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Say a published paper has 4 authors. How many directly contributed to writing the paper?
As in the question, let's say that a paper has four authors. Clearly, all authors contributed something to the work. However, can I deduce that only the first person wrote the physical paper? Or do authors normally write different sections of a paper? For instance, is it possible that author 1 writes the first part of the paper and author 2 writes the second part of the paper? Or, can it be the case that the first author listed contributed the most to the work stated in the paper but did not write the actual paper?
Main questions:

Given a paper with multiple authors, on average, how many people wrote the physical paper?
Is the first author always the person who wrote the physical paper?

You can't deduce anything.  The authors may have split up the writing in any of these ways, and the author ordering doesn't tell you what they did.

How about if it is a Ph.D. student and an advisor? Can I deduce that the student wrote the physical paper under the supervision and assistance of the advisor? I.e., would an advisor ever write a paper with work done with a student?

With four authors, it is generally safe to say 1-12 people were involved in the actual writing, most of the time (not all of whom are authors). I'm not entirely exaggerating - everyone works differently, every collaboration works a bit differently each time, and there is no way of knowing - sometime the authors themselves aren't entirely sure who did what anymore. In many publications there were multiple submissions and changes that at times stretch out over years (sometimes decades), adding and dropping people, and so on. Not to mention fields where author order is alphabetical.

It is not clear to me why this question is getting downvoted. The answer might be obvious to those of us who are established, but for someone new to academia this is legitimately something they might be confused about. As far as I can tell this question doesn't violate any rules of this site.

@Tori: Again, you can deduce nothing.  Even with a PhD student and an advisor, it could have been either of them who actually wrote most of the paper, or they could have shared it in any proportion.

If you really want to know who wrote what, the only reliable way is to ask the authors, though it would be sort of an invasive question coming from a stranger.  I'm kind of curious why you're asking this question.  Why is this something you need to know?

The one who wrote the most or did the most work is the one with the fewest titles after their name...

required reading: http://phdcomics.com/comics/archive.php?comicid=562

@NateEldredge, I am just curious. I am amazed at how many papers are published by some professors, which made me wonder how many papers were actually written by them. Also, I am getting into the phase of my PhD program where I will start writing papers, and I wasn't sure if I was supposed to be the solo writer of the entire paper or not.

@NateEldredge, I have also wondered how some people who do not speak English well have such well-written papers (this is not meant to be taken offensively). I have asked around and the most common response is that some times, if none of the authors feel comfortable writing the paper in English, the authors will ask a graduate student to write it.

Finally, @NateEldredge, my advisor has always told me that it is important to know the order of the authors before starting to work on a paper. I didn't know if this order was an indicator of who was going to write the paper or deal with submitting it/revising it.

This depends on the field and what is customary there. In mathematics you can assume (generally) that all contributed equally. In lab sciences it may not be the case. It is customary in some that the PI who created the lab is on all publications as a co-author but may not have contributed anything to the writing and possibly very little to the ideas. But the PI makes the work of the lab critters possible, so gets on every paper. In some such fields, it is also customary that some fairly low level technicians get on some papers, though didn't contribute to the writing, but they managed the experiments for the actual authors.
Ideally, papers in which people don't contribute "equally" (whatever that means) have a Contributions section to detail the contributions of each as well, possibly of some who weren't listed as co-authors.
The actual writer of the paper may be listed first or not. Sometimes the PI is listed first and sometimes last.
But be aware that intellectual work is hard to measure. Someone may have only spent a few moments thinking about the problem at hand, but provided the crux of the solution that was then written up by others. Flash of insight.
Someone who writes a lot in the field in question can probably answer questions about what is customary in that field. In CS we tend to list authors alphabetically and don't worry much about such things. But we also tend to list only people who actually contribute something meaningful.

What Buffy said.  While you can't tell exactly, the norm in the experimental sciences is that the first author is the student, main writer, and main worker (all three).  You don't even know that for sure. But it is the stereotype.  After the first author, you have no idea the relative contributions and definitely should not assume they are ordered in rank order (often for example if there are collaborating groups, the names are ordered so the groups are together.)

In math, "all contributed equally" is often a polite fiction to cover up the fact that we don't want to attempt to make public comparisons between their contributions.  It certainly does not mean that each of the, say, three authors wrote exactly one third of the paper.  They might have independently written different sections.  Maybe two wrote sketchy notes and the third drafted it into formal English.  Maybe they took turns making extensive revisions to each other's drafts.  There's no way to know unless you ask them.

@NateEldredge, actually, I am referring to the work behind the words, not the one(s) who did the typing. But even there the contributions might be different in kind.

No.  The first author is often (but not always) the person who has done sufficient work to deserve credits for the contents, but this does not necessarily correlated with the actual writing of the paper.
Traditions vary widely with fields but - just like in a parade - the first and last positions are usually the most prestigious, so these are the spots where you should look first a first guess.  However, this is not a hard-and-fast rule:  I have written papers where I’m middle author because the paper had to be rearranged following revisions and I was best placed to deal with this, or because co-authors who obviously did more than I struggled with English.  I myself often work with a true wordsmith to whom I will gladly pass the writing because he can “sell” stuff so much better than I: he is often middle author.
Moreover, traditions have evolved.  When I was a student the first author was with high probability the writer, but this has really changed as the number of authors has increased and the work of each co-author has specialized.


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Is a math degree really useless financially?
I graduated in December 2018 with a double degree in math and electrical engineering and have applied to various graduate schools in pure math and electrical engineering and have been offered admission to phd programs in pure math and electrical engineering.
During my undergraduate studies, I have heard many academic mathematicians say that a math degree opens many opportunities for you and mathematicians can get any job, ...
On the other hand, I have also heard people saying that a math degree is useless financially and you can not get a decent job with a math degree; you either need to become a school teacher or work at a bank, ...
Now that I am about to start my phd program, although I enjoy math a lot, I do not want to get a phd in math and after four years find that it did not worth it financially and I have to do a not well-paid job. I can not also hope that I will end up in academia as I have heard academic jobs are very hard to get.
So that leaves me to think that if I want to get a well-paid job, I should do a phd in electrical engineering and not math.
I am so confused and I do not know which paths am I supposed to take? do a phd in math or electrical engineering?

If you believe working in banking can never be well paid, you may want to research the position of quantitative analyst (quant) more carefully.

Where do you live?

"I have [heard] a math degree is useless financially and [you] need to...work at a bank," isn't that a contradiction? The finance sector pays well.

@CJ59 Indeed, a PhD doesn't generally maximize income (at least not with the figures I've seen), it might even reduce it.

NASA like smart people with maths and probably even more if you have electrical Eng as well... But you should have been aware of that...

@J.Doe: she isn't switching, she has a double degree.

To reinforce @henning's question with an example, in the UK actuaries are well paid (though not as crazily so as the top quants) and require either maths or actuarial science degrees before starting their professional exams.

Just pointing out that you handwaved 'work at a bank' but you've got a great option with a math PhD to go into quantitative finance and be a millionaire.

So if you can’t get a decent job, where does working as a teacher or in banking fit in (personal dislikes aside, working in finance is extremely well paid, and working as a teacher is a respectable, fulfilling, worthwhile profession).

A math degree is very useful financially. See this answer.

Getting a degree in math is certainly not "worthless". Getting a PhD on top of the degree may be a different situation, though, unless your final objective to to become an academic. If you want money, in any field (not just math) sooner or later your have to start working not learning.

Take a note on who gives this kind of feedback. It lacks context. Math requires skills that many people leave underdeveloped, but are very useful in many fields. However, practical skills are necessary too! I would suspect that someone with background on pure abstract math who did all his work on paper may not be as flexible as someone who beyond a mathematician is an experienced programmer (which most mathematicians should be in my opinion). While a programmer with great math skills can be easily employed on several industries, and software developers are in very high demand pure math isn't.

Is taking no PhD at all also an option?

@user2768 I suspect the asker thinks that "work at a bank" means "be a teller at a local bank branch" rather than "work in finance".

If you have any interest in artificial intelligence, math and thus algorithm creation, would be a great fit.

what do you call "not well-paid job"?

Please use comments only for clarification of the question, don't answer question here

Would anyone say a finance or economics degree is financially useless? What are those fields other than applied mathematics? There are people who even use data modeling and statistical analysis examining literature, the field you'd least expect that would use math. Sure, you need to do a little bit of work in specializing, but if all you care about is money go work in the corporate world in a large bank. The pay is ridiculously good.

"after four years" is quite optimistic.

The best response here^ is the person asking where do you live? If you provide a bit more context for the "I have also heard people saying " part, then the advice might be more pertinent. When I did my degrees in the UK I only ever heard "wow math, you're going to do very well" and quantitative analyst jobs in banks were exploding at that time, with increasingly silly job adverts saying "must have PhD math from Oxbridge", with ridiculous base salaries of 1million+ GBP

Also: In my experience the wow factor placed on maths PhD is varies from country to country.  For example, US values it more than UK, and UK more than Australia.

