︠7a452932-5a47-4c23-ac1b-d6f4ca8b10ef︠
3+4
︡efb4c9a4-2ae6-4367-8734-154061209be4︡{"stdout":"7\n"}︡{"done":true}︡
︠a771a35d-118c-4e03-aa27-c1eb58224827︠
3==4
︡216f27d1-9b63-4eeb-88b6-165f9c03710a︡{"stdout":"False\n"}︡{"done":true}︡
︠0a1641da-7eef-4610-ba53-18c26a39fc29︠
3+4
︡72bfa413-b300-447f-a473-0be9741dbd27︡{"stdout":"7\n"}︡{"done":true}︡
︠64359d2e-0a0b-442e-943b-1359ef7314a1︠
is_prime(47)
︡3af461f8-497c-4e02-8b7b-9aef673ad010︡{"stdout":"True\n"}︡{"done":true}︡
︠856ffba5-67a8-48ba-a618-fa878781deb3︠
is_prime(1111111111111111111)
︡ec948707-bc12-42e0-8f2f-9a378518ab38︡{"stdout":"True\n"}︡{"done":true}︡
︠a1c739da-b465-4292-8642-78b049f1e92e︠
is_prime(12345678910987654321)
︡a05a7bbc-1d1b-40c3-ad8c-28c1f985b617︡{"stdout":"True\n"}︡{"done":true}︡
︠ed2ceabf-ded5-41c1-a821-4fc0392fd895︠
diff(x^x)
︡7a0ba455-8afd-4fa5-b940-d8bd9425ceb8︡{"stdout":"x^x*(log(x) + 1)\n"}︡{"done":true}︡
︠e919b22c-7970-4d67-a1fd-0d734558a225︠
for x in [4,5,6,7]:
print x^2
︡52b1de61-91fb-4469-90d4-4fbd177ffd1a︡{"stdout":"16\n25\n36\n49\n"}︡{"done":true}︡
︠2f75a7b9-9c55-44c5-8115-5a3a51557f0a︠
pete = graphs.PetersenGraph()
︡5657ca93-39bb-4546-908d-236e7b859b9b︡{"done":true}︡
︠38c97f51-62f3-4c8f-85e0-43dfae1d8b60︠
show(pete)
︡8dddd259-fffd-46e1-be4c-fff240050a14︡{"d3":{"data":{"charge":0,"directed":false,"edge_labels":false,"edge_thickness":2,"gravity":0,"height":null,"link_distance":50,"link_strength":0,"links":[{"color":"#aaa","curve":0,"name":"","source":0,"strength":0,"target":1},{"color":"#aaa","curve":0,"name":"","source":0,"strength":0,"target":4},{"color":"#aaa","curve":0,"name":"","source":0,"strength":0,"target":5},{"color":"#aaa","curve":0,"name":"","source":1,"strength":0,"target":2},{"color":"#aaa","curve":0,"name":"","source":1,"strength":0,"target":6},{"color":"#aaa","curve":0,"name":"","source":2,"strength":0,"target":3},{"color":"#aaa","curve":0,"name":"","source":2,"strength":0,"target":7},{"color":"#aaa","curve":0,"name":"","source":3,"strength":0,"target":4},{"color":"#aaa","curve":0,"name":"","source":3,"strength":0,"target":8},{"color":"#aaa","curve":0,"name":"","source":4,"strength":0,"target":9},{"color":"#aaa","curve":0,"name":"","source":5,"strength":0,"target":7},{"color":"#aaa","curve":0,"name":"","source":5,"strength":0,"target":8},{"color":"#aaa","curve":0,"name":"","source":6,"strength":0,"target":8},{"color":"#aaa","curve":0,"name":"","source":6,"strength":0,"target":9},{"color":"#aaa","curve":0,"name":"","source":7,"strength":0,"target":9}],"loops":[],"nodes":[{"group":"0","name":"0"},{"group":"0","name":"1"},{"group":"0","name":"2"},{"group":"0","name":"3"},{"group":"0","name":"4"},{"group":"0","name":"5"},{"group":"0","name":"6"},{"group":"0","name":"7"},{"group":"0","name":"8"},{"group":"0","name":"9"}],"pos":[[6.123233995736766e-17,-1],[-0.9510565162951535,-0.3090169943749475],[-0.5877852522924732,0.8090169943749473],[0.5877852522924729,0.8090169943749476],[0.9510565162951536,-0.3090169943749472],[1.5308084989341916e-16,-0.5],[-0.4755282581475767,-0.1545084971874739],[-0.29389262614623674,0.4045084971874736],[0.2938926261462364,0.40450849718747384],[0.4755282581475769,-0.15450849718747348]],"vertex_labels":true,"vertex_size":7,"width":null},"viewer":"graph"}}︡{"done":true}︡
︠9df02f7a-2fe3-4f20-84c9-06ec23a3ad98︠
pete.order()
︡33424fb8-ef7b-4b50-a68a-f3dfd88624b2︡{"stdout":"10\n"}︡{"done":true}︡
︠e30beea0-c7fd-47c4-a6d8-43d1dffb1a76︠
pete.size()
︡e0b41e72-eeb6-4bdc-93c9-d4f0a7125fd7︡{"stdout":"15\n"}︡{"done":true}︡
︠84cf802a-f451-4629-aa9b-cc204b88b54d︠
cory = Graph(7)
︡5e81faa6-addb-4c33-b364-41063d9ff28b︡{"done":true}︡
︠3284c38c-3063-4397-97d7-9ce3e7ae2051︠
show(cory)
︡b5e26dd8-4665-4f80-b20e-c28913fd71e1︡{"d3":{"data":{"charge":-120,"directed":false,"edge_labels":false,"edge_thickness":2,"gravity":0.04,"height":null,"link_distance":50,"link_strength":1,"links":[],"loops":[],"nodes":[{"group":"0","name":"0"},{"group":"0","name":"1"},{"group":"0","name":"2"},{"group":"0","name":"3"},{"group":"0","name":"4"},{"group":"0","name":"5"},{"group":"0","name":"6"}],"pos":[],"vertex_labels":true,"vertex_size":7,"width":null},"viewer":"graph"}}︡{"done":true}︡
