Lectures # 5 and 6: The Prime Number Theorem.


 Kristian Newman
 2 years ago
 Views:
Transcription
1 Lecures # 5 and 6: The Prime Number Theorem Noah Snyder July 8, 22 Riemann s Argumen Riemann used his analyically coninued ζfuncion o skech an argumen which would give an acual formula for π( and sugges how o prove he prime number heorem This argumen is highly unrigorous a poins, bu i is crucial o undersanding he developmen of he res of he heory Noice ha log ζ(s = p n n p ns for Re(s > Leing J( = p k k, noice ha log ζ(s = s dj( again for Re(s > Now use inegraion by pars o ge log ζ(s = s J( s d Now his is a Mellin ransform, so, assuming some echnical resuls, we should be able o use Melin inversion Thus, J( = σ+i log ζ(s s ds 2πi σ i s This converges when σ > Thus in order o find a formula for J( we need only ge a beer formula for log ζ(s Riemann claimed ha ξ(s = ξ( ( ρ s ρ, where he produc is aken over all roos of he ξ funcion (ha is, over all nonrivial zeroes of he ζfuncion This produc does no converge absoluely, and we should pair any erms wih im(ρ posiive wih a corresponding erm wih negaive imaginary par o ge a convergen produc The proof of his produc formula basically depends on geing nice bounds on he growh of he number of zeroes Now we noice ha, ζ(s = 2 s(s πs/2 Γ(s/2 ξ(s = 2 s(s πs/2 Γ(s/2 ξ( ρ ( s ρ log ζ(s = log 2 log s log(s + s 2 log π log Γ(s/2 + log ξ( + ρ ( log s ρ We wan o subsiue his ino our inegral formula and evaluae ermwise, however doing so would lead o divergen inegrals (for eample in he s 2 log π erm Thus Riemann firs inegraed by pars o ge, J( = 2πi σ+i ( d log ζ(s s ds log σ i ds s Now we can subsiue our formula for ζ(s and evaluae erm by erm Wih a good bi of work, Riemann evaluaed hese inegrals and go he formula, J( = Li( ρ Li( ρ + ( 2 d log 2 log
2 Noice ha J( = n= n π( n We can inver his formula o ge, π( = n= µ(n n J(/n This gives us a formula for π( Is dominan erm is n= µ(n n Li(/n This would show he prime number heorem if we could acually prove ha his erm was dominan The key o proving his is o show ha he ρ Li(ρ erms are each smaller, ha is o say we need o show ha Re(ρ < 2 Chebyshev s Funcions Before Riemann s work he only significan progress owards he prime number heorem was made by Chebyshev who proved ha, for sufficienly large and some consans c < < c 2, c log π( c 2 log To prove his he inroduced wo funcions which are crucial in laer proofs of prime number heory Recall ha we conjecure ha he chances ha a number n is prime is roughly log n Thus, if we couned each prime as log p insead of as, hen we would ge a beer behaved funcion Definiion 2 Le θ(n = p n log p (where, as usual, a jumps we define he funcion o be halfway in beween he wo values As we ve seen from Riemann s argumen i is ofen simpler o coun prime powers insead of primes Definiion 22 Le ψ(n = p k n log p There is anoher way of wriing ψ in erms of Von Mangold s Λ funcion Definiion 23 Le { log n if n is a prime power Λ(n = else Clearly ψ( = n Λ(n Firs we noice ha one can epress each of he funcions ψ, θ, π, and J in erms of any of he ohers Proposiion 24 J( = π( = n= n π(/n µ(n n π(/n n= ψ( = θ( = θ( /n n= µ(nψ( /n n= Proof We ve already shown he firs wo, and he proof of he second wo are eacly he same Proposiion 25 π( = θ( log + θ( (log 2 d J( = ψ( log + ψ( = J( log θ( = π( log ψ( (log 2 d J( d π( 2
3 Proof Noice ha π( = log dθ( The heorem follows from inegraion by pars Similarly J( = log dψ(, and we inegrae by pars again Conversely, θ( = log dπ( and ψ( = log dj( Inegraing hese by pars gives he second wo equaions Since θ and ψ are rivially O( log and π and J are rivially O( we can rewrie hese equaions in erms of error esimaes The long and shor of all of his is ha o prove he prime number heorem i is enough o prove any of π Li(, J( Li(, θ(, or ψ( Furhermore, given any eplici error erms in he above approimaions we can find eplici error erms for all of he oher approimaions As i urns ou ψ is he easies funcion o deal wih Proposiion 26 For any n, d n Λ(d = log n Proof Noice ha n = p n pk where k is he larges number such ha p k n Thus, n = p k np Taking he logarihm shows ha log n = d n Λ(d There is anoher way of looking a his ideniy In shorhand his proposiion claims Λ = log n Thus i is equivalen o some ideniy involving Dirichle series Noice ha n= Λ(nn s = p (log pp ms = ζ (s ζ(s m= Also log n = ζ (s f(s, Λζ(s = f(s, log n eacly as we had hoped o show n= Before leaving his las proof we noice ha one of he equaions can be rewrien 3 Chebyshev s Theorem ζ (s ζ(s = s dψ( Theorem 3 For sufficienly large and some consans c < < c 2, c ψ( c 2 Proof Chebyshev noiced ha if we sum d n Λ(d = log n over all n, hen T ( = Λ(m = m log n = log! m n By Sirling s formula Noice ha by Möbius inversion, T ( = log! = log + O(log T ( = Λ(m = Λ(m = ( ψ n m n m n m n n ψ( = µ(nt n= ( n This suggess ha finie epressions which have several erms from n= µ(nt ( n will give good approimaions o ψ Bu we also wan good cancellaions when we plug in he approimaion from Sirling s formula For eample, i would be informaive o look a epressions of he following form: ( ( T ( T T, 2 2 3
