An approach for designing a surface pencil through a given geodesic curve


 Stephany Jefferson
 2 years ago
 Views:
Transcription
1 An approach for deigning a urface pencil hrough a given geodeic curve Gülnur SAFFAK ATALAY, Fama GÜLER, Ergin BAYRAM *, Emin KASAP Ondokuz Mayı Univeriy, Faculy of Ar and Science, Mahemaic Deparmen ABSTRACT Surface and curve play an imporan role in geomeric deign. In recen year, problem of finding a urface paing hrough a given curve have araced much inere. In he preen paper, we propoe a new mehod o conruc a urface inerpolaing a given curve a he geodeic curve of i. Alo, we analyze he condiion when he reuling urface i a ruled urface. In addiion, developabliy along he common geodeic of he member of urface family are dicued. Finally, we illurae hi mehod by preening ome example. 1. Inroducion A roaion minimizing adaped frame (RMF) {T,U,V} of a pace curve conain he curve angen T and he normal plane vecor U, V which how no inananeou roaion abou T. Becaue of heir minimum wi RMF are very inereing in compuer graphic, including freeform deformaion wih curve conrain [16], weep urface modeling [710], modeling of generalized cylinder and ree branche [1115], viualizaion of reamline and ube [1517], imulaion of rope and ring [18], and moion deign and conrol [19]. There are infiniely many adaped frame on a given pace curve [20]. One can produce oher adaped frame from an exiing one by conrolling he orienaion of he frame vecor U and V in he normal plane of he curve. In differenial geomery he mo familiar adaped frame i Frene frame {T, N, B}, where T i he curve angen, N i he principal normal vecor and B T N i he binormal vecor (ee [21] for deail). Beide of i fame, he Frene frame i no a RMF and i i unuiable fopecifying he orienaion of a rigid body along a given curve in applicaion uch a moion planning, animaion, geomeric deign, and roboic, ince i incur unneceary roaion of he body [22]. Furhermore, Frene frame i undefined if he curvaure vanihe. One of mo ignifican curve on a urface i geodeic curve. Geodeic are imporan in he relaiviic decripion of graviy. Einein principle of equivalence ell u ha geodeic repreen he pah of freely falling paricle in a given pace. (Freely falling in hi conex mean moving only under he influence of graviy, wih no oher force involved). The geodeic principle ae ha he free rajecorie are he geodeic of pace. I play a very imporan role in a geomericrelaiviy heory, ince i mean ha he fundamenal equaion of dynamic i compleely deermined by he geomery of pace, and herefore ha no o be e a an independen equaion. Moreover, in uch a heory he acion idenifie (up o a conan) wih he fundamenal lengh invarian, o ha he aionary acion principle and he geodeic principle become idenical. The concep of geodeic alo find i place in variou indurial applicaion, uch a en manufacuring, cuing and paining pah, fibergla ape winding in pipe manufacuring, exile manufacuring [23 28]. In archiecure, ome pecial curve have nice properie in erm of rucural funcionaliy and manufacuring co. One example i planar curve in verical plane, whichcan be ued a uppor elemen. Anoher example i geodeic curve, [29]. Deng, B., decribed mehod o creae paern of pecial curve on urface, which find applicaion in deign and realizaion
2 of freeform archiecure. He preened an evoluion approach o generae a erie of curve which are eiher geodeic or piecewie geodeic, aring from a given ource curve on a urface. In [29], he inveigaed familie of pecial curve (uch a geodeic) on freeform urface, and propoe compuaional ool o creae uch familie. Alo, he inveigaed paern of pecial curve on urface, which find applicaion in deign and realizaion of freeform archiecural hape (for deail, ee [29] ). Mo people have heard he phrae; a raigh line i he hore diance beween wo poin. Bu in differenial geomery, hey ay hi ame hing in a differen language. They ay inead geodeic for he Euclidean meric are raigh line. A geodeic i a curve ha repreen he exremal value of a diance funcion in ome pace. In he Euclidean pace, exremal mean 'minimal',o geodeic are pah of minimal arc lengh. In general relaiviy, geodeic generalize he noion of "raigh line" o curved pace ime. Thi concep i baed on he mahemaical concep of a geodeic. Imporanly, he world line of a paricle free from all exernal force i a paricular ype of geodeic. In oher word, a freely moving paricle alway move along a geodeic. Geodeic are curve along which geodeic curvaure vanihe. Thi i of coure where he geodeic curvaure ha i name from. In recen year, fundamenal reearch ha focued on he revere problem or backward analyi: given a 3D curve, how can we characerize hoe urface ha poe hi curve a a pecial curve, raher han finding and claifying curve on analyical curved urface. The concep of family of urface having a given characeriic curve wa fir inroduced by Wang e.al. [28] in Euclidean 3pace. Kaap e.al. [30] generalized he work of Wang by inroducing new ype of marchingcale funcion, coefficien of he Frene frame appearing in he parameric repreenaion of urface. Alo, urface wih common geodeic in Minkowki 3pace have been he ubjec of many udie. In [31] Kaap and Akyıldız defined urface wih a common geodeic in Minkowki 3pace and gave he ufficien condiion on marchingcale funcion o ha he given curve i a common geodeic on ha urface. Şaffak and Kaap [32] udied family of urface wih a common null geodeic. Lie e al. [33] derived he neceary and ufficien condiion for a given curve o be he line of curvaure on a urface. Bayram e al. [34] udied parameric urface which poe a given curve a a common aympoic. However, hey olved he problem uing Frene frame of he given curve. In hi paper, we obain he neceary and ufficien condiion for a given curve o be boh ioparameric and geodeic on a parameric urface depending on he RMF. Furhermore, we how ha here exi ruled urface poeing a given curve a a common geodeic curve and preen a crieria for hee ruled urface o be developable one. We only udy curve wih an arc lengh parameer becaue uch a udy i eay o follow; if neceary, one can obain imilar reul for arbirarily parameeried regular curve. 2. Background A parameric curve 1 2, L L, ha a conan or parameer value. In hi paper, repec o arc lengh paramee and we aume ha i a curve on a urface r r For every poin of r, if r 0, he e,, along r, where T r, P P, in 3 ha denoe he derivaive of r wih i a regular curve, i.e. r 0 T N B i called he Frene frame / and N B T N. are he uni angen, principal normal, and binormal vecor of he curve a he poin, repecively. Derivaive formula of he Frene frame i governed by he relaion
