Lainiotis filter implementation. via Chandrasekhar type algorithm


 Katherine Cross
 1 years ago
 Views:
Transcription
1 Joural of Computatios & Modellig, vol.1, o.1, 2011, ISSN: prit, olie Iteratioal Scietific Press, 2011 Laiiotis filter implemetatio via Chadrasehar type algorithm Nicholas Assimais 1 ad Maria Adam 2 Abstract A implemetatio of the time ivariat Laiiotis filter usig a Chadrasehar type algorithm is preseted ad compared to the classical oe. he size of model determies which algorithm is faster; a method is proposed to apriori decide, which implemetatio is faster. I the ifiite measuremet oise case, the proposed method is always faster tha the classical oe. Mathematics Subject Classificatio : 93E11, 93C55, 68Q25 Keywords: Laiiotis filter, Chadrasehar algorithm 1 Itroductio Estimatio plays a importat role i may fields of sciece. he estimatio problem has bee solved by meas of a recursive algorithm based o Riccati 1 Departmet of Electroics, echological Educatioal Istitute of Lamia, Lamia 35100, Greece, 2 Departmet of Computer Sciece ad Biomedical Iformatics, Uiversity of Cetral Greece, Lamia 35100, Greece, Article Ifo: Revised : July 14, Published olie : August 31, 2011
2 116 Laiiotis filter implemetatio via Chadrasehar type algorithm type differece equatios. I the last decades, several authors have proposed faster algorithms to solve the estimatio problems by substitutig the Riccati equatios by a set of Chadrasehar type differece equatios [1, 5, 7, 8, 9]. he discrete time Laiiotis filter [6] is a well ow algorithm that solves the estimatio/filterig problem. I this paper, we propose a implemetatio of the time ivariat Laiiotis filter usig Chadrasehar type recursive algorithm to solve the estimatio/filterig problem. It is established that the classical ad the proposed implemetatios are equivalet with respect to their behavior. It is also developed a method to apriori before the Laiiotis filter s implemetatio decide which implemetatio is faster. his is very importat due to the fact that, i most realtime applicatios, it is essetial to obtai the estimate i the shortest possible time. he paper is orgaized as follows: I Sectio 2 the classical implemetatio of Laiiotis filter is preseted. I Sectio 3 the Chadrasehar type algorithm is preseted ad the proposed implemetatio of Laiiotis filter via Chadrasehar type algorithm is itroduced. I Sectio 4 the computatioal requiremets both implemetatios of Laiiotis filter are established ad comparisos are carried out. It is poited out that the proposed implemetatio may be faster tha the classical oe. I additio, a rule is established i order to decide which implemetatio is faster. 2 Classical Implemetatio of Laiiotis filter he estimatio problem arises i liear estimatio ad is associated with time ivariat systems described for 0 by the followig state space equatios: x +1 = F x + w z = Hx + v where x is the 1 state vector at time, z is the m 1 measuremet vector, F is the system trasitio matrix, H is the m output matrix, {w } ad {v } are idepedet Gaussia zeromea white ad ucorrelated radom processes, respectively, Q is the plat oise covariace matrix, R is the m m measuremet oise covariace matrix ad x 0 is a Gaussia
3 N. Assimais ad M. Adam 117 radom process with mea x 0 ad covariace P 0. I the sequel, cosider that Q, R are positive defiite matrices ad deote Q, R > O. he filterig problem is to produce a estimate at time L of the state vector usig measuremets till time L, i.e. the aim is to use the measuremets set {z 1, z 2,..., z L } i order to calculate a estimate value x L L of the state vector x L. he discrete time Laiiotis filter [6] is a well ow algorithm that solves the filterig problem. he estimatio x ad the correspodig estimatio error covariace matrix P at time are computed by the followig equatios, cosistig the Laiiotis filter, [ ] P = P + F I + P O P F 1 [ ] x = F I + P O x + [ ] + K + F I + P O P K m z +1 2 for 0, with iitial coditios P 0 0 = P 0 ad x 0 0 = x 0, where the followig costat matrices are calculated offlie: A = [ HQH + R ] 3 K = QH A 4 K m = F H A 5 P = Q K HQ = Q QH AHQ 6 F = F K HF = F QH AHF 7 O = K m HF = F H AHF 8 with F is a matrix, while K, K m are m matrices. he matrices P ad O are symmetric. Also, the m m matrix HQH + R is osigular, sice R > O, which meas that o measuremet is exact; this is reasoable i physical problems. Moreover, sice Q, R > O, A is a well defied m m symmetric ad positive defiite matrix as well as O is positive defiite. Furthermore, sice Q > O usig the matrix iversio Lemma 3 ad substitutig the matrix A by 3 i 6 we may rewrite P as P = Q QH [ HQH + R ] HQ = [ Q + H R H ] 9 3 Let A, C be osigular matrices, the holds: A + BCD = A A BC + DA B DA
4 118 Laiiotis filter implemetatio via Chadrasehar type algorithm from which it is clear that the symmetric P is a positive defiite matrix. Equatio 1 is the Riccati equatio emaatig from Laiiotis filter. I the case of ifiite measuremet oise R, we have A = O, K = O, K m = O, P = Q, F = F, O = O ad the Laiiotis filter becomes: P = P + F P F = Q + F P F 10 x = F x = F x 11 Equatio 10 is the Lyapuov equatio emaatig from Laiiotis filter. 3 Implemetatio of Laiiotis filter via Chadrasehar type algorithm For time ivariat systems, it is well ow [2] that if the sigal process model is asymptotically stable i.e. all eigevalues of F lie iside the uit circle, the there exists a steady state value P of the estimatio error covariace matrix. he steady state solutio P is calculated by recursively implemetig the Riccati equatio emaatig from Laiiotis filter 1 for = 0, 1,..., with iitial coditio P 0 0 = P 0. he steady state or limitig solutio of the Riccati equatio is idepedet of the iitial coditio [2]. he discrete time Riccati equatio emaatig from the Laiiotis filter equatios has attracted eormous attetio. I view of the importace of the Riccati equatio, there exists cosiderable literature o its recursive solutios [4, 7], cocerig per step or doublig algorithms. he Chadrasehar type algorithm has bee used [7, 8] to solve the Riccati equatio 1. he Chadrasehar type algorithm cosists of the recursio P = P + Y S Y usig recursios for the suitable quatities Y ad S. Hece, the algorithm is based o the idea of defiig the differece equatio ad its factorizatio δp = P P, 12 δp = Y S Y, 13
5 N. Assimais ad M. Adam 119 where Y is a r matrix ad S is a r r matrix, with For every = 0, 1,..., deotig 0 r = raδp 0. O = P + 14 we ote that O is a symmetric ad positive defiite matrix due to the presece of O, recallig that O i 8 is a positive defiite matrix ad P is a positive semidefiite as estimatio error covariace matrix. Also, sice O is a osigular matrix for every = 0, 1,..., the equatio 14 may be writte: P = I O 15 Usig the above otatios ad substitutig the equatios of the Laiiotis filter by a set of Chadrasehar type differece equatios, a recursive filterig algorithm is proposed, as established i the followig theorem, which presets computatioal advatage compared to the classical filterig algorithm, see 4 ad 5 statemets i the ext Sectio 4. heorem 3.1. Let the measuremet oise R be a positive defiite matrix, the plat oise Q be a positive defiite matrix ad P is a osigular matrix, for every = 0, 1, 2,... he set of the followig recursive equatios compose the ew algorithm for the solutio of the discrete time Laiiotis filter, O +1 = O + Y S Y 16 Y +1 = F O Y 17 S +1 = S S Y +1 Y S 18 P = P + Y S Y 19 x = F with iitial coditios: x + K + F P K m z+1, 20 P 0 0 = P 0 x 0 0 = x 0 O 0 = P Y 0 S 0 Y 0 = P + F [I + P 0 O ] P 0 F P 0 22 where F, O, K, K m, P are the matrices i 38.
