Fractals - the ultimate art of mathematics. Adam Kozak
|
|
- August Hart
- 6 years ago
- Views:
Transcription
1 Fratals - the ultimate art of mathematis Adam Koak
2 Outlie What is fratal? Self-similarit dimesio Fratal tpes Iteratio Futio Sstems (IFS) L-sstems Itrodutio to omple umbers Madelbrot sets Julia ad Fatou sets Madelbulbs
3 What is fratal? Wh should I pa attetio to it? Geometri objet with propert of self-similarit i a sale fator i eat maer, approimate or stohasti Similarit dimesio ma be ot equal to topologi dimesio (o-iteger value) Relativel simple reursive defiitios Appliatios: Fratal ompressio Fratal art Ideas i egieerig, eletrois, hemistr, mediie, urba plaig whih have self-similarit patters Fratal atea i mobile phoes apable of apturig muh wider sope of frequeies i muh smaller areas tha lassi atea 3
4 Soure: Wikipedia Fratals i ature Romaeso brooli Fer High voltage breakdow withi a 4 blok of arli Coast with rivers 4
5 Kolmogorov ompleit Evertig what a be desribed, a be desribed as a strig of haraters over a alphabet of sie >. E.g. ifiite strig Ala ma kota, Ala ma kota, Ala ma kota, To eode suh a strig literall we would eed ifiite memor, however we kow that we a rereate its a fiite substrig simpl usig a omputer THIS STRING IS COMPUTABLE Kolmogorov ompleit of a fiite strig is a legth of the shortest omputer program whih rereates the strig (this is a uomputable futio there is o algorithm to evaluate it!) Kolmogorov( Ala ma kota, Ala ma kota, Ala ma kota, Ala ma kota, Ala ma kota, ) = Legth( while (true) prit( Ala ma kota, ); ) It is also alled iformatioal ompleit 5
6 Hausdorff similarit dimesio Similarit dimesio ma be ot equal to topologi dimesio (o-iteger value) N For ormal geometri objet if we sale it b fator (<<), we eed d opies of this objet to fill the area of origial objet where d is dimesio d, lim N d log 3 log d 3 lim log 3 log log log N,
7 Fratal tpes Fratals ma be obtaied from differet oepts: Atrators of Iterated Futio Sstems (IFS) Julia & Fatou sets Madelbrot sets L-sstem (Lidemaer sstem) 7
8 Cotratig mappig Let (X, d) be a metri spae, the f: X X is a otratig mappig if:, : a, a X : d f ( a ), f ( a ) d a a, Baah fied poit theorem: There eists eatl oe poit px suh, that f(p)=p (fied poit of otratig mappig) Reursive eeutio of otratig mappig: f(,)=(/3,/3) lim f os osos osos f os 8
9 Iterated Futio Sstems (IFS) Reursive trasformatios of geometri objet whih sum produt of a set of affie otratig mappigs (ompositios of rotatio, refletio, traslatio ad otratig salig): {F i : X X } (i ) S S Sk i F i Sk S lim Sk k S is a o empt set of poits i a give spae X S is a fratal a attrator of IFS, it s idepedet of iitial S (S is a fied poit of set of otratig mappigs {F i } i metri spae (H, h) where H is set of all ompat subsets of X ad h is Hausdorff distae) 9
10 Iterated Futio Sstems (IFS) A affie otratig mappig F i i spae has the followig formula: os si si os ' ', ' ' i t t F f e d b a 3 4 t t
11 A eample of IFS Sierpiński triagle IFS: {F i : } (i=..3): Sierpiński triagle is a fied poit (attrator) of Iterated Futio Sstem {F, F, F 3 } 4 3, 4, 4, 3 F F F F, F, F, 3
12 IFS workshop Task: loate, out ad defie otratig mappigs Barsle fer with some lues ;) [sr: Wikipedia] Sierpiński arpet [sr: Wikipedia] Sierpiński triagle i 3D spae (pramid) [sr: Wikipedia]
13 L-sstem (Lidemaer sstem) L-sstems are based o reursive grammar with defied variables, ostats, rules, aiom ad geeratig parameters; we a assig some operatios to eah smbol eg.: variables : X F ostats : + [ ] aiom: X rules : (X F-[[X]+X]+F[+FX]-X), (F FF) parameter - agle: 5 Assiged meaig of smbols for above L-sstem: ( F ) draw forward ( - ) tur left 5 ( + ) tur right 5 ( X ) does othig, just otrols evolutio of the urve ( [ ) saves oordiates ad agle o stak (push) ( ] ) reovers oordiates ad agle from stak (pop) Eemplar geerator: 3
14 Quik itrodutio to omple umbers There is o a real umber suh, that = Ok, so let s reate a umber whih is two-dimesioal, ad put suh a umber o imagiar ais, let s all it i Comple plae i Imagiar umbers +i - Real umbers -i Let s preserve additio ad multipliatio like for real umbers keepig i mid, that i = : a + bi + + di = a + + b + d i a + bi + di = a + ad + b i + bdi = a bd + ad + b i 4
