qMh6‰h7Kh!h#h8h%h&‰h9‰ub(h oqN}qO(h*UE.name()qPhU!qQh-hh.‰h‰h/U h0Mh‰hUaos.chdir("/Volumes/HOME/talks/2006-12-waterloo/sage/sage_notebook/worksheets/_/cells/5")
E.name()qRh2U"sage_notebook/worksheets/_/cells/5qSh4U
'_sage_[7]'
qTh6‰h7Kh!h#h8h%h&‰h9‰ub(h oqU}qV(h*UE.Rank()qWhU!qXh-hh.‰h‰h/U h0Mh‰hUaos.chdir("/Volumes/HOME/talks/2006-12-waterloo/sage/sage_notebook/worksheets/_/cells/6")
E.Rank()qYh2U"sage_notebook/worksheets/_/cells/6qZh4U
1
q[h6‰h7Kh!h#h8h%h&‰h9‰ub(h oq\}q](h*Ua = maple('2007')q^hU!q_h-hh.‰h‰h/U h0Mh‰hUjos.chdir("/Volumes/HOME/talks/2006-12-waterloo/sage/sage_notebook/worksheets/_/cells/7")
a = maple('2007')q`h2U"sage_notebook/worksheets/_/cells/7qah4U
qbh6‰h7Kh!h#h8h%h&‰h9‰ub(h oqc}qd(h*UahU!qeh-hh.‰h‰h/U h0Mh‰hUZos.chdir("/Volumes/HOME/talks/2006-12-waterloo/sage/sage_notebook/worksheets/_/cells/8")
aqfh2U"sage_notebook/worksheets/_/cells/8qgh4U
2007
qhh6‰h7Kh!h#h8h%h&‰h9‰ub(h oqi}qj(h*U a.ifactor()qkhU!qlh-hh.‰h‰h/U h0M h‰hUdos.chdir("/Volumes/HOME/talks/2006-12-waterloo/sage/sage_notebook/worksheets/_/cells/9")
a.ifactor()qmh2U"sage_notebook/worksheets/_/cells/9qnh4U
``(3)^2*``(223)
qoh6‰h7Kh!h#h8h%h&‰h9‰ub(h oqp}qq(h*Ua.name()qrhU!qsh-hh.‰h‰h/U h4U
'sage1'
qth0M
h‰hUbos.chdir("/Volumes/HOME/talks/2006-12-waterloo/sage/sage_notebook/worksheets/_/cells/10")
a.name()quh2U#sage_notebook/worksheets/_/cells/10qvU_word_being_completedqwUa.ifacqxh6‰h7Kh!h#h8h%h&‰h9‰ub(h oqy}qz(h*U for i in range(5):
print iq{hU!q|h-hh.‰h‰h/U h0M h‰hUxos.chdir("/Volumes/HOME/talks/2006-12-waterloo/sage/sage_notebook/worksheets/_/cells/11")
for i in range(5):
print iq}h2U#sage_notebook/worksheets/_/cells/11q~h4U
0
1
2
3
4
qh6‰h7Kh!h#h8h%h&‰h9‰ub(h oq€}q(h*U h-hh.‰h/U h0M h2U#sage_notebook/worksheets/_/cells/12q‚h4U h6‰h7K h8h%h9‰ubeU_Worksheet__synchroqƒKU_Worksheet__comp_is_runningq„‰U_Worksheet__dirq…Usage_notebook/worksheets/_q†U_Worksheet__attachedq‡}qˆU_Worksheet__queueq‰]qŠU_Worksheet__next_idq‹M
U_Worksheet__passcryptqŒˆU_Worksheet__nameqU U_Worksheet__saltqŽU1165187731.576083qU_Worksheet__notebookqhU_Worksheet__idq‘KU_Worksheet__next_block_idq’KU_Worksheet__systemq“NU_Worksheet__passcodeq”U
11G3BJNEUV/Pkq•ubUgraphq–(hoq—}q˜(h Ugraphq™h
]qš((h oq›}qœ(U _Cell__inqU*time b = bernoulli(10000, algorithm='gap')qžU_Cell__introspect_htmlqŸU!q U_Cell__worksheetq¡h—U_Cell__completionsq¢‰U_Cell__introspectq£‰U_Cell__out_htmlq¤U U _Cell__idq¥M0U_Cell__is_htmlq¦‰U_before_preparseq§U®os.chdir("/Volumes/HOME/talks/2006-12-waterloo/sage/sage_notebook/worksheets/graph/cells/14")
__SAGE_t__=cputime()
__SAGE_w__=walltime()
b = bernoulli(10000, algorithm='gap')q¨U_word_being_completedq©UbernqªU
_Cell__dirq«U'sage_notebook/worksheets/graph/cells/14q¬U
