UniversitaireWiskundeCompetitie. Problem 2005/4A We have k=1. Show that for every q Q satisfying 0 < q < 1, there exists a finite subset K N so that


 Gerard Burns
 1 years ago
 Views:
Transcription
1 Problemen/UWC NAW 5/7 nr juni Problemen/UWC UniversitireWiskundeCompetitie Edition 005/4 For Session 005/4 we received submissions from Peter Vndendriessche, Vldislv Frnk, Arne Smeets, Jn vn de Lune, en PG Kluit Problem 005/4A We hve k= = Consequently prtil sums must stisfy k(k+) k(k + ) < Show tht for every q Q stisfying 0 < q <, there eists finite subset K N so tht k(k + ) = q Solution This problem ws solved by Peter Vndendriessche, Vldislv Frnk nd Arne Smeets The solution below is bsed on tht of Vldislv Frnk First note tht k(k+) = k k+ Hence k(k+) + (k+)(k+) + + (k+n )(k+n) = k k+ + k+ + k+n k+n = k k+n Consequently it suffices to represent every rtionl number between 0 nd s + k, where 3 k If two consecutive numbers re equl, they simply cncel out, so we llow equl numbers This will be useful in finl step of proof Let b be our rtionl number There is nturl number n such tht n+ < b n Consider = n b = b n bn The numertor of this frction is nonnegtive becuse b n, but less thn, the numertor of b, becuse (n + ) < 0 We hve b = n We now pply the sme lgorithm to Let m be nturl number such tht m+ < m b n The clim is tht m n Nmely, = bn < bn, hence n m n + > n If we continue this lgorithm, we obtin b = ( ( 3 ( ( ) ))) Notice tht the lgorithm cn only be repeted finitely mny times, s the numertor decreses t ech step We now hve b = + 3 ± If is even we re done In other cse we my ssume tht > nd chnge into ( ) Here ( ) nd we re done Of course =, s otherwise b = = which is impossible As generliztion, V Frnk shows tht for ny irrtionl number in the intervl [0,] there eists n infinite sum Problem 005/4B We consider the progressive rithmetic nd geometric mens of the function sequence f n () = n, n N, > 0, = These re nd A n = A n () = n ( n ) = n n( ) G n = G n () = ( ++ +(n ) ) n = n The Mrtinsproperty reds A n+ /A n G n+ /G n In our cse this gives n n+ n + n Eindredctie: Mtthijs Coster Redctiedres: UWC/NAW Mthemtisch Instituut Postbus RA Leiden Prove, more generlly, tht + + for >, > 0, = Solution This problem ws solved by Jn vn de Lune, Peter Vndendriessche, Vldislv Frnk nd Arne Smeets The solution below is bsed on tht of Peter Vndendriessche
2 48 NAW 5/7 nr juni 006 Problemen/UWC Let f (t) be (smooth) nonnegtive function tht is conve on [, b] nd let [, y] [, b] such tht + y = + b We then hve b f (t)dt f (t)dt y To prove this, consider, for given f (t),, nd y, the function g(t) defined by g(t) = f () + ( f (y) f ())(t ) y g(t) is the line through the points (, f ()) nd (y, f (y)) Notice tht the conveity of f gives g(t)dt f (t)dt Let h(t) = g(t) g(t)dt + f (t)dt, then y h(t)dt = f (t)dt By conveity we hve f (t) g(t) h(t) for t [, ] [y, b] Since h(t) is the eqution of line nd + y = + b, we hve Combining these results we find: b h(t)dt = h(t)dt y b f (t)dt f (t)dt + b y f (t)dt = + h(t)dt + b y h(t)dt + b h(t)dt b f (t)dt = = y y f (t)dt y h(t)dt y Problem B is specil cse of this result For R + 0, =, let f (t) = t Then f (t) = t log () 0 Therefore f (t) is conve We hve to distinguish two cses: (, 0) Then 0 < + < + Apply the lemm to the intervl [ +, ] [0, + ] (0, ) Then 0 < < + < + Apply the lemm to the intervl [, + ] [0, + ] Notice tht in the first cse the sign in both numertor nd denomintor chnges on the right side of the eqution: from which we cn deduce + 0 t dt + + t dt, + + ( ) It is esy to prove the generliztion z y b b z y y, where y + z = + b nd < < y < z < b Problem 005/4C A finite geometry is geometric system tht hs only finite number of points For n ffine plne geometry, the ioms re s follows: Given ny two distinct points, there is ectly one line tht includes both points The prllel postulte: Given line L nd point P not on L, there eists ectly one line through P tht is prllel to L 3 There eists set of four points, no three colliner
3 Problemen/UWC NAW 5/7 nr juni We denote the set of points by P, nd the set of lines by L Let σ be n utomorphism of (P, L ) (mening tht three colliner points of P re mpped onto three colliner points of P nd three noncolliner points of P re mpped onto three noncolliner points of P ) Prove tht there eists point P P with σ(p) = P or line L L with σ(l) = L or σ(l) L = Solution This problem hs been solved by Leendert Bleijeng nd Peter Vndendriessche The solution below is bsed on their solutions First we will prove the following lemm: Lemm Let M, L L, then M = L Proof Suppose tht M L > then M = L Therefore we my ssume tht M L = Let M = m nd L = l By Aiom 3 we know tht there eists P P such tht P L nd p M Through P we cn construct line prllel to L nd l lines tht intersect L in its l points In the sme wy we cn construct, through P, line prllel to M nd m lines tht intersect M in its m points Let us now determine the number of lines through P; this