The unified equations to obtain the exact solutions of piezoelectric plane beam subjected to arbitrary loads

Μέγεθος: px
Εμφάνιση ξεκινά από τη σελίδα:

Download "The unified equations to obtain the exact solutions of piezoelectric plane beam subjected to arbitrary loads"

Transcript

1 Journal of Mecancal Scence Tecnology (7 ( 8~8 wwwsprngerlnkcom/content/78-9 DOI 7/s--- Te unfed equatons to obtan te eact solutons of peoelectrc plane beam subjected to arbtrary loads Zang ang *, Gao Puyun, Dongu Wang Xue Department of Astronautcal Scence ngneerng, Scool of Aerospace Materals ngneerng, Natonal Unversty of Defense Tecnology, Hunan Cangsa, 7, Cna (Manuscrpt Receved February, ; Revsed Aprl 8, ; Accepted Aprl, Abstract Te unfed equatons to obtan te eact solutons for peoelectrc plane beam subjected to arbtrary mecancal electrcal loads wt varous ends supported condtons s founded by solvng functonal equatons Comparng ts general metod wt tradtonal tral-error metod, te most advantage s t can obtan te eact solutons drectly does not need to guess modfy te form of stress functon or electrc dsplacement functon repeatedly Frstly, te governng equaton for peoelectrc plane beam s derved Te general soluton for te governng equaton s epressed by s unknown functons Secondly, n terms of boundary condtons of te two longtudnal sdes of te beam, s functonal equatons are yelded Tese equatons are smplfed to derve te unfed equatons to solve te boundary value problems of peoelectrc plane beam Fnally, several eamples sow te correctness generalaton of ts metod Keywords: Functonal equaton; Peoelectrc plane beam; act soluton; Arbtrary load; lastcty teory Introducton Peoelectrc materals ave been wdely used as actuators sensors n deformaton vbraton control due to te couplng between mecancal electrcal felds Snce ts couplng caracterstcs, t s more complcated for analyss desgn of suc ntellgent structural system as compared wt tradtonal structural system Tat s te reason wy tere are so many numercal models for peoelectrc structure as can be seen n revew papers [-] Snce te elastcty solutons for smple form of peoelectrc structure can be regarded as te bencmark for verfyng te varous numercal models, t as also attracted many scentsts engneers to do wt ts problem Te necessty to guess modfy te form of stress functon electrc dsplacement functon to obtan te elastcty solutons for peoelectrc plane beam subjected to varous smple form of loads can be seen n almost eac paper tat deal wt tese tess or related topcs We only lst te paper tat publsed n recent years, suc as stress functon electrc dsplacement functon of qs ( ( n Ref [], q ( n Ref [], q (7 n Ref [], q ( n Ref [7], qs ( ( n Ref [8], q ( n Ref [9], q (9 n Ref [], Ts paper was recommended for publcaton n revsed form by Assocate dtor Maengyo Co * Correspondng autor Tel: , Fa: mal address: anglang8@gmalcom KSM & Sprnger q ( n Ref [], q ( n Ref [], q ( n Ref [] Usually, tey all need to guess te form of stress functon electrc dsplacement functon before carry out ter soluton procedure can only deal wt one specfc problem If te boundary condtons are canged, weter boundary condtons of te two longtudnal sdes or boundary condtons of te ends supported condtons, te prevous assumptons can not be used It s also te reason tat most of te model gven n tese papers are smply supported beam or cantlever beam For oter type of ends supported condtons, suc as Fed end Fed end peoelectrc plane beam, tere are lttle report about t Huang [] only gves a metod of ow to solve te problem wen te peoelectrc beam acted by pressure load tat can be transferred nto snusodal seres need te value of functon wc represent te load must be equated to ero at te two ends of te beam Te assumpton about load n Ref [] does not accord wt practcal condtons Moreover, te ypotess of stress functon electrcal dsplacement functon n Ref [] are not sute for oter type of loads, suc as sear force or electrcal loads Ts paper consders te beavor of peoelectrc plane beam subjected to arbtrary mecancal electrcal loads Comparng ts general metod wt te metod gven n te paper tat lst n Refs [-], te most advantage s t can obtan te solutons drectly does not need to guess modfy te form of stress functon or electrcal dsplacement functon Furtermore, t can deal wt arbtrary mecancal

2 8 Zang et al / Journal of Mecancal Scence Tecnology (7 ( 8~8 q ( q ( q ( q ( q ( ( w q ( ( u were Φ represent electrcal potental Te equlbrum equatons are σ σ σ σ, D D ( Fg Peoelectrc plane beam subjected to arbtrary mecancal electrcal loads From q ( - te followng compatblty equaton s obtaned electrcal loads, wc can not be realed n any open lterature to te best knowledge of te autors ε ε ε ( Teoretcal formulatons Consder a peoelectrc plane beam wt rectangular cross secton subjected to arbtrary loads as sown n Fg Suppose te wdt of beam s unt, te lengt egt are, respectvely, In plane, te consttutve equatons for peoelectrc materal can be epressed as Ref [] ε s s σ d ε s s σ d ε s σ d σ D d δ σ D d d δ σ were σ, σ, σ denote te stress components, ε, ε, ε are te stran components, u w dsplacement components, D D electrc dsplacement components, electrc feld components s, s, s s denote coeffcents of elastc complance d, d d are peoelectrcty coeffcents δ δ are delectrc mpermeablty coeffcents Consderng te followng boundary condtons of te two longtudnal sdes of te beam σ, q,, q σ, q,, q D, q, D, q σ σ were q ( (,,, (, ( ( q represent te arbtrary mecancal electrcal loads, respectvely Te knematc relaton are u w u w ε, ε, ε Φ Φ, ( By vrtue of q (, obtan Φ ε sσ sσ d Φ Φ ε sσ sσ d, ε sσ d Φ Φ D dσ δ, D dσ dσ δ Te stress components can be epressed by usng stress U as functon, U U U σ, σ, σ Substtutng q (8 nto q ( applyng q ( obtans U U U s s s s ( Φ Φ d ( d d Substtutng q (8 nto q (7 applyng q ( obtan U U d d d Φ Φ ( δ δ ( (7 (8 (9 ( Dfferentatng q ( wt respect to once com- Φ obtans bnng q (9 to elmnate Φ U U U a a a In ts secton, te concrete epressons of Append A ettng U ( U (,, ( a j are gven n (

3 Zang et al / Journal of Mecancal Scence Tecnology (7 ( 8~8 8 Substtutng q ( nto qs (9 ( obtans U U U s ( s s s Φ Φ d ( d d U U d d d Φ Φ δ ( δ Integratng q ( wt respect to once obtans U U U s s s s Φ Φ d ( d d f ( ( ( ( ( were f ( are arbtrary dfferentable functon, ( f j ( denote te -t dervatve of f j ( Combnng qs ( ( obtans Φ U U U a a a a f ( Φ U U U a a a a f ( ( (7 Dfferentatng qs ( (7 wt respect to twce, respectvely, lettng tem equate to eac oter obtans U ( s s δ s δ d ( d d U sδ d ( s s δ sδ ( d d U sδ d sδ U sδ d were U (, U (, f ( s (8 f can be regarded as a assstant functon wen we derve q (8 take no effect ereafter q (8 s te governng equaton for peoelectrc beam Te caracterstc equaton of q (8 s λ λ λ λ ( s s δ s δ d ( d d sδ d ( s s δ sδ ( d d sδ d sδ sδ d Te general soluton of q (8 s (, ( λ ( λ ( λ ( λ ( λ ( λ U were (,,, are s arbtrary functons By usng qs (, ( (8 obtan σ λ λ λ ( ( λ λ ( λ ( ( λ λ ( λ ( ( σ λ λ ( λ ( ( λ λ ( λ ( ( λ λ ( λ ( ( σ λ λ ( λ ( ( λ λ ( λ ( ( λ λ ( λ ( ( denote te -t dervatve of ( ( ( ( ( were j j By applyng q (, q ( ntegratng q ( wt respect to trce obtans Φ a 7 λ λ ( ( a 7 λ ( λ ( a ( ( 7 λ λ f f f were f, f ( f ( are arbtrary functons Te dsplacement electrc dsplacement components can be yelded by usng q ( -, qs ( (7 as λ λ λ λ λ λ λ λ λ (9 were λ, λ, λ are caracterstc roots λ λ λ Te relatonsp between te caracterstc roots materal propertes can be obtaned by comparng q (9 wt q (8 of te coeffcents of λ as follows: u a λ λ a λ λ a λ λ d f d d f d g (

4 8 Zang et al / Journal of Mecancal Scence Tecnology (7 ( 8~8 w a λ λ ( ( a λ ( λ ( a ( ( λ λ d f d f f ( ( D a λ ( λ ( ( a λ ( λ ( ( a λ ( λ ( ( δ f δ f ( δ f ( ( D a λ ( λ ( ( a λ ( λ ( ( a λ ( λ δ f δ f were f ( g ( are arbtrary functons By usng qs (, ( (8 (-(9 obtan ( ( d d f d d f ( f d f d f d g ( ( ( (7 (8 (9 ( δ f δ f δ f δ f ( q ( can be regarded as te quadratc algebra equaton wt respect to Usually, q ( can only ave two roots However, q ( must be satsfed for arbtrary value of n te regon [, ] Te only possble stuaton s te coeffcents of wt any degree equate to ero, wc mply tat δ δ f, f, f f ( Integratng q ( wt respect to twce obtan, f C C f C C δ 7 8 δ f C C C C Substtutng q ( nto q ( obtan ( g ( ( d d C C f ( δ d d C C dc dc7 δ ( ( Snce f ( g are te sngle varable functons of, respectvely, wc sow tat δ δ f d d C C d C d C C C 7 ( 9 g d d C C C C were (,,, ( ( C are unknown ntegral constants Solutons must satsfy te boundary condtons at te two longtudnal sdes of te beam eactly By means of q (, qs (, ( (9 yeld s functonal equatons ( ( λ λ λ ( ( λ λ λ ( ( λ λ λ q( ( ( λ λ λ ( ( λ λ λ ( ( λ λ λ q( ( ( λ λ λ ( ( λ λ λ ( ( λ λ λ q( ( ( λ λ λ ( ( λ λ λ ( ( λ λ λ q( ( ( a λ λ ( ( a λ λ ( ( a λ λ qc ( ( ( a λ λ ( ( a λ λ ( ( a λ λ qc ( (7 (8 (9 ( ( (