@DavidRicherby Really?! Wow! I had considered that, but thought they could surely only mean finance.

Well if you goal is to get paid well, why do you even consider doing a Phd, just go for the industry ?

Just a personal anecdote, not really an answer, too long for a comment.
I have a master degree in mathematics and now I'm doing physics simulations for visual effects in Hollywood blockbusters. I believe that a math degree gives you a massive upper hand in any field with reasonable amount mathematics in it. However, you have to be interested in that field and study it in your spare time. I studied computer graphics in my spare time and my math background allowed me to comprehend things in computer graphics orders of magnitudes faster and deeper compared to fellow computer science students.
Take it with a grain of salt, master degree is completely different kind of a beast compared to a PhD degree.

Btw, the pay is probably not as good as in a bank but the job is lot of fun!

I don't know if a maths degree would give you an upper hand vs a physics degree fir your specific example, but I agree that a degree in "not computer science but strong in maths" is very helpful in the world of computers (background, I've a masters in physics, and work in computing).  Maths and physics also do very well in banking, if the maths preference is statistics rather than pure.

Similar here: BS degree was math & physics (the school didn't offer a CS degree back then), but my career has been mostly programming scientific & engineering simulations - in fields as different as meteorology, seismology, and neurobiology.  I can fairly quickly learn enough of the science to figure out how to apply the math to it.

Where I live, a lot of math degree holders work in software development and are generally doing quite well ... especially since many computer science degree holders are not really good at maths. Other opportunities for mathematicians are simulations (i.e. in engineering) or insurance companies.
In fact, maths teacher is also a quite decent occupation where I live, but that is of course subject to supply and demand.
Were the people who told you you might not find a good job also math degree holders? I have found that non-mathematicians sometimes do not really know what mathematicians work as.

There's a big difference between a math degree and a PhD though. While I'm sure that you can get a good job with a PhD in math, I'd wager that for many jobs the PhD won't get you much more money, but will cost you a lot from seniority and simply the 3-4 years that you spend doing a badly paid PhD instead of working in the industyr.

You are if course correct. However, on closer reading it seems as if the OP is already sure she wants to take a PhD and is just unsure which one. If that is the case, maths might still be the more logical option.

@Voo This is correct, but doing a PhD in engineering (as stated in the question) will be "equally bad" from the point of view of seniority. If then the point is to not do a PhD at all, then I would agree.

@gented Somehow missed that when reading the question. Agreed that there's probably very little difference between the two PhDs. Mea culpa.

I'd be more inclined to comment - but I get more space in an answer:
The premise that a degree is a path to a job is blatantly false. Even the best most sought after degree might leave you jobless: wrong region, potential employer doesn't like you, you aren't as good as the rest (some subjects are apparently very competitive, for better or worse).
So you should really pick a degree that you find personally interesting that has a potential of being useful for work.
In the case of mathematics this leaves very must two main routes:

Pure maths in academia, or close to it, teaching
Applied maths in academia or industry

Mathematics is very much the foundation of much of our modern life. Banking, electronics, engineering - BUT this is applied mathematics, not pure mathematics (but even that has its fans).
Some people may joke that an applied mathematician is a physicist ;).
In your specific case, the combination of mathematics with electrical engineering can be quite valuable. You would possibly be better at the calculation that your peers if they did not study mathematics.
Then again, if you hate electrics and love maths, it may not be an advisable choice.
In addition, paths can change. Even a PhD you should chose to some extent by interest if you can - it will make it more interesting.
To bring in a personal note:
I studied mathematics. In the PhD I ended up with some limited experimental and analytical chemistry, chemical kinetics, ab initio quantum chemistry and programming. Then in two post docs I ended up programming more and working more with chemical kinetics and quantum chemistry.
Unfortunately I am not in academia any more - still, for now a very interesting job. And what is it? Numerical simulations (where my previous experience is useful.)
Now in some countries subject mobility isn't very large (Germany springs to mind, maybe also France). In England you come from subject A and do a PhD on sbject B in the end...
So to bring this to a conclusion:
Study what you like - a PhD is also funded. A degree is NOT a "path to money" - follow your interest. Mathematics is the basis of most of the things we need today. If possible, focus on the applied side and there should be many opportunities for an applied mathematician. And don't necessarily reject the pure maths: Developping new functionals in quantum chemistry or new models for say CFD, working on cryptography is well beyong what I would be able to do in maths - especially given that my paths capabilities have been rusting away the past 7.5 years....
Nowadays I do less "actual maths" and more programming, number crunching and data processing.

From my experience (BS in Math, MS in Statistics, incomplete Computational  Statistics PhD, now working in Healthcare industry), this answer is spot on.  I do very little pure math or theoretical statistics, but I apply math and statistics every day.  People may not pay you to do math proofs, but the programming, number crunching, and data processing is in high demand.  If you want to make the degree a better financial decision, focus on how you can apply math.  Practice solving real problems with applications/programs popular outside of academia.

While I agree that the work you do will be more like applied mathematics, I don't think that studying pure mathematics is only a route to academia - I know several who studied pure, but ended up working in meterology, astrophysics, finance, etc.

"So you should really pick a degree that..." —that was an unexpected twist; I thought you'd say, "So you should reconsider whether you need a degree at all or whether you should just start working now."  :)  Or maybe even, "...consider whether you're avoiding work by studying endlessly."

@Stobor Of course there are. But there are fewer (obvious) opportunities for pure mathematicians than for applied mathematicians.

@Wildcard There are people that think like that unfortunately... - I would say everybody who wants to continue studies due to interest in th esubject should be encouraged. It is an opportunity to learn new things and develop a better understanding. - If the interest is in an area that is useful, even more so.

The answer depends very much, as already outlined, on the particular job career you choose to pursue. It is generally true that nowadays it might be quite hard to find a very well-paid tenure position in Academia, therefore the chances that one might end up being financially wealthy with that particular choice are not the best; likewise, industry does not seek many pure mathematicians either.
However, mathematicians and physicist are extremely well employed (and probably consitute the majority) in quantitative analysis, machine learning and data science related jobs, not to mention the subset who choose to continue in software development. It is often said (as you yourself stated) that the ideal solution is to move to the financial sector, nevertheless that is not the only one; as already mentioned, nowadays the field of artificial intelligence and data science is particularly filled with academics: the main trait that pursuing a PhD offers is the capability to solve problems, in general, and to apply a broader set of knowledge to investigate different scenarios. Such skill set can be easily applied elsewhere and as a matter of fact most academics do not stay in Academia (at least nowadays); to make a long story short and answer your question

On the other hand, I have also heard people saying that a math degree is useless financially and you can not get a decent job with a math degree; you either need to become a school teacher or work at a bank

a scientific PhD is not financially rewarding if you decide to stay in Academia (for the majority, with the due exceptions); it can be if you move to industry, to the right places. Moreover I do not see a PhD in electrical engineering opening so many more doors that a PhD in maths cannot.

With either a PhD in Math or EE, you have a good chance to get a well-paying job.  So my advice is: study what you interests you (if neither interests you, a PhD is probably the wrong choice).
To give you some numbers, from the US, the AMS Survey gives data on starting salaries for Math PhDs.  You can also get some data from Payscale on average salaries (e.g., Math PhDs, EE PhDs and Engineering Bachelors):

AMS Median industry starting salary (2016 Math PhD):  ~$106,000
Payscale Average Math PhD salary: $98,000
Payscale Average EE PhD salary: $116,000
Payscale Average Bachelor of Engineering salary with 5-9 years of experience: $83,568

Some words about this data: The AMS data is rather reliable.  Payscale does not have have a lot of data on Math and EE PhDs, and I am not sure if the salaries used are all starting salaries.  The Payscale Math PhD includes a number of academic salaries which are significantly lower.  AMS reports medians and quartiles, Payscale reports averages and ranges.  If we compare the Payscale Median EE PhD salary (by eyeball, about $110k) with the AMS Median (about $106k) we see they are not so different (and the latter is certainly only starting salaries).
Ignoring the relative cost of getting a PhD (which is typically relatively small in the long term), it appears that your general salary prospects are better with a PhD in EE or Math than just a Bachelors, and slightly better with a PhD in EE than a PhD in Math.
That said, there are some areas of math that are considerably more marketable outside of academia than others, such as Statistics and Data Science.

