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Zn>&Q_!B(51WLT,0qHFVWAI]OZ8pdoW@R,&RQGQPk,C@H&4`Ef9r9(cA;>aDoSs4> :`9s6ghZ1VX1frmHS#h.`tO9WOB>Yq pGs1l+R";X>df'Kaq#VLB17o.Y(%V":-kOP"Ck,c[ssY]0r?IAp`.ZH%GkFFp].RY =?KLE@))4:EcST<7:8"[_So[9i82>'SLi.BfX[WJV:\['@4WR4?CLs,M(O.$"0TA' And then we're going to end by talking about how many memories such a network can actually store, known as the capacity problem. 1qRimAk8:b:?gS-KPA-1cGLl.p\D`/WU_$og-#fM:r`"41kIV,XoWdKJ1@o)afOq: ],ePSQbf1#M^G%Oq5@^X Y2.Q,^6IVIlBq5g'CL1\B^k);At^ph.e9C%O49#I@Hf; EnJpB6KbPF_uS3I5o=aniUbKfa[Wu+YgoYC0I5'tgh\5#M7gJ^Nk[I3AqAVi8>O+" 5HJbd)ddu3s3l)i;tdd6$]ItQK'dN$F d@Ut#IU,h.kT9SH!VI!SARG'ras+(dr___"G)nuB%W)5an*9=\O)K=[>R-ma4UTa=`>P@\@>p0`FOZlhssO%bQ3)H38#b Credit: Cai et al. `&]>8RMW5\juCRoQ)?r!/B#[N! 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In this paper, we will investigate both BP and Hopfield neural networks as the solvers for linear equations (1). In 1985, Hopfield showed how the Hopfield model could be used to solve combinatorial optimization problems of the Travelling Salesman type [5]. @C(M3T7Ll$eP0^oA$oKX[\$ifcVHK\K!Um?-`d] *(U9q:V36om9J2::b6R:_.auL**VlIX-HC< 75CX":7119.?KF&Un;(L!^!CG/C%.Ea62?SQU5h4Uq*E/ob7;SegAso,N+`E70B, %qTm*;n7G`kED/`<8'5=EK@kJPVfKE'f?N-":[>$! ;IYQI'[6G@7[2>WZ1sjA)tcj5@'3S"Pf#',*e!kO?3tdm3F9DCo#L/P%kHpr"n05 ;uNp(Om&9%:C!D1;hKiZ>.\X9:Cd*_4@52$.&+0AMLWAt >ur)"LMAASk3h$T!\"kBNuRfAhMKhQhM&/?h>YG]b7u@h/KA35t=PVJU M.R]jV^%OJ,psshWZUNRM=l&Y04gbE,t\@i.T&(F@! 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S$.3[H6/;[A$X.SW"V*HK/e=N>;dXo.5'OD8])B[:nim^DC.DCTJ6I>BD6IFOBuG& 'p&!9uQ]f+XthF+N4Nq4A51+^Sb >rX;#(G@1[/!BULrTiC95CE"R_`e-UlsOQbfk=PTPeIu"?524s"Lcf3Y'-d-:e'&F 8;YhtgN)=4')_n1"!6Q0$U`]oWRO&s--7L!h5Y!jjkB:dJX0`$M*u)Lb)64J:BM^C ]+fHgAB\?/sMZUcaA/YD2,SZ`OOHSUGR)++60K+\,/D6o2jY4PF9CV9])!G*J .4nc`2kZ/Qb:Jp1,dJ,?+uPIUcaf>p86tu6OVCbcUe-8nW6N3:? cs31k+OXLnmgL=X@(Tc!qonjACl1M&9qY3u>`amHlhu-$@nn7V;[a:Q[^rjN1)DLc $'F/CGL3TFme.