5 Zang et al / Journal of Mecancal Scence Tecnology (7 ( 8~8 87 were δ ( δ ( δ ( δ ( qc q C C C C, qc q C C C C Multplyng q (7 by λ ten mnus q (8 obtans λ ( ( λ λ λ ( ( λ λ λ λ λ ( λq( q( λ λ λ λ λ Multplyng q (9 by λ ten mnus q ( obtans λ ( ( λ λ λ ( ( λ λ λ λ λ ( λq( q( λ λ λ λ λ ( ( Replacng wt λ n q ( wt λ n q (, respectvely, obtans λ λ λ ( ( ( λ λ ( λ λ λ λ ( λ λ λ λ ( λ λ λ ( ( λq λ q λ λ λ λ λ ( ( ( λ λ ( λ λ λ λ ( λ λ λ λ ( λ λ λ ( ( λq λ q λ λ q ( mnus q ( obtans ( ( ( ( λ λ λ λ λ λ ( ( λ λ λ λ λ λ λ λ ( ( ( ( ( ( ( ( ( ( ( ( q ( q ( λ( λ λ ( ( λ ( λ λ λ λ λ λ λ λ λ λ λ λq λ q λ Multplyng q (7 by λ ten plus q (8 obtan λ ( ( λ λ λ ( ( λ λ λ λ λ ( λq( q( λ λ λ λ λ Multplyng q ( by λ ten plus q ( obtan λ ( ( λ λ λ ( ( λ λ λ λ λ ( λq( q( λ λ λ λ λ (7 (8 (9 Replacng wt λ n q (8 wt λ n q (9, respectvely, obtan λ λ λ ( ( ( λ λ ( λ λ λ λ ( λ λ λ λ ( λ λ λ ( ( λq λ q λ λ (

6 88 Zang et al / Journal of Mecancal Scence Tecnology (7 ( 8~8 λ λ λ ( ( ( λ λ ( λ λ λ λ ( λ λ λ λ ( λ λ λ ( ( λq λ q λ λ q ( mnus q ( obtans ( ( λ λ λ λ λ λ ( ( λ λ λ λ λ λ λ λ ( ( ( ( ( ( ( ( ( ( ( ( q ( q ( λ( λ λ ( ( λ ( λ λ λ λ λ λ λ λ λ λ λ λq λ q λ ( ( aλ aλ ( ( Multplyng q (7 by a λ ten mnus q ( obtans ( ( λ λ a λ a λ λ λ λqc aq a λ a λ ( ( aλ aλ ( Multplyng q (9 by a λ ten mnus q ( obtans ( ( λ λ a λ a λ λ λ λqc aq a λ a λ ( Replacng wt λ n q ( wt λ n q (, respectvely, obtan ( ( λ λ λ λ a λ a λ λ λ ( aλ aλ a λ a λ λ λ ( aλ aλ ( ( λqc λ aq λ a λ a λ ( ( λ λ λ λ a λ a λ λ λ ( aλ aλ a λ a λ λ λ ( aλ aλ ( ( λqc λ aq λ a λ a λ q ( mnus q ( obtans ( ( λ λ λ λ ( ( λ λ λ λ a λ a λ λ λ ( aλ aλ a λ a λ λ λ ( aλ aλ a λ a λ λ λ ( aλ aλ a λ a λ λ λ ( aλ aλ ( λ λ ( λ aq qc a λ a λ ( λ λ ( λ aq qc a λ a λ ( ( (7 q (7 mnus q (7 obtans ( ( a8 λ λ λ λ ( ( a8 λ λ λ λ ( ( a8 λ λ λ λ k ( (8 were

7 Zang et al / Journal of Mecancal Scence Tecnology (7 ( 8~8 89 k ( ( λ λ ( λ aq qc a λ a λ ( λ λ ( λ aq qc a λ a λ ( ( λ( λ λ ( ( λ ( λ λ λq λ q λ λq λ q λ (9 Replacng wt λ n q ( wt λ n q (, respectvely, obtan ( ( λ λ λ λ a λ a λ λ λ ( aλ aλ a λ a λ λ λ ( aλ aλ ( ( λqc λ aq λ a λ a λ ( ( λ λ λ λ a λ a λ λ λ ( aλ aλ a λ a λ λ λ ( aλ aλ ( ( λqc λ aq λ a λ a λ q ( mnus q ( obtans ( ( λ λ λ λ ( ( λ λ λ λ a λ a λ λ λ ( aλ aλ a λ a λ λ λ ( aλ aλ aλ aλ ( λ λ a λ a λ a λ a λ λ λ ( aλ aλ ( λ λ ( λ aq qc a λ a λ ( λ λ ( λ aq qc a λ a λ q ( mnus q ( obtans ( ( ( ( ( a8 λ λ λ λ ( ( a8 λ λ λ λ ( ( a8 λ λ λ λ k ( ( were k ( ( λ λ ( λ aq qc a λ a λ ( λ λ ( λ aq qc a λ a λ ( ( λ( λ λ ( ( λ ( λ λ λq λ q λ λq λ q λ ( Replacng wt λ n q ( wt λ n q (, respectvely, obtan ( λ a λ a λ λ λ ( aλ aλ a λ a λ λ λ ( aλ aλ ( ( λqc λ aq λ a λ a λ λ a λ a λ λ λ ( aλ aλ a λ a λ λ λ ( aλ aλ ( ( λqc λ aq λ a λ a λ q ( mnus q ( obtans ( ( ( λ ( λ ( ( a8 λ λ λ λ ( ( a8 λ λ λ λ k ( ( (7 were

8 8 Zang et al / Journal of Mecancal Scence Tecnology (7 ( 8~8 k ( ( λ λ ( λ aq qc a λ a λ ( λ λ ( λ aq qc a λ a λ (8 qs (8, ( (7 are te unfed equatons to solve te boundary value problem of peoelectrc plane beam acted by arbtrary mecancal electrcal loads We can construct (, ( ( n terms of k ( (,, Once (, ( ( are constructed, ( can be constructed by replacng wt λ n q ( ntegratng q ( wt respect to trce ( ( λ a8 λ λ a8 λ λ λ qc λ d d d aλ aλ a q λ d d d a λ a λ A A A (9 can be constructed by ntegratng q ( wt respect to trce λ λ λ ( λ λ λ λ( λ λ λ λ λ λ( λ λ λ λ λ λ( λ λ λ λ λ q λ d d d λ q λ d d d λ A A A (7 can be constructed by ntegratng q ( wt respect to trce ( λ λ λ λ ( λ λ λ λ λ (7 λ λ λ λ( λ λ λ λ λ q λ d d d λ q λ d d d λ A A A were A j are unknown constants Te epressons of σ, σ, σ, u, w, D, D Φ can be epressed by usng ( (,,, at present Snce q ( are satsfed, te reamng unknown constants can be determned by usng te boundary condtons of two ends of te beam obtan te epressons of stress, dsplacement, electrcal dsplacement electrcal potental fnally From te soluton procedure mentoned above, ow to construct (, ( ( s te key pont to fgure out te problem Te conclusons gven below can be confrmed by substtutng te epressons of loads epressons of (, ( ( nto qs (8, ( (7 Wen te plane beam acted by loads m qj( qj qj ( j, (7 (, ( ( can be constructed as m m m ( A, ( A, ( A (7 were A,,, m are unknown constants Wen te plane beam acted by loads A m qj( qj qj ( j, (7 (, ( ( can be constructed as m m m ( A, ( A, ( A (7 were A,,, m are unknown constants Wen te plane beam acted by loads A

9 Zang et al / Journal of Mecancal Scence Tecnology (7 ( 8~8 8 m qj ( qj qj ( j, (7 (, ( ( can be constructed as m m m ( A, ( A, ( A (77 were A,,, m are unknown constants Wen te plane beam acted by loads A sn( δ or cos( δ (, (78 q q q q j j j j j were can be constructed as ( δ s a known coeffcent,, δ δ δ δ δ δ ( A cos( F sn( ( A cos( F sn ( ( A cos( F sn ( (79 were A,,,,, F,, F, F are unknown constants Smlarly, f we need to deal wt A sn( δ or cos( δ (, (8 q q q q j j j j j we can smply let te polynomal part of q (79 wt degree of aganst for followng type of loads sn ( δ or cos( δ (, (8 q q q q j j j j j In te case wen te loads are complcated, suc as tey are not contnuous n one of te longtudnal sde, we can translate functons, wc represent te loads, nto Fourer seres as q H H j were η η cos ψ sn (, j j jn n jn n n η j q j d nπ η jn q cos, j H n d H n ψ jn q sn j H n d (, ( ( can be constructed as (8 (8 ( A ncos( H n Fncos( H n n n ( A ncos( H n Fncos( H n n n ( A ncos( H n Fncos( H n n n (8 were A,,, m, n, F n, n, F n, n F n are unknown constants Analogcally, we can deal wt dscontnuous dstrbuton of qj ( ( j, qj ( ( j, by only cange polynomal part of q (8 wt degree of aganst, respectvely A Applcatons In ts secton, varous boundary condtons, ncludng two longtudnal sdes of te beam two ends supported condtons of te beam, are consdered to valdatng te correctness generalaton of ts metod In Secton, te soluton procedure based on te dea gven n Secton s presented n detal Ten te solutons for tree addtonal eamples are gven As te loads become complcated, te fnal epressons become very lengt, In Sectons, 7 te numercal results are gven n te form of surface Te materal constants are all derved from Ref [] For all te numercal eamples, are taken as m m, respectvely Hnged end roller end beam subjected to unform sear force Ts eample s used to compare te solutons wt te results gven n Ref [] Te boundary condtons of te two longtudnal sdes are,,, q (, q ( q q q q q Te boundary condtons of two ends of beam are u, w, w ( σ d, Φ ( σ d ( σ, d ( ( D d, D d Step : Constructng te form of (, ( ( used, are constructed as (8 (8 n terms of te type of load In ts case, q (7 s ( A, ( A, ( A (87

