The Amoroso Distribution
|
|
- Darren Gardner
- 5 years ago
- Views:
Transcription
1 Tch. Not 003v ( ) Th Amoroso Distribution Gavin E. Crooks Th Amoroso distribution, is a continuous, univariat, unimodal probability distribution with a smiinfinit rang. A surprisingly larg mnagri of intrsting, univariat probability distributions ar spcial cass or limiting forms of th Amoroso distribution. Amoroso(x ν,, α, ) = ( ) { α ( ) } x ν x ν Γ(α) xp ν,, + α > 0 x ν ( > 0) x ν ( < 0) (a) This distribution has four ral paramtrs; a location paramtr ν, a scal paramtr, and two shap paramtrs α and. Anothr usful paramtrization is Amoroso (x µ,, α, λ) ( = αα + λ x µ Γ(α) = Amoroso(µ λ,, α, /λ) λαλ (b) ) α { λ ( xp α + λ x µ In th limit that λ 0, th rang bcoms x [, + ] and Amoroso (x µ,, α, 0) { ( ) ( )} = αα x µ x µ Γ(α) xp α α xp (c) (Rcall that lim a 0 ( + ax) /a = x ) W will dfin th standard Amoroso distribution as StdAmoroso(x) = x x = Amoroso(x 0,,, ) = Amoroso (x,,, ) (d) Stting to yilds Parson s typ V (March) distribution 3,4 ParsonV(x µ,, α) = ( ) α+ Γ(α) x ν =Amoroso(x ν,, α, ) () If w st th shap paramtr to unity w obtain Parson s typ III (Vinci) distribution 5 7. ParsonIII(x ν,, α) = ( ) α x ν ( ) Γ(α) (3) =Amoroso(x ν,, α, ) ) λ } () Amoroso ν α µ α λ () Parson typ V (3) Parson typ III (4) Nakagami (6) gnralizd Frécht.. n <0.. n <0 (8) gnralizd Gumbl.. n 0 (5) gnralizd Wibull.. n >0.. n >0 (7) gnralizd xtrm valu (4) Frécht.. <0.. <0 (33) gnralizd log gamma... 0 (9) Gumbl.. 0 (3) BHP.. π 0 (5) Wibull.. >0.. >0 (9) shiftd xponntial.... (3) log gamma x.. 0 (5) gnralizd gamma 0... (9) scald invrs-chi (6) invrs gamma (34) Jffrys (6) gamma 0.. () scald chi 0.. (0) strtchd xponntial 0.. () Lévy 0. - (3) half normal 0. () invrs Rayligh 0. - (0) invrs xponntial 0. - (8) xponntial 0. (4) Rayligh 0. (5) Maxwll 0. 3 (6) Win 0. 4 (7) invrs chi-squar 0. - (8) invrs chi 0. - () chi 0. (7) chi-squar 0. (8) standard xponntial 0 (30) standard Gumbl 0 0 (d) standard Amoroso 0
2 With = w obtain th Nakagami (gnralizd normal) distribution. Nakagami(x ν,, k/, ) (4) ( ) { k ( ) } x ν x ν = xp Γ(k/) If w drop th location paramtr from Amoroso, thn w obtain th gnralizd gamma (hypr gamma, gnralizd Wibull) distribution, th parnt of th gamma family of distributions 8,9. GnGamma(x, α, ) = ( x ) α ( x ) (5) Γ(α) x > 0, > 0 =Amoroso(x 0,, α, ) If th is ngativ thn th distribution is gnralizd invrs gamma. Not surprisingly th gamma (scald-chi-squar) distribution 5,7 is a spcial cas of th gnralizd gamma, whr th scond shap paramtr is st to unity. Gamma(x, α) = Γ(α) ( x ) α x/ = ParsonIII(x 0,, α) = GnGamma(x, α, ) = Amoroso(x 0,, α, ) Instancs of th gamma distribution oftn appar in statistical physics. For xampl th Win (Vinna) distribution Win(x T ) = Gamma(x T, 4) (An approximation to th rlativ intnsity of black body radiations as a function of th frquncy). Th Erlang distribution is a gamma distribution with intgr α. Not that w obtain Amoroso by adding to th gamma distribution both a location (as in Parson typ III) and an additional shap paramtr (as in th gnralizd gamma). Important spcial cass of th gamma distribution includ th chi-squar (χ, chi squard) distribution ( x ) k/ ChiSqr(x k) = x/ (7) Γ(k/) = Gamma(x, k/) = GnGamma(x, k/, ) = Amoroso(x 0,, k/, ) and th xponntial (Parson typ X) distribution (6) Exp(x ) = x (8) = Gamma(x, ) = Amoroso(x 0,,, ) W can also obtain a shiftd xponntial distribution as a spcial cas of th Parson typ III distribution ShiftExp(x ν, ) = (9) = ParsonIII(x ν,, ) = Amoroso(x ν,,, ) Strtchd xponntial 0 StrtchdExp = ( x ) ( x ) (0) = Amoroso(x 0,,, ) Additional spcial cass of th gnralizd gamma distribution includ th chi (χ) distribution ( ) k x Chi(x k) = Γ(k/) x / = GnGamma(x, k/, ) = Amoroso(x 0,, k/, ) and scald-chi (gnralizd Rayligh) distribution. ( ) k x ScaldChi(x s, k) = Γ(k/) x s s s = GnGamma(x s, k/, ) = Amoroso(x 0, s, k/, ) () () Spcial cass of th scald-chi distribution includ th half-normal (smi-normal, positiv dfinit normal ) distribution, HalfNormal(x s) = th Rayligh distribution x s (3) πs = ScaldChi(x s, ) = GnGamma(x s, /, ) = Amoroso(x 0, s, /, ) Rayligh(x s) = x x s s (4) = ScaldChi(x s, ) = GnGamma(x s,, ) = Amoroso(x 0, s,, ) and th Maxwll (Maxwll-Boltzmann) distribution Maxwll(x s) = πs 3 x x s (5) = ScaldChi(x s, 3) = GnGamma(x s, 3/, ) = Amoroso(x 0, s, 3/, ) With ngativ shap paramtrs, th gnralizd gamma gnrats various invrs distributions, including th invrs gamma (scald invrs chi-squar ) distribu-
