Complex Numbers Alpha, Round 1 Test #123

Size: px
Start display at page:

Download "Complex Numbers Alpha, Round 1 Test #123"

Transcription

1 Complex Numbers Alpha, Round Test #3. Wrte your 6-dgt ID# n the I.D. NUMBER grd, left-justfed, and bubble. Check that each column has only one number darkened.. In the EXAM NO. grd, wrte the 3-dgt Test # on ths test cover and bubble. 3. In the Name blank, prnt your name; n the Subject blank, prnt the name of the test; n the Date blank, prnt your school name (no abbrevatons).. Scorng for ths test s 5 tmes the number correct + the number omtted. 5. You may not st adjacent to anyone from your school. 6. TURN OFF ALL CELL PHONES OR OTHER PORTABLE ELECTRONIC DEVICES NOW. 7. No calculators may be used on ths test. 8. Any napproprate behavor or any form of cheatng wll lead to a ban of the student and/or school from future natonal conventons, dsqualfcaton of the student and/or school from ths conventon, at the dscreton of the Mu Alpha Theta Governng Councl. 9. If a student beleves a test tem s defectve, select E) NOTA and fle a Dspute Form explanng why. 0. If a problem has multple correct answers, any of those answers wll be counted as correct. Do not select E) NOTA n that nstance.. Unless a queston asks for an approxmaton or a rounded answer, gve the exact answer.

2 03 ΜΑΘ Natonal Conventon Note: For all questons, answer means none of the above answers s correct. Furthermore, assume that, cs cos sn, and Re( z ) and Im( z ) are the real and magnary parts of z, respectvely, unless otherwse specfed.. Compute 3 6. Assume all answer choces have the correct unts. Good luck! Compute Compute Compute. 3 cs k, k 0, 8 cs k, k 0, 8 cs k, k 0, 8 cs k, k 0, 8 5. Compute 5 n 5 n Re cs((n ) ) Im cs((n ) ). Page of 7

3 03 ΜΑΘ Natonal Conventon k3 6. Let a k k k, where each k s randomly chosen from the set {,,3,}. What s the probablty that a 0? The solutons to the equaton 6 z 79 can be wrtten n the form zk 3(cos k sn k ) where k {,,3,,5,6} and Compute the value of z z Consder two complex numbers z a b and w c d, a, b, c, d, as well as the vectors v Re( z),im( z) and v Re( w),im( w). Whch of the followng s equal to v v? Re( z w) Re z w Re zw Re z w 9. The bnomal coeffcent, n n!, r r! n r! can be wrtten as n n( n )( n ) ( n r ) r r! n order to accommodate for all complex n. Usng ths, compute Defne the operaton as cs( ) cs( ) cos cs. There exst real numbers 0 90 and 0 90 (both n degrees) such that Compute, gnorng unts n your answer. cs +cs cs3 cs Page of 7

4 03 ΜΑΘ Natonal Conventon. Regon R s bounded by the set of all complex numbers on the Argand plane z a b where a and b are real numbers satsfyng the equaton a a b b. Compute the area of R A complex number z satsfes the equaton z z. Gven that Re( z) a, determne the value of z n terms of a. a a a a a a a 3. Consder two vectors v x, y and v,3, where xy,. Gven that v v 5 6, compute x y Consder the matrx A. Determne all possble values of such that the 0 determnant of B A I s 0, where I The fourth roots of unty are the solutons to the equaton x 0. Graphng these roots on the complex plane gves a square whch we call S. Form a sequence of squares S, where the n th square S n s formed by connectng the mdponts of S. n For example, by connectng the mdponts of S, we obtan S. Now, let f ( x ) be the polynomal wth roots equal to the vertces of S n. Compute n f (0). n n Page 3 of 7

5 03 ΜΑΘ Natonal Conventon 6. Consder El s functon, f ( x) 9x 5 x k, where x and k are real. Determne the nterval for k such that the graph of f( x ) does not ntersect the x -axs. 5, 6 5, 6 5 5, ,, n f ( x) x nx. f has 03 roots; one of whch n0 s equal to, whle the other 0 are magnary. Let S equal the sum of the products of the roots of f( x ) taken two at a tme. Compute S. 7. Consder Patrck s functon, Consder Laura s magnary, sx-sded, far dce. The sdes on the frst de show the numbers,,3,,5,6, whle the sdes of the second de show the numbers,,3,,5,6. Laura takes rolls both the dce once and multples the numbers shown. If the result s real, the player gets the absolute value of the result n dollars. If the result s magnary, the player loses the absolute value of the result n dollars. What s the expected value of ths game, n dollars? $.5 $.5 $.5 $.5 C 3. Evaluate C. 9. Consder Caleb s favorte expresson, Consder Wll s geometrc seres. He doesn t care what the frst term s, as long as t s not zero. He would, however, lke a common rato such that at some pont n the geometrc seres, the n th term equals the frst term. What s the smallest value of n such that there are exactly 50 possble common ratos for the seres located n the second quadrant of the Argand plane? Do not consder ponts on the axes Page of 7

6 03 ΜΑΘ Natonal Conventon. The cosne functon can be approxmated by the functon f( ). x x f( x). Evaluate!! Calculate 03 cs. cs cs cs0 03 cs03 0 k k 3. A sequence of ponts zk s plotted on the Argand plane. A bug begns at z and travels along the segments zz, zz3,, znzn,..., z0z03. Let D be the total dstance the bug travels. Fnd the remander when D s dvded by Consder the matrx A 37 z. 3 A, 3 and the complex vector, 3 z. Calculate Gven that e x k x, compute the value of k! k 0 k 0 k n k. nn (!) e e e e Page 5 of 7

