In the spirit of Barry Simon's five volume series: "For a real number $a$, we will use the terms positive and strictly positive for $a\ge 0$ and $a > 0$, respectively. It is not so much that we find nonnegative bad, but the phrase "monotone nondecreasing" for $x > y \implies f(x) \ge f(y)$ is downright confusing..."
i tended not to have a lot of input on papers when i was asked to referee them, but a common issue i noted was someone using a term like "positive" and not being clear about what they meant and it maybe complicating a definition or adding a corner case depending on what it meant.
and my note would be, would be great to see a definition or definitional remark this.
i hate the game practiced by some mathematicians, including those on the continent, where, if you can make some series of logical inferences where they can only mean one thing if the result is true, then they don't need to say what they mean.
math is not a guessing game and not one of those logic puzzles where you figure out who did what by crossing out x's in a grid.
there was a guy i used to sometimes work with who, whenever you pointed out an opportunity for clarification, would launch into some 5 minute explanation of why no clarification was necessary. like a logical proof that he'd won at the game of exposition.
@XanderHenderson Sorry to bother again, and maybe being picky, but when you write $|f(x)| < C |g(x)|$ you exclude the possibility of $|f(x)| = C |g(x)|$ and thus $f \in O(g)$ according to the definition in the .pdf, no?
he was insufferable. he used to contribute to math.SE and nit picked a number of my posts without knowing i was me. he seems to have found other pursuits.
On the other hand, by the least upper bound property of the real numbers, if there exists $D$ such that $|f(x)| < D g(x)$ for all appropriate $x$, then there is $D'$ such that $|f(x)| \le D' |g(x)|$. (Just take $D' = D$; or, if you want something sharp, take $D' = \inf\{ D : |f(x)| < D |g(x)|\}$.)
The point is that throughout most of analysis, the distinction between $<$ and $\le$ is irrelevant, because it all gets smoothed out in the limit.
i don't think he was intentionally being rude, he was just wired to behave in a way that most people found rude. and i am generally accepting of a wide array of personality and communication styles but sometimes he pushed my personal line.
in my postdoc we had prospective grad students visit, and one of our profs began talking to an admitee about potential research areas, and when the admitee said a few things suggesting that algebraic geometry wasn't her focus, began subtly correcting her, like "what you just described in that five word summary isn't algebraic geometry. algebraic geometry is very different from that." i was stunned, thinking, what are we trying to do at this admitted students event.
some people get into math because they like keeping track of details and correcting other people and maybe to some extent being known as being good at something that a lot of other people think is hard.
and not for the real reason to get into math, which is the insane amount of money you get paid to study it.
one time they were surveying grad students about the level of compensation for teaching. i was called into the chair's office because i had more than average teaching experience. how are you doing? do you get by?
the literal answer to that question was yes. i think i profited, i don't know, $200-500 a year. i was making out like a bandit.
i understood the goal of the exercise was to target the teaching pay at exactly whatever would be needed to keep teachers alive and teaching.
i remember some ra/tas whining about pay when i was in grad school and attended some sort of unionish meeting. i asked is this about the $10/mo discrepancy? yes, indeed, it is a travesty! oh, i see, what about the non resident tuition payments that foreign students need to make? oh, screw them.
there's a mathematician i don't like very much, for only the reason that when the grad student union was trying to negotiate for a dental plan, he argued in print in a weird leaflet that he left in our mailboxes that we were all privileged to be where we were and our parents could cover the slack.
i think a lot of people internalize the concept that 'foreign students have everything paid for them by their governments,' which is not true, but might seem closer to true than false because of what some governments do do for their students that the USA doesn't.
the flip side of that is also a very colonialist and insulting assumption that someone not from the US should be destitute. i heard someone say something once that was well intentioned but suggestive of sympathy for the downtrodden, using X as an example.
i said, i don't think you know how many servants X's family has in his home country. which was true. the idea that there might be gobs of money outside of the united states is unfathomable to some people.
Every non abelian group of order $6$ is isomorphic to $\displaystyle S_{3}$ (the symmetry group of order $6$).
