Cyclic Averages of Regular Polygons


 Mamuka Meskhishvili
 7 months ago
 Views:
Transcription
1 Cyclic Averages of Regular Polygos Maua Meshishvili Abstract We cosider a regular plae polygo with vertices ad a arbitrary poit i the plae Let R be the circuscribed radius of the polygo ad L a distace fro the poit to the cetroid of the polygo The the averages of the th powers of distaces fro the poit to the polygo vertices satisfy the relatios = R L, S = R L where =,, R L RL, Itroductio I his boo Matheatical Circus, Marti Garder wrote p 65: There is a beautifully syetric equatio for fidig the side of a equilateral triagle whe give the distaces of a poit fro its three corers: a 4 b 4 c 4 d 4 = a b c d This result was geeralized by J Beti [] fro a equilateral triagle to a regular polygo Cosider a regular plae polygo with vertices ad a arbitrary poit i the plae Deote by s ad q 4 respectively the averages of the squares ad the averages of the fourth powers of the distaces fro the poit to the vertices of the polygo s = d i ad q 4 = d 4 i The i= q 4 R 4 = s R is satisfied, where R is the circuscribed radius of the polygo Keywords ad phrases: Cetroid, regular polygo, average of distaces, polygoal, siplicial, polytopic distaces 00Matheatics Subject Classificatio: 5M04, 5N0, 5N5 i=
2 This result was geeralized to regular siplicial ad regular polytopic distaces i [] ad [], respectively I the aboveetioed papers the distaces are cosidered to the secod ad fourth powers oly Naturally, we are iterested to ow what happes if we cosider the distaces to higher ore tha 4 powers I the preset paper, for a regular polygo we itroduce a special id of averages of the distaces to the eve powers the cyclic averages ad by usig their properties establish the etrical relatios for regular polygos Geeral case Let us cosider a regular plae sided polygo A A A with the circuscribed radius R ad a arbitrary poit P i the plae Deote the distaces betwee P ad the cetroid O of the polygo by L, ad the distaces betwee P ad the ivertex by d i Fig Figure We use the followig otatio for the average of the th powers of the distaces: S = d i Theore For a regular polygo with vertices ad a arbitrary poit i the plae, let S be the averages of the th powers of the distaces fro the poit to the vertices If R is the circuscribed radius ad L the distace betwee the poit ad the cetroid, the = R L, S = R L where =,, i= R L RL,
3 First we eed to prove two leas Lea For arbitrary positive itegers ad, such that <, the followig coditio cos α π = 0 is satisfied, where α is a arbitrary agle Deote T = e iα e iα π e iα π e iα π The real part of T is ReT = cos α π The forula of the su of geoetric progressio gives T = e iα e i π e i π e i π = π iα e i = e, e i π e i π = cos π i π = Sice <, e i π So T = 0, ie ReT = 0, which proves Lea Rear If, the su always cotais α Lea For arbitrary positive itegers ad, such that < ad for a arbitrary agle α the followig coditios are satisfied: if is odd if is eve cos α π = 0; cos α π = Whe is odd, usig the powerreductio forula for cosie cos θ = =0 cos θ,
4 4 we obtai cos α π = = cos α cos α π = = [ 0 cos α 0 cos = cos α π cos α α π cos cos α π 0 cos α π α π cos cos α π [ cos α cos α π 0 cos α α π ] = cos α π cos αcos α π cos α cos cos α π α π cos α π ] Sice <, fro Lea it follows that each su equals zero, which proves the first part of Lea Whe is eve, the powerreductio forula for cosie is cos θ = =0 cos θ Aalogously to the case with odd, the su of the secod addeda vaishes, ad sice the uber of the first addeda is, the total su equals,
5 5 which proves Lea Proof of the theore We itroduce the ew otatios A = R L ad B = RL The π S = A B cos α A B cos α A B cos π α A B cos π α If =, by Lea we have π S = A B cos α A B cos α A B cos π α = A Therefore If >, we have S = R L π = A A B cos α cos α cos π α A B cos α cos π α ± cos π α A B cos α cos π α B cos π α cos α cos π α cos π α Accordig to Lea, all sus with the egative sig vaishes because they cotai odd powers ad there reai oly the sus with eve powers
