Computational Fluid Dynamics. Numerical Methods for Parabolic Equations. Numerical Methods for One-Dimensional Heat Equations
|
|
|
- Joshua Woods
- 5 years ago
- Views:
Transcription
1 Compuaoal Flud Dyamcs p:// Compuaoal Flud Dyamcs p:// Compuaoal Flud Dyamcs Numecal Meods o Paabolc Equaos Lecue Mac 6 7 Géa Tyggvaso Compuaoal Flud Dyamcs Oule Compuaoal Flud Dyamcs Soluo Meods o Paabolc Equaos Oe-Dmesoal Poblems Explc mplc Cak-Ncolso Accuacy sably Vaous scemes Mul-Dmesoal Poblems Aleag Deco Implc (ADI Appoxmae Facozao o Cak-Ncolso Numecal Meods o Oe-Dmesoal Hea Equaos Splg Compuaoal Flud Dyamcs Compuaoal Flud Dyamcs Cosde e duso equao ; x > a < x < b wc s a paabolc equao equg ( x ( x Ial Codo ad ( a φ a ( ; ( b φb ( Bouday Codo (Dcle o ( a ϕ ( ; ( b ϕ ( Bouday Codo a b x x (Neuma Paabolc equaos ca be vewed as e lm o a ypebolc equao w wo caacescs as e sgal speed goes o y Iceasg sgal speed x
2 Compuaoal Flud Dyamcs Explc Meod: FTCS - Compuaoal Flud Dyamcs Explc Meod: FTCS - Explc: FTCS x ( - x Moded Equao ( 6 x wee - Accuacy O( - No odd devaves; dsspave O( Compuaoal Flud Dyamcs Explc Meod: FTCS - 3 Sably: vo Neuma Aalyss ε ε s k < < < G s (β / Foue Codo wee β k BC Compuaoal Flud Dyamcs Explc Meod: FTCS - Doma o Depedece o Explc Sceme Bouday eec s o el a P o may me seps Ts may esul upyscal soluo beavo P Ial Daa BC x I e lm o ad x gog o zeo e compuaoal doma coas e pyscal doma o depedece sce ~ x o sably Compuaoal Flud Dyamcs Compuaoal Flud Dyamcs Implc Meod Implc Meod o e Oe-Dmesoal Hea Equao Implc Meod: Backwad Eule T-dagoal max sysem ( ( a d c b -
3 Compuaoal Flud Dyamcs Implc Meod a d c b We ou d c b I e edpos ae gve a d c 3 b a N N d N N c N N b N a N N d N N b N b a b N c N N Gaussa Elmao Compuaoal Flud Dyamcs Implc Meod - Pvog: eaagg equaos o pu e lages coece o e ma dagoal. - Elmae e colum below ma dagoal. - Repea ul e las equao s eaced. - Back-subsuo a a c a a c a a c è a a c a c a c - Specal case: -dagoal max - Tomas algom Compuaoal Flud Dyamcs Implc Meod % solvg a dagoal sysem x5;.; azeos(x(;dzeos(x-;czeos(x(-; bzeos(x; xzeos(x; b(x-(-; % owad elmao o :x d(d(-(a(/d(-*c(-; b(b(-(a(/d(-*b(-; x(b(/d(; ed % backwad subsuo o x-:-: x((b(-c(*x(/d(; ed Malab ucos Compuaoal Flud Dyamcs Implc Meod Fo smple poblems MATLAB as a umbe o ucos o deal w maces. Help mau: geeal Help spau:spase maces plo(x Compuaoal Flud Dyamcs Implc Meod Moded Equao x ( 6 O( - Te sg suggess a mplc meod may be less accuae a a caeully mplemeed explc meod. Amplcao Faco (vo Neuma aalyss G [ ( cosβ ] Ucodoally sable Compuaoal Flud Dyamcs Cak-Ncolso - Cak-Ncolso Meod (97 ( T-dagoal max sysem - (
4 Compuaoal Flud Dyamcs Cak-Ncolso - Moded Equao 3 x - Secod-ode accuacy O( 36 Amplcao Faco (vo Neuma aalyss G ( cos β ( cos β Ucodoally sable Geealzao Compuaoal Flud Dyamcs Combed Meod A - θ ( θ Explc (FTCS θ Implc / Cak Ncolso θ ( θ - Compuaoal Flud Dyamcs Combed Meod B - Geealzed Tee-Tme-Level Implc Sceme: Rcmye ad Moo (967 ( θ θ θ / - - Implc Tee - level ully mplc ( θ θ Moded Equao: Compuaoal Flud Dyamcs Combed Meod B - θ O( Specal Cases: θ O( θ ( O Compuaoal Flud Dyamcs Rcadso Meod Compuaoal Flud Dyamcs DuFo-Fakel - Rcadso s Meod Smla o Leapog O( bu ucodoally usable! - - Te Rcadso meod ca be made sable by splg by me aveage - - ( / ( (
5 Compuaoal Flud Dyamcs 3 Moded Equao DuFo-Fakel Amplcao aco G s cos ± β β Ucodoally sable Codoally cosse Compuaoal Flud Dyamcs FTCS Sable o BTCS Ucodoally Sable Cak-Ncolso Ucodoally Sable Rcadso Ucodoally Usable ( 6 Paabolc Equao Elemeay Scemes ( 6 ( ( 3 ( O Compuaoal Flud Dyamcs DuFo-Fakel Ucodoally Sable Codoally Cosse 3-Level Implc Ucodoally Sable ( 3 Ad oes! Paabolc Equao Elemeay Scemes Compuaoal Flud Dyamcs Numecal Meods o Mul-Dmesoal Hea Equaos Compuaoal Flud Dyamcs Two-dmesoal gd - - y x Cosde a -D ea equao Compuaoal Flud Dyamcs y x Applyg owad Eule sceme: y x -D ea equao y x I ( Explc Meod -
6 J I J I J I max om J I J I J Compuaoal Flud Dyamcs Explc Meod - J I J I J Vo Neuma Aalyss Compuaoal Flud Dyamcs Explc Meod - 3 ε ε e e kx my ε ε ( e k e k e m e m ε ε ε Wos case ( cosk cosm k m s s 8 Compuaoal Flud Dyamcs Sably Compuaoal Flud Dyamcs Sably lms deped o e dmeso o e poblems 6 Oe-dmesoal low Two-dmesoal low Tee-dmesoal low Implc me egao Dee umecal algoms usually ave dee sably lms Compuaoal Flud Dyamcs Implc Meods Recall owad me meod ( Evaluae e spaal devaves a e ew me ( sead o a ( A ( Ts gves a se o lea equaos o e ew empeaues: ( A Compuaoal Flud Dyamcs Implc Meods Isolae e ew ad solve by eao ( ( A A Te mplc meod s ucodoally sable bu s ecessay o solve a sysem o lea equaos a eac me sep. Oe e me sep mus be ake o be small due o accuacy equemes ad a explc meod s compeve. Kow souce em
7 Compuaoal Flud Dyamcs Implc Meods Secod ode accuacy me ca be obaed by usg e Cak-Ncolso meod - - Compuaoal Flud Dyamcs Cak-Ncolso Cak-Ncolso Meod o -D Hea Equao x y x y I x y Te max equao s expesve o solve ( ( Compuaoal Flud Dyamcs Compuaoal Flud Dyamcs Expesve o solve max equaos. Ca lage me-sep be aceved wou avg solve e max equao esulg om e wo-dmesoal sysem? Te beak oug came w e Aleao-Deco- Implc (ADI meod (Peacema & Racod-md95 s Te ADI Meod ADI cosss o s eag oe ow mplcly w backwad Eule ad e evesg oles ad eag e oe by backwads Eule. Peacema D. ad Racod M. (955. Te umecal soluo o paabolc ad ellpc deeal equaos J. SIAM 3 8- Compuaoal Flud Dyamcs Compuaoal Flud Dyamcs Aleag Deco Implc (ADI Facoal Sep: Sep : / Sep : / ( x y / / / [( ( ] / / / [( ( ] Combg e wo becomes equvale o: / x y y Mdpo Tapsodal Compuaoal Molecules o e ADI Meod / - -
