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<p class="MsoPlainText"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">Hi,<o:p></o:p></span></span></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><o:p> </o:p></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">The solution returned by
<span class="SpellE">fsolve</span> seems to be a valid one because:<o:p></o:p></span></span></font></p>
<p class="MsoPlainText"><font size="2" color="#0070c0" face="Times New Roman"><span style="font-size:11.0pt;font-family:"Times New Roman",serif;color:#0070C0"><span style="mso-bidi-font-size:10.5pt">-->Q([1.496447;0.9574371 ],0)<o:p></o:p></span></span></font></p>
<p class="MsoPlainText"><font size="2" color="#0070c0" face="Times New Roman"><span style="font-size:11.0pt;font-family:"Times New Roman",serif;color:#0070C0"><span style="mso-bidi-font-size:10.5pt"><span class="SpellE">ans</span><span style="mso-spacerun:yes">
</span>= <o:p></o:p></span></span></font></p>
<p class="MsoPlainText"><font size="2" color="#0070c0" face="Times New Roman"><span style="font-size:11.0pt;font-family:"Times New Roman",serif;color:#0070C0"><span style="mso-bidi-font-size:10.5pt"><span style="mso-spacerun:yes"> </span>- 0.0000002<span style="mso-spacerun:yes">
</span><o:p></o:p></span></span></font></p>
<p class="MsoPlainText"><font size="2" color="#0070c0" face="Times New Roman"><span style="font-size:11.0pt;font-family:"Times New Roman",serif;color:#0070C0"><span style="mso-bidi-font-size:10.5pt"><span style="mso-spacerun:yes"> </span>- 0.0000001<span style="mso-spacerun:yes">
</span><o:p></o:p></span></span></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><o:p> </o:p></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">You say that the initial guess of [1.5;-0.9] is close to the expected values, however:<o:p></o:p></span></span></font></p>
<p class="MsoPlainText"><font size="2" color="#0070c0" face="Times New Roman"><span style="font-size:11.0pt;font-family:"Times New Roman",serif;color:#0070C0"><span style="mso-bidi-font-size:10.5pt">-->Q([1.5;-0.9],0)<o:p></o:p></span></span></font></p>
<p class="MsoPlainText"><font size="2" color="#0070c0" face="Times New Roman"><span style="font-size:11.0pt;font-family:"Times New Roman",serif;color:#0070C0"><span style="mso-bidi-font-size:10.5pt"><span class="SpellE">ans</span><span style="mso-spacerun:yes">
</span>= <o:p></o:p></span></span></font></p>
<p class="MsoPlainText"><font size="2" color="#0070c0" face="Times New Roman"><span style="font-size:11.0pt;font-family:"Times New Roman",serif;color:#0070C0"><span style="mso-bidi-font-size:10.5pt"><span style="mso-spacerun:yes"> </span>- 7.54<span style="mso-spacerun:yes">
</span><o:p></o:p></span></span></font></p>
<p class="MsoPlainText"><font size="2" color="#0070c0" face="Times New Roman"><span style="font-size:11.0pt;font-family:"Times New Roman",serif;color:#0070C0"><span style="mso-bidi-font-size:10.5pt"><span style="mso-spacerun:yes"> </span>- 0.077934<span style="mso-spacerun:yes">
</span><o:p></o:p></span></span></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">is far from being zero. So there is probably a typo in your equations.<o:p></o:p></span></span></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><o:p> </o:p></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">PS:<o:p></o:p></span></span></font></p>
<p class="MsoPlainText"><i style="mso-bidi-font-style:normal"><font size="2" face="Calibri"><span style="font-size:10.0pt;mso-bidi-font-size:10.5pt;font-style:italic;mso-bidi-font-style:normal">Another trivial solution is:<o:p></o:p></span></font></i></p>
<p class="MsoPlainText"><font size="2" color="#0070c0" face="Times New Roman"><span style="font-size:10.0pt;mso-bidi-font-size:10.5pt;font-family:"Times New Roman",serif;color:#0070C0">-->Q([-0.9574371,-1.496447],0)<o:p></o:p></span></font></p>
<p class="MsoPlainText"><font size="2" color="#0070c0" face="Times New Roman"><span style="font-size:10.0pt;mso-bidi-font-size:10.5pt;font-family:"Times New Roman",serif;color:#0070C0"><span class="SpellE">ans</span><span style="mso-spacerun:yes">
</span>=<o:p></o:p></span></font></p>
<p class="MsoPlainText"><font size="2" color="#0070c0" face="Times New Roman"><span style="font-size:10.0pt;mso-bidi-font-size:10.5pt;font-family:"Times New Roman",serif;color:#0070C0"><span style="mso-spacerun:yes">
</span>- 0.0000002<span style="mso-spacerun:yes"> </span><o:p></o:p></span></font></p>
<p class="MsoPlainText"><font size="2" color="#0070c0" face="Times New Roman"><span style="font-size:10.0pt;mso-bidi-font-size:10.5pt;font-family:"Times New Roman",serif;color:#0070C0"><span style="mso-spacerun:yes"> </span>- 0.0000001<span style="mso-spacerun:yes">
</span><o:p></o:p></span></font></p>
<p class="MsoPlainText"><i style="mso-bidi-font-style:normal"><font size="2" face="Calibri"><span style="font-size:10.0pt;mso-bidi-font-size:10.5pt;font-style:italic;mso-bidi-font-style:normal">Such non-linear problems should have multiple solutions, including
complex ones.<o:p></o:p></span></font></i></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><o:p> </o:p></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">Regards,<o:p></o:p></span></span></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">Rafael<o:p></o:p></span></span></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><o:p> </o:p></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">-----Original Message-----<br>
