A Note on Sums of Greatest (Least) Prime Factors


 Owen Bell
 4 years ago
 Views:
Transcription
1 It. J. Cotemp. Math. Scieces, Vol. 8, 203, o. 9, HIKARI Ltd, A Note o Sums of Greatest (Least Prime Factors Rafael Jakimczuk Divisio Matemática, Uiversidad Nacioal de Luá Bueos Aires, Argetia Copyright c 203 Rafael Jakimczuk. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. Abstract Let a m ( be the mth power of the least prime factor i the prime factorizatio of. We prove the asymptotic formula a m (i m+ m +log. Let b m ( be the mth power of the greatest prime factor i the prime factorizatio of. We prove the asymptotic formula b m (i ζ(m + m+ m + log, where ζ(s is the Riema s Zeta Fuctio. Cosequetly b m (i lim ζ(m +. a m (i I particular if m we obtai b (i π2 lim ζ(2 a (i 6. Mathematics Subect Classificatio: A4 Keywords: Sums of greatest (least prime factors
2 424 R. Jakimczuk Itroductio ad Lemmas Let m be a positive iteger ad let b m ( be the mth power of the greatest prime factor i the prime factorizatio of. For example if 2 the b (2 3 ad b 4 (2 3 4,if 8 the b (8 3 ad b 2 (8 3 2,if 5 the b (5 5 ad b 4 ( I this ote we prove the asymptotic formula b m (i ζ(m + m+ m + log. ( If m is asymptotic formula is wellkow (see either [] or [4]. I the proof of ( we use a similar method of proof already used i the proof of other theorems (see [3]. The followig lemma is a cosequece of the prime umber theorem (see for example [2]. Lemma. Let m be a oegative iteger ad let s m (x be the sum of the mth powers of the primes ot exceedig x. We have the followig asymptotic formula s m (x p m x m+ h(xxm+ p x (m + log x + log x, (2 where p deotes a positive prime ad h(x 0. Note that h(x depeds of m. Note that if m 0 equatio (2 becomes the Prime Number Theorem. That is, s 0 (x π(x, where π(x is the prime coutig fuctio. Let m be a positive iteger ad let a m ( be the mth power of the least prime factor i the prime factorizatio of. For example if 2 the a (2 2 ad a 4 (2 2 4,if 8 the a (8 2 ad a 2 (8 2 2,if 5 the a (5 5 ad a 4 ( I this ote we prove the asymptotic formula m+ a m (i (m + log. (3 We also shall eed the followig lemma. Lemma.2 Let m be a positive iteger. We have the followig formula m+ ( + m+ where ζ(s is the Riema s Zeta Fuctio. ζ(m +, m ( + m+
3 Sums of greatest (least prime factors 425 Proof. We have ( m ( ( + m+ (( + m+ m+ ( + m+ ( m+ ( +. m+ ( + m+ Therefore m ( + m+ m+ ( + m+. Now The lemma is proved. ζ(m + lim m+ m+. Note that a cosequece of equatio (2 is the followig iequality x m+ s m (x p m <h p x (m + log x, (4 where h>. This iequality hods for x x 0, where x 0 deped of m. 2 Mai Results Now, we shall prove the metioed results. Namely, formulas ( ad (3. Theorem 2. We have the followig asymptotic formula where m is a arbitrary but fixed positive iteger. a m (i m+ m +log, (5 Proof. Let A(, p be the umber of positive itegers ot exceedig such that their least prime factor is the prime p. Therefore A(, p. We have a m (i 2 p 2 p p m A(, p 2 p k p m A(, p+ p m A(, p, (6 k <p
4 426 R. Jakimczuk where k 2 is a positive iteger. Cosider the first sum i (6. Namely p m A(, p. 2 p k We have the followig trivial iequality A(, p p p. Therefore (see (4 That is where 2 p k p m A(, p h(m + mk m log k log h(m + + λ mk m 2 p k 0 <g( < 2 p k p m p m+ (m + log m+ (m + log 2 p k (λ >0. m+ p m h ( k m m log k p m A(, p g( (m + log, (7 Cosider the secod iequality i (6. Namely h(m + mk m + λ (λ >0. (8 p m A(, p. k <p If is large the k<p. O the other had kp >. Cosequetly the uique multiple of p less tha or equal to such that p is its least prime factor is p. That is, we have A(, p. Therefore (see lemma. m+ (m + log + h( p m A(, p p m p m k <p 2 p 2 p k m+ m+ k (m + log k h k log k k + h( h k k m+ log k log + p k ( m+ log m+ (m + log m+ k m+ (m + log + r k( k m+ log k log m+ log ( k m+ + q k( m+ log m+ (m + log m+ (m + log m+ (m + log,
5 Sums of greatest (least prime factors 427 where h( 0, q k ( 0, p k ( 0 ad r k ( 0. That is p m A(, p k <p where r k ( 0. We have m+ (m + log m+ k m+ m+ (m + log + r k( (m + log, (9 a m (i m+ m +log + f( m+ m +log. (0 Substitutig equatios (7 ad (9 ito (6 we obtai a m (i m+ ( (m + log + k + r k(+g( m+ m+ (m + log. Cosequetly f( k + r k(+g(. ( m+ Let ɛ>0. If we choose k sufficietly large the Therefore we have (see ( k m+ < ɛ 3, r k( < ɛ 3, 0 <g( < ɛ 3. f( <ɛ, if is sufficietly large. Now, ɛ is arbitrarily little. Therefore lim f( 0. (2 Equatios (0 ad (2 give (5. The theorem is proved. Theorem 2.2 We have the followig asymptotic formula b m (i ζ(m + m+ m + log, (3 where m is a arbitrary but fixed positive iteger. Proof. Let B(, p be the umber of positive itegers ot exceedig such that their greatest prime factor is the prime p. Therefore 2 p B(, p.
