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| Deletions are marked like this. | Additions are marked like this. |
| Line 51: | Line 51: |
| 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)], |
| Line 79: | Line 79: |
| html(s) | pretty_print(html(s)) |
| Line 89: | Line 89: |
| html("<font color='red'>$\pi(x)$</font> and <font color='blue'>$x/(\log(x)-1)$</font> for $x < %s$"%N) | pretty_print(html("<font color='red'>$\pi(x)$</font> and <font color='blue'>$x/(\log(x)-1)$</font> for $x < %s$"%N)) |
| Line 101: | Line 101: |
| REFERENCES: | REFERENCES: |
| Line 106: | Line 106: |
| 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 |
| Line 114: | Line 114: |
| elif y<0 and -x >= y and y<x: return 4*(y+1)^2 -11*(y+1) + (start+7) +x else: print 'NaN' #Takes in an n and the start value of the spiral and gives its (x,y) coordinate |
elif y<0 and -x >= y and y<x: return 4*(y+1)^2 -11*(y+1) + (start+7) +x else: print('NaN') #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 |
| Line 121: | Line 121: |
| top = ceil(sqrt(num)) | top = ceil(sqrt(num)) |
| Line 126: | Line 126: |
| else: | else: |
| Line 133: | Line 133: |
| else: | else: |
| Line 140: | Line 140: |
| if start < 1 or end <=start: print "invalid start or end value" if n > end: print "WARNING: n is larger than the end value" |
if start < 1 or end <=start: print("invalid start or end value") if n > end: print("WARNING: n is larger than the end value") |
| Line 145: | Line 145: |
| N = M.copy() | N = copy(M) |
| Line 149: | Line 149: |
| #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 |
| Line 162: | Line 162: |
| 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 172: | Line 172: |
| #print x_cord, y_cord if show_lines: for t in [(-size-1)..size+1]: |
if show_lines: for t in [(-size-1)..size+1]: |
| Line 176: | Line 175: |
| if m.is_pseudoprime(): main_list.add(m) | if m.is_pseudoprime(): main_list.add(m) |
| Line 181: | Line 180: |
| #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 183: | Line 182: |
| #print x, "=x y=", y, " num =", num | |
| Line 186: | Line 184: |
| else: x-=1 | else: x-=1 |
| Line 188: | Line 186: |
| elif county < overcount: |
elif county < overcount: |
| Line 191: | Line 189: |
| else: y-=1 | else: y-=1 |
| Line 193: | Line 191: |
| else: | else: |
| Line 199: | Line 197: |
| if not invert and num in main_list: |
if not invert and num in main_list: |
| Line 207: | Line 205: |
| if n != 0: print '(to go from x,y coords to an n, reset by setting n=0)' |
if n != 0: print('(to go from x,y coords to an n, reset by setting n=0)') |
| Line 211: | Line 209: |
| #print 'if n =', n, 'then (x,y) =', (x_cord, y_cord) print '(x,y) =', (x_cord, y_cord), '<=> n =', find_n(x_cord, y_cord, start) print ' ' print "SW/NE line" if -y_cord<x_cord: print '4*t^2 + 2*t +', -x_cord+y_cord+start else: print '4*t^2 + 2*t +', +x_cord-y_cord+start print "NW/SE line" if x_cord<y_cord: print '4*t^2 +', -x_cord-y_cord+start else: print '4*t^2 + 4*t +', +x_cord+y_cord+start |
print('(x,y) =', (x_cord, y_cord), '<=> n =', find_n(x_cord, y_cord, start)) print(' ') print("SW/NE line") if -y_cord<x_cord: print('4*t^2 + 2*t +', -x_cord+y_cord+start) else: print('4*t^2 + 2*t +', +x_cord-y_cord+start) print("NW/SE line") if x_cord<y_cord: print('4*t^2 +', -x_cord-y_cord+start) else: print('4*t^2 + 4*t +', +x_cord+y_cord+start) |
| Line 224: | Line 221: |
| else: | else: |
| Line 233: | Line 230: |
| {{{#!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. if start < 1 or end <=start: print "invalid start or end value" if n > end: print "WARNING: n is greater than end value" |