8 Compuaoal Flud Dyamcs Compuaoal Flud Dyamcs Isead o solvg oe se o lea equaos o e wo-dmesoal sysem solve D equaos o eac gd le. Te decos ca be aleaed o peve ay bas I max om o eac ow / / / N N / / / N souce Ts equao s easly solved by owad elmao ad back-subsuo ADI Meod s Sably Aalyss: Smlaly Compuaoal Flud Dyamcs ε O( ε ε e accuae e kx my ε / ε ε / e k e k ε / k s ε ε / ( ε ( e m e m m s m s k s Combg Compuaoal Flud Dyamcs k m s s ε < ε m s k s Ucodoally sable! Te 3-D veso does o ave e same desable sably popees. Howeve s possble o geeae smla meods o 3D poblems Compuaoal Flud Dyamcs Compuaoal Flud Dyamcs Implc meods o paabolc equaos Allow muc lage me sep (bu mus be balaced agas accuacy! Peseve e paabolc aue o e equao Bu eque e soluo o a lea se o equaos ad ae eeoe muc moe expesve a explc meods Appoxmae acozao Splg ADI povdes a way o cove muldmesoal poblems o a sees o D poblems
9 Dee δ ( Compuaoal Flud Dyamcs Appoxmae Facozao - x ( ( ( δ ( ( y ( ( Te Cak-Ncolso o ea equao becomes δ ( δ ( O( x y wc ca be ewe as δ δ δ δ O( x y Facog eac sde Compuaoal Flud Dyamcs Appoxmae Facozao - δ δ δ δ δ δ δ δ O( x y o δ δ δ δ Cak-Ncolso δ δ ( O( x y Compuaoal Flud Dyamcs Appoxmae Facozao - 3 Te ADI meod ca be we as δ / δ δ δ / Elmag / Up o a aco: Compuaoal Flud Dyamcs Appoxmae Facozao - δ δ ( ADI s a appoxmae acozao o e Cak-Ncolso meod 3 δ δ O( 3 δ δ δ δ Compuaoal Flud Dyamcs Appoxmae Facozao - 5 Te Peacema-Racod meods does o ave e same sably popees 3D as D. A meod w smla popees s: ( δ * δ δ δ zz δ ** * δ δ zz ** δ zz Seveal oe splg meods ave bee poposed e leaue. Compuaoal Flud Dyamcs Smla deas ca be used o me splg: ( / δ ( δ Elmae e al sep: Appoxmae Facozao - 6 δ / ( δ ( Wc s equvale o e sadad FTCS meod excep o e δ δ em wc s ge ode.
10 Compuaoal Flud Dyamcs Compuaoal Flud Dyamcs Wy splg?. Sably lms o -D case apply.. Dee ca be used x ad y decos. Implc me macg by as ellpc solves Backwad Eule Reaage o Compuaoal Flud Dyamcs λ S Compuaoal Flud Dyamcs Oe-Dmesoal Poblems Explc mplc Cak-Ncolso Accuacy sably Vaous scemes Mul-Dmesoal Poblems Aleag Deco Implc (ADI Appoxmae Facozao o Cak-Ncolso Splg Oule Soluo Meods o Paabolc Equaos Ca be solved by ellpc solves Compuaoal Flud Dyamcs p:// Te Adveco-Duso Equao Nave-Sokes equaos Compuaoal Flud Dyamcs Summay u uu x v u y P ρ x µ u ρ x u y Hypebolc pa u x v y Ellpc equao Paabolc pa Te Nave-Sokes equaos coa ee equao ypes a ave e ow caacesc beavo Depedg o e goveg paamees oe beavo ca be doma Te dee equao ypes eque dee soluo ecques Fo vscd compessble lows oly e ypebolc pa suvves
11 Compuaoal Flud Dyamcs Compuaoal Flud Dyamcs D Adveco/duso equao U x D x Fowad me/ceeed space (FTCS U D Sably: We: ε ε ε ε U ε ε e kx ε D ad use ε ε ε ± ε e kx e ±k Gves ε ε Fo sably o U k k D ( e e ( e k e k D s k U sk k G ( As B s 8As 6A s B s ( B cos s 8As As A ( B cos As Mus be <- Mus be < A A B A D B U D U D Compuaoal Flud Dyamcs Compuaoal Flud Dyamcs D Adveco/duso equao U x D x Fowad me/ceeed space (FTCS Sably lms U U D & D D U & D Fo g ad low D D R UL D FTCS Upwd L-W C-N U x D x O( O( O( O( U D & D U D U D Ucodoally sable Compuaoal Flud Dyamcs Compuaoal Flud Dyamcs Seady sae soluo o e adveco/duso equao U x D x U L R L R L 5 R L R L Exac soluo ( ( exp R L x /L exp R L R L UL D
12 Compuaoal Flud Dyamcs Compuaoal Flud Dyamcs Te Cell Reyolds umbe Numecal soluo o: U U x D x Ceeed deece appoxmao Upwd D U D Compuaoal Flud Dyamcs Compuaoal Flud Dyamcs Ceeed deece appoxmao U D Reaage: U D Reaage: Wee: ( ( R ( R R U D Soluo Subsue: Dvde by ( R ( R q ( R q q ( R q q ( R q q ( R Compuaoal Flud Dyamcs Compuaoal Flud Dyamcs ( R q q ( R Solvg o q gves wo soluos: q ad Te geeal soluo s: o C q C q q R R R C C R Apply e bouday codos R C C R R N C C R N C C Te al soluo s: R R R R N
13 Compuaoal Flud Dyamcs Compuaoal Flud Dyamcs Upwd o Soluo U D ( R ( R Ty soluos gvg q q ( R q ( R ( ( N R R R U D Exac soluo ( ( exp R L x /L exp R L Ceeed deeces Upwd R R R R ( R ( R N N R L UL D R U D Compuaoal Flud Dyamcs Compuaoal Flud Dyamcs Upwd Exac Ceeed Upwd Exac Ceeed Compuaoal Flud Dyamcs Compuaoal Flud Dyamcs Upwd Exac Ceeed We ceeed deecg s used o e adveco/duso equao oscllaos may appea we e Cell Reyolds umbe s ge a. Fo upwdg o oscllaos appea. I mos cases e oscllaos ae small ad e cell Reyolds umbe s equely allowed o be ge a w elavely mo eecs o e esul. R U D <
14 Compuaoal Flud Dyamcs Compuaoal Flud Dyamcs D example U x V y D x y D..588 Re cell 3.58 Flow Compuaos usg ceeed deeces o a 3 by 3 gd Compuaoal Flud Dyamcs Compuaoal Flud Dyamcs D..588 Re cell 6.56 D Re cell.93 Compuaoal Flud Dyamcs Compuaoal Flud Dyamcs.5 Fe gd Re cell 3.58 D..5.5 Coase gd Re cell Te Cell Reyolds umbe lmao becomes a ssue we e adveco ems ae dscezed usg ceeed deeces. I may cases e oscllaos oly appea solaed egos wee duso ad adveco ae o compaable magude.5.5 Upwd avods e poblem bu geeally adds dsspao
Suppose we have observed values t 1, t 2, t n of a random variable T.