equls l + nd m + If M L = 0, pick points L nd b M Let N be the line through nd b Then by the previous rgument L = N nd M = N We conclude tht ll lines consist of n equl number of points, sy s Lemm P < L Proof Let P = p nd L = l Every two points define line, nd there re p(p ) pirs of points Ech line hs s points nd is counted s(s ) times Therefore l = p(p ) In order to show tht p < l we hve to prove tht s(s ) < p or p > s(s ) s s + The third iom tells us tht there eist three noncolliner points, b, c P Let L be the line through nd b, M the line through nd c By the prllel postulte, through every point on L there is ectly one line prllel to M Strting with s points on L, we find s lines, ll consisting of s points Therefore p s Suppose tht σ(p) = p, for ll p P, nd tht σ(l) = L nd σ(l) L = for ll L L Consider the function µ : L P given by µ(l) = σ(l) L µ is well defined since σ(l) L is lwys unique point Now suppose tht µ(l) = µ(m) or σ(l) L = σ(m) M = p, nd σ(q) = p Then q L nd q M We know tht q = p Therefore L = M nd µ is injective However, if µ is injective, then P L, which contrdicts the previous lemm Problem 005/4* stisfy We hve k= /k = (π /6) Consequently prtil sums must k < π 6 Given ny q Q stisfying 0 < q < (π /6), does there eist finite subset K N \{} so tht k = q? Solution This problem ws solved by PG Kluit The solution below is bsed on his solution Let q = ki, where k i re different integers Let m be the lest common multiple of ll k i in the sum For ech such k i number k i eists such tht k ik i = m Then q = m (k i ), tht is, q cn be written s frction with denomintor m nd the numertor sum of squres of different divisors of m This rises the question: given m, which numbers cn be written s sums of squres of different divisors of m? We will show tht for highly composite numbers m, more specificlly m = n!, the nswer will be tht sufficiently mny integers cn be written s sums of squres to prove the problem
4 50 NAW 5/7 nr juni 006 Problemen/UWC Lemm Let n 5 be n integer nd let 3 = d < d < d m = n!/3 be ll divisors of n! between 3 nd n!/3 Then d k > d k+ for k < m Proof Let us prove this by induction For n = 5 the divisors d,, d re 3, 4, 5, 6, 8, 0,, 5, 0, 4, 30, nd 40 It is esy to verify the lemm We ssume the lemm is true for n We hve to prove tht the Lemm holds for the divisors of (n + )! The divisors of (n + )! tht re less thn n!/3 clerly stisfy the lemm Even though there my be more divisors, this cnnot influence the inequlity Suppose tht d k nd d k+ re two successive divisors of (n + )!, with n!/3 d k < d k+ (n + )!/3 Let d k d k = d k+d k+ = (n + )! Then d k+ nd d k re two successive divisors of (n + )! with 3 d k+ < d k 3(n + ) As for n 5 we hve 3(n + ) < n!/3, this suffices to conclude the proof Lemm Let n [9, 56] be n integer Then n cn be represented s sum of different squres d + + d k, where d < < d k 0 Proof The proof cn be found by the enumertion of 8 representtions There is slightly shorter proof which will be left to the reder Lemm Let n N, n Then every integer [9,σ (n!) n! 9] cn be represented s = d k, where the d k re different divisors of n! Here σ m () = d d m Proof Let L kn = [9, t] be the longest intervl in [9, ) whose integers cn ll be represented s sum of different squres of some of the first k divisors of n! Let l kn = L kn, the length of the intervl In the proof the nottion will be bbrevite to l k = L k if it is cler which n is ment In the second lemm we sw tht l 0 = 8 Notice tht l = 49 (= 8 + ) Any 56 is represented by the divisors lesser thn or equl to 0, while the integers re represented using We will show in generl tht l k+ = l k + d k+ by induction, s long s d k+ < n!/ The proof will be given in two steps In the first step we prove tht d k+ < l k+ given d k < l k nd l k+ = l k + d k+ In the second step we will prove tht l k+ = l k + d k+ given d k < l k Using these two steps nd the bsic ssumption (k = ) we cn prove for rbitrry k tht l k+ = l k + d k+ First step Given d k < l k nd l k+ = l k + d k+ we find tht d k+ < l k+ Proof The first lemm tells us tht d k+ < d k Therefore we hve d k+ < d k+ + d k < d k+ + l k < l k+ Second step Given d k < l k we find tht l k+ = l k + d k+ Proof The proof is comprble to the proof bove For L k, it is cler tht L k+ s well, while for the numbers L k+ \L k notice tht l k + 8 < l k + d k+ + 8 If we use the number d k+ to represent the sum, we find for the rest y = d k+ tht l k d k+ + 8 < y l k + 8 Using Lemm gin we hve l k d k+ + 8 > l k d k + 8 > 8 Therefore y L k We cn rewrite the results l k+ = l k + d k+ s for k m, where d m = n!/3 l k = k di 8, i=
