A Study of Some Integral Problems Using Maple
|
|
- Abel White
- 6 years ago
- Views:
Transcription
1 Mthemtis n Sttistis (): -, 0 DOI: 0.89/ms A Stuy of Some Integl Poblems Ug Mple Chii-Huei Yu Deptment of Mngement n Infomtion, Nn Jeon Univesity of Siene n Tehnology, Tinn City, 776, Tiwn *Coesponing Autho: hiihuei@mil.njt.eu.tw Copyight 0 Hoizon Reseh Publishing All ights eseve. Abstt This ppe tes the mthemtil softwe Mple s the uiliy tool to stuy fou types of integls. We n obtin the Fouie seies epnsions of these fou types of integls by ug integtion tem by tem theoem. On the othe hn, we povie two emples to o lultion ptilly. The eseh methos opte in this stuy involve fining solutions though mnul lultions n veifying these solutions by ug Mple. Keywos Integls, Fouie Seies Epnsions, Integtion Tem By Tem Theoem, Mple. Intoution In lulus n engineeing mthemtis ouses, we lent mny methos to solve the integl poblems inluing hnge of vibles metho, integtion by pts metho, ptil ftions metho, tigonometi substitution metho, n so on. In this ppe, we minly stuy the following fou types of integls whih e not esy to obtin thei nswes ug the methos mentione bove. [ os( osh[ () os[ os( h[ () os[ os( t + b)osh[ () [ os( t + b)h[ () Whee,, b e el numbes, 0. We n obtin the Fouie seies epnsions of these fou types of integls by ug integtion tem by tem theoem; these e the mjo esults of this ppe (i.e., Theoems, ). As fo the stuy of elte integl poblems n efe to [-]. On the othe hn, we popose some integls to o lultion ptilly. The eseh methos opte in this stuy involve fining solutions though mnul lultions n veifying these solutions by ug Mple. This type of eseh metho not only llows the isovey of lultion eos, but lso helps moify the oiginl ietions of thining fom mnul n Mple lultions. Fo this eson, Mple povies insights n guine eging poblem-solving methos.. Min Results Fistly, we intoue nottion n some fomuls use in this stuy... Nottion Let z + ib be omple numbe, whee i,, b e el numbes. We enote the el pt of z by Re( z), n b the imginy pt of z by Im(z)... Fomuls... Eule's fomul θ e i osθ + iθ, whee θ is ny el numbe.... DeMoive's fomul n (os θ + i θ ) osnθ + i nθ, whee n is ny intege, θ is ny el numbe.... ([6, p]) ( u + iv) uoshv + i osuhv, whee u, v e el numbes.... ([6, p]) os( u + iv) osu osh v i uh v, whee u, v e el numbes.... ([6, p6]) + z ( ) z 0 ( + )!, whee z is ny omple numbe...6. ([6, p6]) os z ( ) z, whee z is ny omple 0 ()! numbe.
2 A Stuy of Some Integl Poblems Ug Mple Net, we intoue n impotnt theoem use in this ppe... Integtion tem by tem theoem ([7, p69]) Suppose { g ( t) } 0 is sequene of Lebesgue integble funtions efine on n intevl [, ]. If g t ( ) 0 is onvegent, then g t ( ) g ( ) t. 0 0 The following is the fist mjo esult of this stuy, we obtin the Fouie seies epnsions of the integls () n ()... Theoem Assume,, b, e el numbes, 0. Then thee eists onstnt C suh tht fo ll R, the integl 0 ( ) [ os( osh[ + [( + )( + b)] ( + )!( + ) An thee eists onstnt C suh tht fo ll the integl 0... Poof Beuse ( ) (By Fomul..) () R, os[ os( h[ Re + os[( + )( + b)] ( + )!( + ) (6) 0 [ os( + b)]osh[ + b)] ( ) (Ug Fomul..) Re{[ epi( + b)]} [ epi( + b)] ( + )! + + Re ( ) epi[( + )( + b)] 0 ( + )! (By DeMoive's fomul) 0 ( ) + (By Eule's fomul) os[( + )( + b)] ( + )! (7) Thus, fo ll R, the integl [ os( osh[ + ( ) 0 ( + + os[( + )( )! ( ) os[( + )( ( + )! 0 (By integtion tem by tem theoem) 0 ( ) + [( + )( + b)] ( + )!( + ) Whee C is some onstnt. On the othe hn, ug Eule's fomul, DeMoive's fomul n Fomul..,.., we hve os[ os( + b)]h[ + b)] Im{[ epi( + b)]} 0 ( ) + [( + )( + b)] ( + )! Theefoe, by integtion tem by tem theoem, we n show tht thee eists onstnt C suh tht fo ll R, the integl 0 ( ).. Rem (8) os[ os( h[ + In Theoem, beuse fo eh 0 0 os[( + )( + b)] ( + )!( + ) q.e.. ( ) + R, os[( + )( ( + )! + ( + )!( + ) os[( + )( 0 + < (+ )!(+ ) It follows tht we n use integtion tem by tem theoem to show tht () hols. The sme eson tht we n pove (6) by ug integtion tem by tem theoem. Net, we etemine the Fouie seies epnsions of the integls () n ()..6. Theoem If the ssumptions e the sme s Theoem. Then thee
3 Mthemtis n Sttistis (): -, 0 eists onstnt C suh tht fo ll R, the integl os[ os( osh[ + ( ) [()( + b)] ()! An thee eists onstnt C suh tht fo ll the integl ( ).6.. Poof (9) R, [ os( h[ os[()( + b)] ()! (0) By Eule's fomul, DeMoive's fomul n Fomul..,..6, we hve + Thus, fo ll os[ os( + b)]osh[ + b)] R 0 ( ) ( ) Re{os[ epi( + b)]} os[()( + b)] ()! os[()( + b)] ()!, the integl () os[ os( osh[ + ( ) os[()( ()! (By integtion tem by tem theoem) + ( ) [()( + b)] ()! Whee C is some onstnt. Similly, by Eule's fomul, DeMoive's fomul n Fomul..,..6, we obtin [ os( + b)]h[ + b)] ( ) Im{os[ epi( + b)]} [()( + b)] ()! () By integtion tem by tem theoem, it follows tht fo ll R, the integl [ os( h[ ( ) Whee C is some onstnt. os[()( + b)] ()! q.e...7. Rem In Theoem, the eson tht we n use integtion tem by tem theoem to pove (9) n (0) is the sme s Rem.. Emples In the following, fo the fou types of integls in this stuy, we popose some integls n use Theoems, to etemine thei Fouie seies epnsions. In ition, we evlute some efinite integls n employ Mple to lulte the ppoimtions of these efinite integls n thei solutions fo veifying ou nswes... Emple In Theoem, ting,, b π / into (), we obtin the following integl π π os t + osh t + + ( ) ( ) 0 ( )!( ) π () Thus, we n etemine the efinite integl fom t π / to t π / 6, π π os t + osh t + π / 6 π / ( ) 0 ( ) 0 + ( + ) π ( + )!( + ) + ( + ) π () ( + )!( + ) We use Mple to veify the oetness of (). >evlf(int((*os(*t+pi/))*osh(*(*t+pi/)),t Pi/..Pi/6),); >evlf(/*sum((-)^*^(*+)/((*+)!*(*+))*s in((*+)*pi/),0..infinity)- /*sum((-)^*^(*+)/(( *+)!*(*+))*((*+)*Pi/),0..infinity),); The bove nswes obtine by Mple ppes I
