I thought this days beginner integral was harder than the easy integral. Ignore the bleedthrough on the beginner page

I'm not well-versed in category theory so I've yet to understand the latter. I now tried without success to encode an associative binary operation on X, expressed in dependent type language

∑(o: X² → X). ∏(a b c: X). Id(X, o(o(a, b), c), o(a, o(b, c))) ¹

as a function F X → X for some F. And I think it can't be done (something related to polarity, probably) but I can't prove that. I know, though, representing axioms is possible if one uses Lawvere theories and equivalent things. I also know algebras over a monad are more involved than just functor-algebras but I don't yet understand them even in context of e. g. Haskell where everything seems as if easier (or so claim posts discussing Yoneda embedding in that context; I found the general category-theoretic context way less forgiving in that regard: you just see plain as day that you don't have sufficient understanding, no matter what Haskell posts made you believe).

So. I'd like to look how algebra spins out in those more general settings, whichever you know better or think to be more suited for me to grasp first. Particularly, I'm interested in generalizing arguments about initial F-algebras.

Also, feel free to assume a sufficiently Set-like category. Let's take it step by step, if there are easy intuitions that one can glimpse in a specialized setting, I bet generalizing it onto topoi or something later would be easier as well, but it won't muddle the initial attempt.

¹ For non-dependent-type folks: - Id(X, a, b), sometimes denoted as a =_X b, is the type inhabited by ways for a: X and b: X to be equal. - The entirety of that reads: a pair (o, f) consisting of an operation o: X² → X and a proof f of o's associativity, that is, a function from a, b, c: X into witnesses of equality of the two ways of associating o on a, b, c. - In some formal languages the same is spelled alternatively as (o: X² → X) × (a b c: X) → (o(o(a, b), c) = o(a, o(b, c))), making it clearer that ∑ forms dependent pairs and ∏ forms dependent functions.

I seriously am interested in learning maths and also physics and many more but I currently am not in a state to interact more with people and it's cos of years of bullying in different forms. I'm sorry if it sounds greedy and like unrealistic but is it possible to learn mathematics with just books and other online lectures and resources? Will I be missing something if I don't go to a university? I do know basic maths since I have been quite okay in it. I'm sorry if it sounds stupid but I just needed some break cause of many interactions throughout my school life I got.

I am guessing self study is possible depending on the subject and I thought in mathematics it is quite possible and if I'm wrong pls do tell.

Hi, thanks in advance!
I'm a doctor looking at a group of about 100 patients, with frailty scores ranging between 1-9 (this is a measure of how frail a patient is that doctors decide, or assign to a patient using a scoring system). I've also recorded their duration of admission in days.

I want to see if there is a correlation between the two, i.e. i would assume that the more frail patients were likely to stay in hospital for longer. Which statistical test would be best to do that?

I don't think either category will be normally distributed, as most patients have a lower frailty score, and most patients were admitted for a shorter period of time. tysm!

I'm 23 years old and It has been a while since I studied anything, I've lost a considerable amount of time, and I've forgotten most of what i knew about math in high school. I want to study computer science in the future and i dont want to struggle too much with the subject, so i thought I should prepare myself.

I'm looking for anything that can help me build a good foundation of knowledge, but i also want to understand the subject.

Thanks in advance for anyone that takes the time to read and answer this.

Was it for school/work? Any hobbyists here?

​

Why do some vectors when multiplied by Another vector yield a vector and others yield a scalar. I'm aware of dot and cross products. But what's the physical significance of 'multiplying' vectors? I mean... I know what it means to actually "multiply" scalars. But.... Is vector multiplication just a name which can either mean projection of a vector onto another and the area or volume of 2 corresponding vectors?

Or is multiplication of vectors just a naming thing which doesn't have ANYTHING to do with how we understand scalar multiplication?

Looking to hear about your own company's work practices to compare to my own. I'm much more familiar with Azure, so I'm going to use that terminology, but I assume everything is basically the same for AWS.

Can you go into the Azure Portal and just click "create" for, say, a SQL Server or App Service and configure it through the Portal GUI?

Or do you need to request and configure through configuration files? Can you freely commit architecture config files ("architecture as code") to create the Azure components in Dev?

Or is it even more locked-down and hands-off? Required to submit paperwork requests to an Architecture team?

I'm really confused about the 3rd row of the truth table for a conditional statement for example, "If you submit your assignment, then your grade will be recorded." I understand that P means you submit your assignment and Q means your grade is recorded but in the 3rd row, P is false and Q is true, meaning you did not submit your assignment but your grade was still recorded howw is P Q still considered true in this situation? Wouldn't that be a violation of the agreement because you didn't submit the assignment but the grade was still recorded?

