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How to ask someone to collaborate on a joint paper?
So far I have worked on papers by myself and have been the sole author in all of them. There is one project that I am interested in and have been working on it for a while with no significant progress. It seems, I'm  definitely going to need some fresh thoughts. In our department there is absolutely no one interested in that subject.
My question is how exactly am I supposed to ask someone to collaborate with me on a joint paper?
In case it matters, I work in field of Mathematics.

Related question :How to find someone online to collaborate on a joint paper?

How mathematicians collaborate depends (among other things) on their career stage.  At the student level, the most common kind of collaboration could be more accurately described as a form of mentorship: you are working with a more senior guide, and that they mean to help you with your work and professional development is a key feature.  This is different from the kind of collaboration you do (in most cases) one you get past the student / postdoc stage.  At that point it is fair and only natural for someone who is considering collaborating with you to ask "What's in it for me?"
The path of most graduate students is one of decreasing collaboration of the mentorship type (in math, mentorship collaborations do not always result in the mentor being the coauthor, so e.g. it is not necessarily viewed as inconsistent for an advisor to talk about "our work" when the final product is a paper written only by the student) and increasing independence from your mentors.  Your trajectory sounds a bit different in that you already have solo papers (although, as in the previous parenthetical, that may not describe the whole story).  Another feature of student research -- the key feature, really -- is that whether your advisor is joining you in doing the mathematics or not, she is supposed to be involved in helping you choose problems to work on that she finds, by virtue of her greater knowledge / experience / cultural familiarity / political savvy, to be especially worthwhile and significant.  A student who works on a problem for a while with no significant progress raises a little red flag in my head. One must ask: why are you not working solely on problems that either you know you are making progress on or your advisor is there to help you make progress on?

In our department there is absolutely no one interested in that subject.

Another little red flag.  What you are doing is not inherently bad but seems more risky than normal.  I fear you are getting less than the standard amount of "mentorship collaboration" and I think you should consider moving somewhere else (as a student, or possibly graduating ASAP and taking a postdoc) where you can benefit from that.
Anyway, there is a different kind of mathematical collaboration.  (By the way,  @Significance has done a good job of describing it in her answer.  But I want to write as a mathematician and in some more detail.)  If you want to start a collaboration with someone of this other sort, then typically this is founded on either significantly overlapping research interests or complementary skill sets, approaches and/or partial results.

Maybe while reading a draft of a paper you see how to strengthen one of the results.  In this case you can (perhaps) get added as a coauthor on a later draft, and especially if you are joining more than one other author this can be a good way to piggyback onto a longer term collaboration already in progress.  Here is an example of mine.
Maybe you have a draft of a paper and you see that your Theorem 2 is a variant of their Theorem 5, so joining forces is a natural choice, if only for reasons of smoothing out the publication process.  Here is an example of such a collaboration of my own, in which we each had overlapping manuscripts that we combined.  I was happy with the resulting paper, which is a little more than the sum of its parts, and remain on good terms with the coauthor, but that was the extent of the collaboration.
Maybe you discover that someone whom you previously did not know (e.g. a recent PhD) is working in the same area as you, or even on the same problem.  Learning that someone else has been working on the same problem as you and that you each have partial results which have been obtained using (even mildly) different ideas and techniques can be immensely helpful.  If each person's perspective is rich enough so that the other cannot readily incorporate it just by reading the other's work, then this is often the beginning of a very fruitful collaboration.  Here is an example of mine.  Or maybe you run into "someone new" and discover that you both have aspirations to work on the same thing, and though you don't have finished product, you each like the sound of the other's ideas and think it will be better to work together.  This is one of the purer forms of collaboration, but it seems to require a rather fortuitous set of circumstances.  Here is an example of mine.
Maybe you have a really attractive half of a result, and you're looking for someone with distinctly complementary skills and/or results to form an attractive whole.  Again, this can be hard to swing, but it's satisfying when it works.  Here is an example of mine.  In this case, I had 2/3 of a new proof of a theorem, so I wrote a mass email to my own mathematics department asking if everyone could help me out.  One of the other students in the department wrote back within a few weeks with an answer.  I added a lot of other material, wrote the entire paper myself, and still feel like I really lucked out.
Maybe you have a question that's truly attractive: a question that no one else thought to ask and that you happen to know that the answer would be very interesting...but you really can't answer (or anyway, aren't trying to answer: it comes to the same!) yourself.  In this case you can try asking the question as widely as you can, and if the answer comes, you can try write a paper together.  This is the least likely scenario yet.  It seems that a lot of amateur mathematicians think mathematical research works this way...but it usually doesn't.  "Here's what I want to do, just tell me how to do it" is usually not a route to publication in mathematics.  (Maybe this is what people on this site mean when they say "An idea is worthless" and I start to flip out in response.  In mathematics "an idea" doesn't mean an idea for a problem; it means an idea for a solution!)  I do have an example of this.  In fact this originated with a question I asked on math.SE. This was quite an unusual situation.  Among other things, my coauthor is not a professional mathematician or academic, so he would never have written this up for publication on his own.  The journal we published in accompanies the articles by little author bios.  His was: "[NJD] is a private citizen of the Netherlands."  But he is a private citizen of the Netherlands who has the skills and abilities of a clever research mathematician.

I don't claim that the above list is exhaustive, but it is certainly long enough.  When I look back over it, I am struck by how many highly positive collaborative experiences I have had, but also how large a role chance and serendipity played in allowing these collaborations to be realized (and it makes me wonder how many collaborations didn't come to pass because I was not luckily connected with the right person at the right time.)  I hope it gives you some ideas about how non-mentorship collaborations can get started.  Good luck.

Amazingly insightful.I glanced at your post on overflow.(interesting problem by the way)well in my case as you probably noticed, the advisor of mine and I never could relate to each other.In fact he is not interested in things I like and vice versa. Long story short, I improved someone's result( the author of original paper was very nice and helped by giving comments on my manuscript) a little bit and published it.Now im trying to do it for general case which is an open problem.

I'm not a mathematician, and in my field sole-authored papers are unusual, so my answer might not be entirely relevant, but for what it's worth:
You don't usually ask someone to collaborate on a paper because you are stuck. Either the collaboration arises naturally through a joint project, or you ask someone to join your paper because they have some specific expertise or experience that you think can add to what is already going to constitute a good, publishable paper. In the latter case, I have had success by sending an email briefly outlining the goals of the paper, its importance, and the timeline on which I intend to have the paper completed. I finish the email by stating what I think they could add to the paper and asking them if they are interested in a co-authorship on this basis. I might already have a draft of the paper ready, and if so, I will attach it so that they can see that I am serious and the paper has a good prospect of timely completion.
Also: a bit of flattery doesn't go astray. Refer to "your important work on..." or "your extensive experience in..." among the reasons for approaching them.
If in fact, you are not approaching someone to collaborate because of something specific that they can add to an already solid paper, but merely looking for someone to help with a work in progress, I suggest approaching someone at a junior level who already knows you, and asking for help.

The other answers are good, but I'd like to make a few other points.  (Like Pete, I'm also a mathematician.)

Some problems are just hard, and you may be ill-equipped to attack them.  Sometimes is just best to put them on the back-burner.  Lots of projects get abandoned, or at least put to the side, and if people didn't do this, many would be stuck forever.
If you're a grad student/postdoc, maybe you should try to work on problems people around you are interested in.
If you do want to still pursue this problem, try to either visit people who might have ideas and talk to them about it or invite them to come visit you.  Special semesters/summer schools/conferences are also great.  Then talk to them about your project, and see if they're interested or if they have any ideas.  Many times, a good suggestion from an expert, rather than a full-fledged collaboration, is all you need to make a break-through.

I know the first two are non-answers to your question, and I can't tell how much energy you are putting into this particular problem, but it's good to take a step back and think about what you are thinking about doing.


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Do I need to make a Bachelor's thesis to apply to a master's degree from a top 100 University in the world?
So let me tell you my story: I studied Industrial and Systems Engineering in the 24th best university in my country, which is around number 1683 in the world. I am from Mexico and thankfully our government is eager to pay our full tuition if we are accepted to one of the best 200 Universities in the World to study a Master's Degree, mainly in engineering.
My main challenge at this moment is to be accepted in a masters' program at an International University. My grades are good, I have a "promedio" of 92, which is pretty high, although not the highest of my class, and I had outstanding results in the professional test. Because of these results, I was exempted of having to do a bachelor's thesis.

Should I sacrifice my opportunity of earning working experience to dedicate myself to make a good Thesis? I am eager to do so if I really need to.
Thanks a lot to the people that took the time to read this.

Ideally, you should have one question per post... And some of these questions, only you can answer. You wrote a fair amount of text (don't forget the commas!), and you described a delicate situation. If you could restrain the questions and provide only the relevant information, the answers should be more informative... In the meanwhile, check those universities to see what is usually involved in their masters'. You should get a better idea of what is a masters' and what you'll need to do...

Thanks a lot, I just erased the questions and made it a little bit more brief.

