What's the purpose of Jupyt-nb ?¶

Jupyt-nb is designed to study Mathematical and Physical Constants properties of ratios as detailed in http://vixra.org/pdf/1904.0218v4.pdf

It uses 2 Julia packages:

MathPhysicalConstants.SI
PhysicalConstant.CODATA2019

it can be linked to numpy using PyCall

Reference for Continuous Improvement¶

Reference for offline visualization¶

Pkg.clone("https://github.com/laguer/MathPhysicalConstants.jl")

In [1]:

versioninfo()

Julia Version 0.6.4
Commit 9d11f62bcb (2018-07-09 19:09 UTC)
Platform Info:
  OS: Linux (x86_64-pc-linux-gnu)
  CPU: Intel(R) Xeon(R) CPU @ 2.20GHz
  WORD_SIZE: 64
  BLAS: libopenblas (USE64BITINT DYNAMIC_ARCH NO_AFFINITY Haswell MAX_THREADS=16)
  LAPACK: libopenblas64_
  LIBM: libopenlibm
  LLVM: libLLVM-3.9.1 (ORCJIT, broadwell)

In [3]:

Pkg.init()

INFO: Initializing package repository /srv/julia/pkg/v0.6
INFO: Package directory /srv/julia/pkg/v0.6 is already initialized.

In [14]:

Pkg.add("Unitful")

INFO: Package Unitful is already installed

In [ ]:

#import Pkg; 
Pkg.status()

In [ ]:

Pkg.clone("https://github.com/laguer/PhysicalConstant.jl.git")

In [17]:

Pkg.add("PhysicalConstant")

INFO: Package PhysicalConstant is already installed

Pkg.dir("PhysicalConstant")

In [2]:

using Unitful
using Measurements
using MathPhysicalConstants.SI

Let's experiment MathPhysConstants by looking at the reduced electron Compton wavelength Basic formula is given by ƛ_e ≡ ħ/m_e.c

In [3]:

big(SI.PlanckConstantHbar)/(big(SI.MassElectron)*big(SI.SpeedOfLight))

Out[3]:

3.861592046068737688702740558503376825155847784672545991504355454736855198364958e-13

In [4]:

big(SI.SanchezElectrConstant)

Out[4]:

1.370359991390000118371972348541021347045898437500000000000000000000000000000000e+02

In [5]:

measurement(SI.PlanckConstantHbar)

Out[5]:

$1.05457162825e-34 \pm 0.0$

In [6]:

big(measurement(SI.PlanckConstantHbar))

Out[6]:

$1.054571628249999983241389640919049903556150033962677651299952084762545619284539e-34 \pm 0.000000000000000000000000000000000000000000000000000000000000000000000000000000$

In [7]:

using Unitful
using Measurements
using MathPhysicalConstants.SI

In [8]:

a1 = big(e^e^e)

Out[8]:

3.814279104760214220732450485229492187500000000000000000000000000000000000000000e+06

In [9]:

a1^e

Out[9]:

7.764865179158049315959156489518625436695868996929735868921423813247391069467881e+17

In [10]:

a = SI.SanchezElectrConstant

Out[10]:

137.035999139

$ln137/ln(a/137) ≈ (2 + 135/d_e)^2$ ¶

In [11]:

using Unitful
using Measurements
using Cosmology
using PyCall
@pyimport numpy as np

np.log(137)/ np.log(a/137)

Out[11]:

18726.168436946326

$exp(2^{e^{2+1/2}})$ ≈ $exp(e^{2e} + e^2)$ = 8.820166056085355e102¶

In [12]:

using Unitful
using Measurements
using Cosmology
using PyCall
@pyimport numpy as np
@pyimport sympy as sy
@pyimport decimal

#np.exp(2^e^2.5)
np.exp(2^e)

Out[12]:

721.178003510984

In [13]:

exp(2^(e^2.5))

Out[13]:

Inf

$exp(e^{2e} + e^2)$ ¶

In [14]:

using Unitful
using Measurements
using Cosmology
using PyCall
@pyimport numpy as np
@pyimport sympy as sy
@pyimport decimal

# decimal.Decimal(np.exp(2^e^2.5))
decimal.Decimal(np.exp(e^(2*e)+e^2))

Out[14]:

8.820166056085355e102

The economic number : $e^{e^e}$ ¶

In [27]:

using Unitful
using Measurements
using Cosmology
using PyCall
@pyimport numpy as np
@pyimport sympy as sy
@pyimport scipy as sci
@pyimport matplotlib.pyplot as plt
@pyimport decimal
println(np.e^np.e^np.e)

3.8142791047601975e6

Wien Reduced Constant

See https://en.wikipedia.org/wiki/Wien%27s_displacement_law

$\omega = hc/kTλ_{Wien}$

NIST tells b=2.897 7729 x 10-3

In [26]:

(SI.PlanckConstantH*SI.SpeedOfLight)/(SI.Boltzmann * 1 * 2.8977729e-3)

Out[26]:

4.9651126128756795

$f(26) ≈ 6(2π^2a^3)^5$ , where $2π^2a^3$ is the area of a 4-sphere of radius a

In [ ]:

In [25]:

using Unitful
using Measurements
using Cosmology
using PyCall
@pyimport math
println(math.pi)

3.141592653589793

This notebook is a preview demo of IJulia: a Julia-language backend combined with the IPython interactive environment. This combination allows you to interact with the Julia language using IPython's powerful graphical notebook, which combines code, formatted text, math, and multimedia in a single document.

Note: this is a preview, because it relies on pre-release bleeding-edge versions of Julia, IPython, and several Julia packages, as explained on the IJulia github page, and functionality is evolving rapidly. We hope to have a more polished release soon.

Basic Julia interaction¶

Basic mathematical expressions work like you expect:

In [52]:

i = e
exp(i*pi)

Out[52]:

5113.984999751927

$\exp(i \pi)$

You can define variables, write loops, and execute arbitrary multiline code blocks. Here is an example of an alternating harmonic series $\sum_{n=1}^\infty \frac{(-1)^n}{n}$ from a Julia tutorial by Homer Reid:

In [26]:

Sum=0.0
for n = 1:2:10000
    Sum += 1/n - 1/(n+1)
end
Sum # an expression on the last line (if it doesn't end with ";") is printed as "Out"

Out[26]:

0.6930971830599458

Previous outputs can be referred to via Out[n], following the IPython, for example Out[2] for the result above. You can also use the shorthand _2, or _ for the previous result, as in IPython. Like in Matlab, ans can also be used to refer to the previous result, even if it was not printed (when the command ended with ;).

For example, the harmonic series above should be converging (slowly) to $\ln 2$ , and we can check this:

In [28]:

Out[26] - log(2)

Out[28]:

-4.9997499999454575e-5

In [29]:

println(math.e^128-1)

3.8877084059945687e55

In [ ]:

using PhysConsts
α.quantity

In [22]:

big((α.value)^-1)

In [23]:

big(log((α.value)^-1))

In [24]:

big(math.exp(699))

In [ ]:

import Pkg; Pkg.status()

In [26]:

using PhysConsts.DATA

In [27]:

c.quantity

Out[27]:

Speed of light in vacuum (c)
Value                         = 2.99792458e8 m s^-1
Standard uncertainty          = (exact)
Relative standard uncertainty = (exact)
Reference                     = CODATA 2019

In [29]:

a0.value

Out[29]:

Bohr radius (a0)
Value                         = 5.2917721067e-11 m
Standard uncertainty          = 1.2e-20 m
Relative standard uncertainty = 2.3e-10
Reference                     = CODATA 2019

In [31]:

G.value

Out[31]:

Newtonian constant of gravitation (Gg)
Value                         = 6.67408e-11 m^3 kg^-1 s^-2
Standard uncertainty          = 3.1e-15 m^3 kg^-1 s^-2
Relative standard uncertainty = 4.6e-5
Reference                     = CODATA 2019

In [33]:

g.quantity

Out[33]:

Proton mass (m_p)
Value                         = 1.672621898e-27 kg
Standard uncertainty          = 2.1e-35 kg
Relative standard uncertainty = 1.3e-8
Reference                     = CODATA 2019

Like Matlab or Scipy + Numpy, Julia has lots of mathematical functions and linear algebra built in. For example, we can define a $500\times500$ random matrix $R$ and form the positive-definite matrix $R^* R$ :