From: users [mailto:users-bounces@lists.scilab.org] On Behalf Of Hermes<br>
Sent: Sunday, September 17, 2017 8:25 AM<br>
To: users@lists.scilab.org<br>
Subject: Re: [Scilab-users] solve a system of equations in <span class="SpellE">scilab</span></span></span></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><o:p> </o:p></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">Hello,<o:p></o:p></span></span></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">Please I need your comments.<o:p></o:p></span></span></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">I have tried to solve the system with different initial conditions for<o:p></o:p></span></span></font></p>
<p class="MsoPlainText"><span class="SpellE"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">fsolve</span></span></font></span> and I always get the same wrong result for the second variable.<o:p></o:p></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><o:p> </o:p></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">function Sys=Q(<span class="SpellE">x,z</span>)<o:p></o:p></span></span></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">Sys(1)=(x(1)-2)^2+(x(2)+2)^2+z^2-9;<o:p></o:p></span></span></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">Sys(2)=x(1)^6+x(2)^6+z^6-12;<o:p></o:p></span></span></font></p>
<p class="MsoPlainText"><span class="SpellE"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">endfunction</span></span></font></span><o:p></o:p></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">z=0;<o:p></o:p></span></span></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">[<span class="SpellE">j,v,info</span>]=<span class="SpellE">fsolve</span>([*1.5;-0.9*],list(<span class="SpellE">Q,z</span>))<o:p></o:p></span></span></font></p>
<p class="MsoPlainText"><span class="SpellE"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">disp</span></span></font></span>(<span class="SpellE">j,v,info</span>)<o:p></o:p></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><o:p> </o:p></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">Even when putting these initial values, very similar to the expected values,<o:p></o:p></span></span></font></p>
<p class="MsoPlainText"><span class="SpellE"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">fsolve</span></span></font></span> returns
<o:p></o:p></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">1.496447<o:p></o:p></span></span></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">0.9574371 the second positive value<o:p></o:p></span></span></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><o:p> </o:p></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">By using these values as initial values in this script:<o:p></o:p></span></span></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><o:p> </o:p></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">//<span class="SpellE">Draghilev's</span> Method<o:p></o:p></span></span></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">clear;<o:p></o:p></span></span></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">function Sys=Q(<span class="SpellE">x,z</span>)<o:p></o:p></span></span></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">Sys(1)=(x(1)-2)^2+(x(2)+2)^2+z^2-9;<o:p></o:p></span></span></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">Sys(2)=x(1)^6+x(2)^6+z^6-12;<o:p></o:p></span></span></font></p>
<p class="MsoPlainText"><span class="SpellE"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">endfunction</span></span></font></span><o:p></o:p></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">z=0;<o:p></o:p></span></span></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">[<span class="SpellE">j,v,info</span>]=<span class="SpellE">fsolve</span>([1.5;-0.9],list(<span class="SpellE">Q,z</span>))<o:p></o:p></span></span></font></p>
<p class="MsoPlainText"><span class="SpellE"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">disp</span></span></font></span>(<span class="SpellE">j,v,info</span>)<o:p></o:p></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><o:p> </o:p></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">N=100;<o:p></o:p></span></span></font></p>
<p class="MsoPlainText"><span class="SpellE"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">smin</span></span></font></span>=0.0;<o:p></o:p></p>
<p class="MsoPlainText"><span class="SpellE"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">smax</span></span></font></span>=0.046;<o:p></o:p></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">h=0.001;<o:p></o:p></span></span></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><o:p> </o:p></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><o:p> </o:p></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">x01=1.496447;<o:p></o:p></span></span></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">//3.9691163496*10^-12;<o:p></o:p></span></span></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">x02=0.9574371;<o:p></o:p></span></span></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">//0.353580783;<o:p></o:p></span></span></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">x03=0;<o:p></o:p></span></span></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">//-1.5130442808;<o:p></o:p></span></span></font></p>