︠e3e18c4f-f9dd-49f0-a8d7-61eaa7586bfd︠
cory.add_edge(0,5)
︡9e3b99ef-5972-402d-935a-64586ec0648f︡{"done":true}︡
︠7fa544a1-6194-4f43-9958-185d0e210455︠
show(cory)
︡64a16648-8526-4c83-b39c-d28d6d84f604︡{"d3":{"data":{"charge":-120,"directed":false,"edge_labels":false,"edge_thickness":2,"gravity":0.04,"height":null,"link_distance":50,"link_strength":1,"links":[{"color":"#aaa","curve":0,"name":"","source":0,"strength":0,"target":5}],"loops":[],"nodes":[{"group":"0","name":"0"},{"group":"0","name":"1"},{"group":"0","name":"2"},{"group":"0","name":"3"},{"group":"0","name":"4"},{"group":"0","name":"5"},{"group":"0","name":"6"}],"pos":[],"vertex_labels":true,"vertex_size":7,"width":null},"viewer":"graph"}}︡{"done":true}︡
︠6808a1bb-7306-4254-ae0c-2f719c6dfe75︠
load("conjecturing.py")
︡d31d12d6-954f-4d7c-bc80-b05cc879b02d︡{"done":true}︡
︠21a55922-2296-4ce1-b760-ab0f3400dc48︠
objects = [2,3,5,60]
︡df92c8bf-97b8-420f-ae3b-8c33862bc318︡{"done":true}︡
︠430907f5-0ac4-4a50-8d43-0199756f493d︠
factor(60)
︡5812e76a-fd07-47fb-840b-54a5fb85bdc3︡{"stdout":"2^2 * 3 * 5\n"}︡{"done":true}︡
︠53f74192-7737-43ff-b325-e1bfeedadcbe︠
len(factor(60))
︡f02bcc64-6a6c-4e68-95d2-82ddb65a8c82︡{"stdout":"3\n"}︡{"done":true}︡
︠bb0d812f-bf40-4878-8710-fb18af6f9b30︠
%lisp
(* (+ 3 4) (- 5 3))
︡071bd6e9-343a-45d8-8bff-a1e13970368a︡{"stdout":"14"}︡{"done":true}︡
︠2247e8ef-98be-452b-86b1-3432f826f024︠
def distinct_primes(n):
return len(factor(n))
︡2ad59b88-8f50-4d2b-b63f-9df256ed134c︡{"done":true}︡
︠ffa2a19e-a7e8-4ef4-8051-9f868845e7ae︠
distinct_primes(60)
︡8ba13234-9a26-4b5c-81a4-c02fa76ede4f︡{"stdout":"3\n"}︡{"done":true}︡
︠b0146cd7-73a2-4fa2-9605-577d1edc1d2f︠
47.digits()
︡a78109b7-f3e9-4873-8fef-5164ae69c22c︡{"stdout":"[7, 4]\n"}︡{"done":true}︡
︠742526e6-8df0-4f45-bbd2-1d2cff54cc36︠
sum(47.digits())
︡855a752b-3ef7-4d0f-b06c-e2b494478364︡{"stdout":"11\n"}︡{"done":true}︡
︠9362c804-050a-4f8a-b255-cc125456f76e︠
def sum_digits(n):
return sum(n.digits())
︡aba822da-635c-4195-92d5-ae4016c19c16︡{"done":true}︡
︠467caec0-3a0b-42ff-9331-94ede19bb8ad︠
sum_digits(47)
︡d2b5aaa5-fa1a-4b4c-a1c6-0a14d26561f4︡{"stdout":"11\n"}︡{"done":true}︡
︠3f958b7e-5170-4577-9121-b16d8412d990︠
def number(n):
return n
︡c0fdf4b9-aaae-4d8c-a245-02b1550c547a︡{"done":true}
︠41fb7a54-f0f5-4e14-970e-6e6d9c2c26e9︠
number(135)
︡05009bcf-d51f-47f7-b0a8-f743bd580d4a︡{"stdout":"135\n"}︡{"done":true}︡
︠002ebf37-46af-4606-96eb-50846c7d2116︠
L=[5,6,7]
L.index(6)
︡99f40481-f4e4-42ac-be84-21bbb7fa07f4︡{"stdout":"1\n"}︡{"done":true}︡
︠a584c803-809d-4d5e-922e-f470ddf22489︠
load("conjecturing.py")
︡f1fdbafd-c848-49a1-b7e4-ad18881c2cac︡
︠42ac56d8-2de3-4c10-a839-e7e7ce4fb708︠
n(ln(10))
︡03b48507-06dd-4f8e-bd43-cf0a640fe0ed︡{"stdout":"2.30258509299405\n"}︡{"done":true}︡
︠dd30e564-c2d0-45ab-8f92-0c597e198e52︠
log(10,10)
︡1bc15258-62f3-4a8c-9a68-e44084bcc2c0︡{"stdout":"1\n"}︡{"done":true}︡
︠955c8956-9aa6-4c59-8784-0041216017db︠
log(100,10)
︡a9582f83-3896-4840-bf41-8e62e18e2cea︡{"stdout":"2\n"}︡{"done":true}︡
︠7ff00b5c-4d55-4737-8703-bc62aeb1d247︠
n(log(10))
︡4c40d862-302f-403f-98f2-648f59bb3a30︡{"stdout":"2.30258509299405\n"}︡{"done":true}︡
︠1d73f80f-0e09-4a9d-b4c6-6c8609a33e9d︠
load("conjecturing.py")
def sum_digits(n):
return sum(n.digits())
def number(n):
return n
def distinct_primes(n):
return len(factor(n))
objects = [2,3,5,60]
invariants = [number, sum_digits, distinct_primes]
conjecture(objects,invariants,invariants.index(number))
︡54335615-8988-454d-ade8-dc7b6dcd21c8︡{"stdout":"[number(x) <= 10^(log(sum_digits(x))/log(10) + 1), number(x) <= sum_digits(x)^distinct_primes(x)]\n"}︡{"done":true}︡
︠5f5671d9-1127-4f43-af80-570cbaafe22c︠
def conj1(n):
return numerical_approx(10^(log(sum_digits(n))/log(10) + 1))