In [32]:

R = rand(50,50)
R' * R

Out[32]:

50×50 Array{Float64,2}:
 14.8401  11.0159   11.0398  12.1371  12.9914  …  10.0666   10.2719   11.5613
 11.0159  14.5508   10.6115  12.8797  13.3489      9.09651  11.5845   12.8904
 11.0398  10.6115   14.664   12.1591  12.6995     11.3192   11.2392   12.7697
 12.1371  12.8797   12.1591  18.0848  15.3832     10.9588   12.952    12.7643
 12.9914  13.3489   12.6995  15.3832  18.8978     11.3417   13.766    13.7376
 10.3582  12.1922   10.8784  12.5355  14.0756  …   9.75837  11.2291   12.7569
 11.403   11.4861   12.1963  12.413   13.4622     11.4271   12.3969   12.9923
 11.9348  11.3753   11.5956  13.6172  13.1759     10.7935   11.2115   12.2721
 12.2734  12.3439   11.7942  14.8864  14.2113     11.7685   12.4404   12.9881
 12.611   12.6692   12.6645  14.3057  15.212      12.4616   12.4764   13.661 
 12.46    10.9597   11.9727  12.0599  12.9428  …  11.0595   11.6294   13.2982
 10.7607  12.462    11.4551  12.9311  14.3571     10.629    12.2371   14.6529
 11.5589  12.314    12.3473  12.9978  13.8131     10.5067   12.1406   13.694 
  ⋮                                            ⋱                             
 11.3496  11.8048   11.2844  13.7931  13.7687     11.7626   12.937    12.6045
 11.0356  12.8355   11.1324  13.4014  13.5355      9.83385  11.9688   12.6278
 12.3802  14.0015   12.4311  13.7834  14.4353  …  12.0277   12.6047   14.6398
 11.4193  12.8167   12.8603  12.5936  14.013      10.8855   12.3162   14.082 
 11.9126  11.3802   10.747   12.2219  12.3567     10.434    10.8304   12.1167
 11.8378  12.2769   11.7172  14.8255  14.3169     11.4519   13.129    13.469 
 12.6271  12.8352   12.74    13.0265  15.1861     12.46     13.2996   15.0121
 12.1822  13.1224   11.2227  13.8452  14.8181  …   9.94071  12.3738   13.4795
 11.6848  10.9105   10.2393  11.041   12.3348      9.44363   9.89868  10.8772
 10.0666   9.09651  11.3192  10.9588  11.3417     14.6154   10.5976   10.1157
 10.2719  11.5845   11.2392  12.952   13.766      10.5976   16.6939   12.2948
 11.5613  12.8904   12.7697  12.7643  13.7376     10.1157   12.2948   18.5953

(Notice that, by default, only a portion of a large matrix is shown. You didn't really want to read $500^2 = 250,000$ numbers, did you?)

Standard output from Julia is also captured and sent to the IJulia notebook as you'd expect:

In [33]:

println("Hello world!\n")
println(STDERR, "Testing, some unicode output on stderr...\n")

Hello world!

Testing, some unicode output on stderr...

IJulia even captures output from external C libraries (and notice how easy it is using Julia's ccall intrinsic):

In [34]:

ccall(:printf, Cint, (Ptr{UInt8},), "Hello from C!!\n");

Hello from C!!

We can define functions, of course, and use them in later input cells:

In [50]:

using Unitful
using Measurements
using Cosmology
using PyCall

@pyimport math
@pyimport numpy as np
f(x) = np.power(np.log(2), (x / 4 ))

Out[50]:

f (generic function with 1 method)

In [51]:

using Unitful
using Measurements
using Cosmology
using PyCall
@pyimport math
@pyimport numpy as np
println(f(2))
f([1.1,1.2,2,3,5,8])

0.8325546111576977

Out[51]:

6-element Array{Float64,1}:
 0.904122
 0.895875
 0.832555
 0.75966 
 0.632458
 0.480453

Notice that the input above both printed an scalar to STDOUT and also returned a vector, the latter using Julia's ability to write polymorphic functions and built-in array operations.