4 A Stuy of Some Integl Poblems Ug Mple ( ), it is beuse Mple lultes by ug speil funtions built in. Both the imginy pts of the bove nswes e zeo, so n be ignoe. On the othe hn, in Theoem, ting,, b π / into (6), we hve the following integl os π os t h π t + ( ) ( ) os ( ) 0 ( )!( ) Hene, we n etemine the efinite integl fom t π /0 to t π /, os π os t h π () π t π / π / 0 ( ) 0 + ( ) (6 + ) π os ( + )!( + ) + ( ) ( + ) π + ( ) os 0 ( + )!( + ) (6) Net, we use Mple to veify the oetness of (6). >evlf(int(os(sqt()*os(*t-*pi/))*h(sqt()*( *t-*pi/)),tpi/0..*pi/),); >evlf(-/*sum((-)^*sqt()^(*+)/((*+)!*(* + ))*os((6*+)*pi/),0..infinity)+/*sum((-)^*sqt ()^(*+)/((*+)!*(*+))*os((*+)*pi/),0.. infinity),); Also, both the imginy pts of the bove nswes obtine by Mple e zeo, so n be ignoe... Emple In Theoem, ting /, 7, b π / 6 into (9), we n etemine the following integl os t t os 7 + osh (/) + ( ) () 7 7 ()! + 6 π (7) Thus, we obtin the efinite integl fom t π / to t π / 7, os os 7t + osh 7t + 6 π / 7 π / 6 π ( / ) π + ( ) 7 ()! ( / ) π ( ) (8) 7 ()! We use Mple to veify the oetness of (8) s follows: >evlf(int(os(/*os(7*t+*pi/6))*osh(/*(7*t+ *Pi/6)),tPi/..*Pi/7),); >evlf(*pi/+/7*sum((-)^*(/)^(*)/((*)!*(* ))*(**Pi/),..infinity)-/7*sum((-)^*(/)^(* )/((*)!*(*))*(**Pi/),..infinity),); Also, both the imginy pts of the bove nswes obtine by Mple e zeo, so n be ignoe. On the othe hn, in Theoem, if ting 9,, b π / into (0), we obtin the following integl π π 9os t h 9 t 9 ( ) os ()! π (9) Theefoe, we hve the efinite integl fom t π / 6 to t π /, π π 9os t h 9 t π / π / 6 9 π ( ) os ()! 9 ( ) (0) ()! Ug Mple to veify the oetness of (0) s follows: >evlf(int((9*os(*t-*pi/))*h(9*(*t-*pi/)),t Pi/6..Pi/),); >evlf(/*sum((-)^*9^(*)/((*)!*(*))*os(** Pi/),..infinity)-/*sum((-)^*9^(*)/((*)!*(*)),..infinity),);
5 Mthemtis n Sttistis (): -, 0 The imginy pts of the bove nswes obtine by Mple e eithe zeo o vey smll, so n be ignoe.. Conlusion Fom the bove isussion, we now the integtion tem by tem theoem plys signifint ole in the theoetil infeenes of this stuy. In ft, the pplition of this theoem is etensive, n n be use to esily solve mny iffiult poblems; we enevo to onut futhe stuies on elte pplitions. On the othe hn, Mple lso plys vitl ssistive ole in poblem-solving. In the futue, we will eten the eseh topi to othe lulus n engineeing mthemtis poblems n solve these poblems by ug Mple. These esults will be use s tehing mteils fo Mple on eution n eseh to enhne the onnottions of lulus n engineeing mthemtis. REFERENCES [] A. A. Ams, H. Gottliebsen, S. A. Linton, n U. Mtin, Automte theoem poving in suppot of ompute lgeb: symboli efinite integtion s se stuy, Poeeings of the 999 Intentionl Symposium on Symboli n Algebi Computtion, pp. -60, Vnouve, Cn, 999. [] C. Oste, Limit of efinite integl, SIAM Review, Vol., No., pp. -6, 99. [] M. A. Nyblom, On the evlution of efinite integl involving neste sque oot funtions, Roy Mountin Jounl of Mthemtis, Vol. 7, No., pp. 0-0, 007. [] C. -H. Yu, A stuy on integl poblems by ug Mple, Intentionl Jounl of Avne Reseh in Compute Siene n Softwe Engineeing, Vol., Issue. 7, pp. -6, 0. [] C. -H. Yu, Evluting some integls with Mple, Intentionl Jounl of Compute Siene n Mobile Computing, Vol., Issue. 7, pp. 66-7, 0. [6] C.-H. Yu, Applition of Mple on evluting the lose foms of two types of integls, Poeeings of the 7th Mobile Computing Woshop, ID6, 0. [7] C.-H. Yu, Applition of Mple on some integl poblems, Poeeings of the Intentionl Confeene on Sfety & Seuity Mngement n Engineeing Tehnology 0, pp. 90-9, 0. [8] C.-H. Yu, Applition of Mple on the integl poblem of some type of tionl funtions, Poeeings of the Annul Meeting n Aemi Confeene fo Assoition of IE, D7-D6, 0. [9] C. -H. Yu, Ug Mple to stuy two types of integls, Intentionl Jounl of Reseh in Compute Applitions n Robotis, Vol., Issue., pp. -, 0. [0] C.-H. Yu, Applition of Mple on evlution of efinite integls, Applie Mehnis n Mteils, in pess. [] C. -H. Yu, Solving some integls with Mple, Intentionl Jounl of Reseh in Aeonutil n Mehnil Engineeing, Vol., Issue., pp. 9-, 0. [] C. -H. Yu, Ug Mple to stuy the integls of tigonometi funtions, Poeeings of the 6th IEEE/Intentionl Confeene on Avne Infoomm Tehnology, No. 009, 0. [] C. -H. Yu, A stuy of the integls of tigonometi funtions with Mple, Poeeings of the Institute of Inustil Enginees A Confeene 0, Spinge, Vol., pp. 60-, 0. [] C.-H. Yu, Applition of Mple on some type of integl poblem, Poeeings of Ubiquitous-Home Confeene 0, pp.06-0, 0. [] C.-H. Yu, Applition of Mple: ting two speil integl poblems s emples, Poeeings of the 8th Intentionl Confeene on Knowlege Community, pp.80-8, 0. [6] W. R. Dei, Intoutoy Comple Anlysis n Applitions, New Yo: Aemi Pess, 97. [7] T. M. Apostol, Mthemtil Anlysis, n e., Boston: Aison-Wesley, 97.
