I m quite fluent doing these operations... But what is it m actually doing??

I mean, when we do dot product, we simply used the formula ab cosθ but, what does this quantity means??

I already tons of people saying, "dot product is the measure of how closely 2 vectors r, and cross product is just the opposite"

But I can't get the intuition, why does it matter and why do we have to care about how closely 2 vectors r?

Also, there r better ways... Let's say I have 2 vectors of length 2 and 6 unit with an angle of 60°

Now, by the defination the dot product should be 6 (261/2)

But, if I told u, "2 vector have dot product of 6", can u really tell how closely this 2 vectors r? No!

The same is true for cross product

Along with that, I can't get what closeness of 2 vectors have anything to do with the formula of work

W= f.s

Why is there a cross product over here!? I mean I get it, but what it represents in terms of closeness of 2 vectors?

And why is it a scalar quantity while cross product is a vector?

From where did the idea of cross and dot fundamentally came from???

And finally.. is it really related to closeness of a vectors or is just there for intuition?

Hello Friends,

I am teaching a class on Bayesian belief networks and relevant sampling techniques. I've always found this to be a pretty dry subject compared to others that we study, so to make it more fun I designed a video game to play with the concept. In brief, you are a paranormal investigator trying to determine if visiting aliens are hostile or friendly. To do this, you have a relatively complex (15 nodes and about 25 edges) BBN, and the first part of the game is to query the BBN to get a sense of when the aliens tend to visit the town. The second part is an investigation where you interview people who claim to have seen the aliens and describe their behavior as friendly or hostile. Your job is, using the insights you gained from the first step, to determine if the eye-witness report is credible or dubious, and your judgment on the aliens if determined by a majority vote, ie did most credible witnesses describe them as hostile or friendly?

My question is about defining credibility. I have two possible answers to this:

A witness is credible iff P(Aliens|evidence) > P(Aliens) - or in other words, the posterior probability given their account of events is greater than the prior probability of alien visitation.

OR

A witness is credible iff P(evidence|Aliens) > P(evidence|~Aliens) - relating the probability of their account to aliens being present or not being present.

These two conditions are clearly related by Bayes rule:

P(A|evidence) = P(evidence|Aliens)P(Aliens)/P(evidence) =

P(evidence|Aliens)P(Aliens)/(P(evidence|Aliens)*P(Aliens)+P(evidence|~Aliens)*P(~Aliens))

All the terms are there and related to each other, but it need not be the case that if one condition is met then the other is necessarily met.

One assumption about this is that we are trusting the evidence the NPC is giving us, but we doubt their claim that they actually saw the aliens. That assumption is fine for me. We also are not evaluating the probability that someone saw aliens given that they say they saw aliens, and that is also fine with me.

What do you think? Or could there be another way we can evaluate credibility?

(Tangent) The game in its more simple form (without the interview mechanic) was a real hit last year, really transforming one of the most boring lectures into one of the most fun ones. The students also learned a lot because they get to actually see and explore things that they previously only heard about - like we say rejection samplers are wasteful because most of their samples are not used. Ok, how many samples are wasted? We say Hamiltonian Monte Carlo samplers are extremely expensive compared to other approaches - ok, how long do they take to run on a graph like this? With algorithms like these, getting to actually explore them and see them at scale is key, and I think that actually using these objects and algorithms does a lot for learning.

I know this is not an appropriate place to ask this, but please here me out. I am planning to start my mathematics studies (again). I have studied basic introductory mathematics, and I have no problems grasping the basic concepts of Linear Algebra, Single-Variable Calculus, Probability and Statistics, Computation and Algorithms, and so on. But I don't have *in-depth* knowledge of any of these. The reason is simple, while I was going through college, I had severe health issues and I only studied enough to get a First Class (>60%) with specialization in theoretical chemistry, which I chose due to peer influence, but I always dreamed I would become a great mathematician.

here's to forgotten dreams!

Anyway, now that I have (somewhat) gotten a handle on my health, I think I am ready to begin my studies again, and I think it might also be greatly productive in me regaining my health.

Now, I know there is MIT OCW and other great courses out there, but I have always been a rather inwardly person, finding solace in textbooks rather than video courses, which seem to drain me out. I have read rigorous textbooks like Apostol, Feller, Ross, Strang, etc. but I think some of these books are *too challenging* (explained later). I am looking for a good understanding of the subject in a concise, easy-going manner, where I can actually solve the exercises. Sorry, I have OCD and not being able to solve exercises piles up and haunts me in my sleep xD I am not looking for school textbooks which don't delve into the finer points making the textbooks rather *drab*. I am looking for something that

• is easy going,
• also develops critical thinking and not just problem solving,
• is not too open ended (else I may just wander off), and
• embraces rigor.

