If I were a Springer-Verlag Graduate Text in Mathematics, I would be Saunders Mac Lane’s Categories for the Working Mathematician. I provide an array of general ideas useful in a wide variety of fields. Starting from foundations, I illuminate the concepts of category, functor, natural transformation, and duality. I then turn to adjoint functors, which provide a description of universal constructions, an analysis of the representation of functors by sets of morphisms, and a means of manipulating direct and inverse limits.…

The adjective mental describes anything having to do with the mind, just as “physical” has to do with the body. Mental has to do with the intellect, the mind, or the brain. Source - Vocabulary.com The Question Does a person’s cognition accounts as a subset of his/her/their mental? If the answer is yes, then what is the need of the bellow statement? “There are mental and cognitive benefits to recovery.…

Exploring what cognition means and how it relates to human thought processes

Why measurement is important? Let’s say we set a goal. It just acts like a location pointer in the cognitive? map which is temporal as well as spatial? in nature. Now you basically need one thing i.e. some measurable quantity which have the following properties. Property 1: There exist a comparable function and the set of all the possible measurable quantity is a totally ordered set $S$. Property 2: If $a, b \in S$ and $a \leq b$ then we say that $b$ is an indicator that you are in the right direction.…

YOUR LOVE RELATIONSHIP. This is the measure of how happy you are in your current state of relationship—whether you’re single and loving it, in a relationship, or desiring one. YOUR FRIENDSHIPS. This is the measure of how strong a support network you have. Do you have at least five people who you know have your back and whom you love being around? YOUR ADVENTURES. How much time do you get to travel, experience the world, and do things that open you to new experiences and excitement?…

Idea Can there exist a turing machine which decides the halting problem i.e. given a Turing Machine $M$ and input $x$, would it halt and spit out output? Let’s consider the cases where the input space excludes all the possible Turing Machines and input ~ encoding of the same Turing Machine (i.e. the input which leads to Barber kind of paradox). Will such a Turing Machine exists?…

Quotes Following are some of quotes by Marcus Aurelius that I like: The first step: Don’t be anxious. Nature controls it all. And before long you’ll be no one, nowhere like Hadrian, like Augustus ---Marcus Aurelius Being a stoic doesn’t mean not feeling emotions, it means not letting yourself be controlled by them ---Marcus Aurelius Some words Twinge: A sudden, sharp localized pain Pragmatism: An approach that evaluates theories or beliefs in terms of the success of their practical application Future Ideas Developing a web app for curating quotes which I come accross.…

Introduction Consider a situation in which we want to calculate the minimum distance we can move in pre-determined amount of steps, let’s say $k$ for all the nodes in a graph. How can we go around with solving this efficiently? Bellman Ford Considering the point of $k$ steps, one can think of bellman ford which have an outer-most loop for iterations. And in each iteration - each edge is considered. Thus moving step by step with each iteration.…

Motivation Consider a given situation in which the number of edges in a given graph is greater than $n^2$ and hence it’s not viable to iterate over all these edges for dfs for $n > 10^5$. And we somehow want to deal with the connected components of this graph. Assume we are given the edges which are not present in the graph and they are of the order of $n$, i.…

This post is regarding a recent problem from Codeforces Round 716 Div2 which can be solved using a randomized approach. This is the first problem I solved using randomization approach, so it’s special. Problem Problem: Cut and Stick This problem requires to find out the existence of the super-most frequent element in a given range $[l, r]$ of an array. Definition: Super-most frequent element in a range $[l,r]$ is the element with frequency greater than $\lceil \frac{r-l+1}{2} \rceil$.…