6 428 R. Jakimczuk We have b m (i p m B(, p p m B(, p+ 2 p 2 p k+ + p m B(, p+ + p m B(, p k <p k + 3 <p 2 k+ <p k p m B(, p p m B(, p. (4 2 <p Cosider the first sum i (4. Namely 2 p k+ We have the followig trivial iequality B(, p p m B(, p. p p. As i theorem 2. we obtai p m m+ B(, p g( (m + log, (5 2 p k+ where h(m + 0 <g( < + λ m(k + m (λ >0. (6 Now, cosider the sum (see (4 p m B(, p (, 2,...,k. (7 + <p If is large the k<p. O the other had p ad ( +p>. Cosequetly the multiples of p less tha or equal to such that p is their greatest prime factor are p, 2p,..., p. That is, we have A(, p. Cosequetly (see (7 p m B(, p p m (, 2,...,k. (8 + <p Lemma. gives + <p + <p p m ( m+ (m + log + h ( m+ log ( m+ + (m + log ( +
7 Sums of greatest (least prime factors 429 m+ + h + log + m+ m+ log ( + log m+ log(+ (m + log log + h h m+ log + ( + log m+ log(+ log m+ log m+ ( + + q m+ ( m+ (m + log + p ( m+ log m+ m+ ( + m+ (m + log m+ + r ( (m + log, where h( 0, q ( 0, p ( 0 ad r ( 0. That is p m m+ m+ ( + m+ (m + log + r m+ ( (m + log, (9 + <p where r ( 0. Substitutig (9 ito (8 we obtai + <p p m B(, p m+ ( m ( + m+ m+ (m + log + r ( (, 2,...,k, (20 (m + log where r ( r ( 0. We have ζ(m + m+ b m (i m + log + f( m+ (m + log. (2 Substitutig (5 ad (20 ito (4 we fid that (see lemma.2 k b m (i g(+ k + r m ( + m+ ( ζ(m + m+ m + log + g( k + m ( + m+ m+ (m + log. k+ m+ (m + log r (
8 430 R. Jakimczuk Cosequetly f( g( k+ k+ + m ( + m+ Let ɛ>0. If we choose k sufficietly large the 0 < < ɛ m ( + m+ 3, 0 <g( < ɛ 3. O the other had, sice k r (. (22 r ( 0 (, 2,...,k, if is sufficietly large the we have r ( ɛ < (, 2,...,k. 3k Therefore we have (see (22 f( <ɛ. Now, ɛ is arbitrarily little. Hece lim f( 0. (23 Equatios (2 ad (23 give (3. The theorem is proved. Corollary 2.3 The followig limits hold lim b m (i a m (i I particular if m we obtai b (i lim a (i ζ(m +. ζ(2 π2 6. Proof. It is a immediate cosequece of Theorem 2. ad Theorem 2.2. The corollary is proved. Let c be a composite umber. If we cosider oly composite umbers i Corollary 2.3 the we have the followig corollary. Corollary 2.4 We have the followig limit c b m (c lim c a m (c.
9 Sums of greatest (least prime factors 43 Proof. Let p be a prime umber. We have a m (p b m (p p m. Theorefore (Theorem 2., Theorem 2.2 ad Lemma. c b m (c c a m (c ζ(m+ m+ + o m+ m+ log log o m+ log The corollary is proved. b m (i p b m (p a m (i p a m (p b m (i p p m a m (i p p m ζ(m+ m+ + o( o( Let p k be a prime power. We have a m (p k b m (p k p m. O the other had, if d is ot a prime power the a m (d <b m (d. We have the followig corollary Corollary 2.5 The followig limit holds d b m (d lim d a m (d. Proof. We have p m p m a m (i. p p k Cosequetly (Lemma. ad Theorem 2. p k. p m m+ m +log. (24 Therefore (Theorem 2., Theorem 2.2 ad equatio (24 d b m (d d a m (d ζ(m+ m+ + o m+ m+ log log o m+ log The corollary is proved. b m (i p k b m (p k a m (i p k a m (p k b m (i p k p m a m (i p k p m ζ(m+ m+ + o( o( ACKNOWLEDGEMENTS. The author is very grateful to Uiversidad Nacioal de Luá. Refereces [] K. Alladi ad P. Erdős, O a additive arithmetic fuctio, Pacific Joural of Mathematics, 7 (977,
10 432 R. Jakimczuk [2] R. Jakimczuk, A ote o sums of powers which have a fixed umber of prime factors, Joural of Iequalities i Pure ad Applied Mathematics, 6 (2005, Article 3. [3] R. Jakimczuk, Sums of prime umbers, the zeta fuctio ad the π umber, Iteratioal Mathematical Forum, 3 (2008, [4] J. Kemey, Largest prime factor, Joural of Pure ad Applied Algebra, 89 (993, Received: February 0, 203
1. MATHEMATICAL INDUCTION
1. MATHEMATICAL INDUCTION EXAMPLE 1: Prove that for ay iteger 1. Proof: 1 + 2 + 3 +... + ( + 1 2 (1.1 STEP 1: For 1 (1.1 is true, sice 1 1(1 + 1. 2 STEP 2: Suppose (1.1 is true for some k 1, that is 1
More informationFactors of sums of powers of binomial coefficients
ACTA ARITHMETICA LXXXVI.1 (1998) Factors of sums of powers of biomial coefficiets by Neil J. Cali (Clemso, S.C.) Dedicated to the memory of Paul Erdős 1. Itroductio. It is well ow that if ( ) a f,a = the
More informationIrreducible polynomials with consecutive zero coefficients
Irreducible polyomials with cosecutive zero coefficiets Theodoulos Garefalakis Departmet of Mathematics, Uiversity of Crete, 71409 Heraklio, Greece Abstract Let q be a prime power. We cosider the problem
