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| Deletions are marked like this. | Additions are marked like this. |
| Line 7: | Line 7: |
| {{{ | {{{#!sagecell |
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| {{{ | {{{#!sagecell |
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| g += line([(j*2-len(cur),-i), ((k*2)-len(rows[i-1]),-i+1)], | g += line([(j*2-len(cur),-i), ((k*2)-len(rows[i-1]),-i+1)], |
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| {{{ | {{{#!sagecell |
| Line 86: | Line 86: |
| {{{ | {{{#!sagecell |
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| == Prime Spiral - Square == | == Prime Spiral - Square FIXME == |
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| {{{ | {{{#!sagecell |
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| REFERENCES: | REFERENCES: |
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| Weisstein, Eric W. "Prime-Generating Polynomial." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Prime-GeneratingPolynomial.html | Weisstein, Eric W. "Prime-Generating Polynomial." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Prime-GeneratingPolynomial.html |
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| elif y<0 and -x >= y and y<x: return 4*(y+1)^2 -11*(y+1) + (start+7) +x | elif y<0 and -x >= y and y<x: return 4*(y+1)^2 -11*(y+1) + (start+7) +x |
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| #Takes in an n and the start value of the spiral and gives its (x,y) coordinate | #Takes in an n and the start value of the spiral and gives its (x,y) coordinate |
| Line 119: | Line 119: |
| num = num - start +1 | num = num - start +1 |
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| top = ceil(sqrt(num)) | top = ceil(sqrt(num)) |
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| else: | else: |
| Line 133: | Line 133: |
| else: | else: |
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| N = M.copy() | N = copy(M) |
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| #These functions return an int based on where the t is located in the spiral |
#These functions return an int based on where the t is located in the spiral |
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| if n !=0: x_cord, y_cord = find_xy(n, start) #Overrides the user given x and y coordinates | if n !=0: x_cord, y_cord = find_xy(n, start) #Overrides the user given x and y coordinates |
| Line 170: | Line 170: |
| Line 173: | Line 173: |
| if show_lines: for t in [(-size-1)..size+1]: |
if show_lines: for t in [(-size-1)..size+1]: |
| Line 176: | Line 176: |
| if m.is_pseudoprime(): main_list.add(m) | if m.is_pseudoprime(): main_list.add(m) |
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| #This for loop changes the matrix by spiraling out from the center and changing each entry as it goes. It is faster than the find_xy function above. | #This for loop changes the matrix by spiraling out from the center and changing each entry as it goes. It is faster than the find_xy function above. |
| Line 186: | Line 186: |
| else: x-=1 | else: x-=1 |
| Line 188: | Line 188: |
| elif county < overcount: |
elif county < overcount: |
| Line 191: | Line 191: |
| else: y-=1 | else: y-=1 |
| Line 193: | Line 193: |
| else: | else: |
| Line 199: | Line 199: |
| if not invert and num in main_list: |
if not invert and num in main_list: |
