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[[TableOfContents]] |
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== Differential Equations == === Euler's Method in one variable === by Marshall Hampton. This needs some polishing but its usable as is. {{{ def tab_list(y, headers = None): ''' Converts a list into an html table with borders. ''' s = '<table border = 1>' if headers: for q in headers: s = s + '<th>' + str(q) + '</th>' for x in y: s = s + '<tr>' for q in x: s = s + '<td>' + str(q) + '</td>' s = s + '</tr>' s = s + '</table>' return s var('x y') @interact def euler_method(y_exact_in = input_box('-cos(x)+1.0', type = str, label = 'Exact solution = '), y_prime_in = input_box('sin(x)', type = str, label = "y' = "), start = input_box(0.0, label = 'x starting value: '), stop = input_box(6.0, label = 'x stopping value: '), startval = input_box(0.0, label = 'y starting value: '), nsteps = slider([2^m for m in range(0,10)], default = 10, label = 'Number of steps: '), show_steps = slider([2^m for m in range(0,10)], default = 8, label = 'Number of steps shown in table: ')): y_exact = lambda x: eval(y_exact_in) y_prime = lambda x,y: eval(y_prime_in) stepsize = float((stop-start)/nsteps) steps_shown = max(nsteps,show_steps) sol = [startval] xvals = [start] for step in range(nsteps): sol.append(sol[-1] + stepsize*y_prime(xvals[-1],sol[-1])) xvals.append(xvals[-1] + stepsize) sol_max = max(sol + [find_maximum_on_interval(y_exact,start,stop)[0]]) sol_min = min(sol + [find_minimum_on_interval(y_exact,start,stop)[0]]) show(plot(y_exact(x),start,stop,rgbcolor=(1,0,0))+line([[xvals[index],sol[index]] for index in range(len(sol))]),xmin=start,xmax = stop, ymax = sol_max, ymin = sol_min) if nsteps < steps_shown: table_range = range(len(sol)) else: table_range = range(0,floor(steps_shown/2)) + range(len(sol)-floor(steps_shown/2),len(sol)) html(tab_list([[i,xvals[i],sol[i]] for i in table_range], headers = ['step','x','y'])) }}} attachment:eulermethod.png |
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=== Continued Fraction Plotter === by William Stein {{{ @interact 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 show(line([(i,z) for i, z in enumerate(c)],rgbcolor=clr),ymax=ymax,figsize=[10,2]) }}} attachment:contfracplot.png |
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=== Computing Generalized Bernoulli Numbers === by William Stein (Sage-2.10.3) {{{ @interact def _(m=selector([1..15],nrows=2), n=(7,(3..10))): G = DirichletGroup(m) s = "<h3>First n=%s Bernoulli numbers attached to characters with modulus m=%s</h3>"%(n,f) s += '<table border=1>' s += '<tr bgcolor="#edcc9c"><td align=center>$\\chi$</td><td>Conductor</td>' + \ ''.join('<td>$B_{%s,\chi}$</td>'%k for k in [1..n]) + '</tr>' for eps in G.list(): v = ''.join(['<td align=center bgcolor="#efe5cd">$%s$</td>'%latex(eps.bernoulli(k)) for k in [1..n]]) s += '<tr><td bgcolor="#edcc9c">%s</td><td bgcolor="#efe5cd" align=center>%s</td>%s</tr>\n'%( eps, eps.conductor(), v) s += '</table>' html(s) }}} attachment:bernoulli.png === Computing modular forms === by William Stein {{{ j = 0 @interact def _(N=[1..100], k=selector([2,4,..,12],nrows=1), prec=(3..40), group=[(Gamma0, 'Gamma0'), (Gamma1, 'Gamma1')]): M = CuspForms(group(N),k) print j; global j; j += 1 print M; print '\n'*3 print "Computing basis...\n\n" if M.dimension() == 0: print "Space has dimension 0" else: prec = max(prec, M.dimension()+1) for f in M.basis(): view(f.q_expansion(prec)) print "\n\n\nDone computing basis." }}} attachment:modformbasis.png |
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Sage Interactions
Post code that demonstrates the use of the interact command in Sage here. It should be easy for people to just scroll through and paste examples out of here into their own sage notebooks.
We'll likely restructure and reorganize this once we have some nontrivial content and get a sense of how it is laid out.