6 6 If is eve, we have S = A = If is odd, we write A A B cos α cos π α B cos π α cos α cos π α cos π α = A B S = A A B cos α cos π α cos π α AB cos α cos π α = A cos π α = A B Usig the floor fuctio the iteger part, the obtaied results ca be cobied ito a sigle forula as follows S = A which proves the theore A B, The values of the averages S, S 4,, S reai costat whe the poit P oves o the circle CO, L, ie if we cosider ay poit o the circle P Fig, these averages will retai the sae values So we ca forulate
7 Defiitio The cyclic average of a regular polygo is the average of the power of the distaces fro the poit to the vertices, the value of which is costat for ay poit o the circle CO, L, where O is the cetroid of the polygo ad L is the distace betwee the poit ad the cetroid The properties of the cyclic average are as follows: Property Each regular go has a uber of cyclic averages, S 4,, S Property Cyclic averages ca be expressed oly i ters of the circuscribed radius R ad the distace L Property The expressios of the ocyclic averages cotai α, ie deped o the directio OP Fig Property 4 For fixed R ad L, the cyclic averages of equal powers of differet regular gos are the sae: = 4 = 5 = 6 =, S 4 = S 4 4 = S 4 5 = S 4 6 =, S 6 4 = S 6 5 = S 6 6 =, S 8 5 = S 8 6 = Property 5 Ay relatios i ters of the cyclic averages S, the circuscibed radius R ad the distace L, which are satisfied for a regular go, are at the sae tie satisfied for ay regular go, where, ie S ca be replaced by S 7 Equilateral triagle There are cyclic averages: Special cases = d d d = R L, S 4 = d4 d 4 d 4 = R L R L By eliiatig L, we obtai the forula itroduced by Garde d 4 d4 d4 d R 4 = d d R I ters of the cyclic averages S 4 R 4 = R By Property 5, for ay S 4 ad S, where Beti s result we have S 4 R 4 = R
8 8 Square There are cyclic averages: 4 = 4 d d d d 4 = R L, S 4 4 = 4 d4 d 4 d 4 d 4 4 = R L R L, S 6 4 = 4 d6 d 6 d 6 d 6 4 = R L 6R L R L Eliiatig L fro the cyclic averages 4 ad S 6 4 we obtai the secod relatio betwee the distaces ad the circuscribed radius Propositio For ay regular go, where 4, we have S 6 = S S R 5R 4 Substitutig R L = 4 ad R L = S ito S 6 4, we establish the direct correspodece betwee the distaces Propositio For ay regular go, where 4, S 6 = S S 4 S For the square, fro Propositio it follows that 8d 6 d 6 d 6 d 6 4d d d d 4 = 6d d d d 4d 4 d 4 d 4 d 4 4, which is equivalet to d d d d 4d d d d 4d d 4 d d = 0 Euerate the vertices of the square: A A A A 4 The oly d d = d d 4 holds, which together with the cyclic averages 4 ad S 4 4 iplies d d = d d 4 = R L, d d d d 4 = R 4 L 4 Refereces [] Beti, J, Regular polygoal distaces, Math Gaz, 8 July 997, pp [] Beti, J, Regular siplicial distaces, Math Gaz 79 July 995, p 06 [] Par, PS, Regular polytopic distaces, Foru Geo 6 06, 7 DEPARTMENT OF MATHEMATICS GEORGIANAMERICAN HIGH SCHOOL 8 CHKONDIDELI STR, TBILISI 080, GEORGIA Eail address:
Jacobi symbols. p 1. Note: The Jacobi symbol does not necessarily distinguish between quadratic residues and nonresidues. That is, we could have ( a
Jacobi sybols efiitio Let be a odd positive iteger If 1, the Jacobi sybol : Z C is the costat fuctio 1 1 If > 1, it has a decopositio ( as ) a product of (ot ecessarily distict) pries p 1 p r The Jacobi
More informationBertrand s postulate Chapter 2
Bertrad s postulate Chapter We have see that the sequece of prie ubers, 3, 5, 7,... is ifiite. To see that the size of its gaps is ot bouded, let N := 3 5 p deote the product of all prie ubers that are
More informationOrthogonal Functions
Royal Holloway Uiversity of odo Departet of Physics Orthogoal Fuctios Motivatio Aalogy with vectors You are probably failiar with the cocept of orthogoality fro vectors; two vectors are orthogoal whe they
More informationBinomial transform of products
Jauary 02 207 Bioial trasfor of products Khristo N Boyadzhiev Departet of Matheatics ad Statistics Ohio Norther Uiversity Ada OH 4580 USA boyadzhiev@ouedu Abstract Give the bioial trasfors { b } ad {
More informationx !1! + 1!2!
