The Beginner Programmer

One of the perks of knowing a programming language is that you can build your own tools and applications. Depending on what you need, it may even be a fast process since you usually do not need to write production grade code and a detailed documentation (although it might still be helpful in the future).

I’ve got used to read news stuff on Reddit, however, it sometimes can be a bit time consuming since it tends to keep you wandering through every and each rabbit hole that pops up. This is fine if you are commuting and have just some spare time to spend on browsing the web but sometimes I just need a quick glance at what’s new and relevant to my interests.

In order to automate this search process, I’ve written a bot that takes as input a list of subreddits, a list of keywords and flags and browses each subreddit looking for the given keywords.

If a keyword is found inside either the body or in the title of a post which has been submitted in one of the selected subreddits, the post title and the links are either printed in the console or saved in a file (in this case the file name must be supplied when starting the search).

The bot is written using praw.

Recently I’ve been getting familiar with AutoCAD and at the same time I’m trying to improve my Python skills. Odd mix, huh?!

While trying to improve my Python skills I thought I could exercise myself on automating boring tasks. I remembered that a few years ago I was given a boring job which involved printing a lot of drawings directly in PDF format from Autocad. I was too lazy to print the drawings one by one and I knew the process could be automated. At the time I put together a Python script that did the job fine, but it was a bit messy. I thought I could make an improved version.

When nice APIs are not available, such as in the case of AutoCAD (at least that was the case a few years ago, nowdays things may have changed), using Pyautogui may help in the task of automating boring tasks.

More than 2 years ago I wrote a short post on Taylor series. The post featured a simple script that took a single variable function (a sine in the example), printed out the Taylor expansion up to the nth term and plotted the approximation along with the original function. As you can see on the right on the “Popular posts” bar, that post is one of the most popular and I’m told it appears among the first results on Google.

The script I wrote originally was a bit clunky, and there surely was room for improvement. Last week I received an email from a reader, Josh, who sent me an improved version of the original post.

Short circuits are one of the most common failures that can happen when dealing with electrical circuits.

Short circuits can be accidental, think of a tree branch leaning onto a high voltage powerline or due to the breakdown of the isolating material (this is often the case as it gets older and loses its isolating property). Whatever the cause, short circuits are, for sure, an enemy of your electrical systems mainly because the following effects:

Electrical transmission systems are something we all take for granted. They work in a reliable manner ensuring high quality of service and as few minutes lost per year as possible.

Low voltage lines and most of medium voltage lines are radial lines, that is they are like a branch of a tree with a unique power supply location. Radial lines are relatively easy to work with, you can solve a radial line problem (ie you can get currents and voltages) by applying Boucherot’s theorem to each section of the line. This is a good news for a Python enthusiast as myself, since repetitive tasks lends themselves to be automated with programming.

Since Boucherot’s theorem uses the absolute values of electrical quantities, it is useful when you are working with AC lines and you would like to know the magnitude (rms) of currents and voltages while at the same time you do not really care about the phase differences. The rms values are used, for instance, when you need to choose the protection systems to install and what kind of electrical wires to use.

Suppose you are given the following three phase balanced radial line:

You can think of C1, C2 and C3 as industrial motors or any other kind of three phased balanced loads.

Recently, a friend went skydiving and, me being me, the first thing I could think about was making some physical considerations on his adventure :)

If you think of a falling object, at first you would think of it as falling with a constant acceleration of g. That is, you would neglect air drag. However, if you think of it, air drag is not exactly neglectable when describing the fall of a skydiver. If you neglect air drag, you would get an ever increasing speed which is not at all the case.

Let’s make some physical considerations:

We’ll assume that the only two forces acting on the skydiver are the force of gravity and the air drag. This is the resulting free body diagram:

 

Yesterday I found an “old” script I wrote during a morning in the last semester. I remember being a little bored and interested in the concept of Q-learning. That was about the time Alpha-Go had beaten the world champion of Go and by reading here and there I found out that a bit of Q-learning mixed with deep learning might have been involved.

I’ve been experimenting for more than two months with Tensorflow, and while I find it a bit more “low level” if compared to other libraries for machine learning, I like it and hopefully I am getting better at using it. During the learning process I found some minor “obstacles” so I decided to write a short tutorial on how to use this amazing deep learning library.