I know I said concise, and that would be helpful, but I don't mind reading large texts or spending a lot of time (I expect 2-3 years spent in this endeavor). But I don't mean Thomas' Calculus (2000 pages without the epsilon-delta definition), I am past that (but not past Apostol's Calculus, as I still struggle with it). There is fine grey area where I feel somewhat comfortable and that is what I am looking for.

How do I begin? Can someone help me gather resources and create a solid plan? Your guidance, opinions on the matter, and personal advice are just as welcome. Thanks in advance :)

I have spent a bit thinking about the problem of meshing topological skeletons and I came up with a solution I kinda like. So I am sharing here in case other people are interested. This is perhaps a bit too applied for most people here. But I think that the relationship between the dual polytope and the meshing structure I cam up with might be interesting to some of you.

https://gitlab.com/dryad1/documentation/-/blob/master/src/math_blog/Parametric%20Polytopology/parametric_polytopology.pdf?ref_type=heads

I just finished the aforementioned book and I enjoyed it. Excluding the sections on non-European mathematics, which were outdated and quite xenophobic, I thought it was generally easy to parse and quite informative. After finishing it, I’m curious if there are any good books out there that have a narrower scope, but provide more information about a specific period. Particularly I’m interested in mathematics from 1700-present. Thank you in advamce

I'm currently living in Chile, I'm an inmigrant studying mathematics ingeneering, and I want to learn PDE theory and that sort of stuff. The thing is that I'm really afraid about not finding proper jobs.

I've seem that academy is really competitive, specialy in South America couse there are not to much options. I want to know, how is to find a job beeing a mathematician? It's true that the only two options is machine learning and teaching? at least in the begining? How is about to emigrate to US or some other developed country with job oportunities doing reserch?

I am a new faculty at a university and have been given the task of coaching our Putnam team. I wasn't big into the Putnam exam when I was a student, so I feel a bit clueless. Besides telling students to work on practice problems, what are things I could organize / suggest for the students?

I see this word coming up a lot but I don't really know what it is. I've read a few equivalent definitions but they're all given by ugly formulas that's made it hard for me to appreciate them. I think the cleanest definition I've seen so far is in terms of exterior algebras and wedge products but the bigger picture is still unclear to me.

What exactly is a Pfaffian, why do we care about it, and why is it important?

A young man lives in Manhattan near a subway express station. He is dating two women: one in Brooklyn; one in the Bronx. To visit the woman in Brooklyn he takes a train on the downtown side of the platform; to visit the woman in the Bronx he takes a train on the uptown side of the same platform. Since he likes both women equally well, he simply takes the first train that comes along. In this way, he lets chance determine whether he rides to the Bronx or to Brooklyn. The young man reaches the subway platform at a random moment each Saturday afternoon. Brooklyn and Bronx trains arrive at the station equally often—every 10 minutes. Yet for some obscure reason he finds himself spending most of his time with the woman in Brooklyn: in fact, on the average, he goes there nine times out of 10. Can you decide why the odds so heavily favor Brooklyn?

This Martin Gardner puzzle was originally published in the February 1957 issue of Scientific American.

Find the solution: https://www.scientificamerican.com/game/math-puzzle-subway-conundrum/

Scientific American has weekly math puzzles! We’ll be posting some of them this week to get a sense for what the math enthusiasts on this subreddit find engaging. In the meantime, enjoy our whole collection! https://www.scientificamerican.com/games/math-puzzles/ 

Posted with moderator permission.

This recurring thread will be for general discussion on whatever math-related topics you have been or will be working on this week. This can be anything, including:

• math-related arts and crafts,
• what you've been learning in class,
• books/papers you're reading,
• preparing for a conference,
• giving a talk.

All types and levels of mathematics are welcomed!

If you are asking for advice on choosing classes or career prospects, please go to the most recent Career & Education Questions thread.

I'm curious in what progress in mathematics consists of.

Is it about creating an ever higher tower of abstraction? Is it about inventing new concepts that make what was once hard to achieve now possible? Is it about discovering unusual interesting properties of mathematical forms we've already created? Or something else...?

Any individual case studies or examples of how you think this process unfolds would be super useful.

Would love your personal thoughts or recommendations to books / articles on the topic.

There are lots of resources on the internet that explian various derivations of faulhaber's formula and the bernouli numbers and so on. But I haven't seen any one of them use eulers formula for the difference of nth powers (https://www.jstor.org/stable/2320064) to do so, which would result in a simple derivation, why is it so?

I love math olympiad- as a student I have pored hours into this subject, loving it for the creativity and logic needed to solve each problem.

When I heard about AlphaGeom that was referred to early 2024, I was shocked- AI has developed to the stage that it can solve questions with international olympiad level of difficulty.

Then came an article in July, praising AI for being able to achieve a high silver medal in the IMO.