Sppose we have obseved vales, 2, of a adom vaable T. The dsbo of T s ow o belog o a cea ype (e.g., expoeal, omal, ec.) b he veco θ ( θ, θ2, θp ) of ow paamees assocaed wh s ow (whee p s he mbe of ow paamees).
( m is the length of columns of A ) spanned by the columns of A : . Select those columns of B that contain a pivot; say those are Bi
Assgmet /MATH 47/Wte Due: Thusday Jauay The poblems to solve ae umbeed [] to [] below Fst some explaatoy otes Fdg a bass of the colum-space of a max ad povg that the colum ak (dmeso of the colum space)
( ) ( ) Weibull Distribution: k ti. u u. Suppose t 1, t 2, t n are times to failure of a group of n mechanisms. The likelihood function is
Webll Dsbo: Des Bce Dep of Mechacal & Idsal Egeeg The Uvesy of Iowa pdf: f () exp Sppose, 2, ae mes o fale of a gop of mechasms. The lelhood fco s L ( ;, ) exp exp MLE: Webll 3//2002 page MLE: Webll 3//2002
Cyclone. Anti-cyclone
Adveco Cycloe A-cycloe Lorez (963) Low dmesoal aracors. Uclear f hey are a good aalogy o he rue clmae sysem, bu hey have some appealg characerscs. Dscusso Is he al codo balaced? Is here a al adjusme
Chapter 5. Long Waves
ape 5. Lo Waes Wae e s o compaed ae dep: < < L π Fom ea ae eo o s s ; amos ozoa moo z p s ; dosac pesse Dep-aeaed coseao o mass
ANSWERS TO ODD NUMBERED EXERCISES IN CHAPTER 2
Joh Rley Novembe ANSWERS O ODD NUMBERED EXERCISES IN CHAPER Seo Eese -: asvy (a) Se y ad y z follows fom asvy ha z Ehe z o z We suppose he lae ad seek a oado he z Se y follows by asvy ha z y Bu hs oads
1. INTRODUCTION In this paper, we consider a general ninth order linear boundary value problem (1) subject to boundary conditions
NUMERICAL SOLUTION OF NINTH ORDER BOUNDARY VALUE PROBLEMS BY PETROV-GALERKIN METHOD WITH QUINTIC B-SPLINES AS BASIS FUNCTIONS AND SEXTIC B-SPLINES AS WEIGHT FUNCTIONS K. N. S. Kas Vswaaham a S. V. Kamay
= y and Normed Linear Spaces
304-50 LINER SYSTEMS Lectue 8: Solutos to = ad Nomed Lea Spaces 73 Fdg N To fd N, we eed to chaacteze all solutos to = 0 Recall that ow opeatos peseve N, so that = 0 = 0 We ca solve = 0 ecusvel backwads
EMA5001 Lecture 3 Steady State & Nonsteady State Diffusion - Fick s 2 nd Law & Solutions
EMA5 Lecue 3 Seady Sae & Noseady Sae ffuso - Fck s d Law & Soluos EMA 5 Physcal Popees of Maeals Zhe heg (6) 3 Noseady Sae ff Fck s d Law Seady-Sae ffuso Seady Sae Seady Sae = Equlbum? No! Smlay: Sae fuco
XII. Addition of many identical spins
XII. Addto of may detcal sps XII.. ymmetc goup ymmetc goup s the goup of all possble pemutatos of obects. I total! elemets cludg detty opeato. Each pemutato s a poduct of a ceta fte umbe of pawse taspostos.
