HADAMARD MEANDER

Meander of Hadamard Family Matrices

The existance of Hadamard matrices H, it follows from existance of Euler matrices: they exists for all orders n =4k –2, if meander is true, the Hadamard conjecture* is theorem. Transitions between E-, M-, H-matrices illustrated by on line algorithm are placed below.

*) See the conjecture 1 Kotsireas, Koukouvinos, Seberry, 2005 also, it commented by big set of samples calculated by generalised Legendre pairs: Fletcher, Gysin, Seberry, 2001 .

MERSENNE MATRIX LEVEL b n=line(41); bH=one(n);
{{q=n+1; b=4*q}} b=sqrtm(b);
{{b=(q-b)./(q-4); bM=b; M=(n+1+(n-1).*b.*b)/2; M=n./M; }}
M=sqrtm(M); M[0]=1; M[1]=1; M[3]=2/sqrt(3); bM[3]=0.5;
Max=1.1; plotb(n,Max);
ex=zero(n); for (i=1;i<rows(n);i++)
if (i%4==0) ex[i]=1; ex[1]=2; ex[2]=1; plotb(n,Max,ex,bH,"#FF0000",'H');
ex=zero(n); for (i=1;i<rows(n);i++)
if (i%4==3) ex[i]=1; ex[3]=2; plotb(n,Max,ex,bM,"#00DD00",'M');
function plotb(n,Max,ex,A,col,name) {
var i,j,W,H,X,Y,x,x1,x2,y,y1,y2,y3,Wx,Wy,w,N,F,F2,bgcol,xycol;
W=530; H=300; X=40; Y=30; w=3; ws=10; // y1=0.2; // UP 1
bgcol="#000000"; xycol="#EEEEEE"; N=rows(n); Wx=W-2*X; Wy=H-2*Y;
OpenCanvas('S',W,H); with (S) {
if (arguments.length==2) { S.clear(); setColor(xycol);
for (i=1;i<N;i++) fillRect(X+Wx*i/N,Y,w,Wy); y=10*Max; y2=Wy/y;
for (j=1;j<y;j++) fillRect(X,H-Y-y2*j,Wx,w);
setColor("#FF0000"); fillRect(X,H-Y-10*y2,Wx,w);
setColor(bgcol); fillRect(X,Y,w,Wy); fillRect(X,H-Y,Wx,w);
setFont('NewTimeRoman',20,1);
for (j=1;j<10;j++) drawString('0.'+j,X-30,H-Y-y2*j-11);
for (j=10;j<y;j++) drawString('1.'+(j-10),X-30,H-Y-y2*j-11);
for (i=0;i<N;i++) { j=3; if (i>9) j=8; if (i==N-1) {
setFont('NewTimeRoman',22,1); drawString('n',X+Wx*i/N-j+5,H-Y-3);
}else{ if (i%2==0) drawString(i,X+Wx*i/N-j,H-Y);}}
drawString('b',X-25,H-Y-10*y2-40);
}else{
X=X-(ws-w)/2; Y=H-Y; y1=y1; y2=Wy/Max; F=false; F2=false;
for (i=1;i<N;i++) { x=X+Wx*i/N; y3=y2*A[i]; y=Y-y3;
if (ex[i]>0) { F=true; setColor("#444444");
fillRect(x-2,y,ws,y3); setColor(col); fillRect(x,y,ws,y3);
if (arguments.length==6) if (i>1)
if (ex[i]>0) drawString(name,x-6,y-30); }
// if (F2) if (i>1) for (j=0;j<w;j++) drawLine(x1,x2+j,x,y+j);
if (F) F2=true; x1=x; x2=y;
}}
paint();
}}
b =(q –(4q )^{1/2} )/(q –4), q =n +1, n =4k –1.

EULER MATRIX LEVEL b

b =(q –(8q )^{1/2} )/(q –8), q =n +2, n =4k –2.

DETERMINANT GRAPH D=1/m ^{n} =n ^{n/2} /h ^{n}

Hadamard H, Belevitch C, Euler E, Mersenne M, Fermat F matrices, Cyan is Proclus.