Needs fix for show_factors {{{#!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. if start < 1 or end <=start: print("invalid start or end value") if n > end: print("WARNING: n is greater than end value") |
| Line 243: | Line 242: |
| Line 251: | Line 250: |
| R = points(list2, alpha = .1) #Faded Composites else: |
R = points(list2, alpha = .1) #Faded Composites else: |
| Line 259: | Line 258: |
| R=points(list2, hue = .1, pointsize = p_size) |
R=points(list2, hue = .1, pointsize = p_size) |
| Line 262: | Line 261: |
| print 'n =', factor(n) |
print('n = {}'.format(factor(n))) |
| Line 270: | Line 269: |
| Q = plot(W1+W2+W3+W4, alpha = .1) | Q = plot(W1+W2+W3+W4, alpha = .1) |
| Line 273: | Line 272: |
| if show_curves: | if show_curves: |
| Line 278: | Line 277: |
| 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)): |
| Line 281: | Line 280: |
| else: | else: |
| Line 284: | Line 283: |
| print 'Pink Curve: n^2 +', c print 'Green Curve: n^2 + n +', c2 def g(m): return (a*m^2+b*m+c); |
print('Pink Curve: n^2 +', c) print('Green Curve: n^2 + n +', c2) def g(m): return (a*m^2+b*m+c); |
| Line 292: | Line 291: |
| c= c2; | c= c2; |
| Line 308: | Line 307: |
| j = 0 @interact def _(N=[1..100], k=selector([2,4,..,12],nrows=1), prec=(3..40), |
@interact def _(N=[1..100], k=selector([2,4,..,12],nrows=1), prec=(3..40), |
| Line 313: | Line 311: |
| print j; global j; j += 1 print M; print '\n'*3 print "Computing basis...\n\n" |
print(M) print('\n' * 3) print("Computing basis...\n\n") |
| Line 317: | Line 315: |
| print "Space has dimension 0" | print("Space has dimension 0") |
| Line 319: | Line 317: |
| prec = max(prec, M.dimension()+1) | prec = max(prec, M.dimension() + 1) |
| Line 322: | Line 320: |
| print "\n\n\nDone computing basis." | print("\n\n\nDone computing basis.") |
| Line 330: | Line 328: |
| {{{#!sagecell html('<h1>Cuspidal Subgroups of Modular Jacobians J0(N)</h1>') |
ncols not working {{{#!sagecell pretty_print(html('<h1>Cuspidal Subgroups of Modular Jacobians J0(N)</h1>')) |
| Line 335: | Line 335: |
| print A.cuspidal_subgroup() | print(A.cuspidal_subgroup()) |
| Line 351: | Line 351: |
| 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 366: |
| == Quadratic Residue Table == | == Quadratic Residue Table FIXME == |
| Line 417: | Line 419: |
| == Cubic Residue Table == | == Cubic Residue Table FIXME == |
| Line 472: | Line 474: |
| MP += line([(i,0),(i,r)], rgbcolor='black') | MP += line([(i,0),(i,r)], rgbcolor='black') |
| Line 475: | Line 477: |
| MP += text('$\omega^2$',(i+.5,r-j-.5),rgbcolor='black') | MP += text(r'$\omega^2$',(i+.5,r-j-.5),rgbcolor='black') |
| Line 477: | Line 479: |
| MP += text('$\omega $',(i+.5,r-j-.5),rgbcolor='black') | MP += text(r'$\omega $',(i+.5,r-j-.5),rgbcolor='black') |
| Line 486: | Line 488: |
| MP += text('$ \pi_1$',(r/2,r+2), rgbcolor='black', fontsize=25) MP += text('$ \pi_2$',(-2.5,r/2), rgbcolor='black', fontsize=25) html('Symmetry of Primary Cubic Residues mod ' \ + '%d primary primes in $ \mathbf Z[\omega]$.'%r) |
MP += text(r'$ \pi_1$',(r/2,r+2), rgbcolor='black', fontsize=25) MP += text(r'$ \pi_2$',(-2.5,r/2), rgbcolor='black', fontsize=25) pretty_print(html('Symmetry of Primary Cubic Residues mod ' \ + r'%d primary primes in $ \mathbf Z[\omega]$.'%r)) |
| Line 549: | Line 551: |