Is a math degree useless financially? Yes, if its just an undergraduate degree.
What about a masters in math? More useful, but still quite useless comparatively to other degrees, unless you back it up with other skills (e.g. I know software developers who've gone into data science/machine learning positions, but that requires self-learning software development on the side).
What about a PhD in math? Much more useful than either a masters or an undergraduate, but mostly for the purpose of enjoying a career in Academia. Since that's not usually well-paid for most people, that doesn't seem to be what you are aiming for (and yes, the fact that Academia is also highly competitive with loads of pressure is yet another reason to avoid that path if you aren't extremely passionated about research. That's the only way you'll be good in Academia, except if you are some sort of genius). There are however other alternatives: you can get a job in finance (again, if you back it up with other skills, like learning financial theory and/or programming), and a job in banking does pay quite well, but if you aren't interested in that sort of thing (don't blame you), then there aren't really that many options left in the industry. People will tell you fairy tale stories about how their math PhD got them a job working with this or that, but the truth is that none of those jobs are guaranteed and your math degree isn't structured for it. At best, you'll still have to spend an enormous amount of time self-studying whatever stuff you need to work in whatever random industry you got hired into. The truth is you won't walk into any industry apart from banking or teaching and be able to dazzle people with your math knowledge. Your math knowledge is and always will remain a nice bonus. Nothing more. You have to be able to offer more.
Personally, if financial gains were my main worry and I wasn't particularly fond of math (which you don't seem to be), Electrical Engineering is a complete no-brainer. Lots of jobs, salary's decent.
Oh, and please don't listen to people who say that a degree in math "improves your analytical thinking" and blah blah blah. As if a degree in Electrical Engineering doesn't do the same.

No, a math degree is not useless financially.  It will not bring you riches by itself, but you can get your foot in the door in a lot of different areas with a math degree.  Generally speaking, the value of a degree is to get your foot in the door.  Unless you go to a top institution, and get top marks in a highly sought after degree, financial well-being will largely be based on what you do after your enter the workforce.

Unless you go to a top institution, and get top marks in highly sought after degree, financial well-being will largely be based on what you do after your enter the workforce. If this were true, then adjusting for people who got high marks in degrees from top institution, we ought to see no difference in the salary earnings of workers when clustered by degrees... but we do. So the only explanation is either a) Mathematicians are just dumber and lazier than others once they enter the workforce which I don't think is true, which leaves option b) math earns less.

@Marqui I think you may be conflating degree with actual field of work... Only a small percentage of people (per Google--slightly less than a third) actually end up working in the same field their degree was in. For the rest, what field you end up in is a bigger indicator of pay than what your original degree was.

@Marqui Do you have any factual support or sources for your argument that "math earns less?"

I know nothing about EE, but what you write about maths seems right to me.

To give a slightly different take than the other answers:
Employers hire people to do a job for them. Many companies have situations slightly more complex than "hire the best for the job," but at the end of the day they want the person who can meet their need as best as possible.
As a math major, you have a skill set that people want. Unfortunately, they rarely are looking for people with only that skill; they likely also have other requirements. For example, in the United States, the NSA explicitly lists mathematics on their jobs page. The mathematicians they hire, though, are rarely only mathematicians; they also have at least some skills in computer science, programming, and other related areas (e.g., linguistics, cryptography).
Basically, check out jobs you may be interested in and make sure that math isn't the only skill you bring to the table. Outside of academia mathematicians typically need to be polyglots to be successful, but polyglot mathematicians can be successful indeed.

Have you considered the ROI of higher education?
If you want to make more money then start working now and building up your work experience instead of taking on more debt and not having several years of work experience.
You will also find out from working that maybe you don't like EE as much as Math for example, and this can help you decide.
If you are a curious person and want to learn, then go for the phd in math and lean towards academia.
I faced the same problem when I was deciding my undergrad, I love astronomy and astrophysics and cosmology. I was concerned about the time and money investment and the questionable returns. I have heard how there is a lot of politics involved in securing grants and funding, and also the small number of jobs are very competitive.
Instead of physics, I went into engineering because it would be more practical to get a BS and begin working right after. Now I don't have a lot of debt, a good entry level job, and I can explore different industries and my own interests and continue growing. I would like to continue my education and get a masters eventually but I'm not sure what it will be in yet.

This can depend on more than just the mathematics. You don't say where you are, nor anything about the nature of the educational system. But, for the US, at least, a BS in mathematics is a great basis for many careers, both in and out of academia. But that is because the baccalaureate is, in the US, a very general degree. One studies more than just mathematics, which amounts to only about a third of the courses taken.
But mathematics is about thinking and you can apply that thinking, especially thinking in a formal and disciplined way to other things. But if you don't really know anything except mathematics, then you aren't in such a good place, except to do more mathematics.
But any company that needs analysis of things, such as products or strategies or, well, anything, can use people that have a disciplined way of organizing and presenting information. Mathematics helps with that, as long as you also have the more specific knowledge that the job requires. But that knowledge is easier to obtain, in most cases, with the discipline of thought that mathematics brings you.
A degree in EE would teach you to think in a different way, whether that is better for you is up to you, however.
But, to give some perspective here, I also think that a degree in Philosophy is also a good launching point for a career, and for much the same reason that mathematics is. It teaches discipline of thought that can be generally applied.
However, you are starting a doctoral program. Such programs prepare you to do research and they are very specialized. They are not general education in any sense. Most, or at least, many, PhD holders (guessing a bit here) stay in academia since the doctorate is a good preparation for that. There are other opportunities, of course, but like the degree itself, those opportunities are very specialized. But mathematics is, even at the highest levels, still good mental training that can be applied in other domains.

Is it really true that most PhD holders stay in academia? I'm pretty sure that's not true within the fields I'm most familiar with, but I don't know if it's counterbalanced by other fields.

@DavidZ I was just about to say the same thing. See, for example, this question and its answers here on Academia.SE.

@DavidZ, It's just a guess, really, but I'm thinking beyond technical fields. I don't have research to back it up. Made a small edit.

It's statistically not possible for most new PhDs to become academics; advisors produce too many academic "children" in their careers to be counterbalanced by the growing demand for education.

@Ian, in the US there are a lot of colleges that want mathematicians. They aren't R1 universities, of course, but math teaching is still highly regarded and necessary. Most students in college study at least some mathematics as a requirement. Getting a doctorate in the US in mathematics usually offers some experience in teaching as well as research. But it is true that not all math PhDs will become primarily researchers.

Moreover, most such small institutions need to be regionally accredited (or their students can't get federal loans) and such accreditation requires that most faculty hold doctorates.

@Buffy Here I really mean professors. I would venture to say that going from a PhD into an adjunct position is not a very desirable outcome for most people.

@Ian or financially tenable, at least in the US.

@Ian, nor was I writing about adjuncts, but regular positions. Accreditation disfavors part timers.

Is a math degree really useless financially?

I guess you mean PhD., as you seem to have already double majored in EE/math.

I am so confused and I do not know which paths am I supposed to take? do a phd in math or electrical engineering?

From a financial perspective...
the path to the greatest in lifetime earnings is probably to get an EE (or math) job now.
That's four (+/-) years more of income and you won't have to pay to go to school anymore.
Normally my advice is follow your passion, but you don't seem to have found your passion yet (from your post at least).

I am so confused and I do not know which paths am I supposed to take?

Only you can answer this. Whatever path, it has to be one that you will be motivated to stick with through your PhD.
With that said, assuming math is what you enjoy, why not consider combining a math PhD with another subject? In other words, you focus your work on math, but are integrating it with another field that interests you and opens up employment opportunities that a pure math degree might not have. In effect, consider being coadvised by someone in another department where math can be used in a more applied research context (The caveat being that if you are already in a program, you may have an advisor that may not allow it.)
The reality is that math has applications everywhere, whether it's physics, chemistry, engineering, economics, biology, ecology, computer science, geology, etc. In my case, I'm doing the reverse. I'm just beginning a PhD in wildlife ecology, but I've always had a strong interest in math, so I'm actually going to try and take several graduate level courses in math. The benefit being that most people in my field dislike math, yet there is a strong need for people capable of integrating advanced mathematical concepts into ecology. For example, we use things like network theory for modelling landscape connectivity for different species, which has real world uses for conservation, but this integration is only relatively recent in ecology, and is still be developed and built on.
If your biggest concern is financial success, then integrating computer science into your program could really set you up for success. Computer science, like math, has applications everywhere, so the two combined will give you a lot of flexibility. You can look at the type of research being done in computer science to get an idea of the different things people are working on. Depending on the approach you take here, you could probably set yourself up for a nice future as a data scientist.
Or you could do something entirely different. I know someone that was a mechanical engineer working on space shuttle related projects before starting a successful PhD program studying bears. Despite not having a biology related degree, they were probably able to pitch their math and computer background as strong selling point.

I got a PhD in pure math, got a job at a financial company, and I am very well paid.
For full disclosure, I did have a small amount of trouble getting a job at a financial company. I interviewed at maybe 5 companies. I was first hired by this company as a lowly programmer (in the last year of my PhD program), then after I got my PhD I applied for a job on the modeling team. You could say I snuck in.

Why were (or weren't) you hired?

@Xen For most of the companies I wasn't hired due to lack of financial knowledge. For JP Morgan, they wanted a C++ maven, which I didn't qualify as. I was hired at my current company for my initial job because of my Java programming skills. The modeling team then hired me because of my intimate knowledge of the internal system. They were really looking for someone who knew finance in the posting, but they accepted someone who they don't have to teach about the internal system who has a PhD in math as the next best thing.