10 8 Zang et al / Journal of Mecancal Scence Tecnology (7 ( 8~8 Step : Substtutng q (87 nto te unfed equatons of qs (8, ( (7 obtan tree lnear algebra equatons wt respect to For te same reason mentoned below q (, te coeffcents of wt any degree must be equated to ero to satsfy te unfed equatons It means tat te unfed equatons wll provde lnear algebra equatons wt respect to unknown constants A, A A, wc wll sow us te relatonsp between ( A A ( A n ts case Step : Substtutng q (87 nto qs (9-(7 to obtan te eplct epressons of (, ( ( It s noted tat at present te epressons of ( (,,, are all eplct polynomal epressons Step : Substtutng ( (,,, nto qs (-(9 combnng qs (, (-( obtan te eplct epressons of σ, σ, σ, u, w, D, D Φ wt some unknown constants We can verfy tat boundary condtons of te two longtudnal sdes of te beam ave been satsfed Step : Te remanng unknown constants can be determned by usng te two ends supported condtons of te beam obtan te fnal solutons By usng te soluton procedure gven above, te epressons for stress, dsplacement, electrcal dsplacement electrcal potental are ( ( q q σ, σ, σ (88 q ( dδ dδ D, D δ (89 q ( dδ dδ dδ Φ δδ (9 q ( sδ dd ( w δ (9 q ( sδ d( qs ( u δ q sδδ δd δd ( d d δ δ (9 We can confrm tat qs (88-(9 satsfy qs (8 (8 ettng te coeffcents of peoelectrcty delectrc mpermeablty equate to ero, te soluton degenerate to te Hnged end Roller end ortotropc plane beam subjected to unform sear force as can be seen n Ref [], wc sow te correctness of ts metod Hnged end roller end beam subjected to unform electrc dsplacement Te boundary condtons of te two longtudnal sdes are q (, q ( q q, q, q, q (9 Te ends supported condtons are te same to q (8 By usng te soluton metod gven above, te solutons are σ, σ, σ (9, D ( D q q (9 q δ δ ( Φ δδ (9 qd q ( δd δd w δ δδ q ( δd δd δd qd ( δ δ δ (97 We can confrm tat qs (9-(97 satsfy qs (9 (8 Te epresson for u are very lengt wll not lst ere for concse It s noted tat wen peoelectrc beam acted by unform electrcal dsplacement te components of w arse but te components of stress equate to ero, wc can be utled for deformaton control Fed end free end beam subjected to unform sear force Te boundary condtons of te two longtudnal sdes are,,, q (, q ( q q q q q Te boundary condtons of two ends of beam are u d w, w,, Φ d ( σ d ( σ d ( σ d ( D d Te solutons are ( (,, (98 (99 q σ, σ q ( ( σ ( q ( δd δd( ( D, D δ ( q ( d Φ δ ( ( δ ( q s d w δ ( δ ( ( q s dd δ (

11 Zang et al / Journal of Mecancal Scence Tecnology (7 ( 8~8 8 We can confrm tat qs (-( satsfy qs (98 (99 Fed end fed end beam subjected to unform sear force Te boundary condtons of te two longtudnal sdes are,,, q (, q ( q q q q q Te boundary condtons of two ends of beam are u, w, u, w dw d dw,, Φ d ( ( D d, D d ( ( (a (b Fg Dstrbuton of σ σ for wole peoelectrc beam Te solutons are ( ( q q σ, σ, σ ( q ( δd δd D, D δ (7 q ( δd δddδ Φ δδ (8 q ( sδ dd ( w δ (9 We can confrm tat qs (-(9 satsfy qs ( ( Fed end roller end beam subjected to lnear sear force Te numercal results for dstrbuton of σ σ n wole peoelectrc beam are gven n Fg (a (b, respectvely, wt loads parameters taken as q q Pa It s noted from Fg (a te ma value of σ take place near te regon of coordnate (, But for σ, te ma value arses near te regon of Fed end nged end beam subjected to quadratc electrc dsplacement Te boundary condtons of te two longtudnal sdes are q, q, q, q q (, q ( q q q Te boundary condtons of two ends of beam are ( Te boundary condtons of te two longtudnal sdes are q, q, q, q q( q q, q ( Te boundary condtons of two ends of beam are ( u, w, u, w d w d, Φ, ( σ d ( ( D d, D d ( u, w, u d w d, Φ ( σ d ( σ, d ( ( D d, D d ( Te numercal results for dstrbuton of w u n wole peoelectrc beam are gven n Fg (a (b, respectvely, wt loads parameters taken as q q q C m We can fnd from Fg (a tat te ma value of w do not arses at te center of te beam also not te bottom upper surface of te beam As to te dstrbuton of u, te cange nterval of ts value are te same to w, but te ma value of u take place at te borderlne of te beam

12 8 Zang et al / Journal of Mecancal Scence Tecnology (7 ( 8~8 Te ends supported condtons are te same to q ( Te numercal results for dstrbuton of D D n wole peoelectrc beam are gven n Fg (a (b, respectvely, wt loads parameters taken as q Pa Comparng Fg (a wt (b, te dstrbuton rule of D D s totally dfferent Te dstrbuton of D can be regarded as antsymmetry wt respect to te center poston (, of beam Te dstrbuton of D can be treated as antsymmetry aganst te as lne of te beam (a Conclusons Te formulae presented n ts paper can consder peoelectrc plane beam subjected to arbtrary mecancal electrcal loads wt varous ends supported condtons Comparng ts general metod wt tradtonal tral--error metod, te most advantage s t can obtan te eact solutons drectly does not need to guess modfy te form of stress functon or electrcal dsplacement functon amples sow te correctness generalaton of ts metod Fg Dstrbuton of w u for wole peoelectrc beam Fg Dstrbuton of D 7 Fed end fed end beam subjected to unform pressure force Te boundary condtons of te two longtudnal sdes are q (, q ( q, q, q q, q (b (a (b D for wole peoelectrc beam ( References [] A Benjeddoudvances n peoelectrc fnte element modelng of adaptve structural elements a survey, Computers Structures, 7 ( ( 7- [] D A Saravanos P R Heylger, Mecancs computatonal models for lamnated peoelectrc beams plates sellsppled Mecancs Revews, ( (999 - [] H A Irsck, Revew on statc dynamc sape control of structures by peoelectrc actuaton, ngneerng Structures, (7 ( - [] D J Huang, H J Dng W Q Cen unfed soluton for an ansotropc functonally graded peoelectrc beam subject to snusodal transverse loads, Journal of Intellgent Materal Systems Structures, (8 (9 - [] D J Huang, H J Dng W Q Cen, Statc analyss of ansotropc functonally graded magneto-electro-elastc beams subjected to arbtrary loadng, uropean Journal of Mecancs A/Solds, 9 ( ( -9 [] D J Huang, H J Dng W Q Cennalyss of functonally graded lamnated peoelectrc cantlever actuators subjected to constant voltage, Smart Materals Structures, 7 (9 (8 - [7] D J Huang, H J Dng W Q Cennalytcal soluton for functonally graded magneto-electro-elastc plane beams, Internatonal Journal of ngneerng Scence, ( (7 7-8 [8] T T Zang Z F S, Bendng beavor of - multlayered peoelectrc curved actuators, Smart Materals Structures, ( (7 - [9] Z F S, Bendng beavor of peoelectrc curved actuator, Smart Materals Structures, ( ( 8-8 [] H J Xang Z F S, lectrostatc analyss of functonally graded peoelectrc cantlevers, Journal of Intellgent Materal Systems Structures, 8 ( (7 7-9 [] Z F S, H J Xang B F Spencer, act analyss of

13 Zang et al / Journal of Mecancal Scence Tecnology (7 ( 8~8 8 mult-layer peoelectrc/composte cantlevers, Smart Materals Structures, (9 ( 7-8 [] Y Cen Z F S, act solutons of functonally gradent peotermoelastc cantlevers parameter dentfcaton, Journal of Intellgent Materal Systems Structures, ( ( -9 [] H J Xang Z F S, Statc analyss for mult-layered peoelectrc cantlevers, Internatonal Journal of Solds Structures, (9 (8-8 [] T Yu, Z Zong, Bendng analyss of a cantlever peoelectrc functonally graded beam, Scence n Cna Seres G: Pyscs, Mecancs & Astronomy, ( ( [] D J Huang, H J Dng H M Wangnalytcal soluton for fed-end ortotropc beams subjected to unform load, Journal of Zejang Unversty (ngneerng Scence, ( ( - Append a a a a a a a a d sδ δ ( d d dδ ( s s δ d ( d d δ ( d d dδ sδ δ ( d d dδ δ δ ( d d dδ sδ d ( d d δ ( d d dδ ( s s δ ( d d δ ( d d dδ sδ δ ( d d dδ δ δ ( d d d δ a ( s d a λ ( s d a λ d a s d a s d a d a s d a s d a d λ a ( λ ( λ λ a ( λ ( λ λ a (A (A (A a a ( s da λ s da d λ a a ( s da λ s da d λ a a ( s da λ s da d λ a a ( d δa λ δa δ λ a a ( d δa λ δa δ λ a a ( d δa λ δa δ λ a a ( d δa λ ( d δa λ δ λ a a ( d δa λ ( d δa λ δ λ a a ( d δa λ ( d δa λ δ λ a a7 aλ a λ a a7 aλ a λ a a7 aλ a λ λ( λ λ λ( a8 a8, a8 a8 λ( λ λ λ( λ λ λ( λ λ λ( a8 a8, a8 a8 λ( λ( aλ aλ λ λ a8, a8, a8 aλ aλ λ (A (A (A (A7 (A8 Zang ang receved te B MS degrees n cvl engneerng from ogstcal ngneerng Unversty, Congqng, Cna n 7, respectvely He s currently a doctor cdate at te Department of Astronautcal Scence ngneerng, Natonal Unversty of Defense Tecnology Hs researc focuses on elastc analyss control of peoelectrc structures

α & β spatial orbitals in

α & β spatial orbitals in The atrx Hartree-Fock equatons The most common method of solvng the Hartree-Fock equatons f the spatal btals s to expand them n terms of known functons, { χ µ } µ= consder the spn-unrestrcted case. We

Διαβάστε περισσότερα

Πανεπιστήµιο Κρήτης - Τµήµα Επιστήµης Υπολογιστών. ΗΥ-570: Στατιστική Επεξεργασία Σήµατος. ιδάσκων : Α. Μουχτάρης. εύτερη Σειρά Ασκήσεων.