6 120 Laiiotis filter implemetatio via Chadrasehar type algorithm Proof. Combiig 14 ad 12 we write O +1 = P = P O P = O + δp, i.e., for every = 0, 1, 2,..., holds δp = O +1 O, 23 i which substitutig δp by 13 the recursio equatio i 16 is obvious. Moreover, usig elemetary algebraic operatios ad properties we may write M + N = NM + N M, M N = N N MM, 24 whe M, N are osigular matrices, as well as [ I + P O ] P = [ ] P P + O P = [P + O ], 25 due to the osigularity of P for every = 0, 1,.... Combiig 12, 1, 25, the first equality i 24, 14 ad 15, we derive: δp +1 = P P = F [I + P+1 +1 O ] P [I + P O ] P F = F [P O ] [P + O ] F [ ] [ = F O P O P O P + O = F P+1 +1 [P O ] O P [P + = F P+1 +1 [P O ] P [P + O ] O = F P P O F = F O +1 O F P ] ] O F Usig the secod equality of 24 ad 23 the last equatio may be writte as: δp +1 = F = F = F = F +1 O +1 O O O O +1 δp O +1 δp δp O O F O F +1 δp F F O F F δp +1 δp F O F
7 N. Assimais ad M. Adam 121 From 13 the last equatio yields δp +1 = F Y S Y O F F i which settig the matrix Y +1 = F Y S Y +1 Y S Y Y by 17 immediately arises O F δp +1 = Y +1 S Y +1 Y +1 S Y +1 Y S Y By 13 we have δp +1 formulated as : = Y +1 S +1 Y+1, thus the equatio i 26 may be Y +1 S +1 Y +1 = Y +1 S S Y +1 Y S Y +1 Multiplyig with Y +1 o the left ad Y +1 o the right the last equality the recursio equatio i 18 is derived. Furthermore, rewritig x i 2 with differet way ad due to 14 we coclude [ ] x = F O I + P O x + [ ] + K + F O I + P O P K m z +1 [ ] = F O I + P O O x + [ ] + K + F O I + P O O P K m = F = F [ P + showig thus the equatio 20. ] x + K + F x + K + F P K m z+1 z +1 [ P + ] P K m z +1 Moreover, P 0 0 = P 0 ad x 0 0 = x 0 are give as the iitial coditios of the problem; by 14 O 0 is computed for = 0 ad the matrices Y 0, S 0 are computed by the factorizatio of the matrix P + F [I + P 0 O ] P 0 F P 0 i 22 i order to used as iitial coditios. Remar For the boudary values of r = raδp 0 we ote that: If r = 0, the, from 12 arises that the estimatio covariace matrix remais costat, i.e. P = P 0, ad equatio 23 yields O = O 0, for every = 0, 1, 2,... hus the algorithm of heorem 3.1 computes iteratively oly the estimatio x taig the form: x = F O O0 x + K + F O 0 P 0 K m z+1
8 122 Laiiotis filter implemetatio via Chadrasehar type algorithm If r = ad P 0 0 = P 0 = O, the we are able to use the iitial coditios Y 0 = I ad S 0 = P. 2. For the zero iitial coditio P 0 0 = P 0 = O, by 1 we derive P 1 1 = P ; recallig that by 9 holds P > O, it is evidet that for every = 1, 2,... arises P > O, that guaratees P be a osigular matrix. Hece heorem 3.1 is applicable for iitial coditio P 0 0 = P 0 = O; i this case by we are able to use the followig iitial coditios: O 0 = ad Y 0 S 0 Y 0 = P. 3.1 Ifiite measuremet oise R I the followig, the special case of ifiite measuremet oise is preseted. I this case P = Q, F = F ad O = O, the the Riccati equatio 1 becomes the Lyapuov equatio 10. Usig 12 ad combiig 10 with 13 we have δp +1 = P P = F P P F = F δp F = F Y S Y F, where settig Y +1 = F Y the above equality is formulated Y +1 S +1 Y δp +1 = Y +1 S Y+1 ad after some algebra arises : S +1 = S +1 = Sice the last equality of the matrices holds for every = 1, 2,..., without loss of geerality, we cosider a arbitrary r r symmetric matrix S = S, 27 with ras = r ad 0 < r. hus, usig 19, 11 ad 27 the followig filterig algorithm, which is based o the Chadrasehar type algorithm, is established. ad with iitial coditios: Y +1 = F Y P = P + Y SY x = F x, P 0 0 = P 0, x 0 0 = x 0, Y 0 SY 0 = Q + F P 0 F P 0
9 N. Assimais ad M. Adam 123 Sice i 27 the matrix S ca be arbitrarily chose, we propose as S the r r idetity matrix; thus we are able to establish the proposed algorithm, which is formulated i the ext theorem. heorem 3.2. Let R be the ifiite measuremet oise R, the plat oise Q be a positive defiite matrix ad F be a trasitio matrix. he set of the followig recursive equatios compose the algorithm for the solutio of the discrete time Laiiotis filter, for = 1, 2,..., with iitial coditios: Y +1 = F Y 28 P = P + Y Y 29 x = F x, 30 P 0 0 = P 0, x 0 0 = x 0, Y 0 Y 0 = Q + F P 0 F P 0 31 Remar 3.2. I the special case P 0 0 = P 0 = O, the equatio 31 becomes Y 0 Y 0 = Q. 4 Computatioal compariso of algorithms he two implemetatios of the Laiiotis filter preseted above are equivalet with respect to their behavior: they calculate theoretically the same estimates, due to the fact that equatios 12 are equivalet to equatios i heorem 3.1 i.e ad equatios are equivalet to equatios for the case of ifiite measuremet oise. he, it is reasoable to assume that both implemetatios of the Laiiotis filter compute the estimate value x L L of the state vector x L, executig the same umber of recursios. hus, i order to compare the algorithms, we have to compare their per recursio calculatio burde required for the olie calculatios; the calculatio burde of the offlie calculatios iitializatio process is ot tae ito accout. he computatioal aalysis is based o the aalysis i [3]: scalar operatios are ivolved i matrix maipulatio operatios, which are eeded for the implemetatio of the filterig algorithms. able 1 summarizes the calculatio burde of eeded matrix operatios.