Solving Some Definite Integrals Using Parseval s Theorem
Ameican Jounal of Numeical Analysis 4 Vol. No. 6-64 Available online at http://pubs.sciepub.com/ajna///5 Science and Education Publishing DOI:.69/ajna---5 Solving Some Definite Integals Using Paseval s
More informationApplication of Parseval s Theorem on Evaluating Some Definite Integrals
Tukish Jounal of Analysis and Numbe Theoy, 4, Vol., No., -5 Available online at http://pubs.sciepub.com/tjant/// Science and Education Publishing DOI:.69/tjant--- Application of Paseval s Theoem on Evaluating
More informationUsing Laplace Transform to Evaluate Improper Integrals Chii-Huei Yu
Available at https://edupediapublicationsog/jounals Volume 3 Issue 4 Febuay 216 Using Laplace Tansfom to Evaluate Impope Integals Chii-Huei Yu Depatment of Infomation Technology, Nan Jeon Univesity of
More informationIllustrating the space-time coordinates of the events associated with the apparent and the actual position of a light source
Illustting the spe-time oointes of the events ssoite with the ppent n the tul position of light soue Benh Rothenstein ), Stefn Popesu ) n Geoge J. Spi 3) ) Politehni Univesity of Timiso, Physis Deptment,
More informationPrerna Tower, Road No 2, Contractors Area, Bistupur, Jamshedpur , Tel (0657) ,
R Pen Towe Rod No Conttos Ae Bistupu Jmshedpu 8 Tel (67)89 www.penlsses.om IIT JEE themtis Ppe II PART III ATHEATICS SECTION I (Totl ks : ) (Single Coet Answe Type) This setion ontins 8 multiple hoie questions.
More informationApplication of Poisson Integral Formula on Solving Some Definite Integrals
Jounal of Copute and Electonic Sciences Available online at jcesblue-apog 015 JCES Jounal Vol 1(), pp 4-47, 30 Apil, 015 Application of Poisson Integal Foula on Solving Soe Definite Integals Chii-Huei
More informationInfluence of the Magnetic Field in the Solar Interior on the Differential Rotation
Influene of the gneti Fiel in the Sol Inteio on the Diffeentil ottion Lin-Sen Li * Deptment of Physis Nothest Noml Univesity Chnghun Chin * Coesponing utho: Lin-Sen Li Deptment of Physis Nothest Noml Univesity
More informationMathematical Reflections, Issue 5, INEQUALITIES ON RATIOS OF RADII OF TANGENT CIRCLES. Y.N. Aliyev
themtil efletions, Issue 5, 015 INEQULITIES ON TIOS OF DII OF TNGENT ILES YN liev stt Some inequlities involving tios of dii of intenll tngent iles whih inteset the given line in fied points e studied
More informationMath 4318 : Real Analysis II Mid-Term Exam 1 14 February 2013
Mth 4318 : Rel Anlysis II Mid-Tem Exm 1 14 Febuy 2013 Nme: Definitions: Tue/Flse: Poofs: 1. 2. 3. 4. 5. 6. Totl: Definitions nd Sttements of Theoems 1. (2 points) Fo function f(x) defined on (, b) nd fo
More informationCHAPTER 7 Applications of Integration
CHAPTER 7 Applitions of Integtion Setion 7. Ae of Region Between Two Cuves.......... Setion 7. Volume: The Disk Method................. Setion 7. Volume: The Shell Method................ Setion 7. A Length
More informationReview of Mathematical Concepts
ENEE 322: Signls nd Systems view of Mthemticl Concepts This hndout contins ief eview of mthemticl concepts which e vitlly impotnt to ENEE 322: Signls nd Systems. Since this mteil is coveed in vious couses
More information1 Using Integration to Find Arc Lengths and Surface Areas
Novembe 9, 8 MAT86 Week Justin Ko Using Integtion to Find Ac Lengths nd Sufce Aes. Ac Length Fomul: If f () is continuous on [, b], then the c length of the cuve = f() on the intevl [, b] is given b s
More informationLecture 10. Solution of Nonlinear Equations - II
Fied point Poblems Lectue Solution o Nonline Equtions - II Given unction g : R R, vlue such tht gis clled ied point o the unction g, since is unchnged when g is pplied to it. Whees with nonline eqution
More informationSTD: XI MATHEMATICS Total Marks: 90. I Choose the correct answer: ( 20 x 1 = 20 ) a) x = 1 b) x =2 c) x = 3 d) x = 0
STD: XI MATHEMATICS Totl Mks: 90 Time: ½ Hs I Choose the coect nswe: ( 0 = 0 ). The solution of is ) = b) = c) = d) = 0. Given tht the vlue of thid ode deteminnt is then the vlue of the deteminnt fomed
More informationThe Area of a Triangle
The e of Tingle tkhlid June 1, 015 1 Intodution In this tile we will e disussing the vious methods used fo detemining the e of tingle. Let [X] denote the e of X. Using se nd Height To stt off, the simplest
More informationCOMPUTER AIDED ANALYSIS OF KINEMATICS AND KINETOSTATICS OF SIX-BAR LINKAGE MECHANISM THROUGH THE CONTOUR METHOD
SINTIFI PROINGS XIV INTRNTIONL ONGRSS "MHINS. THNOLОGIS. MTRILS." 17 - SUMMR SSSION W ISSN 55-X PRINT ISSN 55-1 OMPUTR I NLYSIS OF KINMTIS N KINTOSTTIS OF SIX-R LINKG MHNISM THROUGH TH ONTOUR MTHO Pof.so..