| S = circle((0,0),1,rgbcolor='yellow') \ + line([e_pt,e_gs_pt], rgbcolor='red', thickness=4) \ + line([f_pt,f_gs_pt], rgbcolor='blue', thickness=3) \ + line([ef_pt,ef_gs_pt], rgbcolor='purple',thickness=2) \ + point(e_pt,pointsize=50, rgbcolor='red') \ + point(f_pt,pointsize=50, rgbcolor='blue') \ + point(ef_pt,pointsize=50,rgbcolor='purple') \ + point(f_gs_pt,pointsize=75, rgbcolor='black') \ + point(e_gs_pt,pointsize=75, rgbcolor='black') \ + point(ef_gs_pt,pointsize=75, rgbcolor='black') \ + point(js_pt,pointsize=100,rgbcolor='green') |
S = circle((0,0),1,rgbcolor='yellow') S += line([e_pt,e_gs_pt], rgbcolor='red', thickness=4) S += line([f_pt,f_gs_pt], rgbcolor='blue', thickness=3) S += line([ef_pt,ef_gs_pt], rgbcolor='purple',thickness=2) S += point(e_pt,pointsize=50, rgbcolor='red') S += point(f_pt,pointsize=50, rgbcolor='blue') S += point(ef_pt,pointsize=50,rgbcolor='purple') S += point(f_gs_pt,pointsize=75, rgbcolor='black') S += point(e_gs_pt,pointsize=75, rgbcolor='black') S += point(ef_gs_pt,pointsize=75, rgbcolor='black') S += point(js_pt,pointsize=100,rgbcolor='green') |
| Line 561: | Line 563: |
| S += text('$J(%s,%s) = %s$'%(latex2(e),latex2(f),latex(js)), \ | S += text('$J(%s,%s) = %s$'%(latex2(e),latex2(f),latex(js)), |
| Line 629: | Line 631: |
| S = circle((0,0),1,rgbcolor='yellow') \ + line([e_pt,e_gs_pt], rgbcolor='red', thickness=4) \ + line([f_pt,f_gs_pt], rgbcolor='blue', thickness=3) \ + line([ef_pt,ef_gs_pt], rgbcolor='purple',thickness=2) \ + point(e_pt,pointsize=50, rgbcolor='red') \ + point(f_pt,pointsize=50, rgbcolor='blue') \ + point(ef_pt,pointsize=50,rgbcolor='purple') \ + point(f_gs_pt,pointsize=75, rgbcolor='black') \ + point(e_gs_pt,pointsize=75, rgbcolor='black') \ + point(ef_gs_pt,pointsize=75, rgbcolor='black') \ + point(js_pt,pointsize=100,rgbcolor='green') |
S = circle((0,0),1,rgbcolor='yellow') S += line([e_pt,e_gs_pt], rgbcolor='red', thickness=4) S += line([f_pt,f_gs_pt], rgbcolor='blue', thickness=3) S += line([ef_pt,ef_gs_pt], rgbcolor='purple',thickness=2) S += point(e_pt,pointsize=50, rgbcolor='red') S += point(f_pt,pointsize=50, rgbcolor='blue') S += point(ef_pt,pointsize=50,rgbcolor='purple') S += point(f_gs_pt,pointsize=75, rgbcolor='black') S += point(e_gs_pt,pointsize=75, rgbcolor='black') S += point(ef_gs_pt,pointsize=75, rgbcolor='black') S += point(js_pt,pointsize=100,rgbcolor='green') |
| Line 641: | Line 643: |
| S += text('$J(%s,%s) = %s$'%(latex2(e),latex2(f),latex(js)), \ | S += text('$J(%s,%s) = %s$'%(latex2(e),latex2(f),latex(js)), |
| Line 644: | Line 646: |
| html('$$J(%s,%s) = %s$$'%(latex2(e),latex2(f),latex(js))) | pretty_print(html('$$J(%s,%s) = %s$$'%(latex2(e),latex2(f),latex(js)))) |
| Line 653: | Line 655: |
| 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 659: | Line 661: |
| html('<table bgcolor=lightgrey cellpadding=2>') | s='<table bgcolor=lightgrey cellpadding=2>' |
| Line 661: | Line 663: |
| 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>' pretty_print(html(s)) |
| Line 678: | Line 681: |
| else: | else: |
| Line 685: | Line 688: |
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) |
|
| Line 687: | Line 721: |
| 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 713: | Line 747: |
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) |
|
| Line 753: | Line 757: |
| print "p = %s"%p show(E.change_ring(GF(p)).plot(),xmin=0,ymin=0) |
print("p = %s" % p) show(E.change_ring(GF(p)).plot(), xmin=0, ymin=0) |
| Line 777: | Line 781: |
| print """ <html> |
html(""" |
| Line 802: | Line 805: |
| </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 816: | Line 819: |
| def _(number=e, ymax=selector([None,5,20,..,400],nrows=2), clr=Color('purple'), prec=[500,1000,..,5000]): c = list(continued_fraction(RealField(prec)(number))); print c |