7 03 ΜΑΘ Natonal Conventon 6. Let 03 0 f ( x) x x x. Denote Rn ( ) as the remander when f( x ) s dvded by x n. Compute the remander when 03 k R ( ) s dvded by 000. k Consder a sequence of functons n j f ( x) x. Let the set contan all arguments n 0 such that cs s a soluton to f ( x) 0 for a gven n. Determne the smallest postve nteger n such that the sum of the entres n s greater than or equal to j0 n 8. A Gaussan Integer s a complex number z a b where ab,. Consder the Gaussan Integer z m 3 n. Whch of the followng s not a possble value for z? Use the followng nformaton for Problems 9 and 30: In Problem 8, we defned the Gaussan Integers. Another set of ntegers, called the Esensten Integers, are defned, for ab, ntegers, as z a b; 3 e 3. Because the argument of s, or 60, the Esensten Integers form a trangular 3 lattce on the Argand Plane, whereas the Gaussan Integers form a square lattce. 9. The trangular regon T has vertces located at the ponts z, z 3, and z3. Compute the area of T. (Hnt: Graph the ponts wth shfted axes by what angle would you shft them by?) Page 6 of 7

8 03 ΜΑΘ Natonal Conventon 30. The absolute value of the Gaussan Integers s smply z a b. Ths s not the case for the Esensten Integers. For the general Esensten Integer z a b, deduce the absolute value n terms of a and b. a b ab a b a b a b ab Page 7 of 7

For all questions, answer choice E) NOTA" means none of the above answers is correct.

For all questions, answer choice E) NOTA means none of the above answers is correct. 0 MA Natonal Conventon For all questons, answer choce " means none of the above answers s correct.. In calculus, one learns of functon representatons that are nfnte seres called power 3 4 5 seres. For

More information

Lectures - Week 4 Matrix norms, Conditioning, Vector Spaces, Linear Independence, Spanning sets and Basis, Null space and Range of a Matrix

Lectures - Week 4 Matrix norms, Conditioning, Vector Spaces, Linear Independence, Spanning sets and Basis, Null space and Range of a Matrix Lectures - Week 4 Matrx norms, Condtonng, Vector Spaces, Lnear Independence, Spannng sets and Bass, Null space and Range of a Matrx Matrx Norms Now we turn to assocatng a number to each matrx. We could

More information

COMPLEX NUMBERS AND QUADRATIC EQUATIONS

COMPLEX NUMBERS AND QUADRATIC EQUATIONS COMPLEX NUMBERS AND QUADRATIC EQUATIONS INTRODUCTION We know that x 0 for all x R e the square of a real number (whether postve, negatve or ero) s non-negatve Hence the equatons x, x, x + 7 0 etc are not

More information

8.6 The Complex Number System

8.6 The Complex Number System 8.6 The Complex Number System Earler n the chapter, we mentoned that we cannot have a negatve under a square root, snce the square of any postve or negatve number s always postve. In ths secton we want

More information

NUMERICAL DIFFERENTIATION

NUMERICAL DIFFERENTIATION NUMERICAL DIFFERENTIATION 1 Introducton Dfferentaton s a method to compute the rate at whch a dependent output y changes wth respect to the change n the ndependent nput x. Ths rate of change s called the

More information

Affine transformations and convexity

Affine transformations and convexity Affne transformatons and convexty The purpose of ths document s to prove some basc propertes of affne transformatons nvolvng convex sets. Here are a few onlne references for background nformaton: http://math.ucr.edu/

More information

Math1110 (Spring 2009) Prelim 3 - Solutions

Math1110 (Spring 2009) Prelim 3 - Solutions Math 1110 (Sprng 2009) Solutons to Prelm 3 (04/21/2009) 1 Queston 1. (16 ponts) Short answer. Math1110 (Sprng 2009) Prelm 3 - Solutons x a 1 (a) (4 ponts) Please evaluate lm, where a and b are postve numbers.

More information

= z 20 z n. (k 20) + 4 z k = 4

= z 20 z n. (k 20) + 4 z k = 4 Problem Set #7 solutons 7.2.. (a Fnd the coeffcent of z k n (z + z 5 + z 6 + z 7 + 5, k 20. We use the known seres expanson ( n+l ( z l l z n below: (z + z 5 + z 6 + z 7 + 5 (z 5 ( + z + z 2 + z + 5 5

More information

U.C. Berkeley CS294: Beyond Worst-Case Analysis Luca Trevisan September 5, 2017

U.C. Berkeley CS294: Beyond Worst-Case Analysis Luca Trevisan September 5, 2017 U.C. Berkeley CS94: Beyond Worst-Case Analyss Handout 4s Luca Trevsan September 5, 07 Summary of Lecture 4 In whch we ntroduce semdefnte programmng and apply t to Max Cut. Semdefnte Programmng Recall that

More information

= = = (a) Use the MATLAB command rref to solve the system. (b) Let A be the coefficient matrix and B be the right-hand side of the system.

= = = (a) Use the MATLAB command rref to solve the system. (b) Let A be the coefficient matrix and B be the right-hand side of the system. Chapter Matlab Exercses Chapter Matlab Exercses. Consder the lnear system of Example n Secton.. x x x y z y y z (a) Use the MATLAB command rref to solve the system. (b) Let A be the coeffcent matrx and

More information

First Year Examination Department of Statistics, University of Florida

First Year Examination Department of Statistics, University of Florida Frst Year Examnaton Department of Statstcs, Unversty of Florda May 7, 010, 8:00 am - 1:00 noon Instructons: 1. You have four hours to answer questons n ths examnaton.. You must show your work to receve

More information

Section 8.3 Polar Form of Complex Numbers

Section 8.3 Polar Form of Complex Numbers 80 Chapter 8 Secton 8 Polar Form of Complex Numbers From prevous classes, you may have encountered magnary numbers the square roots of negatve numbers and, more generally, complex numbers whch are the