4 EulerMaclauri Suatio Forula 4. Beroulli Nuber & Beroulli Polyoial 4.. Defiitio of Beroulli Nuber Beroulli ubers B (,,3,) are defied as coefficiets of the followig equatio. x e x  B x! 4.. Expreesio
More information(s)h(s) = K( s + 8 ) = 5 and one finite zero is located at z 1
ROOT LOCUS TECHNIQUE 93 should be desiged differetly to eet differet specificatios depedig o its area of applicatio. We have observed i Sectio 6.4 of Chapter 6, how the variatio of a sigle paraeter like
More informationECE Spring Prof. David R. Jackson ECE Dept. Notes 20
ECE 6341 Sprig 016 Prof. David R. Jackso ECE Dept. Notes 0 1 Spherical Wave Fuctios Cosider solvig ψ + k ψ = 0 i spherical coordiates z φ θ r y x Spherical Wave Fuctios (cot.) I spherical coordiates we
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 informationDifferent kinds of Mathematical Induction
Differet ids of Mathematical Iductio () Mathematical Iductio Give A N, [ A (a A a A)] A N () (First) Priciple of Mathematical Iductio Let P() be a propositio (ope setece), if we put A { : N p() is true}
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 informationChapter 4. Fourier Series
Chapter 4. Fourier Series At this poit we are ready to ow cosider the caoical equatios. Cosider, for eample the heat equatio u t = u, < (4.) subject to u(, ) = si, u(, t) = u(, t) =. (4.) Here,
More informationThe Hypergeometric Coupon Collection Problem and its Dual
Joural of Idustrial ad Systes Egieerig Vol., o., pp 7 Sprig 7 The Hypergeoetric Coupo Collectio Proble ad its Dual Sheldo M. Ross Epstei Departet of Idustrial ad Systes Egieerig, Uiversity of Souther
More informationName Period ALGEBRA II Chapter 1B and 2A Notes Solving Inequalities and Absolute Value / Numbers and Functions
Nae Period ALGEBRA II Chapter B ad A Notes Solvig Iequalities ad Absolute Value / Nubers ad Fuctios SECTION.6 Itroductio to Solvig Equatios Objectives: Write ad solve a liear equatio i oe variable. Solve
More informationJORGE LUIS AROCHA AND BERNARDO LLANO. Average atchig polyoial Cosider a siple graph G =(V E): Let M E a atchig of the graph G: If M is a atchig, the a
MEAN VALUE FOR THE MATCHING AND DOMINATING POLYNOMIAL JORGE LUIS AROCHA AND BERNARDO LLANO Abstract. The ea value of the atchig polyoial is coputed i the faily of all labeled graphs with vertices. We dee
More informationProblem. Consider the sequence a j for j N defined by the recurrence a j+1 = 2a j + j for j > 0
GENERATING FUNCTIONS Give a ifiite sequece a 0,a,a,, its ordiary geeratig fuctio is A : a Geeratig fuctios are ofte useful for fidig a closed forula for the eleets of a sequece, fidig a recurrece forula,
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 informationCALCULATION 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 4707 Spring 2018 (Darij Grinberg): midterm 2 page 1. Math 4707 Spring 2018 (Darij Grinberg): midterm 2 with solutions [preliminary version]
Math 4707 Sprig 08 Darij Griberg: idter page Math 4707 Sprig 08 Darij Griberg: idter with solutios [preliiary versio] Cotets 0.. Coutig firsteve tuples......................... 3 0.. Coutig legal paths
More informationBernoulli Polynomials Talks given at LSBU, October and November 2015 Tony Forbes
Beroulli Polyoials Tals give at LSBU, October ad Noveber 5 Toy Forbes Beroulli Polyoials The Beroulli polyoials B (x) are defied by B (x), Thus B (x) B (x) ad B (x) x, B (x) x x + 6, B (x) dx,. () B 3
More informationLecture 10: Bounded Linear Operators and Orthogonality in Hilbert Spaces
Lecture : Bouded Liear Operators ad Orthogoality i Hilbert Spaces 34 Bouded Liear Operator Let ( X, ), ( Y, ) i i be ored liear vector spaces ad { } X Y The, T is said to be bouded if a real uber c such
More informationSEQUENCES AND SERIES
Sequeces ad 6 Sequeces Ad SEQUENCES AND SERIES Successio of umbers of which oe umber is desigated as the first, other as the secod, aother as the third ad so o gives rise to what is called a sequece. Sequeces
More informationCALCULUS BASIC SUMMER REVIEW
