FAR FIELD SOLUTION OF SH-WAVE BY CIRCULAR INCLUSION AND LINEAR CRACK
|
|
- Augustus Wiggins
- 5 years ago
- Views:
Transcription
1 The 4 th Wold Cofeece o Eathquake Egieeig Octobe -7, 8, Beijig, Chia FAR FIELD SOLUTION OF SH-WAVE BY CIRCULAR INCLUSION AND LINEAR CRACK HogLiag Li,GuoHui Wu, Associate Pofesso, Depatmet of Egieeig Mechaics, Habi Egieeig Uivesity, Habi. Chia leehl@sia.com ABSTRACT : Cicula iclusio is used widely i stuctue egieeig. I this pape, the method of Gee s fuctio is used to ivestigate the poblem of fa field solutio of cicula iclusio ad liea cack impacted by icidet SH-wave. Fistly, a Gee s fuctio is costucted fo the poblem, which is a fudametal solutio of displacemet field fo a elastic space possessig a cicula iclusio while beaig out-of-plae hamoic lie souce foce at ay poit; Secodly, i tems of the solutio of SH-wave s scatteig by a elastic space with a cicula iclusio, ati-plae stesses which ae the same i quatity but opposite i diectio to those metioed befoe, ae loaded at the egio whee the liea cack is i existet actually, we called this pocess cack-divisio ; Fially, the expessios of the displacemet ad stesses ae give whe the cicula iclusio ad liea cack exist at the same time. The, whe the special Gee s fuctio has bee costucted ad close field solutio has bee illustated, the fa field of scatteed wave is studied. The displacemet mode of scatteed wave at fa field ad scatteig coss-sectio ae give. Numeical esults ae illustated ad the ifluece of wave umbe, icidet agles of SH-wave, ad the combiatio of diffeet media paametes ae discussed. The esults ca be applied i the study of factue, ad udamaged fame cack detectio. KEYWORDS: cack, cicula iclusio, Gee s Fuctio, SH-wave scatteig, displacemet mode of scatteed wave at fa field, scatteed coss-sectio. INTRODUCTION Cicula iclusio exists widely i atual media, egieeig mateials ad stuctues, ad defects ae usually foud aoud the iclusio. Whe a composite mateial with cicula iclusio ad cacks is impacted by the dyamic load, o the oe had, the scatteig field poduced by the cicula iclusio ad cacks detemies the dyamic stess cocetatio facto aoud the cicula iclusio, ad theefoe detemies whethe the mateial is damaged o ot; o the othe had, the scatteig field also pesets may chaacteistic paametes of the iclusio ad cacks such as defect compositio, locatio ad shape, so the eseach o the scatteig fa-field is impotat to the geological pospects, seismological ivestigatio, o-destuctio evaluatio ad the othe fields. I the ocea acoustics, the scatteig fa-field of the acoustic wave is also used i the ude-wate suvey, object distiguishig ad so o. I theoy, the scatteig solutio of elastic waves is oe of the basic topics of evese poblems o elastic wave. O the basis of liteatue, few pape cocetates o the scatteig fa-field solutio of SH-wave by a cicula iclusio ad a liea cack aoud the iclusio. I the pape a ew model ad a ew method ae peseted i ode to ivestigate deeply o this kid poblem. At peset, to obtai the theoetical solutio of the poblem coceed i this thesis is of geat iteest ad cetaily it has some difficulties. The developmet of computatioal mechaics has povided may methods to solve the poblem, but a theoetical aswe is still expected i ode to ivestigate the chaacteistics of the cicula iclusio ad cack. The pape uses the Gee s fuctio to study the scatteig fa-field of elastic wave by a cicula iclusio ad a liea cack. The Gee s fuctio should be a fudametal solutio of displacemet field fo a elastic space possessig a cicula iclusio while beaig out-of-plae hamoic lie souce foce at ay poit. I tems of the solutio of SH-wave s scatteig by a elastic space with a cicula iclusio, ati-plae stesses which ae the same i quatity but opposite i diectio to those metioed befoe,
2 The 4 th Wold Cofeece o Eathquake Egieeig Octobe -7, 8, Beijig, Chia ae loaded at the egio whee the liea cack is i existet actually, we called this pocess cack-divisio. The, the expessios of the displacemet ad stesses ae give whe the cicula iclusio ad liea cack exist at the same time. The, whe the special Gee s fuctio has bee costucted ad close field solutio has bee illustated, the fa field of scatteed wave is studied. The displacemet mode of scatteed wave at fa field ad scatteig coss-sectio ae give. At last, a example is give ad its umeical esults ae discussed.. MODEL AND GOVERNING EQUATION The model is show as Fig., a elastic space cotaiig a cicula iclusio ad a liea cack aoud the iclusio. I this pape, the ati-plae shea SH wave model is studied. The displacemet i the elastic space is expessed as W( x, y, t),the displacemet i the iclusio is expessed as W( x, y, t). The goveig equatio of Wi ca be witte i the pola coodiate system as: W W W kw = θ W W W kw = θ (.) (.) ω μ whee ki =, C i Si =,ω C ρ is the cicula fequecy of the displacemet W( i x, y, t), Csi stads fo the shea Si i wave velocity, ρi ad μ i ae the mass desity ad the shea modulus of elasticity espectively. Figue Model of Poblem 3. GREEN S FUNCTION The Gee s fuctio used i this pape is egaded as the displacemet espose to the elastic space cotaiig a cicula iclusio impacted by ati-plae hamoic liea souce foce at ay poit. The depedece of the i t displacemet fuctiog i o time t is e ω.i the pola coodiate system, the goveig equatio of Gi ca be witte as: G G G ( ), G k G G = δ + + G + k G = (3.) θ θ stads fo the positio of the liea souce foce i pola coodiates. The bouday coditios ca be expessed as below:
