solution Using Matrix Techniques
A SERIES OF CLASS NOTES FOR 2005-2006 TO INTRODUCE LINEAR AND NONLINEAR PROBLEMS TO ENGINEERS, SCIENTISTS, AND APPLIED MATHEMATICIANS DE CLASS NOTES 3 A COLLECTION OF HANDOUTS ON SYSTEMS OF ORDINARY DIFFERENTIAL
More informationAbstract. 1. Introduction This note is a supplement to part I ([4]). Let. F x (1.1) x n (1.2) Then the moments L x are the Catalan numbers
Abstract Some elemetary observatios o Narayaa polyomials ad related topics II: -Narayaa polyomials Joha Cigler Faultät für Mathemati Uiversität Wie ohacigler@uivieacat We show that Catala umbers cetral
More informationNumber of Spanning Trees of Circulant Graphs C 6n and their Applications
Joural of Mathematics ad Statistics 8 (): 4-3, 0 ISSN 549-3644 0 Sciece Publicatios Number of Spaig Trees of Circulat Graphs C ad their Applicatios Daoud, S.N. Departmet of Mathematics, Faculty of Sciece,
More informationlim za n n = z lim a n n.
Lecture 6 Sequeces ad Series Defiitio 1 By a sequece i a set A, we mea a mappig f : N A. It is customary to deote a sequece f by {s } where, s := f(). A sequece {z } of (complex) umbers is said to be coverget
More informationHarmonic Number Identities Via Euler s Transform
1 2 3 47 6 23 11 Joural of Iteger Sequeces, Vol. 12 2009), Article 09.6.1 Harmoic Number Idetities Via Euler s Trasform Khristo N. Boyadzhiev Departmet of Mathematics Ohio Norther Uiversity Ada, Ohio 45810
More informationMIDTERM 3 CALCULUS 2. Monday, December 3, :15 PM to 6:45 PM. Name PRACTICE EXAM SOLUTIONS
MIDTERM 3 CALCULUS MATH 300 FALL 08 Moday, December 3, 08 5:5 PM to 6:45 PM Name PRACTICE EXAM S Please aswer all of the questios, ad show your work. You must explai your aswers to get credit. You will
More information3. Z Transform. Recall that the Fourier transform (FT) of a DT signal xn [ ] is ( ) [ ] = In order for the FT to exist in the finite magnitude sense,
3. Z Trasform Referece: Etire Chapter 3 of text. Recall that the Fourier trasform (FT) of a DT sigal x [ ] is ω ( ) [ ] X e = j jω k = xe I order for the FT to exist i the fiite magitude sese, S = x [
More informationAssignment 5: Solutions
McGill Uiversity Departmet of Mathematics ad Statistics MATH 54 Aalysis, Fall 05 Assigmet 5: Solutios. Let y be a ubouded sequece of positive umbers satisfyig y + > y for all N. Let x be aother sequece
More informationk-generalized FIBONACCI NUMBERS CLOSE TO THE FORM 2 a + 3 b + 5 c 1. Introduction
Acta Math. Uiv. Comeiaae Vol. LXXXVI, 2 (2017), pp. 279 286 279 k-generalized FIBONACCI NUMBERS CLOSE TO THE FORM 2 a + 3 b + 5 c N. IRMAK ad M. ALP Abstract. The k-geeralized Fiboacci sequece { F (k)
More informationSubject: Differential Equations & Mathematical Modeling-III
Power Series Solutios of Differetial Equatios about Sigular poits Subject: Differetial Equatios & Mathematical Modelig-III Lesso: Power series solutios of differetial equatios about Sigular poits Lesso
More informationChimica Inorganica 3
himica Iorgaica Irreducible Represetatios ad haracter Tables Rather tha usig geometrical operatios, it is ofte much more coveiet to employ a ew set of group elemets which are matrices ad to make the rule
More informationA CONTINUED FRACTION EXPANSION FOR A q-tangent FUNCTION
Sémiaire Lotharigie de Combiatoire 45 001, Article B45b A CONTINUED FRACTION EXPANSION FOR A q-tangent FUNCTION MARKUS FULMEK Abstract. We prove a cotiued fractio expasio for a certai q taget fuctio that
More informationCentral limit theorem and almost sure central limit theorem for the product of some partial sums
Proc. Idia Acad. Sci. Math. Sci. Vol. 8, No. 2, May 2008, pp. 289 294. Prited i Idia Cetral it theorem ad almost sure cetral it theorem for the product of some partial sums YU MIAO College of Mathematics
More informationSupplemental Material: Proofs
Proof to Theorem Supplemetal Material: Proofs Proof. Let be the miimal umber of traiig items to esure a uique solutio θ. First cosider the case. It happes if ad oly if θ ad Rak(A) d, which is a special
More informationSection 11.8: Power Series
Sectio 11.8: Power Series 1. Power Series I this sectio, we cosider geeralizig the cocept of a series. Recall that a series is a ifiite sum of umbers a. We ca talk about whether or ot it coverges ad i
More informationProperties and Tests of Zeros of Polynomial Functions
Properties ad Tests of Zeros of Polyomial Fuctios The Remaider ad Factor Theorems: Sythetic divisio ca be used to fid the values of polyomials i a sometimes easier way tha substitutio. This is show by
More information