CALCULUS BASIC SUMMER REVIEW NAME rise y y y Slope of a o vertical lie: m ru Poit Slope Equatio: y y m( ) The slope is m ad a poit o your lie is, ). ( y SlopeItercept Equatio: y m b slope= m yitercept=
More informationCommon Fixed Points for Multifunctions Satisfying a Polynomial Inequality
BULETINUL Uiversităţii Petrol Gaze di Ploieşti Vol LXII No /00 6065 Seria Mateatică  Iforatică  Fizică Coo Fixed Poits for Multifuctios Satisfyig a Polyoial Iequality Alexadru Petcu Uiversitatea PetrolGaze
More informationdistinct distinct n k n k n! n n k k n 1 if k n, identical identical p j (k) p 0 if k > n n (k)
THE TWELVEFOLD WAY FOLLOWING GIANCARLO ROTA How ay ways ca we distribute objects to recipiets? Equivaletly, we wat to euerate equivalece classes of fuctios f : X Y where X = ad Y = The fuctios are subject
More informationSome remarks on the paper Some elementary inequalities of G. Bennett
Soe rears o the paper Soe eleetary iequalities of G. Beett Dag Ah Tua ad Luu Quag Bay Vieta Natioal Uiversity  Haoi Uiversity of Sciece Abstract We give soe couterexaples ad soe rears of soe of the corollaries
More informationSimple Polygons of Maximum Perimeter Contained in a Unit Disk
Discrete Comput Geom (009) 1: 08 15 DOI 10.1007/s00500890937 Simple Polygos of Maximum Perimeter Cotaied i a Uit Disk Charles Audet Pierre Hase Frédéric Messie Received: 18 September 007 / Revised:
More informationZeros of Polynomials
Math 160 www.timetodare.com 4.5 4.6 Zeros of Polyomials I these sectios we will study polyomials algebraically. Most of our work will be cocered with fidig the solutios of polyomial equatios of ay degree
More information6.4 Binomial Coefficients
64 Bioial Coefficiets Pascal s Forula Pascal s forula, aed after the seveteethcetury Frech atheaticia ad philosopher Blaise Pascal, is oe of the ost faous ad useful i cobiatorics (which is the foral ter
More information#A18 INTEGERS 11 (2011) THE (EXPONENTIAL) BIPARTITIONAL POLYNOMIALS AND POLYNOMIAL SEQUENCES OF TRINOMIAL TYPE: PART I
#A18 INTEGERS 11 (2011) THE (EXPONENTIAL) BIPARTITIONAL POLYNOMIALS AND POLYNOMIAL SEQUENCES OF TRINOMIAL TYPE: PART I Miloud Mihoubi 1 Uiversité des Scieces et de la Techologie Houari Bouediee Faculty
More informationSequences A sequence of numbers is a function whose domain is the positive integers. We can see that the sequence
Sequeces A sequece of umbers is a fuctio whose domai is the positive itegers. We ca see that the sequece 1, 1, 2, 2, 3, 3,... is a fuctio from the positive itegers whe we write the first sequece elemet
More informationA Recurrence Formula for Packing HyperSpheres
A Recurrece Formula for Packig HyperSpheres DokeyFt. Itroductio We cosider packig of D hyperspheres of uit diameter aroud a similar sphere. The kissig spheres ad the kerel sphere form cells of equilateral
More informationDouble Derangement Permutations
Ope Joural of iscrete Matheatics, 206, 6, 9904 Published Olie April 206 i SciRes http://wwwscirporg/joural/ojd http://dxdoiorg/04236/ojd2066200 ouble erageet Perutatios Pooya aeshad, Kayar Mirzavaziri
More informationFACTORS OF SUMS OF POWERS OF BINOMIAL COEFFICIENTS. Dedicated to the memory of Paul Erdős. 1. Introduction. n k. f n,a =
FACTORS OF SUMS OF POWERS OF BINOMIAL COEFFICIENTS NEIL J. CALKIN Abstract. We prove divisibility properties for sus of powers of bioial coefficiets ad of bioial coefficiets. Dedicated to the eory of
More informationAvailable online at J. Math. Comput. Sci. 4 (2014), No. 3, ISSN:
Available olie at http://scik.org J. Math. Coput. Sci. (1, No. 3, 95 ISSN: 197537 ON SYMMETRICAL FUNCTIONS WITH BOUNDED BOUNDARY ROTATION FUAD. S. M. AL SARARI 1,, S. LATHA 1 Departet of Studies i Matheatics,
More informationOn Order of a Function of Several Complex Variables Analytic in the Unit Polydisc
ISSN 7467659, Eglad, UK Joural of Iforatio ad Coutig Sciece Vol 6, No 3, 0, 9506 O Order of a Fuctio of Several Colex Variables Aalytic i the Uit Polydisc Rata Kuar Dutta + Deartet of Matheatics, Siliguri
