Solution to HW 3, Ma 1a Fall 2016
|
|
- Roy Norris
- 6 years ago
- Views:
Transcription
1 Solution to HW 3, Ma a Fall 206 Section 2. Execise 2: Let C be a subset of the eal numbes consisting of those eal numbes x having the popety that evey digit in the decimal expansion of x is, 3, 5, o 7. Let {c n } be a sequence of elements of C so that c j < fo evey natual numbe j. Show that thee is a subsequence of {c n } which conveges to an element of C. Solution: Take C to be defined as above. We will begin ou poof with a shot lemma. Lemma: Given any x R \ C, thee exists an ɛ > 0 such that fo evey y (x ɛ, x + ɛ), one has that y / C as well. poof We suppose without loss of geneality that x (0, ) \ C, fo if not, we may apply essentially the same poof stating with the highest (nondecimal) place and coecting fo sign. Now, in paticula, we may wite that x =.x x 2 x 3... = x n 0, whee x n n lies between 0 and 9, inclusive, fo any n N. Given that x / C, we have that thee must exist a decimal place such that the value at that decimal place does not lie within {, 3, 5, 7}. Now, by well odeing, thee must be a minimal such decimal place, which we take to be the k-th decimal place. In othe wods, one has that c k / {, 3, 5, 7}, while fo any l such that 0 < l k, one has c l {, 3, 5, 7}. Now, conside ɛ =, 0 k+2 and select y =.y y 2 y 3... = y n 0 n (x 0 k+2, x + 0 k+2 ). We may suppose without loss of geneality that y < x, fo if y = x, we have automatically that y / C, and if y > x, then the poof woks moe o less identically with the subtaction in evese. Then we note that 0 < x y = x y = 0 n < 0 k+2. In paticula, we note that we must have fo all l such that 0 < l k that y l = x l, o y =.x x 2...x k y k y k+..., fo othewise we would have a contadiction, as the distance between them would be geate than. Thus, we in fact 0 k+2 have that x y = n=k 0 n < 0 k+2.
2 Now, futhemoe, if we have that y k = x k, then we have automatically that y / C, so suppose y k x k. Then, in fact, one must have that x k y k =, given that x > y and given that, if the diffeence between the x k and y k wee any lage, we would have x y > 0 k+2 which would be a contadiction. Thus, x y = 0 k + n=k+ 0 n < 0 k+2, which may only be the case if x k+ y k+ = 9. Given that x k+ and y k+ lie between 0 and 9, inclusive, this may only be the case if x k+ = 0 and y k+ = 9. Thus, as y k+ / {, 3, 5, 7}, we have that y / C. Indeed, we have that fo any y (x, x + ), one has y / C. 0 k+2 0 k+2 Now,taking C and {c n } as above, we note that by ou supposition, c n (, ) [, ] fo evey n N, so by the Bolzano-Weiestass theoem, thee exists a convegent subsequence of {c n }. Take {c nk } to be one such convegent subsequence, conveging to a limit c [.]. Now, suppose fo contadiction that c / C. By the lemma we poved above (estated slightly), thee must exist an ɛ > 0 such that fo evey d R such that c d < ɛ, one must have that d / C. Howeve, we note that since lim k c nk = c, thee exists a K N such that fo all k > K, one has that c c nk < ɛ, which immediately implies that fo evey k > K that c nk / C, which is a contadiction. Thus, one must have that c C. In fact, we have poven that, not only must thee exist a convegent subsequence of {c n }, but also that fo EVERY such convegent subsequence of {c n }, its limit must also lie in C. Section 2.2 Execise 2: Pove eithe that the following seies conveges o diveges: n n. Solution: Consideing the seies n n =, we note that = n n. fom hee, we apply the atio test. We see that + = (n + ) n+ /(n + )! n n = (n + ) n+ (n + )206 / (n + )!n 206 = n (n + )n 206. Now, we know that (n + ) 206 = 206 ( 206 ) k=0 k n k is a polynomial in n of degee 206, wheeas (n + )n 206 = n n 206 is a polynomial in n of degee 207. Thus, as the degee of the polynomial in the denominato is geate than that of the polynomial in the numeato, we immediately see that lim + = lim 206 (n + ) 206 = 206 lim n n (n + )n206 2 n (n + ) 206 = 0. (n + )n206