More information

Complex Numbers. x = B B 2 4AC 2A. or x = x = 2 ± 4 4 (1) (5) 2 (1)

Complex Numbers. x = B B 2 4AC 2A. or x = x = 2 ± 4 4 (1) (5) 2 (1) Complex Numbers If you have not yet encountered complex numbers, you wll soon do so n the process of solvng quadratc equatons. The general quadratc equaton Ax + Bx + C 0 has solutons x B + B 4AC A For

More information

1. Estimation, Approximation and Errors Percentages Polynomials and Formulas Identities and Factorization 52

1. Estimation, Approximation and Errors Percentages Polynomials and Formulas Identities and Factorization 52 ontents ommonly Used Formulas. Estmaton, pproxmaton and Errors. Percentages. Polynomals and Formulas 8. Identtes and Factorzaton. Equatons and Inequaltes 66 6. Rate and Rato 8 7. Laws of Integral Indces

More information

Formulas for the Determinant

Formulas for the Determinant page 224 224 CHAPTER 3 Determnants e t te t e 2t 38 A = e t 2te t e 2t e t te t 2e 2t 39 If 123 A = 345, 456 compute the matrx product A adj(a) What can you conclude about det(a)? For Problems 40 43, use

More information

More metrics on cartesian products

More metrics on cartesian products More metrcs on cartesan products If (X, d ) are metrc spaces for 1 n, then n Secton II4 of the lecture notes we defned three metrcs on X whose underlyng topologes are the product topology The purpose of

More information

Problem Set 9 Solutions

Problem Set 9 Solutions Desgn and Analyss of Algorthms May 4, 2015 Massachusetts Insttute of Technology 6.046J/18.410J Profs. Erk Demane, Srn Devadas, and Nancy Lynch Problem Set 9 Solutons Problem Set 9 Solutons Ths problem

More information

Bernoulli Numbers and Polynomials

Bernoulli Numbers and Polynomials Bernoull Numbers and Polynomals T. Muthukumar tmk@tk.ac.n 17 Jun 2014 The sum of frst n natural numbers 1, 2, 3,..., n s n n(n + 1 S 1 (n := m = = n2 2 2 + n 2. Ths formula can be derved by notng that

More information

Review of Taylor Series. Read Section 1.2

Review of Taylor Series. Read Section 1.2 Revew of Taylor Seres Read Secton 1.2 1 Power Seres A power seres about c s an nfnte seres of the form k = 0 k a ( x c) = a + a ( x c) + a ( x c) + a ( x c) k 2 3 0 1 2 3 + In many cases, c = 0, and the

More information

332600_08_1.qxp 4/17/08 11:29 AM Page 481

332600_08_1.qxp 4/17/08 11:29 AM Page 481 336_8_.qxp 4/7/8 :9 AM Page 48 8 Complex Vector Spaces 8. Complex Numbers 8. Conjugates and Dvson of Complex Numbers 8.3 Polar Form and DeMovre s Theorem 8.4 Complex Vector Spaces and Inner Products 8.5

More information

(c) (cos θ + i sin θ) 5 = cos 5 θ + 5 cos 4 θ (i sin θ) + 10 cos 3 θ(i sin θ) cos 2 θ(i sin θ) 3 + 5cos θ (i sin θ) 4 + (i sin θ) 5 (A1)

(c) (cos θ + i sin θ) 5 = cos 5 θ + 5 cos 4 θ (i sin θ) + 10 cos 3 θ(i sin θ) cos 2 θ(i sin θ) 3 + 5cos θ (i sin θ) 4 + (i sin θ) 5 (A1) . (a) (cos θ + sn θ) = cos θ + cos θ( sn θ) + cos θ(sn θ) + (sn θ) = cos θ cos θ sn θ + ( cos θ sn θ sn θ) (b) from De Movre s theorem (cos θ + sn θ) = cos θ + sn θ cos θ + sn θ = (cos θ cos θ sn θ) +

More information

Math 261 Exercise sheet 2

Math 261 Exercise sheet 2 Math 261 Exercse sheet 2 http://staff.aub.edu.lb/~nm116/teachng/2017/math261/ndex.html Verson: September 25, 2017 Answers are due for Monday 25 September, 11AM. The use of calculators s allowed. Exercse

More information

Applied Stochastic Processes

Applied Stochastic Processes STAT455/855 Fall 23 Appled Stochastc Processes Fnal Exam, Bref Solutons 1. (15 marks) (a) (7 marks) The dstrbuton of Y s gven by ( ) ( ) y 2 1 5 P (Y y) for y 2, 3,... The above follows because each of

More information

Problem Set 6: Trees Spring 2018

Problem Set 6: Trees Spring 2018 Problem Set 6: Trees 1-29 Sprng 2018 A Average dstance Gven a tree, calculate the average dstance between two vertces n the tree. For example, the average dstance between two vertces n the followng tree

More information

ACTM State Calculus Competition Saturday April 30, 2011

ACTM State Calculus Competition Saturday April 30, 2011 ACTM State Calculus Competton Saturday Aprl 30, 2011 ACTM State Calculus Competton Sprng 2011 Page 1 Instructons: For questons 1 through 25, mark the best answer choce on the answer sheet provde Afterward

More information

Section 3.6 Complex Zeros

Section 3.6 Complex Zeros 04 Chapter Secton 6 Comple Zeros When fndng the zeros of polynomals, at some pont you're faced wth the problem Whle there are clearly no real numbers that are solutons to ths equaton, leavng thngs there

More information

n α j x j = 0 j=1 has a nontrivial solution. Here A is the n k matrix whose jth column is the vector for all t j=0

n α j x j = 0 j=1 has a nontrivial solution. Here A is the n k matrix whose jth column is the vector for all t j=0 MODULE 2 Topcs: Lnear ndependence, bass and dmenson We have seen that f n a set of vectors one vector s a lnear combnaton of the remanng vectors n the set then the span of the set s unchanged f that vector

