€csage.server.notebook.notebook Notebook q)q}q(U_Notebook__worksheetsq}q(Usieveq(csage.server.notebook.worksheet Worksheet qoq}q (U_Worksheet__filenameq UsieveqU_Worksheet__cellsq]q ((csage.server.notebook.cell TextCell qoq}q(U_TextCell__worksheetqhU_TextCell__textqU

Sieve Worksheet

qU _TextCell__idqMp`ub(csage.server.notebook.cell Cell qoq}q(U _Cell__inqTÅ%hideall #auto def number_grid(c, box_args={}, font_args={},offset=0): """ INPUT: c -- list of length n^2, where n is an integer. The entries of c are RGB colors. box_args -- additional arguments passed to box. font_args -- all additional arguments are passed to the text function, e.g., fontsize. offset -- use to fine tune text placement in the squares OUTPUT: Graphics -- a plot of a grid that illustrates those n^2 numbers colored according to c. """ try: n = square_root(ZZ(len(c))) except ArithmeticError: raise ArithmeticError, "c must have square length" G = Graphics() k = 0 for j in reversed(range(n)): for i in range(n): col = c[int(k)] R = line([(i,j),(i+1,j),(i+1,j+1),(i,j+1),(i,j)], thickness=.2, **box_args) P = polygon([(i,j),(i+1,j),(i+1,j+1),(i,j+1),(i,j)], rgbcolor=col, **box_args) G += P + R if col != (1,1,1): G += text(str(k+1), (i+.5+offset,j+.5), **font_args) k += 1 G.axes(False) G.range(0,n,0,n) return GqU_Cell__introspect_htmlqU!

qU_Cell__worksheetqhU_Cell__completionsq‰U_Cell__out_htmlqUU _Cell__idqMq`U_Cell__is_htmlq ‰U_before_preparseq!TJos.chdir("/Users/was/talks/2007-01-15-sage-cse/sage_notebook/worksheets/sieve/cells/113") __SAGE_t__=cputime() __SAGE_w__=walltime() %hideall #auto def number_grid(c, box_args={}, font_args={},offset=0): """ INPUT: c -- list of length n^2, where n is an integer. The entries of c are RGB colors. box_args -- additional arguments passed to box. font_args -- all additional arguments are passed to the text function, e.g., fontsize. offset -- use to fine tune text placement in the squares OUTPUT: Graphics -- a plot of a grid that illustrates those n^2 numbers colored according to c. """ try: n = square_root(ZZ(len(c))) except ArithmeticError: raise ArithmeticError, "c must have square length" G = Graphics() k = 0 for j in reversed(range(n)): for i in range(n): col = c[int(k)] R = line([(i,j),(i+1,j),(i+1,j+1),(i,j+1),(i,j)], thickness=.2, **box_args) P = polygon([(i,j),(i+1,j),(i+1,j+1),(i,j+1),(i,j)], rgbcolor=col, **box_args) G += P + R if col != (1,1,1): G += text(str(k+1), (i+.5+offset,j+.5), **font_args) k += 1 G.axes(False) G.range(0,n,0,n) return Gq"U _Cell__dirq#U(sage_notebook/worksheets/sieve/cells/113q$U _Cell__outq%U' CPU time: 0.00 s, Wall time: 0.00 s q&Uhas_new_outputq'‰U_Cell__versionq(KU_Cell__sageq)csage.interfaces.sage0 reduce_load_Sage q*)Rq+U_Cell__interruptedq,‰ub(hoq-}q.(hT%hideall #auto def sieve_step(p, n, gone=(1,1,1), prime=(1,0,0), \ multiple=(.6,.6,.6), remaining=(.9,.9,.9), fontsize=11,offset=0): """ Return graphics that illustrates sieving out multiples of p. Numbers that are a nontrivial multiple of primes < p are shown in the gone color. Numbers that are a multiple of p are shown in a different color. The number p is shown in yet a third color. INPUT: p -- a prime (or 1, in which case all points are colored "remaining") n -- a SQUARE integer gone -- rgb color for integers that have been sieved out prime -- rgb color for p multiple -- rgb color for multiples of p remaining -- rgb color for integers that have not been sieved out yet and are not multiples of p. """ if p == 1: c = [remaining]*n else: exclude = prod(prime_range(p)) # exclude multiples of primes < p c = [] for k in range(1,n+1): if k <= p and is_prime(k): c.append(prime) elif k == 1 or (gcd(k,exclude) != 1 and not is_prime(k)): c.append(gone) elif k%p == 0: c.append(multiple) else: c.append(remaining) # end for # end if return number_grid(c,{},{'fontsize':fontsize},offset=offset)q/hU!

q0hhh‰hUhMr`h ‰h!T2os.chdir("/Users/was/talks/2007-01-15-sage-cse/sage_notebook/worksheets/sieve/cells/114") __SAGE_t__=cputime() __SAGE_w__=walltime() %hideall #auto def sieve_step(p, n, gone=(1,1,1), prime=(1,0,0), \ multiple=(.6,.6,.6), remaining=(.9,.9,.9), fontsize=11,offset=0): """ Return graphics that illustrates sieving out multiples of p. Numbers that are a nontrivial multiple of primes < p are shown in the gone color. Numbers that are a multiple of p are shown in a different color. The number p is shown in yet a third color. INPUT: p -- a prime (or 1, in which case all points are colored "remaining") n -- a SQUARE integer gone -- rgb color for integers that have been sieved out prime -- rgb color for p multiple -- rgb color for multiples of p remaining -- rgb color for integers that have not been sieved out yet and are not multiples of p. """ if p == 1: c = [remaining]*n else: exclude = prod(prime_range(p)) # exclude multiples of primes < p c = [] for k in range(1,n+1): if k <= p and is_prime(k): c.append(prime) elif k == 1 or (gcd(k,exclude) != 1 and not is_prime(k)): c.append(gone) elif k%p == 0: c.append(multiple) else: c.append(remaining) # end for # end if return number_grid(c,{},{'fontsize':fontsize},offset=offset)q1h#U(sage_notebook/worksheets/sieve/cells/114q2h%U' CPU time: 0.00 s, Wall time: 0.00 s q3h'‰h(Kh)h+h,‰ub(hoq4}q5(hhhUV

The integers up to 100

Each square contains one of the integers up to 100.q6hMs`ub(hoq7}q8(hUshow(sieve_step(1,100))q9hU!

q:hhh‰U_Cell__introspectq;‰hU@

qh#U(sage_notebook/worksheets/sieve/cells/116q?h%U q@h'‰h(Kh)h+U_Cell__typeqAUwrapqBh,‰ub(hoqC}qD(hhhUc

We sieve out the even integers

We remove each integer (besides 2) that is divisible by 2.qEhMu`ub(hoqF}qG(hUshow(sieve_step(2,100))qHhU!

qIhhh‰h;‰hU@

qJhMv`h=‰h ‰h!Uqos.chdir("/Users/was/talks/2007-01-15-sage-cse/sage_notebook/worksheets/sieve/cells/118") show(sieve_step(2,100))qKh#U(sage_notebook/worksheets/sieve/cells/118qLh%U qMh'‰h(Kh)h*)RqNhAUwrapqOh,‰ub(hoqP}qQ(hhhU.

We sieve out the multiples of 2 and 3

qRhMw`ub(hoqS}qT(hUshow(sieve_step(3,100))qUhU!

qVhhh‰h;‰hU@

qWhMx`h=‰h ‰h!Uqos.chdir("/Users/was/talks/2007-01-15-sage-cse/sage_notebook/worksheets/sieve/cells/120") show(sieve_step(3,100))qXh#U(sage_notebook/worksheets/sieve/cells/120qYh%U qZh'‰h(Kh)hNhAhOh,‰ub(hoq[}q\(hhhU1

We sieve out the multiples of 2, 3 and 5

q]hMy`ub(hoq^}q_(hUshow(sieve_step(5,100))q`hU!

qahhh‰h;‰hU@

qbhMz`h=‰h ‰h!Uqos.chdir("/Users/was/talks/2007-01-15-sage-cse/sage_notebook/worksheets/sieve/cells/122") show(sieve_step(5,100))qch#U(sage_notebook/worksheets/sieve/cells/122qdh%U qeh'‰h(Kh)hNhAhOh,‰ub(hoqf}qg(hhhU4

We sieve out the multiples of 2, 3, 5 and 7

qhhM{`ub(hoqi}qj(hUshow(sieve_step(7,100))qkhU!

qlhhh‰h;‰hU@

qmhM|`h=‰h ‰h!Uqos.chdir("/Users/was/talks/2007-01-15-sage-cse/sage_notebook/worksheets/sieve/cells/124") show(sieve_step(7,100))qnh#U(sage_notebook/worksheets/sieve/cells/124qoh%U qph'‰h(Kh)hNhAhOh,‰ub(hoqq}qr(hhhU7

There are no multiples of 11 left to sieve out

qshM}`ub(hoqt}qu(hUshow(sieve_step(11,100))qvhU!

qwhhh‰h;‰hU@

qxhM~`h=‰h ‰h!Uros.chdir("/Users/was/talks/2007-01-15-sage-cse/sage_notebook/worksheets/sieve/cells/126") show(sieve_step(11,100))qyh#U(sage_notebook/worksheets/sieve/cells/126qzh%U q{h'‰h(Kh)hNhAhOh,‰ub(hoq|}q}(hhhU

Sieving up to 100

q~hM`ub(hoq}q€(hUXs = [sieve_step(p,100) for p in [2,3,5,7]] G = graphics_array(s, 2,2) G.show(axes=False)qhU!

q‚hhh‰h;‰hU@

qƒhM€`h=‰h ‰h!U²os.chdir("/Users/was/talks/2007-01-15-sage-cse/sage_notebook/worksheets/sieve/cells/128") s = [sieve_step(p,100) for p in [2,3,5,7]] G = graphics_array(s, 2,2) G.show(axes=False)q„h#U(sage_notebook/worksheets/sieve/cells/128q…h%U q†h'‰h(Kh)hNhAhOh,‰ub(hoq‡}qˆ(hU#show(sieve_step(13,400,fontsize=6))q‰hU!

qŠhhh‰h;‰hU@

q‹hM‡`h=‰h ‰h!U}os.chdir("/Users/was/talks/2007-01-15-sage-cse/sage_notebook/worksheets/sieve/cells/135") show(sieve_step(13,400,fontsize=6))qŒh#U(sage_notebook/worksheets/sieve/cells/135qh%U qŽh'‰h(Kh)hNhAUwrapqh,‰ub(hoq}q‘(hUhhh‰hUhM`h#U(sage_notebook/worksheets/sieve/cells/141q’h%Uh'‰h(Kh)NhAhh,‰ubeU_Worksheet__synchroq“K5U_Worksheet__comp_is_runningq”‰U_Worksheet__attachedq•}q–U/Users/was/.sage/init.sageq—GAÑjÊesU_Worksheet__variablesq˜]q™(Unumber_grid-functionqšUsieve_step-functionq›eU_Worksheet__dirqœUsage_notebook/worksheets/sieveqU_Worksheet__queueqž]qŸU_Worksheet__next_idq MŽ`U_Worksheet__notebookq¡hU_Worksheet__sageq¢h+U_Worksheet__passcryptq£ˆU_Worksheet__nameq¤Usieveq¥U_Worksheet__saltq¦U1168947677.540541q§U_Worksheet__passcodeq¨U 11G3BJNEUV/Pkq©U_Worksheet__idqªKU_Worksheet__next_block_idq«KU_Worksheet__systemq¬NubUfeaturesq(hoq®}q¯(h Ufeaturesq°h]q±((hoq²}q³(hh®hT«

Some Notebook Features

Images.
Multiple input cells can be queued up for computation, and cancelled at any time.
Persistent:
- Close browser and come back later to see what is happening
- Everything saved every few seconds
List of currently defined variables.
Save/load individual objects

q´hMPub(hoqµ}q¶(hU>show(plot(sin(x),-pi,6*pi,rgbcolor=(0.5,0,0.5)),figsize=[8,3])q·hU!

q¸hh®h‰h;‰hUB

q¹hMPh=‰h ‰h!Ušos.chdir("/Users/was/talks/2007-01-15-sage-cse/sage_notebook/worksheets/features/cells/24") show(plot(sin(x),-pi,6*pi,rgbcolor=(0.5,0,0.5)),figsize=[8,3])qºh#U*sage_notebook/worksheets/features/cells/24q»h%U q¼h'‰h(Kh)h*)Rq½hAhBh,‰ub(hoq¾}q¿(hT>%hide t = Tachyon(xres=700, yres=500, camera_center=(2,7,4), look_at=(2,0,0), raydepth=4) t.light((10,3,2), 1, (1,1,1)) t.light((10,-3,2), 1, (1,1,1)) t.texture('black', color=(0,0,0)) t.texture('red', color=(1,0,0)) t.texture('grey', color=(.9,.9,.9)) t.plane((0,0,0),(0,0,1),'grey') t.cylinder((0,0,0),(1,0,0),.01,'black') t.cylinder((0,0,0),(0,1,0),.01,'black') E = EllipticCurve('37a'); show(E) P = E([0,0]) Q = P n = 60 for i in range(n): Q = Q + P c = i/n + .1 t.texture('r%s'%i,color=(float(i/n),0,0)) t.sphere((Q[0], -Q[1], .01), .04, 'r%s'%i) show(t)qÀhU!

qÁhh®h‰h;‰hUB

qÂhMPh=‰h ‰h!Tšos.chdir("/Users/was/talks/2007-01-15-sage-cse/sage_notebook/worksheets/features/cells/25") %hide t = Tachyon(xres=700, yres=500, camera_center=(2,7,4), look_at=(2,0,0), raydepth=4) t.light((10,3,2), 1, (1,1,1)) t.light((10,-3,2), 1, (1,1,1)) t.texture('black', color=(0,0,0)) t.texture('red', color=(1,0,0)) t.texture('grey', color=(.9,.9,.9)) t.plane((0,0,0),(0,0,1),'grey') t.cylinder((0,0,0),(1,0,0),.01,'black') t.cylinder((0,0,0),(0,1,0),.01,'black') E = EllipticCurve('37a'); show(E) P = E([0,0]) Q = P n = 60 for i in range(n): Q = Q + P c = i/n + .1 t.texture('r%s'%i,color=(float(i/n),0,0)) t.sphere((Q[0], -Q[1], .01), .04, 'r%s'%i) show(t)qÃh#U*sage_notebook/worksheets/features/cells/25qÄU_word_being_completedqÅUTachyonqÆh'‰h(Kh)hNh,‰hAhh%U:

y^2 + y = x^3 - x

qÇub(hoqÈ}qÉ(hUa = (x-10)^5; show(a)qÊhU!

qËhh®h‰h;‰hUhMPh=‰h ‰h!Uqos.chdir("/Users/was/talks/2007-01-15-sage-cse/sage_notebook/worksheets/features/cells/26") a = (x-10)^5; show(a)qÌh#U*sage_notebook/worksheets/features/cells/26qÍh%Uc

x^{5} - 50x^{4} + 1000x^{3} - 10000x^{2} + 50000x - 100000

qÎh'‰h(Kh)h½hAhBh,‰ub(hoqÏ}qÐ(U _Cell__inqÑUlatex(a)qÒU_Cell__introspect_htmlqÓU!

qÔU_Cell__worksheetqÕh®U_Cell__completionsqÖ‰U_Cell__introspectq×‰U_Cell__out_htmlqØUU _Cell__idqÙMPU_Cell__is_htmlqÚ‰U_before_preparseqÛUdos.chdir("/Users/was/talks/2007-01-15-sage-cse/sage_notebook/worksheets/features/cells/31") latex(a)qÜU _Cell__dirqÝU*sage_notebook/worksheets/features/cells/31qÞU _Cell__outqßU= x^{5} - 50x^{4} + 1000x^{3} - 10000x^{2} + 50000x - 100000 qàUhas_new_outputqá‰U_Cell__sageqâh½U_Cell__versionqãKU_Cell__typeqähBU_Cell__timeqå‰U_Cell__interruptedqæ‰ub(hoqç}qè(hUsave aqéhU!

qêhh®h‰h;‰hUhMPh=‰h ‰h!Ubos.chdir("/Users/was/talks/2007-01-15-sage-cse/sage_notebook/worksheets/features/cells/27") save aqëh#U*sage_notebook/worksheets/features/cells/27qìh%U qíh'‰h(Kh)hNhAhh,‰ub(hoqî}qï(hUidef foo(): print """

2 + 2

"""qðhÓU!

qñhh®h‰h×‰hUhMPhå‰hÚ‰hÛUÅos.chdir("/Users/was/talks/2007-01-15-sage-cse/sage_notebook/worksheets/features/cells/28") def foo(): print """

2 + 2

"""qòh#U*sage_notebook/worksheets/features/cells/28qóh%U qôh'‰h(Kh)h½hAhBh,‰ub(hoqõ}qö(hÑUtclass Tensor: def __repr__(self): return "I'm a tensor" def __mul__(self, right): return "abc"q÷hÓU!

qøhÕh®hÖ‰h×‰hØUhÙM"PhÚ‰hÛUÐos.chdir("/Users/was/talks/2007-01-15-sage-cse/sage_notebook/worksheets/features/cells/34") class Tensor: def __repr__(self): return "I'm a tensor" def __mul__(self, right): return "abc"qùhÝU*sage_notebook/worksheets/features/cells/34qúhßU qûhá‰hâh½hãKhähBhå‰hæ‰ub(hoqü}qý(hÑUt = Tensor()qþhÓU!

qÿhÕh®hÖ‰h×‰hØUhÙM!PhÚ‰hÛUhos.chdir("/Users/was/talks/2007-01-15-sage-cse/sage_notebook/worksheets/features/cells/33") t = Tensor()rhÝU*sage_notebook/worksheets/features/cells/33rhßU rhá‰hâh½hãKhähBhå‰hæ‰ub(hor}r(hÑUhÕh®hÖ‰hØUhÙM PhÝU*sage_notebook/worksheets/features/cells/32rhßUhá‰hãKhähBhæ‰ub(hor}r(hÑUfoo()rhÓU!

r hÕh®hÖ‰h×‰hØUhÙMPhÚ‰hÛUaos.chdir("/Users/was/talks/2007-01-15-sage-cse/sage_notebook/worksheets/features/cells/29") foo()r hÝU*sage_notebook/worksheets/features/cells/29rhßUQ

2 + 2

rhá‰hâh½hãKhähBhå‰hæ‰ub(hor }r(hÑUhÕh®hÖ‰hØUhÙMPhÝU*sage_notebook/worksheets/features/cells/30rhßUhá‰hãKhähBhæ‰ubeh“K$h”‰h•}rU/Users/was/.sage/init.sagerGAÑjÊesh˜]r(UTensor-classobjrU9a-sage.rings.polynomial_element.Polynomial_rational_denserUfoo-functionrU t-instancerehœU!sage_notebook/worksheets/featuresrhž]rh M#Ph¡hh¢h½h£ˆh¤Ufeaturesrh¦U1168945214.888470rh¨U 11G3BJNEUV/PkrhªKh«Kh¬NubU bernoullir(hor}r(U_Worksheet__filenamerU bernoullir U_Worksheet__cellsr!]r"((hor#}r$(U_TextCell__worksheetr%jU_TextCell__textr&Uv

Bernoulli Numbers

$$ \frac{x}{e^x - 1} = \sum_{n=0}^{\infty} B_n \frac{x^n}{n!}$$ r'U _TextCell__idr(Mpub(hor)}r*(U _Cell__inr+Utime a = bernoulli(10^4)r,hÓU!

r-U_Cell__worksheetr.jU_Cell__completionsr/‰h×‰U_Cell__out_htmlr0UU _Cell__idr1Mphå‰hÚ‰hÛU›os.chdir("/Users/was/talks/2007-01-15-sage-cse/sage_notebook/worksheets/bernoulli/cells/18") __SAGE_t__=cputime() __SAGE_w__=walltime() a = bernoulli(10^4)r2U _Cell__dirr3U+sage_notebook/worksheets/bernoulli/cells/18r4U _Cell__outr5U' CPU time: 0.34 s, Wall time: 0.36 s r6Uhas_new_outputr7‰U_Cell__versionr8KU_Cell__sager9h*)Rr:U_Cell__typer;hBU_Cell__interruptedr<‰ub(hor=}r>(j+Ulen(str(a))r?hÓU!

r@j.jj/‰h×‰j0Uj1Mphå‰hÚ‰hÛUhos.chdir("/Users/was/talks/2007-01-15-sage-cse/sage_notebook/worksheets/bernoulli/cells/19") len(str(a))rAj3U+sage_notebook/worksheets/bernoulli/cells/19rBj5U 27706 rCj7‰j8Kj9j:j;hBj<‰ub(horD}rE(j+U*time v = bernoulli_mod_p(next_prime(10^4))rFhÓU!

rGj.jj/‰h×‰j0Uj1MphåˆhÚ‰hÛUos.chdir("/Users/was/talks/2007-01-15-sage-cse/sage_notebook/worksheets/bernoulli/cells/20") __SAGE_t__=cputime() __SAGE_w__=walltime() v = bernoulli_mod_p(next_prime(10^4))rHj3U+sage_notebook/worksheets/bernoulli/cells/20rIj5U' CPU time: 0.06 s, Wall time: 0.06 s rJj7‰j8Kj9j:j;hBj<‰ub(horK}rL(j+U,time v = bernoulli_mod_p(next_prime(2*10^5))rMhÓU!

rNj.jj/‰h×‰j0Uj1MphåˆhÚ‰hÛU¯os.chdir("/Users/was/talks/2007-01-15-sage-cse/sage_notebook/worksheets/bernoulli/cells/21") __SAGE_t__=cputime() __SAGE_w__=walltime() v = bernoulli_mod_p(next_prime(2*10^5))rOj3U+sage_notebook/worksheets/bernoulli/cells/21rPj5U' CPU time: 1.57 s, Wall time: 1.58 s rQj7‰j8Kj9j:j;hBj<‰ub(horR}rS(j+Uv[:10]rThÓU!

rUj.jj/‰h×‰j0Uj1Mphå‰hÚ‰hÛUcos.chdir("/Users/was/talks/2007-01-15-sage-cse/sage_notebook/worksheets/bernoulli/cells/22") v[:10]rVj3U+sage_notebook/worksheets/bernoulli/cells/22rWj5U< [1, 1668, 7672, 4527, 7672, 8036, 3603, 1669, 5899, 6162] rXj7‰j8Kj9j:j;hBj<‰ub(horY}rZ(j+U!bernoulli(2) % next_prime(2*10^5)r[j.jj/‰j0Uj1Mpj3U+sage_notebook/worksheets/bernoulli/cells/23r\j5U33334r]j7‰j8Kj9Nj;hOj<‰ub(hor^}r_(j+Uj.jj/‰j0Uj1Mpj3U+sage_notebook/worksheets/bernoulli/cells/24r`j5Uj7‰j8Kj9Nj;hOj<‰ubeU_Worksheet__synchroraKU_Worksheet__namerbU bernoullircU_Worksheet__attachedrd}reU/Users/was/.sage/init.sagerfGAÑjÊesU_Worksheet__saltrgU1168977577.570778rhh˜]ri(Ua-sage.rings.rational.RationalrjUv-listrkeU_Worksheet__dirrlU"sage_notebook/worksheets/bernoullirmU_Worksheet__queuern]roU_Worksheet__next_idrpMph¢j:U_Worksheet__passcryptrqˆU_Worksheet__comp_is_runningrr‰U_Worksheet__passcodersU 11G3BJNEUV/PkrtU_Worksheet__notebookruhU_Worksheet__idrvKU_Worksheet__next_block_idrwKU_Worksheet__systemrxNubUriemannry(horz}r{(h Uriemannr|h]r}((hor~}r(hjzhTË

What is Riemann's Hypothesis?