10 124 Laiiotis filter implemetatio via Chadrasehar type algorithm able 1. Calculatio burde of matrix operatios Matrix Operatio Calculatio Burde A m + B m = C m m A + B = S S : symmetric I + A = B I : idetity A m Bm = C 2m A m Bm = S S : symmetric 2 m + m [A ] = B he per recursio calculatio burde of the Laiiotis filter implemetatios are summarized i able 2. he details are give i the Appedix. able 2. Per recursio calculatio burde of algorithms Implemetatio Noise Per recursio calculatio burde Classical R > O CB c,1 = m + 2m Classical R CB c,2 = Proposed R > O CB p,1 = r 2 Proposed R CB p,2 = 3 2 r From able 2, we derive the followig coclusios: 2r r m + 2m 1. he per recursio calculatio burde of the classical implemetatio depeds o the state vector dimesio. 2. he per recursio calculatio burde of the proposed implemetatio depeds o the state vector dimesio ad o r = raδp Cocerig the oifiite measuremet oise case R > O ad defiig q, r = CB c,1 CB p,1, from able 2 the respective calculatio burdes yield the relatio: q, r = r 2 + 2r 7 2 r 32 From Remar 3.1 the case r = 0 gives degeerated algorithm; thus cosider r 1 we ivestigate two cases : a r =, ad b r <. a 1 r =. I this case, it is obvious that q, = ad sice q, is a decreasig fuctio, we compute q, q1, 1 = 6 < 0. Hece, if r =, the the classical implemetatio is faster tha the proposed oe.
11 N. Assimais ad M. Adam 125 b 1 r <. I this case, we rewrite the equality i 32 as q, r = 6 8r r = fr,,33 6 with fr, = 8r r he discrimiat of fr, is ad its zeros are : = > 0, r 1 = , r 2 = Hece, the factorizatio of fr, is fr, = 8r r 1 r r 2, thus, the equality of q, r i 33 ca bee writte as q, r = 3r r 1 r r Also, it is easily proved that for = 1, 2,... holds > 42 12, from which immediately arises r 1 < 0 ad r 2 > 0; thus, due to the fact r 1, it is obvious r r 1 > 0. Cosequetly, i 35 the sig of q, r depeds o the sig of r r 2, with r 2 i 34, i.e., the choice of implemetatio of the suitable algorithm is related to the compariso of quatities r, r 2 ; if r > r 2 q, r < 0, thus the classical implemetatio is faster tha the proposed oe. if r < r 2 q, r > 0, thus the proposed implemetatio is faster tha the classical oe. 4. Figure 1 depicts the relatio betwee ad r that may hold i order to decide, which implemetatio is faster. I fact r is plotted as fuctio of usig r 2 i 34. he, we are able to establish the followig Rule of humb: the proposed Laiiotis filter implemetatio via Chadrasehar type algorithm is faster tha the classical implemetatio if the followig relatio holds: r <
12 126 Laiiotis filter implemetatio via Chadrasehar type algorithm Figure 1: Proposed algorithm may be faster tha the classical oe. hus, we are able to choose i advace the implemetatio of the faster algorithm comparig oly the quatities r ad by Cocerig the ifiite measuremet oise case R, the calculatio burde of the classical implemetatio is greater tha or equal to the calculatio burde of the proposed implemetatio; the equality holds for r =. hus, the proposed implemetatio is faster tha the classical oe. ACKNOWLEDGEMENS. he authors are deeply grateful to referees for suggestios that have cosiderably improved the quality of the paper. Refereces [1] Abdelhaim Aouche ad Fayçal Hamdi, Calculatig the autocovariaces ad the lielihood for periodic V ARMA models, Joural of Statistical Computatio ad Simulatio, 793, 2009, [2] B.D.O. Aderso, J.B. Moore, Optimal Filterig, Pretice Hall ic., [3] N. Assimais, M. Adam, Discrete time Kalma ad Laiiotis filters compariso, It. Joural of Mathematical Aalysis IJMA, , 2007, [4] N.D. Assimais, D.G. Laiiotis, S.K. Katsias, F.L. Saida, A survey of recursive algorithms for the solutio of the discrete time Riccati equatio, Noliear Aalysis, heory, Methods ad Applicatios, 30, 1997, [5] J.S. Baras ad D.G. Laiiotis, Chadrasehar algorithms for liear time varyig distributed systems, Iformatio Scieces, 172, 1979, [6] D.G. Laiiotis, Discrete Riccati Equatio Solutios: Partitioed Algorithms, IEEE rasactios o AC, AC20, 1975,
13 N. Assimais ad M. Adam 127 [7] D.G. Laiiotis, N.D. Assimais, S.K. Katsias, A ew computatioally effective algorithm for solvig the discrete Riccati equatio, Joural of Mathematical Aalysis ad Applicatios, 1863, 1994, [8] S. Naamori, A. HermosoCarazo, J. JiméezLópez ad J. LiaresPérez, Chadrasehartype filter for a widesese statioary sigal from ucertai observatios usig covariace iformatio, Applied Mathematics ad Computatio, 151, 2004, [9] S. Naamori, Chadrasehartype recursive Wieer estimatio techique i liear discretetime stochastic systems, Applied Mathematics ad Computatio, 1882, 2007,
14 128 Laiiotis filter implemetatio via Chadrasehar type algorithm Appedix Calculatio burdes of algorithms A Measuremet oise is a positive defiite matrix R > O A.1 Classical implemetatio of Laiiotis filter Matrix Operatio Matrix Dimesios Calculatio Burde P O I + P O + [I + P O ] [I + P O ] P F [I + P O ] P F [I + P O ] P F P = P + F [I + P O ] P F F [I + P O ] P K m m 2 2 m m K + F [I + P O ] P K m m + m m K + F [I + P O ] P K m z+1 m m 1 2m F [I + P O ] F [I + P O ] x x = F [I + P O ] x + + K + F [I + P O ] P K m z otal I idetity matrix symmetric matrix CB c,1 = m + 2m
15 N. Assimais ad M. Adam 129 A.2 Proposed implemetatio via Chadrasehar type algorithm Matrix Operatio Matrix Dimesios Calculatio Burde Y S r r r 2r 2 r Y S Y r r 2 r + r O +1 = O + Y S Y Y r 2 2 r r Y +1 = F O Y r 2 2 r r Y S r 2 2 r r S Y O +1 Y S r r r 2 + r 1 2 r2 + r S +1 = S S Y +1 Y S r r + r r 1 2 r2 + r P = P + Y S Y F O F O x F O P F O P K m m 2 2 m m K + F O P K m m + m m K + F O P K m z+1 m m 1 2m x = F x + + K + F O P K m z+1 otal CB p,1 = r 2 2r r m + 2m symmetric matrix
16 130 Laiiotis filter implemetatio via Chadrasehar type algorithm B Ifiite measuremet oise R B.1 Classical implemetatio of Laiiotis filter Matrix Operatio Matrix Dimesios Calculatio Burde F P F P F P = Q + F P F x = F x otal CB c,2 = symmetric matrix B.2 Proposed implemetatio via Chadrasehar type algorithm Matrix Operatio Matrix Dimesios Calculatio Burde Y +1 = F Y r 2 2 r r Y Y r r 2 r + r P = P + Y Y x = F x symmetric matrix otal CB p,2 = 3 2 r + 2 2
On Nonsingularity of Saddle Point Matrices. with Vectors of Ones
Iteratioal Joural of Algebra, Vol. 2, 2008, o. 4, 197204 O Nosigularity of Saddle Poit Matrices with Vectors of Oes Tadeusz Ostrowski Istitute of Maagemet The State Vocatioal Uiversity 400 Gorzów, Polad
More information6. Kalman filter implementation for linear algebraic equations. KarhunenLoeve decomposition
6. Kalma filter implemetatio for liear algebraic equatios. KarhueLoeve decompositio 6.1. Solvable liear algebraic systems. Probabilistic iterpretatio. Let A be a quadratic matrix (ot obligatory osigular.
