Differences between revisions 23 and 66 (spanning 43 versions)
⇤ ← Revision 23 as of 2012-05-09 04:54:12 →
Size: 30463
Editor: jason
Comment:
← Revision 66 as of 2020-06-04 19:10:25 → ⇥
Size: 48486
Editor: kcrisman
Comment:
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))
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            else:              else:
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            else:              else:
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    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
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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:
    
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    #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)
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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 183: Line 182:
        #print x, "=x y=", y, " num =", num
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            else: x-=1              else: x-=1
Line 188: Line 186:
        
        elif county < overcount: 
        elif county < overcount:
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            else: y-=1              else: y-=1
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        else:          else:
Line 199: Line 197:
    
        if not invert and num in main_list: 
        if not invert and num in main_list:
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    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)')
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        #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)
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    else:      else:
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{{{#!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:
    
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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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        print 'n =', factor(n)
        		         print('n = {}'.format(factor(n)))
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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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            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);
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            c= c2;              c= c2;
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== Computing modular forms FIXME == == Computing modular forms ==
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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),
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    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")
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         print "Space has dimension 0"          print("Space has dimension 0")
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        prec = max(prec, M.dimension()+1)         prec = max(prec, M.dimension() + 1)
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    print "\n\n\nDone computing basis."     print("\n\n\nDone computing basis.")
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{{{#!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>'))
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    print A.cuspidal_subgroup()     print(A.cuspidal_subgroup())
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== A Charpoly and Hecke Operator Graph FIXME == == A Charpoly and Hecke Operator Graph ==
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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()
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        MP += line([(i,0),(i,r)], rgbcolor='black')          MP += line([(i,0),(i,r)], rgbcolor='black')
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                MP += text('$\omega^2$',(i+.5,r-j-.5),rgbcolor='black')                 MP += text(r'$\omega^2$',(i+.5,r-j-.5),rgbcolor='black')
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                MP += text('$\omega $',(i+.5,r-j-.5),rgbcolor='black')                 MP += text(r'$\omega $',(i+.5,r-j-.5),rgbcolor='black')
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    MP += text('$ \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]$.'%r))
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    S += point(e_pt,pointsize=50, rgbcolor='red')      S += point(e_pt,pointsize=50, rgbcolor='red')
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    S += point(f_gs_pt,pointsize=75, rgbcolor='black')             S += point(f_gs_pt,pointsize=75, rgbcolor='black')
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        S += text('$J(%s,%s) = %s$'%(latex2(e),latex2(f),latex(js)),          S += text('$J(%s,%s) = %s$'%(latex2(e),latex2(f),latex(js)),
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    S += point(e_pt,pointsize=50, rgbcolor='red')      S += point(e_pt,pointsize=50, rgbcolor='red')
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        html('$$J(%s,%s) = %s$$'%(latex2(e),latex2(f),latex(js)))         pretty_print(html('$$J(%s,%s) = %s$$'%(latex2(e),latex2(f),latex(js))))
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    html(s)}}}     pretty_print(html(s))
}}}
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== Adding points on an elliptic curve FIXME == == Adding points on an elliptic curve ==
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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)
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)
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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)
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 802: Line 806:
    """ % (bits, p, g, a, g, a, p, (g^a), b, g, b, p, (g^b), (g^b), a, p,      """ % (bits, p, g, a, g, a, p, (g^a), b, g, b, p, (g^b), (g^b), a, p,
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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 845: 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 852: 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 870: 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

  1. Integer Factorization
    1. Divisibility Poset
    2. Factor Trees
    3. Factoring an Integer
  2. Prime Numbers
    1. Illustrating the prime number theorem
    2. Prime Spiral - Square FIXME
    3. Prime Spiral - Polar
  3. Modular Forms
    1. Computing modular forms
    2. Computing the cuspidal subgroup
    3. A Charpoly and Hecke Operator Graph
  4. Modular Arithmetic
    1. Quadratic Residue Table FIXME
    2. Cubic Residue Table FIXME
  5. Cyclotomic Fields
    1. Gauss and Jacobi Sums in Complex Plane
    2. Exhaustive Jacobi Plotter
  6. Elliptic Curves
    1. Adding points on an elliptic curve
    2. Plotting an elliptic curve over a finite field
  7. Cryptography
    1. The Diffie-Hellman Key Exchange Protocol
  8. Other
    1. Continued Fraction Plotter
    2. Computing Generalized Bernoulli Numbers
    3. Fundamental Domains of SL_2(ZZ)
  9. Multiple Zeta Values
    1. Computing Multiple Zeta values
      1. Word Input
      2. Composition Input
    2. Program to Compute Integer Relation between Multiple Zeta Values
    3. Word to composition
    4. Composition to Word
    5. Dual of a Word
    6. Shuffle product of two Words
    7. Shuffle Regularization at 0
    8. Shuffle Regularization at 1

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

Needs fix for show_factors

PolarSpiral.PNG

Modular Forms

Computing modular forms

by William Stein

modformbasis.png

Computing the cuspidal subgroup

by William Stein

ncols not working

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)