I've never been on an (academic) admissions committee, but I can hardly imagine that, instead of reading the title of still another bachelor thesis, I find "exempted by university from writing bachelor thesis because of my course performance near the top of class," and consider that a negative. So I don't think you should sweat that. In most fields, undergraduates are not expected to have done research prior to taking up graduate studies; exceptions, as I have learned here, include math where this appears to be more commonplace. For some of your target fields (eg, supply chain management), I know it isn't, or at least wasn't, as some of my research was arguably on the most theoretical end of it, but I had no relevant experience with it prior to writing my paper; but that could have changed. I don't know about the others.
In terms of being accepted at a good school, the ranking of your alma mater is probably what I would worry about most. You'll need strong letters, ideally from people known internationally (check their publications; ask them). You should definitely approach someone from your U.S. experience for a letter.
Finally, your first, second, and last question (as well as part one of your question 4) are highly personal decisions that you are unlikely to get feedback on here. They are off-topic, and even if they weren't, I wouldn't ask such questions on an anonymous internet forum. Ask friends, family, any mentor, and yourself.

This doesn't quite fit your edited question. I'll keep it up a bit; maybe some is of use.

Your answer is of a lot of use, I edited my questions because I had asked too many and I thank you for answering them.

While it probably really won't be considered that negatively, one should still keep in mind that "course performance near the top of class" is still course performance, which - depending on what kinds of tasks courses in the respective university system include - is still something very different from the experience of writing a thesis (and I'm not even referring to theses that require any real research), and at least top performers in exams can nonetheless have a very hard time when writing a thesis (or the other way round).

@O.R.Mapper: I agree, and there are certainly fields where it will count negatively. I just think that if it is university policy to exempt extremely high performers from writing a thesis, it would be unfair to hold it against an applicant; and I want to believe most departments will see that similarly, That said, you are certainly right: it's a different skill set with evidence missing.

@gnometorule: Absolutely, I'm aware the student has no influence on such a policy. (And it is of course still open whether courses maybe included something somewhat similar to a thesis, anyway, maybe at a smaller scale at least.)

Different countries have difference expectations at the Bachelor's level, which means they have different requirements for entry into Masters. For example, it is unusual to do research as part of an engineering degree in Australia. I therefore suspect you could apply to an Australian university to do a Masters in Engineering without complication, and there are several in the top 200. In addition, you would be coming with funding, which (I am sad to say, but good for you) makes you a more attractive applicant.
I suggest you approach it from the other direction. Get the list from your government of which universities it considers acceptable, then look at a couple from each country in the list to see which ones seem to not require research. Once you have identified suitable countries, look at all the universities on the list from that country and find a program you like.

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Alternatives to sewing (or any habit) to stay focused in class
I struggle with staying focused during lectures. However, while listening to online lectures, I started sewing and it has made life so much easier. I am able to listen and follow along. When the time comes to do practice questions, I do a lot better than I usually do.
So the problem is, I cant take my needle and thread into an in-person lecture. Are there any alternatives that I can try? Or should I just talk to my professor and see if I can sit in the back of the class and sew? I don't want to seem disrespectful, but I've tried so much to improve and this has been the most effective thing by far.
Taking notes doesn't usually help. I get so caught up in taking notes that I miss the material. If I just try to listen, I zone out.
Edit regarding duplication:
I saw a similar question someone posted asking how to focus in class. I found the way to do that. My question is more like, "What is an alternative I can do that won't make my professor think I'm being disrespectful? (Looking for something that is academically acceptable) " Or is it appropriate to go to my professor, explain the situation, and ask to do what helps me?

Comments are not for extended discussion; this conversation has been moved to chat.

If sewing helps you learn, consider bringing it to an in-person lecture.

In a large lecture hall, I seriously doubt anyone will notice/care
In a more intimate class, it could be worth asking the professor if they mind.

While you might have a bad experience with a grouchy professor, I think the vast majority will either understand or not care, particularly if you are clearly engaging. Personally, I would rather have a student paying attention while sewing than on their phone.

I would also add to this that you should really not put any consideration into how you are perceived by your classmates for sewing - although I’d be mindful of how distracting it could be, so choose your seat with that in mind as well. It’s unorthodox, admittedly, but you are paying for your education and if it helps you learn then try it out - if it doesn’t work in a bigger setting, then now you know.

Thank you both for your feedback! I really appreciate it. It has given me enough to figure out what I should do moving forward.

One tip based on what I've heard/seen before -- it may come across better if you phrase it as "I focus better if I have something to do with my hands". That is, you're not paying attention to it (instead of the lecture), and you're not doing it because you're trying to multitask -- you're using it as kind of accommodation. I think this sort of thing more generally is pretty common (someone mentioned knitting below, which I think is not unusual, but there are other things people do. I am a doodler, myself, which conveniently looks a lot like notetaking so it tends to pass unremarked.)

I've known several people who would sew or knit in class.  Never seemed to bother anyone.

Courtesy to your lecturer is only common sense & good manners; but also remember that as a student you are the paying customer of a service. You deserve to have your reasonable needs accommodated. The university is working for you, not the other way around. A "grouchy" professor is an unprofessional one, in this context

@UuDdLrLrSs: Students being paying customers is only true in some countries; but, regardless, thinking of yourself as a paying customer is a guaranteed way to get the worst out of university.

@JackAidley education is paid for regardless; if its paid for in taxes that just means the "customer" is something like the whole society who pays for it on the student's behalf. So while the student then has a different moral burden, nonetheless the university still owes them a quality service, and you could even argue that this obligation is even more important in that circumstance. As to acting like a paying customer - based on my own personal experience I have to disagree.

As a professor, I would be fine with someone doing an activity like this in class, and even more fine if they approached me about it at some point to help me understand the context. (as long as it doesn't disturb people sitting around them—if knitting needles regularly clacked together for instance)

My wife knits in faculty and committee meetings. She can do that without looking at the needles, and so can look at whoever is speaking. I suspect you can't do that with sewing, so maybe that solves your problem.

I feel uncomfortable maintaining a deep conversation with a person knitting. Similar to a person chewing a gum, listening to music through one earpod, or writing messages on a smartphone. Perhaps, they are 100% in the conversation, but to me it feels that they are not fully interested.

@DmitrySavostyanov: I think you are correct. When your interest is fixed, the focus stays fixed too on the object of your interest. So a person doing a secondary task conveys that the primary task is not of much interest to him/her. (1/2)

However, the world today is such that we have the obligation to do numerous things in which we are not interested at all. For those cases, if doing a secondary task makes someone’s life easier, then I think as kind individual ourselves, we can let that go. (2/2)

@DmitrySavostyanov Not all of these activities take the same amount or kind of attention. Activities like crocheting or knitting, which don't involve language processing, can help someone keep more focus on the conversation and keep attention from wandering off.

@DmitrySavostyanov Obviously she stops moving her hands when she actually says something. But listening to a seminar or the department head appears to be something that she can proficiently do without being distracted while knitting. Others are reading their phones, and that's definitely distracting.

@WolfgangBangerth Not trying to justify the behaviour of others or teach anyone how to behave. Just sharing my own feelings. I might be wrong and all.

As a lecturer I think I'd be okay with a student knitting in class.  I believe it's less distracting than laptop and cell phone screens, which probably half of the rest of the class are looking at.  (I assume you're not doing both.)  Sewing seems like it would require moving your materials around periodically, whereas knitting is kind of a steady process.

Doodling has already been mentioned in the comments but in particular colouring in the squares in grid paper could be helpful. Or indeed drawing. From a distance it looks very much like note taking and you can add any noted necessary.

It's quite simple. You're at a university: the lecturer is there to deliver a lecture to the hall not to teach you. You're not at school anymore, the lecturer isn't going to (or at least shouldn't) tell you off for how much attention (or lack thereof) that they perceive you are paying.
You (or your government) are paying (for you) to be there so as long as you're not disrupting other people, what you do and don't do is entirely up to you. If such an innocuous thing upsets the lecturer, that's on them not on you.
If you're in a more intimate setting like a tutorial where you're actually being taught, just ask the lecturer if they mind.

To answer you question: You can tie knots, for example surgical knots like a one-handed knot. This can be done below your desk.

Not to detract from other useful answers, but: Have you considered other instruments which occupy your hands during class, but are less involved/bulky/pointy than sewing?
Some people use fidget spinners, fidget cubes - but apparently, those are noisy. You would need something to manipulate silently. There might be other manufactured "fidget gadget" which are silent, or as @LinkBerest and @Brian suggest, some sort of stone you could use.

These seem strictly worse than sewing. Perhaps you could give a reason why you think that this is a better choice?

@JackAidley: 1. Sewing is much more visible - the sewing needles, the surface being sewed, the material dangling to the side and possibly other affects. These items are just something which fits in the palm of your hand, or slightly beyond it. 2. Sewing involves sharp objects - needles. This doesn't.

Fidget spinners are fast moving and distracting, the little cube thing makes audible clicks as it is used. Both are substantially more distracting than sewing. I don't really see why the sharpness of needles is particularly relevant; it's not like they're credible weapons or any risk to people nearby?

@JackAidley: Didn't realize the cube clicks; are you sure? As for spinners - they're not distracting when they're under your desk, or for lack of one, a piece of fabric. I don't know that they're distracting at all but I guess that's subjective.

There are silent fidget toys.  The simplest is a worry stone.