15 Quik itrodutio to omple umbers But there is aother represetatio! Comple plae Imagiar umbers i r + i = r os + isi = os45 + isi45 = + i - Real umbers -i Now applig the rules for trgoometri futios we see that multipliatio is atuall related to rotatio o a plae! Comple plae is a field. a + bi + di = r os + isi r os + isi = r r os( + ) + isi( + ) 5
16 Riema sphere Let s map whole omple plae oto a spehere, where ifilit orrespods to a oth pole 6
17 Madelbrot sets. Madelbrot sets are defied for ratioal futios over losed set of omple umbers (* orrespods to ifiit). Ratioal futio is a divisio of two polomials: 3. Let W deote a ratioal futio depedet o parameter 4. Let 5. Madelbrot set M(W ) of a ratioal futio W is a set of suh poits that is ot overget to *: 7 ) ( ) ( ) ( b b b b a a a a l w W m m m m k k k k ) ( ) ( W W W () W *} () lim : { W C M W i where bi a C {*} C C C C C W : C
18 Madelbrot sets { C : lim W () *} M W This ma be satisfied i two was: Reursio is overget to some poit lim W () where C Reursio fiall falls ito a le (umber of stable les is related to degree of W) Orbit of poit Orbit of poit 8
19 Orbit of poit Madelbrot set - eample The first ad best kow Madelbrot set was defied for polomial futio ( ) W Thus we eed to hek for eah poit i C if sequee, +, ( +) + +, goes to ifiit or ot Workshop: Chek, if poit =+i belogs to Madelbrot set for this futio W () i i W () i i i 3 W () i i i i i 4 W () i i i i... 9
20 Madelbrot set joure
21 Does Madelbrot set eist? Take a look visual ompleit ver low Kolmogorov ompleit of its image for (it = ; < HEIGHT; ++) { for (it = ; < WIDTH; ++) { double = = ; double X = ( - WIDTH/) / ZOOM; double Y = ( - HEIGHT/) / ZOOM; for (it it = MAX_ITER; * + * < 4 && it > ; it--) { tmp = * - * + X; =. * * + Y; = tmp; } image[][] = olor(it); } }
22 Newto method for fidig futio root f ( f '( ) )
23 Julia ad Fatou sets Are based o the same ratioal futios as Madelbrot sets ad are stritl related to them (Julia set is oeted for parameters belogig to Madelbrot set). Fatou sets are areas i C whih are attrated b some poits (here olors red, blue ad gree) for ratioal futio W() W( ) 3 Julia set is a,,border betwee Fatou set areas whih is attrated b ifiit poit (*). f ( ), 3 f ( ) W ( ) f '( ) 3 i 3, i k,, 3 e 3 k i, Here is Julia/Fatou set for futio W() obtaied from Newto s method for futio f() = 3 -. Thus attratig poits for W () orrespod to roots of f(). Gree olor is attratig basi of, red of, ad blue of. 3
24 Madelbulbs Madelbrot sets i 3D Defied b Daiel White ad Paul Nlader usig double rotatio trasformatio for spherial oordiates, sie there is o 3D equivalee to D omple umbers havig all properties of field 4
25 Thak ou for attetio Referees: T. Mart. Fraktale i obiektowe algortm ih wiualiaji. Nakom, Poań, 996. J. Kudrewi. Fraktale i haos. WNT, Warsawa, 7. P. Prusikiewi ad A. Lidemaer. The Algorithmi Beaut of Plats. The Virtual Laborator Series, Spriger 996. B. Madelbrot. The fratal geometr of ature. W.H. Freeme ad Co. New York, delbulb.html Bakgroud soure: 5
Fractal geometry extends classical geometry and is more fun too!
Fractal Geometr Reasos for studig this geometr : Fractal geometr eteds classical geometr ad is more fu too! (i) (ii) etesio of classical geometr such as Euclidea geometr, projective geometr I classical
More informationImage Spaces. What might an image space be
Image Spaces What might a image space be Map each image to a poit i a space Defie a distace betwee two poits i that space Mabe also a shortest path (morph) We have alread see a simple versio of this, i
More informationSx [ ] = x must yield a
Math -b Leture #5 Notes This wee we start with a remider about oordiates of a vetor relative to a basis for a subspae ad the importat speial ase where the subspae is all of R. This freedom to desribe vetors
More informationANOTHER PROOF FOR FERMAT S LAST THEOREM 1. INTRODUCTION
ANOTHER PROOF FOR FERMAT S LAST THEOREM Mugur B. RĂUŢ Correspodig author: Mugur B. RĂUŢ, E-mail: m_b_raut@yahoo.om Abstrat I this paper we propose aother proof for Fermat s Last Theorem (FLT). We foud
More informationAn application of a subset S of C onto another S' defines a function [f(z)] of the complex variable z.