<p class="MsoPlainText"><span class="SpellE"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">ics</span></span></font></span>=[x01; x02; x03];<o:p></o:p></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><o:p> </o:p></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">disp(["x(3)^2+(x(2)-2)^2+(x(1)-2)^2-9";"x(3)^6+x(2)^6+x(1)^6-12"],"systema")<o:p></o:p></span></span></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><o:p> </o:p></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><o:p> </o:p></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">disp(["2*(x(1)-2),2*(x(2)-2),2*x(3)";"6*x(1)^5,6*x(2)^5,6*x(3)^5"],"jacobian(Vm,[x(1),x(2),x(3)])");<o:p></o:p></span></span></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">//<o:p></o:p></span></span></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><o:p> </o:p></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">//D1=12*x(2)^5*x(3)-12*(x(2)-2)*x(3)^5;<o:p></o:p></span></span></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">//D2=12*(x(1)-2)*x(3)^5-12*x(1)^5*x(3);<o:p></o:p></span></span></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">//DD=12*x(1)^5*(x(2)-2)-12*(x(1)-2)*x(2)^5;<o:p></o:p></span></span></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">function
<span class="SpellE">dydt</span>=odes(<span class="SpellE">t,x</span>)<o:p></o:p></span></span></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">//<span class="SpellE">dydt</span>=zeros(x);<o:p></o:p></span></span></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><o:p> </o:p></font></p>
<p class="MsoPlainText"><span class="SpellE"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">dydt</span></span></font></span>(1)=-12*x(2)^5*x(3)-12*(x(2)-2)*x(3)^5;
<o:p></o:p></p>
<p class="MsoPlainText"><span class="SpellE"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">dydt</span></span></font></span>(2)=12*(x(1)-2)*x(3)^5-12*x(1)^5*x(3);<o:p></o:p></p>
<p class="MsoPlainText"><span class="SpellE"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">dydt</span></span></font></span>(3)=12*x(1)^5*(x(2)-2)-12*(x(1)-2)*x(2)^5;<o:p></o:p></p>
<p class="MsoPlainText"><span class="SpellE"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">endfunction</span></span></font></span><o:p></o:p></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">step=<span class="SpellE">smax</span>/N;<o:p></o:p></span></span></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">t=[<span class="SpellE">smin:step:smax</span>*N];<o:p></o:p></span></span></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">t0=0<o:p></o:p></span></span></font></p>
<p class="MsoPlainText"><span class="SpellE"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">atol</span></span></font></span>=h/100000;
<o:p></o:p></p>
<p class="MsoPlainText"><span class="SpellE"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">rtol</span></span></font></span>=h/100000;<o:p></o:p></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">LL= ode("<span class="SpellE">rk</span>", ics,t0, t',<span class="SpellE">atol,rtol,odes</span>)<o:p></o:p></span></span></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">LL=LL'<o:p></o:p></span></span></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">a=<span class="SpellE">gca</span>();//get the handle of the newly created axes<o:p></o:p></span></span></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">param3d1(LL(:,3),LL(:,2),LL(:,1),alpha=35,theta=45,"X@Y@Z",flag=[2,2])<o:p></o:p></span></span></font></p>
<p class="MsoPlainText"><span class="SpellE"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">a.rotation_angles</span></span></font></span>=[65,75];<o:p></o:p></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><o:p> </o:p></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">The calculations have taken a long time without converging to the solution.<o:p></o:p></span></span></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">I have interrupted the execution of the same.<o:p></o:p></span></span></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">By entering the correct values, almost instantaneously, you get the<o:p></o:p></span></span></font></p>
<p class="MsoPlainText"><span class="SpellE"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">soclucion</span></span></font></span> of the system.<o:p></o:p></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">Help Me!!!<o:p></o:p></span></span></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">gracias<o:p></o:p></span></span></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><o:p> </o:p></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><o:p> </o:p></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><o:p> </o:p></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><o:p> </o:p></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><o:p> </o:p></font></p>
<p class="MsoPlainText"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">--<o:p></o:p></span></span></font></p>
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<p class="MsoPlainText"><font size="2" face="Calibri"><span style="font-size:11.0pt"><span style="mso-bidi-font-size:10.5pt">_______________________________________________<o:p></o:p></span></span></font></p>
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