3 where r and he curve T 0 0 T d N 0 N d 0 0 B B, repecively [35].,, rr r (2.1) are called he curvaure and orion of Anoher ueful frame along a curve i roaion minimizing frame. They are ueful in animaion, moion planning, wep urface conrucion and relaed applicaion where he Frene frame may prove unuiable or undefined. A frame among he frame on he curve around he angen vecor T.,, T U V i called roaion minimizing if i i he frame of minimum wi,, T U V i an RMF if, U U r V V r r where denoe he andard inner produc in [36]. Oberve ha uch a pair U and V i no unique; here exi a one parameer family of RMF correponding o differen e of iniial value of U and V. According o Bihop [20], a frame i an RMF if and only if each of and i parallel o. Equivalenly, U ' V ' T 3 U ' V 0, V ' U 0, (2.2) i he neceary and ufficien condiion for he frame o be roaion minimizing [37]. There i a relaion beween he Frene frame (if he Frene frame i defined) and RMF, ha i, U and V are he roaion of N and B of he curve in he normal plane. Then, where U co in N V in co B i he angle beween he vecor N and U (ee Fig. 2), [38]., (2.3) Fig. 2 The Frene frame (T(), N(), B()) and he vecor U(), V(). Eqn. (2.3) implie ha,, T U V i an RMF if i aifie he following relaion U ' co T, V ' in T, '. (2.4)
4 Noe ha,, T U V where he Frene frame i undefined. i defined along he curve even if he curvaure vanihe 3. Surface pencil wih an geodeic curve Suppoe we are given a 3dimenional parameric curve i he arc lengh (regular and form a where 1 Surface pencil ha inerpolae, L L 1 2,,,, P a T b U c V a,, b, and c, a,, b, and c, are 1 C ). 1 2, L L, in which a a common curve i given in he parameric, L L, T T , (3.1) funcion. The value of he marchingcale funcion indicae, repecively, he exenionlike, flexionlike, and reorionlike effec caued by he poin uni hrough ime, aring from r. Remark 3.1 : Oberve ha chooing differen marchingcale funcion yield differen urface poeing a a common curve. Our goal i o find he neceary and ufficien condiion for which he curve ioparameric and geodeic on he urface on he urface parameer P, Secondly he curve 0 T, T 1 2 P,, here exi a parameer 0. Firly, a T, T 1 2 uch ha a, b, c, 0, L L, T T uch ha i an ioparameric curve r i an geodeic curve on he urface P, According o he geodeic heory [39], geodeic curvaure along geodeic. Thu, if we ge: where n, The normal vecor of 0 n(, ) 0 i. (3.2) k de r ', r '', n g here exi a vanihe // N(). (3.3) i he urface normal along he curve P P, can be wrien a n, r P, P, and N i a normal vecor of From Eqn. (2.1) and (2.3), he normal vecor can be expreed a c(, ) b(, ) b(, ) c(, ) n, a(, ) ( )co ( ) a(, ) ( )in ( ) T( ) Thu,. r a(, ) c(, ) c(, ) a(, ) a(, ) ( )in ( ) 1 b(, ) ( )co ( ) c(, ) ( )in ( ) U() b(, ) a(, ) a(, ) b(, ) 1 b(, ) ( )co ( ) c(, ) ( )in ( ) a(, ) ( )co ( ) V ( )
5 where n,, T, U, V c b b c 1, 0 (, 0 ) a(, 0 ) ( )co ( ) (, 0) (, 0) (, 0) a(, 0) ( )in ( ), a c c a 2, 0 (, 0 ) (, 0) a(, 0) ( )in ( ) (, 0) 1 (, 0) b(, 0) ( )co ( ) 0 c(, ) ( )in ( ), b a a b 3, 0 (, 0 ) 1 (, 0) b(, 0) ( )co ( ) c(, 0) ( )in ( ) (, 0) a(, 0) ( )co ( ) (, 0 ). Remark 3.2 : Becaue, a, 0 b, 0 c, 0 0, 0 T1, T2, L1 L2, (3.4) along he curve r, by he definiion of parial differeniaion we have According o remark above, we hould have 1, 0 0, c 2, 0 (, 0), b 3, 0 (, 0). a b c, 0, 0, 0 0, 0 T1, T2, L1 L2. (3.5) Thu, from (3.4) we obain,,, n U V from Eqn. (2.3), we ge n, co ( ), in ( ), N in ( ), co ( ), B( ) from Eqn. (3.3), we know ha r i a geodeic curve if and only if co ( ), in ( ), 0 (3.6) and in ( ), co ( ), 0 (3.7) From (3.6) and (3.7), we obain 1 in ( ) 3 0, 0 here uing (3.5),we have
6 b (, 0) 0 and c in Thu we have following heorem: b, co, 0 0 Theorem 3.3 : The neceary and ufficien condiion for he curve ioparameric and geodeic on he urface P, c b b,0,0 0. i a, b, c, 0, in co, (, ) 0. Corollary 3.4 : The ufficien condiion for he curve geodeic on he urface P, i o be boh (3.8) o be boh ioparameric and b, f, in, c, f, co, f, 0,. 4. Ruled urface pencil wih a common aympoic curve Theorem 4.1 : Given an arclengh curve, here exi a ruled urface poeing r a a common geodeic curve. Proof : Chooing marchingcale funcion a and a, g, b, in, c, co Eqn. (3.1) ake he following form of a ruled urface, in co P 0 g T U V which aifie Eqn. (3.8) inerpolaing r a a common geodeic curve. Corollary 4.2 : Ruled urface (3.11) i developable if and only Proof : and only if 0 g, in co P 0 g T U V (2.1) (2.4) give r, d, d 0, where d gt gt cou inu inv cov d g T in U co V gt g cou inv cou in cot in V in cot gt g co co U g in in V. Employing Eqn. (3.12) in he deerminan we ge g, which complee proof. (3.9) P, (3.10), (3.11) i o be. i developable if. Uing Eqn. (3.12) 5. Example of generaing urface wih a common geodeic curve Example 5.1. Le how ha in( ), co, be a uni peed curve. Then i i eay o
7 3 3 4 T co, in( ), , 5 5 If we chooe 4 5 +c and c= U ' co co( ), in co( ), co( ) V ' coin( ), in in( ), in( ) hen Eqn. (2.5) i aified. By inegraion, we obain Now, U in( ) in( ), co( ) co( ), in( ), V co( ) co( ), in( ) in( ), co( ) ,, T U V i an RMF ince i aifie Eqn. (2.2). If we ake 4, 0,, in( )(in 1),, co( 4 a b c )co 5 5 and 0, 2, hen Eqn. (3.5) i aified. Thu, we obain a member of he urface pencil wih a common geodeic curve r a P co( ) in 1 in co co co 1 (, ) in( ) in 1in in in co co co co, co in in, in 2 in 1 co 2 co where 0 2, 0 2 (Fig. 1).