3 The 4 th Wold Cofeece o Eathquake Egieeig Octobe -7, 8, Beijig, Chia G = G τ = τ (3.), = R z z = R = R = R The basic solutio which satisfies the cotol equatio (3.) ad the bouday coditios (3.) should iclude two pats of motio: the distubace of ati-plae liea souce foce ad the scatteig wave icited by the cicula iclusio. The wave displacemet of the complete elastic space due to the lie souce load δ ( ) o the abitay positio ca be give: () i i () G = H ( k ) 4μ (3.3) () Whee H () is the fist kid of Hakel fuctio ad zeo-ode. The scatteig wave i the elastic space ad i the cicula iclusio ca be witte as: ( s) () ( i) = m m θ θ = m m θ θ m= m= (3.4) G A H ( k ) cos[ m( )], G B J ( k ) cos[ m( )] whee A, B ae ukow coefficiets. m m Theefoe, G G G () i ( s) = +, wave field G of this poblem ca be obtaied. G ( i) = G.Accodig to the bouday coditios, we ca obtai m A, B. So, the m 4. EXPRESSION OF DISPLACEMENT AND STRESS FOR THE MODEL The stess o the cack aoud the iclusio poduced by icidet SH-wave ad the scatteig wave icited by the cicula iclusio ca be obtaied. A pai of opposite foces is applied to the cack; theefoe the esultat foce o the cack is zeo, which ca be thought as cack.. The above costuctig pocess is called cack-divisio techique which ca be used to obtai the expessio of displacemet ad stess fo the model. The detail ca be discussed as follows. Fistly, we coside the icidece of SH-wave o the ifiite liea-elastic space cotaiig a cicula iclusio. () i The icidet displacemet field W hamoic to time ca be witte as follows: () i W = W εi cos[ ( θ α)] J( k) (4.) = whee α is the icidet agle. =, ε = ;, ε =. The scatteig wave i the elastic space ad i the cicula iclusio ca be witte as: ( s) () ( i) = = (4.) W = A H ( k)cos[ ( θ α)], W = B J ( k )cos[ ( θ α)] By usig the bouday coditios, we ca obtai A, B. The displacemet field W ca be give as: () i ( s) W = W + W (4.3)
4 The 4 th Wold Cofeece o Eathquake Egieeig Octobe -7, 8, Beijig, Chia The, we coside the scatteig poblem of icidet SH-wave whe the cicula iclusio ad cack exist at the same time. Accodig to icidet field ad scatteig field i the elastic space cotaiig oly a iclusio, the cack-divisio techique is used to costuct the model of SH-wave scatteig by a elastic space cotaiig a cicula iclusio ad a liea cack. The costuctig pocess is that: the space is sepaated alog the cack ad a pai of ati-plae opposite foces with the multitude τ θ z ae applied to up ad dow sectio of the egio whee cack will appea, theefoe the esultat foce o up (o dow) sectio of the egio is zeo, which ca be thought as cack. The above costucted Gee s fuctio idicates that the basic displacemet solutio ca be obtaied wheeve the ati-plae liea souce foce s positio it is. Cosequetly, we ca obtai the total displacemet field ad stess field ude the iteactio of the cicula iclusio ad the cack fo icidet SH-wave. The foce is applied o the cack ad the tectoic additioal displacemet field ca be obtaied: τ θ z = τ = θ z G (,, θθ, ) (4.4) Itegatig alog the lie of cack, we ca obtai: G (,, θ, θ ) d τ θ z = (4.5) Hece, the total displacemet field ca be witte as follows: W = W G (,, θ, θ ) d τ θ z = (4.6) 5. THE SCATTERING DISPLACEMENT MODE AT FAR FIELD ( s) The total scatteig wave field icludes the scatteig wave W poduced by the cicula iclusio, ad τ G (,, θ, θ ) d poduced by the liea cack, that is θ z = W = W G (,, θ, θ ) d ( zs) ( s) ( t ) τ θ z = (5.) The scatteig wave ca be expessed as a seies of Hakel fuctio, ad thei commo item H () ( K) ca be abstacted. Make use of the asymptotic expessio of Hakel fuctio as the idepedet vaiable is lage eough: () ( / 4) ( ) ( ) i z π H z = i e (5.) π z The scatteig fa-field displacemet ca be expessed as: π ( ) ( ) 8 ik zs π 4 W (, θ ) = e F( θ ) (5.3) k whee
5 The 4 th Wold Cofeece o Eathquake Egieeig Octobe -7, 8, Beijig, Chia i A [ ] θ z = = μ = A μw m A () θ α ε m α m m = = m= A F( θ) = { W ( i) A cos ( θ α ) τ ε (cos θ)[ J ( k ) + ] d } π 4 = { W ( i) A cos [ ( )] i (si m ) m[ J ( k ) H ( k )] π i 4 A A ( i) ε(cos θ)[ J( k) ] d } μ + (5.4) 6.THE SCATTERING CROSS SECTION (SCS) The time aveage eegy flow of the wave ove oe peiod T ca be defied as: ( ) Ave( E ) = iω i ij j ij j 4 σ u σ u da A ω = Im iσiju j da A (6.) Im( ) is the imagiay pat of a complex fuctio. The above-metioed fomula ca be used to calculate the time aveage eegy flow of the elastic wave. The time aveage eegy flow passig though the suface = R( which is axis Z diectio is a uit log) is: (s) z π W (s) z μ Ave( E ω ) = Im( W ) Rdθ ωμ = Im π ( zs) W ( zs) W Rd θ (6.) Substitute (5.) ito fomula(6.), ad make use of ()' () iπ ( m ) / H ( k) Hm ( k) e, (6.3) π we ca obtai the esult at fa distace R: Ave( E ωμ ) = W Im( E) (6.4) π The scatteig coss sectio is the atio of the total eegy of fa field scatteig wave to the time aveage eegy flow pe uit aea of the icidet wave. Fo the plae icidet SH wave, the time aveage of eegy flow pe uit aea is: Ave( E ) Ave( e ) = = μkωw = σωw (6.5) A ad letγ expesses the atio of these two eegy, we ca obtai the followig esult: γ = Im( E),whee, γ is the π k scatteig coss sectio.