More informationPAijpam.eu ON TENSOR PRODUCT DECOMPOSITION
Iteratioal Joural of Pure ad Applied Mathematics Volume 103 No 3 2015, 537545 ISSN: 13118080 (prited versio); ISSN: 13143395 (olie versio) url: http://wwwijpameu doi: http://dxdoiorg/1012732/ijpamv103i314
More informationThe standard deviation of the mean
Physics 6C Fall 20 The stadard deviatio of the mea These otes provide some clarificatio o the distictio betwee the stadard deviatio ad the stadard deviatio of the mea.. The sample mea ad variace Cosider
More informationPOWER SERIES SOLUTION OF FIRST ORDER MATRIX DIFFERENTIAL EQUATIONS
Joural of Applied Mathematics ad Computatioal Mechaics 4 3(3) 38 POWER SERIES SOLUION OF FIRS ORDER MARIX DIFFERENIAL EQUAIONS Staisław Kukla Izabela Zamorska Istitute of Mathematics Czestochowa Uiversity
More informationA Simplified Derivation of Scalar Kalman Filter using Bayesian Probability Theory
6th Iteratioal Workshop o Aalysis of Dyamic Measuremets Jue 3 0 Göteorg Swede A Simplified Derivatio of Scalar Kalma Filter usig Bayesia Proaility Theory Gregory Kyriazis Imetro  Electrical Metrology
More informationComparison Study of Series Approximation. and Convergence between Chebyshev. and Legendre Series
Applied Mathematical Scieces, Vol. 7, 03, o. 6, 3337 HIKARI Ltd, www.mhikari.com http://d.doi.org/0.988/ams.03.3430 Compariso Study of Series Approimatio ad Covergece betwee Chebyshev ad Legedre Series
More informationInfinite Sequences and Series
Chapter 6 Ifiite Sequeces ad Series 6.1 Ifiite Sequeces 6.1.1 Elemetary Cocepts Simply speakig, a sequece is a ordered list of umbers writte: {a 1, a 2, a 3,...a, a +1,...} where the elemets a i represet
More informationDiscreteTime Signals and Systems. DiscreteTime Signals and Systems. Signal Symmetry. Elementary DiscreteTime Signals.
Discreteime Sigals ad Systems Discreteime Sigals ad Systems Dr. Deepa Kudur Uiversity of oroto Referece: Sectios. .5 of Joh G. Proakis ad Dimitris G. Maolakis, Digital Sigal Processig: Priciples, Algorithms,
More informationComplex Analysis Spring 2001 Homework I Solution
Complex Aalysis Sprig 2001 Homework I Solutio 1. Coway, Chapter 1, sectio 3, problem 3. Describe the set of poits satisfyig the equatio z a z + a = 2c, where c > 0 ad a R. To begi, we see from the triagle
More informationStability Analysis of the Euler Discretization for SIR Epidemic Model
Stability Aalysis of the Euler Discretizatio for SIR Epidemic Model Agus Suryato Departmet of Mathematics, Faculty of Scieces, Brawijaya Uiversity, Jl Vetera Malag 6545 Idoesia Abstract I this paper we
More informationLecture 6 Chi Square Distribution (χ 2 ) and Least Squares Fitting
Lecture 6 Chi Square Distributio (χ ) ad Least Squares Fittig Chi Square Distributio (χ ) Suppose: We have a set of measuremets {x 1, x, x }. We kow the true value of each x i (x t1, x t, x t ). We would
More informationDiscrete Orthogonal Moment Features Using Chebyshev Polynomials
Discrete Orthogoal Momet Features Usig Chebyshev Polyomials R. Mukuda, 1 S.H.Og ad P.A. Lee 3 1 Faculty of Iformatio Sciece ad Techology, Multimedia Uiversity 75450 Malacca, Malaysia. Istitute of Mathematical
More informationTHE SYSTEMATIC AND THE RANDOM. ERRORS  DUE TO ELEMENT TOLERANCES OF ELECTRICAL NETWORKS
R775 Philips Res. Repts 26,414423, 1971' THE SYSTEMATIC AND THE RANDOM. ERRORS  DUE TO ELEMENT TOLERANCES OF ELECTRICAL NETWORKS by H. W. HANNEMAN Abstract Usig the law of propagatio of errors, approximated
More informationOscillation and Property B for Third Order Difference Equations with Advanced Arguments
Iter atioal Joural of Pure ad Applied Mathematics Volume 3 No. 0 207, 352 360 ISSN: 38080 (prited versio); ISSN: 343395 (olie versio) url: http://www.ijpam.eu ijpam.eu Oscillatio ad Property B for Third
More informationIn number theory we will generally be working with integers, though occasionally fractions and irrationals will come into play.
Number Theory Math 5840 otes. Sectio 1: Axioms. I umber theory we will geerally be workig with itegers, though occasioally fractios ad irratioals will come ito play. Notatio: Z deotes the set of all itegers
More informationProperties and Hypothesis Testing
Chapter 3 Properties ad Hypothesis Testig 3.1 Types of data The regressio techiques developed i previous chapters ca be applied to three differet kids of data. 1. Crosssectioal data. 2. Time series data.