5 Problemen/UWC NAW 5/7 nr juni In order to complete the proof of Lemm 3 we need to prove tht l (m+)n = l mn + d n!/, where d m+ = n!/ Notice tht 4 < Therefore for rbitrry L (m+)n, we find either L mn or 4 n! L mn Now we find l k = k di 8, i= for k m +, where d m+ = n!/ This concludes the proof of this lemm Theorem For every q Q such tht 0 < q < π 6 k, finite subset K N eists, such tht Proof Let q Q with 0 < q < π 6 We cn find n n N fulfilling ech of the three following properties by choosing n sufficiently lrge Moreover ech of these properties is monotonic, mening tht if it is true for some n 0, it will be true for ll n > n 0 n, If q = /b, where nd b hve no common divisors, then b divides n!, If q = /(n!), then n is chosen such tht 8 < < σ (n!) (n!) 8 To prove the eistence of 3) notice tht = q σ lim (n!) (n!) 8 n n! = π 6 Now Lemm 3 my be pplied, showing tht cn be represented s sum of squres of different divisors of n! This gives us the sought for representtion of q Remrk A solution of the Str Problem turns out to hve been published in Ron Grhm s On Finite Sums of Unit Frctions, Proc London Mth Soc(4), 964, pp The bsic ides behind the two solutions re similr Grhm strts with multiplictive set S, which in the Str Problem is the set of squres Grhm then defines P(S), the set of sums of elements of S Using the nottion S for the set of inverses of the elements of S, Grhm shows tht if P(S) contins ll positive integers, up to finite number, S is finite, nd s n+ /s n is bounded, then p q P(S ) whenever q s for some s S Moreover, for every ɛ > 0 there is n s P(S ) such tht s p q < ɛ
Farey Fractions. Rickard Fernström. U.U.D.M. Project Report 2017:24. Department of Mathematics Uppsala University
U.U.D.M. Project Report 07:4 Frey Frctions Rickrd Fernström Exmensrete i mtemtik, 5 hp Hledre: Andres Strömergsson Exmintor: Jörgen Östensson Juni 07 Deprtment of Mthemtics Uppsl University Frey Frctions
More informationapproaches as n becomes larger and larger. Since e > 1, the graph of the natural exponential function is as below
. Eponentil nd rithmic functions.1 Eponentil Functions A function of the form f() =, > 0, 1 is clled n eponentil function. Its domin is the set of ll rel f ( 1) numbers. For n eponentil function f we hve.
More informationConvex Sets and Functions
B Convex Sets nd Functions Definition B1 Let L, +, ) be rel liner spce nd let C be subset of L The set C is convex if, for ll x,y C nd ll [, 1], we hve 1 )x+y C In other words, every point on the line
More informationMath 360: A primitive integral and elementary functions
Mth 360: A primitive integrl nd elementry functions D. DeTurck University of Pennsylvni October 16, 2017 D. DeTurck Mth 360 001 2017C: Integrl/functions 1 / 32 Setup for the integrl prtitions Definition:
More informationLecture 1. Functional series. Pointwise and uniform convergence.
1 Introduction. Lecture 1. Functionl series. Pointwise nd uniform convergence. In this course we study mongst other things Fourier series. The Fourier series for periodic function f(x) with period 2π is
More informationMath 113 Exam 2 Practice
Mth Em Prctice Februry, 8 Em will cover sections 6.5, 7.7.5 nd 7.8. This sheet hs three sections. The first section will remind you bout techniques nd formuls tht you should know. The second gives number
More informationPolynomial Approximations for the Natural Logarithm and Arctangent Functions. Math 230
Polynomil Approimtions for the Nturl Logrithm nd Arctngent Functions Mth 23 You recll from first semester clculus how one cn use the derivtive to find n eqution for the tngent line to function t given
More informationQUADRATIC EQUATIONS OBJECTIVE PROBLEMS
QUADRATIC EQUATIONS OBJECTIVE PROBLEMS +. The solution of the eqution will e (), () 0,, 5, 5. The roots of the given eqution ( p q) ( q r) ( r p) 0 + + re p q r p (), r p p q, q r p q (), (d), q r p q.
More informationMATH 101A: ALGEBRA I PART B: RINGS AND MODULES 35
MATH 101A: ALGEBRA I PART B: RINGS AND MODULES 35 9. Modules over PID This week we re proving the fundmentl theorem for finitely generted modules over PID, nmely tht they re ll direct sums of cyclic modules.
More informationHandout: Natural deduction for first order logic
MATH 457 Introduction to Mthemticl Logic Spring 2016 Dr Json Rute Hndout: Nturl deduction for first order logic We will extend our nturl deduction rules for sententil logic to first order logic These notes
More informationImproper Integrals. Introduction. Type 1: Improper Integrals on Infinite Intervals. When we defined the definite integral.
Improper Integrls Introduction When we defined the definite integrl f d we ssumed tht f ws continuous on [, ] where [, ] ws finite, closed intervl There re t lest two wys this definition cn fil to e stisfied:
More informationPARTIAL FRACTION DECOMPOSITION
PARTIAL FRACTION DECOMPOSITION LARRY SUSANKA 1. Fcts bout Polynomils nd Nottion We must ssemble some tools nd nottion to prove the existence of the stndrd prtil frction decomposition, used s n integrtion
More informationUSA Mathematical Talent Search Round 1 Solutions Year 21 Academic Year
1/1/21. Fill in the circles in the picture t right with the digits 18, one digit in ech circle with no digit repeted, so tht no two circles tht re connected by line segment contin consecutive digits.