Diola Bagaoko (1 ELEMENTARY FNCTIONS OFA COMPLEX VARIABLES I Basic Defiitio of a Fuctio of a Comple Variable A applicatio of a subset S of C oto aother S' defies a fuctio [f(] of the comple variable z
More informationPolynomial and Rational Functions. Polynomial functions and Their Graphs. Polynomial functions and Their Graphs. Examples
Polomial ad Ratioal Fuctios Polomial fuctios ad Their Graphs Math 44 Precalculus Polomial ad Ratioal Fuctios Polomial Fuctios ad Their Graphs Polomial fuctios ad Their Graphs A Polomial of degree is a
More informationAfter the completion of this section the student. V.4.2. Power Series Solution. V.4.3. The Method of Frobenius. V.4.4. Taylor Series Solution
Chapter V ODE V.4 Power Series Solutio Otober, 8 385 V.4 Power Series Solutio Objetives: After the ompletio of this setio the studet - should reall the power series solutio of a liear ODE with variable
More informationPrincipal Component Analysis. Nuno Vasconcelos ECE Department, UCSD
Priipal Compoet Aalysis Nuo Vasoelos ECE Departmet, UCSD Curse of dimesioality typial observatio i Bayes deisio theory: error ireases whe umber of features is large problem: eve for simple models (e.g.
More informationApply change-of-basis formula to rewrite x as a linear combination of eigenvectors v j.
Eigevalue-Eigevector Istructor: Nam Su Wag eigemcd Ay vector i real Euclidea space of dimesio ca be uiquely epressed as a liear combiatio of liearly idepedet vectors (ie, basis) g j, j,,, α g α g α g α
More informationChapter 1. Complex Numbers. Dr. Pulak Sahoo
Chapter 1 Complex Numbers BY Dr. Pulak Sahoo Assistat Professor Departmet of Mathematics Uiversity Of Kalyai West Begal, Idia E-mail : sahoopulak1@gmail.com 1 Module-2: Stereographic Projectio 1 Euler
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 Slope-Itercept Equatio: y m b slope= m y-itercept=
More informationAPPENDIX F Complex Numbers
APPENDIX F Complex Numbers Operatios with Complex Numbers Complex Solutios of Quadratic Equatios Polar Form of a Complex Number Powers ad Roots of Complex Numbers Operatios with Complex Numbers Some equatios
More informationMEI Conference 2009 Stretching students: A2 Core
MEI Coferece 009 Stretchig studets: A Core Preseter: Berard Murph berard.murph@mei.org.uk Workshop G How ca ou prove that these si right-agled triagles fit together eactl to make a 3-4-5 triagle? What
More informationFluids Lecture 2 Notes
Fluids Leture Notes. Airfoil orte Sheet Models. Thi-Airfoil Aalysis Problem Readig: Aderso.,.7 Airfoil orte Sheet Models Surfae orte Sheet Model A aurate meas of represetig the flow about a airfoil i a
More informationAlgorithms. Elementary Sorting. Dong Kyue Kim Hanyang University
Algorithms Elemetary Sortig Dog Kyue Kim Hayag Uiversity dqkim@hayag.a.kr Cotets Sortig problem Elemetary sortig algorithms Isertio sort Merge sort Seletio sort Bubble sort Sortig problem Iput A sequee
More informationAxioms of Measure Theory
MATH 532 Axioms of Measure Theory Dr. Neal, WKU I. The Space Throughout the course, we shall let X deote a geeric o-empty set. I geeral, we shall ot assume that ay algebraic structure exists o X so that
More informationMathematical Statistics - MS
Paper Specific Istructios. The examiatio is of hours duratio. There are a total of 60 questios carryig 00 marks. The etire paper is divided ito three sectios, A, B ad C. All sectios are compulsory. Questios
More informationSOME NOTES ON INEQUALITIES
SOME NOTES ON INEQUALITIES Rihard Hoshio Here are four theorems that might really be useful whe you re workig o a Olympiad problem that ivolves iequalities There are a buh of obsure oes Chebyheff, Holder,
More informationSummation Method for Some Special Series Exactly
The Iteratioal Joural of Mathematis, Siee, Tehology ad Maagemet (ISSN : 39-85) Vol. Issue Summatio Method for Some Speial Series Eatly D.A.Gismalla Deptt. Of Mathematis & omputer Studies Faulty of Siee
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 informationCalculus 2 TAYLOR SERIES CONVERGENCE AND TAYLOR REMAINDER