Some time ago I uploaded a short script on modelling pressure drop in a circular diameter pipe. While at the time I felt I did a good job, I knew there sure was room for improvement. After having attended a (very tough) course on fluid dynamics and applied physics, I feel a lot more confident on this topic and therefore I would like to share a little model I built for a project I am working on.

Since the beginning of this summer I have been practicing a lot with Scikit-learn to improve my knowledge of Machine Learning both on theory and practice. Last week I also tried to tackle some of the Kaggle competitions although they are really tough if you wish to get into the top 50 best scores. Perhaps I will post something about the experience in the future.

Scikit-learn is a (almost) ready to use package for Machine Learning in Python. It is so very user friendly and in some cases not that much coding around is needed to achieve interesting results such as in the case of the cars dataset.

The cars dataset, from the UCI Machine Learning Repository, is a collection of about 1700 entries of cars each with 6 features that can be easily recognized by the name (buying, maint, doors, persons, lug_boot, safety). Check the dataset description for more detailed information. The feature to be predicted is “class” and the possible values are unacc,acc,good,v-good. Most of the features are categorical, therefore they need to be encoded into numbers. Pandas is great for quick features encoding.

Please make sure to read the disclaimer at the bottom of the page before continuing reading.

Plain vanilla call and put european options are one of the simplest financial derivatives existing. A european call (put) option is essentially a contract which by paying a fee gives you the right to buy (sell) a stock at a predetermined price, the strike price, at a future date. This kind of contracts was originally intended as an insurance tool for companies to fix the selling price of their goods and hedge the price risk.

Options have some interesting features, for starters, if you are the buyer of an option the payoff is potentially unlimited and the loss is limited to the option price as you can see by the payoff diagram below

of course the reverse is true for the seller (huge downside and limited upside).

A hidden Markov model is a statistical model which builds upon the concept of a Markov chain.

The idea behind the model is simple: imagine your system can be modeled as a Markov chain and the signals emitted by the system depend only on the current state of the system. If the states of the system are not visible and what you can observe are only the emitted signals, then this is a Hidden Markov model.

While writing the scripts for the past articles I thought it might be fun to implement the 2D version of the heat and wave equations and then plot the results on a 3D graph.
As for the wave equation, Wolfram has a great page which describes the problem and explains the solution carefully describing each parameter. Here is a little animation I made using the solution they described



The code for this example is available to be downloaded here.
As far as the heat equation is concerned, I opted for a classical example where you have a hot spot in the middle of a square membrane and given an initial temperature, the heat will flow away as expected. Note that in the heat equation animation the colorbar does not change as time goes on. This can be edited by moving the colorbar into the animation function.



download the code for the heat equation.

On my youtube channel you can find many other animations of the wave equation.

Imagine you have an ideal string of length L and would like to find an equation that describes the oscillation of the string. Assuming the string is fixed at its ends and starts its motion in a known position f(x) the simplest assumption one can make is that the acceleration of each piece of the string is somewhat proportional to the curvature of the string as such:



We can express the considerations above in the following way:



The equation above is a partial differential equation (PDE) called the wave equation and can be used to model different phenomena such as vibrating strings and propagating waves.The constant term C has dimensions of m/s and can be interpreted as the wave speed.
It turns out that the problem above has the following general solution



The thing that strikes me about this equation is how powerful the solution is. To think about it, any function that has the argument x-ct or x+ct or a combination of both is a solution to the wave equation. This means that we can model a lot of different waves!  Furthermore, as you could probably spot, the general solution is a combination of a wave travelling to the left and one travelling to the right.
Assuming let’s try the following solution



By implementing the equation in Python for a string of length 2pi and of speed 1 m/s we obtain the following animation:




The range of animation you can run is infinite. I tried out a couple of solutions:

Shortening the string, L=pi

Click here to download the code for the above video.

Trying out f(x-ct) + g(x+ct)= cos(x-ct)**3 + cos(x+ct)**3

Click here to download the code for the above video.

The solution


yields the following:



Click here to download the code for the above video.