More informationOptimization. x = 22 corresponds to local maximum by second derivative test
Optimiztion Lectue 17 discussed the exteme vlues of functions. This lectue will pply the lesson fom Lectue 17 to wod poblems. In this section, it is impotnt to emembe we e in Clculus I nd e deling one-vible
More informationAbout Some Inequalities for Isotonic Linear Functionals and Applications
Applied Mthemticl Sciences Vol. 8 04 no. 79 8909-899 HIKARI Ltd www.m-hiki.com http://dx.doi.og/0.988/ms.04.40858 Aout Some Inequlities fo Isotonic Line Functionls nd Applictions Loedn Ciudiu Deptment
More informationWeek 8. Topic 2 Properties of Logarithms
Week 8 Topic 2 Popeties of Logithms 1 Week 8 Topic 2 Popeties of Logithms Intoduction Since the esult of ithm is n eponent, we hve mny popeties of ithms tht e elted to the popeties of eponents. They e
More informationClass Summary. be functions and f( D) , we define the composition of f with g, denoted g f by
Clss Summy.5 Eponentil Functions.6 Invese Functions nd Logithms A function f is ule tht ssigns to ech element D ectly one element, clled f( ), in. Fo emple : function not function Given functions f, g:
More informationHomework 3 MAE 118C Problems 2, 5, 7, 10, 14, 15, 18, 23, 30, 31 from Chapter 5, Lamarsh & Baratta. The flux for a point source is:
. Homewok 3 MAE 8C Poblems, 5, 7, 0, 4, 5, 8, 3, 30, 3 fom Chpte 5, msh & Btt Point souces emit nuetons/sec t points,,, n 3 fin the flux cuent hlf wy between one sie of the tingle (blck ot). The flux fo
More informationEECE 260 Electrical Circuits Prof. Mark Fowler
EECE 60 Electicl Cicuits Pof. Mk Fowle Complex Numbe Review /6 Complex Numbes Complex numbes ise s oots of polynomils. Definition of imginy # nd some esulting popeties: ( ( )( ) )( ) Recll tht the solution
More informationTopics for Review for Final Exam in Calculus 16A
Topics fo Review fo Finl Em in Clculus 16A Instucto: Zvezdelin Stnkov Contents 1. Definitions 1. Theoems nd Poblem Solving Techniques 1 3. Eecises to Review 5 4. Chet Sheet 5 1. Definitions Undestnd the
More informationIntegrals and Polygamma Representations for Binomial Sums
3 47 6 3 Jounl of Intege Sequences, Vol. 3 (, Aticle..8 Integls nd Polygmm Repesenttions fo Binomil Sums Anthony Sofo School of Engineeing nd Science Victoi Univesity PO Box 448 Melboune City, VIC 8 Austli
More information7.5-Determinants in Two Variables
7.-eteminnts in Two Vibles efinition of eteminnt The deteminnt of sque mti is el numbe ssocited with the mti. Eve sque mti hs deteminnt. The deteminnt of mti is the single ent of the mti. The deteminnt
More informationMark Scheme (Results) January 2008
Mk Scheme (Results) Jnuy 00 GCE GCE Mthemtics (6679/0) Edecel Limited. Registeed in Englnd nd Wles No. 4496750 Registeed Office: One90 High Holbon, London WCV 7BH Jnuy 00 6679 Mechnics M Mk Scheme Question
More informationMichael Rotkowitz 1,2
Novembe 23, 2006 edited Line Contolles e Unifomly Optiml fo the Witsenhusen Counteexmple Michel Rotkowitz 1,2 IEEE Confeence on Decision nd Contol, 2006 Abstct In 1968, Witsenhusen intoduced his celebted
More informationThis immediately suggests an inverse-square law for a "piece" of current along the line.
