My Perspective

Courting to survive, and to learn: an active-inference reading of the courtship–survival trade-off A classic life-history problem asks how an organism with a finite life should divide effort between surviving and reproducing. My earlier mathematical model of courtship versus survival [1] solved a two-year version by maximizing lifetime fitness over mating effort m , giving the optimal effort. So the key idea is invest hard now when survival is poor or mates are abundant now; conserve when the future is bright.  Natural selection shapes organisms whose internal models encode adaptive prior preferences. Expected free energy minimization therefore provides a mechanistic process theory for achieving fitness rather than replacing evolutionary theory. Hence, I recast the same problem as active inference i.e. action selection by minimization of expected free energy (EFE) and show two things. First, the classic optimum is recovered almost exactly as the pragmatic (preference-satisfying) l...

I won a prize for my photo submission entitled "Hollow Mask Illusion" under "the 'why' of shapes" category (adult level) in the "Maths Inside 2023" photo competition that aims to popularize scientific thinking in the general public. My submission discusses why people do not perceive the obverse side of a mask as a concave face but as a regular face instead. This illusion is robust in general and everyone can be surprised at this trick on the brain unless they suffer from Schizophrenia. Here is my winning photo and the caption. This image shows the obverse side of a mask that has been painted to showcase a face. When people see the reverse side of a mask, they are usually unable to perceive it as concave or a hollow face, but instead see a convex regular face as seen in this image. This illusion is a classic example of how brain uses its top-down expectations to shape visual perception. The "Predictive coding account of perceptual inference...

In the 1968 book entitled "The Pythagorean Proposition" by Elisha Scott Loomis, which has documented 371 distinct proofs of Pythagoras theorem, it is claimed that there is no possible proof using trigonometry since all identities of trigonometry are proved using Pythagoras theorem and hence a trigonometric proof results in a circular logic. However this is not the case. In 2009 Jason Zimba has proved [1] Pythagoras theorem using trigonometry avoiding circular logic. His simple proof involves defining sine and cosine angles using similar right angled triangles and then presenting a geometric proof of sine and cosine of the subtraction angle formula and finally leading to the Pythagoras theorem. Reference: [1] Zimba, J., 2009. On the possibility of trigonometric proofs of the Pythagorean theorem. In Forum geometricorum (Vol. 9, pp. 1-4).  Link:  https://forumgeom.fau.edu/FG2009volume9/FG200925.pdf  

My friend Sooraj (Wistler) and I have developed a simple game. We recently launched it on Android titled "Arithmagic". We also have a web version of this game. The game is played on a virtual board consisting of numbered squares arranged in a rectangular fashion (akin to a matrix) where the numbers for each square are randomly chosen single digits from 0 to 9. At the start of every turn the player is given a target number. The task is to find 3 consecutive numbered squares either horizontal, vertical or oblique such that arithmetic operations as indicated (like addition, subtraction, multiplication) applied on these three chosen numbers result in the target number while following BODMAS principle of evaluation. The task can either be timed or untimed. The difficulty of the task is titrated based on time allowed or the scarcity of attaining the target number on the given board game. The goal is to earn as many points as possible awarded based on time to completion of task a...

In this post, I will try to come up with examples from set theory as well as generic examples to help you understand necessity and sufficiency conditions better. Case 1 : A is both necessary and sufficient condition for B Set Theory example: Two identical sets A & B. If an element belongs to B it implies it belongs to A (so A is necessary for B) and if an element belongs to A it implies it belongs to B. (so A is sufficient for B) Generic Example:

As I was thinking about PI, I felt may be I could compute its value from a limiting case. Could PI could be defined as the ratio of perimeter of a "cyclic equilateral polygon" to its "circum-diameter" under the limits of inifinite number of sides for this polygon? Let me explain.