Yeah, the cubes are worse than someone clicking their pen over & over. The spinners are worse: louder if spun on a desk - not how they are suppose to be used but if you allow them someone will try, very visually distracting to others (your suggestion leads to emails from other students about people "doing something with their hand in their pocket or under the table), more if you need.... I do like the idea of worry stones & other silent "fidgeter" objects but the spinners and cubes I ban from every class based on experience.

I'm sorry but I would feel uncomfortable if I saw someone knitting in my lecture.  It looks like what you are saying is not interesting to the person or the person thinks they can do something else whilst listening to you talk (ie. they can multi-task whilst listening to whatever trivial stuff you are saying).  It's eccentric at best and I've never seen anyone do that ever at the math department where I work.
My advice is it's fine but don't sit in the front row where the lecturer can see you doing it as they might get annoyed.  In terms of sewing, I doubt you can do that without looking so you are going to get some comments probably.  I have had comments before even about having my laptop out even though I was listening to the lecture, I'm just warning you lecturers can be extremely touchy about things like that and might take it half as an insult.

This can easily be dealt with in a classroom setting by letting the teacher/professor know about it ahead of time. Talk to them and tell them, "I am able to listen and learn better when I have something to keep my hands occupied". This is not an uncommon practice and if they know to expect it they won't see it as potentially rude. Also if it really does help you learn better the teacher will notice

Yes true, OK if you let the lecturer know before hand no problem.

I used to have a very quiet 15-puzzle, solving the thing was not the point, it was simply keeping my hands occupied. And this was something very small so less likely to be distracting to others.
I had one civics teacher who if she wanted to have an actual conversation with me would make sure my hands were busy with something, she had a number of small objects for the purpose, mostly puzzles of one form or another. A different 15-puzzle, rubik's cubes and the like.

Perhaps only slightly more than a comment, or echoing certain other answers: just a few years ago I had a very good student ask me if it was ok if they knitted during my lectures (which had notes available for later, etc., so note-taking was not strictly essential, perhaps), since otherwise their minor narcolepsy would sabotage them.
I was taken aback at the issue (which I'd not really encountered before in that context, although I did have a narcoleptic colleagues some years ago...), but said "Uh, sure".
The student was able to knit (furiously!) while looking at me and the slides, apparently attentively!
Given my own impatience with lectures and such, this made me rethink several things... E.g., how to "sit still" for 50+ minutes? Play along with a regimented agenda? Is this essential to actual learning?
Sure, some people rationalize their disinclinations to engage... but, in my experience (at least with people more mature than the 18-year-old middle-class kids in the U.S. first-time away from home... at college), in the U.S., most students are acting in good faith. Even if misguidedly in some details, at least "good faith" sets a good common basis for discussion about how to accomplish our goals.
Yes, years ago I did believe in a much harsher, conformist picture of "how things should be". Well, years ago, it was hard (in the U.S.) to "succeed" without such conformity, and it would only take more energy to push back. So, as usual, the people who managed (through gender, skin color, socio-economic class) to "succeed" well enough... had no pressing reason to complain or push back.
So, "no knitting or sewing in class". Sure, why not censure things, if you can get away with it?
So, yes, the issue of "what/how to help/allow students to really benefit from interactions" is subtle. The subordinate issue of "what offends the instructor" is also subtle, but subordinate.
More pointedly, if I know the student who needs something to keep their hands busy, etc., and trust in their "good faith" (and blanket respect for me and my attempts at teaching), I have no problem with any such thing.
No, it's not that simple, generally, because in these somewhat-corporatized-education times, we are not reliably put into situation where we can understand all our students as individuals.

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Skills Difference between Theoretical Physicists and Mathematicians
As I am preparing to start my PhD in theoretical physics, I have come to the realization that the more math you know (as a physics person), the better. But, I also realized that my physics brain is sometimes struggling to fully appreciate the subtle points in abstract math, in part because I was taught as a physicist to not pay too much attention to mathematical details. I am currently trying to get rid of this bad habit.
However, I am curious to know, what do you think are some skills that theoretical physicists have (beyond facts, formulas etc.), that mathematicians would have difficulty learning? In other words, if a mathematician attempted to become a theoretical physicist, and since physics is described through a mathematical framework, do you believe that the mathematician would have a straightforward transition?
My Opinion:
I think that theoretical physics is a combination of making good assumptions and mathematics. So I would assume that mathematicians would have a relatively hard time making sense of the assumptions in physics.
I am very interested to hear opinions from people that transitioned between the fields.

I work in the department of applied mathematics and theoretical physics... I think much of theoretical physics is just math, you just happen to be trying so describe physics with it. Some other mathematicians solve PDEs for modelling biology, but those are just applied mathematicians, not theoretical biologists. There is a bit of everything everywhere, but the Venn diagram between the two is very joint.

Your assumption is on the right track. I would only change one word: "...theoretical physics is a combination of making good assumptions and calculations". Calculations are not the same thing as mathematics. I just wrote an answer with my two cents!

my physics brain is sometimes struggling to fully appreciate the subtle points in abstract math - This is also true for my math brain, as a professional pure mathematician.

You might also want to read Trefethen's short book An Applied Mathematician's Apology. Mathematicians themselves can differ substantially from each other, quite apart from the differences between physicists and mathematicians.

From my experience as a physicist in math lectures, math students seemed to be  worse in doing explicit math  (evaluating integrals, solving differential equations, ...) than physicists compared to writing proofs.

Here’s a way to start: When you go to bed, are you thinking about physical models or a better basis of axioms?

My opinion departs rather significantly from the main line seen in most of the answers here, so I'll write another answer.
I have a PhD in physics and transitioned between physics and math many times, finally settling down with physics. I always say, jokingly, that there are no two fields as far away from each other as physics and mathematics. While this is obviously an exaggeration, let me tell a true story that happened to me in 2012:

I decided to take a course on measure theory while transitioning to the applied math program. I liked (and still like) mathematics a lot more than physics, loved the teacher and the subject, and was finally able to understand what was going on behind the Lebesgue integrals used in quantum mechanics. The first exam contained a question that looked absolutely trivial to me, regarding properties of a set in view of Carathéodory's criterion (if I remember well). My solution was only three or four lines long. Once I finished and left the room I met with math students arguing about how hard that question was, to which I replied "well, you know that the set does this and this, so there's the theorem that allows you to do that and that". Everyone stopped talking and looked at me, so I felt like a genius. Then one of them said "of course, but did you prove that the set exists?".

I got a zero for this solution. The teacher knew I was coming from physics, so he left a comment: "no pieces can be left hanging". It took me some time to understand that mathematics is about flawlessly connecting A and B, often with little regard to either A or B - the connection is the most important thing, and needs to be made with no pieces left hanging. In physics one is also connecting A and B, but often little regard is given to the connection - the focus is mostly placed on A and B. The mathematician will therefore spend their life perfecting their connection skills (think of a gardener adorning and taking care of a path with beautiful flowers), while the physicist will dedicate their time to coming up with interesting events that seem connected (think of an explorer looking for the highest peak in the middle of a dense forest).
Epistemologically I believe these differences stem from the fact that physics is a science, while mathematics isn't. Mathematics is not concerned with nature (although the mathematician might be); it is concerned with following a logical path (any path!) and figuring out its consequences. These consequences might be aligned with something useful to the study of nature (like the case of many PDE's or ODE's), but can also be completely disconnected from reality (like some weird field in propositional logic where negation is not defined [1]). This means that the As and Bs from the previous paragraph can be literally anything in math, while in physics they will most of the time be connected to an experiment [2], with the whole process fitting within the paradigm of the scientific method (which is irrelevant in math). Physicists, therefore, have a luxury unavailable to mathematicians: They can cheat and circle around a rigorous proof by directly comparing their guesses to outcomes of experiments. In mathematics the equivalent of this process would be to declare that your statement was elevated to the status of theorem by the will of a higher power and that's it, period.
Practically, a big consequence of these extremely different approaches is that mathematicians understand a few things in great detail, while physicists tend to have a much wider view (albeit coarse grained). A mathematician will also become stuck once the desired level of rigor cannot be sustained, and will waste time with minor details that a physicist will often skip over due to some glorified, hand waving argument that cannot be put into rigorous form. The physicist, on the other hand, will get stuck trying to connect events that, after talking to a mathematician, turns out could never be connected due to some obscure theorem no one knows about in physics.

[1] Funny enough it's hard to find a field in mathematics that is not relevant to physics, no matter how useless it looks. Turns out explaining nature requires a very high level of abstraction.
[2] There are many physicists working with As and Bs that are not directly connected to any experiment. Some of them are very close to mathematicians, but most of them are not. They are still using the often lax lines of justification adopted in physics, but now dealing with more abstract matters.

but did you prove that the set exists --- This is a good example of different approaches in math and physics, one that is especially prominent in diff. eqs. after the roughly elementary calculus level course. In math there is, by physics standards, an excessive focus on existence theorems. In physics there is, by math standards, an excessive focus on specialized techniques for solving differential equations and in being very familiar with the solutions of certain differential equations (e.g. Bessel functions, elliptic integrals, spherical harmonics, Hankel functions, etc.).

This answer goes to the heart of the matter. Physicists care how the world works. Mathematicians don't.