| Line 207: | Line 207: |
| if n != 0: |
if n != 0: |
| Line 224: | Line 224: |
| else: | else: |
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| {{{ @interact def polar_prime_spiral(start=1, end=2000, show_factors = false, highlight_primes = false, show_curves=true, n = 0): #For more information about the factors in the spiral, visit http://www.dcs.gla.ac.uk/~jhw/spirals/index.html by John Williamson. |
{{{#!sagecell @interact def polar_prime_spiral(start=1, end=2000, show_factors = false, highlight_primes = false, show_curves=true, n = 0): #For more information about the factors in the spiral, visit http://www.dcs.gla.ac.uk/~jhw/spirals/index.html by John Williamson. |
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| R = points(list2, alpha = .1) #Faded Composites else: |
R = points(list2, alpha = .1) #Faded Composites else: |
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| R=points(list2, hue = .1, pointsize = p_size) |
R=points(list2, hue = .1, pointsize = p_size) |
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| Q = plot(W1+W2+W3+W4, alpha = .1) | Q = plot(W1+W2+W3+W4, alpha = .1) |
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| if show_curves: | if show_curves: |
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| if n > (floor(sqrt(n)))^2 and n <= (floor(sqrt(n)))^2 + floor(sqrt(n)): | if n > (floor(sqrt(n)))^2 and n <= (floor(sqrt(n)))^2 + floor(sqrt(n)): |
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| else: | else: |
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| def g(m): return (a*m^2+b*m+c); | def g(m): return (a*m^2+b*m+c); |
| Line 292: | Line 292: |
| c= c2; | c= c2; |
| Line 307: | Line 307: |
| {{{ j = 0 @interact def _(N=[1..100], k=selector([2,4,..,12],nrows=1), prec=(3..40), |
{{{#!sagecell @interact def _(N=[1..100], k=selector([2,4,..,12],nrows=1), prec=(3..40), |
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| print j; global j; j += 1 | |
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| {{{ | {{{#!sagecell |
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| {{{ | {{{#!sagecell |
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| G = Graph(T, multiedges=True, loops=not three_d) | G = DiGraph(T, multiedges=not three_d) if three_d: G.remove_loops() |
| Line 364: | Line 364: |
| == Quadratic Residue Table == | == Quadratic Residue Table FIXME == |
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| {{{ | {{{#!sagecell |
| Line 417: | Line 417: |
| == Cubic Residue Table == | == Cubic Residue Table FIXME == |
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| {{{ | {{{#!sagecell |
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| if Mod(a,3)!=0 and Mod(b,3)==0: return True else: return False |
return Mod(a,3)!=0 and Mod(b,3)==0 |
| Line 475: | Line 472: |
| MP += line([(i,0),(i,r)], rgbcolor='black') | MP += line([(i,0),(i,r)], rgbcolor='black') |
| Line 503: | Line 500: |
| {{{ | {{{#!sagecell |
| Line 552: | Line 549: |
| S = circle((0,0),1,rgbcolor='yellow') \ + line([e_pt,e_gs_pt], rgbcolor='red', thickness=4) \ + line([f_pt,f_gs_pt], rgbcolor='blue', thickness=3) \ + line([ef_pt,ef_gs_pt], rgbcolor='purple',thickness=2) \ + point(e_pt,pointsize=50, rgbcolor='red') \ + point(f_pt,pointsize=50, rgbcolor='blue') \ + point(ef_pt,pointsize=50,rgbcolor='purple') \ + point(f_gs_pt,pointsize=75, rgbcolor='black') \ + point(e_gs_pt,pointsize=75, rgbcolor='black') \ + point(ef_gs_pt,pointsize=75, rgbcolor='black') \ + point(js_pt,pointsize=100,rgbcolor='green') |