_Cell__outqU Uhas_new_outputq®‰U _Cell__sageq¯NU_Cell__versionq°KU _Cell__typeq±Uwrapq²U _Cell__timeq³ˆU_Cell__interruptedq´ˆub(h oqµ}q¶(hU+time b = bernoulli(10000, algorithm='pari')q·hŸU!q¸h¡h—h¢‰h£‰h¤U h¥M0h¦‰h§U¯os.chdir("/Volumes/HOME/talks/2006-12-waterloo/sage/sage_notebook/worksheets/graph/cells/15")
__SAGE_t__=cputime()
__SAGE_w__=walltime()
b = bernoulli(10000, algorithm='pari')q¹h«U'sage_notebook/worksheets/graph/cells/15qºhU'
CPU time: 0.24 s, Wall time: 0.26 s
q»h®‰h¯h")Rq¼h°Kh±h²h³ˆh´‰ub(h oq½}q¾(hUE = EllipticCurve('389a')q¿hŸU!qÀh¡h—h¢‰h£‰h¤U h¥M
0h¦‰h§Uwos.chdir("/Volumes/HOME/talks/2006-12-waterloo/sage/sage_notebook/worksheets/graph/cells/10")
E = EllipticCurve('389a')qÁh©UEllqÂh«U'sage_notebook/worksheets/graph/cells/10qÃhU
qÄh®‰h¯h¼h°Kh±h²h³‰h´‰ub(h oqÅ}qÆ(hUL = E.Lseries_zeros(5)qÇhŸU!qÈh¡h—h¢‰h£‰h¤U h¥M 0h¦‰h§Utos.chdir("/Volumes/HOME/talks/2006-12-waterloo/sage/sage_notebook/worksheets/graph/cells/11")
L = E.Lseries_zeros(5)qÉh©U
E.Lseries_zerqÊh«U'sage_notebook/worksheets/graph/cells/11qËhU
qÌh®‰h¯h¼h°Kh±h²h³‰h´‰ub(h oqÍ}qÎ(hULhŸU!qÏh¡h—h¢‰h£‰h¤U h¥M 0h¦‰h§U_os.chdir("/Volumes/HOME/talks/2006-12-waterloo/sage/sage_notebook/worksheets/graph/cells/12")
LqÐh«U'sage_notebook/worksheets/graph/cells/12qÑhUA
[0.000000000, 0.000000000, 2.87609907, 4.41689608, 5.79340263]
qÒh®‰h¯h¼h°Kh±h²h³‰h´‰ub(h oqÓ}qÔ(hUg = graphs.PetersenGraph()qÕU_Cell__introspect_htmlqÖU!q×hh—h‰U_Cell__introspectq؉hU U_Cell__is_htmlqÙ‰hM 0U_word_being_completedqÚU
graphs.PetqÛU_before_preparseqÜUwos.chdir("/Volumes/HOME/talks/2006-12-waterloo/sage/sage_notebook/worksheets/graph/cells/0")
g = graphs.PetersenGraph()qÝhU&sage_notebook/worksheets/graph/cells/0qÞh U
qßh ‰h KU _Cell__sageqàh#h$UwrapqáU _Cell__timeqâ‰h'‰ub(h oqã}qä(hUshow(g)qåhÖU!qæhh—h‰h؉hU>
qçhM0hÙ‰hÜUdos.chdir("/Volumes/HOME/talks/2006-12-waterloo/sage/sage_notebook/worksheets/graph/cells/1")
show(g)qèhU&sage_notebook/worksheets/graph/cells/1qéh U
qêh ‰h Khàh#h$háhâ‰h'‰ub(h oqë}qì(hU(g = graphs.CompleteBipartiteGraph(5,100)qíhU!qîhh—h‰h‰hU h U
qïhM0h‰hU…os.chdir("/Volumes/HOME/talks/2006-12-waterloo/sage/sage_notebook/worksheets/graph/cells/2")
g = graphs.CompleteBipartiteGraph(5,100)qðhU&sage_notebook/worksheets/graph/cells/2qñhwUgraphs.CompleteBiqòh ‰h Kh!h¼h$h²h&‰h'‰ub(h oqó}qô(h*U graphs.CompleteBipartiteGraphqõhT? File: /Volumes/HOME/s/local/lib/python2.5/site-packages/sage/graphs/graph_database.py
Source Code (starting at line 566):
def CompleteBipartiteGraph(self, n1, n2):
"""
Returns a Complete Bipartite Graph sized n1+n2, with each of the
nodes [0,(n1-1)] connected to each of the nodes [n1,(n2-1)] and
vice versa.
A Complete Bipartite Graph is a graph with its vertices partitioned
into two groups, V1 and V2. Each v in V1 is connected to every v
in V2, and vice versa.