Calulus TAYLO SEIES CONVEGENCE AND TAYLO EMAINDE Let the differee betwee f () ad its Taylor polyomial approimatio of order be (). f ( ) P ( ) + ( ) Cosider to be the remaider with the eat value ad the
More informationa is some real number (called the coefficient) other
Precalculus Notes for Sectio.1 Liear/Quadratic Fuctios ad Modelig http://www.schooltube.com/video/77e0a939a3344194bb4f Defiitios: A moomial is a term of the form tha zero ad is a oegative iteger. a where
More informationObserver Design with Reduced Measurement Information
Observer Desig with Redued Measuremet Iformatio I pratie all the states aot be measured so that SVF aot be used Istead oly a redued set of measuremets give by y = x + Du p is available where y( R We assume
More informationAppendix F: Complex Numbers
Appedix F Complex Numbers F1 Appedix F: Complex Numbers Use the imagiary uit i to write complex umbers, ad to add, subtract, ad multiply complex umbers. Fid complex solutios of quadratic equatios. Write
More informationε > 0 N N n N a n < ε. Now notice that a n = a n.
4 Sequees.5. Null sequees..5.. Defiitio. A ull sequee is a sequee (a ) N that overges to 0. Hee, by defiitio of (a ) N overges to 0, a sequee (a ) N is a ull sequee if ad oly if ( ) ε > 0 N N N a < ε..5..
More informationQuiz. Use either the RATIO or ROOT TEST to determine whether the series is convergent or not.
Quiz. Use either the RATIO or ROOT TEST to determie whether the series is coverget or ot. e .6 POWER SERIES Defiitio. A power series i about is a series of the form c 0 c a c a... c a... a 0 c a where
More informationA) is empty. B) is a finite set. C) can be a countably infinite set. D) can be an uncountable set.
M.A./M.Sc. (Mathematics) Etrace Examiatio 016-17 Max Time: hours Max Marks: 150 Istructios: There are 50 questios. Every questio has four choices of which exactly oe is correct. For correct aswer, 3 marks
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 informationNon Linear Dynamics of Ishikawa Iteration
Iteratioal Joural of Computer Appliatios (975 8887) Volume 7 No. Otober No Liear Dyamis of Ishiawa Iteratio Rajeshri Raa Asst. Professor Applied Siee ad Humaities Departmet G. B. Pat Egg. College Pauri
More informationTHE SOLUTION OF NONLINEAR EQUATIONS f( x ) = 0.
THE SOLUTION OF NONLINEAR EQUATIONS f( ) = 0. Noliear Equatio Solvers Bracketig. Graphical. Aalytical Ope Methods Bisectio False Positio (Regula-Falsi) Fied poit iteratio Newto Raphso Secat The root of
More informationChapter 5: Take Home Test
Chapter : Take Home Test AB Calulus - Hardtke Name Date: Tuesday, / MAY USE YOUR CALCULATOR FOR THIS PAGE. Roud aswers to three plaes. Sore: / Show diagrams ad work to justify eah aswer.. Approimate the
More informationExplicit and closed formed solution of a differential equation. Closed form: since finite algebraic combination of. converges for x x0
Chapter 4 Series Solutios Epliit ad losed formed solutio of a differetial equatio y' y ; y() 3 ( ) ( 5 e ) y Closed form: sie fiite algebrai ombiatio of elemetary futios Series solutio: givig y ( ) as
More informationCOMP26120: Introducing Complexity Analysis (2018/19) Lucas Cordeiro
COMP60: Itroduig Complexity Aalysis (08/9) Luas Cordeiro luas.ordeiro@mahester.a.uk Itroduig Complexity Aalysis Textbook: Algorithm Desig ad Appliatios, Goodrih, Mihael T. ad Roberto Tamassia (hapter )
More informationSYNTHESIS OF SIGNAL USING THE EXPONENTIAL FOURIER SERIES
SYNTHESIS OF SIGNAL USING THE EXPONENTIAL FOURIER SERIES Sadro Adriao Fasolo ad Luiao Leoel Medes Abstrat I 748, i Itrodutio i Aalysi Ifiitorum, Leohard Euler (707-783) stated the formula exp( jω = os(
More informationLecture 7: Polar representation of complex numbers
Lecture 7: Polar represetatio of comple umbers See FLAP Module M3.1 Sectio.7 ad M3. Sectios 1 ad. 7.1 The Argad diagram I two dimesioal Cartesia coordiates (,), we are used to plottig the fuctio ( ) with
More informationEE / EEE SAMPLE STUDY MATERIAL. GATE, IES & PSUs Signal System. Electrical Engineering. Postal Correspondence Course