@user2705196: True to some extent. Although applied mathematicians occupy a position closer to that of physicists - they care how well mathematics models the world, for instance.

One of the most important things about getting to point B from point A in physics is understanding how it relates to the physical world. Specifically, is the relationship between A and B observable/testable in a real experiment...even if only notionally?

@djphd: Regarding your comment about "observable/testable in a real experiment", the theorem in The Large Scale Structure of Space-Time by Hawking/Ellis mentioned in my comments to this MSE answer may be of interest.

You say "In mathematics the equivalent of this process would  .. ." Actually, the notion of a "conjecture" is well-established in mathematical literature. Conjecture A can be used to prove Conjecture B provided A is more "expected" than B. However, conjectures are supposed to be clearly demarcated in mathematical literature, and such proofs rarely make it to textbooks. The latter are meant to be "unimpeachable".

" I always say, jokingly, that there are no two fields as far away from each other as physics and mathematics." I think math is by far the closest field to physics. Maybe chemistry in some cases depending on the sub-field.

@AccidentalTaylorExpansion In my opinion Computer Science is the closest field to mathematics. My statement about the distance between physics and math is due to biologists thinking similarly to chemists, then chemists to physicists, and the comes math and there's a big phase transition. Math is the closest to physics in this scale, but it's simultaneously so far away that is almost discontinuous.

mandatory XKCD @AccidentalTaylorExpansion I couldn't disagree more: chemistry and physics are much closer. As stated in the answer, the probably most fundamental difference appears that math is not science. Physics is, such is chemistry, bioliogy etc. They are all driven by observations and aim to explain phenomena. They all use different level of quantitative abstractions, with Physics surely using the highest level of math.

@Mayou36 I love the cartoon and it's funny that it basically illustrates the little hierarchy I mentioned in my comment above

I love the cartoon --- In case you missed it, see the circles made of this type of observation in the comments to this Academia answer from about a month ago, such as this extended version of the XKCD that @Mayou36 cited.

Extrapolating from coarse-grained wider views, we arrive at Electrical Engineering! Can I bring up Simon Bradshaw's breakdown of the wonderful Eric Laithwaite and his Brain-Eater Syndrome? https://www-users.york.ac.uk/~ss44/sf/laith.htm

@SusanW I have seen so many real-world examples of this phenomenon described in the story. So relatable! Physicists do it as well when they decide to talk about other areas of physics that are not their own.

This is probably a better fit for Mathematics Stack Exchange or Physics Stack Exchange, and might be too opinion-oriented anyway, but for what it's worth I'll give some comments about my experience of mainly studying pure mathematics (i.e. not any mathematics-for-physicists courses [*]) with substantial physics taken both as an undergraduate and during my first (of several) graduate enrollments. However, I've never actually transitioned between the fields.
[*] I have taken an undergraduate "separation of variables" PDE course and undergraduate complex variables, but no vector analysis (I often wanted to remedy this, but never really did) or numerical methods.
When reading physics texts I had much more difficulty in determining whether some random statement being made was supposed to be taken on face-value as an assumption/comment/observation or to be taken as something the reader is expected to logically deduce from previous results.
I also found that certain things that are of primary concern in mathematics (e.g. manipulation of infinite series) were almost always passed over without comment in physics texts. It would have helped me a lot if physics authors had sometimes inserted a few words here and there, such as "formal series manipulation results in", to let the reader know at what level of rigor the reader is expected to use in following a derivation of something. Moreover, I remember one explanation given by an author for truncating an infinite series to get an approximation was that the n-th term approaches zero -- had the author never heard of the harmonic series that is so prominent in every beginning calculus text? This puzzled me for a long time until I realized it was a Taylor series expansion, and the error in dropping (for example) all terms past the 2nd degree (quadratic order of smallness) is roughly speaking proportional to a cubic order of smallness. So the approximation was OK, but the reason for why it was an approximation was very misleading, at least to me initially.
I also had a lot of difficulty early on with the fact that words in math often have very specific meanings (e.g. series is a sum, sequence is an ordered list, vector is an element of a vector space that has been explicitly defined, etc.) and in physics these meanings were often different and certain distinctions were ignored without comment.

I relate to this. Very often I felt strange being one of the few people in class that actually cared about the "math tricks" physicists use to get the right answer.

@CosminA: I suppose you mean problem-solving techniques, such as differentiating under the integral (see my comments here). I suspect those with a math bent would be more interested in studying the method itself -- its limitations, generalizing it, etc. -- than those with a physics bent. I learned several such "math tricks" first in physics -- see comments here -- but this was before the internet and increased math competitions that has led to a huge increase in their familiarity.

@DaveLRenfro This probably isn't a good fit for Math SE, as it seems to run afoul of both the prohibition against asking for personal advice, and the prohibition against opinion-based questions.  It is marginal in either regard, but I suspect that it would not be well-received.

@Xander Henderson: When I saw the question I thought it would soon be closed, but decided to go ahead and make a comment that is now the paragraph with "difficulty in determining whether some random statement being made". As I was putting the finishing touches on my comment, I decided that it was really an answer, considered not saying anything (I assumed the question would be closed), then decided to go ahead and give an answer (because I comment a lot in this group but don't answer much) by throwing in some other stuff. I'm utterly shocked that I'm getting so many upvotes for this.

The "small-angle approximation" for trigonometric functions is often used in physics (and also engineering) without saying it explicitly, at least when the reader is assumed to have seen it before.

@StefanH: These approximations were OK with me, as it was usually clear that approximations were being made. Incidentally, several such examples that I first encountered in physics are given in #6 in this Mathematics Educators answer. The “random statements” stuff I mentioned refers more to textbook exposition stuff, not calculations, although even regarding calculations, I often did not feel that I knew what was “fair game” when making approximations to solve a textbook problem or in trying to anticipate a textbook derivation.

They are OK with me as well, I just noticed that engineers use these approximations a lot, even in "proofs", which might not be permissable from a mathematicians point of view (or the mathematician would use a lot of fancy language, like let x be contained in a sufficiently small neighborhood etc...). So, I thought I might bring it up. Your answers are great btw!

Here are my opinions based on my experience as a student and as a professor.
For as long as I can remember, I always wanted to understand how things worked at a fundamental level. Naturally this led me to study physics in university. I had an epiphany in my first abstract algebra course: the process of starting from a collection of axioms, building a rich theory out of them, and the ensuing sense of certainty gave me immense intellectual satisfaction. At the same time I was struggling in my physics courses, things went too fast, and my understanding never felt quite rock solid. I switched to pure math over the course of a year and I am glad I did.
Skill difference #1: Although it's not a skill per se to succeed in either math or physics you need to feel inspired and motivated by the field.
As a professor, I've had graduate physics students attend advanced math courses. Typically, such students struggled with writing proofs. It was not their fault though: unlike their math student counterparts, they had no previous proof writing experience. The ability to perform long and complex derivations, as is typical in physics, doesn't immediately transfer to proof writing.
Skill difference #2: Proof writing (as a skill) is important in mathematics and doesn't necessarily follow from being good in physics.
Another reason I quit physics was that I could not carry through the complicated derivations (I'm wired to make frequent calculation mistakes.) Also physics is taught the way it is for a reason: the underlying math of even the simplest situations gets difficult very quickly. Let me illustrate. I taught a mathematics course in Functional Analysis and, after one semester, I got to the point where I could discuss the theory of self-adjoint bounded linear operators. Now in quantum theory, the simplest model is the wave function for single particle moving in 1D, but the most basic operators, the position and momentum operators, are unbounded. Basically, I would have needed to give two whole Functional Analysis courses to adequately (for a pure mathematician) address the mathematics of the simplest model in quantum theory.
I can totally understand that a physicist may have  different priorities. For a mathematician transitioning to physics, to do it right would requires a tremendous amount of effort. There would be just too much stuff to learn for me to make a transition.
Skill difference #3: The ability to correctly perform complicated calculations and go with the flow (I really can't think of a better way to say this) is crucial to doing well in physics.
It is because I lack this third skill that I have so  much admiration for physicists. That said, there is no reason why a particularly talented person can't have all these skills and it is also wrong to think that there is a fundamental incompatibility between mathematics and physics.
For my part, over the years, as I pick up more math as a researcher, I frequently find myself revisiting concepts in physics with new insights. Each time I do this, I find physics even cooler.

Somewhat generalizing your perspective, my recollection (when during 1982-1984 I was a (prospective) Ph.D. candidate in math at Indiana Univ. with the goal of doing a Ph.D. in mathematical physics, a program one only entered after passing appropriate math and physics Ph.D. qualifying exams) was that primarily physics students with some pure mathematics background as an undergraduate were much more likely to succeed in getting into IU's math-physics program than primarily math students with some advanced undergraduate physics background. (continued)

This was from students I knew, from my initial advisor (Lenard), and from someone (Newton) my advisor and I talked to a couple of times. (FYI, I was extremely burned-out academically at the time, leaving after 2 years with an incredibly poor academic performance, with maybe 2 or 3 exceptions where some effort was expended.)