4. Runge-Kutta Formula For Differential Equations
NCTU Deprme o Elecrcl d Compuer Egeerg 5 Sprg Course by Pro. Yo-Pg Ce. Ruge-Ku Formul For Derel Equos To solve e derel equos umerclly e mos useul ormul s clled Ruge-Ku ormul
4. Runge-Kutta Formula For Differential Equations. A. Euler Formula B. Runge-Kutta Formula C. An Example for Fourth-Order Runge-Kutta Formula
NCTU Deprme o Elecrcl d Compuer Egeerg Seor Course By Pro. Yo-Pg Ce. Ruge-Ku Formul For Derel Equos A. Euler Formul B. Ruge-Ku Formul C. A Emple or Four-Order Ruge-Ku Formul
Quantum Mechanics II Lecture 11 Time-dependent perturbation theory. Time-dependent perturbation theory (degenerate or non-degenerate starting state)
Pro. O. B. Wrgh, Auum Quaum Mechacs II Lecure Tme-depede perurbao heory Tme-depede perurbao heory (degeerae or o-degeerae sarg sae) Cosder a sgle parcle whch, s uperurbed codo wh Hamloa H, ca exs a superposo
A Survey on Model Reduction Methods to Reduce Degrees of Freedom of Linear Damped Vibrating Systems
opdaa Aavakom 460767 Mah, pg 003 A uvey o Model Reduco Mehods o Reduce Degees o Feedom o Lea Damped Vbag ysems ABRAC hs epo descbes he deals o he model educo mehods o educe degees o eedom o he dyamc aalyss
(1) Cov(, ) E[( E( ))( E( ))]
![(1) Cov(, ) E[( E( ))( E( ))]](/thumbs/86/94405490.jpg "(1) Cov(, ) E[( E( ))( E( ))]")
Impac of Auocorrelao o OLS Esmaes ECON 3033/Evas Cosder a smple bvarae me-seres model of he form: y 0 x The four key assumpos abou ε hs model are ) E(ε ) = E[ε x ]=0 ) Var(ε ) =Var(ε x ) = ) Cov(ε, ε )
A NEW FORMULAE OF VARIABLE STEP 3-POINT BLOCK BDF METHOD FOR SOLVING STIFF ODES
Joual of Pue ad Appled Maemac: Advace ad Applcao Volume Numbe Page 9-7 A NEW ORMULAE O VARIABLE STEP -POINT BLOCK BD METHOD OR SOLVING STI ODES NAGHMEH ABASI MOHAMED BIN SULEIMAN UDZIAH ISMAIL ZARINA BIBI
FORCED VIBRATION of MDOF SYSTEMS
FORCED VIBRAION of DOF SSES he respose of a N DOF sysem s govered by he marx equao of moo: ] u C] u K] u 1 h al codos u u0 ad u u 0. hs marx equao of moo represes a sysem of N smulaeous equaos u ad s me
FALL HOMEWORK NO. 6 - SOLUTION Problem 1.: Use the Storage-Indication Method to route the Input hydrograph tabulated below.
Jorge A. Ramírez HOMEWORK NO. 6 - SOLUTION Problem 1.: Use he Sorage-Idcao Mehod o roue he Ipu hydrograph abulaed below. Tme (h) Ipu Hydrograph (m 3 /s) Tme (h) Ipu Hydrograph (m 3 /s) 0 0 90 450 6 50
Computation of an Over-Approximation of the Backward Reachable Set using Subsystem Level Set Functions
Compuao o a Ove-Appomao o he Backwad Reachable Se usg Subsysem Level Se Fucos Duša M Spaov, Iseok Hwag, ad Clae J oml Depame o Aeoaucs ad Asoaucs Saod Uvesy Saod, CA 94305-4035, USA E-mal: {dusko, shwag,
Computational Fluid Dynamics CFD. Solving system of equations, Grid generation
Compaoal ld Dyamcs CD Solvg sysem of eqaos, Grd geerao Basc seps of CD Problem Dscrezao Resl Gov. Eq. BC I. Cod. Solo OK??,,... Solvg sysem of eqaos he ype of eqaos decdes solo sraegy Marchg problems Eqlbrm
The Poisson Process Properties of the Poisson Process
Posso Processes Summary The Posso Process Properes of he Posso Process Ierarrval mes Memoryless propery ad he resdual lfeme paradox Superposo of Posso processes Radom seleco of Posso Pos Bulk Arrvals ad
4. THE DENSITY MATRIX
4. THE DENSTY MATRX The desy marx or desy operaor s a alerae represeao of he sae of a quaum sysem for whch we have prevously used he wavefuco. Alhough descrbg a quaum sysem wh he desy marx s equvale o
Hydrodynamic Modeling: Hydrodynamic Equations. Prepared by Dragica Vasileska Professor Arizona State University
Hyoyamc Moelg: Hyoyamc Equaos Peae by Dagca Vasleska Poesso Azoa Sae Uesy D-Duso Aoaches Val he use aso omaes Exeso o DD Aoaches Valy Iouco o he el-eee mobly Velocy Sauao Imlcaos Velocy Sauao Dece Scalg
PULSATILE BLOOD FLOW IN CONSTRICTED TAPERED ARTERY USING A VARIABLE-ORDER FRACTIONAL OLDROYD-B MODEL
Bakh, H., e al.: Pulsale Blood Flo Cosced Tapeed Aey Usg... THERMAL SCIENCE: Yea 7, Vol., No. A, pp. 9-4 9 PULSATILE BLOOD FLOW IN CONSTRICTED TAPERED ARTERY USING A VARIABLE-ORDER FRACTIONAL OLDROYD-B
ECSE Partial fraction expansion (m<n) 3 types of poles Simple Real poles Real Equal poles
ECSE- Lecue. Paial facio expasio (m
Continuous Time Markov Chains
Couous me Markov chas have seay sae probably soluos f a oly f hey are ergoc, us lke scree me Markov chas. Fg he seay sae probably vecor for a couous me Markov cha s o more ffcul ha s he scree me case,
Lagrangian & Hamiltonian Mechanics:
XII AGRANGIAN & HAMITONIAN DYNAMICS Iouco Hamlo aaoal Pcple Geealze Cooaes Geealze Foces agaga s Euao Geealze Momea Foces of Cosa, agage Mulples Hamloa Fucos, Cosevao aws Hamloa Dyamcs: Hamlo s Euaos agaga
8. Queueing systems lect08.ppt S Introduction to Teletraffic Theory - Fall
8. Queueg sysems lec8. S-38.45 - Iroduco o Teleraffc Theory - Fall 8. Queueg sysems Coes Refresher: Smle eleraffc model M/M/ server wag laces M/M/ servers wag laces 8. Queueg sysems Smle eleraffc model
Noncommutative Solitons and Quasideterminants
Nocommutatve Soltos ad Quasdetemats asas HNK Nagoya Uvesty ept. o at. Teoetcal Pyscs Sema Haove o eb.8t ased o H ``NC ad's cojectue ad tegable systems NP74 6 368 ep-t/69 H ``Notes o eact mult-solto solutos
School of Mathematics and Actuarial Science, Jaramogi Oginga Odinga University of Science and Technology, Bondo, Kenya