More informationSession 5. (1) Principal component analysis and KarhunenLoève transformation
200 Autum semester Patter Iformatio Processig Topic 2 Image compressio by orthogoal trasformatio Sessio 5 () Pricipal compoet aalysis ad KarhueLoève trasformatio Topic 2 of this course explais the image
More informationSection 1 of Unit 03 (Pure Mathematics 3) Algebra
Sectio 1 of Uit 0 (Pure Mathematics ) Algebra Recommeded Prior Kowledge Studets should have studied the algebraic techiques i Pure Mathematics 1. Cotet This Sectio should be studied early i the course
More informationA Simplified Binet Formula for kgeneralized Fibonacci Numbers
A Simplified Biet Formula for kgeeralized Fiboacci Numbers Gregory P. B. Dresde Departmet of Mathematics Washigto ad Lee Uiversity Lexigto, VA 440 dresdeg@wlu.edu Zhaohui Du Shaghai, Chia zhao.hui.du@gmail.com
More information6.867 Machine learning, lecture 7 (Jaakkola) 1
6.867 Machie learig, lecture 7 (Jaakkola) 1 Lecture topics: Kerel form of liear regressio Kerels, examples, costructio, properties Liear regressio ad kerels Cosider a slightly simpler model where we omit
More informationRecurrence Relations
Recurrece Relatios Aalysis of recursive algorithms, such as: it factorial (it ) { if (==0) retur ; else retur ( * factorial()); } Let t be the umber of multiplicatios eeded to calculate factorial(). The
More informationMOMENTMETHOD ESTIMATION BASED ON CENSORED SAMPLE
Vol. 8 o. Joural of Systems Sciece ad Complexity Apr., 5 MOMETMETHOD ESTIMATIO BASED O CESORED SAMPLE I Zhogxi Departmet of Mathematics, East Chia Uiversity of Sciece ad Techology, Shaghai 37, Chia. Email:
More informationw (1) ˆx w (1) x (1) /ρ and w (2) ˆx w (2) x (2) /ρ.
2 5. Weighted umber of late jobs 5.1. Release dates ad due dates: maximimizig the weight of otime jobs Oce we add release dates, miimizig the umber of late jobs becomes a sigificatly harder problem. For
More informationMAT1026 Calculus II Basic Convergence Tests for Series
MAT026 Calculus II Basic Covergece Tests for Series Egi MERMUT 202.03.08 Dokuz Eylül Uiversity Faculty of Sciece Departmet of Mathematics İzmir/TURKEY Cotets Mootoe Covergece Theorem 2 2 Series of Real
More information62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 +
62. Power series Defiitio 16. (Power series) Give a sequece {c }, the series c x = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + is called a power series i the variable x. The umbers c are called the coefficiets of
More informationPhysics 324, Fall Dirac Notation. These notes were produced by David Kaplan for Phys. 324 in Autumn 2001.
Physics 324, Fall 2002 Dirac Notatio These otes were produced by David Kapla for Phys. 324 i Autum 2001. 1 Vectors 1.1 Ier product Recall from liear algebra: we ca represet a vector V as a colum vector;
More informationThe Choquet Integral with Respect to FuzzyValued Set Functions
The Choquet Itegral with Respect to FuzzyValued Set Fuctios Weiwei Zhag Abstract The Choquet itegral with respect to realvalued oadditive set fuctios, such as siged efficiecy measures, has bee used i
More informationSolving a Nonlinear Equation Using a New TwoStep Derivative Free Iterative Methods
Applied ad Computatioal Mathematics 07; 6(6): 384 http://www.sciecepublishiggroup.com/j/acm doi: 0.648/j.acm.070606. ISSN: 385605 (Prit); ISSN: 38563 (Olie) Solvig a Noliear Equatio Usig a New TwoStep
More informationDefinition 4.2. (a) A sequence {x n } in a Banach space X is a basis for X if. unique scalars a n (x) such that x = n. a n (x) x n. (4.
4. BASES I BAACH SPACES 39 4. BASES I BAACH SPACES Sice a Baach space X is a vector space, it must possess a Hamel, or vector space, basis, i.e., a subset {x γ } γ Γ whose fiite liear spa is all of X ad
More informationA NOTE ON THE TOTAL LEAST SQUARES FIT TO COPLANAR POINTS
A NOTE ON THE TOTAL LEAST SQUARES FIT TO COPLANAR POINTS STEVEN L. LEE Abstract. The Total Least Squares (TLS) fit to the poits (x,y ), =1,,, miimizes the sum of the squares of the perpedicular distaces
More informationECE 901 Lecture 12: Complexity Regularization and the Squared Loss
ECE 90 Lecture : Complexity Regularizatio ad the Squared Loss R. Nowak 5/7/009 I the previous lectures we made use of the Cheroff/Hoeffdig bouds for our aalysis of classifier errors. Hoeffdig s iequality
More informationRearranging the Alternating Harmonic Series
Rearragig the Alteratig Harmoic Series Da Teague C School of Sciece ad Mathematics teague@cssm.edu 00 TCM Coferece CSSM, Durham, C Regroupig Ifiite Sums We kow that the Taylor series for l( x + ) is x
More informationA Risk Comparison of Ordinary Least Squares vs Ridge Regression
Joural of Machie Learig Research 14 (2013) 15051511 Submitted 5/12; Revised 3/13; Published 6/13 A Risk Compariso of Ordiary Least Squares vs Ridge Regressio Paramveer S. Dhillo Departmet of Computer
More informationA collocation method for singular integral equations with cosecant kernel via Semitrigonometric interpolation
Iteratioal Joural of Mathematics Research. ISSN 09765840 Volume 9 Number 1 (017) pp. 4551 Iteratioal Research Publicatio House http://www.irphouse.com A collocatio method for sigular itegral equatios
More informationTeaching Mathematics Concepts via Computer Algebra Systems
Iteratioal Joural of Mathematics ad Statistics Ivetio (IJMSI) EISSN: 4767 PISSN:  4759 Volume 4 Issue 7 September. 6 PP Teachig Mathematics Cocepts via Computer Algebra Systems Osama Ajami Rashaw,
More informationQBINOMIALS AND THE GREATEST COMMON DIVISOR. Keith R. Slavin 8474 SW Chevy Place, Beaverton, Oregon 97008, USA.
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 8 2008, #A05 QBINOMIALS AND THE GREATEST COMMON DIVISOR Keith R. Slavi 8474 SW Chevy Place, Beaverto, Orego 97008, USA slavi@dsloly.et Received:
More information17 Phonons and conduction electrons in solids (Hiroshi Matsuoka)
7 Phoos ad coductio electros i solids Hiroshi Matsuoa I this chapter we will discuss a miimal microscopic model for phoos i a solid ad a miimal microscopic model for coductio electros i a simple metal.