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 publisherprovided material Overview May mathematical
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 informationA New Type of qszászmirakjan Operators
Filoat 3:8 07, 567 568 https://doi.org/0.98/fil7867c Published by Faculty of Scieces ad Matheatics, Uiversity of Niš, Serbia Available at: http://www.pf.i.ac.rs/filoat A New Type of SzászMiraka Operators
More informationBeurling Integers: Part 2
Beurlig Itegers: Part 2 Isomorphisms Devi Platt July 11, 2015 1 Prime Factorizatio Sequeces I the last article we itroduced the Beurlig geeralized itegers, which ca be represeted as a sequece of real umbers
More informationR is a scalar defined as follows:
Math 8. Notes o Dot Product, Cross Product, Plaes, Area, ad Volumes This lecture focuses primarily o the dot product ad its may applicatios, especially i the measuremet of agles ad scalar projectio ad
More informationABOUT CHAOS AND SENSITIVITY IN TOPOLOGICAL DYNAMICS
ABOUT CHAOS AND SENSITIVITY IN TOPOLOGICAL DYNAMICS EDUARD KONTOROVICH Abstract. I this work we uify ad geeralize some results about chaos ad sesitivity. Date: March 1, 005. 1 1. Symbolic Dyamics Defiitio
More informationAMS Mathematics Subject Classification : 40A05, 40A99, 42A10. Key words and phrases : Harmonic series, Fourier series. 1.
J. Appl. Math. & Computig Vol. x 00y), No. z, pp. A RECURSION FOR ALERNAING HARMONIC SERIES ÁRPÁD BÉNYI Abstract. We preset a coveiet recursive formula for the sums of alteratig harmoic series of odd order.
More informationA sequence of numbers is a function whose domain is the positive integers. We can see that the sequence
Sequeces A sequece of umbers is a fuctio whose domai is the positive itegers. We ca see that the sequece,, 2, 2, 3, 3,... is a fuctio from the positive itegers whe we write the first sequece elemet as
More informationStanford Math Circle January 21, Complex Numbers
Staford Math Circle Jauary, 007 Some History Tatiaa Shubi (shubi@mathsjsuedu) Complex Numbers Let us try to solve the equatio x = 5x + x = is a obvious solutio Also, x 5x = ( x )( x + x + ) = 0 yields
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 DaaPicard Departmet of Applied Mathematics Jerusalem College of Techology
More informationThe ztransform. 7.1 Introduction. 7.2 The ztransform Derivation of the ztransform: x[n] = z n LTI system, h[n] z = re j
The Trasform 7. Itroductio Geeralie the complex siusoidal represetatio offered by DTFT to a represetatio of complex expoetial sigals. Obtai more geeral characteristics for discretetime LTI systems. 7.
More informationONE MODULO THREE GEOMETRIC MEAN LABELING OF SOME FAMILIES OF GRAPHS
ONE MODULO THREE GEOMETRIC MEAN LABELING OF SOME FAMILIES OF GRAPHS A.Maheswari 1, P.Padiaraj 2 1,2 Departet of Matheatics,Kaaraj College of Egieerig ad Techology, Virudhuagar (Idia) ABSTRACT A graph G
More informationHow to Maximize a Function without Really Trying
How to Maximize a Fuctio without Really Tryig MARK FLANAGAN School of Electrical, Electroic ad Commuicatios Egieerig Uiversity College Dubli We will prove a famous elemetary iequality called The Rearragemet
More information6 Integers Modulo n. integer k can be written as k = qn + r, with q,r, 0 r b. So any integer.
6 Itegers Modulo I Example 2.3(e), we have defied the cogruece of two itegers a,b with respect to a modulus. Let us recall that a b (mod ) meas a b. We have proved that cogruece is a equivalece relatio
More informationMathematical Induction
Mathematical Iductio Itroductio Mathematical iductio, or just iductio, is a proof techique. Suppose that for every atural umber, P() is a statemet. We wish to show that all statemets P() are true. I a
More informationMath 257: Finite difference methods
Math 257: Fiite differece methods 1 Fiite Differeces Remember the defiitio of a derivative f f(x + ) f(x) (x) = lim 0 Also recall Taylor s formula: (1) f(x + ) = f(x) + f (x) + 2 f (x) + 3 f (3) (x) +...