Miscellaneous
Evaluate a bit of code in a given system
by William Stein (there is no way yet to make the text box big):
@interact def _(system=selector([('sage0', 'Sage'), ('gp', 'PARI'), ('magma', 'Magma')]), code='2+2'): print globals()[system].eval(code)
attachment:evalsys.png
Graph Theory
Automorphism Groups of some Graphs
by William Stein (I spent less than five minutes on this):
@interact def _(graph=['CycleGraph', 'CubeGraph', 'RandomGNP'], n=selector([1..10],nrows=1), p=selector([10,20,..,100],nrows=1)): print graph if graph == 'CycleGraph': print "n (=%s): number of vertices"%n G = graphs.CycleGraph(n) elif graph == 'CubeGraph': if n > 8: print "n reduced to 8" n = 8 print "n (=%s): dimension"%n G = graphs.CubeGraph(n) elif graph == 'RandomGNP': print "n (=%s) vertices"%n print "p (=%s%%) probability"%p G = graphs.RandomGNP(n, p/100.0) print G.automorphism_group() show(plot(G))
attachment:autograph.png
Calculus
A contour map and 3d plot of two inverse distance functions
by William Stein
@interact def _(q1=(-1,(-3,3)), q2=(-2,(-3,3)), cmap=['autumn', 'bone', 'cool', 'copper', 'gray', 'hot', 'hsv', 'jet', 'pink', 'prism', 'spring', 'summer', 'winter']): x,y = var('x,y') f = q1/sqrt((x+1)^2 + y^2) + q2/sqrt((x-1)^2+(y+0.5)^2) C = contour_plot(f, (-2,2), (-2,2), plot_points=30, contours=15, cmap=cmap) show(C, figsize=3, aspect_ratio=1) show(plot3d(f, (x,-2,2), (y,-2,2)), figsize=5, viewer='tachyon')
attachment:mountains.png
A simple tangent line grapher
by Marshall Hampton
html('<h2>Tangent line grapher</h2>') @interact def tangent_line(f = input_box(default=sin(x)), xbegin = slider(0,10,1/10,0), xend = slider(0,10,1/10,10), x0 = slider(0, 1, 1/100, 1/2)): prange = [xbegin, xend] x0i = xbegin + x0*(xend-xbegin) var('x') df = diff(f) tanf = f(x0i) + df(x0i)*(x-x0i) fplot = plot(f, prange[0], prange[1]) print 'Tangent line is y = ' + tanf._repr_() tanplot = plot(tanf, prange[0], prange[1], rgbcolor = (1,0,0)) fmax = f.find_maximum_on_interval(prange[0], prange[1])[0] fmin = f.find_minimum_on_interval(prange[0], prange[1])[0] show(fplot + tanplot, xmin = prange[0], xmax = prange[1], ymax = fmax, ymin = fmin)
attachment:tangents.png
Differential Equations
Euler's Method in one variable
by Marshall Hampton. This needs some polishing but its usable as is.
def tab_list(y, headers = None): ''' Converts a list into an html table with borders. ''' s = '<table border = 1>' if headers: for q in headers: s = s + '<th>' + str(q) + '</th>' for x in y: s = s + '<tr>' for q in x: s = s + '<td>' + str(q) + '</td>' s = s + '</tr>' s = s + '</table>' return s var('x y') @interact def euler_method(y_exact_in = input_box('-cos(x)+1.0', type = str, label = 'Exact solution = '), y_prime_in = input_box('sin(x)', type = str, label = "y' = "), start = input_box(0.0, label = 'x starting value: '), stop = input_box(6.0, label = 'x stopping value: '), startval = input_box(0.0, label = 'y starting value: '), nsteps = slider([2^m for m in range(0,10)], default = 10, label = 'Number of steps: '), show_steps = slider([2^m for m in range(0,10)], default = 8, label = 'Number of steps shown in table: ')): y_exact = lambda x: eval(y_exact_in) y_prime = lambda x,y: eval(y_prime_in) stepsize = float((stop-start)/nsteps) steps_shown = max(nsteps,show_steps) sol = [startval] xvals = [start] for step in range(nsteps): sol.append(sol[-1] + stepsize*y_prime(xvals[-1],sol[-1])) xvals.append(xvals[-1] + stepsize) sol_max = max(sol + [find_maximum_on_interval(y_exact,start,stop)[0]]) sol_min = min(sol + [find_minimum_on_interval(y_exact,start,stop)[0]]) show(plot(y_exact(x),start,stop,rgbcolor=(1,0,0))+line([[xvals[index],sol[index]] for index in range(len(sol))]),xmin=start,xmax = stop, ymax = sol_max, ymin = sol_min) if nsteps < steps_shown: table_range = range(len(sol)) else: table_range = range(0,floor(steps_shown/2)) + range(len(sol)-floor(steps_shown/2),len(sol)) html(tab_list([[i,xvals[i],sol[i]] for i in table_range], headers = ['step','x','y']))
attachment:eulermethod.png
Linear Algebra
Numerical instability of the classical Gram-Schmidt algorithm
by Marshall Hampton (tested by William Stein, who thinks this is really nice!)