3 3 tion, InvGamma(x, α) = Γ(α) th invrs-chi-squar distribution, InvChiSqr(x k) = th invrs-chi distribution, ( ) α+ /x (6) x = GnGamma(x, α, ) = ParsonV(x 0,, α) = Amoroso(x 0,, α, ) ( ) k + x (7) Γ(k/) x = InvGamma(x /, k/) = GnGamma(x /, k/, ) = ParsonV(x 0, /, k/) = Amoroso(x 0, /, k/, ) InvChi(x k) = ( ) k+ x Γ(k/) (8) x scald invrs-chi distribution, = GnGamma(x /, k/, ) = Amoroso(x 0, /, k/, ) ScaldInvChi(x s, k) = s invrs xponntial, Γ(k/) ( ) k+ s x s x (9) = GnGamma(x / s, k/, ) = Amoroso(x 0, / s, k/, ) InvExp(x ) = x /x (0) and invrs Rayligh. = InvGamma(x, ) = GnGamma(x,, ) = Amoroso(x 0,,, ) InvRayligh(x ) = ( ) 3 8s s x x () = GnGamma(x / s,, ) = Amoroso(x 0, / s,, ) Th Lévy distribution (Van dr Waals profil) is a spcial cas of th invrs gamma distribution. Th Lévy distribution is notabl for bing stabl; a linar combination of idntically distributd Lévy distributions is again a Lévy distribution. Th othr stabl distributions with analytic forms ar th normal (which w ncountr blow) and th Cauchy distribution, which is not a mmbr of th Amoroso family. c c/x Lévy(x c) = () π x 3/ = InvGamma(x c/, /) = GnGamma(x c/, /, ) = ParsonV(x 0, c/, /) = Amoroso(x 0, c/, /, ) Th Wibull (xtrm valu typ III, Fishr-Tipptt typ III, Gumbl typ III) distribution,3 occurs with th shap paramtr α =. This is th limiting distribution of th minimum of a larg numbr idntically distributd random variabls that ar at last ν. (Maximum if is ngativ.) Wibull(x ν,, ) = ( x ν = Amoroso(x ν,,, ) ) ( ) (3) Spcial cass of th Wibull distribution includ th xponntial ( = ) and Rayligh ( = ) distributions, and th standard Wibull (ν = 0). Th Frécht (xtrm valu typ II, Fishr-Tipptt typ II, Gumbl typ II, invrs Wibull) distribution is th limiting distribution of th largst of a larg numbr idntically distributd random variabls whos momnts ar not all finit and ar boundd from blow by ν. (If th shap paramtr is ngativ thn minimum rathr than maxima.) ( ) Frécht(x ν,, ) = + x ν ( ) = Amoroso(x ν,,, ) (4) Spcial cass of th Frécht distribution includ th invrs xponntial ( = ) and invrs Rayligh ( = ) and th standard Frécht (ν = 0) distribution. Instad of asking for th minimum or maximum of a larg numbr of random variabls, w instad ask for th nth largst w obtain th gnralizd Wibull distribution GnWibull(x ν,, n, ) = ( ) n x ν and th gnralizd Frécht distribution. GnFrécht(x ν,, n, ) = ( )n = Amoroso(x ν,, n, ) (5) ( ) n + x ν ( )n = Amoroso(x ν,, n, ) (6) Th gnralizd xtrm valu (GEV, von Miss- Jnkinson) distribution is th suprclass of typ I, II and
4 4 III xtrm valu distributions. GnExtrmValu(x µ,, λ) (7) = ( + λ x µ ) { λ ( xp + λ x µ ) } λ = Amoroso (x µ,,, λ) = Amoroso(µ λ,,, /λ) λαλ Th gnralizd Gumbl (gnralizd log-gamma) distribution is th limiting distribution of th nth largst valu of a larg numbr of unboundd idntically distributd random variabls. GnGumbl(x µ,, n) (8) = ( ) ( )} µ x µ x {n xp n xp = Amoroso (x µ,, n, 0) If w limit n = thn w obtain th Gumbl (Fishr- Tipptt (typ I), Fishr-Tipptt-Gumbl, FTG, Gumbl- Fishr-Tipptt, log-wibull, xtrm valu (typ I), doubly xponntial) distribution Gumbl(x µ, ) (9) = {( ) ( )} x µ x µ xp xp = Amoroso (x µ,,, 0) With ngativ scal < 0, this is an xtrm valu distribution of maximum, with > 0 an xtrm valu distribution of minima. (Not that oftn th Gumbl is dfind with th ngativ of th scal usd hr.) A Gomprtz distribution is a truncatd Gumbl. Th standard Gumbl (Gumbl) distribution is StdGumbl(x) = xp {x x } (30) =Amoroso (x 0,,, 0) Anothr spcial cas of th gnralizd Gumbl is th BHP (Bramwll-Holdsworth-Pinton) distribution 4,5 BHP(x µ, ) (3) = { ( ) π x µ xp π ( )} x µ xp = Amoroso (x µ,, π, 0) Log-gamma LogGamma(x, α) = { ( x ) ( x )} Γ(α) xp α xp =Amoroso (x ln α,, α, 0) (3) Gnralizd Log-Gamma(Coal-McNil 6,7 ) GnLogGamma(x µ,, α) (33) = { ( ) ( )} x µ x µ Γ(α) xp α xp = Amoroso (x µ + ln α,, α, 0) If w lt = 0 thn w obtain Jffrys distribution 8, an impropr (unnormalizabl) distribution widly usd as an uninformativ prior in Baysian probability 9. Jffrys(x) x = Amoroso(0,, α, 0)) (34) If and α ar finit, thir xact valus ar irrlvant. If w tak th limit α but kp th product α = p constant thn w can obtain a varity of impropr powrlaw (Parson typ XI 0, fractal) distributions. PowrLaw(x p) x p (35) = lim Amoroso(0,, α, ( p)/α) α If p = 0 w obtain th half-uniform distribution ovr th positiv numbrs. Th normal (Gauss, Gaussian, bll curv) distribution can b obtaind in svral limits. For xampl, Normal(x µ, ) (36) { } = xp (x µ) π = lim α Amoroso (x µ, / α, α, 0) In th limit that w obtain an unboundd uniform distribution, and in th limit 0 w obtain a dlta function distribution. Proprtis ( ) n x ν E[ ] = Γ(α + n ) Γ(α) man = ν + Γ(α + ) Γ(α) [ Γ(α + varianc = ) Γ(α + ) Γ(α) Γ(α) ) Entropy = log Γ(α) + α + ( α ] ψ(α) (37) (38) (39) (40)
