Differences between revisions 15 and 64 (spanning 49 versions)
Revision 15 as of 2012-05-09 04:32:21
Size: 30447
Editor: jason
Comment:
Revision 64 as of 2019-04-06 06:41:25
Size: 48347
Editor: chapoton
Comment: py3 print
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 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 235: Line 232:
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"
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 240:
    
Line 251: Line 248:
        R = points(list2, alpha = .1) #Faded Composites 
    else: 
        R = points(list2, alpha = .1) #Faded Composites
    else:
Line 259: Line 256:
        R=points(list2, hue = .1, pointsize = p_size) 
    
        R=points(list2, hue = .1, pointsize = p_size)
Line 262: Line 259:
        print 'n =', factor(n)
        
        print('n = {}'.format(factor(n)))
Line 270: Line 267:
        Q = plot(W1+W2+W3+W4, alpha = .1)                   Q = plot(W1+W2+W3+W4, alpha = .1)
Line 273: Line 270:
        if show_curves:          if show_curves:
Line 278: Line 275:
            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 278:
            else:              else:
Line 284: Line 281:
            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 289:
            c= c2;              c= c2;
Line 308: Line 305:
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 309:
    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 313:
         print "Space has dimension 0"          print("Space has dimension 0")
Line 319: Line 315:
        prec = max(prec, M.dimension()+1)         prec = max(prec, M.dimension() + 1)
Line 322: Line 318:
    print "\n\n\nDone computing basis."     print("\n\n\nDone computing basis.")
Line 335: Line 331:
    print A.cuspidal_subgroup()     print(A.cuspidal_subgroup())
Line 351: Line 347:
    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 362:
== Quadratic Residue Table == == Quadratic Residue Table FIXME ==
Line 417: Line 415:
== Cubic Residue Table == == Cubic Residue Table FIXME ==
Line 472: Line 470:
        MP += line([(i,0),(i,r)], rgbcolor='black')          MP += line([(i,0),(i,r)], rgbcolor='black')
Line 549: Line 547:
    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 559:
        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 627:
    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 639:
        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 653: Line 651:
        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 657:
    html('<table bgcolor=lightgrey cellpadding=2>')     s='<table bgcolor=lightgrey cellpadding=2>'
Line 661: Line 659:
        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)}}}
Line 678: Line 676:
    else:      else:
Line 685: Line 683:

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 716:
 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 742:

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 752:
    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 776:
    print """
<
html>
    html("""
Line 802: Line 800:
</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 814:
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 853: Line 851:
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 869:

= 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}}

Integer Factorization

Divisibility Poset

by William Stein

divposet.png

Factor Trees

by William Stein

factortree.png

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

primes.png

Prime Spiral - Square FIXME

by David Runde

SquareSpiral.PNG

Prime Spiral - Polar

by David Runde

PolarSpiral.PNG

Modular Forms

Computing modular forms

by William Stein

modformbasis.png

Computing the cuspidal subgroup

by William Stein

cuspgroup.png

A Charpoly and Hecke Operator Graph

by William Stein

heckegraph.png

Modular Arithmetic

Quadratic Residue Table FIXME

by Emily Kirkman

quadres.png

quadresbig.png

Cubic Residue Table FIXME

by Emily Kirkman

cubres.png

Cyclotomic Fields

Gauss and Jacobi Sums in Complex Plane

by Emily Kirkman

jacobising.png

Exhaustive Jacobi Plotter

by Emily Kirkman

jacobiexh.png

Elliptic Curves

Adding points on an elliptic curve

by David Møller Hansen

PointAddEllipticCurve.png

Plotting an elliptic curve over a finite field

ellffplot.png

Cryptography

The Diffie-Hellman Key Exchange Protocol

by Timothy Clemans and William Stein

dh.png

Other

Continued Fraction Plotter

by William Stein

contfracplot.png

Computing Generalized Bernoulli Numbers

by William Stein (Sage-2.10.3)

bernoulli.png

Fundamental Domains of SL_2(ZZ)

by Robert Miller

fund_domain.png

Multiple Zeta Values

by Akhilesh P.

Computing Multiple Zeta values

Word Input

akhi1.png

Composition Input

akhi5.png

Program to Compute Integer Relation between Multiple Zeta Values

akhi10.png

Word to composition

akhi2.png

Composition to Word

akhi3.png

Dual of a Word

akhi4.png

Shuffle product of two Words

akhi6.png

Shuffle Regularization at 0

akhi7.png

Shuffle Regularization at 1

akhi8.png

interact/number_theory (last edited 2020-06-14 09:10:48 by chapoton)