More informationTEACHER CERTIFICATION STUDY GUIDE
COMPETENCY 1. ALGEBRA SKILL 1.1 1.1a. ALGEBRAIC STRUCTURES Kow why the real ad complex umbers are each a field, ad that particular rigs are ot fields (e.g., itegers, polyomial rigs, matrix rigs) Algebra
More informationIntroduction to Machine Learning DIS10
CS 189 Fall 017 Itroductio to Machie Learig DIS10 1 Fu with Lagrage Multipliers (a) Miimize the fuctio such that f (x,y) = x + y x + y = 3. Solutio: The Lagragia is: L(x,y,λ) = x + y + λ(x + y 3) Takig
More informationMachine Learning Theory Tübingen University, WS 2016/2017 Lecture 12
Machie Learig Theory Tübige Uiversity, WS 06/07 Lecture Tolstikhi Ilya Abstract I this lecture we derive risk bouds for kerel methods. We will start by showig that Soft Margi kerel SVM correspods to miimizig
More informationSEQUENCES AND SERIES
9 SEQUENCES AND SERIES INTRODUCTION Sequeces have may importat applicatios i several spheres of huma activities Whe a collectio of objects is arraged i a defiite order such that it has a idetified first
More informationON POINTWISE BINOMIAL APPROXIMATION
Iteratioal Joural of Pure ad Applied Mathematics Volume 71 No. 1 2011, 5766 ON POINTWISE BINOMIAL APPROXIMATION BY wfunctions K. Teerapabolar 1, P. Wogkasem 2 Departmet of Mathematics Faculty of Sciece
More informationChapter 6 Principles of Data Reduction
Chapter 6 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 0 Chapter 6 Priciples of Data Reductio Sectio 6. Itroductio Goal: To summarize or reduce the data X, X,, X to get iformatio about a
More informationPolynomial Multiplication and Fast Fourier Transform
Polyomial Multiplicatio ad Fast Fourier Trasform Com S 477/577 Notes YaBi Jia Sep 19, 2017 I this lecture we will describe the famous algorithm of fast Fourier trasform FFT, which has revolutioized digital
More informationSubject: Differential Equations & Mathematical ModelingIII
Power Series Solutios of Differetial Equatios about Sigular poits Subject: Differetial Equatios & Mathematical ModeligIII Lesso: Power series solutios of differetial equatios about Sigular poits Lesso
More informationGoodnessOfFit For The Generalized Exponential Distribution. Abstract
GoodessOfFit For The Geeralized Expoetial Distributio By Amal S. Hassa stitute of Statistical Studies & Research Cairo Uiversity Abstract Recetly a ew distributio called geeralized expoetial or expoetiated
More informationDisjoint unions of complete graphs characterized by their Laplacian spectrum
Electroic Joural of Liear Algebra Volume 18 Volume 18 (009) Article 56 009 Disjoit uios of complete graphs characterized by their Laplacia spectrum Romai Boulet boulet@uivtlse.fr Follow this ad additioal
More informationCALCULATION OF FIBONACCI VECTORS
CALCULATION OF FIBONACCI VECTORS Stuart D. Aderso Departmet of Physics, Ithaca College 953 Daby Road, Ithaca NY 14850, USA email: saderso@ithaca.edu ad Dai Novak Departmet of Mathematics, Ithaca College
More informationDISTRIBUTION LAW Okunev I.V.
1 DISTRIBUTION LAW Okuev I.V. Distributio law belogs to a umber of the most complicated theoretical laws of mathematics. But it is also a very importat practical law. Nothig ca help uderstad complicated
More informationA RANK STATISTIC FOR NONPARAMETRIC KSAMPLE AND CHANGE POINT PROBLEMS
J. Japa Statist. Soc. Vol. 41 No. 1 2011 67 73 A RANK STATISTIC FOR NONPARAMETRIC KSAMPLE AND CHANGE POINT PROBLEMS Yoichi Nishiyama* We cosider ksample ad chage poit problems for idepedet data i a
More informationComputational Tutorial of Steepest Descent Method and Its Implementation in Digital Image Processing
Computatioal utorial of Steepest Descet Method ad Its Implemetatio i Digital Image Processig Vorapoj Pataavijit Departmet of Electrical ad Electroic Egieerig, Faculty of Egieerig Assumptio Uiversity, Bagkok,
More informationNumber of fatalities X Sunday 4 Monday 6 Tuesday 2 Wednesday 0 Thursday 3 Friday 5 Saturday 8 Total 28. Day
LECTURE # 8 Mea Deviatio, Stadard Deviatio ad Variace & Coefficiet of variatio Mea Deviatio Stadard Deviatio ad Variace Coefficiet of variatio First, we will discuss it for the case of raw data, ad the
More informationCHAPTER 5. Theory and Solution Using Matrix Techniques
A SERIES OF CLASS NOTES FOR 20052006 TO INTRODUCE LINEAR AND NONLINEAR PROBLEMS TO ENGINEERS, SCIENTISTS, AND APPLIED MATHEMATICIANS DE CLASS NOTES 3 A COLLECTION OF HANDOUTS ON SYSTEMS OF ORDINARY DIFFERENTIAL
More informationDimensionfree PACBayesian bounds for the estimation of the mean of a random vector
Dimesiofree PACBayesia bouds for the estimatio of the mea of a radom vector Olivier Catoi CREST CNRS UMR 9194 Uiversité Paris Saclay olivier.catoi@esae.fr Ilaria Giulii Laboratoire de Probabilités et
More informationGeneralization of Samuelson s inequality and location of eigenvalues
Proc. Idia Acad. Sci. Math. Sci.) Vol. 5, No., February 05, pp. 03. c Idia Academy of Scieces Geeralizatio of Samuelso s iequality ad locatio of eigevalues R SHARMA ad R SAINI Departmet of Mathematics,
More informationSequences of Definite Integrals, Factorials and Double Factorials
47 6 Joural of Iteger Sequeces, Vol. 8 (5), Article 5.4.6 Sequeces of Defiite Itegrals, Factorials ad Double Factorials Thierry DaaPicard Departmet of Applied Mathematics Jerusalem College of Techology
More informationNotes on iteration and Newton s method. Iteration
Notes o iteratio ad Newto s method Iteratio Iteratio meas doig somethig over ad over. I our cotet, a iteratio is a sequece of umbers, vectors, fuctios, etc. geerated by a iteratio rule of the type 1 f
More information1 Covariance Estimation
Eco 75 Lecture 5 Covariace Estimatio ad Optimal Weightig Matrices I this lecture, we cosider estimatio of the asymptotic covariace matrix B B of the extremum estimator b : Covariace Estimatio Lemma 4.