5 5 Indx of distributions Distribution Equation χ S chi χ S chi-squar Γ S gamma Amaroso (a) bll curv S normal BHP (3) Bramwll-Holdsworth-Pinton S BHP chi () chi-squar (7) chi-squard S chi-squar Coal-McNil S gnralizd log-gamma dlta (36) doubly xponntial S Gumbl Erlang S gamma xponntial (8) xtrm valu typ N S Fishr-Tipptt typ N Fishr-Tipptt typ I S Gumbl Fishr-Tipptt typ II S Frécht Fishr-Tipptt typ III S Wibull Fishr-Tipptt-Gumbl S Gumbl fractal S powr law flat! s uniform Frécht (4) FTG S Fishr-Tipptt-Gumbl gamma (6) Gaussian S normal Gauss S normal gnral gnralizd gamma S Amoroso gnralizd gamma (5) gnralizd log-gamma (33) gnralizd Gumbl (8) gnralizd xtrm valu (7) gnralizd Frécht (6) gnralizd invrs gamma..... S gnralizd gamma gnralizd normal S Nakagami gnralizd Rayligh S scald-chi gnralizd Wibull (5) GEV S gnralizd xtrm valu Gomprtz S Gumbl Gumbl (9) Gumbl-Fishr-Tipptt S Gumbl Gumbl typ N S Fishr-Tipptt typ N half-normal (3) half-gaussian s half-normal half-uniform (35) hypr gamma S gnralizd gamma invrs chi (8) invrs chi-squar (7) invrs xponntial (0) invrs gamma (6) invrs Rayligh () invrs Wibull S Frécht Jffrys (34) Lévy () log-gamma (3) log-wibull s Gumbl March S Parson typ V Nakagami (4) normal (36) Parson typ III (3) Parson typ V () Parson typ X S xponntial Parson typ XI S powr law positiv dfinit normal S half-normal powr law (35) Rayligh (4) Maxwll (5) Maxwll-Boltzmann S Maxwll shiftd xponntial (9) scald chi () scald chi-squar S gamma scald invrs chi (9) scald invrs chi-squar s invrs gamma smi-normal S half-normal standard Amoroso (d) standard Gumbl (30) strtchd xponntial (0) uniform (36) Van dr Waals S Lévy Vinna S Win Vinci S Parson Typ V von Miss-Jnkinson gnralizd xtrm valu Wibull (3) Win S gamma Amoroso, Annali di Mathmatica, 3 (95). N. L. Johnson, S. Kotz, and N. Balakrishnan, Continuous Univariat Distributions, vol. (Wily, Nw York, 994), nd d. 3 K. Parson, Philos. Trans. R. Soc. A (894). 4 K. Parson, Philos. Trans. R. Soc. A 97, 443 (90). 5 M. Abramowitz and I. A. Stgun, Handbook of Mathmatical Functions with Formulas, Graphs, and Mathmatical Tabls (Dovr, Nw York, 964). 6 K. Parson, Philos. Trans. R. Soc. A (893). 7 K. Parson, Philos. Trans. R. Soc. A 86, 343 (895). 8 E. W. Stacy, Ann. Math. Stat 33, 87 (96). 9 A. Dadpay, E. S. Soofi, and R. Soyr, J. Economtrics 38, 568 (007). 0 J. Lahrrèr and D. Sorntt, Eur. Phys. J. B, 55 (998). A. Glman, J. B. Carlin, H. S. Strn, and D. B. Rubin, Baysian Data Analysis (Chapman & Hall/CRC, Nw York, 004), nd d.
6 6 N. L. Johnson, S. Kotz, and N. Balakrishnan, Continuous Univariat Distributions, vol. (Wily, Nw York, 995), nd d. 3 W. Wibull, J. Appl. Mch.-Trans. ASME 8, 93 (95). 4 S. T. Bramwll, P. C. W. Holdsworth, and J.-F. Pinton, Natur 396 (998). 5 S. T. Bramwll, K. Christnsn, J.-Y. Fortin, P. C. W. Holdsworth, H. J. Jnsn, S. Lis, J. M. Lópz, M. Nicodmi, J.-F. Pinton, and M. Sllitto, Phys. Rv. Ltt. 84, 3744 (000). 6 A. Coal and D. R. McNil, J. Amr. Statist. Assoc. 67, 743 (97). 7 R. Kanko, Dm. Rs. 9, 3 (003). 8 H. Jffrys, Thory of probability (Oxford Univrsity Prss, 948), nd d. 9 E. T. Jayns, Probability Thory: Th Logic of Scinc (Cambridg Univrsity Prss, Cambridg, 003). 0 K. Parson, Philos. Trans. R. Soc. A 6, 49 (96).
Continuous probability distributions
Continuous probability distributions Many continuous probability distributions, including: Uniform Normal Gamma Eponntial Chi-Squard Lognormal Wibull EGR 5 Ch. 6 Uniform distribution Simplst charactrizd
More informationCOMPUTER GENERATED HOLOGRAMS Optical Sciences 627 W.J. Dallas (Monday, April 04, 2005, 8:35 AM) PART I: CHAPTER TWO COMB MATH.
C:\Dallas\0_Courss\03A_OpSci_67\0 Cgh_Book\0_athmaticalPrliminaris\0_0 Combath.doc of 8 COPUTER GENERATED HOLOGRAS Optical Scincs 67 W.J. Dallas (onday, April 04, 005, 8:35 A) PART I: CHAPTER TWO COB ATH
More informationText: WMM, Chapter 5. Sections , ,
Lcturs 6 - Continuous Probabilit Distributions Tt: WMM, Chaptr 5. Sctions 6.-6.4, 6.6-6.8, 7.-7. In th prvious sction, w introduc som of th common probabilit distribution functions (PDFs) for discrt sampl
More informationThe Amoroso Distribution
The Amoroso Distribution Gavin E. Crooks Physical Biosciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA 9470 Abstract: The Amoroso distribution is the natural unification of the gamma
More informationA Propagating Wave Packet Group Velocity Dispersion
Lctur 8 Phys 375 A Propagating Wav Packt Group Vlocity Disprsion Ovrviw and Motivation: In th last lctur w lookd at a localizd solution t) to th 1D fr-particl Schrödingr quation (SE) that corrsponds to
More informationExercise 1. Sketch the graph of the following function. (x 2
Writtn tst: Fbruary 9th, 06 Exrcis. Sktch th graph of th following function fx = x + x, spcifying: domain, possibl asymptots, monotonicity, continuity, local and global maxima or minima, and non-drivability
More informationperm4 A cnt 0 for for if A i 1 A i cnt cnt 1 cnt i j. j k. k l. i k. j l. i l
h 4D, 4th Rank, Antisytric nsor and th 4D Equivalnt to th Cross Product or Mor Fun with nsors!!! Richard R Shiffan Digital Graphics Assoc 8 Dunkirk Av LA, Ca 95 rrs@isidu his docunt dscribs th four dinsional
More informationABEL TYPE THEOREMS FOR THE WAVELET TRANSFORM THROUGH THE QUASIASYMPTOTIC BOUNDEDNESS
Novi Sad J. Math. Vol. 45, No. 1, 2015, 201-206 ABEL TYPE THEOREMS FOR THE WAVELET TRANSFORM THROUGH THE QUASIASYMPTOTIC BOUNDEDNESS Mirjana Vuković 1 and Ivana Zubac 2 Ddicatd to Acadmician Bogoljub Stanković
More informationSECTION where P (cos θ, sin θ) and Q(cos θ, sin θ) are polynomials in cos θ and sin θ, provided Q is never equal to zero.