︡39e14815-7ddf-4802-a6fa-d9c46a01e83c︡{"done":true}︡
︠fffcd81f-4a92-46c3-acb3-0d2648c3d51f︠
def conj2(n):
return numerical_approx(sum_digits(n)^distinct_primes(n))
︡cbfda71e-b4dd-429b-9269-6a659cadf642︡{"done":true}︡
︠c4025041-9b92-4912-8c6c-280394b3bd25︠
conj2(2)
︡ee21ebdb-f507-4312-aef9-217af446469d︡{"stdout":"2.00000000000000\n"}︡{"done":true}︡
︠ce8e6127-7f84-4c2f-8c29-797865207d2a︠
conj2(3)
︡ad86d762-229d-436d-b585-ff7d4eb746d0︡{"stdout":"3.00000000000000\n"}︡{"done":true}︡
︠e4502fe3-32b8-4bec-8c4a-04fe1cfa56e8︠
conj2(5)
︡d118b958-cd83-4f3c-9ab8-29a4b99886fc︡{"stdout":"5.00000000000000\n"}︡{"done":true}︡
︠08a50c2d-9e88-45d3-90e8-efa24636b6d1︠
conj2(60)
︡7a885af7-945d-46f0-8116-bb7daeb4c83f︡{"stdout":"216.000000000000\n"}︡{"done":true}︡
︠5e9364cb-aa0f-4e48-8286-cd2ff9102857︠
n
︡a2f1f77b-e19b-48ae-985f-a7a560f4c2de︡{"stdout":"\n"}︡{"done":true}︡
︠0feac7dd-2bb5-4b03-8ee5-d2eddbc16c83︠
conj1(2)
︡c6dbdcd4-db12-47a7-af09-f35dc08bd962︡{"stdout":"20.0000000000000\n"}︡{"done":true}︡
︠9f7f29f9-3f32-4df2-be76-9473f03d37fe︠
conj1(3)
︡7442c8b0-6b70-4d6d-b051-588145c660b6︡{"stdout":"30.0000000000000\n"}︡{"done":true}︡
︠acc564b5-d3c4-483d-8ea4-402724d08c0b︠
conj1(5)
︡5fce67e7-17a2-42c6-a00b-f590e41949ec︡{"stdout":"50.0000000000000\n"}︡{"done":true}︡
︠de7cf538-e9d9-4420-a5ee-ca6e20869245︠
conj1(60)
︡ce4d90d0-be36-4eeb-ad0b-5577b06b2866︡{"stdout":"60.0000000000000\n"}︡{"done":true}︡
︠be8ef026-6cac-4693-95f9-ff33805d08e6︠
def number(n):
return n
︡b46dd86a-8895-4a4a-b089-4f232d83288d︡
︠30824efa-79bb-4bc8-8277-be4ab7a13e0c︠
#47 is a counterexample to conjecture 2
#add to the program and run new conjectures
load("conjecturing.py")
def sum_digits(n):
return sum(n.digits())
def number(n):
return n
def distinct_primes(n):
return len(factor(n))
objects = [2,3,5,60,47]
invariants = [number, sum_digits, distinct_primes]
conjecture(objects,invariants,invariants.index(number))
︡8f735730-cd88-40c6-ac0c-04866e9cf455︡{"stdout":"[number(x) <= log(log(10^e^sum_digits(x))/log(10)), number(x) <= ceil((sum_digits(x) + 1)^cosh(distinct_primes(x))), number(x) <= 10^(log(sum_digits(x))/log(10) + 1), number(x) <= floor(e^(sum_digits(x) - 1)), number(x) <= floor(cosh(distinct_primes(x))^sum_digits(x))]"}︡{"stdout":"\n"}︡{"done":true}︡
︠cc686431-967b-4dc3-90b7-619bbb799160︠
3+4
︡0ed5a426-5e28-4af7-8dee-1c5a18f56cad︡{"stdout":"7\n"}︡{"done":true}︡
︠288603a3-3597-455c-927e-69663e9ee46c︠
x=7
def test(y):
y = y+1
return y
︡a31f8a90-6de6-40ff-b7bc-26876eb4203c︡{"done":true}︡
︠9b1d85e9-fb17-4fd7-84c4-1b32c5069a19︠
test(x)
x
︡01ba4281-cbc3-462b-872a-c8a06ed8cc5f︡{"stdout":"8\n"}︡{"stdout":"7\n"}︡{"done":true}︡
︠795ebcf4-e413-45ec-a2f0-5a0df6895615︠
L=[1,2,3]
def test2(X):
X[0]=47
return X
︡688a10f3-3ba9-4618-85a3-522f9dbe9432︡{"done":true}︡
︠9637a5b1-ce56-4628-9b46-aab6b00652ab︠
test2(L)
L
︡574f6c37-156a-4d65-ac5b-edca83c45ebd︡{"stdout":"[47, 2, 3]\n"}︡{"stdout":"[47, 2, 3]\n"}︡{"done":true}︡
︠7e25e201-abfb-47a1-9e88-3f7fb3074619︠
sage: import sage.libs.mpmath.all as mpmath
sage: mpmath.mp.dps = 30
sage: mpmath.pslq([sqrt(n) for n in range(2, 8+1)])
︡6352350b-cff9-4cda-b29a-d8382552759a︡{"stdout":"[2, 0, 0, 0, 0, 0, -1]\n"}︡{"done":true}︡
︠669e9ff7-77cf-4010-a9db-f0d614fb1123︠
integral(x^2*log(x)/((x^2-1)*(x^4+1)),x,0,1)
︡54df5a11-e0c7-4d15-82f5-6034502495d9︡{"stdout":"1/32*sqrt(2)*pi^2*(sqrt(2) - 1)\n"}︡{"done":true}︡
︠8949d730-2013-48ab-9c8b-dc50514e3e17︠
numerical_integral(x^2*log(x)/((x^2-1)*(x^4+1)),0,1)
︡427f390a-470f-451e-8c89-734229720722︡{"stdout":"(0.18067126259059388, 8.044808609022901e-11)\n"}︡{"done":true}︡
︠aaa8ec0d-77b2-41a6-b169-1cbd28a2c980︠
mpmath.mp.identify(0.881373587019543)
︡ebfb073b-17c1-41fc-9d60-eef5cf200206︡{"done":true}︡
︠9c757ca5-4480-4889-91cb-3b70e69c0938︠
mpmath.mp.identify(0.22222222222222)
︡509c61c8-a96b-4250-accc-452a61699c6a︡{"stdout":"'((36-sqrt(0))/162)'\n"}︡{"done":true}︡
︠f9352226-3ac7-42e2-9c06-0275f779b5ab︠
mpmath.mp.identify?
