328 .. W period HORN comma J period SOKO L-suppress OWSKI \noindentInitial conditions328 \quad on u areW given . HORN as follows , J . : SOKO \L OWSKI InitialEquation: conditions open parenthesis on 1 $ 3 u closing $ are parenthesis given .. as u openfollows parenthesis : x comma y comma 0 closing parenthesis = u sub 0 open parenthesis x comma y closing parenthesis on Capital Omega comma Equation: open parenthesis 1 4 closing parenthesis .. u sub t open parenthesis x comma y comma\ begin 0{ closinga l i g n ∗} parenthesis = u sub 1 open parenthesis x comma y closing parenthesis on Capital Omega period uWe ( will assume x , that y the initial , 0 conditions ) = for v u and{ theta0 } are( compatible x , with y those ) on \Omega , \ tag ∗{$ ( 1 3 ) $}\\ u { t } ( x , y , 0 ) = u { 1 } ( x , y ) on \Omega . \ tag ∗{$ ( 1 4 ) $} for328 u commaW . HORN i period , J . SOKO e periodLOWSKI Initial conditions on u are given as follows : \endEquation:{ a l i g n ∗} open parenthesis 1 5 closing parenthesis .. v open parenthesis x comma 0 closing parenthesis = v sub 0 open parenthesis x closing parenthesis = u sub 0 open parenthesis x comma y closing parenthesis bar sub Q comma Equation: open parenthesis 1 6 closing parenthesis .. v sub \noindent We will assume that the initial conditions for $ v $ and $ \theta $ are compatible with those t open parenthesis x comma 0 closing parenthesisu(x, y, =0) v sub = 1u open0(x, y parenthesis) on Ω, x closing parenthesis = u sub 1 open(13) parenthesis x comma y closing for $u ,$ i.e. parenthesis bar sub Q comma Equation: openut parenthesis(x, y, 0) = 1 7u closing1(x, y) parenthesis on Ω. .. theta open parenthesis x comma(14) 0 closing parenthesis = theta sub 0 period \ beginWeWe conclude will{ a l assumei g n ∗} this that section the by initial formally conditions showing for thatv and theθ are energy compatible of this system with those for u, i . e . vdecreases ( x period , .. Formally 0 ) comma = v one{ can0 compute} ( the x energy ) of = this u structure{ 0 } as follows( x period , y ) \mid { Q } , \ tag ∗{$ ( 1After 5 multiplying ) $}\\ v open{ parenthesist } ( 4 x closing , parenthesis 0 ) by = u sub v t{ and1 integrating} ( x the result ) = over Capital u { 1 Omega} ( one x receives , : y ) \mid { Q } , \ tag ∗{$ ( 1 6 ) $}\\\theta ( x , 0 ) = \theta { 0 } . \ tag ∗{$ ( 1 7 ) $} Equation: open parenthesis 1 8 closing parenthesisv(x, 0) = .. 2 tov0( thex) = poweru0(x, of y) 1|Q dt, to the power of d integral sub Capital(15) Omega open parenthesis bar u \end{ a l i g n ∗} sub t bar to the power of 2 plus bar nabla u barvt(x, to0) the = powerv1( ofx) 2 = closingu1(x, y parenthesis) |Q, dxdy = integral sub Q Row 1(16) partialdiff u Row 2 partialdiff nu . u sub t dx period θ(x, 0) = θ0. (17) WeMultiplying conclude open this parenthesis section 7 by closing formally parenthesis showing by v sub that t and the integrating energy over of Q this and adding system the result to the integral d e c r e a s e s . \quad Formally , one can compute the energy of this structure as follows . of openWe concludeparenthesis