A Note on Sums of Greatest (Least) Prime Factors
|
|
|
- Owen Bell
- 9 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 [email protected] 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 m-th 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 m-th 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 m-th 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 well-kow (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 m-th 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 m-th 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
Factors 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
Section 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
Properties 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
WHEN 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
Infinite 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...
Asymptotic 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
On Formula to Compute Primes. and the n th Prime
Applied Mathematical cieces, Vol., 0, o., 35-35 O Formula to Compute Primes ad the th Prime Issam Kaddoura Lebaese Iteratioal Uiversity Faculty of Arts ad cieces, Lebao [email protected] amih Abdul-Nabi
A 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
3. 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
CS103X: 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 six-figure salaries i whole dollar amouts are there that cotai at least
ON 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
Convexity, 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
Our aim is to show that under reasonable assumptions a given 2π-periodic 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
Trigonometric 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
Lecture 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.;
CS103A 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
4.3. 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
In 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
Lecture 4: Cauchy sequences, Bolzano-Weierstrass, and the Squeeze theorem
Lecture 4: Cauchy sequeces, Bolzao-Weierstrass, 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
PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY AN ALTERNATIVE MODEL FOR BONUS-MALUS 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 BONUS-MALUS SYSTEM A. G. GULYAN Chair of Actuarial Mathematics
SAMPLE 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
FIBONACCI 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
Research 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
THIN 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
Chapter 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
Lecture 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 d-regular
Chapter 6: Variance, the law of large numbers and the Monte-Carlo method
Chapter 6: Variace, the law of large umbers ad the Mote-Carlo 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
Approximating 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.
Department of Computer Science, University of Otago
Departmet of Computer Sciece, Uiversity of Otago Techical Report OUCS-2006-09 Permutatios Cotaiig May Patters Authors: M.H. Albert Departmet of Computer Sciece, Uiversity of Otago Micah Colema, Rya Fly
Sampling 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,
Chapter 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
Subsets 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 Sciece-1 Lebaese Uiversity Hadath, Lebao email:
Theorems 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 o-egative umber R, called the radius
Overview 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
Factoring x n 1: cyclotomic and Aurifeuillian polynomials Paul Garrett <[email protected]>
(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
Chapter 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
Annuities 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
ON 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
How To Solve An 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
Basic Elements of Arithmetic Sequences and Series
MA40S PRE-CALCULUS 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
Building 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
Note 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 Vassilev-Missana 5 V. Hugo Str., 4 Sofia, Bulgaria e-mail:
I. Chi-squared Distributions
1 M 358K Supplemet to Chapter 23: CHI-SQUARED DISTRIBUTIONS, T-DISTRIBUTIONS, AND DEGREES OF FREEDOM To uderstad t-distributios, we first eed to look at aother family of distributios, the chi-squared distributios.
A Faster Clause-Shortening Algorithm for SAT with No Restriction on Clause Length
Joural o Satisfiability, Boolea Modelig ad Computatio 1 2005) 49-60 A Faster Clause-Shorteig Algorithm for SAT with No Restrictio o Clause Legth Evgey Datsi Alexader Wolpert Departmet of Computer Sciece
5.3. 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
THE HEIGHT OF q-binary SEARCH TREES
THE HEIGHT OF q-binary 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
How To Solve The Homewor Problem Beautifully
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
Sequences 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
Permutations, 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
UC 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
Elementary 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
Partial 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
The Stable Marriage Problem
The Stable Marriage Problem William Hut Lae Departmet of Computer Sciece ad Electrical Egieerig, West Virgiia Uiversity, Morgatow, WV [email protected] 1 Itroductio Imagie you are a matchmaker,
Ekkehart Schlicht: Economic Surplus and Derived Demand
Ekkehart Schlicht: Ecoomic Surplus ad Derived Demad Muich Discussio Paper No. 2006-17 Departmet of Ecoomics Uiversity of Muich Volkswirtschaftliche Fakultät Ludwig-Maximilias-Uiversität Müche Olie at http://epub.ub.ui-mueche.de/940/
A 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,
2-3 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
Cooley-Tukey. Tukey FFT Algorithms. FFT Algorithms. Cooley
Cooley Cooley-Tuey Tuey FFT Algorithms FFT Algorithms Cosider a legth- sequece x[ with a -poit DFT X[ where Represet the idices ad as +, +, Cooley Cooley-Tuey Tuey FFT Algorithms FFT Algorithms Usig these
SECTION 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,
CME 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
THE 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
Running 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 step-by-step procedure for solvig a problem i a fiite amout of time. Most algorithms trasform iput objects ito output objects.
Example 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
THE 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,
where: 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
CHAPTER 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:
Soving 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
Normal 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
Class 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,
SEQUENCES 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
Integer 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
The 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
PART TWO. Measure, Integration, and Differentiation
PART TWO Measure, Itegratio, ad Differetiatio Émile Félix-Édouard-Justi Borel (1871 1956 Émile Borel was bor at Sait-Affrique, Frace, o Jauary 7, 1871, the third child of Hooré Borel, a Protestat miister,
An 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;
Universal 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
0.7 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 Kolmogorov-Smirov 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.
Lecture 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: Courat-Fischer formula ad Rayleigh quotiets The
Solving 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:
NATIONAL 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
INFINITE 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
3. 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
Observation on Sums of Powers of Integers Divisible by Four
Applied Mathematical Sciences, Vol. 8, 2014, no. 45, 2219-2226 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2014.4140 Observation on Sums of Powers of Integers Divisible by Four Djoko Suprijanto
EGYPTIAN 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
A 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
S. 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,
Entropy of bi-capacities
Etropy of bi-capacities Iva Kojadiovic LINA CNRS FRE 2729 Site école polytechique de l uiv. de Nates Rue Christia Pauc 44306 Nates, Frace [email protected] Jea-Luc Marichal Applied Mathematics
Plug-in martingales for testing exchangeability on-line
Plug-i martigales for testig exchageability o-lie 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
NATIONAL 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
Analysis 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 [email protected] Jue 22, 2004 2 Cotets 1 Prelimiaries 9 1.1 Biary Operatio...........................