Sigal-EE Postal Correspodece Course 1 SAMPLE STUDY MATERIAL Electrical Egieerig EE / EEE Postal Correspodece Course GATE, IES & PSUs Sigal System Sigal-EE Postal Correspodece Course CONTENTS 1. SIGNAL
More informationSEQUENCE AND SERIES NCERT
9. Overview By a sequece, we mea a arragemet of umbers i a defiite order accordig to some rule. We deote the terms of a sequece by a, a,..., etc., the subscript deotes the positio of the term. I view of
More informationDigital Signal Processing. Homework 2 Solution. Due Monday 4 October Following the method on page 38, the difference equation
Digital Sigal Proessig Homework Solutio Due Moda 4 Otober 00. Problem.4 Followig the method o page, the differee equatio [] (/4[-] + (/[-] x[-] has oeffiiets a0, a -/4, a /, ad b. For these oeffiiets A(z
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 informationBernoulli Numbers. n(n+1) = n(n+1)(2n+1) = n(n 1) 2
Beroulli Numbers Beroulli umbers are amed after the great Swiss mathematiia Jaob Beroulli5-705 who used these umbers i the power-sum problem. The power-sum problem is to fid a formula for the sum of the
More informationPhysics 3 (PHYF144) Chap 8: The Nature of Light and the Laws of Geometric Optics - 1
Physis 3 (PHYF44) Chap 8: The Nature of Light ad the Laws of Geometri Optis - 8. The ature of light Before 0 th etury, there were two theories light was osidered to be a stream of partiles emitted by a
More informationCalculus. Ramanasri. Previous year Questions from 2016 to
++++++++++ Calculus Previous ear Questios from 6 to 99 Ramaasri 7 S H O P NO- 4, S T F L O O R, N E A R R A P I D F L O U R M I L L S, O L D R A J E N D E R N A G A R, N E W D E L H I. W E B S I T E :
More informationU8L1: Sec Equations of Lines in R 2
MCVU U8L: Sec. 8.9. Equatios of Lies i R Review of Equatios of a Straight Lie (-D) Cosider the lie passig through A (-,) with slope, as show i the diagram below. I poit slope form, the equatio of the lie
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 informationBasic Probability/Statistical Theory I
Basi Probability/Statistial Theory I Epetatio The epetatio or epeted values of a disrete radom variable X is the arithmeti mea of the radom variable s distributio. E[ X ] p( X ) all Epetatio by oditioig
More information+ {JEE Advace 03} Sept 0 Name: Batch (Day) Phoe No. IT IS NOT ENOUGH TO HAVE A GOOD MIND, THE MAIN THING IS TO USE IT WELL Marks: 00. If A (α, β) = (a) A( α, β) = A( α, β) (c) Adj (A ( α, β)) = Sol : We
More informationComplex Numbers Solutions
Complex Numbers Solutios Joseph Zoller February 7, 06 Solutios. (009 AIME I Problem ) There is a complex umber with imagiary part 64 ad a positive iteger such that Fid. [Solutio: 697] 4i + + 4i. 4i 4i
More informationTHE MEASUREMENT OF THE SPEED OF THE LIGHT
THE MEASUREMENT OF THE SPEED OF THE LIGHT Nyamjav, Dorjderem Abstrat The oe of the physis fudametal issues is a ature of the light. I this experimet we measured the speed of the light usig MihelsoÕs lassial
More informationMATH 10550, EXAM 3 SOLUTIONS
MATH 155, EXAM 3 SOLUTIONS 1. I fidig a approximate solutio to the equatio x 3 +x 4 = usig Newto s method with iitial approximatio x 1 = 1, what is x? Solutio. Recall that x +1 = x f(x ) f (x ). Hece,
More informationU8L1: Sec Equations of Lines in R 2
MCVU Thursda Ma, Review of Equatios of a Straight Lie (-D) U8L Sec. 8.9. Equatios of Lies i R Cosider the lie passig through A (-,) with slope, as show i the diagram below. I poit slope form, the equatio
More informationFundamental Concepts: Surfaces and Curves
UNDAMENTAL CONCEPTS: SURACES AND CURVES CHAPTER udametal Cocepts: Surfaces ad Curves. INTRODUCTION This chapter describes two geometrical objects, vi., surfaces ad curves because the pla a ver importat
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 informationPrincipal Component Analysis
Priipal Compoet Aalysis Nuo Vasoelos (Ke Kreutz-Delgado) UCSD Curse of dimesioality Typial observatio i Bayes deisio theory: Error ireases whe umber of features is large Eve for simple models (e.g. Gaussia)