S = circle((0,0),1,rgbcolor='yellow') S += line([e_pt,e_gs_pt], rgbcolor='red', thickness=4) S += line([f_pt,f_gs_pt], rgbcolor='blue', thickness=3) S += line([ef_pt,ef_gs_pt], rgbcolor='purple',thickness=2) S += point(e_pt,pointsize=50, rgbcolor='red') S += point(f_pt,pointsize=50, rgbcolor='blue') S += point(ef_pt,pointsize=50,rgbcolor='purple') S += point(f_gs_pt,pointsize=75, rgbcolor='black') S += point(e_gs_pt,pointsize=75, rgbcolor='black') S += point(ef_gs_pt,pointsize=75, rgbcolor='black') S += point(js_pt,pointsize=100,rgbcolor='green') |
| Line 564: | Line 561: |
| S += text('$J(%s,%s) = %s$'%(latex2(e),latex2(f),latex(js)), \ | S += text('$J(%s,%s) = %s$'%(latex2(e),latex2(f),latex(js)), |
| Line 583: | Line 580: |
| {{{ | {{{#!sagecell |
| Line 632: | Line 629: |
| S = circle((0,0),1,rgbcolor='yellow') \ + line([e_pt,e_gs_pt], rgbcolor='red', thickness=4) \ + line([f_pt,f_gs_pt], rgbcolor='blue', thickness=3) \ + line([ef_pt,ef_gs_pt], rgbcolor='purple',thickness=2) \ + point(e_pt,pointsize=50, rgbcolor='red') \ + point(f_pt,pointsize=50, rgbcolor='blue') \ + point(ef_pt,pointsize=50,rgbcolor='purple') \ + point(f_gs_pt,pointsize=75, rgbcolor='black') \ + point(e_gs_pt,pointsize=75, rgbcolor='black') \ + point(ef_gs_pt,pointsize=75, rgbcolor='black') \ + point(js_pt,pointsize=100,rgbcolor='green') |
S = circle((0,0),1,rgbcolor='yellow') S += line([e_pt,e_gs_pt], rgbcolor='red', thickness=4) S += line([f_pt,f_gs_pt], rgbcolor='blue', thickness=3) S += line([ef_pt,ef_gs_pt], rgbcolor='purple',thickness=2) S += point(e_pt,pointsize=50, rgbcolor='red') S += point(f_pt,pointsize=50, rgbcolor='blue') S += point(ef_pt,pointsize=50,rgbcolor='purple') S += point(f_gs_pt,pointsize=75, rgbcolor='black') S += point(e_gs_pt,pointsize=75, rgbcolor='black') S += point(ef_gs_pt,pointsize=75, rgbcolor='black') S += point(js_pt,pointsize=100,rgbcolor='green') |
| Line 644: | Line 641: |
| S += text('$J(%s,%s) = %s$'%(latex2(e),latex2(f),latex(js)), \ | S += text('$J(%s,%s) = %s$'%(latex2(e),latex2(f),latex(js)), |
| Line 656: | Line 653: |
| ga[i].save('j%d.PNG'%i,figsize=4,aspect_ratio=1, \ | ga[i].save('j%d.png'%i,figsize=4,aspect_ratio=1, |
| Line 662: | Line 659: |
| html('<table bgcolor=lightgrey cellpadding=2>') | s='<table bgcolor=lightgrey cellpadding=2>' |
| Line 664: | Line 661: |
| html('<tr><td align="center"><img src="cell://j%d.PNG"></td>'%(2*i)) html('<td align="center"><img src="cell://j%d.PNG"></td></tr>'%(2*i+1)) html('</table>') }}} |
s+='<tr><td align="center"><img src="cell://j%d.png"></td>'%(2*i) s+='<td align="center"><img src="cell://j%d.png"></td></tr>'%(2*i+1) s+='</table>' html(s)}}} |
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| {{{ | {{{#!sagecell |
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| else: | else: |