PLOTTING:
Upon construction, the position dictionary is filled to override
the spring-layout algorithm. By convention, each complete bipartite
graph will be displayed with the first n1 nodes on the top row (at
y=1) from left to right. The remaining n2 nodes appear at y=0,
also from left to right. The shorter row (partition with fewer
nodes) is stretched to the same length as the longer row, unless
the shorter row has 1 node; in which case it is centered. The x
values in the plot are in domain [0,max{n1,n2}].
In the Complete Bipartite graph, there is a visual difference in
using the spring-layout algorithm vs. the position dictionary used
in this constructor. The position dictionary flattens the graph
and separates the partitioned nodes, making it clear which nodes
an edge is connected to. The Complete Bipartite graph plotted with
the spring-layout algorithm tends to center the nodes in n1 (see
spring_med in examples below), thus overlapping its nodes and edges,
making it typically hard to decipher.
Filling the position dictionary in advance adds O(n) to the
constructor. Feel free to race the constructors below in the
examples section. The much larger difference is the time added by
the spring-layout algorithm when plotting. (Also shown in the
example below). The spring model is typically described as O(n^3),
as appears to be the case in the NetworkX source code.
EXAMPLES:
# The following examples require NetworkX (to use default)
sage: import networkx as NX
# Compare the constructors (results will vary)
sage.: time n = NX.complete_bipartite_graph(389,157); spring_big = Graph(n)
# CPU time: 9.28 s, Wall time: 11.02 s
sage.: time posdict_big = graphs.CompleteBipartiteGraph(389,157)
# CPU time: 10.72 s, Wall time: 13.84 s
# Compare the plotting speeds (results will vary)
sage: n = NX.complete_bipartite_graph(11,17)
sage: spring_med = Graph(n)
sage: posdict_med = graphs.CompleteBipartiteGraph(11,17)
# Notice here how the spring-layout tends to center the nodes of n1
sage.: time spring_med.show()
# CPU time: 3.84 s, Wall time: 4.49 s
sage.: time posdict_med.show()
# CPU time: 0.96 s, Wall time: 1.14 s
# View many Complete Bipartite graphs with a SAGE Graphics Array
# With this constructor (i.e., the position dictionary filled)
sage: g = []
sage: j = []
sage: for i in range(9):
... k = graphs.CompleteBipartiteGraph(i+1,4)
... g.append(k)
...
sage: for i in range(3):
... n = []
... for m in range(3):
... n.append(g[3*i + m].plot(node_size=50, with_labels=False))
... j.append(n)
...
sage: G = sage.plot.plot.GraphicsArray(j)
sage.: G.show()
# Compared to plotting with the spring-layout algorithm
sage: g = []
sage: j = []
sage: for i in range(9):
... spr = NX.complete_bipartite_graph(i+1,4)
... k = Graph(spr)
... g.append(k)
...
sage: for i in range(3):
... n = []
... for m in range(3):
... n.append(g[3*i + m].plot(node_size=50, with_labels=False))
... j.append(n)
...
sage: G = sage.plot.plot.GraphicsArray(j)
sage.: G.show()
"""
pos_dict = {}
c1 = 1 # scaling factor for top row
c2 = 1 # scaling factor for bottom row
c3 = 0 # pad to center if top row has 1 node
c4 = 0 # pad to center if bottom row has 1 node
if n1 > n2:
if n2 == 1:
c4 = (n1-1)/2
else:
c2 = ((n1-1)/(n2-1))
elif n2 > n1:
if n1 == 1:
c3 = (n2-1)/2
else:
c1 = ((n2-1)/(n1-1))
for i in range(n1):
x = c1*i + c3
y = 1
pos_dict[i] = [x,y]
for i in range(n1+n2)[n1:]:
x = c2*(i-n1) + c4
y = 0
pos_dict[i] = [x,y]
G = NX.complete_bipartite_graph(n1,n2)
return graph.Graph(G, pos=pos_dict, name="Complete bipartite graph on %d vertices"%(n1+n2))qöh-h—h.‰h]q÷(U graphs.CompleteBipartiteGraph??qøU eh/U>
qùh4U
qúh0M0h‰hU|os.chdir("/Volumes/HOME/talks/2006-12-waterloo/sage/sage_notebook/worksheets/graph/cells/3")
graphs.CompleteBipartiteGraph??qûh2U&sage_notebook/worksheets/graph/cells/3qühwUgraphs.CompleteBiqýh6‰h7Kh!h#h8h%h&‰h9‰ub(h oqþ}qÿ(hUghŸU!r h¡h—h¢‰h£‰h¤U h¥M0h¦‰h§U^os.chdir("/Volumes/HOME/talks/2006-12-waterloo/sage/sage_notebook/worksheets/graph/cells/5")
