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@mayusthings
Monday, 3rd April 2023
Quantum Notes - Time-Dependent Schrƶdinger Equation
Back at the notes this morning after a chilled weekend; Iām hoping to finish quantum today, as the quicker I make my way through notes and flashcards and onto proper practising, the better - fingers crossed for a productive day :)
šµ Deadwing - Porcupine Tree
Zero to the power of anything is zero. Anything to the zeroth power is one. So what is zero to the zeroth power?
The world's most cited mathematicians would seem to be in disagreement about the issue
I think the supermajority says 1, but a determined minority says 0. I agree with Wolfram Alpha and say you haven't given me enough information to say yet.
It really depends on which question you're actually asking. Obviously we "generally" treat exponentiation as an algorithm, but most mathematicians with interest in foundations will tell you that integer exponentiation is counting something.
From a combinatorial perspective, we say a^b is the number of ways you can make a choice from a options b times in a row. And normally there are no ways to make choices from zero things (and thus 0^1 or 0^5 is 0), but if the number of choices you need to make is zero then it's not a problem. How many ways can you make an impossible choice zero times? One: you don't make any choices.
Oddly enough, this logic extends deep into fundamentals. A set theorist, category theorist, algebraist, or foundations person will tell you that A^B is the set of functions from the set B to the set A. If B is a finite set with b elements and A is a finite set with a elements, then A^B has a^b elements. (Why? To determine a function from B to A we need to decide where the first element of B goes, and where the second goes, etc. So we have to make b choices from a options.)
But this logic and notation extends to places you weren't really thinking about it. We use R^3 to represent triples of real numbers. But we can also think of that as a function from {1,2,3} to R! The first coordinate is f(1), the second coordinate is f(2), and the third coordinate is f(3). And that means we can write things like 2^N for the set of binary sequences, or R^N for the set of real sequences.
And in this framework, we still have 0^0=1. In general 0^n is 0, because you can't have any functions from a non-empty set to an empty set. But there is one function from the empty set to the empty set: the empty function. (This sounds like bullshit but it's a real thing and this is unambiguously right once you get into rigorous foundations stuff.)
---
Okay, so why isn't there a total consensus? If you do analysis (read: formal, fancy calculus), it's not that clear cut. Analysts don't generally want to count things; they want to make things continuous. And the original question makes total sense in an analysis framework.
If you look at the function f(a) = a^0, then f(a) = 1 for any non-zero a. To make this function continuous, you set f(0) = 1. But if you look at the function g(b) = 0^b, then this is undefined for b<0, and is 0 if b>0. For continuity, you'd want to set g(b) = 0.
We can take a step more general and look at h(a,b) = a^b. There is no way to make this continuous at (0,0), so there's no unique definition for that point. Depending on what path you use to approach (0,0), the limit can be any non-negative real number (including zero) or infinity. So from an analysis perspective it really is deeply undefined.
Samajh mein kuch nahi aaya lekin padh k accha lagaāŗ
Omg this is so cool! Thank You sm to @chashniii for sending me this ^_^ It's so so nice to read such stuff where people actually explain their thought process so well!!!
Some group theory notes that I took today.
šSummer Studying Challenge š
August 8 2021: What books are you currently reading?
Yesterday I started reading 1984 by George Orwell.But I have also started two other books namely 'Men of Mathematics' by E.T.Bell and 'Music Of Primes' by Marcus du Sautoy.I have also decided to buy Anna karenina by Tolstoy after reading a post about it by @theclassicalmind.
š3rd August 2021 š
Summer Studying Challenge by @myhoneststudyblr
Q. What is your least favourite thing about the beach?
Probably getting sand in my hair lol
š· The picture attached is my bujo spread for the first week of August.
šCurrenly Reading : Men of Mathematics by E T Bell.
Hey guys! I wanted to make a challenge that felt achievable to folks while still giving that sweet sense of accomplishment. My mental health has been rough lately so I kind of designed it for myself as I try to find balance between taking care of myself and taking care of my goals.
The rules: Every day for 14 days, study and post a picture of what you worked on. It can be for two minutes or for hours. Doesnāt matterāyou did something! There are also mental health prompts for each day Iād love if you answered. They range from pretty light to more thoughtful. Do your best!