4 ( ( ( T ( T T + T, ( ( ( ( T ( T T T + T, ec We will look a he firs epression T ( 2T ( 2 Chebyshev looked a he hird epression and was able o ge consans c and c 2 closer o Noice, ( T ( 2T = ( Λ(m 2 m 2m m The lefhand side is log 2 + O(log The righhand side is ( Λ(m m Λ(m = ψ( 2m m m for large and any consan ε >, In paricular, we can ake c = 69 Similarly, he righhand side is Λ(m m ( m log 2 ε ψ( ( Λ(m = ψ( ψ 2m 2 2 m ψ( ψ ( 2 log 2 + O(log Summing hese esimaes yields, In paricular, we can ake c 2 = 38 ψ( 2 log 2 + O(log 2 By our previous resuls relaing ψ and π, we also ge ha 79 π( 38 log log 4 Reducing he Prime Number Theorem o Facs Abou ζ(s Recall ha Inegrae by pars o see ha ζ (s ζ(s = ζ (s ζ(s = s s dψ( ψ( s d Our general mehod of aack is o rewrie his as a Mellin ransform and hen use Mellin inversion o rerieve ψ( in erms of ζ(s However, o make cerain inegrals behave well laer on, we firs make a sligh change Inegraes by pars again o noice ha Definiion 4 Le φ( = ζ (s ζ(s = s2 ψ( d ( ψ( d s d 4
5 Therefore we have ζ (s ζ(s = s2 φ( s d To wrie his as a Mellin ransform we make he change of variables s s ζ ( s ζ( s ( s 2 = φ( s d In order o apply Mellin inversion we mus check o see ha he echnical condiions of ha heorem are saisfied Noice ha since ψ( = O( log we have φ( = O(log Therefore he inegrand in he Mellin ransform converges absoluely for R(s < Also, ζ (s ζ(s (log nn σ Thus, for any posiive ε, in he region Re(s + ε, he funcion ζ (s ζ(s consan he inegral σ+i ζ ( s ζ( s ( s 2 d σ i n= is bounded by an absolue converges absoluely for any σ < Therefore he condiions of Mellin inversion are saisfied and, φ( = σ+i 2πi σ i ζ ( s ζ( s ( s 2 s ds, for any Re(s < Now we can change variables back s s and muliply boh sides by o ge, Proposiion 42 For any s wih Re(s > he following inegral converges absoluely and φ( = σ+i 2πi σ i ζ (s ζ(s s 2 s d Noice ha hus far we could have gone hrough he argumen wih ψ( insead of φ( and he resuling formula would have a /s insead of /s 2 Our argumen from here on in consiss of several pars Firs we will assume ha here are no zeroes of he ζ funcion on he line Re(s = We will prove his in he ne secion Thus he only pole of he inegrand in he halfplane Re(s is s = We can subrac off his pole o ge a erm which conribues he dominan erm The remaining inegral we can move all he way o he line Re(s = Then we will ge an eplici bound on his inegral This will give us an approimaion for φ( Finally we will need o erac an esimae for ψ( from our knowledge concerning ψ( So noice ha φ( = 2πi σ+i σ i s s 2 s d 2πi σ+i σ i ( ζ (s ζ(s + s s 2 s d The firs inegral can be wrien as he limi of an inegral abou he recangle wih corners +/T ±it and T ± it The inegrals along all bu he righ side die very quickly Thus our inegral is he sum of he residues o he lef of Re(s = 2 The only poles are a s = and s = To his end epand s = e s log = + s log + s 2 log 2 + Thus he residue a s = is log A s = he residue is Therefore his inegral conribues he erm log (The noes ha I am basing his on say ha his inegral is log I canno find ou where he comes from, bu I do no rus my abiliy o do comple analysis very well, and so ha is probably righ Noneheless since we are only ineresed in approimaion he will no maer given our assumpion ha ζ( + i, we have proved: 5
6 Proposiion 43 φ( = log 2πi i ( ζ ( + i ζ( + i + i ( + i 2 ei log d In order o esimae his las inegral we will need a few esimaes on he size of ζ(s and ζ (s These will be proved in he ne secion Thus we will make he following assumpions: Proposiion 44 Leing s = σ + i as usual, we have he bound ζ ( k(s = O(log k in he region σ > log and > 2 Also we have ζ(s = O(log7 in he region σ and > 2 ( Proposiion 45 For any ineger k, φ( = + O Proof Le (log k f( = ( ζ ( + i 2πi ζ( + i + i ( + i 2 Recall ha φ( = R f(ei log Since he second erm is rapidly oscillaing, if we can ge a decen bound on f( we should ge a very good bound on φ( From our esimaes concerning ζ and is derivaives, ( log f ( k( = O ( + 2 Therefore for each k here is a consan C(k wih f ( k( d C(k Now we inegrae by pars k imes o see ha, f(e i log d = ( i log k R R f(e i log C(k R log k R f ( k(e i log Combining his wih our earlier resuls yields our required resuls Noice ha had we aemped o run hrough he above argumen wih ψ he final inegral would no have converged absoluely One would sill epec he oscillaory erm o cancel hings ou, bu proving his would be more difficul All ha remains o do (oher han he analyic resuls pu off ill ne secion is o urn his esimae for φ ino an esimae for ψ I is perhaps surprising ha one can do his, since we are essenially differeniaing an approimaion Bu since ψ behaves so nicely we can in fac do his Theorem 46 For any ineger k, ψ( = + O( log k/2 Proof Suppose he ε( is any funcion saisfying < ε( 2 Le g k( = ha for all sufficienly large and some consan C, since g k (2 g k (, Cg k ( φ( Cg k ( φ( + ε( φ( ε( + Cg k ( + ε( + Cg k ( ε( + 3Cg k ( We have proved log k 6