6 The 4 th Wold Cofeece o Eathquake Egieeig Octobe -7, 8, Beijig, Chia 7.EXAMPLE I this pape, we pay attetio to a epesetative kid of models, which is show as Figue. The adius of the cicula iclusio is, ad the legth of the cack is. I Fig. ad Fig.3, the distace betwee the ie tip of the cack ad the cete of the iclusio is. Fig. ad Fig.3 show that sice thee is a liea cack compaed with the displacemet mode of fa field poduced by the cicula iclusio scatteig wave, the displacemet mode of fa field poduced by the cicula iclusio ad cack scatteig wave is chaged a lot. Whe the icidet wave is vetical to the cack thee is the most chage. I Fig.4, it shows the ifluece of μ μ to the Displacemet Mode whe k = α = 9. It ca be foud that the moe diffeet the mateial of elastic space with the mateial of iclusio is, the bigge ifulece the cack have. I Fig.5, the chage of the scatteig coss sectio goig togethe with the chage of the icidet wave umbe is give. It ca be foud that whe thee is a liea cack low fequecy sympathetic vibatio come ito beig. (a) α = 3 (b) α = 6 (c) α = 9 Figue Ifluece of Cack to the Displacemet Mode whe k = μ μ =
7 The 4 th Wold Cofeece o Eathquake Egieeig Octobe -7, 8, Beijig, Chia (a) α = 3 (b) α = 6 (c) α = 9 Figue 3 Ifluece of Cack to the Displacemet Mode whe k = μ μ = (a) μ μ = 8 (b) μ μ =.5 Figue 4 Ifluece of μ μ to the Displacemet Mode whe k = α = 9
8 The 4 th Wold Cofeece o Eathquake Egieeig Octobe -7, 8, Beijig, Chia Figue 5 Vaiatio of SCS vs. kr whe μ μ = Fom the istaces above-said, it ca be show that the ifluece of the cack should ot be eglected, ad by usig the coclusio we should be capable to estimate the positio of the cack though aalysig test data of the displacemet mode. 8.SUMMARY I this pape, by usig the techique of cack-divisio, fa field solutio of cicula iclusio ad liea cack impacted by icidet SH-wave is give.. By usig the method a example is solved, ad some ew coclusio is give. The method i the pape could be used to study some othe coelative poblem. REFERENCES Pao Y. H. (983).Elastic Waves i Solids. ASME Joual of Applied Mechaics 5:4, Zheg Zhemi, Zhou Heg, Zhag Haxi. (995) Teds of Developmet i Mechaics i the Ealy st Cetuy. Advaces i Mechaics 5:4, Wag Duo, Wag Yuesheg. (993). Recet Pogess i Dyamics of Iteface. Shaghai Joual of Mechaics 4:4, -5 Liu Diakui, Liu Hogwei. (999).Scatteig of SH-wave by Cacks Oigiatig at A Cicula Hole Edge ad Dyamic Stess Itesity Facto. Acta Mechaica Siica 3:3, 9 99 Liu Diakui, Liu Hogwei. (998).Scatteig ad Dyamic Stess Cocetatio of SH-wave by Iteface Cicula Hole. Acta Mechaica Siica 3:5, Liu hogwei, Liu Diakui. (999).Fa Field Solutio of SH-wave Scatteed by Iteface Cicula Hole. Acta Mechaica Solida Siica :4, Li HogLiag, Liu DiaKui. (4).Iteactio of SH-Waves by Cacks with Cicula Iclusio.Joual of Habi Egieeig Uivesity 5:5, Li HogLiagi. (4).The Iteactio of Cicula Cavity, Iclusio with Beelie Cacks by SH-wave, Habi Egieeig Uivesity.
On composite conformal mapping of an annulus to a plane with two holes
O composite cofomal mappig of a aulus to a plae with two holes Mila Batista (July 07) Abstact I the aticle we coside the composite cofomal map which maps aulus to ifiite egio with symmetic hole ad ealy
More informationTechnical Report: Bessel Filter Analysis
Sasa Mahmoodi 1 Techical Repot: Bessel Filte Aalysis 1 School of Electoics ad Compute Sciece, Buildig 1, Southampto Uivesity, Southampto, S17 1BJ, UK, Email: sm3@ecs.soto.ac.uk I this techical epot, we
More informationCHAPTER 5 : SERIES. 5.2 The Sum of a Series Sum of Power of n Positive Integers Sum of Series of Partial Fraction Difference Method
CHAPTER 5 : SERIES 5.1 Seies 5. The Sum of a Seies 5..1 Sum of Powe of Positive Iteges 5.. Sum of Seies of Patial Factio 5..3 Diffeece Method 5.3 Test of covegece 5.3.1 Divegece Test 5.3. Itegal Test 5.3.3
More informationMultivector Functions
I: J. Math. Aal. ad Appl., ol. 24, No. 3, c Academic Pess (968) 467 473. Multivecto Fuctios David Hestees I a pevious pape [], the fudametals of diffeetial ad itegal calculus o Euclidea -space wee expessed
More informationa) The average (mean) of the two fractions is halfway between them: b) The answer is yes. Assume without loss of generality that p < r.
Solutios to MAML Olympiad Level 00. Factioated a) The aveage (mea) of the two factios is halfway betwee them: p ps+ q ps+ q + q s qs qs b) The aswe is yes. Assume without loss of geeality that p
More information2012 GCE A Level H2 Maths Solution Paper Let x,
GCE A Level H Maths Solutio Pape. Let, y ad z be the cost of a ticet fo ude yeas, betwee ad 5 yeas, ad ove 5 yeas categoies espectively. 9 + y + 4z =. 7 + 5y + z = 8. + 4y + 5z = 58.5 Fo ude, ticet costs
More informationUsing Difference Equations to Generalize Results for Periodic Nested Radicals
Usig Diffeece Equatios to Geealize Results fo Peiodic Nested Radicals Chis Lyd Uivesity of Rhode Islad, Depatmet of Mathematics South Kigsto, Rhode Islad 2 2 2 2 2 2 2 π = + + +... Vieta (593) 2 2 2 =
More informationMapping Radius of Regular Function and Center of Convex Region. Duan Wenxi
d Iteatioal Cofeece o Electical Compute Egieeig ad Electoics (ICECEE 5 Mappig adius of egula Fuctio ad Cete of Covex egio Dua Wexi School of Applied Mathematics Beijig Nomal Uivesity Zhuhai Chia 363463@qqcom
More informationAdvanced Physical Geodesy
Supplemetal Notes Review of g Tems i Moitz s Aalytic Cotiuatio Method. Advaced hysical Geodesy GS887 Chistophe Jekeli Geodetic Sciece The Ohio State Uivesity 5 South Oval Mall Columbus, OH 4 7 The followig
More informationAuchmuty High School Mathematics Department Sequences & Series Notes Teacher Version
equeces ad eies Auchmuty High chool Mathematics Depatmet equeces & eies Notes Teache Vesio A sequece takes the fom,,7,0,, while 7 0 is a seies. Thee ae two types of sequece/seies aithmetic ad geometic.