3 Finally, as lim n an+ = 0 <, one has by the atio test that n n = conveges (absolutely). Section 2.2 Execise 6: A 30 yea fixed ate motgage is a loan taken out ove a peiod of 360 months. The initial loan amount is M. Each month, the boowe pays a fixed payment p. We define a function f(j) whee j is the numbe of months that have passed. We let f(0) = M, and we let f(j) = ( + )f(j ) p, fo j 360, whee is the fixed monthly inteest ate. Futhe, we equie that f(360) = 0. Deive and pove a fomula fo p in tems of M and in closed fom. Hint: You ll have to use the fomula fo the sum of a finite geometic seies. It helps to eaange things so that you e setting equal the motgage amount M with inteest on it compounded ove 30 yeas and the steam of monthly payments each compounded fom the moment it is made. Solution: Recalling ou definition of f(j), we have that f(0) = M and f(j) = ( + )f(j ) p fo any j 0, whee M is ou initial loan amount, is ou fixed monthly inteest ate, j is the numbe of months that has passed, and p is ou fixed payment amount. Fo any j N, we may continue out the expansion fom befoe as f(j) = ( + )f(j ) p = ( + )(( + )f(j 2) p) p = ( + ) 2 f(j 2) [( + ) + ]p. = ( + ) 2 (( + )f(j 3) p) [( + ) + ]p = ( + ) 3 f(j 3) [( + ) 2 + ( + ) + ]p j = ( + ) j f(0) [( + ) j + ( + ) j ( + ) + ]p = ( + ) j M ( ( + ) k )p Thus, invoking the fomula fo a finite geometic seies, we see that, in fact f(j) = ( + ) j M ( + )j p. Now, one has that f(360) = 0, so we have that ( + ) 360 M ( + )360 p = 0. Reaanging, we get that ( + ) 360 p = ( + ) 360 M, so dividing out fo the tem in font of p, we get ou desied expession: p = ( + )360 M ( + ) 360, 3 k=0
4 o, to simplify it a little, p = M ( + ) 360. Section 2.3 Execise : Show that the seies n n x n diveges fo all x such that x > 0. Solution: Letting S(x) = nn x n, one clealy has that S(0) = 0. Now, let x be such that x > 0. The easiest way to demonstate that nn x n diveges is actually to go back to the pevious section and invoke the ation test. Note that we have, in this case, = n n x n. Thus, we have that + = (n + )n+ x n+ n n x n = (n + )n+ n n x. Now, noting that n n < (n + ) n fo any n N, one has that, in fact + (n + )n+ = n n x = (n + )n n n (n + ) x > (n + ) x. Thus, using the Infinite Squeeze Theoem, one has that since and fo all n N, one has that lim (n + ) x = x lim (n + ) =, n n in fact, one must have that + > (n + ) x, lim + =. n Thus, invoking the atio test, we see that, since the ation diveges, and consequently lim n an+ >, one has that n n x n diveges fo all x such that x > 0. Section 2.3 Execise 3: Let { } be a sequence satisfying = fo n > 2 with a = and a 2 = 2. Following Example 3, find a ational function epesenting the powe seies x n.
5 What is the adius of convegence of this seies? Justify you answe. Hint: The sequence is just the sum of two geometic sequences. Solution: Conside the function x n, defined only fomally at the moment. Using the defining elations among the, we obtain that ( 2x 3x 2 ) ( 2x 3x 2 ) x n = x n 2 x n+ 3 x n+2 n=2 *eindexing* = x n 2 x n 3 2 x n *pulling out leading tems* = a x + a 2 x 2 + x n 2a x 2 2 x n 3 2 x n *plugging in a and a 2 values* = x + 2x 2 2x 2 + *noting the sum is zeo* = x. ( )x n Thus, dividing though by 2x 3x 2, we see that x 2x 3x 2 = x ( 3x)( + x), and, afte applying the patial factions method, that ( 3x) ( + x). We may now apply the fomula fo the sum of a geometic seies to see that ( 3x) ( + x) = ( 3x + x ) = ( 3 n x n ( ) n x n ) 3 n ( ) n = x n 3 n ( ) n = x n, which implies that = 3n ( ) n, so is the sum of two geometic sequences, both of whose esulting seies convege if and only if x = x < and 3x = 3 x <, so one must have x < 3. Thus, the adius of convegence of the powe seies in question is R = 3. 5
Method for Approximating Irrational Numbers
Method fo Appoximating Iational Numbes Eic Reichwein Depatment of Physics Univesity of Califonia, Santa Cuz June 6, 0 Abstact I will put foth an algoithm fo poducing inceasingly accuate ational appoximations
More informationA proof of the binomial theorem
A poof of the binomial theoem If n is a natual numbe, let n! denote the poduct of the numbes,2,3,,n. So! =, 2! = 2 = 2, 3! = 2 3 = 6, 4! = 2 3 4 = 24 and so on. We also let 0! =. If n is a non-negative
More informationAuchmuty High School Mathematics Department Advanced Higher Notes Teacher Version
The Binomial Theoem Factoials Auchmuty High School Mathematics Depatment The calculations,, 6 etc. often appea in mathematics. They ae called factoials and have been given the notation n!. e.g. 6! 6!!!!!
More informationΣk=1. g r 3/2 z. 2 3-z. g 3 ( 3/2 ) g r 2. = 1 r = 0. () z = ( a ) + Σ. c n () a = ( a) 3-z -a. 3-z. z - + Σ. z 3, 5, 7, z ! = !