gr h«U&sage_notebook/worksheets/graph/cells/5r hUK
Complete bipartite graph on 105 vertices: a simple graph on 105 vertices
r h®‰h¯h¼h°Kh±h²h³‰h´‰ub(h or }r (h*Ug.save('mygraph')r hŸU!r h-h—h.‰h£‰h/U h0M0h¦‰h§Unos.chdir("/Volumes/HOME/talks/2006-12-waterloo/sage/sage_notebook/worksheets/graph/cells/4")
g.save('mygraph')r h2U&sage_notebook/worksheets/graph/cells/4r h4U
r
h6‰h¯h¼h7Kh8h²h³‰h9‰ub(h or }r (hU5P = plot(sin, 0, 10, rgbcolor=(0,0.5,0), thickness=5)r
hŸU!r h¡h—h¢‰h£‰h¤U h¥M0h¦‰h§U’os.chdir("/Volumes/HOME/talks/2006-12-waterloo/sage/sage_notebook/worksheets/graph/cells/6")
P = plot(sin, 0, 10, rgbcolor=(0,0.5,0), thickness=5)r h«U&sage_notebook/worksheets/graph/cells/6r hU
r h®‰h¯h¼h°Kh±h²h³‰h´‰ub(h or }r (hUPhŸU!r h¡h—h¢‰h£‰h¤U h¥M0h¦‰h§U^os.chdir("/Volumes/HOME/talks/2006-12-waterloo/sage/sage_notebook/worksheets/graph/cells/7")
Pr h«U&sage_notebook/worksheets/graph/cells/7r hU5
Graphics object consisting of 1 graphics primitive
r h®‰h¯h¼h°Kh±h²h³‰h´‰ub(h or }r (hUP.show()r hŸU!r h¡h—h¢‰h£‰h¤U>
r h¥M0h¦‰h§Ueos.chdir("/Volumes/HOME/talks/2006-12-waterloo/sage/sage_notebook/worksheets/graph/cells/8")
P.show()r h«U&sage_notebook/worksheets/graph/cells/8r hU
r h®‰h¯h¼h°Kh±h²h³‰h´‰ub(h or }r! (hU h¡h—h¢‰h¤U h¥M 0h«U&sage_notebook/worksheets/graph/cells/9r" hU h®‰h°K h±h²h´‰ubeU_Worksheet__synchror# Kr; hj4 h‰h؉hU hÙ‰hM hÚU bernoulli_mor< hÜUwos.chdir("/Volumes/HOME/talks/2006-12-waterloo/sage/sage_notebook/worksheets/bernoull_mod/cells/0")
bernoulli_mod_p(37)r= hU-sage_notebook/worksheets/bernoull_mod/cells/0r> h UF
[1, 31, 16, 15, 16, 4, 17, 32, 22, 31, 15, 15, 17, 12, 29, 2, 0, 2]
r? h ‰h KU _Cell__sager@ h#h$háhâ‰h'‰ub(h orA }rB (U _Cell__inrC U)time v=bernoulli_mod_p(next_prime(50000))rD U_Cell__introspect_htmlrE U!rF U_Cell__worksheetrG j4 U_Cell__completionsrH ‰U_Cell__introspectrI ‰U_Cell__out_htmlrJ U U_Cell__is_htmlrK ‰U _Cell__idrL M U_word_being_completedrM U bernoulli_morN U_before_preparserO U³os.chdir("/Volumes/HOME/talks/2006-12-waterloo/sage/sage_notebook/worksheets/bernoull_mod/cells/1")
__SAGE_t__=cputime()
__SAGE_w__=walltime()
v=bernoulli_mod_p(next_prime(50000))rP U
_Cell__dirrQ U-sage_notebook/worksheets/bernoull_mod/cells/1rR U
_Cell__outrS U'
CPU time: 0.44 s, Wall time: 0.47 s
rT Uhas_new_outputrU ‰U_Cell__versionrV Kj@ h#U _Cell__typerW háU _Cell__timerX ˆU_Cell__interruptedrY ‰ub(h orZ }r[ (jC Ulen(v)r\ jE U!r] jG j4 jH ‰jI ‰jJ U jL M jK ‰jO Ujos.chdir("/Volumes/HOME/talks/2006-12-waterloo/sage/sage_notebook/worksheets/bernoull_mod/cells/2")
len(v)r^ jQ U-sage_notebook/worksheets/bernoull_mod/cells/2r_ jS U
25010
r` jU ‰jV Kj@ h#jW hájX ‰jY ‰ub(h ora }rb (jC U jG j4 jH ‰jJ U jL M jQ U-sage_notebook/worksheets/bernoull_mod/cells/4rc jS U jU ‰jV K jW hájY ‰ubej# K
h„‰h‡}rd h…U%sage_notebook/worksheets/bernoull_modre h‰]rf h‹M hŒˆhU bernoull_modrg hŽU1165187752.734722rh hhh‘Kj1 K h“Nh”U
11G3BJNEUV/Pkri ubUfactorrj (hork }rl (U_Worksheet__filenamerm Ufactorrn U_Worksheet__cellsro ]rp ((h orq }rr (U _Cell__inrs U U_Cell__worksheetrt jk U_Cell__completionsru ‰U_Cell__out_htmlrv U U _Cell__idrw M@U
_Cell__dirrx U'sage_notebook/worksheets/factor/cells/8ry U
_Cell__outrz U Uhas_new_outputr{ ‰U_Cell__versionr| K U _Cell__typer} Uwrapr~ U_Cell__interruptedr ‰ub(h or€ }r (js T %hide%html
Factor Tree Worksheet
This worksheet uses the SAGE Notebook to illustrate integer factorization as a
product of primes using the ``factor trees'' that students learn about in school.r‚ jt jk ju ‰jv U jw M@jx U'sage_notebook/worksheets/factor/cells/1rƒ jz T
Factor Tree Worksheet
This worksheet uses the SAGE Notebook to illustrate integer factorization as a
product of primes using the ``factor trees'' that students learn about in school.r„ j{ ‰j| KU _Cell__sager… Nj} j~ j ‰ub(h or† }r‡ (js TÄ %hide
print "CLICK ABOVE FOR THE SOURCE CODE"