8 In Eqn. (3.8), if we ake Fig. 1. P (, ) 1 a a member of urface pencil and i geodeic curve. g() 4 3 urface wih a common geodeic curve, hen we obain he following developable ruled a in( ) co( ) in in in co co co, (, ) P co( ) in( ) in co co co in in, where 2 2, 0 2 (Fig. 2). Fig. 2. P (, ) 2 a a member of he developable ruled urface pencil and i geodeic curve.
9 Example 5.2. Le If we ake co,in( ),0 T in,co( ),0, 1, 0.,, T U V U co, in( ),0 b, 1 co, c, in be a uni peed curve. I i obviou ha and V 0,0,1 i an RMF. By chooing marchingcale funcion a and 0 0, hen Eqn. (2.2) i aified and a, 0,, hen Eqn. (3.5) i aified. Thu, we immediaely obain a member of he urface pencil wih a common geodeic curve 3 P (,) co co,in co,in where 0 2, 0 2 (Fig. 3). a Fig. 3. P (, ) 3 a a member of urface pencil and i geodeic curve. For he ame curve le u find a ruled urface. In Eqn. (3.8), if we ake we obain he following developable ruled urface wih a common geodeic curve where 4 P (, ) co,in,, 1 1 (Fig. 4). g() 0, hen a
10 Fig. 4. P (, ) 4 If we ake g () = 0 and a a member of he developable ruled urface pencil and i geodeic curve. 0 3 common geodeic curve r () a hen, we obain he developable ruled urface wih a (,) P5 co 1,in 1, where 0 2, 0 2 (Fig. 5). Fig. 5. P (, ) 5 a a member of he developable ruled urface pencil and i geodeic curve.
11 Reference [1] Bechmann D., Gerber D. Arbirary haped deformaion wih dogme, Viual Compu. 19, 2 3, 2003, pp [2] Peng Q, Jin X, Feng J. Arclenghbaed axial deformaion and lengh preerving deformaion. In Proceeding of Compuer Animaion 1997; [3] Lazaru F, Coquillar S, Jancène P. Ineracive axial deformaion, In Modeling in Compuer Graphic: Springer Verlag 1993; [4] Lazaru F., Verrou A. Feaurebaed hape ranformaion for polyhedral objec. In Proceeding of he 5h Eurographic Workhop on Animaion and Simulaion 1994; [5] Lazaru F, Coquillar S, Jancène P. Axial deformaion: an inuiive echnique. Compu. Aid De 1994; 26, 8: [6] Llama I, Powell A, Roignac J, Shaw CD. Bender: A virual ribbon for deforming 3d hape in biomedical and yling applicaion. In Proceeding of Sympoium on Solid and Phyical Modeling 2005; [7] Bloomenhal M, Rieenfeld RF. Approximaion of weep urface by enor produc NURBS. In SPIE Proceeding Curve and Surface in Compuer Viion and Graphic II 1991; Vol 1610: [8] Pomann H, Wagner M. Conribuion o moion baed urface deign. In J Shape Model 1998; 4, 3&4: [9] Silanen P, Woodward C. Normal orienaion mehod for 3D offe curve, weep urface, kinning. In Proceeding of Eurographic 1992; [10] Wang W, Joe B. Robu compuaion of roaion minimizing frame foweep urface modeling. Compu.Aid De 1997; 29: [11] Shani U, Ballard DH. Spline a embedding for generalized cylinder. Compu Viion Graph Image Proce 1984; 27: [12] Bloomenhal J. Modeling he mighy maple. In Proceeding of SIGGRAPH 1985: [13] Bronvoor WF, Klok F. Ray racing generalized cylinder. ACM Tran Graph 1985; 4, 4: [14] Semwal SK, Hallauer J. Biomedical modeling: implemening lineofacion algorihm for human mucle and bone uing generalized cylinder. Compu Graph 1994:18, 1: [15] Bank DC, Singer BA. A predicorcorrecor echnique for viualizing uneady flow. IEEE Tran on Viualiz Compu Graph 1995; 1, 2: [16] Hanon AJ, Ma H. A quaernion approach o reamline viualizaion. IEEE Tran Viualiz Compu Graph 1995; 1, 2:
12 [17] Hanon A. Conrained opimal framing of curve and urface uing quaernion gau map. In Proceeding of Viulizaion 1998; [18] Barzel R. Faking dynamic of rope and pring. IEEE Compu Graph Appl 1997; 17, 3: [19] Jüler B. Raional approximaion of roaion minimizing frame uing Pyhagoreanhodograph cubic. J Geom Graph 1999; 3: [20] Bihop RL. There i more han one way o frame a curve. Am Mah Mon 1975; 82: [21] O Neill B. Elemenary Differenial Geomery. New York: Academic Pre Inc; [22] Farouki RT, Sakkali T. Raional roaionminimizing frame on polynomial pace curve of arbirary degree. J Symbolic Compu 2010; 45: [23] Brond R, Jeulin D, Gaeau P, Jarrin J, Serpe G. Eimaion of he ranpor properie ofpolymer compoie by geodeic propagaion. J Microc 1994;176: [24] Bryon S. Virual paceime: an environmen for he viualizaion of curved paceimevia geodeic flow. Technical Repor, NASA NAS, Number RNR ; March [25] Grundig L, Eker L, Moncrieff E. Geodeic and emigeodeic line algorihm for cuingpaern generaion of archiecural exile rucure. In:Lan TT, edior. Proceeding of heaiapacific Conference on Shell and Spaial Srucure, Beijing [26] Haw RJ. An applicaion of geodeic curve o ail deign. Compu Graphic Forum1985;4(2): [27] Haw RJ, Munchmeyer RC. Geodeic curve on pached polynomial urface. CompuGraphic Forum 1983;2(4): [28] Wang G. J, Tang K, Tai C. L. Parameric repreenaion of a urface pencil wih a common paial geodeic. Compu. Aided De. 36 (5)(2004) [29] Deng, B., Special Curve Paern for Freeform Archiecure Ph.D. hei, Eingereich an der Technichen Univeria Wien,Fakula für Mahemaik und Geoinformaion von. [30] Kaap E, Akyıldız FT, Orbay K. A generalizaion of urface family wih common paial geodeic. Appl Mah Compu 2008; 201: [31] Kaap E, Akyildiz F.T. Surface wih common geodeic in Minkowki 3pace. Applied Mahemaic and Compuaion, 177 (2006) [32] Şaffak G, Kaap E. Family of urface wih a common null geodeic. Inernaional Journalof Phyical Science Vol. 4(8), pp , Augu, 2009.
