2015 Kansas MAA Undergraduate Mathematics Competition: Solutions! 1! + 2! + 3! ! 1! + 2! + 3! ! mod 10 = 1! + 2! + 3! + 4! mod 10 = 3.
|
|
- Maximilian Geoffrey Bryant
- 7 years ago
- Views:
Transcription
1 215 Kasas MAA Udergraduate Mathematcs Competto: Solutos! 1. Show that the equato x x + 216y 3 = 3 has o teger solutos x, y Z. Soluto: Gve ay tegers x, y Z, t s clear that 216y 3 wll be a eve teger. Further, f x s eve the x x s the sum of two eve tegers, ad hece eve, whle f x s odd the x x s the sum of two odd umbers, ad hece also eve. I ay case, the sum x x + 216y 3 wll be eve for ay x, y Z, ad hece the gve equato ca have o teger solutos. 2. Fd the frst dgt (the oes dgt the sum 1! + 2! + 3! ! Soluto: Notce that the oes dgt the above sum s gve by the sum mod 1. Sce! = mod 1 for all 5, t follows that 1! + 2! + 3! ! mod 1 = 1! + 2! + 3! + 4! mod 1 = 3. Thus, the oes dgt of the above sum s Determe whether there exsts a fte sequece (a of postve real umbers such that the seres a 1 + a a coverges. =1 Soluto: Sce all of the a are postve, t follows that, for each N, a 1 + a a a 1 ad hece the seres dverges by the comparso test. 4. Let f(x be a strctly postve cotuous fucto. Evaluate the tegral f(x f(x + f(4 x dx. Soluto: Lettg I deote the value of the above tegral, we have I = f(x + f(4 x f(4 x f(x + f(4 x dx = 4 f(4 x f(x + f(4 x dx.
2 Mag the substtuto y = 4 x we fd f(4 x f(x + f(4 x dx = f(y 4 f(4 y + f(y dy = I so that, combg wth the prevous detty, we fd I = 4 I I = Suppose that f, g : N N, f s oto, g s oe-to-oe, ad f( g( for all N. Prove that f( = g( for all N. Soluto 1: Sce f s oto, there exsts a 1 N such that f( 1 = 1. Sce g( 1 f( 1 ad g( 1 N, t follows that g( 1 = 1 = f( 1 Furthermore, sce g s oe-to-oe, t follows that 1 s the uque atural umber wth ths property. Smlarly, sce f s oto there exsts a 2 N such that f( 2 = 2. Sce g( 2 f( 2 t follows that g( 2 {1, 2}. Usg that g s oeto-oe, t follows that g( 2 1 (sce 1 2 ad hece that g( 2 = 2. Thus, g( 2 = f( 2 = 2 ad, aga by the fact that g s oe-to-oe, 2 s the uque atural umber wth ths property. Cotug by ducto, t follows that, for each N there exsts a uque N such that f( = g( =. It ow remas to show that { } =1 = N, but ths s clear sce f m N \ { } =1 the g(m N ad hece, by above, there exsts a uque N such that g( = f( = g(m. Sce g s oe-to-oe, t follows that m = ad hece that { } =1 = N, as clamed. Together the, t follows that f( = g( for every N. Soluto 2: By cotradcto. Assume there exsts m N wth g(m < f(m. Sce f s oto, there exsts m 1 N wth f(m 1 = g(m. Sce g s oe-to-oe, ad g(m = f(m 1, g(m 1 < f(m 1 = g(m. Now assume that there exsts postve tegers m, m 1,... m wth g(m < g(m 1 < < g(m. Sce f s oto, there exsts m +1 N wth f(m +1 = g(m. Sce g s oe-to-oe, g(m +1 < f(m +1 = g(m. By ducto, we obta a sequece of postve tegers {m } wth the property that {g(m } s a strctly decreasg sequece. Sce {g(m } N, ths s a cotradcto. 6. For each N, show that ( = = ( 3 + = 7.