You've already gotten a lot of answers, but many seem to be from the point of view of mathematicians, or people who do both mathematics and physics. This answer is from the point of view of someone who did a PhD and some postdocs in theoretical physics, and very strongly believes "physics is not math" and shouldn't be math.
Here are some skills a good theoretical physicist needs:

Look at a real physical system and identify which features are key to understanding the system and which are second order details.
Develop toy models that can be solved analytically, or approximately, and give insight into how those key features explain some phenomena.
Understand the ways in which the toy model gives insight, and which ways the toy model oversimplifies the real world. And, understanding how complications in the real world that cannot be treated analytically affect the behavior of the real system.
A sense of numbers and scales -- being able to quickly do dimensional analysis and mental arithmetic to determine if some effect is likely to be important or not.
The ability to calculate -- correctly perform a long sequence of steps to start from first principles to the answer in a usable form. Calculation is different from proof -- for example, the goal is usually to understand in detail a special case that is relevant to a specific physical problem, rather than make a general statement about all cases in some class.
Intuition and ability to guess new laws. New scientific laws cannot be derived logically from first principles, they need to be guessed, usually based on keeping some physical principles from previous theory and modifying others in a specific, motivated way.
Knowledge of phenomenology (specific experiments and bounds) and a sense of which bounds/results are the most likely to be relevant to constrain some new model.
Ability to keep in mind multiple possible contradictory explanations for a currently unexplained phenomena, and maintain a "scorecard" as new discoveries happen and evidence builds for or against different theoretical explanations.

Now, I suspect that mathematicians also do many of these same kinds of things when they are working at an intuitive level before they settle on a particular theorem they want to prove.
I think much of the difference comes down to what physicists are trying to communicate to each other, vs what mathematicians are trying to communicate.
I think mathematicians want to convey absolute certainty about a logically well defined statement, with some kind of optimal tradeoff between having general assumptions and a powerful conclusion that is determined by the specific field being studied. The "absolute certainty" means a proof is necessary, and finding this "optimal tradeoff" means that a lot of work is put into defining precisely the assumptions that are being made.
Physicists are more interested in describing Nature than in making absolute mathematical statements. A general assumption about Nature is that various functions (like density of a fluid) are smooth (for example as a function of space and time). Therefore, when doing calculus, often proof the exact assumption on a function needed to perform some step is justified is not so important, because functions are usually "sufficiently smooth" than any reasonable assumption is going to end up being satisfied if you check it.
I understand why that probably sounds maddening to a mathematician. But there is a physical reason for it. Suppose that it turns out that this assumption is not correct, that is, that the density of a fluid behaves in a way that we can no longer assume it is arbitrarily smooth. From a mathematical point of view, we can assume that we are working in some well-defined mathematical framework, like the Navier-Stokes PDE, and carefully study solutions which do not obey typical smoothness requirements. From a physics point of view, the lack of smoothness may be pointing to a breakdown in the mathematical model we are using to describe the situation. It may point to the fact that in this case, we cannot treat the fluid as a fluid, and may need to consider the particle nature of the underlying atoms, governed by some other equations. Of course, you'll find exceptions -- physicists who are interested in the mathematical details. But, historically, understanding approximate solutions and physical principles has proven much more fruitful than understanding exact solutions or physically unmotivated but logically possible pathological cases.
Another version of this idea is that one of Hilbert's problems from the early 20th century was to "formalize physics." Most theoretical physicists I know would consider this a waste of time. The reason isn't that it's bad to have a logical foundation for your subject. The reason is that the physics Hilbert was talking about was classical mechanics. All of that physics was replaced, since the he proposed that problem, by quantum mechanics. We never know if our theories are complete, we typically believe our theories are effective theories that are superseded at some deeper level of understanding, and may even contain logical inconsistencies that we punt to a future unknown theory to resolve (like singularities in GR). So, from a physicist's perspective, it's much more interesting to calculate consequences of theories in regimes where everything is well behaved, then to consider the kinds of "pathological" cases you would need to make the theories fully mathematically rigorous, given that these cases might not be of interest physically and even if they do represent something that occurs in Nature, might need to be treated in a different framework.
To try to summarize this in one overly-simplified sentence: I believe a lot of the difference between math and physics boils down to the fact that mathematics is deductive -- it starts from a known, logical starting point and studies the consequences in a logically rigorous way -- whereas physics (like any natural science) is inductive -- the basic principles are always provisional and most of the work is either trying to calculate consequences of the basic principles in specific situations or trying to find holes in the currently accepted framework to get clues on what might lie beneath it.

The distinction you make is one that a former advisor (mentioned in another comment) explained as the difference between a mathematical physicist and a theoretical physicist. A mathematical physicist is more interested in extracting precise and rigorous mathematical deductions from a mathematical model, giving as an example his work with Freeman Dyson on the stability of matter problem. A theoretical physicist is more interested in using mathematics as a tool for insight into physical phenomena, giving his colleague R. Newton as an example.

@DaveLRenfro I would agree with that characterization and the examples you gave.

I sit (as an applied mathematician) on a few PhD committees in Physics and I would say the line is quite blurred, many times. The main difference, I would say, lies in the "requirement for proof". As a mathematician, I want things to be nice and tidy. My physicist colleagues focus more on results: who cares about ensuring all cases are covered, if the one that's of main focus to you is.
So I don't think it is very hard for math people to do work in theoretical physics, if they can overlook some untidy ends. And it's not hard for theoretical physicists to do work in math if they dot their i's. As a matter of fact, you often find a high level of interactions between researchers in both fields, locally, such as sitting on one another's students committees, attending seminars, etc.

As a math person, I do get pretty frustrated trying to read physics material by all the unstated assumptions and vague definitions floating around the work (much as the OP says). Here's an article I found to be very insightful on the subject:
Redish, E.F., Kuo, E. Language of Physics, Language of Math: Disciplinary Culture and Dynamic Epistemology. Sci & Educ 24, 561–590 (2015). https://doi.org/10.1007/s11191-015-9749-7

In this paper, we suggest that a fundamental issue has received
insufficient exploration: the fact that in science, we don’t just use
math, we make meaning with it in a different way than mathematicians
do. In this reflective essay, we explore math as a language and
consider the language of math in physics through the lens of cognitive
linguistics. We begin by offering a number of examples that show how
the use of math in physics differs from the use of math as typically
found in math classes.

The standout example is a fairly simple question to write a function which most mathematicians answer one way, and most physicists a different way. This is from an online primer by Dray and Manogour (2002), now called Corinne’s Shibboleth (and more commentary by Gethyn Jones here):

Suppose the temperature on a slab of metal is given by T(x, y) = k(x ² + y ²) where k is a constant. What is T(r, θ)?

There's a somewhat related question on SE Math Overflow regarding Examples where physical heuristics led to incorrect answers. One of the answers sparked this incisive comment by Noah Snyder:

This is a good example of what seems to be quite common with results
based on physical intuition: they're true in many cases but ultimately
rely on an additional technical assumption. This is the tradeoff
inherent in having a lower standard of rigor, you get better results
faster but often then have to go back and rethink the exceptional
cases.

For me, having physics material that doesn't bother to mention what those exceptional cases are, or what the prerequisites or limitations are, is overwhelmingly hard to digest. Or things that are literally mathematically false statements. I can't even tell if the authors knew but didn't bother to express them properly, or were actually unaware of them altogether. Either way it's hard to know to what extent we can trust a work like that.

Your last paragraph expresses exactly how I often felt (late 1970s to mid 1980s, when I was trying to be a mathematical physicist - see 2nd comment here. I didn't care that rigor was sacrificed. What bothered me was that often I didn't know whether it was, and when I did know it was being sacrificed, I didn't know whether it was on purpose or because the author simply wasn't aware. As an example, I never had any difficulties with Dirac delta function stuff, because it was always made clear that the manipulations were to be treated formally.

A question near and dear to my heart. I arrived at college self-taught in several subjects which, except to those with self-discipline of a type unknown to me, meant my skills were strong in those particular subareas that caught my imagination and much less sturdy in others. I qualified for advanced placement in Physics, but had to grind through Calculus in spite of being bored with significant chunks. Still, I pushed on through both in Physics and Math. I found my interests in Math gradually increasing beyond those in Physics but the tension between the disciplines was a serious issue for me. As mentioned above, I was expected to make essential unstated assumptions in Physics and to rigorously justify every step in Math. It wasn't until after I graduated that I really grasped that I should have explicitly developed two different methods to approach each field.
While I was wrestling with the problems of trying to use a single framework for both Physics and Math, I was also expanding on my computer hardware and software skills, also self-taught. This field was much more direct than the other two and so less painful. I still love Physics and try to follow enough Math to respect a deeply felt love for pure Math as a spectator, but I eventually wound up combining much of my STEM skills and passion working in the space industry, particularly on lunar landers.
I suppose it is too much to expect educators to recognize oddball students struggling with this kind of cognitive dissonance; I think not too many people have so much interest in both subjects that they get into this kind of trouble. Still, I wonder if I couldn't have felt less of a drowning sensation if someone had spent an hour helping me to understand explicitly to think this way over here and think this other way over there.

How does this answer the "Skills Difference between Theoretical Physicists and Mathematicians"? It seems more like a reflection that doesn't reach any conclusion.