Iāll be tracking and reblogging from the tag #TwoWeekStudyChallenge. Please donāt worry about your pics not being aesthetic enough or that what youāre doing isnāt interesting enough. It is, I promise!
This is set up like the 100 Days of Productivity Challenge: no start/end dates. Itās a personal challenge thatās there when you need it!
Week One Prompts
Day 1: What self care did you practice today?
Day 2: What did you do that made you feel proud today?
Day 3: Have you experienced burnout from studying? How did you handle it?
Day 4: How can you improve on taking care of your mental health?
Day 5: What kind of support do you wish you received from your school/university?
Day 6: What popular self-care ritual does nothing for you?
Day 7: What small thing brings you the most comfort?
Week Two Prompts
Day 8: Whatās your comfort song?
Day 9: Do you think your long term goals are achievable and healthy? Why or why not?
Day 10: What subject brings you the most joy to study, either for school or on your own?
Day 11: Do you take mental health days? Do you think you should?
Day 12: Whatās your creative outlet? Whatās a new one youād like to try?
Day 13: What makes you feel inadequate? Would you think it made a friend inadequate?
Day 14: What makes you feel strong and capable?
Iām so tired today,Btw my august spread.
Happy August you guys ^_^
30th July 2020
šæWhat would be your perfect summer day? šæ
Wake up late,have a light breakfast,pack a good lunch and go to a nearby scenic water place with my family.My family includes my parents,my cousins,my uncles and aunts.Eat a ton of fruits,play a lot of games,make a truck load of memories and fall asleep on my mommy's shoulder in the car on the way back home.
30th July 2020
šæWhat would be your perfect summer day? šæ
Wake up late,have a light breakfast,pack a good lunch and go to a nearby scenic water place with my family.My family includes my parents,my cousins,my uncles and aunts.Eat a ton of fruits,play a lot of games,make a truck load of memories and fall asleep on my mommy's shoulder in the car on the way back home.
july 16, 2021
Hey everyone! For the past few days (and since I found out about energy management) Iāve actually been super productive and getting all my work done! Itās very satisfying tbh. And Iāve already started planning out my senior year again. Iām aiming for all Aās this semester! And itās gonna be difficult because all of my courses are either honor level, accelerated honor level or college level courses because of the school I go to (besides stuff like PE, student life courses, etc) but Iām confident that I can do it! :)
16th July - Do you have any summer traditions?
I usually travel out of the country each summer, but COVID -_-
Do you have any favourite theorem(s)? I'd love to know xx
oh okay i like this. and of course i have favorite theorems, mostly based on how useful they are. here you go :
fermat's theorem (for congruences)
lagrange's theorem
fundamental theorem of finite abelian groups
density theorem
fundamental theorem of calculus (the one which uses riemann integrals)
wilson's theorem (for congruences)
special mention : squeeze theorem (mainly cause of the name)
Heya! I finished chapter 12 of Analysis 1.It was I think the toughest chapter so far but it's not that tough that I'll loose my sleep over it.I also got the above book from Amazon today.The title is sexist but so is history š¤·š»āāļøI started from the last chapter and made some notes on it.I also revised chapter 8 today.Otherthan that it was a pretty chill day.I also answered an ask on Tumblr.I thought it was important to answer the question correctly and so I tried to the best of my ability š°
Now Nighty Nighty you guys
I hope you had a good day <3
Hello! I really like your blog and it somehow makes me feel good and safe, your notes are really pretty! When I look at them they seem so interesting and I wish to understand them but the problem is I haven't even started university (I have just finished high school) and I'm kind of scared thinking it would be too difficult for me to grasp. It is just so different from high school math but at the same much more fascinating. Do you perhaps have any advice on how to introduce myself to it, where to start? Or should I just wait for the classes to start and then study?Best wishes to you!! Thank you
Hey!
I just want you to know that mathematics is difficult at the University level but it is not more difficult than any other university course.It is just different.š
So how is mathematics different and why you shouldn't get discouraged by it?