this 8 closing section parenthesis by formally over showing Q yields that the energy of this system decreases . Formally , one can 2compute to the power the energy of 1 dt of to this the structure power of as d follows integral . sub Q open parenthesis bar v sub t bar to the power of 2 plus R bar v sub xx bar to the power \noindent After multiplying ( 4 ) by $ u { t }$ and integrating the result over $ \Omega $ one receives : of 2After plus psi multiplying theta sub ( 1 4 bar ) by vu subt and x barintegrating to the power the result of 2 over minus Ω betaone receives bar v sub : x bar to the power of 4 plus alpha bar v sub x bar to the power of 6 plus bar theta bar closing parenthesis dx Equation: open parenthesis 1 9 closing parenthesis .. = integral sub Q fv t dx plus k integral sub Q open parenthesis\ begin { a l theta i g n ∗} sub 0 minus theta closing parenthesis dx period 2 ˆ{ 1 } dt ˆ{ d }\ int {\OmegaZ } ( \mid uZ { t }\ mid ˆ{ 2 } + \mid \nabla u \mid ˆ{ 2 } This computation can be found in1 Sprekelsd and2 Zheng open2 parenthesis∂u 1 989 closing parenthesis period Next we add open parenthesis 1 8 closing ) dxdy = \ int { Q }\2 ldt e f t [ \(|beginut | +{|array ∇u | )}{dxdyc}\= partial utdx.u \\\ partial \nu \end(18){ array }\ right ] u { t } parenthesis Ω Q ∂ν dxand . open\ tag parenthesis∗{$ ( 1 1 9 closing 8 )parenthesis $} and see that the terms integral sub Q bracketleftbig sub partialdiff nu to the power of partialdiff u bracketrightbig\endMultiplying{ a l i g n ∗} u ( sub7 ) by t dxvt commaand integrating and integral over subQ and Q fv adding t dx cancel the result period to Wethe getintegral of ( 8 ) over Q yields 2 to the power of 1 dt to the power of d integral sub Capital Omega open parenthesis bar u sub t bar to the power of 2 plus bar nabla u bar to the power\noindent of 2 closingMultiplying parenthesis ( dxdyZ 7 ) Equation: by $ v open{ parenthesist }$ and 20 integrating closing parenthesis over .. plus $ Q2 to $ the and power adding of 1 dt the to the result power of to d integral the integral sub Q of ( 8 ) over $Q$1 d yields 2 2 2 4 6 open parenthesis bar v sub2 tdt bar to(| thevt | power+R | ofvxx 2| plus+ψθ R1 bar| vx v| sub−β xx| vx bar| + toα the| vx power| + | ofθ |) 2dx plus psi theta sub 1 bar v sub x bar to the power of 2 minus beta bar v sub x bar to theQ power of 4 plus alpha bar v sub x bar to the power of 6 plus bar theta bar closing parenthesis dx = k integral sub \ begin { a l i g n ∗} Z Z Q open parenthesis theta sub 0 minus theta closing parenthesis dx= periodfvtdx + k (θ0 − θ)dx. (19) 2The ˆ{ left1 } handdt side ˆ{ ofd this}\ equationint { isQ the} time( derivative\mid ofv t { tQ }\mid ˆQ{ 2 } + R \mid v { xx }\mid ˆ{ 2 } + \ psi \theta { 1 }\mid v { x }\mid ˆ{ 2 } − \beta \mid v { x }\mid ˆ{ 4 } + \alpha \mid v This{ x computation}\mid ˆ{ can6 be} found+ in\mid Sprekels\ andtheta Zheng\ (mid 1 989 )) . Next dx we\\ add= ( 1 8\ int ) and ({ 1Q 9 )} andfv see that t the dx + k \ int { Q } R ∂u R ( terms\thetaQ[∂ν ]{utdx,0 }and −Q fvtdx \thetacancel .) We get dx . \ tag ∗{$ ( 1 9 ) $} \end{ a l i g n ∗} Z 1 d 2 2 \noindent This computation can be found in Sprekels2 dt and(| Zhengut | + | ( ∇ 1u | 989)dxdy ) . Next we add ( 1 8 ) Ω and ( 1 9 ) and see thatZ the terms $ \ int { Q } [ ˆ{\ partial u } {\ partial \nu } ] u { t } dx 1 d 2 2 2 4 6 , $ and $ \ int {+2Q dt} fv(| vt | t+R dx$| vxx | + cancelψθ1 | vx .Weget| −β | vx | +α | vx | + | θ |)dx (20) Q \ begin { a l i g n ∗} Z = k (θ0 − θ)dx. 2 ˆ{ 1 } dt ˆ{ d }\ int {\Omega } ( \mid u { t }\midQ ˆ{ 2 } + \mid \nabla u \mid ˆ{ 2 } ) dxdy \\ + 2 ˆ{ 1 } dt ˆ{ d }\ int { Q } ( \mid v { t }\mid ˆ{ 2 } + R \mid v { xx } \midTheˆ{ left2 hand} + side of\ psi this equation\theta is the{ time1 }\ derivativemid ofv t { x }\mid ˆ{ 2 } − \beta \mid v { x }\mid ˆ{ 4 } + \alpha \mid v { x }\mid ˆ{ 6 } + \mid \theta \mid ) dx \ tag ∗{$ ( 20 ) $}\\ = k \ int { Q } ( \theta { 0 } − \theta ) dx . \end{ a l i g n ∗}

\noindent The left hand side of this equation is the time derivative of t 330 .. W period HORN comma J period SOKO L-suppress OWSKI \noindentSimilarly we330 integrate\quad byW parts . HORN over Capital, J . SOKO Omega\ toL theOWSKI power of minus to get : SimilarlyLine 1 minus we integral integrate sub 0 to by the partspower of over T integral $ \ subOmega Capitalˆ{ Omega − }$ minus to get Capital : Delta u phi dxdt = integral sub 0 to the power of T integral sub Capital Omega minus nabla u nabla phi dxdt minus integral sub 0 to the power of T integral sub partialdiff Capital Omega minus partialdiff to the\ [ \ powerbegin of{ a partialdiff l i g n e d } sub − n to\ int the powerˆ{ T of} u{ phi0 }\ d sigmaint dt Line{\Omega 2 = minus− integral } \Delta sub 0 to theu power\phi of T integraldxdt sub = Capital\ int Omegaˆ{ T minus} { 0 u } Capital\ int Delta{\Omega phi dxdt− plus } integral \nabla sub 0 tou the\ powernabla of T\ integralphi subdxdt partialdiff− Capital \ int Omegaˆ{ T } minus{ 0 u}\ sub partialdiffint {\ npartial to the power\ ofOmega partialdiff− } \ partial ˆ{\ partial }ˆ{ u } { n }\phi d \sigma −dt \\ phi d330 sigmaW dt . HORN minus , J integral . SOKOLOWSKI sub 0 toSimilarly the power we of integrate T integral by sub parts partialdiff over Ω Capitalto get : Omega minus partialdiff to the power of partialdiff sub n to the power= of− u phi \ dint sigmaˆ{ dtT period} { 0 }\ int {\Omega − } u \Delta \phi dxdt + \ int ˆ{ T } { 0 }\ int {\ partial \Omega − } u ˆ{\ partialZ T Z \phi } {\ZpartialT Z n } dZ T Z\sigma dt − \ int ˆ{ T } { 0 }\ int {\ partial As a result we have ∂ u \Omega − } \ partial −ˆ{\ partial∆uφdxdt}ˆ{=u } { n∇}\u∇φdxdtphi − d \sigma∂ nφdσdtdt . \end{ a l i g n e d }\ ] Line 1 minus integral sub 0 to the0 powerΩ− of T integral0 subΩ− Capital Omega Capital0 ∂Ω− Delta u phi dxdt = minus integral sub 0 to the power of T integral sub Capital Omega u Capital DeltaZ phiT Z dxdt minus integralZ T Z sub 0 to the powerZ T Z of T integral sub Q Row 1 partialdiff u Row 2 partialdiff n . Q phi d ∂φ ∂ u sigma dt Line 2 minus integral= sub− 0 to theu power∆φdxdt of+ T integral subu∂n Capitaldσdt − Gamma partialdiff∂ nφdσdt. to the power of partialdiff sub n