More informationTHE LEAST COMMON MULTIPLE OF A QUADRATIC SEQUENCE
THE LEAST COMMON MULTIPLE OF A QUADRATIC SEQUENCE JAVIER CILLERUELO Abstract. We obtai, for ay irreducible quadratic olyomial f(x = ax 2 + bx + c, the asymtotic estimate log l.c.m. {f(1,..., f(} log. Whe
More informationSection 11.3: The Integral Test
Sectio.3: The Itegral Test Most of the series we have looked at have either diverged or have coverged ad we have bee able to fid what they coverge to. I geeral however, the problem is much more difficult
More informationProperties of MLE: consistency, asymptotic normality. Fisher information.
Lecture 3 Properties of MLE: cosistecy, asymptotic ormality. Fisher iformatio. I this sectio we will try to uderstad why MLEs are good. Let us recall two facts from probability that we be used ofte throughout
More informationWHEN IS THE (CO)SINE OF A RATIONAL ANGLE EQUAL TO A RATIONAL NUMBER?
WHEN IS THE (CO)SINE OF A RATIONAL ANGLE EQUAL TO A RATIONAL NUMBER? JÖRG JAHNEL 1. My Motivatio Some Sort of a Itroductio Last term I tought Topological Groups at the Göttige Georg August Uiversity. This
More informationInfinite Sequences and Series
CHAPTER 4 Ifiite Sequeces ad Series 4.1. Sequeces A sequece is a ifiite ordered list of umbers, for example the sequece of odd positive itegers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29...
More informationAsymptotic Growth of Functions
CMPS Itroductio to Aalysis of Algorithms Fall 3 Asymptotic Growth of Fuctios We itroduce several types of asymptotic otatio which are used to compare the performace ad efficiecy of algorithms As we ll
More informationOn Formula to Compute Primes. and the n th Prime
Applied Mathematical cieces, Vol., 0, o., 3535 O Formula to Compute Primes ad the th Prime Issam Kaddoura Lebaese Iteratioal Uiversity Faculty of Arts ad cieces, Lebao issam.kaddoura@liu.edu.lb amih AbdulNabi
More informationA probabilistic proof of a binomial identity
A probabilistic proof of a biomial idetity Joatho Peterso Abstract We give a elemetary probabilistic proof of a biomial idetity. The proof is obtaied by computig the probability of a certai evet i two
More information3. Greatest Common Divisor  Least Common Multiple
3 Greatest Commo Divisor  Least Commo Multiple Defiitio 31: The greatest commo divisor of two atural umbers a ad b is the largest atural umber c which divides both a ad b We deote the greatest commo gcd
More informationCS103X: Discrete Structures Homework 4 Solutions
CS103X: Discrete Structures Homewor 4 Solutios Due February 22, 2008 Exercise 1 10 poits. Silico Valley questios: a How may possible sixfigure salaries i whole dollar amouts are there that cotai at least
More informationON AN INTEGRAL OPERATOR WHICH PRESERVE THE UNIVALENCE
Proceedigs of the Iteratioal Coferece o Theory ad Applicatios of Mathematics ad Iformatics ICTAMI 3, Alba Iulia ON AN INTEGRAL OPERATOR WHICH PRESERVE THE UNIVALENCE by Maria E Gageoea ad Silvia Moldoveau
More informationA note on the boundary behavior for a modiﬁed Green function in the upperhalf space
Zhag ad Pisarev Boudary Value Problems (015) 015:114 DOI 10.1186/s136610150363z RESEARCH Ope Access A ote o the boudary behavior for a modiﬁed Gree fuctio i the upperhalf space Yulia Zhag1 ad Valery
More informationConvexity, Inequalities, and Norms
Covexity, Iequalities, ad Norms Covex Fuctios You are probably familiar with the otio of cocavity of fuctios. Give a twicedifferetiable fuctio ϕ: R R, We say that ϕ is covex (or cocave up) if ϕ (x) 0 for
More informationLecture 13. Lecturer: Jonathan Kelner Scribe: Jonathan Pines (2009)
18.409 A Algorithmist s Toolkit October 27, 2009 Lecture 13 Lecturer: Joatha Keler Scribe: Joatha Pies (2009) 1 Outlie Last time, we proved the BruMikowski iequality for boxes. Today we ll go over the
More informationOur aim is to show that under reasonable assumptions a given 2πperiodic function f can be represented as convergent series
8 Fourier Series Our aim is to show that uder reasoable assumptios a give periodic fuctio f ca be represeted as coverget series f(x) = a + (a cos x + b si x). (8.) By defiitio, the covergece of the series
More informationTrigonometric Form of a Complex Number. The Complex Plane. axis. ( 2, 1) or 2 i FIGURE 6.44. The absolute value of the complex number z a bi is