Πανεπιστήµιο Κρήτης - Τµήµα Επιστήµης Υπολογιστών. ΗΥ-570: Στατιστική Επεξεργασία Σήµατος. ιδάσκων : Α. Μουχτάρης. εύτερη Σειρά Ασκήσεων. Πανεπιστήµιο Κρήτης - Τµήµα Επιστήµης Υπολογιστών ΗΥ-570: Στατιστική Επεξεργασία Σήµατος 2015 ιδάσκων : Α. Μουχτάρης εύτερη Σειρά Ασκήσεων Λύσεις Ασκηση 1. 1. Consder the gven expresson for R 1/2 : R 1/2

Διαβάστε περισσότερα

V. Finite Element Method. 5.1 Introduction to Finite Element Method

V. Finite Element Method. 5.1 Introduction to Finite Element Method V. Fnte Element Method 5. Introducton to Fnte Element Method 5. Introducton to FEM Rtz method to dfferental equaton Problem defnton k Boundary value problem Prob. Eact : d d, 0 0 0, 0 ( ) ( ) 4 C C * 4

Διαβάστε περισσότερα

One and two particle density matrices for single determinant HF wavefunctions. (1) = φ 2. )β(1) ( ) ) + β(1)β * β. (1)ρ RHF

One and two particle density matrices for single determinant HF wavefunctions. (1) = φ 2. )β(1) ( ) ) + β(1)β * β. (1)ρ RHF One and two partcle densty matrces for sngle determnant HF wavefunctons One partcle densty matrx Gven the Hartree-Fock wavefuncton ψ (,,3,!, = Âϕ (ϕ (ϕ (3!ϕ ( 3 The electronc energy s ψ H ψ = ϕ ( f ( ϕ

Διαβάστε περισσότερα

A Class of Orthohomological Triangles

A Class of Orthohomological Triangles A Class of Orthohomologcal Trangles Prof. Claudu Coandă Natonal College Carol I Craova Romana. Prof. Florentn Smarandache Unversty of New Mexco Gallup USA Prof. Ion Pătraşcu Natonal College Fraţ Buzeşt

Διαβάστε περισσότερα

1 Complete Set of Grassmann States

1 Complete Set of Grassmann States Physcs 610 Homework 8 Solutons 1 Complete Set of Grassmann States For Θ, Θ, Θ, Θ each ndependent n-member sets of Grassmann varables, and usng the summaton conventon ΘΘ Θ Θ Θ Θ, prove the dentty e ΘΘ dθ

Διαβάστε περισσότερα

Solutions for Mathematical Physics 1 (Dated: April 19, 2015)

Solutions for Mathematical Physics 1 (Dated: April 19, 2015) Solutons for Mathematcal Physcs 1 Dated: Aprl 19, 215 3.2.3 Usng the vectors P ê x cos θ + ê y sn θ, Q ê x cos ϕ ê y sn ϕ, R ê x cos ϕ ê y sn ϕ, 1 prove the famlar trgonometrc denttes snθ + ϕ sn θ cos

Διαβάστε περισσότερα

Symplecticity of the Störmer-Verlet algorithm for coupling between the shallow water equations and horizontal vehicle motion

Symplecticity of the Störmer-Verlet algorithm for coupling between the shallow water equations and horizontal vehicle motion Symplectcty of the Störmer-Verlet algorthm for couplng between the shallow water equatons and horzontal vehcle moton by H. Alem Ardakan & T. J. Brdges Department of Mathematcs, Unversty of Surrey, Guldford

Διαβάστε περισσότερα

Stochastic Finite Element Analysis for Composite Pressure Vessel

Stochastic Finite Element Analysis for Composite Pressure Vessel * ** ** Stochastc Fnte Element Analyss for Composte Pressure Vessel Tae Kyung Hwang Young Dae Doh and Soon Il Moon Key Words : Relablty Progressve Falure Pressure Vessel Webull Functon Abstract ABAQUS

Διαβάστε περισσότερα

Aerodynamics & Aeroelasticity: Eigenvalue analysis

Aerodynamics & Aeroelasticity: Eigenvalue analysis Εθνικό Μετσόβιο Πολυτεχνείο Natonal Techncal Unversty of Athens Aerodynamcs & Aeroelastcty: Egenvalue analyss Σπύρος Βουτσινάς / Spyros Voutsnas Άδεια Χρήσης Το παρόν εκπαιδευτικό υλικό υπόκειται σε άδειες

Διαβάστε περισσότερα

Constant Elasticity of Substitution in Applied General Equilibrium

Constant Elasticity of Substitution in Applied General Equilibrium Constant Elastct of Substtuton n Appled General Equlbru The choce of nput levels that nze the cost of producton for an set of nput prces and a fed level of producton can be epressed as n sty.. f Ltng for

Διαβάστε περισσότερα

CHAPTER 25 SOLVING EQUATIONS BY ITERATIVE METHODS

CHAPTER 25 SOLVING EQUATIONS BY ITERATIVE METHODS CHAPTER 5 SOLVING EQUATIONS BY ITERATIVE METHODS EXERCISE 104 Page 8 1. Find the positive root of the equation x + 3x 5 = 0, correct to 3 significant figures, using the method of bisection. Let f(x) =

Διαβάστε περισσότερα

Variance of Trait in an Inbred Population. Variance of Trait in an Inbred Population

Variance of Trait in an Inbred Population. Variance of Trait in an Inbred Population Varance of Trat n an Inbred Populaton Varance of Trat n an Inbred Populaton Varance of Trat n an Inbred Populaton Revew of Mean Trat Value n Inbred Populatons We showed n the last lecture that for a populaton

Διαβάστε περισσότερα

Neutralino contributions to Dark Matter, LHC and future Linear Collider searches

Neutralino contributions to Dark Matter, LHC and future Linear Collider searches Neutralno contrbutons to Dark Matter, LHC and future Lnear Collder searches G.J. Gounars Unversty of Thessalonk, Collaboraton wth J. Layssac, P.I. Porfyrads, F.M. Renard and wth Th. Dakonds for the γz

Διαβάστε περισσότερα

Non polynomial spline solutions for special linear tenth-order boundary value problems

Non polynomial spline solutions for special linear tenth-order boundary value problems ISSN 746-7233 England UK World Journal of Modellng and Smulaton Vol. 7 20 No. pp. 40-5 Non polynomal splne solutons for specal lnear tenth-order boundary value problems J. Rashdna R. Jallan 2 K. Farajeyan

Διαβάστε περισσότερα

Multi-dimensional Central Limit Theorem

Multi-dimensional Central Limit Theorem Mult-dmensonal Central Lmt heorem Outlne () () () t as () + () + + () () () Consder a sequence of ndependent random proceses t, t, dentcal to some ( t). Assume t 0. Defne the sum process t t t t () t tme

Διαβάστε περισσότερα

ΠΤΥΧΙΑΚΗ/ ΙΠΛΩΜΑΤΙΚΗ ΕΡΓΑΣΙΑ

ΠΤΥΧΙΑΚΗ/ ΙΠΛΩΜΑΤΙΚΗ ΕΡΓΑΣΙΑ ΑΡΙΣΤΟΤΕΛΕΙΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΘΕΣΣΑΛΟΝΙΚΗΣ ΣΧΟΛΗ ΘΕΤΙΚΩΝ ΕΠΙΣΤΗΜΩΝ ΤΜΗΜΑ ΠΛΗΡΟΦΟΡΙΚΗΣ ΠΤΥΧΙΑΚΗ/ ΙΠΛΩΜΑΤΙΚΗ ΕΡΓΑΣΙΑ «ΚΛΑ ΕΜΑ ΟΜΑ ΑΣ ΚΑΤΑ ΠΕΡΙΠΤΩΣΗ ΜΕΣΩ ΤΑΞΙΝΟΜΗΣΗΣ ΠΟΛΛΑΠΛΩΝ ΕΤΙΚΕΤΩΝ» (Instance-Based Ensemble

Διαβάστε περισσότερα

Multi-dimensional Central Limit Theorem

Multi-dimensional Central Limit Theorem Mult-dmensonal Central Lmt heorem Outlne () () () t as () + () + + () () () Consder a sequence of ndependent random proceses t, t, dentcal to some ( t). Assume t 0. Defne the sum process t t t t () t ();

Διαβάστε περισσότερα

8.1 The Nature of Heteroskedasticity 8.2 Using the Least Squares Estimator 8.3 The Generalized Least Squares Estimator 8.