3 Soluto: Frst, rewrte the sum as = ( 3 = ( 3. By the bomal theorem, we ow ( = 3 = (1 + 3 = 4 ad hece, usg the bomal theorem aga, the gve sum s equal to as clamed. = ( 3 4 = (4 + 3 = 7, 7. Let f : [, 1] R be a tegrable fucto (ot ecessarly cotuous. Prove that f f(t 2 for all t [, 1] the there exsts a uque soluto x [, 1] of the equato 3x 1 = x f(tdt. Soluto: Defe the fucto g : [, 1] R by g(x = 3x 1 x f(tdt. Sce f s tegrable, t follows by the fudametal theorem of calculus that g s cotuous o [, 1]. Now, otce that g( = 1 ad that g(1 = 2 f(tdt 2 2 dt =. If g(1 =, the x = 1 s a soluto to the equato. If g(1 >, by the termedate value theorem, we have that there exsts some c (, 1 such that g(c =. To see that the soluto s uque, we clam that g s a strctly creasg fucto o [, 1]. Ideed, gve ay a, b [, 1] wth a < b we have g(b g(a = 3(b a b a f(tdt 3(b a b a 2 dt = b a >. Thus, g s strctly creasg o [, 1] ad hece c s uque, as clamed. 8. Suppose f : [, R s a cotuous, o-egatve fucto. Suppose that f(x + 1 = (1/2f(x for all x, ad tegral f(x dx exsts ad fd ts value. f(x dx = 1. Show that the
4 Soluto: Frst, we clam that the gve mproper tegral coverges. Ideed, ote that sce f s o-egatve t follows that the fucto g : R [, gve by g(t = t f(xdx s a o-decreasg fucto ad, furthermore, gve ay t 1 we have mples g(t t = t f(xdt = = f(x + dx t = (1/2 f(xdx 1 (1/2 = 2. Thus, g s a mootoe o-decreasg fucto that s bouded above. It follows that lm t g(t exsts, whch proves the gve mproper tegral coverges, as clamed. Now, to evaluate the tegral, t suffces to calculate the lmt lm t g(t. Sce we ow the lmt exst, we ca tae the lmt alog the atural umbers. To ths ed, for each N otce that g( = f(x + dx = (1/2 f(xdx = 1 (1/2 ad hece that It follows that = = = lm g( = 1 (1/2 = 2. = f(xdx = Determe, wth proof, all polyomals satsfyg satsfyg P ( = ad P (x = (P (x for all x. Soluto: We clam the oly polyomal satsfyg the gve propertes s P (x = x. To see ths, frst otce that the codto P ( = mples so that P (1 = 1. Smlarly, we fd P ( = P ( = 1. P ( = P ( = P (( = P ( = ( =
5 By ducto, t follows that f we defe the recursve sequece a 1 = 1, a +1 = a for all N the P (a = a for all N. Furthermore, the sequece {a } =1 s strctly creasg sce for all N we have a +1 a = a a = (a 1/ /4 >. It follows that the fucto G(x = P (x x s a polyomal wth ftely may dstct real roots. Sce a o-trval polyomal of degree m ca have at most m real roots by the fudametal theorem of algebra, t follows that G(x = for all x R,.e. P (x = x for all x R, as clamed. 1. Suppose that you have a 2 2 grd wth a sgle 1 1 square removed. Prove that the remag squares ca be tled wth L-shaped tles cosstg of three 1 1 tles - that s, a 2 2 tle wth a sgle 1 1 square removed. Soluto: We prove ths by ducto o. Clearly, ths s possble f = 1, sce a 2 2 grd wth exactly oe square removed s precsely the shape of a gve L-shaped tle. Now, suppose that, for some gve N t s ow that that ths s possble for ay gve 2 2 grd wth exactly oe square removed ca be covered by such L-shaped tles. Cosder the a grd ad otce ths ca be decomposed to a sub-grds a uque way. Furthermore, we ca cosder the mddle 2 2 sub-grd of the grd made up of exactly oe square from each of the four 2 2 sub-grds. Sce there s exactly oe square mssg from the gve grd, t follows that exactly oe of these 2 2 sub-grds has exactly oe square mssg (by hypothess ad hece ca be tled by such L-shaped tles by the ducto hypothess. After coverg ths 2 2 subgrd, t follows that the ceter 2 2 grd has exactly oe square mssg, ad hece ca be tled wth exactly oe L-shaped tle. The remag sub-grds ow each have exactly oe square covered, ad hece ca each be tled by the gve L-shaped tles by the ducto hypothess. Thus, the gve grd ca be tled by such L-shaped tles. The proof s thus complete by mathematcal ducto.
Sequences and Series
Secto 9. Sequeces d Seres You c thk of sequece s fucto whose dom s the set of postve tegers. f ( ), f (), f (),... f ( ),... Defto of Sequece A fte sequece s fucto whose dom s the set of postve tegers.
More informationANOVA Notes Page 1. Analysis of Variance for a One-Way Classification of Data
ANOVA Notes Page Aalss of Varace for a Oe-Wa Classfcato of Data Cosder a sgle factor or treatmet doe at levels (e, there are,, 3, dfferet varatos o the prescrbed treatmet) Wth a gve treatmet level there
More informationPreprocess a planar map S. Given a query point p, report the face of S containing p. Goal: O(n)-size data structure that enables O(log n) query time.