More informationFundamental Theorem of Algebra. Yvonne Lai March 2010
Fudametal Theorem of Algebra Yvoe Lai March 010 We prove the Fudametal Theorem of Algebra: Fudametal Theorem of Algebra. Let f be a ocostat polyomial with real coefficiets. The f has at least oe complex
More informationOrthogonal transformations
Orthogoal trasformatios October 12, 2014 1 Defiig property The squared legth of a vector is give by takig the dot product of a vector with itself, v 2 v v g ij v i v j A orthogoal trasformatio is a liear
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 informationMATH spring 2008 lecture 3 Answers to selected problems. 0 sin14 xdx = x dx. ; (iv) x +
MATH  sprig 008 lecture Aswers to selected problems INTEGRALS. f =? For atiderivatives i geeral see the itegrals website at http://itegrals.wolfram.com. (5vi (0 i ( ( i ( π ; (v π a. This is example
More informationChapter 7: The ztransform. ChihWei Liu
Chapter 7: The Trasform ChihWei Liu Outlie Itroductio The Trasform Properties of the Regio of Covergece Properties of the Trasform Iversio of the Trasform The Trasfer Fuctio Causality ad Stability
More informationRiemann Sums y = f (x)
Riema Sums Recall that we have previously discussed the area problem I its simplest form we ca state it this way: The Area Problem Let f be a cotiuous, oegative fuctio o the closed iterval [a, b] Fid
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 informationAnalytic Continuation
Aalytic Cotiuatio The stadard example of this is give by Example Let h (z) = 1 + z + z 2 + z 3 +... kow to coverge oly for z < 1. I fact h (z) = 1/ (1 z) for such z. Yet H (z) = 1/ (1 z) is defied for
More informationPolynomial Functions and Their Graphs
Polyomial Fuctios ad Their Graphs I this sectio we begi the study of fuctios defied by polyomial expressios. Polyomial ad ratioal fuctios are the most commo fuctios used to model data, ad are used extesively
More informationAn Algorithmist s Toolkit October 20, Lecture 11
18.409 A Algorithist s Toolkit October 20, 2009 Lecture 11 Lecturer: Joatha Keler Scribe: Chaithaya Badi 1 Outlie Today we ll itroduce ad discuss Polar of a covex body. Correspodece betwee or fuctios ad
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 informationCOMPLEX NUMBERS AND DE MOIVRE'S THEOREM SYNOPSIS. Ay umber of the form x+iy where x, y R ad i =  is called a complex umber.. I the complex umber x+iy, x is called the real part ad y is called the imagiary
More informationDiscrete Mathematics: Lectures 8 and 9 Principle of Inclusion and Exclusion Instructor: Arijit Bishnu Date: August 11 and 13, 2009
Discrete Matheatics: Lectures 8 ad 9 Priciple of Iclusio ad Exclusio Istructor: Arijit Bishu Date: August ad 3, 009 As you ca observe by ow, we ca cout i various ways. Oe such ethod is the ageold priciple
More informationTHE KENNESAW STATE UNIVERSITY HIGH SCHOOL MATHEMATICS COMPETITION PART II Calculators are NOT permitted Time allowed: 2 hours
THE 0607 KENNESAW STATE UNIVERSITY HIGH SCHOOL MATHEMATICS COMPETITION PART II Calculators are NOT permitted Time allowed: hours Let x, y, ad A all be positive itegers with x y a) Prove that there are
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 informationTHE GREATEST ORDER OF THE DIVISOR FUNCTION WITH INCREASING DIMENSION
MATHEMATICA MONTISNIGRI Vol XXVIII (013) 175 THE GREATEST ORDER OF THE DIVISOR FUNCTION WITH INCREASING DIMENSION GLEB V. FEDOROV * * Mechaics ad Matheatics Faculty Moscow State Uiversity Moscow, Russia
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 informationChapter 2. Periodic points of toral. automorphisms. 2.1 General introduction
Chapter 2 Periodic poits of toral automorphisms 2.1 Geeral itroductio The automorphisms of the twodimesioal torus are rich mathematical objects possessig iterestig geometric, algebraic, topological ad
More informationAn analog of the arithmetic triangle obtained by replacing the products by the least common multiples
arxiv:10021383v2 [mathnt] 9 Feb 2010 A aalog of the arithmetic triagle obtaied by replacig the products by the least commo multiples Bair FARHI bairfarhi@gmailcom MSC: 11A05 Keywords: AlKaraji s triagle;
More informationComplex Analysis Spring 2001 Homework I Solution
Complex Aalysis Sprig 2001 Homework I Solutio 1. Coway, Chapter 1, sectio 3, problem 3. Describe the set of poits satisfyig the equatio z a z + a = 2c, where c > 0 ad a R. To begi, we see from the triagle
More informationANSWERS SOLUTIONS iiii i. and 1. Thus, we have. i i i. i, A.
