An Unexpected Awakening

There are 3 topics that have been under my radar recently: an improved band-gap circuit design that uses curvature correction, stochastic calculus and it's application to finance - particularly continuous time stochastic processes, and a decent review of current mode control architectures. During my forays on stochastic calculus, in search of an intuitive explanation for inner products (the 'bra' and 'ket' operations/more general form of a dot product), I stumbled upon the YT channel 'MaththeBeautiful' and found an enlightening discussion on the involved concept below:

 

It is a captivating innovation of our perspective on length and how we see it from a geometric point of view. The speaker builds the argument that a set of operational characteristics (i.e. the inner product) has the power to define many forms of measurements (such as length) flexibly (and not the other way around).

After a few moments of reflection on the video, a vivid recollection led me to an epiphany. A concept quite vaguely and not satisfyingly unraveled in digital signal processing and computer architecture - the energy of a signal.

We describe the energy of a signal by Parseval's theorem:

(Snippet courtesy of Wikipedia)


The energy of a signal is intuitively captured through the cross-product terms of the Fourier transform. But how can we really conclude that it is the energy of the signal? What is the intuition behind?

These questions made Parseval's theorem a tad bit opaque to me for years, until I happened on the video above. Linear algebra, a concept I've taken for granted all too often seems to be of utmost importance to a lot of applied mathematics in engineering. The formula that yields from Parseval's theorem is simply the 'length' of the signal ( i.e. <x(t) , x(t)> ). With the dot product definition used in the YT video, we can say that the energy of a signal is the dot product, or more correctly, the inner product of the signal to itself.

Apparently, the inner product is a cosmopolitan tool to a lot of applications. Aside from signal processing, the inner product finds application too in stochastic calculus. Inner products sometimes help ease the complexity of proving theorems in stochastic calculus (or more importantly, proving that the stochastic process is an F-martingale) to apply the correct tools for a proper valuation of assets.

Moving on, I hope to find more 'epiphanies' or 'Aha!' moments from my dabbles in linear algebra and measure theory. Mathematics never fails to astonish me with its elegance and beauty, increasing my appreciation and admiration for it forevermore.



Making Things Easier for Others at Work

Just recently, I found this rant article on the web about having the worst customer/colleague ever. (Link below)
https://www.eeweb.com/profile/aubrey-kagan/articles/mr-mux-the-customer-from-hell

Here, the author Ms. Aubrey describes a difficult co-worker with a funny name (of which she rhymes to the word 'multiplexer' and calls him/her that for the rest of the article). She proceeds to elaborate  Mr. Mux's incompetent failures and oversights, who all the while remains stubborn and unreconstructed to the situation at hand.

At the end of the article, she ends with the question:
"Have you ever come into contact with a Mr. Mux in your professional life?"

... to which I would respond, "That question can be a double-edged sword."

Why? Because no matter the situation our co-worker can thrust us into, it is our indisputable obligation to finesse that situation toward success. For example, if the Android Ligma were to be released in 2030 containing a bug that crashed every mobile device at start-up, Mr. Mux's pink slip won't be enough to satiate the public's outcry over the damages.

Now I'm not saying we should just sit through the situation with a stolid resolve, but rather approach the conflict with the best of everyone's interest in mind. The slightest sign of a red flag should spur us into proactive (and not reactive) action. If the problem can be remedied early, the better. As the famous maxim goes - an ounce of prevention is worth more than a pound of cure.

At this point, I would like to bring up a principle of mine I hold dearly - and that is making things easier for others (not just for our customers but for our boss, colleagues, and heck even janitors too). We all live in a chaotic world and we do not know what other people are going through. Hence, it is my belief that the best course of action is to reserve a reasonable amount of patience and sacrifice for everyone (I mention reasonable as not to overstate my case). In addition, it's the most humane thing to do, and it puts a smile to everyone's faces. It sounds cheesy, yes, but I think it is surprisingly the habit that yields proven optimal results (from my personal experience).

So to answer the question of the previously mentioned article - Yes, I have met and dealt with many Mr. and Ms. Mux's even when I was a wee little elementary/grade school pupil (in class projects) but over the years I have learned to see these people not with harsh acerbic disdain but as a challenge to grow into better individuals of a team.