Computatoal Geometry Chapter 6 Pot Locato 1 Problem Defto Preprocess a plaar map S. Gve a query pot p, report the face of S cotag p. S Goal: O()-sze data structure that eables O(log ) query tme. C p E
More informationSTATISTICAL PROPERTIES OF LEAST SQUARES ESTIMATORS. x, where. = y - ˆ " 1
STATISTICAL PROPERTIES OF LEAST SQUARES ESTIMATORS Recall Assumpto E(Y x) η 0 + η x (lear codtoal mea fucto) Data (x, y ), (x 2, y 2 ),, (x, y ) Least squares estmator ˆ E (Y x) ˆ " 0 + ˆ " x, where ˆ
More informationRelaxation Methods for Iterative Solution to Linear Systems of Equations
Relaxato Methods for Iteratve Soluto to Lear Systems of Equatos Gerald Recktewald Portlad State Uversty Mechacal Egeerg Departmet gerry@me.pdx.edu Prmary Topcs Basc Cocepts Statoary Methods a.k.a. Relaxato
More informationOnline Appendix: Measured Aggregate Gains from International Trade
Ole Appedx: Measured Aggregate Gas from Iteratoal Trade Arel Burste UCLA ad NBER Javer Cravo Uversty of Mchga March 3, 2014 I ths ole appedx we derve addtoal results dscussed the paper. I the frst secto,
More informationOn formula to compute primes and the n th prime
Joural's Ttle, Vol., 00, o., - O formula to compute prmes ad the th prme Issam Kaddoura Lebaese Iteratoal Uversty Faculty of Arts ad ceces, Lebao Emal: ssam.addoura@lu.edu.lb amh Abdul-Nab Lebaese Iteratoal
More information1. The Time Value of Money
Corporate Face [00-0345]. The Tme Value of Moey. Compoudg ad Dscoutg Captalzato (compoudg, fdg future values) s a process of movg a value forward tme. It yelds the future value gve the relevat compoudg
More informationOur 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
More informationOn Savings Accounts in Semimartingale Term Structure Models
O Savgs Accouts Semmartgale Term Structure Models Frak Döberle Mart Schwezer moeyshelf.com Techsche Uverstät Berl Bockehemer Ladstraße 55 Fachberech Mathematk, MA 7 4 D 6325 Frakfurt am Ma Straße des 17.
More informationCurve Fitting and Solution of Equation
UNIT V Curve Fttg ad Soluto of Equato 5. CURVE FITTING I ma braches of appled mathematcs ad egeerg sceces we come across epermets ad problems, whch volve two varables. For eample, t s kow that the speed
More informationAverage Price Ratios
Average Prce Ratos Morgstar Methodology Paper August 3, 2005 2005 Morgstar, Ic. All rghts reserved. The formato ths documet s the property of Morgstar, Ic. Reproducto or trascrpto by ay meas, whole or
More informationChapter 3 0.06 = 3000 ( 1.015 ( 1 ) Present Value of an Annuity. Section 4 Present Value of an Annuity; Amortization
Chapter 3 Mathematcs of Face Secto 4 Preset Value of a Auty; Amortzato Preset Value of a Auty I ths secto, we wll address the problem of determg the amout that should be deposted to a accout ow at a gve
More informationAPPENDIX III THE ENVELOPE PROPERTY
Apped III APPENDIX III THE ENVELOPE PROPERTY Optmzato mposes a very strog structure o the problem cosdered Ths s the reaso why eoclasscal ecoomcs whch assumes optmzg behavour has bee the most successful
More informationAbraham Zaks. Technion I.I.T. Haifa ISRAEL. and. University of Haifa, Haifa ISRAEL. Abstract
Preset Value of Autes Uder Radom Rates of Iterest By Abraham Zas Techo I.I.T. Hafa ISRAEL ad Uversty of Hafa, Hafa ISRAEL Abstract Some attempts were made to evaluate the future value (FV) of the expected
More informationInfinite Sequences and Series
CHAPTER 4 Ifiite Sequeces ad Series 4.1. Sequeces A sequece is a ifiite ordered list of umbers, for example the sequece of odd positive itegers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29...
More informationSTOCHASTIC approximation algorithms have several
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL 60, NO 10, OCTOBER 2014 6609 Trackg a Markov-Modulated Statoary Degree Dstrbuto of a Dyamc Radom Graph Mazyar Hamd, Vkram Krshamurthy, Fellow, IEEE, ad George
More informationNumerical Methods with MS Excel
TMME, vol4, o.1, p.84 Numercal Methods wth MS Excel M. El-Gebely & B. Yushau 1 Departmet of Mathematcal Sceces Kg Fahd Uversty of Petroleum & Merals. Dhahra, Saud Araba. Abstract: I ths ote we show how
More informationAn Effectiveness of Integrated Portfolio in Bancassurance
A Effectveess of Itegrated Portfolo Bacassurace Taea Karya Research Ceter for Facal Egeerg Isttute of Ecoomc Research Kyoto versty Sayouu Kyoto 606-850 Japa arya@eryoto-uacp Itroducto As s well ow the
More informationClassic Problems at a Glance using the TVM Solver
C H A P T E R 2 Classc Problems at a Glace usg the TVM Solver The table below llustrates the most commo types of classc face problems. The formulas are gve for each calculato. A bref troducto to usg the
More informationOn Error Detection with Block Codes
BULGARIAN ACADEMY OF SCIENCES CYBERNETICS AND INFORMATION TECHNOLOGIES Volume 9, No 3 Sofa 2009 O Error Detecto wth Block Codes Rostza Doduekova Chalmers Uversty of Techology ad the Uversty of Gotheburg,
More informationTwo Fundamental Theorems about the Definite Integral
Two Fundamental Theorems about the Definite Integral These lecture notes develop the theorem Stewart calls The Fundamental Theorem of Calculus in section 5.3. The approach I use is slightly different than
More informationHow To Value An Annuity
Future Value of a Auty After payg all your blls, you have $200 left each payday (at the ed of each moth) that you wll put to savgs order to save up a dow paymet for a house. If you vest ths moey at 5%
More informationLecture 7. Norms and Condition Numbers
Lecture 7 Norms ad Codto Numbers To dscuss the errors umerca probems vovg vectors, t s usefu to empo orms. Vector Norm O a vector space V, a orm s a fucto from V to the set of o-egatve reas that obes three
More information16. Mean Square Estimation
6 Me Sque stmto Gve some fomto tht s elted to uow qutty of teest the poblem s to obt good estmte fo the uow tems of the obseved dt Suppose epeset sequece of dom vbles bout whom oe set of obsevtos e vlble
More information6.7 Network analysis. 6.7.1 Introduction. References - Network analysis. Topological analysis
6.7 Network aalyss Le data that explctly store topologcal formato are called etwork data. Besdes spatal operatos, several methods of spatal aalyss are applcable to etwork data. Fgure: Network data Refereces
More informationIn nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008
I ite Sequeces Dr. Philippe B. Laval Keesaw State Uiversity October 9, 2008 Abstract This had out is a itroductio to i ite sequeces. mai de itios ad presets some elemetary results. It gives the I ite Sequeces
More informationof the relationship between time and the value of money.