to the power of u phi dsdt\noindent plus integralAs a sub result 0 to the we power have0 ofΩ T− integral sub Capital0 ∂Ω− Gamma partialdiff0 ∂Ω to− the power of partialdiff sub n to the power of phi u dsdt period BecauseAs a result of the we boundary have conditions on u and phi on Capital Gamma the last two terms vanish period \ [ \Thisbegin allows{ a l i us g n now e d } to − establish \ int theˆ{ weakT } form{ 0 of}\ the waveint equation{\Omega semicolon}\Delta u \phi dxdt = − \ int ˆ{ T } { 0 } \ int {\Omega } u \ZDeltaT Z \phi dxdtZ T Z − \ intZ Tˆ{Z T }∂u{ 0 }\ int { Q }\ l e f t [ \ begin { array }{ c}\ partial Line 1 integral sub 0 to the− power∆ ofuφdxdt T integral= − sub Capitalu∆φdxdt Omega− u open parenthesisQφdσdt phi tt minus Capital Delta phi closing parenthesis dxdt minusu \\\ integralpartial sub Capitaln \end Omega{ array u sub}\ 1right phi open]Q parenthesis\phi 0 closingd \ parenthesissigma∂n dt dx Line\\ 2 plus integral sub Capital Omega u sub 0 phi t open − \ int ˆ{ T } { 0 }\0 intΩ {\Gamma 0}\Ω partial ˆ{\0 partialQ }ˆ{ u } { n }\phi dsdt + \ int ˆ{ T } { 0 } parenthesis 0 closing parenthesis dx minus integral sub 0 toZ theT Z power of T integralZ T Z sub Q Row 1 partialdiff u Row 2 partialdiff n . Q phi d sigma dt \ int {\Gamma }\ partial ˆ{\ partial }ˆ{\phi }∂ u{ n } u dsdt∂ φ . \end{ a l i g n e d }\ ] = 0 − ∂ nφdsdt + ∂ nudsdt. for all phi in Y sub 1 period The term bracketleftbig sub0 partialdiffΓ n to the0 powerΓ of partialdiff u bracketrightbig Q can now be replaced by the balanceBecause of momen of the hyphen boundary conditions on u and φ on Γ the last two terms vanish . This allows us now to establish the \noindenttumweak for form shapeBecause of the memory wave of alloys equation the as boundary follows ; conditions on $ u $ and $ \phi $ on $ \Gamma $ the last two terms vanish . Thisv sub allows tt minus us open now parenthesis to establish sigma open the parenthesis weak form theta of comma the wave v sub equation x closing parenthesis ; closing parenthesis sub x plus Rv sub xxxx = Z T Z Z minus Row 1 partialdiff u Row 2 partialdiff n . Q comma \ [ \ begin { a l i g n e d }\ int ˆ{ T } { 0 }\ intu(φtt −{\∆φOmega)dxdt −} u1φ(0) (dx \phi t t − \Delta \phi ) dxdt − where the function sigma open parenthesis theta0 Ω comma v sub x closingΩ parenthesis is given in open parenthesis 9 closing parenthesis period After \ int {\Omega } u { 1 }\phi ( 0 ) dx \\ this open parenthesis 23 closing parenthesisZ becomes Z T Z  ∂u  +Equation:\ int open{\ parenthesisOmega } 23u closing{ +0 parenthesis}\u0φtphi(0)dx ..−t integral ( sub 0 0 toQφdσdt ) the power dx= 0 of− T integral \ int ˆ sub{ T Capital} { 0 Omega}\ uint open{ parenthesisQ }\ l e f phi t [ \ ttbegin minus{ array }{ c}\ partial Ω 0 Q ∂n Capitalu \\\ Deltapartial phi closingn parenthesis\end{ array dxdt}\ right minus]Q integral sub\phi Capitald Omega\sigma u sub 1dt phi open = parenthesis 0 \end{ 0a l closing i g n e d parenthesis}\ ] dx plus integral sub ∂u Capitalfor all Omegaφ ∈ Y u1 sub. The 0 phi term t open [∂n]Q parenthesiscan now be 0 replacedclosing