︡d4e20afa-71ca-498c-8d1d-06fffc801b11︡{"code":{"lineno":-1,"mode":"text/x-rst","source":"File: /projects/sage/sage-7.5/local/lib/python2.7/site-packages/mpmath/identification.py\nSignature : mpmath.mp.identify(ctx, x, constants=[], tol=None, maxcoeff=1000, full=False, verbose=False)\nDocstring :\nGiven a real number x, \"identify(x)\" attempts to find an exact\nformula for x. This formula is returned as a string. If no match is\nfound, \"None\" is returned. With \"full=True\", a list of matching\nformulas is returned.\n\nAs a simple example, \"identify()\" will find an algebraic formula\nfor the golden ratio:\n\n >>> from mpmath import *\n >>> mp.dps = 15; mp.pretty = True\n >>> identify(phi)\n '((1+sqrt(5))/2)'\n\n\"identify()\" can identify simple algebraic numbers and simple\ncombinations of given base constants, as well as certain basic\ntransformations thereof. More specifically, \"identify()\" looks for\nthe following:\n\n 1. Fractions\n\n 2. Quadratic algebraic numbers\n\n 3. Rational linear combinations of the base constants\n\n 4. Any of the above after first transforming x into f(x)\n where f(x) is 1/x, sqrt x, x^2, log x or exp x, either\n directly or with x or f(x) multiplied or divided by one of\n the base constants\n\n 5. Products of fractional powers of the base constants and\n small integers\n\nBase constants can be given as a list of strings representing\nmpmath expressions (\"identify()\" will \"eval\" the strings to\nnumerical values and use the original strings for the output), or\nas a dict of formula:value pairs.\n\nIn order not to produce spurious results, \"identify()\" should be\nused with high precision; preferably 50 digits or more.\n\n**Examples**\n\nSimple identifications can be performed safely at standard\nprecision. Here the default recognition of rational, algebraic, and\nexp/log of algebraic numbers is demonstrated:\n\n >>> mp.dps = 15\n >>> identify(0.22222222222222222)\n '(2/9)'\n >>> identify(1.9662210973805663)\n 'sqrt(((24+sqrt(48))/8))'\n >>> identify(4.1132503787829275)\n 'exp((sqrt(8)/2))'\n >>> identify(0.881373587019543)\n 'log(((2+sqrt(8))/2))'\n\nBy default, \"identify()\" does not recognize pi. At standard\nprecision it finds a not too useful approximation. At slightly\nincreased precision, this approximation is no longer accurate\nenough and \"identify()\" more correctly returns \"None\":\n\n >>> identify(pi)\n '(2**(176/117)*3**(20/117)*5**(35/39))/(7**(92/117))'\n >>> mp.dps = 30\n >>> identify(pi)\n >>>\n\nNumbers such as pi, and simple combinations of user-defined\nconstants, can be identified if they are provided explicitly:\n\n >>> identify(3*pi-2*e, ['pi', 'e'])\n '(3*pi + (-2)*e)'\n\nHere is an example using a dict of constants. Note that the\nconstants need not be \"atomic\"; \"identify()\" can just as well\nexpress the given number in terms of expressions given by formulas:\n\n >>> identify(pi+e, {'a':pi+2, 'b':2*e})\n '((-2) + 1*a + (1/2)*b)'\n\nNext, we attempt some identifications with a set of base constants.\nIt is necessary to increase the precision a bit.\n\n>>> mp.dps = 50\n>>> base = ['sqrt(2)','pi','log(2)']\n>>> identify(0.25, base)\n'(1/4)'\n>>> identify(3*pi + 2*sqrt(2) + 5*log(2)/7, base)\n'(2*sqrt(2) + 3*pi + (5/7)*log(2))'\n>>> identify(exp(pi+2), base)\n'exp((2 + 1*pi))'\n>>> identify(1/(3+sqrt(2)), base)\n'((3/7) + (-1/7)*sqrt(2))'\n>>> identify(sqrt(2)/(3*pi+4), base)\n'sqrt(2)/(4 + 3*pi)'\n>>> identify(5**(mpf(1)/3)*pi*log(2)**2, base)\n'5**(1/3)*pi*log(2)**2'\n\nAn example of an erroneous solution being found when too low\nprecision is used:\n\n >>> mp.dps = 15\n >>> identify(1/(3*pi-4*e+sqrt(8)), ['pi', 'e', 'sqrt(2)'])\n '((11/25) + (-158/75)*pi + (76/75)*e + (44/15)*sqrt(2))'\n >>> mp.dps = 50\n >>> identify(1/(3*pi-4*e+sqrt(8)), ['pi', 'e', 'sqrt(2)'])\n '1/(3*pi + (-4)*e + 2*sqrt(2))'\n\n**Finding approximate solutions**\n\nThe tolerance \"tol\" defaults to 3/4 of the working precision.\nLowering the tolerance is useful for finding approximate matches.\nWe can for example try to generate approximations for pi:\n\n >>> mp.dps = 15\n >>> identify(pi, tol=1e-2)\n '(22/7)'\n >>> identify(pi, tol=1e-3)\n '(355/113)'\n >>> identify(pi, tol=1e-10)\n '(5**(339/269))/(2**(64/269)*3**(13/269)*7**(92/269))'\n\nWith \"full=True\", and by supplying a few base constants, \"identify\"\ncan generate almost endless lists of approximations for any number\n(the output below has been truncated to show only the first few):\n\n >>> for p in identify(pi, ['e', 'catalan'], tol=1e-5, full=True):\n ... print(p)\n ... # doctest: +ELLIPSIS\n e/log((6 + (-4/3)*e))\n (3**3*5*e*catalan**2)/(2*7**2)\n sqrt(((-13) + 1*e + 22*catalan))\n log(((-6) + 24*e + 4*catalan)/e)\n exp(catalan*((-1/5) + (8/15)*e))\n catalan*(6 + (-6)*e + 15*catalan)\n sqrt((5 + 26*e + (-3)*catalan))/e\n e*sqrt(((-27) + 2*e + 25*catalan))\n log(((-1) + (-11)*e + 59*catalan))\n ((3/20) + (21/20)*e + (3/20)*catalan)\n ...