13 [33] Bayram E, Güler F, Kaap E. Parameric repreenaion of a urface pencil wih a common aympoic curve. Compu Aided De 2012; 44: [34] Li CY, Wang RH, Zhu CG. Parameric repreenaion of a urface pencil wih a common line of curvaure. Compu Aided De 2011;43(9): [35] do Carmo MP. Differenial geomery of curve and urface. Englewood Cliff (New Jerey): Prenice Hall, Inc.; [36] Klok F. Two moving coordinae frame along a 3D rajecory. Compu Aided Geom Deign 1986; 3: [37] Han CY. Nonexience of raional roaionminimizing frame on cubic curve. Compu Aided Geom Deign 2008; 25: [38] Li CY, Wang RH, Zhu CG. An approach for deigning a developable urface hrough a given line of curvaure. Compu Aided De 2013; 45: [39] Naar H. A, Rahad A. B, Hamdoon F.M. Ruled urface wih imelike ruling. Appl. Mah. Compu. 147 (2004)
Topic: Applications of Network Flow Date: 9/14/2007
CS787: Advanced Algorihm Scribe: Daniel Wong and Priyananda Shenoy Lecurer: Shuchi Chawla Topic: Applicaion of Nework Flow Dae: 9/4/2007 5. Inroducion and Recap In he la lecure, we analyzed he problem
More information6.003 Homework #4 Solutions
6.3 Homewk #4 Soluion Problem. Laplace Tranfm Deermine he Laplace ranfm (including he region of convergence) of each of he following ignal: a. x () = e 2(3) u( 3) X = e 3 2 ROC: Re() > 2 X () = x ()e d
More information2.4 Network flows. Many direct and indirect applications telecommunication transportation (public, freight, railway, air, ) logistics
.4 Nework flow Problem involving he diribuion of a given produc (e.g., waer, ga, daa, ) from a e of producion locaion o a e of uer o a o opimize a given objecive funcion (e.g., amoun of produc, co,...).
More informationNewton's second law in action
Newon's second law in acion In many cases, he naure of he force acing on a body is known I migh depend on ime, posiion, velociy, or some combinaion of hese, bu is dependence is known from experimen In
More informationThe Transport Equation
The Transpor Equaion Consider a fluid, flowing wih velociy, V, in a hin sraigh ube whose cross secion will be denoed by A. Suppose he fluid conains a conaminan whose concenraion a posiion a ime will be
More informationCSE202 Greedy algorithms
CSE0 Greedy algorihm . Shore Pah in a Graph hore pah from Princeon CS deparmen o Einein' houe . Shore Pah in a Graph hore pah from Princeon CS deparmen o Einein' houe Tree wih a mo edge G i a ree on n
More informationDIFFERENTIAL EQUATIONS with TI89 ABDUL HASSEN and JAY SCHIFFMAN. A. Direction Fields and Graphs of Differential Equations
DIFFERENTIAL EQUATIONS wih TI89 ABDUL HASSEN and JAY SCHIFFMAN We will assume ha he reader is familiar wih he calculaor s keyboard and he basic operaions. In paricular we have assumed ha he reader knows
More informationChapter 13. Network Flow III Applications. 13.1 Edge disjoint paths. 13.1.1 Edgedisjoint paths in a directed graphs
Chaper 13 Nework Flow III Applicaion CS 573: Algorihm, Fall 014 Ocober 9, 014 13.1 Edge dijoin pah 13.1.1 Edgedijoin pah in a direced graph 13.1.1.1 Edge dijoin pah queiong: graph (dir/undir)., : verice.
More informationFourier series. Learning outcomes
Fourier series 23 Conens. Periodic funcions 2. Represening ic funcions by Fourier Series 3. Even and odd funcions 4. Convergence 5. Halfrange series 6. The complex form 7. Applicaion of Fourier series
More informationFortified financial forecasting models: nonlinear searching approaches
0 Inernaional Conference on Economic and inance Reearch IPEDR vol.4 (0 (0 IACSIT Pre, Singapore orified financial forecaing model: nonlinear earching approache Mohammad R. Hamidizadeh, Ph.D. Profeor,
More informationRenewal processes and Poisson process
CHAPTER 3 Renewal processes and Poisson process 31 Definiion of renewal processes and limi heorems Le ξ 1, ξ 2, be independen and idenically disribued random variables wih P[ξ k > 0] = 1 Define heir parial
More informationMath 201 Lecture 12: CauchyEuler Equations
Mah 20 Lecure 2: CauchyEuler Equaions Feb., 202 Many examples here are aken from he exbook. The firs number in () refers o he problem number in he UA Cusom ediion, he second number in () refers o he problem
More informationChapter 7. Response of FirstOrder RL and RC Circuits
Chaper 7. esponse of FirsOrder L and C Circuis 7.1. The Naural esponse of an L Circui 7.2. The Naural esponse of an C Circui 7.3. The ep esponse of L and C Circuis 7.4. A General oluion for ep and Naural
More informationAppendix A: Area. 1 Find the radius of a circle that has circumference 12 inches.
Appendi A: Area workedou s o OddNumbered Eercises Do no read hese workedou s before aemping o do he eercises ourself. Oherwise ou ma mimic he echniques shown here wihou undersanding he ideas. Bes wa
More informationMathematics in Pharmacokinetics What and Why (A second attempt to make it clearer)
Mahemaics in Pharmacokineics Wha and Why (A second aemp o make i clearer) We have used equaions for concenraion () as a funcion of ime (). We will coninue o use hese equaions since he plasma concenraions
More informationPhysical Topology Discovery for Large MultiSubnet Networks
Phyical Topology Dicovery for Large MuliSubne Nework Yigal Bejerano, Yuri Breibar, Mino Garofalaki, Rajeev Raogi Bell Lab, Lucen Technologie 600 Mounain Ave., Murray Hill, NJ 07974. {bej,mino,raogi}@reearch.belllab.com
More informationApplication of kinematic equation:
HELP: See me (office hours). There will be a HW help session on Monda nigh from 78 in Nicholson 109. Tuoring a #10 of Nicholson Hall. Applicaion of kinemaic equaion: a = cons. v= v0 + a = + v + 0 0 a
More informationFourier Series Solution of the Heat Equation
Fourier Series Soluion of he Hea Equaion Physical Applicaion; he Hea Equaion In he early nineeenh cenury Joseph Fourier, a French scienis and mahemaician who had accompanied Napoleon on his Egypian campaign,
More informationChapter 6. First Order PDEs. 6.1 Characteristics The Simplest Case. u(x,t) t=1 t=2. t=0. Suppose u(x, t) satisfies the PDE.