More informationKEY. Math 334 Midterm II Fall 2007 section 004 Instructor: Scott Glasgow
KEY Math 334 Midtem II Fall 7 sectio 4 Istucto: Scott Glasgow Please do NOT wite o this exam. No cedit will be give fo such wok. Rathe wite i a blue book, o o you ow pape, pefeably egieeig pape. Wite you
More informationMATH Midterm Solutions
MATH 2113 - Midtem Solutios Febuay 18 1. A bag of mables cotais 4 which ae ed, 4 which ae blue ad 4 which ae gee. a How may mables must be chose fom the bag to guaatee that thee ae the same colou? We ca
More informationLecture 24: Observability and Constructibility
ectue 24: Obsevability ad Costuctibility 7 Obsevability ad Costuctibility Motivatio: State feedback laws deped o a kowledge of the cuet state. I some systems, xt () ca be measued diectly, e.g., positio
More informationSOME ARITHMETIC PROPERTIES OF OVERPARTITION K -TUPLES
#A17 INTEGERS 9 2009), 181-190 SOME ARITHMETIC PROPERTIES OF OVERPARTITION K -TUPLES Deick M. Keiste Depatmet of Mathematics, Pe State Uivesity, Uivesity Pak, PA 16802 dmk5075@psu.edu James A. Selles Depatmet
More information= 5! 3! 2! = 5! 3! (5 3)!. In general, the number of different groups of r items out of n items (when the order is ignored) is given by n!
0 Combiatoial Aalysis Copyight by Deiz Kalı 4 Combiatios Questio 4 What is the diffeece betwee the followig questio i How may 3-lette wods ca you wite usig the lettes A, B, C, D, E ii How may 3-elemet
More informationBy 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
More informationApplications of the Dirac Sequences in Electrodynamics
Poc of the 8th WSEAS It Cof o Mathematical Methods ad Computatioal Techiques i Electical Egieeig Buchaest Octobe 6-7 6 45 Applicatios of the Diac Sequeces i Electodyamics WILHELM W KECS Depatmet of Mathematics
More informationCRACK DETECTION IN EULER-BERNOULLI BEAMS ON ELASTIC FOUNDATION USING GENETIC ALGORITHM BASED ON DISCRETE ELEMENT TECHNIQUE
Idia J.Sci.Res.() : 48-5, 04 ISSN:50-08(Olie) ISSN : 0976-876 (Pit) CRACK DEECION IN EULER-BERNOULLI BEAMS ON ELASIC FOUNDAION USING GENEIC ALGORIHM BASED ON DISCREE ELEMEN ECHNIQUE MOJABA GHASEMI a, ALIREZA
More informationLecture 2: Stress. 1. Forces Surface Forces and Body Forces
Lectue : Stess Geophysicists study pheomea such as seismicity, plate tectoics, ad the slow flow of ocks ad mieals called ceep. Oe way they study these pheomea is by ivestigatig the defomatio ad flow of
More informationGround Rules. PC1221 Fundamentals of Physics I. Uniform Circular Motion, cont. Uniform Circular Motion (on Horizon Plane) Lectures 11 and 12
PC11 Fudametals of Physics I Lectues 11 ad 1 Cicula Motio ad Othe Applicatios of Newto s Laws D Tay Seg Chua 1 Goud Rules Switch off you hadphoe ad page Switch off you laptop compute ad keep it No talkig
More informationSupplementary materials. Suzuki reaction: mechanistic multiplicity versus exclusive homogeneous or exclusive heterogeneous catalysis
Geeal Pape ARKIVOC 009 (xi 85-03 Supplemetay mateials Suzui eactio: mechaistic multiplicity vesus exclusive homogeeous o exclusive heteogeeous catalysis Aa A. Kuohtia, Alexade F. Schmidt* Depatmet of Chemisty
More informationModelling rheological cone-plate test conditions
ANNUAL TRANSACTIONS OF THE NORDIC RHEOLOGY SOCIETY, VOL. 16, 28 Modellig heological coe-plate test coditios Reida Bafod Schülle 1 ad Calos Salas-Bigas 2 1 Depatmet of Chemisty, Biotechology ad Food Sciece,
More informationGreatest term (numerically) in the expansion of (1 + x) Method 1 Let T
BINOMIAL THEOREM_SYNOPSIS Geatest tem (umeically) i the epasio of ( + ) Method Let T ( The th tem) be the geatest tem. Fid T, T, T fom the give epasio. Put T T T ad. Th will give a iequality fom whee value
More information( ) 1 Comparison Functions. α is strictly increasing since ( r) ( r ) α = for any positive real number c. = 0. It is said to belong to
Compaiso Fuctios I this lesso, we study stability popeties of the oautoomous system = f t, x The difficulty is that ay solutio of this system statig at x( t ) depeds o both t ad t = x Thee ae thee special
More informationChapter 8 Complex Numbers
Chapte 8 Complex Numbes Motivatio: The ae used i a umbe of diffeet scietific aeas icludig: sigal aalsis, quatum mechaics, elativit, fluid damics, civil egieeig, ad chaos theo (factals. 8.1 Cocepts - Defiitio
More informationEffect of Material Gradient on Stresses of Thick FGM Spherical Pressure Vessels with Exponentially-Varying Properties
M. Zamai Nejad et al, Joual of Advaced Mateials ad Pocessig, Vol.2, No. 3, 204, 39-46 39 Effect of Mateial Gadiet o Stesses of Thick FGM Spheical Pessue Vessels with Expoetially-Vayig Popeties M. Zamai
More informationSome Properties of the K-Jacobsthal Lucas Sequence
Deepia Jhala et. al. /Iteatioal Joual of Mode Scieces ad Egieeig Techology (IJMSET) ISSN 349-3755; Available at https://www.imset.com Volume Issue 3 04 pp.87-9; Some Popeties of the K-Jacobsthal Lucas
More informationGROUND MOTION OF NON-CIRCULAR ALLUVIAL VALLEY FOR INCIDENT PLANE SH-WAVE. Hui QI, Yong SHI, Jingfu NAN
The th World Coferece o Earthquake Egieerig October -7, 8, Beiig, Chia GROUND MOTION OF NON-CIRCULAR ALLUVIAL VALLEY FOR INCIDENT PLANE SH-WAVE Hui QI, Yog SHI, Jigfu NAN ABSTRACT : Professor, Dept. of
More informationEinstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road New Delhi , Ph. : ,
MB BINOMIAL THEOREM Biomial Epessio : A algebaic epessio which cotais two dissimila tems is called biomial epessio Fo eample :,,, etc / ( ) Statemet of Biomial theoem : If, R ad N, the : ( + ) = a b +
More informationL8b - Laplacians in a circle
L8b - Laplacias i a cicle Rev //04 `Give you evidece,' the Kig epeated agily, `o I'll have you executed, whethe you'e evous o ot.' `I'm a poo ma, you Majesty,' the Hatte bega, i a temblig voice, `--ad
More informationProgression. CATsyllabus.com. CATsyllabus.com. Sequence & Series. Arithmetic Progression (A.P.) n th term of an A.P.