Electomgnetic Theoy (EMT) Pof Rui, UNC Asheville, doctophys on YouTube Chpte T Notes The iot-svt Lw T nvese-sque Lw fo Mgnetism Compe the mgnitude of the electic field t distnce wy fom n infinite line
More informationCHAPTER 18: ELECTRIC CHARGE AND ELECTRIC FIELD
ollege Physics Student s Mnul hpte 8 HAPTR 8: LTRI HARG AD LTRI ILD 8. STATI LTRIITY AD HARG: OSRVATIO O HARG. ommon sttic electicity involves chges nging fom nnocoulombs to micocoulombs. () How mny electons
More informationPreviously. Extensions to backstepping controller designs. Tracking using backstepping Suppose we consider the general system
436-459 Advnced contol nd utomtion Extensions to bckstepping contolle designs Tcking Obseves (nonline dmping) Peviously Lst lectue we looked t designing nonline contolles using the bckstepping technique
More informationFourier-Bessel Expansions with Arbitrary Radial Boundaries
Applied Mthemtics,,, - doi:./m.. Pulished Online My (http://www.scirp.og/jounl/m) Astct Fouie-Bessel Expnsions with Aity Rdil Boundies Muhmmd A. Mushef P. O. Box, Jeddh, Sudi Ai E-mil: mmushef@yhoo.co.uk
More informationGeneral Physics II. number of field lines/area. for whole surface: for continuous surface is a whole surface
Genel Physics II Chpte 3: Guss w We now wnt to quickly discuss one of the moe useful tools fo clculting the electic field, nmely Guss lw. In ode to undestnd Guss s lw, it seems we need to know the concept
More information10.3 The Quadratic Formula
. Te Qudti Fomul We mentioned in te lst setion tt ompleting te sque n e used to solve ny qudti eqution. So we n use it to solve 0. We poeed s follows 0 0 Te lst line of tis we ll te qudti fomul. Te Qudti
More informationSPA7010U/SPA7010P: THE GALAXY. Solutions for Coursework 1. Questions distributed on: 25 January 2018.
SPA7U/SPA7P: THE GALAXY Solutions fo Cousewok Questions distibuted on: 25 Jnuy 28. Solution. Assessed question] We e told tht this is fint glxy, so essentilly we hve to ty to clssify it bsed on its spectl
More informationResearch Article Hermite-Hadamard-Type Inequalities for r-preinvex Functions
Hindwi Publishing Copotion Jounl of Applied Mthemtics Volume 3, Aticle ID 6457, 5 pges http://dx.doi.og/.55/3/6457 Resech Aticle Hemite-Hdmd-Type Inequlities fo -Peinvex Functions Wsim Ul-Hq nd Jved Iqbl
More informationData Compression LZ77. Jens Müller Universität Stuttgart
Dt Compession LZ77 Jens Mülle Univesität Stuttgt 2008-11-25 Outline Intoution Piniple of itiony methos LZ77 Sliing winow Exmples Optimiztion Pefomne ompison Applitions/Ptents Jens Mülle- IPVS Univesität
More information10 Statistical Distributions Solutions
Communictions Engineeing MSc - Peliminy Reding 1 Sttisticl Distiutions Solutions 1) Pove tht the vince of unifom distiution with minimum vlue nd mximum vlue ( is ) 1. The vince is the men of the sques
More informationData Structures. Element Uniqueness Problem. Hash Tables. Example. Hash Tables. Dana Shapira. 19 x 1. ) h(x 4. ) h(x 2. ) h(x 3. h(x 1. x 4. x 2.
Element Uniqueness Poblem Dt Stuctues Let x,..., xn < m Detemine whethe thee exist i j such tht x i =x j Sot Algoithm Bucket Sot Dn Shpi Hsh Tbles fo (i=;i
More informationOn Natural Partial Orders of IC-Abundant Semigroups
Intentionl Jounl of Mthemtics nd Computtionl Science Vol. No. 05 pp. 5-9 http://www.publicsciencefmewok.og/jounl/ijmcs On Ntul Ptil Odes of IC-Abundnt Semigoups Chunhu Li Bogen Xu School of Science Est
More informationdefined on a domain can be expanded into the Taylor series around a point a except a singular point. Also, f( z)
08 Tylo eie nd Mcluin eie A holomophic function f( z) defined on domin cn be expnded into the Tylo eie ound point except ingul point. Alo, f( z) cn be expnded into the Mcluin eie in the open dik with diu
More informationSolution of fuzzy multi-objective nonlinear programming problem using interval arithmetic based alpha-cut
Intentionl Jounl of Sttistics nd Applied Mthemtics 016; 1(3): 1-5 ISSN: 456-145 Mths 016; 1(3): 1-5 016 Stts & Mths www.mthsounl.com Received: 05-07-016 Accepted: 06-08-016 C Lognthn Dept of Mthemtics
More informationA NOTE ON THE POCHHAMMER FREQUENCY EQUATION
A note on the Pohhmme feqeny eqtion SCIENCE AND TECHNOLOGY - Reseh Jonl - Volme 6 - Univesity of Mitis Rédit Mitis. A NOTE ON THE POCHHAMMER FREQUENCY EQUATION by F.R. GOLAM HOSSEN Deptment of Mthemtis
More informationSchool of Electrical and Computer Engineering, Cornell University. ECE 303: Electromagnetic Fields and Waves. Fall 2007
School of Electicl nd Compute Engineeing, Conell Univesity ECE 303: Electomgnetic Fields nd Wves Fll 007 Homewok 3 Due on Sep. 14, 007 by 5:00 PM Reding Assignments: i) Review the lectue notes. ii) Relevnt
More informationChapter Direct Method of Interpolation More Examples Mechanical Engineering
Chpte 5 iect Method o Intepoltion Moe Exmples Mechnicl Engineeing Exmple Fo the pupose o shinking tunnion into hub, the eduction o dimete o tunnion sht by cooling it though tempetue chnge o is given by
More informationr r E x w, y w, z w, (1) Where c is the speed of light in vacuum.