09 Maclauin Seies of Completed Riemann Zeta 9. Maclauin Seies of Lemma 9.. ( Maclauin seies of gamma function ) When is the gamma function, n is the polygamma function and B n,kf, f, ae Bell polynomials,
More informationMath 301: The Erdős-Stone-Simonovitz Theorem and Extremal Numbers for Bipartite Graphs
Math 30: The Edős-Stone-Simonovitz Theoem and Extemal Numbes fo Bipatite Gaphs May Radcliffe The Edős-Stone-Simonovitz Theoem Recall, in class we poved Tuán s Gaph Theoem, namely Theoem Tuán s Theoem Let
More informationMarkscheme May 2017 Calculus Higher level Paper 3
M7/5/MATHL/HP3/ENG/TZ0/SE/M Makscheme May 07 Calculus Highe level Pape 3 pages M7/5/MATHL/HP3/ENG/TZ0/SE/M This makscheme is the popety of the Intenational Baccalaueate and must not be epoduced o distibuted
More informationk. s k=1 Part of the significance of the Riemann zeta-function stems from Theorem 9.2. If s > 1 then 1 p s
9 Pimes in aithmetic ogession Definition 9 The Riemann zeta-function ζs) is the function which assigns to a eal numbe s > the convegent seies k s k Pat of the significance of the Riemann zeta-function
More informationOn a quantity that is analogous to potential and a theorem that relates to it
Su une quantité analogue au potential et su un théoème y elatif C R Acad Sci 7 (87) 34-39 On a quantity that is analogous to potential and a theoem that elates to it By R CLAUSIUS Tanslated by D H Delphenich
More informationPhysics 2A Chapter 10 - Moment of Inertia Fall 2018
Physics Chapte 0 - oment of netia Fall 08 The moment of inetia of a otating object is a measue of its otational inetia in the same way that the mass of an object is a measue of its inetia fo linea motion.
More information(n 1)n(n + 1)(n + 2) + 1 = (n 1)(n + 2)n(n + 1) + 1 = ( (n 2 + n 1) 1 )( (n 2 + n 1) + 1 ) + 1 = (n 2 + n 1) 2.
Paabola Volume 5, Issue (017) Solutions 151 1540 Q151 Take any fou consecutive whole numbes, multiply them togethe and add 1. Make a conjectue and pove it! The esulting numbe can, fo instance, be expessed
More information6 PROBABILITY GENERATING FUNCTIONS
6 PROBABILITY GENERATING FUNCTIONS Cetain deivations pesented in this couse have been somewhat heavy on algeba. Fo example, detemining the expectation of the Binomial distibution (page 5.1 tuned out to
More informationChapter Eight Notes N P U1C8S4-6
Chapte Eight Notes N P UC8S-6 Name Peiod Section 8.: Tigonometic Identities An identit is, b definition, an equation that is alwas tue thoughout its domain. B tue thoughout its domain, that is to sa that
More informationAlgebra. Substitution in algebra. 3 Find the value of the following expressions if u = 4, k = 7 and t = 9.
lgeba Substitution in algeba Remembe... In an algebaic expession, lettes ae used as substitutes fo numbes. Example Find the value of the following expessions if s =. a) s + + = = s + + = = Example Find
More informationLecture 28: Convergence of Random Variables and Related Theorems
EE50: Pobability Foundations fo Electical Enginees July-Novembe 205 Lectue 28: Convegence of Random Vaiables and Related Theoems Lectue:. Kishna Jagannathan Scibe: Gopal, Sudhasan, Ajay, Swamy, Kolla An
More informationPROBLEM SET #1 SOLUTIONS by Robert A. DiStasio Jr.
POBLM S # SOLUIONS by obet A. DiStasio J. Q. he Bon-Oppenheime appoximation is the standad way of appoximating the gound state of a molecula system. Wite down the conditions that detemine the tonic and
More informationChapter 3: Theory of Modular Arithmetic 38
Chapte 3: Theoy of Modula Aithmetic 38 Section D Chinese Remainde Theoem By the end of this section you will be able to pove the Chinese Remainde Theoem apply this theoem to solve simultaneous linea conguences
More information16 Modeling a Language by a Markov Process
K. Pommeening, Language Statistics 80 16 Modeling a Language by a Makov Pocess Fo deiving theoetical esults a common model of language is the intepetation of texts as esults of Makov pocesses. This model
More informationworking pages for Paul Richards class notes; do not copy or circulate without permission from PGR 2004/11/3 10:50
woking pages fo Paul Richads class notes; do not copy o ciculate without pemission fom PGR 2004/11/3 10:50 CHAPTER7 Solid angle, 3D integals, Gauss s Theoem, and a Delta Function We define the solid angle,
More informationWhen two numbers are written as the product of their prime factors, they are in factored form.