\n\nThe numerical values are roughly as close to pi as permitted by\nthe specified tolerance:\n\n>>> e/log(6-4*e/3)\n3.14157719846001\n>>> 135*e*catalan**2/98\n3.14166950419369\n>>> sqrt(e-13+22*catalan)\n3.14158000062992\n>>> log(24*e-6+4*catalan)-1\n3.14158791577159\n\n**Symbolic processing**\n\nThe output formula can be evaluated as a Python expression. Note\nhowever that if fractions (like '2/3') are present in the formula,\nPython's \"eval()\" may erroneously perform integer division. Note\nalso that the output is not necessarily in the algebraically\nsimplest form:\n\n >>> identify(sqrt(2))\n '(sqrt(8)/2)'\n\nAs a solution to both problems, consider using SymPy's \"sympify()\"\nto convert the formula into a symbolic expression. SymPy can be\nused to pretty-print or further simplify the formula symbolically:\n\n >>> from sympy import sympify # doctest: +SKIP\n >>> sympify(identify(sqrt(2))) # doctest: +SKIP\n 2**(1/2)\n\nSometimes \"identify()\" can simplify an expression further than a\nsymbolic algorithm:\n\n >>> from sympy import simplify # doctest: +SKIP\n >>> x = sympify('-1/(-3/2+(1/2)*5**(1/2))*(3/2-1/2*5**(1/2))**(1/2)') # doctest: +SKIP\n >>> x # doctest: +SKIP\n (3/2 - 5**(1/2)/2)**(-1/2)\n >>> x = simplify(x) # doctest: +SKIP\n >>> x # doctest: +SKIP\n 2/(6 - 2*5**(1/2))**(1/2)\n >>> mp.dps = 30 # doctest: +SKIP\n >>> x = sympify(identify(x.evalf(30))) # doctest: +SKIP\n >>> x # doctest: +SKIP\n 1/2 + 5**(1/2)/2\n\n(In fact, this functionality is available directly in SymPy as the\nfunction \"nsimplify()\", which is essentially a wrapper for\n\"identify()\".)\n\n**Miscellaneous issues and limitations**\n\nThe input x must be a real number. All base constants must be\npositive real numbers and must not be rationals or rational linear\ncombinations of each other.\n\nThe worst-case computation time grows quickly with the number of\nbase constants. Already with 3 or 4 base constants, \"identify()\"\nmay require several seconds to finish. To search for relations\namong a large number of constants, you should consider using\n\"pslq()\" directly.\n\nThe extended transformations are applied to x, not the constants\nseparately. As a result, \"identify\" will for example be able to\nrecognize \"exp(2*pi+3)\" with \"pi\" given as a base constant, but not\n\"2*exp(pi)+3\". It will be able to recognize the latter if \"exp(pi)\"\nis given explicitly as a base constant."}}︡{"done":true}
︠de78f1cd-986f-4839-bf63-96abd5b9d834︠
#a function that returns the first digit of an integer
def first_digit(n):
return n.digits().pop()
#a function that counts the first digitso of each kind from a list of numbers L
def first_digit_counts(L):
data = [0]*10
for x in L:
#print x, first_digit(x), data[first_digit(x)]
first = first_digit(x)
data[first] = data[first] + 1
return data
#a function that gives the (empirical) probabilities that the integer from a list L starts with digit d
def first_digit_distribution(L):
counts = first_digit_counts(L)
length = len(L)
return [n(counts[i]/length, digits=3) for i in [0..9]]
︡646bf116-7e81-457f-befd-fe8bb684e91e︡{"done":true}︡
︠73f3e59d-d0db-42ea-bb73-b4575a93b321︠
counts=first_digit_counts(srange(1,1000000000))
︡ab5fbb51-6d41-4d17-b9ac-776ce1d9054b︡
︠8c79e891-ba13-49c6-b56a-b672b53b8dd3︠
bar_chart(counts)
︡260737f7-c381-4934-baec-0d5c162176f2︡{"file":{"filename":"/projects/4599068b-1689-4a77-81c7-9cd760e0f06d/.sage/temp/compute5-us/31883/tmp_TYhuRv.svg","show":true,"text":null,"uuid":"24d8b75a-659b-4208-a4a9-ad449cfac606"},"once":false}︡{"done":true}︡
︠f24ed497-0a7f-4dad-9d94-b22a275606aa︠
mpmath.mp.identify(0.180671262590654)
︡7932f6d1-2965-4313-82d1-83242fe698cb︡
︠8e942a98-d756-4d78-a511-8bc950face76︠
3+4
︡7708ead0-ad82-4904-9626-b1b929626402︡{"stdout":"7\n"}︡{"done":true}︡
︠46ee0639-a42b-4ac3-bedb-f95c67ab8d0f︠
@interact
def _(f=x^2, a=-3, b=3):
show(plot(f,(x,a,b)))
︡ceb2726c-a5cf-48b8-8cf1-aa635121eb1e︡{"interact":{"controls":[{"control_type":"input-box","default":"x^2","label":"f","nrows":1,"readonly":false,"submit_button":null,"type":null,"var":"f","width":null},{"control_type":"input-box","default":-3,"label":"a","nrows":1,"readonly":false,"submit_button":null,"type":null,"var":"a","width":null},{"control_type":"input-box","default":3,"label":"b","nrows":1,"readonly":false,"submit_button":null,"type":null,"var":"b","width":null}],"flicker":false,"id":"1d88c6c0-26e6-4bea-803b-0bf47381bead","layout":[[["f",12,null]],[["a",12,null]],[["b",12,null]],[["",12,null]]],"style":"None"}}︡{"done":true}︡