ISSN: 77-754 ISO 900:008 Cetified Intentionl Jonl of Engineeing nd Innovtive Tehnology (IJEIT) olme, Isse 0, Apil 04 The Replement of the Potentils s Conseene of the Limittions Set by the Lw of the Self
More informationA Crash Course in (2 2) Matrices
A Cash Couse in ( ) Matices Seveal weeks woth of matix algeba in an hou (Relax, we will only stuy the simplest case, that of matices) Review topics: What is a matix (pl matices)? A matix is a ectangula
More informationThe Double Integral. The Riemann sum of a function f (x; y) over this partition of [a; b] [c; d] is. f (r j ; t k ) x j y k
The Double Integrl De nition of the Integrl Iterted integrls re used primrily s tool for omputing double integrls, where double integrl is n integrl of f (; y) over region : In this setion, we de ne double
More informationELECTROSTATICS. 4πε0. E dr. The electric field is along the direction where the potential decreases at the maximum rate. 5. Electric Potential Energy:
LCTROSTATICS. Quntiztion of Chge: Any chged body, big o smll, hs totl chge which is n integl multile of e, i.e. = ± ne, whee n is n intege hving vlues,, etc, e is the chge of electon which is eul to.6
More informationRadial geodesics in Schwarzschild spacetime
Rdil geodesics in Schwzschild spcetime Spheiclly symmetic solutions to the Einstein eqution tke the fom ds dt d dθ sin θdϕ whee is constnt. We lso hve the connection components, which now tke the fom using
More informationElectronic Companion for Optimal Design of Co-Productive Services: Interaction and Work Allocation
Submitted to Mnufctuing & Sevice Oetions Mngement mnuscit Electonic Comnion fo Otiml Design of Co-Poductive Sevices: Intection nd Wok Alloction Guillume Roels UCLA Andeson School of Mngement, 110 Westwood
More informationNumbers and indices. 1.1 Fractions. GCSE C Example 1. Handy hint. Key point
GCSE C Emple 7 Work out 9 Give your nswer in its simplest form Numers n inies Reiprote mens invert or turn upsie own The reiprol of is 9 9 Mke sure you only invert the frtion you re iviing y 7 You multiply
More informationChapter Seven Notes N P U1C7
Chpte Seven Notes N P UC7 Nme Peiod Setion 7.: Angles nd Thei Mesue In fling, hitetue, nd multitude of othe fields, ngles e used. An ngle is two diffeent s tht hve the sme initil (o stting) point. The
More informationπ,π is the angle FROM a! TO b
Mth 151: 1.2 The Dot Poduct We hve scled vectos (o, multiplied vectos y el nume clled scl) nd dded vectos (in ectngul component fom). Cn we multiply vectos togethe? The nswe is YES! In fct, thee e two
More informationAlgebra Based Physics. Gravitational Force. PSI Honors universal gravitation presentation Update Fall 2016.notebookNovember 10, 2016
Newton's Lw of Univesl Gvittion Gvittionl Foce lick on the topic to go to tht section Gvittionl Field lgeb sed Physics Newton's Lw of Univesl Gvittion Sufce Gvity Gvittionl Field in Spce Keple's Thid Lw
More informationEigenvectors and Eigenvalues
MTB 050 1 ORIGIN 1 Eigenvets n Eigenvlues This wksheet esries the lger use to lulte "prinipl" "hrteristi" iretions lle Eigenvets n the "prinipl" "hrteristi" vlues lle Eigenvlues ssoite with these iretions.
More informationChapter 3: Theory of Modular Arithmetic 38
Chapte 3: Theoy of Modula Aithmetic 38 Section D Chinese Remainde Theoem By the end of this section you will be able to pove the Chinese Remainde Theoem apply this theoem to solve simultaneous linea conguences
More informationAnswers to test yourself questions
Answes to test youself questions opic Descibing fields Gm Gm Gm Gm he net field t is: g ( d / ) ( 4d / ) d d Gm Gm Gm Gm Gm Gm b he net potentil t is: V d / 4d / d 4d d d V e 4 7 9 49 J kg 7 7 Gm d b E
More informationSolutions to Problems : Chapter 19 Problems appeared on the end of chapter 19 of the Textbook
Solutions to Poblems Chapte 9 Poblems appeae on the en of chapte 9 of the Textbook 8. Pictue the Poblem Two point chages exet an electostatic foce on each othe. Stategy Solve Coulomb s law (equation 9-5)
More informationSchool of Electrical and Computer Engineering, Cornell University. ECE 303: Electromagnetic Fields and Waves. Fall 2007
School of Electicl nd Compute Engineeing, Conell Univesity ECE 303: Electomgnetic Fields nd Wves Fll 007 Homewok 4 Due on Sep. 1, 007 by 5:00 PM Reding Assignments: i) Review the lectue notes. ii) Relevnt
More informationab b. c 3. y 5x. a b 3ab. x xy. p q pq. a b. x y) + 2a. a ab. 6. Simplify the following expressions. (a) (b) (c) (4x
. Simplif the following epessions. 8 c c d. Simplif the following epessions. 6b pq 0q. Simplif the following epessions. ( ) q( m n) 6q ( m n) 7 ( b c) ( b c) 6. Simplif the following epessions. b b b p
More informationStanford University CS259Q: Quantum Computing Handout 8 Luca Trevisan October 18, 2012
Stanfod Univesity CS59Q: Quantum Computing Handout 8 Luca Tevisan Octobe 8, 0 Lectue 8 In which we use the quantum Fouie tansfom to solve the peiod-finding poblem. The Peiod Finding Poblem Let f : {0,...,
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 informationSECTION A STUDENT MATERIAL. Part 1. What and Why.?