I tried to prove it but got stuck mid-way.
Let $\displaystyle G$ be a non-abelian group of order $6$. In $\displaystyle G,$ not all elements (non-identity) have order 2 because then the ...
@Koro it is sort of. my son (17) graduated yesterday but had a completely different maths experience from my daughter (20) who attended the same school, so i do not want to generalise
I've slowly become disillusioned by, for lack of a better word, the "promise" of academia to its students. It was bound to happen after leaving my faculty position.
my daughter's was classical, and had my coxeter out so to speak to help her. my son other other hand had a 9th grade teacher who gave impossible problems (among other issues): math.stackexchange.com/questions/2816411/looking-for-solution-of-a-9th-graders-problem
(Nevada requires that students take four years of math, but many students were not on a calculus-bound path, so they had to come up with something for those who didn't want/need precalculus or calculus).
in a lot of regions of the US, there might be a theoretical opportunity to offer calculus, but someone with the subject matter experience is not there. which means either someone is just randomly brought in and told to read out of the book (they just need to stay one lesson ahead of the students!) or it's not available.
I have a student enrolled in my calculus class right now. He came to office hours on Thursday and asked if he should drop the class (he is a high school student taking the class for dual enrollment credit, and the pace is... intimidating... to say the least).
I am asking this Leslie, copper and Xander because I remember once watching a physics lecture online by Walter Levin if I recall correctly and one student asked what is sin theta.
i sometimes bristle at jokes about americans being ignorant and not speaking other languages. there are places in america where the county doesn't have a fluent speaker of another language. it's a big place. i can't get in my car and drive to germany and find a ton of germans.
@XanderHenderson i had met with the teacher earlier in the year to discuss some other issues and it was the first time in a while that i came close to popping someone in the nose.
IMO I wish I didn't have to waste time going through the calc sequence and wished I could've just started off in analysis. I felt like I was going through the motions for 1-2 years until linear algebra and then taking analysis afterward.
if you ask what the bare minimum is, in terms of what does every high school in the country require, it is low. it might just be some algebra and geometry without sines of angles in it.
My response is to (1) make no changes or admission of error in the moment ("Just do your best, y'all," (2) remove the question from the exam while grading, and (3) apologize profusely when returning the exams.
koro state to state and even city to city in terms of what people have access to. on the order of 10,000-20,000 slightly and sometimes widely different sets of rules.
The Mathematics Common Core is an attempt by mathematics educators to articulate something which could act as the scaffolding for a standardized curriculum, but every jurisdiction can implement it as they see fit (or ignore it entirely).
my sister had an enormous amount of trouble graduating high school. geometry was a requirement and she couldn't pass it. her teacher just let her graduate.
xander that is a good point about common core. there's nobody imposing it from the top down. it's kind of an opt-in, do it as you see it kind of model.
koro, as an attorney, it's not clear to me that our federal government would have the authority to tell the states what to do in the realm of education.
it can fiddle with funding and that obviously can fiddle with a lot of other things. but i don't think it could say, here's the textbook everybody must use for math or you can't graduate high school.
there's also a political element of this. many people, i would say, largely wealthier people, are suspicious of public education in the US. the local control that people in their own communities have over what goes on in their schools is maybe a big part of what some people see as making public education, as a concept, legitimate.
that maybe also reflects our decentralized method of education. to address 50+ sets of standards, you might very well want to be sure that you have 50+ sections in the textbook.
@Thorgott Because we can only say that aH=H if and only if a$\in $H where H is a subgroup of G. So $a<b>\ne <b>$ we can only say this much. But the answer goes one step ahead and says a<b>=G\<b>. This is what I don't understand.
in large states there is some degree of oversight over this, in small states you literally have enterprising textbook authors codifying their textbooks into the state standards.
@leslietownes This comes from the ESSA, which was renewed during the Obama administration. I don't believe there has been a renewal since then. https://www.congress.gov/bill/114th-congress/senate-bill/1177/text
You can CTRL + F on "Mathematics" to read all of the text
I don't see anything addressing mathematics teaching standards in particular, but it's worth reading
skimming the TOC it looks like a classic example of using certain controls over money in an attempt to influence structure, but not directly mandating structure.