The Riemann Hypothesis approximates $\pi(X)$, the number of primes $ Riemann Hypothesis: We have $ |\pi(x) - \mbox{Li}(x)| < \sqrt{x}\cdot \log(x) $

Here is a graph of the three functions $X/\log X$, ${\rm Li}(X)$, and $\pi(X)$ for $X<1000$: We first define two SAGE functions (Li is built into SAGE):

def f(x):
    return x / log(x)
def p(x):
    return prime_pi(x)

r€hM@ub(hor}r‚(hUG#auto def f(x): return x / log(x) def p(x): return prime_pi(x)rƒhU!

r„hjzh‰h;‰hUhM@h=‰h ‰h!U¡os.chdir("/Users/was/talks/2007-01-15-sage-cse/sage_notebook/worksheets/riemann/cells/2") #auto def f(x): return x / log(x) def p(x): return prime_pi(x)r…h#U(sage_notebook/worksheets/riemann/cells/2r†h%U r‡h'‰h(Kh)h*)RrˆhAhBh,‰ub(hor‰}rŠ(hjzhUûNow we plot them:

xmax = 1000
G = plot(f,2,xmax, linestyle=':', label='x/log(x)') + \
    plot(Li,2,xmax, marker='.', plot_points=30, plot_division=0, label='Li(x)') + \
    plot(p, 2,xmax, label='pi(x)', rgbcolor=(1,0,0))
G.show()

r‹hM@ub(horŒ}r(hUÑxmax = 20 G = plot(f, 2,xmax, linestyle='--', label='x/log(x)') + \ plot(Li,2,xmax, plot_points=30, plot_division=0, label='Li(x)') + \ plot(p, 2,xmax, label='P(X)', rgbcolor=(1,0,0)) G.show(xmax=xmax)rŽhU!

rhjzh‰h;‰hU@

rhM@h=‰h ‰h!T'os.chdir("/Users/was/talks/2007-01-15-sage-cse/sage_notebook/worksheets/riemann/cells/4") xmax = 20 G = plot(f, 2,xmax, linestyle='--', label='x/log(x)') + plot(Li,2,xmax, plot_points=30, plot_division=0, label='Li(x)') + plot(p, 2,xmax, label='P(X)', rgbcolor=(1,0,0)) G.show(xmax=xmax)r‘h#U(sage_notebook/worksheets/riemann/cells/4r’h%U r“h'‰h(Kh)hNhAhh,‰ub(hor”}r•(hUÒxmax = 500 G = plot(f, 2,xmax, linestyle='--', label='x/log(x)') + \ plot(Li,2,xmax, plot_points=30, plot_division=0, label='Li(x)') + \ plot(p, 2,xmax, label='P(X)', rgbcolor=(1,0,0)) G.show(xmax=xmax)r–hU!

r—hjzh‰h;‰hU@

r˜hM@h=‰h ‰h!T(os.chdir("/Users/was/talks/2007-01-15-sage-cse/sage_notebook/worksheets/riemann/cells/5") xmax = 500 G = plot(f, 2,xmax, linestyle='--', label='x/log(x)') + plot(Li,2,xmax, plot_points=30, plot_division=0, label='Li(x)') + plot(p, 2,xmax, label='P(X)', rgbcolor=(1,0,0)) G.show(xmax=xmax)r™h#U(sage_notebook/worksheets/riemann/cells/5ršh%U r›h'‰h(Kh)hNhAhh,‰ub(horœ}r(hUÔxmax = 20000 G = plot(f, 2,xmax, linestyle='--', label='x/log(x)') + \ plot(Li,2,xmax, plot_points=30, plot_division=0, label='Li(x)') + \ plot(p, 2,xmax, label='P(X)', rgbcolor=(1,0,0)) G.show(xmax=xmax)ržhU!

rŸhjzh‰h;‰hU@

r hM@h=‰h ‰h!T*os.chdir("/Users/was/talks/2007-01-15-sage-cse/sage_notebook/worksheets/riemann/cells/6") xmax = 20000 G = plot(f, 2,xmax, linestyle='--', label='x/log(x)') + plot(Li,2,xmax, plot_points=30, plot_division=0, label='Li(x)') + plot(p, 2,xmax, label='P(X)', rgbcolor=(1,0,0)) G.show(xmax=xmax)r¡h#U(sage_notebook/worksheets/riemann/cells/6r¢h%U r£h'‰h(Kh)hNhAhh,‰ub(hor¤}r¥(hUhjzh‰hUhM@h#U(sage_notebook/worksheets/riemann/cells/7r¦h%Uh'‰h(Kh)NhAhh,‰ubeh“Kh”‰h•}r§U/Users/was/.sage/init.sager¨GAÑjÊesh˜]r©(U f-functionrªU p-functionr«ehœU sage_notebook/worksheets/riemannr¬hž]rh M@h¡hh¢jˆh£ˆh¤Uriemannr®h¦U1168948561.443285r¯h¨U 11G3BJNEUV/Pkr°hªKh«Kh¬NubU interfacer±(hor²}r³(h U interfacer´h]rµ((hor¶}r·(hj²hU3

Example: Interface to GP/PARI and Singular

r¸hM ub(hor¹}rº(hÑUf = maple('sin(x^2)')r»hÓU!

r¼hÕj²hÖ‰h×‰hØUhÙM hÚ‰hÛUros.chdir("/Users/was/talks/2007-01-15-sage-cse/sage_notebook/worksheets/interface/cells/27") f = maple('sin(x^2)')r½hÝU+sage_notebook/worksheets/interface/cells/27r¾hßU r¿há‰hâh*)RrÀhãKhähBhå‰hæ‰ub(horÁ}rÂ(hÑUfhÓU!

rÃhÕj²hÖ‰h×‰hØUhÙM hÚ‰hÛU^os.chdir("/Users/was/talks/2007-01-15-sage-cse/sage_notebook/worksheets/interface/cells/28") frÄhÝU+sage_notebook/worksheets/interface/cells/28rÅhßU sin(x^2) rÆhá‰hâjÀhãKhähBhå‰hæ‰ub(horÇ}rÈ(hÑUs = str(f.integrate('x'))rÉhÓU!

rÊhÕj²hÖ‰h×‰hØUhÙM hÚ‰hÛUvos.chdir("/Users/was/talks/2007-01-15-sage-cse/sage_notebook/worksheets/interface/cells/29") s = str(f.integrate('x'))rËhÝU+sage_notebook/worksheets/interface/cells/29rÌhßU rÍhá‰hâjÀhãKhähBhå‰hæ‰ub(horÎ}rÏ(hÑUshÓU!

rÐhÕj²hÖ‰h×‰hØUhÙM hÚ‰hÛU^os.chdir("/Users/was/talks/2007-01-15-sage-cse/sage_notebook/worksheets/interface/cells/30") srÑhÝU+sage_notebook/worksheets/interface/cells/30rÒhßU6 '1/2*2^(1/2)*Pi^(1/2)*FresnelS(2^(1/2)/Pi^(1/2)*x)' rÓhá‰hâjÀhãKhähBhå‰hæ‰ub(horÔ}rÕ(hÑUgp.eval('b = 2^3')rÖhÓU!

r×hÕj²hÖ‰h×‰hØUhÙM hÚ‰hÛUoos.chdir("/Users/was/talks/2007-01-15-sage-cse/sage_notebook/worksheets/interface/cells/25") gp.eval('b = 2^3')rØhÝU+sage_notebook/worksheets/interface/cells/25rÙhßU '8' rÚhá‰hâjÀhãKhähBhå‰hæ‰ub(horÛ}rÜ(hÑUgp.eval('b^2')rÝhÓU!

rÞhÕj²hÖ‰h×‰hØUhÙM hÚ‰hÛUkos.chdir("/Users/was/talks/2007-01-15-sage-cse/sage_notebook/worksheets/interface/cells/26") gp.eval('b^2')rßhÝU+sage_notebook/worksheets/interface/cells/26ràhßU '64' ráhá‰hâjÀhãKhähBhå‰hæ‰ub(horâ}rã(hUa = gp('x^3 - y^3'); aräU_Cell__introspect_htmlråU!

ræhj²h‰U_Cell__introspectrç‰hUhM U_Cell__timerè‰U_Cell__is_htmlré‰U_before_preparserêUsos.chdir("/Users/was/talks/2007-01-15-sage-cse/sage_notebook/worksheets/interface/cells/16") a = gp('x^3 - y^3'); arëh#U+sage_notebook/worksheets/interface/cells/16rìh%U x^3 - y^3 ríh'‰h(Kh)jÀhAhBh,‰ub(horî}rï(hUa.newtonpoly()rðjåU!

rñhj²h‰jç‰hUhM jè‰jé‰jêUkos.chdir("/Users/was/talks/2007-01-15-sage-cse/sage_notebook/worksheets/interface/cells/17") a.newtonpoly()ròU_word_being_completedróUa.rôh#U+sage_notebook/worksheets/interface/cells/17rõh%T# Traceback (most recent call last): File "", line 1, in File "/Users/was/talks/2007-01-15-sage-cse/sage_notebook/worksheets/interface/code/6.py", line 4, in exec compile(ur'a.newtonpoly()' + '\n', '', 'single') File "/Users/was/talks/2007-01-15-sage-cse/", line 1, in File "/Users/was/s/local/lib/python2.5/site-packages/sage/interfaces/expect.py", line 684, in __call__ return self._obj.parent().function_call(self._name, [self._obj] + list(args)) File "/Users/was/s/local/lib/python2.5/site-packages/sage/interfaces/expect.py", line 631, in function_call return self.new("%s(%s)"%(function, ",".join([s.name() for s in args]))) File "/Users/was/s/local/lib/python2.5/site-packages/sage/interfaces/expect.py", line 533, in new return self(code) File "/Users/was/s/local/lib/python2.5/site-packages/sage/interfaces/expect.py", line 489, in __call__ return cls(self, x) File "/Users/was/s/local/lib/python2.5/site-packages/sage/interfaces/expect.py", line 718, in __init__ raise TypeError, x TypeError: Error executing code in GP/PARI: CODE: sage[3]=newtonpoly(sage[2]); GP/PARI ERROR: *** expected character: ',' instead of: ...age[3]=newtonpoly(sage[2]); ^--röh'‰h(Kh)jÀhAUhiddenr÷h,‰ub(horø}rù(hUtype(a)rújåU!

rûhj²h‰jç‰hUhM jè‰jé‰jêUdos.chdir("/Users/was/talks/2007-01-15-sage-cse/sage_notebook/worksheets/interface/cells/18") type(a)rüh#U+sage_notebook/worksheets/interface/cells/18rýh%U) rþh'‰h(Kh)hNhAhOh,‰ub(horÿ}r(hU parent(a)rjåU!

rhj²h‰jç‰hUhM jè‰jé‰jêUfos.chdir("/Users/was/talks/2007-01-15-sage-cse/sage_notebook/worksheets/interface/cells/19") parent(a)rh#U+sage_notebook/worksheets/interface/cells/19rh%U GP/PARI interpreter rh'‰h(Kh)hNhAhOh,‰ub(hor}r(hU a.factor()rjåU!

r hj²h‰jç‰hUhM jè‰jé‰jêUgos.chdir("/Users/was/talks/2007-01-15-sage-cse/sage_notebook/worksheets/interface/cells/20") a.factor()r h#U+sage_notebook/worksheets/interface/cells/20rh%TÑ Traceback (most recent call last): File "", line 1, in File "/Users/was/talks/2007-01-15-sage-cse/sage_notebook/worksheets/interface/code/16.py", line 4, in exec compile(ur'a.factor()' + '\n', '', 'single') File "/Users/was/talks/2007-01-15-sage-cse/", line 1, in File "/Users/was/s/local/lib/python2.5/site-packages/sage/interfaces/expect.py", line 684, in __call__ return self._obj.parent().function_call(self._name, [self._obj] + list(args)) File "/Users/was/s/local/lib/python2.5/site-packages/sage/interfaces/expect.py", line 631, in function_call return self.new("%s(%s)"%(function, ",".join([s.name() for s in args]))) File "/Users/was/s/local/lib/python2.5/site-packages/sage/interfaces/expect.py", line 533, in new return self(code) File "/Users/was/s/local/lib/python2.5/site-packages/sage/interfaces/expect.py", line 489, in __call__ return cls(self, x) File "/Users/was/s/local/lib/python2.5/site-packages/sage/interfaces/expect.py", line 718, in __init__ raise TypeError, x TypeError: Error executing code in GP/PARI: CODE: sage[5]=factor(sage[4]); GP/PARI ERROR: *** factor: sorry, factor for general polynomials is not yet implemented.rh'‰h(Kh)hNhAhOh,‰ub(hor }r(hUR. = QQ[x,y] b = R(str(a))rjåU!

rhj²h‰jç‰hUhM jè‰jé‰jêU|os.chdir("/Users/was/talks/2007-01-15-sage-cse/sage_notebook/worksheets/interface/cells/21") R. = QQ[x,y] b = R(str(a))rh#U+sage_notebook/worksheets/interface/cells/21rh%U rh'‰h(Kh)hNhAhOh,‰ub(hor}r(hU b.factor()rjåU!

rhj²h‰jç‰hUhM jè‰jé‰jêUgos.chdir("/Users/was/talks/2007-01-15-sage-cse/sage_notebook/worksheets/interface/cells/22") b.factor()rh#U+sage_notebook/worksheets/interface/cells/22rh%U! (-1*y + x) * (y^2 + x*y + x^2) rh'‰h(Kh)hNhAhOh,‰ub(hor}r(hUjåU!

rhj²h‰jç‰hUhM jè‰jé‰jêU]os.chdir("/Users/was/talks/2007-01-15-sage-cse/sage_notebook/worksheets/interface/cells/23") rh#U+sage_notebook/worksheets/interface/cells/23rh%U r h'‰h(Kh)hNhAhOh,‰ubeh“K;h”‰h•}r!U/Users/was/.sage/init.sager"GAÑjÊesh˜]r#(Ua-sage.interfaces.gp.GpElementr$U$f-sage.interfaces.maple.MapleElementr%Us-strr&ehœU"sage_notebook/worksheets/interfacer'hž]r(h M h¡hh¢jÀh£ˆh¤U interfacer)h¦U1168948633.588002r*h¨U 11G3BJNEUV/Pkr+hªKh«Kh¬NubU _scratch_r,(hor-}r.(U_Worksheet__filenamer/U _scratch_r0U_Worksheet__cellsr1]r2((hor3}r4(hj-hT›

SAGE: Software for Algebra and Geometry Experimentation

r5hM6ub(hor6}r7(hUhj-h‰hUhM7h#U,sage_notebook/worksheets/_scratch_/cells/311r8h%Uh'‰h(Kh)NhAhh,‰ub(hor9}r:(hUhj-h‰hUhM8h#U,sage_notebook/worksheets/_scratch_/cells/312r;h%Uh'‰h(Kh)NhAhh,‰ubeU_Worksheet__synchror<K†U_Worksheet__namer=U _scratch_r>U_Worksheet__dirr?U"sage_notebook/worksheets/_scratch_r@U_Worksheet__attachedrA}rBU/Users/was/.sage/init.sagerCGAÑjÊesU_Worksheet__queuerD]rEU_Worksheet__next_idrFM9U_Worksheet__passcryptrGˆU_Worksheet__comp_is_runningrH‰U_Worksheet__saltrIU1168543793.388489rJU_Worksheet__notebookrKhU_Worksheet__idrLKU_Worksheet__next_block_idrMKU_Worksheet__systemrNNU_Worksheet__passcoderOU 11G3BJNEUV/PkrPubUtalkrQ(horR}rS(U_Worksheet__filenamerTUtalkrUU_Worksheet__cellsrV]rW((horX}rY(hjRhU(

A Few Introductory Calculations

rZhMZub(hor[}r\(hU122 + 3939^100r]hU!

r^hjRh‰h;‰hUhM[h=‰h ‰h!Ufos.chdir("/Users/was/talks/2007-01-15-sage-cse/sage_notebook/worksheets/talk/cells/91") 122 + 3939^100r_h#U&sage_notebook/worksheets/talk/cells/91r`h%Tk 345619375323669975367860807124255037076153194188885931584246755620617168037547598722532539129249459087563155698072067777782990662920343395325000445341613889629673957434153020923227407467827494067876849537309882945198261109518655022861231720822535378266349041507766551309275258117413203506790245934260655896057550180870173070732758020223605128239775421756626123 rah'‰h(Kh)h*)RrbhAUwraprch,‰ub(hord}re(hUtime v = prime_range(10000)rfhU!

rghjRh‰h;‰hUhM\h=ˆh ‰h!U™os.chdir("/Users/was/talks/2007-01-15-sage-cse/sage_notebook/worksheets/talk/cells/92") __SAGE_t__=cputime() __SAGE_w__=walltime() v = prime_range(10000)rhh#U&sage_notebook/worksheets/talk/cells/92rih%U' CPU time: 0.00 s, Wall time: 0.01 s rjh'‰h(Kh)jbhAhBh,‰ub(hork}rl(hÑUhÕjRhÖ‰hØUhÙMehÝU'sage_notebook/worksheets/talk/cells/101rmhßUhá‰hãKhähBhæ‰ub(horn}ro(hUpreparse('2 ^ 3')rphU!

rqhjRh‰h;‰hUhM]h=‰h ‰h!Uios.chdir("/Users/was/talks/2007-01-15-sage-cse/sage_notebook/worksheets/talk/cells/93") preparse('2 ^ 3')rrh#U&sage_notebook/worksheets/talk/cells/93rsh%U 'Integer(2) ** Integer(3)' rth'‰h(Kh)jbhAhBh,‰ub(horu}rv(hU%python 2 ** 3rwhU!

rxhjRh‰h;‰hUhM^h=‰h ‰h!Ufos.chdir("/Users/was/talks/2007-01-15-sage-cse/sage_notebook/worksheets/talk/cells/94") %python 2 ** 3ryh#U&sage_notebook/worksheets/talk/cells/94rzh%U 8 r{h'‰h(Kh)jbhAhBh,‰ub(hor|}r}(hUtime n=factorial(10^6)r~hU!

rhjRh‰h;‰hUhM_h=ˆh ‰h!U”os.chdir("/Users/was/talks/2007-01-15-sage-cse/sage_notebook/worksheets/talk/cells/95") __SAGE_t__=cputime() __SAGE_w__=walltime() n=factorial(10^6)r€h#U&sage_notebook/worksheets/talk/cells/95rh%U' CPU time: 2.20 s, Wall time: 2.24 s r‚h'‰h(Kh)jbhAhBh,‰ub(horƒ}r„(hUshow(factor( factorial(40) ))r…hU!

r†hjRh‰h;‰hUhM`h=‰h ‰h!Uuos.chdir("/Users/was/talks/2007-01-15-sage-cse/sage_notebook/worksheets/talk/cells/96") show(factor( factorial(40) ))r‡h#U&sage_notebook/worksheets/talk/cells/96rˆh%U¬

2^{38} \cdot 3^{18} \cdot 5^{9} \cdot 7^{5} \cdot 11^{3} \cdot 13^{3} \cdot 17^{2} \cdot 19^{2} \cdot 23 \cdot 29 \cdot 31 \cdot 37

r‰h'‰h(Kh)jbhAhBh,‰ub(horŠ}r‹(hU:m = matrix(5,5,[randint(0,1) for _ in range(25)]); show(m)rŒhU!

rhjRh‰h;‰hUhMah=‰h ‰h!U’os.chdir("/Users/was/talks/2007-01-15-sage-cse/sage_notebook/worksheets/talk/cells/97") m = matrix(5,5,[randint(0,1) for _ in range(25)]); show(m)rŽh#U&sage_notebook/worksheets/talk/cells/97rh%U

\left(\begin{array}{rrrrr} 1&1&1&1&0\\ 0&1&0&0&1\\ 0&0&0&1&0\\ 1&0&0&1&1\\ 1&0&0&1&1 \end{array}\right)

rh'‰h(Kh)jbhAhBh,‰ub(hor‘}r’(hUf = m.charpoly('T'); show(f)r“hU!

r”hjRh‰h;‰hUhMbh=‰h ‰h!Utos.chdir("/Users/was/talks/2007-01-15-sage-cse/sage_notebook/worksheets/talk/cells/98") f = m.charpoly('T'); show(f)r•h#U&sage_notebook/worksheets/talk/cells/98r–h%UM

T^{5} - 4T^{4} + 4T^{3} - 3T^{2} + T

r—h'‰h(Kh)jbhAhBh,‰ub(hor˜}r™(hUshow(factor(f))ršhU!