SETION 6. 57 6. Evaluation of Dfinit Intgrals Exampl 6.6 W hav usd dfinit intgrals to valuat contour intgrals. It may com as a surpris to larn that contour intgrals and rsidus can b usd to valuat crtain
More information3 Finite Element Parametric Geometry
3 Finit Elmnt Paramtric Gomtry 3. Introduction Th intgral of a matrix is th matrix containing th intgral of ach and vry on of its original componnts. Practical finit lmnt analysis rquirs intgrating matrics,
More informationThe graph of y = x (or y = ) consists of two branches, As x 0, y + ; as x 0, y +. x = 0 is the
Copyright itutcom 005 Fr download & print from wwwitutcom Do not rproduc by othr mans Functions and graphs Powr functions Th graph of n y, for n Q (st of rational numbrs) y is a straight lin through th
More informationFunctions of Two Random Variables
Functions of Two Random Variabls Maximum ( ) Dfin max, Find th probabilit distributions of Solution: For an pair of random variabls and, [ ] [ ] F ( w) P w P w and w F hn and ar indpndnt, F ( w) F ( w)
More informationCramér-Rao Inequality: Let f(x; θ) be a probability density function with continuous parameter
WHEN THE CRAMÉR-RAO INEQUALITY PROVIDES NO INFORMATION STEVEN J. MILLER Abstract. W invstigat a on-paramtr family of probability dnsitis (rlatd to th Parto distribution, which dscribs many natural phnomna)
More informationThe Matrix Exponential
Th Matrix Exponntial (with xrciss) by D. Klain Vrsion 207.0.05 Corrctions and commnts ar wlcom. Th Matrix Exponntial For ach n n complx matrix A, dfin th xponntial of A to b th matrix A A k I + A + k!
More informationThe Matrix Exponential
Th Matrix Exponntial (with xrciss) by Dan Klain Vrsion 28928 Corrctions and commnts ar wlcom Th Matrix Exponntial For ach n n complx matrix A, dfin th xponntial of A to b th matrix () A A k I + A + k!
More informationDerangements and Applications
2 3 47 6 23 Journal of Intgr Squncs, Vol. 6 (2003), Articl 03..2 Drangmnts and Applications Mhdi Hassani Dpartmnt of Mathmatics Institut for Advancd Studis in Basic Scincs Zanjan, Iran mhassani@iasbs.ac.ir
More informationBINOMIAL COEFFICIENTS INVOLVING INFINITE POWERS OF PRIMES
BINOMIAL COEFFICIENTS INVOLVING INFINITE POWERS OF PRIMES DONALD M. DAVIS Abstract. If p is a prim (implicit in notation and n a positiv intgr, lt ν(n dnot th xponnt of p in n, and U(n n/p ν(n, th unit
More informationSOME PARAMETERS ON EQUITABLE COLORING OF PRISM AND CIRCULANT GRAPH.
SOME PARAMETERS ON EQUITABLE COLORING OF PRISM AND CIRCULANT GRAPH. K VASUDEVAN, K. SWATHY AND K. MANIKANDAN 1 Dpartmnt of Mathmatics, Prsidncy Collg, Chnnai-05, India. E-Mail:vasu k dvan@yahoo.com. 2,
More informationFunctions of Two Random Variables
Functions of Two Random Variabls Maximum ( ) Dfin max, Find th probabilit distributions of Solution: For an pair of random variabls and, [ ] F ( w) P w [ and ] P w w F, ( w, w) hn and ar indpndnt, F (
More informationEEO 401 Digital Signal Processing Prof. Mark Fowler
EEO 401 Digital Signal Procssing Prof. Mark Fowlr Dtails of th ot St #19 Rading Assignmnt: Sct. 7.1.2, 7.1.3, & 7.2 of Proakis & Manolakis Dfinition of th So Givn signal data points x[n] for n = 0,, -1
More informationLinear-Phase FIR Transfer Functions. Functions. Functions. Functions. Functions. Functions. Let
It is impossibl to dsign an IIR transfr function with an xact linar-phas It is always possibl to dsign an FIR transfr function with an xact linar-phas rspons W now dvlop th forms of th linarphas FIR transfr
More informationSelf-interaction mass formula that relates all leptons and quarks to the electron
Slf-intraction mass formula that rlats all lptons and quarks to th lctron GERALD ROSEN (a) Dpartmnt of Physics, Drxl Univrsity Philadlphia, PA 19104, USA PACS. 12.15. Ff Quark and lpton modls spcific thoris
More informationRandom Process Part 1
Random Procss Part A random procss t (, ζ is a signal or wavform in tim. t : tim ζ : outcom in th sampl spac Each tim w rapat th xprimnt, a nw wavform is gnratd. ( W will adopt t for short. Tim sampls
More information6. The Interaction of Light and Matter
6. Th Intraction of Light and Mattr - Th intraction of light and mattr is what maks lif intrsting. - Light causs mattr to vibrat. Mattr in turn mits light, which intrfrs with th original light. - Excitd
More informationFourier Transforms and the Wave Equation. Key Mathematics: More Fourier transform theory, especially as applied to solving the wave equation.