0_0605.qxd /5/05 0:45 AM Page 470 470 Chapter 6 Additioal Topics i Trigoometry 6.5 Trigoometric Form of a Complex Number What you should lear Plot complex umbers i the complex plae ad fid absolute values
More informationLecture 5: Span, linear independence, bases, and dimension
Lecture 5: Spa, liear idepedece, bases, ad dimesio Travis Schedler Thurs, Sep 23, 2010 (versio: 9/21 9:55 PM) 1 Motivatio Motivatio To uderstad what it meas that R has dimesio oe, R 2 dimesio 2, etc.;
More informationCS103A Handout 23 Winter 2002 February 22, 2002 Solving Recurrence Relations
CS3A Hadout 3 Witer 00 February, 00 Solvig Recurrece Relatios Itroductio A wide variety of recurrece problems occur i models. Some of these recurrece relatios ca be solved usig iteratio or some other ad
More information4.3. The Integral and Comparison Tests
4.3. THE INTEGRAL AND COMPARISON TESTS 9 4.3. The Itegral ad Compariso Tests 4.3.. The Itegral Test. Suppose f is a cotiuous, positive, decreasig fuctio o [, ), ad let a = f(). The the covergece or divergece
More informationIn nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008
I ite Sequeces Dr. Philippe B. Laval Keesaw State Uiversity October 9, 2008 Abstract This had out is a itroductio to i ite sequeces. mai de itios ad presets some elemetary results. It gives the I ite Sequeces
More informationLecture 4: Cauchy sequences, BolzanoWeierstrass, and the Squeeze theorem
Lecture 4: Cauchy sequeces, BolzaoWeierstrass, ad the Squeeze theorem The purpose of this lecture is more modest tha the previous oes. It is to state certai coditios uder which we are guarateed that limits
More informationPROCEEDINGS OF THE YEREVAN STATE UNIVERSITY AN ALTERNATIVE MODEL FOR BONUSMALUS SYSTEM
PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY Physical ad Mathematical Scieces 2015, 1, p. 15 19 M a t h e m a t i c s AN ALTERNATIVE MODEL FOR BONUSMALUS SYSTEM A. G. GULYAN Chair of Actuarial Mathematics
More informationSAMPLE QUESTIONS FOR FINAL EXAM. (1) (2) (3) (4) Find the following using the definition of the Riemann integral: (2x + 1)dx
SAMPLE QUESTIONS FOR FINAL EXAM REAL ANALYSIS I FALL 006 3 4 Fid the followig usig the defiitio of the Riema itegral: a 0 x + dx 3 Cosider the partitio P x 0 3, x 3 +, x 3 +,......, x 3 3 + 3 of the iterval
More informationFIBONACCI NUMBERS: AN APPLICATION OF LINEAR ALGEBRA. 1. Powers of a matrix
FIBONACCI NUMBERS: AN APPLICATION OF LINEAR ALGEBRA. Powers of a matrix We begi with a propositio which illustrates the usefuless of the diagoalizatio. Recall that a square matrix A is diogaalizable if
More informationResearch Article Sign Data Derivative Recovery
Iteratioal Scholarly Research Network ISRN Applied Mathematics Volume 0, Article ID 63070, 7 pages doi:0.540/0/63070 Research Article Sig Data Derivative Recovery L. M. Housto, G. A. Glass, ad A. D. Dymikov
More informationTHIN SEQUENCES AND THE GRAM MATRIX PAMELA GORKIN, JOHN E. MCCARTHY, SANDRA POTT, AND BRETT D. WICK
THIN SEQUENCES AND THE GRAM MATRIX PAMELA GORKIN, JOHN E MCCARTHY, SANDRA POTT, AND BRETT D WICK Abstract We provide a ew proof of Volberg s Theorem characterizig thi iterpolatig sequeces as those for
More informationChapter 7 Methods of Finding Estimators
Chapter 7 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 011 Chapter 7 Methods of Fidig Estimators Sectio 7.1 Itroductio Defiitio 7.1.1 A poit estimator is ay fuctio W( X) W( X1, X,, X ) of
More informationLecture 4: Cheeger s Inequality
Spectral Graph Theory ad Applicatios WS 0/0 Lecture 4: Cheeger s Iequality Lecturer: Thomas Sauerwald & He Su Statemet of Cheeger s Iequality I this lecture we assume for simplicity that G is a dregular
More informationChapter 6: Variance, the law of large numbers and the MonteCarlo method
Chapter 6: Variace, the law of large umbers ad the MoteCarlo method Expected value, variace, ad Chebyshev iequality. If X is a radom variable recall that the expected value of X, E[X] is the average value
More informationApproximating Area under a curve with rectangles. To find the area under a curve we approximate the area using rectangles and then use limits to find
1.8 Approximatig Area uder a curve with rectagles 1.6 To fid the area uder a curve we approximate the area usig rectagles ad the use limits to fid 1.4 the area. Example 1 Suppose we wat to estimate 1.