8.1 The Nature of Heteroskedasticity 8.2 Using the Least Squares Estimator 8.3 The Generalized Least Squares Estimator 8. 8.1 The Nature of Heteroskedastcty 8. Usng the Least Squares Estmator 8.3 The Generalzed Least Squares Estmator 8.4 Detectng Heteroskedastcty E( y) = β+β 1 x e = y E( y ) = y β β x 1 y = β+β x + e 1 Fgure

Διαβάστε περισσότερα

Journal of Theoretics Vol.4-5

Journal of Theoretics Vol.4-5 Journal of Theoretcs Vol.4- A Unfed Feld Theory Peter Hckman peter.hckman@ntlworld.com Abstract: In ths paper, the extenson of Remann geometry to nclude an asymmetrc metrc tensor s presented. A new co-varant

Διαβάστε περισσότερα

Phys460.nb Solution for the t-dependent Schrodinger s equation How did we find the solution? (not required)

Phys460.nb Solution for the t-dependent Schrodinger s equation How did we find the solution? (not required) Phys460.nb 81 ψ n (t) is still the (same) eigenstate of H But for tdependent H. The answer is NO. 5.5.5. Solution for the tdependent Schrodinger s equation If we assume that at time t 0, the electron starts

Διαβάστε περισσότερα

HOMEWORK 4 = G. In order to plot the stress versus the stretch we define a normalized stretch:

HOMEWORK 4 = G. In order to plot the stress versus the stretch we define a normalized stretch: HOMEWORK 4 Problem a For the fast loading case, we want to derive the relationship between P zz and λ z. We know that the nominal stress is expressed as: P zz = ψ λ z where λ z = λ λ z. Therefore, applying

Διαβάστε περισσότερα

8.324 Relativistic Quantum Field Theory II

8.324 Relativistic Quantum Field Theory II Lecture 8.3 Relatvstc Quantum Feld Theory II Fall 00 8.3 Relatvstc Quantum Feld Theory II MIT OpenCourseWare Lecture Notes Hon Lu, Fall 00 Lecture 5.: RENORMALIZATION GROUP FLOW Consder the bare acton

Διαβάστε περισσότερα

ΗΥ537: Έλεγχος Πόρων και Επίδοση σε Ευρυζωνικά Δίκτυα,

ΗΥ537: Έλεγχος Πόρων και Επίδοση σε Ευρυζωνικά Δίκτυα, ΗΥ537: Έλεγχος Πόρων και Επίδοση σε Ευρυζωνικά Δίκτυα Βασίλειος Σύρης Τμήμα Επιστήμης Υπολογιστών Πανεπιστήμιο Κρήτης Εαρινό εξάμηνο 2008 Economcs Contents The contet The basc model user utlty, rces and

Διαβάστε περισσότερα

THE SECOND WEIGHTED MOMENT OF ζ. S. Bettin & J.B. Conrey

THE SECOND WEIGHTED MOMENT OF ζ. S. Bettin & J.B. Conrey THE SECOND WEIGHTED MOMENT OF ζ by S. Bettn & J.B. Conrey Abstract. We gve an explct formula for the second weghted moment of ζs) on the crtcal lne talored for fast computatons wth any desred accuracy.

Διαβάστε περισσότερα

derivation of the Laplacian from rectangular to spherical coordinates

derivation of the Laplacian from rectangular to spherical coordinates derivation of the Laplacian from rectangular to spherical coordinates swapnizzle 03-03- :5:43 We begin by recognizing the familiar conversion from rectangular to spherical coordinates (note that φ is used

Διαβάστε περισσότερα

Some generalization of Cauchy s and Wilson s functional equations on abelian groups

Some generalization of Cauchy s and Wilson s functional equations on abelian groups Aequat. Math. 89 (2015), 591 603 c The Author(s) 2013. Ths artcle s publshed wth open access at Sprngerlnk.com 0001-9054/15/030591-13 publshed onlne December 6, 2013 DOI 10.1007/s00010-013-0244-4 Aequatones

Διαβάστε περισσότερα

Noriyasu MASUMOTO, Waseda University, Okubo, Shinjuku, Tokyo , Japan Hiroshi YAMAKAWA, Waseda University

Noriyasu MASUMOTO, Waseda University, Okubo, Shinjuku, Tokyo , Japan Hiroshi YAMAKAWA, Waseda University A Study on Predctve Control Usng a Short-Term Predcton Method Based on Chaos Theory (Predctve Control of Nonlnear Systems Usng Plural Predcted Dsturbance Values) Noryasu MASUMOTO, Waseda Unversty, 3-4-1

Διαβάστε περισσότερα

Derivation for Input of Factor Graph Representation

Derivation for Input of Factor Graph Representation Dervaton for Input of actor Graph Representaton Sum-Product Prmal Based on the orgnal LP formulaton b x θ x + b θ,x, s.t., b, b,, N, x \ b x = b we defne V as the node set allocated to the th core. { V

Διαβάστε περισσότερα

Phasor Diagram of an RC Circuit V R

Phasor Diagram of an RC Circuit V R ESE Lecture 3 Phasor Dagram of an rcut VtV m snt V t V o t urrent s a reference n seres crcut KVL: V m V + V V ϕ I m V V m ESE Lecture 3 Phasor Dagram of an L rcut VtV m snt V t V t L V o t KVL: V m V

Διαβάστε περισσότερα

The Simply Typed Lambda Calculus

The Simply Typed Lambda Calculus Type Inference Instead of writing type annotations, can we use an algorithm to infer what the type annotations should be? That depends on the type system. For simple type systems the answer is yes, and

Διαβάστε περισσότερα

LECTURE 4 : ARMA PROCESSES

LECTURE 4 : ARMA PROCESSES LECTURE 4 : ARMA PROCESSES Movng-Average Processes The MA(q) process, s defned by (53) y(t) =µ ε(t)+µ 1 ε(t 1) + +µ q ε(t q) =µ(l)ε(t), where µ(l) =µ +µ 1 L+ +µ q L q and where ε(t) s whte nose An MA model

Διαβάστε περισσότερα

2 Composition. Invertible Mappings

2 Composition. Invertible Mappings Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan Composition. Invertible Mappings In this section we discuss two procedures for creating new mappings from old ones, namely,

Διαβάστε περισσότερα

Example Sheet 3 Solutions

Example Sheet 3 Solutions Example Sheet 3 Solutions. i Regular Sturm-Liouville. ii Singular Sturm-Liouville mixed boundary conditions. iii Not Sturm-Liouville ODE is not in Sturm-Liouville form. iv Regular Sturm-Liouville note

Διαβάστε περισσότερα

2 Lagrangian and Green functions in d dimensions

2 Lagrangian and Green functions in d dimensions Renormalzaton of φ scalar feld theory December 6 Pdf fle generated on February 7, 8. TODO Examne ε n the two-pont functon cf Sterman. Lagrangan and Green functons n d dmensons In these notes, we ll use

Διαβάστε περισσότερα

4.6 Autoregressive Moving Average Model ARMA(1,1)

4.6 Autoregressive Moving Average Model ARMA(1,1) 84 CHAPTER 4. STATIONARY TS MODELS 4.6 Autoregressive Moving Average Model ARMA(,) This section is an introduction to a wide class of models ARMA(p,q) which we will consider in more detail later in this

Διαβάστε περισσότερα

Lecture 2: Dirac notation and a review of linear algebra Read Sakurai chapter 1, Baym chatper 3

Lecture 2: Dirac notation and a review of linear algebra Read Sakurai chapter 1, Baym chatper 3 Lecture 2: Dirac notation and a review of linear algebra Read Sakurai chapter 1, Baym chatper 3 1 State vector space and the dual space Space of wavefunctions The space of wavefunctions is the set of all

Διαβάστε περισσότερα

Finite Field Problems: Solutions

Finite Field Problems: Solutions Finite Field Problems: Solutions 1. Let f = x 2 +1 Z 11 [x] and let F = Z 11 [x]/(f), a field. Let Solution: F =11 2 = 121, so F = 121 1 = 120. The possible orders are the divisors of 120. Solution: The

Διαβάστε περισσότερα

Other Test Constructions: Likelihood Ratio & Bayes Tests

Other Test Constructions: Likelihood Ratio & Bayes Tests Other Test Constructions: Likelihood Ratio & Bayes Tests Side-Note: So far we have seen a few approaches for creating tests such as Neyman-Pearson Lemma ( most powerful tests of H 0 : θ = θ 0 vs H 1 :

Διαβάστε περισσότερα

6.1. Dirac Equation. Hamiltonian. Dirac Eq.

6.1. Dirac Equation. Hamiltonian. Dirac Eq. 6.1. Dirac Equation Ref: M.Kaku, Quantum Field Theory, Oxford Univ Press (1993) η μν = η μν = diag(1, -1, -1, -1) p 0 = p 0 p = p i = -p i p μ p μ = p 0 p 0 + p i p i = E c 2 - p 2 = (m c) 2 H = c p 2

Διαβάστε περισσότερα

ΠΤΥΧΙΑΚΗ ΕΡΓΑΣΙΑ "ΠΟΛΥΚΡΙΤΗΡΙΑ ΣΥΣΤΗΜΑΤΑ ΛΗΨΗΣ ΑΠΟΦΑΣΕΩΝ. Η ΠΕΡΙΠΤΩΣΗ ΤΗΣ ΕΠΙΛΟΓΗΣ ΑΣΦΑΛΙΣΤΗΡΙΟΥ ΣΥΜΒΟΛΑΙΟΥ ΥΓΕΙΑΣ "

ΠΤΥΧΙΑΚΗ ΕΡΓΑΣΙΑ ΠΟΛΥΚΡΙΤΗΡΙΑ ΣΥΣΤΗΜΑΤΑ ΛΗΨΗΣ ΑΠΟΦΑΣΕΩΝ. Η ΠΕΡΙΠΤΩΣΗ ΤΗΣ ΕΠΙΛΟΓΗΣ ΑΣΦΑΛΙΣΤΗΡΙΟΥ ΣΥΜΒΟΛΑΙΟΥ ΥΓΕΙΑΣ ΤΕΧΝΟΛΟΓΙΚΟ ΕΚΠΑΙΔΕΥΤΙΚΟ ΙΔΡΥΜΑ ΚΑΛΑΜΑΤΑΣ ΣΧΟΛΗ ΔΙΟΙΚΗΣΗΣ ΟΙΚΟΝΟΜΙΑΣ ΤΜΗΜΑ ΜΟΝΑΔΩΝ ΥΓΕΙΑΣ ΚΑΙ ΠΡΟΝΟΙΑΣ ΠΤΥΧΙΑΚΗ ΕΡΓΑΣΙΑ "ΠΟΛΥΚΡΙΤΗΡΙΑ ΣΥΣΤΗΜΑΤΑ ΛΗΨΗΣ ΑΠΟΦΑΣΕΩΝ. Η ΠΕΡΙΠΤΩΣΗ ΤΗΣ ΕΠΙΛΟΓΗΣ ΑΣΦΑΛΙΣΤΗΡΙΟΥ ΣΥΜΒΟΛΑΙΟΥ

Διαβάστε περισσότερα

Matrices and Determinants

Matrices and Determinants Matrices and Determinants SUBJECTIVE PROBLEMS: Q 1. For what value of k do the following system of equations possess a non-trivial (i.e., not all zero) solution over the set of rationals Q? x + ky + 3z

Διαβάστε περισσότερα

EE512: Error Control Coding

EE512: Error Control Coding EE512: Error Control Coding Solution for Assignment on Finite Fields February 16, 2007 1. (a) Addition and Multiplication tables for GF (5) and GF (7) are shown in Tables 1 and 2. + 0 1 2 3 4 0 0 1 2 3

Διαβάστε περισσότερα

Exercises 10. Find a fundamental matrix of the given system of equations. Also find the fundamental matrix Φ(t) satisfying Φ(0) = I. 1.