parenthesis by the balance dx plus of integral momen sub- tum 0 to for the shape power memory of T integral alloys as sub Q open parenthesis v sub tt minusfollows open parenthesis sigma open parenthesis theta comma v sub x closing parenthesis closing parenthesis sub x plus Rv sub xxxx closing parenthesis phi\noindent d sigma dtf o = r 0 aperiod l l $ \phi \ in Y { 1 } . $ Theterm $ [ ˆ{\ partial u } {\ partial n } ] Q $ can now be replaced by the balance of momen − tum for shape memory alloys as follows  ∂u  In the last term we can again integratevtt by− parts(σ(θ, v tox)) getx + Rvxxxx = − Q, integral sub 0 to the power of T integral sub Q v sub tt phi d sigma dt∂n = integral sub 0 to the power of T integral sub Q v phi tt d sigma dt plus \ [ v { t t } − ( \sigma ( \theta , v { x } )) { x } + Rv { xxxx } = − \ l e f t [ \ begin { array }{ c}\ partial integralwhere sub the Q function v sub 0 phiσ(θ, t v open) is givenparenthesis in ( 9 0 ) .closing After parenthesisthis ( 23 ) becomes d sigma minus u \\\ partial n \endx { array }\ right ]Q, \ ]

Z T Z Z Z \noindent where the function $u(φtt\sigma− ∆φ)dxdt( − \thetau1φ(0)dx +,u v0φt{(0)xdx} ) $ is given in(23) ( 9 ) . After this ( 23 ) becomes 0 Ω Ω Ω Z T Z \ begin { a l i g n ∗} + (vtt − (σ(θ, vx))x + Rvxxxx)φdσdt = 0. \ int ˆ{ T } { 0 }\ int {\Omega0 }Q u ( \phi t t − \Delta \phi ) dxdt − \ int {\Omega } u { 1 }\phi ( 0 ) dx + \ int {\Omega } u { 0 }\phi t ( 0 ) dx \ tag ∗{$ ( 23 ) $}\\ + In the last term we can again integrate by parts to get \ int ˆ{ T } { 0 }\ int { Q } ( v { t t } − ( \sigma ( \theta , v { x } )) { x } + Rv { xxxx } ) \phi d \sigma dtZ T =Z 0 . Z T Z Z \end{ a l i g n ∗} vttφdσdt = vφttdσdt + v0φt(0)dσ− 0 Q 0 Q Q \noindent In the last term we can again integrate by parts to get

\ [ \ int ˆ{ T } { 0 }\ int { Q } v { t t }\phi d \sigma dt = \ int ˆ{ T } { 0 }\ int { Q } v \phi t t d \sigma dt + \ int { Q } v { 0 }\phi t ( 0 ) d \sigma − \ ] A model for passive damping of a membrane .... 337 \noindentHorn commaA model W period for and passive Soko l-suppress damping owski of comma a membrane J period ..\ h open f i l l parenthesis337 2 0 0 0 closing parenthesis Models for Adaptive Structures using \noindentShape MemoryHorn Actuators , W . and period Soko .. Procedings\ l owski of MTNS , J . 2000\quad comma( 2 Perpignan 0 0 0 ) open Models parenthesis for Adaptive elec hyphen Structures using Shapetronic Memory closing parenthesis Actuators period . \quad Procedings of MTNS 2000 , Perpignan ( elec − t rHorn o n i c comma ) . W period and Soko l-suppress owski comma J period open parenthesis 2002 closing parenthesis .. An elastic membrane .. with .. an .. atA hyphen model for passive damping of a membrane 337 \noindent Horn , W . and Soko \ l owski , J . ( 2002 ) \quad An elastic membrane \quad with \quad an \quad at − tachedHorn nonlinear, W . and thermoelasticSokolowski rod , J period . ( ..20 Applied 0 0 ) Models Mathematics for Adaptive and Computer Structures using Shape Memory Actuators tached nonlinear thermoelastic rod . \quad Applied Mathematics and