Introduction In an earlier earlier post we saw how to get started with ESP8266 on a NodeMCU board using Micropython. The setup required many steps including the installation of the driver, the bootloader. Following that, the setup required installation of separate applications for serial communications and code upload. Going through these steps can help us appreciate the various parts of the…
Introduction In this post we will demonstrate a minimal example of interfacing ESP8266 with an external webserver. The ESP-to-Webserver interfacing will allow us to send commands to the ESP using our computer or cellphone, with the webserver as a mediator. This in turn allows us to build autonomous systems (using our computer as the main device and the ESP as the edge device). It also lets us send…
Introduction [image] Microcontrollers sit between the abstract world of bytes and the real world objects you can see and touch. Programming microcontrollers is a way to connect these two worlds. The satisfaction of getting a sensor reading flashing on a cheap LCD is known only to those who have spent the hours fighting the electronics. Many low-cost (< $10) microcontrollers and development boards…
Introduction Can a Gaussian function be approximated with a set of simpler functions? This is the question we’re going to explore in the present article. What exactly is a ‘simpler function’ and why might we want such an approximation? Let us explore these questions first. To keep things simple, we will restrict ourselves to the simple one dimensional Gaussian function: \(f(x) = e^{-x^2}\). This…
The Fourier transform The Fourier transform tells us that any signal can be represented in time or frequency. Imagine we have a pure tone of 1000 Hz sounding with a strength of 1 decibel. We can represent this tone by tabulating its values at all times (in an infinitely large table). This is the time representation , also known as the time-domain signal and written as \(x(t)\). Alternatively, we…
Introduction Surely you’re joking, Mr Feynman! brought differentiation under the integral sign (DUI) from relative obscurity into folklore. Most of us get introduced to DUI in the context of evaluation of tricky definite integrals . But outside of this magical application, DUI is an important tool in the theory integral transforms . Integral transforms themselves are central to Physics and…
Introduction The Central Limit Theorem (CLT) states that the distribution of the average value of \(n\) random variables drawn from any distribution (with a finite mean \(\mu\) and a variance \(\sigma^2\)) approaches a normal distribution. In notation, if \(x_1, x_2, \ldots, x_n\) are drawn from a distribution, and \[\bar{x} = \frac{1}{n} \left(x_1 + x_2 + \cdots + x_n\right)\] then \[\bar{x} \sim…
Laplace’s method is a general technique for approximating functions of the form $g(x) = e^{M f(x)}$ around global maximum of $f(x)$. It may seem like we’re studying a rare special case of functions, but functions of this particular form appear in combinatorics and probability theory (and related areas in physics which make use of probability, such as quantum electrodynamics and quantum…
Solving problems by generalization Expanding the scope of a problem can sometimes be a crucial step in its solution. In mathematics, this pattern translates to expanding the definition of a function or an expression from positive to negative integers, from integers to reals or from real to complex numbers. These generalizations often lead to deep insights and form “bridges” between unconnected…
Optimization is all around us. Various physical entities are constantly solving some form of optimization problem: water running downhill (finding state of least energy), light rays trying traveling between two points (finding route of shortest time), or electric currents finding paths of least resistance. Optimization is also used extensively in industry. In explicit forms, it is used to maximize…