On the other hand adding a string to a number is not defined (there is no + method defined for those types, although we could easily add one), and attempting to do so will throw an exception:

f("Hello?")¶

Interactive Plotting in IJulia¶

Below we'll show off some plotting using the excellent Gadfly package. The plots are heavily inspired by this blog post.

✨✨✨✨✨

NOTE: This will take a while to compile the first time you import packages. If you know a bit of Julia and have a good solution for how Binder can pre-compile packages before the user runs them, let us know in the repo2docker Julia issue here: https://github.com/jupyter/repo2docker/issues/23.

✨✨✨✨✨

In [ ]:

using RDatasets
sleep = dataset("lme4","sleepstudy")

In [ ]:

using Gadfly
plot(sleep, x = "Days", y = "Reaction", Geom.point)

In [ ]:

plot(sleep, x = "Reaction", Geom.density, color = "Subject", Scale.x_continuous(minvalue= 0, maxvalue= 500))

In [ ]:

plot(x = sleep[:Reaction], Geom.histogram(bincount = 30), Scale.x_continuous(minvalue = 200), color = sleep[:Days])

Multimedia display in IJulia¶

Like most programming languages, Julia has a built-in print(x) function for outputting an object x as text, and you can override the resulting text representation of a user-defined type by overloading Julia's show function. The next version of Julia, however, will extend this to a more general mechanism to display arbitrary multimedia representations of objects, as defined by standard MIME types. More specifically, the Julia multimedia I/O API provides:

A display(x) function requests the richest available multimedia display of a Julia object x (with a text/plain fallback).
Overloading writemime allows one to indicate arbitrary multimedia representations (keyed by standard MIME types) of user-defined types.
Multimedia-capable display backends may be registered by subclassing a generic Display type. IJulia provides one such backend which, thanks to the IPython notebook, is capable of displaying HTML, LaTeX, SVG, PNG, and JPEG media formats.

The last two points are critical, because they separate multimedia export (which is defined by functions associated with the originating Julia data) from multimedia display (defined by backends which know nothing about the source of the data).

Precisely these mechanism were used to create the inline PyPlot plots above. To start with, the simplest thing is to provide the MIME type of the data when you call display, which allows you to pass "raw" data in the corresponding format:

In [ ]:

display("text/html", """Hello <b>world</b> in <font color="red">HTML</font>!""")

However, it will be more common to attach this information to types, so that they display correctly automatically. For example, let's define a simple HTML type in Julia that contains a string and automatically displays as HTML (given an HTML-capable backend such as IJulia):

In [58]:

type HTML
   s::String
end
import Base.show
show(io::IO, MIME, x::HTML) = print(io, x.s)

Out[58]:

show (generic function with 428 methods)

Here, show is just a function that writes x in the corresponding format (text/html) to the I/O stream io. The @MIME is a bit of magic to allow Julia's multiple dispatch to automatically select the correct writemime function for a given MIME type (here "text/html") and object type (here HTML). We also needed an import statement in order to add new methods to an existing function from another module.

This show definition is all that we need to make any object of type HTML display automatically as HTML text in IJulia:

In [31]:

x = HTML("<ul> <li> Hello from a bulleted list! </ul>")

Out[31]:

Hello from a bulleted list!

In [ ]:

display(x)
#println(x)

In [ ]:

What's the purpose of Jupyt-nb ?¶

Reference for Continuous Improvement¶

Reference for offline visualization¶

ln137/ln(a/137)≈(2+135/de)2ln137/ln(a/137) ≈ (2 + 135/d_e)^2¶

exp(2e2+1/2)exp(2^{e^{2+1/2}}) ≈ exp(e2e+e2)exp(e^{2e} + e^2) = 8.820166056085355e102¶

exp(e2e+e2)exp(e^{2e} + e^2)¶

The economic number : eeee^{e^e}¶

Basic Julia interaction¶

f("Hello?")¶

Interactive Plotting in IJulia¶

Multimedia display in IJulia¶

$ln137/ln(a/137) ≈ (2 + 135/d_e)^2$ ¶

$exp(2^{e^{2+1/2}})$ ≈ $exp(e^{2e} + e^2)$ = 8.820166056085355e102¶

$exp(e^{2e} + e^2)$ ¶

The economic number : $e^{e^e}$ ¶