r›hjRh‰h;‰hUhMch=‰h ‰h!Ugos.chdir("/Users/was/talks/2007-01-15-sage-cse/sage_notebook/worksheets/talk/cells/99") show(factor(f))rœh#U&sage_notebook/worksheets/talk/cells/99rh%US

T \cdot (T^{4} - 4T^{3} + 4T^{2} - 3T + 1)

ržh'‰h(Kh)jbhAhBh,‰ub(horŸ}r (hUhjRh‰hUhMdh#U'sage_notebook/worksheets/talk/cells/100r¡h%Uh'‰h(Kh)NhAhh,‰ubeU_Worksheet__synchror¢KªU_Worksheet__comp_is_runningr£‰U_Worksheet__attachedr¤}r¥U/Users/was/.sage/init.sager¦GAÑjÊesh˜]r§(U_-intr¨U8f-sage.rings.polynomial_element.Polynomial_integer_denser©U7m-sage.matrix.matrix_integer_dense.Matrix_integer_denserªUn-sage.rings.integer.Integerr«Uv-listr¬eU_Worksheet__dirrUsage_notebook/worksheets/talkr®U_Worksheet__queuer¯]r°U_Worksheet__next_idr±Mfh¢jbU_Worksheet__passcryptr²ˆU_Worksheet__namer³Utalkr´U_Worksheet__saltrµU1168942707.757150r¶U_Worksheet__notebookr·hU_Worksheet__idr¸KU_Worksheet__next_block_idr¹KU_Worksheet__systemrºNU_Worksheet__passcoder»U 11G3BJNEUV/Pkr¼ubUsagexr½(hor¾}r¿(h UsagexrÀh]rÁ((horÂ}rÃ(hj¾hU0

Compiled code... right in the notebook!

rÄhM0ub(horÅ}rÆ(hUSdef is2pow(n): while n != 0 and n%2 == 0: n = n >> 1 return n == 1rÇhÓU!

rÈhj¾h‰h×‰hUhM0hå‰hÚ‰hÛU«os.chdir("/Users/was/talks/2007-01-15-sage-cse/sage_notebook/worksheets/sagex/cells/6") def is2pow(n): while n != 0 and n%2 == 0: n = n >> 1 return n == 1rÉh#U&sage_notebook/worksheets/sagex/cells/6rÊh%U rËh'‰h(Kh)h*)RrÌhAhBh,‰ub(horÍ}rÎ(hU1time v = [n for n in range(10^5) if is2pow(n)]; vrÏhÓU!

rÐhj¾h‰h×‰hUhM0håˆhÚ‰hÛU¯os.chdir("/Users/was/talks/2007-01-15-sage-cse/sage_notebook/worksheets/sagex/cells/7") __SAGE_t__=cputime() __SAGE_w__=walltime() v = [n for n in range(10^5) if is2pow(n)]; vrÑh#U&sage_notebook/worksheets/sagex/cells/7rÒh%U| [1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536] CPU time: 3.29 s, Wall time: 3.31 s rÓh'‰h(Kh)jÌhAhBh,‰ub(horÔ}rÕ(hUp%sagex def is2pow_compiled(unsigned int n): while n != 0 and n%2 == 0: n = n >> 1 return n == 1rÖhÓU!

r×hj¾h‰h×‰hUf_sage5.crØhM0hå‰hÚ‰hÛUÈos.chdir("/Users/was/talks/2007-01-15-sage-cse/sage_notebook/worksheets/sagex/cells/8") %sagex def is2pow_compiled(unsigned int n): while n != 0 and n%2 == 0: n = n >> 1 return n == 1rÙh#U&sage_notebook/worksheets/sagex/cells/8rÚh%U rÛh'‰h(Kh)jÌhAhBh,‰ub(horÜ}rÝ(hUrßhj¾h‰h×‰hUhM 0håˆhÚ‰hÛU¸os.chdir("/Users/was/talks/2007-01-15-sage-cse/sage_notebook/worksheets/sagex/cells/9") __SAGE_t__=cputime() __SAGE_w__=walltime() v = [n for n in range(10^6) if is2pow_compiled(n)]; vràh#U&sage_notebook/worksheets/sagex/cells/9ráh%U” [1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288] CPU time: 0.40 s, Wall time: 0.40 s râh'‰h(Kh)jÌhAhBh,‰ub(horã}rä(hÑUhÕj¾hÖ‰hØUhÙM 0hÝU'sage_notebook/worksheets/sagex/cells/10råhßUhá‰hãKhähBhæ‰ubeU_Worksheet__synchroræKh”‰hœUsage_notebook/worksheets/sagexrçh˜]rè(Uis2pow-functionréU*is2pow_compiled-builtin_function_or_methodrêUn-intrëUv-listrìeh•}ríU/Users/was/.sage/init.sagerîGAÑjÊeshž]rïh M0h¡hh£ˆh¤Usagexrðh¦U1168948540.675805rñh¨U 11G3BJNEUV/PkròU_Worksheet__next_block_idróKhªKh¢jÌh¬NubuU_Notebook__historyrô]rõ(U;# Worksheet '_scratch_' (2007-01-11 at 11:33) sage: 2 + 3 5röU¦# Worksheet '_scratch_' (2007-01-11 at 11:33) sage: m = matrix(5,5, range(25)); m [ 0 1 2 3 4] [ 5 6 7 8 9] [10 11 12 13 14] [15 16 17 18 19] [20 21 22 23 24]r÷UE# Worksheet '_scratch_' (2007-01-11 at 11:34) sage: f = m.charpoly() røUL# Worksheet '_scratch_' (2007-01-11 at 11:34) sage: f x^5 - 60*x^4 - 250*x^3rùU{# Worksheet '_scratch_' (2007-01-11 at 11:34) sage: show(f)

x^{5} - 60x^{4} - 250x^{3}

rúUˆ# Worksheet '_scratch_' (2007-01-11 at 11:34) sage: show(factor(f))

(x^{2} - 60x - 250) \cdot x^{3}

rûU;# Worksheet '_scratch_' (2007-01-11 at 11:37) sage: 2 ^ 3 8rüUI# Worksheet '_scratch_' (2007-01-11 at 11:37) sage: %python sage: 2 ^ 3 1rýUb# Worksheet '_scratch_' (2007-01-11 at 11:38) sage: %maple sage: integrate(Sin[x],x) int(Sin[x],x)rþU\# Worksheet '_scratch_' (2007-01-11 at 11:38) sage: %maple sage: integrate(sin(x),x) -cos(x)rÿUˆ# Worksheet '_scratch_' (2007-01-11 at 11:38) sage: %maple sage: integrate(sin(x^2),x) 1/2*2^(1/2)*Pi^(1/2)*FresnelS(2^(1/2)/Pi^(1/2)*x)rT## Worksheet '_scratch_' (2007-01-11 at 11:40) sage: t = Tachyon(xres=800,yres=800, camera_center=(2,5,2), look_at=(2.5,0,0)) sage: t.light((0,0,100), 1, (1,1,1)) sage: t.texture('r', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(1,0,0)) sage: for i in srange(0,50,0.1): ... t.sphere((i/10,sin(i),cos(i)), 0.05, 'r') ... sage: t.texture('white', color=(1,1,1), opacity=1, specular=1, diffuse=1) sage: t.plane((0,0,-100), (0,0,-100), 'white') sage: t.save('sage.png') rTõ# Worksheet '_scratch_' (2007-01-11 at 11:41) A beautiful picture of rational points on a rank 1 elliptic curve. sage: t = Tachyon(xres=1000, yres=800, camera_center=(2,7,4), look_at=(2,0,0), raydepth=4) sage: t.light((10,3,2), 1, (1,1,1)) sage: t.light((10,-3,2), 1, (1,1,1)) sage: t.texture('black', color=(0,0,0)) sage: t.texture('red', color=(1,0,0)) sage: t.texture('grey', color=(.9,.9,.9)) sage: t.plane((0,0,0),(0,0,1),'grey') sage: t.cylinder((0,0,0),(1,0,0),.01,'black') sage: t.cylinder((0,0,0),(0,1,0),.01,'black') sage: E = EllipticCurve('37a') sage: P = E([0,0]) sage: Q = P sage: n = 100 sage: for i in range(n): ... Q = Q + P ... c = i/n + .1 ... t.texture('r%s'%i,color=(float(i/n),0,0)) ... t.sphere((Q[0], -Q[1], .01), .04, 'r%s'%i) ... ... sage: t.save('sage.png') # long time, e.g., 10-20 seconds rTô# Worksheet '_scratch_' (2007-01-11 at 11:41) A beautiful picture of rational points on a rank 1 elliptic curve. sage: t = Tachyon(xres=1000, yres=800, camera_center=(2,7,4), look_at=(2,0,0), raydepth=4) sage: t.light((10,3,2), 1, (1,1,1)) sage: t.light((10,-3,2), 1, (1,1,1)) sage: t.texture('black', color=(0,0,0)) sage: t.texture('red', color=(1,0,0)) sage: t.texture('grey', color=(.9,.9,.9)) sage: t.plane((0,0,0),(0,0,1),'grey') sage: t.cylinder((0,0,0),(1,0,0),.01,'black') sage: t.cylinder((0,0,0),(0,1,0),.01,'black') sage: E = EllipticCurve('37a') sage: P = E([0,0]) sage: Q = P sage: n = 40 sage: for i in range(n): ... Q = Q + P ... c = i/n + .1 ... t.texture('r%s'%i,color=(float(i/n),0,0)) ... t.sphere((Q[0], -Q[1], .01), .04, 'r%s'%i) ... ... sage: t.save('sage.png') # long time, e.g., 10-20 seconds rT# Worksheet '_scratch_' (2007-01-11 at 11:43) sage: t = Tachyon(xres=1000, yres=800, camera_center=(2,7,4), look_at=(2,0,0), raydepth=4) sage: t.light((10,3,2), 1, (1,1,1)) sage: t.light((10,-3,2), 1, (1,1,1)) sage: t.texture('black', color=(0,0,0)) sage: t.texture('red', color=(1,0,0)) sage: t.texture('grey', color=(.9,.9,.9)) sage: t.plane((0,0,0),(0,0,1),'grey') sage: t.cylinder((0,0,0),(1,0,0),.01,'black') sage: t.cylinder((0,0,0),(0,1,0),.01,'black') sage: E = EllipticCurve('37a'); show(E) sage: P = E([0,0]) sage: Q = P sage: n = 40 sage: for i in range(n): ... Q = Q + P ... c = i/n + .1 ... t.texture('r%s'%i,color=(float(i/n),0,0)) ... t.sphere((Q[0], -Q[1], .01), .04, 'r%s'%i) ... sage: show(t)

y^2 + y = x^3 - x

rT# Worksheet '_scratch_' (2007-01-11 at 11:43) sage: t = Tachyon(xres=1000, yres=800, camera_center=(2,7,4), look_at=(2,0,0), raydepth=4) sage: t.light((10,3,2), 1, (1,1,1)) sage: t.light((10,-3,2), 1, (1,1,1)) sage: t.texture('black', color=(0,0,0)) sage: t.texture('red', color=(1,0,0)) sage: t.texture('grey', color=(.9,.9,.9)) sage: t.plane((0,0,0),(0,0,1),'grey') sage: t.cylinder((0,0,0),(1,0,0),.01,'black') sage: t.cylinder((0,0,0),(0,1,0),.01,'black') sage: E = EllipticCurve('37a'); show(E) sage: P = E([0,0]) sage: Q = P sage: n = 60 sage: for i in range(n): ... Q = Q + P ... c = i/n + .1 ... t.texture('r%s'%i,color=(float(i/n),0,0)) ... t.sphere((Q[0], -Q[1], .01), .04, 'r%s'%i) ... sage: show(t)

y^2 + y = x^3 - x

rT# Worksheet '_scratch_' (2007-01-11 at 11:43) sage: t = Tachyon(xres=700, yres=500, camera_center=(2,7,4), look_at=(2,0,0), raydepth=4) sage: t.light((10,3,2), 1, (1,1,1)) sage: t.light((10,-3,2), 1, (1,1,1)) sage: t.texture('black', color=(0,0,0)) sage: t.texture('red', color=(1,0,0)) sage: t.texture('grey', color=(.9,.9,.9)) sage: t.plane((0,0,0),(0,0,1),'grey') sage: t.cylinder((0,0,0),(1,0,0),.01,'black') sage: t.cylinder((0,0,0),(0,1,0),.01,'black') sage: E = EllipticCurve('37a'); show(E) sage: P = E([0,0]) sage: Q = P sage: n = 60 sage: for i in range(n): ... Q = Q + P ... c = i/n + .1 ... t.texture('r%s'%i,color=(float(i/n),0,0)) ... t.sphere((Q[0], -Q[1], .01), .04, 'r%s'%i) ... sage: show(t)

y^2 + y = x^3 - x

rUA# Worksheet '_scratch_' (2007-01-11 at 11:45) sage: f = sin(x^2) rU»# Worksheet '_scratch_' (2007-01-11 at 11:45) sage: g = f.integrate(x); g Traceback (most recent call last): ... AttributeError: 'Function_composition' object has no attribute 'integrate'rU~# Worksheet '_scratch_' (2007-01-11 at 11:45) sage: g = f.integral(x); g Traceback (most recent call last): ... AssertionErrorr UT# Worksheet '_scratch_' (2007-01-11 at 11:45) sage: from sage.calculus.all import * r UA# Worksheet '_scratch_' (2007-01-11 at 11:45) sage: f = sin(x^2) rTž# Worksheet '_scratch_' (2007-01-11 at 11:45) sage: g = f.integral(x); g Traceback (most recent call last): ... SyntaxError: invalid syntax (, line 1) Error using SAGE to evaluate 'sqrt(%pi)*((sqrt(Integer(2))*%i + sqrt(Integer(2)))*erf((sqrt(Integer(2))*%i + sqrt(Integer(2)))*x/Integer(2)) + (sqrt(Integer(2))*%i - sqrt(Integer(2)))*erf((sqrt(Integer(2))*%i - sqrt(Integer(2)))*x/Integer(2)))/Integer(8)'rUH# Worksheet '_scratch_' (2007-01-11 at 11:47) sage: I 1.00000000000000*Ir T# Worksheet '_scratch_' (2007-01-11 at 11:48) sage: %hideall sage: #auto sage: sage: # The source code to draw factorization trees... sage: sage: import random sage: sage: def factor_tree(n, font=10): ... rows = [] ... v = [(n,None,0)] ... ftree(rows, v, 0, factor(n)) ... return draw_ftree(rows, font) ... sage: sage: def ftree(rows, v, i, F): ... if len(v) > 0: ... # add a row to g at the ith level. ... rows.append(v) ... w = [] ... for i in range(len(v)): ... k, _,_ = v[i] ... if k is None or is_prime(k): ... w.append((None,None,None)) ... else: ... d = random.choice(divisors(k)[1:-1]) ... w.append((d,k,i)) ... e = k//d ... if e == 1: ... w.append((None,None)) ... else: ... w.append((e,k,i)) ... if len(w) > len(v): ... ftree(rows, w, i+1, F) ... sage: sage: sage: def draw_ftree(rows,font): ... g = Graphics() ... for i in range(len(rows)): ... cur = rows[i] ... for j in range(len(cur)): ... e, f, k = cur[j] ... if not e is None: ... if is_prime(e): ... c = (1,0,0) ... else: ... c = (0,0,.4) ... g += text(str(e), (j*2-len(cur),-i), fontsize=font, rgbcolor=c) ... if not k is None and not f is None: ... g += line([(j*2-len(cur),-i), (k*2-len(rows[i-1]),-i+1)], alpha=0.4, thickness=1) ... return g ... sage: sage: def factor_trees(n, cols=4, rows=3, font=10): ... return graphics_array([[factor_tree(n,font) for _ in range(cols)] for _ in range(rows)]) CPU time: 0.00 s, Wall time: 0.00 srUT# Worksheet '_scratch_' (2007-01-11 at 11:48) sage: from sage.calculus.all import * rUA# Worksheet '_scratch_' (2007-01-11 at 11:48) sage: f = sin(x^2) rT# Worksheet '_scratch_' (2007-01-11 at 11:49) sage: %hideall sage: #auto sage: sage: # The source code to draw factorization trees... sage: sage: import random sage: sage: def factor_tree(n, font=10): ... rows = [] ... v = [(n,None,0)] ... ftree(rows, v, 0, factor(n)) ... return draw_ftree(rows, font) ... sage: sage: def ftree(rows, v, i, F): ... if len(v) > 0: ... # add a row to g at the ith level. ... rows.append(v) ... w = [] ... for i in range(len(v)): ... k, _,_ = v[i] ... if k is None or is_prime(k): ... w.append((None,None,None)) ... else: ... d = random.choice(divisors(k)[1:-1]) ... w.append((d,k,i)) ... e = k//d ... if e == 1: ... w.append((None,None)) ... else: ... w.append((e,k,i)) ... if len(w) > len(v): ... ftree(rows, w, i+1, F) ... sage: sage: sage: def draw_ftree(rows,font): ... g = Graphics() ... for i in range(len(rows)): ... cur = rows[i] ... for j in range(len(cur)): ... e, f, k = cur[j] ... if not e is None: ... if is_prime(e): ... c = (1,0,0) ... else: ... c = (0,0,.4) ... g += text(str(e), (j*2-len(cur),-i), fontsize=font, rgbcolor=c) ... if not k is None and not f is None: ... g += line([(j*2-len(cur),-i), (k*2-len(rows[i-1]),-i+1)], alpha=0.4, thickness=1) ... return g ... sage: sage: def factor_trees(n, cols=4, rows=3, font=10): ... return graphics_array([[factor_tree(n,font) for _ in range(cols)] for _ in range(rows)]) CPU time: 0.00 s, Wall time: 0.00 srUT# Worksheet '_scratch_' (2007-01-11 at 11:49) sage: from sage.calculus.all import * rUA# Worksheet '_scratch_' (2007-01-11 at 11:49) sage: f = sin(x^2) rU•# Worksheet '_scratch_' (2007-01-11 at 11:49) sage: g = f.integral(x); g Traceback (most recent call last): ... ValueError: too many values to unpackrT# Worksheet '_scratch_' (2007-01-11 at 11:49) sage: %hideall sage: #auto sage: sage: # The source code to draw factorization trees... sage: sage: import random sage: sage: def factor_tree(n, font=10): ... rows = [] ... v = [(n,None,0)] ... ftree(rows, v, 0, factor(n)) ... return draw_ftree(rows, font) ... sage: sage: def ftree(rows, v, i, F): ... if len(v) > 0: ... # add a row to g at the ith level. ... rows.append(v) ... w = [] ... for i in range(len(v)): ... k, _,_ = v[i] ... if k is None or is_prime(k): ... w.append((None,None,None)) ... else: ... d = random.choice(divisors(k)[1:-1]) ... w.append((d,k,i)) ... e = k//d ... if e == 1: ... w.append((None,None)) ... else: ... w.append((e,k,i)) ... if len(w) > len(v): ... ftree(rows, w, i+1, F) ... sage: sage: sage: def draw_ftree(rows,font): ... g = Graphics() ... for i in range(len(rows)): ... cur = rows[i] ... for j in range(len(cur)): ... e, f, k = cur[j] ... if not e is None: ... if is_prime(e): ... c = (1,0,0) ... else: ... c = (0,0,.4) ... g += text(str(e), (j*2-len(cur),-i), fontsize=font, rgbcolor=c) ... if not k is None and not f is None: ... g += line([(j*2-len(cur),-i), (k*2-len(rows[i-1]),-i+1)], alpha=0.4, thickness=1) ... return g ... sage: sage: def factor_trees(n, cols=4, rows=3, font=10): ... return graphics_array([[factor_tree(n,font) for _ in range(cols)] for _ in range(rows)]) CPU time: 0.00 s, Wall time: 0.00 srUT# Worksheet '_scratch_' (2007-01-11 at 11:49) sage: from sage.calculus.all import * rUA# Worksheet '_scratch_' (2007-01-11 at 11:49) sage: f = sin(x^2) rT# Worksheet '_scratch_' (2007-01-11 at 11:49) sage: g = f.integral(x); g sqrt(_Pi_)*((sqrt(2)*_I_ + sqrt(2))*erf((sqrt(2)*_I_ + sqrt(2))*x/2) + (sqrt(2)*_I_ - sqrt(2))*erf((sqrt(2)*_I_ - sqrt(2))*x/2))/8 Traceback (most recent call last): File "", line 1, in File "/Users/was/talks/2007-01-11-uw-undergrads/sage_notebook/worksheets/_scratch_/code/4.py", line 4, in exec compile(ur'g = f.integral(x); g' + '\n', '', 'single') File "/Users/was/talks/2007-01-11-uw-undergrads/", line 1, in ...rUM# Worksheet '_scratch_' (2007-01-11 at 11:50) sage: SER(I) 1.00000000000000*IrT# Worksheet '_scratch_' (2007-01-11 at 11:50) sage: %hideall sage: #auto sage: sage: # The source code to draw factorization trees... sage: sage: import random sage: sage: def factor_tree(n, font=10): ... rows = [] ... v = [(n,None,0)] ... ftree(rows, v, 0, factor(n)) ... return draw_ftree(rows, font) ... sage: sage: def ftree(rows, v, i, F): ... if len(v) > 0: ... # add a row to g at the ith level. ... rows.append(v) ... w = [] ... for i in range(len(v)): ... k, _,_ = v[i] ... if k is None or is_prime(k): ... w.append((None,None,None)) ... else: ... d = random.choice(divisors(k)[1:-1]) ... w.append((d,k,i)) ... e = k//d ... if e == 1: ... w.append((None,None)) ... else: ... w.append((e,k,i)) ... if len(w) > len(v): ... ftree(rows, w, i+1, F) ... sage: sage: sage: def draw_ftree(rows,font): ... g = Graphics() ... for i in range(len(rows)): ... cur = rows[i] ... for j in range(len(cur)): ... e, f, k = cur[j] ... if not e is None: ... if is_prime(e): ... c = (1,0,0) ... else: ... c = (0,0,.4) ... g += text(str(e), (j*2-len(cur),-i), fontsize=font, rgbcolor=c) ... if not k is None and not f is None: ... g += line([(j*2-len(cur),-i), (k*2-len(rows[i-1]),-i+1)], alpha=0.4, thickness=1) ... return g ... sage: sage: def factor_trees(n, cols=4, rows=3, font=10): ... return graphics_array([[factor_tree(n,font) for _ in range(cols)] for _ in range(rows)]) CPU time: 0.00 s, Wall time: 0.00 srUT# Worksheet '_scratch_' (2007-01-11 at 11:50) sage: from sage.calculus.all import * rUA# Worksheet '_scratch_' (2007-01-11 at 11:50) sage: f = sin(x^2) rT# Worksheet '_scratch_' (2007-01-11 at 11:50) sage: g = f.integral(x); g sqrt(_Pi_)*((sqrt(2)*_I_ + sqrt(2))*erf((sqrt(2)*_I_ + sqrt(2))*x/2) + (sqrt(2)*_I_ - sqrt(2))*erf((sqrt(2)*_I_ - sqrt(2))*x/2))/8 Traceback (most recent call last): File "", line 1, in File "/Users/was/talks/2007-01-11-uw-undergrads/sage_notebook/worksheets/_scratch_/code/4.py", line 4, in exec compile(ur'g = f.integral(x); g' + '\n', '', 'single') File "/Users/was/talks/2007-01-11-uw-undergrads/", line 1, in ...rT# Worksheet '_scratch_' (2007-01-11 at 11:51) sage: %hideall sage: #auto sage: sage: # The source code to draw factorization trees... sage: sage: import random sage: sage: def factor_tree(n, font=10): ... rows = [] ... v = [(n,None,0)] ... ftree(rows, v, 0, factor(n)) ... return draw_ftree(rows, font) ... sage: sage: def ftree(rows, v, i, F): ... if len(v) > 0: ... # add a row to g at the ith level. ... rows.append(v) ... w = [] ... for i in range(len(v)): ... k, _,_ = v[i] ... if k is None or is_prime(k): ... w.append((None,None,None)) ... else: ... d = random.choice(divisors(k)[1:-1]) ... w.append((d,k,i)) ... e = k//d ... if e == 1: ... w.append((None,None)) ... else: ... w.append((e,k,i)) ... if len(w) > len(v): ... ftree(rows, w, i+1, F) ... sage: sage: sage: def draw_ftree(rows,font): ... g = Graphics() ... for i in range(len(rows)): ... cur = rows[i] ... for j in range(len(cur)): ... e, f, k = cur[j] ... if not e is None: ... if is_prime(e): ... c = (1,0,0) ... else: ... c = (0,0,.4) ... g += text(str(e), (j*2-len(cur),-i), fontsize=font, rgbcolor=c) ... if not k is None and not f is None: ... g += line([(j*2-len(cur),-i), (k*2-len(rows[i-1]),-i+1)], alpha=0.4, thickness=1) ... return g ... sage: sage: def factor_trees(n, cols=4, rows=3, font=10): ... return graphics_array([[factor_tree(n,font) for _ in range(cols)] for _ in range(rows)]) CPU time: 0.00 s, Wall time: 0.00 srUT# Worksheet '_scratch_' (2007-01-11 at 11:51) sage: from sage.calculus.all import * rUA# Worksheet '_scratch_' (2007-01-11 at 11:51) sage: f = sin(x^2) r T# Worksheet '_scratch_' (2007-01-11 at 11:51) sage: g = f.integral(x); g sqrt(_Pi_)*((sqrt(2)*_I_ + sqrt(2))*erf((sqrt(2)*_I_ + sqrt(2))*x/2) + (sqrt(2)*_I_ - sqrt(2))*erf((sqrt(2)*_I_ - sqrt(2))*x/2))/8 {'_I_': 1.00000000000000*I, 'cos': cos, '_Pi_': , 'sec': sec, 'x': x, '_E_': , 'sin': sin} Traceback (most recent call last): File "", line 1, in File "/Users/was/talks/2007-01-11-uw-undergrads/sage_notebook/worksheets/_scratch_/code/4....r!UF# Worksheet '_scratch_' (2007-01-11 at 11:51) sage: f = sin(x)*cos(y) r"T# Worksheet '_scratch_' (2007-01-11 at 11:51) sage: g = f.integral(x); g -cos(x)*cos(y) {'_I_': 1.00000000000000*I, 'cos': cos, '_Pi_': , 'sec': sec, 'x': x, 'y': y, '_E_': , 'sin': sin} ((-1)*cos(x))*cos(y)r#Uh# Worksheet '_scratch_' (2007-01-11 at 11:52) sage: %maple sage: integrate(sin(x)*cos(x),x) 1/2*sin(x)^2r$Uk# Worksheet '_scratch_' (2007-01-11 at 11:52) sage: %maple sage: integrate(sin(x)*cos(x)^2,x) -1/3*cos(x)^3r%Ur# Worksheet '_scratch_' (2007-01-11 at 11:52) sage: %maple sage: integrate(sin(x)*cos(x)+1/x,x) 1/2*sin(x)^2+ln(x)r&UL# Worksheet '_scratch_' (2007-01-11 at 11:52) sage: f = sin(x)*cos(x) + 1/x r'T# Worksheet '_scratch_' (2007-01-11 at 11:52) sage: g = f.integral(x); g log(x) - cos(x)^2/2 {'_I_': 1.00000000000000*I, 'cos': cos, '_Pi_': , 'sec': sec, 'x': x, 'y': y, '_E_': , 'sin': sin} Traceback (most recent call last): File "", line 1, in File "/Users/was/talks/2007-01-11-uw-undergrads/sage_notebook/worksheets/_scratch_/code/11.py", line 4, in exec compile(ur'g = f.integral(x); g' + '\n', '', 'single') File "/User...r(T# Worksheet '_scratch_' (2007-01-11 at 11:53) sage: %hideall sage: #auto sage: sage: # The source code to draw factorization trees... sage: sage: import random sage: sage: def factor_tree(n, font=10): ... rows = [] ... v = [(n,None,0)] ... ftree(rows, v, 0, factor(n)) ... return draw_ftree(rows, font) ... sage: sage: def ftree(rows, v, i, F): ... if len(v) > 0: ... # add a row to g at the ith level. ... rows.append(v) ... w = [] ... for i in range(len(v)): ... k, _,_ = v[i] ... if k is None or is_prime(k): ... w.append((None,None,None)) ... else: ... d = random.choice(divisors(k)[1:-1]) ... w.append((d,k,i)) ... e = k//d ... if e == 1: ... w.append((None,None)) ... else: ... w.append((e,k,i)) ... if len(w) > len(v): ... ftree(rows, w, i+1, F) ... sage: sage: sage: def draw_ftree(rows,font): ... g = Graphics() ... for i in range(len(rows)): ... cur = rows[i] ... for j in range(len(cur)): ... e, f, k = cur[j] ... if not e is None: ... if is_prime(e): ... c = (1,0,0) ... else: ... c = (0,0,.4) ... g += text(str(e), (j*2-len(cur),-i), fontsize=font, rgbcolor=c) ... if not k is None and not f is None: ... g += line([(j*2-len(cur),-i), (k*2-len(rows[i-1]),-i+1)], alpha=0.4, thickness=1) ... return g ... sage: sage: def factor_trees(n, cols=4, rows=3, font=10): ... return graphics_array([[factor_tree(n,font) for _ in range(cols)] for _ in range(rows)]) CPU time: 0.00 s, Wall time: 0.00 sr)UT# Worksheet '_scratch_' (2007-01-11 at 11:53) sage: from sage.calculus.all import * r*UL# Worksheet '_scratch_' (2007-01-11 at 11:53) sage: f = sin(x)*cos(x) + 1/x r+U\# Worksheet '_scratch_' (2007-01-11 at 11:53) sage: g = f.integral(x); g log(x) - cos(x)^2/2r,Uœ# Worksheet '_scratch_' (2007-01-11 at 11:53) sage: show(g)