7 On he oher hand, since ψ is an increasing funcion, φ( + ε( φ( = Combining hese wo equaions shows ha +ε( ψ( + ε( + 3Cg k ( + ε( ε( ψ( Considering φ( φ( ε( in he same way yields φ( 2Cg k( ε( ε( d ψ( + ε( + ε( + 6Cg k( ε( Now we can choose ε( in such a way o minimize he error erm The bes such choice is ε( = c g k ( where we choose c small enough so ha we sill have ε( 2 Plugging his epression ino our previous resuls yields he heorem This is equivalen o he prime number heorem Plugging our esimae for ψ ino our previous relaions, π( = ( log + 2 log 2 d + O log k However, by inegraion by pars, Li( = log + 2 ( π( = Li( + O log k d + O( we have log 2 ( Noice ha he approimaion π( = log only holds, a priori, up o O 5 Some Facs Abou ζ(s Proposiion 5 For any real, ζ( + i log 2 Proof Throughou his proof any ime we use he symbol c i means a paricular consan which may change from equaion o equaion Recall ha log ζ(s p s + c p Reζ(s p cos log p p σ + c If s = + i were a zero of he zea funcion, hen lim σ + log ζ(σ + i = cos log p lim σ + p σ = This implies ha he vas majoriy of numbers cos log p are near nearly all he numbers log p would lie near he poins of he arihmeic progression (2n + π This is impossible because his regulariy would sugges ha cos(2 log p were nearly for he vas majoriy of primes This in urn suggess ha ζ(s has a pole a s = + 2i Now we make his argumen rigorous Suppose ζ(s had a zero a s = + i, hen ζ(s/(s i would be analyic near s = + i In paricular, aking he real par of log of ζ(s/(s i, we see ha cos( log p p σ < log(σ + c p 7
8 Le δ > be some small posiive number Le S be he sum of p σ over all primes which saisfy (2n + π log p < δ for some ineger n, and le S 2 be he sum over primes which do no saisfy his condiion For erms in he second sum cos( log p > cos δ S (cos δs 2 < log(σ + K On he oher hand, since here is a simple pole a, we have S + S 2 < log(σ + c S (cos δs 2 < S S 2 + c S 2 < c cos δ However, since +2πi is no a pole of ζ(s, he real par of log ζ(s is bounded above near s = +2i Therefore cos 2 log p p σ < c p Again we can spli his sum up over he wo ses of primes For primes of he firs ype cos(2 log p > cos 2δ > S cos 2δ S 2 < c S < c ( cos δ cos 2δ Hence, for some consan depending on δ, S + S 2 < C(δ Leing σ approach makes he lefhand side blow up which is a conradicion For a more clever bu perhaps less informaive proof ha a zero a ζ( + i would force a pole a ζ( + 2i look a he proof of his resul on one of he ne few copied pages The proofs of he following wo resuls are on he ne few phoocopied pages Proposiion 52 Leing s = σ + i as usual, we have he bound ζ ( k(s = O(log k in he region σ > log and > 2 Proposiion 53 Leing s = σ + i as usual, we have he bound ζ(s = O(log7 in he region σ and > 2 8
ANALYTIC PROOF OF THE PRIME NUMBER THEOREM
ANALYTIC PROOF OF THE PRIME NUMBER THEOREM RYAN SMITH, YUAN TIAN Conens Arihmeical Funcions Equivalen Forms of he Prime Number Theorem 3 3 The Relaionshi Beween Two Asymoic Relaions 6 4 Dirichle Series
More informationRepresenting Periodic Functions by Fourier Series. (a n cos nt + b n sin nt) n=1
Represening Periodic Funcions by Fourier Series 3. Inroducion In his Secion we show how a periodic funcion can be expressed as a series of sines and cosines. We begin by obaining some sandard inegrals
More informationThe Transport Equation
The Transpor Equaion Consider a fluid, flowing wih velociy, V, in a hin sraigh ube whose cross secion will be denoed by A. Suppose he fluid conains a conaminan whose concenraion a posiion a ime will be
More informationcooking trajectory boiling water B (t) microwave 0 2 4 6 8 101214161820 time t (mins)
Alligaor egg wih calculus We have a large alligaor egg jus ou of he fridge (1 ) which we need o hea o 9. Now here are wo accepable mehods for heaing alligaor eggs, one is o immerse hem in boiling waer
More informationDifferential Equations. Solving for Impulse Response. Linear systems are often described using differential equations.
Differenial Equaions Linear sysems are ofen described using differenial equaions. For example: d 2 y d 2 + 5dy + 6y f() d where f() is he inpu o he sysem and y() is he oupu. We know how o solve for y given
More informationMathematics in Pharmacokinetics What and Why (A second attempt to make it clearer)
Mahemaics in Pharmacokineics Wha and Why (A second aemp o make i clearer) We have used equaions for concenraion () as a funcion of ime (). We will coninue o use hese equaions since he plasma concenraions
More informationChapter 7. Response of FirstOrder RL and RC Circuits
Chaper 7. esponse of FirsOrder L and C Circuis 7.1. The Naural esponse of an L Circui 7.2. The Naural esponse of an C Circui 7.3. The ep esponse of L and C Circuis 7.4. A General oluion for ep and Naural
More informationMTH6121 Introduction to Mathematical Finance Lesson 5
26 MTH6121 Inroducion o Mahemaical Finance Lesson 5 Conens 2.3 Brownian moion wih drif........................... 27 2.4 Geomeric Brownian moion........................... 28 2.5 Convergence of random
More informationAppendix A: Area. 1 Find the radius of a circle that has circumference 12 inches.