Chaper 6 Firs Order PDEs 6.1 Characerisics 6.1.1 The Simples Case Suppose u(, ) saisfies he PDE where b, c are consan. au + bu = 0 If a = 0, he PDE is rivial (i says ha u = 0 and so u = f(). If a = 0,
More informationOptimal Investment and Consumption Decision of Family with Life Insurance
Opimal Invesmen and Consumpion Decision of Family wih Life Insurance Minsuk Kwak 1 2 Yong Hyun Shin 3 U Jin Choi 4 6h World Congress of he Bachelier Finance Sociey Torono, Canada June 25, 2010 1 Speaker
More informationInductance and Transient Circuits
Chaper H Inducance and Transien Circuis Blinn College  Physics 2426  Terry Honan As a consequence of Faraday's law a changing curren hrough one coil induces an EMF in anoher coil; his is known as muual
More informationOn the Connection Between MultipleUnicast Network Coding and SingleSource SingleSink Network Error Correction
On he Connecion Beween MulipleUnica ework Coding and SingleSource SingleSink ework Error Correcion Jörg Kliewer JIT Join work wih Wenao Huang and Michael Langberg ework Error Correcion Problem: Adverary
More informationComplex Fourier Series. Adding these identities, and then dividing by 2, or subtracting them, and then dividing by 2i, will show that
Mah 344 May 4, Complex Fourier Series Par I: Inroducion The Fourier series represenaion for a funcion f of period P, f) = a + a k coskω) + b k sinkω), ω = π/p, ) can be expressed more simply using complex
More information1.2 Goals for Animation Control
A Direc Manipulaion Inerface for 3D Compuer Animaion Sco Sona Snibbe y Brown Universiy Deparmen of Compuer Science Providence, RI 02912, USA Absrac We presen a new se of inerface echniques for visualizing
More information2.6 Limits at Infinity, Horizontal Asymptotes Math 1271, TA: Amy DeCelles. 1. Overview. 2. Examples. Outline: 1. Definition of limits at infinity
.6 Limis a Infiniy, Horizonal Asympoes Mah 7, TA: Amy DeCelles. Overview Ouline:. Definiion of is a infiniy. Definiion of horizonal asympoe 3. Theorem abou raional powers of. Infinie is a infiniy This
More informationAP Calculus BC 2010 Scoring Guidelines
AP Calculus BC Scoring Guidelines The College Board The College Board is a noforprofi membership associaion whose mission is o connec sudens o college success and opporuniy. Founded in, he College Board
More informationsdomain Circuit Analysis
Domain ircui Analyi Operae direcly in he domain wih capacior, inducor and reior Key feaure lineariy i preered c decribed by ODE and heir Order equal number of plu number of Elemenbyelemen and ource ranformaion
More informationOn the degrees of irreducible factors of higher order Bernoulli polynomials
ACTA ARITHMETICA LXII.4 (1992 On he degrees of irreducible facors of higher order Bernoulli polynomials by Arnold Adelberg (Grinnell, Ia. 1. Inroducion. In his paper, we generalize he curren resuls on
More informationHeat demand forecasting for concrete district heating system
Hea demand forecaing for concree diric heaing yem Bronilav Chramcov Abrac Thi paper preen he reul of an inveigaion of a model for horerm hea demand forecaing. Foreca of hi hea demand coure i ignifican
More informationANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS
ANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS R. Caballero, E. Cerdá, M. M. Muñoz and L. Rey () Deparmen of Applied Economics (Mahemaics), Universiy of Málaga,
More informationCHAPTER 11 NONPARAMETRIC REGRESSION WITH COMPLEX SURVEY DATA. R. L. Chambers Department of Social Statistics University of Southampton
CHAPTER 11 NONPARAMETRIC REGRESSION WITH COMPLEX SURVEY DATA R. L. Chamber Deparmen of Social Saiic Univeriy of Souhampon A.H. Dorfman Office of Survey Mehod Reearch Bureau of Labor Saiic M.Yu. Sverchkov
More informationStochastic Optimal Control Problem for Life Insurance
Sochasic Opimal Conrol Problem for Life Insurance s. Basukh 1, D. Nyamsuren 2 1 Deparmen of Economics and Economerics, Insiue of Finance and Economics, Ulaanbaaar, Mongolia 2 School of Mahemaics, Mongolian
More informationCapacitors and inductors
Capaciors and inducors We coninue wih our analysis of linear circuis by inroducing wo new passive and linear elemens: he capacior and he inducor. All he mehods developed so far for he analysis of linear
More informationThe Chase Problem (Part 2) David C. Arney
The Chae Problem Par David C. Arne Inroducion In he previou ecion, eniled The Chae Problem Par, we dicued a dicree model for a chaing cenario where one hing chae anoher. Some of he applicaion of hi kind
More informationA Comparative Study of Linear and Nonlinear Models for Aggregate Retail Sales Forecasting
A Comparaive Sudy of Linear and Nonlinear Model for Aggregae Reail Sale Forecaing G. Peer Zhang Deparmen of Managemen Georgia Sae Univeriy Alana GA 30066 (404) 6514065 Abrac: The purpoe of hi paper i
More informationJournal Of Business & Economics Research September 2005 Volume 3, Number 9
Opion Pricing And Mone Carlo Simulaions George M. Jabbour, (Email: jabbour@gwu.edu), George Washingon Universiy YiKang Liu, (yikang@gwu.edu), George Washingon Universiy ABSTRACT The advanage of Mone Carlo
More informationThe Torsion of Thin, Open Sections
EM 424: Torsion of hin secions 26 The Torsion of Thin, Open Secions The resuls we obained for he orsion of a hin recangle can also be used be used, wih some qualificaions, for oher hin open secions such
More informationA Curriculum Module for AP Calculus BC Curriculum Module
Vecors: A Curriculum Module for AP Calculus BC 00 Curriculum Module The College Board The College Board is a noforprofi membership associaion whose mission is o connec sudens o college success and opporuniy.
More informationGLAS Team Member Quarterly Report. June , Golden, Colorado (Colorado School of Mines)
GLAS Team ember Quarerly Repor An Nguyen, Thomas A Herring assachuses Insiue of Technology Period: 04/01/2004 o 06/30//2004 eeings aended Tom Herring aended he eam meeing near GSFC a he end of June, 2004.
