Mathysteria

We have all been familiar with the 'sine' graph in our 10th-grade math. Nice and simple to plot, with no complications. However, a spin-off version of this graph exists, laying the foundations for topological spaces. Let S = {(x, sin 1/x): 0<x<=1}, then, S' ={(x, sin 1/x): 0<x<=1}U { 0 × [ − 1 , 1 ]}  has the capacity to oscillate infinitely in a finite domain! This occurs as 1/x becomes larger as x approaches zero.  This curve is clearly connected on the real line; however, it is not path-connected. That means if we want to go from a certain point to (0,0), we can surely walk in its direction on the curve, but we would still be a long way from the origin because of infinite oscillations near it. It tries to settle down at the origin, but keeps spiraling into existential jitters instead. Even though sin(1/x) is connected, it is not locally connected, as we cannot find connected open sets on the y-axis (recall S is the union of S with {0 x [-1,1]}). Consider ma...

Following our last expedition of  Turing Machine, let us now see it apply to DNA computing.  The information in the DNA is represented using the four-character genetic alphabet- A [adenine], G [guanine], C [cytosine], and T [thymine]. These four are known as bases and are linked through deoxyribose. This connection between the bases follows a direction such that the lock-and-key principle is achieved -  (A)(T) and (C)(G) This means Adenine (A) pairs with Thymine (T), and Cytosine (C) pairs with Guanine (G). These bases make DNA a computational medium. Drawing the analogy from traditional computers to DNA computing, we see that while the former process formation sequentially, DNA computing allows for parallelism, the ability to perform many computations simultaneously by leveraging the vast number of DNA molecules. This significantly speeds up the process.  DNA computing uses biochemical reactions and DNA molecules instead of silicon-based computing like conven...