Since no one seems to have defended the physics approach yet let me just add that giving up some rigor does have its advantages. Theoretical physics is usually very much ahead of rigorous math. The most famous example is of course that the path integral in quantum field theory which is rarely well defined. Yet when we perform logically seeming manipulations on it we manage to make precise and accurate predictions that have never turned out to be wrong. (Except in cases where they were wrong ;), but in those cases we have again - in our physicsy way - learned why the specific manipulations were incorrect.)
There is more of an attitude of doing something because it looks right and then checking the result (against experiment or consistency with principles we hold dear). This probably comes from the fact that historically physicist were able to check any computation/result against experiment. Of course modern theoretical physics does not always have that luxury anymore (mainly because it has outraced experiment quite a bit with little expectation that experiments will for example reach the energy scales that are of most interest to (high energy) theoretical physicists.
If you are interested in finding the right answer (rather then proving that it is the right answer) the physics approach is usually the faster one. (Of course you might sometimes, I would say very rarely, be eventually surprised  that you found the wrong answer.)

I see you wanted replies from people who switched between fields and I actually did not. (Although I am now in a Math department, I'm very much still a physicist). Perhaps there is something telling that all the answers from people who moved between fields say they now identify as a mathematician.

IMO you are short-changing physics. The differences between the two approaches to learning go significantly deeper than the usage of formal logic.

Mathematicians have no difficulty working with things (like path integrals) that a conjectural (have a look at the Langlands programme or the Birch and Swinnerton-Dyer conjecture). However, things which are conjectural are explicitly identified as such. To paraphrase a famous mathematician, "There is nothing called a theorem-within-quotes."

I have not studied math or physics at anything like an advanced level. But I have read a lot of biographies of mathematicians and physicists. Of course, physicists vary. But quite a few of the very best have a visual and spatial intuition about real objects that many mathematicians don't have. Mathematicians are better at imagining things that don't exist, while physicists are better at imagining things that do exist!

I like your comments about imagining things that exist versus those that do not. Also, I you're 're a statistician from your profile. I've read bio of Fisher that say he was great at visualizing and imagining data in higher dimensions because of childhood sickness that left in in bed much of the time.

"Mathematicians are better at imagining things that don't exist, while physicists are better at imagining things that do exist!" This is an important point. Counter-examples that show that the proposed theory does not yield what you want are important (in Mathematics) to demarcate the assumptions that are required in order to find the appropriate definitions.

There are many great comparisons given, and I would like to add another along side them. In physics many assumptions are often made to simplify things. That is necessary, and perfectly reasonable.  HOWEVER, one thing that has been driving me crazy is those assumptions are often not listed and tracked.  That makes it much easier to forget something is an assumption.
Also, math is valid if it holds together logically.  In physics reality can break your beautiful model.

This answer isn't complete and is only meant to be complementary to the other answers here.
One thing I have noticed is that physics curricula, on average, do a much better job emphasizing geometric visualisation than mathematics curricula.   Mechanical engineering curricula, are similarly strong as physics programs for this.
IMO it's really a shame how poor mathematics curricula tend to be when it comes to geometry.
Mathematics programs tend to emphasize the literal, so they tend to attract people who are verbal thinkers, and scare-away geometric/spatial thinkers.  This is much in-line with the criticisms Temple Grandin has of curricula in western nations.

A comment by a famous mathematician during a lecture may clarify why mathematicians do this: "Please provide the mathematics, we can provide our own motivation." There are different kinds of mathematical intuition. There is something like "algebraic intuition". Mathematicians with this kind of thinking get annoyed when too much effort is spent on giving geometric explanations. Perhaps this does not entirely excuse teachers who do not tell the students their own motivation and intuition, provided they do not force students to follow their way of thinking.

@Kapil: which again is an expression of the narrow-mindedness of mathematics.  That geometry is seen as "outside" is a serious flaw in the endeavor.

I report a conversation I had with an exceptionally talented physics professor.
He said, at some point you realise that it is not possible to know everything and the skill set to build proof in mathematics is very different from the kind of intuitive understanding of phenomena you need to do physics. There is only one person who knows everything and is (insert name of the head of department).
He would also say that one has to have the courage to drop irrelevant terms like flies (eg in an expansion) and make bold approximations without fear.
We're talking about a guy who does work with very abstract topological stuff in soft matter physics, for context.

Yeah, but physicists will drop terms based on intuition, have and compute ill defined integrals, assume solutions to a problem exists or assume convergence to some limit and not bother about it and leave it to the mathematical physicists to deal with

This kind of stuff you find in books, see Van Kampen

I have not looked at all the all the answers, but I will give my story.
The high school and undergraduate skills are about the same.  It is not until you take the advanced undergraduate or graduate courses in real and complex 'analysis' that you know whether you want to be a physicist or a mathematician.  First year graduate courses in electromagnetism and 'mathematical physics' will also help you decide.  I think I liked math more than physics until I took courses in 'analysis'. The point is that the distinguished math teacher felt insulted when I asked, "What is the point? How can this be used?"  A good example is the three names Poincare, Einstein, Minkowski.  All three independently derived special relativity within a couple of years of each other.  Poincare was the first. He wrote down most of the important equations, which is why the name 'Poincare group' is used.  He didn't care much about using it.  As you have guessed, he was a mathematician.  Einstein wrote as a physicist, using gedanken experiments as examples.  A problem with this method is that all the pseudo paradoxes of special relativity are based on gedanken experiments.
Minkowski was a mathematics professor who possibly would have been the greatest physicist of the 20th century had he not come down with appendicitis at an early age.
He invented and applied Minkowski space to develop the special relativity we use today.  However, apart from him, the more modern example of von Neumann, and possibly Dirac, there is a wide gulf between physicists and mathematicians.  From what you have written, you sound like a physicist. It would be a good idea for you to audit, rather than take, the mathematics analysis courses, because you will find the detours into unimportant theorems will annoy you. As a physicist, start with a purpose, develop it and work it out, and then present your conclusions.

I think there's a big difference between mathematical detail and pedantry. Very often the validity of a theorem relies on many details, and one has to be very careful with the assumptions. Sometimes as a physicists we assume things that may be obvious, but it's better to spell them out. As concerns the question, pure mathematicians with no training in physics may have hard time in developing physical intuition, notably if they don't know the fundamental principles of physics (which are legion). Mastering these principles and developing the physical intuition requires a lot of thinking and training. Theoretical phycisists may invent whatever maths are needed to solve a problem, even if those maths are not superrigorous or lack formalization. This is the way physics has been done for years or centuries.
In my opinion, (hyper)formalization of a physical idea sometimes hinders the progress of it. I think that once the general framework (or theory) proves to be on the right path, one can start formalizing things. Otherwise it can be a total waste of time from the physics point of view.
Moreover, if one has a strong physical intuition one can guess the answer of a problem even without solving it in detail. And even if we make many mistakes in a particular calculation we can get back to them and correct them if the outcome contradicts the physical intuition.
I agree that the more maths a theoretical phycisist knows the better, but knowing all the maths of the world is NOT a guarantee to become a good physicist.

I think not (hyper)formalizing too fast is a good point that ties back to what @Andrew was mentioning about physics having provisional principles that you play with until you seem to be on the right track.

It is not true that mathematicians work only with rigorous frameworks while doing mathematics, forget (hyper)formalsation (whatever that is!). They will not publish until they have something they consider rigorous.

In my experience as a mathematician who has started to study physics at various points, there is a level of "physical intuition" that I lack compared to my colleagues who studied physics the whole time.
Like, I can see all the equations, and understand them in a vacuum, but it's unclear to me which situations warrant which equations.
Another similar issue I have is trying to pin down the exact "axioms" or assumptions that are being made in each context. I tend to go down rabbit holes chasing details that were never there to begin with, because (as you say) physicists tend to ignore them.
My whole ability to understand math comes from my ability to trace everything down to its foundation, or at least a foundation that I'm comfortable with; in physics, this often eludes me, causing me to feel as though my understanding of the subject is fractured and shaky.

It is not so much a difference in skills, but the philosophical approach that determines their use.
A physicist would primarily be interested in questions about what reality is and how we perceive it.
A mathematician would primarily be interested in how we reason about things: in abstraction and logic, in linguistic and non-linguistic mental constructs.
This difference in philosophy changes the kinds of questions one asks and the kinds of answers that are considered "satisfactory".
It also leads to differences in the aspects of the skills (be they computational, relational, spatial or some other) that one hones even when the basic skills are from the same set.
In summary, the difference is not so much in the skill sets, but in the value/onus placed by the "owner" of these skills on them. As a consequence, their use can be different.

Update: Looking through the answers, I realised that Andrew has given a somewhat similar, but more detailed answer. However, he has written it as a physicist, and I have written it as a mathematician.

An Example: Occam's Razor is commonly used in physics research and in mathematical thinking.
In physics, it would typically be used to remove things which have low impact on the phenomenon under consideration. To find out "which terms to neglect".
In mathematics, it would be used to remove hypothesis that are not used in a certain proof or in typical uses of a certain definition.
In the former case, the reality being described determines what can be "shaved off". In the latter case, the argument or calculation determines what can be excised.