TIME AND THE VALUE OF MONEY Most agrbusess maagers are famlar wth the terms compoudg, dscoutg, auty, ad captalzato. That s, most agrbusess maagers have a tutve uderstadg that each term mples some relatoshp
More informationThe Gompertz-Makeham distribution. Fredrik Norström. Supervisor: Yuri Belyaev
The Gompertz-Makeham dstrbuto by Fredrk Norström Master s thess Mathematcal Statstcs, Umeå Uversty, 997 Supervsor: Yur Belyaev Abstract Ths work s about the Gompertz-Makeham dstrbuto. The dstrbuto has
More informationSection 11.3: The Integral Test
Sectio.3: The Itegral Test Most of the series we have looked at have either diverged or have coverged ad we have bee able to fid what they coverge to. I geeral however, the problem is much more difficult
More informationOn Cheeger-type inequalities for weighted graphs
O Cheeger-type equaltes for weghted graphs Shmuel Fredlad Uversty of Illos at Chcago Departmet of Mathematcs 851 S. Morga St., Chcago, Illos 60607-7045 USA Rehard Nabbe Fakultät für Mathematk Uverstät
More informationStatistical Pattern Recognition (CE-725) Department of Computer Engineering Sharif University of Technology
I The Name of God, The Compassoate, The ercful Name: Problems' eys Studet ID#:. Statstcal Patter Recogto (CE-725) Departmet of Computer Egeerg Sharf Uversty of Techology Fal Exam Soluto - Sprg 202 (50
More informationLecture 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
More information4.3. The Integral and Comparison Tests
4.3. THE INTEGRAL AND COMPARISON TESTS 9 4.3. The Itegral ad Compariso Tests 4.3.. The Itegral Test. Suppose f is a cotiuous, positive, decreasig fuctio o [, ), ad let a = f(). The the covergece or divergece
More informationFinite Dimensional Vector Spaces.
Lctur 5. Ft Dmsoal Vctor Spacs. To b rad to th musc of th group Spac by D.Maruay DEFINITION OF A LINEAR SPACE Dfto: a vctor spac s a st R togthr wth a oprato calld vctor addto ad aothr oprato calld scalar
More informationIDENTIFICATION OF THE DYNAMICS OF THE GOOGLE S RANKING ALGORITHM. A. Khaki Sedigh, Mehdi Roudaki
IDENIFICAION OF HE DYNAMICS OF HE GOOGLE S RANKING ALGORIHM A. Khak Sedgh, Mehd Roudak Cotrol Dvso, Departmet of Electrcal Egeerg, K.N.oos Uversty of echology P. O. Box: 16315-1355, ehra, Ira sedgh@eetd.ktu.ac.r,
More informationThe simple linear Regression Model
The smple lear Regresso Model Correlato coeffcet s o-parametrc ad just dcates that two varables are assocated wth oe aother, but t does ot gve a deas of the kd of relatoshp. Regresso models help vestgatg
More informationCH. V ME256 STATICS Center of Gravity, Centroid, and Moment of Inertia CENTER OF GRAVITY AND CENTROID
CH. ME56 STTICS Ceter of Gravt, Cetrod, ad Momet of Ierta CENTE OF GITY ND CENTOID 5. CENTE OF GITY ND CENTE OF MSS FO SYSTEM OF PTICES Ceter of Gravt. The ceter of gravt G s a pot whch locates the resultat
More informationPlastic Number: Construction and Applications
Scet f c 0 Advaced Advaced Scetfc 0 December,.. 0 Plastc Number: Costructo ad Applcatos Lua Marohć Polytechc of Zagreb, 0000 Zagreb, Croata lua.marohc@tvz.hr Thaa Strmeč Polytechc of Zagreb, 0000 Zagreb,
More informationON SLANT HELICES AND GENERAL HELICES IN EUCLIDEAN n -SPACE. Yusuf YAYLI 1, Evren ZIPLAR 2. yayli@science.ankara.edu.tr. evrenziplar@yahoo.