More information1 Hash tables. 1.1 Implementation
Lecture 8 Hash Tables, Uiversal Hash Fuctios, Balls ad Bis Scribes: Luke Johsto, Moses Charikar, G. Valiat Date: Oct 18, 2017 Adapted From Virgiia Williams lecture otes 1 Hash tables A hash table is a
More informationProbability, Expectation Value and Uncertainty
Chapter 1 Probability, Expectatio Value ad Ucertaity We have see that the physically observable properties of a quatum system are represeted by Hermitea operators (also referred to as observables ) such
More informationa for a 1 1 matrix. a b a b 2 2 matrix: We define det ad bc 3 3 matrix: We define a a a a a a a a a a a a a a a a a a
Math Sb Lecture # Notes This wee is all about determiats We ll discuss how to defie them, how to calculate them, lear the allimportat property ow as multiliearity, ad show that a square matrix A is ivertible
More informationRademacher Complexity
EECS 598: Statistical Learig Theory, Witer 204 Topic 0 Rademacher Complexity Lecturer: Clayto Scott Scribe: Ya Deg, Kevi Moo Disclaimer: These otes have ot bee subjected to the usual scrutiy reserved for
More informationFIXED POINTS OF nvalued MULTIMAPS OF THE CIRCLE
FIXED POINTS OF VALUED MULTIMAPS OF THE CIRCLE Robert F. Brow Departmet of Mathematics Uiversity of Califoria Los Ageles, CA 900951555 email: rfb@math.ucla.edu November 15, 2005 Abstract A multifuctio
More informationA PROOF OF THE TWIN PRIME CONJECTURE AND OTHER POSSIBLE APPLICATIONS
A PROOF OF THE TWI PRIME COJECTURE AD OTHER POSSIBLE APPLICATIOS by PAUL S. BRUCKMA 38 Frot Street, #3 aaimo, BC V9R B8 (Caada) email : pbruckma@hotmail.com ABSTRACT : A elemetary proof of the Twi Prime
More informationSeries III. Chapter Alternating Series
Chapter 9 Series III With the exceptio of the Null Sequece Test, all the tests for series covergece ad divergece that we have cosidered so far have dealt oly with series of oegative terms. Series with
More informationSome New Iterative Methods for Solving Nonlinear Equations
World Applied Scieces Joural 0 (6): 870874, 01 ISSN 1818495 IDOSI Publicatios, 01 DOI: 10.589/idosi.wasj.01.0.06.830 Some New Iterative Methods for Solvig Noliear Equatios Muhammad Aslam Noor, Khalida
More informationA Block Cipher Using Linear Congruences
Joural of Computer Sciece 3 (7): 556560, 2007 ISSN 15493636 2007 Sciece Publicatios A Block Cipher Usig Liear Cogrueces 1 V.U.K. Sastry ad 2 V. Jaaki 1 Academic Affairs, Sreeidhi Istitute of Sciece &
More informationTrue Nature of Potential Energy of a Hydrogen Atom
True Nature of Potetial Eergy of a Hydroge Atom Koshu Suto Key words: Bohr Radius, Potetial Eergy, Rest Mass Eergy, Classical Electro Radius PACS codes: 365Sq, 365w, 33+p Abstract I cosiderig the potetial
More informationChapter 11 Output Analysis for a Single Model. Banks, Carson, Nelson & Nicol DiscreteEvent System Simulation
Chapter Output Aalysis for a Sigle Model Baks, Carso, Nelso & Nicol DiscreteEvet System Simulatio Error Estimatio If {,, } are ot statistically idepedet, the S / is a biased estimator of the true variace.
More informationUsing An Accelerating Method With The Trapezoidal And MidPoint Rules To Evaluate The Double Integrals With Continuous Integrands Numerically
ISSN 50 (Paper) ISSN 505 (Olie) Vol.7, No., 017 Usig A Acceleratig Method With The Trapezoidal Ad MidPoit Rules To Evaluate The Double Itegrals With Cotiuous Itegrads Numerically Azal Taha Abdul Wahab
More informationFrequency Response of FIR Filters
EEL335: DiscreteTime Sigals ad Systems. Itroductio I this set of otes, we itroduce the idea of the frequecy respose of LTI systems, ad focus specifically o the frequecy respose of FIR filters.. Steadystate
More informationLecture 3 The Lebesgue Integral
Lecture 3: The Lebesgue Itegral 1 of 14 Course: Theory of Probability I Term: Fall 2013 Istructor: Gorda Zitkovic Lecture 3 The Lebesgue Itegral The costructio of the itegral Uless expressly specified
More informationBasis for simulation techniques
Basis for simulatio techiques M. Veeraraghava, March 7, 004 Estimatio is based o a collectio of experimetal outcomes, x, x,, x, where each experimetal outcome is a value of a radom variable. x i. Defiitios
More informationThis is an introductory course in Analysis of Variance and Design of Experiments.
1 Notes for M 384E, Wedesday, Jauary 21, 2009 (Please ote: I will ot pass out hardcopy class otes i future classes. If there are writte class otes, they will be posted o the web by the ight before class
More informationK. Grill Institut für Statistik und Wahrscheinlichkeitstheorie, TU Wien, Austria
MARKOV PROCESSES K. Grill Istitut für Statistik ud Wahrscheilichkeitstheorie, TU Wie, Austria Keywords: Markov process, Markov chai, Markov property, stoppig times, strog Markov property, trasitio matrix,
More informationAddition: Property Name Property Description Examples. a+b = b+a. a+(b+c) = (a+b)+c
Notes for March 31 Fields: A field is a set of umbers with two (biary) operatios (usually called additio [+] ad multiplicatio [ ]) such that the followig properties hold: Additio: Name Descriptio Commutativity
More informationInternational Journal of Mathematical Archive3(4), 2012, Page: Available online through ISSN
Iteratioal Joural of Mathematical Archive3(4,, Page: 544553 Available olie through www.ima.ifo ISSN 9 546 INEQUALITIES CONCERNING THE BOPERATORS N. A. Rather, S. H. Ahager ad M. A. Shah* P. G. Departmet
More information5.1. The Rayleigh s quotient. Definition 49. Let A = A be a selfadjoint matrix. quotient is the function. R(x) = x,ax, for x = 0.