Lur 7 Fourir Transforms and th Wav Euation Ovrviw and Motivation: W first discuss a fw faturs of th Fourir transform (FT), and thn w solv th initial-valu problm for th wav uation using th Fourir transform
More informationJournal of Modern Applied Statistical Methods May, 2007, Vol. 6, No. 1, /07/$ On the Product of Maxwell and Rice Random Variables
Journal of Modrn Applid Statistical Mthods Copyright 7 JMASM, Inc. May, 7, Vol. 6, No., 538 947/7/$95. On th Product of Mawll and Ric Random Variabls Mohammad Shail Miami Dad Collg B. M. Golam Kibria Florida
More informationEvaluating Reliability Systems by Using Weibull & New Weibull Extension Distributions Mushtak A.K. Shiker
Evaluating Rliability Systms by Using Wibull & Nw Wibull Extnsion Distributions Mushtak A.K. Shikr مشتاق عبذ الغني شخير Univrsity of Babylon, Collg of Education (Ibn Hayan), Dpt. of Mathmatics Abstract
More informationBINOMIAL COEFFICIENTS INVOLVING INFINITE POWERS OF PRIMES. 1. Statement of results
BINOMIAL COEFFICIENTS INVOLVING INFINITE POWERS OF PRIMES DONALD M. DAVIS Abstract. If p is a prim and n a positiv intgr, lt ν p (n dnot th xponnt of p in n, and u p (n n/p νp(n th unit part of n. If α
More informationTypes of Transfer Functions. Types of Transfer Functions. Types of Transfer Functions. Ideal Filters. Ideal Filters
Typs of Transfr Typs of Transfr x[n] X( LTI h[n] H( y[n] Y( y [ n] h[ k] x[ n k] k Y ( H ( X ( Th tim-domain classification of an LTI digital transfr function is basd on th lngth of its impuls rspons h[n]:
More informationY 0. Standing Wave Interference between the incident & reflected waves Standing wave. A string with one end fixed on a wall
Staning Wav Intrfrnc btwn th incint & rflct wavs Staning wav A string with on n fix on a wall Incint: y, t) Y cos( t ) 1( Y 1 ( ) Y (St th incint wav s phas to b, i.., Y + ral & positiv.) Rflct: y, t)
More informationCh. 24 Molecular Reaction Dynamics 1. Collision Theory
Ch. 4 Molcular Raction Dynamics 1. Collision Thory Lctur 16. Diffusion-Controlld Raction 3. Th Matrial Balanc Equation 4. Transition Stat Thory: Th Eyring Equation 5. Transition Stat Thory: Thrmodynamic
More information(Upside-Down o Direct Rotation) β - Numbers
Amrican Journal of Mathmatics and Statistics 014, 4(): 58-64 DOI: 10593/jajms0140400 (Upsid-Down o Dirct Rotation) β - Numbrs Ammar Sddiq Mahmood 1, Shukriyah Sabir Ali,* 1 Dpartmnt of Mathmatics, Collg
More informationSER/BER in a Fading Channel
SER/BER in a Fading Channl Major points for a fading channl: * SNR is a R.V. or R.P. * SER(BER) dpnds on th SNR conditional SER(BER). * Two prformanc masurs: outag probability and avrag SER(BER). * Ovrall,
More informationANALYSIS IN THE FREQUENCY DOMAIN
ANALYSIS IN THE FREQUENCY DOMAIN SPECTRAL DENSITY Dfinition Th spctral dnsit of a S.S.P. t also calld th spctrum of t is dfind as: + { γ }. jτ γ τ F τ τ In othr words, of th covarianc function. is dfind
More informationHomotopy perturbation technique
Comput. Mthods Appl. Mch. Engrg. 178 (1999) 257±262 www.lsvir.com/locat/cma Homotopy prturbation tchniqu Ji-Huan H 1 Shanghai Univrsity, Shanghai Institut of Applid Mathmatics and Mchanics, Shanghai 272,
More informationOn the irreducibility of some polynomials in two variables
ACTA ARITHMETICA LXXXII.3 (1997) On th irrducibility of som polynomials in two variabls by B. Brindza and Á. Pintér (Dbrcn) To th mmory of Paul Erdős Lt f(x) and g(y ) b polynomials with intgral cofficints
More information1 Isoparametric Concept
UNIVERSITY OF CALIFORNIA BERKELEY Dpartmnt of Civil Enginring Spring 06 Structural Enginring, Mchanics and Matrials Profssor: S. Govindj Nots on D isoparamtric lmnts Isoparamtric Concpt Th isoparamtric
More informationSearch sequence databases 3 10/25/2016
Sarch squnc databass 3 10/25/2016 Etrm valu distribution Ø Suppos X is a random variabl with probability dnsity function p(, w sampl a larg numbr S of indpndnt valus of X from this distribution for an
More information10. The Discrete-Time Fourier Transform (DTFT)
Th Discrt-Tim Fourir Transform (DTFT Dfinition of th discrt-tim Fourir transform Th Fourir rprsntation of signals plays an important rol in both continuous and discrt signal procssing In this sction w
More informationProblem Set 6 Solutions
6.04/18.06J Mathmatics for Computr Scinc March 15, 005 Srini Dvadas and Eric Lhman Problm St 6 Solutions Du: Monday, March 8 at 9 PM in Room 3-044 Problm 1. Sammy th Shark is a financial srvic providr
More informationSolution of Assignment #2
olution of Assignmnt #2 Instructor: Alirza imchi Qustion #: For simplicity, assum that th distribution function of T is continuous. Th distribution function of R is: F R ( r = P( R r = P( log ( T r = P(log
More informationTypes of Transfer Functions. Types of Transfer Functions. Ideal Filters. Ideal Filters. Ideal Filters
Typs of Transfr Typs of Transfr Th tim-domain classification of an LTI digital transfr function squnc is basd on th lngth of its impuls rspons: - Finit impuls rspons (FIR) transfr function - Infinit impuls
More informationHigher order derivatives
Robrto s Nots on Diffrntial Calculus Chaptr 4: Basic diffrntiation ruls Sction 7 Highr ordr drivativs What you nd to know alrady: Basic diffrntiation ruls. What you can larn hr: How to rpat th procss of
More informationEstimation of apparent fraction defective: A mathematical approach
Availabl onlin at www.plagiarsarchlibrary.com Plagia Rsarch Library Advancs in Applid Scinc Rsarch, 011, (): 84-89 ISSN: 0976-8610 CODEN (USA): AASRFC Estimation of apparnt fraction dfctiv: A mathmatical
More informationLecture 37 (Schrödinger Equation) Physics Spring 2018 Douglas Fields
Lctur 37 (Schrödingr Equation) Physics 6-01 Spring 018 Douglas Filds Rducd Mass OK, so th Bohr modl of th atom givs nrgy lvls: E n 1 k m n 4 But, this has on problm it was dvlopd assuming th acclration
More informationEEO 401 Digital Signal Processing Prof. Mark Fowler
EEO 401 Digital Signal Procssing Prof. Mark Fowlr ot St #18 Introduction to DFT (via th DTFT) Rading Assignmnt: Sct. 7.1 of Proakis & Manolakis 1/24 Discrt Fourir Transform (DFT) W v sn that th DTFT is
More informationHydrogen Atom and One Electron Ions
Hydrogn Atom and On Elctron Ions Th Schrödingr quation for this two-body problm starts out th sam as th gnral two-body Schrödingr quation. First w sparat out th motion of th cntr of mass. Th intrnal potntial
More informationEquidistribution and Weyl s criterion
Euidistribution and Wyl s critrion by Brad Hannigan-Daly W introduc th ida of a sunc of numbrs bing uidistributd (mod ), and w stat and prov a thorm of Hrmann Wyl which charactrizs such suncs. W also discuss
More informationHardy-Littlewood Conjecture and Exceptional real Zero. JinHua Fei. ChangLing Company of Electronic Technology Baoji Shannxi P.R.