ON SLANT HELICES AND ENERAL HELICES IN EUCLIDEAN -SPACE Yusuf YAYLI Evre ZIPLAR Departmet of Mathematcs Faculty of Scece Uversty of Akara Tadoğa Akara Turkey yayl@sceceakaraedutr Departmet of Mathematcs
More informationThe Digital Signature Scheme MQQ-SIG
The Dgtal Sgature Scheme MQQ-SIG Itellectual Property Statemet ad Techcal Descrpto Frst publshed: 10 October 2010, Last update: 20 December 2010 Dalo Glgorosk 1 ad Rue Stesmo Ødegård 2 ad Rue Erled Jese
More informationn + d + q = 24 and.05n +.1d +.25q = 2 { n + d + q = 24 (3) n + 2d + 5q = 40 (2)
MATH 16T Exam 1 : Part I (In-Class) Solutons 1. (0 pts) A pggy bank contans 4 cons, all of whch are nckels (5 ), dmes (10 ) or quarters (5 ). The pggy bank also contans a con of each denomnaton. The total
More informationThe analysis of annuities relies on the formula for geometric sums: r k = rn+1 1 r 1. (2.1) k=0
Chapter 2 Autes ad loas A auty s a sequece of paymets wth fxed frequecy. The term auty orgally referred to aual paymets (hece the ame), but t s ow also used for paymets wth ay frequecy. Autes appear may
More informationSecurity Analysis of RAPP: An RFID Authentication Protocol based on Permutation
Securty Aalyss of RAPP: A RFID Authetcato Protocol based o Permutato Wag Shao-hu,,, Ha Zhje,, Lu Sujua,, Che Da-we, {College of Computer, Najg Uversty of Posts ad Telecommucatos, Najg 004, Cha Jagsu Hgh
More informationWe are now ready to answer the question: What are the possible cardinalities for finite fields?
Chapter 3 Fnte felds We have seen, n the prevous chapters, some examples of fnte felds. For example, the resdue class rng Z/pZ (when p s a prme) forms a feld wth p elements whch may be dentfed wth the
More informationChapter Eight. f : R R
Chapter Eght f : R R 8. Itroducto We shall ow tur our atteto to the very mportat specal case of fuctos that are real, or scalar, valued. These are sometmes called scalar felds. I the very, but mportat,
More informationThe Mean Value Theorem
The Mean Value Theorem THEOREM (The Extreme Value Theorem): If f is continuous on a closed interval [a, b], then f attains an absolute maximum value f(c) and an absolute minimum value f(d) at some numbers
More informationMDM 4U PRACTICE EXAMINATION
MDM 4U RCTICE EXMINTION Ths s a ractce eam. It does ot cover all the materal ths course ad should ot be the oly revew that you do rearato for your fal eam. Your eam may cota questos that do ot aear o ths
More informationBayesian Network Representation
Readgs: K&F 3., 3.2, 3.3, 3.4. Bayesa Network Represetato Lecture 2 Mar 30, 20 CSE 55, Statstcal Methods, Sprg 20 Istructor: Su-I Lee Uversty of Washgto, Seattle Last tme & today Last tme Probablty theory
More informationarxiv:math/0510414v1 [math.pr] 19 Oct 2005
A MODEL FOR THE BUS SYSTEM IN CUERNEVACA MEXICO) JINHO BAIK ALEXEI BORODIN PERCY DEIFT AND TOUFIC SUIDAN arxv:math/05044v [mathpr 9 Oct 2005 Itroducto The bus trasportato system Cuerevaca Mexco has certa
More informationCHAPTER 2. Time Value of Money 6-1
CHAPTER 2 Tme Value of Moey 6- Tme Value of Moey (TVM) Tme Les Future value & Preset value Rates of retur Autes & Perpetutes Ueve cash Flow Streams Amortzato 6-2 Tme les 0 2 3 % CF 0 CF CF 2 CF 3 Show
More informationFIBONACCI NUMBERS: AN APPLICATION OF LINEAR ALGEBRA. 1. Powers of a matrix
FIBONACCI NUMBERS: AN APPLICATION OF LINEAR ALGEBRA. Powers of a matrix We begi with a propositio which illustrates the usefuless of the diagoalizatio. Recall that a square matrix A is diogaalizable if
More informationChapter 3. AMORTIZATION OF LOAN. SINKING FUNDS R =
Chapter 3. AMORTIZATION OF LOAN. SINKING FUNDS Objectves of the Topc: Beg able to formalse ad solve practcal ad mathematcal problems, whch the subjects of loa amortsato ad maagemet of cumulatve fuds are
More informationDIRAC s BRA AND KET NOTATION. 1 From inner products to bra-kets 1
DIRAC s BRA AND KET NOTATION B. Zwebach October 7, 2013 Cotets 1 From er products to bra-kets 1 2 Operators revsted 5 2.1 Projecto Operators..................................... 6 2.2 Adjot of a lear operator.................................