10 1 Study Guide Pages 420 425 Factos Because 3 4 12, we say that 3 and 4 ae factos of 12. In othe wods, factos ae the numbes you multiply to get a poduct. Since 2 6 12, 2 and 6 ae also factos of 12. The
More informationNumerical approximation to ζ(2n+1)
Illinois Wesleyan Univesity Fom the SelectedWoks of Tian-Xiao He 6 Numeical appoximation to ζ(n+1) Tian-Xiao He, Illinois Wesleyan Univesity Michael J. Dancs Available at: https://woks.bepess.com/tian_xiao_he/6/
More informationDivisibility. c = bf = (ae)f = a(ef) EXAMPLE: Since 7 56 and , the Theorem above tells us that
Divisibility DEFINITION: If a and b ae integes with a 0, we say that a divides b if thee is an intege c such that b = ac. If a divides b, we also say that a is a diviso o facto of b. NOTATION: d n means
More informationCHAPTER 3. Section 1. Modeling Population Growth
CHAPTER 3 Section 1. Modeling Population Gowth 1.1. The equation of the Malthusian model is Pt) = Ce t. Apply the initial condition P) = 1. Then 1 = Ce,oC = 1. Next apply the condition P1) = 3. Then 3
More informationON INDEPENDENT SETS IN PURELY ATOMIC PROBABILITY SPACES WITH GEOMETRIC DISTRIBUTION. 1. Introduction. 1 r r. r k for every set E A, E \ {0},
ON INDEPENDENT SETS IN PURELY ATOMIC PROBABILITY SPACES WITH GEOMETRIC DISTRIBUTION E. J. IONASCU and A. A. STANCU Abstact. We ae inteested in constucting concete independent events in puely atomic pobability
More informationof the contestants play as Falco, and 1 6
JHMT 05 Algeba Test Solutions 4 Febuay 05. In a Supe Smash Bothes tounament, of the contestants play as Fox, 3 of the contestants play as Falco, and 6 of the contestants play as Peach. Given that thee
More informationarxiv: v1 [math.nt] 12 May 2017
SEQUENCES OF CONSECUTIVE HAPPY NUMBERS IN NEGATIVE BASES HELEN G. GRUNDMAN AND PAMELA E. HARRIS axiv:1705.04648v1 [math.nt] 12 May 2017 ABSTRACT. Fo b 2 and e 2, let S e,b : Z Z 0 be the function taking
More informationJournal of Inequalities in Pure and Applied Mathematics
Jounal of Inequalities in Pue and Applied Mathematics COEFFICIENT INEQUALITY FOR A FUNCTION WHOSE DERIVATIVE HAS A POSITIVE REAL PART S. ABRAMOVICH, M. KLARIČIĆ BAKULA AND S. BANIĆ Depatment of Mathematics
More informationA Bijective Approach to the Permutational Power of a Priority Queue
A Bijective Appoach to the Pemutational Powe of a Pioity Queue Ia M. Gessel Kuang-Yeh Wang Depatment of Mathematics Bandeis Univesity Waltham, MA 02254-9110 Abstact A pioity queue tansfoms an input pemutation
More information9.1 The multiplicative group of a finite field. Theorem 9.1. The multiplicative group F of a finite field is cyclic.
Chapte 9 Pimitive Roots 9.1 The multiplicative goup of a finite fld Theoem 9.1. The multiplicative goup F of a finite fld is cyclic. Remak: In paticula, if p is a pime then (Z/p) is cyclic. In fact, this
More information3.1 Random variables
3 Chapte III Random Vaiables 3 Random vaiables A sample space S may be difficult to descibe if the elements of S ae not numbes discuss how we can use a ule by which an element s of S may be associated
More informationMotithang Higher Secondary School Thimphu Thromde Mid Term Examination 2016 Subject: Mathematics Full Marks: 100
Motithang Highe Seconday School Thimphu Thomde Mid Tem Examination 016 Subject: Mathematics Full Maks: 100 Class: IX Witing Time: 3 Hous Read the following instuctions caefully In this pape, thee ae thee
More informationA generalization of the Bernstein polynomials
A genealization of the Benstein polynomials Halil Ouç and Geoge M Phillips Mathematical Institute, Univesity of St Andews, Noth Haugh, St Andews, Fife KY16 9SS, Scotland Dedicated to Philip J Davis This
More informationHopefully Helpful Hints for Gauss s Law
Hopefully Helpful Hints fo Gauss s Law As befoe, thee ae things you need to know about Gauss s Law. In no paticula ode, they ae: a.) In the context of Gauss s Law, at a diffeential level, the electic flux
More informationQuestion 1: The dipole
Septembe, 08 Conell Univesity, Depatment of Physics PHYS 337, Advance E&M, HW #, due: 9/5/08, :5 AM Question : The dipole Conside a system as discussed in class and shown in Fig.. in Heald & Maion.. Wite
More informationF-IF Logistic Growth Model, Abstract Version
F-IF Logistic Gowth Model, Abstact Vesion Alignments to Content Standads: F-IFB4 Task An impotant example of a model often used in biology o ecology to model population gowth is called the logistic gowth
More information2.5 The Quarter-Wave Transformer
/3/5 _5 The Quate Wave Tansfome /.5 The Quate-Wave Tansfome Reading Assignment: pp. 73-76 By now you ve noticed that a quate-wave length of tansmission line ( λ 4, β π ) appeas often in micowave engineeing
More informationA Short Combinatorial Proof of Derangement Identity arxiv: v1 [math.co] 13 Nov Introduction