# The source code to draw factorization trees...
import random
def factor_tree(n, font=10):
rows = []
v = [(n,None,0)]
ftree(rows, v, 0, factor(n))
return draw_ftree(rows, font)
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)
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
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)])rˆ U_Cell__introspect_htmlr‰ U!rŠ jt jk ju ‰U_Cell__introspectr‹ ‰jv U jw M@U _Cell__timerŒ ‰U_Cell__is_htmlr ‰U_before_preparserŽ T" os.chdir("/Volumes/HOME/talks/2006-12-waterloo/sage/sage_notebook/worksheets/factor/cells/2")
%hide
print "CLICK ABOVE FOR THE SOURCE CODE"
# The source code to draw factorization trees...
import random
def factor_tree(n, font=10):
rows = []
v = [(n,None,0)]
ftree(rows, v, 0, factor(n))
return draw_ftree(rows, font)
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)
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
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)])r jx U'sage_notebook/worksheets/factor/cells/2r jz U"
CLICK ABOVE FOR THE SOURCE CODE
r‘ j{ ‰j| Kj… h#j} háj ‰ub(h or’ }r“ (js UVF = factor_trees(300000000, rows=1, cols=1, font=8)
F.show(axes=False, figsize=[10,6])r” j‰ U!r• jt jk ju ‰j‹ ‰jv U?
r– jw M@jŒ ‰j ‰jŽ U´os.chdir("/Volumes/HOME/talks/2006-12-waterloo/sage/sage_notebook/worksheets/factor/cells/3")
F = factor_trees(300000000, rows=1, cols=1, font=8)
F.show(axes=False, figsize=[10,6])r— jx U'sage_notebook/worksheets/factor/cells/3r˜ jz U
r™ j{ ‰j| Kj… h#j} háj ‰ub(h orš }r› (js UbF = factor_tree(prod(primes(40)))
F.show(xmin=-3,xmax=7,ymin=-5,ymax=0.5,figsize=[8,2],axes=False)rœ j‰ U!r jt jk ju ‰j‹ ‰jv U?
rž jw M@jŒ ‰j ‰jŽ UÀos.chdir("/Volumes/HOME/talks/2006-12-waterloo/sage/sage_notebook/worksheets/factor/cells/4")
F = factor_tree(prod(primes(40)))
F.show(xmin=-3,xmax=7,ymin=-5,ymax=0.5,figsize=[8,2],axes=False)rŸ jx U'sage_notebook/worksheets/factor/cells/4r jz U
r¡ j{ ‰j| Kj… h#j} háj ‰ub(h or¢ }r£ (js U9show(factor_trees(1200,cols=3,rows=3),axes=False,dpi=100)r¤ j‰ U!r¥ jt jk ju ‰j‹ ‰jv U?