N XXXX XXXX 6 E esearc Arcle olume 6 ssue No. Two mesoal Maemacal Models for ovecve-spersve Flow of Pescdes Porous Meda e H. W. Adams N. Omolo Oga M. E. Oduor Okoa 3 T.. O. Amer cool of Maemacal ceces
Least Squares Fitting (LSQF) with a complicated function Theexampleswehavelookedatsofarhavebeenlinearintheparameters
Leas Squares Fg LSQF wh a complcaed fuco Theeampleswehavelookedasofarhavebeelearheparameers ha we have bee rg o deerme e.g. slope, ercep. For he case where he fuco s lear he parameers we ca fd a aalc soluo
Fractional Integrals Involving Generalized Polynomials And Multivariable Function
IOSR Joual of ateatcs (IOSRJ) ISSN: 78-578 Volue, Issue 5 (Jul-Aug 0), PP 05- wwwosoualsog Factoal Itegals Ivolvg Geealzed Poloals Ad ultvaable Fucto D Neela Pade ad Resa Ka Deatet of ateatcs APS uvest
Minimizing spherical aberrations Exploiting the existence of conjugate points in spherical lenses
Mmzg sphecal abeatos Explotg the exstece of cojugate pots sphecal leses Let s ecall that whe usg asphecal leses, abeato fee magg occus oly fo a couple of, so called, cojugate pots ( ad the fgue below)
k of the incident wave) will be greater t is too small to satisfy the required kinematics boundary condition, (19)
TOTAL INTRNAL RFLTION Kmacs pops Sc h vcos a coplaa, l s cosd h cd pla cocds wh h X pla; hc 0. y y y osd h cas whch h lgh s cd fom h mdum of hgh dx of faco >. Fo cd agls ga ha h ccal agl s 1 ( /, h hooal
Professor Wei Zhu. 1. Sampling from the Normal Population
AMS570 Pofesso We Zhu. Samplg fom the Nomal Populato *Example: We wsh to estmate the dstbuto of heghts of adult US male. It s beleved that the heght of adult US male follows a omal dstbuto N(, ) Def. Smple
EE 6885 Statistical Pattern Recognition
EE 6885 Sascal Paer Recogo Fall 005 Prof. Shh-Fu Chag hp://www.ee.columba.edu/~sfchag Lecure 5 (9//05 4- Readg Model Parameer Esmao ML Esmao, Chap. 3. Mure of Gaussa ad EM Referece Boo, HTF Chap. 8.5 Teboo,
Determination of Antoine Equation Parameters. December 4, 2012 PreFEED Corporation Yoshio Kumagae. Introduction
refeed Soluos for R&D o Desg Deermao of oe Equao arameers Soluos for R&D o Desg December 4, 0 refeed orporao Yosho Kumagae refeed Iroduco hyscal propery daa s exremely mpora for performg process desg ad
QR factorization. Let P 1, P 2, P n-1, be matrices such that Pn 1Pn 2... PPA
QR facorzao Ay x real marx ca be wre as AQR, where Q s orhogoal ad R s upper ragular. To oba Q ad R, we use he Householder rasformao as follows: Le P, P, P -, be marces such ha P P... PPA ( R s upper ragular.
A PATRA CONFERINŢĂ A HIDROENERGETICIENILOR DIN ROMÂNIA,
A PATRA ONFERINŢĂ A HIDROENERGETIIENILOR DIN ROMÂNIA, Do Pael MODELLING OF SEDIMENTATION PROESS IN LONGITUDINAL HORIZONTAL TANK MODELAREA PROESELOR DE SEPARARE A FAZELOR ÎN DEANTOARE LONGITUDINALE Da ROBESU,
Key words: Fractional difference equation, oscillatory solutions,
OSCILLATION PROPERTIES OF SOLUTIONS OF FRACTIONAL DIFFERENCE EQUATIONS Musafa BAYRAM * ad Ayd SECER * Deparme of Compuer Egeerg, Isabul Gelsm Uversy Deparme of Mahemacal Egeerg, Yldz Techcal Uversy * Correspodg
dm dt = 1 V The number of moles in any volume is M = CV, where C = concentration in M/L V = liters. dcv v
Mg: Pcess Aalyss: Reac ae s defed as whee eac ae elcy lue M les ( ccea) e. dm he ube f les ay lue s M, whee ccea M/L les. he he eac ae beces f a hgeeus eac, ( ) d Usually s csa aqueus eeal pcesses eac,
NUMERICAL SOLUTIONS TO ORDINARY DIFFERENTIAL EQUATIONS
NUMERICAL SOLUTIONS TO ORDINARY DIFFERENTIAL EQUATIONS If e eqao coas dervaves of a - order s sad o be a - order dffereal eqao. For eample a secod-order eqao descrbg e oscllao of a weg aced po b a sprg
AML710 CAD LECTURE 12 CUBIC SPLINE CURVES. Cubic Splines Matrix formulation Normalised cubic splines Alternate end conditions Parabolic blending
CUIC SLINE CURVES Cubc Sples Marx formulao Normalsed cubc sples Alerae ed codos arabolc bledg AML7 CAD LECTURE CUIC SLINE The ame sple comes from he physcal srume sple drafsme use o produce curves A geeral
Technical Appendix for Inventory Management for an Assembly System with Product or Component Returns, DeCroix and Zipkin, Management Science 2005.
Techc Appedx fo Iveoy geme fo Assemy Sysem wh Poduc o Compoe eus ecox d Zp geme Scece 2005 Lemm µ µ s c Poof If J d µ > µ he ˆ 0 µ µ µ µ µ µ µ µ Sm gumes essh he esu f µ ˆ > µ > µ > µ o K ˆ If J he so
Applying Eyring s Model to Times to Breakdown of Insulating Fluid
Ieaoal Joual of Pefomably Egeeg, Vol. 8, No. 3, May 22, pp. 279-288. RAMS Cosulas Ped Ida Applyg Eyg s Model o Tmes o Beakdow of Isulag Flud DANIEL I. DE SOUZA JR. ad R. ROCHA Flumese Fed. Uvesy, Cvl Egeeg
ON TOTAL TIME ON TEST TRANSFORM ORDER ABSTRACT
V M Chacko E CONVE AND INCREASIN CONVE OAL IME ON ES RANSORM ORDER R&A # 4 9 Vol. Decembe ON OAL IME ON ES RANSORM ORDER V. M. Chacko Depame of Sascs S. homas Collee hss eala-68 Emal: chackovm@mal.com
Department of Chemical Engineering University of Tennessee Prof. David Keffer. Course Lecture Notes SIXTEEN
D. Keffe - ChE 40: Hea Tansfe and Fluid Flow Deamen of Chemical Enee Uniesi of Tennessee Pof. Daid Keffe Couse Lecue Noes SIXTEEN SECTION.6 DIFFERENTIL EQUTIONS OF CONTINUITY SECTION.7 DIFFERENTIL EQUTIONS
Density estimation III. Linear regression.
