Dissecting The Riemann Zeta Function

1 mainThe Riemann Zeta function is an extremely important function in mathematics and physics. It is intimately related to very deep results surrounding the prime numbers. Now why would we want to care about prime numbers? Well, the entire concept of web security is built around prime numbers. Most of the algorithms for banking security, cryptography, networking, communication, etc are constructed using these prime numbers and the related theorems. The reason we do this is because of the inherently sporadic nature of prime numbers. You never know where the next one is going to appear on the number line! So what does that have to do with the Reimann zeta function? If prime numbers are random, what’s the point of looking into them?   Continue reading

Random Number Generators In Programming

If you have fiddled around enough with C/C++/Objective-C, you must have noticed that if you use rand() function on its own, it gives the exact same numbers every time. Now how is that possible? Isn’t rand() function supposed to generated completely random numbers in a given range? The reason for this has something to do with the seed value. If you use the same seed value every time, then you will get the same numbers.   Continue reading

Derandomization Of RANSAC

mainLet’s say you are a clothes designer and you want to design a pair of jeans. Since you are new to all this, you go out and collect a bunch of measurements from people to see how to design your jeans as far as sizing is concerned. One aspect of this project would be to see how the height of a person relates to the size of the jeans you are designing. From the measurements you took from those people, you notice a certain pattern that relates height of a person to the overall size of the jeans. Now you generalize this pattern and say that for a given height, a particular size is recommended. To deduce the pattern, you just took a bunch of points and drew a line through them so that it is close to all those points. Pretty simple right! What if there are a few points that are way off from all the other points? Would you consider them while deducing your pattern? You will probably discard them because they are outliers. This was a small sample set, so you could notice these outliers manually. What if there were a million points?   Continue reading

Reimann Hypothesis And Its Connection To Cryptography

Over the centuries, mathematicians have been involved in solving some of most complex problems. But what is the motivation behind that? The pursuit of truth! But The Clay Mathematics Institute thought that there should be a little more than that. So to celebrate mathematics in the new millennium, they established seven Millennium Prize Problems. The prize money for each problem is one million dollars. That’s pretty exciting! These were some of the most difficult problems over which many mathematicians were racking their brains. Reimann Hypothesis is one of them. The interesting thing about this particular problem is that it has far reaching consequences in the field of modern cryptography and internet security. Now how can an obscure and complex mathematical problem affect cryptography and internet security?   Continue reading

The Butterfly Effect

This blog post is a continuation of my previous post on Chaos Theory. Although it is not required for you to read that post to understand this post, it would be better if you glance through it once. All of us have heard about the Butterfly Effect. It is one of the very famous examples given in the field of chaos theory. I should also give credit to the movie “The Butterfly Effect” for popularizing this term. So what exactly is butterfly effect? Is it just a theory? Where does it happen in real life?   Continue reading

Chaos Theory

Chaos Theory is a mathematical sub-discipline that attempts to explain the fact that complex and unpredictable results can and will occur in systems that are sensitive to their initial conditions. Some common examples of systems that chaos theory helped understand are earth’s weather system, the behavior of water boiling on a stove, migratory patterns of birds, or the spread of vegetation across a continent. The Butterfly Effect is one of more famous examples of chaos theory. I have discussed more about it here. Chaos occurs in nature and it manifests itself in various forms. Chaos-based graphics show up all the time, wherever flocks of little space ships sweep across the movie screen in highly complex ways, or whenever amazing landscapes are displayed in some dramatic movie scene. It is used a lot in movies to generate obscure background using computer-generated chaos art. So what exactly is chaos? How does it work?   Continue reading

Probabilistic Randomness Of Stochasticity

Do you see what I did with the title there? Anyway, you must have heard the term ‘probability’ being used around you. People use it in different contexts and in different forms – “What is the probability that Spain will win the next world cup?” or “I will probably finish reading the book by midnight” or “It’s quite probable that she won’t return until tomorrow”. When people talk about probability as a mathematical concept, all they think of is the percentage chance of something happening. But is that all there is to it? If that is the case, then why did they have to dedicate an entire branch of study to this? Probability theory is much more than just calculating the likeliness of something happening. It’s used almost everywhere, by almost everyone, for almost everything. Surprised? Well let’s find out then. Continue reading