Appendi A: Area workedou s o OddNumbered Eercises Do no read hese workedou s before aemping o do he eercises ourself. Oherwise ou ma mimic he echniques shown here wihou undersanding he ideas. Bes wa
More informationComplex Fourier Series. Adding these identities, and then dividing by 2, or subtracting them, and then dividing by 2i, will show that
Mah 344 May 4, Complex Fourier Series Par I: Inroducion The Fourier series represenaion for a funcion f of period P, f) = a + a k coskω) + b k sinkω), ω = π/p, ) can be expressed more simply using complex
More information#A81 INTEGERS 13 (2013) THE AVERAGE LARGEST PRIME FACTOR
#A8 INTEGERS 3 (03) THE AVERAGE LARGEST PRIME FACTOR Eric Naslund Dearmen of Mahemaics, Princeon Universiy, Princeon, New Jersey naslund@mahrinceonedu Received: /8/3, Revised: 7/7/3, Acceed:/5/3, Published:
More information23.3. Even and Odd Functions. Introduction. Prerequisites. Learning Outcomes
Even and Odd Funcions 3.3 Inroducion In his Secion we examine how o obain Fourier series of periodic funcions which are eiher even or odd. We show ha he Fourier series for such funcions is considerabl
More informationHANDOUT 14. A.) Introduction: Many actions in life are reversible. * Examples: Simple One: a closed door can be opened and an open door can be closed.
Inverse Funcions Reference Angles Inverse Trig Problems Trig Indeniies HANDOUT 4 INVERSE FUNCTIONS KEY POINTS A.) Inroducion: Many acions in life are reversible. * Examples: Simple One: a closed door can
More informationRandom Walk in 1D. 3 possible paths x vs n. 5 For our random walk, we assume the probabilities p,q do not depend on time (n)  stationary
Random Walk in D Random walks appear in many cones: diffusion is a random walk process undersanding buffering, waiing imes, queuing more generally he heory of sochasic processes gambling choosing he bes
More information23.3. Even and Odd Functions. Introduction. Prerequisites. Learning Outcomes
Even and Odd Funcions 23.3 Inroducion In his Secion we examine how o obain Fourier series of periodic funcions which are eiher even or odd. We show ha he Fourier series for such funcions is considerabl
More informationChapter 6. First Order PDEs. 6.1 Characteristics The Simplest Case. u(x,t) t=1 t=2. t=0. Suppose u(x, t) satisfies the PDE.
Chaper 6 Firs Order PDEs 6.1 Characerisics 6.1.1 The Simples Case Suppose u(, ) saisfies he PDE where b, c are consan. au + bu = 0 If a = 0, he PDE is rivial (i says ha u = 0 and so u = f(). If a = 0,
More information17 Laplace transform. Solving linear ODE with piecewise continuous right hand sides
7 Laplace ransform. Solving linear ODE wih piecewise coninuous righ hand sides In his lecure I will show how o apply he Laplace ransform o he ODE Ly = f wih piecewise coninuous f. Definiion. A funcion
More informationAP Calculus BC 2010 Scoring Guidelines
AP Calculus BC Scoring Guidelines The College Board The College Board is a noforprofi membership associaion whose mission is o connec sudens o college success and opporuniy. Founded in, he College Board
More informationAP Calculus AB 2007 Scoring Guidelines
AP Calculus AB 7 Scoring Guidelines The College Board: Connecing Sudens o College Success The College Board is a noforprofi membership associaion whose mission is o connec sudens o college success and
More informationFACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING. Lectures
FACULY OF MAHEMAICAL SUDIES MAHEMAICS FOR PAR I ENGINEERING Lecures MODULE 3 FOURIER SERIES Periodic signals Wholerange Fourier series 3 Even and odd uncions Periodic signals Fourier series are used in
More information2.5 Life tables, force of mortality and standard life insurance products
Soluions 5 BS4a Acuarial Science Oford MT 212 33 2.5 Life ables, force of moraliy and sandard life insurance producs 1. (i) n m q represens he probabiliy of deah of a life currenly aged beween ages + n
More information2.6 Limits at Infinity, Horizontal Asymptotes Math 1271, TA: Amy DeCelles. 1. Overview. 2. Examples. Outline: 1. Definition of limits at infinity
.6 Limis a Infiniy, Horizonal Asympoes Mah 7, TA: Amy DeCelles. Overview Ouline:. Definiion of is a infiniy. Definiion of horizonal asympoe 3. Theorem abou raional powers of. Infinie is a infiniy This
More informationFourier series. Learning outcomes
Fourier series 23 Conens. Periodic funcions 2. Represening ic funcions by Fourier Series 3. Even and odd funcions 4. Convergence 5. Halfrange series 6. The complex form 7. Applicaion of Fourier series
More information4.2 Trigonometric Functions; The Unit Circle
4. Trigonomeric Funcions; The Uni Circle Secion 4. Noes Page A uni circle is a circle cenered a he origin wih a radius of. Is equaion is as shown in he drawing below. Here he leer represens an angle measure.