@Koro If you hover over the message, a triangle will appear on the left. Click on this, and note the "permalink" link in the box which pops up. That permalink will have a numerical string near the end (probably something like 58319xxx). This is the comments ID, which I believe is assigned sequentially, across all SE chat, since the dawn of time in 2009 or so.
here's a funny alternative argument: if there is only one element of order 2, it is central. the order 2 subgroup it generates is necessarily the entire center. the center is a normal subgroup and the quotient has order 3, so is necessarily cyclic. however, the quotient of a group by its center cannot be cyclic and non-trivial. contradiction.
Continuing my work through Dummit & Foote's "Abstract Algebra", 3.1.36 asks the following (which is exactly the same as exercise 5 in this related MSE answer):
Prove that if $G/Z(G)$ is cyclic, then $G$ is abelian. [If $G/Z(G)$ is cyclic with generator $xZ(G)$, show that every element of $G$ ...
i really like rose's book, in dover reprint. there's a bit of a learning curve, i think the exercise above is on page 2, or something. but it's got great exercises.
Back when I taught, I had written my own notes to teach a class for 4 semesters because every textbook was abysmal. My colleagues asked me why didn't I bother to publish it.
1) there was nothing novel about it and everyone in industry knows that material 2) why in the world would you want students paying $100+ for code they could easily Google
my favorite book on linear algebra? "Linear Algebra," of course.
math might actually be less confusing if copyright could be asserted in titles.
clarinetist another side of that is if it's not going to be in every classroom in the country the money involved for you is relatively low. bringing the focus to the only important thing, which is money.
Thus more work for wages that would be less than I would be getting in retail.
Someone tried to justify that to me by saying that I could be famous and thus get people to enroll in my class (enrollment was a struggle, thus cutting my adjunct pay)
"you could be famous" is the story of the 21st century.
if you won't do it for anything else, do it for attention. get an A in this.
so much of academia is premised on the idea that attention or involvement is currency.
they might make my wife chair of her department next year. she's trying any way out of that. it's just more work, no upside, people only get mad at you.
but you're chair of something. don't you want to be chair of something? all eyes will be on you!
Being an adjunct for two years and working with administration on data work for two years has given me quite a negative outlook on academia. Have you seen the Spring 2021 enrollment figures from the US National Center for Education Statistics?
The narrative that was going around was that people would be lacking the "college experience" at the 4-years, so the 2-year schools would see increased enrollment
Public 2-years saw a 9.5% drop year over year
whereas, for undergrad enrollment, public 4-years saw a 1.9% drop year over year
That's what I suspect. Though I think it's just delaying the inevitable - job searching has been tough all around, as I've been in touch with former students
i knew a guy in grad school who worked at radio shack (maybe dating myself a bit here) throughout grad school and then he just kept doing that instead of taking a next academic step.
my wife worked as a fitness instructor at a YMCA throughout her phd education. i don't think any of her advisors had any experience in this kind of world, where you're also doing this other work. maybe some of them did.
she could go back to that now and maybe not earn that much less than she's getting from being a professor of something.
I mean, what's the point of doing anything else when the only clear aim of an education is to prepare people to get their PhDs and certify them as researchers?
I say this as a former faculty member. I haven't been able to justify why else the education system is the way it is.
I've heard that academics will talk poorly of people who are pursuing non-academic, "lesser" interests
it was a lot of characterization of alternate routes as opting out, quitting, failing at something. and yes if you haven't internalized academia it's harder to understand
my wife is on a short list for a non academic job now, because she is under the impression that she may have failed at something by having a tenured position in an academic department.
i think this varies enormously from department to department, but broadly, anything super low is going to raise eyebrows anywhere, and anything that's just static in the mix, who cares.