def GS_classic(a_list): ''' Given a list of vectors or a matrix, returns the QR factorization using the classical (and numerically unstable) Gram-Schmidt algorithm. ''' if type(a_list) != list: cols = a_list.cols() a_list = [x for x in cols] indices = range(len(a_list)) q = [] r = [[0 for i in indices] for j in indices] v = [a_list[i].copy() for i in indices] for i in indices: for j in range(0,i): r[j][i] = q[j].inner_product(a_list[i]) v[i] = v[i] - r[j][i]*q[j] r[i][i] = (v[i]*v[i])^(1/2) q.append(v[i]/r[i][i]) q = matrix([q[i] for i in indices]).transpose() return q, matrix(r) def GS_modern(a_list): ''' Given a list of vectors or a matrix, returns the QR factorization using the 'modern' Gram-Schmidt algorithm. ''' if type(a_list) != list: cols = a_list.cols() a_list = [x for x in cols] indices = range(len(a_list)) q = [] r = [[0 for i in indices] for j in indices] v = [a_list[i].copy() for i in indices] for i in indices: r[i][i] = v[i].norm(2) q.append(v[i]/r[i][i]) for j in range(i+1, len(indices)): r[i][j] = q[i].inner_product(v[j]) v[j] = v[j] - r[i][j]*q[i] q = matrix([q[i] for i in indices]).transpose() return q, matrix(r) html('<h2>Numerical instability of the classical Gram-Schmidt algorithm</h2>') @interact def gstest(precision = slider(range(3,53), default = 10), a1 = input_box([1,1/1000,1/1000]), a2 = input_box([1,1/1000,0]), a3 = input_box([1,0,1/1000])): myR = RealField(precision) displayR = RealField(5) html('precision in bits: ' + str(precision) + '<br>') A = matrix([a1,a2,a3]) A = [vector(myR,x) for x in A] qn, rn = GS_classic(A) qb, rb = GS_modern(A) html('Classical Gram-Schmidt:') show(matrix(displayR,qn)) html('Stable Gram-Schmidt:') show(matrix(displayR,qb))
attachment:GramSchmidt.png
Number Theory
Continued Fraction Plotter
by William Stein
@interact 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 show(line([(i,z) for i, z in enumerate(c)],rgbcolor=clr),ymax=ymax,figsize=[10,2])
attachment:contfracplot.png
Illustrating the prime number thoerem
by William Stein
@interact def _(N=(100,(2..2000))): html("<font color='red'>$\pi(x)$</font> and <font color='blue'>$x/(\log(x)-1)$</font> for $x < %s$"%N) show(plot(prime_pi, 0, N, rgbcolor='red') + plot(x/(log(x)-1), 5, N, rgbcolor='blue'))
attachment:primes.png
Computing Generalized Bernoulli Numbers
by William Stein (Sage-2.10.3)
@interact def _(m=selector([1..15],nrows=2), n=(7,(3..10))): G = DirichletGroup(m) s = "<h3>First n=%s Bernoulli numbers attached to characters with modulus m=%s</h3>"%(n,f) s += '<table border=1>' s += '<tr bgcolor="#edcc9c"><td align=center>$\\chi$</td><td>Conductor</td>' + \ ''.join('<td>$B_{%s,\chi}$</td>'%k for k in [1..n]) + '</tr>' for eps in G.list(): v = ''.join(['<td align=center bgcolor="#efe5cd">$%s$</td>'%latex(eps.bernoulli(k)) for k in [1..n]]) s += '<tr><td bgcolor="#edcc9c">%s</td><td bgcolor="#efe5cd" align=center>%s</td>%s</tr>\n'%( eps, eps.conductor(), v) s += '</table>' html(s)
attachment:bernoulli.png
Computing modular forms
by William Stein
j = 0 @interact def _(N=[1..100], k=selector([2,4,..,12],nrows=1), prec=(3..40), group=[(Gamma0, 'Gamma0'), (Gamma1, 'Gamma1')]): M = CuspForms(group(N),k) print j; global j; j += 1 print M; print '\n'*3 print "Computing basis...\n\n" if M.dimension() == 0: print "Space has dimension 0" else: prec = max(prec, M.dimension()+1) for f in M.basis(): view(f.q_expansion(prec)) print "\n\n\nDone computing basis."