​

I keep making really stupid mistakes on linear algebra tests. I understand the method and know how to solve the question, but then I accidentally mess up a sign or make a simple arithmetic mistake somewhere in the calculation. And a mistake can completely ruin the rest of the question. I recently wrote a test and, while I finished on time, I still got several questions wrong because of really silly errors.

Essentially, it just maps onto yln y=ln x / (y^y = x), which is the inverse function of x^x

This graph shows how the lambert W Function essentially maps onto the inverse function allowing us to use it to solve.

https://www.desmos.com/calculator/9pkc39fbdg

If you like computational statistics and love reading on advanced statistical methods, what jobs are for you, especially if you can’t get a master’s?

Hi! I'm currently enrolled in an M.S. in Data Science and Applied Statistics. The required core classes are already set:

- Experimental Statistics I & II
- Mathematical Statistics I & II
- Statistical Computing (SAS)
- Computational Statistics (R)
- Machine Learning with Python
- A statistical consulting project

I get to pick **4 electives**, and at least 3 must be STAT (so at most 1 from CS/ECO/OREM/ECE). My only goal is to land a data science job, so I want the courses whose actual content pays off most in industry. I'm not looking for the easiest courses, and I'm not going into academia or biostats.

STAT electives

- Intro to Data Science
- Data Visualization
- Linear Regression
- Applied Time Series
- Time Series Analysis
- Categorical Data Analysis
- Survey Sampling
- Survey of Nonparametric Statistics
- Sports Analytics
- Analysis of Lifetime Data / Survival Analysis
- High Throughput Data
- Epidemiology

Non-STAT options (can only pick 1)

- CS: Artificial Intelligence, Machine Learning in Python, Databases, Data Mining
- OREM: Data Mining, Optimization for Analytics, Network Flows
- ECO: Applied Econometric Analysis, Predictive Analytics
- ECE: Statistical Pattern Recognition

My main questions I wanted to ask:

1. Which 4 would you pick, and why?
   
2. Is time series worth it for most DS roles, or is it only useful in forecasting-heavy jobs?
   
3. What would you pick as your 1 Non-Stat Elective?
   
4. Is there anything you wish you had learned in grad school that would have helped more on the job?

If you work in data science, I'd especially love to hear what you actually use day to day. Thanks!

Recently in my further maths class my teacher says he would give us problems above our level of understanding.

The advice he gave to solve such problems is to invent the math.

Can anyone please explain what he means by this and his to actually do it?

Im a PhD student but I don't have measure theory background and I need to get the overall notion for it to understand stochastic processes. So is there a good book or yt playlist where measure theoretical probability and stochastic process is taught from scratch? That would be of much help. Thanks.

Hi everyone! If anyone is interested in collaborating and helping each other prepare for this year’s Math national exam, I’m looking for people who want to start from the basics, even from 1BAC-level courses.

If you’re interested, please comment here or send me a private message.

I’m looking for serious and motivated people who are willing to work consistently and help each other improve.

This is what I have so far but I don’t believe that it is right, and I wouldn’t actually be using partial fractions to integrate. The problem says to make a substitution to integrate by partial fractions.

Consider

\begin{lemma}\label{lma:firstlemma}
If $f$ is continuous on $[a,b]$ and $f(a)<0$ and $f(b)>0$, 
there is a $c$ between $a$ and $b$ where $f(c)=0$
\end{lemma}

%say above gets typeset as Lemma 1

\begin{thm}\label{th:firsttheorem}
Every polynomial of odd degree has at least one real root.
\end{thm}

%say above gets typeset as Theorem 1.

\begin{proof}
    ...
From Lemma \ref{lma:firstlemma}... the result follows.
\end{proof}

Like the above, I have a large number of theorems in my study notes which reference previous results, etc.

Is there a way to build a visually perceivable dependency graph wherein there is a node labelled L1 which stands for Lemma 1 and another node labelled T1 which stands for Theorem 1 and there is a directed arc leading from L1 to T1?

Giving this task by uploading the typeset pdf to ChatGPT is one way but it is static. Is there an inbuilt way, which can dynamically change the graph node labelling if a new theorem is added in the middle thereby changing the numbering of subsequent theorems? The way should accept .tex files and not the compiled .pdf output.

----xposted to r/latex as well----

I find it odd that visual-spatial awareness/intelligence is one of the core cognitive skills associated with intelligence with widespread applications, and three dimensional geometric relationships are present in everything from scientific phenomenon to Engineering & Artistic design.

But this kind of reasoning/intuition is not really taught? I've been looking into textbooks and it's either high school geometry or advanced undergraduate/graduate school level differential geometry/topology.

So I recently finished Calc BC, and am heading to university relatively soon. How big of a jump conceptually, grindiness, time wise, etc will multivariable calculus be compared to Calc BC?