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Size: 30457
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Size: 48347
Comment: py3 print
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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 101: | Line 101: |
| REFERENCES: | REFERENCES: |
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| Weisstein, Eric W. "Prime-Generating Polynomial." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Prime-GeneratingPolynomial.html | Weisstein, Eric W. "Prime-Generating Polynomial." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Prime-GeneratingPolynomial.html |
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| elif y<0 and -x >= y and y<x: return 4*(y+1)^2 -11*(y+1) + (start+7) +x 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: |
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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: |
| 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]: |
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| 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 | |
| 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: |
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| 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") |
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| 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 305: | Line 302: |
| == Computing modular forms FIXME == | == Computing modular forms == |
| 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 340: | Line 336: |
| == 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() |
| Line 472: | Line 470: |
| MP += line([(i,0),(i,r)], rgbcolor='black') | MP += line([(i,0),(i,r)], rgbcolor='black') |
| Line 553: | Line 551: |
| S += point(e_pt,pointsize=50, rgbcolor='red') | S += point(e_pt,pointsize=50, rgbcolor='red') |
| Line 556: | Line 554: |
| S += point(f_gs_pt,pointsize=75, rgbcolor='black') | S += point(f_gs_pt,pointsize=75, rgbcolor='black') |
| 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 633: | Line 631: |
| S += point(e_pt,pointsize=50, rgbcolor='red') | S += point(e_pt,pointsize=50, rgbcolor='red') |
| 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) |
|
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| curve = E.plot(rgbcolor = (0,0,1),xmin=25,xmax=25,plot_points=300) | curve = E.plot(rgbcolor = (0,0,1),xmin=-5,xmax=5,plot_points=300) |
| Line 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) |
|
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| 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 801: |
| """ % (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, |
| Line 815: | 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 852: | 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 870: | 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}} |
Contents
Integer Factorization
Divisibility Poset
by William Stein
Factor Trees
by William Stein
More complicated demonstration using Mathematica: http://demonstrations.wolfram.com/FactorTrees/
Factoring an Integer
by Timothy Clemans
Sage implementation of the Mathematica demonstration of the same name. http://demonstrations.wolfram.com/FactoringAnInteger/
Prime Numbers
Illustrating the prime number theorem
by William Stein
Prime Spiral - Square FIXME
by David Runde
Prime Spiral - Polar
by David Runde
Modular Forms
Computing modular forms
by William Stein
Computing the cuspidal subgroup
by William Stein
A Charpoly and Hecke Operator Graph
by William Stein
Modular Arithmetic
Quadratic Residue Table FIXME
by Emily Kirkman
Cubic Residue Table FIXME
by Emily Kirkman
Cyclotomic Fields
Gauss and Jacobi Sums in Complex Plane
by Emily Kirkman
Exhaustive Jacobi Plotter
by Emily Kirkman
Elliptic Curves
Adding points on an elliptic curve
by David Møller Hansen
Plotting an elliptic curve over a finite field
Cryptography
The Diffie-Hellman Key Exchange Protocol
by Timothy Clemans and William Stein
Other
Continued Fraction Plotter
by William Stein
Computing Generalized Bernoulli Numbers
by William Stein (Sage-2.10.3)
Fundamental Domains of SL_2(ZZ)
by Robert Miller
Multiple Zeta Values
by Akhilesh P.
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