More informationMaintenance scheduling and process optimization under uncertainty
Compuers and Chemical Engineering 25 (2001) 217 236 www.elsevier.com/locae/compchemeng ainenance scheduling and process opimizaion under uncerainy C.G. Vassiliadis, E.N. Piikopoulos * Deparmen of Chemical
More informationLecture III: Finish Discounted Value Formulation
Lecure III: Finish Discouned Value Formulaion I. Inernal Rae of Reurn A. Formally defined: Inernal Rae of Reurn is ha ineres rae which reduces he ne presen value of an invesmen o zero.. Finding he inernal
More informationAP Calculus AB 2013 Scoring Guidelines
AP Calculus AB 1 Scoring Guidelines The College Board The College Board is a missiondriven noforprofi organizaion ha connecs sudens o college success and opporuniy. Founded in 19, he College Board was
More informationAnswer, Key Homework 2 David McIntyre 45123 Mar 25, 2004 1
Answer, Key Homework 2 Daid McInyre 4123 Mar 2, 2004 1 This prinou should hae 1 quesions. Muliplechoice quesions may coninue on he ne column or page find all choices before making your selecion. The
More informationOptimal Path Routing in Single and Multiple Clock Domain Systems
IEEE TRANSACTIONS ON COMPUTERAIDED DESIGN, TO APPEAR. 1 Opimal Pah Rouing in Single and Muliple Clock Domain Syem Soha Haoun, Senior Member, IEEE, Charle J. Alper, Senior Member, IEEE ) Abrac Shrinking
More informationProcess Modeling for Object Oriented Analysis using BORM Object Behavioral Analysis.
Proce Modeling for Objec Oriened Analyi uing BORM Objec Behavioral Analyi. Roger P. Kno Ph.D., Compuer Science Dep, Loughborough Univeriy, U.K. r.p.kno@lboro.ac.uk 9RMW FKMerunka Ph.D., Dep. of Informaion
More information17 Laplace transform. Solving linear ODE with piecewise continuous right hand sides
7 Laplace ransform. Solving linear ODE wih piecewise coninuous righ hand sides In his lecure I will show how o apply he Laplace ransform o he ODE Ly = f wih piecewise coninuous f. Definiion. A funcion
More informationHANDOUT 14. A.) Introduction: Many actions in life are reversible. * Examples: Simple One: a closed door can be opened and an open door can be closed.
Inverse Funcions Reference Angles Inverse Trig Problems Trig Indeniies HANDOUT 4 INVERSE FUNCTIONS KEY POINTS A.) Inroducion: Many acions in life are reversible. * Examples: Simple One: a closed door can
More informationRandom Walk in 1D. 3 possible paths x vs n. 5 For our random walk, we assume the probabilities p,q do not depend on time (n)  stationary
Random Walk in D Random walks appear in many cones: diffusion is a random walk process undersanding buffering, waiing imes, queuing more generally he heory of sochasic processes gambling choosing he bes
More informationTerm Structure of Prices of Asian Options
Term Srucure of Prices of Asian Opions Jirô Akahori, Tsuomu Mikami, Kenji Yasuomi and Teruo Yokoa Dep. of Mahemaical Sciences, Risumeikan Universiy 111 Nojihigashi, Kusasu, Shiga 5258577, Japan Email:
More informationBSplines and NURBS Week 5, Lecture 9
CS 430/536 Compuer Graphics I BSplines an NURBS Wee 5, Lecure 9 Davi Breen, William Regli an Maxim Peysahov Geomeric an Inelligen Compuing Laboraory Deparmen of Compuer Science Drexel Universiy hp://gicl.cs.rexel.eu
More informationThree Dimensional Grounding Grid Design
Three Dimenional Grounding Grid Deign Fikri Bari Uzunlar 1, Özcan Kalenderli 2 1 Schneider Elecric Turkey, Ianbul, Turkey bari.uzunlar@r.chneiderelecric.com 2 Ianbul Technical Univeriy, ElecricalElecronic
More informationA Brief Introduction to the Consumption Based Asset Pricing Model (CCAPM)
A Brief Inroducion o he Consumpion Based Asse Pricing Model (CCAPM We have seen ha CAPM idenifies he risk of any securiy as he covariance beween he securiy's rae of reurn and he rae of reurn on he marke
More informationNewton s Laws of Motion
Newon s Laws of Moion MS4414 Theoreical Mechanics Firs Law velociy. In he absence of exernal forces, a body moves in a sraigh line wih consan F = 0 = v = cons. Khan Academy Newon I. Second Law body. The
More informationFormulating CyberSecurity as Convex Optimization Problems
Formulaing CyberSecuriy a Convex Opimizaion Problem Kyriako G. Vamvoudaki, João P. Hepanha, Richard A. Kemmerer, and Giovanni Vigna Univeriy of California, Sana Barbara Abrac. Miioncenric cyberecuriy
More informationTime variant processes in failure probability calculations
Time varian processes in failure probabiliy calculaions A. Vrouwenvelder (TUDelf/TNO, The Neherlands) 1. Inroducion Acions on srucures as well as srucural properies are usually no consan, bu will vary
More information1. y 5y + 6y = 2e t Solution: Characteristic equation is r 2 5r +6 = 0, therefore r 1 = 2, r 2 = 3, and y 1 (t) = e 2t,
Homework6 Soluions.7 In Problem hrough 4 use he mehod of variaion of parameers o find a paricular soluion of he given differenial equaion. Then check your answer by using he mehod of undeermined coeffiens..