\log \left( x \right) - \frac{\cos \left( x \right)^{2}}{2}

r-U¹# Worksheet '_scratch_' (2007-01-11 at 11:53) sage: f = sin(x)*cos(x) + 1/x; show(f)

\sin \left( x \right) \cdot \cos \left( x \right) + \frac{1}{x}

r.U¯# Worksheet '_scratch_' (2007-01-11 at 11:53) sage: g = f.integral(x); show(g)

\log \left( x \right) - \frac{\cos \left( x \right)^{2}}{2}

r/UJ# Worksheet '_scratch_' (2007-01-11 at 11:53) sage: f = sin(x+y)*cos(x+z) r0Uƒ# Worksheet '_scratch_' (2007-01-11 at 11:54) sage: f.trig_expand() (cos(x)*sin(y) + sin(x)*cos(y))*(cos(x)*cos(z) - sin(x)*sin(z))r1TG# Worksheet '_scratch_' (2007-01-11 at 11:54) sage: h = f.trig_expand(); show(h)

\cos \left( x \right) \cdot \sin \left( y \right) + \sin \left( x \right) \cdot \cos \left( y \right) \cdot \cos \left( x \right) \cdot \cos \left( z \right) - \sin \left( x \right) \cdot \sin \left( z \right)

r2U°# Worksheet '_scratch_' (2007-01-11 at 11:54) sage: h.trig_simplify() ((((-1)*cos(x))*sin(x))*sin(y) - sin(x)^2*cos(y))*sin(z) + (cos(x)^2*sin(y) + cos(x)*sin(x)*cos(y))*cos(z)r3U±# Worksheet '_scratch_' (2007-01-11 at 11:54) sage: f = sin(x+y)*cos(x+z); show(f)

\sin \left( x + y \right) \cdot \cos \left( x + z \right)

r4TG# Worksheet '_scratch_' (2007-01-11 at 11:54) sage: h = f.trig_expand(); show(h)

r5U/# Worksheet '_scratch_' (2007-01-11 at 11:54) r6T# Worksheet '_scratch_' (2007-01-11 at 12:04) sage: %hideall sage: #auto sage: sage: # The source code to draw factorization trees... sage: sage: import random sage: sage: def factor_tree(n, font=10): ... rows = [] ... v = [(n,None,0)] ... ftree(rows, v, 0, factor(n)) ... return draw_ftree(rows, font) ... sage: sage: def ftree(rows, v, i, F): ... if len(v) > 0: ... # add a row to g at the ith level. ... rows.append(v) ... w = [] ... for i in range(len(v)): ... k, _,_ = v[i] ... if k is None or is_prime(k): ... w.append((None,None,None)) ... else: ... d = random.choice(divisors(k)[1:-1]) ... w.append((d,k,i)) ... e = k//d ... if e == 1: ... w.append((None,None)) ... else: ... w.append((e,k,i)) ... if len(w) > len(v): ... ftree(rows, w, i+1, F) ... sage: sage: sage: def draw_ftree(rows,font): ... g = Graphics() ... for i in range(len(rows)): ... cur = rows[i] ... for j in range(len(cur)): ... e, f, k = cur[j] ... if not e is None: ... if is_prime(e): ... c = (1,0,0) ... else: ... c = (0,0,.4) ... g += text(str(e), (j*2-len(cur),-i), fontsize=font, rgbcolor=c) ... if not k is None and not f is None: ... g += line([(j*2-len(cur),-i), (k*2-len(rows[i-1]),-i+1)], alpha=0.4, thickness=1) ... return g ... sage: sage: def factor_trees(n, cols=4, rows=3, font=10): ... return graphics_array([[factor_tree(n,font) for _ in range(cols)] for _ in range(rows)]) CPU time: 0.00 s, Wall time: 0.00 sr7T# Worksheet '_scratch_' (2007-01-11 at 12:04) sage: %hideall sage: #auto sage: sage: # The source code to draw factorization trees... sage: sage: import random sage: sage: def factor_tree(n, font=10): ... rows = [] ... v = [(n,None,0)] ... ftree(rows, v, 0, factor(n)) ... return draw_ftree(rows, font) ... sage: sage: def ftree(rows, v, i, F): ... if len(v) > 0: ... # add a row to g at the ith level. ... rows.append(v) ... w = [] ... for i in range(len(v)): ... k, _,_ = v[i] ... if k is None or is_prime(k): ... w.append((None,None,None)) ... else: ... d = random.choice(divisors(k)[1:-1]) ... w.append((d,k,i)) ... e = k//d ... if e == 1: ... w.append((None,None)) ... else: ... w.append((e,k,i)) ... if len(w) > len(v): ... ftree(rows, w, i+1, F) ... sage: sage: sage: def draw_ftree(rows,font): ... g = Graphics() ... for i in range(len(rows)): ... cur = rows[i] ... for j in range(len(cur)): ... e, f, k = cur[j] ... if not e is None: ... if is_prime(e): ... c = (1,0,0) ... else: ... c = (0,0,.4) ... g += text(str(e), (j*2-len(cur),-i), fontsize=font, rgbcolor=c) ... if not k is None and not f is None: ... g += line([(j*2-len(cur),-i), (k*2-len(rows[i-1]),-i+1)], alpha=0.4, thickness=1) ... return g ... sage: sage: def factor_trees(n, cols=4, rows=3, font=10): ... return graphics_array([[factor_tree(n,font) for _ in range(cols)] for _ in range(rows)]) CPU time: 0.00 s, Wall time: 0.00 sr8T# Worksheet '_scratch_' (2007-01-11 at 12:05) sage: %hideall sage: #auto sage: sage: # The source code to draw factorization trees... sage: sage: import random sage: sage: def factor_tree(n, font=10): ... rows = [] ... v = [(n,None,0)] ... ftree(rows, v, 0, factor(n)) ... return draw_ftree(rows, font) ... sage: sage: def ftree(rows, v, i, F): ... if len(v) > 0: ... # add a row to g at the ith level. ... rows.append(v) ... w = [] ... for i in range(len(v)): ... k, _,_ = v[i] ... if k is None or is_prime(k): ... w.append((None,None,None)) ... else: ... d = random.choice(divisors(k)[1:-1]) ... w.append((d,k,i)) ... e = k//d ... if e == 1: ... w.append((None,None)) ... else: ... w.append((e,k,i)) ... if len(w) > len(v): ... ftree(rows, w, i+1, F) ... sage: sage: sage: def draw_ftree(rows,font): ... g = Graphics() ... for i in range(len(rows)): ... cur = rows[i] ... for j in range(len(cur)): ... e, f, k = cur[j] ... if not e is None: ... if is_prime(e): ... c = (1,0,0) ... else: ... c = (0,0,.4) ... g += text(str(e), (j*2-len(cur),-i), fontsize=font, rgbcolor=c) ... if not k is None and not f is None: ... g += line([(j*2-len(cur),-i), (k*2-len(rows[i-1]),-i+1)], alpha=0.4, thickness=1) ... return g ... sage: sage: def factor_trees(n, cols=4, rows=3, font=10): ... return graphics_array([[factor_tree(n,font) for _ in range(cols)] for _ in range(rows)]) CPU time: 0.00 s, Wall time: 0.00 sr9Ur# Worksheet '_scratch_' (2007-01-11 at 12:05) sage: %maple sage: integrate(sin(x)*cos(x)+1/x,x) 1/2*sin(x)^2+ln(x)r:UR# Worksheet '_scratch_' (2007-01-11 at 12:05) sage: magma('Factorization(-2006)') r;UR# Worksheet '_scratch_' (2007-01-11 at 12:05) sage: magma('Factorization(-2006)') r<UR# Worksheet '_scratch_' (2007-01-11 at 12:06) sage: magma('Factorization(-2006)') r=T# Worksheet '_scratch_' (2007-01-11 at 12:06) sage: %hideall sage: #auto sage: sage: # The source code to draw factorization trees... sage: sage: import random sage: sage: def factor_tree(n, font=10): ... rows = [] ... v = [(n,None,0)] ... ftree(rows, v, 0, factor(n)) ... return draw_ftree(rows, font) ... sage: sage: def ftree(rows, v, i, F): ... if len(v) > 0: ... # add a row to g at the ith level. ... rows.append(v) ... w = [] ... for i in range(len(v)): ... k, _,_ = v[i] ... if k is None or is_prime(k): ... w.append((None,None,None)) ... else: ... d = random.choice(divisors(k)[1:-1]) ... w.append((d,k,i)) ... e = k//d ... if e == 1: ... w.append((None,None)) ... else: ... w.append((e,k,i)) ... if len(w) > len(v): ... ftree(rows, w, i+1, F) ... sage: sage: sage: def draw_ftree(rows,font): ... g = Graphics() ... for i in range(len(rows)): ... cur = rows[i] ... for j in range(len(cur)): ... e, f, k = cur[j] ... if not e is None: ... if is_prime(e): ... c = (1,0,0) ... else: ... c = (0,0,.4) ... g += text(str(e), (j*2-len(cur),-i), fontsize=font, rgbcolor=c) ... if not k is None and not f is None: ... g += line([(j*2-len(cur),-i), (k*2-len(rows[i-1]),-i+1)], alpha=0.4, thickness=1) ... return g ... sage: sage: def factor_trees(n, cols=4, rows=3, font=10): ... return graphics_array([[factor_tree(n,font) for _ in range(cols)] for _ in range(rows)]) CPU time: 0.00 s, Wall time: 0.00 sr>Un# Worksheet '_scratch_' (2007-01-11 at 12:06) sage: magma('Factorization(-2006)') [ <2, 1>, <17, 1>, <59, 1> ]r?Ur# Worksheet '_scratch_' (2007-01-11 at 12:06) sage: %maple sage: integrate(sin(x)*cos(x)+1/x,x) 1/2*sin(x)^2+ln(x)r@T# Worksheet '_scratch_' (2007-01-11 at 12:07) sage: %hideall sage: #auto sage: sage: # The source code to draw factorization trees... sage: sage: import random sage: sage: def factor_tree(n, font=10): ... rows = [] ... v = [(n,None,0)] ... ftree(rows, v, 0, factor(n)) ... return draw_ftree(rows, font) ... sage: sage: def ftree(rows, v, i, F): ... if len(v) > 0: ... # add a row to g at the ith level. ... rows.append(v) ... w = [] ... for i in range(len(v)): ... k, _,_ = v[i] ... if k is None or is_prime(k): ... w.append((None,None,None)) ... else: ... d = random.choice(divisors(k)[1:-1]) ... w.append((d,k,i)) ... e = k//d ... if e == 1: ... w.append((None,None)) ... else: ... w.append((e,k,i)) ... if len(w) > len(v): ... ftree(rows, w, i+1, F) ... sage: sage: sage: def draw_ftree(rows,font): ... g = Graphics() ... for i in range(len(rows)): ... cur = rows[i] ... for j in range(len(cur)): ... e, f, k = cur[j] ... if not e is None: ... if is_prime(e): ... c = (1,0,0) ... else: ... c = (0,0,.4) ... g += text(str(e), (j*2-len(cur),-i), fontsize=font, rgbcolor=c) ... if not k is None and not f is None: ... g += line([(j*2-len(cur),-i), (k*2-len(rows[i-1]),-i+1)], alpha=0.4, thickness=1) ... return g ... sage: sage: def factor_trees(n, cols=4, rows=3, font=10): ... return graphics_array([[factor_tree(n,font) for _ in range(cols)] for _ in range(rows)]) CPU time: 0.00 s, Wall time: 0.00 srAUr# Worksheet '_scratch_' (2007-01-11 at 12:07) sage: %maple sage: integrate(sin(x)*cos(x)+1/x,x) 1/2*sin(x)^2+ln(x)rBT# Worksheet '_scratch_' (2007-01-11 at 13:10) sage: %hideall sage: #auto sage: sage: # The source code to draw factorization trees... sage: sage: import random sage: sage: def factor_tree(n, font=10): ... rows = [] ... v = [(n,None,0)] ... ftree(rows, v, 0, factor(n)) ... return draw_ftree(rows, font) ... sage: sage: def ftree(rows, v, i, F): ... if len(v) > 0: ... # add a row to g at the ith level. ... rows.append(v) ... w = [] ... for i in range(len(v)): ... k, _,_ = v[i] ... if k is None or is_prime(k): ... w.append((None,None,None)) ... else: ... d = random.choice(divisors(k)[1:-1]) ... w.append((d,k,i)) ... e = k//d ... if e == 1: ... w.append((None,None)) ... else: ... w.append((e,k,i)) ... if len(w) > len(v): ... ftree(rows, w, i+1, F) ... sage: sage: sage: def draw_ftree(rows,font): ... g = Graphics() ... for i in range(len(rows)): ... cur = rows[i] ... for j in range(len(cur)): ... e, f, k = cur[j] ... if not e is None: ... if is_prime(e): ... c = (1,0,0) ... else: ... c = (0,0,.4) ... g += text(str(e), (j*2-len(cur),-i), fontsize=font, rgbcolor=c) ... if not k is None and not f is None: ... g += line([(j*2-len(cur),-i), (k*2-len(rows[i-1]),-i+1)], alpha=0.4, thickness=1) ... return g ... sage: sage: def factor_trees(n, cols=4, rows=3, font=10): ... return graphics_array([[factor_tree(n,font) for _ in range(cols)] for _ in range(rows)]) CPU time: 0.00 s, Wall time: 0.00 srCU;# Worksheet '_scratch_' (2007-01-11 at 13:10) sage: 2 ^ 3 8rDUI# Worksheet '_scratch_' (2007-01-11 at 13:10) sage: %python sage: 2 ^ 3 1rEUr# Worksheet '_scratch_' (2007-01-11 at 13:10) sage: %maple sage: integrate(sin(x)*cos(x)+1/x,x) 1/2*sin(x)^2+ln(x)rFU¦# Worksheet '_scratch_' (2007-01-11 at 13:11) sage: m = matrix(5,5, range(25)); m [ 0 1 2 3 4] [ 5 6 7 8 9] [10 11 12 13 14] [15 16 17 18 19] [20 21 22 23 24]rGUE# Worksheet '_scratch_' (2007-01-11 at 13:11) sage: f = m.charpoly() rHUL# Worksheet '_scratch_' (2007-01-11 at 13:11) sage: f x^5 - 60*x^4 - 250*x^3rIU{# Worksheet '_scratch_' (2007-01-11 at 13:11) sage: show(f)

x^{5} - 60x^{4} - 250x^{3}

rJUˆ# Worksheet '_scratch_' (2007-01-11 at 13:11) sage: show(factor(f))