Lecure 6 Mlos Hauskrec mlos@cs.p.eu 539 Seo Square Des esmao III. Lear regresso. Daa: Des esmao D { D D.. D} D a vecor of arbue values Obecve: r o esmae e uerlg rue probabl srbuo over varables X px usg
Fakultas Matematika dan Ilmu Pengetahuan Alam, Institut Teknologi Bandung, Bandung, 40132, Indonesia *
MacWllams Equvalece Theoem fo he Lee Wegh ove Z 4 leams Baa * Fakulas Maemaka da Ilmu Pegeahua lam, Isu Tekolog Badug, Badug, 403, Idoesa * oesodg uho: baa@mahbacd BSTRT Fo codes ove felds, he MacWllams
Exponential Synchronization of the Hopfield Neural Networks with New Chaotic Strange Attractor
ITM Web of Cofeeces, 0509 (07) DOI: 0.05/ mcof/070509 ITA 07 Expoeal Sychozao of he Hopfeld Neual Newos wh New Chaoc Sage Aaco Zha-J GUI, Ka-Hua WANG* Depame of Sofwae Egeeg, Haa College of Sofwae Techology,qogha,
CHAPTER 4 RADICAL EXPRESSIONS
6 CHAPTER RADICAL EXPRESSIONS. The th Root of a Real Number A real umber a s called the th root of a real umber b f Thus, for example: s a square root of sce. s also a square root of sce ( ). s a cube
On EPr Bimatrices II. ON EP BIMATRICES A1 A Hence x. is said to be EP if it satisfies the condition ABx
Iteatoal Joual of Mathematcs ad Statstcs Iveto (IJMSI) E-ISSN: 3 4767 P-ISSN: 3-4759 www.jms.og Volume Issue 5 May. 4 PP-44-5 O EP matces.ramesh, N.baas ssocate Pofesso of Mathematcs, ovt. ts College(utoomous),Kumbakoam.
Reinforcement learning
Lecue 3 Reinfocemen leaning Milos Hauskech milos@cs.pi.edu 539 Senno Squae Reinfocemen leaning We wan o lean he conol policy: : X A We see examples of x (bu oupus a ae no given) Insead of a we ge a feedback
Introduction to Matrices and Matrix Approach to Simple Linear Regression
Itroducto to Matrces ad Matrx Approach to Smple Lear Regresso Matrces Defto: A matrx s a rectagular array of umbers or symbolc elemets I may applcatos, the rows of a matrx wll represet dvduals cases (people,
Density estimation III.
Lecure 4 esy esmao III. Mlos Hauskrec mlos@cs..edu 539 Seo Square Oule Oule: esy esmao: Mamum lkelood ML Bayesa arameer esmaes MP Beroull dsrbuo. Bomal dsrbuo Mulomal dsrbuo Normal dsrbuo Eoeal famly Eoeal
The ray paths and travel times for multiple layers can be computed using ray-tracing, as demonstrated in Lab 3.
C. Trael me cures for mulple reflecors The ray pahs ad rael mes for mulple layers ca be compued usg ray-racg, as demosraed Lab. MATLAB scrp reflec_layers_.m performs smple ray racg. (m) ref(ms) ref(ms)
P a g e 5 1 of R e p o r t P B 4 / 0 9
P a g e 5 1 of R e p o r t P B 4 / 0 9 J A R T a l s o c o n c l u d e d t h a t a l t h o u g h t h e i n t e n t o f N e l s o n s r e h a b i l i t a t i o n p l a n i s t o e n h a n c e c o n n e
Phys 2310 Fri. Oct. 23, 2017 Today s Topics. Begin Chapter 6: More on Geometric Optics Reading for Next Time
Py F. Oct., 7 Today Topc Beg Capte 6: Moe o Geometc Optc eadg fo Next Tme Homewok t Week HW # Homewok t week due Mo., Oct. : Capte 4: #47, 57, 59, 6, 6, 6, 6, 67, 7 Supplemetal: Tck ee ad e Sytem Pcple
Conservative and Easily Implemented Finite Volume Semi-Lagrangian WENO Methods for 1D and 2D Hyperbolic Conservation Laws
Joural of Appled Mahemacs ad Physcs, 07, 5, 59-8 hp://www.scrp.org/oural/amp ISSN Ole: 37-4379 ISSN Pr: 37-435 Coservave ad Easly Implemeed Fe Volume Sem-Lagraga WENO Mehods for D ad D Hyperbolc Coservao
Application Of Alternating Group Explicit Method For Parabolic Equations
WSEAS RANSACIONS o INFORMAION SCIENCE ad APPLICAIONS Qghua Feg Applcato Of Alteatg oup Explct Method Fo Paabolc Equatos Qghua Feg School of Scece Shadog uvesty of techology Zhagzhou Road # Zbo Shadog 09
THIS PAGE DECLASSIFIED IAW E
THS PAGE DECLASSFED AW E0 2958 BL K THS PAGE DECLASSFED AW E0 2958 THS PAGE DECLASSFED AW E0 2958 B L K THS PAGE DECLASSFED AW E0 2958 THS PAGE DECLASSFED AW EO 2958 THS PAGE DECLASSFED AW EO 2958 THS
A Generalized Recursive Coordinate Reduction Method for Multibody System Dynamics
eaoal Joual fo Mulscale Compuaoal Egeeg, 1(2&3181 199 (23 A Geealzed Recusve Coodae Reduco Mehod fo Mulbody Sysem Dyamcs J. H. Cchley & K. S. Adeso Depame of Mechacal, Aeoaucal, ad Nuclea Egeeg, Resselae
Lecture 3 summary. C4 Lecture 3 - Jim Libby 1
Lecue su Fes of efeece Ivce ude sfoos oo of H wve fuco: d-fucos Eple: e e - µ µ - Agul oeu s oo geeo Eule gles Geec slos cosevo lws d Noehe s heoe C4 Lecue - Lbb Fes of efeece Cosde fe of efeece O whch
Phys Nov. 3, 2017 Today s Topics. Continue Chapter 2: Electromagnetic Theory, Photons, and Light Reading for Next Time
Phys 31. No. 3, 17 Today s Topcs Cou Chap : lcomagc Thoy, Phoos, ad Lgh Radg fo Nx Tm 1 By Wdsday: Radg hs Wk Fsh Fowls Ch. (.3.11 Polazao Thoy, Jos Macs, Fsl uaos ad Bws s Agl Homwok hs Wk Chap Homwok
CIS 800/002 The Algorithmic Foundations of Data Privacy October 13, Lecture 9. Database Update Algorithms: Multiplicative Weights
CIS 800/002 The Algorthmc Foudatos of Data Prvacy October 13, 2011 Lecturer: Aaro Roth Lecture 9 Scrbe: Aaro Roth Database Update Algorthms: Multplcatve Weghts We ll recall aga) some deftos from last tme:
Partial Molar Properties of solutions
Paral Molar Properes of soluos A soluo s a homogeeous mxure; ha s, a soluo s a oephase sysem wh more ha oe compoe. A homogeeous mxures of wo or more compoes he gas, lqud or sold phase The properes of a
For the plane motion of a rigid body, an additional equation is needed to specify the state of rotation of the body.