At that point, I was just coping- I thought "Oh, it solved the Algebra, Number theory, Geometry questions by blindly bashing and trying many options, but its unable to solve the combinatorics questions as it is bad at logic. Humans will forever beat AI in terms of logical reasoning." It was true that this model solved all the questions in the IMO except the two combinatorics questions.

But I was wrong. In late 2024, OpenAI developed a new AI chatbot model o1 that, on average, solved 11/15 questions in the AIME (A math competition in the US). That's already better than me: and most of those problems are combinatorics questions.

Of course, right now these models aren't that good in combinatorics yet, but it will be able to surpass even most IMO medalists in combinatorics, in the coming years.

If AI models already beat most humans in the area of logical reasoning, whats the point of studying so much math olympiad if an AI will destroy you in it?

Why even study math in the first place then, if an AI would be much more effective than us? I'm losing motivation to study because of this...

I'll preface with I don't study differential equations and have at best a scattered understanding of parts of the theory.

When teaching or studying intro DE's, we pretty universally cover the Laplace transform as a method of solving constant coefficient linear IVPs. Some courses will also go over power series solutions to equations with nonconstant coefficients and, if they're lucky, possibly the Method of Frobenius.

Here's what I'm curious about: The motivation and ideas leading to the development of the Laplace transform itself are almost never taught. Things like the historical study of various integral forms and the extension of power series to a continuous indexing variable.

Is there any well-developed study of solutions to DEs where, instead of a power series solution, we look for a solution in the form of an integral transform?

I tried working out a few possibilities, but it seems to fail for various reasons depending on the form of the differential operator and even the form of the inhomogeneous term. For example, if we take something like a second order operator with polynomial coefficients and some forcing term g,

y''-2xy'+x^(2)y = g(x), y(0)=a, y'(0)=b

we can guess a solution of the form y=∫_0^∞ f(t)x^(t) dt where f is an unknown function. This would be a continuum-indexed analogue of a power series solution. After substituting this into the DE, we can do some simplifying calculations and write the left side as the Laplace transform of some polynomial multiple of f. Using the properties of ℒ, we can recast the original DE as a new DE whose solution is the Laplace transform of this unknown function f.

What seems to happen in some surprisingly simple cases is that this simply leads nowhere. It seems to be the case that if the function g is not chosen fairly carefully, then the equation expressing g as a Laplace transform of f simply has no solution. The issue is that the function g(e^(-s)) must tend towards 0 as s approaches ∞ in order to be in the range of ℒ and this simply is not the case for many reasonable choices of g.

So what gives? Why is it that a power series solution to the above equation is perfectly viable, but this integral transform solution appears not to be? And is there a better guess for a transform that will work? Could we perhaps try something like a "basis" of delta functions? I'd really like to know more about this sort of thing if it's out there.

Hey everyone (im 19yo), I’ve always been someone who didn’t like math at all. I used to find it confusing, and honestly, I was pretty bad at it. But for some reason, all of a sudden, I feel this urge to understand math on a deeper level. Along with math, I’ve also started feeling interested in physics and philosophy fields I never really cared about before.

The problem is, even with this new curiosity, I’m struggling to stay motivated. I’m not sure where to start, and it’s a bit overwhelming since I don’t have a strong foundation in math. Do you have any advice on how I can dive into these subjects in a way that builds a solid understanding and keeps me engaged? Any tips for overcoming that mental block and finding joy in learning math would be amazing. Thanks in advance!

Hi everyone!

Do you have any tips for working with LaTeX? I’m a master’s student and have been using it for a few years, but I still find it pretty exhausting. For instance, yesterday I completed an assignment in about an hour, but it took almost two more hours to type it up in LaTeX, mainly because I constantly loose focus of what I was writing.

Any advice would be greatly appreciated!

Hi there! I am a statistics post graduate with an interest in cool problems and beautiful solutions.

I have seen many recreational math books and resources(magazines like Cambridge The Mathematical Gazette) however i was wondering if something same exists within the statistics domain?

It's frustrating that such an exciting tool is not available to us

I'm currently and undergraduate student doing a joint biology-math major. I'm taking my second class on modeling in biology, and it seems like everything we have learned has seemed very pointless. Every discrete model we looked at was just 2 pages of algebra that ended with "real life testing showed that this model was not satisfactory" or "The system either approaches 0 or approaches the carrying capacity defined by these constants" or "now that you've done a bunch of algebra and got the eigenvalues you can raise this matrix to an arbitrary power to show that your population will approach infinity". All the non-discrete models just involve taking the integral and raising e to some power and it somehow works out and gives you a line which it approaches or doesn't approach. Either that or we are just doing poisson distributions with nucleotides instead of other variables.