More informationOn the L p conjecture for locally compact groups
Arch. Math. 89 (2007), 237 242 c 2007 Birkhäuser Verlag Basel/Switzerlad 0003/889X/0302376, ublished olie 2007080 DOI 0.007/s0003007993x Archiv der Mathematik O the L cojecture for locally comact
More informationDepartment of Computer Science, University of Otago
Departmet of Computer Sciece, Uiversity of Otago Techical Report OUCS200609 Permutatios Cotaiig May Patters Authors: M.H. Albert Departmet of Computer Sciece, Uiversity of Otago Micah Colema, Rya Fly
More informationSampling Distribution And Central Limit Theorem
() Samplig Distributio & Cetral Limit Samplig Distributio Ad Cetral Limit Samplig distributio of the sample mea If we sample a umber of samples (say k samples where k is very large umber) each of size,
More informationChapter 5 O A Cojecture Of Erdíos Proceedigs NCUR VIII è1994è, Vol II, pp 794í798 Jeærey F Gold Departmet of Mathematics, Departmet of Physics Uiversity of Utah Do H Tucker Departmet of Mathematics Uiversity
More informationSubsets of Prime Numbers
Iteratioal Joural of Mathematics ad Computer Sciece, 7(2012), o. 2, 101 112 M CS Subsets of Prime Numbers Badih Ghusayi Departmet of Mathematics Faculty of Sciece1 Lebaese Uiversity Hadath, Lebao email:
More informationTheorems About Power Series
Physics 6A Witer 20 Theorems About Power Series Cosider a power series, f(x) = a x, () where the a are real coefficiets ad x is a real variable. There exists a real oegative umber R, called the radius
More informationOverview of some probability distributions.
Lecture Overview of some probability distributios. I this lecture we will review several commo distributios that will be used ofte throughtout the class. Each distributio is usually described by its probability
More informationFactoring x n 1: cyclotomic and Aurifeuillian polynomials Paul Garrett <garrett@math.umn.edu>
(March 16, 004) Factorig x 1: cyclotomic ad Aurifeuillia polyomials Paul Garrett Polyomials of the form x 1, x 3 1, x 4 1 have at least oe systematic factorizatio x 1 = (x 1)(x 1
More informationChapter 5: Inner Product Spaces
Chapter 5: Ier Product Spaces Chapter 5: Ier Product Spaces SECION A Itroductio to Ier Product Spaces By the ed of this sectio you will be able to uderstad what is meat by a ier product space give examples
More informationAnnuities Under Random Rates of Interest II By Abraham Zaks. Technion I.I.T. Haifa ISRAEL and Haifa University Haifa ISRAEL.
Auities Uder Radom Rates of Iterest II By Abraham Zas Techio I.I.T. Haifa ISRAEL ad Haifa Uiversity Haifa ISRAEL Departmet of Mathematics, Techio  Israel Istitute of Techology, 3000, Haifa, Israel I memory
More informationON THE DENSE TRAJECTORY OF LASOTA EQUATION
UNIVERSITATIS IAGELLONICAE ACTA MATHEMATICA, FASCICULUS XLIII 2005 ON THE DENSE TRAJECTORY OF LASOTA EQUATION by Atoi Leo Dawidowicz ad Najemedi Haribash Abstract. I preseted paper the dese trajectory
More informationTangent circles in the ratio 2 : 1. Hiroshi Okumura and Masayuki Watanabe. In this article we consider the following old Japanese geometry problem
116 Taget circles i the ratio 2 : 1 Hiroshi Okumura ad Masayuki Wataabe I this article we cosider the followig old Japaese geometry problem (see Figure 1), whose statemet i [1, p. 39] is missig the coditio
More informationBasic Elements of Arithmetic Sequences and Series
MA40S PRECALCULUS UNIT G GEOMETRIC SEQUENCES CLASS NOTES (COMPLETED NO NEED TO COPY NOTES FROM OVERHEAD) Basic Elemets of Arithmetic Sequeces ad Series Objective: To establish basic elemets of arithmetic
More informationBuilding Blocks Problem Related to Harmonic Series
TMME, vol3, o, p.76 Buildig Blocks Problem Related to Harmoic Series Yutaka Nishiyama Osaka Uiversity of Ecoomics, Japa Abstract: I this discussio I give a eplaatio of the divergece ad covergece of ifiite
More informationNote on some explicit formulae for twin prime counting function
Notes on Number Theory and Discrete Mathematics Vol. 9, 03, No., 43 48 Note on some explicit formulae for twin prime counting function Mladen VassilevMissana 5 V. Hugo Str., 4 Sofia, Bulgaria email:
More informationI. Chisquared Distributions
1 M 358K Supplemet to Chapter 23: CHISQUARED DISTRIBUTIONS, TDISTRIBUTIONS, AND DEGREES OF FREEDOM To uderstad tdistributios, we first eed to look at aother family of distributios, the chisquared distributios.