︠b2b933bb-9f9f-4a5a-a0c2-65958acb6d1b︠
@interact
def _(f=input_box(x^2,width=20),
color=color_selector(widget='colorpicker', label=""),
axes=True,
fill=True,
zoom=range_slider(-3,3,default=(-3,3))):
show(plot(f,(x,zoom[0], zoom[1]), color=color, axes=axes,fill=fill))
︡34223fb4-2c9b-4562-87f0-f3ca7ee23c6e︡{"done":true,"error":"killed"}︡{"done":true,"error":"killed"}
︠154abdd8-6659-49b0-be90-d189d58ee444︠
@interact
def _(f=input_box(x^2,width=20),
color=color_selector(widget='colorpicker', label=""),
axes=True,
fill=True,
zoom=range_slider(-3,3,default=(-3,3))):
show(plot(f,(x,zoom[0], zoom[1]), color=color, axes=axes,fill=fill))
︡71fbe8a5-b01c-4a20-afce-9c548643a1f8︡{"done":true,"error":"killed"}︡{"done":true,"error":"killed"}
︠6bbb9ab1-6b45-429c-afca-e19001060669︠
3+4
︡f6d6a9b7-70ec-47a5-b232-4958341aa08c︡{"stdout":"7\n"}︡{"done":true}︡
︠9c856fb1-11ac-4b02-b3a0-1127c8c50ad0s︠
@interact
def _(f=input_box(x^2,width=20),
axes=True,
fill=True,
zoom=range_slider(-3,3,default=(-3,3))):
show(plot(f,(x,zoom[0], zoom[1]), axes=axes,fill=fill))
︡5cc6fe5b-24e9-42d8-93af-04e9e542d87a︡{"done":true,"error":"killed"}︡{"done":true,"error":"killed"}
︠8fe4faa9-3e7b-4481-b78a-b3edf0170fb3ss︠
@interact
def _(f=input_box(x^2,width=20),
axes=True,
fill=True,
zoom=range_slider(-3,3,default=(-3,3))):
show(plot(f,(x,zoom[0], zoom[1]), axes=axes,fill=fill))
︡93d34391-c394-4e31-a1dc-8bb79f89382e︡{"interact":{"controls":[{"control_type":"input-box","default":"x^2","label":"f","nrows":1,"readonly":false,"submit_button":null,"type":null,"var":"f","width":20},{"control_type":"checkbox","default":true,"label":"axes","readonly":false,"var":"axes"},{"control_type":"checkbox","default":true,"label":"fill","readonly":false,"var":"fill"},{"animate":true,"control_type":"range-slider","default":[0,500],"display_value":true,"label":"zoom","vals":["-3.0","-2.988","-2.976","-2.964","-2.952","-2.94","-2.928","-2.916","-2.904","-2.892","-2.88","-2.868","-2.856","-2.844","-2.832","-2.82","-2.808","-2.796","-2.784","-2.772","-2.76","-2.748","-2.736","-2.724","-2.712","-2.7","-2.688","-2.676","-2.664","-2.652","-2.64","-2.628","-2.616","-2.604","-2.592","-2.58","-2.568","-2.556","-2.544","-2.532","-2.52","-2.508","-2.496","-2.484","-2.472","-2.46","-2.448","-2.436","-2.424","-2.412","-2.4","-2.388","-2.376","-2.364","-2.352","-2.34","-2.328","-2.316","-2.304","-2.292","-2.28","-2.268","-2.256","-2.244","-2.232","-2.22","-2.208","-2.196","-2.184","-2.172","-2.16","-2.148","-2.136","-2.124","-2.112","-2.1","-2.088","-2.076","-2.064","-2.052","-2.04","-2.028","-2.016","-2.004","-1.992","-1.98","-1.968","-1.956","-1.944","-1.932","-1.92","-1.908","-1.896","-1.884","-1.872","-1.86","-1.848","-1.836","-1.824","-1.812","-1.8","-1.788","-1.776","-1.764","-1.752","-1.74","-1.728","-1.716","-1.704","-1.692","-1.68","-1.668","-1.656","-1.644","-1.632","-1.62","-1.608","-1.596","-1.584","-1.572","-1.56","-1.548","-1.536","-1.524","-1.512","-1.5","-1.488","-1.476","-1.464","-1.452","-1.44","-1.428","-1.416","-1.404","-1.392","-1.38","-1.368","-1.356","-1.344","-1.332","-1.32","-1.308","-1.296","-1.284","-1.272","-1.26","-1.248","-1.236","-1.224","-1.212","-1.2","-1.188","-1.176","-1.164","-1.152","-1.14","-1.128","-1.116","-1.104","-1.092","-1.08","-1.068","-1.056","-1.044","-1.032","-1.02","-1.008","-0.996","-0.984","-0.972","-0.96","-0.948","-0.936","-0.924","-0.912","-0.9","-0.888","-0.876","-0.864","-0.852","-0.84","-0.828","-0.816","-0.804","-0.792","-0.78","-0.768","-0.756","-0.744","-0.732","-0.72","-0.708","-0.696","-0.684","-0.672","-0.66","-0.648","-0.636","-0.624","-0.612","-0.6","-0.588","-0.576","-0.564","-0.552","-0.54","-0.528","-0.516","-0.504","-0.492","-0.48","-0.468","-0.456","-0.444","-0.432","-0.42","-0.408","-0.396","-0.384","-0.372","-0.36","-0.348","-0.336","-0.324","-0.312","-0.3","-0.288","-0.276","-0.264","-0.252","-0.24","-0.228","-0.216","-0.204","-0.192","-0.18","-0.168","-0.156","-0.144","-0.132","-0.12","-0.108","-0.096","-0.084","-0.072","-0.06","-0.048","-0.036","-0.024","-0.012","2.52228793407e-15","0.012","0.024","0.036","0.048","0.06","0.072","0.084","0.096","0.108","0.12","0.132","0.144","0.156","0.168","0.18","0.192","0.204","0.216","0.228","0.24","0.252","0.264","0.276","0.288","0.3","0.312","0.324","0.336","0.348","0.36","0.372