The kecs of rgd bodes reas he relaoshps bewee he exeral forces acg o a body ad he correspodg raslaoal ad roaoal moos of he body. he kecs of he parcle, we foud ha wo force equaos of moo were requred o defe
Relativity and Astrophysics Lecture 38 Terry Herter. Rain fall source to distance observer Distance source to rain fall frame
Light and Tides Relativity and Astophysics Lectue 38 Tey Hete Outline etic in the Rain Fame Inside the hoizon One-way motion Rain Fall Light Cones Photon Exchange Rain all souce to distance obseve Distance
Solution set Stat 471/Spring 06. Homework 2
oluo se a 47/prg 06 Homework a Whe he upper ragular elemes are suppressed due o smmer b Le Y Y Y Y A weep o he frs colum o oba: A ˆ b chagg he oao eg ad ec YY weep o he secod colum o oba: Aˆ YY weep o
are positive, and the pair A, B is controllable. The uncertainty in is introduced to model control failures.
Lectue 4 8. MRAC Desg fo Affe--Cotol MIMO Systes I ths secto, we cosde MRAC desg fo a class of ult-ut-ult-outut (MIMO) olea systes, whose lat dyacs ae lealy aaetezed, the ucetates satsfy the so-called
T T V e g em D e j ) a S D } a o "m ek j g ed b m "d mq m [ d, )
. ) 6 3 ; 6 ;, G E E W T S W X D ^ L J R Y [ _ ` E ) '" " " -, 7 4-4 4-4 ; ; 7 4 4 4 4 4 ;= : " B C CA BA " ) 3D H E V U T T V e g em D e j ) a S D } a o "m ek j g ed b m "d mq m [ d, ) W X 6 G.. 6 [ X
ON THE EXTENSION OF WEAK ARMENDARIZ RINGS RELATIVE TO A MONOID
wwweo/voue/vo9iue/ijas_9 9f ON THE EXTENSION OF WEAK AENDAIZ INGS ELATIVE TO A ONOID Eye A & Ayou Eoy Dee of e Nowe No Uvey Lzou 77 C Dee of e Uvey of Kou Ou Su E-: eye76@o; you975@yooo ABSTACT Fo oo we
Lecture 10: Condensed matter systems
Lectue 0: Codesed matte systems Itoducg matte ts codesed state.! Ams: " Idstgushable patcles ad the quatum atue of matte: # Cosequeces # Revew of deal gas etopy # Femos ad Bosos " Quatum statstcs. # Occupato
AN EXPLICIT WAVELET-BASED FINITE DIFFERENCE SCHEME FOR SOLVING ONE-DIMENSIONAL HEAT EQUATION
ISSN 0853-8697 AN EXPLICI WAVELE-BASED FINIE DIFFERENCE SCHEME FOR SOLVING ONE-DIMENSIONAL HEA EQUAION Mahmmod Azz Muhammed 1 Adh Susao 1 F. Soesao 1 Soerso 1Deparme of Elecrcal Egeerg Faculy of Egeerg
SIMULATIUON OF SEISMIC ACTION FOR TBILISI CITY WITH LOCAL SEISMOLOGICAL PARTICULARITIES AND SITE EFFECTS
SIMULTIUON OF SEISMIC CTION FOR TBILISI CITY WITH LOCL SEISMOLOGICL PRTICULRITIES ND SITE EFFECTS Paaa REKVV ad Keeva MDIVNI Geoga Naoal ssocao fo Egeeg Sesmology ad Eahquake Egeeg Tbls Geoga ekvavapaaa@yahoo.com
TESTS BASED ON MAXIMUM LIKELIHOOD
ESE 5 Toy E. Smth. The Basc Example. TESTS BASED ON MAXIMUM LIKELIHOOD To llustrate the propertes of maxmum lkelhood estmates ad tests, we cosder the smplest possble case of estmatg the mea of the ormal
Homonuclear Diatomic Molecule
Homouclea Datomc Molecule Eegy Dagam H +, H, He +, He A B H + eq = Agstom Bg Eegy kcal/mol A B H eq = Agstom Bg Eegy kcal/mol A B He + eq = Agstom Bg Eegy kcal/mol A He eq = Bg Eegy B Kcal mol 3 Molecula
Advection! Discontinuous! solutions shocks! Shock Speed! ! f. !t + U!f. ! t! x. dx dt = U; t = 0
p://www.d.edu/~gryggva/cfd-course/ Advecio Discoiuous soluios socks Gréar Tryggvaso Sprig Discoiuous Soluios Cosider e liear Advecio Equaio + U = Te aalyic soluio is obaied by caracerisics d d = U; d d
VECTOR MECHANICS FOR ENGINEERS: Vector Mechanics for Engineers: Dynamics. In the current chapter, you will study the motion of systems of particles.