Pogessio Sequece & Seies A set of umbes whose domai is a eal umbe is called a SEQUENCE ad sum of the sequece is called a SERIES. If a, a, a, a 4,., a, is a sequece, the the expessio a + a + a + a 4 + a
More informationA NOTE ON DOMINATION PARAMETERS IN RANDOM GRAPHS
Discussioes Mathematicae Gaph Theoy 28 (2008 335 343 A NOTE ON DOMINATION PARAMETERS IN RANDOM GRAPHS Athoy Boato Depatmet of Mathematics Wilfid Lauie Uivesity Wateloo, ON, Caada, N2L 3C5 e-mail: aboato@oges.com
More informationMATH /19: problems for supervision in week 08 SOLUTIONS
MATH10101 2018/19: poblems fo supevisio i week 08 Q1. Let A be a set. SOLUTIONS (i Pove that the fuctio c: P(A P(A, defied by c(x A \ X, is bijective. (ii Let ow A be fiite, A. Use (i to show that fo each
More informationGeneralized Fibonacci-Lucas Sequence
Tuish Joual of Aalysis ad Numbe Theoy, 4, Vol, No 6, -7 Available olie at http://pubssciepubcom/tjat//6/ Sciece ad Educatio Publishig DOI:6/tjat--6- Geealized Fiboacci-Lucas Sequece Bijeda Sigh, Ompaash
More informationLESSON 15: COMPOUND INTEREST
High School: Expoeial Fuctios LESSON 15: COMPOUND INTEREST 1. You have see this fomula fo compoud ieest. Paamete P is the picipal amou (the moey you stat with). Paamete is the ieest ate pe yea expessed
More informationUsing Counting Techniques to Determine Probabilities
Kowledge ticle: obability ad Statistics Usig outig Techiques to Detemie obabilities Tee Diagams ad the Fudametal outig iciple impotat aspect of pobability theoy is the ability to detemie the total umbe
More informationOn ARMA(1,q) models with bounded and periodically correlated solutions
Reseach Repot HSC/03/3 O ARMA(,q) models with bouded ad peiodically coelated solutios Aleksade Weo,2 ad Agieszka Wy oma ska,2 Hugo Steihaus Cete, Woc aw Uivesity of Techology 2 Istitute of Mathematics,
More information2. Characteristics of Synchrotron Radiation
. Chaacteistics of Schoto Radiatio. Itoductio The adiatio i geeal is chaacteized b the followig tems: spectal age, photo flu, photo flu desit, billiace, ad the polaizatio. The photo flu is the oveall flu
More informationBINOMIAL THEOREM An expression consisting of two terms, connected by + or sign is called a
BINOMIAL THEOREM hapte 8 8. Oveview: 8.. A epessio cosistig of two tems, coected by + o sig is called a biomial epessio. Fo eample, + a, y,,7 4 5y, etc., ae all biomial epessios. 8.. Biomial theoem If
More informationAIEEE 2004 (MATHEMATICS)
AIEEE 004 (MATHEMATICS) Impotat Istuctios: i) The test is of hous duatio. ii) The test cosists of 75 questios. iii) The maimum maks ae 5. iv) Fo each coect aswe you will get maks ad fo a wog aswe you will
More informationBINOMIAL THEOREM NCERT An expression consisting of two terms, connected by + or sign is called a
8. Oveview: 8.. A epessio cosistig of two tems, coected by + o sig is called a biomial epessio. Fo eample, + a, y,,7 4, etc., ae all biomial 5y epessios. 8.. Biomial theoem BINOMIAL THEOREM If a ad b ae
More informationLatticed pentamode acoustic cloak (supplementary Info)
Lattied petamode aousti loak (supplemetay Ifo) Yi Che, Xiaoig Liu ad Gegkai Hu Key Laboatoy of yamis ad Cotol of Flight Vehile, Miisty of Eduatio, Shool of Aeospae Egieeig, Beiig Istitute of Tehology,
More information( ) ( ) ( ) ( ) Solved Examples. JEE Main/Boards = The total number of terms in the expansion are 8.
Mathematics. Solved Eamples JEE Mai/Boads Eample : Fid the coefficiet of y i c y y Sol: By usig fomula of fidig geeal tem we ca easily get coefficiet of y. I the biomial epasio, ( ) th tem is c T ( y )
More informationThe Pigeonhole Principle 3.4 Binomial Coefficients
Discete M athematic Chapte 3: Coutig 3. The Pigeohole Piciple 3.4 Biomial Coefficiets D Patic Cha School of Compute Sciece ad Egieeig South Chia Uivesity of Techology Ageda Ch 3. The Pigeohole Piciple
More informationTHE ANALYTIC LARGE SIEVE
THE ANALYTIC LAGE SIEVE 1. The aalytic lage sieve I the last lectue we saw how to apply the aalytic lage sieve to deive a aithmetic fomulatio of the lage sieve, which we applied to the poblem of boudig
More informationDANIEL YAQUBI, MADJID MIRZAVAZIRI AND YASIN SAEEDNEZHAD
MIXED -STIRLING NUMERS OF THE SEOND KIND DANIEL YAQUI, MADJID MIRZAVAZIRI AND YASIN SAEEDNEZHAD Abstact The Stilig umbe of the secod id { } couts the umbe of ways to patitio a set of labeled balls ito
More informationEVALUATION OF SUMS INVOLVING GAUSSIAN q-binomial COEFFICIENTS WITH RATIONAL WEIGHT FUNCTIONS
EVALUATION OF SUMS INVOLVING GAUSSIAN -BINOMIAL COEFFICIENTS WITH RATIONAL WEIGHT FUNCTIONS EMRAH KILIÇ AND HELMUT PRODINGER Abstact We coside sums of the Gaussia -biomial coefficiets with a paametic atioal
More informationIntegral Problems of Trigonometric Functions
06 IJSRST Volume Issue Pit ISSN: 395-60 Olie ISSN: 395-60X Themed Sectio: Sciece ad Techology Itegal Poblems of Tigoometic Fuctios Chii-Huei Yu Depatmet of Ifomatio Techology Na Jeo Uivesity of Sciece
More informationEffects of Some Structural Parameters on the Vibration of a Simply Supported Non-prismatic Double-beam System