More informationMTH 505: Number Theory Spring 2017
MTH 505: Numer Theory Spring 207 Homework 2 Drew Armstrong The Froenius Coin Prolem. Consider the eqution x ` y c where,, c, x, y re nturl numers. We cn think of $ nd $ s two denomintions of coins nd $c
More informationA basic logarithmic inequality, and the logarithmic mean
Notes on Number Theory nd Discrete Mthemtics ISSN 30 532 Vol. 2, 205, No., 3 35 A bsic logrithmic inequlity, nd the logrithmic men József Sándor Deprtment of Mthemtics, BbeşBolyi University Str. Koglnicenu
More informationAnonymous Math 361: Homework 5. x i = 1 (1 u i )
Anonymous Mth 36: Homewor 5 Rudin. Let I be the set of ll u (u,..., u ) R with u i for ll i; let Q be the set of ll x (x,..., x ) R with x i, x i. (I is the unit cube; Q is the stndrd simplex in R ). Define
More informationMORE FUNCTION GRAPHING; OPTIMIZATION. (Last edited October 28, 2013 at 11:09pm.)
MORE FUNCTION GRAPHING; OPTIMIZATION FRI, OCT 25, 203 (Lst edited October 28, 203 t :09pm.) Exercise. Let n be n rbitrry positive integer. Give n exmple of function with exctly n verticl symptotes. Give
More informationa n+2 a n+1 M n a 2 a 1. (2)
Rel Anlysis Fll 004 Tke Home Finl Key 1. Suppose tht f is uniformly continuous on set S R nd {x n } is Cuchy sequence in S. Prove tht {f(x n )} is Cuchy sequence. (f is not ssumed to be continuous outside
More informationUNIFORM CONVERGENCE MA 403: REAL ANALYSIS, INSTRUCTOR: B. V. LIMAYE
UNIFORM CONVERGENCE MA 403: REAL ANALYSIS, INSTRUCTOR: B. V. LIMAYE 1. Pointwise Convergence of Sequence Let E be set nd Y be metric spce. Consider functions f n : E Y for n = 1, 2,.... We sy tht the sequence
More information13.3 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS
33 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS As simple ppliction of the results we hve obtined on lgebric extensions, nd in prticulr on the multiplictivity of extension degrees, we cn nswer (in
More informationHomework Solution  Set 5 Due: Friday 10/03/08
CE 96 Introduction to the Theory of Computtion ll 2008 Homework olution  et 5 Due: ridy 10/0/08 1. Textook, Pge 86, Exercise 1.21. () 1 2 Add new strt stte nd finl stte. Mke originl finl stte nonfinl.
More informationThe First Fundamental Theorem of Calculus. If f(x) is continuous on [a, b] and F (x) is any antiderivative. f(x) dx = F (b) F (a).
The Fundmentl Theorems of Clculus Mth 4, Section 0, Spring 009 We now know enough bout definite integrls to give precise formultions of the Fundmentl Theorems of Clculus. We will lso look t some bsic emples
More informationThe final exam will take place on Friday May 11th from 8am 11am in Evans room 60.
Mth 104: finl informtion The finl exm will tke plce on Fridy My 11th from 8m 11m in Evns room 60. The exm will cover ll prts of the course with equl weighting. It will cover Chpters 1 5, 7 15, 17 21, 23
More informationMATH FIELD DAY Contestants Insructions Team Essay. 1. Your team has forty minutes to answer this set of questions.
MATH FIELD DAY 2012 Contestnts Insructions Tem Essy 1. Your tem hs forty minutes to nswer this set of questions. 2. All nswers must be justified with complete explntions. Your nswers should be cler, grmmticlly
More informationBest Approximation in the 2norm
Jim Lmbers MAT 77 Fll Semester 111 Lecture 1 Notes These notes correspond to Sections 9. nd 9.3 in the text. Best Approximtion in the norm Suppose tht we wish to obtin function f n (x) tht is liner combintion
More informationCSCI FOUNDATIONS OF COMPUTER SCIENCE
1 CSCI 2200 FOUNDATIONS OF COMPUTER SCIENCE Spring 2015 My 7, 2015 2 Announcements Homework 9 is due now. Some finl exm review problems will be posted on the web site tody. These re prcqce problems not
More information20 MATHEMATICS POLYNOMIALS
0 MATHEMATICS POLYNOMIALS.1 Introduction In Clss IX, you hve studied polynomils in one vrible nd their degrees. Recll tht if p(x) is polynomil in x, the highest power of x in p(x) is clled the degree of
More informationMATH 573 FINAL EXAM. May 30, 2007
MATH 573 FINAL EXAM My 30, 007 NAME: Solutions 1. This exm is due Wednesdy, June 6 efore the 1:30 pm. After 1:30 pm I will NOT ccept the exm.. This exm hs 1 pges including this cover. There re 10 prolems.
More information8 Laplace s Method and Local Limit Theorems
8 Lplce s Method nd Locl Limit Theorems 8. Fourier Anlysis in Higher DImensions Most of the theorems of Fourier nlysis tht we hve proved hve nturl generliztions to higher dimensions, nd these cn be proved
More informationSTRAND B: NUMBER THEORY
Mthemtics SKE, Strnd B UNIT B Indices nd Fctors: Tet STRAND B: NUMBER THEORY B Indices nd Fctors Tet Contents Section B. Squres, Cubes, Squre Roots nd Cube Roots B. Inde Nottion B. Fctors B. Prime Fctors,
More informationMATHEMATICS PART A. 1. ABC is a triangle, right angled at A. The resultant of the forces acting along AB, AC
FIITJEE Solutions to AIEEE MATHEMATICS PART A. ABC is tringle, right ngled t A. The resultnt of the forces cting long AB, AC with mgnitudes AB nd respectively is the force long AD, where D is the AC foot
More informationMath 554 Integration
Mth 554 Integrtion Hndout #9 4/12/96 Defn. A collection of n + 1 distinct points of the intervl [, b] P := {x 0 = < x 1 < < x i 1 < x i < < b =: x n } is clled prtition of the intervl. In this cse, we
More informationNew Expansion and Infinite Series
Interntionl Mthemticl Forum, Vol. 9, 204, no. 22, 06073 HIKARI Ltd, www.mhikri.com http://dx.doi.org/0.2988/imf.204.4502 New Expnsion nd Infinite Series Diyun Zhng College of Computer Nnjing University
More informationTHE QUADRATIC RECIPROCITY LAW OF DUKEHOPKINS. Circa 1870, G. Zolotarev observed that the Legendre symbol ( a p
THE QUADRATIC RECIPROCITY LAW OF DUKEHOPKINS PETE L CLARK Circ 1870, Zolotrev observed tht the Legendre symbol ( p ) cn be interpreted s the sign of multipliction by viewed s permuttion of the set Z/pZ
More informationset is not closed under matrix [ multiplication, ] and does not form a group.