(x^{2} - 60x - 250) \cdot x^{3}

rKU–# Worksheet '_scratch_' (2007-01-11 at 13:11) sage: F = factor_trees(factorial(10), rows=1, cols=1, font=14) sage: F.show(axes=False, figsize=[10,6]) rLU–# Worksheet '_scratch_' (2007-01-11 at 13:11) sage: F = factor_trees(factorial(10), rows=1, cols=1, font=14) sage: F.show(axes=False, figsize=[10,6]) rMT# Worksheet '_scratch_' (2007-01-11 at 13:12) sage: t = Tachyon(xres=700, yres=500, camera_center=(2,7,4), look_at=(2,0,0), raydepth=4) sage: t.light((10,3,2), 1, (1,1,1)) sage: t.light((10,-3,2), 1, (1,1,1)) sage: t.texture('black', color=(0,0,0)) sage: t.texture('red', color=(1,0,0)) sage: t.texture('grey', color=(.9,.9,.9)) sage: t.plane((0,0,0),(0,0,1),'grey') sage: t.cylinder((0,0,0),(1,0,0),.01,'black') sage: t.cylinder((0,0,0),(0,1,0),.01,'black') sage: E = EllipticCurve('37a'); show(E) sage: P = E([0,0]) sage: Q = P sage: n = 60 sage: for i in range(n): ... Q = Q + P ... c = i/n + .1 ... t.texture('r%s'%i,color=(float(i/n),0,0)) ... t.sphere((Q[0], -Q[1], .01), .04, 'r%s'%i) ... sage: show(t)

y^2 + y = x^3 - x

rNUT# Worksheet '_scratch_' (2007-01-11 at 13:13) sage: from sage.calculus.all import * rOU¹# Worksheet '_scratch_' (2007-01-11 at 13:13) sage: f = sin(x)*cos(x) + 1/x; show(f)

\sin \left( x \right) \cdot \cos \left( x \right) + \frac{1}{x}

rPU¯# Worksheet '_scratch_' (2007-01-11 at 13:13) sage: g = f.integral(x); show(g)

\log \left( x \right) - \frac{\cos \left( x \right)^{2}}{2}

rQU±# Worksheet '_scratch_' (2007-01-11 at 13:13) sage: f = sin(x+y)*cos(x+z); show(f)

\sin \left( x + y \right) \cdot \cos \left( x + z \right)

rRTG# Worksheet '_scratch_' (2007-01-11 at 13:13) sage: h = f.trig_expand(); show(h)

rSUü# Worksheet '_scratch_' (2007-01-11 at 13:15) sage: for i in range(10): ... print "hello world ", i hello world 0 hello world 1 hello world 2 hello world 3 hello world 4 hello world 5 hello world 6 hello world 7 hello world 8 hello world 9rTT‚# Worksheet '_scratch_' (2007-01-11 at 13:17) sage: t = Tachyon(xres=800,yres=800, camera_center=(2,5,2), look_at=(2.5,0,0)) sage: t.light((0,0,100), 1, (1,1,1)) sage: t.texture('r', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(1,0,0)) sage: for i in srange(0,50,0.1): ... t.sphere((i/10,sin(i),cos(i)), 0.05, 'r') ... sage: t.texture('white', color=(1,1,1), opacity=1, specular=1, diffuse=1) sage: t.plane((0,0,-100), (0,0,-100), 'white') sage: t.save('sage.png') Traceback (most recent call last): ... TypeError: float() argument must be a string or a numberrUT# Worksheet '_scratch_' (2007-01-11 at 13:18) sage: %hideall sage: #auto sage: sage: # The source code to draw factorization trees... sage: sage: import random sage: sage: def factor_tree(n, font=10): ... rows = [] ... v = [(n,None,0)] ... ftree(rows, v, 0, factor(n)) ... return draw_ftree(rows, font) ... sage: sage: def ftree(rows, v, i, F): ... if len(v) > 0: ... # add a row to g at the ith level. ... rows.append(v) ... w = [] ... for i in range(len(v)): ... k, _,_ = v[i] ... if k is None or is_prime(k): ... w.append((None,None,None)) ... else: ... d = random.choice(divisors(k)[1:-1]) ... w.append((d,k,i)) ... e = k//d ... if e == 1: ... w.append((None,None)) ... else: ... w.append((e,k,i)) ... if len(w) > len(v): ... ftree(rows, w, i+1, F) ... sage: sage: sage: def draw_ftree(rows,font): ... g = Graphics() ... for i in range(len(rows)): ... cur = rows[i] ... for j in range(len(cur)): ... e, f, k = cur[j] ... if not e is None: ... if is_prime(e): ... c = (1,0,0) ... else: ... c = (0,0,.4) ... g += text(str(e), (j*2-len(cur),-i), fontsize=font, rgbcolor=c) ... if not k is None and not f is None: ... g += line([(j*2-len(cur),-i), (k*2-len(rows[i-1]),-i+1)], alpha=0.4, thickness=1) ... return g ... sage: sage: def factor_trees(n, cols=4, rows=3, font=10): ... return graphics_array([[factor_tree(n,font) for _ in range(cols)] for _ in range(rows)]) CPU time: 0.00 s, Wall time: 0.00 srVT## Worksheet '_scratch_' (2007-01-11 at 13:18) sage: t = Tachyon(xres=800,yres=800, camera_center=(2,5,2), look_at=(2.5,0,0)) sage: t.light((0,0,100), 1, (1,1,1)) sage: t.texture('r', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(1,0,0)) sage: for i in srange(0,50,0.1): ... t.sphere((i/10,sin(i),cos(i)), 0.05, 'r') ... sage: t.texture('white', color=(1,1,1), opacity=1, specular=1, diffuse=1) sage: t.plane((0,0,-100), (0,0,-100), 'white') sage: t.save('sage.png') rWUT# Worksheet '_scratch_' (2007-01-11 at 13:26) sage: from sage.calculus.all import * rXU¹# Worksheet '_scratch_' (2007-01-11 at 13:26) sage: f = sin(x)*cos(x) + 1/x; show(f)

\sin \left( x \right) \cdot \cos \left( x \right) + \frac{1}{x}

rYT# Worksheet '_scratch_' (2007-01-16 at 02:06) sage: %hideall sage: #auto sage: sage: # The source code to draw factorization trees... sage: sage: import random sage: sage: def factor_tree(n, font=10): ... rows = [] ... v = [(n,None,0)] ... ftree(rows, v, 0, factor(n)) ... return draw_ftree(rows, font) ... sage: sage: def ftree(rows, v, i, F): ... if len(v) > 0: ... # add a row to g at the ith level. ... rows.append(v) ... w = [] ... for i in range(len(v)): ... k, _,_ = v[i] ... if k is None or is_prime(k): ... w.append((None,None,None)) ... else: ... d = random.choice(divisors(k)[1:-1]) ... w.append((d,k,i)) ... e = k//d ... if e == 1: ... w.append((None,None)) ... else: ... w.append((e,k,i)) ... if len(w) > len(v): ... ftree(rows, w, i+1, F) ... sage: sage: sage: def draw_ftree(rows,font): ... g = Graphics() ... for i in range(len(rows)): ... cur = rows[i] ... for j in range(len(cur)): ... e, f, k = cur[j] ... if not e is None: ... if is_prime(e): ... c = (1,0,0) ... else: ... c = (0,0,.4) ... g += text(str(e), (j*2-len(cur),-i), fontsize=font, rgbcolor=c) ... if not k is None and not f is None: ... g += line([(j*2-len(cur),-i), (k*2-len(rows[i-1]),-i+1)], alpha=0.4, thickness=1) ... return g ... sage: sage: def factor_trees(n, cols=4, rows=3, font=10): ... return graphics_array([[factor_tree(n,font) for _ in range(cols)] for _ in range(rows)]) CPU time: 0.00 s, Wall time: 0.00 srZU;# Worksheet '_scratch_' (2007-01-16 at 02:06) sage: 2 ^ 3 8r[UI# Worksheet '_scratch_' (2007-01-16 at 02:07) sage: %python sage: 2 ^ 3 1r\U_# Worksheet '_scratch_' (2007-01-16 at 02:07) sage: factorial r]Ur# Worksheet '_scratch_' (2007-01-16 at 02:07) sage: %maple sage: integrate(sin(x)*cos(x)+1/x,x) 1/2*sin(x)^2+ln(x)r^Un# Worksheet '_scratch_' (2007-01-16 at 02:07) sage: magma('Factorization(-2006)') [ <2, 1>, <17, 1>, <59, 1> ]r_U¦# Worksheet '_scratch_' (2007-01-16 at 02:07) sage: m = matrix(5,5, range(25)); m [ 0 1 2 3 4] [ 5 6 7 8 9] [10 11 12 13 14] [15 16 17 18 19] [20 21 22 23 24]r`UE# Worksheet '_scratch_' (2007-01-16 at 02:07) sage: f = m.charpoly() raUL# Worksheet '_scratch_' (2007-01-16 at 02:07) sage: f x^5 - 60*x^4 - 250*x^3rbU{# Worksheet '_scratch_' (2007-01-16 at 02:07) sage: show(f)

x^{5} - 60x^{4} - 250x^{3}

rcUˆ# Worksheet '_scratch_' (2007-01-16 at 02:07) sage: show(factor(f))

(x^{2} - 60x - 250) \cdot x^{3}

rdU6# Worksheet 'talk' (2007-01-16 at 02:20) sage: 2 + 2 4reUe# Worksheet 'talk' (2007-01-16 at 02:25) sage: v = vector(RDF, range(5)); v (0.0, 1.0, 2.0, 3.0, 4.0)rfTê# Worksheet 'talk' (2007-01-16 at 02:25) sage: v.fft() ------------------------------------------------------------ Unhandled SIGBUS: A bus error occured in SAGE. This probably occured because a *compiled* component of SAGE has a bug in it (typically accessing invalid memory) or is not properly wrapped with _sig_on, _sig_off. You might want to run SAGE under gdb with 'sage -gdb' to debug this. SAGE will now terminate (sorry). ------------------------------------------------------------rgUc# Worksheet 'talk' (2007-01-16 at 02:26) sage: v = vector(CDF, range(5)); v (0, 1.0, 2.0, 3.0, 4.0)rhUŸ# Worksheet 'talk' (2007-01-16 at 02:26) sage: v.fft() (10.0, -2.5 + 3.44095468521*I, -2.5 + 0.812299251556*I, -2.5 - 0.812299251556*I, -2.5 - 3.44095468521*I)riUŸ# Worksheet 'talk' (2007-01-16 at 02:35) sage: v.fft() (10.0, -2.5 + 3.44095468521*I, -2.5 + 0.812299251556*I, -2.5 - 0.812299251556*I, -2.5 - 3.44095468521*I)rjT# Worksheet 'talk' (2007-01-16 at 02:35) sage: v = vector(CDF, range(500)); v (0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0, 13.0, 14.0, 15.0, 16.0, 17.0, 18.0, 19.0, 20.0, 21.0, 22.0, 23.0, 24.0, 25.0, 26.0, 27.0, 28.0, 29.0, 30.0, 31.0, 32.0, 33.0, 34.0, 35.0, 36.0, 37.0, 38.0, 39.0, 40.0, 41.0, 42.0, 43.0, 44.0, 45.0, 46.0, 47.0, 48.0, 49.0, 50.0, 51.0, 52.0, 53.0, 54.0, 55.0, 56.0, 57.0, 58.0, 59.0, 60.0, 61.0, 62.0, 63.0, 64.0, 65.0, 66.0, 67.0, 68.0, 69.0, 70.0, 71.0, 72.0, 73.0, 74.0, 75.0, 76.0, 77.0, 78....rkT# Worksheet 'talk' (2007-01-16 at 02:35) sage: v.fft() (124750.0, -250.0 + 39788.2109375*I, -250.0 + 19893.3203125*I, -250.0 + 13261.3408203*I, -250.0 + 9945.08984375*I, -250.0 + 7955.12890625*I, -250.0 + 6628.31396484*I, -250.0 + 5680.43945312*I, -250.0 + 4969.40234375*I, -250.0 + 4416.25732422*I, -250.0 + 3973.63623047*I, -250.0 + 3611.39648438*I, -250.0 + 3309.44238281*I, -250.0 + 3053.86206055*I, -250.0 + 2834.71850586*I, -250.0 + 2644.72363281*I, -250.0 + 2478.4128418*I, -250.0 + 2331.60595703*I, -250.0 +...rlT5# Worksheet 'talk' (2007-01-16 at 02:35) sage: v = vector(CDF, [sin(x/500.0) for x in range(500)]); v (0, 0.00199999866667, 0.00399998933334, 0.00599996400006, 0.00799991466694, 0.00999983333417, 0.0119997120021, 0.0139995426711, 0.0159993173421, 0.0179990280157, 0.0199986666933, 0.0219982253763, 0.0239976960664, 0.0259970707657, 0.0279963414768, 0.0299955002025, 0.0319945389463, 0.033993449712, 0.0359922245039, 0.0379908553269, 0.0399893341866, 0.041987653089, 0.0439858040409, 0.0459837790496, 0.0479815701232, 0.0499791692707, 0.0519765685015, 0.053973759...rmT# Worksheet 'talk' (2007-01-16 at 02:35) sage: v = vector(CDF, [sin(x/500.0) for x in range(500)]) sage: v[:10] (0, 0.00199999866667, 0.00399998933334, 0.00599996400006, 0.00799991466694, 0.00999983333417, 0.0119997120021, 0.0139995426711, 0.0159993173421, 0.0179990280157)rnU;# Worksheet 'talk' (2007-01-16 at 02:36) sage: w = v.fft() roT|# Worksheet 'talk' (2007-01-16 at 02:36) sage: w[:10] (229.428034957, -6.39426063905 + 68.7015075684*I, -1.88562296191 + 33.6926765442*I, -1.06954231846 + 22.3810653687*I, -0.785272818889 + 16.7635536194*I, -0.653933897994 + 13.4016036987*I, -0.582652096172 + 11.1629257202*I, -0.539692762675 + 9.56479740143*I, -0.511819093419 + 8.3665304184*I, -0.492712905594 + 7.43463230133*I)rpUi# Worksheet 'talk' (2007-01-16 at 02:36) sage: line(w) Graphics object consisting of 1 graphics primitiverqU=# Worksheet 'talk' (2007-01-16 at 02:36) sage: show(line(w)) rrUÅ# Worksheet 'talk' (2007-01-16 at 02:36) sage: L = line([(i,w[i]) for i in range(len(w))]) sage: show(L) Traceback (most recent call last): ... TypeError: can't convert complex to float; use abs(z)rsUn# Worksheet 'talk' (2007-01-16 at 02:37) sage: L = line([(i,abs(w[i])) for i in range(len(w))]) sage: show(L) rtT# Worksheet 'talk' (2007-01-16 at 02:37) sage: v = vector(CDF, [sin(x/100.0) for x in range(100)]) sage: v[:10] (0, 0.00999983333417, 0.0199986666933, 0.0299955002025, 0.0399893341866, 0.0499791692707, 0.0599640064794, 0.0699428473375, 0.0799146939692, 0.089878549198)ruU;# Worksheet 'talk' (2007-01-16 at 02:37) sage: w = v.fft() rvT|# Worksheet 'talk' (2007-01-16 at 02:37) sage: w[:10] (45.5486508387, -1.61580835603 + 13.7360706329*I, -0.714081047275 + 6.730073452*I, -0.550865296841 + 4.46351623535*I, -0.494011927483 + 3.3357732296*I, -0.467744827165 + 2.65913581848*I, -0.453489305266 + 2.20714426041*I, -0.444898433232 + 1.88325178623*I, -0.439324852151 + 1.63931918144*I, -0.435504927678 + 1.44864630699*I)rwUn# Worksheet 'talk' (2007-01-16 at 02:37) sage: L = line([(i,abs(w[i])) for i in range(len(w))]) sage: show(L) rxUn# Worksheet 'talk' (2007-01-16 at 02:37) sage: L = line([(i,abs(v[i])) for i in range(len(w))]) sage: show(L) ryUu# Worksheet 'talk' (2007-01-16 at 02:37) sage: L = line([(i,abs(v[i])) for i in range(len(w))]) sage: show(L,ymin=0) rzUu# Worksheet 'talk' (2007-01-16 at 02:37) sage: L = line([(i,abs(w[i])) for i in range(len(w))]) sage: show(L,ymin=0) r{Uá# Worksheet 'talk' (2007-01-16 at 02:37) sage: v = vector(CDF, [sin(x/100.0) for x in range(100*2*pi)]) sage: v[:10] Traceback (most recent call last): ... TypeError: range() integer end argument expected, got Constant_arith.r|T# Worksheet 'talk' (2007-01-16 at 02:38) sage: v = vector(CDF, [sin(x/100.0) for x in range(600)]) sage: v[:10] (0, 0.00999983333417, 0.0199986666933, 0.0299955002025, 0.0399893341866, 0.0499791692707, 0.0599640064794, 0.0699428473375, 0.0799146939692, 0.089878549198)r}Uu# Worksheet 'talk' (2007-01-16 at 02:38) sage: L = line([(i,abs(v[i])) for i in range(len(w))]) sage: show(L,ymin=0) r~T# Worksheet 'talk' (2007-01-16 at 02:38) sage: v = vector(CDF, [sin(x/(2*pi*100.0)) for x in range(100)]) sage: v[:10] (0, 0.00159154875901, 0.00318309348658, 0.00477463015129, 0.00636615472172, 0.00795766316649, 0.00954915145427, 0.0111406155538, 0.0127320514338, 0.0143234550631)rUu# Worksheet 'talk' (2007-01-16 at 02:38) sage: L = line([(i,abs(v[i])) for i in range(len(w))]) sage: show(L,ymin=0) r€T# Worksheet 'talk' (2007-01-16 at 02:38) sage: v = vector(CDF, [sin(x/(2*pi*100.0)) for x in range(100)]) sage: v[:10] (0, 0.00159154875901, 0.00318309348658, 0.00477463015129, 0.00636615472172, 0.00795766316649, 0.00954915145427, 0.0111406155538, 0.0127320514338, 0.0143234550631)rUÌ# Worksheet 'talk' (2007-01-16 at 02:38) sage: L = line([(i,v[i]) for i in range(len(w))]) sage: show(L,ymin=0) Traceback (most recent call last): ... TypeError: can't convert complex to float; use abs(z)r‚Uu# Worksheet 'talk' (2007-01-16 at 02:38) sage: L = line([(i,abs(v[i])) for i in range(len(w))]) sage: show(L,ymin=0) rƒT # Worksheet 'talk' (2007-01-16 at 02:39) sage: v = vector(CDF, [sin(2*pi*x/100.0) for x in range(100)]) sage: v[:10] (0, 0.0627905195293, 0.125333233564, 0.187381314586, 0.248689887165, 0.309016994375, 0.368124552685, 0.425779291565, 0.481753674102, 0.535826794979)r„Uu# Worksheet 'talk' (2007-01-16 at 02:39) sage: L = line([(i,abs(v[i])) for i in range(len(w))]) sage: show(L,ymin=0) r…Uv# Worksheet 'talk' (2007-01-16 at 02:39) sage: L = line([(i,real(v[i])) for i in range(len(w))]) sage: show(L,ymin=0) r†U;# Worksheet 'talk' (2007-01-16 at 02:39) sage: w = v.fft() r‡T¯# Worksheet 'talk' (2007-01-16 at 02:39) sage: w[:10] (-1.92901250529e-15, -7.11698499758e-15 - 50.0*I, 8.55143376465e-16 - 2.61196161813e-15*I, 4.62540636536e-16 - 3.12059007988e-15*I, 4.04775994251e-16 + 1.33226762955e-15*I, 2.87932893683e-15 - 1.19700578524e-15*I, 1.86054878335e-15 - 9.78839744177e-16*I, 1.88105518864e-15 - 4.9454193326e-16*I, 3.29664787776e-16 + 1.44959725698e-15*I, -4.85178122793e-15 + 1.18897961867e-15*I)rˆUu# Worksheet 'talk' (2007-01-16 at 02:39) sage: L = line([(i,abs(w[i])) for i in range(len(w))]) sage: show(L,ymin=0) r‰Uo# Worksheet 'talk' (2007-01-16 at 02:39) sage: L = line([(i,real(v[i])) for i in range(len(w))]) sage: show(L) rŠU|# Worksheet 'talk' (2007-01-16 at 02:40) sage: L = line([(i,abs(w[i])) for i in range(len(w))]) sage: show(L,ymin=0,xmax=5) r‹T # Worksheet 'talk' (2007-01-16 at 02:40) sage: v = vector(CDF, [sin(2*pi*x/100.0) for x in range(100)]) sage: v[:10] (0, 0.0627905195293, 0.125333233564, 0.187381314586, 0.248689887165, 0.309016994375, 0.368124552685, 0.425779291565, 0.481753674102, 0.535826794979)rŒUo# Worksheet 'talk' (2007-01-16 at 02:40) sage: L = line([(i,real(v[i])) for i in range(len(w))]) sage: show(L) rU;# Worksheet 'talk' (2007-01-16 at 02:40) sage: w = v.fft() rŽT¯# Worksheet 'talk' (2007-01-16 at 02:40) sage: w[:10] (-1.92901250529e-15, -7.11698499758e-15 - 50.0*I, 8.55143376465e-16 - 2.61196161813e-15*I, 4.62540636536e-16 - 3.12059007988e-15*I, 4.04775994251e-16 + 1.33226762955e-15*I, 2.87932893683e-15 - 1.19700578524e-15*I, 1.86054878335e-15 - 9.78839744177e-16*I, 1.88105518864e-15 - 4.9454193326e-16*I, 3.29664787776e-16 + 1.44959725698e-15*I, -4.85178122793e-15 + 1.18897961867e-15*I)rU|# Worksheet 'talk' (2007-01-16 at 02:40) sage: L = line([(i,abs(w[i])) for i in range(len(w))]) sage: show(L,ymin=0,xmax=5) rT# Worksheet 'li pi' (2007-01-16 at 02:42) sage: #auto sage: def f(x): ... return x / log(x) ... sage: sage: def Li(x): ... return float(gp('intnum(x=2,%s,1/log(x))'%float(x))) ... sage: sage: def p(x): ... return prime_pi(x) CPU time: 0.00 s, Wall time: 0.00 sr‘T# Worksheet 'li pi' (2007-01-16 at 02:42) sage: xmax = 1000 sage: G = plot(f, 2,xmax, linestyle='--', label='x/log(x)') + \ ... plot(Li,2,xmax, plot_points=30, plot_division=0, label='Li(x)') + \ ... plot(p, 2,xmax, label='P(X)', rgbcolor=(1,0,0)) ... sage: G.show(xmax=xmax) r’T# Worksheet 'li pi' (2007-01-16 at 02:42) sage: xmax = 10000 sage: G = plot(f, 2,xmax, linestyle='--', label='x/log(x)') + \ ... plot(Li,2,xmax, plot_points=30, plot_division=0, label='Li(x)') + \ ... plot(p, 2,xmax, label='P(X)', rgbcolor=(1,0,0)) ... sage: G.show(xmax=xmax) r“U# Worksheet 'li pi' (2007-01-16 at 02:43) sage: #auto sage: def f(x): ... return x / log(x) ... sage: def p(x): ... return prime_pi(x) r”T# Worksheet 'li pi' (2007-01-16 at 02:43) sage: xmax = 1000 sage: G = plot(f, 2,xmax, linestyle='--', label='x/log(x)') + \ ... plot(Li,2,xmax, plot_points=30, plot_division=0, label='Li(x)') + \ ... plot(p, 2,xmax, label='P(X)', rgbcolor=(1,0,0)) ... sage: G.show(xmax=xmax) r•T# Worksheet 'li pi' (2007-01-16 at 02:43) sage: xmax = 100 sage: G = plot(f, 2,xmax, linestyle='--', label='x/log(x)') + \ ... plot(Li,2,xmax, plot_points=30, plot_division=0, label='Li(x)') + \ ... plot(p, 2,xmax, label='P(X)', rgbcolor=(1,0,0)) ... sage: G.show(xmax=xmax) r–T# Worksheet 'li pi' (2007-01-16 at 02:43) sage: xmax = 50 sage: G = plot(f, 2,xmax, linestyle='--', label='x/log(x)') + \ ... plot(Li,2,xmax, plot_points=30, plot_division=0, label='Li(x)') + \ ... plot(p, 2,xmax, label='P(X)', rgbcolor=(1,0,0)) ... sage: G.show(xmax=xmax) r—T# Worksheet 'li pi' (2007-01-16 at 02:43) sage: xmax = 500 sage: G = plot(f, 2,xmax, linestyle='--', label='x/log(x)') + \ ... plot(Li,2,xmax, plot_points=30, plot_division=0, label='Li(x)') + \ ... plot(p, 2,xmax, label='P(X)', rgbcolor=(1,0,0)) ... sage: G.show(xmax=xmax) r˜T# Worksheet 'li pi' (2007-01-16 at 02:44) sage: xmax = 20000 sage: G = plot(f, 2,xmax, linestyle='--', label='x/log(x)') + \ ... plot(Li,2,xmax, plot_points=30, plot_division=0, label='Li(x)') + \ ... plot(p, 2,xmax, label='P(X)', rgbcolor=(1,0,0)) ... sage: G.show(xmax=xmax) r™T# Worksheet 'li pi' (2007-01-16 at 02:44) sage: xmax = 20000 sage: G = plot(f, 2,xmax, linestyle='--', label='x/log(x)') + \ ... plot(Li,2,xmax, plot_points=30, plot_division=0) + \ ... plot(p, 2,xmax, label='P(X)', rgbcolor=(1,0,0)) ... sage: G.show(xmax=xmax) ršT# Worksheet 'li pi' (2007-01-16 at 02:44) sage: xmax = 20000 sage: G = plot(f, 2,xmax, linestyle='--', label='x/log(x)') + \ ... plot(Li,2,xmax, plot_points=30, plot_division=0, label='Li(x)') + \ ... plot(p, 2,xmax, label='P(X)', rgbcolor=(1,0,0)) ... sage: G.show(xmax=xmax) r›T# Worksheet 'li pi' (2007-01-16 at 02:48) sage: xmax = 50 sage: G = plot(f, 2,xmax, linestyle='--', label='x/log(x)') + \ ... plot(Li,2,xmax, plot_points=30, plot_division=0, label='Li(x)') + \ ... plot(p, 2,xmax, label='P(X)', rgbcolor=(1,0,0)) ... sage: G.show(xmax=xmax) rœT# Worksheet 'li pi' (2007-01-16 at 02:48) sage: xmax = 500 sage: G = plot(f, 2,xmax, linestyle='--', label='x/log(x)') + \ ... plot(Li,2,xmax, plot_points=30, plot_division=0, label='Li(x)') + \ ... plot(p, 2,xmax, label='P(X)', rgbcolor=(1,0,0)) ... sage: G.show(xmax=xmax) rT# Worksheet 'li pi' (2007-01-16 at 02:48) sage: xmax = 25 sage: G = plot(f, 2,xmax, linestyle='--', label='x/log(x)') + \ ... plot(Li,2,xmax, plot_points=30, plot_division=0, label='Li(x)') + \ ... plot(p, 2,xmax, label='P(X)', rgbcolor=(1,0,0)) ... sage: G.show(xmax=xmax) ržT# Worksheet 'li pi' (2007-01-16 at 02:48) sage: xmax = 10 sage: G = plot(f, 2,xmax, linestyle='--', label='x/log(x)') + \ ... plot(Li,2,xmax, plot_points=30, plot_division=0, label='Li(x)') + \ ... plot(p, 2,xmax, label='P(X)', rgbcolor=(1,0,0)) ... sage: G.show(xmax=xmax) rŸT# Worksheet 'li pi' (2007-01-16 at 02:48) sage: xmax = 20 sage: G = plot(f, 2,xmax, linestyle='--', label='x/log(x)') + \ ... plot(Li,2,xmax, plot_points=30, plot_division=0, label='Li(x)') + \ ... plot(p, 2,xmax, label='P(X)', rgbcolor=(1,0,0)) ... sage: G.show(xmax=xmax) r U6# Worksheet 'talk' (2007-01-16 at 02:50) sage: 2 + 2 4r¡U\# Worksheet 'talk' (2007-01-16 at 02:50) sage: eratosthenes(20) [2, 3, 5, 7, 11, 13, 17, 19]r¢U># Worksheet 'talk' (2007-01-16 at 02:50) sage: n = pari(1000) r£U\# Worksheet 'talk' (2007-01-16 at 02:53) sage: eratosthenes(20) [2, 3, 5, 7, 11, 13, 17, 19]r¤U[# Worksheet 'talk' (2007-01-16 at 02:53) sage: prime_range(20) [2, 3, 5, 7, 11, 13, 17, 19]r¥UK# Worksheet 'features' (2007-01-16 at 03:02) sage: show(plot(sin),dpi=100) r¦UM# Worksheet 'features' (2007-01-16 at 03:02) sage: show(plot(sin(x)),dpi=20) r§UQ# Worksheet 'features' (2007-01-16 at 03:02) sage: show(plot(sin(x),0,5),dpi=20) r¨UT# Worksheet 'features' (2007-01-16 at 03:03) sage: show(plot(sin(x),0,6*pi),dpi=20) r©UT# Worksheet 'features' (2007-01-16 at 03:03) sage: show(plot(sin(x),0,6*pi),dpi=30) rªUT# Worksheet 'features' (2007-01-16 at 03:03) sage: show(plot(sin(x),0,6*pi),dpi=50) r«UT# Worksheet 'features' (2007-01-16 at 03:03) sage: show(plot(sin(x),0,6*pi),dpi=60) r¬U[# Worksheet 'features' (2007-01-16 at 03:03) sage: show(plot(sin(x),0,6*pi),figsize=[8,3]) rU]# Worksheet 'features' (2007-01-16 at 03:03) sage: show(plot(sin(x),-pi,6*pi),figsize=[8,3]) r®T‹# Worksheet 'features' (2007-01-16 at 03:39) %hide sage: t = Tachyon(xres=512,yres=512, camera_center=(3,0.3,0)) sage: t.light((4,3,2), 0.2, (1,1,1)) sage: t.texture('t0', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(1.0,0,0)) sage: t.texture('t1', ambient=0.1, diffuse=0.9, specular=0.3, opacity=1.0, color=(0,1.0,0)) sage: t.texture('t2', ambient=0.2,diffuse=0.7, specular=0.5, opacity=0.7, color=(0,0,1.0)) sage: k=0 sage: for i in srange(-1,1,0.05): ... k += 1 ... t.sphere((i,i^2-0.5,i^3), 0.1, 't%s'%(k%3)) ... sage: t.save('sage.png') r¯UA# Worksheet 'features' (2007-01-16 at 03:39) sage: a = (x-10)^10 r°U:# Worksheet 'features' (2007-01-16 at 03:39) sage: save a r±T# Worksheet 'features' (2007-01-16 at 03:39) sage: a = (x-10)^10; show(a)