There is an entire book written on it.It is called 'Alex's Adventures in Numberland' written by Alex Bellos.There is one particular page that I really like.It talks about how are brain perceives the world around us on a logarithmic scale but mathematics is linear and that is why we have to put the extra effort into it (To convert logarithmic to linear) but according to me this extra effort is far less than wading through hundreds of textbooks often contradicting each other that many other university courses require. Here is the cover page of the book I recommended if you would like to read it.
Apart from having a positive attitude towards it I think it'll be great if you could start studying it before your university begins.
So from where should you start?
If you know where you'll be going for your university studies then you can check if your university has uploaded their notes online and start accordingly.Some old students also usually post their notes,so see if you can get your hands on them.If these two options fail then there are two more.Oxford university has made all their mathematics notes avalilable.You can start with their Introduction to mathematics and complex numbers course (I'll put the link at the end).A student from Cambridge University named Dexter Chua also has made his notes available.You can start with Numbers and sets if you use his notes (Again I'll put the link at the end).
Oxford notes:
https://courses.maths.ox.ac.uk/overview/undergraduate
Dexter's notes:
https://dec41.user.srcf.net/notes/
You should be fine with these but the beginning steps into mathematics needs a little more help.You can use the book called as 'How to prove it:A structured approach' by Daniel J. Velleman.If you can read this with the first course that you are taking (your university notes or Oxbridge notes) then I think you'll enjoy the course more just like I did.Here is the image of the book
If you want to know more then I'm just a message away so feel free to ask your doubts.
And thank you for all your compliments hehe ^_^
Heya!!
So today I just revised the first four easy chapters of analysis 1 and then I started chapter 12 (Analysis 1).Chapter 12 is about tests for convergent series like the leibinz test for alternating series,comparison tests (there are two of them) and a lot more which I will be doing tomorrow.I am feeling very tired today.I didnāt even do much.I just had 2 hours of statistics exam and have been feeling tired ever since.Maybe I did not rest enough after the exam or maybe I just have to go and sleep now.
I did place an order for some multivitamin pills though.This is the first time Iāll be using the brand so letās just see how that goes and apart from that I telephoned two of my school teachers.Here in India itās guru Poornima (Basically Teacherās Day according to the Hindu calendar) and it was lovely talking to them.I really do miss them a lot.I also put a sheet mask on in the evening hoping that itāll make me feel better.It did but Iām still tired so Iām just gonna go sleep.
Aww look at you cutie reading everything hehe I hope your day went better^_^
See you tomorrow <3
Folder
Hi everyone, hereās the folder :)
It contains PDFs of A-level textbooks for:
OCR Physics AĀ
OCR Chemistry A
Edexcel Further Maths - Mechanics
Edexcel Further Maths - Core Pure 1 & 2
Edexcel Further Maths - Decision
Edexcel Maths - Pure 1 & 2
Edexcel Maths - Stats/Mechanics 1 & 2
Hope you find it useful!
july 21, 2021
Hello! Hereās a picture of some of the books in my bookshelf that I havenāt read yet :). Today is kind of ok, besides the stomach cramps Iām having ;-;. But Iām trying not to let them ruin my mood anyways :)
21st July - What is an unpopular opinion that you have about summer?
Sorry but summer is very overrated. And I donāt particularly enjoy baking in my own sweatĀ
Heya!!
Today I revised all the topics I covered in Analysis 1 so far and did chapter 8 and 9.
Chapter 8 is about the algebra of limits.In there I learnt how to write proofs using Īµ-N method.There were quite a number of cases (with examples!).Most of them were good and could be deduced easily from previous theorems and axioms but I did find 4 proofs a little challenging.I have noted them down and Iāll be discussing those with my prof soon.
Chapter 9 was about monotonic sequences.In there I learnt about the monotonic sequence theorem and the proof of this theorem is very cute.This theorem's proof is very intresting because it connects to the theorem where square root of 2 is proved as irrational but it is more general.So it patches all the gaps on the real number line quit nicely.There were some examples too which I liked and I also learnt the steps of proving a theorem when monotonicity is involved.
It was a pretty cool day.
I hope you had a good day too^_^
***
The Mayuriit Project
Will I make it? Stay tuned to find out!
206 days leftā¦