More information

Unit 5: Quadratic Equations & Functions

Unit 5: Quadratic Equations & Functions Date Perod Unt 5: Quadratc Equatons & Functons DAY TOPIC 1 Modelng Data wth Quadratc Functons Factorng Quadratc Epressons 3 Solvng Quadratc Equatons 4 Comple Numbers Smplfcaton, Addton/Subtracton & Multplcaton

More information

8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS

8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS SECTION 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS 493 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS All the vector spaces you have studed thus far n the text are real vector spaces because the scalars

More information

Example: (13320, 22140) =? Solution #1: The divisors of are 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 27, 30, 36, 41,

Example: (13320, 22140) =? Solution #1: The divisors of are 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 27, 30, 36, 41, The greatest common dvsor of two ntegers a and b (not both zero) s the largest nteger whch s a common factor of both a and b. We denote ths number by gcd(a, b), or smply (a, b) when there s no confuson

More information

2.3 Nilpotent endomorphisms

2.3 Nilpotent endomorphisms s a block dagonal matrx, wth A Mat dm U (C) In fact, we can assume that B = B 1 B k, wth B an ordered bass of U, and that A = [f U ] B, where f U : U U s the restrcton of f to U 40 23 Nlpotent endomorphsms

More information

Lecture 3: Probability Distributions

Lecture 3: Probability Distributions Lecture 3: Probablty Dstrbutons Random Varables Let us begn by defnng a sample space as a set of outcomes from an experment. We denote ths by S. A random varable s a functon whch maps outcomes nto the

More information

HMMT February 2016 February 20, 2016

HMMT February 2016 February 20, 2016 HMMT February 016 February 0, 016 Combnatorcs 1. For postve ntegers n, let S n be the set of ntegers x such that n dstnct lnes, no three concurrent, can dvde a plane nto x regons (for example, S = {3,

More information

THE CHINESE REMAINDER THEOREM. We should thank the Chinese for their wonderful remainder theorem. Glenn Stevens

THE CHINESE REMAINDER THEOREM. We should thank the Chinese for their wonderful remainder theorem. Glenn Stevens THE CHINESE REMAINDER THEOREM KEITH CONRAD We should thank the Chnese for ther wonderful remander theorem. Glenn Stevens 1. Introducton The Chnese remander theorem says we can unquely solve any par of

More information

Math 217 Fall 2013 Homework 2 Solutions

Math 217 Fall 2013 Homework 2 Solutions Math 17 Fall 013 Homework Solutons Due Thursday Sept. 6, 013 5pm Ths homework conssts of 6 problems of 5 ponts each. The total s 30. You need to fully justfy your answer prove that your functon ndeed has

More information

Expected Value and Variance

Expected Value and Variance MATH 38 Expected Value and Varance Dr. Neal, WKU We now shall dscuss how to fnd the average and standard devaton of a random varable X. Expected Value Defnton. The expected value (or average value, or

More information

Foundations of Arithmetic

Foundations of Arithmetic Foundatons of Arthmetc Notaton We shall denote the sum and product of numbers n the usual notaton as a 2 + a 2 + a 3 + + a = a, a 1 a 2 a 3 a = a The notaton a b means a dvdes b,.e. ac = b where c s an

More information

8.1 Arc Length. What is the length of a curve? How can we approximate it? We could do it following the pattern we ve used before

8.1 Arc Length. What is the length of a curve? How can we approximate it? We could do it following the pattern we ve used before .1 Arc Length hat s the length of a curve? How can we approxmate t? e could do t followng the pattern we ve used before Use a sequence of ncreasngly short segments to approxmate the curve: As the segments

More information

Exercises. 18 Algorithms

Exercises. 18 Algorithms 18 Algorthms Exercses 0.1. In each of the followng stuatons, ndcate whether f = O(g), or f = Ω(g), or both (n whch case f = Θ(g)). f(n) g(n) (a) n 100 n 200 (b) n 1/2 n 2/3 (c) 100n + log n n + (log n)

More information

First day August 1, Problems and Solutions

First day August 1, Problems and Solutions FOURTH INTERNATIONAL COMPETITION FOR UNIVERSITY STUDENTS IN MATHEMATICS July 30 August 4, 997, Plovdv, BULGARIA Frst day August, 997 Problems and Solutons Problem. Let {ε n } n= be a sequence of postve

More information

Errors for Linear Systems

Errors for Linear Systems Errors for Lnear Systems When we solve a lnear system Ax b we often do not know A and b exactly, but have only approxmatons  and ˆb avalable. Then the best thng we can do s to solve ˆx ˆb exactly whch

More information

Rao IIT Academy/ SSC - Board Exam 2018 / Mathematics Code-A / QP + Solutions JEE MEDICAL-UG BOARDS KVPY NTSE OLYMPIADS SSC - BOARD

Rao IIT Academy/ SSC - Board Exam 2018 / Mathematics Code-A / QP + Solutions JEE MEDICAL-UG BOARDS KVPY NTSE OLYMPIADS SSC - BOARD Rao IIT Academy/ SSC - Board Exam 08 / Mathematcs Code-A / QP + Solutons JEE MEDICAL-UG BOARDS KVPY NTSE OLYMPIADS SSC - BOARD - 08 Date: 0.03.08 MATHEMATICS - PAPER- - SOLUTIONS Q. Attempt any FIVE of

More information

U.C. Berkeley CS294: Spectral Methods and Expanders Handout 8 Luca Trevisan February 17, 2016