x^{10} - 100x^{9} + 4500x^{8} - 120000x^{7} + 2100000x^{6} - 25200000x^{5} + 210000000x^{4} - 1200000000x^{3} + 4500000000x^{2} - 10000000000x + 10000000000

r²U¨# Worksheet 'features' (2007-01-16 at 03:39) sage: a = (x-10)^5; show(a)

x^{5} - 50x^{4} + 1000x^{3} - 10000x^{2} + 50000x - 100000

r³U:# Worksheet 'features' (2007-01-16 at 03:39) sage: save a r´T‰# Worksheet 'sieve' (2007-01-16 at 03:41) sage: %hideall sage: #auto sage: sage: def number_grid(c, box_args={}, font_args={},offset=0): ... """ ... INPUT: ... c -- list of length n^2, where n is an integer. ... The entries of c are RGB colors. ... box_args -- additional arguments passed to box. ... font_args -- all additional arguments are passed ... to the text function, e.g., fontsize. ... offset -- use to fine tune text placement in the squares ... ... OUTPUT: ... Graphics -- a plot of a grid that illustrates ... those n^2 numbers colored according to c. ... """ ... try: ... n = square_root(ZZ(len(c))) ... except ArithmeticError: ... raise ArithmeticError, "c must have square length" ... G = Graphics() ... k = 0 ... for j in reversed(range(n)): ... for i in range(n): ... col = c[int(k)] ... R = line([(i,j),(i+1,j),(i+1,j+1),(i,j+1),(i,j)], thickness=.2, **box_args) ... P = polygon([(i,j),(i+1,j),(i+1,j+1),(i,j+1),(i,j)], rgbcolor=col, **box_args) ... G += P + R ... if col != (1,1,1): ... G += text(str(k+1), (i+.5+offset,j+.5), **font_args) ... k += 1 ... G.axes(False) ... G.range(0,n,0,n) ... return G CPU time: 0.00 s, Wall time: 0.00 srµT€# Worksheet 'sieve' (2007-01-16 at 03:41) sage: %hideall sage: #auto sage: sage: def sieve_step(p, n, gone=(1,1,1), prime=(1,0,0), \ ... multiple=(.6,.6,.6), remaining=(.9,.9,.9), ... fontsize=11,offset=0): ... """ ... Return graphics that illustrates sieving out ... multiples of p. Numbers that are a nontrivial ... multiple of primes < p are shown in the gone color. ... Numbers that are a multiple of p are shown in ... a different color. The number p is shown in ... yet a third color. ... ... INPUT: ... p -- a prime (or 1, in which case all points are colored "remaining") ... n -- a SQUARE integer ... gone -- rgb color for integers that have been sieved out ... prime -- rgb color for p ... multiple -- rgb color for multiples of p ... remaining -- rgb color for integers that have not been sieved out yet ... and are not multiples of p. ... """ ... if p == 1: ... c = [remaining]*n ... else: ... exclude = prod(prime_range(p)) # exclude multiples of primes < p ... c = [] ... for k in range(1,n+1): ... if k <= p and is_prime(k): ... c.append(prime) ... elif k == 1 or (gcd(k,exclude) != 1 and not is_prime(k)): ... c.append(gone) ... elif k%p == 0: ... c.append(multiple) ... else: ... c.append(remaining) ... # end for ... # end if ... return number_grid(c,{},{'fontsize':fontsize},offset=offset) CPU time: 0.00 s, Wall time: 0.00 sr¶UH# Worksheet 'sieve' (2007-01-16 at 03:41) sage: show(sieve_step(1,100)) r·UH# Worksheet 'sieve' (2007-01-16 at 03:41) sage: show(sieve_step(2,100)) r¸UH# Worksheet 'sieve' (2007-01-16 at 03:41) sage: show(sieve_step(3,100)) r¹UH# Worksheet 'sieve' (2007-01-16 at 03:41) sage: show(sieve_step(5,100)) rºUH# Worksheet 'sieve' (2007-01-16 at 03:41) sage: show(sieve_step(7,100)) r»UI# Worksheet 'sieve' (2007-01-16 at 03:41) sage: show(sieve_step(11,100)) r¼U•# Worksheet 'sieve' (2007-01-16 at 03:41) sage: s = [sieve_step(p,100) for p in [2,3,5,7]] sage: G = graphics_array(s, 2,2) sage: G.show(axes=False) r½UN# Worksheet 'sieve' (2007-01-16 at 03:41) sage: G.save('sieve_boxes_100.pdf') r¾US# Worksheet 'sieve' (2007-01-16 at 03:41) sage: show(sieve_step(2,400,fontsize=6)) r¿US# Worksheet 'sieve' (2007-01-16 at 03:41) sage: show(sieve_step(3,400,fontsize=6)) rÀUS# Worksheet 'sieve' (2007-01-16 at 03:41) sage: show(sieve_step(5,400,fontsize=6)) rÁUS# Worksheet 'sieve' (2007-01-16 at 03:41) sage: show(sieve_step(7,400,fontsize=6)) rÂUT# Worksheet 'sieve' (2007-01-16 at 03:41) sage: show(sieve_step(11,400,fontsize=6)) rÃUr# Worksheet 'features' (2007-01-16 at 03:42) sage: show(plot(sin(x),-pi,6*pi,rgbcolor=(0.5,0,0.5)),figsize=[8,3]) rÄUr# Worksheet 'features' (2007-01-16 at 03:43) sage: show(plot(sin(x),-pi,6*pi,rgbcolor=(0.5,0,0.5)),figsize=[8,3]) rÅT‹# Worksheet 'features' (2007-01-16 at 03:43) %hide sage: t = Tachyon(xres=512,yres=512, camera_center=(3,0.3,0)) sage: t.light((4,3,2), 0.2, (1,1,1)) sage: t.texture('t0', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(1.0,0,0)) sage: t.texture('t1', ambient=0.1, diffuse=0.9, specular=0.3, opacity=1.0, color=(0,1.0,0)) sage: t.texture('t2', ambient=0.2,diffuse=0.7, specular=0.5, opacity=0.7, color=(0,0,1.0)) sage: k=0 sage: for i in srange(-1,1,0.05): ... k += 1 ... t.sphere((i,i^2-0.5,i^3), 0.1, 't%s'%(k%3)) ... sage: t.save('sage.png') rÆU:# Worksheet 'features' (2007-01-16 at 03:43) sage: save a rÇU¨# Worksheet 'features' (2007-01-16 at 03:43) sage: a = (x-10)^5; show(a)

x^{5} - 50x^{4} + 1000x^{3} - 10000x^{2} + 50000x - 100000

rÈU:# Worksheet 'features' (2007-01-16 at 03:43) sage: save a rÉT# Worksheet 'features' (2007-01-16 at 03:45) sage: %hide sage: t = Tachyon(xres=700, yres=500, camera_center=(2,7,4), look_at=(2,0,0), raydepth=4) sage: t.light((10,3,2), 1, (1,1,1)) sage: t.light((10,-3,2), 1, (1,1,1)) sage: t.texture('black', color=(0,0,0)) sage: t.texture('red', color=(1,0,0)) sage: t.texture('grey', color=(.9,.9,.9)) sage: t.plane((0,0,0),(0,0,1),'grey') sage: t.cylinder((0,0,0),(1,0,0),.01,'black') sage: t.cylinder((0,0,0),(0,1,0),.01,'black') sage: E = EllipticCurve('37a'); show(E) sage: P = E([0,0]) sage: Q = P sage: n = 60 sage: for i in range(n): ... Q = Q + P ... c = i/n + .1 ... t.texture('r%s'%i,color=(float(i/n),0,0)) ... t.sphere((Q[0], -Q[1], .01), .04, 'r%s'%i) ... sage: show(t)

y^2 + y = x^3 - x

rÊT# Worksheet 'features' (2007-01-16 at 03:46) sage: %hide sage: t = Tachyon(xres=700, yres=500, camera_center=(2,7,4), look_at=(2,0,0), raydepth=4) sage: t.light((10,3,2), 1, (1,1,1)) sage: t.light((10,-3,2), 1, (1,1,1)) sage: t.texture('black', color=(0,0,0)) sage: t.texture('red', color=(1,0,0)) sage: t.texture('grey', color=(.9,.9,.9)) sage: t.plane((0,0,0),(0,0,1),'grey') sage: t.cylinder((0,0,0),(1,0,0),.01,'black') sage: t.cylinder((0,0,0),(0,1,0),.01,'black') sage: E = EllipticCurve('37a'); show(E) sage: P = E([0,0]) sage: Q = P sage: n = 60 sage: for i in range(n): ... Q = Q + P ... c = i/n + .1 ... t.texture('r%s'%i,color=(float(i/n),0,0)) ... t.sphere((Q[0], -Q[1], .01), .04, 'r%s'%i) ... sage: show(t)

y^2 + y = x^3 - x

rËUú# Worksheet 'talk' (2007-01-16 at 03:50) sage: 2 + 2 4 Traceback (most recent call last): File "", line 1, in File "/Users/was/talks/2007-01-15-sage-cse/sage_notebook/worksheets/talk/code/2.py", line 5, in print "\n\nrÌU6# Worksheet 'talk' (2007-01-16 at 03:50) sage: 2 + 2 4rÍU[# Worksheet 'talk' (2007-01-16 at 03:50) sage: prime_range(20) [2, 3, 5, 7, 11, 13, 17, 19]rÎU6# Worksheet 'talk' (2007-01-16 at 03:50) sage: 2 ^ 3 8rÏUD# Worksheet 'talk' (2007-01-16 at 03:50) sage: %python sage: 2 ^ 3 1rÐUD# Worksheet 'talk' (2007-01-16 at 03:50) sage: factorial(10) 3628800rÑUP# Worksheet 'talk' (2007-01-16 at 03:52) sage: factorial(20) 2432902008176640000rÒUq# Worksheet 'talk' (2007-01-16 at 03:52) sage: factor( factorial(20) ) 2^18 * 3^8 * 5^4 * 7^2 * 11 * 13 * 17 * 19rÓU # Worksheet 'talk' (2007-01-16 at 03:52) sage: factor( factorial(50) ) 2^47 * 3^22 * 5^12 * 7^8 * 11^4 * 13^3 * 17^2 * 19^2 * 23^2 * 29 * 31 * 37 * 41 * 43 * 47rÔUŽ# Worksheet 'talk' (2007-01-16 at 03:52) sage: factor( factorial(40) ) 2^38 * 3^18 * 5^9 * 7^5 * 11^3 * 13^3 * 17^2 * 19^2 * 23 * 29 * 31 * 37rÕUõ# Worksheet 'talk' (2007-01-16 at 03:52) sage: show(factor( factorial(40) ))

2^{38} \cdot 3^{18} \cdot 5^{9} \cdot 7^{5} \cdot 11^{3} \cdot 13^{3} \cdot 17^{2} \cdot 19^{2} \cdot 23 \cdot 29 \cdot 31 \cdot 37

rÖU¦# Worksheet 'talk' (2007-01-16 at 03:53) sage: %sagex sage: def is2pow(unsigned int n): ... while n != 0 and n%2 == 0: ... n = n >> 1 ... return n == 1 r×U‚# Worksheet 'talk' (2007-01-16 at 03:53) sage: time v = [n for n in range(10^5) if is2pow(n)] CPU time: 0.04 s, Wall time: 0.07 srØUŒ# Worksheet 'talk' (2007-01-16 at 03:54) sage: def is2pow(n): ... while n != 0 and n%2 == 0: ... n = n >> 1 ... return n == 1 rÙU¯# Worksheet 'talk' (2007-01-16 at 03:54) sage: %sagex sage: def is2pow_compiled(unsigned int n): ... while n != 0 and n%2 == 0: ... n = n >> 1 ... return n == 1 rÚU‚# Worksheet 'talk' (2007-01-16 at 03:54) sage: time v = [n for n in range(10^5) if is2pow(n)] CPU time: 3.39 s, Wall time: 4.54 srÛU‹# Worksheet 'talk' (2007-01-16 at 03:54) sage: time v = [n for n in range(10^5) if is2pow_compiled(n)] CPU time: 0.04 s, Wall time: 0.04 srÜU…# Worksheet 'talk' (2007-01-16 at 03:55) sage: v [1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536]rÝUã# Worksheet 'talk' (2007-01-16 at 03:55) sage: time v = [n for n in range(10^5) if is2pow_compiled(n)]; v [1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536] CPU time: 0.04 s, Wall time: 0.05 srÞUÚ# Worksheet 'talk' (2007-01-16 at 03:55) sage: time v = [n for n in range(10^5) if is2pow(n)]; v [1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536] CPU time: 3.29 s, Wall time: 3.39 srßUA# Worksheet 'talk' (2007-01-16 at 03:56) sage: a = gp('2+3'); a 5ràUA# Worksheet 'talk' (2007-01-16 at 03:56) sage: a = gp('2+3'); a 5ráU6# Worksheet 'talk' (2007-01-16 at 03:57) sage: a^3 125râU]# Worksheet 'talk' (2007-01-16 at 03:57) sage: type(a) rãU6# Worksheet 'talk' (2007-01-16 at 04:03) sage: 2 + 2 4räU[# Worksheet 'talk' (2007-01-16 at 04:03) sage: prime_range(20) [2, 3, 5, 7, 11, 13, 17, 19]råU6# Worksheet 'talk' (2007-01-16 at 04:03) sage: 2 ^ 3 8ræUD# Worksheet 'talk' (2007-01-16 at 04:03) sage: %python sage: 2 ^ 3 1rçUP# Worksheet 'talk' (2007-01-16 at 04:03) sage: factorial(20) 2432902008176640000rèUõ# Worksheet 'talk' (2007-01-16 at 04:03) sage: show(factor( factorial(40) ))