A math degree provides evidence that you are a strong problem solver, that you can grasp and apply logic, and that you are comfortable with
algorithms. These skills apply to many areas.  You have the opportunity to find what you are passionate about and pursue it.  Many jobs say something
like candidates must have degree in X, or other related field.  Math is almost always accepted in the other related field category based on the skills
it demands. However, a math degree in itself may not always be enough, so you should be prepared to develop specialized skills.
What you are asking is really a personal question that you will be able to answer for yourself better than any of us can do.  That said, I do would
not recommend a PHD in math if you do not know how you want to use it.  You need to find your passion. It's OK if you don't know and if you don't get it right in your first
job, but by entering the real world you will get exposure to real life things that you like and dislike.
I was a math major and found myself in a similar position. I started out at an actuarial company and switched to a software
development. Both paths required time outside of work to learn the specialized skills required by the job, but the problem-solving skills demanded of a
math major have provided an excellent foundation.

My undergraduate degree is in pure mathematics and it opens many doors. First, remember that there are a lot of jobs which require a degree but are not overly picky about which one. For instance, I started my career as an Army Officer after receiving my bachelors. A degree is a requirement and a math degree in particular opened the door to Military Intelligence.
Math is also an acceptable starting point for many careers that are tangentially related. After leaving the army, I became a programmer and database administrator for a while. I know others with math degrees that went into financial analysis
Math also serves as a basis for many related graduate degrees. I went into law after I decided I did not want to continue in programming.
And that is to answer your direct question about the value of a pure math degree. You said you also have an electrical engineering degree. While a masters is very helpful if you want to go into work as a true electrical engineer, a EE degree opens a huge array of doors by itself. I know many other programmers that got an EE degree to start.

As already said, the answer is No, it isn’t.
Among the thousand or so software engineers and systems engineers at my first civilian job, many were math majors.

I did a PhD, left academia and have worked a career in areas for which that is an asset. Never had to use very advanced maths, but 'mathematical thinking' has been essential.
A key thing to observe is that maths comes under two well known labels:

Pure
Applied

When in academia this can sometimes seen a trite distinction, commonly with a bit of snobbery thrown in.
Outside of that, 'applied' also means that you have to take a problem and translate into mathematical form. No longer can you 'given such and such' or 'assuming that'. You need to take the data that you are given and determine the conditions it satisfies, or fix it so that it does. You typically need to translate the business problem into abstract form, solve it abstractly, then implement in software. In fact, even to get the first of these involves a ton of talking to people and language skills.
Think of such roles such as:

Data Scientist/ Analyst : A huge  growth area. The Analyst very well paid and mathematical
Operations Research Analyst : Cool, but not that may positions available.
Modeller - finance: Still a lot of jobs, well paid, but you may have reasons not to
Modeller - defence: Still a lot of jobs, quite diverse, but you may have reasons not to
Modeller - other: Fewer jobs, can be very interesting and valuable.

So, if you think you'll want exit academia after PhD, the specific additional knowledge a employer will look for then is programming. If you have EE, you most likely already have some knowledge. Continue to build on that during your PhD. Maybe take some courses through institution or online. Learn about OR and Data Science. You'll then be well set up to find a good position on a good entry salary.

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text:
How do journals remain objective, if they are for-profit entities (with for-profit interest)?
How do journals remain objective, if they are for-profit entities (with for-profit interest)?
Occasionally, something being for-profit is seen to influence the goals and interests. This creates some believability problems like:

If the publication has to produce money, then why do we trust its
objective?

If the publication produces money, then why do we need to measure its
utility, if it already produces money (i.e. it has a capitalistic
need)?

In fact, if we'd trust the economic measure, then just like in other things, "whether someone buys it" is a measure for quality. However, it's certainly possible to sell what people want to read, which is not necessarily correlated with information quality (compare to e.g. tabloid magazines).
Technically it would also be possible that "influencers" would be bought so that people who write and selected people who publish would both win, just like in some other industry where those who profit can also be those who control.

It is the editorial board that decides what papers are accepted. In my field, math, editors and referees do their work for free.

I am not finding this post very clear - "compensated" by who, for what purpose? What does "compensated" mean? Are you contrasting journals with different publication models, for-profit versus non-profit publishing, what?

@BryanKrause Well, compare to newspapers. They sometimes sell, because they have things that people want to read about, which is not necessarily the same as being most objective, .... So certainly it's then possible to publish things that have high sales, but which are not high-quality in information.

I mean that to strengthen quality, one may then have to reduce the incentive for selling what people want to read, like above.

The market for a journal is completely different from the market for a general-circulation newspaper. And, these days, most publishing is happening "open access", where the authors pay, not the readers. I think it's going to be difficult for you to write a question about these things if you're not familiar with these concepts in academic publishing. It might help to start by mentioning your assumptions and understanding. I fear, though, that the question will end up far too broad and unfocused.

@BryanKrause Is it? If publications are used as a measure to teach in public universities?

Huh? The typical audience for journal articles are researchers in the field. Subscriptions to journals are typically purchased by university libraries so academics at that institution can read them; that's the audience. I don't know what you're talking about.

There is enough grift to go around in not-for-profit endeavors as well.  Even for an open access journal, money -- of course -- is part of the picture, even for open access.

Let me give it a try.
What you described might be something a disreputable publisher might try, and maybe has, but a reputable publisher can't do that and maintain their reputation.
In fact the same situation applies to many other businesses. As a business, it is responsible to its shareholders to make a profit, and it is responsible to its customers to maintain trust in its quality. It is a balance, but without the trust of the customers there is nothing that can be done to satisfy shareholders in the long term.
If you make widgets for sale and the customers think you make junk, then you don't sell many widgets, so the shareholders get after you to make better widgets.
The practices you describe couldn't be carried out in secret for very long. Word would get out. Academics wouldn't trust the journal to provide good scholarship, science or whatever. So, publishing there wouldn't enhance your reputation and sales of the journal would soon fall. Librarians, who buy a lot of academic stuff, are smart enough to know these things.
So, while money is a driver, it isn't the only driver. Reputation matters. Think about what has happened to the profitability of the social media site formerly known as Twitter as a current example.

This is a good answer, don't know why the question is downvoted, because this is what is asked for.

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text:
How does one find an advisor doing research on a niche interdiscipinary subject?
I am very interested in in-situ resource utilization (ISRU), the use of lunar/martian/asteroidal resources (generally regolith, atmosphere, and water ice) to produce propellant or construction materials on those bodies. It's a new & growing topic--the first instrument to test production of rocket fuel from the Martian atmosphere is due to launch a month from today (June 20th, 2020) aboard the Perseverance rover--and it's an interdisciplinary one, lying at the intersection of geology, mineral extraction, and aerospace and chemical engineering.
How do I find potential doctorate advisors working in this field? I have been looking through Google Scholar & conference catalogs (eg.: AGU) to find the names of those writing & presenting on ISRU techniques. However, the majority of the papers authored on ISRU implementations are by industry & government authors, and there are no dedicated graduate programs I am aware of aside from Colorado School of Mines' Space Resources program. The majority of academic work in ISRU seems to be by faculty in either the aerospace or mining engineering departments.
This concerns me. I am not an engineering student: I am a geology & math undergraduate, and I am worried about the quantity of deficiencies I may need to take entering into an engineering graduate program (if I can get into ISRU research, it's absolutely worth it though). More importantly, I doubt I am exactly the type of student engineering programs are looking for. My grades are good, I have a lot of undergrad research experience, but like I said, at the end of the day, I have no engineering background. I anticipate rejection, and I anticipate a lot of it. Even if it's not outright rejection, odds are there won't be that many doctorate positions available with funding when I am applying (for next fall).
Therefore, I'm trying to track down almost everybody I can find who is active in both academia and ISRU research, to maximize my chances of making a connection with a potential advisor and finding a place for me in their department. But as I mentioned previously, it's difficult to do so because a) authors on IRSU in academia are few & far between, and b) the departments in which people are doing ISRU research are much broader than my interests.
Do you experienced people have any advisor-finding advice for a lowly undergrad?

You could try to identify where those government / industry professionals did their PhD’s or reach out to them. They’ll have significantly more domain knowledge and can guide you better.

You could write to the government/industry professionals and ask them, they likely know more.

If you're particularly interested in a specific topic, I'd guess you've read some scientific research in that field. Who published it?

Just an fyi, ISRU and ISRF are not new. Many works have been done of these topics since the 80s. You might want to read on the work commissioned by NASA or that done at Rutgers University, University of Alabama or Clemson University. You might want to reach out to some of these schools/researchers if you are interested in a school in the US.