Prolem 2.3: Which of the following collections of 2 2 mtrices with rel entries form groups under [ mtrix ] multipliction? i) Those of the form for which c d 2 Answer: The set of such mtrices is not closed
More information38 Riemann sums and existence of the definite integral.
38 Riemnn sums nd existence of the definite integrl. In the clcultion of the re of the region X bounded by the grph of g(x) = x 2, the xxis nd 0 x b, two sums ppered: ( n (k 1) 2) b 3 n 3 re(x) ( n These
More informationLecture 14: Quadrature
Lecture 14: Qudrture This lecture is concerned with the evlution of integrls fx)dx 1) over finite intervl [, b] The integrnd fx) is ssumed to be relvlues nd smooth The pproximtion of n integrl by numericl
More informationON THE ENTRY SUM OF CYCLOTOMIC ARRAYS. Don Coppersmith IDACCR. John Steinberger UC Davis
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 6 (2006), #A26 ON THE ENTRY SUM OF CYCLOTOMIC ARRAYS Don Coppersmith IDACCR John Steinberger UC Dvis Received: 3/29/05, Revised: 0/8/06, Accepted:
More information4402 Geometry/Topology: Differentiable Manifolds Northwestern University Solutions of Practice Problems for Final Exam
4402 Geometry/Topology: Differentible Mnifolds Northwestern University Solutions of Prctice Problems for Finl Exm 1) Using the cnonicl covering of RP n by {U α } 0 α n, where U α = {[x 0 : : x n ] RP
More informationFinite Automata Theory and Formal Languages TMV027/DIT321 LP4 2018
Finite Automt Theory nd Forml Lnguges TMV027/DIT321 LP4 2018 Lecture 10 An Bove April 23rd 2018 Recp: Regulr Lnguges We cn convert between FA nd RE; Hence both FA nd RE ccept/generte regulr lnguges; More
More informationChapter 7 Notes, Stewart 8e. 7.1 Integration by Parts Trigonometric Integrals Evaluating sin m x cos n (x) dx...
Contents 7.1 Integrtion by Prts................................... 2 7.2 Trigonometric Integrls.................................. 8 7.2.1 Evluting sin m x cos n (x)......................... 8 7.2.2 Evluting
More informationSection 14.3 Arc Length and Curvature
Section 4.3 Arc Length nd Curvture Clculus on Curves in Spce In this section, we ly the foundtions for describing the movement of n object in spce.. Vector Function Bsics In Clc, formul for rc length in
More information63. Representation of functions as power series Consider a power series. ( 1) n x 2n for all 1 < x < 1
3 9. SEQUENCES AND SERIES 63. Representtion of functions s power series Consider power series x 2 + x 4 x 6 + x 8 + = ( ) n x 2n It is geometric series with q = x 2 nd therefore it converges for ll q =
More informationHarvard University Computer Science 121 Midterm October 23, 2012
Hrvrd University Computer Science 121 Midterm Octoer 23, 2012 This is closedook exmintion. You my use ny result from lecture, Sipser, prolem sets, or section, s long s you quote it clerly. The lphet is
More informationMore Properties of the Riemann Integral
More Properties of the Riemnn Integrl Jmes K. Peterson Deprtment of Biologil Sienes nd Deprtment of Mthemtil Sienes Clemson University Februry 15, 2018 Outline More Riemnn Integrl Properties The Fundmentl
More informationMath 211A Homework. Edward Burkard. = tan (2x + z)
Mth A Homework Ewr Burkr Eercises 5C Eercise 8 Show tht the utonomous system: 5 Plne Autonomous Systems = e sin 3y + sin cos + e z, y = sin ( + 3y, z = tn ( + z hs n unstble criticl point t = y = z =
More informationOn the degree of regularity of generalized van der Waerden triples
On the degree of regulrity of generlized vn der Werden triples Jcob Fox Msschusetts Institute of Technology, Cmbridge, MA 02139, USA Rdoš Rdoičić Deprtment of Mthemtics, Rutgers, The Stte University of
More informationQuantum Physics II (8.05) Fall 2013 Assignment 2
Quntum Physics II (8.05) Fll 2013 Assignment 2 Msschusetts Institute of Technology Physics Deprtment Due Fridy September 20, 2013 September 13, 2013 3:00 pm Suggested Reding Continued from lst week: 1.
More informationThe mth Ratio Convergence Test and Other Unconventional Convergence Tests
The mth Rtio Convergence Test nd Other Unconventionl Convergence Tests Kyle Blckburn My 14, 2012 Contents 1 Introduction 2 2 Definitions, Lemms, nd Theorems 2 2.1 Defintions.............................