More informationA Faster ClauseShortening Algorithm for SAT with No Restriction on Clause Length
Joural o Satisfiability, Boolea Modelig ad Computatio 1 2005) 4960 A Faster ClauseShorteig Algorithm for SAT with No Restrictio o Clause Legth Evgey Datsi Alexader Wolpert Departmet of Computer Sciece
More information5.3. Generalized Permutations and Combinations
53 GENERALIZED PERMUTATIONS AND COMBINATIONS 73 53 Geeralized Permutatios ad Combiatios 53 Permutatios with Repeated Elemets Assume that we have a alphabet with letters ad we wat to write all possible
More informationTHE HEIGHT OF qbinary SEARCH TREES
THE HEIGHT OF qbinary SEARCH TREES MICHAEL DRMOTA AND HELMUT PRODINGER Abstract. q biary search trees are obtaied from words, equipped with the geometric distributio istead of permutatios. The average
More informationEngineering 323 Beautiful Homework Set 3 1 of 7 Kuszmar Problem 2.51
Egieerig 33 eautiful Homewor et 3 of 7 Kuszmar roblem.5.5 large departmet store sells sport shirts i three sizes small, medium, ad large, three patters plaid, prit, ad stripe, ad two sleeve legths log
More informationSequences and Series
CHAPTER 9 Sequeces ad Series 9.. Covergece: Defiitio ad Examples Sequeces The purpose of this chapter is to itroduce a particular way of geeratig algorithms for fidig the values of fuctios defied by their
More informationPermutations, the Parity Theorem, and Determinants
1 Permutatios, the Parity Theorem, ad Determiats Joh A. Guber Departmet of Electrical ad Computer Egieerig Uiversity of Wiscosi Madiso Cotets 1 What is a Permutatio 1 2 Cycles 2 2.1 Traspositios 4 3 Orbits
More informationUC Berkeley Department of Electrical Engineering and Computer Science. EE 126: Probablity and Random Processes. Solutions 9 Spring 2006
Exam format UC Bereley Departmet of Electrical Egieerig ad Computer Sciece EE 6: Probablity ad Radom Processes Solutios 9 Sprig 006 The secod midterm will be held o Wedesday May 7; CHECK the fial exam
More informationElementary Theory of Russian Roulette
Elemetary Theory of Russia Roulette iterestig patters of fractios Satoshi Hashiba Daisuke Miematsu Ryohei Miyadera Itroductio. Today we are goig to study mathematical theory of Russia roulette. If some
More information. P. 4.3 Basic feasible solutions and vertices of polyhedra. x 1. x 2
4. Basic feasible solutios ad vertices of polyhedra Due to the fudametal theorem of Liear Programmig, to solve ay LP it suffices to cosider the vertices (fiitely may) of the polyhedro P of the feasible
More informationPartial Di erential Equations
Partial Di eretial Equatios Partial Di eretial Equatios Much of moder sciece, egieerig, ad mathematics is based o the study of partial di eretial equatios, where a partial di eretial equatio is a equatio
More informationThe Stable Marriage Problem
The Stable Marriage Problem William Hut Lae Departmet of Computer Sciece ad Electrical Egieerig, West Virgiia Uiversity, Morgatow, WV William.Hut@mail.wvu.edu 1 Itroductio Imagie you are a matchmaker,
More informationEkkehart Schlicht: Economic Surplus and Derived Demand
Ekkehart Schlicht: Ecoomic Surplus ad Derived Demad Muich Discussio Paper No. 200617 Departmet of Ecoomics Uiversity of Muich Volkswirtschaftliche Fakultät LudwigMaximiliasUiversität Müche Olie at http://epub.ub.uimueche.de/940/
More informationA Recursive Formula for Moments of a Binomial Distribution
A Recursive Formula for Momets of a Biomial Distributio Árpád Béyi beyi@mathumassedu, Uiversity of Massachusetts, Amherst, MA 01003 ad Saverio M Maago smmaago@psavymil Naval Postgraduate School, Moterey,
More information23 The Remainder and Factor Theorems
 The Remaider ad Factor Theorems Factor each polyomial completely usig the give factor ad log divisio 1 x + x x 60; x + So, x + x x 60 = (x + )(x x 15) Factorig the quadratic expressio yields x + x x
More informationCooleyTukey. Tukey FFT Algorithms. FFT Algorithms. Cooley
Cooley CooleyTuey Tuey FFT Algorithms FFT Algorithms Cosider a legth sequece x[ with a poit DFT X[ where Represet the idices ad as +, +, Cooley CooleyTuey Tuey FFT Algorithms FFT Algorithms Usig these
More informationSECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES
SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES Read Sectio 1.5 (pages 5 9) Overview I Sectio 1.5 we lear to work with summatio otatio ad formulas. We will also itroduce a brief overview of sequeces,
More informationCME 302: NUMERICAL LINEAR ALGEBRA FALL 2005/06 LECTURE 8
CME 30: NUMERICAL LINEAR ALGEBRA FALL 005/06 LECTURE 8 GENE H GOLUB 1 Positive Defiite Matrices A matrix A is positive defiite if x Ax > 0 for all ozero x A positive defiite matrix has real ad positive
More informationTHE ABRACADABRA PROBLEM
THE ABRACADABRA PROBLEM FRANCESCO CARAVENNA Abstract. We preset a detailed solutio of Exercise E0.6 i [Wil9]: i a radom sequece of letters, draw idepedetly ad uiformly from the Eglish alphabet, the expected
More informationRunning Time ( 3.1) Analysis of Algorithms. Experimental Studies ( 3.1.1) Limitations of Experiments. Pseudocode ( 3.1.2) Theoretical Analysis
Ruig Time ( 3.) Aalysis of Algorithms Iput Algorithm Output A algorithm is a stepbystep procedure for solvig a problem i a fiite amout of time. Most algorithms trasform iput objects ito output objects.