From my perspective (as an advisor), you sound like the ideal candidate: someone who is self-driven and motivated to work outside the regular comfort zones.
Assuming you can't find anyone at the exact intersection you're looking for, my advice would be to find a potential advisor who values interdisciplinary science like you do, and works in one or more of the disciplines involved. I would start by asking the most relevant faculty at your current university if they might know of someone who fits the bill that you could email.
For an advisor that values interdisciplinarity, your background shouldn't matter so much - they will be looking for initiative, drive, and decent grades. Argue those three things as concisely as you can when you send those emails.


text:
What to do if your research idea has already been published?
I am an amateur researcher (no PhD, Masters, etc...) and just want to put something out there so I can put it on my resume to demonstrate I know how to perform research and hopefully get a job.  I did a Google search and found a paper that discusses the same idea I had.  However, the paper approaches the problem differently than I would have done (although the more I read the harder it is for me to separate the two in my head), and does not go as in depth as I plan to.  With that said, the underlying concept/theory are the same.
It seems like what I can add to the field are additional evidence that the technique works and maybe (most likely not) an expansion of the technique/theory to allow for it to be used more broadly.  Realistically, I'll probably only be able to offer additional evidence that his technique works, as I am not educated enough and in the field - but one never knows.  I'd have to dig deeper, and will only know after I run my simulations anyways to see if the concept can be expanded.
Is there a case for me to construct a research paper?  Or should I just do a bunch of simulations, which is what I was planning on doing, to demonstrate that his technique works and show how it could be applied.  In the latter case, if my work is deemed unoriginal, I would just publish the results either via a blog or Twitter and reference the original paper or just keep it for personal use/knowledge.  In the former case, if my work is deemed original, I would actually write up a research paper, but still obviously reference his paper.
Thanks!

Email the author(s), setup a call, and explore your options, which may include co-authoring an extended version of their work or a new work.

That's a good idea.

It is hard to give advice specific to your case without knowing more, so I'll mostly restrict to general comments here. What you do depends on the field, to some extent, and to the individual paper and kind of research.
But, if you use a known technique to reach a known result, your results are probably not going to be published. It is, at best, a learning experience.
Independent researchers have a number of problems, but they often come down to lack of professional advice in getting started in research. Working with a guide can help you avoid dead ends. The big problem is that there is a lot to know about any given narrow subject and it can take a long time to absorb it. A guide can help you find an appropriate path to learning what is already known and what is still valuable to pursue. Ideally, you'd like to know about the earlier work before you spend time that might not be productive.
You don't suggest that your research is in mathematics, but there, it is possible to publish new proofs of known theorems under certain circumstances. A proof give insight into why something is true as well as the fact that it is. Sometimes those insights can be transferred to other problems yet unsolved. So, a new proof, using a novel technique, can be valuable in itself and might get published.
Another kind of research depends on statistical methods. Social science and some of the "hard" sciences can use such methods. Some of the published results here are worth replicating, since statistics alone only give evidence, not proof, that something might be true. But, again, you need some variation for the new study to be valuable. Sometimes a paper will contain assumptions that may be unwarranted (normal distribution of a variable in a population, say) and a new study making fewer assumptions might be worth doing. Likewise, some medical studies are extremely important and may affect people's health. So, a lot of evidence for what is really going on is needed and replicated studies can be essential. And some studies actually refute the results of others, thus requiring more study. Statistical studies, if done well, have a measurable probability of producing misleading results. Replication reduces the probability of error, provided that things are done properly, and probably with some variation on the original.

Thanks.  Maybe I'll be able to demonstrate a expansion of his techniques (through simulation), since he does mention it only applies to specific situations.


text:
Recomendation letter
Can a lecturer from another faculty write a reference letter for me for my masters degree.
This lecturer has introduced me to many parent to teach their student based on my field and the result has been so good and he us willing to write me a reference letter.
Can he?

Actually, anyone can write you a letter. The best letters are from those who can help a reader evaluate your suitability for a degree and your likelihood of success in it. Usually that is people closest to your studies, but others can also be helpful.
Evaluate what you think each potential writer will say about you. It may actually be good if they say different, but positive, things.
The only letters that might be discounted, I think, are those that might be judged to be biased in your favor for non academic reasons. A letter from one of your parents or siblings, for example.


text:
Short postdoc abroad
I'm looking for some career/life advice. I am currently 9 months into my first postdoc job in the US (originally 1 year contract) in the life sciences, after finishing a PhD in the UK. My contract ends later this year so I'll have been here 1.5 years and I'm not sure I want to renew it. I know most postdocs are usually 2-3 years in length, but I think I am ready to go back to the UK once the contract ends. The project is great, the lab is amazing and I've gained so many new skills. I know that finishing the project would be great for my career. But I've had some bouts of bad luck in my personal life and it has really affected me. I miss my family and friends at home and now I'm torn between my career and mental health...
Talking to my supervisor, she is understanding but also doesn't think I will have enough data by August for a paper as it's a huge ongoing project. I'm wondering what the damage would be from having that 1.5 year postdoc abroad on my CV without any papers. Or would employers look at it and be understanding that it might be difficult to be far from home or that I have at least gained valuable skills despite no papers as it was less than 2-3 years?
Anyone experienced similar situations or have any advice?

I would not worry about the "1.5 year postdoc with no papers" specifically, but more just about the "no papers" part.  Is there any possibility of continuing to work on the project after you leave, enough for at least authorship on the resulting paper?

@NateEldredge I haven't discussed with the PI yet, but I believe I would get some co-authorship on the resulting papers as I have done a lot of work to push the project forward so far. But as it is a large project with several years still left on the grant, it could take some time until the papers are written...

If you want to go, go.  But try to line something good to go to (not just go away).  Also, please try not to get into things where you have to wait so long and are so dependent on others.  Postdocs are supposed to be little paper factories.  It is nornal in life sciences for many projects to not have these huge deays or organizations.  Try to get something out of this one, but if you are unhappy bail now.  [Seems a little flawed that you went into a 12 month proect (before extension) with no quick papers expected.  Avoid these traps.]

@guest surely postdocs are meant to produce good research results and not just be "paper factories"...

Gotta produce.  Better papers is fine.  But you need to have overall production.  Especially with short appointments.  People want to see some numbers.

would employers look at
it and be understanding that it might be difficult to be far from home
  or that I have at least gained valuable skills despite no papers as it
  was less than 2-3 years?

They do not automatically understand. You need to convince them.
There can be multiple reasons for 1.5 years without a paper. A big project that requires a lot of engineering effort is a perfectly reasonable reason. In this case, a letter of reference from your postdoc advisor is very important. You may help her to sketch out what she needs to write:

Your responsibilities and skills needed to perform in the project
Your contribution/achievements after 1.5 years
...

IMHO, you should not mention "be difficult to be far from home etc". It sounds like an excuse for poor performance, or at least it gives an impression that you are not able to overcome difficulties.


text:
Putting a scholarship received for upcoming summer school in CV
I applied for a summer school and received a scholarship to attend. In the meantime, I am planning on applying for a Masters, in a field related to summer school, before attending the summer school. Is it acceptable to put the scholarship in the CV?
What would be the appropriate place to put the scholarship? I have a honours and awards section in my CV. But I'm not sure that's the proper place to include the scholarship for the summer school.

Yes. The important date is the date that the scholarship was awarded, not when its associated event will occur.
Let’s take this further, and say that subsequently the summer school is cancelled, and so you don’t actually receive any money to attend it. It would still be perfectly fine to put the scholarship on your CV, as the achievement you are listing is the independent validation of your capability and talents. That judgment stands, regardless of whether the associated event has occurred. Attendance at the summer school itself would be another item on your CV.
A common example is the following: you simultaneously apply for and are offered doctoral scholarships from two different universities. You can only take up one of them of course, but it is still fine to list the other in your CV, with a note that you didn’t take it up. The offers themselves are each evidence of independent validation of your attributes, regardless of whether you actually accept them. Let’s go further and say that you decide not to take up either offer, but leave academia entirely. You would still list both scholarship offers, as they provide useful evidence to a potential employer (i.e. that you chose to leave, rather than that you couldn’t cut it).
So congratulations on your achievement, and never be afraid to claim justifiable credit for it.

Thanks. What would be the proper place to include it? I have "honours and awards" section but I'm not sure that's the proper place..

It is an honour that you were awarded, so sure, it fits there.