Seeth Edto CHPTER 4 VECTOR MECHNICS FOR ENINEERS: DYNMICS Fedad P. ee E. Russell Johsto, J. Systems of Patcles Lectue Notes: J. Walt Ole Texas Tech Uesty 003 The Mcaw-Hll Compaes, Ic. ll ghts eseed. Seeth
Numerical Analysis Formulae Booklet
Numercal Aalyss Formulae Booklet. Iteratve Scemes for Systems of Lear Algebrac Equatos:.... Taylor Seres... 3. Fte Dfferece Approxmatos... 3 4. Egevalues ad Egevectors of Matrces.... 3 5. Vector ad Matrx
BINOMIAL THEOREM OBJECTIVE PROBLEMS in the expansion of ( 3 +kx ) are equal. Then k =
wwwskshieduciocom BINOMIAL HEOREM OBJEIVE PROBLEMS he coefficies of, i e esio of k e equl he k /7 If e coefficie of, d ems i e i AP, e e vlue of is he coefficies i e,, 7 ems i e esio of e i AP he 7 7 em
MOLECULAR VIBRATIONS
MOLECULAR VIBRATIONS Here we wsh to vestgate molecular vbratos ad draw a smlarty betwee the theory of molecular vbratos ad Hückel theory. 1. Smple Harmoc Oscllator Recall that the eergy of a oe-dmesoal
By Joonghoe Dho. The irradiance at P is given by
CH. 9 c CH. 9 c By Joogo Do 9 Gal Coao 9. Gal Coao L co wo po ouc, S & S, mg moocomac wav o am qucy. L paao a b muc ga a. Loca am qucy. L paao a b muc ga a. Loca po obvao P a oug away om ouc o a a P wavo
RECAPITULATION & CONDITIONAL PROBABILITY. Number of favourable events n E Total number of elementary events n S
Fomulae Fo u Pobablty By OP Gupta [Ida Awad We, +91-9650 350 480] Impotat Tems, Deftos & Fomulae 01 Bascs Of Pobablty: Let S ad E be the sample space ad a evet a expemet espectvely Numbe of favouable evets
6. Introduction to Transistor Amplifiers: Concepts and Small-Signal Model
6. ntoucton to anssto mples: oncepts an Small-Sgnal Moel Lectue notes: Sec. 5 Sea & Smth 6 th E: Sec. 5.4, 5.6 & 6.3-6.4 Sea & Smth 5 th E: Sec. 4.4, 4.6 & 5.3-5.4 EE 65, Wnte203, F. Najmaba Founaton o
[ ] 2. [ ]3 + (Δx i + Δx i 1 ) / 2. Δx i-1 Δx i Δx i+1. TPG4160 Reservoir Simulation 2018 Lecture note 3. page 1 of 5
![[ ] 2. [ ]3 + (Δx i + Δx i 1 ) / 2. Δx i-1 Δx i Δx i+1. TPG4160 Reservoir Simulation 2018 Lecture note 3. page 1 of 5](/thumbs/85/92299540.jpg "[ ] 2. [ ]3 + (Δx i + Δx i 1 ) / 2. Δx i-1 Δx i Δx i+1. TPG4160 Reservoir Simulation 2018 Lecture note 3. page 1 of 5")
TPG460 Reservor Smulaon 08 page of 5 DISCRETIZATIO OF THE FOW EQUATIOS As we already have seen, fne dfference appromaons of he paral dervaves appearng n he flow equaons may be obaned from Taylor seres
Chapter Simpson s 1/3 Rule of Integration. ( x)
Cper 7. Smpso s / Rule o Iegro Aer redg s per, you sould e le o. derve e ormul or Smpso s / rule o egro,. use Smpso s / rule o solve egrls,. develop e ormul or mulple-segme Smpso s / rule o egro,. use
Fault Tolerant Computing. Fault Tolerant Computing CS 530 Probabilistic methods: overview
Probably 1/19/ CS 53 Probablsc mehods: overvew Yashwa K. Malaya Colorado Sae Uversy 1 Probablsc Mehods: Overvew Cocree umbers presece of uceray Probably Dsjo eves Sascal depedece Radom varables ad dsrbuos
By the end of this section you will be able to prove the Chinese Remainder Theorem apply this theorem to solve simultaneous linear congruences
Chapte : Theoy of Modula Aithmetic 8 Sectio D Chiese Remaide Theoem By the ed of this sectio you will be able to pove the Chiese Remaide Theoem apply this theoem to solve simultaeous liea cogueces The
Anouncements. Conjugate Gradients. Steepest Descent. Outline. Steepest Descent. Steepest Descent
oucms Couga Gas Mchal Kazha (6.657) Ifomao abou h Sma (6.757) hav b pos ol: hp://www.cs.hu.u/~msha Tch Spcs: o M o Tusay afoo. o Two paps scuss ach w. o Vos fo w s caa paps u by Thusay vg. Oul Rvw of Sps
Chapter 9 Jordan Block Matrices
Chapter 9 Jorda Block atrces I ths chapter we wll solve the followg problem. Gve a lear operator T fd a bass R of F such that the matrx R (T) s as smple as possble. f course smple s a matter of taste.
Kinematics. Redundancy. Task Redundancy. Operational Coordinates. Generalized Coordinates. m task. Manipulator. Operational point
Mapulato smatc Jot Revolute Jot Kematcs Base Lks: movg lk fed lk Ed-Effecto Jots: Revolute ( DOF) smatc ( DOF) Geealzed Coodates Opeatoal Coodates O : Opeatoal pot 5 costats 6 paametes { postos oetatos
Mathematical Formulation
Mahemacal Formulao The purpose of a fe fferece equao s o appromae he paral ffereal equao (PE) whle maag he physcal meag. Eample PE: p c k FEs are usually formulae by Taylor Seres Epaso abou a po a eglecg
Lecture Y4: Computational Optics I
Phooc ad opolcoc chologs DPMS: Advacd Maals Udsadg lgh ma acos s cucal fo w applcaos Lcu Y4: Compuaoal Opcs I lfos Ldoks Room Π, 65 746 ldok@cc.uo.g hp://cmsl.maals.uo.g/ldoks Rflco ad faco Toal al flco
DUALITY IN MULTIPLE CRITERIA AND MULTIPLE CONSTRAINT LEVELS LINEAR PROGRAMMING WITH FUZZY PARAMETERS
Ida Joual of Fudameal ad ppled Lfe Sceces ISSN: 223 6345 (Ole) Ope ccess, Ole Ieaoal Joual valable a www.cbech.o/sp.ed/ls/205/0/ls.hm 205 Vol.5 (S), pp. 447-454/Noua e al. Reseach cle DULITY IN MULTIPLE
Solution of Impulsive Differential Equations with Boundary Conditions in Terms of Integral Equations
Joural of aheacs ad copuer Scece (4 39-38 Soluo of Ipulsve Dffereal Equaos wh Boudary Codos Ters of Iegral Equaos Arcle hsory: Receved Ocober 3 Acceped February 4 Avalable ole July 4 ohse Rabba Depare
MULTI-OBJECTIVE GEOMETRIC PROGRAMMING PROBLEM AND ITS APPLICATIONS
Yugoslav Joual of Opeaos Reseach Volume (), Numbe, -7 DOI:.98/YJORI MULTI-OBJECTIVE GEOMETRIC PROGRAMMING PROBLEM AND ITS APPLICATIONS Sahdul ISLAM Depame of Mahemacs, Guskaa Mahavdyalaya, Guskaa, Budwa
Exponential Generating Functions - J. T. Butler
Epoetal Geeatg Fuctos - J. T. Butle Epoetal Geeatg Fuctos Geeatg fuctos fo pemutatos. Defto: a +a +a 2 2 + a + s the oday geeatg fucto fo the sequece of teges (a, a, a 2, a, ). Ep. Ge. Fuc.- J. T. Butle
CONTROL ROUTH ARRAY AND ITS APPLICATIONS
3 Asa Joual of Cool, Vol 5, No, pp 3-4, Mach 3 CONTROL ROUTH ARRAY AND ITS APPLICATIONS Dazha Cheg ad TJTa Bef Pape ABSTRACT I hs pape he Rouh sably ceo [6] has bee developed o cool Rouh aay Soe foulas
Lecture 02: Bounding tail distributions of a random variable
CSCI-B609: A Theorst s Toolkt, Fall 206 Aug 25 Lecture 02: Boudg tal dstrbutos of a radom varable Lecturer: Yua Zhou Scrbe: Yua Xe & Yua Zhou Let us cosder the ubased co flps aga. I.e. let the outcome