A Shot Combinatoial Poof of Deangement Identity axiv:1711.04537v1 [math.co] 13 Nov 2017 Ivica Matinjak Faculty of Science, Univesity of Zageb Bijenička cesta 32, HR-10000 Zageb, Coatia and Dajana Stanić
More informationA Relativistic Electron in a Coulomb Potential
A Relativistic Electon in a Coulomb Potential Alfed Whitehead Physics 518, Fall 009 The Poblem Solve the Diac Equation fo an electon in a Coulomb potential. Identify the conseved quantum numbes. Specify
More informationarxiv: v1 [math.co] 4 May 2017
On The Numbe Of Unlabeled Bipatite Gaphs Abdullah Atmaca and A Yavuz Ouç axiv:7050800v [mathco] 4 May 207 Abstact This pape solves a poblem that was stated by M A Haison in 973 [] This poblem, that has
More information< 1. max x B(0,1) f. ν ds(y) Use Poisson s formula for the ball to prove. (r x ) x y n ds(y) (x B0 (0, r)). 1 nα(n)r n 1
7 On the othe hand, u x solves { u n in U u on U, so which implies that x G(x, y) x n ng(x, y) < n (x B(, )). Theefoe u(x) fo all x B(, ). G ν ds(y) + max g G x ν ds(y) + C( max g + max f ) f(y)g(x, y)
More informationThe Substring Search Problem
The Substing Seach Poblem One algoithm which is used in a vaiety of applications is the family of substing seach algoithms. These algoithms allow a use to detemine if, given two chaacte stings, one is
More informationSTUDY OF SOLUTIONS OF LOGARITHMIC ORDER TO HIGHER ORDER LINEAR DIFFERENTIAL-DIFFERENCE EQUATIONS WITH COEFFICIENTS HAVING THE SAME LOGARITHMIC ORDER
UNIVERSITATIS IAGELLONICAE ACTA MATHEMATICA doi: 104467/20843828AM170027078 542017, 15 32 STUDY OF SOLUTIONS OF LOGARITHMIC ORDER TO HIGHER ORDER LINEAR DIFFERENTIAL-DIFFERENCE EQUATIONS WITH COEFFICIENTS
More informationLecture 8 - Gauss s Law
Lectue 8 - Gauss s Law A Puzzle... Example Calculate the potential enegy, pe ion, fo an infinite 1D ionic cystal with sepaation a; that is, a ow of equally spaced chages of magnitude e and altenating sign.
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department. Problem Set 10 Solutions. r s
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Depatment Physics 8.033 Decembe 5, 003 Poblem Set 10 Solutions Poblem 1 M s y x test paticle The figue above depicts the geomety of the poblem. The position
More informationSolutions to Problem Set 8
Massachusetts Institute of Technology 6.042J/18.062J, Fall 05: Mathematics fo Compute Science Novembe 21 Pof. Albet R. Meye and Pof. Ronitt Rubinfeld evised Novembe 27, 2005, 858 minutes Solutions to Poblem
More informationBrief summary of functional analysis APPM 5440 Fall 2014 Applied Analysis
Bief summay of functional analysis APPM 5440 Fall 014 Applied Analysis Stephen Becke, stephen.becke@coloado.edu Standad theoems. When necessay, I used Royden s and Keyzsig s books as a efeence. Vesion
More informationarxiv: v1 [math.nt] 28 Oct 2017
ON th COEFFICIENT OF DIVISORS OF x n axiv:70049v [mathnt] 28 Oct 207 SAI TEJA SOMU Abstact Let,n be two natual numbes and let H(,n denote the maximal absolute value of th coefficient of divisos of x n
More informationVanishing lines in generalized Adams spectral sequences are generic
ISSN 364-0380 (on line) 465-3060 (pinted) 55 Geomety & Topology Volume 3 (999) 55 65 Published: 2 July 999 G G G G T T T G T T T G T G T GG TT G G G G GG T T T TT Vanishing lines in genealized Adams spectal
More informationSingle Particle State AB AB
LECTURE 3 Maxwell Boltzmann, Femi, and Bose Statistics Suppose we have a gas of N identical point paticles in a box of volume V. When we say gas, we mean that the paticles ae not inteacting with one anothe.
More informationStanford University CS259Q: Quantum Computing Handout 8 Luca Trevisan October 18, 2012
Stanfod Univesity CS59Q: Quantum Computing Handout 8 Luca Tevisan Octobe 8, 0 Lectue 8 In which we use the quantum Fouie tansfom to solve the peiod-finding poblem. The Peiod Finding Poblem Let f : {0,...,
More informationEM Boundary Value Problems
EM Bounday Value Poblems 10/ 9 11/ By Ilekta chistidi & Lee, Seung-Hyun A. Geneal Desciption : Maxwell Equations & Loentz Foce We want to find the equations of motion of chaged paticles. The way to do
More information4. Some Applications of first order linear differential
August 30, 2011 4-1 4. Some Applications of fist ode linea diffeential Equations The modeling poblem Thee ae seveal steps equied fo modeling scientific phenomena 1. Data collection (expeimentation) Given
More informationInternet Appendix for A Bayesian Approach to Real Options: The Case of Distinguishing Between Temporary and Permanent Shocks
Intenet Appendix fo A Bayesian Appoach to Real Options: The Case of Distinguishing Between Tempoay and Pemanent Shocks Steven R. Genadie Gaduate School of Business, Stanfod Univesity Andey Malenko Gaduate
More informationPhysics 161 Fall 2011 Extra Credit 2 Investigating Black Holes - Solutions The Following is Worth 50 Points!!!