40 RODICA D. COSTIN 5. The Rayleigh s priciple ad the i priciple for the eigevalues of a selfadjoit matrix Eigevalues of selfadjoit matrices are easy to calculate. This sectio shows how this is doe usig
More informationResearch Article A New SecondOrder Iteration Method for Solving Nonlinear Equations
Abstract ad Applied Aalysis Volume 2013, Article ID 487062, 4 pages http://dx.doi.org/10.1155/2013/487062 Research Article A New SecodOrder Iteratio Method for Solvig Noliear Equatios Shi Mi Kag, 1 Arif
More informationThe Adomian Polynomials and the New Modified Decomposition Method for BVPs of nonlinear ODEs
Mathematical Computatio March 015, Volume, Issue 1, PP.1 6 The Adomia Polyomials ad the New Modified Decompositio Method for BVPs of oliear ODEs Jusheg Dua # School of Scieces, Shaghai Istitute of Techology,
More informationProof of Fermat s Last Theorem by Algebra Identities and Linear Algebra
Proof of Fermat s Last Theorem by Algebra Idetities ad Liear Algebra Javad Babaee Ragai Youg Researchers ad Elite Club, Qaemshahr Brach, Islamic Azad Uiversity, Qaemshahr, Ira Departmet of Civil Egieerig,
More informationMatsubaraGreen s Functions
MatsubaraGree s Fuctios Time Orderig : Cosider the followig operator If H = H the we ca trivially factorise this as, E(s = e s(h+ E(s = e sh e s I geeral this is ot true. However for practical applicatio
More informationZeros of Polynomials
Math 160 www.timetodare.com 4.5 4.6 Zeros of Polyomials I these sectios we will study polyomials algebraically. Most of our work will be cocered with fidig the solutios of polyomial equatios of ay degree
More informationCLOSED FORM FORMULA FOR THE NUMBER OF RESTRICTED COMPOSITIONS
Submitted to the Bulleti of the Australia Mathematical Society doi:10.1017/s... CLOSED FORM FORMULA FOR THE NUMBER OF RESTRICTED COMPOSITIONS GAŠPER JAKLIČ, VITO VITRIH ad EMIL ŽAGAR Abstract I this paper,
More information( ) (( ) ) ANSWERS TO EXERCISES IN APPENDIX B. Section B.1 VECTORS AND SETS. Exercise B.11: Convex sets. are convex, , hence. and. (a) Let.
Joh Riley 8 Jue 03 ANSWERS TO EXERCISES IN APPENDIX B Sectio B VECTORS AND SETS Exercise B: Covex sets (a) Let 0 x, x X, X, hece 0 x, x X ad 0 x, x X Sice X ad X are covex, x X ad x X The x X X, which
More informationA note on selfnormalized DickeyFuller test for unit root in autoregressive time series with GARCH errors
Appl. Math. J. Chiese Uiv. 008, 3(): 970 A ote o selformalized DickeyFuller test for uit root i autoregressive time series with GARCH errors YANG Xiaorog ZHANG Lixi Abstract. I this article, the uit
More information6.883: Online Methods in Machine Learning Alexander Rakhlin
6.883: Olie Methods i Machie Learig Alexader Rakhli LECTURES 5 AND 6. THE EXPERTS SETTING. EXPONENTIAL WEIGHTS All the algorithms preseted so far halluciate the future values as radom draws ad the perform
More informationMatrices and vectors
Oe Matrices ad vectors This book takes for grated that readers have some previous kowledge of the calculus of real fuctios of oe real variable It would be helpful to also have some kowledge of liear algebra
More informationCALCULATING FIBONACCI VECTORS
THE GENERALIZED BINET FORMULA FOR CALCULATING FIBONACCI VECTORS Stuart D Aderso Departmet of Physics, Ithaca College 953 Daby Road, Ithaca NY 14850, USA email: saderso@ithacaedu ad Dai Novak Departmet
More informationStudy on Coal Consumption Curve Fitting of the Thermal Power Based on Genetic Algorithm
Joural of ad Eergy Egieerig, 05, 3, 43437 Published Olie April 05 i SciRes. http://www.scirp.org/joural/jpee http://dx.doi.org/0.436/jpee.05.34058 Study o Coal Cosumptio Curve Fittig of the Thermal Based
More informationChapter 3. Strong convergence. 3.1 Definition of almost sure convergence
Chapter 3 Strog covergece As poited out i the Chapter 2, there are multiple ways to defie the otio of covergece of a sequece of radom variables. That chapter defied covergece i probability, covergece i
More informationIntegrable Properties Associated with a Discrete ThreebyThree Matrix Spectral Problem
Commu. Theor. Phys. Beijig Chia 52 2009 pp. 981 986 c Chiese Physical Society ad IOP Publishig Ltd Vol. 52 No. 6 December 15 2009 Itegrable Properties Associated with a Discrete ThreebyThree Matrix Spectral
More informationPower Comparison of Some Goodnessoffit Tests
Florida Iteratioal Uiversity FIU Digital Commos FIU Electroic Theses ad Dissertatios Uiversity Graduate School 762016 Power Compariso of Some Goodessoffit Tests Tiayi Liu tliu019@fiu.edu DOI: 10.25148/etd.FIDC000750
More informationPHY4905: NearlyFree Electron Model (NFE)
PHY4905: NearlyFree Electro Model (NFE) D. L. Maslov Departmet of Physics, Uiversity of Florida (Dated: Jauary 12, 2011) 1 I. REMINDER: QUANTUM MECHANICAL PERTURBATION THEORY A. Nodegeerate eigestates
More informationLøsningsførslag i 4M
Norges tekisk aturviteskapelige uiversitet Istitutt for matematiske fag Side 1 av 6 Løsigsførslag i 4M Oppgave 1 a) A sketch of the graph of the give f o the iterval [ 3, 3) is as follows: The Fourier
More information3/8/2016. Contents in latter part PATTERN RECOGNITION AND MACHINE LEARNING. Dynamical Systems. Dynamical Systems. Linear Dynamical Systems
Cotets i latter part PATTERN RECOGNITION AND MACHINE LEARNING CHAPTER 13: SEQUENTIAL DATA Liear Dyamical Systems What is differet from HMM? Kalma filter Its stregth ad limitatio Particle Filter Its simple
More informationComplex Algorithms for Lattice Adaptive IIR Notch Filter
4th Iteratioal Coferece o Sigal Processig Systems (ICSPS ) IPCSIT vol. 58 () () IACSIT Press, Sigapore DOI:.7763/IPCSIT..V58. Complex Algorithms for Lattice Adaptive IIR Notch Filter Hog Liag +, Nig Jia
More informationDe la Vallée Poussin Summability, the Combinatorial Sum 2n 1
J o u r a l of Mathematics ad Applicatios JMA No 40, pp 520 (2017 De la Vallée Poussi Summability, the Combiatorial Sum 1 ( 2 ad the de la Vallée Poussi Meas Expasio Ziad S. Ali Abstract: I this paper
More informationAn analog of the arithmetic triangle obtained by replacing the products by the least common multiples
arxiv:10021383v2 [mathnt] 9 Feb 2010 A aalog of the arithmetic triagle obtaied by replacig the products by the least commo multiples Bair FARHI bairfarhi@gmailcom MSC: 11A05 Keywords: AlKaraji s triagle;
More information