SECTION A STUDENT MATERIAL Prt Wht nd Wh.? Student Mteril Prt Prolem n > 0 n > 0 Is the onverse true? Prolem If n is even then n is even. If n is even then n is even. Wht nd Wh? Eploring Pure Mths Are
More informationPerturbative and Non-perturbative Aspects of the Chern-Simons-Witten Theory
Indonesin Jounl of Physis Vol 9 No Jnuy 8 Petubtive nd Non-petubtive Aspets of the Chen-Simons-itten Theoy Asep Yoyo dy F P Zen b Jus Sli Kossih Tiynt d Deptment of Physis Diponegoo Univesity Semng Indonesi
More informationComparative Studies of Law of Gravity and General Relativity. No.1 of Comparative Physics Series Papers
Comptive Studies of Lw of Gvity nd Genel Reltivity No. of Comptive hysics Seies pes Fu Yuhu (CNOOC Resech Institute, E-mil:fuyh945@sin.com) Abstct: As No. of comptive physics seies ppes, this ppe discusses
More informationMethod for Approximating Irrational Numbers
Method fo Appoximating Iational Numbes Eic Reichwein Depatment of Physics Univesity of Califonia, Santa Cuz June 6, 0 Abstact I will put foth an algoithm fo poducing inceasingly accuate ational appoximations
More informationMiskolc Mathematical Notes HU e-issn Tribonacci numbers with indices in arithmetic progression and their sums. Nurettin Irmak and Murat Alp
Miskolc Mathematical Notes HU e-issn 8- Vol. (0), No, pp. 5- DOI 0.85/MMN.0.5 Tibonacci numbes with indices in aithmetic pogession and thei sums Nuettin Imak and Muat Alp Miskolc Mathematical Notes HU
More informationSection 35 SHM and Circular Motion
Section 35 SHM nd Cicul Motion Phsics 204A Clss Notes Wht do objects do? nd Wh do the do it? Objects sometimes oscillte in simple hmonic motion. In the lst section we looed t mss ibting t the end of sping.
More informationPhysics 2A Chapter 10 - Moment of Inertia Fall 2018
Physics Chapte 0 - oment of netia Fall 08 The moment of inetia of a otating object is a measue of its otational inetia in the same way that the mass of an object is a measue of its inetia fo linea motion.
More information(a) Counter-Clockwise (b) Clockwise ()N (c) No rotation (d) Not enough information
m m m00 kg dult, m0 kg bby. he seesw stts fom est. Which diection will it ottes? ( Counte-Clockwise (b Clockwise ( (c o ottion ti (d ot enough infomtion Effect of Constnt et oque.3 A constnt non-zeo toque
More informationu(r, θ) = 1 + 3a r n=1
Mth 45 / AMCS 55. etuck Assignment 8 ue Tuesdy, Apil, 6 Topics fo this week Convegence of Fouie seies; Lplce s eqution nd hmonic functions: bsic popeties, computions on ectngles nd cubes Fouie!, Poisson
More informationGeneralized Kronecker Product and Its Application
Vol. 1, No. 1 ISSN: 1916-9795 Generlize Kroneker Prout n Its Applition Xingxing Liu Shool of mthemtis n omputer Siene Ynn University Shnxi 716000, Chin E-mil: lxx6407@163.om Astrt In this pper, we promote
More informationSOME GENERAL NUMERICAL RADIUS INEQUALITIES FOR THE OFF-DIAGONAL PARTS OF 2 2 OPERATOR MATRICES
italian jounal of pue and applied mathematics n. 35 015 (433 44) 433 SOME GENERAL NUMERICAL RADIUS INEQUALITIES FOR THE OFF-DIAGONAL PARTS OF OPERATOR MATRICES Watheq Bani-Domi Depatment of Mathematics
More informationResearch Article On Alzer and Qiu s Conjecture for Complete Elliptic Integral and Inverse Hyperbolic Tangent Function
Abstact and Applied Analysis Volume 011, Aticle ID 697547, 7 pages doi:10.1155/011/697547 Reseach Aticle On Alze and Qiu s Conjectue fo Complete Elliptic Integal and Invese Hypebolic Tangent Function Yu-Ming
More informationChapter 7. Kleene s Theorem. 7.1 Kleene s Theorem. The following theorem is the most important and fundamental result in the theory of FA s:
Chpte 7 Kleene s Theoem 7.1 Kleene s Theoem The following theoem is the most impotnt nd fundmentl esult in the theoy of FA s: Theoem 6 Any lnguge tht cn e defined y eithe egul expession, o finite utomt,
More informationPhysics 217 Practice Final Exam: Solutions
Physis 17 Ptie Finl Em: Solutions Fll This ws the Physis 17 finl em in Fll 199 Twenty-thee students took the em The vege soe ws 11 out of 15 (731%), nd the stndd devition 9 The high nd low soes wee 145
More informationk. s k=1 Part of the significance of the Riemann zeta-function stems from Theorem 9.2. If s > 1 then 1 p s
9 Pimes in aithmetic ogession Definition 9 The Riemann zeta-function ζs) is the function which assigns to a eal numbe s > the convegent seies k s k Pat of the significance of the Riemann zeta-function
More informationDouble sequences of interval numbers defined by Orlicz functions
ACTA ET COENTATIONES UNIVERSITATIS TARTUENSIS DE ATHEATICA Volume 7, Numbe, June 203 Available online at www.math.ut.ee/acta/ Double sequences of inteval numbes efine by Olicz functions Ayhan Esi Abstact.
More information9.4 The response of equilibrium to temperature (continued)
9.4 The esponse of equilibium to tempetue (continued) In the lst lectue, we studied how the chemicl equilibium esponds to the vition of pessue nd tempetue. At the end, we deived the vn t off eqution: d
More informationModule 4: Moral Hazard - Linear Contracts
Module 4: Mol Hzd - Line Contts Infomtion Eonomis (E 55) Geoge Geogidis A pinipl employs n gent. Timing:. The pinipl o es line ontt of the fom w (q) = + q. is the sly, is the bonus te.. The gent hooses
More information(n 1)n(n + 1)(n + 2) + 1 = (n 1)(n + 2)n(n + 1) + 1 = ( (n 2 + n 1) 1 )( (n 2 + n 1) + 1 ) + 1 = (n 2 + n 1) 2.