More informationPART 1 MULTIPLE CHOICE Circle the appropriate response to each of the questions below. Each question has a value of 1 point.
PART MULTIPLE CHOICE Circle the pproprite response to ech of the questions below. Ech question hs vlue of point.. If in sequence the second level difference is constnt, thn the sequence is:. rithmetic
More informationPRIMES AND QUADRATIC RECIPROCITY
PRIMES AND QUADRATIC RECIPROCITY ANGELICA WONG Abstrct We discuss number theory with the ultimte gol of understnding udrtic recirocity We begin by discussing Fermt s Little Theorem, the Chinese Reminder
More informationNWI: Mathematics. Various books in the library with the title Linear Algebra I, or Analysis I. (And also Linear Algebra II, or Analysis II.
NWI: Mthemtics Literture These lecture notes! Vrious books in the librry with the title Liner Algebr I, or Anlysis I (And lso Liner Algebr II, or Anlysis II) The lecture notes of some of the people who
More informationSOLUTIONS FOR ADMISSIONS TEST IN MATHEMATICS, COMPUTER SCIENCE AND JOINT SCHOOLS WEDNESDAY 5 NOVEMBER 2014
SOLUTIONS FOR ADMISSIONS TEST IN MATHEMATICS, COMPUTER SCIENCE AND JOINT SCHOOLS WEDNESDAY 5 NOVEMBER 014 Mrk Scheme: Ech prt of Question 1 is worth four mrks which re wrded solely for the correct nswer.
More information1 Nondeterministic Finite Automata
1 Nondeterministic Finite Automt Suppose in life, whenever you hd choice, you could try oth possiilities nd live your life. At the end, you would go ck nd choose the one tht worked out the est. Then you
More informationOn the Generalized Weighted QuasiArithmetic Integral Mean 1
Int. Journl of Mth. Anlysis, Vol. 7, 2013, no. 41, 20392048 HIKARI Ltd, www.mhikri.com http://dx.doi.org/10.12988/ijm.2013.3499 On the Generlized Weighted QusiArithmetic Integrl Men 1 Hui Sun School
More informationSection 5.1 #7, 10, 16, 21, 25; Section 5.2 #8, 9, 15, 20, 27, 30; Section 5.3 #4, 6, 9, 13, 16, 28, 31; Section 5.4 #7, 18, 21, 23, 25, 29, 40
Mth B Prof. Audrey Terrs HW # Solutions by Alex Eustis Due Tuesdy, Oct. 9 Section 5. #7,, 6,, 5; Section 5. #8, 9, 5,, 7, 3; Section 5.3 #4, 6, 9, 3, 6, 8, 3; Section 5.4 #7, 8,, 3, 5, 9, 4 5..7 Since
More informationProblem Set 3
14.102 Problem Set 3 Due Tuesdy, October 18, in clss 1. Lecture Notes Exercise 208: Find R b log(t)dt,where0
More informationMath 324 Course Notes: Brief description
Brief description These re notes for Mth 324, n introductory course in Mesure nd Integrtion. Students re dvised to go through ll sections in detil nd ttempt ll problems. These notes will be modified nd
More informationVariational Techniques for SturmLiouville Eigenvalue Problems
Vritionl Techniques for SturmLiouville Eigenvlue Problems Vlerie Cormni Deprtment of Mthemtics nd Sttistics University of Nebrsk, Lincoln Lincoln, NE 68588 Emil: vcormni@mth.unl.edu Rolf Ryhm Deprtment
More informationAbstract factorials. Angelo B. Mingarelli
Notes on Number Theory nd Discrete Mthemtics Vol. 9, 203, No. 4, 43 76 Abstrct fctorils Angelo B. Mingrelli School of Mthemtics nd Sttistics Crleton University, Ottw, Ontrio, Cnd, KS 5B6 emils: mingre@mth.crleton.c
More information42nd International Mathematical Olympiad
nd Interntionl Mthemticl Olympid Wshington, DC, United Sttes of Americ July 8 9, 001 Problems Ech problem is worth seven points. Problem 1 Let ABC be n cutengled tringle with circumcentre O. Let P on
More informationLecture 19: Continuous Least Squares Approximation
Lecture 19: Continuous Lest Squres Approximtion 33 Continuous lest squres pproximtion We begn 31 with the problem of pproximting some f C[, b] with polynomil p P n t the discrete points x, x 1,, x m for
More informationAMATH 731: Applied Functional Analysis Fall Some basics of integral equations
AMATH 731: Applied Functionl Anlysis Fll 2009 1 Introduction Some bsics of integrl equtions An integrl eqution is n eqution in which the unknown function u(t) ppers under n integrl sign, e.g., K(t, s)u(s)
More information1.3 Regular Expressions
56 1.3 Regulr xpressions These hve n importnt role in describing ptterns in serching for strings in mny pplictions (e.g. wk, grep, Perl,...) All regulr expressions of lphbet re 1.Ønd re regulr expressions,
More informationREVIEW Chapter 1 The Real Number System
Mth 7 REVIEW Chpter The Rel Number System In clss work: Solve ll exercises. (Sections. &. Definition A set is collection of objects (elements. The Set of Nturl Numbers N N = {,,,, 5, } The Set of Whole
More informationMath Calculus with Analytic Geometry II
orem of definite Mth 5.0 with Anlytic Geometry II Jnury 4, 0 orem of definite If < b then b f (x) dx = ( under f bove xxis) ( bove f under xxis) Exmple 8 0 3 9 x dx = π 3 4 = 9π 4 orem of definite Problem
More informationalong the vector 5 a) Find the plane s coordinate after 1 hour. b) Find the plane s coordinate after 2 hours. c) Find the plane s coordinate