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def line_from_curve_points(E,P,Q,style='-',rgb=(1,0,0),length=25): """ P,Q two points on an elliptic curve. Output is a graphic representation of the straight line intersecting with P,Q. """ # The function tangent to P=Q on E if P == Q: if P[2]==0: return line([(1,-length),(1,length)],linestyle=style,rgbcolor=rgb) else: # Compute slope of the curve E in P l=-(3*P[0]^2 + 2*E.a2()*P[0] + E.a4() - E.a1()*P[1])/((-2)*P[1] - E.a1()*P[0] - E.a3()) f(x) = l * (x - P[0]) + P[1] return plot(f(x),-length,length,linestyle=style,rgbcolor=rgb) # Trivial case of P != R where P=O or R=O then we get the vertical line from the other point elif P[2] == 0: return line([(Q[0],-length),(Q[0],length)],linestyle=style,rgbcolor=rgb) elif Q[2] == 0: return line([(P[0],-length),(P[0],length)],linestyle=style,rgbcolor=rgb) # Non trivial case where P != R else: # Case where x_1 = x_2 return vertical line evaluated in Q if P[0] == Q[0]: return line([(P[0],-length),(P[0],length)],linestyle=style,rgbcolor=rgb) #Case where x_1 != x_2 return line trough P,R evaluated in Q" l=(Q[1]-P[1])/(Q[0]-P[0]) f(x) = l * (x - P[0]) + P[1] return plot(f(x),-length,length,linestyle=style,rgbcolor=rgb) |
|
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| curve = E.plot(rgbcolor = (0,0,1),xmin=25,xmax=25,plot_points=300) | curve = E.plot(rgbcolor = (0,0,1),xmin=-5,xmax=5,plot_points=300) |
| Line 716: | Line 744: |
def line_from_curve_points(E,P,Q,style='-',rgb=(1,0,0),length=25): """ P,Q two points on an elliptic curve. Output is a graphic representation of the straight line intersecting with P,Q. """ # The function tangent to P=Q on E if P == Q: if P[2]==0: return line([(1,-length),(1,length)],linestyle=style,rgbcolor=rgb) else: # Compute slope of the curve E in P l=-(3*P[0]^2 + 2*E.a2()*P[0] + E.a4() - E.a1()*P[1])/((-2)*P[1] - E.a1()*P[0] - E.a3()) f(x) = l * (x - P[0]) + P[1] return plot(f(x),-length,length,linestyle=style,rgbcolor=rgb) # Trivial case of P != R where P=O or R=O then we get the vertical line from the other point elif P[2] == 0: return line([(Q[0],-length),(Q[0],length)],linestyle=style,rgbcolor=rgb) elif Q[2] == 0: return line([(P[0],-length),(P[0],length)],linestyle=style,rgbcolor=rgb) # Non trivial case where P != R else: # Case where x_1 = x_2 return vertical line evaluated in Q if P[0] == Q[0]: return line([(P[0],-length),(P[0],length)],linestyle=style,rgbcolor=rgb) #Case where x_1 != x_2 return line trough P,R evaluated in Q" l=(Q[1]-P[1])/(Q[0]-P[0]) f(x) = l * (x - P[0]) + P[1] return plot(f(x),-length,length,linestyle=style,rgbcolor=rgb) |
|
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| {{{ | {{{#!sagecell |
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| {{{ | {{{#!sagecell |
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| print """ <html> |
html(""" |
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| </ol></html> """ % (bits, p, g, a, g, a, p, (g^a), b, g, b, p, (g^b), (g^b), a, p, (g^ b)^a, g^a, b, p, (g^a)^b) |
</ol> """ % (bits, p, g, a, g, a, p, (g^a), b, g, b, p, (g^b), (g^b), a, p, (g^ b)^a, g^a, b, p, (g^a)^b)) |
| Line 817: | Line 814: |
| {{{ @interact def _(number=e, ymax=selector([None,5,20,..,400],nrows=2), clr=Color('purple'), prec=[500,1000,..,5000]): |
{{{#!sagecell @interact def _(number=e, ymax=selector([5,20,..,400],nrows=2), clr=Color('purple'), prec=[500,1000,..,5000]): |
| Line 827: | Line 824: |
| {{{ | {{{#!sagecell |
| Line 848: | Line 845: |
| {{{ | {{{#!sagecell |
| Line 856: | Line 853: |
| def _(gen = selector(['t+1', 't-1', '-1/t'], nrows=1)): | def _(gen = selector(['t+1', 't-1', '-1/t'], buttons=True,nrows=1)): |
| Line 874: | Line 871: |