Physics 161 Fall 011 Exta Cedit Investigating Black Holes - olutions The Following is Woth 50 Points!!! This exta cedit assignment will investigate vaious popeties of black holes that we didn t have time
More informationA Crash Course in (2 2) Matrices
A Cash Couse in ( ) Matices Seveal weeks woth of matix algeba in an hou (Relax, we will only stuy the simplest case, that of matices) Review topics: What is a matix (pl matices)? A matix is a ectangula
More informationAST 121S: The origin and evolution of the Universe. Introduction to Mathematical Handout 1
Please ead this fist... AST S: The oigin and evolution of the Univese Intoduction to Mathematical Handout This is an unusually long hand-out and one which uses in places mathematics that you may not be
More informationLab 10: Newton s Second Law in Rotation
Lab 10: Newton s Second Law in Rotation We can descibe the motion of objects that otate (i.e. spin on an axis, like a popelle o a doo) using the same definitions, adapted fo otational motion, that we have
More informationLecture 16 Root Systems and Root Lattices
1.745 Intoduction to Lie Algebas Novembe 1, 010 Lectue 16 Root Systems and Root Lattices Pof. Victo Kac Scibe: Michael Cossley Recall that a oot system is a pai (V, ), whee V is a finite dimensional Euclidean
More information763620SS STATISTICAL PHYSICS Solutions 2 Autumn 2012
763620SS STATISTICAL PHYSICS Solutions 2 Autumn 2012 1. Continuous Random Walk Conside a continuous one-dimensional andom walk. Let w(s i ds i be the pobability that the length of the i th displacement
More informationDetermining solar characteristics using planetary data
Detemining sola chaacteistics using planetay data Intoduction The Sun is a G-type main sequence sta at the cente of the Sola System aound which the planets, including ou Eath, obit. In this investigation
More informationProbablistically Checkable Proofs
Lectue 12 Pobablistically Checkable Poofs May 13, 2004 Lectue: Paul Beame Notes: Chis Re 12.1 Pobablisitically Checkable Poofs Oveview We know that IP = PSPACE. This means thee is an inteactive potocol
More informationQIP Course 10: Quantum Factorization Algorithm (Part 3)
QIP Couse 10: Quantum Factoization Algoithm (Pat 3 Ryutaoh Matsumoto Nagoya Univesity, Japan Send you comments to yutaoh.matsumoto@nagoya-u.jp Septembe 2018 @ Tokyo Tech. Matsumoto (Nagoya U. QIP Couse
More informationC/CS/Phys C191 Shor s order (period) finding algorithm and factoring 11/12/14 Fall 2014 Lecture 22
C/CS/Phys C9 Sho s ode (peiod) finding algoithm and factoing /2/4 Fall 204 Lectue 22 With a fast algoithm fo the uantum Fouie Tansfom in hand, it is clea that many useful applications should be possible.
More informationChapter 5 Linear Equations: Basic Theory and Practice
Chapte 5 inea Equations: Basic Theoy and actice In this chapte and the next, we ae inteested in the linea algebaic equation AX = b, (5-1) whee A is an m n matix, X is an n 1 vecto to be solved fo, and
More informationMacro Theory B. The Permanent Income Hypothesis
Maco Theoy B The Pemanent Income Hypothesis Ofe Setty The Eitan Beglas School of Economics - Tel Aviv Univesity May 15, 2015 1 1 Motivation 1.1 An econometic check We want to build an empiical model with
More information. Using our polar coordinate conversions, we could write a
504 Chapte 8 Section 8.4.5 Dot Poduct Now that we can add, sutact, and scale vectos, you might e wondeing whethe we can multiply vectos. It tuns out thee ae two diffeent ways to multiply vectos, one which
More informationDo Managers Do Good With Other People s Money? Online Appendix
Do Manages Do Good With Othe People s Money? Online Appendix Ing-Haw Cheng Haison Hong Kelly Shue Abstact This is the Online Appendix fo Cheng, Hong and Shue 2013) containing details of the model. Datmouth
More information10/04/18. P [P(x)] 1 negl(n).
Mastemath, Sping 208 Into to Lattice lgs & Cypto Lectue 0 0/04/8 Lectues: D. Dadush, L. Ducas Scibe: K. de Boe Intoduction In this lectue, we will teat two main pats. Duing the fist pat we continue the
More informationSuggested Solutions to Homework #4 Econ 511b (Part I), Spring 2004
Suggested Solutions to Homewok #4 Econ 5b (Pat I), Sping 2004. Conside a neoclassical gowth model with valued leisue. The (epesentative) consume values steams of consumption and leisue accoding to P t=0
More informationScattering in Three Dimensions
Scatteing in Thee Dimensions Scatteing expeiments ae an impotant souce of infomation about quantum systems, anging in enegy fom vey low enegy chemical eactions to the highest possible enegies at the LHC.