More informationExample 2 Find the square root of 0. The only square root of 0 is 0 (since 0 is not positive or negative, so those choices don t exist here).
BEGINNING ALGEBRA Roots ad Radicals (revised summer, 00 Olso) Packet to Supplemet the Curret Textbook  Part Review of Square Roots & Irratioals (This portio ca be ay time before Part ad should mostly
More informationTHE ARITHMETIC OF INTEGERS.  multiplication, exponentiation, division, addition, and subtraction
THE ARITHMETIC OF INTEGERS  multiplicatio, expoetiatio, divisio, additio, ad subtractio What to do ad what ot to do. THE INTEGERS Recall that a iteger is oe of the whole umbers, which may be either positive,
More informationwhere: T = number of years of cash flow in investment's life n = the year in which the cash flow X n i = IRR = the internal rate of return
EVALUATING ALTERNATIVE CAPITAL INVESTMENT PROGRAMS By Ke D. Duft, Extesio Ecoomist I the March 98 issue of this publicatio we reviewed the procedure by which a capital ivestmet project was assessed. The
More informationCHAPTER 7: Central Limit Theorem: CLT for Averages (Means)
CHAPTER 7: Cetral Limit Theorem: CLT for Averages (Meas) X = the umber obtaied whe rollig oe six sided die oce. If we roll a six sided die oce, the mea of the probability distributio is X P(X = x) Simulatio:
More informationSoving Recurrence Relations
Sovig Recurrece Relatios Part 1. Homogeeous liear 2d degree relatios with costat coefficiets. Cosider the recurrece relatio ( ) T () + at ( 1) + bt ( 2) = 0 This is called a homogeeous liear 2d degree
More informationNormal Distribution.
Normal Distributio www.icrf.l Normal distributio I probability theory, the ormal or Gaussia distributio, is a cotiuous probability distributio that is ofte used as a first approimatio to describe realvalued
More informationClass Meeting # 16: The Fourier Transform on R n
MATH 18.152 COUSE NOTES  CLASS MEETING # 16 18.152 Itroductio to PDEs, Fall 2011 Professor: Jared Speck Class Meetig # 16: The Fourier Trasform o 1. Itroductio to the Fourier Trasform Earlier i the course,
More informationSEQUENCES AND SERIES
Chapter 9 SEQUENCES AND SERIES Natural umbers are the product of huma spirit. DEDEKIND 9.1 Itroductio I mathematics, the word, sequece is used i much the same way as it is i ordiary Eglish. Whe we say
More informationInteger Factorization Algorithms
Iteger Factorizatio Algorithms Coelly Bares Departmet of Physics, Orego State Uiversity December 7, 004 This documet has bee placed i the public domai. Cotets I. Itroductio 3 1. Termiology 3. Fudametal
More informationThe analysis of the Cournot oligopoly model considering the subjective motive in the strategy selection
The aalysis of the Courot oligopoly model cosiderig the subjective motive i the strategy selectio Shigehito Furuyama Teruhisa Nakai Departmet of Systems Maagemet Egieerig Faculty of Egieerig Kasai Uiversity
More informationPART TWO. Measure, Integration, and Differentiation
PART TWO Measure, Itegratio, ad Differetiatio Émile FélixÉdouardJusti Borel (1871 1956 Émile Borel was bor at SaitAffrique, Frace, o Jauary 7, 1871, the third child of Hooré Borel, a Protestat miister,
More informationAn Efficient Polynomial Approximation of the Normal Distribution Function & Its Inverse Function
A Efficiet Polyomial Approximatio of the Normal Distributio Fuctio & Its Iverse Fuctio Wisto A. Richards, 1 Robi Atoie, * 1 Asho Sahai, ad 3 M. Raghuadh Acharya 1 Departmet of Mathematics & Computer Sciece;
More informationActa Acad. Paed. Agriensis, Sectio Mathematicae 29 (2002) 77 87. ALMOST SURE FUNCTIONAL LIMIT THEOREMS IN L p( ]0, 1[ ), WHERE 1 p <
Acta Acad. Paed. Agriesis, Sectio Mathematicae 29 22) 77 87 ALMOST SUR FUNCTIONAL LIMIT THORMS IN L ], [ ), WHR < József Túri Nyíregyháza, Hugary) Dedicated to the memory of Professor Péter Kiss Abstract.
More informationUniversal coding for classes of sources
Coexios module: m46228 Uiversal codig for classes of sources Dever Greee This work is produced by The Coexios Project ad licesed uder the Creative Commos Attributio Licese We have discussed several parametric
More information0.7 0.6 0.2 0 0 96 96.5 97 97.5 98 98.5 99 99.5 100 100.5 96.5 97 97.5 98 98.5 99 99.5 100 100.5
Sectio 13 KolmogorovSmirov test. Suppose that we have a i.i.d. sample X 1,..., X with some ukow distributio P ad we would like to test the hypothesis that P is equal to a particular distributio P 0, i.e.
More informationLecture 3. denote the orthogonal complement of S k. Then. 1 x S k. n. 2 x T Ax = ( ) λ x. with x = 1, we have. i = λ k x 2 = λ k.