r¦ jw M@jŒ ‰j ‰jŽ U—os.chdir("/Volumes/HOME/talks/2006-12-waterloo/sage/sage_notebook/worksheets/factor/cells/5")
show(factor_trees(1200,cols=3,rows=3),axes=False,dpi=100)r§ jx U'sage_notebook/worksheets/factor/cells/5r¨ jz U
r© j{ ‰j| Kj… h#j} j~ j ‰ubeU_Worksheet__synchrorª K
U_Worksheet__comp_is_runningr« ‰U_Worksheet__attachedr¬ }r U_Worksheet__dirr® U sage_notebook/worksheets/factorr¯ U_Worksheet__queuer° ]r± U_Worksheet__next_idr² M
@U_Worksheet__passcryptr³ ˆU_Worksheet__namer´ Ufactorrµ U_Worksheet__saltr¶ U1165225547.833440r· U_Worksheet__notebookr¸ hU_Worksheet__idr¹ KU_Worksheet__next_block_idrº KU_Worksheet__systemr» NU_Worksheet__passcoder¼ U
11G3BJNEUV/Pkr½ ubUtestr¾ (hor¿ }rÀ (U_Worksheet__filenamerÁ Utestr U_Worksheet__cellsrà ]rÄ (h orÅ }rÆ (jC U jG j¿ jH ‰jJ U jL M PjQ U%sage_notebook/worksheets/test/cells/0rÇ jS U jU ‰jV K jW hájY ‰ubaU_Worksheet__namerÈ UtestrÉ U_Worksheet__attachedrÊ }rË U_Worksheet__dirrÌ U sage_notebook/worksheets/testrÍ U_Worksheet__queuerÎ ]rÏ U_Worksheet__next_idrÐ MPU_Worksheet__passcryptrÑ ˆU_Worksheet__comp_is_runningrÒ ‰U_Worksheet__saltrÓ U1165250401.859302rÔ U_Worksheet__notebookrÕ hU_Worksheet__idrÖ KU_Worksheet__systemr× NU_Worksheet__passcoderØ U
11G3BJNEUV/PkrÙ ubU _scratch_rÚ (horÛ }rÜ (h U _scratch_rÝ h
]rÞ (h orß }rà (hU hjÛ h‰hK hU*sage_notebook/worksheets/_scratch_/cells/0rá h U h ‰h K h'‰ubah„‰h…U"sage_notebook/worksheets/_scratch_râ h”U
11G3BJNEUV/Pkrã h‰]rä h‹KhŒˆhU _scratch_rå hŽU1165187727.127047ræ hhh‘K h“NubuU_Notebook__historyrç ]rè (U¨# Worksheet 'graph' (2006-12-03 at 15:35)
sage: g = graphs.StarGraph()
Traceback (most recent call last):
...
TypeError: StarGraph() takes exactly 2 arguments (1 given)ré UK# Worksheet 'graph' (2006-12-03 at 15:35)
sage: g = graphs.PetersenGraph()
rê U8# Worksheet 'graph' (2006-12-03 at 15:35)
sage: show(g)
rë T¶ # Worksheet 'factor' (2006-12-04 at 01:46)
sage: %hide
sage: print "CLICK ABOVE FOR THE SOURCE CODE"
sage: # The source code to draw factorization trees...
sage: import random
sage: def factor_tree(n, font=10):
... rows = []
... v = [(n,None,0)]
... ftree(rows, v, 0, factor(n))
... return draw_ftree(rows, font)
...
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: 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=0.4)
... return g
...
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)])
CLICK ABOVE FOR THE SOURCE CODErì UŽ# Worksheet 'factor' (2006-12-04 at 01:46)
sage: F = factor_trees(300000000, rows=1, cols=1, font=8)
sage: F.show(axes=False, figsize=[10,6])
rí Uš# Worksheet 'factor' (2006-12-04 at 01:46)
sage: F = factor_tree(prod(primes(30)))
sage: F.show(xmin=-3,xmax=7,ymin=-5,ymax=0.5,figsize=[8,2],axes=False)
rî Uk# Worksheet 'factor' (2006-12-04 at 01:46)
sage: show(factor_trees(1200,cols=3,rows=3),axes=False,dpi=100)
rï U,# Worksheet 'factor' (2006-12-04 at 01:46)
rð U,# Worksheet 'factor' (2006-12-04 at 01:46)
rñ UK# Worksheet 'graph' (2006-12-04 at 08:38)
sage: g = graphs.PetersenGraph()
rò U8# Worksheet 'graph' (2006-12-04 at 08:38)
sage: show(g)
ró UK# Worksheet 'graph' (2006-12-04 at 08:38)
sage: g = graphs.PetersenGraph()
rô U8# Worksheet 'graph' (2006-12-04 at 08:38)
sage: show(g)
rõ T´ # Worksheet 'factor' (2006-12-04 at 08:38)
sage: %hide
sage: print "CLICK ABOVE FOR THE SOURCE CODE"
sage: # The source code to draw factorization trees...
sage: import random
sage: def factor_tree(n, font=10):
... rows = []
... v = [(n,None,0)]
... ftree(rows, v, 0, factor(n))
... return draw_ftree(rows, font)
...