More informationDifferential Equations and Linear Superposition
Differenial Equaions and Linear Superposiion Basic Idea: Provide soluion in closed form Like Inegraion, no general soluions in closed form Order of equaion: highes derivaive in equaion e.g. dy d dy 2 y
More informationEconomics Honors Exam 2008 Solutions Question 5
Economics Honors Exam 2008 Soluions Quesion 5 (a) (2 poins) Oupu can be decomposed as Y = C + I + G. And we can solve for i by subsiuing in equaions given in he quesion, Y = C + I + G = c 0 + c Y D + I
More informationMath 201 Lecture 12: CauchyEuler Equations
Mah 20 Lecure 2: CauchyEuler Equaions Feb., 202 Many examples here are aken from he exbook. The firs number in () refers o he problem number in he UA Cusom ediion, he second number in () refers o he problem
More informationStochastic Optimal Control Problem for Life Insurance
Sochasic Opimal Conrol Problem for Life Insurance s. Basukh 1, D. Nyamsuren 2 1 Deparmen of Economics and Economerics, Insiue of Finance and Economics, Ulaanbaaar, Mongolia 2 School of Mahemaics, Mongolian
More informationChapter 4: Exponential and Logarithmic Functions
Chaper 4: Eponenial and Logarihmic Funcions Secion 4.1 Eponenial Funcions... 15 Secion 4. Graphs of Eponenial Funcions... 3 Secion 4.3 Logarihmic Funcions... 4 Secion 4.4 Logarihmic Properies... 53 Secion
More informationGraphing the Von Bertalanffy Growth Equation
file: d:\b1732013\von_beralanffy.wpd dae: Sepember 23, 2013 Inroducion Graphing he Von Beralanffy Growh Equaion Previously, we calculaed regressions of TL on SL for fish size daa and ploed he daa and
More informationAP Calculus AB 2013 Scoring Guidelines
AP Calculus AB 1 Scoring Guidelines The College Board The College Board is a missiondriven noforprofi organizaion ha connecs sudens o college success and opporuniy. Founded in 19, he College Board was
More informationInductance and Transient Circuits
Chaper H Inducance and Transien Circuis Blinn College  Physics 2426  Terry Honan As a consequence of Faraday's law a changing curren hrough one coil induces an EMF in anoher coil; his is known as muual
More informationFourier Series & The Fourier Transform
Fourier Series & The Fourier Transform Wha is he Fourier Transform? Fourier Cosine Series for even funcions and Sine Series for odd funcions The coninuous limi: he Fourier ransform (and is inverse) The
More information4.8 Exponential Growth and Decay; Newton s Law; Logistic Growth and Decay
324 CHAPTER 4 Exponenial and Logarihmic Funcions 4.8 Exponenial Growh and Decay; Newon s Law; Logisic Growh and Decay OBJECTIVES 1 Find Equaions of Populaions Tha Obey he Law of Uninhibied Growh 2 Find
More informationGraduate Macro Theory II: Notes on Neoclassical Growth Model
Graduae Macro Theory II: Noes on Neoclassical Growh Model Eric Sims Universiy of Nore Dame Spring 2011 1 Basic Neoclassical Growh Model The economy is populaed by a large number of infiniely lived agens.
More informationFourier Series Solution of the Heat Equation
Fourier Series Soluion of he Hea Equaion Physical Applicaion; he Hea Equaion In he early nineeenh cenury Joseph Fourier, a French scienis and mahemaician who had accompanied Napoleon on his Egypian campaign,
More informationImagine a Source (S) of sound waves that emits waves having frequency f and therefore
heoreical Noes: he oppler Eec wih ound Imagine a ource () o sound waes ha emis waes haing requency and hereore period as measured in he res rame o he ource (). his means ha any eecor () ha is no moing
More informationEntropy: From the Boltzmann equation to the Maxwell Boltzmann distribution
Enropy: From he Bolzmann equaion o he Maxwell Bolzmann disribuion A formula o relae enropy o probabiliy Ofen i is a lo more useful o hink abou enropy in erms of he probabiliy wih which differen saes are
More informationnonlocal conditions.
ISSN 17493889 prin, 17493897 online Inernaional Journal of Nonlinear Science Vol.11211 No.1,pp.39 Boundary Value Problem for Some Fracional Inegrodifferenial Equaions wih Nonlocal Condiions Mohammed
More informationImproper Integrals. Dr. Philippe B. laval Kennesaw State University. September 19, 2005. f (x) dx over a finite interval [a, b].
Improper Inegrls Dr. Philippe B. lvl Kennesw Se Universiy Sepember 9, 25 Absrc Noes on improper inegrls. Improper Inegrls. Inroducion In Clculus II, sudens defined he inegrl f (x) over finie inervl [,
More informationChapter 2 Problems. s = d t up. = 40km / hr d t down. 60km / hr. d t total. + t down. = t up. = 40km / hr + d. 60km / hr + 40km / hr
Chaper 2 Problems 2.2 A car ravels up a hill a a consan speed of 40km/h and reurns down he hill a a consan speed of 60 km/h. Calculae he average speed for he rip. This problem is a bi more suble han i
More information4 Convolution. Recommended Problems. x2[n] 1 2[n]
4 Convoluion Recommended Problems P4.1 This problem is a simple example of he use of superposiion. Suppose ha a discreeime linear sysem has oupus y[n] for he given inpus x[n] as shown in Figure P4.11.
More information11. Tire pressure. Here we always work with relative pressure. That s what everybody always does.