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text:
Improving my GPA after two disastrous years
I am a computer science student and  had disastrous first two years.There are many F's on my transcript and my overall GPA was 2.44/4.00.Then I retook all the courses that I had failed and improved my GPA to 3.34/4.00.I have another year before graduation and will keep improving but I am worried about the F's and D's on my transcript even if I retook the courses and passed with A's.Will they be problem in Msc and PhD applications? There is no special reason for that poor performance.I just didn't plan a career in academia but then I changed my opinion and decided to make a come-back.Is this a valid excuse?
In my country,only the most recent grade is counted in GPA calculation but all grades for the specific course appear on transcript.

I don't understand what answer you are looking for. If admissions are competitive, you will be competing against people that did not have any disasterous years of grades.

Don’t look for excuses - they’re just signs you did something wrong. If your GPA is sufficient then that’s it, and if they will look at the details you can’t change that either. No advice from us will allow you to change the past.

In most schools and most departments, admission to grad school is highly competitive. "A lot of Fs" is definitely going to be a problem. Maybe that's not the answer you want, but it's the only answer there is, especially since there is no real reason for those Fs.

Your most recent new courses -- have you passed those, the first time, with A's?

Yes, very poor grades will be a concern for some applications. However, many (most?) people recognize that young people "grow up" and change attitude and direction. This is part of the reason that only the latest grade is counted in GPA. In the US, perhaps especially, people, while still having concerns, will look at the latest grade as most important for indicating your current status.
Also, in the US, letters of recommendation can attest to the fact that, academically, you are a different person than you were at the start.
Yes, a concern, more serious some places than others, but not necessarily a block going forward as long as you continue good performance.

I can't speak for German institutions, but in the US, starting from scratch won't help as all prior records need to be submitted. You are where you are. Make the best of it and cast a wide net for future study. And continue to outperform your earlier self and not just in retaking courses but in new ones.

Sorry, but I won't be part of any committee or system that considers you. And the grades in the field you want to enter are most important. Get letters that mention your transformation if possible.


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What are uncited references?
I came across this article which has a section named uncited references.
Why it is there and what is the purpose of that section?

This is an unnecessary originality of the authors of that article.

Note that the article you linked to is a review article. A review article summarizes primary literature, often reporting on the most important findings regarding a particular question or questions being addressed. In reviewing many articles, there may be papers that the author(s) reviewed in order to derive the results and major findings of the paper, but that didn't have anything interesting or exemplary of the major findings to report/cite directly in the manuscript. In that case, the author(s) would still need to cite the paper somewhere because it was part of the data collection component of the manuscript.
Journals may deal with those references in different ways. When feasible, authors may have few enough references such that they fit into a table in the manuscript, or they may report papers used in the data collection as supplementary data. If neither of those options is used, the author must cite those papers somewhere in the manuscript because they were a part of the study, and an "Uncited References" section may be one option for doing that.

all_licenses: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/3.0/


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Is it okay to address "Prof. X" as "Mr. X"?
I am from the Netherlands.
I have seen quite a few threads which address the question of whether or not it is acceptable to address a professor by their first name. However, none of them seem to address the point of whether it is still permissible to refer to them as Mr/Mrs X?
For example, if you know that someone's full title is "Prof. Dr. X" is it acceptable to write them an email starting with "Dear Mr/Mrs X"?

Comments are not for extended discussion; this conversation has been moved to chat.

This will depend on the norm within your country to some extent, but I'm inclined to say that at least in the US, calling a professor 'Mr.' is insulting, provided you are in an academic context.
My neighbors, many friends, and the cashier at the cafe know me as Mr. X. But on campus, I'm Dr. X...and you know I'm a professor. Using 'Mr.' seems strangely intentional in a way to diminish ones accomplishments.

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The Netherlands has a democratic but also a meritocratic culture.  Many people do not use their title.  Sweden is similar (but I believe Germany is not).  I am from The Netherlands (but I live in the UK).  Although I have a PhD, I find that it feels strange to be referred to as Dr. Holl.  I don't think my title makes me any better than anybody else, and certainly I don't feel insulted if someone refers to be as Mr. Holl.  In all likelihood I will not even notice.  Many people in The Netherlands will feel the same.
However, academia is very international.  There are certainly people working at universities in The Netherlands who would expect to be addressed to with their titles.  Mr.¹ Dr. Ir. X.  So, although chances are it will be no problem to address someone as Mr. X, if you don't know the person, you might make a faux pas.  Many people don't care; some do.  Better safe than sorry; better to be too formal than too informal.  Therefore, even in The Netherlands, I would avoid Mr. X.

¹Incidentally, in Dutch, Mr. is the title for Master of Laws.

In think this is a very good description of the specific situation in the Netherlands.

This indeed, in the Netherlands you only use a title if and only if it's directly relevant. And even there it's not obligatory.

Agreed. Since getting my PhD I've lived in Spain, France and Greece and have never been addressed as "Doctor". I would even say that it feels pretentious and fake to me. Then again, the aforementioned countries all use "Doctor" as a form of address almost exclusively for medical doctors and not PhDs. Even in English though, I would only expect to be addressed as Doctor in the most formal of situations.

@gerrit I agree with you. You're a human like the rest of us and as I said in my other comment I agree using it to prove your credibility in some cases.
If I acknowledge you as a Ph.D. it can only make you feel better about yourself it's not going to help me let's say study better because I won't feel you're that smart transferring knowledge to me telepathically.
Therefore there is no point at least I feel that way.

No. For professional correspondence, Dr. Smith is correct, Ms./Mr. Smith is incorrect.  Miss and Mrs. should be avoided in all correspondence.
If you know the person well, or the person has asked for you to do so, first names might be appropriate.  Of course, all of this is contingent on regional (even institutional) culture.

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In The Netherlands this is not completely correct for formal correspondence. The official title of a full professor is Prof. dr. while for anyone with a PhD it is Dr..

I have to say that the claim that "Miss and Mrs. should be avoided in all correspondence" is a little bit of an oversimplification. In the event that someone had emailed me and signed "Mrs Smith", I would address my reply to "Mrs Smith" -- it would seem odd (and rude) to go out of one's way to ignore an (implied) statement on how someone would prefer to be addressed.

Short and sweet: Use "professor" or "doctor" unless/until the addressee faculty member allows you to talk to them on a first-name basis. My experience is that, most faculty members don't care about how they are addressed. The latter is mostly true within the departmental boundaries.

If in doubt I'd go with "Dr Surname". This is appropriate for most academics as all Professors are also Doctors. However use Prof (or their full academic title, they're often not hard to find online or in their emails) if you know it. Mr or Mrs is no less formal in English and academic titles are also gender neutral which is more professional in my opinion.
In my experience, Professors won't mind being mistaken for Dr as much as Mr/Mrs. Once we know each other, they often go to first name terms. This is especially common in my country (New Zealand) where formal titles are seldom used.
Of course this all depends on your cultural context. For instance, Japanese academics will often use "Family-name San (Mr/Mrs)" with each other but this is because they use honorifics in contexts Westerners would use first names. They still expect to be addressed by their academic rank by their students and people meeting for the first time. Even in a formal culture like this the English titles are ok.
Although most academics are fairly relaxed about it, I would encourage you to learn about the culture they're from (not yours--international research teams are common) and how they prefer to be addressed. I cannot speak for academic environments in Europe but "Dr" is the most commonly used in academia, although there are  exceptions.

I will preface this with my country being a place where it is common to use first names or even hypocorism in the workplace. These may be considered very informal but I go by "Tom" in all contexts and address my supervisor as "Mik", even though historically these are nicknames. I even met a senior member of government  (now our PM) who introduced himself as "Bill". I still encourage use of formal titles with someone you don't know that well, especially more senior faculty until they drop formalities with you.

I would never use the full academic title in addressing a professor in a day-to-day basis, e.g. an e-mail. "Dear Prof. Dr. Dr. Ing habil. X" does not sound good. For any official documents, sure, but not in communication. Still, I like this answer the most, as it addresses the cultural differences, +1.

Sorry for the ambiguity, by full title I mean their highest title and family name (e.g., Professor Spencer or Lord Winston) rather than abbreviated titles or "Professor" without a name (this isn't Hogwarts). These formalities are more typical in email correspondence whereas in person you get a more immediate response to guage how formal they wish to be (e.g., Kiwis introducing themselves using firstnames or the Japanese exchanging bows and namecards). I've advised mainly for a meeting for the first time. For everyday interactions that is highly depending on working environment and culture.

For instance, Japanese academics will often use "Family-name San (Mr/Mrs)" with each other but this is because they use honorifics in contexts Westerners would use first names. misunderstands what Japanese people are doing. Japanese people call university professors surname-sensei, but they realize "teacher surname" is wrong. They then mistakenly believe "mr surname" is the polite equivalent.

Similarly, They still expect to be addressed by their academic rank by their students and people meeting for the first time.  Japanese students at most institutions never call their instructors professor (kyoujyu) or assistant professor (jyun-kyoujyu), instead they are always surname-sensei. Only the most westernized departments and individuals would ever use first names even among faculty members. It's more a behavior they imagine (correctly or mistakenly) is correct in the West and thus emulate when talking with Westerners.

Precisely, what I was referring to is academics who hold PhD's addressing their peers as with San in a less formal context (where westerners use first names). Although doing so with Mr in English speaking countries is not polite. They still expect to be formally addressed with sensei (Prof) or hakase (Dr) in a working environment, especially by those of lower rank.