Sure. Just make clear the scholarship it is for a date/summer school in the future. You could also add sth. like planned/forthcoming/... or the like, to make this point clear.


text:
Would leaving my current PhD program ABD be a red flag when applying to other universities?
tl;dr: I am considering leaving my current PhD program ABD, after four years, several publications and a strained relationship with my bully advisor. How, when and do I explain why I'm leaving if I apply to other PhD programs (in different European countries) and would it be considered a red flag and mean an outright rejection?
Wider context: I am a PhD student in an STEM field at a European university, currently in my fourth year.
Over time, I have become increasingly disappointed and disillusioned, and I have been pushed around, bullied and slaved away for minimum pay. My relationship with my advisor is becoming worse all the time. They are a very self-absorbed person and a bully, but very well connected in the University. I have worked very hard for them not only on my PhD but on numerous other tasks that are not even related to my job. I completed all the formal requirements for graduation but my advisor keeps stalling because they need someone to work for them cheaply and the PhD dissertation is strong leverage.
Now I decided I am ready to leave my job and my PhD without graduation as it has become apparent that they will not let me finish no matter how much more I do.
I was thinking of applying to other programs, but it will be obvious that I was already a PhD student, as I can't just leave it off my CV and the publication track record will also be there. My question is, will quitting my program ABD and applying to another be a red flag? Do I mention anything about why I'm leaving and when? I have no intention of bad-mouthing my advisor to anyone (except obviously you strangers on the internet), so how do I explain my leaving without coming across as a whiner and a quitter? Or should I just give up and get a regular job?
P.S. I read this question which might be a duplicate, but there the student quit while having a relationship and contact with a specific advisor at the new university, while I am just thinking of applying to several calls where I do not have any personal connection with the people hiring. If people here think it's still a duplicate, I will agree.

your post is far too long

@AzorAhai--hehim I edited, hope it's better now.

The first order of business is making sure that you cannot graduate. If you have more than 10 papers published in non-predatory journals, you are objectively ready to wrap it up. I would suggest contacting former PhD students of your advisor to know how they "escaped".
Assuming you will not be able to graduate, look at your exit options. Your division of the job market between academia and "regular 9-5 job[s]" is to black-and-white.
Most large companies and some midsize companies have R&D laboratoires where research jobs is at least as intellectually stimulating as in academia; compared to academia, you trade some freedom, both intellectual (the cool topic you worked on is not funded this year because of company priorities) and physical (you need to be on premises at fixed hours, and you might even need to follow a dress code) in exchange for better working conditions (on average a better salary, better work/life balance; in experimental sciences you have a better-equipped lab with dedicated workers). That might or might not suit your taste.
For whichever position you end up pursuing, I strongly advise you rehearse what you will say at an interview. You will have to "badmouth" your current advisor; hop over to the Workplace stackexchange and read a few questions on that topic; the quick summary is to keep it as factual as possible. For PhD positions any reasonable interviewer would think that you should have defended with that number of publications, so I would point at that and say you disagree with your advisor's decision not to let you defend; avoir bringing other concerns if not asked. For industry positions the interviewer might not be familiar with PhD programs, so I would rather say something vague about an unsound work environment.
For PhD positions, your CV will probably be tossed out without an interview a few times because it is atypical, but if you get to the interview your chances look better than average since you demonstrated an ability to do research, write papers/grants etc.

The advice of UJM is good, especially the first paragraph. But, there might be an option to help you move.
I left one program for another, but for different reasons. But I was successful in moving primarily because there was another faculty member who supported me in ways that my advisor did not. He expressed confidence in my ability and it was his recommendations that made my move possible and my future a success.
So, my advice, if it applies to you, is to see who you might have for allies at your current place who could strongly promote your candidacy elsewhere. If such people exist, they might have both recommendations and some influence about other universities. Allies elsewhere would be another possibility.
But just leaving and trying to start over would be difficult, as you suggest. It isn't impossible, and others have done it, but look for a springboard to help make the jump.


text:
How do I distribute a survey questionnaire to obtain a reasonable response rate?
I have built a survey questionnaire focused on Software Engineering Researchers and now I need to distribute it, but I am not so sure how. My questions are:

Are there any other means to distribute it to obtain a reasonable response rate while bearing in mind the survey population?
Is it ok to use a electronic mailing list for it? (Regarding both the survey integrity and the mailing list protocols)
If so, are there any appropriate mailing lists to my field?

I have recently subscribed to SBC's mailing list and I got used to receiving questionnaires through it. I have done some research and only found ACM's SIGSOFT.
Editions made to solve any misunderstandings are in bold

What will you do with the results of the questionnaire?  Be sure your procedure is approved in advance by the appropriate authority (such as a university committee) since this constitutes an "experiment with human subjects".  It may sound crazy, but this is required  nowadays.

Contact SBC and get them to send it out.

Sending surveys to mailing lists like that sounds like a great way to annoy your entire field (but you say you've been getting them, so maybe it's ok).

I have the sense that this is the $64,000 question throughout the social sciences.  Welcome to their world.

The University you are affiliated with likely has a 'behavioural research ethics board' or some similar title as was pointed out by @GEdgar. Such surveys constitute behavioural research and require approval by an ethics committee. The ethics application will usually require that you provide information about how participant anonymity will be maintained (YES -- even if that's a 'given'), how/where you will store the data and for how long, and what you will do with the data. The application, in most cases, can be completed online. If you are not the PI, bring your PI in the loop and get them on board. If applicable, you will LIKELY -- not necessarily -- need them to 'sponsor' your endeavour.
Edit
It is okay once 'ethics' says it's okay, which it likely will. This is minimal risk behavioural research that will likely see a quick turn around time. Also, the ethics application will require that you share with them the survey you are distributing. As you likely know, if you are using Qualtrics (or equivalent), it should be straightforward to download the questionnaire/survey. Search "YOUR INSTITUTION" AND "ETHICS" on Google. No doubt, you will find the right contacts, and I'm sure they'd be delighted to provide you with additional information, such as sample applications.

In thinking about anonymity, consider the following: For several years, I was the only female computer science PhD student over age 50 at UCSD. A series of apparently harmless demographic questions would identify me.

I will definitely take into consideration your advice and look more into the ethics issue. But my question was more directed to how one distributes their survey to obtain a reasonable response rate while bearing in mind the survey population.

@unelunatriste What Speleo didn't quite say is that in submitting the application (which let me be clear, is required), you will have to say how you're gong to distribute it. When you look up their guidelines, they may have restrictions you need to look into.

@unelunatriste You are correct, I primarily addressed your second question. The reason for this is that how, where, and to whom you will be able to distribute the survey in order to get a 'reasonable' response rate depends on what the ethics will approve. An email list for your field sounds like a great idea. We don't have much other context to go on, in terms of suggesting other methods of distribution that would be practical for you. Are you a student, professor, are you a member of any scholarly associations in your field, etc.? Also, define reasonable response rate for your purpose.


text:
What does it mean for a (assoc.) professor to have very few first-author publications while directing a lab/group?
I've stumbled upon various professors (R1 universities in CS) having almost no first-author publications for multiple years while directing (or co-directing) their research groups/labs.
From the perspective of a PhD applicant, this might mean two things:

They spend so much time and  effort advising their PhD students that they do not have time for their own first-author research. This is a good thing. (Their students' research is also the professor's research in a way though.)

They are either unproductive, or spending most of  their time doing things outside academia (consulting, starting companies etc.).

I was wondering if this was normal, and whether there were other possible explanations for this type of a publishing record.
In other words,  should this be a "red flag" when looking for potential PhD advisors?
Edit: This is not a theory-focused field. So the author names are ordered based on their contributions.

You may need to specify the field a bit further. If I am not mistaken in the more theoretical parts of CS, never being the first author simply is an indication of having a name with a first letter that is found later in the alphabet. So you may need to look more closely at their publications and check if the author list is sorted or not.

@mlk Thanks for pointing that out. I edited the question.

In other areas, biology for instance, PIs names will appear last in the listing.

Does this answer your question? What does first authorship really mean? (See the CS specific answer in that thread.)

They are usually last author in many fields and countries

The natural followup question is whether this is a field where a prof might actually "lead" their own project. In some fields (CS, epi), profs can have their own "independent" side projects where they're first/only author. In other fields, the logistics are so daunting that this is nearly impossible: it'd be hard to fit biological time points around meetings, teaching, etc.

The authorship standard in my field (neuroscience/biomedical) is that the first author is the person who primarily did the work, often a graduate student or post doc; the last author is the person who supervised the work, often a professor. Middle authors are seen as having more modest contributions, but their actual impact varies quite a bit and it's impossible to judge just from the author list. In this scheme, you would be better off judging a professor's productivity by considering the number and significance of last author publications they have.
CS has varying authorship conventions (in theoretical areas closer to math they tend to follow the math convention of alphabetical authorship, for example), but my understanding is that in at least some applied areas of CS they follow this same convention. It would actually be pretty weird for a professor to have a large percentage of first author publications in my field, it would suggest they are not letting their students get proper credit for the work or are not advising students.
I would not, however, characterize this as "they do not have time for their own first-author research"; rather, their role in their own research has changed and now their own research includes working with students. You don't need to be the first author for it to be your research.

This is certainly the answer as far the disciplines are hard science. In all chemistry, materials science, and, to the best of my knowledge, physics and engineering, the standard is that leading scientists are last authors. Regional exceptions do exist. And other situations as well. The chemistry group where I did my master adopted alphabetical order by default. OP should look at the last author position instead.