More information(8) 1f = f. can be viewed as a real vector space where addition is defined by ( a1+ bi
Geeral Liear Spaes (Vetor Spaes) ad Solutios o ODEs Deiitio: A vetor spae V is a set, with additio ad salig o elemet deied or all elemets o the set, that is losed uder additio ad salig, otais a zero elemet
More informationThe Relationship of the Cotangent Function to Special Relativity Theory, Silver Means, p-cycles, and Chaos Theory
Origial Paper Forma, 8, 49 6, 003 The Relatioship of the Cotaget Futio to Speial Relativity Theory, Silver Meas, p-yles, ad Chaos Theory Jay KAPPRAFF * ad Gary W ADAMSON New Jersey Istitute of Tehology,
More information1. Complex numbers. Chapter 13: Complex Numbers. Modulus of a complex number. Complex conjugate. Complex numbers are of the form
Comple umbers ad comple plae Comple cojugate Modulus of a comple umber Comple umbers Comple umbers are of the form Sectios 3 & 32 z = + i,, R, i 2 = I the above defiitio, is the real part of z ad is the
More informationChapter 8 Hypothesis Testing
Chapter 8 for BST 695: Speial Topis i Statistial Theory Kui Zhag, Chapter 8 Hypothesis Testig Setio 8 Itrodutio Defiitio 8 A hypothesis is a statemet about a populatio parameter Defiitio 8 The two omplemetary
More informationBasic Waves and Optics
Lasers ad appliatios APPENDIX Basi Waves ad Optis. Eletromageti Waves The eletromageti wave osists of osillatig eletri ( E ) ad mageti ( B ) fields. The eletromageti spetrum is formed by the various possible
More informationAP Calculus BC Review Applications of Derivatives (Chapter 4) and f,
AP alculus B Review Applicatios of Derivatives (hapter ) Thigs to Kow ad Be Able to Do Defiitios of the followig i terms of derivatives, ad how to fid them: critical poit, global miima/maima, local (relative)
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 two-dimesioal torus are rich mathematical objects possessig iterestig geometric, algebraic, topological ad
More informationThe Phi Power Series
The Phi Power Series I did this work i about 0 years while poderig the relatioship betwee the golde mea ad the Madelbrot set. I have fially decided to make it available from my blog at http://semresearch.wordpress.com/.
More informationLecture Notes for Analysis Class
Lecture Notes for Aalysis Class Topological Spaces A topology for a set X is a collectio T of subsets of X such that: (a) X ad the empty set are i T (b) Uios of elemets of T are i T (c) Fiite itersectios
More informationInformal Notes: Zeno Contours, Parametric Forms, & Integrals. John Gill March August S for a convex set S in the complex plane.
Iformal Notes: Zeo Cotours Parametric Forms & Itegrals Joh Gill March August 3 Abstract: Elemetary classroom otes o Zeo cotours streamlies pathlies ad itegrals Defiitio: Zeo cotour[] Let gk ( z = z + ηk
More informationM A T H F A L L CORRECTION. Algebra I 1 4 / 1 0 / U N I V E R S I T Y O F T O R O N T O
M A T H 2 4 0 F A L L 2 0 1 4 HOMEWORK ASSIGNMENT #4 CORRECTION Algebra I 1 4 / 1 0 / 2 0 1 4 U N I V E R S I T Y O F T O R O N T O P r o f e s s o r : D r o r B a r - N a t a Correctio Homework Assigmet
More informationFINDING ZEROS OF COMPLEX FUNCTIONS
FINDING ZEROS OF COMPLEX FUNCTIONS It is well kow sice the time of Newto that the zeros of a real fuctio f(x) ca be foud b carrig out the iterative procedure- f ( x[ x[ 1] x[ ] subject to x[0] x f '( x[
More informationThe beta density, Bayes, Laplace, and Pólya
The beta desity, Bayes, Laplae, ad Pólya Saad Meimeh The beta desity as a ojugate form Suppose that is a biomial radom variable with idex ad parameter p, i.e. ( ) P ( p) p ( p) Applyig Bayes s rule, we
More informationFixed Point Approximation of Weakly Commuting Mappings in Banach Space
BULLETIN of the Bull. Malaysia Math. S. So. (Seod Series) 3 (000) 8-85 MALAYSIAN MATHEMATICAL SCIENCES SOCIETY Fied Poit Approimatio of Weakly Commutig Mappigs i Baah Spae ZAHEER AHMAD AND ABDALLA J. ASAD
More informationMATH CALCULUS II Objectives and Notes for Test 4