Poceeigs of the Wol Cogess o Egieeig 017 Vol WCE 017, July 5-7, 017, Loo, U.K. Effects of Some Stuctual Paametes o the Vibatio of a Simply Suppote No-pismatic Double-beam System Olasumbo O. Agboola, Membe,
More informationEDEXCEL NATIONAL CERTIFICATE UNIT 28 FURTHER MATHEMATICS FOR TECHNICIANS OUTCOME 2- ALGEBRAIC TECHNIQUES TUTORIAL 1 - PROGRESSIONS
EDEXCEL NATIONAL CERTIFICATE UNIT 8 FURTHER MATHEMATICS FOR TECHNICIANS OUTCOME - ALGEBRAIC TECHNIQUES TUTORIAL - PROGRESSIONS CONTENTS Be able to apply algebaic techiques Aithmetic pogessio (AP): fist
More informationSome Integral Mean Estimates for Polynomials
Iteatioal Mathematical Foum, Vol. 8, 23, o., 5-5 HIKARI Ltd, www.m-hikai.com Some Itegal Mea Estimates fo Polyomials Abdullah Mi, Bilal Ahmad Da ad Q. M. Dawood Depatmet of Mathematics, Uivesity of Kashmi
More informationConditional Convergence of Infinite Products
Coditioal Covegece of Ifiite Poducts William F. Tech Ameica Mathematical Mothly 106 1999), 646-651 I this aticle we evisit the classical subject of ifiite poducts. Fo stadad defiitios ad theoems o this
More informationApplicability of Four Parameter Viscoelastic Model for Longitudinal Wave Propagation in Non-Homogeneous Rods
Applicability of Fou Paamete Viscoelastic Model fo Logitudial Wave Popagatio i No-Homogeeous Rods KANWALJEET KAUR Faculty of Applied Scieces, BMSCE, Muktsa-56, Idia RAJNEESH KAKAR* Picipal, DIPS Polytechic
More informationON EUCLID S AND EULER S PROOF THAT THE NUMBER OF PRIMES IS INFINITE AND SOME APPLICATIONS
Joual of Pue ad Alied Mathematics: Advaces ad Alicatios Volume 0 Numbe 03 Pages 5-58 ON EUCLID S AND EULER S PROOF THAT THE NUMBER OF PRIMES IS INFINITE AND SOME APPLICATIONS ALI H HAKAMI Deatmet of Mathematics
More informationANSWERS, HINTS & SOLUTIONS HALF COURSE TEST VII (Main)
AIITS-HT-VII-PM-JEE(Mai)-Sol./7 I JEE Advaced 06, FIITJEE Studets bag 6 i Top 00 AIR, 7 i Top 00 AIR, 8 i Top 00 AIR. Studets fom Log Tem lassoom/ Itegated School Pogam & Studets fom All Pogams have qualified
More informationCh 3.4 Binomial Coefficients. Pascal's Identit y and Triangle. Chapter 3.2 & 3.4. South China University of Technology
Disc ete Mathem atic Chapte 3: Coutig 3. The Pigeohole Piciple 3.4 Biomial Coefficiets D Patic Cha School of Compute Sciece ad Egieeig South Chia Uivesity of Techology Pigeohole Piciple Suppose that a
More informationPROGRESSION AND SERIES
INTRODUCTION PROGRESSION AND SERIES A gemet of umbes {,,,,, } ccodig to some well defied ule o set of ules is clled sequece Moe pecisely, we my defie sequece s fuctio whose domi is some subset of set of
More informationTHE GRAVITATIONAL POTENTIAL OF A MULTIDIMENSIONAL SHELL
THE GRAVITATIONAL POTENTIAL OF A MULTIDIMENSIONAL SHELL BY MUGUR B. RĂUŢ Abstact. This pape is a attept to geealize the well-kow expessio of the gavitatioal potetial fo oe tha thee diesios. We used the
More informationRange Symmetric Matrices in Minkowski Space
BULLETIN of the Bull. alaysia ath. Sc. Soc. (Secod Seies) 3 (000) 45-5 LYSIN THETICL SCIENCES SOCIETY Rae Symmetic atices i ikowski Space.R. EENKSHI Depatmet of athematics, amalai Uivesity, amalaiaa 608
More information4. PERMUTATIONS AND COMBINATIONS
4. PERMUTATIONS AND COMBINATIONS PREVIOUS EAMCET BITS 1. The umbe of ways i which 13 gold cois ca be distibuted amog thee pesos such that each oe gets at least two gold cois is [EAMCET-000] 1) 3 ) 4 3)
More informationME 354, MECHANICS OF MATERIALS LABORATORY MECHANICAL PROPERTIES AND PERFORMANCE OF MATERIALS: TORSION TESTING*
ME 354, MECHANICS OF MATEIALS LABOATOY MECHANICAL POPETIES AND PEFOMANCE OF MATEIALS: TOSION TESTING* MGJ/08 Feb 1999 PUPOSE The pupose of this execise is to obtai a umbe of expeimetal esults impotat fo
More informationFIXED POINT AND HYERS-ULAM-RASSIAS STABILITY OF A QUADRATIC FUNCTIONAL EQUATION IN BANACH SPACES
IJRRAS 6 () July 0 www.apapess.com/volumes/vol6issue/ijrras_6.pdf FIXED POINT AND HYERS-UAM-RASSIAS STABIITY OF A QUADRATIC FUNCTIONA EQUATION IN BANACH SPACES E. Movahedia Behbaha Khatam Al-Abia Uivesity
More informationLOCUS OF THE CENTERS OF MEUSNIER SPHERES IN EUCLIDEAN 3-SPACE. 1. Introduction
LOCUS OF THE CENTERS OF MEUSNIER SPHERES IN EUCLIDEAN -SPACE Beyha UZUNOGLU, Yusuf YAYLI ad Ismail GOK Abstact I this study, we ivestigate the locus of the cetes of the Meusie sphees Just as focal cuve
More informationTHE ANALYSIS OF SOME MODELS FOR CLAIM PROCESSING IN INSURANCE COMPANIES
Please cite this atle as: Mhal Matalyck Tacaa Romaiuk The aalysis of some models fo claim pocessig i isuace compaies Scietif Reseach of the Istitute of Mathemats ad Compute Sciece 004 Volume 3 Issue pages
More informationChapter 2 Sampling distribution
[ 05 STAT] Chapte Samplig distibutio. The Paamete ad the Statistic Whe we have collected the data, we have a whole set of umbes o desciptios witte dow o a pape o stoed o a compute file. We ty to summaize
More informationphysicsandmathstutor.com