"ದನ ಸಿಂಗ ಹುಲಿ ಹಕ್ಕಿ ಹಾವು ಮೀನ್ ಗಳಿಗೆಲ್ಲ ।  ಇನಿಸುಣಿಸು, ಬೆದೆ, ಬೆದರು - ಅಷ್ಟೆ ಜೀವಿತವು ॥  ಮನುಜನೆಂತನಿತರಿಂ ತೃಪ್ತಿವಡೆವನವಂಗೆ ।  ಕ್ಷಣಕ್ಷಣವು ಹೊಸ ಹಸಿವು - ಮಂಕುತಿಮ್ಮ ॥ ೨೦೯ ॥" Quoting D.V.Gundappa's Mankutimma Kagga (rigmaroles for a common man) #209, "Cow, lion, tiger, snake, fish and other animals have a simple definition of life. Struggling for food, securing a mate to have a healthy progeny, being scared about life (from predators or nature) - this pretty much completes their life and they are happy if they are able to do just that. But man is not like them. He is not happy just doing the same. His desires grow and change every second." Introduction In natural world, organisms strive to survive long enough to be able to pass on their genes to the next generation. Thus biologically speaking, every individual has two primary tasks - survival and procreation. Since natural selection would work against those individuals who do not strategize in such a manner, so it is pertine...

I have n functions f_i(x){i=1,...,n} which do not preserve the monotonic mapping order i.e. if x_1 < x_2, then in general, f_i(x_1) is not less than f_i(x_2) (for all i = 1,2,...n). So can I construct a function g mapping on fi's such that g(f1,f2,...fn) is monotonic? If yes, please elaborate on the method for the same. Even if g is a unimodal function, it should be fine. Put differently, for every value of x over an interval, I have the (n) outcomes of some physical process denoted by f_i (i = 1,2, ..., n). Now given such a set of f_i's corresponding to an unknown value of x (but known to lie within an interval), I have the task of estimating x. Since these f_i's are not monotonic in nature (no set pattern), I felt it would aid in this estimation task of mine if I could construct a monotonic function g using the f_i's. Please do convey your ideas.

This article is regarding the concept of Adhikamasa and Kshayamasa (intercalary months) used in some luni-solar calendars in India. Luni-solar calendars are in vogue since lunar months determine the high and low tides while the solar year determines the seasons. But the use of  luni-solar calendars create a unique problem. Since the lunar year (12 lunar months) lags behind the solar year (~365.25 days), this creates a need to keep the lunar year and the solar year in synchrony. Hence the need to use Adhika masa (extra month) and Kshaya masa (removed month)!  The reason for their asynchrony is explained next.

I recently came across this interesting article. And I wish to share it with you all. Here's a story about George Dantzig - the famed mathematician who's contributions to Operations Research and systems engineering have made him immortal. As a college student, George studied very hard and often late into the night. So late, that he overslept one morning, arriving 20 minutes late for Prof. Neyman's class. He quickly copied the two maths problems on the board, assuming they were the homework assignment. It took him several days to work through the two problems, but finally he had a breakthrough and dropped the homework on Neyman's desk the next day. Six weeks later, on a Sunday morning, George was awakened at 6 a.m. by his excited professor. Since George was late for class, he hadn't heard the professor announce that the two unsolvable equations on the board were mathematical mind-teasers that even Einstein hadn't been able to answer. But George Dantzig, working w...

Do you want to know How to calculate the day of any given date off hand? Then read on.. For any given date, there is a simple workable formula to calculate the corresponding day. Since for most of the times we would work with dates in 20th and 21st century, So let me just limit myself to explain a simple formula that would work for these 200 years! Method : Given any date between 1900 to 2099, to calculate the corresponding day off the hand, we need to extract the following numbers. Legend: DD - day in the month MM - month CC - century YY - year in the century LY - no. of leap years past since beginning of century Given Date: DD MM CCYY LY - YY / 4 (last 2 digits of the year divided by 4) -> Quotient only Y - YY mod 7 (last 2 digits of the year's remainder with 7) L - LY mod 7 (no. of leap years in this century -> remainder with 7) M - value in the string ( 033 614 625 035 ) corresponding to month D - DD mod 7 (day in the month remainder with 7) C - 0 if CC = 20th century; ...