","0.384","0.396","0.408","0.42","0.432","0.444","0.456","0.468","0.48","0.492","0.504","0.516","0.528","0.54","0.552","0.564","0.576","0.588","0.6","0.612","0.624","0.636","0.648","0.66","0.672","0.684","0.696","0.708","0.72","0.732","0.744","0.756","0.768","0.78","0.792","0.804","0.816","0.828","0.84","0.852","0.864","0.876","0.888","0.9","0.912","0.924","0.936","0.948","0.96","0.972","0.984","0.996","1.008","1.02","1.032","1.044","1.056","1.068","1.08","1.092","1.104","1.116","1.128","1.14","1.152","1.164","1.176","1.188","1.2","1.212","1.224","1.236","1.248","1.26","1.272","1.284","1.296","1.308","1.32","1.332","1.344","1.356","1.368","1.38","1.392","1.404","1.416","1.428","1.44","1.452","1.464","1.476","1.488","1.5","1.512","1.524","1.536","1.548","1.56","1.572","1.584","1.596","1.608","1.62","1.632","1.644","1.656","1.668","1.68","1.692","1.704","1.716","1.728","1.74","1.752","1.764","1.776","1.788","1.8","1.812","1.824","1.836","1.848","1.86","1.872","1.884","1.896","1.908","1.92","1.932","1.944","1.956","1.968","1.98","1.992","2.004","2.016","2.028","2.04","2.052","2.064","2.076","2.088","2.1","2.112","2.124","2.136","2.148","2.16","2.172","2.184","2.196","2.208","2.22","2.232","2.244","2.256","2.268","2.28","2.292","2.304","2.316","2.328","2.34","2.352","2.364","2.376","2.388","2.4","2.412","2.424","2.436","2.448","2.46","2.472","2.484","2.496","2.508","2.52","2.532","2.544","2.556","2.568","2.58","2.592","2.604","2.616","2.628","2.64","2.652","2.664","2.676","2.688","2.7","2.712","2.724","2.736","2.748","2.76","2.772","2.784","2.796","2.808","2.82","2.832","2.844","2.856","2.868","2.88","2.892","2.904","2.916","2.928","2.94","2.952","2.964","2.976","2.988","3.0"],"var":"zoom","width":null}],"flicker":false,"id":"453f6eb1-1a61-42e5-9305-255d82fbbc5d","layout":[[["f",12,null]],[["axes",12,null]],[["fill",12,null]],[["zoom",12,null]],[["",12,null]]],"style":"None"}}︡{"done":true}︡
︠753162a8-18a4-46f0-bebc-acc7c58f361as︠
@interact
def _(m=('matrix', identity_matrix(2)), auto_update=False):
print m.eigenvalues()
︡009e98c4-b821-4481-861a-d1123891c4b0︡{"interact":{"controls":[{"control_type":"input-grid","default":[["1","0"],["0","1"]],"label":"matrix","ncols":2,"nrows":2,"var":"m","width":5},{"control_type":"checkbox","default":false,"label":"auto_update","readonly":false,"var":"auto_update"}],"flicker":false,"id":"24ac10e5-302b-4faf-b286-34d250bfaff0","layout":[[["m",12,null]],[["auto_update",12,null]],[["",12,null]]],"style":"None"}}︡{"done":true}︡
︠ef3bd78b-554a-41d8-b9e6-09057331d734s︠
var("z")
complex_plot(z**2, (-5, 5), (-5, 5))
︡fa619685-76f6-4feb-b59f-2b67ea39f40d︡{"stdout":"z\n"}︡{"file":{"filename":"/projects/4599068b-1689-4a77-81c7-9cd760e0f06d/.sage/temp/compute5-us/32006/tmp_tyWZln.svg","show":true,"text":null,"uuid":"32ae5306-8438-470f-92b5-1c16c1d373d0"},"once":false}︡{"done":true}︡
︠2b2ddc9d-75c7-41ef-8eba-d19b4143ce93s︠
complex_plot?
︡63fd0b90-c1ab-4a7c-bde7-3bdd2166d7df︡{"code":{"filename":null,"lineno":-1,"mode":"text/x-rst","source":"File: /projects/sage/sage-7.5/src/sage/misc/lazy_import.pyx\nSignature : complex_plot(f, xrange, yrange, plot_points=100, interpolation='catrom', **kwds)\nDocstring :\n\"complex_plot\" takes a complex function of one variable, f(z) and\nplots output of the function over the specified \"xrange\" and\n\"yrange\" as demonstrated below. The magnitude of the output is\nindicated by the brightness (with zero being black and infinity\nbeing white) while the argument is represented by the hue (with red\nbeing positive real, and increasing through orange, yellow, ... as\nthe argument increases).\n\n\"complex_plot(f, (xmin, xmax), (ymin, ymax), ...)\"\n\nINPUT:\n\n* \"f\" -- a function of a single complex value x + iy\n\n* \"(xmin, xmax)\" -- 2-tuple, the range of \"x\" values\n\n* \"(ymin, ymax)\" -- 2-tuple, the range of \"y\" values\n\nThe following inputs must all be passed in as named parameters:\n\n* \"plot_points\" -- integer (default: 100); number of points to\n plot in each direction of the grid\n\n* \"interpolation\" -- string (default: \"'catrom'\"), the\n interpolation method to use: \"'bilinear'\", \"'bicubic'\",\n \"'spline16'\", \"'spline36'\", \"'quadric'\", \"'gaussian'\", \"'sinc'\",\n \"'bessel'\", \"'mitchell'\", \"'lanczos'\", \"'catrom'\", \"'hermite'\",\n \"'hanning'\", \"'hamming'\", \"'kaiser'\"\n\nEXAMPLES:\n\nHere we plot a couple of simple functions:\n\n sage: complex_plot(sqrt(x), (-5, 5), (-5, 5))\n Graphics object consisting of 1 graphics primitive\n\n sage: complex_plot(sin(x), (-5, 5), (-5, 5))\n Graphics object