U.C. Berkeley CS294: Spectral Methods and Expanders Handout 8 Luca Trevisan February 17, 2016 U.C. Berkeley CS94: Spectral Methods and Expanders Handout 8 Luca Trevsan February 7, 06 Lecture 8: Spectral Algorthms Wrap-up In whch we talk about even more generalzatons of Cheeger s nequaltes, and

More information

The Number of Ways to Write n as a Sum of ` Regular Figurate Numbers

The Number of Ways to Write n as a Sum of ` Regular Figurate Numbers Syracuse Unversty SURFACE Syracuse Unversty Honors Program Capstone Projects Syracuse Unversty Honors Program Capstone Projects Sprng 5-1-01 The Number of Ways to Wrte n as a Sum of ` Regular Fgurate Numbers

More information

2 Finite difference basics

2 Finite difference basics Numersche Methoden 1, WS 11/12 B.J.P. Kaus 2 Fnte dfference bascs Consder the one- The bascs of the fnte dfference method are best understood wth an example. dmensonal transent heat conducton equaton T

More information

Gravitational Acceleration: A case of constant acceleration (approx. 2 hr.) (6/7/11)

Gravitational Acceleration: A case of constant acceleration (approx. 2 hr.) (6/7/11) Gravtatonal Acceleraton: A case of constant acceleraton (approx. hr.) (6/7/11) Introducton The gravtatonal force s one of the fundamental forces of nature. Under the nfluence of ths force all objects havng

More information

MATH 5707 HOMEWORK 4 SOLUTIONS 2. 2 i 2p i E(X i ) + E(Xi 2 ) ä i=1. i=1

MATH 5707 HOMEWORK 4 SOLUTIONS 2. 2 i 2p i E(X i ) + E(Xi 2 ) ä i=1. i=1 MATH 5707 HOMEWORK 4 SOLUTIONS CİHAN BAHRAN 1. Let v 1,..., v n R m, all lengths v are not larger than 1. Let p 1,..., p n [0, 1] be arbtrary and set w = p 1 v 1 + + p n v n. Then there exst ε 1,..., ε

More information

On the irreducibility of a truncated binomial expansion

On the irreducibility of a truncated binomial expansion On the rreducblty of a truncated bnomal expanson by Mchael Flaseta, Angel Kumchev and Dmtr V. Pasechnk 1 Introducton For postve ntegers k and n wth k n 1, defne P n,k (x = =0 ( n x. In the case that k

More information

Digital Signal Processing

Digital Signal Processing Dgtal Sgnal Processng Dscrete-tme System Analyss Manar Mohasen Offce: F8 Emal: manar.subh@ut.ac.r School of IT Engneerng Revew of Precedent Class Contnuous Sgnal The value of the sgnal s avalable over

More information

1 Matrix representations of canonical matrices

1 Matrix representations of canonical matrices 1 Matrx representatons of canoncal matrces 2-d rotaton around the orgn: ( ) cos θ sn θ R 0 = sn θ cos θ 3-d rotaton around the x-axs: R x = 1 0 0 0 cos θ sn θ 0 sn θ cos θ 3-d rotaton around the y-axs:

More information

and problem sheet 2

and problem sheet 2 -8 and 5-5 problem sheet Solutons to the followng seven exercses and optonal bonus problem are to be submtted through gradescope by :0PM on Wednesday th September 08. There are also some practce problems,

More information

Chapter 5. Solution of System of Linear Equations. Module No. 6. Solution of Inconsistent and Ill Conditioned Systems

Chapter 5. Solution of System of Linear Equations. Module No. 6. Solution of Inconsistent and Ill Conditioned Systems Numercal Analyss by Dr. Anta Pal Assstant Professor Department of Mathematcs Natonal Insttute of Technology Durgapur Durgapur-713209 emal: anta.bue@gmal.com 1 . Chapter 5 Soluton of System of Lnear Equatons

More information

C/CS/Phy191 Problem Set 3 Solutions Out: Oct 1, 2008., where ( 00. ), so the overall state of the system is ) ( ( ( ( 00 ± 11 ), Φ ± = 1

C/CS/Phy191 Problem Set 3 Solutions Out: Oct 1, 2008., where ( 00. ), so the overall state of the system is ) ( ( ( ( 00 ± 11 ), Φ ± = 1 C/CS/Phy9 Problem Set 3 Solutons Out: Oct, 8 Suppose you have two qubts n some arbtrary entangled state ψ You apply the teleportaton protocol to each of the qubts separately What s the resultng state obtaned

More information

APPENDIX A Some Linear Algebra

APPENDIX A Some Linear Algebra APPENDIX A Some Lnear Algebra The collecton of m, n matrces A.1 Matrces a 1,1,..., a 1,n A = a m,1,..., a m,n wth real elements a,j s denoted by R m,n. If n = 1 then A s called a column vector. Smlarly,

More information

Lecture 5 Decoding Binary BCH Codes

Lecture 5 Decoding Binary BCH Codes Lecture 5 Decodng Bnary BCH Codes In ths class, we wll ntroduce dfferent methods for decodng BCH codes 51 Decodng the [15, 7, 5] 2 -BCH Code Consder the [15, 7, 5] 2 -code C we ntroduced n the last lecture

More information

One-sided finite-difference approximations suitable for use with Richardson extrapolation

One-sided finite-difference approximations suitable for use with Richardson extrapolation Journal of Computatonal Physcs 219 (2006) 13 20 Short note One-sded fnte-dfference approxmatons sutable for use wth Rchardson extrapolaton Kumar Rahul, S.N. Bhattacharyya * Department of Mechancal Engneerng,

More information

Problem Solving in Math (Math 43900) Fall 2013

Problem Solving in Math (Math 43900) Fall 2013 Problem Solvng n Math (Math 43900) Fall 2013 Week four (September 17) solutons Instructor: Davd Galvn 1. Let a and b be two nteger for whch a b s dvsble by 3. Prove that a 3 b 3 s dvsble by 9. Soluton:

More information

Problem Do any of the following determine homomorphisms from GL n (C) to GL n (C)?