L8 VECTOR EQUATIONS OF LINES HL Mth  Sntowski Vector eqution of line 1 A plne strts journey t the point (4,1) moves ech hour long the vector. ) Find the plne s coordinte fter 1 hour. b) Find the plne
More informationQuadratic reciprocity
Qudrtic recirocity Frncisc Bozgn Los Angeles Mth Circle Octoer 8, 01 1 Qudrtic Recirocity nd Legendre Symol In the eginning of this lecture, we recll some sic knowledge out modulr rithmetic: Definition
More informationMath 4200: Homework Problems
Mth 4200: Homework Problems Gregor Kovčič 1. Prove the following properties of the binomil coefficients ( n ) ( n ) (i) 1 + + + + 1 2 ( n ) (ii) 1 ( n ) ( n ) + 2 + 3 + + n 2 3 ( ) n ( n + = 2 n 1 n) n,
More informationEulerMaclaurin Summation Formula 1
Jnury 9, EulerMclurin Summtion Formul Suppose tht f nd its derivtive re continuous functions on the closed intervl [, b]. Let ψ(x) {x}, where {x} x [x] is the frctionl prt of x. Lemm : If < b nd, b Z,
More informationDIRECT CURRENT CIRCUITS
DRECT CURRENT CUTS ELECTRC POWER Consider the circuit shown in the Figure where bttery is connected to resistor R. A positive chrge dq will gin potentil energy s it moves from point to point b through
More informationMatrix Eigenvalues and Eigenvectors September 13, 2017
Mtri Eigenvlues nd Eigenvectors September, 7 Mtri Eigenvlues nd Eigenvectors Lrry Cretto Mechnicl Engineering 5A Seminr in Engineering Anlysis September, 7 Outline Review lst lecture Definition of eigenvlues
More informationFinal Exam Study Guide
Finl Exm Study Guide Includes. Integrls & Antiderivtive Rules 2. Definite Integrls (Integrls with bounds) 3. Are Between Two Curves  Region Bounded by Two Curves 4. Consumer nd Producer Surplus. USubstitution.
More informationEntropy and Ergodic Theory Notes 10: Large Deviations I
Entropy nd Ergodic Theory Notes 10: Lrge Devitions I 1 A chnge of convention This is our first lecture on pplictions of entropy in probbility theory. In probbility theory, the convention is tht ll logrithms
More informationPartial Differential Equations
Prtil Differentil Equtions Notes by Robert Piché, Tmpere University of Technology reen s Functions. reen s Function for OneDimensionl Eqution The reen s function provides complete solution to boundry
More informationLecture 3. Limits of Functions and Continuity
Lecture 3 Limits of Functions nd Continuity Audrey Terrs April 26, 21 1 Limits of Functions Notes I m skipping the lst section of Chpter 6 of Lng; the section bout open nd closed sets We cn probbly live
More informationMATH34032: Green s Functions, Integral Equations and the Calculus of Variations 1
MATH34032: Green s Functions, Integrl Equtions nd the Clculus of Vritions 1 Section 1 Function spces nd opertors Here we gives some brief detils nd definitions, prticulrly relting to opertors. For further
More informationMath 3B Final Review
Mth 3B Finl Review Written by Victori Kl vtkl@mth.ucsb.edu SH 6432u Office Hours: R 9:4510:45m SH 1607 Mth Lb Hours: TR 12pm Lst updted: 12/06/14 This is continution of the midterm review. Prctice problems
More informationBIFURCATIONS IN ONEDIMENSIONAL DISCRETE SYSTEMS
BIFRCATIONS IN ONEDIMENSIONAL DISCRETE SYSTEMS FRANCESCA AICARDI In this lesson we will study the simplest dynmicl systems. We will see, however, tht even in this cse the scenrio of different possible
More informationIII. Lecture on Numerical Integration. File faclib/dattab/lecturenotes/numericalinter03.tex /by EC, 3/14/2008 at 15:11, version 9
III Lecture on Numericl Integrtion File fclib/dttb/lecturenotes/numericalinter03.tex /by EC, 3/14/008 t 15:11, version 9 1 Sttement of the Numericl Integrtion Problem In this lecture we consider the
More information4 VECTORS. 4.0 Introduction. Objectives. Activity 1
4 VECTRS Chpter 4 Vectors jectives fter studying this chpter you should understnd the difference etween vectors nd sclrs; e le to find the mgnitude nd direction of vector; e le to dd vectors, nd multiply
More informationChapter 4 Contravariance, Covariance, and Spacetime Diagrams
Chpter 4 Contrvrince, Covrince, nd Spcetime Digrms 4. The Components of Vector in Skewed Coordintes We hve seen in Chpter 3; figure 3.9, tht in order to show inertil motion tht is consistent with the Lorentz
More informationNOTES AND PROBLEMS: INTEGRATION THEORY
NOTES AND PROBLEMS: INTEGRATION THEORY SAMEER CHAVAN Abstrct. These re the lecture notes prepred for prticipnts of AFSI to be conducted t Kumun University, Almor from 1st to 27th December, 2014. Contents
More informationz TRANSFORMS z Transform Basics z Transform Basics Transfer Functions Back to the Time Domain Transfer Function and Stability
TRASFORS Trnsform Bsics Trnsfer Functions Bck to the Time Domin Trnsfer Function nd Stility DSPG 6. Trnsform Bsics The definition of the trnsform for digitl signl is: n X x[ n is complex vrile The trnsform
More informationT b a(f) [f ] +. P b a(f) = Conclude that if f is in AC then it is the difference of two monotone absolutely continuous functions.