18.409 A Algorithmist s Toolkit September 17, 009 Lecture 3 Lecturer: Joatha Keler Scribe: Adre Wibisoo 1 Outlie Today s lecture covers three mai parts: CouratFischer formula ad Rayleigh quotiets The
More informationPerfect Packing Theorems and the AverageCase Behavior of Optimal and Online Bin Packing
SIAM REVIEW Vol. 44, No. 1, pp. 95 108 c 2002 Society for Idustrial ad Applied Mathematics Perfect Packig Theorems ad the AverageCase Behavior of Optimal ad Olie Bi Packig E. G. Coffma, Jr. C. Courcoubetis
More informationSolving Logarithms and Exponential Equations
Solvig Logarithms ad Epoetial Equatios Logarithmic Equatios There are two major ideas required whe solvig Logarithmic Equatios. The first is the Defiitio of a Logarithm. You may recall from a earlier topic:
More informationNATIONAL SENIOR CERTIFICATE GRADE 12
NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P EXEMPLAR 04 MARKS: 50 TIME: 3 hours This questio paper cosists of 8 pages ad iformatio sheet. Please tur over Mathematics/P DBE/04 NSC Grade Eemplar INSTRUCTIONS
More informationINFINITE SERIES KEITH CONRAD
INFINITE SERIES KEITH CONRAD. Itroductio The two basic cocepts of calculus, differetiatio ad itegratio, are defied i terms of limits (Newto quotiets ad Riema sums). I additio to these is a third fudametal
More information3. If x and y are real numbers, what is the simplified radical form
lgebra II Practice Test Objective:.a. Which is equivalet to 98 94 4 49?. Which epressio is aother way to write 5 4? 5 5 4 4 4 5 4 5. If ad y are real umbers, what is the simplified radical form of 5 y
More informationObservation on Sums of Powers of Integers Divisible by Four
Applied Mathematical Sciences, Vol. 8, 2014, no. 45, 22192226 HIKARI Ltd, www.mhikari.com http://dx.doi.org/10.12988/ams.2014.4140 Observation on Sums of Powers of Integers Divisible by Four Djoko Suprijanto
More informationEGYPTIAN FRACTION EXPANSIONS FOR RATIONAL NUMBERS BETWEEN 0 AND 1 OBTAINED WITH ENGEL SERIES
EGYPTIAN FRACTION EXPANSIONS FOR RATIONAL NUMBERS BETWEEN 0 AND OBTAINED WITH ENGEL SERIES ELVIA NIDIA GONZÁLEZ AND JULIA BERGNER, PHD DEPARTMENT OF MATHEMATICS Abstract. The aciet Egyptias epressed ratioal
More informationTO: Users of the ACTEX Review Seminar on DVD for SOA Exam MLC
TO: Users of the ACTEX Review Semiar o DVD for SOA Eam MLC FROM: Richard L. (Dick) Lodo, FSA Dear Studets, Thak you for purchasig the DVD recordig of the ACTEX Review Semiar for SOA Eam M, Life Cotigecies
More informationA Mathematical Perspective on Gambling
A Mathematical Perspective o Gamblig Molly Maxwell Abstract. This paper presets some basic topics i probability ad statistics, icludig sample spaces, probabilistic evets, expectatios, the biomial ad ormal
More informationS. Tanny MAT 344 Spring 1999. be the minimum number of moves required.
S. Tay MAT 344 Sprig 999 Recurrece Relatios Tower of Haoi Let T be the miimum umber of moves required. T 0 = 0, T = 7 Iitial Coditios * T = T + $ T is a sequece (f. o itegers). Solve for T? * is a recurrece,
More informationEntropy of bicapacities
Etropy of bicapacities Iva Kojadiovic LINA CNRS FRE 2729 Site école polytechique de l uiv. de Nates Rue Christia Pauc 44306 Nates, Frace iva.kojadiovic@uivates.fr JeaLuc Marichal Applied Mathematics
More informationA RANDOM PERMUTATION MODEL ARISING IN CHEMISTRY
J. Appl. Prob. 45, 060 070 2008 Prited i Eglad Applied Probability Trust 2008 A RANDOM PERMUTATION MODEL ARISING IN CHEMISTRY MARK BROWN, The City College of New York EROL A. PEKÖZ, Bosto Uiversity SHELDON
More informationPlugin martingales for testing exchangeability online
Plugi martigales for testig exchageability olie Valetia Fedorova, Alex Gammerma, Ilia Nouretdiov, ad Vladimir Vovk Computer Learig Research Cetre Royal Holloway, Uiversity of Lodo, UK {valetia,ilia,alex,vovk}@cs.rhul.ac.uk
More informationNATIONAL SENIOR CERTIFICATE GRADE 11
NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P NOVEMBER 007 MARKS: 50 TIME: 3 hours This questio paper cosists of 9 pages, diagram sheet ad a page formula sheet. Please tur over Mathematics/P DoE/November
More informationAnalysis Notes (only a draft, and the first one!)
Aalysis Notes (oly a draft, ad the first oe!) Ali Nesi Mathematics Departmet Istabul Bilgi Uiversity Kuştepe Şişli Istabul Turkey aesi@bilgi.edu.tr Jue 22, 2004 2 Cotets 1 Prelimiaries 9 1.1 Biary Operatio...........................
More information