11. Tire pressure. The graph You have a hole in your ire. You pump i up o P=400 kilopascals (kpa) and over he nex few hours i goes down ill he ire is quie fla. Draw wha you hink he graph of ire pressure
More informationRelative velocity in one dimension
Connexions module: m13618 1 Relaive velociy in one dimension Sunil Kumar Singh This work is produced by The Connexions Projec and licensed under he Creaive Commons Aribuion License Absrac All quaniies
More informationModule 4. Singlephase AC circuits. Version 2 EE IIT, Kharagpur
Module 4 Singlephase A circuis ersion EE T, Kharagpur esson 5 Soluion of urren in A Series and Parallel ircuis ersion EE T, Kharagpur n he las lesson, wo poins were described:. How o solve for he impedance,
More information4. International Parity Conditions
4. Inernaional ariy ondiions 4.1 urchasing ower ariy he urchasing ower ariy ( heory is one of he early heories of exchange rae deerminaion. his heory is based on he concep ha he demand for a counry's currency
More informationOn the degrees of irreducible factors of higher order Bernoulli polynomials
ACTA ARITHMETICA LXII.4 (1992 On he degrees of irreducible facors of higher order Bernoulli polynomials by Arnold Adelberg (Grinnell, Ia. 1. Inroducion. In his paper, we generalize he curren resuls on
More information1 The basic circulation problem
2WO08: Graphs and Algorihms Lecure 4 Dae: 26/2/2012 Insrucor: Nikhil Bansal The Circulaion Problem Scribe: Tom Slenders 1 The basic circulaion problem We will consider he maxflow problem again, bu his
More informationTechnical Appendix to Risk, Return, and Dividends
Technical Appendix o Risk, Reurn, and Dividends Andrew Ang Columbia Universiy and NBER Jun Liu UC San Diego This Version: 28 Augus, 2006 Columbia Business School, 3022 Broadway 805 Uris, New York NY 10027,
More informationA Mathematical Description of MOSFET Behavior
10/19/004 A Mahemaical Descripion of MOSFET Behavior.doc 1/8 A Mahemaical Descripion of MOSFET Behavior Q: We ve learned an awful lo abou enhancemen MOSFETs, bu we sill have ye o esablished a mahemaical
More information11/6/2013. Chapter 14: Dynamic ADAS. Introduction. Introduction. Keeping track of time. The model s elements
Inroducion Chaper 14: Dynamic DS dynamic model of aggregae and aggregae supply gives us more insigh ino how he economy works in he shor run. I is a simplified version of a DSGE model, used in cuingedge
More information9. Capacitor and Resistor Circuits
ElecronicsLab9.nb 1 9. Capacior and Resisor Circuis Inroducion hus far we have consider resisors in various combinaions wih a power supply or baery which provide a consan volage source or direc curren
More informationSection 7.1 Angles and Their Measure
Secion 7.1 Angles and Their Measure Greek Leers Commonly Used in Trigonomery Quadran II Quadran III Quadran I Quadran IV α = alpha β = bea θ = hea δ = dela ω = omega γ = gamma DEGREES The angle formed
More informationCapacitors and inductors
Capaciors and inducors We coninue wih our analysis of linear circuis by inroducing wo new passive and linear elemens: he capacior and he inducor. All he mehods developed so far for he analysis of linear
More informationThe Torsion of Thin, Open Sections
EM 424: Torsion of hin secions 26 The Torsion of Thin, Open Secions The resuls we obained for he orsion of a hin recangle can also be used be used, wih some qualificaions, for oher hin open secions such
More informationINVESTIGATION OF THE INFLUENCE OF UNEMPLOYMENT ON ECONOMIC INDICATORS
INVESTIGATION OF THE INFLUENCE OF UNEMPLOYMENT ON ECONOMIC INDICATORS Ilona Tregub, Olga Filina, Irina Kondakova Financial Universiy under he Governmen of he Russian Federaion 1. Phillips curve In economics,
More informationDIFFERENTIAL EQUATIONS with TI89 ABDUL HASSEN and JAY SCHIFFMAN. A. Direction Fields and Graphs of Differential Equations
DIFFERENTIAL EQUATIONS wih TI89 ABDUL HASSEN and JAY SCHIFFMAN We will assume ha he reader is familiar wih he calculaor s keyboard and he basic operaions. In paricular we have assumed ha he reader knows
More informationSection 5.1 The Unit Circle
Secion 5.1 The Uni Circle The Uni Circle EXAMPLE: Show ha he poin, ) is on he uni circle. Soluion: We need o show ha his poin saisfies he equaion of he uni circle, ha is, x +y 1. Since ) ) + 9 + 9 1 P
More informationSecond Order Linear Differential Equations
Second Order Linear Differenial Equaions Second order linear equaions wih consan coefficiens; Fundamenal soluions; Wronskian; Exisence and Uniqueness of soluions; he characerisic equaion; soluions of homogeneous
More informationYTM is positively related to default risk. YTM is positively related to liquidity risk. YTM is negatively related to special tax treatment.
. Two quesions for oday. A. Why do bonds wih he same ime o mauriy have differen YTM s? B. Why do bonds wih differen imes o mauriy have differen YTM s? 2. To answer he firs quesion les look a he risk srucure
More informationA NOTE ON THE ALMOST EVERYWHERE CONVERGENCE OF ALTERNATING SEQUENCES WITH DUNFORD SCHWARTZ OPERATORS
C O L L O Q U I U M M A T H E M A T I C U M VOL. LXII 1991 FASC. I A OTE O THE ALMOST EVERYWHERE COVERGECE OF ALTERATIG SEQUECES WITH DUFORD SCHWARTZ OPERATORS BY RYOTARO S A T O (OKAYAMA) 1. Inroducion.