Exercises 10. Find a fundamental matrix of the given system of equations. Also find the fundamental matrix Φ(t) satisfying Φ(0) = I. 1. Exercises 0 More exercises are available in Elementary Differential Equations. If you have a problem to solve any of them, feel free to come to office hour. Problem Find a fundamental matrix of the given

Διαβάστε περισσότερα

On Curvature Tensors in Absolute Parallelism Geometry

On Curvature Tensors in Absolute Parallelism Geometry arxv:gr-qc/0604111 v1 26 Apr 2006 On Curvature Tensors n Absolute Parallelsm Geometry Nabl L. Youssef and Amr M. Sd-Ahmed Department of Mathematcs, Faculty of Scence, Caro Unversty e-mal: nyoussef@frcu.eun.eg

Διαβάστε περισσότερα

Forced vibrations of a two-layered shell in the case of viscous resistance

Forced vibrations of a two-layered shell in the case of viscous resistance Journal of Physcs: Conference Seres PAPER OPEN ACCESS Forced vbratons of a two-layered shell n the case of vscous resstance To cte ths artcle: L A Aghalovyan and L G Ghulghazaryan 08 J. Phys.: Conf. Ser.

Διαβάστε περισσότερα

On a four-dimensional hyperbolic manifold with finite volume

On a four-dimensional hyperbolic manifold with finite volume BULETINUL ACADEMIEI DE ŞTIINŢE A REPUBLICII MOLDOVA. MATEMATICA Numbers 2(72) 3(73), 2013, Pages 80 89 ISSN 1024 7696 On a four-dimensional hyperbolic manifold with finite volume I.S.Gutsul Abstract. In

Διαβάστε περισσότερα

Congruence Classes of Invertible Matrices of Order 3 over F 2

Congruence Classes of Invertible Matrices of Order 3 over F 2 International Journal of Algebra, Vol. 8, 24, no. 5, 239-246 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/.2988/ija.24.422 Congruence Classes of Invertible Matrices of Order 3 over F 2 Ligong An and

Διαβάστε περισσότερα

Generalized Fibonacci-Like Polynomial and its. Determinantal Identities

Generalized Fibonacci-Like Polynomial and its. Determinantal Identities Int. J. Contemp. Math. Scences, Vol. 7, 01, no. 9, 1415-140 Generalzed Fbonacc-Le Polynomal and ts Determnantal Identtes V. K. Gupta 1, Yashwant K. Panwar and Ompraash Shwal 3 1 Department of Mathematcs,

Διαβάστε περισσότερα

Asymptotically Confirmed Hypotheses Method for the Construction of Micropolar and Classical Theories of Elastic Thin Shells

Asymptotically Confirmed Hypotheses Method for the Construction of Micropolar and Classical Theories of Elastic Thin Shells Advances n Pure Mathematcs 5 5 69-64 Publshed Onlne August 5 n ScRes http://wwwscrporg/ournal/apm http://dxdoorg/46/apm5557 Asymptotcally Confrmed Hypotheses Method for the Constructon of Mcropolar and

Διαβάστε περισσότερα

Chapter 7 Transformations of Stress and Strain

Chapter 7 Transformations of Stress and Strain Chapter 7 Transformations of Stress and Strain INTRODUCTION Transformation of Plane Stress Mohr s Circle for Plane Stress Application of Mohr s Circle to 3D Analsis 90 60 60 0 0 50 90 Introduction 7-1

Διαβάστε περισσότερα

Strain gauge and rosettes

Strain gauge and rosettes Strain gauge and rosettes Introduction A strain gauge is a device which is used to measure strain (deformation) on an object subjected to forces. Strain can be measured using various types of devices classified

Διαβάστε περισσότερα

Every set of first-order formulas is equivalent to an independent set

Every set of first-order formulas is equivalent to an independent set Every set of first-order formulas is equivalent to an independent set May 6, 2008 Abstract A set of first-order formulas, whatever the cardinality of the set of symbols, is equivalent to an independent

Διαβάστε περισσότερα

THREE-DIMENSIONAL VISCO-ELASTIC ARTIFICIAL BOUNDARIES IN TIME DOMAIN FOR WAVE MOTION PROBLEMS

THREE-DIMENSIONAL VISCO-ELASTIC ARTIFICIAL BOUNDARIES IN TIME DOMAIN FOR WAVE MOTION PROBLEMS 6 Vol. No.6 005 ENGNEENG MEHANS Dec. 005 000-4750(005)06-0046-06 * (. 00084. 000) O47.4, P5. A THEE-DMENSONAL VSO-ELAST ATFAL BOUNDAES N TME DOMAN FO WAVE MOTON POBLEMS * LU Jng-bo, WANG Zhen-yu, DU Xu-l,

Διαβάστε περισσότερα

b. Use the parametrization from (a) to compute the area of S a as S a ds. Be sure to substitute for ds!

b. Use the parametrization from (a) to compute the area of S a as S a ds. Be sure to substitute for ds! MTH U341 urface Integrals, tokes theorem, the divergence theorem To be turned in Wed., Dec. 1. 1. Let be the sphere of radius a, x 2 + y 2 + z 2 a 2. a. Use spherical coordinates (with ρ a) to parametrize.

Διαβάστε περισσότερα

A domain decomposition method for the Oseen-viscoelastic flow equations

A domain decomposition method for the Oseen-viscoelastic flow equations A doman decomposton method for the Oseen-vscoelastc flow equatons Eleanor Jenkns Hyesuk Lee Abstract We study a non-overlappng doman decomposton method for the Oseen-vscoelastc flow problem. The data on

Διαβάστε περισσότερα

Απόκριση σε Μοναδιαία Ωστική Δύναμη (Unit Impulse) Απόκριση σε Δυνάμεις Αυθαίρετα Μεταβαλλόμενες με το Χρόνο. Απόστολος Σ.

Απόκριση σε Μοναδιαία Ωστική Δύναμη (Unit Impulse) Απόκριση σε Δυνάμεις Αυθαίρετα Μεταβαλλόμενες με το Χρόνο. Απόστολος Σ. Απόκριση σε Δυνάμεις Αυθαίρετα Μεταβαλλόμενες με το Χρόνο The time integral of a force is referred to as impulse, is determined by and is obtained from: Newton s 2 nd Law of motion states that the action

Διαβάστε περισσότερα

Chapter 6: Systems of Linear Differential. be continuous functions on the interval

Chapter 6: Systems of Linear Differential. be continuous functions on the interval Chapter 6: Systems of Linear Differential Equations Let a (t), a 2 (t),..., a nn (t), b (t), b 2 (t),..., b n (t) be continuous functions on the interval I. The system of n first-order differential equations

Διαβάστε περισσότερα

Concrete Mathematics Exercises from 30 September 2016

Concrete Mathematics Exercises from 30 September 2016 Concrete Mathematics Exercises from 30 September 2016 Silvio Capobianco Exercise 1.7 Let H(n) = J(n + 1) J(n). Equation (1.8) tells us that H(2n) = 2, and H(2n+1) = J(2n+2) J(2n+1) = (2J(n+1) 1) (2J(n)+1)

Διαβάστε περισσότερα

Young Kwark. Warrington College of Business, University of Florida, 1454 Union Road, Gainesville, FL U.S.A.

Young Kwark. Warrington College of Business, University of Florida, 1454 Union Road, Gainesville, FL U.S.A. ESEC TICE PTFOM O WOESE? STTEGIC TOO FO ONINE ETIES TO ENEFIT FOM TI-PTY INFOMTION Young Kark Warrngton College of usness, Unversty of Florda, 1454 Unon oad, Ganesvlle, F 3611 U.S.. {young.kark@arrngton.ufl.edu}

Διαβάστε περισσότερα

Approximation of distance between locations on earth given by latitude and longitude

Approximation of distance between locations on earth given by latitude and longitude Approximation of distance between locations on earth given by latitude and longitude Jan Behrens 2012-12-31 In this paper we shall provide a method to approximate distances between two points on earth

Διαβάστε περισσότερα

9 /393 / Downloaded from energy.kashanu.ac.r at 5:3 0330 on Saturday October 0th 08 * hajakbar@grad.kashanu.ac.r mohammad@kashanu.ac.r. (shunt-apf) :... PSIM. : * 3... Downloaded from energy.kashanu.ac.r

Διαβάστε περισσότερα

Duals of the QCQP and SDP Sparse SVM. Antoni B. Chan, Nuno Vasconcelos, and Gert R. G. Lanckriet

Duals of the QCQP and SDP Sparse SVM. Antoni B. Chan, Nuno Vasconcelos, and Gert R. G. Lanckriet Duals of the QCQP and SDP Sparse SVM Anton B. Chan, Nuno Vasconcelos, and Gert R. G. Lanckret SVCL-TR 007-0 v Aprl 007 Duals of the QCQP and SDP Sparse SVM Anton B. Chan, Nuno Vasconcelos, and Gert R.

Διαβάστε περισσότερα

Section 8.3 Trigonometric Equations

Section 8.3 Trigonometric Equations 99 Section 8. Trigonometric Equations Objective 1: Solve Equations Involving One Trigonometric Function. In this section and the next, we will exple how to solving equations involving trigonometric functions.