Problem Do any of the following determine homomorphisms from GL n (C) to GL n (C)? Homework 8 solutons. Problem 16.1. Whch of the followng defne homomomorphsms from C\{0} to C\{0}? Answer. a) f 1 : z z Yes, f 1 s a homomorphsm. We have that z s the complex conjugate of z. If z 1,z 2

More information

Société de Calcul Mathématique SA

Société de Calcul Mathématique SA Socété de Calcul Mathématque SA Outls d'ade à la décson Tools for decson help Probablstc Studes: Normalzng the Hstograms Bernard Beauzamy December, 202 I. General constructon of the hstogram Any probablstc

More information

Module 14: THE INTEGRAL Exploring Calculus

Module 14: THE INTEGRAL Exploring Calculus Module 14: THE INTEGRAL Explorng Calculus Part I Approxmatons and the Defnte Integral It was known n the 1600s before the calculus was developed that the area of an rregularly shaped regon could be approxmated

More information

STUDY PACKAGE. Subject : Mathematics Topic : COMPLEX NUMBER Available Online :

STUDY PACKAGE. Subject : Mathematics Topic : COMPLEX NUMBER Available Online : fo/u fopkjr Hkh# tu] ugha vkjehks dke] fofr ns[k NksM+s rqjar e/;e eu dj ';kea q#"k flag lady dj] lgrs fofr vusd] ^cuk^ u NksM+s /;s; dks] j?kqcj jk[ks VsdAA jfpr% ekuo /kez.ksrk ln~xq# Jh j.knksm+nklth

More information

Difference Equations

Difference Equations Dfference Equatons c Jan Vrbk 1 Bascs Suppose a sequence of numbers, say a 0,a 1,a,a 3,... s defned by a certan general relatonshp between, say, three consecutve values of the sequence, e.g. a + +3a +1

More information

π e ax2 dx = x 2 e ax2 dx or x 3 e ax2 dx = 1 x 4 e ax2 dx = 3 π 8a 5/2 (a) We are considering the Maxwell velocity distribution function: 2πτ/m

π e ax2 dx = x 2 e ax2 dx or x 3 e ax2 dx = 1 x 4 e ax2 dx = 3 π 8a 5/2 (a) We are considering the Maxwell velocity distribution function: 2πτ/m Homework Solutons Problem In solvng ths problem, we wll need to calculate some moments of the Gaussan dstrbuton. The brute-force method s to ntegrate by parts but there s a nce trck. The followng ntegrals

More information

College of Computer & Information Science Fall 2009 Northeastern University 20 October 2009

College of Computer & Information Science Fall 2009 Northeastern University 20 October 2009 College of Computer & Informaton Scence Fall 2009 Northeastern Unversty 20 October 2009 CS7880: Algorthmc Power Tools Scrbe: Jan Wen and Laura Poplawsk Lecture Outlne: Prmal-dual schema Network Desgn:

More information

International Mathematical Olympiad. Preliminary Selection Contest 2012 Hong Kong. Outline of Solutions

International Mathematical Olympiad. Preliminary Selection Contest 2012 Hong Kong. Outline of Solutions Internatonal Mathematcal Olympad Prelmnary Selecton ontest Hong Kong Outlne of Solutons nswers: 7 4 7 4 6 5 9 6 99 7 6 6 9 5544 49 5 7 4 6765 5 6 6 7 6 944 9 Solutons: Snce n s a two-dgt number, we have

More information

Linear Approximation with Regularization and Moving Least Squares

Linear Approximation with Regularization and Moving Least Squares Lnear Approxmaton wth Regularzaton and Movng Least Squares Igor Grešovn May 007 Revson 4.6 (Revson : March 004). 5 4 3 0.5 3 3.5 4 Contents: Lnear Fttng...4. Weghted Least Squares n Functon Approxmaton...

More information

MAE140 - Linear Circuits - Winter 16 Midterm, February 5

MAE140 - Linear Circuits - Winter 16 Midterm, February 5 Instructons ME140 - Lnear Crcuts - Wnter 16 Mdterm, February 5 () Ths exam s open book. You may use whatever wrtten materals you choose, ncludng your class notes and textbook. You may use a hand calculator

More information

Min Cut, Fast Cut, Polynomial Identities

Min Cut, Fast Cut, Polynomial Identities Randomzed Algorthms, Summer 016 Mn Cut, Fast Cut, Polynomal Identtes Instructor: Thomas Kesselhem and Kurt Mehlhorn 1 Mn Cuts n Graphs Lecture (5 pages) Throughout ths secton, G = (V, E) s a mult-graph.

More information

This model contains two bonds per unit cell (one along the x-direction and the other along y). So we can rewrite the Hamiltonian as:

This model contains two bonds per unit cell (one along the x-direction and the other along y). So we can rewrite the Hamiltonian as: 1 Problem set #1 1.1. A one-band model on a square lattce Fg. 1 Consder a square lattce wth only nearest-neghbor hoppngs (as shown n the fgure above): H t, j a a j (1.1) where,j stands for nearest neghbors

More information

Bezier curves. Michael S. Floater. August 25, These notes provide an introduction to Bezier curves. i=0

Bezier curves. Michael S. Floater. August 25, These notes provide an introduction to Bezier curves. i=0 Bezer curves Mchael S. Floater August 25, 211 These notes provde an ntroducton to Bezer curves. 1 Bernsten polynomals Recall that a real polynomal of a real varable x R, wth degree n, s a functon of the