Hardy-Littlwood Conjctur and Excptional ral Zro JinHua Fi ChangLing Company of Elctronic Tchnology Baoji Shannxi P.R.China E-mail: fijinhuayoujian@msn.com Abstract. In this papr, w assum that Hardy-Littlwood
More informationAbstract Interpretation: concrete and abstract semantics
Abstract Intrprtation: concrt and abstract smantics Concrt smantics W considr a vry tiny languag that manags arithmtic oprations on intgrs valus. Th (concrt) smantics of th languags cab b dfind by th funzcion
More informationLecture 6.4: Galois groups
Lctur 6.4: Galois groups Matthw Macauly Dpartmnt of Mathmatical Scincs Clmson Univrsity http://www.math.clmson.du/~macaul/ Math 4120, Modrn Algbra M. Macauly (Clmson) Lctur 6.4: Galois groups Math 4120,
More informationINTEGRATION BY PARTS
Mathmatics Rvision Guids Intgration by Parts Pag of 7 MK HOME TUITION Mathmatics Rvision Guids Lvl: AS / A Lvl AQA : C Edcl: C OCR: C OCR MEI: C INTEGRATION BY PARTS Vrsion : Dat: --5 Eampls - 6 ar copyrightd
More informationLINEAR DELAY DIFFERENTIAL EQUATION WITH A POSITIVE AND A NEGATIVE TERM
Elctronic Journal of Diffrntial Equations, Vol. 2003(2003), No. 92, pp. 1 6. ISSN: 1072-6691. URL: http://jd.math.swt.du or http://jd.math.unt.du ftp jd.math.swt.du (login: ftp) LINEAR DELAY DIFFERENTIAL
More informationDifference -Analytical Method of The One-Dimensional Convection-Diffusion Equation
Diffrnc -Analytical Mthod of Th On-Dimnsional Convction-Diffusion Equation Dalabav Umurdin Dpartmnt mathmatic modlling, Univrsity of orld Economy and Diplomacy, Uzbistan Abstract. An analytical diffrncing
More informationBSc Engineering Sciences A. Y. 2017/18 Written exam of the course Mathematical Analysis 2 August 30, x n, ) n 2
BSc Enginring Scincs A. Y. 27/8 Writtn xam of th cours Mathmatical Analysis 2 August, 28. Givn th powr sris + n + n 2 x n, n n dtrmin its radius of convrgnc r, and study th convrgnc for x ±r. By th root
More informationThat is, we start with a general matrix: And end with a simpler matrix:
DIAGON ALIZATION OF THE STR ESS TEN SOR INTRO DUCTIO N By th us of Cauchy s thorm w ar abl to rduc th numbr of strss componnts in th strss tnsor to only nin valus. An additional simplification of th strss
More informationDeift/Zhou Steepest descent, Part I
Lctur 9 Dift/Zhou Stpst dscnt, Part I W now focus on th cas of orthogonal polynomials for th wight w(x) = NV (x), V (x) = t x2 2 + x4 4. Sinc th wight dpnds on th paramtr N N w will writ π n,n, a n,n,
More informationNEW APPLICATIONS OF THE ABEL-LIOUVILLE FORMULA
NE APPLICATIONS OF THE ABEL-LIOUVILLE FORMULA Mirca I CÎRNU Ph Dp o Mathmatics III Faculty o Applid Scincs Univrsity Polithnica o Bucharst Cirnumirca @yahoocom Abstract In a rcnt papr [] 5 th indinit intgrals
More informationLecture 2: Discrete-Time Signals & Systems. Reza Mohammadkhani, Digital Signal Processing, 2015 University of Kurdistan eng.uok.ac.
Lctur 2: Discrt-Tim Signals & Systms Rza Mohammadkhani, Digital Signal Procssing, 2015 Univrsity of Kurdistan ng.uok.ac.ir/mohammadkhani 1 Signal Dfinition and Exampls 2 Signal: any physical quantity that
More informationLimiting value of higher Mahler measure
Limiting valu of highr Mahlr masur Arunabha Biswas a, Chris Monico a, a Dpartmnt of Mathmatics & Statistics, Txas Tch Univrsity, Lubbock, TX 7949, USA Abstract W considr th k-highr Mahlr masur m k P )
More informationECE 650 1/8. Homework Set 4 - Solutions
ECE 65 /8 Homwork St - Solutions. (Stark & Woods #.) X: zro-man, C X Find G such that Y = GX will b lt. whit. (Will us: G = -/ E T ) Finding -valus for CX: dt = (-) (-) = Finding corrsponding -vctors for
More informationMath 34A. Final Review
Math A Final Rviw 1) Us th graph of y10 to find approimat valus: a) 50 0. b) y (0.65) solution for part a) first writ an quation: 50 0. now tak th logarithm of both sids: log() log(50 0. ) pand th right
More information1973 AP Calculus AB: Section I
97 AP Calculus AB: Sction I 9 Minuts No Calculator Not: In this amination, ln dnots th natural logarithm of (that is, logarithm to th bas ).. ( ) d= + C 6 + C + C + C + C. If f ( ) = + + + and ( ), g=
More informationA Prey-Predator Model with an Alternative Food for the Predator, Harvesting of Both the Species and with A Gestation Period for Interaction
Int. J. Opn Problms Compt. Math., Vol., o., Jun 008 A Pry-Prdator Modl with an Altrnativ Food for th Prdator, Harvsting of Both th Spcis and with A Gstation Priod for Intraction K. L. arayan and. CH. P.