+1. I would add that some fields of biology use "corresponding author" to indicate the "most involved" of the senior authors. Others seem to consider it as denoting nothing more than poor soul stuck dealing with the online submission portal (and, a contact point for questions).

I suggest not putting too much emphasis on this one factor. The reason is that there are just about as many reasons for this as there are associate professors. You've named a couple of them. Let me add a few more.
Some recently tenured faculty need a break, especially if their path to tenure was especially arduous.
Some recently tenured faculty want to change their research focus now that they have the freedom to do so. This takes a while and may seem unproductive from the outside.
Some people contribute collaboratively to the work of others and are happy to let the others take the major part of the credit, especially since their own position is now secure.
In some collaborations it is difficult to say who was the "major" contributor. As you note, some professors are happy to let their students take the lead.
Some professors just aren't enamored of the "paper chase" and are happy to do research for its own rewards.
But if they are leading a lab, and the lab as a whole is productive, they might be ideal advisors. They aren't as likely to step in front of you at the last moment claiming first authorship. I'll guess a lot of "first authorship" by professors is demanded rather than earned. I'm relying for that on questions asked on this site, rather than personal experience, however.
I contributed quite a lot to a lot of the projects of others and was happy to do so. I was secure in my position and didn't need any citation count heat index confirmation of that. The work was good and made contributions. I doubt that in most of it anyone really thought about authorship order (one exception where it was very clear - to all of us who had the lead). In particular, I don't appear on the dissertations/publications of my doctoral students (other than an ack), nor did my advisor ever consider joining me in authorship on my work, though he did contribute to it.
In choosing an advisor, think more about who can be helpful in the short and long term. Your reputation is your own, it doesn't derive from that of your advisor. Ask around (other students) to see what sort of support you are likely to get.

I had a friend on the faculty (tenured) while I was a grad student. He predicted at the time that he would never make full professor. It turned out to be too pessimistic.

In many experimental fields, the first-author papers published by a professor are also different types of papers (reviews, perspectives, op-eds, etc), not the data-driven papers that a trainee would work on.

Maybe there are sufficiently confident in themselves and their work not to need the recognition that comes from being 1st author.  I will certainly give authorship priority to my students whenever it’s reasonable: my contributions tend to be more at the conceptual than the calculational level, and a 1st-author paper will have more i pact on their CV than mine.

I think it is also a matter of attitude and personal choice.
To give you and example, both my PhD and postdoc advisors are well-established with about the same number of publications per year but have a very different publication profile:

My postdoc advisor has no first-author papers from the last 5 years or so: they have lots of students and prefer to supervise lots of projects. Most of their papers are as senior author.
My PhD advisor has less students and prefers to also pursue projects that are interesting to them resulting in first author papers (about one per year) and some last author papers.

From a student's perspective, it is a very different experience:

My postdoc advisor's group is large, hierarchically structured and research lines and projects are very well-defined. Lots of hands-on supervision and help from the supervisor. The PhD students don't have to stress about what to do, it is quite clear to them.
My PhD advisor's group was smaller with much more interaction and collaboration. Students were expected to be more independent and responsible and propose projects. Students could always ask for help or advice but one had to ask.

Which kind of supervisor is better, depends on the student. So, to answer your question, the lack of first-author papers is not an indication of a better/worse supervisor but it could be an indication of a different type of supervisor.


text:
Publishing Unpublished undergraduate research after a decade
I want to see whether I can publish my undergraduate work after a decade.
Context:
This is to support my academic career as I am planning to apply for a PhD program (Biophysics/Chemistry/Structural biology) in the US soon. I do have an MA from a highly ranked Public Ivy (mastered out from PhD due to health issues). I am targetting a few very competitive programs.
The last two years of my career are a bit productive. I got two coauthor papers this year (Angewandte Chemie, Analytical Chemistry). Most probably get two others (Science, Angewandte Chemie). Around 15 conference abstracts published (citable). But on papers, my name is in the middle as I am just a research tech (except conference abstracts where I am the first and presenting author for most).
Thus, I am not satisfied as I don't have any first-author papers. Then I realized that I can publish my undergraduate work.
In my undergraduate research, I synthesized a novel hydroxamic acid hexadentate ligand and then carried out a speciation study of its iron (ІІІ) complexes using UV-Vis spectroscopy. Only one type of ligand-iron (ІІІ) complexes was revealed in spectroscopic studies. Then I was supposed to verify my results by using MD simulation. I have not completed this part. However, I can complete this now.
I did this work a decade ago and only submitted my thesis to fulfill a degree requirement. However, I believed no one synthesized the compound after that.
I am interested to know how to go and determine

work is publishable in quality

If it is good enough, how should I proceed to finish up the work and publish it?

Also, whether I can file a patent for this compound (I can find some useful application for the compound)

Another objective for doing this is to learn the process of publishing a first author paper and process of filing a patent rather than going after any commercial interest.
I got my undergraduate degree in Sri Lanka, and my PI never bothered to publish the work or file for a patent as he does not have an active research group. Moreover, most professors in Sri Lanka are only interested in teaching.
I appreciate any advice on this matter.

Update #01
Thank you @buffy
My priority is to get a first author paper out before the 2021 application session starts ( At least submit it). The patent is secondary, but I would like to get it to strengthen my green card application.
The department I work for now helps with patent filing, and they have dedicated staff for that matter.
My biggest issue right now is to find someone that can assess the quality of the work. My current PI is busy and often delay answering to my questions related to current work. So I do not have high hope there.
Also, I am mostly doing biophysics/physical chemistry/analytical chemistry/Biochemistry related work now. Thus regarding this particular work, I am not sure how to judge the quality or which journal to target.
Would it be appropriate to email a few professors that do similar work for guidance?
Update 02
Thank you @kosmos
I have contacted my supervisor and previous collaborators. I am waiting for their response. I will see where people published similar work and go from there. I am not interested in paying someone to proofread; instead, I would like to get an idea about the quality of the work. My current research is on the modulation of G-Protein–Coupled Receptors (GPCRs) by soft membrane matter. Thus, I have minimal exposure to physical organic chemistry to answer that question myself.

You have a lot of questions to answer before you proceed.
If you intend to patent, you should start with that, or at least note that if you publish first, then in some places (US) a clock starts and you need to file a patent within a fixed time period or you work could become "prior art". You also need to know if the university is the proper party to file the patent. They may have rules about that, which you implicitly agreed to earlier. And patenting can be an expensive proposition, require advice from a patent attorney.
If you intend to publish, you need to determine whether others should be co-authors, and, if so, get them involved. Be careful of issues of plagiarism here if others contributed.
As to the required quality for publishing, the editor and reviewers of a journal will give you that information after you submit your work.

Thank you Buffy!

I got my work published after a decade. I touched base with my previous supervisor and he agreed to proofread my work. There is no stopping you from doing that. But as pointed out in another answer you must talk to your previous collaborators and ask for their permission to include their name if they are interested, or otherwise.
The same applies for a patent. You cannot own it by yourself if there are other collaborators. You would need a patent lawyer and it will cost you money, both for the lawyer and patent fees. Also, your Sri Lankan PI might have other issues, e.g. his university might not allow him to own a patent as they might claim that the work came out of using their resources, which I believe should be the case.
As for getting your paper proofread, you could get that service for pay. I believe there are many freelancers providing that service. You can consider that option, though personally I never did that and I do not prefer it. Since your previous PI does not try to get his papers published it is unlikely that he would be able to do a good job in supporting you even if he agreed to.
Look for the impact factor of the conferences and journals to learn about their quality. And then look at the papers in those journals to gauge the quality of your work. Check if any Nobel laureates or famous chemists have got their papers published with them. It should be fairly easy to determine the quality of the publisher. I generally send my paper to a target journal/conference which is best to start with. If it gets rejected, depending on the comments I downgrade it to the journal of poor quality or keep persisting with quality journals if I get good reviews. Though I do not study in your area, I believe the same process could work for you.
As you might already know this process of getting published requires time and patience, so if you need it by 2021 you must hurry up. Quality journals/conferences take a lot of time to get back to you.
Best of luck!


text:
Should I be concerned that my (successful) application to grad school only barely survived the first cut?
I am starting my PhD in a few days. One of the faculty told one of his students that me and another incoming PhD student just made it in terms of the initial admission ranking, i.e. we were in the bottom of the pile among those who made the cut. The reason that we got in was because the top candidates they sent offers to didn't come and there were professors asking for us specifically.
I know I could report this professor for his behavior but I'm not gonna do that. However, I am wondering if initial admission ranking means anything in terms of actual PhD performances. My field is statistics.

I would be massively surprised if the answer did not entirely depend on institution-specific and other volatile/arbitrary factors (for example: how many strong applications were there in the specific year?)

@cag51 The "first cut" isn't this the last cut?

My thinking was that we know the "initial ranking" was low, and so they barely got in according to this "first" cut. But yeah, there's probably a cleaner way to write the title....