More information

Norms, Condition Numbers, Eigenvalues and Eigenvectors

Norms, Condition Numbers, Eigenvalues and Eigenvectors Norms, Condton Numbers, Egenvalues and Egenvectors 1 Norms A norm s a measure of the sze of a matrx or a vector For vectors the common norms are: N a 2 = ( x 2 1/2 the Eucldean Norm (1a b 1 = =1 N x (1b

More information

Georgia Tech PHYS 6124 Mathematical Methods of Physics I

Georgia Tech PHYS 6124 Mathematical Methods of Physics I Georga Tech PHYS 624 Mathematcal Methods of Physcs I Instructor: Predrag Cvtanovć Fall semester 202 Homework Set #7 due October 30 202 == show all your work for maxmum credt == put labels ttle legends

More information

A be a probability space. A random vector

A be a probability space. A random vector Statstcs 1: Probablty Theory II 8 1 JOINT AND MARGINAL DISTRIBUTIONS In Probablty Theory I we formulate the concept of a (real) random varable and descrbe the probablstc behavor of ths random varable by

More information

Remarks on the Properties of a Quasi-Fibonacci-like Polynomial Sequence

Remarks on the Properties of a Quasi-Fibonacci-like Polynomial Sequence Remarks on the Propertes of a Quas-Fbonacc-lke Polynomal Sequence Brce Merwne LIU Brooklyn Ilan Wenschelbaum Wesleyan Unversty Abstract Consder the Quas-Fbonacc-lke Polynomal Sequence gven by F 0 = 1,

More information

CS-433: Simulation and Modeling Modeling and Probability Review

CS-433: Simulation and Modeling Modeling and Probability Review CS-433: Smulaton and Modelng Modelng and Probablty Revew Exercse 1. (Probablty of Smple Events) Exercse 1.1 The owner of a camera shop receves a shpment of fve cameras from a camera manufacturer. Unknown

More information

18.781: Solution to Practice Questions for Final Exam

18.781: Solution to Practice Questions for Final Exam 18.781: Soluton to Practce Questons for Fnal Exam 1. Fnd three solutons n postve ntegers of x 6y = 1 by frst calculatng the contnued fracton expanson of 6. Soluton: We have 1 6=[, ] 6 6+ =[, ] 1 =[,, ]=[,,

More information

3.1 Expectation of Functions of Several Random Variables. )' be a k-dimensional discrete or continuous random vector, with joint PMF p (, E X E X1 E X

3.1 Expectation of Functions of Several Random Variables. )' be a k-dimensional discrete or continuous random vector, with joint PMF p (, E X E X1 E X Statstcs 1: Probablty Theory II 37 3 EPECTATION OF SEVERAL RANDOM VARIABLES As n Probablty Theory I, the nterest n most stuatons les not on the actual dstrbuton of a random vector, but rather on a number

More information

Finding Dense Subgraphs in G(n, 1/2)

Finding Dense Subgraphs in G(n, 1/2) Fndng Dense Subgraphs n Gn, 1/ Atsh Das Sarma 1, Amt Deshpande, and Rav Kannan 1 Georga Insttute of Technology,atsh@cc.gatech.edu Mcrosoft Research-Bangalore,amtdesh,annan@mcrosoft.com Abstract. Fndng

More information

FE REVIEW OPERATIONAL AMPLIFIERS (OP-AMPS)( ) 8/25/2010

FE REVIEW OPERATIONAL AMPLIFIERS (OP-AMPS)( ) 8/25/2010 FE REVEW OPERATONAL AMPLFERS (OP-AMPS)( ) 1 The Op-amp 2 An op-amp has two nputs and one output. Note the op-amp below. The termnal labeled l wth the (-) sgn s the nvertng nput and the nput labeled wth

More information

NP-Completeness : Proofs

NP-Completeness : Proofs NP-Completeness : Proofs Proof Methods A method to show a decson problem Π NP-complete s as follows. (1) Show Π NP. (2) Choose an NP-complete problem Π. (3) Show Π Π. A method to show an optmzaton problem

More information

Structure and Drive Paul A. Jensen Copyright July 20, 2003

Structure and Drive Paul A. Jensen Copyright July 20, 2003 Structure and Drve Paul A. Jensen Copyrght July 20, 2003 A system s made up of several operatons wth flow passng between them. The structure of the system descrbes the flow paths from nputs to outputs.

More information

A First Order q-difference System for the BC 1 -Type Jackson Integral and Its Applications

A First Order q-difference System for the BC 1 -Type Jackson Integral and Its Applications Symmetry Integrablty and Geometry: Methods and Applcatons SIGMA 5 2009 041 14 pages A Frst Order -Dfference System for the BC 1 -Type Jackson Integral and Its Applcatons Masahko ITO Department of Physcs

More information

Math 426: Probability MWF 1pm, Gasson 310 Homework 4 Selected Solutions

Math 426: Probability MWF 1pm, Gasson 310 Homework 4 Selected Solutions Exercses from Ross, 3, : Math 26: Probablty MWF pm, Gasson 30 Homework Selected Solutons 3, p. 05 Problems 76, 86 3, p. 06 Theoretcal exercses 3, 6, p. 63 Problems 5, 0, 20, p. 69 Theoretcal exercses 2,

More information

Introduction to Vapor/Liquid Equilibrium, part 2. Raoult s Law:

Introduction to Vapor/Liquid Equilibrium, part 2. Raoult s Law: CE304, Sprng 2004 Lecture 4 Introducton to Vapor/Lqud Equlbrum, part 2 Raoult s Law: The smplest model that allows us do VLE calculatons s obtaned when we assume that the vapor phase s an deal gas, and

More information

Exercises of Chapter 2

Exercises of Chapter 2 Exercses of Chapter Chuang-Cheh Ln Department of Computer Scence and Informaton Engneerng, Natonal Chung Cheng Unversty, Mng-Hsung, Chay 61, Tawan. Exercse.6. Suppose that we ndependently roll two standard