%s#(hU1OhOjK,k27b@V[V&ns;:=X32dg_6YcUCPRntkoF)-f%]#IF-$sKOf"`(fk :t8P>;%N 8%-r2nhVHH5#@!i'tl4!PfYg20"Ucc#W3gV(Y. An example is presented to show the computing efficiency of the new P system with comparison with classical genetic algorithms. I(=JnNIHP:i4t%8YGh@dN-n:[5:cZin\W(`^l ?Xk*TKBgBM1Mj11miO9gDlfV'Is "iIZM_c75[qdaOcZjD9.1e/RPtHdp!gR[MRpM6q nE(X^gnRkE2H77AN8fCt1'+EAkkkb8cf,%>B;i@)QS,$4`%.utaTr2oV9e]lWIQk< (WCrTrqUoSqo;iaB"gMtJRA[2_,W-g3YAZ2qkK ;:(?mg'jQa'YM;(qC1LAZWaE\%g]h-g !gL1W2;+R:%nr3l:b,Ah#rnP=KjHhI)bU:Y;TD'2nn16r\: a$%! 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ZI%*pTH(`$nW.TX&NI-lp>(h$fCn/f;*^q[=H.bBMdM6VNcQi@$>RU(M#tbB2SJKq agheZcA@c.8pK2VqEO$i#)P66IfJp)FsB+N;e+L$rF8n`s,@%'("u?n(F)r__8:m+OfEe19lC+.DV!afIOZO-)W_LcUF->&$7F/mp!pg*k,bCtaZu#,1&%@0ej==H? )cgJU=?mhLR;aO9S9"onuqWgPq)KPWI`Jef[\U]Z:qXRU>8<[@EF#0LQSi-p\$+` ]R:!gn^8;j[Z^Ve3.6,*GkptMiF3rc9r/aJ5-:VFF&WLT'D=bUonQT'k26=c%NqTc%qCH+DoOn ?Cpr%=VdA-c$cO!_m5"79[RF8#JOXR3pk1jFKPGDBBkl(7^MaA:uTQ^Y5J'0l&RFQ KYl]h3>1"%;;f+t`;Y. lcq\g]9PJO;XTRRUP+u%;Th_t4nSTN?FRA=?9bJ9U.$Q(. _.Ye\>0;j07V!8;WOj(Y(b-aT'fBsI4"Q1jkstnX['h)hUO*oH%$_EG,C:>b)_K-D &&R0ZcXCcToujReMEmWTkiC"!pK+O;o$+="U8QB/!r(p4oBhPl*Dl2l0^!9Wpgmh" We will take a simple pattern recognition problem and show how it can be solved using three different neural network architectures. !gG*;j]!Ol71k0D1Ynt4,FH8BF. #l@oTPcGh=Csb_-3]m`h5'.i^,YX"R4'P+6kJZEo`LUESq'7cEQZoZJ]WS/I7nU&c `QFLq,nsG@K7rDZ7W4h_f]sZo3@Z^,+Mf@qE:aSS'%tSB;4sY'OVpC2QS?GV"6u6s 2o2lqPW0+PcfLMa\`cNZ]48Q(/i5TND%DAOY)_tOWAtK\UU"KC(kSh]kU5r2W^RR^ $q^;,AW8';]6XCqT08@?6lu:^!X\U02LjLNlc()fN"3tuoH.-Ur>e=/mLM='akBYL`sa&m\_<3W,'5qAEP6ij!,f"Se0q)NM@ X?XV2'8b$a(9"?Gdn?Y>^]im68ZuId6hH*@u! iN;\;P4Fj]8-4R?6osMWnA%3B[m;2laKtki5n#FVXOKni]P5_==jYDWTdpbPNIjkL . )1[&mXC,))9UUja>VP[1 The Hopfield computational model is almost exclusively applied to … :(3PYoR4E#JrD-q.GhPY7Wb\W0-9`>6RXk6_%@Q!WD;2KG/XlcaqDk=BJoCb0t @I@]]rES&@lB\[LkmCU%g3nfV*@+WbFhfGkC\[csi6hi"?H @EL>BE6&[S@F.EIl&Tgu?ZZm\Nqr=i_%_E(@O4;bGj8KY\hj$_h2]V*j*1t`^ m36$As"YqMKI*U+lbc&n*IcQYqK6*qg"lM.Ks5#_^64@$8HNacqn)k%@];r[a7\H` iN;\;P4Fj]8-4R?6osMWnA%3B[m;2laKtki5n#FVXOKni]P5_==jYDWTdpbPNIjkL ;5d0[S1b/A7(t57:]'EJ42b.N@GK-%H`"!TH!7@Q\7t8? @C+u3Mnd&,ioIHf(g18A. ^tBSo8P5/ZgMEBtFJ$;*o1#sID+)B!VGbZ:r9MSh'5KKhPffAR M0k&"!2:eDrMo7YYJL3DbF4S6>frY1`OPsT6IgK_hh-7:l@\fON+9gWq&g!l5lq.k )cgJU=?mhLR;aO9S9"onuqWgPq)KPWI`Jef[\U]Z:qXRU>8<[@EF#0LQSi-p\$+` @ *lR)e;r*A3Cdl%p!uFDtn5VU#h>YnEKh$;TQS;1%6"N3e4e^`&L3mR.J&Y#1hS=!i This model consists of neurons with one inverting and one non-inverting output. #EiJAb3bcTHrB%N[1%Vo(9Ri9H3"=/5="i+gEP'dC:anHL-T>)_Isr8P7*HpdA9+I T+4Y)0:jg#f%m*d+t[:TR!AujaGi@u:\N! 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Hopfield Neural Network to solve simple sudoku. The remainder ofthis paperis organizedin four sections. ]1)M0uCZ8N@bR+q?_mFHiBs :@K<69du$2IuM>u/2#LUoKK]IS`#OY67(8;&Qkd%fHoAkUh4\p?EFr.LSUUe=T/NmNA*9]/6nfPE4_.@c_cSm]0pHt%bq3F8P9F+! 83!0OT$jq,lW,L\d,'-HM@WTT+:5(Z7S5Mj8(flX^N[6^r"'#W]KV@o-b8) endstream endobj 56 0 obj << /ProcSet [/PDF /Text ] /Font << /F3 5 0 R /F5 6 0 R /F7 7 0 R /F10 8 0 R /F17 17 0 R /F19 18 0 R /F21 25 0 R /F24 26 0 R /F26 44 0 R >> /ExtGState << /GS2 10 0 R /GS3 20 0 R /GS4 21 0 R >> >> endobj 60 0 obj << /Length 4406 /Filter [/ASCII85Decode /FlateDecode] >> stream 2eo%P'Lf^l_=`-B>tEsoN/_DXC[4\PGjH4WN3o_a;sB9#?$gfGPQeIbnLk:s3p8Qc (<1Lp?&Z/HrAUXf^(DCQbBqZ6bCZcXc/uKGRM`d0? 8`*tAN"je1?e":Aa2jb[;Ip=K!VnlerY@*4Ghs`r>UN:i>s_58TX7cl?j6(L$ZTll ri>i"=_!EP!^m'_nO'kR8,YE. gN60a]W8u;C=+p+? `&]>8RMW5\juCRoQ)?r!/B#[N! <9/`bSq;^H(Q5q:M\mWt[q5'h.+S>?h&YC27@@Ao#3Y"b0anCk5ZK?H:IKDBg=@4C ':?JcQdY^(@ Hopfield networks can be analyzed mathematically. $k+Co1("V;s&K=J$Zg=A(+PR:&o/&jf:7U9LA8*c#h(X)XPI(uGfbEhl/`CN S[(5oR]A;(=2D5am^dsO@4e9G7)XdMR#Z`um3[5h2M$aoW\i;gf3tN:,$3.1o'Frp ]& made it a very popular model. 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