Computer Science. Procedings 12 open of parenthesis MTNS 2000 4 closing, Perpignan parenthesis ( elec comma - tronic 479 ) . hyphen 487 period Science 12 ( 4 ) , 479 − 487 . KokotovHorn , comma W . and A periodSoko Yu lowski period , .. J and . ( Plamenevsky 2002 ) An comma elastic B membrane period A period with .. open an parenthesis at - tached 2 0 nonlinear 0 0 closing parenthesis On the Cauchy Kokotov , A . Yu . \quad and Plamenevsky , B . A . \quad ( 2 0 0 0 ) On the Cauchy − D i r i c h l e t hyphenthermoelastic Dirichlet rod . Applied Mathematics and Computer Science 12 ( 4 ) , 479 - 487 . Kokotov , A . Yu . Problem for Hyperbolic Systems in a wedge . \quad St . Petersburg Math . \quad J . 1 1 Problemand Plamenevsky for Hyperbolic , B Systems . A . in ( 2 a 0wedge 0 0 ) periodOn the .. CauchySt period - Dirichlet Petersburg Problem Math periodfor Hyperbolic .. J period Systems 1 1 in a wedge ( 3 ) , 497 −− 534 . open. St parenthesis . Petersburg 3 closing Math . parenthesis J . 1 1 comma( 3 ) , 497 endash– 534 . 534Kozlov period , V . A . , Maz ’ ya , V . G . and Rossmann Kozlov , V . A . , Maz ’ ya , V . G . and Rossmann , J . \quad ( 1 997 ) \quad Elliptic Boundary Kozlov, J . comma ( 1 997 V ) periodElliptic A period Boundary comma Value Maz quoterightProblems in ya Domains comma V with period Point G Singularities period and Rossmann. American comma Mathe J period .. open parenthesis 1 997 Value Problems in Domains with Point Singularities . \quad American Mathe − closing- parenthesis .. Elliptic Boundary Valuematical Problems Society in, Providence Domains with , R . Point I . Kozlov Singularities , V . A period . and ..Maz American ’ ya , VMathe . G . hyphen ( 1 999 ) Comparison Principles \noindent matical Society , Providence , R . I . maticalfor Nonlin Society - ear comma Operator Providence Differential comma Equations R period in Banach I period Spaces . Amer . Math . Soc . Transl . 189 , 1 49 Kozlov ,V . A . andMaz ’ ya ,V . G . ( 1 999 ) \quad Comparison Principles for Nonlin − Kozlov– 1 57 . commaM u¨ ller V period , I . Aand period Seelecke and Maz , S quoteright . ( 1 998 ya ) commaAdaptive V period air foil G period with shape open memory parenthesis 1 999 closing parenthesis .. Comparison ear Operator Differential Equationsst in Banach Spaces . Amer . Math . Soc . Principlesa − forht Nonlin− ll − t hyphenp−ay−colonslash−s w − one period − M − wd−ee−mts − ia−nperiod−g−uo − ni − fd − th − periodt−e−islash − T − tM − mr − R − slashR − me−e−esi−e−ta−ng−rcR − hm − Pra−oj−nineeight−eslash−ct−n Transl . \quad 189 , 1. 49 −− 1 57 . earo “ Operator hyphen − Differentialparenright − Equationsde − five in− one Banach Spaces period Amerm − periodseven Math periodse Soc Ta rnperiod M $ \ddot{u} $ ller , I . and Seeleckeeight−eight−period , S.eight . \−quadh ( 1(P 998−periodha ) \quad Adaptive air foil with shape memory C − rr−oy−vcomma−sta−S l l −Translperiod periodA − ine ..