Paabola Volume 5, Issue (017) Solutions 151 1540 Q151 Take any fou consecutive whole numbes, multiply them togethe and add 1. Make a conjectue and pove it! The esulting numbe can, fo instance, be expessed
More informationChapter Gauss Quadrature Rule of Integration
Chpter 7. Guss Qudrture Rule o Integrtion Ater reding this hpter, you should e le to:. derive the Guss qudrture method or integrtion nd e le to use it to solve prolems, nd. use Guss qudrture method to
More informationA Cornucopia of Pythagorean triangles
A onucopi of Pytgoen tingles onstntine Zelto Deptment of temtics 0 ckey Hll 9 Univesity Plce Univesity of Pittsbug Pittsbug PA 60 USA Also: onstntine Zelto PO Bo 80 Pittsbug PA 0 USA e-mil ddesses: ) onstntine_zelto@yoocom
More informationTHEORY OF EQUATIONS OBJECTIVE PROBLEMS. If the eqution x 6x 0 0 ) - ) 4) -. If the sum of two oots of the eqution k is -48 ) 6 ) 48 4) 4. If the poduct of two oots of 4 ) -4 ) 4) - 4. If one oot of is
More informationCore 2 Logarithms and exponentials. Section 1: Introduction to logarithms
Core Logrithms nd eponentils Setion : Introdution to logrithms Notes nd Emples These notes ontin subsetions on Indies nd logrithms The lws of logrithms Eponentil funtions This is n emple resoure from MEI
More informationINTEGRATION. 1 Integrals of Complex Valued functions of a REAL variable
INTEGRATION NOTE: These notes re supposed to supplement Chpter 4 of the online textbook. 1 Integrls of Complex Vlued funtions of REAL vrible If I is n intervl in R (for exmple I = [, b] or I = (, b)) nd
More informationOn Some Hadamard-Type Inequalıtıes for Convex Functıons
Aville t htt://vuedu/ Al Al Mth ISSN: 93-9466 Vol 9, Issue June 4, 388-4 Alictions nd Alied Mthetics: An Intentionl Jounl AAM On Soe Hdd-Tye Inequlıtıes o, Convex Functıons M Ein Özdei Detent o Mthetics
More informationProperties and Formulas
Popeties nd Fomuls Cpte 1 Ode of Opetions 1. Pefom ny opetion(s) inside gouping symols. 2. Simplify powes. 3. Multiply nd divide in ode fom left to igt. 4. Add nd sutt in ode fom left to igt. Identity
More informationChapter Introduction to Partial Differential Equations
hpte 10.01 Intodtion to Ptil Diffeentil Eqtions Afte eding this hpte o shold be ble to: 1. identif the diffeene between odin nd ptil diffeentil eqtions.. identif diffeent tpes of ptil diffeentil eqtions.
More informationSOLUTIONS ( ) ( )! ( ) ( ) ( ) ( )! ( ) ( ) ( ) ( ) n r. r ( Pascal s equation ). n 1. Stepanov Dalpiaz
STAT UIU Pctice Poblems # SOLUTIONS Stepov Dlpiz The followig e umbe of pctice poblems tht my be helpful fo completig the homewo, d will liely be vey useful fo studyig fo ems...-.-.- Pove (show) tht. (
More informationNumerical approximation to ζ(2n+1)
Illinois Wesleyan Univesity Fom the SelectedWoks of Tian-Xiao He 6 Numeical appoximation to ζ(n+1) Tian-Xiao He, Illinois Wesleyan Univesity Michael J. Dancs Available at: https://woks.bepess.com/tian_xiao_he/6/
More informationA NOTE ON VERY WEAK SOLUTIONS FOR A CLASS OF NONLINEAR ELLIPTIC EQUATIONS
SARAJEVO JOURNAL OF MATHEMATICS Vol3 15 2007, 41 45 A NOTE ON VERY WEAK SOLUTIONS FOR A CLASS OF NONLINEAR ELLIPTIC EQUATIONS LI JULING AND GAO HONGYA Abstact We pove a new a pioi estimate fo vey weak
More informationSOME INTEGRAL INEQUALITIES FOR HARMONICALLY CONVEX STOCHASTIC PROCESSES ON THE CO-ORDINATES
Avne Mth Moels & Applitions Vol3 No 8 pp63-75 SOME INTEGRAL INEQUALITIES FOR HARMONICALLY CONVE STOCHASTIC PROCESSES ON THE CO-ORDINATES Nurgül Okur * Imt Işn Yusuf Ust 3 3 Giresun University Deprtment
More informationMarkscheme May 2017 Calculus Higher level Paper 3
M7/5/MATHL/HP3/ENG/TZ0/SE/M Makscheme May 07 Calculus Highe level Pape 3 pages M7/5/MATHL/HP3/ENG/TZ0/SE/M This makscheme is the popety of the Intenational Baccalaueate and must not be epoduced o distibuted
More informationDivisibility. c = bf = (ae)f = a(ef) EXAMPLE: Since 7 56 and , the Theorem above tells us that
Divisibility DEFINITION: If a and b ae integes with a 0, we say that a divides b if thee is an intege c such that b = ac. If a divides b, we also say that a is a diviso o facto of b. NOTATION: d n means
More informationMultiplying and Dividing Rational Expressions
Lesson Peview Pt - Wht You ll Len To multipl tionl epessions To divide tionl epessions nd Wh To find lon pments, s in Eecises 0 Multipling nd Dividing Rtionl Epessions Multipling Rtionl Epessions Check
More informationF / x everywhere in some domain containing R. Then, + ). (10.4.1)
0.4 Green's theorem in the plne Double integrls over plne region my be trnsforme into line integrls over the bounry of the region n onversely. This is of prtil interest beuse it my simplify the evlution
More informationChaos and bifurcation of discontinuous dynamical systems with piecewise constant arguments
Malaya Jounal of Matematik ()(22) 4 8 Chaos and bifucation of discontinuous dynamical systems with piecewise constant aguments A.M.A. El-Sayed, a, and S. M. Salman b a Faculty of Science, Aleandia Univesity,
More informationSection 2.3. Matrix Inverses
Mtri lger Mtri nverses Setion.. Mtri nverses hree si opertions on mtries, ition, multiplition, n sutrtion, re nlogues for mtries of the sme opertions for numers. n this setion we introue the mtri nlogue
More informationFind this material useful? You can help our team to keep this site up and bring you even more content consider donating via the link on our site.
Find this mteil useful? You cn help ou tem to keep this site up nd bing you even moe content conside donting vi the link on ou site. Still hving touble undestnding the mteil? Check out ou Tutoing pge to
More information