Note how the resulting wave (cyan) is the sum of two cosine waves travelling in opposite directions.

What happens if you input the f(x-ct) term only and set g = 0? Basically what you get is a single travelling wave. The same happens if f = 0 and g = g(x+ct).


Click here to download the code for the above video.

Hope this was entertaining and helpful!

RC and RL are one of the most basics examples of electric circuits and yet they are very rich in content. Let’s examine each one carefully

Here is the schematics of an RL circuit

The components used in an RL circuit are a DC voltage source (V1) a resistor (R1) and an inductor (L1). The inductor in the schematics is used to represent physical characteristic of the circuit. If I had to describe what the inductor represents I would say it represents the ‘electrical inertia’ of the circuit. As currents flows into the circuit it generates a magnetic field, that change in the magnetic field causes a change in the flux of the field concatenated to the circuit, this in turn, by the Faraday-Neumann-Lenz law generates a voltage in the circuit that is opposite to the voltage that is generating the magnetic field (V1). This is the reason because the current in the circuit will not jump immediately to its full value of V/R given by Ohm’s law. The current will eventually reach the value V/R after an infinite amount of time, but for practical purposes we can consider it at that level after some time has passed (namely 4 times tau = L/R should be sufficient).

It so happens that, given the materials and the geometry, the inductance coefficients of the circuit I am modelling is 0.0229 H. Note that inductance depends only on these two parameters (materials and geometry). I calculated the coefficient using FEMM since the analytical calculation was not that easy to do. You can download FEMM here: http://www.femm.info/wiki/HomePage . This software is useful for many electromagnetic simulations and fairly user-friendly.

What we would expect is that the current will obey the following differential equation given by Ohm’s law at each point in time:

The equation above yields the following solution

Now, the same phenomenon (reversed) happens when you open the circuit, the inductor opposes the change in flux by generating a voltage that tries to keep up the shrinking flux of the magnetic field through the circuit. That means that, if you were to switch on and off the current at particular intervals you would obtain this behaviour:

As you can see, at time 0 the circuit is closed and the current is reaching its theoretical value, after 4 times tau (time constant of the circuit) the circuit is opened again and the current exponentially decays to zero. By knowing tau, one could find the frequency at which the switch should click in order to maintain this behaviour. This process yields the on and off cycle (in green). Where the green function hits the x axis the switch clicks and opens/closes the circuit.

Here below you can find the script I used to produce this graph and model the circuit.

If you are not satisfied with this crude simulation, perhaps you could use LTspiceIV which is a great free simulation software for electrical and electronic circuits. I used it to produce the schematics above and to simulate the behaviour of the current in the first 40ms after startup. Below is the LTspice simulation graph

As you can see by plotting the current through the resistor in LTspice, it seems our calculation were right!

Now let’s move on to the RC circuit, here it is:

Aside from the voltage source, the RC circuit is composed of a capacitor and a resistor, we therefore assume that self-inductance is negligible. When we close the circuit, there is no inductance here so the current will just jump to the V/R value and since the capacitor is charging up and building a voltage, we can expect the current at the resistor to drop as times goes by. The differential equation we have now is the following:

By solving it, one finds:

Therefore

The current through the resistor drops exponentially, depending on a time constant tau = 1/RC. This behaviour is indeed what we find by running the simulation in Python and LTspice

Python code used for the RC circuit:

Apparently, it looks like I’m all focused on PDEs at this time of the year. There’s a good reason though! I find PDEs extremely fascinating in that they allow you to model complex behaviours of objects such as waves propagating and strings vibrating and so on. If not for their beautiful mathematical content you must love them for their applications in physics!
Take the transport equation, imagine you’d like to model the behaviour of an ocean wave. Well, in an ideal world, once generated, our wave would go on forever at a constant speed, say k. In order to find the function that represents our wave at each point at every time t, one should solve the following problem



It turns out solutions are all in the form of:



For those of you already familiar with the wave equation, this should look extremely familiar. It is indeed the equation of a wave travelling towards the positive x at speed k
If we choose a Gaussian function for instance, then we would obtain the following:



and by running the same code we used for the heat equation, here is what we get:


Since the code is pretty much the same as the heat equation I will not post it here. However, it is downloadable via dropbox by clicking here if you’d like to check it out.
Now a question should arise. Our wave is an ideal wave, it does not take into account friction and if you think of an ocean wave, it dissolves itself quickly as it meets the beach. Can we model this with the transport equation? Sure! We can use the transport equation with exponential decay. The problem we need to solve now looks a little different:



where gamma is a constant which relates to the decay rate. The general solution now looks something like this



Note that it decays and tends to zero as time tends to infinity and that it is still a wave travelling ‘to the right’. If we then use the same Gaussian as before, we obtain a much more realistic wave. Here is a short clip generated using the code at the bottom of the page (the animation is slower since I set dt=0.01 for a better resolution).



You can play with the parameters I set and see how the behaviour of the wave changes!

While writing the code for the previous post I slightly modified the code in order to add 2 ‘peaks of heat’. Now, by looking at what I wrote I am not quite sure that the solution is consistent (physically speaking). At first glance we can see that there are three ‘heat spots’. If you imagine of heating the rod in two spots and cooling it in between, the situation is the following:



By setting L = 3*pi and making some other small changes we obtain:



Physically we could interpret the result as follows:
the heat flows spontaneously towards areas where the temperature is lower, therefore the temperature should go up in the middle of the rod (since it has been cooled down and has a lower temperature compared its surroundings) and go down in the sides. Assuming that the left and right boundaries of the rod are kept at constant temperature through an ideal thermostat.


Why not  checking if this solution is mathematically and physically consistent?

By making some assumptions, I am going to simulate the flow of heat through an ideal rod.
Suppose you have a cylindrical rod whose ends are maintained at a fixed temperature and is heated at a certain x for a certain interval of time. Suppose that the temperature in each section with infinitesimal width dx is uniform so that we can describe the temperature in the rod using a function of only x and t.


Mathematically speaking, problem we are now facing is the following:



where k is a constant called thermal diffusivity and is different according to the different materials. By using the method of separation of variables, we can find the solution we need and by applying the initial conditions we find a particular solution for f(x) = sin(x) and L = pi
Our solution looks something like this:

Now we only need to evaluate our function at each x and t. Remember that if u(x,y) is differentiable, then:



holds. We can throw out the last term and approximate our function using the above relation since partial derivatives of u must exists and we can easily get them. Thinking about it, the second term is useless too, since we are not moving along the x axis, therefore we are left with the following:



By using matplotlib and the animation function I generated this short clip:


Here is the python implementation of the solution and the code used to graph the solution evolving with time.


Hope this was useful!

Magnetism and magnetism related phenomena are fascinating almost for everyone, in fact, I remember being a intrigued by the interaction between magnets since I was a little kid. On the other side, Python is great for plotting vector fields, although now that I am using also Matlab I found out most functions in numpy and matlab are similar to those used in Matlab (I don’t know who got inspired by who though).

Here is a small sketch of a vector field. Remember the Biot-Savart law for a straight infinitely long wire, which (assuming the current is flowing upwards) looks something like this:

Below is a 3d representation of the vector field and the Python code used to generate it. The tricky part (if I’m allowed to say so) is that you need to convert the value of the vector field from cylindrical coordinates into cartesian coordinates in order to plot it. Perhaps there’s a way to plot a vector field using also spherical and cylindrical coordinates however I still do not know how to do that.

Hope this was interesting! :)

Hi everyone! Remember that in the last article I wrote that you can use the FFT to clean a signal from background noise? Well here is an example of signal filtering.

We start by generating a signal and then add some random noise using the random number generator in numpy. Here is the noisy signal:

Now, our signal is made up of two main frequencies: 20 and 30 Hz while the rest is mainly background noise. You can check this in the FFT of the signal:

Suppose we’d like to isolate the 20 Hz bit, that means we’d have to set to 0 each value in the Spectrum which is greater or lower than 20 Hz. Let’s block out everything outside 18 and 22 Hz and then apply the inverse FTT. Here is the signal we found (blue) compared to the exact signal we wanted to find (red). At the bottom of the graph there is the error between the two signals, notice that it is not that high, (at worst we missed 0.08, however on average we missed less).

Here is the python code I used to make this.

Have fun!