MATH 44 - CALCULUS II Objectives ad Notes for Test 4 To do well o this test, ou should be able to work the followig tpes of problems. Fid a power series represetatio for a fuctio ad determie the radius
More informationMa 530 Infinite Series I
Ma 50 Ifiite Series I Please ote that i additio to the material below this lecture icorporated material from the Visual Calculus web site. The material o sequeces is at Visual Sequeces. (To use this li
More informationInformation Theory and Statistics Lecture 4: Lempel-Ziv code
Iformatio Theory ad Statistics Lecture 4: Lempel-Ziv code Łukasz Dębowski ldebowsk@ipipa.waw.pl Ph. D. Programme 203/204 Etropy rate is the limitig compressio rate Theorem For a statioary process (X i)
More informationProduction Test of Rotary Compressors Using Wavelet Analysis
Purdue Uiversity Purdue e-pubs Iteratioal Compressor Egieerig Coferee Shool of Mehaial Egieerig 2006 Produtio Test of Rotary Compressors Usig Wavelet Aalysis Haishui Ji Shaghai Hitahi Eletrial Appliatio
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 informationNonstandard Lorentz-Einstein transformations
Nostadard Loretz-istei trasformatios Berhard Rothestei 1 ad Stefa Popesu 1) Politehia Uiversity of Timisoara, Physis Departmet, Timisoara, Romaia brothestei@gmail.om ) Siemes AG, rlage, Germay stefa.popesu@siemes.om
More informationAlgebra II Notes Unit Seven: Powers, Roots, and Radicals
Syllabus Objectives: 7. The studets will use properties of ratioal epoets to simplify ad evaluate epressios. 7.8 The studet will solve equatios cotaiig radicals or ratioal epoets. b a, the b is the radical.
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 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 informationJEE(Advanced) 2018 TEST PAPER WITH SOLUTION (HELD ON SUNDAY 20 th MAY, 2018)
JEE(Advaced) 08 TEST PAPER WITH SOLUTION (HELD ON SUNDAY 0 th MAY, 08) PART- : JEE(Advaced) 08/Paper- SECTION. For ay positive iteger, defie ƒ : (0, ) as ƒ () j ta j j for all (0, ). (Here, the iverse
More informationMathematics Extension 2
009 HIGHER SCHOOL CERTIFICATE EXAMINATION Mathematics Etesio Geeral Istructios Readig time 5 miutes Workig time hours Write usig black or blue pe Board-approved calculators may be used A table of stadard
More informationDefinition 4.2. (a) A sequence {x n } in a Banach space X is a basis for X if. unique scalars a n (x) such that x = n. a n (x) x n. (4.
4. BASES I BAACH SPACES 39 4. BASES I BAACH SPACES Sice a Baach space X is a vector space, it must possess a Hamel, or vector space, basis, i.e., a subset {x γ } γ Γ whose fiite liear spa is all of X ad
More information2 Geometric interpretation of complex numbers
2 Geometric iterpretatio of complex umbers 2.1 Defiitio I will start fially with a precise defiitio, assumig that such mathematical object as vector space R 2 is well familiar to the studets. Recall that
More informationTHE ZETA FUNCTION AND THE RIEMANN HYPOTHESIS. Contents 1. History 1
THE ZETA FUNCTION AND THE RIEMANN HYPOTHESIS VIKTOR MOROS Abstract. The zeta fuctio has bee studied for ceturies but mathematicias are still learig about it. I this paper, I will discuss some of the zeta
More informationx c the remainder is Pc ().
Algebra, Polyomial ad Ratioal Fuctios Page 1 K.Paulk Notes Chapter 3, Sectio 3.1 to 3.4 Summary Sectio Theorem Notes 3.1 Zeros of a Fuctio Set the fuctio to zero ad solve for x. The fuctio is zero at these
More informationMath 299 Supplement: Real Analysis Nov 2013
Math 299 Supplemet: Real Aalysis Nov 203 Algebra Axioms. I Real Aalysis, we work withi the axiomatic system of real umbers: the set R alog with the additio ad multiplicatio operatios +,, ad the iequality
More informationVector Spaces and Vector Subspaces. Remarks. Euclidean Space
Vector Spaces ad Vector Subspaces Remarks Let be a iteger. A -dimesioal vector is a colum of umbers eclosed i brackets. The umbers are called the compoets of the vector. u u u u Euclidea Space I Euclidea
More informationtoo many conditions to check!!