physicsadmathstuto.com physicsadmathstuto.com Jauay 2009 2 a 7. Give that X = 1 1, whee a is a costat, ad a 2, blak (a) fid X 1 i tems of a. Give that X + X 1 = I, whee I is the 2 2 idetity matix, (b)
More informationAdvanced Higher Formula List
Advaced Highe Fomula List Note: o fomulae give i eam emembe eveythig! Uit Biomial Theoem Factoial! ( ) ( ) Biomial Coefficiet C!! ( )! Symmety Idetity Khayyam-Pascal Idetity Biomial Theoem ( y) C y 0 0
More informationResearch on Modal Parameters Identification of Parallel Manipulator with Flexible Multi-Body System
Reseach Joual of Applied Scieces, Egieeig ad echology 5(): 974-979, 3 ISS: 4-7459; e-iss: 4-7467 Maxwell Scietific Ogaizatio, 3 Submitted: Septembe 6, Accepted: Octobe 3, Published: Mach 5, 3 Reseach o
More informationGRAVITATIONAL FORCE IN HYDROGEN ATOM
Fudametal Joual of Mode Physics Vol. 8, Issue, 015, Pages 141-145 Published olie at http://www.fdit.com/ GRAVITATIONAL FORCE IN HYDROGEN ATOM Uiesitas Pedidika Idoesia Jl DR Setyabudhi No. 9 Badug Idoesia
More informationVolcanic activity and tidal heating of Saturn s moon Enceladus
Volcaic activity ad tidal heatig of Satu s moo celadus P. Vaga Geodetic ad Geophysical eseach Istitute, Seismological Obsevatoy, Budapest, Meedek u. 18-111, ugay (vaga@seismology.hu 1. Pelimiay statemets
More informationOn the Explicit Determinants and Singularities of r-circulant and Left r-circulant Matrices with Some Famous Numbers
O the Explicit Detemiats Sigulaities of -ciculat Left -ciculat Matices with Some Famous Numbes ZHAOLIN JIANG Depatmet of Mathematics Liyi Uivesity Shuaglig Road Liyi city CHINA jzh08@siacom JUAN LI Depatmet
More informationMath 166 Week-in-Review - S. Nite 11/10/2012 Page 1 of 5 WIR #9 = 1+ r eff. , where r. is the effective interest rate, r is the annual
Math 66 Week-i-Review - S. Nite // Page of Week i Review #9 (F-F.4, 4.-4.4,.-.) Simple Iteest I = Pt, whee I is the iteest, P is the picipal, is the iteest ate, ad t is the time i yeas. P( + t), whee A
More informationMinimal order perfect functional observers for singular linear systems
Miimal ode efect fuctioal obseves fo sigula liea systems Tadeusz aczoek Istitute of Cotol Idustial lectoics Wasaw Uivesity of Techology, -66 Waszawa, oszykowa 75, POLAND Abstact. A ew method fo desigig
More informationComplementary Dual Subfield Linear Codes Over Finite Fields
1 Complemetay Dual Subfield Liea Codes Ove Fiite Fields Kiagai Booiyoma ad Somphog Jitma,1 Depatmet of Mathematics, Faculty of Sciece, Silpao Uivesity, Naho Pathom 73000, hailad e-mail : ai_b_555@hotmail.com
More informationOn a Problem of Littlewood
Ž. JOURAL OF MATHEMATICAL AALYSIS AD APPLICATIOS 199, 403 408 1996 ARTICLE O. 0149 O a Poblem of Littlewood Host Alze Mosbache Stasse 10, 51545 Waldbol, Gemay Submitted by J. L. Bee Received May 19, 1995
More informationMATHS FOR ENGINEERS ALGEBRA TUTORIAL 8 MATHEMATICAL PROGRESSIONS AND SERIES
MATHS FOR ENGINEERS ALGEBRA TUTORIAL 8 MATHEMATICAL PROGRESSIONS AND SERIES O completio of this ttoial yo shold be able to do the followig. Eplai aithmetical ad geometic pogessios. Eplai factoial otatio
More informationStrong Result for Level Crossings of Random Polynomials. Dipty Rani Dhal, Dr. P. K. Mishra. Department of Mathematics, CET, BPUT, BBSR, ODISHA, INDIA
Iteatioal Joual of Reseach i Egieeig ad aageet Techology (IJRET) olue Issue July 5 Available at http://wwwijetco/ Stog Result fo Level Cossigs of Rado olyoials Dipty Rai Dhal D K isha Depatet of atheatics
More informationOn the Zeros of Daubechies Orthogonal and Biorthogonal Wavelets *
Applied Mathematics,, 3, 778-787 http://dx.doi.og/.436/am..376 Published Olie July (http://www.scirp.og/joual/am) O the Zeos of Daubechies Othogoal ad Biothogoal Wavelets * Jalal Kaam Faculty of Sciece
More informationLecture 6: October 16, 2017
Ifomatio ad Codig Theoy Autum 207 Lectue: Madhu Tulsiai Lectue 6: Octobe 6, 207 The Method of Types Fo this lectue, we will take U to be a fiite uivese U, ad use x (x, x 2,..., x to deote a sequece of
More informationDamped Vibration of a Non-prismatic Beam with a Rotational Spring
Vibratios i Physical Systems Vol.6 (0) Damped Vibratio of a No-prismatic Beam with a Rotatioal Sprig Wojciech SOCHACK stitute of Mechaics ad Fudametals of Machiery Desig Uiversity of Techology, Czestochowa,
More informationTaylor Transformations into G 2
Iteatioal Mathematical Foum, 5,, o. 43, - 3 Taylo Tasfomatios ito Mulatu Lemma Savaah State Uivesity Savaah, a 344, USA Lemmam@savstate.edu Abstact. Though out this pape, we assume that
More informationConsider unordered sample of size r. This sample can be used to make r! Ordered samples (r! permutations). unordered sample
Uodeed Samples without Replacemet oside populatio of elemets a a... a. y uodeed aagemet of elemets is called a uodeed sample of size. Two uodeed samples ae diffeet oly if oe cotais a elemet ot cotaied
More informationRotational symmetry applied to boundary element computation for nuclear fusion plasma
Bouda Elemets ad Othe Mesh Reductio Methods XXXII 33 Rotatioal smmet applied to bouda elemet computatio fo uclea fusio plasma M. Itagaki, T. Ishimau & K. Wataabe 2 Facult of Egieeig, Hokkaido Uivesit,