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: 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: 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)])
CLICK ABOVE FOR THE SOURCE CODErö UŽ# Worksheet 'factor' (2006-12-04 at 08:38)
sage: F = factor_trees(300000000, rows=1, cols=1, font=8)
sage: F.show(axes=False, figsize=[10,6])
r÷ Uš# Worksheet 'factor' (2006-12-04 at 08:38)
sage: F = factor_tree(prod(primes(30)))
sage: F.show(xmin=-3,xmax=7,ymin=-5,ymax=0.5,figsize=[8,2],axes=False)
rø Uš# Worksheet 'factor' (2006-12-04 at 08:39)
sage: F = factor_tree(prod(primes(40)))
sage: F.show(xmin=-3,xmax=7,ymin=-5,ymax=0.5,figsize=[8,2],axes=False)
rù Uš# Worksheet 'factor' (2006-12-04 at 08:39)
sage: F = factor_tree(prod(primes(40)))
sage: F.show(xmin=-3,xmax=7,ymin=-5,ymax=0.5,figsize=[8,2],axes=False)
rú UŽ# Worksheet 'bernoull_mod' (2006-12-04 at 08:43)
sage: bernoulli_mod_p(37)
[1, 31, 16, 15, 16, 4, 17, 32, 22, 31, 15, 15, 17, 12, 29, 2, 0, 2]rû U„# Worksheet 'bernoull_mod' (2006-12-04 at 08:43)
sage: time v=bernoulli_mod_p(next_prime(5000))
CPU time: 0.03 s, Wall time: 0.07 srü UB# Worksheet 'bernoull_mod' (2006-12-04 at 08:44)
sage: len(v)
2501rý T # Worksheet 'bernoull_mod' (2006-12-04 at 08:44)
sage: v
[1, 834, 2835, 2978, 2835, 3108, 2285, 835, 4888, 2243, 4580, 2458, 756, 496, 1317, 2489, 3478, 4680, 3187, 2262, 2324, 4505, 4911, 2558, 4545, 905, 8, 377, 3917, 4639, 3600, 117, 2518, 229, 2505, 1860, 1165, 3868, 1535, 3272, 3541, 2424, 2600, 4612, 4701, 2935, 709, 2363, 2790, 1538, 1367, 1803, 2090, 4292, 3056, 2545, 2590, 1578, 1042, 3748, 1051, 3430, 125, 3684, 1830, 3890, 3630, 2170, 4351, 50, 2361, 3842, 1265, 2985, 3199, 11, 4662, 2269, 833, 2141, ...rþ U…# Worksheet 'bernoull_mod' (2006-12-04 at 08:44)
sage: time v=bernoulli_mod_p(next_prime(50000))
CPU time: 0.44 s, Wall time: 0.47 srÿ UC# Worksheet 'bernoull_mod' (2006-12-04 at 08:44)
sage: len(v)
25010r UŽ# Worksheet 'bernoull_mod' (2006-12-04 at 08:44)
sage: bernoulli_mod_p(37)
[1, 31, 16, 15, 16, 4, 17, 32, 22, 31, 15, 15, 17, 12, 29, 2, 0, 2]r U0# Worksheet '' (2006-12-04 at 13:44)
sage: 2+3
5r U0# Worksheet '' (2006-12-04 at 13:44)
sage: 2+5
7r UP# Worksheet '' (2006-12-04 at 13:44)
sage: E = magma.EllipticCurve([1,2,3,4,5])
r U‚# Worksheet '' (2006-12-04 at 13:44)
sage: E
Elliptic Curve defined by y^2 + x*y + 3*y = x^3 + 2*x^2 + 4*x + 5 over Rational Fieldr U_# Worksheet '' (2006-12-04 at 13:44)
sage: type(E)
r U?# Worksheet '' (2006-12-04 at 13:45)
sage: E.name()
'_sage_[7]'r U5# Worksheet '' (2006-12-04 at 13:45)
sage: E.Rank()
1r U<# Worksheet '' (2006-12-04 at 13:45)
sage: a = maple('2+3')
r U.# Worksheet '' (2006-12-04 at 13:45)
sage: a
5r
U=# Worksheet '' (2006-12-04 at 13:46)
sage: a = maple('2007')
r U1# Worksheet '' (2006-12-04 at 13:46)
sage: a
2007r UF# Worksheet '' (2006-12-04 at 13:46)
sage: a.ifactor()
``(3)^2*``(223)r
U;# Worksheet '' (2006-12-04 at 13:46)
sage: a.name()
'sage1'r UV# Worksheet '' (2006-12-04 at 13:47)
sage: for i in range(5):
... print i
0
1
2
3
4r UY# Worksheet 'graph' (2006-12-04 at 14:09)
sage: g = graphs.CompleteBipartiteGraph(5,100)
r U8# Worksheet 'graph' (2006-12-04 at 14:09)
sage: show(g)
r U{# Worksheet 'graph' (2006-12-04 at 14:16)
sage: g
Traceback (most recent call last):
...