Διαβάστε περισσότερα

ΕΠΙΧΕΙΡΗΣΙΑΚΗ ΑΛΛΗΛΟΓΡΑΦΙΑ ΚΑΙ ΕΠΙΚΟΙΝΩΝΙΑ ΣΤΗΝ ΑΓΓΛΙΚΗ ΓΛΩΣΣΑ

ΕΠΙΧΕΙΡΗΣΙΑΚΗ ΑΛΛΗΛΟΓΡΑΦΙΑ ΚΑΙ ΕΠΙΚΟΙΝΩΝΙΑ ΣΤΗΝ ΑΓΓΛΙΚΗ ΓΛΩΣΣΑ Ανοικτά Ακαδημαϊκά Μαθήματα στο ΤΕΙ Ιονίων Νήσων ΕΠΙΧΕΙΡΗΣΙΑΚΗ ΑΛΛΗΛΟΓΡΑΦΙΑ ΚΑΙ ΕΠΙΚΟΙΝΩΝΙΑ ΣΤΗΝ ΑΓΓΛΙΚΗ ΓΛΩΣΣΑ Ενότητα 11: The Unreal Past Το περιεχόμενο του μαθήματος διατίθεται με άδεια Creative Commons

Διαβάστε περισσότερα

Homework 3 Solutions

Homework 3 Solutions Homework 3 Solutions Igor Yanovsky (Math 151A TA) Problem 1: Compute the absolute error and relative error in approximations of p by p. (Use calculator!) a) p π, p 22/7; b) p π, p 3.141. Solution: For

Διαβάστε περισσότερα

Universität Regensburg Mathematik

Universität Regensburg Mathematik Unverstät Regensburg Matematk Fnte element approxmaton of a sarp nterface approac for gradent flow dynamcs of two-pase bomembranes Jon W. Barrett, Harald Garcke and Robert Nürnberg Preprnt Nr. 06/2017

Διαβάστε περισσότερα

Statistical Inference I Locally most powerful tests

Statistical Inference I Locally most powerful tests Statistical Inference I Locally most powerful tests Shirsendu Mukherjee Department of Statistics, Asutosh College, Kolkata, India. shirsendu st@yahoo.co.in So far we have treated the testing of one-sided

Διαβάστε περισσότερα

Inverse trigonometric functions & General Solution of Trigonometric Equations. ------------------ ----------------------------- -----------------

Inverse trigonometric functions & General Solution of Trigonometric Equations. ------------------ ----------------------------- ----------------- Inverse trigonometric functions & General Solution of Trigonometric Equations. 1. Sin ( ) = a) b) c) d) Ans b. Solution : Method 1. Ans a: 17 > 1 a) is rejected. w.k.t Sin ( sin ) = d is rejected. If sin

Διαβάστε περισσότερα

A Two Sample Test for Mean Vectors with Unequal Covariance Matrices

A Two Sample Test for Mean Vectors with Unequal Covariance Matrices A Two Sample Test for Mean Vectors wth Unequal Covarance Matrces Tamae Kawasak 1 and Takash Seo 2 1 Department of Mathematcal Informaton Scence Graduate School of Scence, Tokyo Unversty of Scence, Tokyo,

Διαβάστε περισσότερα

Lecture 26: Circular domains

Lecture 26: Circular domains Introductory lecture notes on Partial Differential Equations - c Anthony Peirce. Not to be copied, used, or revised without eplicit written permission from the copyright owner. 1 Lecture 6: Circular domains

Διαβάστε περισσότερα

Right Rear Door. Let's now finish the door hinge saga with the right rear door

Right Rear Door. Let's now finish the door hinge saga with the right rear door Right Rear Door Let's now finish the door hinge saga with the right rear door You may have been already guessed my steps, so there is not much to describe in detail. Old upper one file:///c /Documents

Διαβάστε περισσότερα

Econ 2110: Fall 2008 Suggested Solutions to Problem Set 8 questions or comments to Dan Fetter 1

Econ 2110: Fall 2008 Suggested Solutions to Problem Set 8  questions or comments to Dan Fetter 1 Eon : Fall 8 Suggested Solutions to Problem Set 8 Email questions or omments to Dan Fetter Problem. Let X be a salar with density f(x, θ) (θx + θ) [ x ] with θ. (a) Find the most powerful level α test

Διαβάστε περισσότερα

ΑΝΩΤΑΤΗ ΣΧΟΛΗ ΠΑΙ ΑΓΩΓΙΚΗΣ ΚΑΙ ΤΕΧΝΟΛΟΓΙΚΗΣ ΕΚΠΑΙ ΕΥΣΗΣ ΠΑΡΑΔΟΤΕΟ ΕΠΙΣΤΗΜΟΝΙΚΗ ΕΡΓΑΣΙΑ ΣΕ ΔΙΕΘΝΕΣ ΕΠΙΣΤΗΜΟΝΙΚΟ ΠΕΡΙΟΔΙΚΟ

ΑΝΩΤΑΤΗ ΣΧΟΛΗ ΠΑΙ ΑΓΩΓΙΚΗΣ ΚΑΙ ΤΕΧΝΟΛΟΓΙΚΗΣ ΕΚΠΑΙ ΕΥΣΗΣ ΠΑΡΑΔΟΤΕΟ ΕΠΙΣΤΗΜΟΝΙΚΗ ΕΡΓΑΣΙΑ ΣΕ ΔΙΕΘΝΕΣ ΕΠΙΣΤΗΜΟΝΙΚΟ ΠΕΡΙΟΔΙΚΟ ΑΝΩΤΑΤΗ ΣΧΟΛΗ ΠΑΙ ΑΓΩΓΙΚΗΣ ΚΑΙ ΤΕΧΝΟΛΟΓΙΚΗΣ ΕΚΠΑΙ ΕΥΣΗΣ (Α.Σ.ΠΑΙ.Τ.Ε.) «Αρχιμήδης ΙΙΙ Ενίσχυση Ερευνητικών ομάδων στην Α.Σ.ΠΑΙ.Τ.Ε.» Υποέργο: 8 Τίτλος: «Εκκεντρότητες αντισεισμικού σχεδιασμού ασύμμετρων

Διαβάστε περισσότερα

Homework 8 Model Solution Section

Homework 8 Model Solution Section MATH 004 Homework Solution Homework 8 Model Solution Section 14.5 14.6. 14.5. Use the Chain Rule to find dz where z cosx + 4y), x 5t 4, y 1 t. dz dx + dy y sinx + 4y)0t + 4) sinx + 4y) 1t ) 0t + 4t ) sinx

Διαβάστε περισσότερα

9.09. # 1. Area inside the oval limaçon r = cos θ. To graph, start with θ = 0 so r = 6. Compute dr

9.09. # 1. Area inside the oval limaçon r = cos θ. To graph, start with θ = 0 so r = 6. Compute dr 9.9 #. Area inside the oval limaçon r = + cos. To graph, start with = so r =. Compute d = sin. Interesting points are where d vanishes, or at =,,, etc. For these values of we compute r:,,, and the values

Διαβάστε περισσότερα

DESIGN OF MACHINERY SOLUTION MANUAL h in h 4 0.

DESIGN OF MACHINERY SOLUTION MANUAL h in h 4 0. DESIGN OF MACHINERY SOLUTION MANUAL -7-1! PROBLEM -7 Statement: Design a double-dwell cam to move a follower from to 25 6, dwell for 12, fall 25 and dwell for the remader The total cycle must take 4 sec

Διαβάστε περισσότερα

A Method for Determining Service Level of Road Network Based on Improved Capacity Model

A Method for Determining Service Level of Road Network Based on Improved Capacity Model 30 4 2013 4 Journal of Hghway and Transportaton Research and Development Vol. 30 No. 4 Apr. 2013 do10. 3969 /j. ssn. 1002-0268. 2013. 04. 018 1 1 2 1. 4000742. 201804 2 U491. 1 + 3 A 1002-0268 201304-0101

Διαβάστε περισσότερα

Partial Differential Equations in Biology The boundary element method. March 26, 2013

Partial Differential Equations in Biology The boundary element method. March 26, 2013 The boundary element method March 26, 203 Introduction and notation The problem: u = f in D R d u = ϕ in Γ D u n = g on Γ N, where D = Γ D Γ N, Γ D Γ N = (possibly, Γ D = [Neumann problem] or Γ N = [Dirichlet

Διαβάστε περισσότερα

Pricing of Options on two Currencies Libor Rates

Pricing of Options on two Currencies Libor Rates Prcng o Optons on two Currences Lbor Rates Fabo Mercuro Fnancal Models, Banca IMI Abstract In ths document we show how to prce optons on two Lbor rates belongng to two derent currences the ormer s domestc,

Διαβάστε περισσότερα

Fourier Series. MATH 211, Calculus II. J. Robert Buchanan. Spring Department of Mathematics

Fourier Series. MATH 211, Calculus II. J. Robert Buchanan. Spring Department of Mathematics Fourier Series MATH 211, Calculus II J. Robert Buchanan Department of Mathematics Spring 2018 Introduction Not all functions can be represented by Taylor series. f (k) (c) A Taylor series f (x) = (x c)

Διαβάστε περισσότερα

Stresses in a Plane. Mohr s Circle. Cross Section thru Body. MET 210W Mohr s Circle 1. Some parts experience normal stresses in

Stresses in a Plane. Mohr s Circle. Cross Section thru Body. MET 210W Mohr s Circle 1. Some parts experience normal stresses in ME 10W E. Evans Stresses in a Plane Some parts eperience normal stresses in two directions. hese tpes of problems are called Plane Stress or Biaial Stress Cross Section thru Bod z angent and normal to

Διαβάστε περισσότερα

ANSWERSHEET (TOPIC = DIFFERENTIAL CALCULUS) COLLECTION #2. h 0 h h 0 h h 0 ( ) g k = g 0 + g 1 + g g 2009 =?

ANSWERSHEET (TOPIC = DIFFERENTIAL CALCULUS) COLLECTION #2. h 0 h h 0 h h 0 ( ) g k = g 0 + g 1 + g g 2009 =? Teko Classes IITJEE/AIEEE Maths by SUHAAG SIR, Bhopal, Ph (0755) 3 00 000 www.tekoclasses.com ANSWERSHEET (TOPIC DIFFERENTIAL CALCULUS) COLLECTION # Question Type A.Single Correct Type Q. (A) Sol least

Διαβάστε περισσότερα

ΠΤΥΧΙΑΚΗ ΕΡΓΑΣΙΑ ΒΑΛΕΝΤΙΝΑ ΠΑΠΑΔΟΠΟΥΛΟΥ Α.Μ.: 09/061. Υπεύθυνος Καθηγητής: Σάββας Μακρίδης

ΠΤΥΧΙΑΚΗ ΕΡΓΑΣΙΑ ΒΑΛΕΝΤΙΝΑ ΠΑΠΑΔΟΠΟΥΛΟΥ Α.Μ.: 09/061. Υπεύθυνος Καθηγητής: Σάββας Μακρίδης Α.Τ.Ε.Ι. ΙΟΝΙΩΝ ΝΗΣΩΝ ΠΑΡΑΡΤΗΜΑ ΑΡΓΟΣΤΟΛΙΟΥ ΤΜΗΜΑ ΔΗΜΟΣΙΩΝ ΣΧΕΣΕΩΝ ΚΑΙ ΕΠΙΚΟΙΝΩΝΙΑΣ ΠΤΥΧΙΑΚΗ ΕΡΓΑΣΙΑ «Η διαμόρφωση επικοινωνιακής στρατηγικής (και των τακτικών ενεργειών) για την ενδυνάμωση της εταιρικής

Διαβάστε περισσότερα

Assalamu `alaikum wr. wb.