−period 189 commaa − So−n 1d 49− endashl i P − d 1s 57” l perioda − F e − M − mR − nX − evC − sT − ky − nine8 − zero − Bperiod − twotwo − Anine − period. period−s p amainM u-dieresiss w − llerih − commat P I i periode − c andew Seeleckes − i e S comma m o S- periodoh .. Bounda open parenthesisr − ie 1 998Wa closing l − terde parenthesisG c r .. u Adaptive yt r − e, air foil with shape memory \ hspacea-h t-l∗{\ l-t subf i l p-al } y-colon$ a−h to thet−l power l− oft slash-s{ p−a sub period y−colon w-one ˆ{ toslash the power−s } of{ st period-M-w. }}l w−one sub ˆ d-e{ s e-m t } t s-iperiod sub a-n−M period-g-u−w { d− o-ne i-f e d-t−m Spreke l − s J. a n d Z eng , S . ( 1989 ) G o − l a l Solu ns to t e E q f − o a Ginzb rg h-periodt } s− subi t-e-i{ a−, slash-T-tn period M-m− r-R-slashg−u } R-mo−n sub i e-e-e−f s i-e-td−tb a-n h g-r−period c R-h m-P{ subt−e r− a-ouai }− j-ninetislash ns to the−T− powert M of−m eight-e r−R slash-c−s l a s t-n h o quotedblleft R−m { e−e−e s i−e−t a−n g−r c } R−h m−P { r a−o j−nine ˆ{ eight −e slash −c t−n }}$ o ‘‘ $ hyphen−parenright −d hyphen-parenright-d– L anda u T h o e-five-one r − y for tsub u − eight-eight-periodr ctural P h a − s periodeiTra n eight-h i − si − m-sevenonsinS subape open M parenthesis P-period ha se Ta rn e−fC-r i v e sub−one r-o y-v{ toeight the power−eightory of− Acomma-speriod l o s . P t a-S hs . li l-period− c eighta A-i− Dh sub3} n,m e-period−nineseven− five a-S{ – sub(P 7 6 o-n m− d-lperiod i P-d sub s ha quotedblright}$ se Ta l a-F rn e-M-m R-n X-e sub v C-s T-k y-nineg , 8 hyphen S . ( 1995 zero-B ) period-twoNon i n−er two-A−a P nine-period a r−a bol subc−i periodEqua t−i onsa n dco upled H perb l−oc−hyphenP a− \ centerline { $ C−r { r−o y−v ˆ{ comma−s } t a−S }$ l $ l−period A−i { n e−period } a−S { o−n } arab subl mai− ic nSys s .. t w-i em h-tperiod .. P− i e-cs ew Lo s-i g mae S n.. Ho m o hyphen u − s e oh .. Bounda r-i e to the power of period-s Wa l-t erde to the power of p G c r u yt r-e commad−l $ i $ P−d { s }$ ’ ’ l $ a−Fo en−M− ,m B R− tnM K X−e { v } C−s T−k y−nine 8 − zero−B period−two twoSpreke−A l-s nine sub− commaperiod J period{ . } a$ n d} .. Z eng comma S period open parenthesis 1989 closing parenthesis .. G o-l sub b a l Solu .. ns to t e E Row 1 l Row 2 u a-t i ns . a \ centerlineGinzb rg endash{ $ La anda{ mai u T h o n r-y}$ for t s u-r\quad ctural P$ h w− a-si sub h e− it sub $ Tra\quad n i-s i-oP subi $nsinS e−c .. $ ape ew .. M $ s−i $ e S \quad m o − oh \quad Bounda $ rory−i A l eo s ˆ period{ period P hs− i-cs } a$ .. D Wa 3 comma $ l−t nine-five erde endash ˆ{ p 7 6}$ .. m G c r u yt $ r−e , $ } g comma .. S period open parenthesis 1995 closing parenthesis .. Non i n-e r-a P a r-a bol c-i .. Equa t-i onsa n dco upled H perb l-o c-hyphen sub Pa\noindent hyphen Spreke $ l−s { , }$ J . a n d \quad Zeng , S . ( 1989 ) \quad G $ o−l { b }$ a l Solu \quad ns to t e E $q\rabbegin l-i sub{ array c Sys}{ t emcc } period-sl \\ ..u Lo g a ma−t n Ho i .. & u-s e ns \end{ array } f−o$ a Ginzbo .. n ..rg comma−− LandauTho .. B .. t to the power $r of− My .. $ K f o r t $ u−r$ cturalPh $a−s { e } i { Tra }$ n $ i−s i−o { nsinS }$ \quad ape \quad M

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