More informationEXST Regression Techniques Page 1
EXST704 - Rgrssion Tchniqus Pag 1 Masurmnt rrors in X W hav assumd that all variation is in Y. Masurmnt rror in this variabl will not ffct th rsults, as long as thy ar uncorrlatd and unbiasd, sinc thy
More informationAn Extensive Study of Approximating the Periodic. Solutions of the Prey Predator System
pplid athmatical Scincs Vol. 00 no. 5 5 - n xtnsiv Study of pproximating th Priodic Solutions of th Pry Prdator Systm D. Vnu Gopala Rao * ailing addrss: Plot No.59 Sctor-.V.P.Colony Visahapatnam 50 07
More informationObjective Mathematics
x. Lt 'P' b a point on th curv y and tangnt x drawn at P to th curv has gratst slop in magnitud, thn point 'P' is,, (0, 0),. Th quation of common tangnt to th curvs y = 6 x x and xy = x + is : x y = 8
More informationGEOMETRICAL PHENOMENA IN THE PHYSICS OF SUBATOMIC PARTICLES. Eduard N. Klenov* Rostov-on-Don, Russia
GEOMETRICAL PHENOMENA IN THE PHYSICS OF SUBATOMIC PARTICLES Eduard N. Klnov* Rostov-on-Don, Russia Th articl considrs phnomnal gomtry figurs bing th carrirs of valu spctra for th pairs of th rmaining additiv
More informationThe van der Waals interaction 1 D. E. Soper 2 University of Oregon 20 April 2012
Th van dr Waals intraction D. E. Sopr 2 Univrsity of Orgon 20 pril 202 Th van dr Waals intraction is discussd in Chaptr 5 of J. J. Sakurai, Modrn Quantum Mchanics. Hr I tak a look at it in a littl mor
More informationRecall that by Theorems 10.3 and 10.4 together provide us the estimate o(n2 ), S(q) q 9, q=1
Chaptr 11 Th singular sris Rcall that by Thorms 10 and 104 togthr provid us th stimat 9 4 n 2 111 Rn = SnΓ 2 + on2, whr th singular sris Sn was dfind in Chaptr 10 as Sn = q=1 Sq q 9, with Sq = 1 a q gcda,q=1
More informationEinstein Rosen inflationary Universe in general relativity
PRAMANA c Indian Acadmy of Scincs Vol. 74, No. 4 journal of April 2010 physics pp. 669 673 Einstin Rosn inflationary Univrs in gnral rlativity S D KATORE 1, R S RANE 2, K S WANKHADE 2, and N K SARKATE
More informationAn Application of Hardy-Littlewood Conjecture. JinHua Fei. ChangLing Company of Electronic Technology Baoji Shannxi P.R.China
An Application of Hardy-Littlwood Conjctur JinHua Fi ChangLing Company of Elctronic Tchnology Baoji Shannxi P.R.China E-mail: fijinhuayoujian@msn.com Abstract. In this papr, w assum that wakr Hardy-Littlwood
More informationThomas Whitham Sixth Form
Thomas Whitham Sith Form Pur Mathmatics Unit C Algbra Trigonomtr Gomtr Calculus Vctor gomtr Pag Algbra Molus functions graphs, quations an inqualitis Graph of f () Draw f () an rflct an part of th curv
More informationCOHORT MBA. Exponential function. MATH review (part2) by Lucian Mitroiu. The LOG and EXP functions. Properties: e e. lim.
MTH rviw part b Lucian Mitroiu Th LOG and EXP functions Th ponntial function p : R, dfind as Proprtis: lim > lim p Eponntial function Y 8 6 - -8-6 - - X Th natural logarithm function ln in US- log: function
More informationProbability and Stochastic Processes: A Friendly Introduction for Electrical and Computer Engineers Roy D. Yates and David J.
Probability and Stochastic Procsss: A Frindly Introduction for Elctrical and Computr Enginrs Roy D. Yats and David J. Goodman Problm Solutions : Yats and Goodman,4.3. 4.3.4 4.3. 4.4. 4.4.4 4.4.6 4.. 4..7
More informationExact and Approximate Detection Probability Formulas in Fundamentals of Radar Signal Processing
Exact and Approximat tction robabiity Formuas in Fundamntas of Radar Signa rocssing Mark A. Richards Sptmbr 8 Introduction Tab 6. in th txt Fundamntas of Radar Signa rocssing, nd d. [], is rproducd bow.
More information1.2 Faraday s law A changing magnetic field induces an electric field. Their relation is given by:
Elctromagntic Induction. Lorntz forc on moving charg Point charg moving at vlocity v, F qv B () For a sction of lctric currnt I in a thin wir dl is Idl, th forc is df Idl B () Elctromotiv forc f s any
More informationLecture 19: Free Energies in Modern Computational Statistical Thermodynamics: WHAM and Related Methods
Statistical Thrmodynamics Lctur 19: Fr Enrgis in Modrn Computational Statistical Thrmodynamics: WHAM and Rlatd Mthods Dr. Ronald M. Lvy ronlvy@tmpl.du Dfinitions Canonical nsmbl: A N, V,T = k B T ln Q
More informationHomework #3. 1 x. dx. It therefore follows that a sum of the
Danil Cannon CS 62 / Luan March 5, 2009 Homwork # 1. Th natural logarithm is dfind by ln n = n 1 dx. It thrfor follows that a sum of th 1 x sam addnd ovr th sam intrval should b both asymptotically uppr-
More informationsurface of a dielectric-metal interface. It is commonly used today for discovering the ways in
Surfac plasmon rsonanc is snsitiv mchanism for obsrving slight changs nar th surfac of a dilctric-mtal intrfac. It is commonl usd toda for discovring th was in which protins intract with thir nvironmnt,
More informationSupplementary Materials
6 Supplmntary Matrials APPENDIX A PHYSICAL INTERPRETATION OF FUEL-RATE-SPEED FUNCTION A truck running on a road with grad/slop θ positiv if moving up and ngativ if moving down facs thr rsistancs: arodynamic
More informationChapter 1. Chapter 10. Chapter 2. Chapter 11. Chapter 3. Chapter 12. Chapter 4. Chapter 13. Chapter 5. Chapter 14. Chapter 6. Chapter 7.