I am sorry I misread you answer because I failed to see that you were referring to peer reference. You are absolutely correct that surname-san or even nickname-san is common among co-authors etc. On the other hand, watch what happens if a student first names a male tenured professor.

@Leon Meier Indeed, this matter does highly depend on cultural background and working environment. I do not have experience with academic environments in The Netherlands, although I have travelled there. I have provided examples (of cultural extremes) where "Mr" or firstnames would be used but this is not the norm: my recommendation to use "Dr" or their academic title (eg, Prof) should be relevant to European countries.

Let it be noted that I answered a previous edit of the question which did not emphasise this to be specifically about the Netherlands. I have also tried to give a broad range of cultural examples, should they benefit anyone else using SE in the future--not just the OP. Furthermore, I think it is polite to conform to the cultural norms of whom you are addressing and the OP has not specified (they are not necessarily Dutch).

Not all professors are doctors.

No I would not do that unless you only know them only as a friend outside of the University environment.  I only refer to professors as "Professor".  In my field (chemistry) all professors are doctors so it is a given.  Plus there are plenty of people (PostDocs, teaching staff, research staff) around who are PhDs but aren't professors.  I therefore consider "Professor" to be a higher level of respect in the University environment.

Norms regarding the use of titles vary widely. Here is my understanding of norms in the Australian context.
Typically, the norm in Australia is to address academics by their first name. This applies to many contexts:

When a student greets an academic. "Hi John"
When a student writes an email to an academic. "Dear Jenny"

However, if you are going to use a title in an academic context (i.e., Mr, Ms, Dr, Prof, A/Prof, etc.), then try to use the correct title. In an academic context, it is generally better to leave out titles entirely than to assign someone a lower title. E.g., John Smith is better than Mr John Smith (John has a PhD).
 Titles are particularly relevant for more official forms of correspondence, particularly where the academic status of the academic is of some relevance.
Some examples of relevant contexts include:

Official forms: Ethics, grant, PhD admission applications, etc.
Official correspondence with research participants: e.g., A plain language statement given to participants
Correspondence with the media and press releases
Author descriptions in some journal articles

Rules for what title to use:

If someone has the academic rank of professor, then use "Prof ..."
If someone has the academic rank of Associate Professor (roughly equivalent to North American "professor"), then use "A/Prof"
If someone has a PhD and is not a professor or associate professor (e.g., they are an associate lecturer, lecturer, senior lecturer, various research roles, etc.), then use "Dr"
If you are unsure whether an academic has a PhD or doctorate, then probably err on the side of using "Dr"
If they do not have a PhD or doctorate, then use "Mr" or "Ms" (do not use Mrs or Miss unless you know that the academic has a preference for this)

@Leon. The question title is "Is it okay to address “Prof. X” as “Mr. X”?" The purpose of the site is to provide useful resources for people searching for answers on the internet. Thus, others can address the Netherlands specific issue, but the answers will be more useful for others if general answers are provided to the question that transcend different regional conventions or at least address a range of conventions.

Furthermore, the top answer discusses the United States, which is not the Netherlands.

My answer assumes that you are having a lower academic rank.
Start by using the full and correct academic title, whichever it is. Example: "Dear Prof. Dr. Dr. h.c. Smith: ..."
Next, it is up to them to suggest a shorter form. Some don't say anything, some say "Prof. is enough", some say "Ms. Smith is enough", and still others may say "You can call me Hannah."

At the mathematics faculty of UU, I rarely even use surnames for my professors. What is applicable for you depends heavily on your professors and the way they communicate to you. However: if you feel it needed to write formally; then go through the effort of using the proper pronouns of Prof.

I'm not sure what he means by UU, but if its Utrecht University (https://www.uu.nl/en) then it is in the Netherlands.

Yup, Utrecht university

Mr. and Mrs. are already as formal as Dr.. Moreover, if you are close enough with a faculty member - and they are okay with you calling them by their first name -
 then you have already crossed the bridge from formal to informal.
To be on the safe side, I would refer to those with doctorates or professorships as Dr. or Professor (at least in the US these titles are interchangeable) in professional correspondence (email for example). It would seem disrespectful to call a Dr. a Mr./Mrs. because it is as if you are dismissing their hard work to achieve a doctoral degree. Anyone who holds a PhD should be called Dr. (or professor if in a teaching capacity), and Mr./Mrs. if they hold a Master's degree or less.

In the United States, Doctor or Professor are the appropriate terms of address. One exception is the University of Virginia, where professors are referred to as Mr. or Ms. (except for MDs, who are called "Doctor.") In most institutions in the United States, calling a faculty member "Professor" is somewhat safer than "Doctor," since some faculty who hold non-doctoral terminal degrees, such as an MFA, still have faculty appointments. But Mr. or Ms. is generally seen as a no-no.
Of course, universities in other parts of the world will be different.

Always stick with the context--your position, their position, the topic under discussion, the location/medium of the discussion. What would you expect someone else in your position to do in that context, if you didn't have any other knowledge of the two people involved? Even if the other person doesn't care what you call them, it's also useful to avoid confusing or offending anyone else around you.
Examples:
I grew up next door to a university professor and ended up attending the same school--in the same department, even. (Undergraduate level, which does make some difference.) He is older than my parents, but younger than my grandparents. My grandparents address him as Firstname, my parents address him as Mr. Lastname, and I address him as Mr. Lastname at home and as Professor Lastname or Dr. Lastname at school. If I used Mr. Lastname to a student, it would probably take them a minute to figure out who I meant. If I used Firstname in any of those contexts, there probably would be someone listening who would clutch at their pearls (be offended), even though my neighbor himself does not care.
I also work with my significant other's dad, in a workplace where first names are generally used. At work I call him Firstname, at home I switch between Firstname, Mr. Lastname, and "SO's Dad" depending on who I'm talking to. If I used Mr. Lastname at work, it would definitely take people a bit to figure out who I was talking about!


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Is a research article the most comprehensive information available about a research project?
I sometimes stumble upon an article about an experiment, e.g a medical experiment done in some medical journal, or an article refering to a physics experiment done, and would like to know more exact details about the experiment and just in general as much information as possible about the subject.
I never wrote an article for a journal nor have I participated in the management of an experiment, but I understand documentation is done beyond what is written in the article, a journal article is a summary of the experiment and only elaborates on the important points.
For example a lab book AFAIK is often kept and used during the course of an experiment.
But this question pertains to theoretical research as well, e.g in mathematics; I may wish to find more information about the process by which the author came to his findings.
Is there usually a more complete documentation available which delineates the whole process of the experiment/research-project?
One thing I found is this Wikipedia article about "Open notebook science", but I understand this is not the norm. Still, I suspect I may be missing something (maybe a iece of documentation not as comprehensive as a lab notebook, but more so then a journal article.
Is there anything more for me to do if I wish to dig deeper and understand the subject more thoroughly?

Is there usually a more complete documentation available which delineates the whole process of the experiment/research-project?

Typically not all of the details of the research can feasibly be published. Different areas have different methods to cope with this. Some journals will allow supplementary material to be submitted that's made available online through the publisher's site. Some authors will add links to homepages for the project where more details are available. Sometimes authors will publish an extended "technical report" (e.g., with full proofs or full methodologies that would not be feasible under a page limit) and publish it on their institute homepage or a site like arXiv. I believe some biomedical journals allow for publishing databases containing micro-data on experiments ...
So it varies widely. The most generic answer, however, is simply to get in contact with the authors and ask them if more details are available! Their contact details should be provided in the paper.

+1 for contacting authors. I have never reached out to someone and said "hey I want to use your work, can you tell me about the methodology" and gotten any response other than that they're thrilled that someone wants to use their work for something.


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Do researchers use plagiarism tools before sending their manuscripts to the publishers?
Do student researchers use plagiarism tools before sending their manuscripts to the publishers? If they do, do they spend money on this type of software from their own pockets?
If someone doesn't want to spend money from their own pocket, can they approach  their P.I. or the faculty?

Does this answer your question? What is the acceptable similarity in a mathematics PhD dissertation when checking by Turnitin?

There's a bunch of questions like this, and the answer to all of them is that you should be able to detect plagiarism yourself if you wrote the document, so the software is pointless.

@AnonymousPhysicist This assumes you trust your co-authors 100%.

@FedericoPoloni If you are the coauthor who did not write the document, you should know enough about it to be able to tell if it was plagiarized.  If you do not know that much, you have not earned coauthorship, in my view.

If you wrote your manuscript yourself and did not plagiarise, then you probably won't need to check it using a plagiarism detection software?  The purpose of this software is to detect potential plagiarism in the work submitted by someone else. As a PI, if my student or postdoc asked me to pay for them to check their own writing for plagiarism, it would make me somewhat concerned.

I did some paraphrasing. Now, I am concerned that if they are good enough.

You can show the original text and your paraphrased writing to your adviser/PI and ask what they think about it.

@user366312 Plagiarism is not about insufficient paraphrasing, it is about not giving credit where credit is due: when you read something somewhere and pass it as your own. Paraphrasing exists to present a coherent story and has actually little to do with plagiarism aside from the easiest way to detect it.