More informationRECTIFYING THE CIRCUMFERENCE WITH GEOGEBRA
ECTIFYING THE CICUMFEENCE WITH GEOGEBA A. Matín Dinnbie, G. Matín González and Anthony C.M. O 1 Intoducction The elation between the cicumfeence and the adius of a cicle is one of the most impotant concepts
More informationMATH 417 Homework 3 Instructor: D. Cabrera Due June 30. sin θ v x = v r cos θ v θ r. (b) Then use the Cauchy-Riemann equations in polar coordinates
MATH 417 Homewok 3 Instucto: D. Cabea Due June 30 1. Let a function f(z) = u + iv be diffeentiable at z 0. (a) Use the Chain Rule and the fomulas x = cosθ and y = to show that u x = u cosθ u θ, v x = v
More information2 x 8 2 x 2 SKILLS Determine whether the given value is a solution of the. equation. (a) x 2 (b) x 4. (a) x 2 (b) x 4 (a) x 4 (b) x 8
5 CHAPTER Fundamentals When solving equations that involve absolute values, we usually take cases. EXAMPLE An Absolute Value Equation Solve the equation 0 x 5 0 3. SOLUTION By the definition of absolute
More informationFractional Zero Forcing via Three-color Forcing Games
Factional Zeo Focing via Thee-colo Focing Games Leslie Hogben Kevin F. Palmowski David E. Robeson Michael Young May 13, 2015 Abstact An -fold analogue of the positive semidefinite zeo focing pocess that
More informationBerkeley Math Circle AIME Preparation March 5, 2013
Algeba Toolkit Rules of Thumb. Make sue that you can pove all fomulas you use. This is even bette than memoizing the fomulas. Although it is best to memoize, as well. Stive fo elegant, economical methods.
More informationReview: Electrostatics and Magnetostatics
Review: Electostatics and Magnetostatics In the static egime, electomagnetic quantities do not vay as a function of time. We have two main cases: ELECTROSTATICS The electic chages do not change postion
More informationPDF Created with deskpdf PDF Writer - Trial ::
A APPENDIX D TRIGONOMETRY Licensed to: jsamuels@bmcc.cun.edu PDF Ceated with deskpdf PDF Wite - Tial :: http://www.docudesk.com D T i g o n o m e t FIGURE a A n g l e s Angles can be measued in degees
More informationBetween any two masses, there exists a mutual attractive force.
YEAR 12 PHYSICS: GRAVITATION PAST EXAM QUESTIONS Name: QUESTION 1 (1995 EXAM) (a) State Newton s Univesal Law of Gavitation in wods Between any two masses, thee exists a mutual attactive foce. This foce
More informationSemicanonical basis generators of the cluster algebra of type A (1)
Semicanonical basis geneatos of the cluste algeba of type A (1 1 Andei Zelevinsky Depatment of Mathematics Notheasten Univesity, Boston, USA andei@neu.edu Submitted: Jul 7, 006; Accepted: Dec 3, 006; Published:
More informationNew problems in universal algebraic geometry illustrated by boolean equations
New poblems in univesal algebaic geomety illustated by boolean equations axiv:1611.00152v2 [math.ra] 25 Nov 2016 Atem N. Shevlyakov Novembe 28, 2016 Abstact We discuss new poblems in univesal algebaic
More informationf h = u, h g = v, we have u + v = f g. So, we wish
Answes to Homewok 4, Math 4111 (1) Pove that the following examples fom class ae indeed metic spaces. You only need to veify the tiangle inequality. (a) Let C be the set of continuous functions fom [0,
More informationThe Chromatic Villainy of Complete Multipartite Graphs
Rocheste Institute of Technology RIT Schola Wos Theses Thesis/Dissetation Collections 8--08 The Chomatic Villainy of Complete Multipatite Gaphs Anna Raleigh an9@it.edu Follow this and additional wos at:
More informationA GENERALIZATION OF A CONJECTURE OF MELHAM. 1. Introduction The Fibonomial coefficient is, for n m 1, defined by
A GENERALIZATION OF A CONJECTURE OF MELHAM EMRAH KILIC 1, ILKER AKKUS, AND HELMUT PRODINGER 3 Abstact A genealization of one of Melha s conectues is pesented Afte witing it in tes of Gaussian binoial coefficients,
More informationTANTON S TAKE ON CONTINUOUS COMPOUND INTEREST
CURRICULUM ISPIRATIOS: www.maa.og/ci www.theglobalmathpoject.og IOVATIVE CURRICULUM OLIE EXPERIECES: www.gdaymath.com TATO TIDBITS: www.jamestanton.com TATO S TAKE O COTIUOUS COMPOUD ITEREST DECEMBER 208
More informationDouble-angle & power-reduction identities. Elementary Functions. Double-angle & power-reduction identities. Double-angle & power-reduction identities
Double-angle & powe-eduction identities Pat 5, Tigonomety Lectue 5a, Double Angle and Powe Reduction Fomulas In the pevious pesentation we developed fomulas fo cos( β) and sin( β) These fomulas lead natually
More informationQuasi-Randomness and the Distribution of Copies of a Fixed Graph