Chaptr Binomial Epansion Chaptr 0 Furthr Probability Chaptr Limits and Drivativs Chaptr Discrt Random Variabls Chaptr Diffrntiation Chaptr Discrt Probability Distributions Chaptr Applications of Diffrntiation
More informationInference Methods for Stochastic Volatility Models
Intrnational Mathmatical Forum, Vol 8, 03, no 8, 369-375 Infrnc Mthods for Stochastic Volatility Modls Maddalna Cavicchioli Cá Foscari Univrsity of Vnic Advancd School of Economics Cannargio 3, Vnic, Italy
More informationComputing and Communications -- Network Coding
89 90 98 00 Computing and Communications -- Ntwork Coding Dr. Zhiyong Chn Institut of Wirlss Communications Tchnology Shanghai Jiao Tong Univrsity China Lctur 5- Nov. 05 0 Classical Information Thory Sourc
More informationElements of Statistical Thermodynamics
24 Elmnts of Statistical Thrmodynamics Statistical thrmodynamics is a branch of knowldg that has its own postulats and tchniqus. W do not attmpt to giv hr vn an introduction to th fild. In this chaptr,
More informationWhat are those βs anyway? Understanding Design Matrix & Odds ratios
Ral paramtr stimat WILD 750 - Wildlif Population Analysis of 6 What ar thos βs anyway? Undrsting Dsign Matrix & Odds ratios Rfrncs Hosmr D.W.. Lmshow. 000. Applid logistic rgrssion. John Wily & ons Inc.
More informationProcdings of IC-IDC0 ( and (, ( ( and (, and (f ( and (, rspctivly. If two input signals ar compltly qual, phas spctra of two signals ar qual. That is
Procdings of IC-IDC0 EFFECTS OF STOCHASTIC PHASE SPECTRUM DIFFERECES O PHASE-OLY CORRELATIO FUCTIOS PART I: STATISTICALLY COSTAT PHASE SPECTRUM DIFFERECES FOR FREQUECY IDICES Shunsu Yamai, Jun Odagiri,
More informationa 1and x is any real number.
Calcls Nots Eponnts an Logarithms Eponntial Fnction: Has th form y a, whr a 0, a an is any ral nmbr. Graph y, Graph y ln y y Th Natral Bas (Elr s nmbr): An irrational nmbr, symboliz by th lttr, appars
More informationEinstein Equations for Tetrad Fields
Apiron, Vol 13, No, Octobr 006 6 Einstin Equations for Ttrad Filds Ali Rıza ŞAHİN, R T L Istanbul (Turky) Evry mtric tnsor can b xprssd by th innr product of ttrad filds W prov that Einstin quations for
More informationIntroduction to the Fourier transform. Computer Vision & Digital Image Processing. The Fourier transform (continued) The Fourier transform (continued)
Introduction to th Fourir transform Computr Vision & Digital Imag Procssing Fourir Transform Lt f(x) b a continuous function of a ral variabl x Th Fourir transform of f(x), dnotd by I {f(x)} is givn by:
More informationComplex Powers and Logs (5A) Young Won Lim 10/17/13
Complx Powrs and Logs (5A) Copyright (c) 202, 203 Young W. Lim. Prmission is grantd to copy, distribut and/or modify this documnt undr th trms of th GNU Fr Documntation Licns, Vrsion.2 or any latr vrsion
More informationFull Waveform Inversion Using an Energy-Based Objective Function with Efficient Calculation of the Gradient
Full Wavform Invrsion Using an Enrgy-Basd Objctiv Function with Efficint Calculation of th Gradint Itm yp Confrnc Papr Authors Choi, Yun Sok; Alkhalifah, ariq Ali Citation Choi Y, Alkhalifah (217) Full
More informationTitle. Author(s)Pei, Soo-Chang; Ding, Jian-Jiun. Issue Date Doc URL. Type. Note. File Information. Citationand Conference:
Titl Uncrtainty Principl of th -D Affin Gnralizd Author(sPi Soo-Chang; Ding Jian-Jiun Procdings : APSIPA ASC 009 : Asia-Pacific Signal Citationand Confrnc: -7 Issu Dat 009-0-0 Doc URL http://hdl.handl.nt/5/39730
More informationImage Filtering: Noise Removal, Sharpening, Deblurring. Yao Wang Polytechnic University, Brooklyn, NY11201
Imag Filtring: Nois Rmoval, Sharpning, Dblurring Yao Wang Polytchnic Univrsity, Brooklyn, NY http://wb.poly.du/~yao Outlin Nois rmoval by avraging iltr Nois rmoval by mdian iltr Sharpning Edg nhancmnt
More informationQuasi-Classical States of the Simple Harmonic Oscillator
Quasi-Classical Stats of th Simpl Harmonic Oscillator (Draft Vrsion) Introduction: Why Look for Eignstats of th Annihilation Oprator? Excpt for th ground stat, th corrspondnc btwn th quantum nrgy ignstats
More information4. (5a + b) 7 & x 1 = (3x 1)log 10 4 = log (M1) [4] d = 3 [4] T 2 = 5 + = 16 or or 16.
. 7 7 7... 7 7 (n )0 7 (M) 0(n ) 00 n (A) S ((7) 0(0)) (M) (7 00) 8897 (A). (5a b) 7 7... (5a)... (M) 7 5 5 (a b ) 5 5 a b (M)(A) So th cofficint is 75 (A) (C) [] S (7 7) (M) () 8897 (A) (C) [] 5. x.55
More information(1) Then we could wave our hands over this and it would become:
MAT* K285 Spring 28 Anthony Bnoit 4/17/28 Wk 12: Laplac Tranform Rading: Kohlr & Johnon, Chaptr 5 to p. 35 HW: 5.1: 3, 7, 1*, 19 5.2: 1, 5*, 13*, 19, 45* 5.3: 1, 11*, 19 * Pla writ-up th problm natly and
More informationCHAPTER 5. Section 5-1
SECTION 5-9 CHAPTER 5 Sction 5-. An ponntial function is a function whr th variabl appars in an ponnt.. If b >, th function is an incrasing function. If < b
More informationA. Limits and Horizontal Asymptotes ( ) f x f x. f x. x "±# ( ).
A. Limits and Horizontal Asymptots What you ar finding: You can b askd to find lim x "a H.A.) problm is asking you find lim x "# and lim x "$#. or lim x "±#. Typically, a horizontal asymptot algbraically,
More informationCS 361 Meeting 12 10/3/18
CS 36 Mting 2 /3/8 Announcmnts. Homwork 4 is du Friday. If Friday is Mountain Day, homwork should b turnd in at my offic or th dpartmnt offic bfor 4. 2. Homwork 5 will b availabl ovr th wknd. 3. Our midtrm
More information