More information1. MATHEMATICAL INDUCTION
1. MATHEMATICAL INDUCTION EXAMPLE 1: Prove that for ay iteger 1. Proof: 1 + 2 + 3 +... + ( + 1 2 (1.1 STEP 1: For 1 (1.1 is true, sice 1 1(1 + 1. 2 STEP 2: Suppose (1.1 is true for some k 1, that is 1
More informationFundamentals of Mass Transfer
Chapter Fudametals of Mass Trasfer Whe a sgle phase system cotas two or more speces whose cocetratos are ot uform, mass s trasferred to mmze the cocetrato dffereces wth the system. I a mult-phase system
More informationSimple Linear Regression
Smple Lear Regresso Regresso equato a equato that descrbes the average relatoshp betwee a respose (depedet) ad a eplaator (depedet) varable. 6 8 Slope-tercept equato for a le m b (,6) slope. (,) 6 6 8
More informationSAMPLE QUESTIONS FOR FINAL EXAM. (1) (2) (3) (4) Find the following using the definition of the Riemann integral: (2x + 1)dx
SAMPLE QUESTIONS FOR FINAL EXAM REAL ANALYSIS I FALL 006 3 4 Fid the followig usig the defiitio of the Riema itegral: a 0 x + dx 3 Cosider the partitio P x 0 3, x 3 +, x 3 +,......, x 3 3 + 3 of the iterval
More informationÉditeur Inria, Domaine de Voluceau, Rocquencourt, BP 105 LE CHESNAY Cedex (France) ISSN 0249-6399
Uté de recherche INRIA Lorrae, techopôle de Nacy-Brabos, 615 rue du jard botaque, BP 101, 54600 VILLERS-LÈS-NANCY Uté de recherche INRIA Rees, IRISA, Campus uverstare de Beauleu, 35042 RENNES Cedex Uté
More informationProving the Computer Science Theory P = NP? With the General Term of the Riemann Zeta Function
Research Joural of Mahemacs ad Sascs 3(2): 72-76, 20 ISSN: 2040-7505 Maxwell Scefc Orgazao, 20 Receved: Jauary 08, 20 Acceped: February 03, 20 Publshed: May 25, 20 Provg he ompuer Scece Theory P NP? Wh
More informationCommon p-belief: The General Case
GAMES AND ECONOMIC BEHAVIOR 8, 738 997 ARTICLE NO. GA97053 Commo p-belef: The Geeral Case Atsush Kaj* ad Stephe Morrs Departmet of Ecoomcs, Uersty of Pesylaa Receved February, 995 We develop belef operators
More informationOverview of some probability distributions.
Lecture Overview of some probability distributios. I this lecture we will review several commo distributios that will be used ofte throughtout the class. Each distributio is usually described by its probability
More informationT = 1/freq, T = 2/freq, T = i/freq, T = n (number of cash flows = freq n) are :
Bullets bods Let s descrbe frst a fxed rate bod wthout amortzg a more geeral way : Let s ote : C the aual fxed rate t s a percetage N the otoal freq ( 2 4 ) the umber of coupo per year R the redempto of
More informationADAPTATION OF SHAPIRO-WILK TEST TO THE CASE OF KNOWN MEAN
Colloquum Bometrcum 4 ADAPTATION OF SHAPIRO-WILK TEST TO THE CASE OF KNOWN MEAN Zofa Hausz, Joaa Tarasńska Departmet of Appled Mathematcs ad Computer Scece Uversty of Lfe Sceces Lubl Akademcka 3, -95 Lubl
More informationFractal-Structured Karatsuba`s Algorithm for Binary Field Multiplication: FK
Fractal-Structured Karatsuba`s Algorthm for Bary Feld Multplcato: FK *The authors are worg at the Isttute of Mathematcs The Academy of Sceces of DPR Korea. **Address : U Jog dstrct Kwahadog Number Pyogyag
More informationCredibility Premium Calculation in Motor Third-Party Liability Insurance
Advaces Mathematcal ad Computatoal Methods Credblty remum Calculato Motor Thrd-arty Lablty Isurace BOHA LIA, JAA KUBAOVÁ epartmet of Mathematcs ad Quattatve Methods Uversty of ardubce Studetská 95, 53
More informationStatistical Decision Theory: Concepts, Methods and Applications. (Special topics in Probabilistic Graphical Models)
Statstcal Decso Theory: Cocepts, Methods ad Applcatos (Specal topcs Probablstc Graphcal Models) FIRST COMPLETE DRAFT November 30, 003 Supervsor: Professor J. Rosethal STA4000Y Aal Mazumder 9506380 Part
More informationOptimal multi-degree reduction of Bézier curves with constraints of endpoints continuity
Computer Aded Geometrc Desg 19 (2002 365 377 wwwelsevercom/locate/comad Optmal mult-degree reducto of Bézer curves wth costrats of edpots cotuty Guo-Dog Che, Guo-J Wag State Key Laboratory of CAD&CG, Isttute
More information2 When is a 2-Digit Number the Sum of the Squares of its Digits?