= Multiple Zeta Values = by Akhilesh P. == Word to composition == {{{#!sagecell @interact def _( weight=(7,(2..30))): n=weight a=[0 for i in range(n-1)] a.append(1) @interact def _(v=('word', input_grid(1, n, default=[a], to_value=lambda x: vector(flatten(x))))): a=[v[i] for i in range(len(v))] def bintocomp(a): b=[] count=1 for j in range(len(a)): if(a[j]==0): count=count+1 else: b.append(count) count=1 return(b) print "Composition is ",bintocomp(a) }}} {{attachment:akhi2.png}} == Composition to Word == {{{#!sagecell @interact def _( Depth=(7,(1..30))): n=Depth a=[] a.append(2) a=a+[1 for i in range(1,n)] @interact def _(v=('composition', input_grid(1, n, default=[a], to_value=lambda x: vector(flatten(x))))): a=[v[i] for i in range(len(v))] def comptobin(a): word=[] for i in range(len(a)): word=word+[0]*(a[i]-1)+[1] return(word) print "Word is ",comptobin(a) }}} {{attachment:akhi3.png}} == Dual of a Word == {{{#!sagecell @interact def _( weight=(7,(2..30))): n=weight a=[0 for i in range(n-1)] a.append(1) @interact def _(v=('word', input_grid(1, n, default=[a], to_value=lambda x: vector(flatten(x))))): a=[v[i] for i in range(len(v))] def dual(a): b=list() b=a b=b[::-1] for i in range(len(b)): b[i]=1-b[i] return(b) print "Dual word is ",dual(a) }}} {{attachment:akhi4.png}} == Computing Multiple Zeta values == === Word Input === {{{#!sagecell R=RealField(10) @interact def _( weight=(5,(2..20))): n=weight a=[0 for i in range(n-1)] a.append(1) @interact def _(v=('word', input_grid(1, n, default=[a], to_value=lambda x: vector(flatten(x)))), accuracy=(100..100000)): D=accuracy a=[v[i] for i in range(len(v))] DD=int(3.321928*D)+int(R(log(3.321928*D))/R(log(10)))+4 RIF=RealIntervalField(DD) def Li(word): n=int(DD*log(10)/log(2))+1 B=[] L=[] S=[] count=-1 k=len(word) for i in range(k): B.append(RIF('0')) L.append(RIF('0')) if(word[i]==1 and i<k-1): S.append(RIF('0')) count=count+1 T=RIF('1') for m in range(n): T=T/2 B[k-1]=RIF('1')/(m+1) j=count for i in range(k-2,-1,-1): if(word[i]==0): B[i]=B[i+1]/(m+1) elif(word[i]==1): B[i]=S[j]/(m+1) S[j]=S[j]+B[i+1] j=j-1 L[i]=T*B[i]+L[i] L[k-1]=T*B[k-1]+L[k-1] return(L) def dual(a): b=list() b=a b=b[::-1] for i in range(len(b)): b[i]=1-b[i] return(b) def zeta(a): b=dual(a) l1=Li(a)+[1] l2=Li(b)+[1] Z=RIF('0') for i in range(len(l1)): Z=Z+l1[i]*l2[len(a)-i] return(Z) print zeta(a) }}} {{attachment:akhi1.png}} === Composition Input === {{{#!sagecell R=RealField(10) @interact def _( Depth=(5,(2..20))): n=Depth a=[2] a=a+[1 for i in range(n-1)] @interact def _(v=('Composition', input_grid(1, n, default=[a], to_value=lambda x: vector(flatten(x)))), accuracy=(100..100000)): D=accuracy a=[v[i] for i in range(len(v))] def comptobin(a): word=[] for i in range(len(a)): word=word+[0]*(a[i]-1)+[1] return(word) a=comptobin(a) DD=int(3.321928*D)+int(R(log(3.321928*D))/R(log(10)))+4 RIF=RealIntervalField(DD) def Li(word): n=int(DD*log(10)/log(2))+1 B=[] L=[] S=[] count=-1 k=len(word) for i in range(k): B.append(RIF('0')) L.append(RIF('0')) if(word[i]==1 and i<k-1): S.append(RIF('0')) count=count+1 T=RIF('1') for m in range(n): T=T/2 B[k-1]=RIF('1')/(m+1) j=count for i in range(k-2,-1,-1): if(word[i]==0): B[i]=B[i+1]/(m+1) elif(word[i]==1): B[i]=S[j]/(m+1) S[j]=S[j]+B[i+1] j=j-1 L[i]=T*B[i]+L[i] L[k-1]=T*B[k-1]+L[k-1] return(L) def dual(a): b=list() b=a b=b[::-1] for i in range(len(b)): b[i]=1-b[i] return(b) def zeta(a): b=dual(a) l1=Li(a)+[1] l2=Li(b)+[1] Z=RIF('0') for i in