def _(number=e, ymax=selector([5,20,..,400],nrows=2), clr=Color('purple'), prec=[500,1000,..,5000]): c = list(continued_fraction(RealField(prec)(number))); print(c) |
| Line 846: | Line 849: |
| L = [[-0.5, 2.0^(x/100.0) - 1 + sqrt(3.0)/2] for x in xrange(1000, -1, -1)] R = [[0.5, 2.0^(x/100.0) - 1 + sqrt(3.0)/2] for x in xrange(1000)] xes = [x/1000.0 for x in xrange(-500,501,1)] |
L = [[-0.5, 2.0^(x/100.0) - 1 + sqrt(3.0)/2] for x in range(1000, -1, -1)] R = [[0.5, 2.0^(x/100.0) - 1 + sqrt(3.0)/2] for x in range(1000)] xes = [x/1000.0 for x in range(-500,501,1)] |
| Line 853: | Line 856: |
| 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 871: | Line 874: |
= Multiple Zeta Values = by Akhilesh P. == Computing Multiple Zeta values == === Word Input === {{{#!sagecell R=RealField(10) @interact def _( weight=(5,(2..100))): 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) u=zeta(a) RIF=RealIntervalField(int(3.321928*D)) u=u/1 print(u) }}} {{attachment:akhi1.png}} === Composition Input === {{{#!sagecell R=RealField(10) @interact def _( Depth=(5,(2..100))): 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) u=zeta(a) RIF=RealIntervalField(int(3.321928*D)) u=u/1 print(u) }}} {{attachment:akhi5.png}} == Program to Compute Integer Relation between Multiple Zeta Values == {{{#!sagecell from mpmath import * print("Enter the number of composition") @interact def _( n=(5,(2..100))): a=[] for i in range(n): a.append([i+2,1]) print("In each box Enter composition as an array") @interact def _(v=('Compositions', input_box( default=a, to_value=lambda x: vector(flatten(x)))), accuracy=(100..100000)): D=accuracy R=RealField(10) a=v def comptobin(a): word=[] for i in range(len(a)): word=word+[0]*(a[i]-1)+[1] return(word) DD=int(D)+int(R(log(3.321928*D))/R(log(10)))+4 RIF=RealIntervalField(DD) mp.dps=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(mpf('0')) L.append(mpf('0')) if(word[i]==1 and i<k-1): S.append(mpf('0')) count=count+1 T=mpf('1') for m in range(n): T=T/2 B[k-1]=mpf('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=mpf('0') for i in range(len(l1)): Z=Z+l1[i]*l2[len(a)-i] return(Z) zet=[] for i in range(n): zet.append((zeta(comptobin(a[i])))) mp.dps=D for i in range(n): zet[i]=zet[i]/1 print("zeta(", a[i], ")=", zet[i]) u=pslq(zet,tol=10**-D,maxcoeff=100,maxsteps=10000) print("the Intger Relation between the above zeta values given by the vector") print(u) }}} {{attachment:akhi10.png}} == Word to composition == {{{#!sagecell @interact def _( weight=(7,(2..100))): 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 {}".format(bintocomp(a))) }}} {{attachment:akhi2.png}} == Composition to Word == {{{#!sagecell @interact def _( Depth=(7,(1..100))): 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 {}".format(comptobin(a))) }}} {{attachment:akhi3.png}} == Dual of a Word == {{{#!sagecell @interact def _( weight=(7,(2..100))): 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 {}"?format(dual(a))) }}} {{attachment:akhi4.png}} == Shuffle product of two Words == {{{#!sagecell @interact def _( w1=(2,(2..100)), w2=(2,(2..100))): 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))))): a=[v1[i] for i in range(len(v1))] b=[v2[i] for i in range(len(v2))] 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]) }}} {{attachment:akhi6.png}} == Shuffle Regularization at 0 == {{{#!sagecell @interact def _( w=(2,(2..100))): a=[0] a=a+[1 for i in range(1,w)] import itertools #this program gives the list of all binary words of weight n and depth k @interact def _(v=('word', input_grid(1, w, default=[a], to_value=lambda x: vector(flatten(x))))): a=[v[i] for i in range(len(v))] 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 sort1(a,l,m): b=[] b.append([]) b.append([]) n=len(a[0]) for i in range(n): b[0].append(a[0][i]) b[1].append(a[1][i]) for j in range(l-1,-1,-1): k=0 for t in range(m+1): for i in range(n): if(a[0][i][j]== t): b[0][k]=a[0][i] b[1][k]=a[1][i] k=k+1 for i in range(n): a[0][i]=b[0][i] a[1][i]=b[1][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 count1(a): n=len(a[0]) b=[] b.append([]) b.append([]) b[0].append(a[0][0]) b[1].append(a[1][0]) c=0 for i in range(1,n): if(a[0][i]==a[0][i-1]): b[1][c]=b[1][c]+a[1][i] else: b[0].append(a[0][i]) b[1].append(a[1][i]) c=c+1 return(b) 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) def Regshuf0(a): r=[] r.append([]) r.append([]) t=0 c=1 for i in range(len(a)+1): if(t==0): b=shuffle(a[:len(a)-i],a[len(a)-i:]) for j in range(len(b[0])): r[0].append(b[0][j]) r[1].append(b[1][j]*c) c=-c if(i<len(a)): if(a[len(a)-1-i]==1): t=1 r=sort1(r,len(a),max(a+[0])) r=count1(r) rg=[] rg.append([]) rg.append([]) for i in range(len(r[0])): if(r[1][i] is not 0): rg[0].append(r[0][i]) rg[1].append(r[1][i]) return(rg) c = Regshuf0(a) for i in range(len(c[0])-1): if(c[1][i] != 0): print(c[1][i],"*",c[0][i] ,"+ ") if(c[1][len(c[0])-1] != 0): print(c[1][len(c[0])-1],"*",c[0][len(c[0])-1]) }}} {{attachment:akhi7.png}} == Shuffle Regularization at 1 == {{{#!sagecell @interact def _( w=(2,(2..20))): a=[0] a=a+[1 for i in range(1,w)] import itertools #this program gives the list of all binary words of weight n and depth k @interact def _(v=('word', input_grid(1, w, default=[a], to_value=lambda x: vector(flatten(x))))): a=[v[i] for i in range(len(v))] 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 sort1(a,l,m): b=[] b.append([]) b.append([]) n=len(a[0]) for i in range(n): b[0].append(a[0][i]) b[1].append(a[1][i]) for j in range(l-1,-1,-1): k=0 for t in range(m+1): for i in range(n): if(a[0][i][j]== t): b[0][k]=a[0][i] b[1][k]=a[1][i] k=k+1 for i in range(n): a[0][i]=b[0][i] a[1][i]=b[1][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 count1(a): n=len(a[0]) b=[] b.append([]) b.append([]) b[0].append(a[0][0]) b[1].append(a[1][0]) c=0 for i in range(1,n): if(a[0][i]==a[0][i-1]): b[1][c]=b[1][c]+a[1][i] else: b[0].append(a[0][i]) b[1].append(a[1][i]) c=c+1 return(b) 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) def Regshuf1(a): r=[] r.append([]) r.append([]) t=0 c=1 for i in range(len(a)+1): if(t==0): b=shuffle(a[:i],a[i:]) for j in range(len(b[0])): r[0].append(b[0][j]) r[1].append(b[1][j]*c) c=-c if(i<len(a)): if(a[i]==0): t=1 r=sort1(r,len(a),max(a+[0])) r=count1(r) rg=[] rg.append([]) rg.append([]) for i in range(len(r[0])): if(r[1][i] is not 0): rg[0].append(r[0][i]) rg[1].append(r[1][i]) return(rg) c = Regshuf1(a) for i in range(len(c[0])-1): if(c[1][i] != 0): print(c[1][i],"*",c[0][i] ,"+ ") if(c[1][len(c[0])-1] != 0): print(c[1][len(c[0])-1],"*",c[0][len(c[0])-1]) }}} {{attachment:akhi8.png}} |
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
Needs fix for show_factors
Modular Forms
Computing modular forms
by William Stein
Computing the cuspidal subgroup
by William Stein
ncols not working
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.
Computing Multiple Zeta values
Word Input
Composition Input
Program to Compute Integer Relation between Multiple Zeta Values
Word to composition
Composition to Word
Dual of a Word
Shuffle product of two Words
Shuffle Regularization at 0
Shuffle Regularization at 1