NameError: name 'g' is not definedr UY# Worksheet 'graph' (2006-12-04 at 14:16)
sage: g = graphs.CompleteBipartiteGraph(5,100)
r Uz# Worksheet 'graph' (2006-12-04 at 14:16)
sage: g
Complete bipartite graph on 105 vertices: a simple graph on 105 verticesr UB# Worksheet 'graph' (2006-12-04 at 14:16)
sage: g.save('mygraph')
r UW# Worksheet 'graph' (2006-12-04 at 14:18)
sage: P = plot(sin, 0, 10, rgbcolor=(1,1,0))
r Ud# Worksheet 'graph' (2006-12-04 at 14:18)
sage: P
Graphics object consisting of 1 graphics primitiver U9# Worksheet 'graph' (2006-12-04 at 14:18)
sage: P.show()
r UY# Worksheet 'graph' (2006-12-04 at 14:18)
sage: P = plot(sin, 0, 10, rgbcolor=(0,0.5,0))
r Ud# Worksheet 'graph' (2006-12-04 at 14:18)
sage: P
Graphics object consisting of 1 graphics primitiver U9# Worksheet 'graph' (2006-12-04 at 14:18)
sage: P.show()
r Uf# Worksheet 'graph' (2006-12-04 at 14:18)
sage: P = plot(sin, 0, 10, rgbcolor=(0,0.5,0), thickness=5)
r Ud# Worksheet 'graph' (2006-12-04 at 14:18)
sage: P
Graphics object consisting of 1 graphics primitiver U9# Worksheet 'graph' (2006-12-04 at 14:18)
sage: P.show()
r U™# Worksheet 'graph' (2006-12-04 at 14:25)
sage: bernoulli(100)
-94598037819122125295227433069493721872702841533066936133385696204311395415197247711/33330r UJ# Worksheet 'graph' (2006-12-04 at 14:25)
sage: E = EllipticCurve('389a')
r UG# Worksheet 'graph' (2006-12-04 at 14:25)
sage: L = E.Lseries_zeros(2)
r! UL# Worksheet 'graph' (2006-12-04 at 14:25)
sage: L
[0.000000000, 0.000000000]r" UG# Worksheet 'graph' (2006-12-04 at 14:25)
sage: L = E.Lseries_zeros(5)
r# Up# Worksheet 'graph' (2006-12-04 at 14:26)
sage: L
[0.000000000, 0.000000000, 2.87609907, 4.41689608, 5.79340263]r$ U~# Worksheet 'graph' (2006-12-04 at 14:27)
sage: time b = bernoulli(1000, algorithm='gap')
CPU time: 0.06 s, Wall time: 4.59 sr% U~# Worksheet 'graph' (2006-12-04 at 14:27)
sage: time b = bernoulli(1000, algorithm='gap')
CPU time: 0.01 s, Wall time: 0.01 sr& U~# Worksheet 'graph' (2006-12-04 at 14:27)
sage: time b = bernoulli(1002, algorithm='gap')
CPU time: 0.02 s, Wall time: 0.03 sr' U€# Worksheet 'graph' (2006-12-04 at 14:28)
sage: time b = bernoulli(10000, algorithm='pari')
CPU time: 0.44 s, Wall time: 0.48 sr( U€# Worksheet 'graph' (2006-12-04 at 14:28)
sage: time b = bernoulli(10000, algorithm='pari')
CPU time: 0.24 s, Wall time: 0.27 sr) U€# Worksheet 'graph' (2006-12-04 at 14:28)
sage: time b = bernoulli(10000, algorithm='pari')
CPU time: 0.24 s, Wall time: 0.26 sr* U€# Worksheet 'graph' (2006-12-04 at 14:28)
sage: time b = bernoulli(10000, algorithm='pari')
CPU time: 0.24 s, Wall time: 0.26 sr+ eU_Notebook__defaultsr, }r- (Ucell_output_colorr. U#0000EEr/ Umax_history_lengthr0 MôUcell_input_colorr1 U#0000000r2 Uword_wrap_colsr3 KPuU_Notebook__worksheet_dirr4 Usage_notebook/worksheetsr5 U_Notebook__history_countr6 K U_Notebook__log_serverr7 ‰U_Notebook__filenamer8 Usage_notebook/nb.sobjr9 U _Notebook__default_worksheetr: jÛ U_Notebook__server_logr; ]r< U _Notebook__next_worksheet_idr= KU_Notebook__kill_idler> K U_Notebook__systemr? NU_Notebook__show_debugr@ ‰U_Notebook__dirrA U
sage_notebookrB U_Notebook__authrC U:U_Notebook__colorrD NU_Notebook__object_dirrE Usage_notebook/objectsrF U_default_filenamerG U?/Volumes/HOME/talks/2006-12-waterloo/sage/sage_notebook/nb.sobjrH U_Notebook__splashpagerI ˆub.