Quasi-Randomness and the Distibution of Copies of a Fixed Gaph Asaf Shapia Abstact We show that if a gaph G has the popety that all subsets of vetices of size n/4 contain the coect numbe of tiangles one
More informationA Multivariate Normal Law for Turing s Formulae
A Multivaiate Nomal Law fo Tuing s Fomulae Zhiyi Zhang Depatment of Mathematics and Statistics Univesity of Noth Caolina at Chalotte Chalotte, NC 28223 Abstact This pape establishes a sufficient condition
More informationON LACUNARY INVARIANT SEQUENCE SPACES DEFINED BY A SEQUENCE OF MODULUS FUNCTIONS
STUDIA UNIV BABEŞ BOLYAI, MATHEMATICA, Volume XLVIII, Numbe 4, Decembe 2003 ON LACUNARY INVARIANT SEQUENCE SPACES DEFINED BY A SEQUENCE OF MODULUS FUNCTIONS VATAN KARAKAYA AND NECIP SIMSEK Abstact The
More informationI. CONSTRUCTION OF THE GREEN S FUNCTION
I. CONSTRUCTION OF THE GREEN S FUNCTION The Helmohltz equation in 4 dimensions is 4 + k G 4 x, x = δ 4 x x. In this equation, G is the Geen s function and 4 efes to the dimensionality. In the vey end,
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department Physics 8.07: Electromagnetism II September 15, 2012 Prof. Alan Guth PROBLEM SET 2
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Depatment Physics 8.07: Electomagnetism II Septembe 5, 202 Pof. Alan Guth PROBLEM SET 2 DUE DATE: Monday, Septembe 24, 202. Eithe hand it in at the lectue,
More informationCircular Orbits. and g =
using analyse planetay and satellite motion modelled as unifom cicula motion in a univesal gavitation field, a = v = 4π and g = T GM1 GM and F = 1M SATELLITES IN OBIT A satellite is any object that is
More informationON SPARSELY SCHEMMEL TOTIENT NUMBERS. Colin Defant 1 Department of Mathematics, University of Florida, Gainesville, Florida
#A8 INTEGERS 5 (205) ON SPARSEL SCHEMMEL TOTIENT NUMBERS Colin Defant Depatment of Mathematics, Univesity of Floida, Gainesville, Floida cdefant@ufl.edu Received: 7/30/4, Revised: 2/23/4, Accepted: 4/26/5,
More informationUniversal Gravitation
Chapte 1 Univesal Gavitation Pactice Poblem Solutions Student Textbook page 580 1. Conceptualize the Poblem - The law of univesal gavitation applies to this poblem. The gavitational foce, F g, between
More informationCALCULUS II Vectors. Paul Dawkins
CALCULUS II Vectos Paul Dawkins Table of Contents Peface... ii Vectos... 3 Intoduction... 3 Vectos The Basics... 4 Vecto Aithmetic... 8 Dot Poduct... 13 Coss Poduct... 21 2007 Paul Dawkins i http://tutoial.math.lama.edu/tems.aspx
More informationCHAPTER 5 : SERIES. 5.2 The Sum of a Series Sum of Power of n Positive Integers Sum of Series of Partial Fraction Difference Method
CHAPTER 5 : SERIES 5.1 Seies 5. The Sum of a Seies 5..1 Sum of Powe of Positive Iteges 5.. Sum of Seies of Patial Factio 5..3 Diffeece Method 5.3 Test of covegece 5.3.1 Divegece Test 5.3. Itegal Test 5.3.3
More informationSo, if we are finding the amount of work done over a non-conservative vector field F r, we do that long ur r b ur =
3.4 Geen s Theoem Geoge Geen: self-taught English scientist, 793-84 So, if we ae finding the amount of wok done ove a non-consevative vecto field F, we do that long u b u 3. method Wok = F d F( () t )
More information4/18/2005. Statistical Learning Theory
Statistical Leaning Theoy Statistical Leaning Theoy A model of supevised leaning consists of: a Envionment - Supplying a vecto x with a fixed but unknown pdf F x (x b Teache. It povides a desied esponse
More informationSyntactical content of nite approximations of partial algebras 1 Wiktor Bartol Inst. Matematyki, Uniw. Warszawski, Warszawa (Poland)
Syntactical content of nite appoximations of patial algebas 1 Wikto Batol Inst. Matematyki, Uniw. Waszawski, 02-097 Waszawa (Poland) batol@mimuw.edu.pl Xavie Caicedo Dep. Matematicas, Univ. de los Andes,
More informationOn the ratio of maximum and minimum degree in maximal intersecting families
On the atio of maximum and minimum degee in maximal intesecting families Zoltán Lóánt Nagy Lale Özkahya Balázs Patkós Máté Vize Mach 6, 013 Abstact To study how balanced o unbalanced a maximal intesecting
More information556: MATHEMATICAL STATISTICS I
556: MATHEMATICAL STATISTICS I CHAPTER 5: STOCHASTIC CONVERGENCE The following efinitions ae state in tems of scala anom vaiables, but exten natually to vecto anom vaiables efine on the same obability
More informationIntroduction Common Divisors. Discrete Mathematics Andrei Bulatov
Intoduction Common Divisos Discete Mathematics Andei Bulatov Discete Mathematics Common Divisos 3- Pevious Lectue Integes Division, popeties of divisibility The division algoithm Repesentation of numbes
More information