More informationCointegration: The Engle and Granger approach
Coinegraion: The Engle and Granger approach Inroducion Generally one would find mos of he economic variables o be nonsaionary I(1) variables. Hence, any equilibrium heories ha involve hese variables require
More informationPROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE
Profi Tes Modelling in Life Assurance Using Spreadshees PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE Erik Alm Peer Millingon 2004 Profi Tes Modelling in Life Assurance Using Spreadshees
More informationAP Calculus AB 2010 Scoring Guidelines
AP Calculus AB 1 Scoring Guidelines The College Board The College Board is a noforprofi membership associaion whose mission is o connec sudens o college success and opporuniy. Founded in 1, he College
More information2. Waves in Elastic Media, Mechanical Waves
2. Waves in Elasic Media, Mechanical Waves Wave moion appears in almos ever branch of phsics. We confine our aenion o waves in deformable or elasic media. These waves, for eample ordinar sound waves in
More information3 RungeKutta Methods
3 RungeKua Mehods In conras o he mulisep mehods of he previous secion, RungeKua mehods are singlesep mehods however, muliple sages per sep. They are moivaed by he dependence of he Taylor mehods on he
More informationRC (ResistorCapacitor) Circuits. AP Physics C
(ResisorCapacior Circuis AP Physics C Circui Iniial Condiions An circui is one where you have a capacior and resisor in he same circui. Suppose we have he following circui: Iniially, he capacior is UNCHARGED
More informationDuration and Convexity ( ) 20 = Bond B has a maturity of 5 years and also has a required rate of return of 10%. Its price is $613.
Graduae School of Business Adminisraion Universiy of Virginia UVAF38 Duraion and Convexiy he price of a bond is a funcion of he promised paymens and he marke required rae of reurn. Since he promised
More information1 HALFLIFE EQUATIONS
R.L. Hanna Page HALFLIFE EQUATIONS The basic equaion ; he saring poin ; : wrien for ime: x / where fracion of original maerial and / number of halflives, and / log / o calculae he age (# ears): age (halflife)
More information6.003 Homework #4 Solutions
6.3 Homewk #4 Soluion Problem. Laplace Tranfm Deermine he Laplace ranfm (including he region of convergence) of each of he following ignal: a. x () = e 2(3) u( 3) X = e 3 2 ROC: Re() > 2 X () = x ()e d
More informationLecture III: Finish Discounted Value Formulation
Lecure III: Finish Discouned Value Formulaion I. Inernal Rae of Reurn A. Formally defined: Inernal Rae of Reurn is ha ineres rae which reduces he ne presen value of an invesmen o zero.. Finding he inernal
More information5.8 Resonance 231. The study of vibrating mechanical systems ends here with the theory of pure and practical resonance.
5.8 Resonance 231 5.8 Resonance The sudy of vibraing mechanical sysems ends here wih he heory of pure and pracical resonance. Pure Resonance The noion of pure resonance in he differenial equaion (1) ()
More informationRotational Inertia of a Point Mass
Roaional Ineria of a Poin Mass Saddleback College Physics Deparmen, adaped from PASCO Scienific PURPOSE The purpose of his experimen is o find he roaional ineria of a poin experimenally and o verify ha
More informationNewton s Laws of Motion
Newon s Laws of Moion MS4414 Theoreical Mechanics Firs Law velociy. In he absence of exernal forces, a body moves in a sraigh line wih consan F = 0 = v = cons. Khan Academy Newon I. Second Law body. The
More informationCircuit Types. () i( t) ( )
Circui Types DC Circuis Idenifying feaures: o Consan inpus: he volages of independen volage sources and currens of independen curren sources are all consan. o The circui does no conain any swiches. All
More information= r t dt + σ S,t db S t (19.1) with interest rates given by a mean reverting OrnsteinUhlenbeck or Vasicek process,
Chaper 19 The BlackScholesVasicek Model The BlackScholesVasicek model is given by a sandard imedependen BlackScholes model for he sock price process S, wih imedependen bu deerminisic volailiy σ
More informationA Probability Density Function for Google s stocks
A Probabiliy Densiy Funcion for Google s socks V.Dorobanu Physics Deparmen, Poliehnica Universiy of Timisoara, Romania Absrac. I is an approach o inroduce he Fokker Planck equaion as an ineresing naural
More information1. y 5y + 6y = 2e t Solution: Characteristic equation is r 2 5r +6 = 0, therefore r 1 = 2, r 2 = 3, and y 1 (t) = e 2t,
Homework6 Soluions.7 In Problem hrough 4 use he mehod of variaion of parameers o find a paricular soluion of he given differenial equaion. Then check your answer by using he mehod of undeermined coeffiens..
More informationImpact of Debt on Primary Deficit and GSDP Gap in Odisha: Empirical Evidences
S.R. No. 002 10/2015/CEFT Impac of Deb on Primary Defici and GSDP Gap in Odisha: Empirical Evidences 1. Inroducion The excessive pressure of public expendiure over is revenue receip is financed hrough
More informationPart 1: White Noise and Moving Average Models
Chaper 3: Forecasing From Time Series Models Par 1: Whie Noise and Moving Average Models Saionariy In his chaper, we sudy models for saionary ime series. A ime series is saionary if is underlying saisical
More informationEquation for a line. Synthetic Impulse Response 0.5 0.5. 0 5 10 15 20 25 Time (sec) x(t) m
Fundamenals of Signals Overview Definiion Examples Energy and power Signal ransformaions Periodic signals Symmery Exponenial & sinusoidal signals Basis funcions Equaion for a line x() m x() =m( ) You will
More informationWeek #9  The Integral Section 5.1
Week #9  The Inegral Secion 5.1 From Calculus, Single Variable by HughesHalle, Gleason, McCallum e. al. Copyrigh 005 by John Wiley & Sons, Inc. This maerial is used by permission of John Wiley & Sons,
More informationTHE PRESSURE DERIVATIVE
Tom Aage Jelmer NTNU Dearmen of Peroleum Engineering and Alied Geohysics THE PRESSURE DERIVATIVE The ressure derivaive has imoran diagnosic roeries. I is also imoran for making ye curve analysis more reliable.