013 ΜΑΘ Natioal Covetio ANSWERS (1) C A A A B (6) B D D A B (11) C D D A A (16) D B A A C (1) D B C B C (6) D C B C C 1. We have SOLUTIONS 1 3 11 61 iiii 131161 i 013 013, C.. The powers of i cycle betwee
More informationPAijpam.eu ON DERIVATION OF RATIONAL SOLUTIONS OF BABBAGE S FUNCTIONAL EQUATION
Iteratioal Joural of Pure ad Applied Mathematics Volume 94 No. 204, 920 ISSN: 38080 (prited versio); ISSN: 343395 (olie versio) url: http://www.ijpam.eu doi: http://dx.doi.org/0.2732/ijpam.v94i.2 PAijpam.eu
More informationThe Random Walk For Dummies
The Radom Walk For Dummies Richard A Mote Abstract We look at the priciples goverig the oedimesioal discrete radom walk First we review five basic cocepts of probability theory The we cosider the Beroulli
More informationJEE ADVANCED 2013 PAPER 1 MATHEMATICS
Oly Oe Optio Correct Type JEE ADVANCED 0 PAPER MATHEMATICS This sectio cotais TEN questios. Each has FOUR optios (A), (B), (C) ad (D) out of which ONLY ONE is correct.. The value of (A) 5 (C) 4 cot cot
More informationThe Riemann Zeta Function
Physics 6A Witer 6 The Riema Zeta Fuctio I this ote, I will sketch some of the mai properties of the Riema zeta fuctio, ζ(x). For x >, we defie ζ(x) =, x >. () x = For x, this sum diverges. However, we
More informationNEW FAST CONVERGENT SEQUENCES OF EULERMASCHERONI TYPE
UPB Sci Bull, Series A, Vol 79, Iss, 207 ISSN 227027 NEW FAST CONVERGENT SEQUENCES OF EULERMASCHERONI TYPE Gabriel Bercu We itroduce two ew sequeces of EulerMascheroi type which have fast covergece
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 informationON THE LEHMER CONSTANT OF FINITE CYCLIC GROUPS
ON THE LEHMER CONSTANT OF FINITE CYCLIC GROUPS NORBERT KAIBLINGER Abstract. Results of Lid o Lehmer s problem iclude the value of the Lehmer costat of the fiite cyclic group Z/Z, for 5 ad all odd. By complemetary
More informationOn Topologically Finite Spaces
saqartvelos mecierebata erovuli aademiis moambe, t 9, #, 05 BULLETIN OF THE GEORGIAN NATIONAL ACADEMY OF SCIENCES, vol 9, o, 05 Mathematics O Topologically Fiite Spaces Giorgi Vardosaidze St Adrew the
More informationMAT 271 Project: Partial Fractions for certain rational functions
MAT 7 Project: Partial Fractios for certai ratioal fuctios Prerequisite kowledge: partial fractios from MAT 7, a very good commad of factorig ad complex umbers from Precalculus. To complete this project,
More informationOn the Fibonaccilike Sequences of Higher Order
Article Iteratioal Joural of oder atheatical Scieces, 05, 3(): 559 Iteratioal Joural of oder atheatical Scieces Joural hoepage: wwwoderscietificpressco/jourals/ijsaspx O the Fiboaccilike Sequeces of
More informationMath F215: Induction April 7, 2013
Math F25: Iductio April 7, 203 Iductio is used to prove that a collectio of statemets P(k) depedig o k N are all true. A statemet is simply a mathematical phrase that must be either true or false. Here
More informationProblem Cosider the curve give parametrically as x = si t ad y = + cos t for» t» ß: (a) Describe the path this traverses: Where does it start (whe t =
Mathematics Summer Wilso Fial Exam August 8, ANSWERS Problem 1 (a) Fid the solutio to y +x y = e x x that satisfies y() = 5 : This is already i the form we used for a first order liear differetial equatio,
More informationSOME SEQUENCE SPACES DEFINED BY ORLICZ FUNCTIONS
ARCHIVU ATHEATICU BRNO Tomus 40 2004, 33 40 SOE SEQUENCE SPACES DEFINED BY ORLICZ FUNCTIONS E. SAVAŞ AND R. SAVAŞ Abstract. I this paper we itroduce a ew cocept of λstrog covergece with respect to a Orlicz
More informationON CONVERGENCE OF BASIC HYPERGEOMETRIC SERIES. 1. Introduction Basic hypergeometric series (cf. [GR]) with the base q is defined by