Regardless of whether you report the professor, be very careful with confidential information. He may be an outlier, or the department culture may allow inappropriate communication.

@PatriciaShanahan Yeah, that's how I feel. It depends on the demographics of the faculty as well. Native faculty don't seem to gossip these kind of things. But non-native ones seem to do this a lot. At least I heard a lot of stories as such.

we were in the bottom of the pile.

Not really! You were in the very top of the pile. Many departments get more PhD applications than they can accept. Typical acceptance rates are about 10%, although I don't know about statistics specifically.
So if they offered eight spots, you were applicants #9 and #10. That's good. Unfortunately, when it comes down to choosing eight people out of the top 20 or so, it comes down basically chance. There is no objective way of measuring how good a candidate might be. So there is likely no real difference between #9 and #10, or more critically, between #8 -- the cutoff -- and #9. That a professor was asking for you specifically means a lot more than the overall ranking come up with by a committee.
Because "ranking" in the top 10% or so is meaningless, it doesn't really correlate with performance. Likely, you and you cohort study different enough things that it's hard to compare anyway.
It is quite inappropriate for a professor to reveal students' "rankings," especially to other people, though.

Would the downvoter care to make a suggestion for improvement?

I think the meaninglessness comes from the fact that quite a bit of the scoring system is stochastic. A person who scored 89 could just as easily scored 91 and the difference between these people is quite arbitrary. Those two points could move you from reject to accept or from “just barely accept” to “clearly accept.”

@Dawn Well, yes, that was my point. I've never actually heard of a department that assigned explicit points. (But, I'm just a PhD student!)

I could tell where you were going with it but maybe emphasizing the stochastic nature would make your point even more clear. Just because person #8 had a slightly better draw in some sense doesn’t mean that person #8 is any better than person #10.

"Most graduate departments get many more applications than they can accept."  Not sure I believe that.  Some do.  But perhaps not most.

@AnonymousPhysicist My suspicion is that almost any department will strictly speaking get a lot more applicants that they have positions for, but not qualified applicants. If you filter out the applicants that don't fulfil the formal requirements and those that don't demonstrate the necessary background, you are often not much above the number of positions, if at all.

First, at my R1 math dept in the US, we no longer pretend to linearly rank grad applicants. That's ridiculous and meaningless.
I'd wager that the faculty person who made such remarks has some sort of resentment of grudge about some demographic of students... and tells his/her own students that [some demographic] of students is inferior to them...
(I'll not go into details about my observations of such behaviors... tho' one stereotypical riff is non-US-origin faculty to disparage lazy americans...)
In any case, although some of my colleagues here do definitely like algorithms and numerical rankings, all my (multi-decade) experience indicates that these numbers are not merely not good indicators, but just worthless. They test many things, but not what we want for a PhD program in mathematics, of whatever sort.
So, yes, there are people who have poor enough judgement to say such things, but it's basically disconnected from reality. Still, even deluded people with enough power can make their delusions be operationally real, so be alert... sigh...
EDIT: among other things, in my R1 US math dept, by now we do not pretend to linearly order applicants. It's more like a first-pass that tries to assess whether people would succeed in our program, or not. Obvs don't want to admit people (for their benefit and ours) who are not at all prepared, or maybe don't have the interest or ability. But among those who could meet our expectations ... how to "rank". There's really no sane system. GRE subject test scores have shown their own irrelevance, ... GPA? Whah? Predicting what 22-year olds may accomplish based on the previous few years of their life is ridiculous.
In my 35-year involvement with grad admissions, I find that the only reliable assessment method requires "reading between the lines" in personal statements and letters of recommendation. How in the world could I describe this in algorithmic terms? (Ok, I could make up something, especially if litigated-against, but, c'mon...)
Yes, I am all too well aware that many programs do behave as though some algorithm is sufficient. In fact, our program plays against that by looking at candidates whose "typical algorithmic" status would be not-so-high, but whom we can anticipate would be great students. :)

Thanks for the advice. I have a feeling what you suggested was what the faculty thought. He did his  undergraduate from non-US school and the students in the top of the pile that did not come in the end are from schools such as Tsinghua and Peking University. I myself came from China but I did not do undergraduate study in China. So I have a strong feeling that what he had in mind was that the fact that I did not do my undergrad study at a prestigious school so I am somehow inferior.

Ah, yes, indeed, anthropologically, evidently, many human beings are sensitive to "status"... even if it is disconnected from function.

In fact, I have a feeling that within the ranking algorithm, the presitgiousness of the previous schools has a strong weight in the final scoring

Sure, if some numerical algorithm is being used, surely the "status" of prior schools will enter. Sure, this is not UN-correlated with quality of available coursework and insight of faculty letter writers, but it's really not predictive... in my experience.

I totally agree with this. Fom my own experience, I have a classmate from my undergrad who went to PhD to the same school as me but two years ahead of me. He is now actually the top student in my department..

@paulgarrett: Do you think success in math olympiads plays a role? I suspect that I know your answer. But it is true that a lot of prominent mathematicians were gold and silver medalists at the IMO.

@Mehta, well, of course, any sort of PR is a plus... it will get peoples' attention... but (as I know by now) most of those fun adolescent competitions don't give any kind of indicator about future success in serious mathematics. Sure, yes, contests show a certain sort of enthusiasm, and quickness, ... but that's not quite mathematics is about...

Interesting question. My guess is that 'bottom of the pile' matters, and matters more at universities that have diverse scores. It would be hard to answer directly.
Departments do not generally release this information. This is in part because the process may be far more organic than they would like to broadcast, with individual professors raising their hands for students for a variety of factors that are not directly related to scores. This is almost entirely because of the fidelity of these tests to determine success, and the realities of test sensitivity and specificity at relatively high values. Although scores matter (a scholar search for 'admissions statistics to graduate student success' will show you this), for a good program students tend all to be at the top of the pile.
To sort the top of that pile, advisors look for other metrics of likely success, most notably the idea of 'fit' to the professor's teaching style, personality, and areas of interest. The 'bottom of the pile' your friend ended up on may have been related to scores, but also may have been related to the advisor's understanding of 'fit'. Certainly, this advisor seems not to be happy with the match, and is doing themselves and their student no favors in terms of making it work.
The unsavory impetus for your question aside, the very reason that this professor's behavior is improper (and cruel) is also the reason the question may be challenging to directly answer; it is all a bit taboo. But, you know, if you did answer this taboo question empirically, I think it would be a useful contribution, and statistics would not be a poor field to answer it within...

we were in the bottom of the pile...[does] initial admission ranking means anything in terms of actual PhD performances?

I would clarify this a bit. Let's say there were 100 applicants for 8 spots. What likely happened was that some formula was used to rank applicants from #1 to #100 (the details of how this is done will vary widely). Now this formula is far from perfect, so the program will not just blindly follow it -- there are lots of other factors to consider, including the matching between the advisors and the students. But anyone ranked below, say, #35 with this formula is unlikely to get one of the 8 slots, so those applications will be set aside. This is an easy and objective way to reduce the number of applications that need to be further considered.
It sounds like your rankings according to this initial formula was only just above the cutoff (say, #33 and #34). This puts you in the top 1/3 of applicants, which you should be proud of. Better yet, it got you into the next round, where the professors decided that you would be a good fit for the program.
Now, is the #1 student more qualified than the #35 student? Probably. But between the #25 student and the #30 student, the difference is probably negligible. Students are multidimensional, which is why the final decision was made by humans, not by the formula. Further, the elements in the formula (undergraduate grades, undergraduate research experience, writing skills in the SOP) are only loosely correlated with the skills necessary to be successful in graduate school. So, I suspect the correlation between one's initial formula score and one's graduate school success will be similarly small but positive.

Telling you that only barely survived the initial cut was perhaps tactless, but I don't think it could lead to disciplinary action. The faculty member in question told another student the OP's "rank." I doubt it would lead to disciplinary action, but IMO that's ruder than "tactless."

Ah, good catch, I had misread and thought the faculty member had discussed this with the students in question. Yes, I agree that is definitely "ruder than tactless." I'm still skeptical that "reporting" this incident would lead anywhere, but I have edited to remove that whole section.

In addition to the already good answers:
The criteria used for PhD admission are only weakly related to what it takes to do a successful PhD.
Doing a PhD is about doing research. (In the US, not in its entirety, since you still take courses, but that's what it is really about.)  Admission is based on grades in taking courses and passing the corresponding exams, together with some other criteria.  Good performance in exams is only weakly related to being good at research, as are the other criteria.  So just because other people did better in their exams, it does not mean that they will be better researchers.

It depends on the field, for engineering as long as you have a degree you will benefit. For areas of pure science or mathematics I would be worried, mainly because the jobs just aren't there - even generic teaching jobs.
I guess this is independent of the issue of degree completion but on a practical basis I wouldn't suggest theoretical physics or mathematics to anyone who isn't at the top, top of the field because you're going to end up doing something else after graduation.