More informationCHARGE AND DISCHARGE OF A CAPACITOR
REFERENCES RC Circuis: Elecrical Insrumens: Mos Inroducory Physics exs (e.g. A. Halliday and Resnick, Physics ; M. Sernheim and J. Kane, General Physics.) This Laboraory Manual: Commonly Used Insrumens:
More informationMA261A Calculus III 2006 Fall Homework 4 Solutions Due 9/29/2006 8:00AM
MA6A Calculus III 006 Fall Homework 4 Soluions Due 9/9/006 00AM 97 #4 Describe in words he surface 3 A halflane in he osiive x and y erriory (See Figure in Page 67) 97 # Idenify he surface cos We see
More informationPHYS245 Lab: RC circuits
PHYS245 Lab: C circuis Purpose: Undersand he charging and discharging ransien processes of a capacior Display he charging and discharging process using an oscilloscope Undersand he physical meaning of
More informationThe option pricing framework
Chaper 2 The opion pricing framework The opion markes based on swap raes or he LIBOR have become he larges fixed income markes, and caps (floors) and swapions are he mos imporan derivaives wihin hese markes.
More informationA Brief Introduction to the Consumption Based Asset Pricing Model (CCAPM)
A Brief Inroducion o he Consumpion Based Asse Pricing Model (CCAPM We have seen ha CAPM idenifies he risk of any securiy as he covariance beween he securiy's rae of reurn and he rae of reurn on he marke
More informationA Note on Using the Svensson procedure to estimate the risk free rate in corporate valuation
A Noe on Using he Svensson procedure o esimae he risk free rae in corporae valuaion By Sven Arnold, Alexander Lahmann and Bernhard Schwezler Ocober 2011 1. The risk free ineres rae in corporae valuaion
More informationFullwave rectification, bulk capacitor calculations Chris Basso January 2009
ullwave recificaion, bulk capacior calculaions Chris Basso January 9 This shor paper shows how o calculae he bulk capacior value based on ripple specificaions and evaluae he rms curren ha crosses i. oal
More informationChapter 8: Regression with Lagged Explanatory Variables
Chaper 8: Regression wih Lagged Explanaory Variables Time series daa: Y for =1,..,T End goal: Regression model relaing a dependen variable o explanaory variables. Wih ime series new issues arise: 1. One
More informationWHAT ARE OPTION CONTRACTS?
WHAT ARE OTION CONTRACTS? By rof. Ashok anekar An oion conrac is a derivaive which gives he righ o he holder of he conrac o do 'Somehing' bu wihou he obligaion o do ha 'Somehing'. The 'Somehing' can be
More informationINDEPENDENT MARGINALS OF OPERATOR LÉVY S PROBABILITY MEASURES ON FINITE DIMENSIONAL VECTOR SPACES
Journal of Applied Analysis 1, 1 (1995), pp. 39 45 INDEPENDENT MARGINALS OF OPERATOR LÉVY S PROBABILITY MEASURES ON FINITE DIMENSIONAL VECTOR SPACES A. LUCZAK Absrac. We find exponens of independen marginals
More informationDensity Dependence. births are a decreasing function of density b(n) and deaths are an increasing function of density d(n).
FW 662 Densiydependen populaion models In he previous lecure we considered densiy independen populaion models ha assumed ha birh and deah raes were consan and no a funcion of populaion size. Longerm
More informationTwo Compartment Body Model and V d Terms by Jeff Stark
Two Comparmen Body Model and V d Terms by Jeff Sark In a onecomparmen model, we make wo imporan assumpions: (1) Linear pharmacokineics  By his, we mean ha eliminaion is firs order and ha pharmacokineic
More informationCHAPTER FIVE. Solutions for Section 5.1
CHAPTER FIVE 5. SOLUTIONS 87 Soluions for Secion 5.. (a) The velociy is 3 miles/hour for he firs hours, 4 miles/hour for he ne / hour, and miles/hour for he las 4 hours. The enire rip lass + / + 4 = 6.5
More informationState Machines: Brief Introduction to Sequencers Prof. Andrew J. Mason, Michigan State University
Inroducion ae Machines: Brief Inroducion o equencers Prof. Andrew J. Mason, Michigan ae Universiy A sae machine models behavior defined by a finie number of saes (unique configuraions), ransiions beween
More informationChapter 8 Copyright Henning Umland All Rights Reserved
Chaper 8 Copyrigh 19972004 Henning Umland All Righs Reserved Rise, Se, Twiligh General Visibiliy For he planning of observaions, i is useful o know he imes during which a cerain body is above he horizon
More informationChapter 2: Principles of steadystate converter analysis
Chaper 2 Principles of SeadySae Converer Analysis 2.1. Inroducion 2.2. Inducor volsecond balance, capacior charge balance, and he small ripple approximaion 2.3. Boos converer example 2.4. Cuk converer
More informationRevisions to Nonfarm Payroll Employment: 1964 to 2011
Revisions o Nonfarm Payroll Employmen: 1964 o 2011 Tom Sark December 2011 Summary Over recen monhs, he Bureau of Labor Saisics (BLS) has revised upward is iniial esimaes of he monhly change in nonfarm
More information