Doesn't help that the textbooks we are using are all like 30-40 years old, but this stuff just doesn't seem useful. I'm wondering when this stuff actually gets interesting and what real life applications of population biology looks life?

Due to nerve damage in my hand, it makes it very difficult to grip a pen for extended periods of time. I'm looking to get back into math and eventually go back to uni for a math degree. However, given this problem, and the extreme amount of writing which is required to study math, it's a challenge that I have had difficulty finding a solution for outside of TeX-styled markdown.

Is anyone aware of a more intuitive math typing software which would allow me to to type something like:

2x^2 + 3x + 4

or something similar, and get the equivalent TeX-formatted output in real time so I can study on my computer as opposed to on paper or by typing TeX for each simplification of an equation that I'm performing?

Edit: just for reference, I’ve been using Jupyter notebooks and MathJax but I end up losing my train of thought when having to look up certain syntax to get things to look right. Hence why I thought I would ask, and thank you all for your suggestions! Much appreciated 🙏

This might be a niche question - but I remember around >10 years ago, I frequently visited a website made by Prof. Zachary Tseng who had a lot of notes on differential equations (exercises, explanations, breakdown of different subtopics). I enjoyed them so much that even way into my academic career, I ended up constantly going back to that site to refresh my knowledge on diff eqns. The website had a super easy user interface with wonderful printability. (silly me to not download everything back then)

However, I recently went to the site again but found out that it has been removed by the university he was teaching at with no forwarding address. Does anyone know how can I find that again? Is my only option to contact him on his email and ask him? I just wanted to know if anyone else has ever used the site, or knows what I am talking about?

This is the link that I had bookmarked for all these years but now shows nothing https://www.personal.psu.edu/personal-410.shtml

I made a python program (I got bored) that finds consecutive numbers with the same amount of unique factors. Almost every 'group' of consecutive numbers it outputs (where there are 4 or more consecutive numbers) seem to all have 6 unique factors (I'm excluding the number itself here, but this doesn't really matter). Is there a reason why?

Code used and output of code (up to 300,000) can be found here: https://github.com/matt-xiao/factorFinder/

Is anyone here familiar with the Berggren Tree, which is a complete ternary tree of primitive Pythagorean triples, discovered by Berggren in 1934? I find it quite fascinating, and I'm currently trying to write a math paper on them, in particular, their relationship to the Stern-Brocot tree, which is a complete binary tree. I hope I'm not duplicating any work that may have been previously done, so if anyone here knows anything about this or can suggest any good references, I'd be very grateful!

I'm teaching Linear Algebra for the second time this year. (I teach at a special high school for exceptionally gifted youngsters). This year I committed to getting to the SVD by the end of the semester, and we will be introducing it next week.

As often occurs, I am finding that in needing to find a way to explain things to my students, I've found better ways to explain things to myself. This is the way I plan to arrive at the idea of singular vectors, and I haven't ever quite seen it shown this way before:

https://preview.redd.it/hqk6csjs1zzd1.png?width=1712&format=png&auto=webp&s=0ccb7a8bc485d235462a11d7db7a072dbc9a29f2

Evidently, the "suggestions" lead us to see that Av_i and Av_j have remained orthogonal after transformation by A. We can then re-define the u's to be the resulting orthonormal basis for the column-space of A, and get U \Sigma = AV. From there, it is easy to show that the sigmas are the squareroots of the eigenvalues of A^(T)A and it all falls into place.

For me, this is the way that SVD should be shown to students. Any comments or further suggestions for my approach? Any different approaches that helped SVD "click" for you?

For non-chaotic systems, you can use work-precision diagrams. But with chaotic systems, trajectories diverge exponentially so this approach doesn't work.

I know you can measure statistical quantities instead (mean energy, etc.) but looking for a practical reference/book that walks through the details - how to compute reference values, what quantities to measure, how long to run simulations, etc. More interested in numerical implementation than theoretical analysis.

Anyone have good recommendations that cover this well?

I'm a Master's student and would see myself as an algebraist (at least, I'm interested in algebraic number theory, commutative algebra, algebraic geometry, that stuff). But I always avoided logic and some set-theoretic problems (e.g., is this statement provable without assuming Zorn's lemma?): these questions seem so abstract that I don't want to wrap my head around it and they seem not to be "real math", but "meta-math". Another reason for avoiding logic was mainly due to Logicomix (a really good graphic novel), whose subtext makes the claim that logicians become mad, and I don't want to get mad.

Hence the title: Why should I care about logic? Or at least an introduction to logic?

I know of some very technical and almost absurd results from (real) algebraic geometry which rely on logic, e.g., Lefschetz principle, Tarski-Seidenberg theorem, Krivine-Stengle Positivstellensatz, and some topics on real closed fields in logical nature. Why should I study the proofs?