More informationCircuit Types. () i( t) ( )
Circui Types DC Circuis Idenifying feaures: o Consan inpus: he volages of independen volage sources and currens of independen curren sources are all consan. o The circui does no conain any swiches. All
More informationNew Evidence on Mutual Fund Performance: A Comparison of Alternative Bootstrap Methods. David Blake* Tristan Caulfield** Christos Ioannidis*** and
New Evidence on Muual Fund Performance: A Comparion of Alernaive Boorap Mehod David Blake* Trian Caulfield** Chrio Ioannidi*** and Ian Tonk**** June 2014 Abrac Thi paper compare he wo boorap mehod of Koowki
More informationMTH6121 Introduction to Mathematical Finance Lesson 5
26 MTH6121 Inroducion o Mahemaical Finance Lesson 5 Conens 2.3 Brownian moion wih drif........................... 27 2.4 Geomeric Brownian moion........................... 28 2.5 Convergence of random
More informationEmpirical heuristics for improving Intermittent Demand Forecasting
Empirical heuriic for improving Inermien Demand Forecaing Foio Peropoulo 1,*, Konanino Nikolopoulo 2, Georgio P. Spihouraki 1, Vailio Aimakopoulo 1 1 Forecaing & Sraegy Uni, School of Elecrical and Compuer
More informationNanocubes for RealTime Exploration of Spatiotemporal Datasets
Nanocube for RealTime Exploraion of Spaioemporal Daae Lauro Lin, Jame T Kloowki, and arlo Scheidegger Fig 1 Example viualizaion of 210 million public geolocaed Twier po over he coure of a year The daa
More informationTEMPORAL PATTERN IDENTIFICATION OF TIME SERIES DATA USING PATTERN WAVELETS AND GENETIC ALGORITHMS
TEMPORAL PATTERN IDENTIFICATION OF TIME SERIES DATA USING PATTERN WAVELETS AND GENETIC ALGORITHMS RICHARD J. POVINELLI AND XIN FENG Deparmen of Elecrical and Compuer Engineering Marquee Universiy, P.O.
More informationON SHARP HIGHER ORDER SOBOLEV EMBEDDINGS
Communicaion in Conemporary Mahemaic Vol. 6, No. 3 (24) 495 511 c World Scienific Publihing Company ON SHARP HIGHER ORDER SOBOLEV EMBEDDINGS MARIO MILMAN Deparmen of Mahemaic, Florida Alanic Univeriy,
More informationcooking trajectory boiling water B (t) microwave 0 2 4 6 8 101214161820 time t (mins)
Alligaor egg wih calculus We have a large alligaor egg jus ou of he fridge (1 ) which we need o hea o 9. Now here are wo accepable mehods for heaing alligaor eggs, one is o immerse hem in boiling waer
More informationSOME REMARKS ON TARDIFF S FIXED POINT THEOREM ON MENGER SPACES
PORTUGALIAE MATHEMATICA Vol. 54 Fac. 4 997 SOME REMARKS ON TARDIFF S FIXED POINT THEOREM ON MENGER SPACES E. Pără and V. Rad Inrodcion Le D + be he family of all diribion fncion F : R [0, ] ch ha F (0)
More informationRelative velocity in one dimension
Connexions module: m13618 1 Relaive velociy in one dimension Sunil Kumar Singh This work is produced by The Connexions Projec and licensed under he Creaive Commons Aribuion License Absrac All quaniies
More information11/6/2013. Chapter 14: Dynamic ADAS. Introduction. Introduction. Keeping track of time. The model s elements
Inroducion Chaper 14: Dynamic DS dynamic model of aggregae and aggregae supply gives us more insigh ino how he economy works in he shor run. I is a simplified version of a DSGE model, used in cuingedge
More informationPENSION REFORM IN BELGIUM: A NEW POINTS SYSTEM BETWEEN DB and DC
PENSION REFORM IN BELGIUM: A NEW POINS SYSEM BEWEEN B and C Pierre EVOLER (*) (March 3 s, 05) Absrac More han in oher counries, he Belgian firs pillar of public pension needs urgen and srucural reforms
More information4. International Parity Conditions
4. Inernaional ariy ondiions 4.1 urchasing ower ariy he urchasing ower ariy ( heory is one of he early heories of exchange rae deerminaion. his heory is based on he concep ha he demand for a counry's currency
More information4 Convolution. Recommended Problems. x2[n] 1 2[n]
4 Convoluion Recommended Problems P4.1 This problem is a simple example of he use of superposiion. Suppose ha a discreeime linear sysem has oupus y[n] for he given inpus x[n] as shown in Figure P4.11.
More informationLIFE INSURANCE WITH STOCHASTIC INTEREST RATE. L. Noviyanti a, M. Syamsuddin b
LIFE ISURACE WITH STOCHASTIC ITEREST RATE L. oviyani a, M. Syamsuddin b a Deparmen of Saisics, Universias Padjadjaran, Bandung, Indonesia b Deparmen of Mahemaics, Insiu Teknologi Bandung, Indonesia Absrac.
More informationHow has globalisation affected inflation dynamics in the United Kingdom?
292 Quarerly Bullein 2008 Q3 How ha globaliaion affeced inflaion dynamic in he Unied Kingdom? By Jennifer Greenlade and Sephen Millard of he Bank Srucural Economic Analyi Diviion and Chri Peacock of he
More informationCHARGE AND DISCHARGE OF A CAPACITOR
REFERENCES RC Circuis: Elecrical Insrumens: Mos Inroducory Physics exs (e.g. A. Halliday and Resnick, Physics ; M. Sernheim and J. Kane, General Physics.) This Laboraory Manual: Commonly Used Insrumens:
More informationOption Pricing Under Stochastic Interest Rates
I.J. Engineering and Manufacuring, 0,3, 889 ublished Online June 0 in MECS (hp://www.mecspress.ne) DOI: 0.585/ijem.0.03. Available online a hp://www.mecspress.ne/ijem Opion ricing Under Sochasic Ineres
More informationCalculation of variable annuity market sensitivities using a pathwise methodology
cuing edge Variable annuiie Calculaion of variable annuiy marke eniiviie uing a pahwie mehodology Under radiional finie difference mehod, he calculaion of variable annuiy eniiviie can involve muliple Mone
More informationFormulating CyberSecurity as Convex Optimization Problems Æ
Formulaing CyberSecuriy a Convex Opimizaion Problem Æ Kyriako G. Vamvoudaki,João P. Hepanha, Richard A. Kemmerer 2, and Giovanni Vigna 2 Cener for Conrol, Dynamicalyem and Compuaion (CCDC), Univeriy
More informationAnalogue and Digital Signal Processing. First Term Third Year CS Engineering By Dr Mukhtiar Ali Unar
Analogue and Digial Signal Processing Firs Term Third Year CS Engineering By Dr Mukhiar Ali Unar Recommended Books Haykin S. and Van Veen B.; Signals and Sysems, John Wiley& Sons Inc. ISBN: 073807 Ifeachor
More informationDifferential Equations. Solving for Impulse Response. Linear systems are often described using differential equations.