ON CONVERGENCE OF BASIC HYPERGEOMETRIC SERIES TOSHIO OSHIMA Abstract. We examie the covergece of qhypergeometric series whe q =. We give a coditio so that the radius of the covergece is positive ad get
More informationSZEGO S THEOREM STARTING FROM JENSEN S THEOREM
UPB Sci Bull, Series A, Vol 7, No 3, 8 ISSN 377 SZEGO S THEOREM STARTING FROM JENSEN S THEOREM Cǎli Alexe MUREŞAN Mai îtâi vo itroduce Teorea lui Jese şi uele coseciţe ale sale petru deteriarea uǎrului
More informationChapter 2. Asymptotic Notation
Asyptotic Notatio 3 Chapter Asyptotic Notatio Goal : To siplify the aalysis of ruig tie by gettig rid of details which ay be affected by specific ipleetatio ad hardware. [1] The Big Oh (ONotatio) : It
More informationPUTNAM TRAINING INEQUALITIES
PUTNAM TRAINING INEQUALITIES (Last updated: December, 207) Remark This is a list of exercises o iequalities Miguel A Lerma Exercises If a, b, c > 0, prove that (a 2 b + b 2 c + c 2 a)(ab 2 + bc 2 + ca
More informationSolutions to Final Exam Review Problems
. Let f(x) 4+x. Solutios to Fial Exam Review Problems Math 5C, Witer 2007 (a) Fid the Maclauri series for f(x), ad compute its radius of covergece. Solutio. f(x) 4( ( x/4)) ( x/4) ( ) 4 4 + x. Sice the
More informationx+ 2 + c p () x c p () x is an arbitrary function. ( sin x ) dx p f() d d f() dx = x dx p cosx = cos x+ 2 d p () x + xa r (1.
Super Derivative (Noiteger ties Derivative). Super Derivative ad Super Differetiatio Defitio.. p () obtaied by cotiuig aalytically the ide of the differetiatio operator of Higher Derivative of a fuctio
More informationLargest families without an rfork
Largest families without a rfor Aalisa De Bois Uiversity of Salero Salero, Italy debois@math.it Gyula O.H. Katoa Réyi Istitute Budapest, Hugary ohatoa@reyi.hu Itroductio Let [] = {,,..., } be a fiite
More informationCHAPTER 5 SOME MINIMAX AND SADDLE POINT THEOREMS
CHAPTR 5 SOM MINIMA AND SADDL POINT THORMS 5. INTRODUCTION Fied poit theorems provide importat tools i game theory which are used to prove the equilibrium ad eistece theorems. For istace, the fied poit
More informationAn Introduction to Randomized Algorithms
A Itroductio to Radomized Algorithms The focus of this lecture is to study a radomized algorithm for quick sort, aalyze it usig probabilistic recurrece relatios, ad also provide more geeral tools for aalysis
More informationREVISION SHEET FP1 (MEI) ALGEBRA. Identities In mathematics, an identity is a statement which is true for all values of the variables it contains.
The mai ideas are: Idetities REVISION SHEET FP (MEI) ALGEBRA Before the exam you should kow: If a expressio is a idetity the it is true for all values of the variable it cotais The relatioships betwee
More informationMath 451: Euclidean and NonEuclidean Geometry MWF 3pm, Gasson 204 Homework 3 Solutions
Math 451: Euclidea ad NoEuclidea Geometry MWF 3pm, Gasso 204 Homework 3 Solutios Exercises from 1.4 ad 1.5 of the otes: 4.3, 4.10, 4.12, 4.14, 4.15, 5.3, 5.4, 5.5 Exercise 4.3. Explai why Hp, q) = {x
More informationINMO2018 problems and solutions
Pioeer Educatio The Best Way To Success INMO018 proles ad solutios NTSE Olypiad AIPMT JEE  Mais & Advaced 1. Let ABC e a oequilateral triagle with iteger sides. Let D ad E e respectively the idpoits
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 informationMT5821 Advanced Combinatorics
MT5821 Advaced Combiatorics 1 Coutig subsets I this sectio, we cout the subsets of a elemet set. The coutig umbers are the biomial coefficiets, familiar objects but there are some ew thigs to say about
More informationOptimally Sparse SVMs
A. Proof of Lemma 3. We here prove a lower boud o the umber of support vectors to achieve geeralizatio bouds of the form which we cosider. Importatly, this result holds ot oly for liear classifiers, but
More information