consisting of 1 graphics primitive\n\n sage: complex_plot(log(x), (-10, 10), (-10, 10))\n Graphics object consisting of 1 graphics primitive\n\n sage: complex_plot(exp(x), (-10, 10), (-10, 10))\n Graphics object consisting of 1 graphics primitive\n\nA function with some nice zeros and a pole:\n\n sage: f(z) = z^5 + z - 1 + 1/z\n sage: complex_plot(f, (-3, 3), (-3, 3))\n Graphics object consisting of 1 graphics primitive\n\nHere is the identity, useful for seeing what values map to what\ncolors:\n\n sage: complex_plot(lambda z: z, (-3, 3), (-3, 3))\n Graphics object consisting of 1 graphics primitive\n\nThe Riemann Zeta function:\n\n sage: complex_plot(zeta, (-30,30), (-30,30))\n Graphics object consisting of 1 graphics primitive\n\nExtra options will get passed on to show(), as long as they are\nvalid:\n\n sage: complex_plot(lambda z: z, (-3, 3), (-3, 3), figsize=[1,1])\n Graphics object consisting of 1 graphics primitive\n\n sage: complex_plot(lambda z: z, (-3, 3), (-3, 3)).show(figsize=[1,1]) # These are equivalent"}}︡{"done":true}︡
︠15f74eeb-3026-43a5-b77c-a1dfa11aca4es︠
@interact
def julia_plot(expo = slider(-10,10,0.1,2), \
iterations=slider(1,100,1,30), \
c_real = slider(-2,2,0.01,0.5), \
c_imag = slider(-2,2,0.01,0.5), \
zoom_x = range_slider(-2,2,0.01,(-1.5,1.5)), \
zoom_y = range_slider(-2,2,0.01,(-1.5,1.5))):
var('z')
I = CDF.gen()
f(z) = z^expo + c_real + c_imag*I
ff_j = fast_callable(f, vars=[z], domain=CDF)
def julia(z):
for i in range(iterations):
z = ff_j(z)
if abs(z) > 2:
return z
return z
print 'z <- z^%s + (%s+%s*I)' % (expo, c_real, c_imag)
complex_plot(julia, zoom_x,zoom_y, plot_points=200, dpi=150).show(frame=True, aspect_ratio=1)
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︠c392bbc0-2bd1-448b-b6da-e6a851b61ccas︠
@interact
def _(n=(5..100)):
Poset(([1..n], lambda x, y: y%x == 0) ).show()
︡96bc7e4d-cca8-478a-b641-26b2355c06c5︡{"interact":{"controls":[{"animate":true,"control_type":"slider","default":0,"display_value":true,"label":"n","vals":["5","6","7","8","9","10","11","12","13","14","15","16","17","18","19","20","21","22","23","24","25","26","27","28","29","30","31","32","33","34","35","36","37","38","39","40","41","42","43","44","45","46","47","48","49","50","51","52","53","54","55","56","57","58","59","60","61","62","63","64","65","66","67","68","69","70","71","72","73","74","75","76","77","78","79","80","81","82","83","84","85","86","87","88","89","90","91","92","93","94","95","96","97","98","99","100"],"var":"n","width":null}],"flicker":false,"id":"ed697c40-aa91-4c5a-b890-fecc58c5cdf7","layout":[[["n",12,null]],[["",12,null]]],"style":"None"}}︡{"done":true}︡
︠de6ecec4-4e48-48d3-b9eb-172ad55c900as︠
5//2
︡33630c79-b4d5-4c3b-8b83-798a1a472a3b︡{"stdout":"2\n"}︡{"done":true}︡
︠cdc983a0-4cc4-4f06-8c81-64fbe8de097ds︠
matrix_plot(matrix(2,[1,2,3,4]))
︡4b610621-ff35-4048-b43e-99c7b241ce58︡{"file":{"filename":"/projects/4599068b-1689-4a77-81c7-9cd760e0f06d/.sage/temp/compute5-us/32006/tmp_mnDDpq.svg","show":true,"text":null,"uuid":"ae6cf6eb-b339-40cc-8678-65df764552bd"},"once":false}︡{"done":true}︡
︠9dcac6be-46b9-46b1-9fe2-a8b6e5753c1es︠
def sierpinski(N):
'''Generates the Sierpinski triangle by taking the modulo-2
of each element in Pascal's triangle'''
S=[([0]*(N//2-a//2))+
[binomial(a,b)%2 for b in range(a+1)]+
([0]*(N//2-a//2)) for a in range(0,N,2)]
return S
@interact
def _(N=slider([2 ** a for a in range(12)],
label='Number of iterations', default=64),
size=slider(1, 20, label='Size', step_size=1, default=9)):
M = sierpinski(2 * N)
matrix_plot(M, cmap='binary').show(figsize=[size, size])
︡11ec7989-8469-442d-819b-5dc69f881c1b︡{"interact":{"controls":[{"animate":true,"control_type":"slider","default":6,"display_value":true,"label":"Number of iterations","vals":["1","2","4","8","16","32","64","128","256","512","1024","2048"],"var":"N","width":null},{"animate":true,"control_type":"slider","default":8,"display_value":true,"label":"Size","vals":["1","2","3","4","5","6","7","8","9","10","11","12","13","14","15","16","17","18","19","20"],"var":"size","width":null}],"flicker":false,"id":"a9344e2e-73af-4b25-8b07-e0d32812468b","layout":[[["N",12,null]],[["size",12,null]],[["",12,null]]],"style":"None"}}︡{"done":true}︡
︠428145e4-0756-4732-a72b-890b682bfeb5s︠
@interact
def rwalk3d(n=slider(50,1000,step_size=1), frame=True):
pnt = [0,0,0]
v = [copy(pnt)]
for i in range(n):
pnt[0] += random()-0.5
pnt[1] += random()-0.5
pnt[2] += random()-0.5
v.append(copy(pnt))
show(line3d(v,color='black'),aspect_ratio=[1,1,1],frame=frame)
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