attachment:modformbasis.png
Computing the cuspidal subgroup
by William Stein
html('<h1>Cuspidal Subgroups of Modular Jacobians J0(N)</h1>') @interact def _(N=selector([1..8*13], ncols=8, width=10, default=10)): A = J0(N) print A.cuspidal_subgroup()
attachment:cuspgroup.png
A Charpoly and Hecke Operator Graph
by William Stein
# Note -- in Sage-2.10.3; multiedges are missing in plots; loops are missing in 3d plots @interact def f(N = prime_range(11,400), p = selector(prime_range(2,12),nrows=1), three_d = ("Three Dimensional", False)): S = SupersingularModule(N) T = S.hecke_matrix(p) G = Graph(T, multiedges=True, loops=not three_d) html("<h1>Charpoly and Hecke Graph: Level %s, T_%s</h1>"%(N,p)) show(T.charpoly().factor()) if three_d: show(G.plot3d(), aspect_ratio=[1,1,1]) else: show(G.plot(),figsize=7)
attachment:heckegraph.png
Demonstrating the Diffie-Hellman Key Exchange Protocol
by Timothy Clemans (refereed by William Stein)
@interact def diffie_hellman(button=selector(["New example"],label='',buttons=True), bits=("Number of bits of prime", (8,12,..512))): maxp = 2^bits p = random_prime(maxp) k = GF(p) g = k.multiplicative_generator() a = ZZ.random_element(10, maxp) b = ZZ.random_element(10, maxp) print """ <html> <style> .gamodp { background:yellow } .gbmodp { background:orange } .dhsame { color:green; font-weight:bold } </style> <h2>%s-Bit Diffie-Hellman Key Exchange</h2> <ol style="color:#000;font:12px Arial, Helvetica, sans-serif"> <li>Alice and Bob agree to use the prime number p=%s and base g=%s.</li> <li>Alice chooses the secret integer a=%s, then sends Bob (<span class="gamodp">g<sup>a</sup> mod p</span>):<br/>%s<sup>%s</sup> mod %s = <span class="gamodp">%s</span>.</li> <li>Bob chooses the secret integer b=%s, then sends Alice (<span class="gbmodp">g<sup>b</sup> mod p</span>):<br/>%s<sup>%s</sup> mod %s = <span class="gbmodp">%s</span>.</li> <li>Alice computes (<span class="gbmodp">g<sup>b</sup> mod p</span>)<sup>a</sup> mod p:<br/>%s<sup>%s</sup> mod %s = <span class="dhsame">%s</span>.</li> <li>Bob computes (<span class="gamodp">g<sup>a</sup> mod p</span>)<sup>b</sup> mod p:<br/>%s<sup>%s</sup> mod %s = <span class="dhsame">%s</span>.</li> </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)
attachment:dh.png
Plotting an elliptic curve over a finite field
E = EllipticCurve('37a') @interact def _(p=slider(prime_range(1000), default=389)): show(E) print "p = %s"%p show(E.change_ring(GF(p)).plot(),xmin=0,ymin=0)
attachment:ellffplot.png
Web apps
Bioinformatics: protein browser
by Marshall Hampton (tested by William Stein)
import urllib2 as U @interact def protein_browser(GenBank_ID = input_box('165940577', type = str), file_type = selector([(1,'fasta'),(2,'GenPept')])): if file_type == 2: gen_str = 'http://www.ncbi.nlm.nih.gov/entrez/viewer.fcgi?db=protein&sendto=t&id=' else: gen_str = 'http://www.ncbi.nlm.nih.gov/entrez/viewer.fcgi?db=protein&sendto=t&dopt=fasta&id=' f = U.urlopen(gen_str + GenBank_ID) g = f.read() f.close() html(g)
attachment:biobrowse.png
Miscellaneous Graphics
Somewhat Silly Egg Painter
by Marshall Hampton (refereed by William Stein)
var('s,t') g(s) = ((0.57496*sqrt(121 - 16.0*s^2))/sqrt(10.+ s)) def P(color, rng): return parametric_plot3d((cos(t)*g(s), sin(t)*g(s), s), (s,rng[0],rng[1]), (t,0,2*pi), plot_points = [150,150], rgbcolor=color, frame = False, opacity = 1) colorlist = ['red','blue','red','blue'] @interact def _(band_number = selector(range(1,5)), current_color = Color('red')): html('<h1 align=center>Egg Painter</h1>') colorlist[band_number-1] = current_color egg = sum([P(colorlist[i],[-2.75+5.5*(i/4),-2.75+5.5*(i+1)/4]) for i in range(4)]) show(egg)
attachment:eggpaint.png