More informationSums of Involving the Harmonic Numbers and the Binomial Coefficients
Ameica Joual of Computatioal Mathematics 5 5 96-5 Published Olie Jue 5 i SciRes. http://www.scip.og/oual/acm http://dx.doi.og/.46/acm.5.58 Sums of Ivolvig the amoic Numbes ad the Biomial Coefficiets Wuyugaowa
More informationDiscussion 02 Solutions
STAT 400 Discussio 0 Solutios Spig 08. ~.5 ~.6 At the begiig of a cetai study of a goup of pesos, 5% wee classified as heavy smoes, 30% as light smoes, ad 55% as osmoes. I the fiveyea study, it was detemied
More informationA note on random minimum length spanning trees
A ote o adom miimum legth spaig tees Ala Fieze Miklós Ruszikó Lubos Thoma Depatmet of Mathematical Scieces Caegie Mello Uivesity Pittsbugh PA15213, USA ala@adom.math.cmu.edu, usziko@luta.sztaki.hu, thoma@qwes.math.cmu.edu
More informationSHIFTED HARMONIC SUMS OF ORDER TWO
Commu Koea Math Soc 9 0, No, pp 39 55 http://dxdoiog/03/ckms0939 SHIFTED HARMONIC SUMS OF ORDER TWO Athoy Sofo Abstact We develop a set of idetities fo Eule type sums I paticula we ivestigate poducts of
More informationCounting Functions and Subsets
CHAPTER 1 Coutig Fuctios ad Subsets This chapte of the otes is based o Chapte 12 of PJE See PJE p144 Hee ad below, the efeeces to the PJEccles book ae give as PJE The goal of this shot chapte is to itoduce
More informationECEN 5014, Spring 2013 Special Topics: Active Microwave Circuits and MMICs Zoya Popovic, University of Colorado, Boulder
ECEN 5014, Spig 013 Special Topics: Active Micowave Cicuits ad MMICs Zoya Popovic, Uivesity of Coloado, Boulde LECTURE 7 THERMAL NOISE L7.1. INTRODUCTION Electical oise is a adom voltage o cuet which is
More informationr, this equation is graphed in figure 1.
Washigto Uivesity i St Louis Spig 8 Depatmet of Ecoomics Pof James Moley Ecoomics 4 Homewok # 3 Suggested Solutio Note: This is a suggested solutio i the sese that it outlies oe of the may possible aswes
More informationFINITE ELEMENT ANALYSIS OF A BWR FEED WATER DISTRIBUTOR UNDER EXTREME TRANSIENT PRESSURE LOAD
FINITE ELEMENT ANALYSIS OF A BWR FEED WATER DISTRIBUTOR UNDER EXTREME TRANSIENT PRESSURE LOAD Ebehad Altstadt, Hema Ohlmeye 1, Fak Otemba 1, Fak-Pete Weiss 1. Itoductio The beak of a feed wate lie outside
More informationCalculation of Matrix Elements in the Foldy-Wouthuysen Representation
Calculatio of Matix Elemets i the Foldy-Wouthuyse Repesetatio V.P. Nezamov*, A.A.Sadovoy**, A.S.Ul yaov*** RFNC-VNIIEF, Saov, Russia Abstact The pape compaes the methods used to calculate matix elemets
More informationThe Application of Parseval s Theorem to Integral Problems
Applied Mathematics ad Physics, 0, Vol., No., -9 Available olie at http://pubs.sciepub.com/amp/// Sciece ad Educatio Publishig DOI:0.69/amp--- The Applicatio of Paseval s Theoem to Itegal Poblems Chii-Huei
More information[Dhayabaran*, 5(1): January, 2016] ISSN: (I2OR), Publication Impact Factor: 3.785
[Dhayabaa* 5(): Jauay 206] ISSN: 2277-9655 (I2OR) Publicatio Impact Facto: 3.785 IJESRT INTERNATIONAL JOURNAL OF ENGINEERING SCIENCES & RESEARCH TECHNOLOGY SOLVING FUZZY DIFFERENTIAL EQUATIONS USING RUNGE-KUTTA
More informationLecture 2-5. Optical Fiber
We use qualitative desciptio athe tha quatitative desciptio. What is fibe? - Best aswe is cicula ad vey log waveguide. - Geeal ideas fo guidig i a plaa waveguide is exteded to a cylidical waveguide - Fibe
More informationComputational Methods of Solid Mechanics. Project report
Computational Methods of Solid Mechanics Poject epot Due on Dec. 6, 25 Pof. Allan F. Bowe Weilin Deng Simulation of adhesive contact with molecula potential Poject desciption In the poject, we will investigate
More informationMulti-parameter Analysis of a Rigid Body. Nonlinear Coupled Rotations around
Adv. Theo. Appl. Mech., Vol. 6, 3, o., 9-7 HIKARI Ltd, www.m-hikai.com http://dx.doi.og/.988/atam.3.378 Multi-paamete Aalysis of a Rigid Body Noliea Coupled Rotatios aoud No Itesectig Axes Based o the
More informationSolving Fuzzy Differential Equations Using Runge-Kutta Third Order Method for Three Stages Contra-Harmonic Mean
ISSN (Pit): 347-671 Iteatioal Joual of Iovative Reseach i Sciece, Egieeig ad Techology (A High Impact Facto, Mothly Pee Reviewed Joual) Vol. 5, Issue, Febuay 16 Solvig Fuzzy Diffeetial Equatios Usig Ruge-Kutta
More informationJJMIE Jordan Journal of Mechanical and Industrial Engineering
JJIE Joda Joual o echaical ad Idustial Egieeig Volume 8 Numbe 4, August 4 ISSN 995-6665 Pages 7 - Dyamic Aalysis ad Desig o Steel-Ball Gidig achies Based o No-Slip Cases Jigju Zhag *, Guoguag Li, Ruizhe
More informationMath 7409 Homework 2 Fall from which we can calculate the cycle index of the action of S 5 on pairs of vertices as
Math 7409 Hoewok 2 Fall 2010 1. Eueate the equivalece classes of siple gaphs o 5 vetices by usig the patte ivetoy as a guide. The cycle idex of S 5 actig o 5 vetices is 1 x 5 120 1 10 x 3 1 x 2 15 x 1
More information