2^{38} \cdot 3^{18} \cdot 5^{9} \cdot 7^{5} \cdot 11^{3} \cdot 13^{3} \cdot 17^{2} \cdot 19^{2} \cdot 23 \cdot 29 \cdot 31 \cdot 37

réU¡# Worksheet 'talk' (2007-01-16 at 04:03) sage: m = matrix(5,5, range(25)); m [ 0 1 2 3 4] [ 5 6 7 8 9] [10 11 12 13 14] [15 16 17 18 19] [20 21 22 23 24]rêUˆ# Worksheet 'talk' (2007-01-16 at 04:03) sage: f = m.charpoly(); show(f)

x^{5} - 60x^{4} - 250x^{3}

rëUƒ# Worksheet 'talk' (2007-01-16 at 04:03) sage: show(factor(f))

(x^{2} - 60x - 250) \cdot x^{3}

rìUî# Worksheet 'talk' (2007-01-16 at 04:03) sage: m = matrix(5,5, range(25)); show(m)

\left(\begin{array}{rrrrr} 0&1&2&3&4\\ 5&6&7&8&9\\ 10&11&12&13&14\\ 15&16&17&18&19\\ 20&21&22&23&24 \end{array}\right)

ríU‹# Worksheet 'talk' (2007-01-16 at 04:03) sage: f = m.charpoly('T'); show(f)

T^{5} - 60T^{4} - 250T^{3}

rîUƒ# Worksheet 'talk' (2007-01-16 at 04:03) sage: show(factor(f))

(T^{2} - 60T - 250) \cdot T^{3}

rïUÈ# Worksheet 'talk' (2007-01-16 at 04:04) sage: m = matrix(5,5,[randint(10) for _ in range(25)]); show(m) Traceback (most recent call last): ... TypeError: randint() takes exactly 3 arguments (2 given)rðT# Worksheet 'talk' (2007-01-16 at 04:04) sage: m = matrix(5,5,[randint(-10,10) for _ in range(25)]); show(m)

\left(\begin{array}{rrrrr} 0&-7&-8&-6&7\\ -8&-8&0&-2&1\\ -2&-4&7&0&-6\\ 5&8&-4&0&5\\ 3&-3&2&0&7 \end{array}\right)

rñU£# Worksheet 'talk' (2007-01-16 at 04:04) sage: f = m.charpoly('T'); show(f)

T^{5} - 6T^{4} - 95T^{3} + 497T^{2} + 589T - 12256

ròU˜# Worksheet 'talk' (2007-01-16 at 04:04) sage: show(factor(f))

(T^{5} - 6T^{4} - 95T^{3} + 497T^{2} + 589T - 12256)

róUö# Worksheet 'talk' (2007-01-16 at 04:04) sage: m = matrix(5,5,[randint(0,1) for _ in range(25)]); show(m)

\left(\begin{array}{rrrrr} 0&1&1&1&1\\ 1&1&0&0&1\\ 0&0&0&0&1\\ 0&0&0&0&0\\ 0&0&1&1&0 \end{array}\right)

rôU“# Worksheet 'talk' (2007-01-16 at 04:04) sage: f = m.charpoly('T'); show(f)

T^{5} - T^{4} - 2T^{3} + T^{2} + T

rõU—# Worksheet 'talk' (2007-01-16 at 04:04) sage: show(factor(f))

(T - 1) \cdot T \cdot (T + 1) \cdot (T^{2} - T - 1)

röUö# Worksheet 'talk' (2007-01-16 at 04:04) sage: m = matrix(5,5,[randint(0,1) for _ in range(25)]); show(m)

\left(\begin{array}{rrrrr} 1&0&0&1&0\\ 0&1&0&0&1\\ 1&0&0&0&1\\ 1&1&1&0&0\\ 0&1&1&1&0 \end{array}\right)

r÷U™# Worksheet 'talk' (2007-01-16 at 04:04) sage: f = m.charpoly('T'); show(f)

T^{5} - 2T^{4} - 2T^{3} + T^{2} + 5T - 2

røUŽ# Worksheet 'talk' (2007-01-16 at 04:04) sage: show(factor(f))

(T^{5} - 2T^{4} - 2T^{3} + T^{2} + 5T - 2)

rùUö# Worksheet 'talk' (2007-01-16 at 04:04) sage: m = matrix(5,5,[randint(0,1) for _ in range(25)]); show(m)

\left(\begin{array}{rrrrr} 0&0&1&0&0\\ 1&0&0&1&1\\ 0&1&1&1&1\\ 0&0&0&1&1\\ 0&1&1&0&0 \end{array}\right)

rúU# Worksheet 'talk' (2007-01-16 at 04:04) sage: f = m.charpoly('T'); show(f)

T^{5} - 2T^{4} - T^{3} - T^{2}

rûUˆ# Worksheet 'talk' (2007-01-16 at 04:04) sage: show(factor(f))

T^{2} \cdot (T^{3} - 2T^{2} - T - 1)

rüU‡# Worksheet 'interface' (2007-01-16 at 04:06) sage: parent(a) Traceback (most recent call last): ... NameError: name 'a' is not definedrýUƒ# Worksheet 'interface' (2007-01-16 at 04:06) sage: 5 + a Traceback (most recent call last): ... NameError: name 'a' is not definedrþUF# Worksheet 'interface' (2007-01-16 at 04:06) sage: a = gp('2+3'); a 5rÿU;# Worksheet 'interface' (2007-01-16 at 04:06) sage: a^3 125rUb# Worksheet 'interface' (2007-01-16 at 04:06) sage: type(a) rUQ# Worksheet 'interface' (2007-01-16 at 04:06) sage: parent(a) GP/PARI interpreterrU<# Worksheet 'interface' (2007-01-16 at 04:06) sage: 5 + a 10rUS# Worksheet 'interface' (2007-01-16 at 04:06) sage: parent(5+a) GP/PARI interpreterrUF# Worksheet 'interface' (2007-01-16 at 04:06) sage: pari(5) + gp(5) 10rUk# Worksheet 'interface' (2007-01-16 at 04:06) sage: parent(pari(5) + gp(5)) Interface to the PARI C libraryrUS# Worksheet 'interface' (2007-01-16 at 04:06) sage: parent(5+a) GP/PARI interpreterrUE# Worksheet 'interface' (2007-01-16 at 04:06) sage: gap(5) + gp(5) 10rUN# Worksheet 'interface' (2007-01-16 at 04:06) sage: parent(gap(5) + gp(5)) Gapr U^# Worksheet 'interface' (2007-01-16 at 04:06) sage: parent(gp(5) + gap(5)) GP/PARI interpreterr UN# Worksheet 'interface' (2007-01-16 at 04:06) sage: parent(gap(5) + gp(5)) GaprUT# Worksheet 'interface' (2007-01-16 at 04:07) sage: a = gp('x^3 - y^3'); a x^3 - y^3rUY# Worksheet 'interface' (2007-01-16 at 04:07) sage: a^3 x^9 - 3*y^3*x^6 + 3*y^6*x^3 - y^9r Ub# Worksheet 'interface' (2007-01-16 at 04:07) sage: type(a) rUQ# Worksheet 'interface' (2007-01-16 at 04:07) sage: parent(a) GP/PARI interpreterrT# Worksheet 'interface' (2007-01-16 at 04:07) sage: a.factor() Traceback (most recent call last): ... TypeError: Error executing code in GP/PARI: CODE: sage[22]=factor(sage[20]); GP/PARI ERROR: *** factor: sorry, factor for general polynomials is not yet implemented.rU‹# Worksheet 'interface' (2007-01-16 at 04:07) sage: sage_eval(a) Traceback (most recent call last): ... TypeError: source must be a string.rU–# Worksheet 'interface' (2007-01-16 at 04:07) sage: b = sage_eval(str(a)); b Traceback (most recent call last): ... NameError: name 'y' is not definedrU°# Worksheet 'interface' (2007-01-16 at 04:07) sage: R. = QQ[x,y] sage: b = R(a) Traceback (most recent call last): ... TypeError: unable to convert x^3 - y^3 to a rationalrUZ# Worksheet 'interface' (2007-01-16 at 04:07) sage: R. = QQ[x,y] sage: b = R(str(a)) rU]# Worksheet 'interface' (2007-01-16 at 04:07) sage: b.factor() (-1*y + x) * (y^2 + x*y + x^2)rUH# Worksheet 'sieve' (2007-01-16 at 04:09) sage: show(sieve_step(1,100)) rUH# Worksheet 'sieve' (2007-01-16 at 04:09) sage: show(sieve_step(2,100)) rUH# Worksheet 'sieve' (2007-01-16 at 04:09) sage: show(sieve_step(3,100)) rUH# Worksheet 'sieve' (2007-01-16 at 04:09) sage: show(sieve_step(5,100)) rUH# Worksheet 'sieve' (2007-01-16 at 04:09) sage: show(sieve_step(7,100)) rUI# Worksheet 'sieve' (2007-01-16 at 04:09) sage: show(sieve_step(11,100)) rU•# Worksheet 'sieve' (2007-01-16 at 04:09) sage: s = [sieve_step(p,100) for p in [2,3,5,7]] sage: G = graphics_array(s, 2,2) sage: G.show(axes=False) rT‰# Worksheet 'sieve' (2007-01-16 at 04:10) sage: %hideall sage: #auto sage: sage: def number_grid(c, box_args={}, font_args={},offset=0): ... """ ... INPUT: ... c -- list of length n^2, where n is an integer. ... The entries of c are RGB colors. ... box_args -- additional arguments passed to box. ... font_args -- all additional arguments are passed ... to the text function, e.g., fontsize. ... offset -- use to fine tune text placement in the squares ... ... OUTPUT: ... Graphics -- a plot of a grid that illustrates ... those n^2 numbers colored according to c. ... """ ... try: ... n = square_root(ZZ(len(c))) ... except ArithmeticError: ... raise ArithmeticError, "c must have square length" ... G = Graphics() ... k = 0 ... for j in reversed(range(n)): ... for i in range(n): ... col = c[int(k)] ... R = line([(i,j),(i+1,j),(i+1,j+1),(i,j+1),(i,j)], thickness=.2, **box_args) ... P = polygon([(i,j),(i+1,j),(i+1,j+1),(i,j+1),(i,j)], rgbcolor=col, **box_args) ... G += P + R ... if col != (1,1,1): ... G += text(str(k+1), (i+.5+offset,j+.5), **font_args) ... k += 1 ... G.axes(False) ... G.range(0,n,0,n) ... return G CPU time: 0.00 s, Wall time: 0.00 srT‰# Worksheet 'sieve' (2007-01-16 at 04:10) sage: %hideall sage: #auto sage: sage: def number_grid(c, box_args={}, font_args={},offset=0): ... """ ... INPUT: ... c -- list of length n^2, where n is an integer. ... The entries of c are RGB colors. ... box_args -- additional arguments passed to box. ... font_args -- all additional arguments are passed ... to the text function, e.g., fontsize. ... offset -- use to fine tune text placement in the squares ... ... OUTPUT: ... Graphics -- a plot of a grid that illustrates ... those n^2 numbers colored according to c. ... """ ... try: ... n = square_root(ZZ(len(c))) ... except ArithmeticError: ... raise ArithmeticError, "c must have square length" ... G = Graphics() ... k = 0 ... for j in reversed(range(n)): ... for i in range(n): ... col = c[int(k)] ... R = line([(i,j),(i+1,j),(i+1,j+1),(i,j+1),(i,j)], thickness=.2, **box_args) ... P = polygon([(i,j),(i+1,j),(i+1,j+1),(i,j+1),(i,j)], rgbcolor=col, **box_args) ... G += P + R ... if col != (1,1,1): ... G += text(str(k+1), (i+.5+offset,j+.5), **font_args) ... k += 1 ... G.axes(False) ... G.range(0,n,0,n) ... return G CPU time: 0.00 s, Wall time: 0.00 srUš# Worksheet 'sieve' (2007-01-16 at 04:10) sage: show(sieve_step(1,100)) Traceback (most recent call last): ... NameError: name 'sieve_step' is not definedrUš# Worksheet 'sieve' (2007-01-16 at 04:10) sage: show(sieve_step(2,100)) Traceback (most recent call last): ... NameError: name 'sieve_step' is not definedr T‰# Worksheet 'sieve' (2007-01-16 at 04:11) sage: %hideall sage: #auto sage: sage: def number_grid(c, box_args={}, font_args={},offset=0): ... """ ... INPUT: ... c -- list of length n^2, where n is an integer. ... The entries of c are RGB colors. ... box_args -- additional arguments passed to box. ... font_args -- all additional arguments are passed ... to the text function, e.g., fontsize. ... offset -- use to fine tune text placement in the squares ... ... OUTPUT: ... Graphics -- a plot of a grid that illustrates ... those n^2 numbers colored according to c. ... """ ... try: ... n = square_root(ZZ(len(c))) ... except ArithmeticError: ... raise ArithmeticError, "c must have square length" ... G = Graphics() ... k = 0 ... for j in reversed(range(n)): ... for i in range(n): ... col = c[int(k)] ... R = line([(i,j),(i+1,j),(i+1,j+1),(i,j+1),(i,j)], thickness=.2, **box_args) ... P = polygon([(i,j),(i+1,j),(i+1,j+1),(i,j+1),(i,j)], rgbcolor=col, **box_args) ... G += P + R ... if col != (1,1,1): ... G += text(str(k+1), (i+.5+offset,j+.5), **font_args) ... k += 1 ... G.axes(False) ... G.range(0,n,0,n) ... return G CPU time: 0.00 s, Wall time: 0.00 sr!T€# Worksheet 'sieve' (2007-01-16 at 04:11) sage: %hideall sage: #auto sage: sage: def sieve_step(p, n, gone=(1,1,1), prime=(1,0,0), \ ... multiple=(.6,.6,.6), remaining=(.9,.9,.9), ... fontsize=11,offset=0): ... """ ... Return graphics that illustrates sieving out ... multiples of p. Numbers that are a nontrivial ... multiple of primes < p are shown in the gone color. ... Numbers that are a multiple of p are shown in ... a different color. The number p is shown in ... yet a third color. ... ... INPUT: ... p -- a prime (or 1, in which case all points are colored "remaining") ... n -- a SQUARE integer ... gone -- rgb color for integers that have been sieved out ... prime -- rgb color for p ... multiple -- rgb color for multiples of p ... remaining -- rgb color for integers that have not been sieved out yet ... and are not multiples of p. ... """ ... if p == 1: ... c = [remaining]*n ... else: ... exclude = prod(prime_range(p)) # exclude multiples of primes < p ... c = [] ... for k in range(1,n+1): ... if k <= p and is_prime(k): ... c.append(prime) ... elif k == 1 or (gcd(k,exclude) != 1 and not is_prime(k)): ... c.append(gone) ... elif k%p == 0: ... c.append(multiple) ... else: ... c.append(remaining) ... # end for ... # end if ... return number_grid(c,{},{'fontsize':fontsize},offset=offset) CPU time: 0.00 s, Wall time: 0.00 sr"UH# Worksheet 'sieve' (2007-01-16 at 04:11) sage: show(sieve_step(1,100)) r#UH# Worksheet 'sieve' (2007-01-16 at 04:11) sage: show(sieve_step(2,100)) r$UH# Worksheet 'sieve' (2007-01-16 at 04:11) sage: show(sieve_step(3,100)) r%UH# Worksheet 'sieve' (2007-01-16 at 04:11) sage: show(sieve_step(5,100)) r&UH# Worksheet 'sieve' (2007-01-16 at 04:11) sage: show(sieve_step(7,100)) r'UI# Worksheet 'sieve' (2007-01-16 at 04:11) sage: show(sieve_step(11,100)) r(U•# Worksheet 'sieve' (2007-01-16 at 04:11) sage: s = [sieve_step(p,100) for p in [2,3,5,7]] sage: G = graphics_array(s, 2,2) sage: G.show(axes=False) r)US# Worksheet 'sieve' (2007-01-16 at 04:11) sage: show(sieve_step(2,400,fontsize=6)) r*UT# Worksheet 'sieve' (2007-01-16 at 04:11) sage: show(sieve_step(13,400,fontsize=6)) r+U6# Worksheet 'talk' (2007-01-16 at 04:49) sage: 2 + 2 4r,U[# Worksheet 'talk' (2007-01-16 at 04:49) sage: prime_range(20) [2, 3, 5, 7, 11, 13, 17, 19]r-U6# Worksheet 'talk' (2007-01-16 at 04:49) sage: 2 ^ 3 8r.UD# Worksheet 'talk' (2007-01-16 at 04:49) sage: %python sage: 2 ^ 3 1r/UP# Worksheet 'talk' (2007-01-16 at 04:49) sage: factorial(20) 2432902008176640000r0Uõ# Worksheet 'talk' (2007-01-16 at 04:49) sage: show(factor( factorial(40) ))

2^{38} \cdot 3^{18} \cdot 5^{9} \cdot 7^{5} \cdot 11^{3} \cdot 13^{3} \cdot 17^{2} \cdot 19^{2} \cdot 23 \cdot 29 \cdot 31 \cdot 37

r1Uö# Worksheet 'talk' (2007-01-16 at 04:49) sage: m = matrix(5,5,[randint(0,1) for _ in range(25)]); show(m)

\left(\begin{array}{rrrrr} 1&0&1&0&0\\ 1&0&1&0&1\\ 0&0&0&1&0\\ 1&0&1&0&0\\ 1&0&1&1&1 \end{array}\right)

r2U‡# Worksheet 'talk' (2007-01-16 at 04:49) sage: f = m.charpoly('T'); show(f)

T^{5} - 2T^{4} + T^{2}

r3U# Worksheet 'talk' (2007-01-16 at 04:49) sage: show(factor(f))

(T - 1) \cdot T^{2} \cdot (T^{2} - T - 1)

r4Ur# Worksheet 'features' (2007-01-16 at 04:49) sage: show(plot(sin(x),-pi,6*pi,rgbcolor=(0.5,0,0.5)),figsize=[8,3]) r5T# Worksheet 'features' (2007-01-16 at 04:49) sage: %hide sage: t = Tachyon(xres=700, yres=500, camera_center=(2,7,4), look_at=(2,0,0), raydepth=4) sage: t.light((10,3,2), 1, (1,1,1)) sage: t.light((10,-3,2), 1, (1,1,1)) sage: t.texture('black', color=(0,0,0)) sage: t.texture('red', color=(1,0,0)) sage: t.texture('grey', color=(.9,.9,.9)) sage: t.plane((0,0,0),(0,0,1),'grey') sage: t.cylinder((0,0,0),(1,0,0),.01,'black') sage: t.cylinder((0,0,0),(0,1,0),.01,'black') sage: E = EllipticCurve('37a'); show(E) sage: P = E([0,0]) sage: Q = P sage: n = 60 sage: for i in range(n): ... Q = Q + P ... c = i/n + .1 ... t.texture('r%s'%i,color=(float(i/n),0,0)) ... t.sphere((Q[0], -Q[1], .01), .04, 'r%s'%i) ... sage: show(t)

y^2 + y = x^3 - x

r6U# Worksheet 'riemann' (2007-01-16 at 04:51) sage: #auto sage: def f(x): ... return x / log(x) ... sage: def p(x): ... return prime_pi(x) r7T# Worksheet 'riemann' (2007-01-16 at 04:51) sage: xmax = 20 sage: G = plot(f, 2,xmax, linestyle='--', label='x/log(x)') + \ ... plot(Li,2,xmax, plot_points=30, plot_division=0, label='Li(x)') + \ ... plot(p, 2,xmax, label='P(X)', rgbcolor=(1,0,0)) ... sage: G.show(xmax=xmax) r8T# Worksheet 'riemann' (2007-01-16 at 04:51) sage: xmax = 500 sage: G = plot(f, 2,xmax, linestyle='--', label='x/log(x)') + \ ... plot(Li,2,xmax, plot_points=30, plot_division=0, label='Li(x)') + \ ... plot(p, 2,xmax, label='P(X)', rgbcolor=(1,0,0)) ... sage: G.show(xmax=xmax) r9T# Worksheet 'riemann' (2007-01-16 at 04:51) sage: xmax = 20000 sage: G = plot(f, 2,xmax, linestyle='--', label='x/log(x)') + \ ... plot(Li,2,xmax, plot_points=30, plot_division=0, label='Li(x)') + \ ... plot(p, 2,xmax, label='P(X)', rgbcolor=(1,0,0)) ... sage: G.show(xmax=xmax) r:U6# Worksheet 'talk' (2007-01-16 at 11:11) sage: 2 + 2 4r;U6# Worksheet 'talk' (2007-01-16 at 11:11) sage: 2 + 2 4r<U6# Worksheet 'talk' (2007-01-16 at 11:11) sage: 2 + 3 5r=U[# Worksheet 'talk' (2007-01-16 at 11:11) sage: prime_range(20) [2, 3, 5, 7, 11, 13, 17, 19]r>U6# Worksheet 'talk' (2007-01-16 at 11:12) sage: 2 ^ 3 8r?UD# Worksheet 'talk' (2007-01-16 at 11:12) sage: %python sage: 2 ^ 3 1r@UP# Worksheet 'talk' (2007-01-16 at 11:12) sage: factorial(20) 2432902008176640000rAUõ# Worksheet 'talk' (2007-01-16 at 11:12) sage: show(factor( factorial(40) ))

2^{38} \cdot 3^{18} \cdot 5^{9} \cdot 7^{5} \cdot 11^{3} \cdot 13^{3} \cdot 17^{2} \cdot 19^{2} \cdot 23 \cdot 29 \cdot 31 \cdot 37

rBUö# Worksheet 'talk' (2007-01-16 at 11:12) sage: m = matrix(5,5,[randint(0,1) for _ in range(25)]); show(m)