When Does a Number Equal the Sum of the Squares or Cubes of its Digits? An Exposition and a Call for a More elegant Proof 1 Introduction We will look at theorems of the following form: by William Gasarch
More informationPERRON FROBENIUS THEOREM
PERRON FROBENIUS THEOREM R. CLARK ROBINSON Defnton. A n n matrx M wth real entres m, s called a stochastc matrx provded () all the entres m satsfy 0 m, () each of the columns sum to one, m = for all, ()
More informationLoad Balancing Control for Parallel Systems
Proc IEEE Med Symposum o New drectos Cotrol ad Automato, Chaa (Grèce),994, pp66-73 Load Balacg Cotrol for Parallel Systems Jea-Claude Heet LAAS-CNRS, 7 aveue du Coloel Roche, 3077 Toulouse, Frace E-mal
More informationECONOMIC CHOICE OF OPTIMUM FEEDER CABLE CONSIDERING RISK ANALYSIS. University of Brasilia (UnB) and The Brazilian Regulatory Agency (ANEEL), Brazil
ECONOMIC CHOICE OF OPTIMUM FEEDER CABE CONSIDERING RISK ANAYSIS I Camargo, F Fgueredo, M De Olvera Uversty of Brasla (UB) ad The Brazla Regulatory Agecy (ANEE), Brazl The choce of the approprate cable
More information8.5 UNITARY AND HERMITIAN MATRICES. The conjugate transpose of a complex matrix A, denoted by A*, is given by
6 CHAPTER 8 COMPLEX VECTOR SPACES 5. Fnd the kernel of the lnear transformaton gven n Exercse 5. In Exercses 55 and 56, fnd the mage of v, for the ndcated composton, where and are gven by the followng
More informationLocally Adaptive Dimensionality Reduction for Indexing Large Time Series Databases
Locally Adaptve Dmesoalty educto for Idexg Large Tme Seres Databases Kaushk Chakrabart Eamo Keogh Sharad Mehrotra Mchael Pazza Mcrosoft esearch Uv. of Calfora Uv. of Calfora Uv. of Calfora edmod, WA 985
More informationSpeeding up k-means Clustering by Bootstrap Averaging
Speedg up -meas Clusterg by Bootstrap Averagg Ia Davdso ad Ashw Satyaarayaa Computer Scece Dept, SUNY Albay, NY, USA,. {davdso, ashw}@cs.albay.edu Abstract K-meas clusterg s oe of the most popular clusterg
More information1 Example 1: Axis-aligned rectangles
COS 511: Theoretcal Machne Learnng Lecturer: Rob Schapre Lecture # 6 Scrbe: Aaron Schld February 21, 2013 Last class, we dscussed an analogue for Occam s Razor for nfnte hypothess spaces that, n conjuncton
More informationConstrained Cubic Spline Interpolation for Chemical Engineering Applications
Costraed Cubc Sple Iterpolato or Chemcal Egeerg Applcatos b CJC Kruger Summar Cubc sple terpolato s a useul techque to terpolate betwee kow data pots due to ts stable ad smooth characterstcs. Uortuatel
More informationSHAPIRO-WILK TEST FOR NORMALITY WITH KNOWN MEAN
SHAPIRO-WILK TEST FOR NORMALITY WITH KNOWN MEAN Wojcech Zelńsk Departmet of Ecoometrcs ad Statstcs Warsaw Uversty of Lfe Sceces Nowoursyowska 66, -787 Warszawa e-mal: wojtekzelsk@statystykafo Zofa Hausz,
More informationMathematics of Finance
CATE Mathematcs of ace.. TODUCTO ths chapter we wll dscuss mathematcal methods ad formulae whch are helpful busess ad persoal face. Oe of the fudametal cocepts the mathematcs of face s the tme value of
More informationCS103X: Discrete Structures Homework 4 Solutions
CS103X: Discrete Structures Homewor 4 Solutios Due February 22, 2008 Exercise 1 10 poits. Silico Valley questios: a How may possible six-figure salaries i whole dollar amouts are there that cotai at least
More informationThree Dimensional Interpolation of Video Signals
Three Dmesoal Iterpolato of Vdeo Sgals Elham Shahfard March 0 th 006 Outle A Bref reve of prevous tals Dgtal Iterpolato Bascs Upsamplg D Flter Desg Issues Ifte Impulse Respose Fte Impulse Respose Desged
More informationMath 319 Problem Set #3 Solution 21 February 2002
Math 319 Problem Set #3 Solution 21 February 2002 1. ( 2.1, problem 15) Find integers a 1, a 2, a 3, a 4, a 5 such that every integer x satisfies at least one of the congruences x a 1 (mod 2), x a 2 (mod
More informationA Study of Unrelated Parallel-Machine Scheduling with Deteriorating Maintenance Activities to Minimize the Total Completion Time
Joural of Na Ka, Vol. 0, No., pp.5-9 (20) 5 A Study of Urelated Parallel-Mache Schedulg wth Deteroratg Mateace Actvtes to Mze the Total Copleto Te Suh-Jeq Yag, Ja-Yuar Guo, Hs-Tao Lee Departet of Idustral
More informationInnovation and Production in the Global Economy Online Appendix
Iovato ad Producto te Goba Ecoomy Oe Appedx Costas Aroas Prceto, Yae ad NBER Nataa Ramodo Arzoa State Adrés Rodríguez-Care UC Bereey ad NBER Stepe Yeape Pe State ad NBER December 204 Abstract I ts oe Appedx
More informationConversion of Non-Linear Strength Envelopes into Generalized Hoek-Brown Envelopes
Covero of No-Lear Stregth Evelope to Geeralzed Hoek-Brow Evelope Itroducto The power curve crtero commoly ued lmt-equlbrum lope tablty aaly to defe a o-lear tregth evelope (relatohp betwee hear tre, τ,
More informationRecurrence. 1 Definitions and main statements
Recurrence 1 Defntons and man statements Let X n, n = 0, 1, 2,... be a MC wth the state space S = (1, 2,...), transton probabltes p j = P {X n+1 = j X n = }, and the transton matrx P = (p j ),j S def.