range(len(l1)): Z=Z+l1[i]*l2[len(a)-i] return(Z) print zeta(a) }}} {{attachment:akhi5.png}} == Shuffle product of two Words == {{{#!sagecell @interact def _( w1=(2,(2..20)), w2=(2,(2..20))): a=[0] b=[0 for i in range(w2-1)] a=a+[1 for i in range(1,w1)] b=b+[1] import itertools #this program gives the list of all binary words of weight n and depth k @interact def _(v1=('word1', input_grid(1, w1, default=[a], to_value=lambda x: vector(flatten(x)))), v2=('word2', input_grid(1, w2, default=[b], to_value=lambda x: vector(flatten(x))))): def kbits(n, k): result = [] for bits in itertools.combinations(range(n), k): s = ['0'] * n for bit in bits: s[bit] = '1' result.append(''.join(s)) return result def sort(a,l,m): b=[] n=len(a) for i in range(n): b.append(a[i]) for j in range(l-1,-1,-1): k=0 for t in range(m+1): for i in range(n): if(a[i][j]== t): b[k]=a[i] k=k+1 for i in range(n): a[i]=b[i] return(a) def count(a): n=len(a) b=[] b.append(a[0]) m=[] m.append(1) c=0 for i in range(1,n): if(a[i]==a[i-1]): m[c]=m[c]+1 else: b.append(a[i]) m.append(1) c=c+1 return(b,m) def shuffle(a,b): r=len(a) s=len(b) # Generating an array of strings containing all combinations of weight r+s and depth s M=kbits(r+s,s) n=len(M) a1= [] for i in range(n): a1.append(list(M[i])) # The zeroes are replaced by the entries of a and the ones by the entries of b a2= [] for i in range(n): a2.append([]) count0=0 count1=0 for j in range(s+r): if(a1[i][j]=='0'): a2[i].append(a[count0]) count0=count0+1 if(a1[i][j]=='1'): a2[i].append(b[count1]) count1=count1+1 # Reordering in lexicographic order the entries of a2: this is done by first reordering them according to the last digit, then the next to last digit, etc a3=sort(a2,r+s,max(a+b+[0])) # Getting the same list without repetitions and with multiplicities a4=count(a3) return(a4) c=shuffle(a,b) for i in range(len(c[0])-1): print c[1][i],"*",c[0][i] ,"+ ", print c[1][len(c[0])-1],"*",c[0][len(c[0])-1] }}} |
Contents
Integer Factorization
Divisibility Poset
by William Stein
Factor Trees
by William Stein
More complicated demonstration using Mathematica: http://demonstrations.wolfram.com/FactorTrees/
Factoring an Integer
by Timothy Clemans
Sage implementation of the Mathematica demonstration of the same name. http://demonstrations.wolfram.com/FactoringAnInteger/
Prime Numbers
Illustrating the prime number theorem
by William Stein
Prime Spiral - Square FIXME
by David Runde
Prime Spiral - Polar
by David Runde
Modular Forms
Computing modular forms
by William Stein
Computing the cuspidal subgroup
by William Stein
A Charpoly and Hecke Operator Graph
by William Stein
Modular Arithmetic
Quadratic Residue Table FIXME
by Emily Kirkman
Cubic Residue Table FIXME
by Emily Kirkman
Cyclotomic Fields
Gauss and Jacobi Sums in Complex Plane
by Emily Kirkman
Exhaustive Jacobi Plotter
by Emily Kirkman
Elliptic Curves
Adding points on an elliptic curve
by David Møller Hansen
Plotting an elliptic curve over a finite field
Cryptography
The Diffie-Hellman Key Exchange Protocol
by Timothy Clemans and William Stein
Other
Continued Fraction Plotter
by William Stein
Computing Generalized Bernoulli Numbers
by William Stein (Sage-2.10.3)
Fundamental Domains of SL_2(ZZ)
by Robert Miller
Multiple Zeta Values
by Akhilesh P.
Word to composition
Composition to Word
Dual of a Word
Computing Multiple Zeta values
Word Input
Composition Input
Shuffle product of two Words