\left(\begin{array}{rrrrr} 1&1&0&0&0\\ 0&0&1&0&0\\ 0&0&1&1&0\\ 0&0&1&0&0\\ 1&1&0&1&0 \end{array}\right)

rCU‡# Worksheet 'talk' (2007-01-16 at 11:12) sage: f = m.charpoly('T'); show(f)

T^{5} - 2T^{4} + T^{2}

rDU# Worksheet 'talk' (2007-01-16 at 11:12) sage: show(factor(f))

(T - 1) \cdot T^{2} \cdot (T^{2} - T - 1)

rET‰# Worksheet 'sieve' (2007-01-16 at 11:12) sage: %hideall sage: #auto sage: sage: def number_grid(c, box_args={}, font_args={},offset=0): ... """ ... INPUT: ... c -- list of length n^2, where n is an integer. ... The entries of c are RGB colors. ... box_args -- additional arguments passed to box. ... font_args -- all additional arguments are passed ... to the text function, e.g., fontsize. ... offset -- use to fine tune text placement in the squares ... ... OUTPUT: ... Graphics -- a plot of a grid that illustrates ... those n^2 numbers colored according to c. ... """ ... try: ... n = square_root(ZZ(len(c))) ... except ArithmeticError: ... raise ArithmeticError, "c must have square length" ... G = Graphics() ... k = 0 ... for j in reversed(range(n)): ... for i in range(n): ... col = c[int(k)] ... R = line([(i,j),(i+1,j),(i+1,j+1),(i,j+1),(i,j)], thickness=.2, **box_args) ... P = polygon([(i,j),(i+1,j),(i+1,j+1),(i,j+1),(i,j)], rgbcolor=col, **box_args) ... G += P + R ... if col != (1,1,1): ... G += text(str(k+1), (i+.5+offset,j+.5), **font_args) ... k += 1 ... G.axes(False) ... G.range(0,n,0,n) ... return G CPU time: 0.00 s, Wall time: 0.00 srFT€# Worksheet 'sieve' (2007-01-16 at 11:12) sage: %hideall sage: #auto sage: sage: def sieve_step(p, n, gone=(1,1,1), prime=(1,0,0), \ ... multiple=(.6,.6,.6), remaining=(.9,.9,.9), ... fontsize=11,offset=0): ... """ ... Return graphics that illustrates sieving out ... multiples of p. Numbers that are a nontrivial ... multiple of primes < p are shown in the gone color. ... Numbers that are a multiple of p are shown in ... a different color. The number p is shown in ... yet a third color. ... ... INPUT: ... p -- a prime (or 1, in which case all points are colored "remaining") ... n -- a SQUARE integer ... gone -- rgb color for integers that have been sieved out ... prime -- rgb color for p ... multiple -- rgb color for multiples of p ... remaining -- rgb color for integers that have not been sieved out yet ... and are not multiples of p. ... """ ... if p == 1: ... c = [remaining]*n ... else: ... exclude = prod(prime_range(p)) # exclude multiples of primes < p ... c = [] ... for k in range(1,n+1): ... if k <= p and is_prime(k): ... c.append(prime) ... elif k == 1 or (gcd(k,exclude) != 1 and not is_prime(k)): ... c.append(gone) ... elif k%p == 0: ... c.append(multiple) ... else: ... c.append(remaining) ... # end for ... # end if ... return number_grid(c,{},{'fontsize':fontsize},offset=offset) CPU time: 0.00 s, Wall time: 0.00 srGUH# Worksheet 'sieve' (2007-01-16 at 11:12) sage: show(sieve_step(1,100)) rHUH# Worksheet 'sieve' (2007-01-16 at 11:12) sage: show(sieve_step(1,100)) rIUH# Worksheet 'sieve' (2007-01-16 at 11:13) sage: show(sieve_step(2,100)) rJUH# Worksheet 'sieve' (2007-01-16 at 11:13) sage: show(sieve_step(1,100)) rKUH# Worksheet 'sieve' (2007-01-16 at 11:13) sage: show(sieve_step(2,100)) rLUH# Worksheet 'sieve' (2007-01-16 at 11:13) sage: show(sieve_step(3,100)) rMUH# Worksheet 'sieve' (2007-01-16 at 11:13) sage: show(sieve_step(5,100)) rNUH# Worksheet 'sieve' (2007-01-16 at 11:14) sage: show(sieve_step(7,100)) rOUI# Worksheet 'sieve' (2007-01-16 at 11:14) sage: show(sieve_step(11,100)) rPU•# Worksheet 'sieve' (2007-01-16 at 11:14) sage: s = [sieve_step(p,100) for p in [2,3,5,7]] sage: G = graphics_array(s, 2,2) sage: G.show(axes=False) rQUT# Worksheet 'interface' (2007-01-16 at 11:30) sage: a = gp('x^3 - y^3'); a x^3 - y^3rRU?# Worksheet 'interface' (2007-01-16 at 11:30) sage: a x^3 - y^3rSUb# Worksheet 'interface' (2007-01-16 at 11:30) sage: type(a) rTUQ# Worksheet 'interface' (2007-01-16 at 11:30) sage: parent(a) GP/PARI interpreterrUT# Worksheet 'interface' (2007-01-16 at 11:30) sage: a.factor() Traceback (most recent call last): ... TypeError: Error executing code in GP/PARI: CODE: sage[2]=factor(sage[1]); GP/PARI ERROR: *** factor: sorry, factor for general polynomials is not yet implemented.rVUZ# Worksheet 'interface' (2007-01-16 at 11:30) sage: R. = QQ[x,y] sage: b = R(str(a)) rWU]# Worksheet 'interface' (2007-01-16 at 11:30) sage: b.factor() (-1*y + x) * (y^2 + x*y + x^2)rXUT# Worksheet 'interface' (2007-01-16 at 11:39) sage: a = gp('x^3 - y^3'); a x^3 - y^3rYU?# Worksheet 'interface' (2007-01-16 at 11:39) sage: a x^3 - y^3rZUb# Worksheet 'interface' (2007-01-16 at 11:39) sage: type(a) r[UQ# Worksheet 'interface' (2007-01-16 at 11:39) sage: parent(a) GP/PARI interpreterr\UT# Worksheet 'interface' (2007-01-16 at 11:39) sage: a = gp('x^3 - y^3'); a x^3 - y^3r]U?# Worksheet 'interface' (2007-01-16 at 11:39) sage: a x^3 - y^3r^Ub# Worksheet 'interface' (2007-01-16 at 11:39) sage: type(a) r_UQ# Worksheet 'interface' (2007-01-16 at 11:39) sage: parent(a) GP/PARI interpreterr`T# Worksheet 'interface' (2007-01-16 at 11:39) sage: a.factor() Traceback (most recent call last): ... TypeError: Error executing code in GP/PARI: CODE: sage[5]=factor(sage[4]); GP/PARI ERROR: *** factor: sorry, factor for general polynomials is not yet implemented.raUZ# Worksheet 'interface' (2007-01-16 at 11:39) sage: R. = QQ[x,y] sage: b = R(str(a)) rbU]# Worksheet 'interface' (2007-01-16 at 11:39) sage: b.factor() (-1*y + x) * (y^2 + x*y + x^2)rcU/# Worksheet 'interface' (2007-01-16 at 11:39) rdT# Worksheet 'bernoulli' (2007-01-16 at 12:00) sage: time bernoulli(10^4) -211595838046290940721792738040989085665429357988608368792530057574963977461347659806060988412665049378263206010619011705021028593325296217651385927185658068328963543023779613882052256977186230979132745800171383704596841985854242995142873592500747065854779015122240428012255687613284177651255774757047931080059741927443045700873077308529026131284714239112011242912285115288580651936754872068059186057890381422381552365672295636936897632669791887088510748161499...reUq# Worksheet 'bernoulli' (2007-01-16 at 12:00) sage: time a = bernoulli(10^4) CPU time: 0.19 s, Wall time: 0.19 srfUE# Worksheet 'bernoulli' (2007-01-16 at 12:00) sage: len(str(a)) 27706rgUƒ# Worksheet 'bernoulli' (2007-01-16 at 12:00) sage: time v = bernoulli_mod_p(next_prime(10^4)) CPU time: 0.06 s, Wall time: 0.13 srhU…# Worksheet 'bernoulli' (2007-01-16 at 12:00) sage: time v = bernoulli_mod_p(next_prime(2*10^5)) CPU time: 1.59 s, Wall time: 1.66 sriU# Worksheet 'bernoulli' (2007-01-16 at 12:00) sage: v[:10] [1, 33334, 113335, 4762, 113335, 51516, 177072, 33335, 7444, 183767]rjU[# Worksheet 'bernoulli' (2007-01-16 at 12:01) sage: bernoulli(2) % next_prime(2*10^5) 33334rkU6# Worksheet 'talk' (2007-01-16 at 15:26) sage: 2 + 3 5rlUr# Worksheet 'features' (2007-01-16 at 15:26) sage: show(plot(sin(x),-pi,6*pi,rgbcolor=(0.5,0,0.5)),figsize=[8,3]) rmUT# Worksheet 'interface' (2007-01-16 at 15:26) sage: a = gp('x^3 - y^3'); a x^3 - y^3rnU# Worksheet 'riemann' (2007-01-16 at 15:26) sage: #auto sage: def f(x): ... return x / log(x) ... sage: def p(x): ... return prime_pi(x) roU# Worksheet 'sagex' (2007-01-16 at 15:27) sage: def is2pow(n): ... while n != 0 and n%2 == 0: ... n = n >> 1 ... return n == 1 rpT‰# Worksheet 'sieve' (2007-01-16 at 15:27) sage: %hideall sage: #auto sage: sage: def number_grid(c, box_args={}, font_args={},offset=0): ... """ ... INPUT: ... c -- list of length n^2, where n is an integer. ... The entries of c are RGB colors. ... box_args -- additional arguments passed to box. ... font_args -- all additional arguments are passed ... to the text function, e.g., fontsize. ... offset -- use to fine tune text placement in the squares ... ... OUTPUT: ... Graphics -- a plot of a grid that illustrates ... those n^2 numbers colored according to c. ... """ ... try: ... n = square_root(ZZ(len(c))) ... except ArithmeticError: ... raise ArithmeticError, "c must have square length" ... G = Graphics() ... k = 0 ... for j in reversed(range(n)): ... for i in range(n): ... col = c[int(k)] ... R = line([(i,j),(i+1,j),(i+1,j+1),(i,j+1),(i,j)], thickness=.2, **box_args) ... P = polygon([(i,j),(i+1,j),(i+1,j+1),(i,j+1),(i,j)], rgbcolor=col, **box_args) ... G += P + R ... if col != (1,1,1): ... G += text(str(k+1), (i+.5+offset,j+.5), **font_args) ... k += 1 ... G.axes(False) ... G.range(0,n,0,n) ... return G CPU time: 0.00 s, Wall time: 0.00 srqT€# Worksheet 'sieve' (2007-01-16 at 15:27) sage: %hideall sage: #auto sage: sage: def sieve_step(p, n, gone=(1,1,1), prime=(1,0,0), \ ... multiple=(.6,.6,.6), remaining=(.9,.9,.9), ... fontsize=11,offset=0): ... """ ... Return graphics that illustrates sieving out ... multiples of p. Numbers that are a nontrivial ... multiple of primes < p are shown in the gone color. ... Numbers that are a multiple of p are shown in ... a different color. The number p is shown in ... yet a third color. ... ... INPUT: ... p -- a prime (or 1, in which case all points are colored "remaining") ... n -- a SQUARE integer ... gone -- rgb color for integers that have been sieved out ... prime -- rgb color for p ... multiple -- rgb color for multiples of p ... remaining -- rgb color for integers that have not been sieved out yet ... and are not multiples of p. ... """ ... if p == 1: ... c = [remaining]*n ... else: ... exclude = prod(prime_range(p)) # exclude multiples of primes < p ... c = [] ... for k in range(1,n+1): ... if k <= p and is_prime(k): ... c.append(prime) ... elif k == 1 or (gcd(k,exclude) != 1 and not is_prime(k)): ... c.append(gone) ... elif k%p == 0: ... c.append(multiple) ... else: ... c.append(remaining) ... # end for ... # end if ... return number_grid(c,{},{'fontsize':fontsize},offset=offset) CPU time: 0.00 s, Wall time: 0.00 srrUH# Worksheet 'sieve' (2007-01-16 at 15:27) sage: show(sieve_step(1,100)) rsU6# Worksheet 'talk' (2007-01-16 at 15:27) sage: 2 + 3 5rtU[# Worksheet 'talk' (2007-01-16 at 15:27) sage: prime_range(20) [2, 3, 5, 7, 11, 13, 17, 19]ruU6# Worksheet 'talk' (2007-01-16 at 15:45) sage: 2 + 3 5rvU<# Worksheet 'talk' (2007-01-16 at 15:46) sage: 2 + 3939 3941rwUa# Worksheet 'talk' (2007-01-16 at 15:46) sage: 122 + 3939^10 899207267153464902431575960617242723rxT¦# Worksheet 'talk' (2007-01-16 at 15:46) sage: 122 + 3939^100 345619375323669975367860807124255037076153194188885931584246755620617168037547598722532539129249459087563155698072067777782990662920343395325000445341613889629673957434153020923227407467827494067876849537309882945198261109518655022861231720822535378266349041507766551309275258117413203506790245934260655896057550180870173070732758020223605128239775421756626123ryUb# Worksheet 'talk' (2007-01-16 at 15:47) sage: v = prime_range(20); v [2, 3, 5, 7, 11, 13, 17, 19]rzUn# Worksheet 'talk' (2007-01-16 at 15:47) sage: time v = prime_range(1000) CPU time: 0.00 s, Wall time: 0.00 sr{Uo# Worksheet 'talk' (2007-01-16 at 15:47) sage: time v = prime_range(10000) CPU time: 0.00 s, Wall time: 0.01 sr|UD# Worksheet 'talk' (2007-01-16 at 15:49) sage: %python sage: 2 ^ 3 1r}UE# Worksheet 'talk' (2007-01-16 at 15:49) sage: %python sage: 2 ** 3 8r~U6# Worksheet 'talk' (2007-01-16 at 15:49) sage: 2 ^ 3 8rU[# Worksheet 'talk' (2007-01-16 at 15:49) sage: preparse('2 ^ 3') 'Integer(2) ** Integer(3)'r€UP# Worksheet 'talk' (2007-01-16 at 15:49) sage: factorial(20) 2432902008176640000rUj# Worksheet 'talk' (2007-01-16 at 15:50) sage: time n=factorial(10^6) CPU time: 2.20 s, Wall time: 2.24 sr‚Uõ# Worksheet 'talk' (2007-01-16 at 15:50) sage: show(factor( factorial(40) ))

2^{38} \cdot 3^{18} \cdot 5^{9} \cdot 7^{5} \cdot 11^{3} \cdot 13^{3} \cdot 17^{2} \cdot 19^{2} \cdot 23 \cdot 29 \cdot 31 \cdot 37

rƒUö# Worksheet 'talk' (2007-01-16 at 15:51) sage: m = matrix(5,5,[randint(0,1) for _ in range(25)]); show(m)

\left(\begin{array}{rrrrr} 1&1&1&1&0\\ 0&1&0&0&1\\ 0&0&0&1&0\\ 1&0&0&1&1\\ 1&0&0&1&1 \end{array}\right)

r„U•# Worksheet 'talk' (2007-01-16 at 15:51) sage: f = m.charpoly('T'); show(f)

T^{5} - 4T^{4} + 4T^{3} - 3T^{2} + T

r…UŽ# Worksheet 'talk' (2007-01-16 at 15:51) sage: show(factor(f))

T \cdot (T^{4} - 4T^{3} + 4T^{2} - 3T + 1)

r†UT# Worksheet 'interface' (2007-01-16 at 16:08) sage: a = gp('x^3 - y^3'); a x^3 - y^3r‡U?# Worksheet 'interface' (2007-01-16 at 16:08) sage: a x^3 - y^3rˆT_# Worksheet 'interface' (2007-01-16 at 16:09) sage: a.newtonpoly() Traceback (most recent call last): ... TypeError: Error executing code in GP/PARI: CODE: sage[3]=newtonpoly(sage[2]); GP/PARI ERROR: *** expected character: ',' instead of: ...age[3]=newtonpoly(sage[2]); ^--r‰UF# Worksheet 'interface' (2007-01-16 at 16:10) sage: gp.eval('2^3') '8'rŠUJ# Worksheet 'interface' (2007-01-16 at 16:10) sage: gp.eval('b = 2^3') '8'r‹UG# Worksheet 'interface' (2007-01-16 at 16:10) sage: gp.eval('b^2') '64'rŒT# Worksheet 'interface' (2007-01-16 at 16:11) sage: f = mathematica('Sin[x^2]') Timeout exceeded in read_nonblocking(). version: 2.0 ($Revision: 1.151 $) command: /Users/was/bin/math args: ['/Users/was/bin/math'] patterns: In[[0-9]+]:= buffer (last 100 chars): before (last 100 chars): to http://register.wolfram.com or consult your Getting Started documentation. Enter your name: after: match: None match_index: None exitstatus: None flag_eof: 0 pid: 12758 child_fd: ...rU# Worksheet 'interface' (2007-01-16 at 16:11) sage: f Traceback (most recent call last): ... NameError: name 'f' is not definedrŽUJ# Worksheet 'interface' (2007-01-16 at 16:12) sage: f = maple('sin(x^2)') rU># Worksheet 'interface' (2007-01-16 at 16:12) sage: f sin(x^2)rUv# Worksheet 'interface' (2007-01-16 at 16:12) sage: f.integrate('x') 1/2*2^(1/2)*Pi^(1/2)*FresnelS(2^(1/2)/Pi^(1/2)*x)r‘UN# Worksheet 'interface' (2007-01-16 at 16:12) sage: s = str(f.integrate('x')) r’Ui# Worksheet 'interface' (2007-01-16 at 16:13) sage: s '1/2*2^(1/2)*Pi^(1/2)*FresnelS(2^(1/2)/Pi^(1/2)*x)'r“Uq# Worksheet 'bernoulli' (2007-01-16 at 16:15) sage: time a = bernoulli(10^4) CPU time: 0.34 s, Wall time: 0.36 sr”UE# Worksheet 'bernoulli' (2007-01-16 at 16:15) sage: len(str(a)) 27706r•Uƒ# Worksheet 'bernoulli' (2007-01-16 at 16:16) sage: time v = bernoulli_mod_p(next_prime(10^4)) CPU time: 0.06 s, Wall time: 0.11 sr–U…# Worksheet 'bernoulli' (2007-01-16 at 16:16) sage: time v = bernoulli_mod_p(next_prime(2*10^5)) CPU time: 1.57 s, Wall time: 1.58 sr—Uƒ# Worksheet 'bernoulli' (2007-01-16 at 16:17) sage: time v = bernoulli_mod_p(next_prime(10^4)) CPU time: 0.06 s, Wall time: 0.06 sr˜Ut# Worksheet 'bernoulli' (2007-01-16 at 16:17) sage: v[:10] [1, 1668, 7672, 4527, 7672, 8036, 3603, 1669, 5899, 6162]r™U# Worksheet 'sagex' (2007-01-16 at 16:18) sage: def is2pow(n): ... while n != 0 and n%2 == 0: ... n = n >> 1 ... return n == 1 ršUÛ# Worksheet 'sagex' (2007-01-16 at 16:18) sage: time v = [n for n in range(10^5) if is2pow(n)]; v [1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536] CPU time: 3.29 s, Wall time: 3.31 sr›U°# Worksheet 'sagex' (2007-01-16 at 16:19) sage: %sagex sage: def is2pow_compiled(unsigned int n): ... while n != 0 and n%2 == 0: ... n = n >> 1 ... return n == 1 rœUä# Worksheet 'sagex' (2007-01-16 at 16:19) sage: time v = [n for n in range(10^5) if is2pow_compiled(n)]; v [1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536] CPU time: 0.04 s, Wall time: 0.04 srUü# Worksheet 'sagex' (2007-01-16 at 16:19) sage: time v = [n for n in range(10^6) if is2pow_compiled(n)]; v [1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288] CPU time: 0.40 s, Wall time: 0.40 sržUÈ# Worksheet 'features' (2007-01-16 at 16:40) sage: def foo(): ... print """ ... sage:

sage: 2 + 2 sage:

sage:

sage: sage: """ rŸU†# Worksheet 'features' (2007-01-16 at 16:40) sage: foor() Traceback (most recent call last): ... NameError: name 'foor' is not definedr U…# Worksheet 'features' (2007-01-16 at 16:40) sage: foo()

2 + 2

r¡U¨# Worksheet 'features' (2007-01-16 at 16:41) sage: a = (x-10)^5; show(a)

x^{5} - 50x^{4} + 1000x^{3} - 10000x^{2} + 50000x - 100000

r¢Uv# Worksheet 'features' (2007-01-16 at 16:41) sage: latex(a) x^{5} - 50x^{4} + 1000x^{3} - 10000x^{2} + 50000x - 100000r£U´# Worksheet 'features' (2007-01-16 at 16:44) sage: class Tensor: ... def __repr__(self): ... return "I'm a tensor" ... def __mul__(self, right): ... return "abc" r¤U@# Worksheet 'features' (2007-01-16 at 16:44) sage: t = Tensor() r¥eU_Notebook__defaultsr¦}r§(Ucell_output_colorr¨U#0000EEr©Umax_history_lengthrªMôUcell_input_colorr«U#0000000r¬Uword_wrap_colsrKPuU_Notebook__worksheet_dirr®Usage_notebook/worksheetsr¯U_Notebook__history_countr°KU_Notebook__log_serverr±‰U_Notebook__filenamer²Usage_notebook/nb.sobjr³U_Notebook__default_worksheetr´j-U_Notebook__server_logrµ]r¶U_Notebook__next_worksheet_idr·KU_Notebook__kill_idler¸KU_Notebook__systemr¹NU_Notebook__show_debugrº‰U_Notebook__dirr»U sage_notebookr¼U_Notebook__authr½U:U_Notebook__colorr¾NU_Notebook__object_dirr¿Usage_notebook/objectsrÀU_default_filenamerÁU:/Users/was/talks/2007-01-15-sage-cse/sage_notebook/nb.sobjrÂU_Notebook__splashpagerÃ‰ub.