Rel Vribles, Fll 2014 Problem set 5 Solution suggestions Exerise 1. Let f be bsolutely ontinuous on [, b] Show tht nd T b (f) P b (f) f (x) dx [f ] +. Conlude tht if f is in AC then it is the differene
More information15  TRIGONOMETRY Page 1 ( Answers at the end of all questions )
 TRIGONOMETRY Pge P ( ) In tringle PQR, R =. If tn b c = 0, 0, then Q nd tn re the roots of the eqution = b c c = b b = c b = c [ AIEEE 00 ] ( ) In tringle ABC, let C =. If r is the inrdius nd R is the
More informationMA123, Chapter 10: Formulas for integrals: integrals, antiderivatives, and the Fundamental Theorem of Calculus (pp.
MA123, Chpter 1: Formuls for integrls: integrls, ntiderivtives, nd the Fundmentl Theorem of Clculus (pp. 27233, Gootmn) Chpter Gols: Assignments: Understnd the sttement of the Fundmentl Theorem of Clculus.
More informationIf u = g(x) is a differentiable function whose range is an interval I and f is continuous on I, then f(g(x))g (x) dx = f(u) du
Integrtion by Substitution: The Fundmentl Theorem of Clculus demonstrted the importnce of being ble to find ntiderivtives. We now introduce some methods for finding ntiderivtives: If u = g(x) is differentible
More informationPreSession Review. Part 1: Basic Algebra; Linear Functions and Graphs
PreSession Review Prt 1: Bsic Algebr; Liner Functions nd Grphs A. Generl Review nd Introduction to Algebr Hierrchy of Arithmetic Opertions Opertions in ny expression re performed in the following order:
More informationAnatomy of a Deterministic Finite Automaton. Deterministic Finite Automata. A machine so simple that you can understand it in less than one minute
Victor Admchik Dnny Sletor Gret Theoreticl Ides In Computer Science CS 525 Spring 2 Lecture 2 Mr 3, 2 Crnegie Mellon University Deterministic Finite Automt Finite Automt A mchine so simple tht you cn
More informationMA Handout 2: Notation and Background Concepts from Analysis
MA350059 Hndout 2: Nottion nd Bckground Concepts from Anlysis This hndout summrises some nottion we will use nd lso gives recp of some concepts from other units (MA20023: PDEs nd CM, MA20218: Anlysis 2A,
More informationMath 8 Winter 2015 Applications of Integration
Mth 8 Winter 205 Applictions of Integrtion Here re few importnt pplictions of integrtion. The pplictions you my see on n exm in this course include only the Net Chnge Theorem (which is relly just the Fundmentl
More informationDiscrete Leastsquares Approximations
Discrete Lestsqures Approximtions Given set of dt points (x, y ), (x, y ),, (x m, y m ), norml nd useful prctice in mny pplictions in sttistics, engineering nd other pplied sciences is to construct curve
More informationHomework 4. (1) If f R[a, b], show that f 3 R[a, b]. If f + (x) = max{f(x), 0}, is f + R[a, b]? Justify your answer.
Homework 4 (1) If f R[, b], show tht f 3 R[, b]. If f + (x) = mx{f(x), 0}, is f + R[, b]? Justify your nswer. (2) Let f be continuous function on [, b] tht is strictly positive except finitely mny points
More information5.5 The Substitution Rule
5.5 The Substitution Rule Given the usefulness of the Fundmentl Theorem, we wnt some helpful methods for finding ntiderivtives. At the moment, if n ntiderivtive is not esily recognizble, then we re in
More informationWeek 7 Riemann Stieltjes Integration: Lectures 1921
Week 7 Riemnn Stieltjes Integrtion: Lectures 1921 Lecture 19 Throughout this section α will denote monotoniclly incresing function on n intervl [, b]. Let f be bounded function on [, b]. Let P = { = 0
More informationIntegrals along Curves.
Integrls long Curves. 1. Pth integrls. Let : [, b] R n be continuous function nd let be the imge ([, b]) of. We refer to both nd s curve. If we need to distinguish between the two we cll the function the
More informationProblem Set 9. Figure 1: Diagram. This picture is a rough sketch of the 4 parabolas that give us the area that we need to find. The equations are:
(x + y ) = y + (x + y ) = x + Problem Set 9 Discussion: Nov., Nov. 8, Nov. (on probbility nd binomil coefficients) The nme fter the problem is the designted writer of the solution of tht problem. (No one
More informationENGI 3424 Engineering Mathematics Five Tutorial Examples of Partial Fractions
ENGI 44 Engineering Mthemtics Five Tutoril Exmples o Prtil Frctions 1. Express x in prtil rctions: x 4 x 4 x 4 b x x x x Both denomintors re liner nonrepeted ctors. The coverup rule my be used: 4 4 4
More informationHow do we solve these things, especially when they get complicated? How do we know when a system has a solution, and when is it unique?
XII. LINEAR ALGEBRA: SOLVING SYSTEMS OF EQUATIONS Tody we re going to tlk out solving systems of liner equtions. These re prolems tht give couple of equtions with couple of unknowns, like: 6= x + x 7=
More information