Projects of Academic Interests

This article will take a look, at Infinite Series, with a special emphasis towards the interesting conundrum arising out of the Ramanujan Summation & Zeta function regularization. Now to begin with, as a prologue... The Summation / Sum of an Infinite Series is defined / exits : if the Partial Sums of that Infinite Series either approaches a finite value (thus resulting in a Convergent series), or the Partial Sums of that Infinite Series goes to ± infinity (resulting in a Divergent series). Two quick examples whould elaborate, further solidify the meaning of above statements ... Example 1: Take the series :  1/2 + 1/4 + 1/8  + 1/16 + .... + 1/2 n Now the sum of above series can be shown to approach 1, as n approaches infinity. One can refer to the many articles / videos on the internet, that clearly illustrates geometrically, as to why the summation of above series approaches 1. (Hint: 1/2 of a square + 1/2 of the remaining square + 1/2 of that remainder + etc etc = full s...

Ok, where do we start...  How about with a small intro... Now one school of thought in mathematics is of the view / advocates that there are multiple theoretical / conceptual   infinities , and thus that some theoretical /   conceptual infinities are greater / larger than some other theoretical / conceptual infinities ... ( aka ∞ * ∞ > ∞) This article will discuss a few points associated with this debate. Point no 1: Realise that what we are talking here / addressing here, is the theoretical / conceptual infinity (or the mathematical concept of infinity), and not regarding infinity we come across in applied physics or engineering.  Thus for example, for all practical purposes in optics, we can consider the sun to be a light source at infinity ( ∞ ). The reason being : the angle between two light rays emanating from our sun, when measured on earth is negligibly small (literarily = 0). Aka the light rays are parralel, to each other for all practical purposes. BUT...

Sypnosis : The seed / root of this hypothesis lies with an attempt to providing a possible explanation to the Quantum Entanglement phenomenon (or in Einstein's words : Spooky Actions at a Distance), whilst at the same time not violating the Rules of Causality. 𐤎* Hypotethis  ( Ξ* Hypotethis )   (Take 1 / Revision 1.0) 𐤎* Hypotethis Postulate I: Einstein's Laws of Relativity, Lorentz  Transformations, Maxwels Equations, Neutonian Mechanics etc. etc. all applies to the Realms & Phenomenons of the / confined to the / bounded by / to the Four Dimensional Space Time [xyz-t] .  Thus the statement that nothing can move faster than the speed of light c , is a valid statement, for all phenomenon inside the confines of the Four Dimensional Space Time [xyz-t]. 𐤎* Hypotethis Postulate II:, There could be phenomenon in nature that falls beyond the confines / not bounded by the Four Dimensional Space Time [xyz-t].  For example certain phenomenon that occurs / tak...

Sypnosis: In the field of science(s), it becomes possible to describe / define phenomenon, via multitudes of theorems. Thus for example, the nature / characteristics of gravity can be defined via Neutonian Mechanics or using Einsteins General Theory of Relativity etc. Thus in Newtonian Mechanics the force of Gravity whould be as follows: F  = G (M 1 . M 2 ) / R 2 Whereas In General Relativity Einstein describes Gravity as the effect of bending of Space Time. g* Hypothesis   (Take 1 / Revision 1.0) The g* hypothesis presented in this article, postulates that photons  develops a mass component in the presence of matter / mass (or in the viscinity of / via interaction with the  gravitons present in a dielectric medium or near a massed object). Thus as Einstein famously and correctly predicted that the path of photons / light whould bend in the presence of the gravitational field of a Star / Galaxy (or the curvature of space time near a Star / Galaxy leading to gravitat...