Differenial Equaions Linear sysems are ofen described using differenial equaions. For example: d 2 y d 2 + 5dy + 6y f() d where f() is he inpu o he sysem and y() is he oupu. We know how o solve for y given
More informationA Note on Renewal Theory for T iid Random Fuzzy Variables
Applied Mahemaical Sciences, Vol, 6, no 6, 97979 HIKARI Ld, wwwmhikaricom hp://dxdoiorg/988/ams6686 A Noe on Renewal Theory for T iid Rom Fuzzy Variables Dug Hun Hong Deparmen of Mahemaics, Myongji
More informationAn empirical analysis about forecasting Tmall airconditioning sales using time series model Yan Xia
An empirical analysis abou forecasing Tmall aircondiioning sales using ime series model Yan Xia Deparmen of Mahemaics, Ocean Universiy of China, China Absrac Time series model is a hospo in he research
More informationPartial Internal Control Recovery on 1D KleinGordon Systems
IB J. Sci. Vol. 4 A, No.,,  Parial Inernal Conrol Recovery on D KleinGordon Sysems Iwan Pranoo Faculy of Mahemaics and Naural Sciences, Insiu eknologi Bandung, Jl. Ganesha No. Bandung 43 Absrac. In
More informationINDEPENDENT MARGINALS OF OPERATOR LÉVY S PROBABILITY MEASURES ON FINITE DIMENSIONAL VECTOR SPACES
Journal of Applied Analysis 1, 1 (1995), pp. 39 45 INDEPENDENT MARGINALS OF OPERATOR LÉVY S PROBABILITY MEASURES ON FINITE DIMENSIONAL VECTOR SPACES A. LUCZAK Absrac. We find exponens of independen marginals
More informationTechnical Appendix to Risk, Return, and Dividends
Technical Appendix o Risk, Reurn, and Dividends Andrew Ang Columbia Universiy and NBER Jun Liu UC San Diego This Version: 28 Augus, 2006 Columbia Business School, 3022 Broadway 805 Uris, New York NY 10027,
More informationa. Defining set of equations. Create separate mfile nanme.m containing the set of first order equations.
Signals and Sysems. Eperimen. Hins Problems: a) Solving sae equaions using sandard Malab 5. procedures b) Solving ordinary differenial nh order equaions wihou Symbolic Mah Toolbo (Malab 5.) a.) Solving
More informationRealtime Particle Filters
Realime Paricle Filers Cody Kwok Dieer Fox Marina Meilă Dep. of Compuer Science & Engineering, Dep. of Saisics Universiy of Washingon Seale, WA 9895 ckwok,fox @cs.washingon.edu, mmp@sa.washingon.edu Absrac
More informationDependent Interest and Transition Rates in Life Insurance
Dependen Ineres and ransiion Raes in Life Insurance Krisian Buchard Universiy of Copenhagen and PFA Pension January 28, 2013 Absrac In order o find marke consisen bes esimaes of life insurance liabiliies
More informationCointegration: The Engle and Granger approach
Coinegraion: The Engle and Granger approach Inroducion Generally one would find mos of he economic variables o be nonsaionary I(1) variables. Hence, any equilibrium heories ha involve hese variables require
More informationModule 3 Design for Strength. Version 2 ME, IIT Kharagpur
Module 3 Design for Srengh Lesson 2 Sress Concenraion Insrucional Objecives A he end of his lesson, he sudens should be able o undersand Sress concenraion and he facors responsible. Deerminaion of sress
More informationMeasuring macroeconomic volatility Applications to export revenue data, 19702005
FONDATION POUR LES ETUDES ET RERS LE DEVELOPPEMENT INTERNATIONAL Measuring macroeconomic volailiy Applicaions o expor revenue daa, 1970005 by Joël Cariolle Policy brief no. 47 March 01 The FERDI is a
More informationDistributing Human Resources among Software Development Projects 1
Disribuing Human Resources among Sofware Developmen Proecs Macario Polo, María Dolores Maeos, Mario Piaini and rancisco Ruiz Summary This paper presens a mehod for esimaing he disribuion of human resources
More informationOption PutCall Parity Relations When the Underlying Security Pays Dividends
Inernaional Journal of Business and conomics, 26, Vol. 5, No. 3, 22523 Opion Puall Pariy Relaions When he Underlying Securiy Pays Dividends Weiyu Guo Deparmen of Finance, Universiy of Nebraska Omaha,
More informationTanaka formula and Levy process
Tanaka formula and Levy process Simply speaking he Tanaka formula is an exension of he Iô formula while Lévy process is an exension of Brownian moion. Because he Tanaka formula and Lévy process are wo
More informationAP Calculus AB 2010 Scoring Guidelines
AP Calculus AB 1 Scoring Guidelines The College Board The College Board is a noforprofi membership associaion whose mission is o connec sudens o college success and opporuniy. Founded in 1, he College
More informationDYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS
DYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS Hong Mao, Shanghai Second Polyechnic Universiy Krzyszof M. Osaszewski, Illinois Sae Universiy Youyu Zhang, Fudan Universiy ABSTRACT Liigaion, exper
More informationCommunication Networks II Contents
3 / 1  Communicaion Neworks II (Görg)  www.comnes.unibremen.de Communicaion Neworks II Conens 1 Fundamenals of probabiliy heory 2 Traffic in communicaion neworks 3 Sochasic & Markovian Processes (SP
More informationNiche Market or Mass Market?
Niche Marke or Mass Marke? Maxim Ivanov y McMaser Universiy July 2009 Absrac The de niion of a niche or a mass marke is based on he ranking of wo variables: he monopoly price and he produc mean value.
More information3 RungeKutta Methods
3 RungeKua Mehods In conras o he mulisep mehods of he previous secion, RungeKua mehods are singlesep mehods however, muliple sages per sep. They are moivaed by he dependence of he Taylor mehods on he
More informationCrosssectional and longitudinal weighting in a rotational household panel: applications to EUSILC. Vijay Verma, Gianni Betti, Giulio Ghellini
Croecional and longiudinal eighing in a roaional houehold panel: applicaion o EUSILC Viay Verma, Gianni Bei, Giulio Ghellini Working Paper n. 67, December 006 CROSSSECTIONAL AND LONGITUDINAL WEIGHTING
More informationSubsistence Consumption and Rising Saving Rate
Subience Conumpion and Riing Saving Rae Kenneh S. Lin a, HiuYun Lee b * a Deparmen of Economic, Naional Taiwan Univeriy, Taipei, 00, Taiwan. b Deparmen of Economic, Naional Chung Cheng Univeriy, ChiaYi,
More information9. Capacitor and Resistor Circuits
ElecronicsLab9.nb 1 9. Capacior and Resisor Circuis Inroducion hus far we have consider resisors in various combinaions wih a power supply or baery which provide a consan volage source or direc curren
More information