More informationWe investigate a simple adaptive approach to optimizing seat protection levels in airline
Reveue Maagemet Wthout Forecastg or Optmzato: A Adaptve Algorthm for Determg Arle Seat Protecto Levels Garrett va Ryz Jeff McGll Graduate School of Busess, Columba Uversty, New York, New York 10027 School
More informationREVISTA INVESTIGACION OPERACIONAL Vol. 25, No. 1, 2004. k n ),
REVISTA INVESTIGACION OPERACIONAL Vol 25, No, 24 RECURRENCE AND DIRECT FORMULAS FOR TE AL & LA NUMBERS Eduardo Pza Volo Cero de Ivesgacó e Maemáca Pura y Aplcada (CIMPA), Uversdad de Cosa Rca ABSTRACT
More informationApproximation Algorithms for Scheduling with Rejection on Two Unrelated Parallel Machines
(ICS) Iteratoal oural of dvaced Comuter Scece ad lcatos Vol 6 No 05 romato lgorthms for Schedulg wth eecto o wo Urelated Parallel aches Feg Xahao Zhag Zega Ca College of Scece y Uversty y Shadog Cha 76005
More informationSection 8.3 : De Moivre s Theorem and Applications
The Sectio 8 : De Moivre s Theorem ad Applicatios Let z 1 ad z be complex umbers, where z 1 = r 1, z = r, arg(z 1 ) = θ 1, arg(z ) = θ z 1 = r 1 (cos θ 1 + i si θ 1 ) z = r (cos θ + i si θ ) ad z 1 z =
More informationThe Method of Partial Fractions Math 121 Calculus II Spring 2015
Rational functions. as The Method of Partial Fractions Math 11 Calculus II Spring 015 Recall that a rational function is a quotient of two polynomials such f(x) g(x) = 3x5 + x 3 + 16x x 60. The method
More informationGeneralizations of Pauli channels
Acta Math. Hugar. 24(2009, 65 77. Geeralzatos of Paul chaels Dées Petz ad Hromch Oho 2 Alfréd Réy Isttute of Mathematcs, H-364 Budapest, POB 27, Hugary 2 Graduate School of Mathematcs, Kyushu Uversty,
More informationChapter 7 - Sampling Distributions. 1 Introduction. What is statistics? It consist of three major areas:
Chapter 7 - Samplig Distributios 1 Itroductio What is statistics? It cosist of three major areas: Data Collectio: samplig plas ad experimetal desigs Descriptive Statistics: umerical ad graphical summaries
More informationOPTIMAL KNOWLEDGE FLOW ON THE INTERNET
İstabul Tcaret Üverstes Fe Blmler Dergs Yıl: 5 Sayı:0 Güz 006/ s. - OPTIMAL KNOWLEDGE FLOW ON THE INTERNET Bura ORDİN *, Urfat NURİYEV ** ABSTRACT The flow roblem ad the mmum sag tree roblem are both fudametal
More informationChapter 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
More information0.7 0.6 0.2 0 0 96 96.5 97 97.5 98 98.5 99 99.5 100 100.5 96.5 97 97.5 98 98.5 99 99.5 100 100.5
Sectio 13 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.
More informationwhere p is the centroid of the neighbors of p. Consider the eigenvector problem
Vrtual avgato of teror structures by ldar Yogja X a, Xaolg L a, Ye Dua a, Norbert Maerz b a Uversty of Mssour at Columba b Mssour Uversty of Scece ad Techology ABSTRACT I ths project, we propose to develop
More informationProperties of MLE: consistency, asymptotic normality. Fisher information.
Lecture 3 Properties of MLE: cosistecy, asymptotic ormality. Fisher iformatio. I this sectio we will try to uderstad why MLEs are good. Let us recall two facts from probability that we be used ofte throughout
More informationChapter 5: Inner Product Spaces
Chapter 5: Ier Product Spaces Chapter 5: Ier Product Spaces SECION A Itroductio to Ier Product Spaces By the ed of this sectio you will be able to uderstad what is meat by a ier product space give examples
More informationTheorems About Power Series
Physics 6A Witer 20 Theorems About Power Series Cosider a power series, f(x) = a x, () where the a are real coefficiets ad x is a real variable. There exists a real o-egative umber R, called the radius
More information