Anything else they nitpick beyond that is extremely subject-specific and department-specific
For example, you are not going to get into a good stats school if you don't have excellent grades in multivariable calc, real analysis, and linear algebra. I know they definitely didn't care about my abstract algebra grades.
my worst grade in college was a C+ in linear algebra. i later got a phd, in what you might describe as linear algebra, from the same department that employed the guy who gave me the C+. i think they understood that guy. a different department may have looked at that differently.
it's very hard, if you're looking across a group of candidates, to focus on a candidate who has a sort of repeated pattern of lower grades in various things, vs. someone who doesn't. even if the someone who doesn't went to a "worse school" (quotes meaning i don't endorse the concept).
I am asking my friend is on grad school was like get good grades and classes interest you but are hard as hell. Then he says they mostly look at your math gpa.
but there's the practical question of, if it's between X who has this record and didn't get a C in basketweaving, and Y who has the same record and did, that's where maybe you see the cut being made.
i knew several students in my grad program who were barely verbal. they would have had awful scores on the GRE in that department. i don't think anyone cared.
I would *strongly* suggest, then, not committing to enroll in grad school unless you are sure what your next step would be.
I see way, way too many people going to grad school because they think it's the next logical step after an undergrad, end up dropping out, and then get a job that someone with a bachelor's degree could get. Grad school is more than that. If you want an opportunity to really zero into something, grad school is the time to do it.
I know what I'm saying sounds harsh, but my undergraduate advisor gave me similar advice in my undergrad - and it was honestly the best choice I made to go straight into industry and figure out where I wanted to take my education afterward
i'd been burned by a startup as an undergrad so i just went into grad school, as clarinetist describes, as a kind of default, f this kind of situation. i don't use any of what i learned then now.
one of my friends once had a boyfriend for a few weeks who had a phd in astro something and when we fact checked it, the school did not offer that degree.
Yes. I know several people who got accepted from a research-university-that-shall-not-be-named to do a PhD in statistics, specializing in astrostatistics
it sounds almost real, but i'm not convinced that it's real.
my wife runs into this a lot where it's like, how many areas do you say your work could impact, before you're just talking garbage. i always say, less is more. but i'm also the person who doesn't work in academia anymore.
Overspecialization is a real problem in academia, to the point where people get overly protective and feel the need to justify themselves constantly
@EM4 Quantitative finance is one example, but I don't claim to know how to get hired into that field. Do some searching into stochastic differential equations in finance.
But note: the differential equations aspect is only one small part of it. You need a substantial background in probability (and yes, coding) to even know where to begin with that material.
i do not have a huge amount of finance interview experience. one time i did have an interview where the guy pushed some code to me and asked what i'd add to it. before i added to it, i said, 'this is going to freeze if you push the empty list to it, there's no initial condition.'
@EM4 Have you taken measure-theoretic probability? (Fine if you haven't; in the US, it's usually a graduate-level class.) That's usually an assumed prereq prior to learning about SDEs
@EM4 The prerequisite is analysis at the level of Rudin. Exposure to complex analysis, topology, and functional analysis would be helpful.
Basically, imagine re-constructing probability using axioms, theorems, and proofs like you do in analysis. Example textbook: Probability Theory by Klenke.
functional analysis prerequisites, to my mind, would be, something resembling a course in finite-dimensional linear algebra, and something resembling a course in real analysis, not just on R but with some kind of abstract understanding of topology or convergence outside of sequences of real numbers.
from your problem set. I've done the first part. I am just trying to deduce what the linear map should be to satisfy the difference quotient. It appears to me that the linear map should be $\frac{x}{\|x\|}$, but when attempting to perform a sanity check that this is a linear map, things fall apart.
In the $3$ case, first vector varies over a sphere, so 2 degrees of freedom. Second vector varies over a circle orthogonal to first. So 1 degree of freedom. Third vector uniquely determined. So $2+1=3$.
Why did you change from the original limit notation? Fit it into the definition of the derivative.
Continuing my work through Dummit & Foote's "Abstract Algebra", 3.1.36 asks the following (which is exactly the same as exercise 5 in this related MSE answer):
Prove that if $G/Z(G)$ is cyclic, then $G$ is abelian. [If $G/Z(G)$ is cyclic with generator $xZ(G)$, show that every element of $G$ ...