Vector Spaces Aioms of a Vector Space closre Defiitio : Let V be a o empty set of vectors with operatios : i. Vector additio :, v є V + v є V ii. Scalar mltiplicatio: li є V k є V where k is scalar. The,
More informationThe Stokes Theorem. (Sect. 16.7) The curl of a vector field in space
The tokes Theorem. (ect. 6.7) The curl of a vector field i space. The curl of coservative fields. tokes Theorem i space. Idea of the proof of tokes Theorem. The curl of a vector field i space Defiitio
More informationSequences I. Chapter Introduction
Chapter 2 Sequeces I 2. Itroductio A sequece is a list of umbers i a defiite order so that we kow which umber is i the first place, which umber is i the secod place ad, for ay atural umber, we kow which
More informationReview of Discrete-time Signals. ELEC 635 Prof. Siripong Potisuk
Review of Discrete-time Sigals ELEC 635 Prof. Siripog Potisuk 1 Discrete-time Sigals Discrete-time, cotiuous-valued amplitude (sampled-data sigal) Discrete-time, discrete-valued amplitude (digital sigal)
More informationContext-free grammars and. Basics of string generation methods
Cotext-free grammars ad laguages Basics of strig geeratio methods What s so great about regular expressios? A regular expressio is a strig represetatio of a regular laguage This allows the storig a whole
More information(A sequence also can be thought of as the list of function values attained for a function f :ℵ X, where f (n) = x n for n 1.) x 1 x N +k x N +4 x 3
MATH 337 Sequeces Dr. Neal, WKU Let X be a metric space with distace fuctio d. We shall defie the geeral cocept of sequece ad limit i a metric space, the apply the results i particular to some special
More informationOutline. CS38 Introduction to Algorithms 5/23/2014. Linear programming. Lecture 16 May 22, coping with intractibility
Outlie CS38 Itroductio to Algorithms Lecture 6 Ma, 04 Liear programmig L dualit ellipsoid algorithm * slides from Kevi Wae copig with itractibilit N-completeess Ma, 04 CS38 Lecture 6 Ma, 04 CS38 Lecture
More information4.1 Sigma Notation and Riemann Sums
0 the itegral. Sigma Notatio ad Riema Sums Oe strategy for calculatig the area of a regio is to cut the regio ito simple shapes, calculate the area of each simple shape, ad the add these smaller areas
More informationChapter 5.4 Practice Problems
EXPECTED SKILLS: Chapter 5.4 Practice Problems Uderstad ad kow how to evaluate the summatio (sigma) otatio. Be able to use the summatio operatio s basic properties ad formulas. (You do ot eed to memorize
More informationCertain inclusion properties of subclass of starlike and convex functions of positive order involving Hohlov operator
Iteratioal Joural of Pure ad Applied Mathematial Siees. ISSN 0972-9828 Volume 0, Number (207), pp. 85-97 Researh Idia Publiatios http://www.ripubliatio.om Certai ilusio properties of sublass of starlike
More informationPRELIM PROBLEM SOLUTIONS
PRELIM PROBLEM SOLUTIONS THE GRAD STUDENTS + KEN Cotets. Complex Aalysis Practice Problems 2. 2. Real Aalysis Practice Problems 2. 4 3. Algebra Practice Problems 2. 8. Complex Aalysis Practice Problems
More informationPartial Differential Equations
EE 84 Matematical Metods i Egieerig Partial Differetial Eqatios Followig are some classical partial differetial eqatios were is assmed to be a fctio of two or more variables t (time) ad y (spatial coordiates).
More informationClass #25 Wednesday, April 19, 2018
Cla # Wedesday, April 9, 8 PDE: More Heat Equatio with Derivative Boudary Coditios Let s do aother heat equatio problem similar to the previous oe. For this oe, I ll use a square plate (N = ), but I m
More informationThe Use of Filters in Topology
The Use of Filters i Topology By ABDELLATF DASSER B.S. Uiversity of Cetral Florida, 2002 A thesis submitted i partial fulfillmet of the requiremets for the degree of Master of Siee i the Departmet of Mathematis
More informationAlternating Series. 1 n 0 2 n n THEOREM 9.14 Alternating Series Test Let a n > 0. The alternating series. 1 n a n.
0_0905.qxd //0 :7 PM Page SECTION 9.5 Alteratig Series Sectio 9.5 Alteratig Series Use the Alteratig Series Test to determie whether a ifiite series coverges. Use the Alteratig Series Remaider to approximate
More informationReal Numbers R ) - LUB(B) may or may not belong to B. (Ex; B= { y: y = 1 x, - Note that A B LUB( A) LUB( B)
Real Numbers The least upper boud - Let B be ay subset of R B is bouded above if there is a k R such that x k for all x B - A real umber, k R is a uique least upper boud of B, ie k = LUB(B), if () k is
More information