More information

Polynomials. 1 More properties of polynomials

Polynomials. 1 More properties of polynomials Polynomals 1 More propertes of polynomals Recall that, for R a commutatve rng wth unty (as wth all rngs n ths course unless otherwse noted), we defne R[x] to be the set of expressons n =0 a x, where a

More information

Solutions to the 71st William Lowell Putnam Mathematical Competition Saturday, December 4, 2010

Solutions to the 71st William Lowell Putnam Mathematical Competition Saturday, December 4, 2010 Solutons to the 7st Wllam Lowell Putnam Mathematcal Competton Saturday, December 4, 2 Kran Kedlaya and Lenny Ng A The largest such k s n+ 2 n 2. For n even, ths value s acheved by the partton {,n},{2,n

More information

Linear, affine, and convex sets and hulls In the sequel, unless otherwise specified, X will denote a real vector space.

Linear, affine, and convex sets and hulls In the sequel, unless otherwise specified, X will denote a real vector space. Lnear, affne, and convex sets and hulls In the sequel, unless otherwse specfed, X wll denote a real vector space. Lnes and segments. Gven two ponts x, y X, we defne xy = {x + t(y x) : t R} = {(1 t)x +

More information

Statistics II Final Exam 26/6/18

Statistics II Final Exam 26/6/18 Statstcs II Fnal Exam 26/6/18 Academc Year 2017/18 Solutons Exam duraton: 2 h 30 mn 1. (3 ponts) A town hall s conductng a study to determne the amount of leftover food produced by the restaurants n the

More information

SUMS PROBLEM COMPETITION, 2001

SUMS PROBLEM COMPETITION, 2001 SUMS PROBLEM COMPETITION, 200 SOLUTIONS Suppose that after n vsts to Aunt Joylene (and therefore also n vsts to Uncle Bruce Lnda has t n ten cent peces and d n dollar cons After a vst to Uncle Bruce she

More information

Ph 219a/CS 219a. Exercises Due: Wednesday 23 October 2013

Ph 219a/CS 219a. Exercises Due: Wednesday 23 October 2013 1 Ph 219a/CS 219a Exercses Due: Wednesday 23 October 2013 1.1 How far apart are two quantum states? Consder two quantum states descrbed by densty operators ρ and ρ n an N-dmensonal Hlbert space, and consder

More information

10-701/ Machine Learning, Fall 2005 Homework 3

10-701/ Machine Learning, Fall 2005 Homework 3 10-701/15-781 Machne Learnng, Fall 2005 Homework 3 Out: 10/20/05 Due: begnnng of the class 11/01/05 Instructons Contact questons-10701@autonlaborg for queston Problem 1 Regresson and Cross-valdaton [40

More information

The Geometry of Logit and Probit

The Geometry of Logit and Probit The Geometry of Logt and Probt Ths short note s meant as a supplement to Chapters and 3 of Spatal Models of Parlamentary Votng and the notaton and reference to fgures n the text below s to those two chapters.

More information

Analytical Chemistry Calibration Curve Handout

Analytical Chemistry Calibration Curve Handout I. Quck-and Drty Excel Tutoral Analytcal Chemstry Calbraton Curve Handout For those of you wth lttle experence wth Excel, I ve provded some key technques that should help you use the program both for problem

More information

Probability Theory. The nth coefficient of the Taylor series of f(k), expanded around k = 0, gives the nth moment of x as ( ik) n n!

Probability Theory. The nth coefficient of the Taylor series of f(k), expanded around k = 0, gives the nth moment of x as ( ik) n n! 8333: Statstcal Mechancs I Problem Set # 3 Solutons Fall 3 Characterstc Functons: Probablty Theory The characterstc functon s defned by fk ep k = ep kpd The nth coeffcent of the Taylor seres of fk epanded

More information

2013 ΜΑΘ National Convention

2013 ΜΑΘ National Convention Gemini Mu Test #911 1. In the Name blank, print the names of both members of the Gemini Team; in the Subject blank, print the name of the test; in the Date blank, print your school name (no abbreviations).

More information

DUE: WEDS FEB 21ST 2018

DUE: WEDS FEB 21ST 2018 HOMEWORK # 1: FINITE DIFFERENCES IN ONE DIMENSION DUE: WEDS FEB 21ST 2018 1. Theory Beam bendng s a classcal engneerng analyss. The tradtonal soluton technque makes smplfyng assumptons such as a constant

More information

Modeling curves. Graphs: y = ax+b, y = sin(x) Implicit ax + by + c = 0, x 2 +y 2 =r 2 Parametric:

Modeling curves. Graphs: y = ax+b, y = sin(x) Implicit ax + by + c = 0, x 2 +y 2 =r 2 Parametric: Modelng curves Types of Curves Graphs: y = ax+b, y = sn(x) Implct ax + by + c = 0, x 2 +y 2 =r 2 Parametrc: x = ax + bxt x = cos t y = ay + byt y = snt Parametrc are the most common mplct are also used,

More information

9. Complex Numbers. 1. Numbers revisited. 2. Imaginary number i: General form of complex numbers. 3. Manipulation of complex numbers

9. Complex Numbers. 1. Numbers revisited. 2. Imaginary number i: General form of complex numbers. 3. Manipulation of complex numbers 9. Comple Numbers. Numbers revsted. Imagnar number : General form of comple numbers 3. Manpulaton of comple numbers 4. The Argand dagram 5. The polar form for comple numbers 9.. Numbers revsted We saw

More information

Report on Image warping

Report on Image warping Report on Image warpng Xuan Ne, Dec. 20, 2004 Ths document summarzed the algorthms of our mage warpng soluton for further study, and there s a detaled descrpton about the mplementaton of these algorthms.

More information