Assalamu `alaikum wr. wb. LUMP SUM Assalamu `alaikum wr. wb. LUMP SUM Wassalamu alaikum wr. wb. Assalamu `alaikum wr. wb. LUMP SUM Wassalamu alaikum wr. wb. LUMP SUM Lump sum lump sum lump sum. lump sum fixed price lump sum lump

Διαβάστε περισσότερα

Vidyamandir Classes. Solutions to Revision Test Series - 2/ ACEG / IITJEE (Mathematics) = 2 centre = r. a

Vidyamandir Classes. Solutions to Revision Test Series - 2/ ACEG / IITJEE (Mathematics) = 2 centre = r. a Per -.(D).() Vdymndr lsses Solutons to evson est Seres - / EG / JEE - (Mthemtcs) Let nd re dmetrcl ends of crcle Let nd D re dmetrcl ends of crcle Hence mnmum dstnce s. y + 4 + 4 6 Let verte (h, k) then

Διαβάστε περισσότερα

Higher-order Spatial Accuracy in Diffeomorphic Image Registration

Higher-order Spatial Accuracy in Diffeomorphic Image Registration Geometry, Imagng and Computng Volume, Number, 1, 214 Hger-order Spatal Accuracy n Dffeomorpc Image Regstraton Henry O. Jacobs and Stefan Sommer We dscretze a cost functonal for mage regstraton problems

Διαβάστε περισσότερα

Vol. 34 ( 2014 ) No. 4. J. of Math. (PRC) : A : (2014) Frank-Wolfe [7],. Frank-Wolfe, ( ).

Vol. 34 ( 2014 ) No. 4. J. of Math. (PRC) : A : (2014) Frank-Wolfe [7],. Frank-Wolfe, ( ). Vol. 4 ( 214 ) No. 4 J. of Math. (PRC) 1,2, 1 (1., 472) (2., 714) :.,.,,,..,. : ; ; ; MR(21) : 9B2 : : A : 255-7797(214)4-759-7 1,,,,, [1 ].,, [4 6],, Frank-Wolfe, Frank-Wolfe [7],.,,.,,,., UE,, UE. O-D,,,,,

Διαβάστε περισσότερα

ST5224: Advanced Statistical Theory II

ST5224: Advanced Statistical Theory II ST5224: Advanced Statistical Theory II 2014/2015: Semester II Tutorial 7 1. Let X be a sample from a population P and consider testing hypotheses H 0 : P = P 0 versus H 1 : P = P 1, where P j is a known

Διαβάστε περισσότερα

8. ΕΠΕΞΕΡΓΑΣΊΑ ΣΗΜΆΤΩΝ. ICA: συναρτήσεις κόστους & εφαρμογές

8. ΕΠΕΞΕΡΓΑΣΊΑ ΣΗΜΆΤΩΝ. ICA: συναρτήσεις κόστους & εφαρμογές 8. ΕΠΕΞΕΡΓΑΣΊΑ ΣΗΜΆΤΩΝ ICA: συναρτήσεις κόστους & εφαρμογές ΚΎΡΤΩΣΗ (KUROSIS) Αθροιστικό (cumulant) 4 ης τάξεως μίας τ.μ. x με μέσο όρο 0: kurt 4 [ x] = E[ x ] 3( E[ y ]) Υποθέτουμε διασπορά=: kurt[ x]

Διαβάστε περισσότερα

PARTIAL NOTES for 6.1 Trigonometric Identities

PARTIAL NOTES for 6.1 Trigonometric Identities PARTIAL NOTES for 6.1 Trigonometric Identities tanθ = sinθ cosθ cotθ = cosθ sinθ BASIC IDENTITIES cscθ = 1 sinθ secθ = 1 cosθ cotθ = 1 tanθ PYTHAGOREAN IDENTITIES sin θ + cos θ =1 tan θ +1= sec θ 1 + cot

Διαβάστε περισσότερα

6.3 Forecasting ARMA processes

6.3 Forecasting ARMA processes 122 CHAPTER 6. ARMA MODELS 6.3 Forecasting ARMA processes The purpose of forecasting is to predict future values of a TS based on the data collected to the present. In this section we will discuss a linear

Διαβάστε περισσότερα

[1], [2] - (Danfoss, Rexroth, Char-Lynn. [3, 4, 5]), .. [6]. [7]

[1], [2] - (Danfoss, Rexroth, Char-Lynn. [3, 4, 5]), .. [6]. [7] OTROL. COISSION OF OTORIZATION AND ENERGETICS IN AGRICULTURE 0, Vol. 6, No. 5, 87 98 -,,, 008,.,., e-mal: mosgv@ukr.net. -,... -. :, -,. [],,.,,.., []. - (Danoss, Rexroth, Char-Lynn. [,, 5]),. -,.. [6]..,

Διαβάστε περισσότερα

Dr. D. Dinev, Department of Structural Mechanics, UACEG

Dr. D. Dinev, Department of Structural Mechanics, UACEG Lecture 4 Material behavior: Constitutive equations Field of the game Print version Lecture on Theory of lasticity and Plasticity of Dr. D. Dinev, Department of Structural Mechanics, UACG 4.1 Contents

Διαβάστε περισσότερα

MABUCHI AND AUBIN-YAU FUNCTIONALS OVER COMPLEX THREE-FOLDS arxiv: v1 [math.dg] 27 Mar 2010

MABUCHI AND AUBIN-YAU FUNCTIONALS OVER COMPLEX THREE-FOLDS arxiv: v1 [math.dg] 27 Mar 2010 MABUCHI AND AUBIN-YAU FUNCTIONALS OVER COMPLE THREE-FOLDS arv:1.57v1 [math.dg] 27 Mar 21 YI LI Abstract. In ths paper we construct Mabuch L M ω functonal and Aubn- Yau functonals Iω AY,J AY ω on any compact

Διαβάστε περισσότερα

Συστήματα Διαχείρισης Βάσεων Δεδομένων

Συστήματα Διαχείρισης Βάσεων Δεδομένων ΕΛΛΗΝΙΚΗ ΔΗΜΟΚΡΑΤΙΑ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΡΗΤΗΣ Συστήματα Διαχείρισης Βάσεων Δεδομένων Φροντιστήριο 9: Transactions - part 1 Δημήτρης Πλεξουσάκης Τμήμα Επιστήμης Υπολογιστών Tutorial on Undo, Redo and Undo/Redo

Διαβάστε περισσότερα

ΤΕΧΝΟΛΟΓΙΚΟ ΕΚΠΑΙΔΕΥΤΙΚΟ ΙΔΡΥΜΑ ΚΡΗΤΗΣ ΣΧΟΛΗ ΔΙΟΙΚΗΣΗΣ ΚΑΙ ΟΙΚΟΝΟΜΙΑΣ ΤΜΗΜΑ ΛΟΓΙΣΤΙΚΗΣ ΠΤΥΧΙΑΚΗ ΕΡΓΑΣΙΑ

ΤΕΧΝΟΛΟΓΙΚΟ ΕΚΠΑΙΔΕΥΤΙΚΟ ΙΔΡΥΜΑ ΚΡΗΤΗΣ ΣΧΟΛΗ ΔΙΟΙΚΗΣΗΣ ΚΑΙ ΟΙΚΟΝΟΜΙΑΣ ΤΜΗΜΑ ΛΟΓΙΣΤΙΚΗΣ ΠΤΥΧΙΑΚΗ ΕΡΓΑΣΙΑ ΤΕΧΝΟΛΟΓΙΚΟ ΕΚΠΑΙΔΕΥΤΙΚΟ ΙΔΡΥΜΑ ΚΡΗΤΗΣ ΣΧΟΛΗ ΔΙΟΙΚΗΣΗΣ ΚΑΙ ΟΙΚΟΝΟΜΙΑΣ ΤΜΗΜΑ ΛΟΓΙΣΤΙΚΗΣ ΠΤΥΧΙΑΚΗ ΕΡΓΑΣΙΑ ΛΟΓΙΣΤΙΚΗ ΚΑΙ ΦΟΡΟΛΟΓΙΑ Ο.Ε. ΕΙΣΗΓΗΤΡΙΑ ΚΑΘΗΓΗΤΡΙΑ: κ. ΟΥΡΑΝΟΥ ΕΡΜΙΟΝΗ ΣΠΟΥΔΑΣΤΡΙΕΣ: ΔΕΜΕΤΖΟΥ ΑΓΛΑΪΑ

Διαβάστε περισσότερα

Reminders: linear functions

Reminders: linear functions Reminders: linear functions Let U and V be vector spaces over the same field F. Definition A function f : U V is linear if for every u 1, u 2 U, f (u 1 + u 2 ) = f (u 1 ) + f (u 2 ), and for every u U

Διαβάστε περισσότερα

Lecture 2. Soundness and completeness of propositional logic

Lecture 2. Soundness and completeness of propositional logic Lecture 2 Soundness and completeness of propositional logic February 9, 2004 1 Overview Review of natural deduction. Soundness and completeness. Semantics of propositional formulas. Soundness proof. Completeness

Διαβάστε περισσότερα

Modelling Lifetime Dependence for Older Ages using a Multivariate Pareto Distribution

Modelling Lifetime Dependence for Older Ages using a Multivariate Pareto Distribution Modellng Lfetme Dependence for Older Ages usng a Multvarate Pareto Dstrbuton Danel H Ala Znovy Landsman Mchael Sherrs 3 School of Mathematcs, Statstcs and Actuaral Scence Unversty of Kent, Canterbury,

Διαβάστε περισσότερα